Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

Stabilit y structures, motivic Donaldson-Thomas in v arian ts and cluster transformations Maxim Kon tsevi c h, Y an Soib elman No vem b er 26, 2024 Con ten ts 1 In tro duction 3 1.1 Coun ting problems for 3-dimensional Calabi-Y au v arieties . . . 3 1.2 Non-comm utativ e v arieties with p olarization . . . . . . . . . . 5 1.3 Donaldson-Thomas in v a ria n ts for non-comm utativ e 3 d Calabi- Y au v arieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Multiplicativ e w all-crossing form ula . . . . . . . . . . . . . . . 10 1.5 Some analogies a nd sp eculations . . . . . . . . . . . . . . . . . 15 1.6 Ab out the con ten t of the pap er . . . . . . . . . . . . . . . . . 17 2 Stabilit y conditions for gra ded Lie algebras 20 2.1 Stabilit y data . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Reform ulation of the stabilit y data . . . . . . . . . . . . . . . 22 2.3 T op olo g y and the wall-cross ing form ula . . . . . . . . . . . . . 25 2.4 Crossing the w all of second kind . . . . . . . . . . . . . . . . . 30 2.5 In v aria n ts Ω( γ ) and the dilogarithm . . . . . . . . . . . . . . . 32 2.6 Symplectic double torus . . . . . . . . . . . . . . . . . . . . . 33 2.7 Complex in tegrable systems and stabilit y data . . . . . . . . . 35 2.8 Relation wi th the w orks of Jo yce, and of Bridgeland and T oledano- Laredo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.9 Stabilit y data on gl ( n ) . . . . . . . . . . . . . . . . . . . . . . 43 1 3 Ind-constructible categories a nd stabilit y structures 45 3.1 Ind-constructible categor ies . . . . . . . . . . . . . . . . . . . 45 3.2 Stac k of ob jects . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Ind-constructible Calabi- Y au categories and p oten tials . . . . 53 3.4 T op olo g y on the space of stabilit y structures . . . . . . . . . . 56 4 Motivic functions a nd motivic Milnor ﬁber 60 4.1 Recollection on motivic functions . . . . . . . . . . . . . . . . 60 4.2 Motivic f unctions in the equiv arian t setting . . . . . . . . . . . 62 4.3 Motivic Milnor ﬁb er . . . . . . . . . . . . . . . . . . . . . . . 65 4.4 An integral iden tit y . . . . . . . . . . . . . . . . . . . . . . . . 69 4.5 Equiv alence relation on motivic functions . . . . . . . . . . . . 73 4.6 Numerical realization of motivic functions . . . . . . . . . . . 75 5 Orien tation data on odd Calabi-Y au categories 77 5.1 Remarks o n the motivic Milnor ﬁb er of a quadratic form . . . 77 5.2 Orien tation data . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 Orien tation data fro m a splitting of bifunctors . . . . . . . . . 83 6 Motivic Donaldson -Thomas inv arian ts 84 6.1 Motivic Hall algebra and stabilit y data . . . . . . . . . . . . . 84 6.2 Motivic weigh ts and stabilit y data o n motivic quan tum tori . . 92 6.3 F rom motivic Hall algebra to motivic quan tum torus . . . . . 95 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.5 D0-D6 BPS b ound states: an example related to the MacMa- hon function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7 Quasi-clas sical limit and integrali ty conjecture 109 7.1 Quasi-classical limit, n umerical DT-in v arian ts . . . . . . . . . 109 7.2 Deformation inv ariance and in termediate Jacobian . . . . . . . 112 7.3 Absence o f p oles in the series A Hall V . . . . . . . . . . . . . . . 115 7.4 Reduction to the case of catego ry of mo dules . . . . . . . . . . 120 7.5 Evidence for the inte gra lit y conjecture . . . . . . . . . . . . . 125 8 Donaldson-Th omas in v arian ts and cluster trans formations 128 8.1 Spherical collections and mutations . . . . . . . . . . . . . . . 128 8.2 Orien tation data fo r cluster collections . . . . . . . . . . . . . 136 8.3 Quan tum DT-in v arian ts for quiv ers . . . . . . . . . . . . . . . 137 2 8.4 Quiv ers and cluster transformations . . . . . . . . . . . . . . . 137 References 143 1 In tro duction 1.1 Coun ting problems for 3 -dimensional Calabi-Y au v arietie s Let X be a compact complex 3- dimensional K ¨ ahler manifo ld suc h that c 1 ( T X ) = 0 ∈ P ic ( X ) (hence b y Y au theorem X admits a Calabi-Y au metric). W e can asso ciate with X sev eral mo duli spaces whic h hav e the virtual dimension zero: a) mo duli of holomorphic curv es in X with ﬁxed g enus and degree; b) mo duli of holomorphic v ector bundles on X (or, more generally , of coheren t shea v es) with a ﬁxed Chern c haracter; c) mo duli o f sp ecial Lagrangian submanifolds 1 with a ﬁx ed homology class endo w ed with a U (1) lo cal system. In order to ha ve a w ell-deﬁned virtual n um b er o f points of the moduli space one needs compactne ss and a perfect obstruction the ory with virtual dimension zero (see [4], [67], [68]). 2 The compactiﬁ cation is kno wn in the case a). It is give n b y the mo duli of stable maps. The corresponding virtual n um b ers are Gromo v-Witten in v ariants (GW-in v arian ts for short). Donald- son and Thomas in [19],[68] addressed the cases b) and c). Analytic al dif- ﬁculties there are not com pletely resolv ed. The most unde rsto o d example is the one of torsion-free shea v es of rank one with the ﬁxed Chern charac- ter of the form (1 , 0 , a, b ) ∈ H ev ( X ). The corresp onding virtual n umbers are called Donaldson-Thomas inv arian ts (D T-inv arian ts for short). One sees that the n um b er of (discrete) parameters describing GW-inv ar ia nts is equal to 1 + dim H 2 ( X ) (gen us and degree) and coincides with the num b er of parameters des cribing DT-inv arian ts. The conjecture from [47] (pro v ed in man y cases) sa ys that GW-in v a ria nts and D T- inv arian ts can b e expresse d one through a nother. The full putativ e virtual num bers in the case b) should 1 Recall that a Lag rangian submanifold L ⊂ X is called sp e cial iﬀ the restriction to L of a holomor phic volume form on X is a real volume for m on L . 2 The latter means that the deformation theo ry of a point is con tro lled by a diﬀeren tial- graded Lie algebr a g such that H i ( g ) = 0 for i 6 = 1 , 2 and dim H 1 ( g ) = dim H 2 ( g ). 3 dep end on as t wice as man y parameters (i.e. dim H ev ( X )). By mirror sym- metry one reduces the case c) to the case b) for the dual Calabi-Y au manifold. Unlik e to GW-inv arian ts and D T-inv arian ts these virtual nu mbers should de- p end on some ch oices (the K¨ ahler structure in the case b) and the complex structure in the case c), see [68]). In part icular, in the case c), fo r a compact 3 d Calabi-Y au manifold X we should hav e an ev en function Ω S LAG : H 3 ( X , Z ) \ { 0 } → Q , whic h de p ends on the complex struc ture on X in suc h a w a y that for any non-zero γ ∈ H 3 ( X , Z ) the nu mber Ω S LAG ( γ ) is a constructibl e function with respect to a real analytic stratiﬁcation of the mo duli space of complex struc- tures. Moreo v er this num b er is integer for a generic complex structure. The in v a ria n t Ω S LAG ( γ ) is the virtual n umber of sp ecial Lagrangian submanifolds L ⊂ X in the class γ (or more generally , special Lagrangian submanifolds endo w ed with lo cal system s of arbitrary ra nk). Our aim in this pap er is to desc rib e a framew ork for “generaliz ed Donaldson- Thomas in v arian ts” and their w all-crossing formul as in the case of non- comm utativ e compact 3 d Calabi- Y au v arieties. A c hoice of p olarization (“complexiﬁed K¨ ahler structure”) will b e enco ded into a c hoice o f “stabilit y condition” on C . Then w e deﬁne a generalized Donaldson-Thomas in v ari- an t Ω( γ ) as the “n um b er” of stable ob jects in C with a ﬁxed class γ in the K -group. Similar problem f o r ab elian categories w as address ed in the series of pap ers by Jo yce [32][33][34] and in the recen t pap er of Bridgeland and T oledano Laredo [10]. Our pap er can be though t of as a generalization to the case of triangulated categories (the necess ity of suc h a generalization is motiv ated by b oth mathematical and ph ysical applications, see e.g. [67], [21]). One o f motiv a tions for our coun ting f orm ula was the microlocal f o r- m ula b y K. Behrend (see [2]) for the virtual n um b er in the case of so called symmetric obstruction theory (s ee [4]), whic h is the case for ob jects in 3 d Calabi-Y au categories. The ab ov e example b) correspo nds to the b ounded deriv ed category D b ( X ) of coheren t sheav es on X (more precisely to its A ∞ - enric hmen t). The example c) corresp onds to the F uk ay a category . In that case the w all-crossing form ulas describe the b eha vior of Ω S LAG . Ev en in the geometric situation our formalism extends b ey ond the case of smo oth compact Calabi-Y au v arieties. 4 1.2 Non-comm utativ e v arieties with p olarizati on All A ∞ -categories in this pap er will b e ind-constructible. Th is roughly means that their space s o f ob jects are coun table inductiv e limits o f constructible sets (for more details see Section 3) . W e de ﬁne a non-c ommutative pr op er algebr aic variety over a b ase ﬁeld k as an E xt -ﬁnite ind-constructible k - linear triangulated A ∞ -category C . F o r tw o ob j ects E and F we denote b y Hom • ( E , F ) the complex of morphisms a nd b y Ext • ( E , F ) its cohomology . Here are few examples of suc h categories. Example 1 a) A ∞ -version of D b ( X ) , the b ounde d derive d c ate gory of the c a te gory of c oher ent she aves on a sm o oth pr oje ctive algebr aic varie ty X/ k . In this c as e D b ( X ) c o incides with the triangulate d c ate gory P er f ( X ) of p erfe ct c o mplexes on X . b) Mor e gene r al ly, for a (not ne c essarily pr op er) smo oth variety X en- dowe d with a close d pr op er subset X 0 ⊂ X , the c orr esp onding triangulate d c a te gory is the ful l sub c ate gory of P er f ( X ) c o nsisting of c omplexes o f she aves with c o h omolo gy supp orte d on X 0 . c) A lso for a (not ne c essari l y pr op er) smo oth variety X we c an c onsider the the ful l sub c ate gory of P er f ( X ) c onsisting of c omplexes of she a ves with c o mp actly s upp orte d c ohomolo gy. d) The A ∞ -version of the c ate gory P er f ( X ) of p erfe ct c omplexe s on a pr o p er, not ne c essarily smo oth scheme X over k . e) If A is an A ∞ -algebr a with ﬁnite-di m ensional c ohomolo gy then C = P er f ( A ) is the c ate go ry of p erfe ct A -mo dules. f ) If k is the ﬁeld of char acteristic zer o and A is ﬁ n itely gen er a te d in the sense of [71], (in p articular it is homolo gic al ly smo oth, se e [42]) then C is the c a te gory of A -mo dules of ﬁ nite dimension over k . g) If the c ate gory C is ind-c onstructible and E ∈ O b ( C ) then left and right ortho g o nal to the minimal triangulate d sub c ate gory gener ate d b y E ar e also ind-c onstructible (sinc e the c onditions Ext • ( X , E ) = 0 and Ext • ( E , X ) = 0 ar e “ c onstructible”). Let us mak e few commen ts on the list. Example a) is a particular case of examples b),c),d). Using the results of [7] we can reduce geometric examples b),d) to the algebraic ex ample e), and also the example c) to the example f ). Let us discus s a t ypical (and most imp ortan t) example e) at the le ve l of ob jects of the category . W e claim that the set of isomorphism classes of ob jects of C can b e cov ered b y an inductiv e limit of constructib le sets. 5 First, replacing A b y its minimal mo del w e may assume that A is ﬁnite- dimensional. Basic examples of perfect A -mo dules are direct sums of shifts of A , i.e. mo dules of the t yp e M = A [ n 1 ] ⊕ A [ n 2 ] ⊕ · · · ⊕ A [ n r ] , r > 0 and their “upp er-triangular de formatio ns” (a.k .a. t wisted comple xes). The latter are described b y solutions to the Maurer-Cartan equations X 1 ≤ l ≤ r − 1 m l ( α, . . . , α ) = 0 where α = ( a ij ) i 1 M od N . Eac h isomorphism class of an ob ject app ears in the union for inﬁnitely man y v alues of N . In order to a void the “ov ercoun ting” w e deﬁne a subsc heme of ﬁnite ty p e M od 0 N ⊂ M od N consisting of ob jects not isomorphic to ob jects from M od N ′ for N ′ < N . W e conclude that ob jects of P er f ( A ) form an ind-constructible set (more precisely , an ind-cons tructible stac k). One can tak e care about morphism s in the category in a sim ilar wa y . This explains the example e). W e deﬁ ne a p olarization on a non-comm utativ e prop er algebraic v ariet y o v er k (a v ersion of Bridgeland stability condition, see [9]) b y the followin g data and axioms: • an ind-constructible homomorphism cl : K 0 ( C ) → Γ, where Γ ≃ Z n is a free ab elian group of ﬁnite ra nk endow ed with a bilinear form 3 3 In physics litera ture Γ is called the c har ge lattice. 6 h• , •i : Γ ⊗ Γ → Z suc h that for an y tw o ob jects E , F ∈ O b ( C ) we ha v e h cl( E ) , cl( F ) i = χ ( E , F ) := X i ( − 1) i dim Ext i ( E , F ) , • an additiv e map Z : Γ → C , called the central c harge, • a collection C ss of (isomorphism classe s of ) non- zero ob jects in C called the semistable ones, suc h that Z ( E ) 6 = 0 for an y E ∈ C ss , where w e write Z ( E ) for Z (cl( E )), • a c hoice Log Z ( E ) ∈ C o f the logarithm of Z ( E ) deﬁ ned for an y E ∈ C ss . Making a connec tion with [9] w e sa y that t he last three items deﬁne a stability structur e (or stability c ondition) on the category C . F or E ∈ C ss w e denote b y Arg ( E ) ∈ R the imaginary part of Log Z ( E ). The a b o ve data satisfy t he follo wing axioms: • for all E ∈ C ss and fo r all n ∈ Z w e hav e E [ n ] ∈ C ss and Arg Z ( E [ n ]) = Arg Z ( E ) + π n , • for all E 1 , E 2 ∈ C ss with Arg ( E 1 ) > Arg( E 2 ) w e hav e Ext C ≤ 0 ( E 1 , E 2 ) = 0 , • for any ob j ect E ∈ O b ( C ) there exist n > 0 and a c hain of morphisms 0 = E 0 → E 1 → · · · → E n = E (an analog of ﬁltra t io n) suc h that the correspo nding “quotien ts” F i := C one ( E i − 1 → E i ) , i = 1 , . . . , n are semistable and Arg( F 1 ) > Arg( F 2 ) > · · · > Arg( F n ), • for each γ ∈ Γ \ { 0 } the s et of isomorphism classes o f a C ss γ ⊂ O b ( C ) γ consisting of semistable ob jects E suc h that cl( E ) = γ a nd Arg( E ) is ﬁxed, is a constructible set, • (Supp ort Property) Pick a norm k · k on Γ ⊗ R , then there exists C > 0 suc h that for a ll E ∈ C ss one has k E k≤ C | Z ( E ) | . 7 In the ab o v e deﬁnition one can allo w Γ to hav e a torsion. In geo metric examples a), d) for k = C one can tak e Γ = K 0 top ( X ( C )) where K 0 top denotes the top ological K 0 -group. Similarly , in examples b),c) one should take the K 0 -groups with a ppropriate supp orts. Another c hoice for Γ is the image of the algebraic K 0 -group under the Chern ch ara cter. Y et another c hoice is Γ = K num 0 ( C ), whic h is the quotien t o f the group K 0 ( C ) by the in tersection of the left and right k ernels of the Euler form χ ( E , F ). F inally one can pic k a ﬁnite collection of ind-construc tible functors Φ i : C → P er f ( k ) , 1 ≤ i ≤ n and deﬁne cl( E ) = ( χ (Φ 1 ( E )) , . . . , χ (Φ n ( E ))) ∈ Z n =: Γ , where χ : K 0 ( P er f ( k )) → Z is the isomorphism of groups giv en b y the Euler c haracteristic. Remark 1 The origin of t he Supp ort Pr op erty is ge ometric and c an b e ex- plaine d in the c ase of the c ate gory of A -br anes (t he derive d F ukaya c ate gory D b ( F ( X )) ) of a c omp act 3 -dimensio nal Calabi-Y au manifold X . L et us ﬁx a C alabi-Y au metric g 0 on X . Asymptotic al ly, in the lar ge volume limit (as the r esc al e d symple ctic form app r o aches inﬁnity) it g ives rise to the stability c o ndition on D b ( F ( X )) such that stable obje cts ar e sp e cial L agr angian sub- manifolds, and | Z ( L ) | is the volume of L with r esp e ct to g 0 . Then for any harmonic f o rm η one has | R L η | ≤ C | Z ( L ) | . I t fol low s that the norm of the c o homolo gy class of L is b ounde d (up to a sc a l a r fac tor) by the norm of the line ar functional Z . The Suppo r t Prop erty im plies that the set { Z ( E ) ∈ C | E ∈ C ss } is a discrete s ubset of C with at most p olynomially gro wing dens ity at inﬁnit y . It also implies that the stability condition is lo cally ﬁnite in the sense of Bridgeland (see [9]). An y stabilit y condition giv es a b ounded t -structure on C with the corresp onding heart consis ting of semistable ob jects E with Arg( E ) ∈ (0 , π ] and their extensions. Remark 2 The c ase of the classic al Mumfor d no tion of stability with r esp e ct to an ample line bund le (and its r eﬁn e ment for c oher ent she a ves deﬁne d by Simpson) is no t an e xample of the Bridgeland stability c ondi tion , it is r ather a limiting de gener ate c a se of it (s e e [1], [ 7 3] and R e m ark at the end of Se ction 2.1). 8 F or giv en C and a homomorphism cl : K 0 ( C ) → Γ a s ab o v e, let us denote b y S tab ( C ) := S tab ( C , cl) the set of stabilit y conditions ( Z, C ss , (Log Z ( E )) E ∈C ss ). Space S tab ( C ) can b e endo w ed with a Hausdorﬀ top olog y , whic h w e discuss in de tail in Section 3.4. Then w e ha v e an ind-constructib le ve rsion of the follo wing fundamen tal result of Bridgeland (see [9]). Theorem 1 The for g e tting m a p S tab ( C ) → C n ≃ Hom(Γ , C ) given by ( Z , C ss , (Log Z ( E )) E ∈C ss ) 7→ Z , is a lo c al home o morphism. Hence, S tab ( C ) is a complex manifold, not necessarily connected. Under appropriate assumptions one can sho w also that the group of auto equiv a- lences Aut( C ) acts prop erly and disc ontin uously on S tab ( C ). On the qu o- tien t orbifold S tab ( C ) / Aut( C ) there is a natural non-holomorphic action of the group GL + (2 , R ) of orien tation-preserving R -linear a utomorphisms of R 2 ≃ C . 1.3 Donaldson-Thomas in v arian ts for non-comm utativ e 3 d Calabi-Y au v arieties Recall that a non-comm utativ e Calabi-Y au v ariet y of dimens ion d (a.k.a Calabi-Y au category of dimension d ) is giv en by a n E xt -ﬁnite triangulated A ∞ -category C whic h carries a functorial non-degenerate pairing ( • , • ) : Hom • C ( E , F ) ⊗ Hom • C ( F , E ) → k [ − d ] (see e.g. [42], [65], [4 4]), suc h that the p olylinear forms ( m n ( f 0 , . . . , f n ) , f n +1 ) deﬁned on ⊗ 0 ≤ i ≤ n +1 Hom • C ( E i , E i +1 ) by higher comp ositions m n are cyclically in v a ria n t. W e w ill discus s mainly the case d = 3 and assume that our non- comm utativ e 3 d Calabi-Y au v ariety is ind-constructible and endo w ed with p olarization. Under these assumptions w e deﬁne motivic Donald s on-Thomas invaria n ts whic h take v alues in certain Grothendiec k groups of algebraic v arieties (more details are giv en in Se ctions 4 and 6 ) . Assuming some “absence of p oles” conjectures, whic h we discus s in detail in Section 7 one can pass to the “quasi-classical limit” whic h corresponds to the taking of Euler c haracteristic of all relev ant motiv es. In this w ay w e obtain the putative numeric al DT- invariants Ω( γ ) ∈ Q , γ ∈ Γ \ { 0 } . Morally , Ω( γ ) coun ts semistable ob jects of C with a giv en class γ ∈ Γ \ { 0 } . 9 There is a special case when our formulas can b e compared with those from [2] (see Section 7.1). Namely , let us deﬁne a Schur obje ct E ∈ O b ( C ) as suc h that Ext < 0 ( E , E ) = 0 , Ext 0 ( E , E ) = k · I d E . By the Calabi-Y au prop erty in the dimension d = 3 w e kno w that the only p ossibly non- trivial groups Ext i ( E , E ) , i = 0 , 1 , 2 , 3 are Ext 0 ( E , E ) ≃ Ext 3 ( E , E ) ≃ k , Ext 1 ( E , E ) ≃ (Ext 2 ( E , E )) ∗ . In other w ords the ranks are (1 , a, a, 1) , a ∈ Z > 0 . Recall (see [41], [42]) that the deformation theory of an y ob ject E ∈ O b ( C ) is controlle d b y a diﬀeren tial-graded Lie algebra (DGLA for short) g E suc h that H i ( g E ) ≃ Ext i ( E , E ) , i ∈ Z . F o r a giv en Sc hur ob ject E instead of g E w e can use a DGLA b g E = τ ≤ 2 ( g E ) /τ ≤ 0 ( g E ) wh ere τ ≤ i is the truncation functor. This mak es sense since τ ≤ 0 ( g E ) is an ideal (in the homotop y sense) in g E . The mo diﬁed deformation theory giv es rise to a p erfect obstruction theory in the sense of [2], [4]. The corresp onding mo duli space is the same as the original one, although controlling DGLAs are not q uasi-isomorphic. The con tribution of Sc h ur ob jects to Ω( γ ) can b e iden tiﬁed with the Behrend microlo cal f o rm ula for DT-in v aria nts. F rom this p oin t of view ob jects of the category C should b e inte rpreted as critical p oin ts of the function (called the p oten tial), whic h is obtained from the solution to the so- called classical master equation. The latter has a ve ry natural in terpretation in terms of the non-comm utativ e formal sym plectic dg-sc heme deﬁne d b y the A ∞ -category C endo w ed with a Calabi-Y au structure (see [42]). 1.4 Multiplicativ e w all-crossing form ula The w a ll-crossing form ulas for the num erical D o naldson-Thomas in v arian ts do not dep end on the ir origin and can b e ex pressed in terms of graded Lie algebras. This is explained in Section 2. Our main application is the case o f 3 d Calabi-Y a u categories. Let us recall that if C is an E xt -ﬁnite Calabi-Y au category of the o dd dimension d (e.g. d = 3) then the Euler form χ : K 0 ( C ) ⊗ K 0 ( C ) → Z , χ ( E , F ) := X n ∈ Z ( − 1) n dim Ext n ( E , F ) 10 is ske w-symmetric. In this case we also assume that if C is endo w ed with p olarization, then a sk ew-symmetric bilinear for m h• , •i : Γ ⊗ Γ → Z is giv en and satisﬁes h cl( E ) , cl( F ) i = χ ( E , F ) ∀ E , F ∈ O b ( C ) . In general, having a free a b elian group Γ of ﬁnite rank endo w ed w ith an in teger-v a lued sk ew-symmetric form h• , •i , w e deﬁne a Lie algebra ov er Q g Γ := g (Γ , h• , •i ) , with the basis ( e γ ) γ ∈ Γ and the Lie brack et [ e γ 1 , e γ 2 ] = ( − 1) h γ 1 ,γ 2 i h γ 1 , γ 2 i e γ 1 + γ 2 . This Lie algebra is isomorphic (non-canonically) to the Lie algebra of regular functions on the a lg ebraic P oisson torus Hom(Γ , G m ) endo w ed with the natural translation-in v ariant P oisson brac k et. 4 An additiv e map Z : Γ → C is called gen e ric if there are no t w o Q - independen t elemen ts of the lattice Γ whic h are mapp ed b y Z in t o the same straigh t line in R 2 = C . The set of non-generic maps is a countable union of real hypersurfaces in C n = Hom(Γ , C ). These h yp ersurfaces are called wal ls . Let us c ho ose suc h an additiv e map Z and an arbitrary norm k • k on the re al v ector space Γ R = Γ ⊗ R . W e w ill k eep the same notation for the R -linear extension of Z to Γ R . Finally , assume that w e are giv en a n ev en map Ω : Γ \ { 0 } → Z supported on the set of γ ∈ Γ suc h t hat k γ k ≤ C | Z ( γ ) | for some giv en constan t C > 0. Let ( Z t ) t ∈ [0 , 1] b e a generic pie ce-wise smo o th path in C n = Hom(Γ , C ) suc h that Z 0 and Z 1 are generic. The w all-crossing form ula calculates the function Ω 1 correspo nding to Z 1 in terms of the function Ω = Ω 0 correspo nd- ing to Z 0 . This is analogous t o the analytic con tin uation of a holomorp hic function express ed in terms of its T a ylor co eﬃcien ts. The contin uat io n is unique if it exists, and is not changed under a small deformation of t he path with the ﬁxed endp oints . Let us call strict a sector in R 2 with the v ertex at the o r ig in (0 , 0) whic h is less than 180 ◦ . With a strict sector V ⊂ R 2 w e a sso ciate a group elemen t A V giv en b y the inﬁnite product A V := − → Y γ ∈ Z − 1 ( V ) ∩ Γ exp − Ω( γ ) ∞ X n =1 e nγ n 2 ! . 4 Later we will use the multiplication as well. It is given explicitly b y e γ 1 e γ 2 = ( − 1) h γ 1 ,γ 2 i e γ 1 + γ 2 . 11 The pro duct tak es v alue in a certain pro-nilp oten t Lie g roup G V := G V , Γ , h• , • , i , whic h w e will describe b elow. The right arro w in the pro duct sign means that the pro duct is tak en in the clo ckwise order on the set of ra ys R + · Z ( γ ) ⊂ V ⊂ C . F or the pro duct in the anti-clockw ise order w e will use the left arrow. Let us describe the Lie algebra g V = Lie ( G V ) of the pro-nilpo ten t Lie group G V . W e denote b y C ( V ) a con v ex cone in Γ R whic h is the con v ex h ull of the set of p oin ts v ∈ Z − 1 ( V ) s uch that k v k≤ C | Z ( v ) | . The Lie algebra Lie ( G V ) is the inﬁnite pro duct Q γ ∈ Γ ∩ C ( V ) Q · e γ equipped with the ab o v e Lie brac k et. No w w e can formulate the w all-crossing form ula. It says (roughly) that A V do es not c hange as long a s no lattice p oin t γ ∈ Γ with Ω t ( γ ) 6 = 0 crosses the b oundary of the cone Z − 1 t ( V ) (here Ω t correspo nds to the p oin t t ∈ [0 , 1 ]). By our assumptions, if t = t 0 correspo nds to a non-generic cen tral c harge Z t 0 then there exists a 2- dimensional lattice Γ 0 ⊂ Γ suc h that Z t 0 (Γ 0 ) b elongs to a real line R e iα for some α ∈ [0 , π ]. The w all-crossing form ula describes the change o f v alues Ω( γ ) for γ ∈ Γ 0 and de p ends only on the res triction Ω | Γ 0 of Ω to the lattice Γ 0 . V alues Ω( γ ) for γ / ∈ Γ 0 do not c hange at t = t 0 . Denote by k ∈ Z the v alue of the for m h• , •i on a ﬁxed basis γ 1 , γ 2 of Γ 0 ≃ Z 2 suc h that C ( V ) ∩ Γ 0 ⊂ Z > 0 · γ 1 ⊕ Z > 0 · γ 2 . W e assume that k 6 = 0, otherwise there will b e no j ump in v alues of Ω on Γ 0 . The group elemen ts whic h w e are in terested in can be iden tiﬁed with pro ducts of the following automorphisms 5 of Q [[ x, y ]] preserving the sym plectic form k − 1 ( xy ) − 1 dx ∧ dy : T ( k ) a,b : ( x, y ) 7→ 7→  x · (1 − ( − 1) k ab x a y b ) − k b , y · (1 − ( − 1) k ab x a y b ) k a  , a, b > 0 , a + b > 1 . F or γ = aγ 1 + bγ 2 w e ha v e T ( k ) a,b = exp − X n > 1 e nγ n 2 ! in the ab o ve notation. Any exact sympl ectomorphism φ of Q [[ x, y ]] can b e 5 Here w e write an a utomorphism as acting on elemen ts of the alge br a of functions. The corresp onding automo r phism on p oints is given by the inv erse for mula. 12 decomposed uniquely in to a clo c kwise and an an ti-clo ckw ise pro duct: φ = − → Y a,b  T ( k ) a,b  c a,b = ← − Y a,b  T ( k ) a,b  d a,b with certain exp onen ts c a,b , d a,b ∈ Q . These exp onen ts should b e in terpreted as the limiting v alues of the func tions Ω ± t 0 = lim t → t 0 ± 0 Ω t restricted t o Γ 0 . The passage from the clo c kwise order (when the slop e a/b ∈ [0 , + ∞ ] ∩ P 1 ( Q ) decreases ) to the an ti- clo c kwise order ( when the slope increase s) g iv es the c hange of Ω | Γ 0 as we cross t he wall. It will b e con v enien t to denote T (1) a,b simply b y T a,b . The pro-nilp oten t group generated by transformations T ( k ) a,b coincides with the one generated by tra nsformations T a, | k | b . The compatibilit y of the w all- crossing formu la with the inte g r ality of the n um b ers Ω( γ ) is not ob vious but follow s from: Conjecture 1 If for k > 0 on e de c om p oses the p r o duct T 1 , 0 · T 0 ,k in the opp osite or der: T 1 , 0 · T 0 ,k = Y a/b incr e ases ( T a,k b ) d ( a,b,k ) , then d ( a, b, k ) ∈ Z f o r al l a, b, k . An equiv alent form of this conjecture says that if one decomposes T k 1 , 0 · T k 0 , 1 in the opp osite o rder then all exp onen ts will b elong to k Z . Here are decompositions for k = 1 , 2 T 1 , 0 · T 0 , 1 = T 0 , 1 · T 1 , 1 · T 1 , 0 , T (2) 1 , 0 · T (2) 0 , 1 = T (2) 0 , 1 · T (2) 1 , 2 · T (2) 2 , 3 · · · · · ( T (2) 1 , 1 ) − 2 · · · · · T (2) 3 , 2 · T (2) 2 , 1 · T (2) 1 , 0 , or equiv alently T 1 , 0 · T 0 , 2 = T 0 , 2 · T 1 , 4 · T 2 , 6 · · · · · T − 2 1 , 2 · · · · · T 3 , 4 · T 2 , 2 · T 1 , 0 . Greg Mo ore a nd F rederik Denef p oin ted o ut that the factors in the last form ula corresp ond to the BPS sp ectrum of N = 2 , d = 4 s up er Y ang-Mills mo del studied b y Seiberg and Witten in [63]. A “ph ysical” explanation of our formulas in this contex t w as give n in [24], see also our Section 2.7. 13 F or k > 3 or k ≤ − 1 the decomposition of T ( k ) 1 , 0 · T ( k ) 0 , 1 is not y et kno wn completely . Computer experimen ts giv e a conjectural form ula fo r t he diago- nal term with the slope a/b = 1. The corresp onding sy mplectomorphism is giv en by the map ( x, y ) 7→ ( x · F k ( xy ) − k , y · F k ( xy ) k ) , where the series F k = F k ( t ) ∈ 1 + t Z [[ t ]] is an algebraic h yp ergeometric series giv en f o r k > 3 b y the form ulas: ∞ X n =0  ( k − 1) 2 n + k − 1 n  t n ( k − 2) n + 1 = exp ∞ X n =1  ( k − 1) 2 n n  k ( k − 1) 2 t n n ! . The f unction F k satisﬁes the equation F k ( t )  1 − tF k − 2 k ( t )  k − 1 = 0 . Remark 3 The ab ov e example f o r k = 1 is c omp atible with the exp e cte d b e h avior of Donaldson- Thomas inv ariants whe n we h a ve two spheric al obje cts E 1 , E 2 ∈ C (s p hericity me ans that Ext • ( E i , E i ) = H • ( S 3 ) ) such that Ext 1 ( E 2 , E 1 ) = k , Ext n ( E 2 , E 1 ) = 0 for n 6 = 1 . In this c ase on the one side of the wal l we have two semistable obje cts E 1 , E 2 , and on the other side we have thr e e semistable obje cts E 1 , E 2 , E 12 wher e E 12 is the extension of E 2 by E 1 . In the c ase of the derive d of the F ukaya c at- e g o ry t he obje cts E i , i = 1 , 2 c an c orr esp ond to emb e dde d sp e cial L agr angian spher es interse cting tr ansversal ly at one p oi n t. Then E 12 c o rr esp onds to their L ag r angian c onne cte d sum. The automorphisms T a,b are a sp ecial case of the more general ones. Namely , w e can consider the follow ing rational automorphism s of g Γ (consid- ered as a P oisson algebra): T γ : e µ 7→ (1 − e γ ) h γ ,µ i e µ , γ , µ ∈ Γ . The group elemen t A V in the ab ov e notation has the form A V = − → Y γ ∈ Z − 1 ( V ) ∩ Γ T Ω( γ ) γ 14 and acts on a completion of g Γ . It is easy to quan tize this Poiss on alge- bra. The c orresp onding algeb ra (quan tum torus) is additiv ely generated by quan tum generators ˆ e γ , γ ∈ Γ sub ject to the relations ˆ e γ ˆ e µ = q 1 2 h γ ,µ i ˆ e γ + µ , where q is a parameter (with the classical limit q 1 2 → − 1). The n one has form ulas similar to the ab o v e for the “quan tum” analogs of automorphisms T γ , γ ∈ Γ (see Sections 6 .4 and 7.1). F or general k ≥ 2 the decomp osition of the pro duct T 1 , 0 · T 0 ,k as in Con- jecture 1, describ es numeri cal DT-in v ariants of the Calabi-Y au category as- so ciated with the Kronec ke r quiv er consisting of tw o v ertices and k parallel arro ws (see Section 8 for a general theory). Recen t pap er [58] giv es an ex- plicit form ula for d ( a, b, k ) in terms of the Euler c haracteristic of the framed mo duli space of se mistable repres entations of the q uiv er. Moreo v er, a w eak form of the in tegrality Conjecture 1 is pro v ed in [58]. 1.5 Some analogie s and sp eculations The ab ov e form ulas for symplectomorphisms are partially motiv ated b y [40], Section 10, where similar form ulas app eared in a diﬀeren t problem. Both form ulas in v olv e Hamiltonian v ector ﬁelds ass o ciated with the dilogarithm function. The p roblem discus sed in [40] w as the reconstruction of the rigid analytic K3 surface from its ske leton, whic h is a sphere S 2 equipped with an in tegral aﬃne structure, singular at a ﬁnite s et of p oin ts. The group whic h is ve ry similar to the pro-nilp otent group G V w as in tro duced in the lo c. cit. where w e assigned symplec tomorphisms to edges of a certain tree in S 2 . That tree should b e though t of as an analog of the w alls in the space of stability structures. Edges of the tree (we called them “line s” in [40]) corresp ond to pseudo-holomorphic discs with the b oundary o n the La grangian toric ﬁb ers of the dual K3 surface. When w e approac h the “large complex structure limit” cusp in the mo duli space of K3 surfaces, the discs degenerate in to gradien t lines of some smo o t h functions on S 2 , th us deﬁning edges of the tree. Hence the reconstruction problem for K3 surfaces (and for higher- dimensional Calabi-Y au manifolds, see [27], [28]) is gov erne d b y the coun ting of rational curv es in the mirror dual Calabi-Y au manifold. This observ a tion suggests by a nalogy the questions b elow. 1) First, let us note that w e may assume that the bilinear form h• , •i is non-degenerate on Γ b y replacing Γ by a “larger” lattice (e.g. by Γ ⊕ Γ ∨ , 15 where Γ ∨ = Hom(Γ , Z ) is the dual lattice , see Section 2.6). Then the Lie algebra g Γ will be realized as the Lie algebra of exact Hamiltonian v ector ﬁelds on the algebraic symplectic torus Hom(Γ , G m ). The collection of for- mal symplectomorphism s A V deﬁned ab o v e giv e rise to a rigid analytic space X an o v er a ny non-archime dean ﬁeld, similarly to [40]. This space carries an analytic sympl ectic for m and describ es “the b ehavior at inﬁnit y” of a (p ossi- bly non-algebraic) formal smo oth symplectic sc heme o v er Z . String Theory suggests that there exists an actual complex symplectic manifold M ( vec tor or h yp er m ultiplet mo duli space) admitting a (partial) com pactiﬁcation M and suc h that X an ( C (( t ))) = M ( C [[ t ]]) \ ( M ( C [[ t ]]) ∪ ( M \ M )( C [[ t ]])) , i.e. it is the space of fo rmal paths hitting the compactifyin g divisor but not b elonging to it). In the case of the F uk a y a category of a com plex 3 d Calabi-Y au manifold X the space M lo oks “at inﬁnit y” as a deformation of a complex symplectic manifold M cl where dim M = dim M cl = dim H 3 ( X ). The latter is the total space of the bundle M cl → M X , whe re M X is the mo duli space of complex structures on X . The ﬁb er of the bundle is isomor- phic to the space ( H 3 , 0 ( X ) \ { 0 } ) × ( H 3 ( X , C ) /H 3 , 0 ( X ) ⊕ H 2 , 1 ( X ) ⊕ H 3 ( X , Z )) parametrizing pairs (holo morphic volume element, p oi n t of the interme diate Jac o bian) . 6 F urthermore, as w e discuss in Section 7.2, w e expect that there is a complex in tegrable system asso ciated with an arbitrary homolog ically smo oth 3 d Calabi-Y au category and the ﬁber being the “Deligne c ohomology” of the category . 2) Is it true that the coun ting of the inv arian ts Ω( γ ) for C is equiv a len t to the coun ting of (some) holomorphic discs “near inﬁnit y” in M ? Is it p ossible to construc t an A ∞ -category associated with those disc s a nd to pro v e that it is a 3- dimensional Calabi-Y au category? 3) The study of the dependence of BPS states on a point of the moduli space o f vec tor and hy p er m ultiplets giv en in [18] and [13] suggests that M is h yp erk¨ ahler and the in v a rian ts Ω( γ ) for C ( counting of ob jects of C ) c an b e inte rpreted as the coun ting o f some “quaternion curv es” in M . Punctures 6 In a very interesting paper [24] a construction of the hyper k¨ ahler structure on M was suggested by means of our wall-crossing formulas. 16 “at in ﬁnity ” of those curv es can b e interp reted as 4 d black holes. It w ould b e nice to think ab out the problem of coun ting suc h maps as a “quaternionic analog” of the coun ting of rational Gromov - Witten inv a rian ts. Hop efully (b y the analog y with the “Gromov-Witten story”) o ne can deﬁne an a ppro- priate A ∞ -category (“quaternionic F uk a y a category”) and prov e th at it is a 3-dimensional Calabi-Y au category . This w ould relate our in v a r ia nts Ω ( γ ) with “quaternionic” Gromo v-Witten inv arian ts. 4) Geometry similar to the one discus sed in this pap er also appears in the theory of m o duli s paces of holomorphic ab elian d iﬀeren tials (see e.g. [78]). The mo duli spac e of ab elian d iﬀeren tials is a complex manifold, div ided b y real “ walls” of co dimension one in to pieces glued from con v ex cones. It also carries a natural non-holomorphic action of the group GL + (2 , R ). There is an analog of the cen tral c harge Z in the story . It is giv en by the integral of an ab elian diﬀeren tial o ve r a path b et w een mark ed p oin ts in a complex curv e. This mak es plausi ble the idea that the mo duli space o f ab elian diﬀeren tials asso ciated w ith a complex curv e with mark ed points, is isomorphic to the mo duli space of stabilit y structures on the (prop erly deﬁned) F uk a y a category of this curv e. 5) W e exp ect that our wall-crossi ng form ulas are re lated to those in the Donaldson theory of 4 d manifolds with b + 2 = 1 (cf. e.g. recen t paper [52]) as well a s with Borche rds hyperb olic Kac-Mo o dy algebras and m ultiplicativ e automorphic forms. The formulas from [12] also lo o k ve ry similar. 1.6 Ab out the con ten t of the pap er In Section 2 w e w ork out in detail the approac h to the in v ariants Ω( γ ) and the w all-crossing form ula s ke tche d in the In tro duction in the framew ork of graded Lie algebras. It is based on the notion of stability data whic h admits t w o equiv alent descriptions : in terms of a colle ction of elemen ts a ( γ ) of a graded Lie algebra g = ⊕ γ ∈ Γ g γ and in terms of a collection of group ele- men ts A V whic h satisfy the “F actorization Prop ert y”. The latter sa ys that A V 1 A V 2 = A V for an y strict sector V and its decomp osition in to tw o sectors V 1 , V 2 (in the clo c kwise order) b y a r ay emanating from the v ertex. W e de- ﬁne the top ology on the space of stabilit y data. It immediately leads to the w all-crossing fo rm ula. Then we discuss a sp ecial case when the lattice carry an in teger-v alued sk ew-symmetric bilinear form. The s k ew-symmetric fo rm on the la t tice Γ giv es rise to a P oisson structure on the torus Hom(Γ , G m ) of its c haracters. Then w e in tro duce a double symp le c tic torus, whic h cor- 17 respo nds to the lattice Γ ⊕ Γ ∨ . This allo ws us to construc t an em b edding of the pro-nilp oten t groups G V (see Section 1.4) into the group of formal symplec tomorphisms of the double torus. W e show how the “nume rical DT- in v a ria n ts” Ω( γ ) a rise from a collection of elemen ts A V whic h satisfy the F actorization Prop ert y A V 1 A V 2 = A V for any strict sector V . W e in tro- duce the notion of the “w all of s econd kind” s uch that (in the categorical framew ork) crossing su ch a wall corresp onds to a c hange of th e t -structure. Then the m ultiplicativ e w all-crossing formula is e quiv alen t to the trivialit y of t he mono drom y of a “non- linear conn ection” on the space of nume rical stabilit y data. Also w e discuss the relationship with the work s o f Joy ce, and Bridgeland and T o ledano-Laredo b y in tro ducing (unde r ce rtain conditions) a connection with irregular singularitie s on C . In Section 2 .7 . we explain ho w stabilit y data ar ise from complex in tegrable s ystems. W e illustrate our consideration b y an examp le of Seib erg- Witten curv e. Arising g eometry is the same as in the “string junction” interp retation of Seiberg-Witten mo del (see e.g. [53]). The last section is dev oted to stabilit y data on gl ( n, Q ). It is related t o the study of v acua in N = 2 supersymmetric Quan tum F ield Theories (see [11]). Section 3 is dev oted to some basics on ind-constructible categories, in- cluding the deﬁnition of the top ology on the space of stabilit y structures. Also w e discuss the notion of the p oten tial of an ob ject of Calabi-Y au cat- egory and the categorical v ersion of the w all-crossing formu la. The w ay it is form ulated is clos e i ntuitiv ely to the ph ysics considerations: w e lo o k ho w the “motive ” of the mo duli sp ace of semistable ob jects c hanges when some of exact triangles b ecome unstable. Section 4 is dev oted to motivic functions and motivic Milnor ﬁb er. W e start b y recalling basics on motivic functions and motivic in tegration, includ- ing their equiv arian t v ersions (motivic stac k functions, s ee also [35 ]). Then w e discuss t he notion of motivic Milnor ﬁber in tro duced by D enef and Lo eser as w ell as its l -adic incarnation. Rough idea is to use the motivic Milnor ﬁb er of the p oten tial of the 3 d Calabi-Y au category in order to deﬁne in v arian ts of the ind-constructible set of semistable ob jects. The tec hnical question arises: there might b e t w o quadrics with the same rank and determi nant but diﬀeren t Chow motiv es. In order to resolv e this diﬃcult y w e introduce cer- tain equiv a lence relation o n motiv ic functions, so that in the quotien t suc h quadrics are the same. Also, w e discuss an imp ortan t integral iden tity whic h will play the k ey role in Section 6. Section 5 is devote d to an additional structure, whic h w e call o r ientation 18 data. It is a sup er line bundle on the space of ob jects of our category . Roughly , it is a square ro ot o f the sup er line bundle o f cohomology . Although the n umerical DT-in v arian ts do not dep end on the orien tation data, the motivic DT-in v a rian ts in tro duced in Section 6 depend o n it in an essen tial w a y . Section 6 is dev oted to the deﬁnition of motivic DT-in v arian ts. First w e deﬁ ne the motivic Hal l algebr a of an ind-construc tible triangulated A ∞ - category and pro v e its ass o ciativit y . It generalizes the deriv ed Hall algebra in tro duced by T o ¨ en in [70]. W e deﬁne the motivic vers ion A Hall V of the elemen t A V as a n in v ertible elemen t of the completed motivic Hall algebra asso ciated with the sector V . The elemen ts A Hall V satisfy the F actorization Prop erty . Basic idea behind the F actorization Prop erty (and he nce the m ultiplicativ e w all-crossing form ula) is that the inﬁnite product in the latter corresponds to the in tegration o v er the space of al l ob jects of the category C V generated b y exte nsions of semis table o b jects with the cen tra l c harge in V . The latter can b e easily con trolled when w e cross the w all. Motivic DT-in v aria nts a pp ear as elemen ts of a ce rtain quan tum torus with the co eﬃcien t ring giv en by the equiv a lence classes of motivic functions. Ba- sic fact is the theorem whic h say s that in the case of 3 d Calabi- Y au category there is a homomorphism of the motivic Hall algebra into the motivic quan- tum torus deﬁned in terms of th e motivic Milnor ﬁber of the p oten tial. In man y cases the im ages of the elem ents A Hall V can be computed explicitly in terms of the motivic v ersion of the quan tum dilogarithm f unction. The im- ages of A Hall V are denoted b y A mot V . This collection (one eleme nt for ev ery strict sector V ) is called the motivic DT-in v a r ia nt. The collection of thes e elemen ts satisfy the F actorization Prop ert y . Replacing motiv es b y their Serre p olynomials, w e obtain q -analog s of D o naldson-Thomas in v ariants, denoted b y A V , q . W e discuss their prop erties as w ell a s the “quasi-classical limit” A V as q 1 / 2 → − 1. W e f o rm ulate the conjectures ab out the exis tence of the limit (absence o f p oles conjecture) and in tegrality prop ert y of the limits (in- tegralit y conjecture). The latter are related to the Conjecture 1 from Section 1.4. These conjectures are discussed in detail in Section 7, where w e presen t v arious arg umen ts and computations in their fa v or. Presumably , the tec h- nique de ve lop ed b y D. Joy ce can lead to the proof of our conjecture s. The n umerical DT-inv arian ts Ω : Γ \ { 0 } → Z are de ﬁned as co eﬃcien ts in the decomposition of symp lectomorphism A V in to the product of p o w ers T Ω( γ ) γ in the clo c kwise order. 19 In Section 8 we consid er in de tail the case of 3-dimensional Calabi-Y au category en dow ed with a ﬁnite collection of spherical g enerators satisfying some extra prop erty (cluster collection). Suc h categories corresp ond to quiv- ers with p otentials (Theorem 9). Apply ing general consider ations fr o m the previous sections w e form ulate some results ab out quiv ers and m utations. They are almost ob vious in the categorical framew ork, but seem to b e new in the framew ork of quiv ers. Finally we explain that cluster transformations app ear naturally as birational symple ctomorphisms of the double torus in the case when crossing of the w all of second kind corresp onds to a m utation at a v ertex of the quiv er (equi v alen tly , to a m utation at the corresponding spherical ob j ect of the Calabi-Y au catego ry). Sev eral parts of the theory presen ted here hav e t o b e dev eloped in more detail. This concerns ind-constructible categories and motivic stac k func- tions. Also, w e presen t only a sk etc h of the pro of of the l -adic v ersion of the main iden tity in Section 4.4, leav ing aside few tec hnical details (whic h are not diﬃcult to restore), and the deﬁnition of the o rien tation data for cluster categories in Section 8.2 is le ft as a conjecture (although there is no doubt that it should b e true). A ckno w le dgm ents. W e thank to Mina Aganagic, R o ma Bezruk avnik ov, T om Bridgeland, F rederik D enef, Eman uel D iaconescu, Pierre D eligne, Sasha Gonc harov , Mark Gross, Dominic Jo yce, Greg Mo ore, Andrew Neitz ke , Nikita Nekraso v, Andrei Okounk ov, Rah ul P andharipande, Markus Reinek e, Balazs Szendr¨ oi, Don Z agier for useful discussions and corresp ondence. Y.S. thanks to IHES and the Univ ersit y P aris-6 for excellen t researc h conditions. His w ork w as partially supp orted by an NSF grant. 2 Stabilit y conditions for graded Lie alge b ras 2.1 Stabilit y data Let us ﬁx a free abelian g r o up Γ of ﬁnite rank, and a graded Lie algebra g = ⊕ γ ∈ Γ g γ o v er Q . 7 Deﬁnition 1 Stability data on g is a p air σ = ( Z, a ) such that: 1) Z : Γ → R 2 ≃ C is a homomorphism of ab elia n gr oups c al le d the c e n tr al char ge; 7 In e x amples g is a R -linear Lie algebr a, where R is a comm utative unital Q - algebra. 20 2) a = ( a ( γ )) γ ∈ Γ \{ 0 } is a c ol le ction of elements a ( γ ) ∈ g γ , satisfying the fol l o w ing Suppor t P rop erty : Pick a norm k • k on t he ve ctor sp ac e Γ R = Γ ⊗ Z R . Then ther e exists C > 0 such that for any γ ∈ Supp a (i.e. a ( γ ) 6 = 0 ) one has k γ k≤ C | Z ( γ ) | . Ob viously the Supp ort Prop erty does not dep end on the ch oice of the norm. W e will denote the set of all stabilit y data on g b y S tab ( g ). Later we will equip this set with a Hausdorﬀ top o logy . The Supp ort Prop erty is equiv a len t to the followin g condition (whic h w e will also call the Supp ort Prop erty): Ther e exists a quadr atic form Q o n Γ R such that 1) Q | Ker Z < 0 ; 2) Supp a ⊂ { γ ∈ Γ \ { 0 }| Q ( γ ) > 0 } , wher e we use the same no tation Z for the natur al ex tension of Z to Γ R . Indeed, w e ma y assume that the norm k • k is the Euclidean norm in a c hosen basis and tak e Q ( γ ) = − k γ k 2 + C 1 | Z ( γ ) | 2 for suﬃcien tly large p ositiv e constan t C 1 . Generically Q has signature (2 , n − 2), where n = rk Γ. In degenerate cases Q can hav e signature (1 , n − 1) or (0 , n ). F or a give n quadratic form Q on Γ R w e denote by S tab Q ( g ) ⊂ S tab ( g ) the set of stability data satisfying the a b o ve conditions 1) and 2). Obvious ly S tab ( g ) = ∪ Q S tab Q ( g ), where the union is tak en o v er all quadratic forms Q . Remark 4 In the c ase of a 3 -dimensio n al Calabi-Y au mani f o ld X ther e is a natur al c andidate for the quadr atic form Q of the signatur e (2 , n − 2) ne e de d to formulate the Supp ort Pr op e rty. Namely, identifying H 3 ( X , R ) with H 3 , 0 ( X , C ) ⊕ H 2 , 1 ( X , C ) we c an e quip H 3 ( X , R ) w ith the c om plex structur e. F urthermor e , the natur al symp le ctic form c oming fr om the Ho dge structur e gives ris e to a pse udo - h ermitian form on H 3 ( X , R ) of the sig n atur e (2 , n − 2) , wher e n = dim R H 3 ( X , R ) . On e c an ask whether this form is p ositive on ev- ery sp e cial L agr angian submanifold of X . If this is true, then the Supp ort Pr op erty g i ves rise t o a b ound on the supp ort of the function Ω discusse d in Se ction 1.4. Supp ort Prop ert y implies the follow ing estimate for the n um b er of p o in ts in the Supp a with the cen tral c harge inside of the disc of radius R : # ( Z (Supp a ) ∩ { z ∈ C | | z | ≤ R } ) = O ( R n ) , 21 where R → ∞ and n = rk Γ, Therefore the set Z (Supp a ) is discrete in C and do es not contain zero. Remark 5 It se em r e asonable to c onsider “limiting c ases” of stability da ta when the Supp ort Pr op erty is no t satisﬁe d. Then the numb ers Re Z and Im Z ar e al lo we d to take values in arbitr ary total ly or der e d ﬁelds, e.g. R (( t )) (her e t is a formal p ar ameter such that t > 0 and t < x for any x ∈ R > 0 ). Some of our c onsider ations b elow make sense in this situation. I n the fr amework of st abili ty c o nditions on triangulate d c ate gories such structur es a pp e ar e d in [1], [73]. 2.2 Reform u lation of the stabilit y data In what follo ws w e will consid er v arious cones in Γ R and in R 2 i.e. subsets, whic h a re closed under addition and multipli cation by a p ositiv e real n um b er. W e a ssume that the v ertex of the cone (i.e. the zero of the ve ctor space) do es not b elong to the cone. W e will call a cone strict if it is non-empt y a nd do es not contain a straigh t line. In particular, all strict cones on the plane (we will call them strict sec tors) are sectors, whic h a r e smaller than 1 80 degrees (not necessarily closed or op en). W e allo w the sector to b e degenerate (whic h means that it is a ray with the v ertex at the orig in). W e o r ient the plane (and hence all sectors) in the clo c kwise direction. W e write l 1 ≤ l 2 if the ray s l 1 , l 2 b ound a strict closed sector and l 1 precedes l 2 in the clo c kwise order (w e a llo w l 1 = l 2 ). Let us ﬁx a q uadratic form Q on Γ R . W e are going to describ e b elow another set of data and will sho w tha t it is naturally isomorphic to the set S tab Q ( g ). Let S b e the set o f strict sectors in R 2 p ossibly degenerate (ra ys). W e denote b y d S tab Q ( g ) the set of pairs ( Z , A ) suc h that: a) Z : Γ → R 2 is an additive m a p such that Q | Ker Z < 0 ; b) A = ( A V ) V ∈S is a c ol le ctions of ele ments A V ∈ G V , Z,Q , wher e G V , Z,Q is a pr o-nilp otent gr oup with the pr o-nilp otent gr a d e d Lie al g e br a g V , Z,Q = Y γ ∈ Γ ∩ C ( V ,Z ,Q ) g γ , wher e C ( V , Z , Q ) is the c onvex c one gener ate d by the set S ( V , Z , Q ) = { x ∈ Γ R \ { 0 }| Z ( x ) ∈ V , Q ( x ) > 0 } . 22 The ab ov e deﬁnition mak es sense b ecause the cone C ( V , Z , Q ) is strict, as one can easily see by elemen tary linear algebra. Hence for a triangle ∆ whic h is cut from V b y a straigh t line, an y γ ∈ Z − 1 (∆) can b e represen ted as a sum of ot her elemen ts of Γ ∩ C ( V , Z , Q ) in ﬁnitely man y w ays . F urthermore, the triangle ∆ deﬁnes an ideal J ∆ ⊂ g V , Z,Q consisting of elemen ts y = ( y γ ) ∈ g V , Z,Q suc h that for ev ery comp onen t y γ the corresponding γ do es not b elong to the conv ex hull of Z − 1 (∆). Then the quotien t g ∆ := g V , Z,Q /J ∆ is a nilp oten t Lie algebra, and g V , Z,Q = lim ← − ∆ ⊂ V g ∆ . Let G ∆ = exp( g ∆ ) b e the nilp oten t group corres p onding to the Lie algebra g ∆ . Then G V , Z,Q = lim ← − ∆ G ∆ is a pro- nilp oten t group. If V = V 1 ⊔ V 2 (in the clo c kwise order) then there are natural em b eddings G V i ,Z,Q → G V , Z,Q , i = 1 , 2. W e impo se the follo wing axiom on the set of pairs ( Z, A ): F actorization Pr op erty : The elem e nt A V is given by the p r o duct A V = A V 1 A V 2 wher e the e quality is understo o d i n G V , Z,Q . W e remark that if Q 1 ≤ Q a nd b oth f o rms Q, Q 1 are negativ e on Ker Z then G V , Z,Q 1 ⊂ G V , Z,Q for an y V ∈ S . W e sa y that the ( Z , A ) ∈ d S tab Q ( g ) and ( Z ′ , A ′ ) ∈ d S tab Q ′ ( g ) are equiv alent if Z = Z ′ := Z and there exists Q 0 suc h that Q ≤ Q 0 , Q ′ ≤ Q 0 , Q 0 | Ker Z < 0 and moreov er for an y V ∈ S w e ha v e A V = A ′ V as elemen ts of the group G V , Z,Q 0 . Theorem 2 1) F or a ﬁxe d Q ther e is a n atur al bije ction b etwe en sets d S tab Q ( g ) and S tab Q ( g ) . 2) Any two elem e n ts of d S tab Q ( g ) a nd d S tab Q ′ ( g ) ar e e quivalent if and on ly if they deﬁn e the same elem e nt in S tab ( g ) . Pr o of. Supp ose that w e are giv en a pa ir ( Z , A ) ∈ d S tab Q ( g ). In order to construct the corres p onding elemen t in S tab Q ( g ) we tak e the same Z as a homomorphism Γ → R 2 . What is left is to construct a collection a ( γ ) ∈ g γ . W e deﬁne it suc h as follo ws. a) If Z ( γ ) = 0 then w e set a ( γ ) = 0. b) Suppose Z ( γ ) 6 = 0. Let us conside r the ray l = R > 0 Z ( γ ). Then w e ha ve an elemen t log ( A l ) ∈ g l,Z ,Q ⊂ Q γ ∈ Γ g γ . W e denote by a ( γ ) the comp onen t of log ( A l ) whic h b elongs to g γ . This assignmen t give s rise to stability data ( Z , a ) ∈ S tab Q ( g ). In order to sho w that it is injectiv e, we observ e that 23 the F a ctorization Prop erty implies that A V = Q − → l ⊂ V A l , where the pro duct is take n in the clo c kwise order ov er the set of all rays l whic h b elong to V . Indeed, let us consider the image of A V in G ∆ . Then only ﬁnitely many ra ys con tribute to the pro duct Q − → l ⊂ V A l , and the pro duct formula fo llows from the F actorization Prop ert y . Since G V , Z,Q = lim ← − ∆ G ∆ the desired equalit y holds. Con v ersely , if we ha ve stabilit y data ( Z , a ) ∈ S tab Q ( g ) , then w e construct a pair ( Z , A ) ta king the same Z and Q , and for an y ra y l w e set A l = exp   X γ ∈ Γ ∩ C ( l, Z,Q ) a ( γ )   . Notice that A l = 1 if there are no eleme nts γ such that Z ( γ ) ∈ l . W e deﬁne A V for any V ∈ a using the F actorization Prop erty , i.e. A V = Q − → l ⊂ V A l . This pro v es part 1) of the theorem. P art 2) follow s immediately from deﬁnitions. The theorem is prov ed.  Remark 6 We wil l use the same name “stabili ty da ta” for either of the set of data which app e ar in the ab ove the o r em an d w i l l denote either set by S tab ( g ) . Remark 7 L e t R 2 \ { (0 , 0) } = ⊔ 1 ≤ i ≤ n V i , whe r e V i , 1 ≤ i ≤ n ar e strict (semiclose d) se ctors. Then the stability data with a given c en tr al char ge Z ar e uniquely determin e d b y an arbitr ary c ol le ction of e l e ments A V i ∈ G V i ,Z,Q for some quadr atic form Q . There ex ists a generalization of stabilit y data suitable f or motivi c Hall algebras. Namely , let us a ssume that the Lie algebra g carries an automor- phism η suc h that η ( g γ ) = g − γ for any γ ∈ Γ. Deﬁnition 2 Symmetric stability data for ( g , η ) is a p air ( Z, b a ) wher e Z : Γ → C is an additive m ap a nd b a is a map ( γ , ϕ ) 7→ b a ( γ , ϕ ) ∈ g γ wher e ϕ ∈ R , γ ∈ Γ is such that Z ( γ ) ∈ R > 0 e iϕ and b a ( γ , ϕ + π ) = η ( b a ( γ , ϕ )) . All the considerations ab out stabilit y data admit a straightforw ard gener- alization to the symmetric case. W e will use them without further commen ts. 24 Remark 8 L e t H Γ b e a Γ -gr ade d unital asso ciative algebr a c onsider e d as a gr a d e d Lie algebr a. Then the pr o-nilp o tent gr oups G V , Z,Q discusse d ab ove ar e the g r oups o f invertible elements of the form f = 1 + . . . in the pr o-nilp otent asso ciative a lgebr as which ar e c o mpletions of H Γ . Remark 9 De c om p osition g = ⊕ γ ∈ Γ g γ and the Lie a l g ebr as g V , Z,Q ar e si m - ilar to the r o ot de c omp osition and nilp otent sub algebr as in Ka c-Mo o dy Lie algebr as. The involution γ 7→ − γ is similar to the “C artan involution”. These analo gies dese rve further study, sinc e Donaldson-Thomas inva ria nts (mor e pr e cisely, c ounting functions for BPS states) app e ar in physics as a kind of char acter formulas (se e e.g. [18], formula (2.7)). In p articular o ur multiplic ative wal l-cr ossing formulas in the c ase of wal l of se c o n d kind should b e r elate d to automorphic forms of Bor cher ds (se e [8]). T he motivic Hal l al- gebr a deﬁne d b elow in Se ction 6 c ould b e thought of as the motivic version of the algebr a of BPS states (se e [29]). 2.3 T op ology and the w all-crossing form ula Here we are going to in tro duce a Hausdorﬀ top ology on the set of stabilit y data in suc h a w ay that the forgetting map S tab ( g ) → Hom(Γ , C ) ≃ C n , ( Z , a ) 7→ Z will b e a lo cal homeomorphism. In particular S tab ( g ) carries a structure of a complex manifold (in general with an uncoun table nu mber o f comp onen ts, eac h of whic h is para compact). In order to deﬁne the top ology w e deﬁne the notion of a contin uous family of p oints in S tab ( g ). Let X b e a top ological space, x 0 ∈ X b e a p oin t, and ( Z x , a x ) ∈ S tab ( g ) b e a family para metrized b y X . Deﬁnition 3 We say that the family is c ontinuous at x 0 if the fol lowing c o nditions ar e satisﬁe d: a) The map X → Hom(Γ , C ) , x 7→ Z x is c ontinuous at x = x 0 . b) L et us cho ose a quadr atic form Q 0 such that ( Z x 0 , a x 0 ) ∈ S tab Q 0 ( g ) . Then ther e exis ts an o p e n neighb orho o d U 0 of x 0 such that ( Z x , a x ) ∈ S tab Q 0 ( g ) for al l x ∈ U 0 . c) F or any close d strict se ctor V such that Z (Supp a x 0 ) ∩ ∂ V = ∅ the map x 7→ log A V , x,Q x ∈ g V , Z x ,Q x ⊂ Y γ ∈ Γ g γ 25 is c ontinuous at x = x 0 . Her e we endow the ve ctor sp ac e Q γ ∈ Γ g γ with the pr o duct top olo gy of discr ete s e ts, and A V , x,Q x is the gr oup element asso c i - ate d with ( Z x , a x ) , se ctor V and a quadr atic form Q x such that ( Z x , a x ) ∈ S tab Q x ( g ) . Remark 10 Part c) of the Deﬁ nition 3 me ans that for any γ ∈ Γ \ { 0 } the c o mp onent of log A V , x,Q x b e l o nging to g γ is l o c al ly c onstant as a function of x in a neig hb orho o d of x 0 . The elemen t log( A V , x,Q x ) ∈ Q γ ∈ Γ g γ do es not dep end on Q x , e.g. w e can tak e Q x := Q 0 for x close to x 0 . The conti nuit y means informally that fo r an y closed triangle ∆ ⊂ R 2 with one ve rtex at the o r ig in, the pro jection of log A V , Z x ,Q x in to the v ector space ⊕ γ ∈ ∆ g γ do es not dep end on x ∈ X as long as there is no elemen t γ ∈ Supp a x suc h that Z ( γ ) crosses the b oundary ∂ ∆. It is easy to see that the ab ov e deﬁnition giv es rise to a top ology on S tab ( g ). Prop osition 1 This top olo gy is Hausdorﬀ. Pr o of. Let ( Z, a ) and ( Z ′ , a ′ ) b e t w o limits of a sequence ( Z n , a n ) as n → ∞ . W e hav e to pro ve that ( Z , a ) = ( Z ′ , a ′ ). It is clear that Z = Z ′ since Hom(Γ , C ) is Hausdorﬀ. Let us now choose quadratic forms Q and Q ′ whic h are compatible with a and a ′ respectiv ely in the sense of Deﬁnition 3. Then there exists a quadratic form Q 0 suc h that Q 0 is negative on K er Z = Ker Z ′ and also Q ≤ Q 0 , Q ′ ≤ Q 0 . Then for all suﬃcien tly la rge n the form Q 0 is compatible with a n . F or a generic sector V ⊂ R 2 its bo undary ra ys do not in tersect Z (Γ). By part c) of the Deﬁnition 3 we ha ve : A V , Z,Q 0 = A ′ V , Z ′ ,Q 0 since the pro duct Q γ ∈ Γ g γ is Hausdorﬀ. Since any ray in R 2 with the v ertex at the origin can b e obtained as an in tersection of generic sectors then w e conclude that a = a ′ . The Prop osition is prov ed.  Let us ﬁx an elemen t Z 0 ∈ Hom(Γ , C ) and a quadratic form Q 0 compati- ble with Z 0 (i.e. negativ e o n its k ernel). W e denote b y U Q 0 ,Z 0 the connected comp onen t con taining Z 0 in the domain { Z ∈ Hom(Γ , C ) | ( Q 0 ) | Ker Z < 0 } . In what follo ws w e will frequen tly use the follo wing elem entary observ ation. Prop osition 2 If Q is a quadr atic form on a ﬁnite-di m ensional ve ctor sp a c e Γ R and Z : Γ R → C is an R -line ar map such that Q | Ker Z < 0 then the interse ction { x ∈ Γ R | Q ( x ) > 0 } ∩ Z − 1 ( l ) is a c onvex c one (p ossibly empty) for any r ay l ⊂ C with the vertex at the origin. 26 Let γ 1 , γ 2 ∈ Γ \ { 0 } b e t w o Q -linearly indep enden t elemen ts suc h that Q 0 ( γ i ) > 0 , Q 0 ( γ 1 + γ 2 ) > 0 , i = 1 , 2. W e introduce the set W Q 0 γ 1 ,γ 2 = { Z ∈ U Q 0 ,Z 0 | R > 0 Z ( γ 1 ) = R > 0 Z ( γ 2 ) } . In this w a y w e obtain a countable collection of h yp ersurfaces W Q 0 γ 1 ,γ 2 ⊂ U Q 0 ,Z 0 called the w al ls c orr es p onding to Q 0 , γ 1 , γ 2 . W e denote their union b y W 1 := W Q 0 1 and sometimes call it the wal l of ﬁrst kin d (ph ysicists call it the w all of marginal stabilit y). Let us consider a con tin uous path Z t , 0 ≤ t ≤ 1 in U Q 0 ,Z 0 whic h in tersects eac h of these w alls for ﬁnitely many v alues of t ∈ [0 , 1]. Sup p ose that w e hav e a contin uous lifting path ( Z t , a t ) of this path suc h that Q 0 is compatible with eac h a t for all 0 ≤ t ≤ 1. Then for an y γ ∈ Γ \ { 0 } suc h that Q 0 ( γ ) > 0 the elemen t a t ( γ ) do es not c hange as long as t satisﬁes the condition Z t ( γ ) / ∈ ∪ γ 1 ,γ 2 ∈ Γ \{ 0 } , γ 1 + γ 2 = γ W Q 0 γ 1 ,γ 2 . If this condition is not satisﬁed w e sa y tha t t is a discon tin uity p oin t for γ . F or a giv en γ there are ﬁnitely man y discon tinuit y p oints . Notice that for each t ∈ [0 , 1] there exist limits a ± t ( γ ) = lim ε → 0 , ε > 0 a t ± ε ( γ ) (for t = 0 or t = 1 only one of the limits is w ell-deﬁned). Then the con tin uit y of the lifted path ( Z t , a t ) is equiv a lent to the follo wing wal l-c r os s i n g formula whic h holds for a ny t ∈ [0 , 1] and arbitrary γ ∈ Γ \ { 0 } : − → Y µ ∈ Γ pr im , Z t ( µ ) ∈ l γ ,t exp X n > 1 a − t ( nµ ) ! = = exp   X µ ∈ Γ pr im , Z t ( µ ) ∈ l γ ,t , n > 1 a t ( nµ )   = − → Y µ ∈ Γ pr im , Z t ( µ ) ∈ l γ ,t exp X n > 1 a + t ( nµ ) ! , where l γ ,t = R > 0 Z t ( γ ), and Γ pr im ⊂ Γ is the set o f primitiv e v ectors. The ﬁrst and the last pro ducts are taken in the clo c kwise order of Arg ( Z t − ε ) and Arg( Z t + ε ) respectiv ely , where ε > 0 is suﬃcien tly small. Moreo v er, for eac h γ w e ha v e a − t ( γ ) = a + t ( γ ) = a t ( γ ) unless there exist non-zero γ 1 , γ 2 suc h that γ = γ 1 + γ 2 and Z t ∈ W Q 0 γ 1 ,γ 2 . 27 Remark 11 Informal l y sp e aking, the wal l - c r os s i n g fo rmula says that for a very smal l s e c tor V c ontaining the r ay l γ ,t the c orr esp onding e l e ment A V , c o nsider e d as a function of time, is lo c a l ly c onstant in a neighb orho o d of t . F or eac h γ ∈ Γ \ { 0 } the w all-crossing f o rm ula allows us to calculate a 1 ( γ ) is terms of a 0 ( γ ′ ) for a ﬁnite collection of elemen ts γ ′ ∈ Γ \ { 0 } . Morally it is an inductiv e pro cedure on the ordered set of discon tin uit y p oints t i ∈ [0 , 1]. The only thing w e need to c hec k is that for eac h γ ∈ Γ \ { 0 } the computation in v olv es ﬁnitely man y ele men ts of Γ. F o r that w e need some prep ara t io n. First w e in tro duce a partial order on the set S Q 0 = (Γ ∩ Q − 1 0 ( R > 0 )) × [0 , 1] generated by the f ollo wing relations: a) ( γ , t ) > ( γ , t ′ ) if t > t ′ ; b) if γ = P 1 ≤ i ≤ m γ i , Q 0 ( γ i ) > 0 , Z t ( γ i ) ∈ R > 0 Z t ( γ ) , 1 ≤ i ≤ m, m > 2, where not all γ i b elong to Q · γ , then ( γ , t ) > ( γ i , t ) for a ll 1 ≤ i ≤ m . Lemma 1 F or any ( γ , t ) ∈ S Q 0 the set ( γ ′ , t ′ ) ∈ S Q 0 such that ( γ ′ , t ′ ) ≤ ( γ , t ) is a ﬁnite union of sets of the form { γ α } × [0 , t α ] . The Lemma immediately implies the desired result. Corollary 1 The element a t ( γ ) is a ﬁnite Lie expr e ssion of the elem ents a 0 ( γ α ) . Pr o of of the L emma . Let us assume the con trary . Then w e hav e an inﬁnite sequence t 1 > t 2 > t 3 > . . . such that ( γ 1 , t 1 ) > a ) ( γ 2 , t 2 ) > b ) ( γ 3 , t 3 ) > a ) ( γ 4 , t 4 ) > b ) . . . , where the subscri pt a ) or b ) denotes the tw o diﬀeren t p ossibilities for the partial order deﬁned ab ov e. Let t ∞ = lim n →∞ t n . It is easy to see t hat there exists a Euclidean norm k • k on Γ R suc h that for any v 1 , v 2 ∈ Γ R satisfying the prop erties Q 0 ( v i ) > 0 , Z t ∞ ( v 1 ) ∈ R > 0 Z t ∞ ( v 2 ) w e ha v e the inequalities k v i k < k v 1 + v 2 k for i = 1 , 2. Moreo v er the same prop ert y holds if w e replace the map Z t ∞ b y an addi- tiv e map Z wh ich is close to it. Then we conclude that k γ 2 n k > k γ 2 n +1 k = k γ 2 n +2 k for a ll suﬃcien tly large n . This contradic ts to the fact the lattice Γ is discrete in Γ R . The Lemma is pro v ed.  28 The previous discussion allo ws us to lift a generic path Z t , 0 ≤ t ≤ 1 as ab ov e to a unique con tin uous path ( Z t , a t ) ∈ S tab ( g ) , 0 ≤ t ≤ 1 which starts at a giv en point ( Z 0 , a 0 ) ∈ S tab ( g ). In o ther w ords, w e ha v e the notion o f a p ar al lel tr ansp ort along a generic path. This observ a t io n is a part of the follo wing more general statemen t . Theorem 3 F or given quadr atic form Q 0 and ( Z 0 , a 0 ) ∈ S tab Q 0 ( g ) ther e exists a unique c ontinuous map φ : U Z 0 ,Q 0 → S tab ( g ) such that it is a se ction of the natur al pr oje ction S tab ( g ) → U Z 0 ,Q 0 , and φ ( Z 0 ) = ( Z 0 , a 0 ) . Pr o of. W e ha ve already prov ed the existence of a lift ed path ( Z t , a t ) for a g eneric path Z t pro vided the b eginning p oint Z 0 is ﬁxed. What is left to pro v e that the endp oin t ( a 1 , Z 1 ) do es not dep end on a c hoice o f the generic path Z t . W e are going to sk etc h the pro of leavin g the details for the reader. Let us consider an inﬁnitesimally small lo op around the in tersection p oint Z o f tw o or more w alls. W e w ould lik e to pro v e that the mono drom y of the parallel transp o rt along the lo op is trivial. There are t w o p ossibilities: a) there a re tw o diﬀeren t sublattices Γ 1 , Γ 2 ⊂ Γ o f ranks > 2 suc h that Z (Γ i ) , i = 1 , 2 b elong to tw o diﬀeren t lines in the plane R 2 ; b) there exists a sublattice Γ 3 ⊂ Γ suc h that rk Γ 3 > 3 and Z (Γ 3 ) belongs to a line in R 2 . In the case a) the corresponding Lie subalgebras o f the completion of g Γ are gra ded by non-in tersecting subsets of Γ. Hence the corresp onding w all-crossing transformations comm ute. In the case b) let us c ho ose a decomp osition R 2 \ { (0 , 0) } = ⊔ 1 ≤ i ≤ 4 V i , where V i , 1 ≤ i ≤ 4 are strict sectors suc h that R · Z (Γ 3 ) ⊂ V 1 ⊔ V 3 ⊔ { (0 , 0) } . When we mov e around the inﬁ nitesimally s mall lo op the elemen t a ( γ ) can c hange o nly for γ ∈ Γ 3 . Hence w e can replace Γ b y Γ 3 in all computations. The w all-crossing f orm ula implies that the ele men ts A V i , 1 ≤ i ≤ 4 do not c hange along the lo op (moreo v er, by our assu mption w e ha ve A V 2 = A V 4 = 1). By Remark 7 from Section 2.2 w e conclude that the stabilit y data with the cen tral c harge Z ′ whic h is close to Z a re uniquely determined b y Z ′ and the collection of elemen ts A V i , 1 ≤ i ≤ 4. Hence the mono drom y around the lo o p is trivial. Finally one has to c hec k the the glob al mono drom y around a lo op in U Z 0 ,Q 0 is trivial. It follo ws from the fact that the fundamen tal group π 1 ( U Z 0 ,Q 0 ) is generated by the lo op Z 7→ Z e 2 π it , t ∈ [0 , 1]. But the mono drom y around this lo op is trivial for generic Z , b ecause the lo op do es not interse ct the w alls.  . 29 W e can write the wall-crossi ng formu la in the w ay similar to the o ne from the In tro duction. In the case of generic path w e ha v e at a discon tin uity p oint t 0 ∈ [0 , 1] a tw o-dimensional lattice Γ 0 ≃ Z 2 whic h is pro jected b y Z t 0 in to a real line in R 2 . W e c ho ose an isomorphism Γ 0 ≃ Z 2 in suc h a w ay that Q − 1 0 ( R > 0 ) ∩ (Γ 0 \ { 0 } ) is con tained in Z 2 > 0 ∪ Z 2 < 0 . Also w e assume that the orien tation o f Γ 0 ⊗ R deﬁned by Z t agrees with the one on Z 2 > 0 for t = t 0 − ε and is opp osite to it for t = t 0 + ε , where ε > 0 is suﬃcien tly small. Then if γ = ( m, n ) ∈ Z 2 > 0 , and a ± t 0 ( γ ) := a ± ( m, n ), w e can write the the w all-crossing form ula in the following w ay : − → Y ( m,n )=1 exp X k > 1 a − ( k m, k n ) ! = ← − Y ( m,n )=1 exp X k > 1 a + ( k m, k n ) ! , where in the LHS w e tak e the pro duct ov er all coprime m, n in the increasing order of m/n ∈ Q , while in the RHS w e take the pro duct o v er all coprim e m, n in the decreasing order. Bo t h pro ducts are equal to exp( P m,n> 0 a t 0 ( m, n )). 2.4 Crossing the w all of second kind Here we will interpret the parallel transp ort in a diﬀeren t w ay , intro ducing a wall of another kind. W e use the notation f r o m the previous section. In particular, w e ﬁx the quadratic form Q 0 and the connecte d comp onen t U of the set { Z ∈ Hom(Γ , C ) | ( Q 0 ) | Ker Z < 0 } . F or a giv en primitiv e γ ∈ Γ \ { 0 } we in tro duce the set W Q 0 γ = { Z ∈ U | Z ( γ ) ∈ R > 0 } . It is a hy p ersurface in U . W e call it a wal l of s e c ond kind asso ciated with γ . W e call the union ∪ γ W Q 0 γ the wal l of se c ond kind and denote it b y W 2 . Deﬁnition 4 We say t hat a p ath σ = ( Z t ) 0 ≤ t ≤ 1 ⊂ U is short if the c onvex c o ne C σ which is the c onvex hul l of  ∪ 0 ≤ t ≤ 1 Z − 1 t ( R > 0 )  ∩ { Q 0 > 0 } is strict. With a short path w e a sso ciate a pro-nilp otent group G C σ with the Lie algebra g C σ = Q γ ∈ C σ ∩ Γ g γ . The f ollo wing result is obvious . Prop osition 3 F or a generic short p ath σ = ( Z t ) 0 ≤ t ≤ 1 ther e exi s ts no mor e than c ountable set t i ∈ [0 , 1] an d c orr esp o nding primitive γ i ∈ Γ \ { 0 } such that Z t i ∈ W Q 0 γ i . F or e ach i we have : rk Z − 1 t i ( R ) ∩ Γ = 1 . 30 Let us recall the con tin uous lifting m ap φ : U → S tab ( g ) f rom the previous section. In the notation o f the previous Proposition w e deﬁne for any t i a group elemen t A t i = exp ε i X n > 1 a t i ( nγ i ) ! ∈ G C σ , where ε i = ± 1 dep ending on the direction in whic h the path Z t ( γ i ) crosses R > 0 for t suﬃcien tly close t o t i . Theorem 4 F or any short l o op the mono dr omy Q − → t i A t i is e qual to the iden- tity (her e the pr o duct is taken in the incr e asing o r de r of the eleme nts t i ). Pr o of. Here w e also presen t a sk etch of the pro of. Similarly to t he pro of of the Theorem 3 w e consider the case of inﬁnitesimally small lo op σ around a p oint Z suc h that rk Γ 2 = 2 whe re Γ 2 := Z − 1 ( R ) ∩ Γ (i.e. Z is a p oin t where tw o, and hence inﬁnitely man y , w alls of second kind inters ect). Since σ is inﬁnitesimally small w e can replace Γ b y Γ 2 . Then we ha ve the space Hom(Γ 2 , C ) ≃ R 4 whic h contains a coun table collection of w alls consisting of those Z : Γ 2 → C fo r whic h there exists γ ∈ Γ 2 \ { 0 } suc h that Q 0 ( γ ) > 0 and Z ( γ ) ∈ R . All the hypersurfaces con tain R 2 = Hom(Γ 2 , R ) ⊂ Hom(Γ 2 , C ). F actorizing b y this subspace R 2 w e obtain a colle ction of lines with ratio- nal slop es in the union of tw o opp osite strict sectors S ∪ ( − S ) ⊂ R 2 = Hom(Γ 2 , i R ). W e ha v e to pro v e that the pro duct o ve r a lo op surrounding (0 , 0) is the iden tit y elemen t. But it is easy to see that the pro duct o v er the ray s b elonging to eac h of the sectors is equal to the left (resp. right) hand side of the w all- crossing formul a.  Let us no w in tro duce a set X 1 ⊂ Γ × U whic h consists of pairs ( γ , Z ) suc h that γ ∈ Γ \ { 0 } is a non-zero elemen t, Q 0 ( γ ) > 0 , Z ( γ ) ∈ R > 0 and Z − 1 ( R > 0 ) ∩ Γ = ( Q > 0 · γ ) ∩ Γ. Prop osition 4 The set of c ontinuous se c tion s ψ : U → S tab ( g ) such that ψ ( Z ) is c omp atible with Q 0 for a n y Z ∈ U is in one-to-one c orr esp ondenc e with functions ˜ a : X 1 → g such that ˜ a ( γ , Z ) ∈ g γ satisfying the pr o p erty that for any smal l lo op σ the mono dr omy deﬁne d in the pr evious the or em is e qual to the identity. Pr o of. The bijection is giv en by the form ula ˜ a ( γ , Z ) = a ψ ( Z ) ( γ ). By the previous theorem the corresponding mono drom y is trivial. Con v ersely , the 31 trivialit y of the mono drom y is equiv alen t to the w all-crossing form ula in the sp ecial case when a 2-dimensional sublattice of Γ is mapp ed b y Z in to the line R ⊂ C . The general case of an arbitrary line can b e reduced to this one b y a rotation Z 7→ Z e 2 π it (it do es not change the v alues a ( γ ) b ecause w e do not cross the wall of ﬁrst kind).  Let us also in tro duce a set X 2 ⊂ Γ × U w hich consists of suc h pairs ( γ , Z ) that Q 0 ( γ ) > 0 , Z ( γ ) > 0 and there are no non-zero Q -indep enden t elemen ts γ 1 , γ 2 ∈ Γ with the prop erty γ = γ 1 + γ 2 , Q 0 ( γ i ) > 0 , Z ( γ i ) > 0 , i = 1 , 2. Since X 2 is a lo cally-closed h yp ersurface in U × Hom(Γ , C ) it has ﬁnitely man y connected comp onen ts. Obv iously , we ha ve X 1 ⊂ X 2 . It follows from the w all- crossing for mula that for a con tin uous section ψ : U → S tab ( g ) the restriction of the function a to X 2 is lo cally-constan t and uniquely determines the section ψ . Therefore, the v alues of the restric- tion a | π 0 ( X 2 ) pro vides a countable co ordinate system (satisfying a coun table system of equations) on the set o f con tin uous sections { ψ : U → S tab ( g ) } as ab ov e. It can b e compared with another coun table co ordinate system (with no constraints ) give n the v alue ψ ( Z 0 ) fo r Z 0 ∈ U . The latter co ordinate system is not v ery con v enien t since one has to choose a generic Z 0 . 2.5 In v arian ts Ω( γ ) and the dilogarithm Let Γ b e a free ab elian gro up of ﬁnite rank n as b efore, endo w ed with a sk ew-symme tric integer-v alued bilinear form h • , •i : Γ × Γ → Z . Recall the Lie algebra g Γ = g Γ , h• , •i = ⊕ γ ∈ Γ Q · e γ with the Lie brack et [ e γ 1 , e γ 2 ] = ( − 1) h γ 1 ,γ 2 i h γ 1 , γ 2 i e γ 1 + γ 2 . Let us in tro duce a comm utative asso ciativ e pro duct on g Γ b y the form ula e γ 1 e γ 2 = ( − 1) h γ 1 ,γ 2 i e γ 1 + γ 2 . W e denote by T Γ := T Γ , h• , •i the spectrum of this comm utativ e a lg ebra. It is easy to see that T Γ is a torsor o v er the algebraic torus Hom (Γ , G m ). Moreov er T Γ is an algebraic P oisson manifold with the Poiss on brack et { a, b } := [ a, b ] . The Pois son structure on T Γ is inv arian t with res p ect to the action of Hom(Γ , G m ). 32 W e can sp ecify the results of the previous sections to the Lie algebra g Γ . F or stabilit y data ( Z, a ) w e can write uniquely (by the M¨ obius inv ersion form ula) a ( γ ) = − X n > 1 , 1 n γ ∈ Γ \{ 0 } Ω( γ /n ) n 2 e γ , where Ω : Γ \ { 0 } → Q is a function. Then we hav e exp X n > 1 a ( nγ ) ! = exp − X n > 1 Ω( nγ ) X k > 1 e k nγ k 2 ! := exp − X n > 1 Ω( nγ ) Li 2 ( e nγ ) ! , where Li 2 ( t ) = P k > 1 t k k 2 is the dilogarithm function. The Lie algebra g Γ acts on T Γ b y Hamiltonian v ector ﬁelds. Let us denote b y T γ the formal P oisson automorphism T γ = exp( {− Li 2 ( e γ ) , •} ) , T γ ( e µ ) = (1 − e γ ) h γ ,µ i e µ considered as an automorphism of algebra of functions. More precisely for an y strict con v ex cone C ⊂ Γ R con taining γ the eleme nt T γ acts on the formal sc heme S pf ( Q µ ∈ Γ ∩ C Q e µ ). Moreo ve r T γ is the T a ylor expansion of a birational automorphism of T Γ . Finally , in the case when Γ comes from a 3 d Calabi-Y au category the n um b ers Ω( γ ) are (conjecturally) in tegers fo r ( γ , Z ) ∈ X 2 in notation of Section 2.4. They prov ide generalization of DT-inv a rian ts (BPS degeneracies in phy sics language). 2.6 Symplectic double torus If the sk ew-symmetric bilinear form on Γ is degenerate, then the action of g Γ on T Γ is not exact. In order to r emedy the pro blem w e can em b ed (Γ , h• , •i ) in to a larger symplectic lattice. A p ossible choice is Γ ⊕ Γ ∨ , where Γ ∨ = Hom(Γ , Z ). The corresp onding non-degenerate biline ar form is h ( γ 1 , ν 1 ) , ( γ 2 , ν 2 ) i = h γ 1 , γ 2 i + ν 2 ( γ 1 ) − ν 1 ( γ 2 ) . Let us choose a basis e i , 1 ≤ i ≤ n = rk Γ of Γ. It g iv es rise to the co ordinates y i , 1 ≤ i ≤ n on T Γ . The P oisson structure on T Γ can b e written as { y i , y j } = b ij y i y j , 33 where b ij = h e i , e j i . Let us also introduce additional co ordinates x j , 1 ≤ j ≤ rk Γ ∨ in suc h a w a y that ( y i , x j ) , 1 ≤ i, j ≤ n will b e the co ordinates on the double torus D ( T Γ ) with the Poiss on brack ets { x i , x j } = 0 , { y i , x j } = δ ij y i x j . There is a pro jection π : D ( T Γ ) → T Γ , π (( y i ) 1 ≤ i ≤ n , ( x j ) 1 ≤ j ≤ n ) = ( y i ) 1 ≤ i ≤ n . Notice that π is a Poiss on morphism o f the symplectic manifold D ( T Γ ) onto the Pois son manifold T Γ . Let C ⊂ Γ R b e a closed con v ex strict cone. Let us c ho o se a closed con v ex strict cone C 1 ⊂ (Γ ⊕ Γ ∨ ) ⊗ R whic h contains C ⊕ { 0 } . With the cone C 1 w e asso ciate the Poiss on algebra Q [[ C 1 ]] consisting of series P γ ,δ ∈ C 1 ∩ (Γ ⊕ Γ ∨ ) c γ ,δ y γ x δ . The pro-nilp otent group G C = exp( Q γ ∈ C ∩ Γ g γ ) acts b y P oisson automorphisms of Q [[ C 1 ]]. Let us consider a closed algebraic submanifold N ⊂ D ( T Γ ) deﬁned by the equations y i Y j x b ij j = − 1 , 1 ≤ i ≤ n . Lemma 2 The image o f the gr oup G C pr e s e rves the c orr esp o nding c omple- tion of N . Pr o of. It suﬃces to c hec k that the image of the Lie algebra g C preserv es the equations of N . Notice that this image b elongs to the Lie a lg ebra of Hamiltonian v ector ﬁelds on D ( T Γ ) g enerated by { y γ , •} , where γ ∈ Γ and y γ = y γ 1 1 . . . y γ n n . T aking logarithms we see that { log( y γ ) , log( y i ) + X j b ij log( x j ) } = X j γ j b j i + X j b ij γ j = 0 . This concludes the pro o f.  Remark 12 It is cle ar that the action of the image of G C also c omm utes with the map π . Mor e over the image of G C in the gr oup of exact symple cto- morphisms of the c ompletion of T Γ c o rr esp onding to C c an b e char acterize d by the pr op erty that it pr eserves the c ompletion of N an d c o mmutes with π . 34 Let us ﬁnally mak e a remark ab out a p ossible non-arc himedean geometry in terpretation of our construc tion. Let us c ho ose a complete non-arc himede an ﬁeld K with the residue ﬁeld of characteris tic zero. Exte nding scalars we can think of the algebraic v ariet y D ( T Γ ) as of v ar iety ov er K . W e denote by D ( T Γ ) an the corresp onding non-arch imedean K -analytic space in the sense o f Berk o vic h (see [40] for the explanation of the relev ance of Berk o vic h approac h to the large complex structure limit of Calabi- Y au v arieties ). Then the g r o up G C acts o n the analytic subset o f D ( T Γ ) an giv en b y inequalities {| e γ | < 1 , γ ∈ C \ { 0 }} . Here we interpret e γ as a Lauren t monomial on D ( T Γ ). The symplectic double torus together with submanifold N will b e used again only in Section 8. 2.7 Complex in tegrable systems and stabilit y data In this s ection w e ex plain ho w complex in tegrable systems (with some ad- ditional structures) give rise to stabilit y data in the graded Lie algebra g Γ asso ciated with a symplectic lattice. In particular, Seib erg-Witten diﬀeren- tial can be interpre ted as the cen tral c harge for a complex in tegrable system, while the BPS degeneracies are in terpreted via our “n umerical” Donaldson- Thomas in v arian ts as the num b er of certain g radien t trees o n the base of a complex inte gra ble system. Recall that a c omplex inte gr able s ystem is a holomorphic map π : X → B where ( X , ω 2 , 0 X ) is a holomorphic symplectic manifold, dim X = 2 dim B , and the generic ﬁb er o f π is a Lagrangian submanifold, whic h is a p olarized ab elian v ariety . W e assume (in order to simplify the exp osition) that the p olarization is principal. The ﬁbration π is non-singular outside of a closed sub v ariety B sing ⊂ B of co dimension at least one. It fo llows that on the op en subset B sm := B \ B sing w e hav e a lo cal system Γ of symplectic lattices with the ﬁb er ov er b ∈ B sm equal to Γ b := H 1 ( X b , Z ) , X b = π − 1 ( b ) (the symplectic structure on Γ b is give n by the p olarization). F urthermore, the set B sm is lo cally (near eac h p oint b ∈ B sm ) em- b edded as a holomorphic Lagrangian sub v a r iety into a n aﬃne sym plectic space parallel to H 1 ( X b , C ). Namely , let us c ho ose a symplectic basis γ i ∈ Γ b , 1 ≤ i ≤ 2 n . Then w e ha ve a collection of holomorphic closed 1-forms α i = R γ i ω 2 , 0 X , 1 ≤ i ≤ 2 n in a neighborho o d of b . There exists (w ell-deﬁned lo cally up to a n additiv e constan t) holomorphic functions z i , 1 ≤ i ≤ 2 n suc h that α i = dz i , 1 ≤ i ≤ 2 n . They deﬁne a n em b edding of a neigh- b orho o d o f b in to C 2 n . The collection o f 1-forms α i giv es rise to an ele- 35 men t δ ∈ H 1 ( B sm , Γ ∨ ⊗ C ). We assume that δ = 0. This assumption is equiv alen t to an existenc e of a section Z ∈ Γ( B sm , Γ ⊗ O B sm ) suc h that α i = Z ( γ i ) , 1 ≤ i ≤ 2 n . Deﬁnition 5 We c al l Z the c entr al cha r ge o f the inte gr able system. Hence, for ev ery p oint b ∈ B sm w e ha ve a symplectic la t t ice Γ b endo w ed with an additiv e map Z b : Γ b → C . Our goal will b e to deﬁne a con tinuous family of stabilit y data on graded Lie alg ebras g Γ b with cen t r a l charges Z b . First, we sho w an example of section Z . Example 2 (S eib er g-Witten curve) L et B = C b e a c omplex line endowe d with a c omplex c o or din a te u . We denote by X 0 = T ∗ ( C \ { 0 } ) the c otange n t bund le to the punctur e d line. We endow it with the c o or dinates ( x, y ) , y 6 = 0 and the symple ctic fo rm ω 2 , 0 = dx ∧ dy y . Ther e is a pr oje ction π 0 : X 0 → B given by π ( x, y ) = 1 2 ( x 2 − y − c y ) , wher e c is a ﬁxe d c onstant. Fib ers of π 0 ar e punctur e d el liptic curves y + c y = x 2 − 2 u . We denote by X the c omp actiﬁc ation of X 0 obtaine d by the c omp actiﬁc ations of the ﬁb ers. We den ote by π : X → B the c orr esp onding pr oje ction. Then Z u ∈ H 1 ( π − 1 ( u ) , C ) is r epr e s ente d by a mer om orphic 1 -form λ S W = xdy y (Seib er g-Witten form). T he form λ S W has zer o r esidues, henc e it deﬁnes an element of H 1 ( π − 1 ( u ) , C ) for an y u ∈ B sm , wher e B sm = B \ { b − , b + } c o nsists of p oints wher e the ﬁb er of π is a n o n-de gener a te el liptic curve. The dense op en set B sm ⊂ B carries a K¨ ahler form ω 1 , 1 B = Im X 1 ≤ i ≤ n α i ∧ α n + i ! . W e denote b y g B the corresp onding K¨ ahler metric. 36 F or any t ∈ C ∗ w e deﬁne an in tegral aﬃne structure on C ∞ -manifold B sm giv en b y a colle ction of closed 1-forms Re( tα i ) , 1 ≤ i ≤ 2 n . F or an y simply-connec ted open subset U ⊂ B sm and a co v arian tly constan t section γ ∈ Γ( B sm , Γ ) w e ha v e a closed 1-form α γ ,t = Re  t Z γ ω 2 , 0 X  = d Re( tZ ( γ )) , and the corresp onding gradien t v ector ﬁeld v γ ,t = g − 1 B ( α γ ,t ). Notice that this v ector ﬁeld is a constan t ﬁeld w ith in tegral direction in the in tegral aﬃne structure asso ciated with closed 1-forms Im( tα i ) , 1 ≤ i ≤ 2 n . Similarly to [40] we can construct inﬁnite or iented trees lying in B s uch that its external ve rtices b elong to B sing , and edges are p ositively oriente d tra jectories of v ector ﬁelds v γ ,t . All in ternal v ertices hav e v alency at least 3, and ev ery suc h ve rtex should b e thought of as a splitting p o in t: a tra jectory of the v ector ﬁeld v γ ,t is split at a v ertex into sev eral tra jectories of v ector ﬁelds v γ 1 ,t , . . . , v γ k ,t suc h that γ = γ 1 + · · · + γ k . The restriction of the function Z to a tree giv es rise to a C -v alued function suc h that o n the tra jectory of v ector ﬁe ld v γ ,t it is equal to the restriction of Z ( γ ) to this tra jectory . W e assume that this function approac h to z ero as long as w e approach an external v ertex of the tree (whic h b elongs to B sing ). It is easy to see that tZ ( γ ) is a p ositiv e num b er at an y other p oin t of the tree (hence it deﬁnes a length function). W e expect that for any p oin t b ∈ B sm and γ ∈ Γ b there exist ﬁnitely man y suc h trees whic h pass the p o in t b in the direction of γ (w e can think of b as a ro ot of the tree, hence w e can say ab ov e tha t w e consider orien ted trees suc h that all external v ertices except of the ro ot b elong to B sing ). Here w e choose an aﬃne structure with t ∈ R > 0 ( Z ( γ ) − 1 b ). Probably the n um b er of suc h trees for ﬁxed b, γ is ﬁnite, since their lengths should b e b ounded. 8 F or a ﬁxed t ∈ C ∗ the union W t of all trees as ab o ve is in fact a coun table union of real hy p ersurfaces in B sm . They are analogs of the walls o f second kind. The set W t dep ends on Arg t only . The union ∪ θ ∈ [ 0 , 2 π i ) W te iθ sw ap the whole space B sm . Let us denote b y W (1) the union o ve r all t ∈ C ∗ / R > 0 of the sets of in ternal ve rtices of all trees in W (1) (splitting p oin ts of the gr a dien t tra jectories). This is an analog o f the w all o f ﬁrst kind. 8 In [40] w e mo diﬁed th e gradient ﬁelds near B sing in or der to gua rantee the conv erg ence of inﬁnite pro ducts in the adic top ology . It seems that we were to o ca utious, a nd the conv ergence holds without any mo diﬁcation. 37 In [40] w e suggested a pro cedure of assigning rational m ultiplicities to edges of trees (see a lso [27],[28]). This leads to the follo wing picture. Con- sider the total space tot ( Γ ) of the lo cal system Γ . It follows fro m ab o v e assumptions and considerations that we ha ve a lo cally constan t function Ω : tot ( Γ ) → Q whic h jumps at the subset consisting of the lifts of the w all W (1) to tot ( Γ ). Then for a ﬁxed b ∈ B sm the pair ( Z, Ω) deﬁnes stabilit y data on the g raded Lie algebra g Γ b of the group of formal symplectomor- phisms of the symplectic torus T Γ b . In this w a y we obtain a lo cal em b edding B sm ֒ → S tab ( g Γ b ). In the ab o v e example of Seib erg-Witten curv e, the w all W (1) is an ov al- shap ed curv e whic h contains tw o singular p oints b ± ∈ B sing . A typical W t lo oks suc h as follo ws. b − + b The wall-cross ing formula coincides with the one fo r T (2) a,b (see Introduc- tion). Remark 13 1) We exp e ct that the ab ove c onsider ations ar e valid for a lar ge class of c omplex inte gr able systems, e.g. Hitchin system. 2) In the c ase when we have a 3 d c omplex c omp ac t Calabi-Y au mani- fold X , t he mo duli sp ac e M X of c omplex structur es on X is lo c al ly emb e d- de d into the pr oje ctive sp ac e P ( H 3 ( X , C )) as a b ase of a L agr angian c one L X ⊂ H 3 ( X , C ) . It c arries a K¨ ahler metric (Weil-Petersson me tric). We c a n r ep e at the ab ove c onside r ation s given for inte gr able systems, r eplac i n g the gr a d ient ﬂows by the attr actor ﬂow (se e e.g. [13]). The ab ove c a se of i n te- gr a b l e systems is obtaine d in the lim it, when the c one b e c omes “very sharp”. 2.8 Relati on with the w orks of Jo yce, and of Bridge- land and T oledano-Laredo Let g , Γ b e as in Section 2.1 . W e assume that the ground ﬁeld is C . Supp o se that C ⊂ Γ R is a strict con v ex cone. W e are in terested in suc h stabilit y data 38 ( Z , a ) t hat Supp a ⊂ C ∪ ( − C ). W e deﬁne D as an op en subset of Ho m(Γ , C ) whic h consists of additiv e maps such that C ∩ Γ is mapp ed in to the upp er- half plane H + = { z ∈ C | Im ( z ) > 0 } . W e in terpret D a s an op en subset of S tab ( g ). Ev ery α ∈ C ∩ Γ giv es rise to an inv ertible function (co ordinate) z α ∈ O ( D ) × suc h that z α ( Z , a ) = Z ( α ). Recall the pro-nilp oten t Lie algebra g C = Q γ ∈ C ∩ Γ g γ and the corresp ond- ing pro-nilp otent group G C . In the pap er [32] by D. Jo yce the f ollo wing system of diﬀeren tial equations for a collection o f holomorphic functions ( f α ) α ∈ C ∩ Γ , f α ∈ O ( D ) ⊗ g α w as considered: ∀ α ∈ C ∩ Γ d f α = − 1 2 X β + γ = α [ f β , f γ ] d lo g z β z γ . It follow s that if ( f α ) satisﬁes the ab ov e system of equations then the diﬀeren tial 1-form ω = X α f α d log z α ∈ Ω 1 ( D ) b ⊗ g C := Y α ∈ C ∩ Γ (Ω 1 ( g ) ⊗ g α ) giv es rise to the ﬂat connec tion, since dω + 1 2 [ ω , ω ] = 0 . Moreo v er, setting F = P α f α w e o bserv e that dF + [ ω , F ] = 0 , i.e. F is a ﬂat section of this connection in the adjoin t represe ntation. One can c hec k b y induction that there exists a unique solution to the ab ov e sy stem of diﬀeren tial equations (mo dulo constan ts for eac h function f α ). This means that the set of s olutions is is omorphic to g C (non-canonically). F or an y n > 0 and pairwise diﬀeren t n um b ers x i ∈ C \ { 0 , 1 } , i = 1 , . . . , n w e introduce the follow ing f unction (m ultilogarithm) whic h is holomorphic when all x i lie outside of the in terv al [0 , 1]: L n ( x 1 , . . . , x n ) := v .p. Z 0 1 X α 1 + ··· + α n = α f α 1 . . . f α n I n ( z α 1 , z α 2 , . . . , z α n ) , where fo r z 1 , . . . , z n ∈ C suc h that 0 < Im z 1 < Im z 2 < · · · < Im z n w e set 9 I n ( z 1 , z 2 , . . . , z n ) := = 2 π i ( − 1) n − 1 L n − 1  z 1 z 1 + ··· + z n , z 1 + z 2 z 1 + ··· + z n , . . . , z 1 + z 2 + ··· + z n − 1 z 1 + ··· + z n  . One can sho w that in fact E α ∈ g C , and it is a lo cally constan t along strata of the stratiﬁcation deﬁned b y t he w alls z β /z γ ∈ R where α = β + γ with β , γ ∈ C ∩ Γ and β is not parallel to γ . F or a solution ( f α ) of the ab o v e system of diﬀeren tial equations w e deﬁne a diﬀeren tial 1-for m on D × C ∗ suc h that b ω := X α f α e vz α d log( v z α ) , where v is the standard co ordinate o n C ∗ . Then one c hec ks that d b ω + 1 2 [ b ω , b ω ] = 0 . Let M ( f α ) ∈ G C b e the mono drom y of the corresp o nding ﬂat connection computed along a closed lo op in the complex v - plane, whic h starts a t + i ∞ and go es in the an ti-clo c kwise direction around v = 0. The ﬂatness implies that the mono drom y do es not dep end on the p oin t of D . On the other hand let us conside r the elemen t N ( f α ) ∈ G C deﬁned as − → Y l ⊂H + exp X α ∈ C ∩ Γ ,z α ∈ l E α ! , where the pro duct is tak en ov er all ra ys l ⊂ H + with the v ertex at the orig in. 9 This for mul a was prop o sed in [10] as the in version o f the Joyce formula which expressed f α ’s in terms o f E α ’s. 40 Conjecture 2 We have M ( f α ) = N ( f α ) . The conjec ture implies that the elemen ts a α := E α satisfy the w all-crossing form ula. The elemen t M ( f α ) is equal (in our notation) to the elemen t A V , where V is a strict sector in H + con taining Z ( C ). W e will discuss b elow a sequenc e of iden tities whic h imply the conjectu re. But w e need to introduce certain functions ﬁrst. Let ϕ : (0 , 1) → C \ { 0 } b e the inﬁnite con tour whic h starts and ends at + i ∞ , go es in the an ti-clo c kwise dire ction and surrounds the p oint 0 ∈ C . With the con tour ϕ w e asso ciate the follo wing function on ( H + ) n , n > 1: K n ( z 1 , . . . , z n ) := Z 0 1 a nd a collection of complex n um b ers z i ∈ H + , 1 ≤ i ≤ n . W e call a sequ ence 0 = i 0 < i 1 < · · · < i k − 1 < i k admissible if Arg( z 1 + · · · + z i 1 ) > Arg( z i 1 +1 + · · · + z i 2 ) > · · · > Arg( z i k − 1 +1 + · · · + z i k ) . F or a ﬁxed admissible sequenc e we ha v e a partition k = l 1 + · · · + l m where l 1 , l 2 , . . . , l m are the n um b ers of consec utiv e equalities in the ab ov e seque nce of inequalities for the argumen ts. Let Ω k ,l 1 ,...,l m ( z 1 , . . . , z n ) b e the set of all admissible sequences 0 = i 0 < i 1 < · · · < i k − 1 < i k with the give n partition k = l 1 + · · · + l m . Under these assumptions and notation o ne can see that the previous Conjecture 2 is equiv alen t to Conjecture 3 We have K n ( z 1 , . . . , z n ) = X Ω k,l 1 ,...,l m ( z 1 ,...,z n ) Y 1 ≤ j ≤ m 1 l j ! I i 1 ( z 1 , . . . , z i 1 ) · I i 2 − i 1 ( z i 1 +1 , . . . , z i 2 ) · . . . · I i k − i k − 1 ( z i k − 1 +1 , . . . , z n ) . 41 Indeed, for z i = z α i , i = 1 , . . . , n the l.h.s. of the form ula is the contribution of the term f α 1 . . . f α n in the expansion of M ( f α ) . Similarly , t he r.h.s. is the con tribution of the same term in N ( f α ) . Here w e giv e a pro o f of the ab ov e conjecture in the special case: Prop osition 5 If 0 < Arg z 1 < · · · < Arg z n < π then K n ( z 1 , . . . , z n ) = I n ( z 1 , z 2 , . . . , z n ) . Pr o of. F or n = 1 b oth sides are equal to 2 π i . F or n > 2 w e pro ceed by induction. First one c hec ks directly that dK n ( z 1 , . . . , z n ) = − n − 1 X i =1 d log  z i +1 z i  K n − 1 ( z 1 , . . . , z i + z i +1 , . . . , z n ) . The same form ula holds if w e replace K n , K n − 1 b y I n , I n − 1 respectiv ely . Th us w e see b y ind uction that K n − I n = const n . W e w an t to pro v e that const n = 0. In order to do that w e tak e z j = z j ( ε ) , 1 ≤ j ≤ n , suc h as follows : z 1 ( ε ) = i + 1 ε , z n ( ε ) = i − 1 ε , z k ( ε ) = i − k , 2 ≤ k ≤ n − 1 . Here i = √ − 1. Then 0 < Arg z 1 ( ε ) < Arg z 2 ( ε ) < · · · < Arg z n ( ε ) < π and | P 1 ≤ j ≤ k z j ( ε ) | → ∞ as ε → 0 fo r k = 1 , . . . , n − 1, and moreo v er | P 1 ≤ j ≤ n z j ( ε ) | is a constan t function of ǫ . Therefore, I n ( z 1 ( ε ) , . . . , z n ( ε )) → 0 as ε → 0, since all the argumen ts of the function L n − 1 in the deﬁnition of I n approac h inﬁnit y . Hence in order to ﬁnish the pro of it suﬃces to sho w that K n ( z 1 ( ε ) , . . . , z n ( ε )) → 0 as ε → 0. Here is the sk etch o f the pro of. 10 Notice that Z v 2 + i ∞ e v 1 z 1 ( ε ) dv 1 /v 1 = 1 z 1 ( ε ) e v 1 z 1 ( ε ) /v 2 + r 1 ( ε ) , 10 W e thank Andrei O kounk ov for the idea of the pro o f. 42 where r 1 ( ε ) → 0 as ε → 0. Rep eating we obtain that K n ( z 1 ( ε ) , . . . , z n ( ε )) = 1 z 1 ( ε ) 1 z 1 ( ε ) + z 2 ( ε ) . . . 1 z 1 ( ε ) + z 2 ( ε ) + · · · + z n − 1 ( ε ) × × Z ϕ e v n ( z 1 ( ε )+ ··· + z n ( ε )) dv n /v n + r n ( ε ) , where the in tegral is tak en ov er the con tour ϕ described b efore, and r n ( ε ) → 0 as ε → 0. It follo ws from our c hoice of num bers z j ( ε ) , 1 ≤ j ≤ n that K n ( z 1 ( ε ) , . . . , z n ( ε )) → 0 as ε → 0.  One can hop e that the tec hnique dev elop ed in [32] helps in pro ving the general case. A relationship b etw een Joyce form ulas and iterated in tegrals is discuss ed in [10] in a slightly diﬀeren t form. In that pap er the elemen ts N ( f α ) are in- terpreted as Stok es m ultipliers fo r a diﬀeren t system o f diﬀeren tial equations on C (with co ordinate t ) with v alues in the Lie algebra whic h is an exten- sion of g C b y the a b elian Lie algebra Hom(Γ , C ) (an analog of the Cartan subalgebra). It has irregular singularit y at the origin giv en by Z t 2 , where Z is the cen tral c harge of the stability structure. In f act the connection from [10] reduce s to our connection after the c hange of v ariables v = 1 /t and the conjugation by exp( − v Z ). 2.9 Stabilit y data on gl ( n ) Let g = gl ( n, Q ) b e the Lie alg ebra of the general linear group. W e consider it as a Γ-graded Lie algebra g = ⊕ γ ∈ Γ g γ , where Γ = { ( k 1 , . . . , k n ) | k i ∈ Z , X 1 ≤ i ≤ n k i = 0 } is the ro ot lattice. W e endo w g with the Carta n in v olution η . Algebra g has the standard basis E ij ∈ g γ ij consisting of matrices with the single non-zero en try a t the place ( i, j ) equal to 1. Then η ( E ij ) = − E j i . In what follow s w e are going to consider symmetric (with respect to η ) stabilit y dat a on g . W e notice that Hom(Γ , C ) ≃ C n / C · (1 , . . . , 1 ) . W e deﬁne a subs pace Hom ◦ (Γ , C ) ⊂ Hom(Γ , C ) consisting (up to a shift by the multip les o f the v ector (1 , . . . , 1)) of vec tors ( z 1 , . . . , z n ) suc h t hat z i 6 = z j 43 if i 6 = j . Similarly w e deﬁne a subspace Hom ◦◦ (Γ , C ) ⊂ Hom(Γ , C ) consisting (up to the same shift) of suc h ( z 1 , . . . , z n ) that there is no z i , z j , z k b elonging to the same real line as lo ng as i 6 = j 6 = k . Ob viously there is a n inclusion Hom ◦◦ (Γ , C ) ⊂ Hom ◦ (Γ , C ). F or Z ∈ Hom(Γ , C ) w e hav e Z ( γ ij ) = z i − z j . If Z ∈ Hom ◦◦ (Γ , C ) then symmetric stabilit y data with suc h Z is the same as a sk ew-symmetric ma- trix ( a ij ) with rational en tries determined from the equalit y a ( γ ij ) = a ij E ij . Ev ery contin uous path in Hom ◦ (Γ , C ) admits a unique lifting to S tab ( g ) as long as w e ﬁx the lifting of the initial p oint. The matrix ( a ij ) c hanges when w e cross w alls in Hom ◦ (Γ , C ) \ Hom ◦◦ (Γ , C ). A t ypical wall-cross ing corre- sp onds to the case when in the ab ov e notation the p oint z j crosses a straigh t segmen t joining z i and z k , i 6 = j 6 = k . In this case the only c hange in the matrix ( a ij ) is of the form: a ik 7→ a ik + a ij a j k . This follo ws from the m ultiplicativ e w all-crossing form ula whic h is of the form: exp( a ij E ij ) exp( a ik E ik ) exp( a j k E j k ) = = exp( a j k E j k ) exp(( a ik + a ij a j k ) E ik ) exp( a ij E ij ) . Same w all-crossing formulas a pp eared in [11] in the study of the change o f the n um b er of solitons in N = 2 t w o- dimensional sup ersymmetric QFT. In [11] the n um b ers a ij w ere in tegers, a nd the wall-cross ing preserv ed integral- it y . In our considerations, for an y Z ∈ Hom ◦◦ (Γ , C ) the fundamen tal gro up π 1 (Hom ◦ (Γ , C ) , Z ) acts on the space of sk ew-symmetric matrices b y p olyno- mial transformations with in teger co eﬃcien ts. It can b e iden tiﬁed with the w ell-kno wn actions of the pure braid group on the space of upper-triangular matrices in the theory of Ga brielo v bases of isolated singul arities and in the theory of triangulated categories endow ed with exceptional collections. F ur- thermore, the matrices exp( a ij E ij ) = 1 + a ij E ij can b e in terpreted as Stok es matrices of a certain connection in a neighborho o d o f 0 ∈ C , which has irreg- ular singularities ( tt ∗ -connection f r o m [11], s ee also [24]). This observ ation should be compared with the results ab out the irregular connection from the previous section. 44 3 Ind-const ructible categories and stabilit y stru c - tures 3.1 Ind-constructible categories Here we in tro duce a n ind-constructible v ersion of the notion of a (triangu- lated) A ∞ -category . Let k b e a ﬁeld, k b e its algebraic closure. By a v ariet y o v er k (not necessarily irreducible) w e mean a reduced separated sc heme of ﬁnite ty p e o v er k . Recall the follow ing deﬁnition. Deﬁnition 6 L et S b e a v a riety over k . A subset X ⊂ S ( k ) is c al le d c on- structible over k if it b elongs to the Bo ole an alge b r a gener ate d by k -p o i nts o f op en (e quivale n tly close d) s ubsc hemes of S . Equiv alen tly , a constructible set is t he un ion of a ﬁni te collection of k - p oin ts of disjoin t lo cally closed sub v arieties ( S i ⊂ S ) i ∈ I . F or an y ﬁeld extension k ⊂ k ′ ⊂ k w e deﬁne the set of k ′ -p oints X ( k ′ ) of the constructible set X as ( X ∩ S ( k ′ )) ⊂ S ( k ). In particular, X ( k ) = X . W e deﬁne the category C O N k of constructible sets ov er k as a cate- gory with ob jects ( X , S ), where X and S as ab ov e. The set of morphisms Hom C O N k (( X 1 , S 1 ) , ( X 2 , S 2 )) is deﬁned to b e the set of maps f : X 1 → X 2 suc h that the re exists a decomp osition of X 1 in to the ﬁnite disjoin t union of k -p oin ts of v arieties ( S i ⊂ S 1 ) i ∈ I suc h that the res triction of f to eac h S i ( k ) is a morphism of sc hemes S i → S 2 . W e see that there is a natural faithful functor from C ON k to the category of sets equipped with the action of Aut( k / k ). Deﬁnition 7 A n ind-c onstructible set over k is g i v en by a chain of emb e d- dings of c onstructible sets X := ( X 1 → X 2 → X 3 → . . . ) ov e r k . A mor- phism of ind-c onstructible sets is deﬁne d as a map g : ∪ i X i ( k ) → ∪ i Y i ( k ) such that for any i ther e exi s ts n ( i ) such that g | X i ( k ) : X i ( k ) → Y n ( i ) ( k ) c omes fr om a c onstructible map. Ind-constructible sets form a full sub category I C k of the category o f ind- ob jects in C O N k . Remark 14 Equivalently, we c an c onsider a c ountable c o l le ction Z i = X i \ X i − 1 of non-interse cting c onstructible sets. Then a morphism ⊔ i ∈ I Z i → ⊔ j ∈ J Z ′ j is given b y a c ol le ction o f c onstructible maps f i : Z i → ⊔ i ∈ J i Z ′ j , wher e e ach J i is a ﬁnite set. 45 The category of constructible (or ind-constructible) sets has ﬁb ered pro d- ucts. There is a notion of construc tible (or ind-constructibl e) v ector bundle (i.e. the one with the ﬁb ers whic h are a ﬃne spaces o f v arious ﬁnite dimen- sions). Deﬁnition 8 A n ind-c o n structible A ∞ -c ate gory ov er k is d e ﬁ ne d by the fol- lowing data: 1) An ind- c onstructible set M = O b ( C ) = ⊔ i ∈ I X i over k , c al le d the set of obje cts. 2) A c ol le ction of ind-c onstructible ve ctor bund les HO M n → M × M , n ∈ Z c a l le d the bund le s of morphisms of de gr e e n . The r estriction HO M n → X i × X j is a ﬁnite-dimensional c onstructible ve ctor bund le for any n ∈ Z , i, j ∈ I , and the r estriction HO M n → X i × X j is a zer o bund le for n ≤ C ( i, j ) , wher e C ( i, j ) is some c onstant. 3) F or any n > 1 , l 1 , . . . , l n ∈ Z , ind-c onstructible morphisms o f ind- c o nstructible bund les m n : p ∗ 1 , 2 HO M l 1 ⊗ · · · ⊗ p ∗ n,n +1 HO M l n → p ∗ 1 ,n +1 HO M l 1 + ··· + l n +2 − n , wher e p i,i +1 denote natur al pr oje ctions of M n +1 to M 2 . T h ese morphisms ar e c al le d higher c omp osi tion maps. The ab ov e dat a are required to satisfy the follow ing axioms A1)-A3): A1) Higher ass o ciativity pr op erty for m n , n > 1 in the sense of A ∞ - c a te gorie s . W e lea ve for the reader to write do wn the corres p o nding w ell- kno wn iden tities (see [37],[42]). This a xiom implies that w e ha ve a small k -linear non-unital A ∞ -category C ( k ) with the set of ob jects M ( k ) and morphisms H O M • ( k ). A2) (we ak unit) Ther e is a c onstructible s e c tion s of the ind- c onstructible bund le HO M 0 | D iag → M such that the im age of s b elongs to the ke rnel of m 1 and gives rise to the identity morphism s in Z -gr ade d k -line ar c ate gory H • ( C ( k )) . Alternativ ely , instead of A2) one can use the axiomatics of A ∞ -categories with strict units (see [42], [46]). An ind-constructible A ∞ -category C give s rise t o a collection of ind- constructible bundles o v er O b ( C ) × O b ( C ) giv en by E X T i := H i ( HO M • ) , i ∈ Z 46 with the ﬁb er ov er a pair of ob jects ( E , F ) equal to Ext i ( E , F ) := H i ( HO M • E , F ) . The cohomology groups are tak en with resp ect to the diﬀerent ial m 1 . A3) (lo c al r e g ularity) Ther e ex ists a c ountable c ol le ction of schemes ( S i ) i ∈ I of ﬁnite typ e o v e r k , a c ol le ction of algebr aic k -ve ctor bund les HOM n i , n ∈ Z over S i × S i for al l i , and ind - c onstructible identiﬁc ations ⊔ i S i ( k ) ≃ M , HOM n i ≃ H O M n | S i × S i , n ∈ Z such that al l h i g her c omp ositions m n , n > 2 , c onsider e d for o b j e cts fr om S i for any given i ∈ I , b e c ome morphisms of a l g ebr a ic ve ctor b und les. W e will o ften call ind-constructib le A ∞ -categories simply b y ind-constructible categories. The basic example of an ind-constructible category is the category P er f ( A ) o f p erfect A -mo dules where A an A ∞ -algebra o v er k with ﬁni te- dimensional cohomology (see the discussion after the Example 1 in Section 1.2 of In tro duction). W e deﬁne a functor b et we en t wo ind-constructible categories mimic king the usual deﬁnition of an A ∞ -functor. A functor Φ : C 1 → C 2 is called an e quivalenc e if Φ is a full em b edding, i.e. it induces an isomorphism Ext • ( E , F ) ≃ Ext • (Φ( E ) , Φ( F )) ∀ E , F ∈ O b ( C 1 )( k ) and moreo ve r, there exists an ind-constructible over k map s : O b ( C 2 )( k ) → O b ( C 1 )( k ) suc h that for a n y ob ject E ∈ O b ( C 2 )( k ) we ha v e E ≃ Φ( s ( E )). Using the notions of a functor and of an equiv alence w e can de ﬁne the prop ert y of an ind-constructible w eakly unital A ∞ -category C to b e triang u- lated. F or example, the prop erty to hav e exact triangles can b e fo rm ulated as follows . Consider a ﬁnite A ∞ -category C 3 consisting of 3 ob jects E 1 , E 2 , E 3 with non-trivial morphism spaces Hom 0 ( E i , E i ) = k · id E i , Hom 0 ( E 1 , E 2 ) ≃ Hom 0 ( E 2 , E 3 ) ≃ Hom 1 ( E 3 , E 1 ) ≃ k equiv alen t to the full sub category of the category of repres en tations of the quiv er A 2 consisting of mo dules of dimensions (0 , 1) , (1 , 1) , (1 , 0). Let C 2 ⊂ C 3 b e the full sub category consisting of ﬁrst t w o ob j ects. 47 It is easy to see directly from the deﬁnitions that for an y ind-constructible category C there are natural ind-construc tible categories F un( C i , C ) , i = 2 , 3 whose ob jects ov er k are the usual A ∞ -categories of functors from C i ( k ) to C ( k ) as deﬁned e.g. in [41],[42] and [37]. There exists a natural restriction functor r 32 : F un( C 3 , C ) → F un( C 2 , C ) . Similarly to t he setting o f usual A ∞ -categories, the ind-constructible v er- sion of the axiom of exact triangles sa ys that r 32 is an equiv a lence. In the same manner o ne can deﬁne other prop erties of triangulated A ∞ -categories (i.e. the exis tence of shift functors, ﬁnite sums, see [42], [65]) in the ind- constructible setting. In Se ctions 5,6 we will us e a simpliﬁed notation C one ( f ) for a cone of morphism f in C ( k ) “ pretending” that cones are functorial. The precise prescription is to tak e an ob ject in F un( C 2 , C ) corres p o nding to f , ﬁnd an isomorphic ob ject in F un( C 3 , C ), and then tak e the image in C ( k ) of the ob ject E 3 . All this can b e prop erly form ulated using the language of constructible stac ks, see 3.2 and 4.2. Let us call an ind-constructible A ∞ -category minim a l on the diagonal if the restriction of m 1 to the diagonal ∆ ⊂ M × M is trivial. One can sho w that an y ind-constructible A ∞ -category is equiv alent to a o ne whic h is minimal on the diagonal. Remark 15 T ypic al ly in pr actic e one has a de c o mp osition O b ( C ) = ⊔ i ∈ I X i wher e X i ar e sc h emes, not just c onstructible sets. Mor e over, for any E ∈ X i ( k ) ther e is a natur al map T E X i → Ext 1 ( E , E ) . The r e ason for this is the fact that the deformation the ory of the obje ct E should b e c ontr o l le d by the DGLA Ext • ( E , E ) . We did not include the ab ove pr op erty into the list of axioms sinc e it do es not play a ny r ole in our c onstructions. 3.2 Stac k of ob jects In this section we a ssume that the g round ﬁeld k is p erfect, i.e. the k is a Galois ﬁeld o v er k . Our goal in this section is to explain ho w to asso ciate with an ind-constructib le A ∞ -category C ov er k an ordinary k -linear A ∞ -category C ( k ), in suc h a w ay that ind-constructible equiv alences will induce the usual equiv alences. F or an y ﬁeld extension k ′ ⊃ k (e.g. fo r k ′ = k ) one can deﬁne A ∞ -category C naive ( k ′ ) to b e the small k ′ -linear category with t he set of ob jects given b y ( O b ( C ))( k ′ ) and ob vious morphisms and compo sitions. Thi s 48 is not a satisfactory notion b ecause in the deﬁn ition of the equiv alence w e demand o nly the surjectivit y on isomorphism classes of ob jects o ve r k . The naiv e category C naive ( k ) will b e a f ull subcategory o f the “correct” category C ( k ). One should read carefully brac k ets, as in our notation O b ( C ( k )) 6 = ( O b ( C ))( k ) =: O b ( C naive ( k )) , con trary to the case of k where w e ha v e O b ( C ( k )) = ( O b ( C ))( k ) . W e will see also the “set o f isomorphism classes of ob jects” in C should b e b etter unders to o d as an ind-constructibl e stac k 11 . Let k ′ ⊂ k b e a ﬁnite Galois extension of k and consider an elemen t E ′ ∈ ( O b ( C ))( k ′ ) ⊂ O b ( C ( k )) suc h that σ ( E ) is isomorphic to E f o r all σ ∈ Gal( k ′ / k ). W e w ould like to deﬁne the descen t data fo r suc h E ′ , whic h should b e data necessary to deﬁne an ob ject in (not y et deﬁned) k -linear category C ( k ) whic h b ecomes isomorphic to E ′ after the extension of scalars from k to k ′ . First, for a ﬁnite non-empt y collection of ob jects ( E i ) i ∈ I of a ny A ∞ - category C ′ linear o v er a ﬁeld k ′ (not necess arily a p erfect one) w e deﬁne an identiﬁc ation data for ob jects of this collection to b e an A ∞ -functor Φ from the A ∞ -category C I , k ′ describing I copies of the same ob ject: O b ( C I ) = I , Hom • ( i, j ) = Hom 0 ( i, j ) ≃ k ′ to C ′ . In plain terms, to giv e suc h a functor is to giv e a closed morp hism of degree 0 for any pair of ob jects E i , E j (represen ting the identit y id E i in H • ( C ( k )) for i = j ), a homotop y f o r an y triple of ob jects, homotop y b etw een homotopies fo r an y quadruple of ob jects, etc . Th us, w e in a sense iden tify all the ob jects of the collection ( E i ) i ∈ I and hence can treat it is a new ob ject (canonically isomorphic to all ( E i ) i ∈ I ), without choosing an y sp eciﬁc elemen t i ∈ I . Returning to the case o f E ′ ∈ ( O b ( C ))( k ′ ), w e deﬁne the descen t data as the iden tiﬁcation of t he collection o f ob jects ( σ ( E ′ )) σ ∈ Gal ( k ′ / k ) of the category C naive ( k ′ ) equiv arian t with resp ect to the action of Gal( k ′ / k ) acting b oth on the collection and on the co eﬃcien ts in the identi ﬁcation. 11 Even in the ca se when k = k it is impor ta n t to keep track on a utomorphisms groups of ob jects (a nd no t only on the set of isomorphism classes), e.g. for the co rrect deﬁnition of the motivic Hall algebr a in 6 .1. 49 W e deﬁne the set O b ( C ( k )) of ob jects of C ( k ) to b e the inductiv e limit o v er ﬁnite G alois extensions k ′ / k of descen t data as ab ov e. Also one can deﬁne morphisms and higher comp ositions. W e leav e the following Prop osition without a pro of. Prop osition 6 Ther e is a natur al structur e of a k -line ar A ∞ -c ate gory on C ( k ) c ontaining C naive ( k ) as a f ul l sub c ate go ry. Any e quivalenc e Φ : C 1 → C 2 in ind-c onstructible sense induc es an e quivalenc e C 1 ( k ) → C 2 ( k ) . If C is triangulate d in ind-c onstructible sense then C ( k ) is also triangulate d. If E is an ob ject of C ( k ) then an y other ob ject E ′ of C ( k ) whic h is iso- morphic to E after the extension of scalars to C ( k ) is in fact isomorphic to E in C ( k ) (in other w ords, t here are no non-trivial twis ted f o rms). The rea- son is t hat (as follow s directly from deﬁnitions) the set of suc h “ k -for ms” of E is classiﬁed b y H 1 (Gal( k / k ) , G E ), where G E is a simplicial group a sso ci- ated with the A ∞ -algebra End • C ( k ) ( E , E ). The re is a sp ectral sequence whic h con v erges to t his set and has the second term E 2 = ( E pq 2 ) giv en b y H 1 (Gal( k / k ) , (Ext 0 C ( k ) ( E , E )) × ) , H 2 (Gal( k / k ) , Ext − 1 C ( k ) ( E , E )) , H 3 (Gal( k / k ) , Ext − 2 C ( k ) ( E , E )) , . . . . W e observ e that all Galois cohomology g r o ups with co eﬃcien ts in Ext < 0 C ( k ) ( E , E ) are trivial (since Ext i C ( k ) ( E , E ) , i < 0 are just sums of copies of the addi- tiv e gro up G a ( k )). Also the set H 1 (Gal( k / k ) , (Ext 0 C ( k ) ( E , E )) × ) is the one- elemen t set, b ecause for an y ﬁnite-dimensional algebra A o v er k w e ha ve H 1 (Gal( k / k ) , A × ) = 0 (a v ersion of Hilbert 90 theorem, see also section 2.1 in [35]). One can deduce from t he ab ov e sp ectral sequence an imp orta nt corollary: the se t of isomorphism clas ses of ob jects of C ( k ) is in a natural bijection with the set of isomorphism classes of the usual descen t data in category H • ( C ( k )) endo w ed with the strict action of Gal( k / k ). Finally , we will explain how to asso ciate an ind-constructible stac k to an ind-constructible category C o ve r k . First of all, w e can alw a ys assume that C satisﬁes the follo wing axiom A4) Ther e ex i s ts a de c omp osition M = ⊔ i ∈ I X i into the c ountable disjoint union of c onstructible sets over k such that any two isomorphic obje cts of H • ( C ( k )) b elong to the sam e p art X i ( k ) for some i ∈ I . 50 Indeed, if w e choose any decomposition M = ⊔ i ∈ I X ′ i in to disjoin t union of constructible sets o v er k and identify I with the set of natural num b er { 1 , 2 , . . . } , then we can shrink X ′ i ( k ) to the subset consisting of ob jects whic h are not isomorphic to ob jects from ∪ j 0 suc h that for an y E ∈ S i ( k ) the subset of o b jects in S i ( k ) isomorphic to to E has dimension δ ( i ). This can b e achie ve d by sub dividing eac h S i in to smaller pieces, a nd b y remo ving some unnecess ary pieces consisting o f ob jects whic h b elong to other pieces . Then ta king a generic slice of co dimension δ ( i ) (and th us shrinking C to an equiv alent sub category), and ta king further sub divisions, one ma y assume that w e ha ve a lo cally r egular sub division of O b ( C ) suc h that an y isomorphism class of ob jects in S i ( k ) is ﬁnite. Moreo v er, w e ma y assume that the cardinalit y c i of all isomorphis m classes in S i ( k ) dep end only o n i , and also the dimen sion d i of the algebra Ext 0 ( E , E ) fo r E ∈ S i ( k ) also depends only o n i . F or an y giv en i ∈ I let us conside r the constructible set Z i o v er k parametrizing isomorphis m classes of ob j ects in S i ( k ). There is a natural constructible (ov er k ) bundle of ﬁnite-dimensional unital asso ciativ e alge- bras A , with the ﬁb er A x o v er any f ull collection x = ( E 1 , . . . , E c i ) (up to p erm utation) of diﬀeren t isomorphic ob jects equal to ⊕ 1 ≤ j 1 ,j 2 ≤ c i Ext 0 ( E j 1 , E j 2 ) . The ab ov e algebra is Morita equiv alen t to Ext 0 ( E j , E j ) for ev ery j ≤ c i , a nd in fa ct is isomorphic to the matrix algebra A x ≃ Mat( c i × c i , Ext 0 ( E j , E j )) ∀ j ≤ c i . Informally sp eaking, the “stack ” of ob jects from S i is the stac k o f pro jectiv e mo dules M ov er a lg ebra A x for some x ∈ Z i ( k ) whic h a r e isomorphic after Morita equiv alence to a free mo dule of rank one ov er Ext 0 ( E j , E j ) where E j is some represen tativ e of the equiv alence class x , i.e. M is isomorphic to the standard mo dule (Ext 0 ( E j , E j )) ⊕ c i o v er the matrix algebra for eve ry j ≤ c i . W e see that M has dimension N i := c i d i 51 o v er k . This leads to the follow ing construction. D eﬁne a constructible set Y i o v er k to b e the set of pairs ( x, f ) where x ∈ Z i ( k ) is a p oint and f is a homomorphism of A x to the algebra of matrices Mat( N i × N i , k ) suc h that the resulting structure of A x -mo dule on k N i b elongs t o the isomorphism class of pro jectiv e A x -mo dules discusse d ab ov e. The gro up GL ( N i , k ) acts naturally on Y i b y c hanging the basis in t he standard co ordinate space k N i . The quotien t set is naturally iden tiﬁed with Z i ( k ), a nd the stabilizer of ev ery p oin t is isomorphic to Ext 0 ( E j , E j ) × in the ab ov e notation. The essen tial elemen t of the presen ted construction is that ev erything is equiv ariant with resp ect to the action of Gal( k / k ). Hence, w e come to the c onclusion that one asso ciates (making man y c hoices) with an ind- constructible category C ov er k a coun table collection of v arieties ( Y i ) i ∈ I (w e can assume that Y i are not just constructible se ts but v arieties after mak- ing further sub divisions) endow ed with alg ebraic a ctions of aﬃne algebraic groups GL ( N i ) suc h that the g r o up oid of isomorphism classes of C ( k ) is nat- urally equiv alen t to the g r o up oid of the disjoin t union of sets Y i ( k ) endo w ed with GL ( N i , k )-actions. If w e repl ace C by an equiv alen t ind-constructible category , or mak e diﬀeren t c hoices in the construction, w e obtain an equiv- alen t in an o bvious sense “ind-constructible stac k”. W e will discus s ind- constructible stac ks later, in Section 4.2. Moreo v er, using the fact that the ﬁrst G alois cohomology with co eﬃcien ts in GL ( N i ) v a nish, one can see that the same is tr ue for C ( k ) (and replace Y i ( k ), GL ( N i , k )-actions by Y i ( k ) a nd GL ( N i , k )-action ∀ i ∈ I ). In general, for any ﬁeld k ′ , k ⊂ k ′ ⊂ k one can deﬁne the descen t data for k ′ and a k ′ -linear A ∞ -category C ( k ′ ) (whic h is triangulated if C is triangulated in the ind-constructible sense). The gro up oid of isomorphism classes of ob jects of H • ( C ( k ′ )) is equ iv alen t to the groupoid o f the disjoin t union of sets Y i ( k ′ ) endo w ed with GL ( N i , k ′ )-actions. More generally , one can deﬁ ne the category C ( k ′ ) for any ﬁeld extension k ′ ⊃ k , not necessarily an algebraic one. In the case k ′ = k w e get a non-fata l crash o f not a tions, b ecause the A ∞ -category C ( k ) in last sens e is equiv alen t to the previously deﬁned C ( k ). In what follows, we will ass ume for conv enience that O b ( C ) for an ind- constructible category C is described b y sc hemes Y i with GL ( N i )-actions. In particular, for an y exten sion k ′ ⊃ k w e will hav e a bijection I so ( C ( k ′ )) ≃ ⊔ i ∈ I Y i ( k ′ ) /GL ( N i , k ′ ) b et w een the set of isomorphis m classes in C ( k ′ ) and the set of orbits. 52 Remark 16 In fa ct, obje cts of an A ∞ -c ate gory form not a stack but a higher stack, i . e . one should sp e ak ab out isomorphi s m s b etwe e n isomorphi s m s etc. Passing to the level of or dinary stacks w e m ake a trunc ation. Pr esumably, for a pr o p er tr e atment of ind-c onstructible c ate gories and pr oblems like non- functoriality of c ones, one should intr o duc e higher c onstructible stacks. L o ok- ing on the guiding example of identiﬁc ation data for ﬁnite non-empty c ol le c- tions, one c an guess an appr opriate notion of a higher c onstructible stack. Namely, it should b e a simplicial c onstructible set X • which satisﬁe s a c on- structible vers i o n of the Kan pr op erty (i.e. ther e exists a c onstructible lifting fr om horns to simpli c es) and such that 1) for a n y k > 2 the c onstructible map ( ∂ 0 , . . . , ∂ k ) : X k → ( X k − 1 ) k +1 is a c onstructible ve ctor bund le over its image ( i.e. ther e exists a c onstructible identiﬁc ation of non-empty ﬁb ers of this map w ith ve ctor sp ac es ) . 2) ∃ k 0 such that ∀ k > k 0 the ab ove map is an i n clusion. The r e ason for the ﬁrst pr op erty is that in the c ase of identiﬁc ation on e ac h step (exc ept ﬁrst two) we have to so lve line ar e quations. The se c on d pr op erty c o mes fr om the p r op erty HO M n | X i × X i = 0 for n ≪ 0 in our axiomatics of ind-c onstructible c ate gories. 3.3 Ind-constructible Calabi-Y au categories and p o- ten tials Let k b e a ﬁeld of c haracteristic zero. Recall that a Calabi-Y au category of dimension d is a w eakly unital k -linear triangulated A ∞ -category C (see [41], [42], [65]), suc h that fo r an y t w o ob jects E , F the Z -graded v ector space Hom • ( E , F ) = ⊕ n ∈ Z Hom n ( E , F ) is ﬁnite-dimensional (hence the space Ext • ( E , F ) is also ﬁnite-dimensional) and moreov er: 1) W e are giv en a non-degenerate pairing ( • , • ) : Hom • ( E , F ) ⊗ Hom • ( F , E ) → k [ − d ] , whic h is symmetric with resp ect to in terch ange of ob jects E and F ; 2) F or an y N > 2 and a sequenc e of ob j ects E 1 , E 2 , . . . , E N w e are giv en a p olylinear Z / N Z -in v arian t map W N : ⊗ 1 ≤ i ≤ N (Hom • ( E i , E i +1 )[1]) → k [3 − d ] , where [1] means the shift in the category of Z -graded v ector spaces, and w e set E N +1 = E 1 ; 53 3) W e hav e: W N ( a 1 , . . . , a N ) = ( m N − 1 ( a 1 , . . . , a N − 1 ) , a N ) , where m n : ⊗ 1 ≤ i ≤ n Hom • ( E i , E i +1 ) → Hom • ( E 1 , E n +1 )[2 − n ] are higher com- p osition maps. The collection ( W N ) N > 2 is called the p otential of C . If d = 3 then for a n y ob ject E ∈ O b ( C ) w e deﬁne a formal series W tot E at 0 ∈ Hom • ( E , E )[1] by the formula: W tot E ( α ) = X n > 2 W n ( α, . . . , α ) n . W e call W tot E the total (or ful l) p otential of the ob j e ct E . W e call the p otential of E the restriction of W tot E to the subsp ace Hom 1 ( E , E ). W e will denote it b y W E . The notion of a Calabi-Y a u category admits a natural generalization to the ind-constructib le case (the pairing is required to b e a morphism of con- structible v ector bundles). It follows from the Axiom A3) that there exists a decomposition of O b ( C ) ≃ ⊔ S i in to the disjoint union of sc hemes suc h that all T a ylor componen ts W N of t he po ten tial are symmetrizations o f regular sections of cyclic p ow ers of algebraic vec tor bundles on sc hemes S i . There- fore w e can treat the f a mily o f p otentials W C = ( W E ) E ∈ O b ( C ) as a function, whic h is regular with resp ect to the v ariable E and formal in the direction α ∈ Hom 1 ( E , E ) (or α ∈ Ext 1 ( E , E ) if our categor y is minimal on the diag- onal). Also the p oten tial W E considered as a function of E ∈ S j b ecomes a section of the pro-algebraic vec tor bundle Q n > 2 S y m n ( HO M 1 | D iag ( S j ) ⊂ S j × S j ) ⋆ , where D iag denotes here the diagonal em b edding. Prop osition 7 In the c ase of 3 d Calabi-Y au c ate gory C c onsisting of one obje ct E the p otential W E admits (after a fo rmal change of c o or di n ates) a splitting: W E = W min E ⊕ Q E ⊕ N E , wher e W min E is the p otential of the minimal mo del C min (i.e. it is a for- mal series on Ext 1 ( E , E ) ), the quadr atic form Q E is deﬁne d on the ve ctor sp ac e Hom 1 ( E , E ) / K er( m 1 : Ho m 1 ( E , E ) → Hom 2 ( E , E )) by the fo rm ula Q E ( α, α ) = m 2 ( α,α ) 2 , an d N E is the zer o function on the imag e o f the map m 1 : Ho m 0 ( E , E ) → Hom 1 ( E , E ) . In the ab ove s plitting fo rmula we use the notation ( f ⊕ g )( x, y ) = f ( x ) + g ( y ) for the dir e ct sum of forma l functions f and g . 54 The abov e Prop osition follo ws from the m inimal mo del theorem for Calabi- Y au a lg ebras (i.e. Calabi-Y au c ategor ies with only o ne ob j ect). In its for- m ulation b elo w w e are going to use the la nguage of fo rmal no n-commu tative geometry from [42]. W e assume that the ground ﬁeld has c haracteristic zero. Theorem 5 a) L et ( X, x 0 , ω , d X ) b e a Z / 2 Z -gr ade d non-c ommutative for- mal p ointe d manifold ( X, x 0 ) en d owe d with an o dd symple ctic fo rm ω and homolo gic al ve ctor ﬁeld d X which pr ese rv e s ω an d vanishes at x 0 . Th en it i s isomorphic to the pr o duct ( X ′ , x ′ 0 , ω ′ , d X ′ ) × ( X ′′ , x ′′ 0 , ω ′′ , d X ′′ ) , wher e ( X ′ , x ′ 0 , ω ′ , d X ′ ) is minimal in the sense that ( Lie d X ′ ) | T x ′ 0 X ′ = 0 (i.e. d X ′ vanishes quadr atic a l ly at x ′ 0 ), a n d the se c on d factor sa tisﬁ e s the fol low- ing pr op erty: ther e exists a ﬁnite-dimen sional s up er ve ctor sp ac e V endo w e d with an even non-de gener ate quadr atic form Q V such that ( X ′′ , x ′′ 0 , ω ′′ , d X ′′ ) is isomorphic to the non-c ommutative formal p ointe d manifol d asso ciate d with V ⊕ Π V ∗ (her e Π is the change o f p arity functor) end owe d with a c onstant symple ctic form ω V c o ming fr om the natur al p airing b etwe en V and Π V ∗ , and homolo gic al ve ctor ﬁe ld d V is the Hamiltonian ve ctor ﬁeld asso ciate d w ith the pul l-b ack of Q V under the na tur al pr oje ction V ⊕ Π V ∗ → V . b) In the Z -gr ade d c ase when X c orr esp onds to a 3 d Calabi-Y au algebr a (i.e. ω has de gr e e − 1 ) a similar statement holds. I n this c ase V is Z -gr ade d ve ctor sp ac e , Q V has de gr e e 0 , and the tangent sp ac e T x ′′ 0 X ′′ isomorphic t o V ⊕ V ∗ [ − 1] . Pr o of. One can pro v e part a) similarly to t he usual minimal mo del t he- orem for A ∞ -algebras or L ∞ -algebras (it is induction b y the o rder of the T ay lor expansion, see e.g. [41]). P art b) is a G m -equiv ariant v ersion of part a).  The Proposition follow s from part b) of the Th eorem, since w e ha v e a decomposition Hom 1 ( E , E ) ≃ Ext 1 ( E , E ) ⊕ V 0 ⊕ ( V 1 ) ∗ where V i , i ∈ Z are the graded comp onen ts of V . The restriction of W tot E to Hom 1 ( E , E ) is the direct sum of W min E , the restriction of Q V to V 0 (w e iden tify Q V with Q E ) and the zero function on ( V 1 ) ∗ . Corollary 2 The minim al mo del p otential W min E do es not dep end on a choic e of minim al mo del for End • ( E ) , if c onsider e d up to a formal n on-line ar auto- morphism of the bund le HO M 1 r estricte d to the diagon al D iag ( S j ) ⊂ S j × S j . 55 Pr o of. Change of the minimal mo del is a Z -graded c hange of co ordinates. It preserv es the top ological ideal generated b y all co ordinates of non-zero degrees.  W e remark that there is a notion of Calabi-Y au category v a lid o v er a ﬁeld k o f arbitrary c haracteristic. In the case of a category with one ob ject E let us denote b y A the A ∞ -algebra Hom • ( E , E ). W e assume that Ext • ( E , E ) = H • ( A ) is ﬁnite-dimensional. Then a Calabi-Y au structure of dime nsion d on A is giv en b y a functional T r of degree − d o n the cyc lic homology H C • ( A ) suc h that the induced functional on H • ( A ) / [ H • ( A ) , H • ( A )] giv es rise to a non-degenerate bilinear form ( a, b ) 7→ T r ( ab ), where a, b ∈ H • ( A ). In the case of p ositiv e ch ara cteristic the notion o f the p oten tial do es not exist in the conv entional sense. This c an b e seen in the example A = F 3 h ξ i / ( ξ 4 ) , deg ξ = +1. The p otential should hav e the form W ( ξ ) = ξ 3 / 3 + . . . whic h do es not mak e sense ov er the ﬁeld F 3 . In general it seems that although the p oten tial do es not exist, its diﬀer- en tial is w ell-deﬁned as a closed 1- form. Remark 17 In the c ase of char acteristic zer o the cyclic homolo gy H C • ( A ) c a n b e id entiﬁe d with the c ohomolo gy of the c o mplex ⊕ n > 1 C y cl n ( A [1]) of cyclic al ly inva ria nt tensors ( se e [ 4 2]). T her efor e the p otential W b e c omes a functional of de gr e e 3 − d on the latter c omplex , va n ishing on the imag e of the diﬀer ential. Henc e it deﬁnes a class [ W ] in ( H C • ( A )) ∗ . Th e latter sp ac e is a k [[ u ]] -mo dule, wher e u is a variabl e , deg u = +2 (se e lo c. cit.). The class [ W ] is r elate d to the functional T r discusse d ab ove by the form ula [ W ] = uT r . In the c as e of a Ca l a bi-Y au algeb r a of dim ension d = 2 k + 1 it is natur al to intr o duc e a cyclic functional W k with the c orr esp onding c l a ss [ W k ] = u k T r . I t c an b e thought of a s a higher-dimension al C h ern-Simons action. In p articular, it de ﬁnes a formal p ower series W 0 k of de gr e e zer o such that it vanis hes with the ﬁrst k deriva tive s on the formal schem e of solutions to the Maur er-Cartan e quation. 3.4 T op ology on the space of stabilit y structures Let C b e an ind-constructible w eakly unital A ∞ -category o v er a ﬁeld k of arbitrary characteris tic. Let cl : O b ( C ) → Γ ≃ Z n b e a map of ind- constructible sets (where Γ is considered as a coun table set of p oin ts) such that the induced map O b ( C )( k ) → Γ factorizes through a group homomor- phism cl k : K 0 ( C ( k )) → Γ. It is easy to see that for an y ﬁeld extension 56 k ′ ⊃ k w e obtain a homomorphism cl k ′ : K 0 ( C ( k ′ )) → Γ. In the case when C is a Calabi-Y au category w e require that Γ is endo w ed with an in teger-v alued bilinear form h• , •i and the homomorphism cl k is compatible with h• , •i and the Euler fo rm on K 0 ( C ( k )). F or ind-constructible triangulated A ∞ -categories the notion of stabilit y structure admits the following v ersion. Deﬁnition 9 A c onstructible stability structur e on ( C , cl) is given by the fol lowing data (cf . Intr o duction, Se ction 1.2): • an in d -c onstructible subset C ss ⊂ O b ( C ) c o nsisting of obje cts c al le d semistable, an d satisfying the c ondition that with e a c h obje ct it c ontains al l iso morphic obje cts, • an additive map Z : Γ → C c al le d c entr al c h ar g e , such that Z ( E ) := Z (cl( E )) 6 = 0 if E ∈ C ss , • a choic e of the br anch of lo garithm Lo g Z ( E ) ∈ C for any E ∈ C ss which is c onstructible as a function of E . These data ar e r e quir e d to satisfy the c orr esp onding axioms fr om S e ction 1.2 for the c a te gory C ( k ) . In p articular • the set of E ∈ C ss ( k ) ⊂ O b ( C )( k ) with the ﬁxe d cl( E ) ∈ Γ \ { 0 } and ﬁxe d Log Z ( E ) is a c onstructible set. Before w e pro ceed with the top ology let us mak e a comparison with the “Lie-algebraic” story o f Section 2. First, we observ e that the set C ss can b e though t of as an analog of the collection of elemen ts ( a ( γ )) γ ∈ Γ \{ 0 } from Section 2.1. Then w e giv e the follo wing deﬁnition of another data and axioms whic h is equiv alent to the one giv en ab ov e and can b e thought of as an analog of the collection of the group elemen ts A V . Deﬁnition 10 A c o nstructible stability structur e on ( C , cl) is given by the fol lowing data: • an ad ditive map Z : Γ → C , 57 • for any b ounde d c on ne cte d set I ⊂ R a n ind - c o nstructible subset P ( I ) ⊂ O b ( C )( k ) whi c h c o ntains which every o b j e ct al l isomorphic obje cts. These data ar e r e quir e d to satisfy the fol l o w ing axioms 12 : • the zer o obje ct of the c ate g ory C ( k ) b elongs to al l P ( I ) , • ∪ n ∈ Z > 0 P ([ − n, n ]) = O b ( C )( k ) , • if I 1 < I 2 in the sense that every element of I 1 is strictly less than any element of I 2 then for any E 1 ∈ P ( I 1 ) and E 2 ∈ P ( I 2 ) one has Ext ≤ 0 ( E 2 , E 1 ) = 0 , • P ( I + 1) = P ( I )[1] wher e [1] is the shift functor in C ( k ) , • (Extensi o n Pr op erty) If I = I 1 ⊔ I 2 and I 1 < I 2 in the ab ove s e nse then the ind-c onstructible set P ( I ) is isomo rphic to the ind-c onstructible sub- set of such obje cts E ∈ O b ( C )( k ) which ar e extensions E 2 → E → E 1 with E m ∈ P ( I m ) , m = 1 , 2 , • if I is an interval of the length strictly less than one, E ∈ P ( I ) , E 6 = 0 , then Z (cl( E )) b elongs to the strict se ctor V I = { z = r e π iϕ ∈ C ∗ | r > 0 , ϕ ∈ I } , • ther e is a no n -de gener ate quadr atic form Q on Γ R such that Q | Ker Z < 0 , and for any interval I of the le ngth strictly less than 1 the set { cl( E ) ∈ Γ | E ∈ P ( I ) } ⊂ Γ b e l o ngs to the c onvex c one C ( V I , Z, Q ) deﬁne d in Se ction 2.2, • let I b e an interval of the length strictly les s than 1 , and γ ∈ Γ . Then the set { E ∈ P ( I ) | cl( E ) = γ } is c onstructible. The equiv alence of Deﬁnitions 9,1 0 can b e pro v ed similarly to the pro of of Theorem 2. With this equiv alen t descrip tion of a constructible stabilit y condition w e observ e that the collection of sets P ( I ) consid ered for all in terv als I with 12 One s ho uld read expr essions O b ( C ) , P ( I ) etc. in this list of axioms as sets o f k -p oints. 58 the length less than 1 are analogous to the collection of elemen ts A V where V = V I (see Section 2.2) and the Extension Prop ert y is analog o us to the F actorization Prop ert y . One has the following result. Prop osition 8 F or any c onstructible stability structur e on C and any ﬁeld extension k ⊂ k ′ the c ate gory C ( k ′ ) c arries a lo c a l ly ﬁnite stability structur e in the sense of [9] with the c entr al char ge given by Z ◦ cl k ′ and the c ol le ction of additive sub c ate gories P ( I )( k ′ ) , wher e I runs thr ough the set of b ounde d c o nne cte d subsets of R as ab ove. Pr o of. The pro of is straightforw a rd. Lo cal ﬁniteness in the sense of [9] follo ws from our (stronger) assumption on the quadratic form Q .  Let us denote b y S tab ( C , cl) the set o f constructible stabilit y structures on C with a ﬁxed class map cl. Our goal is to in tro duce a top ology on S tab ( C , cl). Let ∆ ⊂ C be a triangle with one vertex at the origin. W e c ho ose a branc h of the function z 7→ Log z for z ∈ ∆. W e denote the corresp onding arg ument function b y Arg ( z ). W e denote by C ∆ , Log an A ∞ -sub category of C generated b y the zero ob ject 0 , semistable o b jects E with Z ( E ) ∈ ∆ , Arg ( E ) ∈ Arg (∆) as w ell as extensions J o f suc h ob jects satisfying t he condition Z ( J ) ∈ ∆. W e allo w the case ∆ = V where V is a sector, in whic h case w e will use the notation C V , Log . It is easy to see that C ∆ , Log is an ind-constructible cat- egory . Notice that in the language of ind-constructible sets P ( I ) w e ha v e O b ( C V I , Log ) = P ( I ) for some c hoice of the branc h Log. The condition of genericit y o f the sector V I correspo nding to a closed in terv al I = [ a, b ] of the length les s than 1 corresponds to the follo wing genericit y condition of the set P ( I ): b oth P ( { a } ) and P ( { b } ) are zero categories (equiv alently P ([ a, b ]) = P (( a, b ))). Let us ﬁx a non-degenerate quadratic form Q on Γ R suc h that Q | Ker Z < 0 and Q (cl( E )) > 0 for an y E ∈ C ss . W e introduce the top ology on S tab ( C , cl) in the fo llowing wa y . Let us consider a family σ x = ( Z x , C ss x , . . . ) , x ∈ X o f stabilit y structures in a neigh b orho o d of x 0 ∈ X . Then for ev ery p oin t x , a generic closed inte rv al I = [ a, b ] of the length less than 1 w e ha ve the full category P ( I ) x ⊂ O b ( C ). F or a given γ ∈ Γ \ { 0 } w e denote by P ( I ) x,γ the constructible subset of ob jects E ∈ P ( I ) x suc h that cl( E ) = γ . W e say that a family σ x = ( Z x , C ss x , . . . ) , x ∈ X of stabilit y structures parametrized b y a top ological space X is c ontinuous at a given stability struc- tur e σ x 0 = ( Z 0 , C ss 0 , . . . ) if: 59 a) The map x → Z x is contin uous a t x = x 0 . b) There exists a neigh bo r ho o d U of x 0 suc h that fo r an y E ∈ C ss x , x ∈ U w e hav e Q (cl ( E )) > 0. c) F or an y generic closed in terv al I of the length strictly less than 1 the constructible set P ( I ) x,γ is lo cally constan t near x 0 (cf. Deﬁnition 3c)). In this w ay w e obtain a Hausdorﬀ top olo g y on S tab ( C , cl ). W e can deﬁne a parallel transpo rt along a path σ t in the space S tab ( C , cl) similarly to the case of stabilit y structures in graded Lie algebras discusse d in Section 2. Eac h time when w e cross the w all of ﬁrst kind w e use the ab ov e prop ert y c) in order to “recalculate” the set of semistable ob jects. In order to do this w e use the follo wing prop ert y: E ∈ O b ( C ∆ , Log ) is semistable iﬀ there is no extension E 2 → E → E 1 where E i , i = 1 , 2 are non-ze ro ob jects of C ∆ , Log suc h that Arg( E 2 ) > Arg ( E 1 ). The se cons iderations also ensure that the Theorem 1 from In tro duction holds (i.e. the natural pro jection of the space of stabilit y conditions to the space of cen tral c harges is a lo cal homeomorphism). 4 Motivic functions and motivic Milnor ﬁb er 4.1 Recollecti on on motivic functions Recall (see [14 ]) that for an y constru ctible s et X ov er k one can deﬁne an ab elian group M ot ( X ) of motivic functions as the group generated b y sym- b ols [ π : S → X ] := [ S → X ] where π is a morphism of constructible sets, sub j ect to the relations [( S 1 ⊔ S 2 ) → X ] = [ S 1 → X ] + [ S 2 → X ] . F or any constructible morphism f : X → Y w e hav e tw o homomorphisms of groups: 1) f ! : M ot ( X ) → M ot ( Y ), such that [ π : S → X ] 7→ [ f ◦ π : S → X ]; 2) f ∗ : M ot ( Y ) → M ot ( X ), suc h that [ S ′ → Y ] 7→ [ S ′ × Y X → X ]. Moreo v er, M ot ( X ) is a comm utativ e ring via the ﬁb er pro duct op eration. W e denote by L ∈ M ot ( S pec ( k )) the elemen t [ A 1 k ] := [ A 1 k → S pec ( k )]. It is customary to add its f ormal in v erse L − 1 to the ring M ot ( S pec ( k )) (or more generally to the ring M ot ( X ) whic h is a M ot ( S pec ( k ))-algebra). There are sev eral “realizations” of the theory of motivic functions whic h w e a re go ing to recall b elo w. 60 (i) There is a homomorphism of rings χ : M ot ( X ) → C onstr ( X , Z ) , where C onstr ( X , Z ) is the ring of intege r- v alued constructible functions on X endo we d with the p oint wise multipl ication. More precisely , the elemen t [ π : Y → X ] is mapp ed in to χ ( π ) , where χ ( π )( x ) = χ ( π − 1 ( x )), whic h is the Euler c haracteristic of the ﬁb er π − 1 ( x ). (ii) Let now X b e a sc heme of ﬁnite t yp e o v er a ﬁeld k , and l 6 = char k b e a prime num b er. There is a homomorphism of rings M ot ( X ) → K 0 ( D b constr ( X , Q l )) , where D b constr ( X , Q l ) is the b ounded derive d category of ´ etale l -adic shea v es on X with constructible cohomology , such that [ π : S → X ] 7→ π ! ( Q l ) , whic h is the direct image of the constan t sheaf Q l . Notice that D b constr ( X , Q l ) is a tensor category , hence G rothendiec k gro up K 0 is naturally a ring. The homomorphisms f ! and f ∗ discuss ed ab ov e corresp ond to the functors f ! (direct image with compact supp ort) and f ∗ (pullbac k), whic h w e will denote b y the same sym b o ls. W e will also us e the notation R X φ := f ! ( φ ) for the canonical map f : X → S pec ( k ). (iii) In the sp ecial case X = S pec ( k ) the ab ov e homomorphism b ecomes a map [ S ] 7→ X i ( − 1) i [ H i c ( S × S p ec ( k ) S pec ( k ) , Q l )] ∈ K 0 (Gal( k / k ) − mod Q l ) , where Gal( k / k ) − mod Q l is the tensor category o f ﬁnite-dime nsional con tin- uous l -adic represen tations of t he Galois group G al( k / k ), and w e tak e the ´ etale cohomology of S with compact suppo rt. (iv) If k = F q is a ﬁnite ﬁeld then for an y n > 1 we ha v e a homomorphism M ot ( X ) → Z X ( F q n ) giv en by [ π : Y → X ] 7→ ( x 7→ # { y ∈ X ( F q n ) | π ( y ) = x } ) . 61 Op erations f ! , f ∗ correspo nd to pullbac ks and pushforw ards o f f unctions on ﬁnite sets. (v) If k ⊂ C then the category of l -adic constructible shea v es on a sc heme of ﬁnite t yp e X can b e replaced in the ab ov e cons iderations b y the Saito’s category of mixed Ho dge mo dules (see [61]). (vi) In the case X = S pec ( k ) one has t w o additional homomorphisms: a) Serre p olynomial M ot ( S pec ( k )) → Z [ q 1 / 2 ] suc h that [ Y ] 7→ X i ( − 1) i X w ∈ Z > 0 dim H i,w c ( Y ) q w / 2 , where H i,w c ( Y ) is the w eigh t w comp onen t in the i -th W eil cohomolog y group with compact support. b) If char k = 0 then w e ha v e the Ho dge p olynomial M ot ( S pec ( k )) → Z [ z 1 , z 2 ] suc h that [ Y ] 7→ X i > 0 ( − 1) i X p, q > 0 dim Gr p F (Gr W p + q H i D R,c ( Y )) z p 1 z q 2 , where Gr W • and Gr • F denote the graded componen ts with resp ect to the w eigh t and Ho dge ﬁltrations, a nd H i D R,c denotes the de Rham cohomology with compact supp ort. Clearly the Ho dge p olynomial determines the Serre p olynomial via the homomorphism Z [ z 1 , z 2 ] → Z [ q 1 / 2 ] suc h that z i 7→ q 1 / 2 , i = 1 , 2. 4.2 Motivic functions in the equiv arian t setting Here w e giv e a short exp osition of the generalization of the theory of motivic functions in the equiv arian t setting (essen tially due to Jo yce [3 5], here we use sligh tly diﬀeren t terms). Let X b e a constructible set o ve r a ﬁeld k and G b e an aﬃne algebraic group acting on X , in the sense that G ( k ) acts on X ( k ) and there exists a G - v ariety S ov er k with a constructible equiv arian t iden tiﬁcation X ( k ) ≃ S ( k ). W e deﬁne the group M ot G ( X ) of G -equiv arian t motivic functions as ab elian group generated b y G -equiv ariant constructible maps [ Y → X ] mo d- ulo the relations 62 • [( Y 1 ⊔ Y 2 ) → X ] = [ Y 1 → X ] + [ Y 2 → X ], • [ Y 2 → X ] = [( Y 1 × A d k ) → X ] if Y 2 → Y 1 is a G - equiv arian t constructible v ector bundle of ra nk d . This gr o up for m a commu tative ring via the ﬁb er pro duct, and a morphism of constructible sets with group actions induces a pullbac k homomorphism of corresp onding rings. There is no natural op eration of a pushforw ard for equiv arian t motivic functions, and for that one has to enlarge ring o f func- tions. Consider the follo wing 2- category of constructible stac ks. First, its ob- jects are pairs ( X , G ) as ab ov e 13 , and the ob jects of the category of 1- morphisms Ho m(( X 1 , G 1 ) , ( X 2 , G 2 )) a r e pairs ( Z , f ) where Z is a G 1 × G 2 - constructible set suc h that { e } × G 2 acts freely on Z in suc h a w a y that w e ha v e the induced G 1 -equiv ariant isomorphism Z / G 2 ≃ X 1 , and f : Z → X 2 is a G 1 × G 2 -equiv ariant map ( G 1 acts trivially on X 2 ). An elemen t of Hom(( X 1 , G 1 ) , ( X 2 , G 2 )) deﬁnes a map of sets X 1 ( k ) /G 1 ( k ) → X 2 ( k ) /G 2 ( k ). F urthermore, elemen ts of Hom(( X 1 , G 1 ) , ( X 2 , G 2 )) form naturally ob jects of a group oid, so w e obtain a 2-category S C O N k of c onstructible stacks over k . The 2-category of constructib le stac ks carries a direct sum op eration induced b y the disjoin t union o f stac ks ( X 1 , G 1 ) ⊔ ( X 2 , G 2 ) = (( X 1 × G 2 ⊔ X 2 × G 1 ) , G 1 × G 2 ) , as wel l as the pro duct induced b y the Cartesian pro duct ( X 1 , G 1 ) × ( X 2 , G 2 ) = ( X 1 × X 2 , G 1 × G 2 ) . The ab elian group of stac k motivic function M ot st (( X , G )) is generated b y the gro up of isomorphism classes of 1-morphisms of stac ks [( Y , H ) → ( X , G )] with the ﬁxed target ( X, G ), sub ject to the relations • [(( Y 1 , G 1 ) ⊔ ( Y 2 , G 2 )) → ( X , G )] = [( Y 1 , G 1 ) → ( X , G )] + [( Y 2 , G 2 ) → ( X , G )] • [( Y 2 , G 1 ) → ( X , G )] = [( Y 1 × A d k , G 1 ) → ( X , G )] if Y 2 → Y 1 is a G 1 - equiv arian t constructible v ector bundle of rank d . 13 Strictly sp eaking,we should denote such stacks a s triples ( X , G, α ) where α is the action o f G on X . 63 The ring M ot G ( X ) maps to M ot st (( X , G )). Notice that eve ry isomorphism class [( Y , H ) → ( X, G )] corresp o nds to an o rdinary morphism of construc tible sets acted by algebraic g r o ups. Indeed, in the notation of the deﬁnition of 1-morphism of stac ks w ith ( X 1 , G 1 ) := ( Y , H ) and ( X 2 , G 2 ) := ( X , G ), w e can replace the source ( Y , H ) by an eq uiv alen t stac k ( Z, G 1 × G 2 ) and get an ordinary morphism ( Z , G 1 × G 2 ) → ( X , G ) of constructible sets acted b y a lg ebraic groups. One can deﬁne t he pullbac k, the pushforw ard and the pro duct of elemen ts of M ot st (( X , G )). Finally , for a c onstructible stac k S = ( X , G ) w e deﬁne its c lass in the ring K 0 ( V ar k )[[ L ] − 1 , ([ GL ( n )] − 1 ) n > 1 ] a s [ S ] = [( X × GL ( n )) /G ] [ GL ( n )] , where w e ha ve chose n an em b edding G → GL ( n ) f o r some n > 1, and ( X × GL ( n )) /G is the ordinary quotien t b y the diagonal free action (th us in the RHS w e hav e the quotien t of motiv es of ordinary v arieties). The result do es not dep end on the c hoice of em b edding (see [3], Lemma 2.3). Th en w e deﬁne the in tegral R S : M ot st ( S ) → K 0 ( V ar k )[[ L ] − 1 , ([ GL ( n )] − 1 ) n > 1 ] as R S [ S ′ → S ] = [ S ′ ]. If k is ﬁnite, one can asso ciate with ev ery constructible stac k S = ( X, G ) a ﬁnite set S ( k ), the set of orbits of GL ( n, k ) acting on (( X × GL ( n )) /G )( k ). There is a homomorphi sm of the alg ebra of stack motivic functions to the algebra of Q -v alued functions on S ( k ). The identi ty function 1 ( X,G ) := [( X, G ) → ( X , G )] represen t ed b y the iden tit y map, when in terpreted as a measure (for push- forw ards) maps to the “stac k coun ting measure” on S ( k ) whic h is equal to #( GL ( n, k )) − 1 times the direct image o f the ordinary coun ting measure 14 on (( X × GL ( n )) /G )( k ). Its densit y with resp ect to the ordinary countin g measure on S ( k ) is giv en by the in v erse to the order of the stabilizer. Our construction in Section 3.2 can b e rephrased as a construction of an ind-constructible stac k of ob jects. Hence w e can sp eak ab o ut stac k motivic functions on an ind-constructible category . 14 F or every ﬁnite set S the counting measure on S has weigh t 1 fo r each element s ∈ S . 64 4.3 Motivic Milnor ﬁb er Let M be a complex manifold, x 0 ∈ M . Recall, that for a germ f of an analytic f unction at x 0 suc h that f ( x 0 ) = 0 one can deﬁne its Milnor ﬁb er M F x 0 ( f ), whic h is a lo cally trivial C ∞ -bundle ov er S 1 of manifolds with the b oundary ( deﬁned only up to a diﬀeomorphism): { z ∈ M | dist ( z , x 0 ) ≤ ε 1 , | f ( z ) | = ε 2 } → S 1 = R / 2 π Z , where z 7→ Arg f ( z ). Here dist is an y s mo oth metric on M near x 0 , and there exists a constan t C = C ( f , dist ) and a p ositiv e in teger N = N ( f ) such that for 0 < ε 1 ≤ C and 0 < ε 2 < ε N 1 the C ∞ t yp e of t he bundle is the same for all ε 1 , ε 2 , dist . In particular, ta king the cohomology of the ﬁb ers we obtain a well-deﬁ ned lo cal system on S 1 . There are sev eral alg ebro- geometric ve rsions of this construction (theories of nearby cycles). They pro duce analogs o f lo cal systems on S 1 , f or example l - adic represen tations of the group Gal( k (( t )) sep / k (( t ))) where l 6 = char k . There is a con v enien t mo del of the Milnor ﬁb er in non-archim edean ge- ometry . In order to describ e it w e note that the ﬁeld K = k (( t )) is a non- arc himedean ﬁeld endow ed with the (standard) v aluation, and with the norm giv en by | a | = c val ( a ) for a giv en constan t c ∈ (0 , 1). Let f ∈ k [[ x 1 , . . . , x n ]] , f (0) = 0 b e a f o rmal series considered as a n elemen t of K [[ x 1 , . . . , x n ]]. Clearly it is con v ergen t in the non-arc himedean sense in the domain U ⊂ ( A n k ) an deﬁned b y inequalities | x i | < 1 , 1 ≤ i ≤ n . The non-arc himedean analog of the Milnor ﬁb er is g iven (at the lev el of p o ints) b y the ﬁbration { x = ( x 1 , . . . , x n ) ∈ U | max i | x i | ≤ ε 1 , 0 < | f ( x ) | ≤ ε 2 } → → { w ∈ ( A 1 k ) an | 0 < | w | ≤ ε 2 } , where ε 1 , ε 2 are p ositiv e n um b ers as a b o ve (cf. [54]). Ideally , we would lik e to hav e the follow ing picture. Let V → X b e a v ector bundle o ve r a sc heme of ﬁnite type X/ k , and f ∈ Y n > 1 Γ( X , S y m n ( V ∗ )) 65 b e a function o n the for mal completion of zero section o f tot( V ) v anishing on X . W e would lik e to associate with suc h data a motivic Milnor ﬁb er M F ( f ) ∈ M o t ( X × G m ) . Here the factor G m replaces the circle S 1 in the analytic picture. Moreov er, the motivic function M F ( f ) should b e “unramiﬁed” in G m -direction (i.e. it should correspond to a G m -in v arian t stratiﬁcation of X × G m ). In the case k = C , assuming that f is con v ergen t near zero secti on, the v a lue M F x ( f ) at a p oin t ( x, ǫ ) ∈ X ( C ) × C ∗ should be thought of as a represen ta tiv e of the alternating sum X i ( − 1) i [ H i ( f − 1 ( ε ) ∩ B 0 ,x )] where ε 6 = 0 is a suﬃc iently small complex n um b er and B 0 ,x is a small op en ball around 0 in the ﬁb er V x . Notice that here w e use the usual cohomology and not the one with compact support. Also, we can consider the case when X is a constructible set and V → X is a constructible v ector bundle. W e say that f ∈ Q n > 1 Γ( X , S y m n ( V ∗ )) is constructible if ( X, V , f ) is constructibly isomorphic to an algebraic family of formal functions ov er a sc heme of ﬁnite t yp e as abov e. This goal w as ac hiev ed b y Denef and Lo eser (see [14]) in the case char k = 0 b y using motivic inte gratio n and resolution of singularities . In this case t he group M ot ( X × G m ) is replaced by M ot µ ( X ), where µ = lim ← − n µ n and µ acts trivially o n X . Here µ n is the gro up of n - th ro ots of 1 in k . W e will alw ay s assume that µ - action is “go o d” in the sense that µ acts via a ﬁnite quotien t µ n and ev ery its orbit is contained in a n aﬃne op en subsc heme. Notice that there is a homomorphism of groups M ot µ ( X ) → M ot ( X × G m ) , [ π : Y → X ] 7→ [ π 1 : ( Y × G m ) /µ n → X × G m ] , where µ acts on Y via its quotien t µ n and π 1 ( y , t ) = π ( y ) t n . As w e w ork with constructible sets, it is suﬃcien t to deﬁne the motivic Milnor ﬁb er not for a family , but for an individu al f o rmal g erm of a f unction. Let M b e a smo oth formal sc heme ov er k with closed p oin t x 0 and f b e a function on M v anishing at x 0 (e.g. M could b e the formal completion at 0 of a ﬁb er of ve ctor bundle V → X in the ab o v e notation). W e assume that f is not iden tically equal to zero near x 0 , otherwise the Milnor ﬁb er w ould b e empt y . Let us ch o ose a simple normal crossing resolution of singularities π : M ′ → M of the h ypersurface in M giv en by the equation f = 0 with 66 exceptional divisors D j , j ∈ J . The explicit form ula for the motivic Milnor ﬁb er from [14] lo oks suc h as follo ws 15 . M F x 0 ( f ) = X I ⊂ J,I 6 = ∅ (1 − L ) # I − 1 [ g D I 0 ∩ π − 1 ( x 0 )] ∈ M ot µ ( S pec ( k )) , where D I = ∩ j ∈ I D j , D 0 I is the complemen t in D I to the union of all other exceptional divisors, and g D I 0 → D 0 I is a certain Galois co ve r with the Ga lois group µ m I , where m I is the g.c.d. of the multip licities of all divis ors D i , i ∈ I (see [14] f or the details). Informally speaking, the ﬁb er of the co ve r g D I 0 → D 0 I is the set of connected comp o nents of a non-zero lev el set of function f ◦ π near a p oin t of D 0 I . The space M ot µ ( X ) carries an ass o ciativ e pro duct in tro duced b y Loo i- jenga (see [45]) whic h is diﬀeren t from the one deﬁned ab ov e. It is essen tial for the motivic Thom-Sebastiani theorem whic h will b e discussed later. Let us sk etc h a construction of this pro duct. First, let us in tro duce the comm utativ e r ing M ot ( X × A 1 k ) conv whic h coincides as an ab elian group with M ot ( X × A 1 k ) but carries the “con v olution pro duct” [ f 1 : S 1 → X × A 1 k ] ◦ [ f 2 : S 2 → X × A 1 k ] = [ f 1 ⊕ f 2 : S 1 × X S 2 → X × A 1 k ] , where ( f 1 ⊕ f 2 )( s 1 , s 2 ) = ( pr X ( f 1 ( s 1 )) , pr A 1 k ( f 1 ( s 1 )) + pr A 1 k ( f 2 ( s 2 ))). The ring M ot ( X × A 1 k ) conv con tains t he ideal I := pr ∗ X ( M ot ( X )) . By deﬁnition w e ha v e an epimorphism of ab elian groups M ot ( X ) → I . Let i : X → X × A 1 k , x 7→ ( x, 0) , j : X × ( A 1 k \ { 0 } ) → X × A 1 k b e natural em b eddings. They giv e rise to an isomorphis m of abelian groups i ∗ ⊕ j ∗ : M ot ( X × A 1 k ) ≃ M ot ( X ) ⊕ M ot ( X × ( A 1 k \ { 0 } )) . Since i ∗ ◦ pr ∗ X = id M ot ( X ) w e see that the restriction of pr ∗ X to I giv es an isomorphism of ab elian g roups I ≃ M ot ( X ) , 15 In [14] it w as assumed that f is a reg ular function on a s mo oth scheme, but the form ula and all the arguments work in the for mal setting as well. 67 and j ∗ induces the isomorphism o f groups M ot ( X × A 1 k ) / I ≃ M ot ( X × ( A 1 k \ { 0 } )) . Using the latter isomorphism w e t ra nsfer the con volution pro duct and endo w M ot ( X × ( A 1 k \ { 0 } )) with an associative pro duct whic h w e will call exotic . Recall that w e ha v e a homomorphism of groups M ot µ ( X ) → M ot ( X × G m ) = M ot ( X × ( A 1 k \ { 0 } )) . One can chec k that the image of M ot µ ( X ) is closed under the exotic pro duct. In tuitiv ely , t he image consists of isotrivial fa milies of v arieties o ve r X × ( A 1 k \ { 0 } ) equipped with a ﬂat connection whic h has ﬁnite (i.e. b elonging to some µ n ) mono drom y . The complicated form ula from [45] coincides with the induced pro duct on M ot µ ( X ). In what follows we will use the notation M µ ( X ) := ( M ot µ ( X ) , exotic pro duct) . Let V → X, V ′ → Y b e tw o constructible v ector bundles endo we d with constructible families f , g of formal p ow er series. W e denote b y f ⊕ g the sum of pullbac ks of f and g to the constructible v ector bundle pr ∗ X V ⊕ pr ∗ Y V ′ → X × Y . Then w e ha ve the follo wing motivic v ersion of Thom-Sebastiani theorem. Theorem 6 ([15]) One has (1 − M F ( f ⊕ g )) = pr ∗ X (1 − M F ( f )) · pr ∗ Y (1 − M F ( g )) ∈ M µ ( X × Y ) . One can mak e similar constructions in the equiv aria nt setting. Let X/ k b e a constructible set end ow ed w ith the go o d action of an a ﬃne algebraic group G . W e endo w X also with the t rivial µ -action. Then, similarly to the ab ov e, w e can equip M o t G × µ ( X ) with the exotic pro duct (b y considering G - equiv arian t families ov er X in the previous considerations). W e will denote the resulting ring b y M G,µ ( X ). Using the canonical resolution of singularities (see e.g. [5]) one can deﬁne the equiv a rian t motivic Milnor ﬁber in the case of equiv ariant families o f functions, and state the correspo nding v ersion of Thom-Sebastiani theorem. In the case of arbitrary k there is an l -adic ve rsion of the ab ov e results. More precisel y , the theory of Milnor ﬁber is replaced by the theory of nearb y 68 cycles (see [64]), with the con v olution deﬁned b y Laumon [43]. The Thom- Sebastiani theorem w as prov ed in this case b y Pierre Deligne a nd probably b y Lei F u (b oth unpublished ). There is an analog of the Ho dge p olynomial in the story (see [14], 3.1.3). Let us assume that k = C for simplicit y . Then w e hav e a homomorphism of rings M µ ( S pec ( k )) → ( X α,β ∈ Q ,α + β ∈ Z c α,β z α 1 z β 2 | c α,β ∈ Z ) . Namely , for a smoo th pro jectiv e µ n -sc heme Y w e set [ Y ] 7→ X p,q > 0 ,p,q ∈ Z ( − 1) p + q dim H p,q , 0 ( Y ) z p 1 z q 2 + + X p,q > 0 ,p,q ∈ Z X 1 ≤ i ≤ n − 1 ( − 1) p + q dim H p,q ,i ( Y ) z p + i/n 1 z q +1 − i/n 2 , where H p,q ,i ( Y ) is the subspace of the cohomology H p,q ( Y ), where an elemen t ξ ∈ µ n acts by mu ltiplication by ξ i . The app earance of rational exp onents w as ﬁrst time observ ed in the Ho dge sp ectrum of a complex isolated singu- larit y . T a king z 1 = z 2 = q 1 / 2 w e obtain the corresponding Serre p olynomial. 4.4 An in tegral iden tit y In this section w e are going to discuss the identi ty whic h w ill b e crucial in the pro of of the main theorem of Section 6. Let k b e a ﬁeld of c haracteristic zero, a nd V 1 , V 2 , V 3 b e ﬁnite-dimensional k -v ector spaces . Conjecture 4 L et W b e a form a l serie s on the p r o duct V 1 × V 2 × V 3 of thr e e ve ctor sp ac es, dep ending in a c onstructible way on ﬁnitely many ex tr a p a r am eters, such that W (0 , 0 , 0) = 0 and W has de gr e e zer o with r esp e c t to the diagonal ac tion of the multiplic ative gr oup G m with the weights (1 , − 1 , 0) . We denote by c W the G m -e quivariant extensio n of W to the formal neighb orho o d b V 1 of V 1 × { 0 } × { 0 } ⊂ V 1 × V 2 × V 3 . Th en we have the fol lowi n g fo rmula (wher e we denote the di r e ct image by the inte gr al): Z v 1 ∈ V 1 (1 − M F ( v 1 , 0 , 0) ( c W )) = L dim V 1 (1 − M F (0 , 0 , 0) ( W | (0 , 0 ,V 3 ) )) , 69 wher e in the RHS we c onsider the motivic Milnor ﬁb er at (0 , 0 , 0) o f the r estriction of W to the subsp ac e (0 , 0 , V 3 ) . Using the ob vious equalit y R V 1 1 = L dim V 1 w e can rewrite the iden tit y as Z v 1 ∈ V 1 M F ( v 1 , 0 , 0) ( c W ) = L dim V 1 · M F (0 , 0 , 0) ( W | (0 , 0 ,V 3 ) ) . Let us discuss the l -adic vers ion of t he Conjecture. F or simplici ty w e assume that the v ector space s do not dep end on extra par a meters. Then w e ha v e a morphism of formal sche mes π : b V 1 → S pf ( k [[ w ]]) suc h that w 7→ c W as w ell as an em b edding i V 1 : V 1 → c W − 1 (0). F or an y morph ism π : X → S pf ( k [[ w ]]) of fo rmal sc hemes to we denote b y R ψ π the functor of nearb y cycles . It acts f rom the b ounded deriv ed category of l -adic constructible shea v es on X to the b ounded deriv ed category of l -adic constructible shea v es on X 0 = π − 1 (0) endow ed with the action of the inertia gr o up (hence they can b e informally though t of as l -adic constructible shea v es on X 0 × S p ec ( k ) k (( w ))). Prop osition 9 The c omp lex R Γ c ( i ∗ V 1 R ψ π ( Q l )) is isomo rp h ic (as a c omp lex of G al( k (( w )) / k (( w ))) -mo dules) to the c omplex R Γ c ( V 1 , Q l ) ⊗ j ∗ R ψ b π ( Q l ) , wher e j : S pec ( k ) → V 3 , j (0) = 0 i s the natur al emb e dd i n g and b π is the morphism o f the fo rmal c om p letion of 0 ∈ { 0 } × { 0 } × V 3 to S pf ( k [[ w ]]) given by the r estriction W | { 0 }×{ 0 }× V 3 . Pr o of. W e will giv e a sk etc h of the pro of based on the non-a rchime dean mo del for the Milnor ﬁb er desc rib ed in Section 4 .3. Let us consider the k (( t ))-analytic space (in the sense of Berk ovic h) as- so ciated with the sc heme ( V 1 × V 2 × V 3 ) × S p ec ( k ) S pec ( k (( t ))). Let us c ho ose suﬃcien tly small positive n um b ers ε 1 , ε 2 , ε 3 , ε 4 (w e will specify them later) and deﬁne an analytic subspac e Y = Y ε 1 ,ε 2 ,ε 3 ,ε 4 b y the following inequalities: | v 1 | ≤ 1 + ε 1 , | v 2 | , | v 3 | ≤ ε 2 , ε 4 ≤ | W ( v 1 , v 2 , v 3 ) | ≤ ε 3 . Notice that the series W ( v 1 , v 2 , v 3 ) is conv ergen t on Y b ecause of homogeneit y prop ert y . W e in tro duce another analytic space Y ′ = Y ′ ε 1 ,ε 2 ,ε 3 ,ε 4 ⊂ Y ε 1 ,ε 2 ,ε 3 ,ε 4 b y c hanging the inequalit y | v 1 | ≤ 1 + ε 1 to the equalit y | v 1 | = 1 + ε 1 (all other inequalities remain unc hanged). There is a natural pro jection pr Y → A (resp. its restriction pr Y ′ → A ) of Y (resp. of Y ′ ) to the ann ulus A = { w | ε 4 ≤ | w | ≤ ε 3 } . Let us now consider the complex C one (( pr Y → A ) ∗ ( Q l ) → ( pr Y ′ → A ) ∗ ( Q l ))[ − 1] . 70 This is a li s se sheaf on the ann ulus, i.e. a contin uous l -adic represen tation of the fundamen tal gr o up of the k (( t ))-analytic space A . There is a tautological em b edding k (( w )) → O an ( A ). It induces the homomorphism of proﬁnite groups π 1 ( A ) → Gal( k (( w )) / k (( w ))). Then one can sho w that the complex C one (( pr Y → A ) ∗ ( Q l ) → ( pr Y ′ → A ) ∗ ( Q l ))[ − 1] ≃ ≃ ( pr Y → A ) ∗ ( C one ( Q l → ( i Y ′ ֒ → Y ) ∗ Q l )) on A is quasi-isomorphic to the pull-bac k of the complex R Γ c ( i ∗ V 1 R ψ π ( Q l )) of Ga l( k (( w )) / k (( w )))-mo dules. W e decomp ose the space Y in to a disjoin t union Y 0 ⊔ Y 1 where for Y 0 w e hav e | v 1 || v 2 | = 0 while for Y 1 w e hav e | v 1 || v 2 | 6 = 0. Similarly , w e ha v e a decomposition Y ′ = Y ′ 0 ⊔ Y ′ 1 . W e claim that the complex ( pr Y 0 → A ) ∗ ( C one ( Q l → ( i Y ′ 0 ֒ → Y 0 ) ∗ Q l )) is quasi-isomorphic to the pull-bac k of the complex R Γ c ( V 1 , Q l ) ⊗ j ∗ R ψ b π ( Q l ) of Gal( k (( w )) / k (( w )))-mo dules. Notice that W | Y 0 dep ends on v 3 only . F ur- thermore, Y 0 and Y ′ 0 can b e decomp osed as t he pro ducts Y 0 = Y 3 × Z 0 , Y ′ 0 = Y 3 × Z ′ 0 , where Z 0 := { ( v 1 , v 2 ) ∈ V an 1 × V an 2 | v 1 = 0 or v 2 = 0 , | v 1 | ≤ 1 + ε 1 , | v 2 | ≤ ε 2 } , Z ′ 0 := { ( v 1 , v 2 ) ∈ V an 1 × V an 2 | v 2 = 0 , | v 1 | = 1 + ε 1 } , Y 3 := { v 3 ∈ V an 3 | | v 3 | ≤ ε 2 , ε 4 ≤ | W (0 , 0 , v 3 ) | ≤ ε 3 } . Here we denote b y V an i , i = 1 , 2 , 3 t he k (( t ))-analytic space asso ciated with the sc heme V i × S p ec ( k ) S pec ( k (( t )) , i = 1 , 2 , 3. Notice that Z 0 is the b ouquet of tw o (non-archim edean) balls. It follo ws that the inclusion of the ball Z ′′ 0 = { ( v 1 , 0) | | v 1 | ≤ 1 + ε 1 } in to Z 0 induces isomorphisms of the Berk o vic h ´ etale cohomology groups of the l - adic shea v es on the analytic spaces. Therefore the cohomology groups of the pair ( Z 0 , Z ′ 0 ) coincide with the cohomology groups of the pair ( Z ′′ 0 , Z ′ 0 ). The latter are equal to R Γ c ( V an 1 , Q l ) (whic h corresp onds to the image of L dim V 1 in K 0 (Gal( k (( t )) / k (( t ))) − mod )). 71 W e ha v e an obv ious morphism of complexes of shea v es on Y : f : C one ( Q l → ( i Y ′ ֒ → Y ) ∗ Q l ) → C one (( i Y 0 ֒ → Y ) ∗ Q l → ( i Y ′ 0 ֒ → Y ) ∗ Q l ) . In order to pro v e the Proposition w e ha v e to prov e that the ( pr Y → A ) ∗ f is a quasi-isomorphism, i.e. ( pr Y → A ) ∗ ( C one ( f )) is zero. The compactne ss of spaces Y , Y ′ , Y 0 , Y ′ 0 implies that ( pr Y → A ) ∗ ( C one ( f )) ≃ ( pr Y 1 → A ) ! ( C one ( Q l → ( i Y ′ 1 → Y 1 ) ∗ Q l )) . The (partially deﬁned) actions of the group G m on Y 1 and Y ′ 1 are free, and the v alue of W do es not change under the a ction. More precisely , one can deﬁne easily analytic “spaces of orbits” e Y 1 ⊃ e Y ′ 1 of G m acting o n Y 1 and Y ′ 1 respectiv ely . The pro jections Y 1 → e Y 1 , Y ′ 1 → e Y ′ 1 are prop er maps, and the map W factors through them. Hence it is enough to ch ec k that ( pr Y 1 → e Y 1 ) ∗ ( C one ( Q l → ( i Y ′ 1 ֒ → Y 1 ) ∗ Q l )) ≃ 0 . This follow s from the fact that ev ery orbit in Y 1 is a closed annu lus, its in tersection with Y ′ 1 is a circle, and the inclusion of a circle in to an ann ulus induces an isomorphism of ´ etale cohomolog y groups. This concludes the sk etc h of the pro of.  Remark 18 1) In the pr o of we did n ot sp e cify the values of ε i , i = 1 , 2 , 3 , 4 . We c a n take ε 4 = O ( ε 3 ) (e.g. take ε 4 = 1 2 ε 3 ), ε 2 = o (1) and ε 3 = O ( ε N 2 ) , ε 1 = O ( ε M 3 ) for some inte gers N , M > 0 . 2) I n the pr o of we use d the c omp a rison of the c ohomolo gy of the she af of n e a rby cycles with the ´ etale c ohomolo gy of subvarieties of k (( t )) -analytic sp ac es (se e [54]). W e strongly b eliev e that the analog of the Propo sition holds at the lev el of motivic rings in the case char k = 0. 72 4.5 Equiv alence relati on on motivic functions W e start with a motiv ation for this section. There are man y examples of pairs of constructib le sets (or ev en v arieties) X 1 , X 2 o v er a ﬁeld k suc h that their classes [ X 1 ] and [ X 2 ] in M ot ( S pec ( k )) are diﬀeren t (or at least not ob viously coincide), but X 1 and X 2 coincide in each realization desc rib ed in Section 4.1 (i)-( vi) (e.g. when X 1 , X 2 are isogeneous ab elian v arieties). In particular w e will b e in terested in the case when X l , l = 1 , 2 are aﬃne quadrics g iv en b y the equations P 1 ≤ i ≤ n a i,l x 2 i = 1, suc h that they ha v e equal determinan ts: Q 1 ≤ i ≤ n a i, 1 = Q 1 ≤ i ≤ n a i, 2 ∈ k × . Here we pro p ose a mo diﬁcation of the notion of motivic function whic h is a vers ion of the G rothendiec k’s approac h to the theory of pure motiv es with numeric al equiv a lence. Let us explain it in the case of X = S pec ( k ), where k is a ﬁeld of characteris tic zero (whic h w e will assume throughout this section). W e start with the s ymmetric monoidal Q -linear c ategor y M ef f ( k ). Its ob jects are smo oth pro jectiv e v arieties o v er k and Hom M ef f ( k ) ( Y 1 , Y 2 ) = Q ⊗ Im( Z dim Y 2 ( Y 1 × Y 2 ) → H 2 dim Y 2 D R ( Y 1 × Y 2 )) , where Z n ( X ) denotes as usual the space of algebraic cycles in X of co di- mension n , a nd w e tak e the image of the natural map in to the alg ebraic de Rham cohomology . The n Hom M ef f ( k ) ( Y 1 , Y 2 ) is a ﬁnite-dimensional Q -v ector space. Instead of de Rham cohomology w e can use Betti cohomology (for an em b edding k ֒ → C ) or l -adic cohomolog y . Comparison theorems imply that the im ag e of the group of cycle s in the cohomology do es not depend on a cohomology theory . Comp osition of morphism s is giv en b y the usual composition of correspon- dences, and the tensor pro duct is giv en by the Cartesian pro duct of v arieties. W e extend the category M ef f ( k ) b y adding formally ﬁnite sums (then it b e- comes an additiv e category), and ﬁnally taking the Karo ubian en ve lop e. The K 0 -ring of the resulting category con tains the elemen t L = [ P 1 k ] − [ S pec ( k )]. Adding formally the in v erse L − 1 w e obtain the ring whic h w e denote b y M ot coh ( S pec ( k )). It is an easy corollary o f Bittner theorem (see [6]) that the natural map M ot ( S pec ( k )) → M o t coh ( S pec ( k )) whic h assigns to a smo ot h pro jectiv e v ariety its class in M o t coh ( S pec ( k )) is a homomorphism of rings. The ab ov e considerations can b e generalized to the case of motiv es ov er constructible sets. 73 Deﬁnition 11 L et X b e a c onstructible set over a ﬁeld k , char k = 0 . A c o nstructible family of sm o oth p r oje ctive varieties ov er X is r epr esente d by a p a ir c onsisting of a smo oth pr o je ctive morphis m h : Y → X 0 of schemes of ﬁnite typ e over k and a c onstructible isomorphism j : X 0 c onstr ≃ X . Two such r ep r esentations h : Y → X 0 , j : X 0 c onstr ≃ X, h ′ : Y ′ → X ′ 0 , j ′ : X ′ 0 c onstr ≃ X ar e identiﬁe d if we ar e given c onstructible isomorphism s f : Y c onstr ≃ Y ′ , g : X 0 c onstr ≃ X ′ 0 such that h ′ ◦ f = g ◦ h, j ′ ◦ g = j , and for any p oint x ∈ X ( k ) the induc e d c onstructible i s omorphism b etwe en smo oth pr oje ctive varieties ( j ◦ h ) − 1 ( x ) and ( j ′ ◦ h ′ ) − 1 ( x ) is an iso morphism of schem es. F or a construc tible family of smo o th pro jectiv e v arieties ov er X and a p oin t x ∈ X there is a w ell-deﬁned smo o th pro jectiv e v ariet y Y x o v er the residue ﬁeld k ( x ) called the ﬁb er over x . Moreo v er, one can deﬁne the notion of constructible family of algebraic cycles of the ﬁxed dimension. W e sa y that suc h a family is homologically equiv alen t to zero if for an y x ∈ X the corre- sp onding cycles in Y x map to zero in H • D R ( Y x ). Also, ha ving tw o constructible families of smoot h pro jectiv e v arieties o v er X one eas ily deﬁnes their pro duct, whic h is again a constructible family of smo oth pro jectiv e v arieties o ve r X . All that allows us to generalize our constructions from the case X = S pec ( k ) to the g eneral case. In this wa y w e obta in the ring M ot coh ( X ) as w ell a s the natural homomorphism of rings M ot ( X ) → M ot coh ( X ). Deﬁnition 12 We say that two elements of M ot ( X ) ar e (c ohomolo gic al ly) e q uiva lent if their im ages in M ot coh ( X ) c oincide. The se t of equiv a lence classes (in fact the ring) will b e denoted b y M ot ( X ). It is isomorphic to the image of M ot ( X ) in M ot coh ( X ). In particular, the ab ov e-men tioned quadrics deﬁne the same elemen t in M ot ( S pec ( k )). Let no w X b e a constructible set ov er k , endo we d w ith a n action of an aﬃne algebraic group G . W e deﬁne an equiv alence relation on M ot G ( X ) in the follow ing wa y . First w e c ho ose an embedding G ֒ → GL ( N ). W e sa y that f , g ∈ M ot G ( X ) are equiv alen t if their pull-bac ks to M ot (( X × GL ( N )) /G ) ha v e the same image in M ot ( X × GL ( N )) /G ). Using the fact that all GL ( N )- torsors o v er a constructible set are trivial it is easy to sho w that this equiv alence relation do es not dep end on the em b edding G ֒ → GL ( N ). 74 The ring of equiv alence classes is denoted b y M ot G ( X ). 16 Similarly one de- ﬁnes the ring M ot G × µ ( X ), where µ = lim ← − n µ n . The exotic pro duct desce nds to M ot G × µ ( X ). Hence w e obtain the ring M G,µ ( X ) of equiv alence classes as w ell as homomorphism of rings M G,µ ( X ) → M G,µ ( X ). 4.6 Numerical realizati on of motivic functions This section is not used in further consideration, its g o al is only to sho w that the abstractly deﬁned ring M G,µ ( X ) can b e realized as certain ring of functions with n umerical v alues. Let Z b e a sc heme of ﬁnite type o ve r a ﬁnite ﬁeld k ≃ F q endo w ed with an action of the group µ n of ro ots of 1 suc h that n < p = char F q . W e c ho ose a prime l 6 = p . Let us deﬁne Y as the quotien t ( Z × ( A 1 k \ { 0 } )) / µ n with respect to the diagonal action of µ n . Then w e ha v e a morphism π : Y → A 1 k \ { 0 } , ( z , t ) 7→ t n ∀ ( z , t ) ∈ ( Z × A 1 k \ { 0 } )( F q ) . Let j : A 1 k \ { 0 } → A 1 k b e the natural em b edding. W e deﬁne the n um b er N Z = T r F r ( F ( j ∗ π ! ( Q l,Y )) | s =1 ) ∈ Q l , whic h is the trace of the F rob enius F r of the ﬁb er at s = 1 o f the F ourier transform of the l -adic sheaf j ∗ π ! ( Q l,Y ), where Q l,Y denotes the constan t sheaf Q l on Y . In fact the n um b er N Z can b e considered as an elemen t of the cyclotomic ﬁeld Q ( η p ), where η p is a primitiv e p - th ro ot of 1: 1 + η p + · · · + η p − 1 p = 0 . W e hav e a canonical non-tr ivial c haracter χ : F q → Q ( η p ) ∗ giv en by t he comp osition of the trace T r F q → F p : F q → F p with the additiv e c haracter χ p : F p → Q ( η p ) ∗ , m mo d p 7→ η m p . Then N Z = X s ∈ ( A 1 k \{ 0 } )( F q ) #( π − 1 ( s )( F q )) χ ( s ) . Notice that the last form ula mak es sense for constructible Z as w ell. Let now X b e a constructi ble set ov er a ﬁeld k , char k = 0, endow ed with an action of an aﬃne algebraic group G . 16 In this way w e hav e c ir cumv ent ed the pro blem of deﬁning the categor y of G -eq uiv a riant motivic sheaves. 75 Deﬁnition 13 We c al l a mo del for ( X , G ) the fo l lowing choic es: • a ﬁni tely gener ate d subring R ⊂ k , • a sch e me X → S pec ( R ) of ﬁnite typ e, • an aﬃne gr oup sche m e G → S pec ( R ) to gether with an emb e dding G ֒ → GL ( N ) R , • an ac tion of G on X , • a c onstructible identiﬁc ation ov e r k of X × S p ec ( R ) S pec ( k ) w ith X , as wel l as an iso m orphism of gr oups G × S p ec ( R ) S pec ( k ) ≃ G over k , c om- p a tible with the ac tion s . Suc h a mo del alwa ys exists, and mo dels f orm a ﬁlterted system. With a giv en mo del fo r ( X, G ) w e associate a comm utativ e unital ring K ( X ) = lim − → { open U ⊂ ( X × GL ( n )) / G } Y { closed x ∈ U } Q ( η char k ( x ) ) , where k ( x ) is the residue ﬁeld of x (whic h is a ﬁnite ﬁeld). Supp ose that we are giv en a mo del for ( X , G ) and let f ∈ M G,µ ( X ). As alw a ys w e a ssume that the µ -action on X is g o o d and factors through the action of some µ n . Deﬁnition 14 A mo del for f c omp atible with the mo d el ( R , X , G ) for ( X , G ) c o nsists of the fol lo wing data: • a ﬁnite set J , numb ers N j , d j , n j ∈ Z , wher e j ∈ J , such that al l numb ers N j ar e p ositive and inve rtible in O ( X ) , • G × µ N j -e quivariant morphisms of c ons tructible sets Y j → X given for e a ch j ∈ J , wher e Y j → X ar e G × µ N j -schemes of ﬁnite typ e, and we endow X with the trivial action of the gr oup µ N j , These data ar e r e quir e d to satisfy the c ondition that f = X j ∈ J n j [ Y j × S p ec ( R ) S pec ( k ) → X ] · L d j . 76 Mo dels for f alw a ys exist. Moreo v er, for a n y ﬁnite collection ( f i ) I ∈ I of ele- men ts of M G,µ ( X ) there exists a mo del for ( X , G ) with compatible mo dels for ( f i ) I ∈ I . Ha ving a mo del for f w e can asso ciate with it a n elemen t f num ∈ K ( X ) in the follow ing w ay . Let x ∈ ( X × GL ( N ) R ) / G be a closed p o int. W e can apply the considerations of the b eginning of this section to the ﬁb er Z j,x o v er the p oin t x , of the map ( Y j × GL ( N ) R ) / G → ( X × GL ( N ) R ) / G , where Y j is the sc heme from the deﬁnition of the mo del for f . Then for eac h j w e obtain an elemen t N Z j,x ∈ Q ( η p ) , p = char k ( x ). Finally , we set f num ( x ) := X j ∈ J n j q d j x N Z j,x , where q x := # k ( x ). Hence w e realize f as a function with v alues in n um b ers. 5 Orien tation data on o d d Calabi-Y au cate- gories Considerations of this section are reminiscen t of those in Quan tum Field Theory when one tries to deﬁne determinan ts fo r the Gaussian integral in a free theory . 5.1 Remarks on the motivic Milnor ﬁb er of a quadratic form Although the theory of motivic Milnor ﬁb er was deﬁned ov er a ﬁeld of ch ar- acteristic zero, an esse ntial part of considerations b elow has meaning o v er an arbitrary ﬁeld k , char k 6 = 2 if w e replace the notion of motivic Milnor ﬁb er b y its l -adic v ersion. Let now V b e a k -v ector space endo w ed with a non-degenerate quadratic form Q . W e deﬁne an eleme nt I ( Q ) = (1 − M F 0 ( Q )) L − 1 2 dim V ∈ M µ ( S pec ( k ))[ L ± 1 / 2 ] , where L 1 / 2 is a formal sym b ol whic h satisﬁes the relation ( L 1 / 2 ) 2 = L , and Q is in terpreted as a function on V . Then the motivic Thom-Sebastiani theorem implies that I ( Q 1 ⊕ Q 2 ) = I ( Q 1 ) I ( Q 2 ) . 77 Also w e hav e I ( Q ) = 1, if Q is a split fo r m: Q = P 1 ≤ i ≤ n x i y i for V = k 2 n . Therefore, we ha v e a homomorphism of gr o ups I : W itt ( k ) → ( M µ ( S pec ( k ))[ L ± 1 / 2 ]) × , where W itt ( k ) is the Witt group of the ﬁeld k . W e can think of it as a m ultiplicativ e c haracter. Let us denote b y J 2 ( k ) := J 2 ( S pec ( k )) the quotien t of the group Z × k × / ( k × ) 2 b y the subgroup generated b y the elemen t (2 , − 1). There is an ob vious homomorphism W itt ( k ) → J 2 ( k ) giv en for a quadratic form Q b y [ Q ] 7→ (rk Q, det ( Q ) mo d( k × ) 2 ) . Notice that all “motivic realizations” of I ( Q ) in the sense of Section 4.1 dep end only on the image of [ Q ] in J 2 ( k ). This is s imilar to the classic al form ula (f or k = R and a p ositiv e deﬁnite form Q ) Z V exp( − Q ( x )) dx = (2 π ) − 1 2 dim V (det( Q )) − 1 / 2 in the s ense that the answ er dep ends on dim V and det( Q ) only . In particular, the homomorphism of rings M µ ( S pec ( k )) → K 0 ( D b constr ( G m , Q l )) (see Section 4.1, (ii), (iii) and Section 4.2) induces (by com bining with the ab ov e c haracter) a homomorphism of ab elian groups W itt ( k ) → ( K 0 ( D b constr ( G m , Q l ))[ L ± 1 / 2 ]) × . It is easy to see that it factors through the homomorphism W itt ( k ) → J 2 ( k ). F or example the elemen t of K 0 -group corresp onding to the pair ( n, a ) , n ∈ 2 Z + 1 , a ∈ k × is represen ted b y L − 1 / 2 [ F ] where F is a lo cal system on G m asso ciated with the double co v er of G m giv en b y y 7→ y 2 a, y ∈ G m ( k ). Question 1 The ab o ve c onsider ations giv e rise to the fol lowing question. L et us c onsider the family of quadr a tic forms Q a 1 ,a 2 ( x, y ) = a 1 x 2 1 + a 2 x 2 2 wher e a 1 , a 2 ∈ k × . Is it true that [ Q a 1 ,a 2 ] = [ Q a ′ 1 ,a ′ 2 ] in K 0 ( V ar k ) as long as a 1 a 2 = a ′ 1 a ′ 2 ? W e ex p ect that the answ er to the Question is negativ e, and this is the main reason for introducing the equiv alence relation for the motivic functions in Section 4.5. 78 The ab ov e considerations can b e generalized t o arbitrary constructible (or ind-constructible) sets. Namely , let X b e a constructible set ov er k . W e deﬁne the group J 2 ( X ) as the quotien t of the group C onstr ( X , Z ) ×  C onstr ( X , G m ) /C onstr ( X , G m ) 2  b y the subgroup consis ting of the elemen ts (2 f , ( − 1) f ) , f ∈ C onstr ( X, Z ), where w e denote b y C onstr ( X , Y ) t he set of constructible maps X → Y . T o a constructible v ector bundle V → X endo we d with a non-degenerate quadratic form Q = ( Q x ) x ∈ X w e a sso ciate the elemen t I ( V , Q ) := (1 − M F 0 ( Q )) L − 1 2 dim V ∈ ( M µ ( X )[ L ± 1 / 2 ]) × . Here w e treat eac h Q x as a formal p ow er series on the ﬁb er V x and dim V ∈ C onstr ( X , Z ). One can sho w that this correspondence giv es rise to a homo- morphism of groups I : J 2 ( X ) → ( M µ ( X )[ L ± 1 / 2 ]) × . This fact has a s imple “n umerical” coun terpart: fo r the case k ≃ F q , t w o aﬃne quadrics giv en by equations Q 1 ( x ) = 0 , Q 2 ( x ) = 0 hav e the same n um b er o f points if Q 1 , Q 2 are tw o non-degenerate quadratic forms of equal rank and determinan t. Let us consider a sy mmetric monoidal category sP ic 2 ( X ) consis ting of constructible super line bundles L → X endo we d with an isomorphism L ⊗ 2 ≃ 1 X , where 1 X is a trivial ev en line bundle on X . It is easy to see that the group J 2 ( X ) is the group of isomorphism classes of ob jects of sP ic 2 ( X ). If V → X is a constructible sup er v ector bundle, V = V eve n ⊕ Π V odd then there is a w ell-deﬁned sup er line bundle (called sup er determinan t bundle) sdet( V ) → X giv en b y sdet( V ) := Π dim V even − dim V odd  ∧ top V eve n ⊗ ( ∧ top V odd ) ∗  , where Π is the parit y c hange functor. Recall canonical isomorphisms : 1. sdet( V ∗ ) ≃ sdet(Π( V )) ≃ (sdet( V )) ∗ , 2. sdet( ⊕ i ∈ I V i ) ≃ ⊗ i ∈ I sdet( V i ), 3. if V carries an o dd diﬀeren tial d then sdet( V ) ≃ sdet( H • ( V , d )), 79 4. for an exact triangle of complexes ( V • 1 , d 1 ) → ( V • 2 , d 2 ) → ( V • 3 , d 3 ) there is a canonical isomorphism sdet( V • 2 ) ≃ sdet( V • 1 ) ⊗ sdet( V • 3 ) . In case if V carries a non-degen erate quadratic form Q = ( Q x ) x ∈ X w e ha v e a canonical isomorphism sdet( V ) ≃ (sdet( V )) ∗ . Therefore in this case w e ha ve a w ell-deﬁned ob ject of sP ic 2 ( X ). Its class in the g r o up J 2 ( X ) is represen t ed b y the pair (dim V , { det( Q x ) } x ∈ X mo d( C onstr ( X , G m )) 2 ) . The ab ov e considerations can b e generalized to the case when X is acted b y an algebraic group G . Then one replaces the category sP ic 2 ( X ) b y the category sP ic 2 ( X , G ) of G - equiv arian t constructible sup er line bundles L endo w ed with a G -equiv arian t isomorphism L ⊗ 2 ≃ 1 X . The group of iso- morphism classes of sP ic 2 ( X , G ) we denote by J 2 ( X , G ). In what follow s w e will often omit the w ord “equiv ariant” in the considerations inv olving the category sP ic 2 ( X , G ). Remark 19 L et us ma ke an additiona l ass umption that √ − 1 ∈ k . I n this c a se the quadr atic form x 2 + y 2 = ( x + √ − 1 y )( x − √ − 1 y ) is spli t. Then M F 0 ( x 2 + y 2 ) = 1 − L and we c an c onsider the elem e nt L 1 / 2 := 1 − M F 0 ( x 2 ) which enjoys the pr op erty ( L 1 / 2 ) 2 = L . F urthermor e, the gr oup J 2 ( X ) c an b e c a nonic al ly identiﬁe d with the pr o duct C onstr ( X , Z / 2 Z ) × C onstr ( X , G m ) / ( C onstr ( X , G m )) 2 . Ther efor e the isomorp h ism classes of obje cts of sP ic 2 ( X ) c a n b e identiﬁe d with p a irs (c onstructible µ 2 -function, c onstructible µ 2 -torsor). 5.2 Orien tatio n data Let C b e an ind-constructible k -linear 3-dimensional Calabi-Y au category . 17 Then w e ha v e a natural ind-constructible sup er line bundle D ov er O b ( C ) with the ﬁber o v er E given b y D E = sdet(Ext • ( E , E )). It follo ws that on 17 There is a notion of Z / 2 Z -graded o dd o r even Ca la bi-Y au categ ory , see [42]. Some considerations o f this section ca n b e generalized to Z / 2 Z -g r aded case. 80 the ind-constructible stac k of exact triangles E 1 → E 2 → E 3 w e hav e an isomorphism of the pull-bac ke d line bundles whic h ﬁb erwise reads as D E 2 ⊗ D − 1 E 1 ⊗ D − 1 E 3 ≃ (sdet(Ext • ( E 1 , E 3 ))) ⊗ 2 . Let us explain this isomorphism. The m ultiplicativit y of sup erdeterminan ts on exact triangles give s rise to a canonical isomorphism sdet(Ext • ( E 2 , E 2 )) ≃ sdet(Ext • ( E 1 , E 1 )) ⊗ sdet(Ext • ( E 1 , E 3 )) ⊗ ⊗ sdet(Ext • ( E 3 , E 1 )) ⊗ sdet(Ext • ( E 3 , E 3 )) . By the Calabi-Y au property w e ha v e sdet(Ext • ( E 3 , E 1 )) ≃ sdet(Π(Ext • ( E 3 , E 1 ))) ∗ ≃ sdet(Ext • ( E 1 , E 3 )) whic h implies the desired form ula. When O b ( C ) = ⊔ i ∈ I Y i is a decomp osition in to the union of GL ( N i )-in v arian t c onstructible sets as at the end of 3.2, then the restriction D | Y i is a GL ( N i )-equiv ariant sup er line bundle and the ab ov e isomorphisms ar e also equiv ariant. Deﬁnition 15 Orientation data on C c onsists of a cho i c e of an i n d-c onstructible sup er line bund le √ D on O b ( C ) such that its r e s triction to e ach Y i , i ∈ I is GL ( N i ) -e quivariant, endowe d on e ach X i with GL ( N i ) -e quivariant isomor- phisms ( √ D ) ⊗ 2 ≃ D and such that for the natur al pul l-b acks to the ind- c o nstructible stack of exact triangles E 1 → E 2 → E 3 we ar e given e quiva ria nt isomorphisms: √ D E 2 ⊗ ( √ D E 1 ) − 1 ⊗ ( √ D E 3 ) − 1 ≃ sdet(Ext • ( E 1 , E 3 )) such that the induc e d e quivariant isomorphism D E 2 ⊗ D − 1 E 1 ⊗ D − 1 E 3 ≃ (sdet(Ext • ( E 1 , E 3 ))) ⊗ 2 c o incides with the one w hich we ha v e a priori. W e deﬁne the group J 2 ( C ) := Q i ∈ I J 2 ( Y i , GL ( N i )). W e ha ve a canonical equiv arian t sup er line bundle D ≤ 1 whose ﬁb er at E ∈ O b ( C ) is D ≤ 1 ,E := sdet( τ ≤ 1 (Ext • ( E , E ))) , 81 where τ ≤ i , i ∈ Z denotes the standard truncation functor. It is easy t o see that we ha ve an equiv arian t isomorphism D ⊗ 2 ≤ 1 ≃ D . Then o n the space of exact triangles E 1 → E 2 → E 3 w e hav e a n equiv arian t isomorphism of sup er line bundles ﬁberwise giv en b y ( D ≤ 1 ,E 2 ⊗ D − 1 ≤ 1 ,E 1 ⊗ D − 1 ≤ 1 ,E 3 ) ⊗ 2 ≃ (sdet(Ext • ( E 1 , E 3 ))) ⊗ 2 . Let no w F u n ( C 3 , C ) b e the ind-construc tible category of A ∞ -functors from the the category C 3 considered in 3.1 . Its ob jects can b e thought of as exact triangles E 1 → E 2 → E 3 = C one ( E 1 → E 2 ) in C . There are three functors F unct ( C 3 , C ) → C whic h asso ciate to an exact triangle E 1 → E 2 → E 3 the ob jects E 1 , E 2 , E 3 respectiv ely . These functors induce three homomorphisms φ i : J 2 ( C ) → J 2 ( F u nct ( C 3 , C )) , i = 1 , 2 , 3 . The sup er line bundle L with the ﬁb er L E 1 → E 2 → E 3 = ( D ≤ 1 ,E 2 ⊗ D − 1 ≤ 1 ,E 1 ⊗ D − 1 ≤ 1 ,E 3 ) ⊗ (sdet(Ext • ( E 1 , E 3 ))) − 1 deﬁnes a n elemen t l ∈ J 2 ( F u nct ( A 2 , C )), since L ⊗ 2 ≃ 1 F unct ( C 3 , C ) . Then a c hoice of orien ta tion data on C is equiv alent to a choice of h ∈ J 2 ( C ) suc h that − φ 1 ( h ) + φ 2 ( h ) − φ 3 ( h ) = l . Indeed a c hoice of orientation data giv es rise to a sup er line bundle √ D suc h that √ D ⊗ 2 ≃ D ⊗ 2 ≤ 1 . Therefore the quotien t h = √ D ⊗ D − 1 ≤ 1 deﬁnes an elemen t in J 2 ( C ), and the condition fo r the tensor squares of the sup er line bundles √ D E i , i = 1 , 2 , 3 on the space of exact triangles is equiv alen t to the equation − φ 1 ( h ) + φ 2 ( h ) − φ 3 ( h ) = l . Remark 20 A l l the ab ove c onsider ations admit a str aightforwar d gener al- ization to the c ase of Cal a bi-Y au c ate gory of arb i tr ary o dd dimension d . In the c ase d = 1 (mo d 4) we have c anonic al orientation data given by √ D E := sdet( τ ≤ d − 1 2 (Ext • ( E , E ))) . This is due to the observation that in the explicit description of the analo g of the ob s truction element l deﬁne d a b ove in terms of a sup er ve ctor bund le endowe d with a symmetric bilin e a r form, the s up er ve ctor bund le turns out to b e pur ely o dd, henc e the biline ar form is split. It fol lows that the obstruction element is trivial. I n the c ase d = 3 c onsider e d in this p ap er the obstruction do es not have to vanis h . 82 5.3 Orien tatio n data from a splitting of bifunctors Let C b e a triangulated ind-constructiv e category ov er a ﬁeld k . W e will assume that all f unctors, bifunctors etc. respect this structure. In this section w e are going to discuss a sp ecial framew ork in whic h orien tation data is easy to construct. Let F : C × C op → P er f ( S pec ( k )) b e a biadditiv e bifunctor and d b e an in teger. W e deﬁne the dual bifunctor of de gr e e d as a bifunctor F ∨ = F ∨ ,d giv en by F ∨ ( E 2 , E 1 ) := F ( E 1 , E 2 ) ∗ [ − d ] . Clearly F 7→ F ∨ is an in v olution. Deﬁnition 16 A self-duality structur e on F of de gr e e d is an isom orphism F → F ∨ of bifunctors such that for any two obje cts E 1 , E 2 the induc e d n o n- de gener ate p airing F ( E 1 , E 2 ) ⊗ F ( E 1 , E 2 ) → k [ − d ] is symmetric o n the leve l of c ohomolo gy H • ( F ( E 1 , E 2 )) . If F is endowe d with a self-duality structur e of de gr e e d then we c al l it self-dual. F or a Calabi-Y au category of dimension d the bifunctor ( E 1 , E 2 ) 7→ Hom • ( E 1 , E 2 ) is self-dual. F or any self-dual bifunctor F of o dd degree d w e can rep eat considerations of Section 5.2. Namely , w e deﬁne an ind-constructible super line bundle D F with the ﬁber D F E := sdet( F ( E , E )). Then for a ny exact triangle E 1 → E 2 → E 3 w e ha v e a canonical isomorphism √ D F E 2 ⊗ ( √ D F E 1 ) − 1 ⊗ ( √ D F E 3 ) − 1 ≃ sdet( F ( E 1 , E 3 )) ⊗ 2 . Then one can ask the same question: is ther e an ind-c onstructible sup er line bund le √ D F which is c omp atible with the a b ove isomorphism in the sense of Deﬁnition 15? The answ er is p o sitiv e for any bifunctor of the fo rm F ≃ H ⊕ H ∨ , with the obvious self-dualit y structure . In this case w e set √ D F E := sdet( H ( E , E )) . 83 More generally w e can use an A 1 -homotop y in this sp ecial case. More pre- cisely , supp ose we are giv en a bifunctor 18 G : C × C op → P er f ( A 1 k ). It can b e thought of as a family G t : C × C op → P er f ( S pec ( k )) of bifunctors, parametrized by t ∈ A 1 ( k ), namely G t = i ∗ t ◦ G , w here i t : S pec ( k ) → A 1 is the em b edding corresp onding to t . Since the category P er f ( A 1 k ) has an ob vious dualit y func tor (taking dual to a complex of ve ctor bundles) then the deﬁnition of self-dualit y structure extends naturally to families. Sup- p ose that w e hav e a fa mily of self-dual bifunctors G t , t ∈ A 1 ( k ) suc h that G 0 ≃ Hom • ( • , • ) a nd G 1 ≃ H ⊕ H ∨ for some bifunctor H , and the iso- morphisms prese rve the se lf- duality structure s. Then w e ha ve a canonical orien tation data on C , since any sup er line bundle o v er A 1 k is trivial and all ﬁb ers are canonically isomorphic. 6 Motivic Donalds o n-Thomas in v arian ts 6.1 Motivic Hall algebra and stabilit y data In this section the ﬁeld k can ha v e arbitrary c haracteristic. Let C b e an ind-constructible triangulated A ∞ -category o ve r a ﬁeld k . W e are going to des crib e a motivic generalization of the deriv ed Hall algebras from [70]. As usual, w e ha v e a constructible coun table decomposition O b ( C ) = ⊔ i ∈ I Y i with group GL ( N i ) acting on Y i . Let us consider a M ot ( S pec ( k ))- mo dule ⊕ i M ot st ( Y i , GL ( N i )) (see section 4.2) and extend it by adding neg- ativ e p ow ers L n , n < 0 o f the motiv e of the aﬃne line L . W e denote the resulting M ot ( S pec ( k ))-mo dule by H ( C ). W e understand elemen ts of H ( C ) as me asur es ( a nd not as functions), because in the deﬁnition of the pro duct w e will use t he pushforw ard maps. W e w ould like to make H ( C ) in to an asso ciativ e algebra, called the motivic Hal l algebr a . W e need some preparations fo r that. First w e observ e that if [ π i : Z i → O b ( C )] , i = 1 , 2 a re t wo elemen ts 19 of H ( C ) then one has a 18 In fa ct we would like to say that G is “ind-constructible” in some s ense. A suﬃcien t, but not necess a ry condition would be the existence of an ind-constructible fu nctor G ′ from C × C op to P er f ( P 1 k ) suc h that G is isomorphic to the compositio n of G ′ and the restriction functor P er f ( P 1 k ) → P er f ( A 1 k ). 19 Here we consider for simplicity the ca se when the gr oups acting on Z 1 , Z 2 are trivial, the g eneralization to the c a se o f non-trivia l gr o ups is str aightforw ard. 84 constructible set tot (( π 1 × π 2 ) ∗ ( E X T 1 )) whic h is the total space of the pull- bac k of the ind-constructible bundle E X T 1 o v er O b ( C ) × O b ( C ). Then the map C one (see Section 3.1) af ter the shift [1 ] maps the total space to O b ( C ). F or an y N ∈ Z w e in tro duce the “truncated” Euler c haracteristic ( E , F ) ≤ N := X i ≤ N ( − 1) i dim Ext i ( E , F ) . In the f uture we will use the notation ( E , F ) i for dim Ext i ( E , F ), hence ( E , F ) ≤ N = P i ≤ N ( − 1) i ( E , F ) i . With the pair [ π i : Y i → X i ] , i = 1 , 2 as ab o v e we can ass o ciate a collection of constructible sets W n =  ( y 1 , y 2 , α ) | y i ∈ Y i , α ∈ Ext 1 ( π 2 ( y 2 ) , π 1 ( y 1 )) , ( π 2 ( y 2 ) , π 1 ( y 1 )) ≤ 0 = n  , where n ∈ Z is arbitrary . Clearly [ tot (( π 1 × π 2 ) ∗ ( E X T 1 )) → O b ( C )] = X n ∈ Z [ W n → O b ( C )] . W e deﬁne the pro duct [ Y 1 → O b ( C )] · [ Y 2 → O b ( C )] = X n ∈ Z [ W n → O b ( C )] L − n , where the map W n → O b ( C ) is give n by the form ula ( y 1 , y 2 , α ) 7→ C one ( α : π 2 ( y 2 )[ − 1] → π 1 ( y 1 )) . Prop osition 10 The ab ove formula make s H ( C ) into an asso ciative a l g ebr a . Pr o of. W e a re going to prov e the result for t he “delta functions” ν E = [ pt → O b ( C )] , pt 7→ E , where E is an ob ject of C ( k ). The case of equiv arian t families is similar. In other w ords, w e w ould lik e to pro v e that ( ν E 1 · ν E 2 ) · ν E 3 = ν E 1 · ( ν E 2 · ν E 3 ) . Replacing the category by its minimal mo del we may replace in all consid- erations Hom • b y Ext • . Let us also remark that an eleme nt α ∈ Ext 1 ( E , F ) 85 deﬁnes an extension E α whic h w e can interp ret as a deformation of the ob- ject E ⊕ F (the trivial extension). Therefore for an y ob ject G the group Ext • ( G, E α ) is equal to the cohomology of the complex (Ext • ( G, E ⊕ F ) , d α ), where d α is the op erator o f m ultiplication (up to a sign) b y α . Notice that ν E 1 · ν E 2 = L − ( E 2 ,E 1 ) ≤ 0 [Ext 1 ( E 2 , E 1 ) → O b ( C )] := L − ( E 2 ,E 1 ) ≤ 0 Z α ∈ Ext 1 ( E 2 ,E 1 ) ν E α , where E α is the ob ject corresp onding to the extension α , i.e. E α = C one ( α : E 2 [ − 1] → E 1 ) . It follow s that ( ν E 1 · ν E 2 ) · ν E 3 = L − ( E 2 ,E 1 ) ≤ 0 Z α ∈ Ext 1 ( E 2 ,E 1 ) , β ∈ Ext 1 ( E 3 ,E α ) L − ( E 3 ,E α ) ≤ 0 ν E β . W e observ e that ( E 3 , E α ) ≤ 0 = ( E 3 , E 2 ⊕ E 1 ) ≤ 0 − l α = ( E 3 , E 2 ) ≤ 0 + ( E 3 , E 1 ) ≤ 0 − l α , where the “error term” l α > 0 can b e computed in terms of the linear map d α . Therefore one can write ( ν E 1 · ν E 2 ) · ν E 3 = L − ( E 2 ,E 1 ) ≤ 0 − ( E 3 ,E 1 ) ≤ 0 − ( E 3 ,E 2 ) ≤ 0 Z α ∈ Ext 1 ( E 2 ,E 1 ) ,β ∈ Ext 1 ( E 3 ,E α ) L l α ν E β . One can write a similar expression for ν E 1 · ( ν E 2 · ν E 3 ). In this case the “error term” will b e denoted by r α instead of l α . Notice that the diﬀeren tial d α : Ext 0 ( E 3 , E 2 ) ⊕ Ext 0 ( E 3 , E 1 ) → Ext 1 ( E 3 , E 1 ) ⊕ Ext 1 ( E 3 , E 2 ) satisﬁes the prop erty that the only non- trivial comp onen t i s the map α R : Ext 0 ( E 3 , E 2 ) → Ext 1 ( E 3 , E 1 ). Here we denote b y α R the linear op erator of m ultiplication b y α ∈ Ext 1 ( E 2 , E 1 ) from the righ t. W e will use the same con v en tion fo r the linear op erator α R : Ext 1 ( E 3 , E 2 ) → Ext 2 ( E 3 , E 1 ). Hence w e see that dim Ext 1 ( E 3 , E α ) = dim Ker  α R : Ext 1 ( E 3 , E 2 ) → Ext 2 ( E 3 , E 1 )  + 86 + dim Cok er  α R : Ext 0 ( E 3 , E 2 ) → Ext 1 ( E 3 , E 1 )  . Let us no w conside r the constructible set X 1 , 2 , 3 = { ( α , γ , δ ) ∈ Ext 1 ( E 2 , E 1 ) ⊕ Ext 1 ( E 3 , E 2 ) ⊕ Ext 1 ( E 3 , E 1 ) | α ◦ γ = 0 } . Notice that a triple ( α , γ , δ ) ∈ X 1 , 2 , 3 deﬁnes the deformatio n of the ob ject E 1 ⊕ E 2 ⊕ E 3 preserving the ﬁltration E 1 ⊂ E 1 ⊕ E 2 ⊂ E 1 ⊕ E 2 ⊕ E 3 . More precisely , the triple giv es rise to a t wisted complex , whic h is de ﬁned b y the corresponding to ( α , γ , δ ) solution to the Maurer-Cartan equation (strictly upp er-triangular matrix acting on E 1 ⊕ E 2 ⊕ E 3 ). The latter obser- v ation means that there is an ind-constructible map X 1 , 2 , 3 → O b ( C ) whic h assigns to a p oint ( α, γ , δ ) the corresponding tw isted complex. Let us no w ﬁx α ∈ Ext 1 ( E 2 , E 1 ) and consider the ind-constructible subset X α 1 , 2 , 3 ⊂ X 1 , 2 , 3 whic h c onsists of the triple s with ﬁxed α . There is a natu- ral pro jection ( α, γ , δ ) 7→ ( γ , δ ), whic h give s rise to the construc tible map X α 1 , 2 , 3 → Ext 1 ( E 3 , E α ). This is a constructible aﬃne bundle with the ﬁb ers isomorphic to Im( α R : Ext 0 ( E 3 , E 2 ) → Ext 1 ( E 3 , E 1 )). Also, one can see directly that the dimension of the latter space is l α = ( E 3 , E 2 ⊕ E 1 ) ≤ 0 − ( E 3 , E α ) ≤ 0 . Hence we hav e the follo wing iden tit y in H ( C ): Z α ∈ Ext 1 ( E 2 ,E 1 ) , β ∈ Ext 1 ( E 3 ,E α ) L l α ν E β = [ X 1 , 2 , 3 → O b ( C )] . Therefore, ( ν E 1 · ν E 2 ) · ν E 3 = L − ( E 2 ,E 1 ) ≤ 0 − ( E 3 ,E 1 ) ≤ 0 − ( E 3 ,E 2 ) ≤ 0 · [ X 1 , 2 , 3 → O b ( C )] . Similar considerations sho w that ν E 1 · ( ν E 2 · ν E 3 ) = L − ( E 2 ,E 1 ) ≤ 0 − ( E 3 ,E 1 ) ≤ 0 − ( E 3 ,E 2 ) ≤ 0 · [ X 1 , 2 , 3 → O b ( C )] . This prov es the asso ciativit y of the pro duct in H ( C ).  F or a constructible stabilit y structure on C with an ind-constructible class map cl : K 0 ( C ) → Γ, a cen tral c harge Z : Γ → C , a strict sector V ⊂ R 2 87 and a branc h Log of the logarithm function on V w e ha v e the category C V := C V , Log deﬁned in Section 3.4. Hence w e hav e the c ompletion b H ( C V ) := Y γ ∈ (Γ ∩ C ( V ,Z,Q )) ∪{ 0 } H ( C V ∩ cl − 1 ( γ )) . Then we ha v e an in v ertible elemen t A Hall V ∈ b H ( C V ) suc h that A Hall V := 1 + · · · = X i ∈ I 1 ( Ob ( C V ) ∩ Y i ,GL ( N i )) , where 1 S is the iden tit y function (see 4.2) but interp reted as a coun ting measure 20 . In short, elemen t A Hall V is giv en b y the countin g measure restricted to C V . The summand 1 comes from zero ob j ect. Prop osition 11 Elements A Hall V satisfy the F ac torization Pr op erty: A Hall V = A Hall V 1 · A Hall V 2 for a strict se ctor V = V 1 ⊔ V 2 (de c omp osition in the clo ckwise or der). Pr o of. The pro of fo llo ws from the follo wing observ ations: 1) F or an y E i ∈ O b ( C V i ( k )) , i = 1 , 2 one has ( E 2 , E 1 ) ≤ 0 = dim Ext 0 ( E 2 , E 1 ) b ecause Ext i ( E 2 , E 1 ) = 0 fo r i < 0 . 2) The set { [ E ] ∈ I so ( C V ( k )) } is in one-to- one correspo ndence with the set of isomorphisms classes o f the triples ( E 1 , E 2 , α ) suc h that E i ∈ O b ( C V i ( k )) , i = 1 , 2 and α ∈ Ext 1 ( E 2 , E 1 ) (the map betw een the sets a ssigns to the triple the extens ion E α ). 3) The automorphism gro up of the triple ( E 1 , E 2 , α ) is the stabilizer of α for the natural action o f the group Aut( E 2 ) × Aut( E 1 ) o n the vec tor space Ext 1 ( E 2 , E 1 ). 4) There is an exact sequence of gro ups 1 → Ext 0 ( E 2 , E 1 ) → Aut( E α ) → Aut( E 1 , E 2 , α ) → 1 In order to apply these o bserv ations one uses the fact that a n ob ject E ∈ C V ( k ) contains a unique sub o b ject E 1 ∈ O b ( C V 1 ( k )) suc h that the quotien t o b ject E 2 b elongs to C V 2 ( k ), and then the f actor L − ( E 2 ,E 1 ) ≤ 0 cancels the ratio b etw een the stabilizer of α and the automorphism g r o up of the extension E α .  20 The same is true if one uses the languag e of higher stac ks beca use for an y E ∈ Ob ( C ( V ))( k ) o ne has Ext < 0 ( E , E ) = 0. 88 Corollary 3 L et us end o w H ( C ) with an automorphism η given b y the shift functor [1] . Then the c ol le ction ( A Hall V ) gives rise to a symmetric stability data on H ( C ) c onsider e d as a gr ade d Lie algebr a (se e De ﬁnition 2 and R emark 8 in S e ction 2 . 2 ). Mor e over we obtain a lo c al home omorp h ism S tab ( C , cl) → S tab ( H ( C ) ) . The ab o ve considerations can be illustrated in t he case of ﬁnite ﬁelds . Namely , let us assume that C is a triangulated category o v er a ﬁnite ﬁeld F q . W e deﬁne the Hall algebra H ( C ) a s an a sso ciativ e unital algebra o v er Q , whic h is a Q -vec tor space spanned b y isomorphism classes [ E ] of o b jects E ∈ O b ( C ). The m ultiplication is giv en b y the formula [ E ] · [ F ] = q − ( F, E ) ≤ 0 X α ∈ Ext 1 ( F, E ) [ E α ] , where E α is the extensi on corresp onding to α ∈ Ext 1 ( F , E ). W e deﬁne a stabilit y condition on C in the same w ay as in the In tro duction (or Section 3.4) without imposing a ny construc tibility condition (since w e do not a ssume that our category is ind-constructible). Inside of the set S tab ( C ) of stabilit y conditions on C w e consider a subset S tab 0 ( C ) consisting of suc h stabilit y conditions that the set { E ∈ C ss γ | Arg( E ) = ϕ } is ﬁnite for any γ ∈ Γ , ϕ ∈ [0 , 2 π ). This prop erty is analogous to the one in the ind-constructible setting whic h sa ys that C ss γ is a constructible set. Then for an y strict sector V and a c hoice of the branc h Log w e ha v e an elemen t A Hall V ∈ b H ( C V ) giv en b y A Hall V = 1 + · · · = X [ E ] ∈ I so ( C V ) [ E ] # Aut( E ) . Similarly to the motivic case the collection o f elemen ts ( A Hall V ) satisﬁes the F actorization prop erty . Hence it deﬁnes a stabilit y data on the space H ( C ) considered as a graded Lie algebra. The relationship of our ve rsion of Hall algebra to the T o¨ en deriv ed Hall algebra from [70] is described in t he follo wing prop osition. Prop osition 12 Ther e is a homomorphis m of ring s H ( C ) → H T o ( C ) , wher e H T o ( C ) is the derive d Hal l algebr a ove r F q deﬁne d by T o¨ en in [70] (se e also [76]), such that [ π : Y → O b ( C )] 7→ X y ∈ Y ( F q ) [ π ( y )]# Aut( y )( F q ) q ( y,y ) < 0 . 89 Mor e over, for any strict se ctor V the ab ove homomo rphism a dmits a natur al extension to the c omplete d Hal l alge br a s such that the element A Hall V ∈ b H ( C V ) is mapp e d to the ele m ent of the c omplete d Hal l algebr a b H T o ( C V ) given by P [ x ] ∈ I s o ( C V ) [ x ] . Pr o of. Straightforw ard.  In fact, in the T o¨ en v ersion of the Hall algebra the fa ctorization prop ert y A Hall V = A Hall V 1 A Hall V 2 is ess en tially trivial. Th e reason is that the structure constan ts in b H T o ( C V ) for the elemen ts of the basis corresp onding to ob jects in a heart of a t -structure are the usual one, i.e. they count the n um b er of 2-step ﬁltrations of a give n ob ject with g iven isomorphism classes of the asso ciate graded factors. The factorization pro p ert y means that an y ob ject in C V has a unique sub ob j ect in C V 1 with quotien t in C V 2 . Remark 21 One c an try to go e v en further in an attempt to “c ate gorify” the motivic Hal l algebr a. Her e one has to assume that obje cts of C form not just an ind-c onstructible stack, but a higher stack of lo c al ly ﬁnite typ e in the sense of T o ¨ en and V ezzosi (se e [72]) . The c orr esp onding c ate gory wil l b e the monoidal c ate g o ry of motivic she aves on O b ( C ) . The motivic Hal l algebr a is the K 0 -ring of this c ate gory. In the c ase of the n on-c ommutative variety endowe d with p ola riz ation one c an deﬁne (for any strict se ctor V ) the sub c at- e g o ry F V of “ motivic she aves with c entr al cha r ges in V ”. Nevertheless, the F actorization Pr op erty c ould fail sinc e the obje ct A Hall V c a n b e non-isomorphic to the obje ct A Hall V 1 ⊗ A Hall V 2 (but their image s in K 0 c o incide). Finally , w e explain how to rephrase the factorization prop ert y in terms of t -structures, without the use of stabilit y conditions. Here w e understand a t - structure α on a small triangulated category C a s a pair of strictly full sub categor ies (i.e. a pair of sets of equiv alence classe s of ob jects) C − ,α , C + ,α ⊆ C suc h that for any E − ∈ C − ,α , E + ∈ C + ,α w e ha ve E xt ≤ 0 ( E − , E + ) = 0, and an y ob j ect E ∈ C can b e repres ente d (uniquely) as an extension τ − ,α ( E ) → E → τ + ,α ( E ) , τ ± ,α ( E ) ∈ C ± ,α . 90 An y stabilit y condition on C deﬁnes tw o t -structures α l , α r suc h that C − ,α l (resp. C − ,α r ) consists of extensions of semistable o b jects E with Arg ( E ) > 0 (resp. with Arg ( E ) > 0 ). These t w o t -struc tures do not c hange under the action of the gro up  a 11 0 a 21 a 22    a 11 , a 22 > 0  ⊂ ] GL + (2 , R ) of transformations pre serving the upp er half- plane. In partic ularly , w e see that while a connected c omp onen t in the S tab ( C ) is a real 2 n -dimensi onal manifold for n := ra nk (Γ), the set o f corresp onding t -structures is at most ( n − 1)-dimensional. In tro duce an order on the set of t -structures b y α 1 ≤ α 2 ⇐ ⇒ C − ,α 1 ⊆ C − ,α 2 ⇐ ⇒ C + ,α 1 ⊇ C + ,α 2 . The shift functor acts on t -structures, and α [1] ≤ α for an y t -structure α . Let no w C b e an ind-constructible category endo we d with an ind-constructible homomorphism cl : K 0 ( C ( k )) → Γ and α 1 , α 2 are t w o ind-constructible t - structures. W e sa y α 1 ≤ constr α 2 iﬀ • α 1 ≤ α 2 ≤ α 1 [ − 1] , • ∀ γ ∈ Γ C + ,α 1 ∩ C − ,α 2 ∩ cl − 1 ( γ ) is constructible, • the cone generated b y { γ ∈ Γ | C + ,α 1 ∩ C − ,α 2 ∩ cl − 1 ( γ ) 6 = 0 } is strict. If α 1 ≤ constr α 2 then w e deﬁn e a n elemen t A α 1 ,α 2 of an appropriately com- pleted Hall algebra as the sum of the “coun ting measure” ov er the ob jects in C + ,α 1 ∩ C − ,α 2 . Obv iously , elemen ts A V (for an op en, or a closed, or a semi-op en strict sector V ) are of the for m A α 1 ,α 2 for appropriate t -structures α 1 , α 2 . The factorization prop ert y generalize s to A α 1 ,α 3 = A α 1 ,α 2 · A α 2 ,α 3 if α 1 ≤ constr α 2 , α 2 ≤ constr α 3 , α 1 ≤ constr α 3 . Notice that in the case of stabilit y conditions the elemen t A V is preserv ed under the action of a subgroup of ] GL + (2 , R ) conjugated to the gr o up of p ositiv e diagonal matrices. This action o n S tab ( C ) / Aut ( C ) has a go o d c hance to b e ergo dic, as indicates a similar example with the mo duli spaces of curve s with ab elian diﬀeren tials (see a review [78 ]). 91 6.2 Motivic w eigh ts and stabilit y data on motivic quan- tum tori Let C b e a 3- dimensional ind-constructib le Calabi-Y au category o ve r a ﬁeld of c haracteristic zero (see Section 3.3). In this section w e are go ing to de- ﬁne motivic D onaldson-Thomas in v ariants a sso ciated with a constructible stabilit y condition and an orien tation data on C . Step 1 . Let us deﬁne the ring D µ = M µ ( S pec ( k ))[ L − 1 , L 1 / 2 , ([ GL ( n )] − 1 ) n > 1 ] , where the ring M µ ( S pec ( k )) w as deﬁned in 4.3, and L = [ A 1 k ] is the motive of the aﬃne line. The elemen t L 1 / 2 is a formal sym b ol satisfying the equation ( L 1 / 2 ) 2 = L . Instead of in v erting motiv es [ GL ( n )] = ( L n − 1 ) ( L n − L ) . . . ( L n − L n − 1 ) of all general linear groups w e can in v ert motiv es o f all pro jectiv e spaces [ P n ] = L n +1 − 1 L − 1 . W e also will consider the ring D µ of equiv a lence classes of functions from D µ b y the equiv alence relation deﬁned in Section 4.5 . The ring D µ will pla y the role of the univ ersal co eﬃcien t ring where motiv ic Donaldson-Thomas in v a ria n ts tak e v alue. Step 2 . W e deﬁne an algebra M µ ( O b ( C )) asso ciated with C whic h will con tain certain canonical elemen t called the motivic weigh t . F irst, w e deﬁne M µ ( O b ( C )) := Y i M GL ( N i ) ,µ ( Y i )[ L − 1 , L 1 / 2 ] , where ( Y i , GL ( N i )) i ∈ I is a decomp osition of the stac k of ob jects of C as at the e nd of 3.2. Algebra M µ ( O b ( C )) is obtained f rom it by passing to the equiv alence classes in the sense of Section 4.5. F or an y GL ( N i )-in v arian t constructib le set Z ⊂ Y i for some i ∈ I , w e ha v e a M µ ( S pec ( k ))[ L − 1 , L 1 / 2 ]-linear map Z Z : M µ ( O b ( C )) → D µ 92 whic h is the µ -equiv ariant v ersion of integral o v er stac k ( Z , GL ( N i )) (see 4.2) of the restriction to Z . Explicitly , if f | Y i is represen ted by a µ × GL ( N i )- equiv arian t map X → Y i then Z Z f = [ X × Y i Z ] / [ GL ( N i )] ∈ D µ where [ X × Y i Z ] is in terpreted a constructible set with µ -action. By additivit y w e extend the in tegral to the case when Z is a ﬁnite union of GL ( N i )-in v arian t constructible set Z i ⊂ Y i for diﬀeren t i ∈ I . Step 3 . No w w e a re going to deﬁne the motivic w eigh t. Recall that for an y E ∈ O b ( C )( k ) we hav e deﬁned the p o ten tial W min E whic h is a formal p o w er series in α ∈ Ext 1 ( E , E ) whic h starts with cubic terms. W e denote b y M F ( E ) := M F 0 ( W min E ) the motivic Milnor ﬁb er of W min E at 0 ∈ Ext 1 ( E , E ). Then the assign- men t E 7→ M F ( E ) can b e in terpreted as the v a lue of some function M F ∈ M µ ( O b ( C )). Let us choose an o rien tation data √ D fo r C . Recall that in Section 5.2 w e deﬁned the elemen t h ∈ J 2 ( C ) represen ted b y the equiv ariant sup er line bundle √ D ⊗ D − 1 ≤ 1 with trivialized tensor square. F or a represen tativ e of h giv en b y a pair ( V , Q ) w e ha v e I ( h ) = (1 − M F 0 ( Q )) L − 1 2 rk Q . Let us c ho o se suc h a represen tativ e. Deﬁnition 17 The motivic weig h t w ∈ M µ ( O b ( C )) is the function deﬁne d on obje cts by the formula w ( E ) = L 1 2 P i ≤ 1 ( − 1) i dim Ext i ( E ,E ) (1 − M F ( E ))(1 − M F 0 ( Q E )) L − 1 2 rk Q E . The image w ∈ M µ ( O b ( C )) do es not dep end on the c hoice of a represen- tativ e of h and is equal to w ( E ) = L 1 2 P i ≤ 1 ( − 1) i dim Ext i ( E ,E ) (1 − M F ( E )) I ( h ( E )) , where h ( E ) is t he v alue of the obstruction h at the p oin t E . Step 4 . Let us no w ﬁx the f ollo wing data: 93 • a triple (Γ , h• , •i , Q ) consisting of a free ab elian group Γ of ﬁnite rank endo w ed with a sk ew-symme tric bilinear f o rm h• , •i : Γ ⊗ Γ → Z , and a quadratic form Q o n Γ R = Γ ⊗ R ; • an ind-constructible , Gal( k / k )-equiv arian t homomorphism cl k : K 0 ( C ( k )) → Γ compatible with the Euler form a nd the sk ew-symmetric bilinear f orm; • a constructible stabilit y structure σ ∈ S tab ( C , cl) compatible with the quadratic form Q in the sens e that Q | Ker Z < 0 and Q (cl k ( E )) > 0 for E ∈ C ss ( k ). In the next section we are going to deﬁne a homomorphism fro m the motivic Hall algebra to the asso ciative unital algebra called motivic quan tum torus. The latter is deﬁned in the followi ng w a y . F or an y comm utativ e unital ring C whic h con tains a n inv ertible sym b ol L 1 / 2 w e introduce a C -linear asso ciativ e algebra R Γ ,C := ⊕ γ ∈ Γ C · ˆ e γ where the generators ˆ e γ , γ ∈ Γ satisfy the relations ˆ e γ 1 ˆ e γ 2 = L 1 2 h γ 1 ,γ 2 i ˆ e γ 1 + γ 2 , ˆ e 0 = 1 . W e will call it the quan tum torus a sso ciated with Γ and C . F or an y strict sector V ⊂ R 2 w e deﬁne R V , C := Y γ ∈ Γ ∩ C 0 ( V ,Z ,Q ) C · ˆ e γ and call it the quan tum torus asso ciated with V . Here w e in tro duce a nota- tion whic h will b e used later: C 0 ( V , Z , Q ) := C ( V , Z , Q ) ∪ { 0 } where the cone C ( V , Z , Q ) was deﬁned in 2.2. Algebra R V , C is the natural completion of the subalgebra R V , C ∩ R Γ ,C ⊂ R Γ ,C . Let us c ho ose as C the ring D µ . W e denote R Γ := R Γ , D µ the correspond- ing quantum to rus and call it the motivic quantum torus a sso ciated with Γ. Similarly , we ha ve motivic quan tum to ri R V asso ciated with strict sectors V . 94 Step 5 . W e deﬁne an elemen t A mot V ∈ R V := R V , D µ in the fo llowing w ay . First, w e ﬁx a branch of the function Log z , where z ∈ V (the result will not dep end on the choice of the branch). Recall the category C V , Log ⊂ C (see Section 3.4). It follows from our assumptions that for any γ ∈ Γ t he set C V , γ = { E ∈ O b ( C V , Log ) | cl( E ) = γ } is constructible. Finally , w e deﬁne the desired elemen t A mot V = X γ Z C V ,γ w ! · ˆ e γ . The elemen t A mot V in fact depends only o n w . Informally , one can write A mot V = X E ∈ I so ( C V , Log ) w ( E ) [Aut( E )] ˆ e cl( E ) = 1 + · · · ∈ R V , where I so ( C V , Log ) denotes the set of isomorphism classes of ob jects of the category C V , Log . Theorem 7 Assuming the i n te gr al identity, the c ol le c tion of elements ( A mot V ) satisﬁes the F actorization Pr op erty: if a strict se ctor V is de c om p ose d into a disjoint union V = V 1 ⊔ V 2 (in the c lo ckwise or der) then A mot V = A mot V 1 A mot V 2 . Mor e over we h ave a lo c al home omorphism S tab ( C ) → S tab  R Γ , D µ  . This theorem follows immediately fro m the statemen t of Prop osition 11 (see 6 .1) ab out the eleme nts A Hall V , and the Theorem 8 f r o m the next section. 6.3 F rom motivic Hal l alge bra to motivic quan tum torus Assume that C is an ind-constructible 3 d Calabi-Y au category endo we d with p olarization and orien tation data √ D . The Hall algebra of C is graded by the corresp onding lattice Γ: H ( C ) = ⊕ γ ∈ Γ H ( C ) γ . Main result of this section is the follo wing theorem. 95 Theorem 8 The map Φ : H ( C ) → R Γ given by the f ormula Φ( ν ) = ( ν, w ) ˆ e γ , ν ∈ H ( C ) γ is a homomo rphism of Γ -gr ade d Q -algebr as. Her e w is the mo tivic w e ight and ( • , • ) is the p airing b etwe en motivic me asur es and functions. In other w ords, the homomorphism H ( C ) → R Γ can b e written as [ π : Y → O b ( C )] 7→ 7→ Z Y (1 − M F ( π ( y ))) (1 − M F 0 ( Q π ( y ) )) L − 1 2 rk Q π ( y ) L 1 2 ( π ( y ) ,π ( y )) ≤ 1 ˆ e cl( π ( y )) , where R Y is understo o d as the direct image functor (see Section 4.2). The natural extension of the ab o v e homomorphism to the completion of b H ( C V ) maps the elemen t A Hall V to the elemen t A mot V deﬁned in Section 6.2. Pr o of. F or simpl icity we will prese nt the pro of o f the Theorem for ν E := [ δ E : p t → O b ( C )] , where δ E ( pt ) = E ∈ O b ( C ( k )) is the “ delta- function”. The general pro of for equiv arian t constructible families is similar. W e will also assume that our category is minimal on the diagonal. The pro of will consists of sev eral steps. Step 1. W e ha v e: ν E 1 · ν E 2 = L − ( E 2 ,E 1 ) ≤ 0 [ π 21 : Ext 1 ( E 2 , E 1 ) → O b ( C )] , the map π 21 is the restriction of the cone map 21 tot (Ker( m 1 : H O M 0 → H O M 1 )) → O b ( C ) to the ﬁber o v er the p oin t ( E 2 [ − 1] , E 1 ). Under this map the eleme nt α ∈ Ext 1 ( E 2 , E 1 ) is mapped to the ob ject E α = C one ( α : E 2 [ − 1] → E 1 ) . 21 Recall that we pretend that such a ma p exists. In fact, it is deﬁned only as a 1 - morphism o f stacks. 96 Let us denote by γ i the class cl( E i ) ∈ Γ , i = 1 , 2. Then w e ha v e: Φ( ν E i ) = L 1 2 ( E i ,E i ) ≤ 1 (1 − M F 0 ( W min E i )) I ( h ( E i )) ˆ e γ i , i = 1 , 2 , where h ( E i ) is the v alue at E i of the elemen t h ∈ J 2 ( C ) (i.e. the image of the restriction map to J 2 ( C )) giv en b y the sup er line bundle √ D ⊗ D − 1 ≤ 1 with trivialized tensor square. W e ha v e: Φ( ν E 1 )Φ( ν E 2 ) = L 1 2 (( E 1 ,E 1 ) ≤ 1 +( E 2 ,E 2 ) ≤ 1 ) × × L 1 2 (( E 1 ,E 2 ) ≤ 1 − ( E 2 ,E 1 ) ≤ 1 ) (1 − M F 0 ( W min E 1 ⊕ W min E 2 )) I ( h ( E 1 )) I ( h ( E 2 )) ˆ e γ 1 + γ 2 . In order to obtain this for mula w e used the Calabi-Y au prop ert y , whic h implies that h γ 1 , γ 2 i = P j ∈ Z ( − 1) j ( E 1 , E 2 ) j = ( E 1 , E 2 ) ≤ 1 + ( E 1 , E 2 ) > 2 = = ( E 1 , E 2 ) ≤ 1 − ( E 2 , E 1 ) ≤ 1 , where we emplo y the notation ( E 1 , E 2 ) > m = P j > m ( − 1) j ( E 1 , E 2 ) j . Also w e used the motivic Thom-Sebastiani theorem for the Milnor ﬁb ers and the pro duct form ula for the basis elemen ts in the motivic quan tum torus R Γ . On the other hand, w e can apply Φ to the pro duct ν E 1 · ν E 2 and o btain: Φ( ν E 1 · ν E 2 ) = = L − ( E 2 ,E 1 ) ≤ 0 R α ∈ Ext 1 ( E 2 ,E 1 ) L 1 2 ( E α ,E α ) ≤ 1 (1 − M F 0 ( W min E α )) I ( h ( E α )) ˆ e γ 1 + γ 2 . Using the ide ntit y ( E 2 , E 1 ) ≤ 0 = ( E 2 , E 1 ) ≤ 1 + ( E 2 , E 1 ) 1 (and also re call that ( E 2 , E 1 ) 1 = dim Ext 1 ( E 2 , E 1 )) and observing that ( E 1 ⊕ E 2 , E 1 ⊕ E 2 ) ≤ 1 = ( E 1 , E 1 ) ≤ 1 + ( E 2 , E 2 ) ≤ 1 + ( E 1 , E 2 ) ≤ 1 + ( E 2 , E 1 ) ≤ 1 w e a rriv e to the follo wing equalit y whic h is equiv alen t to Φ( ν E 1 · ν E 2 ) = Φ( ν E 1 )Φ( ν E 2 ) and hence implies the Theorem: L ( E 2 ,E 1 ) 1 (1 − M F 0 ( W min E 1 ⊕ W min E 2 )) I ( h ( E 1 )) I ( h ( E 2 )) = = Z α ∈ Ext 1 ( E 2 ,E 1 ) L 1 2 (( E α ,E α ) ≤ 1 − ( E 1 ⊕ E 2 ,E 1 ⊕ E 2 ) ≤ 1 ) (1 − M F 0 ( W min E α )) I ( h ( E α )) . 97 Step 2. No w w e w ould lik e to express the diﬀeren ce ( E α , E α ) ≤ 1 − ( E 1 ⊕ E 2 , E 1 ⊕ E 2 ) ≤ 1 as the rank of a certain linear o p erator. Recall that the ob ject E α can b e though t of as a deformation of the ob ject E 0 := E 1 ⊕ E 2 . Therefore, there is a sp ectral sequence whic h starts at Ext • ( E 1 ⊕ E 2 , E 1 ⊕ E 2 ) and con ve rges to Ext • ( E α , E α ). Using the A ∞ -structure one can make it v ery explicit. Namely , let us denote by d α : Ext • ( E 1 ⊕ E 2 , E 1 ⊕ E 2 ) → Ext • ( E 1 ⊕ E 2 , E 1 ⊕ E 2 ) the diﬀeren tial of degree +1 giv en b y the form ula d α = m 2 ( α, • ) + m 2 ( • , α ) + m 3 ( α, • , α ) . Then the gra ded v ector space Ext • ( E α , E α ) is isomorphic to the cohomology of d α (cf. e.g. [42], Remark 10.1.5). It is clear that fo r an y cohomological complex ( C • , d ) of ﬁnite-dimensional v ector spaces w e ha v e the equ ality X i ≤ 1 ( − 1) i dim H i ( C ) − X i ≤ 1 ( − 1) i dim C i = rk d (1) , where d (1) : C 1 → C 2 is the comp onen t of d . Applyin g this observ ation to our complex w e obtain that ( E α , E α ) ≤ 1 − ( E 1 ⊕ E 2 , E 1 ⊕ E 2 ) ≤ 1 = rk d (1) α . Step 3. Let us in tro duce a k -v ector space M E 1 ,E 2 = Ext 1 ( E 1 ⊕ E 2 , E 1 ⊕ E 2 ) = = Ext 1 ( E 1 , E 1 ) ⊕ Ext 1 ( E 2 , E 1 ) ⊕ Ext 1 ( E 1 , E 2 ) ⊕ Ext 1 ( E 2 , E 2 ) . It can b e interpre ted as the tangent space to the mo duli space o f f o rmal deformations o f the ob ject E 1 ⊕ E 2 . W e c ho ose co ordinates ( x, α, β , y ) on this space in suc h a w ay that x denotes t he co ordinates on Ext 1 ( E 1 , E 1 ), α denotes the co ordinates o n Ext 1 ( E 2 , E 1 ), β denotes the co ordinates on Ext 1 ( E 1 , E 2 ) and y denotes the co o r dinates on Ext 1 ( E 2 , E 2 ). Then the p o in t (0 , α, 0 , 0) corresponds (b y abuse of notation) to the isomorphism class α ∈ Ext 1 ( E 2 , E 1 ) of an exact triangle E 1 → E α → E 2 . Later we are go ing to use 98 the in tegral iden tit y f rom Section 4.4 a pplying it to the formal neighborho o d of the subspace consisting of the p oints (0 , α, 0 , 0). In order to do that w e will r elate the p o ten tial of the ob j ect E α with a certain formal function on M E 1 ,E 2 . W e may assume that the full sub category C ( E 1 , E 2 ) consis ting of the pair of ob j ects E 1 , E 2 is minimal. As in the case of one ob ject the p oten tial of C induces a formal p o w er series W E 1 ,E 2 = W ( x, α, β , y ) on M E 1 ,E 2 . It is deﬁned as the ab elianization of a series P n > 3 W n /n in cyclic paths in the quiv er Q E 1 ,E 2 with the v ertices E 1 and E 2 and ( E i , E j ) 1 edges b et w een v ertices E i and E j for i, j ∈ { 1 , 2 } . Since an y cyclic path has the same n um b er of edges in the direction E 1 → E 2 as in the direction E 2 → E 1 w e conclude that the p otential W E 1 ,E 2 is G m -in v arian ts with resp ect to the G m - action o n the graded v ector space M E 1 ,E 2 with the we ights w t x = w t y = 0 and w t α = − w t β = 1. The p oten tial W E 1 ,E 2 is obtained from the p o tential W E 1 ⊕ E 2 b y a formal c hange of v a riables. It f o llo ws from G m -in v ariance of W E 1 ,E 2 that it b elongs to k [ α ][[ x, β , γ ]]. Therefore it deﬁnes a function on the fo rmal neigh b o r ho o d of the aﬃne sub- space { (0 , α , 0 , 0) } ⊂ M E 1 ,E 2 . In particular, for an y α ∈ Ext 1 ( E 2 , E 1 ) w e obtain a formal p ow er series W E 1 ,E 2 ,α on M E 1 ,E 2 whic h is the T a ylor expan- sion of W E 1 ,E 2 at the p oin t (0 , α, 0 , 0). Similarly to the Prop osition 7 fro m Section 3.3 the series W E 1 ,E 2 ,α b ecomes (after a formal change of co ordinates) a direct sum W min E α ⊕ Q E α ⊕ N E α , where Q E α is a non-degenerate quadratic form, N E α is the zero function on a v ector subspace , a nd W min E α do es not con tain terms of degree less than 3 in its T a ylor expansion. By the motivic Thom-Sebastiani theorem w e ha v e (1 − M F 0 ( W E 1 ,E 2 ,α )) = (1 − M F ( E α ))(1 − M F 0 ( Q E α )) . Let us consider the quadratic form (( W E 1 ,E 2 ) ′′ ) | (0 ,α, 0 , 0) on M E 1 ,E 2 , where ( W E 1 ,E 2 ) ′′ denotes the second deriv ativ e of the p otential with resp ect to the aﬃne co ordinates. It follows from the ab o v e discussion that this quadratic form is equal to the direct sum of Q E α and the zero quadratic form. It is easy to ch ec k that  (( W E 1 ,E 2 ) ′′ ) | (0 ,α, 0 , 0)  ( v ) is equal to ( d (1) α v , v ) for an y v ∈ M E 1 ,E 2 . Hence Q E α can b e iden tiﬁed with the q uadratic f o rm on Im( d (1) α ) giv en b y ( u, ( d (1) α ) − 1 u ). Step 4. Recall (see Section 5.2) that for any exact triangle E 1 → E α → E 2 w e ha v e a sup er line bundle L with a canonically trivialized square: 99 L E 1 → E α → E 2 = ( D ≤ 1 ,E α ⊗ D − 1 ≤ 1 ,E 1 ⊗ D − 1 ≤ 1 ,E 2 ) ⊗ (sdet(Ext • ( E 1 , E 2 ))) − 1 . F or a split triangle E α ≃ E 1 ⊕ E 2 (i.e. α = 0) this line bundle is canonically trivialized since • by deﬁnition, fo r E α ≃ E 1 ⊕ E 2 w e ha v e D ≤ 1 ,E α ≃ D ≤ 1 ,E 1 ⊗ D ≤ 1 ,E 2 ⊗ ⊗ sdet(Ext ≤ 1 ( E 1 , E 2 )) ⊗ sdet(Ext ≤ 1 ( E 2 , E 1 )) , • by the Calabi-Y au prop erty w e ha v e sdet(Ext ≤ 1 ( E 2 , E 1 )) ≃ sdet(Ext > 2 ( E 1 , E 2 )) . Therefore, fo r an y exact triangle E 1 → E α → E 2 w e ha v e an isomorphism L E 1 → E α → E 2 ≃ D ≤ 1 ,E α ⊗ D − 1 ≤ 1 ,E 1 ⊕ E 2 . On the other hand, cons iderations sim ilar to those in Step 2 giv e rise to a canonical isomorphism D ≤ 1 ,E α ⊗ D − 1 ≤ 1 ,E 1 ⊕ E 2 ≃ sdet(Im( d (1) α )) . One can see that the trivialization (sdet(Im( d (1) α ))) ⊗ 2 ≃ 1 comes exactly from the non-degenerate quadratic form Q E α . Therefore, for an arbitrary ex act triangle E 1 → E α → E 2 w e ha ve an isomorphism of sup er lines compatible with the trivializations of squares: L E 1 → E α → E 2 ≃ sdet(Im ( d (1) α )) . This implies that I ( Q E α ) = I ( l ( E 1 → E α → E 2 )) , where l ∈ J 2 ( F u nct ( C 3 , C )) w as deﬁned in Section 5.2. Step 5. Let us apply the integral iden tity from Section 4.4 t o the p oten tial W E 1 ,E 2 . W e put V 1 := Ext 1 ( E 2 , E 1 ) , V 2 := Ext 1 ( E 1 , E 2 ) , V 3 := Ext 1 ( E 1 , E 1 ) ⊕ Ext 1 ( E 2 , E 2 ) . 100 W e ha v e: Z α ∈ Ext 1 ( E 2 ,E 1 ) (1 − M F (0 ,α, 0 , 0) ( W E 1 ,E 2 )) = = L ( E 2 ,E 1 ) 1 (1 − M F 0 (( W E 1 ,E 2 ) | Ext 1 ( E 1 ,E 1 ) ⊕ Ext 1 ( E 2 ,E 2 ) )) . On the other hand the LHS of the in tegral identit y is equal to Z α ∈ Ext 1 ( E 2 ,E 1 ) (1 − M F 0 ( Q E α ))(1 − M F ( W min E α ) = = Z α ∈ Ext 1 ( E 2 ,E 1 ) L 1 2 rk Q E α I ( Q E α )(1 − M F ( W min E α )) . Recall that rk Q E α = rk( d (1) α ) = ( E α , E α ) ≤ 1 − ( E 1 ⊕ E 2 , E 1 ⊕ E 2 ) ≤ 1 b y Steps 2 and 3. Then the in tegral identit y b ecomes the follo wing equalit y: Z α ∈ Ext 1 ( E 2 ,E 1 ) L 1 2 (( E α ,E α ) ≤ 1 − ( E 1 ⊕ E 2 ,E 1 ⊕ E 2 ) ≤ 1 ) I ( Q E α )(1 − M F ( W min E α )) = = L ( E 2 ,E 1 ) 1 (1 − M F 0 ( W min E 1 ⊕ W min E 2 )) . Comparing this formula with the one we wan ted to pro v e on Step 1 w e see that they coincide if I ( Q E α ) = I ( h ( E α )) I ( h ( E 1 )) I ( h ( E 2 )) . No w using Step 4 w e observ e that this co cycle condition is equiv alent to the main prop erty of the orien tation dat a on exact triangles. This concludes the pro of of the The orem.  Deﬁnition 18 L et C b e an ind-c onstructible 3 - d imensional Cala bi-Y au c at- e g o ry endowe d with p olarization, σ ∈ S tab ( C , cl) . We c al l the c ol le ction of elements ( A mot V ∈ R V ) of the c omplete d motivic quantum tori ( R V ) (fo r al l strict se ctors V ⊂ R 2 ) the mo tivic Donaldson-T homas i n variant of C . Let us consider the follo wing unital Q -subalgebra of Q ( q 1 / 2 ): D q := Z [ q 1 / 2 , q − 1 / 2 ,  ( q n − 1 ) − 1  n > 1 ] . There is a homomorphism of rings D µ → D q giv en b y the tw isted Serre p olynomial. Na mely , it maps L 1 / 2 7→ q 1 / 2 , and on M µ it is the comp osition 101 of the Serre p olynomial with the in v olution q 1 / 2 7→ − q 1 / 2 . Therefore, we ha v e a homomorphism of algebras R Γ → R Γ ,q , where R Γ ,q is a D q -algebra generated by ˆ e γ , γ ∈ Γ, sub ject to the relations ˆ e γ ˆ e µ = q 1 2 h γ ,µ i ˆ e γ + µ , ˆ e 0 = 1 . Similarly to t he motivic case, w e ha v e the algebra R V , q asso ciated with an y strict sector V . The motivic DT-inv arian ts giv e rise to stabilit y dat a on the graded Lie algebra associated with R Γ ,q . W e will denote b y A V , q ∈ R V , q the eleme nt correspo nding to A mot V . 6.4 Examples 1) Assume tha t a 3- dimensional Calabi-Y a u category C is generated b y one spherical ob ject E deﬁned o v er k . Therefore R := Ext • ( E , E ) ≃ H • ( S 3 , k ). In this case w e tak e Γ = K 0 ( C ( k )) ≃ Z · cl k ( E ), and the s ke w-symmetric form on Γ is trivial. In order to c ho ose an orien tation data , let us ﬁx a basis r 0 = 1 , r 3 in the algebra R (the subsc ript indicates the degree). Consider R as a bimo dule ov er itself and denote this bimo dule by M . The corresp onding to 1 , r 3 bimo dule basis will b e denoted by 1 M , r 3 ,M . Then w e ha v e a f a mily M t , 0 ≤ t ≤ 1 of R -bimo dule structures on M suc h that 1 M · t r 3 = r 3 · t 1 M = (1 − t ) r 3 ,M . Hence M 0 = M and M 1 ≃ N ⊕ N ∨ in notation of Section 5.3. The latt er giv es a decompo sition of the bifunctor Hom • . The ab ov e family of bimo dules deﬁne a homotopy whic h can b e used for deﬁnition of an orien tation data as in Section 5.2. F or any z ∈ C , Im z > 0 w e ha v e a stabilit y condition σ z suc h that E ∈ C ss , Z ( E ) := Z (cl k ( E )) = z , Arg( E ) = Arg ( z ) ∈ (0 , π ). F or a strict sector V suc h that Arg( V ) ⊂ (0 , π ) w e ha v e the category C V whic h is either trivial (if z / ∈ V ) or consis ts o f ob jects 0 , E , E ⊕ E , . . . ( if z ∈ V ). T hen A mot V = 1 in the ﬁrst case and A mot V = X n > 0 L n 2 / 2 [ GL ( n )] ˆ e n γ 1 , in the second case. Here γ 1 := cl k ( E ) is the generator of Γ. 102 Let us commen t on the a nswe r. In this case Ext 1 ( nE , nE ) = 0, where we set nE = E ⊕ n , n > 1. There fo re W nE = 0 whic h implies that M F ( W nE ) = 0. The n umerator is L n 2 / 2 = L 1 2 dim Ext 0 ( nE , nE ) = L 1 2 P i ≤ 1 ( − 1) i dim Ext i ( nE , nE ) , since Ext 6 =0 ( nE , nE ) = 0. Let us consider the “quan tum dilogarithm” series E ( q 1 / 2 , x ) = X n > 0 q n 2 / 2 ( q n − 1) . . . ( q n − q n − 1 ) x n ∈ Q ( q 1 / 2 )[[ x ]] . Since [ GL ( n )] = ( L n − 1) . . . ( L n − L n − 1 ), w e conclude that A mot V = E ( L 1 / 2 , ˆ e γ 1 ) . In order to simplify the notation w e will denote E ( q 1 / 2 , x ) simply by E ( x ). In Section 7.1 we will discuss the quasi-classi cal limit, and will a sso ciate n umerical Donaldson-Thomas in v arian ts Ω( γ ) ∈ Q for any γ ∈ Γ fo r g iven stabilit y structure σ ∈ S tab ( C , cl ) . In our basic example w e ha v e (for an y σ ) Ω( ± γ 1 ) = 1 , Ω( n γ 1 ) = 0 for n 6 = ± 1 . 2) Assume that C is generated b y t w o spherical ob jects E 1 , E 2 deﬁned ov er k suc h that dim Ext i ( E 2 , E 1 ) = 0 if i 6 = 1 and dim Ext 1 ( E 2 , E 1 ) = 1. Notice that the unique (up to isomorphism) non-trivial extension E 12 app ears in the exact triangle E 1 → E 12 → E 2 and it is a sphe rical ob ject. F or any z 1 , z 2 ∈ C , Im z i > 0 , i = 1 , 2 there is a unique stabilit y condition σ z 1 ,z 2 suc h that Z ( E i ) := Z (cl k ( E i )) = z i , i = 1 , 2, and the category C V ( k ) in the case z 1 , z 2 ∈ V , Arg ( V ) ⊂ (0 , π ) consists of subsequ ent extensions of the copies of E 1 and E 2 . If Arg( z 1 ) > Arg( z 2 ) then the only σ z 1 ,z 2 -semistable o b jects are (up to shifts) E 1 , 2 E 1 , . . . , E 2 , 2 E 2 , . . . , w here we use the notation nE for E ⊕ n , as b efore. If Arg ( z 2 ) > Arg ( z 1 ) then we hav e three gro ups of σ z 1 ,z 2 -semistable ob jects: nE 1 , nE 2 , nE 12 , n > 1. 103 ❛ r r r r r r r r r r r r r r ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅          ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅          E 1 2 E 1 3 E 1 E 2 2 E 2 3 E 2 E 2 2 E 2 3 E 2 E 1 2 E 1 3 E 1 E 12 2 E 12 ❛ The w all-crossing form ula implies the following w ell-kno wn iden tit y (see [22]) in the algebra D q hh x 1 , x 2 ii / ( x 1 x 2 − q x 2 x 1 ): E ( x 1 ) E ( x 2 ) = E ( x 2 ) E ( x 12 ) E ( x 1 ) , where x 12 = q − 1 / 2 x 1 x 2 = q 1 / 2 x 2 x 1 and x i correspo nds to ˆ e cl k ( E i ) , i = 1 , 2 , 12. Namely , b oth sides of the ab o v e iden tity are equal to A V big ,q for any sector V big in the upp er half-plane con taining z 1 , z 2 . The LHS and the RHS of the iden tit y come from the decomp ositions A V big ,q = A V 1 ,q A V 2 ,q , A V big ,q = A V 2 ,q A V 12 ,q A V 1 ,q , where V i , i = 1 , 2 , 12 are some na rr ow sectors con taining z i . Remark 22 The function E ( x ) satisﬁe s also the identity E ( x 2 ) E ( x 1 ) = E ( x 1 + x 2 ) for x 1 , x 2 ob eying the r e lations x 1 x 2 = q x 2 x 1 as ab ove. This fol lows fr om the formula E ( x ) = exp q  q 1 / 2 q − 1 x  wher e exp q ( x ) is the usual q -exp onent exp q ( x ) := X n > 0 x n [ n ] q ! , [ n ] q ! := n Y j =1 [ j ] q , [ j ] q := 1 + q + · · · + q j − 1 . The exp onential p r op erty of E ( x ) se ems to play no r ole in our c onsider ations. 104 If we denote b y γ i ∈ Γ ≃ Z 2 the c lasses cl(E i ) , i = 1 , 2, then the only non-trivial n umerical Donaldson-Thomas inv arian ts are Ω( ± γ 1 ) = Ω( ± γ 2 ) = 1 in the case Arg ( z 1 ) > Arg( z 2 ), a nd Ω( ± γ 1 ) = Ω( ± γ 2 ) = Ω( ± ( γ 1 + γ 2 )) = 1 in the case Arg ( z 1 ) < Arg( z 2 ). 6.5 D0-D6 BPS b ound states: an example related to the MacMahon function Let X b e a compact 3 d Calabi-Y au manifold o v er k , suc h that H 1 ( X , O X ) = 0. W e denote by C (0 , 6) the ind-constructible triang ulated category generated b y the structure sheaf O X and torsion shea v es O x , x ∈ X . 22 This category has a t -structure with the heart consisting of coheren t shea v es on X whic h are trivial v ector bundles outside of a ﬁnite set. Then O X is the only spherical ob ject in C (0 , 6) . W e choose Γ = Z γ 1 ⊕ Z γ 2 , whic h is the image o f K 0 ( C (0 , 6) ) under the Chern class in the quotien t of the Chow group b y the n umerical equiv alence, whe re γ 1 = cl k ( O x )) fo r an y p oin t x ∈ X , and γ 2 = cl k ( O X ). W e a re g o ing to consider a stabilit y condition σ = ( Z , ( C (0 , 6) ) ss , . . . ) o n C (0 , 6) with the ab o ve t -structure and suc h that z 1 := Z ( γ 1 ) = − 1 , z 2 = Z ( γ 2 ) = i = √ − 1 . Then σ -semistable ob jects in C (0 , 6) will b e either pure torsion shea v es sup- p orted at ﬁnitely man y p oints or torsion-free shea v es. This corresp onds to t he follo wing picture for Ω( γ ). 22 This category is rela ted to the counting of D0-D6 BP S b ound states, compare with [13], for mula (6.1). 105 ✲ ✻ r r r r r r r r r r r r r r r r r r r r − χ − χ − χ χ 2 +5 χ 2 − χ 3 +15 χ 2 +20 χ 6 ? ? ? ? ? 1 0 0 0 0 0 0 0 0 Let us commen t on the last ﬁgure. a) The v ertical line corresp onds t o the subcatego ry generated by the spherical ob ject O X , f or whic h we know Ω( γ ). Namely , Ω( γ 2 ) = 1 and Ω( nγ 2 ) = 0 , n > 2. b) Horizon tal line z 2 − nz 1 , n > 0 corresponds to shea v es of ideals of 0-dimensional subsc hemes. Then: X n > 0 Ω( γ 2 − nγ 1 ) t n = M ( − t ) χ ( X ) , where χ ( X ) is the Euler c haracteristic of X and M ( x ) := Y n > 1 (1 − x n ) − n ∈ Z [[ x ]] is the MacMahon function (see [47], [4] ab o ut this iden tity). c) The torsion shea ve s O x , x ∈ X are Sc h ur ob j ects in C (0 , 6) . The ir mo duli space is canonically iden tiﬁed with X . By Behrend’s fo r mula (see [2]) their con tribution to the virtual fundamen tal n um b er of ob j ects is Ω( γ 1 ) = ( − 1) dim X χ ( X ) = − χ ( X ) . d) The nu mbers mark ed by “?” corresp ond to (p ossibly non-Sc h ur) o b- jects. Notice that there are no semistable ob jects with the class nγ 2 − mγ 1 with 0 < m < n . They corresp ond to the sector ﬁlled by 0’s. 106 Let us no w c ho ose a path σ z 1 ( τ ) ,z 2 ( τ ) in the space of the ab o v e stabili ty structures suc h that z 1 ( τ ) = − exp( iτ ) , z 2 ( τ ) = i , where τ ∈ [0 , π / 2 + ε ] , i = √ − 1 and ε > 0 is suﬃcien tly small. The heart of the t -structure for τ > 0 consists of complexes of shea v es E suc h that there exists an exact triangle E 1 → E → E 2 [ − 1], where E 1 is a torsion-free sheaf and E 2 is a torsion sheaf (indeed, the new t -structure is obtained from the initial one b y the standard tilting procedure). This heart coincides with the category C (0 , 6) V for any τ ∈ (0 , π / 2 + ε ] where V = { z ∈ C ∗ | 0 ≤ Arg( z ) ≤ π / 2 + ε } . Ob ject O X ∈ C (0 , 6) V can not b e represe nted as a non-trivial extension in C (0 , 6) V , hence it is semistable for any τ ∈ [0 , π / 2 + ε ]. Let us no w consider the case τ ∈ ( π / 2 , π / 2 + ε ]. Then ob ject O X has the minimal argumen t among all non-trivial ob j ects in C (0 , 6) V . The refore, all other indecomposable semistable ob jects E are strictly on the left o f O X , and w e ha v e Ext 0 ( E , O X ) = 0 . T aking the long exact sequence o f E xt -groups to the ob ject O X one easily sho ws that in the decomposition E 1 → E → E 2 [ − 1] w e ha v e E 1 = 0. Hence in this new heart C (0 , 6) V the left orthogonal to O X consists of ob jects F [ − 1], where F is a torsion sheaf. W e conclude that for the stabilit y condition with τ ∈ ( π / 2 , π / 2 + ε ] the only semistable o b jects hav e classes whic h b elong to Z 6 =0 γ 2 ⊔ Z 6 =0 γ 1 . Therefore, all DT-inv arian ts Ω τ ( γ ) for σ z 1 ( τ ) ,z 2 ( τ ) with τ > π / 2 are completely determined b y t he n um b ers a n = Ω τ ( − nγ 1 ) , n > 1 (and known in v arian ts Ω τ ( mγ 2 ) = δ m, 1 , m > 1). Then the w all-crossing form ula determines a ll the inv arian ts Ω( γ ) = Ω 0 ( γ ) for the initial stabilit y condition σ z 1 (0) ,z 2 (0) in terms of the n um b ers a n , n > 1. The wall-crossin g form ula implies that the follo wing ident ity: Y n > 1 T a n − nγ 1 T γ 2 = − → Y m > 1 ,n > 0 T Ω( − nγ 1 + mγ 2 ) − nγ 1 + mγ 2 Y n > 1 T a n − nγ 1 . Using the k nown result for sp ecial v alues Ω( γ 2 − nγ 1 ) , n > 1 (in t erms of the MacMahon f unction), one can deduce that all the num b ers a n = Ω( nγ 1 )) 107 for n > 1 are equal to − χ ( X ). W e don’t know ho w to pro v e this iden tity directly . W e see that in v arian ts Ω τ ( γ ) for τ > π / 2 ha v e a mu ch simple r for m than Ω( γ ) = Ω 0 ( γ ). Moreov er, it is now p ossible (in principle) to work out a form ula f o r Ω( − nγ 1 + mγ 2 ) for an y giv en m > 2. Remark 23 One c an try to gener alize the ab ove c onsider ations to the c ase of D0-D 2-D6 b ound states. Mathematic al ly this me ans that we c onsider the triangulate d c ate gory gener ate d by the she af O X and she av e s with a t most 1 - dimensional supp ort (cf. [47]). A p r oblem arises her e, sinc e fo r the natur al t -structur e ther e is no c entr al char ge which gives a stability c ondition on the c a te gory. Pr esumably, in this c ase one c an use the limit stability c onditions (se e [1], [73]). Remark 24 L et X b e a 3 d c omp l e x C a labi-Y au manifold, C ≃ P 1 ⊂ X a r ationa l curve with normal b und le isomorphic to O ( − 1) ⊕ O ( − 1) and C b e a n ind-c onstructible A ∞ -version of the c ate gory P er f C ( X ) of p erfe ct c o m plexes supp o rte d on C . Then Γ := K 0 ( C ) ≃ Z 2 c a rries a trivial skew-symmetric (Euler) form. The lattic e Γ is gen e r ate d by cl( O pt ) and cl( O C ) . It fol lows that ther e ar e n o wal l-cr ossings in this c ase, and henc e our invariants Ω( γ ) do not change under c ontinuous defo rm ations of a stability c ondition. In or d er to use this ide a for c omputations one c an cho ose two stability c onditions by sp e cifying the c orr esp onding t -structur es and c entr al ch ar g e s : a) cho ose the t -structur e with the he art c ons isting of c oher ent she aves on X supp orte d on C and the c entr al char ge Z such that Z (cl( O pt )) ∈ R < 0 , Im Z ( cl ( O C )) > 0 ; b) cho ose the t -structur e given by the c ate gory o f ﬁnite-dimen sional r ep- r ese ntations of the quiver with two vertic es and two double arr ows in e ach dir e ction and the p otential W = a 1 b 1 a 2 b 2 − a 1 b 2 a 2 b 1 . Then c alculations fr om [66] give the fol lowing formulas for the invariants Ω( γ ) : Ω( n cl( O pt )) = − 2 , n 6 = 0 ; Ω( n cl( O pt ) ± cl( O C )) = 1 , n ∈ Z . In al l other c ases Ω( γ ) = 0 . F or r e c ent gener alization of [66] se e [49],[50],[51]. 108 7 Quasi-classical li mit and in teg r ality co n jec- ture 7.1 Quasi-classical limit, n umerical DT-in v arian ts The elemen ts A V , q ∈ R V , q correspo nding to A mot V are series in ˆ e γ , γ ∈ Γ with co eﬃcien ts whic h are rational functions in q 1 / 2 . They can hav e p oles as q n = 1 for some n > 1. Hence it is not clear how to tak e the quasi-classical limit as q 1 / 2 → − 1 (this corresponds to the taking of Euler c haracteristic of the corresp onding motiv es). Let us assume that the sk ew-symmetric form on Γ is non- degenerate (oth- erwise we can replace Γ by the symplectic lattice Γ ⊕ Γ ∨ ). The elemen t A V , q deﬁnes an automorphism of an appropriate completion of R Γ ,q . More pre- cisely , it acts b y the conjugation x 7→ A V , q xA − 1 V , q on the subring Y γ ∈ C 0 ( V ) ∩ Γ D q ˆ e γ where C 0 ( V ) = C 0 ( V , Z , Q ) is the union o f 0 with the con v ex hull C ( V , Z , Q ) of the set Z − 1 ( V ) ∩ { Q > 0 } (see Section 2). Let us recall the example of the category generated b y t w o spherical ob- jects from Section 6.4. W e will use notation fo r sectors V 1 , V 2 , V big in tro duced there. One has, for quan tum v ariables x 1 x 2 = q x 2 x 1 and A V 1 ,q = E ( x 1 ): x 1 7→ E ( x 1 ) x 1 E ( x 1 ) − 1 = x 1 ; x 2 7→ E ( x 1 ) x 2 E ( x 1 ) − 1 = x 2 (1 + q 1 / 2 x 1 ) . This follows f ro m the formula f ( x 1 ) x 2 = x 2 f ( q x 1 ), where f ( x ) is an arbitrary series as w ell from the form ula E ( x ) = Y n > 0 (1 + q (2 n +1) / 2 x ) − 1 , whic h is v alid for 0 < q < 1. The latter form ula implies the needed identit y in Q ( q 1 / 2 )[[ x ]]: E ( q x ) = (1 + q 1 / 2 x ) E ( x ) . A similar form ula holds for the conjugation b y A V 2 ,q . W e remark that in this example the conjugation b y A V , q for V 1 , V 2 or V big preserv es the subring 109 Q γ ∈ C 0 ( V ) ∩ Γ Z [ q ± 1 / 2 ] ˆ e γ . In particular, one can mak e a sp ecialization at q 1 / 2 = − 1 . Remark 25 R e c al l that at the end of Se ction 6.3 we deﬁ n e d a homomor- phism R Γ → R Γ ,q as the c om p o sition o f Serr e p olynomial with the involution q 1 / 2 7→ − q − 1 / 2 . In p articular, the sp e cializ a tion q 1 / 2 = − 1 is wel l-deﬁn e d on the subring of s e ries in gener ators ˆ e γ with c o eﬃcients in M µ ( S pec ( k ))[ L − 1 / 2 ] (se e als o Se ction 7.3), and it c orr esp onds to the usual Euler char acteristic. We use the twisting q 1 / 2 7→ − q − 1 / 2 in or der to avoid a lot of min us signs in formulas. The “inte ger” quan tum torus M γ ∈ C 0 ( V ) ∩ Γ Z [ q ± 1 / 2 ] ˆ e γ ⊂ R Γ ,q has the quasi-classical limit 23 whic h is the Poiss on algebra with basis e γ , γ ∈ C 0 ( V ) ∩ Γ with the pro duct and P oisson brac k et giv en b y e γ e µ = ( − 1) h γ ,µ i e γ + µ , { e γ , e µ } = ( − 1) h γ ,µ i h γ , µ i e γ + µ . The P oisson brac k et is the limit of a normalized brack et: [ ˆ e γ , ˆ e µ ] =  q 1 / 2 h γ ,µ i − q − 1 / 2 h γ ,µ i  ˆ e γ + µ , lim q 1 / 2 →− 1 ( q − 1) − 1 ·  q 1 / 2 h γ ,µ i − q − 1 / 2 h γ ,µ i  = ( − 1) h γ ,µ i h γ , µ i . One can write informally e γ = lim q 1 / 2 →− 1 ˆ e γ q − 1 . Conjecture 5 F o r any 3 d Calabi-Y au c ate gory wi th p olarization a nd any strict se ctor V the automorphism x 7→ A V , q xA − 1 V , q pr e s e rves the subring Y γ ∈ C 0 ( V ) ∩ Γ D + q ˆ e γ , wher e D + q := Z [ q ± 1 / 2 ] . 23 There is another quasi-classical limit q 1 / 2 → +1 which we do not co nsider here. 110 Later we will presen t argumen ts in fav or of this conjecture as w ell as a stronger v ersion. Assum ing t he Conjecture we can deﬁne “num erical” DT- in v a ria n ts of a 3 d Calabi-Y au category with p olarization in the followin g w a y . Conside r the quasi-classical limit (i.e. specialization at q 1 / 2 = − 1) of the automorphism x 7→ A V , q xA − 1 V , q . W e will presen t (see Section 7.4. and Conjecture 10) an explicit conjectural for mula for this “quasi-classic al limit” whic h do es not depend on the orien tation data. The quasi-classical limit giv es rise to a formal symplec tomorphism of the torus T Γ and therefore induces the stabilit y data on the graded Lie alg ebra g Γ (see Section 2.5). Alternative ly , w e can deﬁne suc h data as a ( γ ) := lim q 1 / 2 →− 1 ( q − 1) a ( γ ) q in the obvious notation. F or a g eneric cen tral c harge Z the symplectomor- phism can be written as A V = − → Y Z ( γ ) ∈ V T Ω( γ ) γ , where T γ ( e µ ) = (1 − e γ ) h γ ,µ i e µ and Ω( γ ) ∈ Q (see Section 2 .5). In the ab o v e example of the Calabi-Y au category generated b y one spherical ob ject E w e hav e Ω( n cl( E )) = 1 if n 6 = 1 and Ω( n cl( E )) = 0 otherwise. Conjecture 6 F o r a generic c en tr al char ge Z al l numb ers Ω( γ ) , γ ∈ Γ \ { 0 } ar e in te gers. The collection (Ω( γ )) γ ∈ Γ seems to b e the correct mathematical deﬁnition of the coun ting o f BPS states in String Theory . Finally , w e mak e a commen t ab out the relationship with the w ork of Kai Behrend (see [2]). Recall that he deﬁned a Z -v alued inv a rian t of a critic al p oin t x of a function f on X whic h is equal to ( − 1) dim X (1 − χ ( M F x ( f ))) , where χ denotes the Euler c haracteristic. By Thom-Sebastiani theorem this n um b er do es not c hange if w e add to f a function with a quadratic singularit y at x (stable equiv alence). 111 Let M b e a sch eme with p erfect obstruction theory (see [4]). Th us M is lo cally represen ted as a sch eme of critical p oin ts of a function f on a manifold X . Then the ab ov e inv arian t giv es rise to a Z -v alued constructible function B on M . The global in v arian t is Z M B dχ := X n ∈ Z nχ ( B − 1 ( n )) , where χ denotes the Euler c haracteristic. Behren d prov ed that for a pr op er M the in v arian t R M B dχ coincides with the degree of the virtual f undamen tal class [ M ] vir t ∈ H 0 ( M ) giv en b y R [ M ] vir t 1. No w let us assume that M ⊂ C ss consists of Sch ur o b jects E (see Section 1.3), suc h that cl( E ) = γ ∈ Γ is a ﬁxed primitiv e class. Let us lo ok at the con tribution of M to the motivic DT-in v arian t a ( γ ) mot . By deﬁnition it is equal to Z M L 1 2 (1 − dim Ext 1 ( E ,E )) L − 1 (1 − M F ( E ))(1 − M F 0 ( Q E )) L − 1 2 rk Q E ˆ e γ . Mapping it to the quan tum torus and taking the quasi-classic al limit q 1 / 2 → − 1, and t a king into accoun t the relation − a ( γ ) = Ω( γ ) for primitiv e γ ∈ Γ (see Section 2.5), w e obtain that Behrend’s formula implies that the contri- bution of M to the v alue Ω( γ ) is equal to R [ M ] vir t 1. 7.2 Deformatio n in v ariance and in termediate Jacobian W e a lso exp ect the follow ing (not v ery precise) conjecture to b e true as we ll. Conjecture 7 Th e c ol le ction (Ω( γ )) γ ∈ Γ is invariant with r e s p e ct to the “p o- larization p r eserving” deformation s of C , in the c ase when C is homolo gic al ly smo oth in the sense of [4 2]. The motiv ation f o r the la st Conjecture is the deformation in v ariance o f the virtual fundam ental class in the “classic al” Donaldson-Thomas theory . Recall that homologically smo oth E xt -ﬁnite categories can b e though t as non-comm utativ e analogs of smo oth prop er sc hemes. Hence, w e can exp ect that the mo duli stac ks of semistable ob jects in suc h categories are also prop er in some sense. Therefore, w e can also exp ect that the degree of the virtual fundamen tal class is in v ariant under deformations. 112 Also, w e exp ect the follo wing generalization of our theory in the case when k = C and the 3 d Calabi-Y au category is homologically smo oth (see [42]). 1) First, w e recall that ev en without imp osing the Calabi-Y au condition one exp ects that a triangulated compact homologically smo oth A ∞ -category C (p ossibly Z / 2 Z -graded) admits (conjecturally) a non-commu tative pure Ho dge structure (see [39], [36], [42] a b out motiv ations, deﬁnitions as well as some conjectures and applications of this notion). In particular, p erio dic cyclic homology groups H P eve n ( C ) (resp. H P odd ( C )) carry descending Ho dge ﬁltrations H P eve n ( C ) · · · ⊃ F i eve n ⊃ F i − 1 eve n ⊃ . . . , i ∈ Z H P odd ( C ) · · · ⊃ F i odd ⊃ F i − 1 odd ⊃ . . . , i ∈ Z + 1 2 . In 3 d Calabi-Y au case w e assume that the smallest non-trivial term o f the ﬁl- tration F • odd is F − 3 / 2 , dim F − 3 / 2 = 1. Moreo v er, in general, it is exp ected that there are lattices K eve n top ( C ) and K odd top ( C ) whic h b elong to the corresp onding p erio dic cyclic homology groups (the y represen t the non-comm utativ e ve rsion of the image of the top ological K -theory in the de Rham cohomology). 2) If C is homologically smo oth Calabi-Y au category then it is easy to see that (assum ing the degeneration of the Ho dge-to-de R ham conjecture , see [42]) the mo duli space M of formal deformations of C is smo oth of dimension dim M = 1 2 dim H P odd ( C ) (this is a corollary of the formalit y of t he little disc op erad as w ell as the fact that the action of the Connes diﬀeren tial is represen t ed by the rotation of the circle, whic h is homotopically trivial under the assumption). It is exp ected that the global mo duli space also exists. Notice that the Calabi-Y au structure on C induces a symplectic structure on the v ector space H P odd ( C ) and in the 3 d case the mo duli space M is lo cally em b edded in to H P odd ( C ) as a La grangian cone. 3) W e exp ect that for an arbitrary triangulated compact homo logically smo oth A ∞ -category C one has a non-comm utative v ersion of the Deligne cohomology H D ( C ) whic h ﬁts in to a short exact seq uence 0 → H P odd ( C ) / ( F 1 / 2 odd + K top odd ( C )) → H D ( C ) → F 0 eve n ∩ K top eve n ( C ) → 0 . 113 Morally , H D ( C ) should b e though t as zero cohomology group of the homotop y colimit of the followin g diagram o f cohomology theories: H C − • ( C )   y K top • ( C ) − − − → H P • ( C ) where H C − • ( C ) is the negativ e cyclic homology . An y ob j ect of C should ha v e its c haracteristic class in H D ( C ). More precisely , there should b e a homomorphism of groups ch D : K 0 ( C ) → H D ( C ) (in the c ase of Calabi-Y au manifold it is related to holomorphic Chern-Simons functional). The reason for this is that ev ery ob ject E ∈ O b ( C ) has natural c haracteristic classes in K top 0 ( C ) and in H C − 0 ( C ) whose images in H P 0 ( C ) coincide with eac h other. The tota l space M tot of the ﬁbration M tot → M with t he ﬁber H D ( C ) ov er the p oint [ C ] ∈ M should b e a holomorphic symplec tic manifold (cf. [2 0 ]). Moreo v er, any ﬁb er of this ﬁbration (i.e. the group H D ( C ) for give n [ C ]) is a countable union of complex Lagrangian tori. By analogy with the comm utativ e case w e exp ect that the lo cus L ⊂ M tot consisting of v a lues of ch D is a coun table union o f L a grangian sub v arieties. Ev ery suc h subv a riet y can b e either a ﬁnite ramiﬁed cov ering of M or a ﬁbration o v er a prop er sub v ariet y of M with the ﬁb ers whic h are ab elian v arieties. 4) F or generic [ C ] ∈ M one can use the triple ( K 0 ( C ) , H D ( C ) , ch D ) in- stead of the triple ( K 0 ( C ) , Γ , cl). Analogs of o ur motivic D onaldson-Thomas in v a ria n ts A mot V ∈ R V will b e formal countable sums of p oin ts in H D ( C ) with “w eigh ts” whic h are elemen ts of the motivic ring D µ . The pushforw ard map from H D ( C ) to Γ = F 0 eve n ∩ K top 0 ( C ) giv es the nume rical D T-in v arian ts. The con tinu ity o f motivic D T-in v a r ia nts means tha t after taking the quasi- classical limit the w eigh ts b ecome inte ger-v alued functions on the set of those irreducible comp onen ts of L whic h are ﬁnite ramiﬁed cov erings on M . These considerations lead to the following Question 2 Is ther e a na tur al extension of the numeri c al D T-invariants to those c omp one n ts of L which pr oje ct to a pr op e r subvariety of M ? Remark 26 L et us notic e the similarity of the ab ove c onsider ations with those in the the ory of Gr omov -Witten invariants. Supp ose X is a 3 d c omplex 114 c o mp act C a labi-Y au manifold with H 1 ( X , Z ) = 0 . Then we have an exact se quenc e 0 → H 3 D R ( X ) / ( F 2 H 3 D R ( X ) + H 3 ( X , Z )) → H 4 D ( X ) → H 4 ( X , Z ) → 0 , wher e H 4 D ( X ) = H 4 ( X , Z → O X → Ω 1 X ) is the Deligne c ohomolo gy. Then any curve C ⊂ X deﬁnes the cl a ss [ C ] ∈ H 4 D ( X ) . F o r a generic c om- plex structur e on X the cla s s is c onstant in any sm o oth c onne cte d f a mily o f curves. Mor e over, a s tabl e map to X deﬁnes a c l a ss in H 4 D ( X ) . Then we h a ve exactly the same pictur e with holomorphic symple ctic ﬁbr ation M tot → M with the L agr angian ﬁ b e rs, as we discusse d ab ov e . Similarly to the c ase of DT-invariants the GW-invariants app e ar as inﬁnite line ar c ombinations of p o ints in H 4 D ( X ) , but this time with r ational c o eﬃc i e nts. We exp e ct that the wel l-known r elationship “GW=DT ” (se e [47]) should b e a statement ab out the e q uali ty of the ab ove-discusse d c ounting function s ( assuming p ositive answer to the ab ove question). 7.3 Absence of p oles in the series A Hall V Here w e are g oing to discuss a stronger ve rsion of the Conjecture 5. Conjecture 8 L et D + := M µ ( S pec ( k ))[ L − 1 / 2 ] b e the ring of e quivalenc e classes of mo tivic functions. Then the automorphism of the motivic q uantum torus giv e n by x 7→ A mot V x ( A mot V ) − 1 pr e s e rves the subring Q γ ∈ C ( V ) ∩ Γ D + ˆ e γ for al l strict s e c tors V ⊂ R 2 . It is enough to c hec k the conjecture f or all x = ˆ e γ , γ ∈ Γ. Moreo v er, b ecause of F actorization Prop ert y it is enough to consider the case when V = l is a ray . In the latter case w e can split the inﬁnite pro duct in to those corresp onding to diﬀeren t arithmetic progression, hence reducing the conjecture to the case when Z (Γ) ∩ l = Z > 0 · γ 0 for some non-zero γ 0 ∈ Γ. Then w e ha ve A mot l = A mot l ( ˆ e γ 0 ) = 1 + X n > 1 c n ˆ e n γ 0 ∈ D µ [[ ˆ e γ 0 ]] . Using the comm utation relations in the motivic quan tum torus w e hav e: A mot l ( ˆ e γ 0 ) ˆ e γ ( A mot l ( ˆ e γ 0 )) − 1 = ˆ e γ A mot l ( L h γ 0 ,γ i ˆ e γ 0 ) A mot l ( ˆ e γ 0 ) − 1 . 115 Since fo r any series f ( t ) = 1 + . . . we hav e f ( L n t ) f ( t ) = f ( L n t ) f ( L n − 1 t ) . . . f ( L t ) f ( t ) , in order to prov e the conjecture it suﬃces to c hec k that A mot l ( L ˆ e γ 0 ) A mot l ( ˆ e γ 0 ) − 1 ∈ D + [[ ˆ e γ 0 ]] . Since in that case w e are dealing with ob jects whose cen tral charges b elong to the r ay l , w e can restrict ourselv es to the subcategory C l . The latter can b e thought of as a heart of the t -structure of an ind-constructible 3 d Calabi-Y au category with vani s hing Euler form. More precisely , C l ( k ) is an ab elian artinian category with Hom C l ( k ) ( E , F ) := Ext 0 C ( k ) ( E , F ). Then K 0 ( C l ( k )) ≃ ⊕ E 6 = 0 Z · [ E ], where the sum runs ov er the set of non-zero simple ob jects of C l ( k ). Next, we can reduce the conjecture to a sp ecial case when cl k ( E ) = γ 0 is a ﬁxed class for all simple ob jects E of C l ( k ). Indeed, let us consider an ind-constructible homomorphism cl ′ k : K 0 ( C l ( k )) → Γ ′ := Z ⊕ Z such that cl ′ k ( E ) = (1 , 0) if cl k ( E ) = γ 0 and cl ′ k ( E ) = (0 , 1) if cl k ( E ) ∈ { 2 γ 0 , 3 γ 0 , . . . } for a simple ob ject E . L et c ho ose tw o complex nu mbers z 1 , z 2 in suc h a wa y that 0 < Arg( z 1 ) < Arg( z 2 ) < π and deﬁne a cen tral c harge Z ′ : Γ ′ → C b y the form ula Z ′ ((1 , 0)) = z 1 , Z ′ ((0 , 1)) = z 2 . In this wa y w e obtain a new stabilit y structure on the triangulated env elop e of C l ( k ) with the same heart. In particular, the elemen t A mot l will b e decomposed in to a n inﬁnite pro duct: A mot l = − → Y A mot l ′ of series A mot l ′ correspo nding to ab elian categories C l ′ ( k ) for the new stability structure. One of these categories will b e the sub category generated by simple ob jects E suc h that cl ′ k ( E ) = γ 0 . Let us call s uch category pur e of class γ 0 . All other categories C l ′ ( k ) do not con tain ob jects with the class γ 0 . Rep eating the pro cedure we reduce the conjecture to the cas e of pure category of the class mγ 0 for some m > 1. Similarly to the ab ov e arguments w e can reduce it further to the case m = 1. In this case the conjecture f o llo ws from the one b elo w whic h concerns Hall algebras of categories whic h are no longer required to carry a Calabi-Y au structure. 116 In order to form ulate this new conjecture w e are going t o use the fol- lo wing set-up. Let ( C , A ) b e a pair consisting of a n ind-constructible tri- angulated A ∞ -category ov er a ground ﬁeld k and A ⊂ O b ( C ) b e an ind- constructible subset suc h that A ( k ) is the heart of a b ounded t -structure in C ( k ). W e assume that simple ob jects of the ab elian catego ry A ( k ) form a constructible subset of O b ( C )( k ) and eve ry ob ject in A ( k ) is a ﬁnite ex - tension of simple ones. These data are equiv alent to a sp ecial kind of an ind-constructible category with a stabilit y structure. Namely , let us take Γ := Z and deﬁne cl k ( E ) = 1 for eve ry simple ob ject of A ( k ). It follows that cl k ( F ) = l eng th ( F ) for an y ob ject of A ( k ). F urthermore, w e choose a complex n um b er z 0 in the upp er-half plane and deﬁne a cen tral c harge Z : Γ → C b y the formu la Z (1) = z 0 . Then A = C l for l = R > 0 · z 0 . There- fore the elemen t A Hall l deﬁned for this stabilit y structure can b e though t o f as a series in one v ariable: A Hall l ( t ) = 1 + X n > 1 c n t n . Let us deﬁne a subalgebra H + ( C ) ⊂ H ( C ) to b e the set of linear com bi- nation of eleme nts o f the form L n · [ Z → O b ( C )] where n ∈ Z and Z → O b ( C ) is a 1 - morphism of ind-constructible stac ks (see Section 4.2) with Z b eing an ordinary constructible set endo w ed with trivial action of the trivial g r o up. The multiplic ation la w in H ( C ) preserv es suc h class of elemen ts. Conjecture 9 Th e element F l ( t ) := A Hall l ( L t ) A Hall l ( t ) − 1 b e l o ngs to the c om- plete d Hal l alg ebr a d H + ( C ) (i. e . w e do not invert motives [ GL ( n )] , n > 1 of the gener al line ar gr oups). Belo w we discuss tw o sp ecial cases in whic h the ab ov e conjecture holds. But ﬁrst we presen t a similar motiv ating statemen t in t he case of ﬁnite ﬁelds. Let R b e ﬁnitely generated alg ebra o v er a ﬁnite ﬁeld F q , and R − mod f denotes the category of ﬁnite-dimensional (ov er F q ) left R -mo dules. W e deﬁne the Hall algebra H ( R − mod f ) as a unital asso ciativ e algebra o v er the ring Z [ 1 q ] generated by the isomorphism classes [ M ] of ob j ects of R − mod f with the 117 m ultiplication [ E ] · [ F ] = q − dim Hom( F ,E ) X α ∈ Ext 1 ( F, E ) [ E α ] , where, as befor e, E α denotes an extension with the class α . Prop osition 13 L et A ( t ) := X [ M ] ∈ I so ( R − mod f ) [ M ] # Aut( M ) t dim M . Then F ( t ) := A ( q t ) A ( t ) − 1 ∈ H ( R − mod f )[[ t ]] . Mor e over, F ( t ) = X I ⊂ R,I = RI , dim R/I < ∞ [ R/I ] t dim R/I . Hence the quotien t F ( t ) do es not hav e denominator ( q n − 1) , n > 1 and can b e represe nted in terms o f the “ non-comm utativ e Hilb ert sc heme” of left ideals in R of ﬁnite co dimension. Pr o of. Let us make use of the basis of “renormalized” elemen ts c [ E ] := [ E ] # Aut( E ) in the Q -algebra H ( R − mod f ) ⊗ Q . Then the pro duct can b e rewritten in a more familiar form: c [ E ] · c [ F ] = X [ G ] c c [ G ] c [ E ] , c [ F ] c [ G ] , where the structure constan t c c [ G ] c [ E ] , c [ F ] ∈ Z denotes the n um b er of subo b jects in G isomorphic to E and suc h that the quotien t is isomorphic to F . In these notation w e hav e: A ( t ) = X [ M ] d [ M ] t dim M . 118 Since ﬁrst statemen t of the Proposition follo ws from the second o ne, w e are going to sho w the latter. In the new notation it b ecomes: X I ⊂ R,I = RI , dim R/I < ∞ [ [ R/I ]# Aut( R/ I ) t dim R/I · X [ M ] d [ M ] t dim M = = X [ N ] c [ N ] q dim N t dim N . Let us ﬁx an ob j ect N , and cons ider the co eﬃcien t of the term c [ N ] t dim N . In the RHS it is equal to q dim N . It is e asy to see that the corresp onding co eﬃcien t in the LHS is of the form X I ⊂ R,I = RI , dim R/I < ∞ X N ′ ⊂ N , N ′ ≃ R/I # Aut( R/I ) = = X I ⊂ R,I = RI , dim R/I < ∞ X R/I ֒ → N 1 = # N = q dim N . Notice t hat in the last sum we consider all p ossible embeddings of R /I to N and ev ery summand corresp onds to a c hoice of a cyclic ve ctor in a cyclic submodule in N . This prov es the Prop osition.  The ab ov e Proposition suggest to interp ret our category as a category of mo dules and then apply similar argumen ts whic h reduce the sum (or e ve n the motivic integral) to the sum ov er all cyclic submo dules. It is useful to k eep this in mind when consid ering tw o examples in the next subsection. Remark 27 The sub algebr a H + ( C ) of the Hal l algebr a has the ad vantage that one c an apply the Euler ch a r ac teristic χ to its elements ﬁb erwise over O b ( C ) , and g e t a c onstructible Z -value d function (with c onstructible supp ort) on the ind-c onstructible set I so ( C ) of iso m orphism classes of obje cts of C ( k ) . The multiplic ation in H + ( C ) desc en d s to a multiplic ation o n the ab elian gr oup of such functions. It is e asy t o se e that this multiplic ation is c ommutative, and one has ν E × ν F = ν E ⊕ F wher e ν E etc. ar e delta-functions (se e Se ction 6.1). This fol lows fr om the fact that for a ny non -zer o α ∈ Ext 1 ( F , E ) al l obje cts E tα ar e isomorphic to e a ch other for t ∈ k × , and the Euler char acteristic of G m is zer o. 119 7.4 Reduction to the case of category of mo dules Here w e presen t t w o sp ecial cases when the conjecture holds. 1) Assume that the ab elian categor y A ( k ) con tains only one 24 (up to an isomorphism) simple ob ject E 6 = 0, and this ob ject is deﬁned o v er the ﬁeld k . Hence Ext 0 C ( k ) ( E , E ) ≃ k . W e also assume that A ∞ -algebra Hom • ( E , E ) is minimal, i.e. m 1 = 0, and hence Hom • ( E , E ) = Ext • ( E , E ). Prop osition 14 The c ate g o ry A ( k ) is e quivalent to the c a te gory B − mod f ,cont of c ontinuous ﬁnite-dimensional r epr esentations of a ﬁnitely gener ate d top o- lo gic al alge b r a B . Pr o of. There is a general w a y to construct the a lg ebra B from the A ∞ - structure. Let x 1 , . . . , x m b e a basis in the v ector space (Ext 1 ( E , E )) ∗ . Then the higher comp o stion maps m n : Ext 1 ( E , E ) ⊗ n → Ext 2 ( E , E ) , n > 2 deﬁne a linear map X n > 2 m n : ( Ext 2 ( E , E )) ∗ → k hh x 1 , . . . , x m ii = Y n > 0 ((Ext 1 ( E , E )) ∗ ) ⊗ n . W e deﬁne a top o logical algebra B E := B as the quotien t of k hh x 1 , . . . , x m ii b y the closure of the 2 -sided ideal generated by the image of P n > 2 m n . Next w e observ e that any ob ject M of A ( k ) is a ﬁnite extens ion of ob j ects isomorphic to E . Henc e, it can b e thought of as deformation of an ob ject mE := E ⊕ E ⊕ · · · ⊕ E ( m summ ands) preserving the ﬁltration E ⊂ E ⊕ E ⊂ · · · ⊂ mE , where m = l eng th ( M ). Ev ery such a deformation is give n by a solution to the Maurer-Cartan equation X n > 2 m n ( α, . . . , α ) = 0 , where α = ( α ij ) is an upper-triangular matrix with co eﬃcien ts from Ext 1 ( E , E ). It is easy to see that suc h a solution giv es rise to a represen tation of the alge- bra B in the upp er-triangular matrices of ﬁnite size. F urthermore one c hec ks that this correspondence pro vides an equiv a lence of categories F : A ( k ) ≃ B − mod f ,cont . 24 The arguments b elow extend immediately to the cas e of ﬁnitely man y such ob jects. 120 This prov es the Prop osition.  Notice that leng th ( M ) = dim F ( M ) for any ob ject M . Using the fr a mew ork of ﬁnite-dimensional con tin uous represen ta tions w e can mo dify the argumen ts from the pro of of the Prop o sition 13 to the case of motivic functions instead of ﬁnite ﬁelds and o btain the form ula A Hall ( L t ) A Hall ( t ) − 1 = X n > 0 [ π : H il b n ( B ) → O b ( C )] t n , where H il b n ( B ) is the sc heme of closed left ideals in B of co dimension n (non-comm utativ e analog of Hilb ert sc heme) and π ( I ) = B /I for an y suc h ideal. 2) Let us assume that k = F q and A is an ab elian k -linear category suc h that ev ery ob ject has ﬁnitely many sub ob jects. W e deﬁne t he map cl : K 0 ( A ) → Z suc h that cl([ E ]) = n if E is sim ple ob j ect and E nd ( E ) ≃ F q n . Prop osition 15 Assume that A i s a he art o f a t -structu r e of a triangulate d Ext -ﬁnite F q -line ar A ∞ -c ate gory C . L et us c onsider the series A ( t ) := X [ M ] ∈ I so ( A ) [ M ] # Aut( M ) t cl( M ) . Then we claim that F ( t ) := A ( q t ) A ( t ) − 1 = X [ M ] , M is cyclic c M [ M ] t cl( M ) , wher e c M ∈ Z [ 1 q ] , and the n otion of a cyclic obje ct is intr o duc e d b elow. W e ar e go ing to reduce the pro of to the case of mo dules o v er a n algebra. Moreo v er we will give an explicit formula for the co eﬃcien ts c M . In order to do that w e need the follo wing categorical deﬁnition of a cyclic ob ject. Deﬁnition 19 We say that an obje ct N in a n artinian a b e lian c ate gory is cyclic if ther e is no epimorphism N → E ⊕ E w her e E 6 = 0 is simple. 121 In the category of ﬁnite-dimensional mo dules o ve r an asso ciativ e algebra (o v er an y ﬁeld), cyclic ob jects a r e the same as cyclic mo dules. An y ob ject M ∈ A admits a decomposition M = ⊕ α M α in to a direct sum o f indecomp osables. F or eac h indecomposable summand M α w e ha v e a decomposition M ss α = ⊕ i E α,i of its max imal sem isimple factor M ss α (called the coso cle of M α ) in to a direct sum of simple ob jects E α,i . Let us assume that M is a cyclic ob ject. It is equiv alent to the condition that all simple factors E α,i are pairwise diﬀeren t. Notice that End( M ss ) = ⊕ α,i End( E α,i ) ≃ ⊕ α,i F q m α,i where m α,i = cl( E α,i ) ∈ Z > 0 . Also, it fo llo ws f r o m the cyc licity of M that End( M ) ss ≃ ⊕ α F q n α for some p ositiv e integers n α . It follo ws from the deﬁnition that m α,i is divisible b y n α for any pair ( α, i ). Observ e that in the ab ov e notation # Aut( M ) = q r Y α ( q n α − 1 ) , where r is the dimension ov er F q of the radical o f End ( M ). No w w e claim that in the ab ov e Prop osition 15 c M = q cl( M ) · Q α,i q m α,i − 1 q m α,i q r · Q α ( q n α − 1 ) . The prop ert y n α | m α,i implies that c M ∈ Z [ 1 q ]. W e are going to pro v e the Prop osition together with the ab o v e form ula for c M . Pr o of. W e ma y assume that A is generated b y ﬁnitely man y simple ob- jects (but they can b e deﬁned ov er diﬀeren t ﬁnite extensions of F q ). First, we claim that A is equiv alen t to the category B − mod f ,cont of ﬁnite-dimens ional con tin uous represen tations of a top ological algebra B , similarly to the pre- vious example. More precise ly , let N = ⊕ i E i b e the direct sum of all simple ob jects E i , and set C := End( N ). Then C is a semisimple asso ciativ e unital F q -algebra, whic h is isomorphic to ⊕ i F q cl( E i ) . Let us consider Ext 1 ( N , N ) as a C -bimo dule and take G := Hom C ⊗ C op − mod (Ext 1 ( N , N ) , C ⊗ C op ) 122 to b e the dual bimo dule. The top ological free algebra Y n > 0 G ⊗ C ⊗ G ⊗ C · · · ⊗ C G ( n tensor factors) con tains a closed tw o- sided ideal generated b y the image of the map P n > 2 m n (here w e use t he “ A ∞ -origin” of our ab elian category). W e denote b y B the quotien t of the f ree algebra by this ideal. Then B can b e thought of as a completed path algebra of the quiv er deﬁned b y simple ob jects E i with the arrows whic h corresp ond to a basis of  Ext 1 ( E i , E j )  ∗ . Similarly to the previously considered example, we hav e an equiv alence o f categories Ψ : A ≃ B − mod f ,cont . Under this equiv alence simple ob ject E i maps to the direct summand F q cl( E i ) of C , hence dim Ψ( E i ) = cl( E i ). It follo ws that for an y ob ject M w e ha v e dim Ψ( M ) = cl( M ). Lemma 3 Mo dule M ∈ B − mod f ,cont is cyclic iﬀ M ss is cyclic. Mor e over v ∈ M is a gener ator iﬀ its pr oje ction v ∈ M ss is a gener ator. Pr o of of lemma. The ﬁrst statemen t follo ws directly from the deﬁnition of a cyclic ob ject. In order to prov e the second statemen t a ssume that v ∈ M ss is a generator. W e w ant to prov e that the quotien t M /B v = 0. If this is not the case then w e ha v e an epimorphism M /B v → E i 0 to a simple mo dule E i 0 . It fo llows that w e hav e an epimorphism M ss → E i 0 suc h that v 7→ 0. This con tradicts to the assumption that v ∈ M ss is a g enerator. The lemma is pro v ed.  In order to ﬁnish the pro of of the Prop osition, it is enough to c hec k that the co eﬃcien t c M giv en b y a pro duct form ula on the previous page, is equal to the num ber of isomorphism clas ses of generators v ∈ M up to an automorphism of M . In order to do that w e observ e that the pro duct Q α,i ( q m α,i − 1) from the fo r mula for c M is in fact eq ual to the n um b er of generators of M ss . F urthermore, the factor q cl( M ) Q α,i q m α,i is equal to the n um b er of liftings of a generator of M ss to a g enerator of M (this n um b er is the n um b er o f elemen ts in the k ernel of the pro jection M → M ss ). Finally , w e recall that q r Q α ( q n α − 1) = # Aut( M ). Applying the ab ov e lemma we ﬁnish the pro of of the Prop osition.  Remark 28 It lo oks plausible that the Pr op os ition holds without the assump- tion that A is a t -structur e o f an A ∞ -c ate gory. 123 W e do not kno w the “motivic” analog of the ab o ve Prop osition. In that case one should replace A b y an ind-constructible ab elian category o v er an y ﬁeld. There is a notion of semisimple and cyclic mo dules, it is preserv ed under ﬁeld extensions 25 . It lo o ks natural to exp ect that an analog of the quotien t Q α,i ( q m α,i − 1) Q α ( q n α − 1) is the motiv e Aut ( M ss ) / Aut ′ ( M ) where Aut ( M ss ) is the aﬃne group sc heme of automorphisms of M ss and Aut ′ ( M ) is the image of the sc heme of auto morphisms o f M in Aut ( M ss ). Both groups sc hemes are algebraic tori. Although the motivic v ersion is not absolutely clear, w e can write do wn the “n umerical” vers ion, whic h is the result of the quasi- classical limit q 1 / 2 → − 1 ( equiv alen tly , this is the result of taking the Euler c haracteristic of the corresp onding motiv es). It f ollo ws from the Prop o sition that in the quasi-class ical limit only those terms in the form ula f or c M are non-zero for whic h Aut ( M ss ) = Aut ′ ( M ). Let us call suc h o b jects sp e c ial cyclic . A cyclic ob ject is sp ecial cyclic iﬀ under the extension of scalars to k the co co cles of all indecomposable sum mands (i.e. ob jects M ss α in our notation) a re simple. In the case of ﬁnite-dimensional mo dules ov er an asso ciativ e algebra A , a cyclic ob ject (or mo dule) M is sp ecial iﬀ the sc heme o f left ideals I ⊂ A suc h that M ≃ A/I has Euler c haracteristic 1. F or non-sp ecial cyclic ob jects the corresp onding Euler characteris tic v anishes . Let us return to our considerations in the case of 3 d ind-constructible Calabi-Y au category o ve r a ﬁel d k of characte ristic zero. W e r educed the main conjecture to the case of a single ray , hence A ( k ) is the heart of a t -structure of C l . In this case isomorphism classes of special cyclic ob jects M with the ﬁxed class cl( M ) fo rm a constructible set S C n . Th us, we arriv e to the following form ula χ Φ ( F l ( t )) = χ Φ ( A mot l ( L t ) A mot l ( t ) − 1 ) = = P n > 0 t n R S C n ( − 1) ( M ,M ) 6 1 (1 − χ ( M F ( M ))) d χ , where R V f dχ = P n ∈ Z nχ ( f − 1 ( n )) denotes the “integral ov er Euler charac- teristic” χ of the map f : V → Z , and χ Φ is the comp osition o f the ho- momorphism Φ from the motivic Hall algebra to the motivic quan tum t o rus (restricted to subalgebra H + ( C ) ⊂ H ( C )), and o f the Euler c haracteristic morphism acting on co eﬃcien t s as χ : D + → Z . W e remark that the RHS do es not depend on the orien tation data. 25 Notice that notions of simple or indecomp os a ble ob jects are not preserved under the ﬁeld ex tension. 124 Conjecture 10 I n c ase i f the c ate gory C is not e ndowe d with orien tation data the ab ove pr o c e dur e gives rise to wel l-deﬁ n e d stability data on the g r ad e d Lie algebr a g Γ of Poisson automorphisms of the algeb r aic Poisson torus Hom(Γ , G m ) as wel l as a c ontinuous lo c al hom e o morphism S tab ( C , cl) → S tab ( g Γ ) . 7.5 Evidence for the in tegrali t y conjecture In this section w e presen t argumen ts in fav or of the in tegralit y of the “ n u- merical” DT-inv arian ts Ω( γ ). Recall that if E is an o b ject of a k -linear triangulated category , then w e sa y that E is a Sc h ur ob ject if Ext < 0 ( E , E ) = 0 , Ext 0 ( E , E ) ≃ k · id E . Let us assume now that C is a n ind-constructible 3 d Calabi-Y au category generated by a Sc h ur ob j ect E ∈ C ( k ) in the sense that the category C ( k ) consists of ﬁnite extensions of the shifts E [ i ] , i ∈ Z . In this case K 0 ( C ( k )) ≃ Z · cl k ( E ). W e tak e Γ = K 0 ( C ( k )) and the trivial sk ew-symmetric form on Γ. F or an y z ∈ C , Im z > 0 our category carries an obvious stabilit y condition σ z suc h that Z ( E ) := Z ( cl k ( E )) = z , Arg ( E ) = Arg( z ) ∈ (0 , π ). All ob jects F ∈ C ss ( k ) with Arg ( F ) = Arg ( E ) are n -fold extensions of copies of E fo r some n > 1. W e denote by l the ra y R > 0 · z . In the previous section w e obtained a form ula for χ Φ ( F l ( t )) in terms of the in tegral ov er Euler characteris tic ov er the mo duli space of special cyclic ob jects of C l ( k ). W e are going to mak e it more explicit further, b y using the p oten tial of E . Let us recall ( see Section 3 .3 ) that with the ob ject E w e asso ciate a collection of cyclically inv arian t p olylinear maps W N : ( Ext 1 ( E , E ) ⊗ N ) Z / N Z → k , N > 3 , a 1 ⊗ · · · ⊗ a N 7→ W N ( a 1 , . . . , a N ) = ( m N − 1 ( a 1 , . . . , a N 1 ) , a N ) . Let us c ho ose a basis x 1 , . . . , x m in Ext 1 ( E , E ). Then to an y n > 0 and collection of matrices X 1 , . . . , X m ∈ M at ( n × n, k ) w e asso ciate the n umber W ( n ) N ( X 1 , . . . , X m ) = 1 N X 1 6 i 1 ,...,i N 6 m W N ( x i 1 , . . . , x i N ) T r( X i 1 . . . X i N ) . Th us w e ha v e a p olynomial on k mn 2 . The series W ( n ) = X N > 3 W ( n ) N 125 is a formal function on the formal neigh b orho o d of the reduced closed sub- sc heme N il p m,n ⊂ A mn 2 whose k -p oin ts are collections o f matrices ( X 1 , . . . , X m ) ∈ M at ( n × n, k ) whic h satisfy the prop ert y that there exists a basis in whic h all ( X i ) i =1 ,...,m are strictly upper triangular. Equiv alen tly , T r( X i 1 . . . X i r ) = 0 for an y sequence of indices i • ∈ { 1 , . . . , m } with r ≥ ( n + 1). This prop erty ensures that W ( n ) N is we ll-deﬁned in a formal neigh bo r ho o d of N ilp m,n . Then χ Φ ( F l ( t )) = 1+ X n > 1 t n Z N ilp S C m,n /P GL ( n ) ( − 1) n 2 (1 − m ) (1 − χ ( M F ( X 1 ,...,X m ) ( W ( n ) ))) dχ , where N il p S C m,n for n > 1 is a subsc heme of N il p m,n whose k -p o ints consis ts of those collections ( X 1 , . . . , X m ) for whic h co dim X i Im( X i ) ! = 1 . Let us commen t on this formula. F irst w e remark that it is suﬃcien t to integrate o ve r the set C r it ( W ( n ) ) of critical p oints of W ( n ) , since for all non-critical p oin ts χ ( M F ( X 1 ,...,X m ) ( W ( n ) )) = 1. Rep eating the argumen ts of the previous section w e o btain that C ( k ) is equiv alen t to the category B W − mod f ,cont of contin uous ﬁnite-dimensional represen tations ov er k of the top ological k - a lgebra B W = k hh x 1 , . . . , x m ii / ( ∂ x i W ) , 1 6 i 6 m , where x i , 1 6 i 6 m are the co ordinates corresp onding to the c hosen basis x i , 1 6 i 6 m , and ( ∂ x i W ) denotes the closure of the 2- sided ideal generated b y the cyclic deriv ativ es o f the c yclic p oten tial W = P N > 3 N − 1 W N . Inde ed, it is straigh tforw ard to see that a p oint ( X 1 , . . . , X m ) ∈ N ilp m,n ( k ) gives rise to a contin uous n - dimensional represen tation of W if an only if it belongs to C r it ( W ( n ) ). In terms of the category C ( k ) these p oin ts corresp ond to n - fold extensions of the Sc h ur o b ject E b y itself. Sp ecial cyclic mo dules correspond to critical p oin ts belonging to N il p S C m,n ⊂ N il p m,n . Considering an ob ject M of length n as an upp er-tria ngular deformatio n of the “free” ob ject nE = E ⊕ · · · ⊕ E ( n -times) w e see that ( M , M ) 6 1 = ( nE , nE ) 6 1 + r , r := dim Im( W ( n ) ) ′′ ( X 1 ,...,X m ) . Then 1 − M F ( X 1 ,...,X m ) ( W ( n ) ) = (1 − M F ( E ))(1 − M F 0 ( Q E )) , 126 where Q E is a quadratic form a nd rk Q E = r . Indeed, W ( n ) coincides with the p oten tial W nE of the ob ject nE under the isomorphism Ext 1 ( nE , nE ) ≃ k mn 2 . Th us w e see that ( − 1) ( M ,M ) 6 1 = ( − 1) n 2 (1 − m )+ r . Since χ (1 − M F 0 ( Q E )) = ( − 1) rk Q E w e obtain the des ired formula for χ Φ ( F l ( t )). Alternativ ely , in the in tegral one can replace the quotien t N il p S C m,n /P GL ( n ) b y N il p cy cl m,n /GL (1) ( n ). Here N il p cy cl m,n ⊂ N il p m,n consists of collection of ma- trices suc h that k hh X 1 , . . . , X m ii v 1 = k n where v 1 := (1 , 0 , . . . , 0) is the ﬁrst base ve ctor, and the g roup GL (1) ( n ) ⊂ GL ( n ) is the stabiliz er of v 1 . Notice that GL (1) ( n ) acts freely on N il p cy cl m,n . The reason is that the con tribution of non-sp ecial cyclic ob jects v anishes as follo ws from the v anishing o f the Euler c haracteristic of corresp onding sc hemes of mo dules with c hosen cyclic generators. Conjecture 11 We have: χ Φ ( F l ( t )) = Y n > 1 (1 − t n ) n Ω( n ) , wher e al l Ω( n ) = Ω( n cl k ( E )) ar e i n te ger numb ers (se e Se ction 1.4). Let us illustrate the conjecture in few examples. 1) Let m = 0 (i.e. the case of just one spherical ob ject). Then W = 0 and B W = k . The only non-trivial cyclic represen tation hav e dimension one, hence χ ( F l ( t )) = 1 − t . Then we ha ve Ω(1) = 1 , Ω( n ) = 0 for n > 1 . 2) Let m = 1 , W ( n ) ( X 1 ) = T r ( X d 1 ) for d = 3 , 4 , . . . . Then B W = k [ x 1 ] / ( x d − 1 1 ). There is a unique isomorphism class o f cyclic B W -mo dules in an dimensi on n = 0 , 1 , 2 , . . . , d − 1. One can sho w directly that χ ( F l ( t )) = (1 − t ) d − 1 , Ω(1) = d − 1 , Ω( n ) = 0 for n > 1 . 3) Let m ≥ 1 b e arbitrary and W = 0. In this case χ ( F l ( t )) = 1 + X n > 1 ( − 1) n 2 (1 − m ) χ ( N il p cy cl m,n /GL (1) ( n )) t n . 127 Euler c haracteristic χ ( N il p cy cl m,n /GL (1) ( n )) coincides with the Euler c har- acteristic of the non-comm utative Hilb ert sc heme H ( m ) n, 1 from [59]. The latter parametrizes left ideals of co dimension n in the free a lgebra k h x 1 , . . . , x m i . The r eason wh y w e can disregard all non-nilp oten t collections ( X 1 , . . . , X m ) of matrices is that the latter carries a free action of the group G m , suc h that X i 7→ λ X i , 1 6 i 6 m whe re λ ∈ G m ( k ). Hence the corresp onding Euler c haracteristic is trivial. Then using explicit form ulas from [59] w e o btain G ( m ) ( t ) := χ ( F l ( t )) = X n > 0 ( − 1) n (1 − m ) ( m − 1) n + 1  mn n  t n . Notice that this series can b e written as exp X n > 1 ( − 1) n (1 − m ) mn  mn n  t n ! . F or m = 1 w e hav e Ω(1) = − 1 , Ω( n ) = 0 , n > 2. In general Ω( n ) = 1 mn 2 X d | n µ ( n/d )  md d  ( − 1) ( m − 1) d +1 , where µ ( k ) is the M¨ obius function (for m = 2 see the entry A 131 8 68 in the online Encyclop edia of in teger sequences ). Remark 29 One c an che ck that the gener ating function G = G ( m ) is a l g e- br a i c : it satisﬁes the e quation 26 G ( t ) + t ( − 1) m ( G ( t )) m − 1 = 0 . A n inter esting question arises: which algebr aic functions admit multiplic a- tive factorization of the form Q n > 1 (1 − t n ) n Ω( n ) , w h e r e al l Ω( n ) ar e inte ger numb ers? 8 Donaldson- Th o mas in v arian ts and clus ter transformations 8.1 Spherical collections and m utations Let C b e a 3- dimensional ind-constructib le Calabi-Y au category o ve r a ﬁeld k of characte ristic zero. Assume that it is endo we d with a ﬁnite collection of 26 Compare with the algebraic series in the Introduction, section 1.4 . 128 spherical generators E = { E i } i ∈ I of C deﬁned ov er k . Then Ext • C ( k ) ( E i , E i ) is isomorphic to H • ( S 3 , k ) , i ∈ I . The matrix of the Euler form (tak en with the min us sign) a ij := − χ  Ext • C ( k ) ( E i , E j )  is in teger and sk ew-symmetric. In fact, the ind-constructible category C can b e canonically reconstruc ted fro m the (plain, i.e. not ind-constructible) k -linear Calabi-Y au A ∞ -category C ( k ), or ev en from its full sub category con- sisting of the collection E . In what follows we will omit the subscript C ( k ) in the notation for Ext • -spaces. Deﬁnition 20 The c ol le c tion E is c al le d cluster if for any i 6 = j the gr ade d sp ac e ⊕ m ∈ Z Ext m ( E i , E j ) is either zer o, or it is c on c entr ate d in one of two de gr e es m = 1 or m = 2 only. W e will assume that our collection is cluster. In that case K 0 ( C ( k )) ≃ Z I with the basis fo rmed b y the isomorphism classes [ E i ] , i ∈ I . With the cluster collection E w e a sso ciate a quiv er Q suc h that Q do es not ha v e or iented cycles of lengths 1 and 2, and a ij > 0 is the n umber of arrows from i to j (we iden tify the set of arrows from i t o j with a basis in Ext 1 ( E i , E j )). Then the p oten tial for the ob ject E = ⊕ i ∈ I E i giv es rise to the po tential W = W Q of the quiv er Q , i.e. the restriction of the p oten tial t o ⊕ i,j ∈ I Ext 1 ( E i , E j ). The latter is an inﬁnite linear com bination of cyclic w ords (see [17], [77] where the p oten tial app ears abstractly without the relation with Calabi-Y au categories). Any suc h linear comb ination is called a p oten tial of Q . In our case the p oten tial is automatically minimal, i.e. all words ha v e length at least 3. T he group of con tin uous automorphisms of the completed path algebra o f Q preserving t he pro jectors pr i , i ∈ I , a cts on the set of p oten tials of Q . W e call it the gauge action. Let us state t he follo wing general result. Theorem 9 L et C b e a 3-dimensi onal k -line ar Cala b i - Y a u c ate gory gener- ate d by a ﬁnite c ol le ction E = { E i } i ∈ I of gener ators satisfying the c ondition that • Ext 0 ( E i , E i ) = k id E i , • Ext 0 ( E i , E j ) = 0 for any i 6 = j , • Ext < 0 ( E i , E j ) = 0 , for any i, j . 129 The e quivalenc e classes o f such c ate gories with r esp e ct to A ∞ -tr ansf o rmations pr e s e rving the C a l a bi-Y au structur e and the c ol le ction E , ar e in one-to-one c o rr esp ondenc e with the gauge e quivalenc e classes of p airs ( Q, W ) wher e Q is a ﬁnite oriente d quiver (p ossibly with cycles of length 1 or 2 ) and W is a m inimal p otential of Q (i.e. its T aylor de c o mp osition starts with terms o f de gr e e at le ast 3 ). The case of cluster collections corresponds to quiv ers without orien ted cycles of length 1 and 2. Pr o of. W e will presen t the proof of the Theorem in the case of the category with single ob ject E (i.e. A = Hom( E , E ) is a 3 d Calabi-Y au algebra). The general case can b e pro v ed in a similar wa y . Let Q b e a quiv er with one vertex and | J | lo ops, where J is a ﬁnite set. W e assume that Q is endo w ed with the po ten tial W 0 . W e would lik e to construct a 3 d Calabi-Y au category with a single o b ject E suc h that the nu mber of lo ops in Q is equal to Ext 1 ( E , E ) and the restriction o f the p oten tial of the category to Ext 1 ( E , E ) coincides with the giv en W 0 . Our considerations pro ceed suc h as follo ws. Assuming that suc h a category exists w e will ﬁnd an explicit f orm ula for the p oten tial on A = Hom( E , E ). Then w e simply take this explicit form ula as the deﬁnition. If the desired category is constructed then w e can consider the gra ded v ector spac e Ext • ( E , E )[1] whic h decomposes as Ext 0 ( E , E )[1] ⊕ Ext 1 ( E , E ) ⊕ Ext 2 ( E , E )[ − 1] ⊕ Ext 3 ( E , E )[ − 2] . The ﬁrst and the last summand are isomorphic to k [1] and k [ − 2] resp ectiv ely , and the middle t wo summands a re dual t wo each other. W e introduce graded co ordinates o n Ext • ( E , E )[1] and denote them suc h as follo ws: a) the co ordinate α of degree +1 on Ext 0 ( E , E )[1]; b) the co ordinate a of degree − 2 on Ext 3 ( E , E )[ − 2]; c) the co ordinates x i , i ∈ J of degree 0 on Ext 1 ( E , E ); d) the co ordinates ξ i , i ∈ J of degree − 1 on Ext 2 ( E , E )[ − 1]. The Calabi-Y au structure on A giv es rise to the minimal p oten tial W = W ( α , x i , ξ i , a ), whic h is a series in cyclic w ords on the space Ext • ( E , E )[1]. If it arises from the pair ( Q, W 0 ), then the restriction W (0 , x i , 0 , 0) mus t coincide with W 0 = W 0 ( x i ). F urthermore, A deﬁnes a non-comm utativ e formal p ointed graded manifold endo w ed with a symplectic structure (see [42]). The potential W satisﬁes the “class ical BV e quation” { W , W } = 0, where {• , •} denotes the correspo nding Pois son brac ke t. 130 With these preliminary considerations we see what problem should b e solv ed. W e need to construct an extension of W 0 to the formal series W of degree 0 in cyclic words on the g ra ded v ector space k [1] ⊕ k J ⊕ k J [ − 1] ⊕ k [ − 2], satisfying the classic al BV-equation with resp ect to the P oisson brac k et { W , W } = X i ∂ W /∂ x i ∂ W /∂ ξ i + ∂ W /∂ α ∂ W /∂ a . Here is the construction. Let us start with the p o ten tial W can = α 2 a + X i ∈ J ( αx i ξ i − α ξ i x i ) . This p oten tial mak es the ab o v e g raded v ector space in to a 3 d Calabi-Y au algebra with a sso ciativ e pro duct a nd the unit. The multip lication v a nishes on the graded comp onen ts Ext 1 ( E , E ) ⊗ Ext 1 ( E , E ) → Ext 2 ( E , E ) and is a non-degenerate bilinear form on comp onen ts Ext 1 ( E , E ) ⊗ Ext 2 ( E , E ) → Ext 3 ( E , E ) ≃ k . No w w e s ee that starting with an arbitrary minimal p otenti al W 0 on Ext 1 ( E , E ) w e can lift it to the minimal potential on Ext • ( E , E ) b y setting c W := W can + W 0 . W e claim that { c W , c W } = 0. Indeed , we hav e { W can , W can } = { W 0 , W 0 } = 0. Moreo v er, { W can , W 0 } = α X i ∈ J [ x i , ∂ W 0 /∂ x i ] = 0 (w e use here t he w ell-kno wn iden tity P i ∈ J [ x i , ∂ W 0 /∂ x i ] = 0). Next w e need to c hec k compatibilit y of the ab o v e construction with the gauge group action. Let G 0 b e the subgroup of the grading preserving auto- morphisms of the group of con tin uous automorphisms of the algebra of formal series k hh α, x i , ξ i , a ii , i ∈ J . Let J ⊂ k hh α , x i , ξ i , a ii b e a closed 2-sided ideal generated b y α, a and ξ i for i ∈ J . Since ev ery generator of J has non-zero degree w e conclude that the group G 0 preserv es J (it can b e deduced from the fact that it transforms generators into series of non-zero degrees). There- fore we obtain a homomorphism of groups G 0 → Aut( k hh x i ii ) , i ∈ J . The 131 restriction of the potential W to Ext 1 ( E , E ) deﬁn es a surjection from the set of A ∞ -equiv alence classes of 3 d Calabi-Y au algebras to the g auge equiv- alence classes of ( Q, W 0 ), where Q is a quiv er with one vertex endow ed with the minimal p oten tial W 0 . Suc h algebras can b e though t of as deformations of the “ansatz”, whic h is a 3 d Calabi-Y au algebra A can correspo nding to the p oten tial W can . Finally we are going to sho w that the ab ov e surjection is in fact a bijection. The latter will follow from the equiv a lence o f the corresp onding deformation theories. The deformation theory of the Calabi-Y au algebra A can is con trolled b y a DG LA g A can = ⊕ n ∈ Z g n A can , whic h is a DG Lie subalgebra of the DG Lie algebra [ g A can = Y n > 1 C y cl n (( A [1]) ∗ ) ! [ − 1] of a ll cyclic series in the v ariables α, x i , ξ i , a, i ∈ J (the Lie brac k et is giv en b y the P oisson brac k et and the diﬀeren tial is giv en by { W can , •} ). Namely , the comp onen t of g A can of degree N consists of all cyclic series whic h con tain at least 2 + N letters α , x i , ξ i , a, i ∈ J . W e will call the degree deﬁned in terms of these letter a cyclic de gr e e in order to distinguish it f rom the c ohomolo gic a l de gr e e of complexes . Notice that the set of A ∞ -equiv alence classes of minimal 3 d C alabi-Y au agebras can b e iden tiﬁed with the s et of gauge equiv a lence classes of solutions γ ∈ g 1 A can to the Maurer-Cartan equation dγ + 1 2 [ γ , γ ] = 0 . Similarly , the set of ga uge equiv a lence classes of minimal p oten tials on A 1 = Ext 1 ( E , E ) can b e iden tiﬁed with the set of gauge equiv a lence classes of solutions to the Maurer-Cartan equation in the DGLA h = h 0 ⊕ h 1 , where h 0 = Y n > 1 (( A 1 ) ∗ ) ⊗ n ⊗ A 1 , h 1 = Y n > 3 C y cl n (( A 1 ) ∗ ) . Here w e iden tify h 0 with the Lie algebra of con tin uous deriv ations of the top o - logical algebra k hh x 1 , . . . , x n ii prese rving the augmen tation ide al ( x 1 , . . . , x n ), and w e iden tify h 1 the h 0 -mo dule of minimal cyclic p otentials o n A 1 . The ab ov e construction of the “lifting” c W = W 0 + W can can b e in terpreted as a homomorphism of DGLAs ψ : h → g A can . Namely , h 0 is iden tiﬁed (after the shift [1]) with the space of such cyclic series in x i , ξ i , i ∈ J whic h contain 132 exactly one of the v ariables ξ i and at least one of the v a r ia bles x j for some i, j ∈ J . Similarly h 1 is iden tiﬁed with the space of cyclic series in x i , i ∈ J whic h has terms o f degree at least 3. W e claim that ψ induces an epimorphism (previous considerations ensure that it is a monomorphism) of cohomology groups in degree 1, and for b oth DGLAs h and g A can there is no cohomology in degree greater or equal than 2. This w ould imply the desired surjectivit y of ψ . Notice that the diﬀeren tial { W can , •} preserv es the diﬀerence b et w een cyclic and cohomological degree. It follows that the complex g A can is a direct summand of the complex [ g A can . The latter is dual to the cyclic complex C C • ( A can ). Let A + can ⊂ A can b e a non-unital A ∞ -subalgebra cons isting of terms of p ositiv e cohomological degree. Then, one has for the cyclic ho- mology: H C • ( A can ) ≃ H C • ( A + can ) ⊕ H C • ( k ). In terms of the dual complex this isomorphism means the decomposition into a direct sum of the space of cyclic series in v aria bles x i , ξ i , a, i ∈ J (corresponds to ( H C • ( A + can )) ∗ ) and the space of cyclic series in the v ariable α of o dd cyclic degree (corresp onds to ( H C • ( k ) ∗ ). It is easy to see that the series in the v ariable α do not con tribute to the cohomology of g A can ⊂ [ g A can . The cohomological degree o f series in v ariables x i , ξ i , a, i ∈ I is non- p ositiv e. Recall that w e shifted the grading in Lie algebras by 1 with respect to the cohomological grading. Hence H > 2 ( g A can ) = 0. Also, it is immediate that H 1 ( g A can ) is isomorphic to the space of cyclic series in the v ariables x i , i ∈ I with terms of degree at least 3. Hence H 1 ( g A can ) ≃ h 1 ≃ H 1 ( h ) (the latter holds since the diﬀeren tial on h is trivial). This concludes the pro of .  Next, w e will in tro duce the notion o f a mutation on the set of cluster collections in a give n category C . Let us c ho o se an elemen t of I whic h we will denote b y 0. W e are going to write i < 0 if a i 0 > 0, and i > 0 if i 6 = 0 and a i 0 6 0. The m utation of E a t the o b ject E 0 is deﬁned as a new spherical collection E ′ = ( E ′ i ) i ∈ I suc h that: E ′ i = E i , i < 0 , E ′ 0 = E 0 [ − 1] , E ′ i = R E 0 ( E i ) , i > 0 . where R E 0 ( E i ) := C one ( E 0 ⊗ Ext • ( E 0 , E i ) → E i ) is the reﬂec tion functor giv en b y the cone of the natural ev aluation map. Explici tly , the ob ject E ′ i 133 for i > 0 ﬁts in the exact triangle E i → E ′ i → E 0 ⊗ Ext 1 ( E 0 , E i ) . Notice that a ll ob jects E ′ i , i 6 = 0 belong to the ab elian category generated b y E i , i ∈ I . W e remark that the spherical collection E ′ is not necessarily a cluster one. A t the lev el of the lattice Γ := Z I the change of the spherical col- lections E → E ′ correspo nds to the follow ing relation b et we en the basis v i := cl k ( E i ) , i ∈ I and the mutated basis v ′ i = cl k ( E ′ i ) , i ∈ I : v ′ i = v i , i < 0 , v ′ 0 = − v 0 , v ′ i = v i − h v 0 , v i i v 0 = v i + a 0 i v 0 , i > 0 . W e recall that a ij = −h v i , v j i . The m utated matrix ( a ′ ij ) is giv en b y a ′ ij = a ij + a i 0 a 0 j if i < 0 < j, a ′ i 0 = − a i 0 , a ′ 0 i = − a 0 i , a ′ ij = a ij , o t herwise . Th us w e see that the m utation at E 0 giv es rise to the m utation o f the matr ix ( a ij ) in the sense of cluster algebras (see [7 7]). Notice that at the categorical lev el the m utation is not an in v olution. The composition of the mutation at E 0 and o f the m utation at E ′ 0 = E 0 [ − 1] is the reﬂection functor R E 0 applied to all eleme nts o f the cluster collection. Iden tifying C alabi- Y au categories endo we d with cluster collections with quiv ers with p oten tials w e obtain the w ell-kno wn notion of mutation of a quiv er with p oten tial (see [77]). Then we ha v e the follo wing result. Theorem 10 In the sch e me (an inﬁnite-dimensio nal aﬃne sp ac e) o f p oten- tials P T ther e is a c ountable set of alg e br a i c hyp ersurfac es X i , i > 1 invariant under the gauge gr oup action, such that for any p otential b elonging to the se t P T \ ∪ i > 1 X i one c an make mutations indeﬁnitely, obtaining e ach time a p o- tential fr om P T \ ∪ i > 1 X i . In p articular, al l c orr esp onding quivers do not have oriente d cycles of length one or two. Sketch of the pr o of. The m utated spherical collection fails to b e cluster if for some i 6 = j w e hav e sim ultaneously Ext 1 ( E ′ i , E ′ j ) 6 = 0 and Ext 2 ( E ′ i , E ′ j ) 6 = 134 0. This prop ert y is not stable under deformations of 3- dimensional Calabi- Y au A ∞ -category , since w e can add a quadratic term to the po ten tial W E ′ i ⊕ E ′ j reducing the dimension of Ext 1 ( E ′ i , E ′ j ) and Ext 2 ( E ′ i , E ′ j ). Therefore, the prop ert y that the m utated collection is also a cluster one holds on a Zariski op en non- empt y subset of the space of all p oten tials. Moreo v er, the mu - tation induces a birational iden tiﬁcation betw een v arieties (may b e inﬁnite- dimensional) of ga uge equiv alence classes of generic p ot entials for quiv ers correspo nding to sk ew-symmetric matrices ( a ij ) and ( a ′ ij ).  An y cluster collection E = { E i } i ∈ I deﬁnes an op en domain U E ⊂ S tab ( C , cl), where Γ = K 0 ( C ( k )) , cl = id . Name ly , for an y collection z i ∈ C , Im z i > 0 , i ∈ I w e ha v e a stabilit y condi tion σ ( z i ) := σ ( z i ) i ∈ I with the t -structure deﬁned b y ( E i ) i ∈ I and the cen tral c harge Z suc h that Z ( E i ) := Z (cl( E i )) = z i , i ∈ I . The heart of the t -structure is an ab elian category A E generated by ( E i ) i ∈ I , whic h is artinian with simple ob j ects E i , i ∈ I . This ab elian category is equiv alen t to the category of con tin uous ﬁnite-dimens ional represen tations of the algebra B W where W is the p oten tial of the path algebra of the quiv er Q . If E ′ is a cluster collection obtained from E b y the m utation at E 0 then the domains U E and U E ′ do not hav e common interior p oints, but hav e a com- mon part of the b oundary whic h is the w all of second kind. The common b oundary corresp onds to the stabilit y structure with Z ( E 0 ) ∈ R < 0 . r ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ② ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ③ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❖ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✍ ✻ Z ( E 0 ) Z ( E 0 [ − 1]) Z ( E i ) , i 6 = 0 135 Category A E ′ is obtained from A E b y tilting. Namely , an y ob ject M of A E admits a unique presen tation as a n extension 0 → nE 0 → M → N → 0 where N ∈ B := { E ∈ A E | Hom( E , E 0 ) = 0 } . Similarly , any ob ject M ′ of A E ′ admits a unique presen tation as an extension 0 → N → M ′ → nE 0 [ − 1] → 0 with N ∈ B . 8.2 Orien tatio n data for cluster collections Let E = ( E i ) i ∈ I b e a cluster collection. W e set R := R E = Ext • ( E , E ), where E = ⊕ i ∈ I E i . Then R is an A ∞ -algebra. W e denote b y M := M E the algebra R considered as R -bimo dule. Using the truncation functors τ 6 i and τ > i w e deﬁne a sub-bimo dule M > 2 = τ > 2 M as w ell a s a quotien t bimo dule M / M > 2 , whic h is isomorphic t o M 6 1 = τ 6 1 M . Then w e can deform the extension M > 2 → M → M 6 1 in to the direct sum of bimo dules M > 2 ⊕ M 6 1 . Moreo v er, one can c hec k that there exists a deformation whic h consists of self-dual bimo dules (i.e. they give rise to self-dual functors in the sense of Section 5.3 ) . Thu s w e w ould lik e to deﬁne an o r ientation data using the splitting giv en by t he bifunctor F whic h corresp onds to the bimo dule M 6 1 , i.e. ( E i , E j ) 7→ τ 6 1 Ext • ( E i , E j ). Let E ′ = ( E ′ i ) i ∈ I b e the cluster colle ction obtained b y a m utation at i = 0 . One can c hec k directly that Z / 2 Z -v alued quadratic form deﬁned on K 0 ( C ( k )) by [ E ] = X I ∈ I n i [ E i ] 7→ X i ∈ I n 2 i − X i,j ∈ I ,a ij > 0 a ij n i n j mo d 2 Z is in v arian t under mutations . This means that the parity of the sup er line bundle √ D F = sdet( τ 6 1 Ext • ( F , F ) ) is preserv ed under mutations. This mak es plausible the follo wing conjecture. Conjecture 12 B i f unc tors M E 6 1 and M E ′ 6 1 deﬁne isom o rphic orientation d ata on C . In o rder t o c hec k the conjecture one needs to ﬁnd a self-dual A 1 -deformation of M E 6 1 ⊕ ( M E ′ 6 1 ) ∨ to a bifunctor o f the t yp e N ⊕ N ∨ (w e identify bifunctors with bimo dules here). 136 8.3 Quan tum DT-in v arian ts for quiv ers F or any σ ∈ U E (recall domain U E in tro duced at the end of Section 8.1) w e ha v e the corresp onding eleme nt A Hall V , where V is an y strict sec tor containi ng all Z ( E i ) , i ∈ I . The elemen t A Hall V do es not de p end on σ . Moreo v er, this elemen t dep ends only on the gauge equiv alence class of the corresponding p oten tial. The asso ciated elem ent A V , q := A E ,q of the quan tum torus R V , q dep ends (f o r a generic p otential) on the matrix ( a ij ) only . Let us asso ciate with our quiv er Q the quantum torus R Q,q . By deﬁnition it is an asso ciative unital a lgebra ov er the ﬁeld Q ( q 1 / 2 ) of rational functions, with inv ertible generators ˆ e ± 1 i , i ∈ I sub ject to the relations ˆ e i ˆ e j = q a j i ˆ e j ˆ e i . W e a re going to use its double D ( R Q,q ), whic h is generated by R Q,q , new set of generators ˆ e ∨ i , i ∈ I sub ject to the additional set of relations: ˆ e ∨ i ˆ e ∨ j = ˆ e ∨ j ˆ e ∨ i , ˆ e ∨ i ˆ e j = q − δ ij ˆ e j ˆ e ∨ i , i, j ∈ I . The corresp onding quasi-classical limits are P oisson tori which w e will denote b y T Q and D ( T Q ) respectiv ely . Iden tifying R Γ ,q with R Q,q in the ob vious w a y w e obtain an eleme nt E Q = 1 + · · · ∈ [ R Q,q correspo nding to A E ,q . W e observ e that E Q is a series in non-negative p o we rs of ˆ e i , i ∈ I . Conjugation with E Q giv es rise to an a utomorphism of the quan tum torus D ( R Q,q ). By t he “absence of p o les” conjecture it do es not hav e p oles at q n = 1 , n > 1. In particular it deﬁnes a formal s ymplectomorphism of the double torus D ( T Q ) (see Section 2.6, with the notation b ij := − a ij ). 8.4 Quiv ers and cluster transformations The formal p ow er series E Q in ˆ e i , i ∈ I deﬁned in the previous section satisfy a nu mber of remark able prop erties. 1) If | I | = 1 then Q is a quiv er with one v ertex i . W e hav e E Q = E ( ˆ e i ) , 137 where E is the quan tum dilogarithm function. 2) Let I = I 1 ⊔ I 2 , a nd w e assume that a i 1 ,i 2 < 0 for an y i 1 ∈ I 1 , i 2 ∈ I 2 . Then we ha v e tw o sub quiv ers Q 1 and Q 2 of Q with the sets of v ertices I 1 and I 2 correspo ndingly , a nd all the arrow s connecting Q 1 and Q 2 go only in the direction from Q 2 to Q 1 (i.e. there is no arro ws f r o m Q 1 to Q 2 ). Prop osition 16 One has: E Q = E Q 1 E Q 2 wher e we emb e d R Q j ,q , j = 1 , 2 into R Q,q in the obvious w ay: ˆ e i 7→ ˆ e i for i ∈ I 1 or i ∈ I 2 . Pr o of. Consider the stabilit y condition σ ∈ U E on the Calabi-Y au category C Q asso ciated with Q and a generic p oten tial. Let E = { E i } i ∈ I b e the correspo nding cluster collection. W e c ho o se a stability condition σ ∈ U E in suc h a w ay that Arg( E i 1 ) > Arg ( E i 2 ) for i 1 ∈ I 1 , i 2 ∈ I 2 . In this case C ss Q = C ss Q 1 ⊔ C ss Q 2 . This implies the desi red iden tity .  Remark 30 It fol low s fr om the Pr op erties 1) and 2) that for any acyclic quiver Q the elemen t E Q c a n b e expr esse d as the pr o duct of E ( ˆ e i ) , i ∈ I . In p a rticular, the c onjugation by E Q has a wel l - d eﬁne d quasi-cl a ssic al limit as q 1 / 2 → − 1 , which is a b i r ational symple ctomorphism of the torus D ( T Q ) . 3) Let Q ′ b e the quiv er obtained f rom Q b y the m utation at 0 ∈ I . W e denote the standard generators of the corresp onding quan tum tori b y ( ˆ e ′ i ) i ∈ I , ˆ e ′ i = ˆ e cl k ( E ′ i ) and ( ˆ e i ) i ∈ I , ˆ e i = ˆ e cl k ( E i ) respectiv ely . Let us introduce the elemen ts R Q = E ( ˆ e 0 ) − 1 · E Q , R Q ′ = E Q ′ · E ( ˆ e ′ 0 ) − 1 . Here R Q is a series in v aria bles ˆ e i for i < 0, and in (dep enden t) v ariables ˆ e i , ˆ e i ˆ e 0 , . . . , ˆ e i ˆ e a 0 i 0 for j > 0. Similarly , R Q ′ is a series in v ariables ˆ e ′ i for i < 0 and ˆ e ′ i , ˆ e ′ i ˆ e ′ 0 . . . , ˆ e ′ i ( ˆ e ′ 0 ) a 0 i for i > 0. Then R Q = R Q ′ under the iden tiﬁcation ˆ e ′ i = ˆ e i , i < 0 , ˆ e ′ 0 = ˆ e − 1 0 , ˆ e ′ i = q − 1 2 a 2 0 i ˆ e i ˆ e a 0 i 0 , i > 0 . 138 This follo ws from the ab ov e-discussed picture of tilting via the w all-crossing, more precisely , from the formul a E ( ˆ e cl k ( E 0 ) ) − 1 A E ,q = A E ′ ,q E ( ˆ e − cl k ( E 0 ) ) − 1 . Elemen t R Q = R Q ′ correspo nds to the in tegral ov er the space of ob jects of category B in notation at the end of Section 8 .1 . F or the conv enience of the r eader w e giv e also the form ulas comparing dual co ordinates on the double quantum torus: ˆ e ∨ i ′ = ˆ e ∨ i , ∀ i 6 = 0 , ˆ e ∨ 0 ′ = ( ˆ e ∨ 0 ) − 1 · Q i> 0 ( ˆ e ∨ i ) a 0 i . Let us now consider the minimal class P of o rien ted ﬁnite quiv ers whic h satisﬁes the follo wing properties: a) the trivial quiv er (one vertex no arro ws) b elongs to P ; b) class P is closed under m utations; c) if Q 1 , Q 2 ∈ P then a quiv er Q obtained f r o m the disjoin t union of Q 1 and Q 2 b y inserting a ﬁn ite num ber of arrow s from Q 2 to Q 1 (without c hanging an ything else for Q 1 and Q 2 ) also b elongs to P . W e will sa y in this case that Q is an extens ion of Q 1 b y Q 2 . At the lev el of categories this means that a ny ob j ect J of the category A ( E ) generated b y E i ∈ E , i ∈ I is an extension F 1 → J → F 2 where F 1 (resp. F 2 ) is an ob ject of the ab elian category generated b y E i , i ∈ I 1 (resp. E i , i ∈ I 2 ). This class P enjo ys the prop erty that the gauge group asso ciated w ith Q ∈ P when acting on the space of po tentials on Q has one op en o rbit (this can b e show n by induction), hence the corresp o nding 3-dimensional Calabi- Y au category is rigid . Moreo v er for a ny Q ∈ P the elemen t E Q is a ﬁnite pro duct of the elemen ts E ( f ), w here f = ˆ e γ is a monomial. In particular, the conjugation with E Q has a quasi-classical limit as q 1 / 2 → − 1, whic h is a birational transformation. One of the ﬁrst non trivial examples of a quiv er Q whic h is not in the class P is the quiv er Q 3 whic h has three v ertices and t wo parallel arro ws b et we en an y t wo ve rtices (see the Figure). This quiv er is stable under mutations. The elemen t E Q 3 satisﬁes an o v erdetermined system of equations. The computer c hec k sho ws that the conjugation with E Q 3 has the quasi-classical limit whic h is not rational. It is not clear whether it admits an analytic con tinuation. 139 r r ✛ ✛ ✡ ✡ ✡ ✡ ✡ ✡ ✣ ✡ ✡ ✡ ✡ ✡ ✡ ✣ ❏ ❏ ❏ ❏ ❏ ❏ ❫ ❏ ❏ ❏ ❏ ❏ ❏ ❫ r The m utation prop erty of Q 3 has the follo wing explicit corollary . Namely , there exist collections c i,j,k , b m 1 ,m 2 ,n ∈ Q ( q 1 / 2 ) , i, j, k ∈ Z > 0 , m 1 , n > 0 , − m 1 6 m 2 6 m 1 suc h that the follo wing system of equations is satisﬁed: c 0 , 0 , 0 = b 0 , 0 , 0 = 1 , c i,j,k = c j,k ,i = c k ,i,j , c n 0 ,n 1 ,n 2 = X l > 0 ε l q l ( n 2 − n 1 ) b n 1 ,n 0 − l − n 1 ,n 2 , c n 0 ,n 1 ,n 2 = X l > 0 ε l q l ( n 2 − n 0 ) b n 0 ,n 0 + l − n 1 ,n 2 , where ε l = q l 2 / 2 ( q l − 1 ) . . . ( q l − q l − 1 ) are coeﬃcien ts o f the series E . T o ha v e a solution of this syste m of equations is the same as to write the elemen t E Q = X i,j,k c i,j,k ˆ e ( i,j,k ) , where w e iden tiﬁed Γ with Z 3 . The ab ov e system of equations follows from the ide ntit y R Q 3 = R Q ′ 3 since Q 3 = Q ′ 3 after the m utation. The elemen ts b m 1 ,m 2 ,n are deriv ed from c i,j,k . Notice that the ab ov e system of equations has a solution whic h is not unique. Therefore the elemen t E Q is determined non-uniquely , but only up to a m ultiplication by a series of the t yp e 1 + X n > 1 a n ˆ e n 1 , 1 , 1 , a n ∈ Q ( q 1 / 2 ) 140 whic h b elongs to the cen ter of the quan tum to rus R Q 3 ,q . Let as b efore E = ( E i ) i ∈ I b e a cluster collection in C suc h that the cor- respo nding p oten tial is generic. W e make an a dditional a ssumption that the conjugation Ad A E ,q : x 7→ A E ,q xA − 1 E ,q is a birational transformation of the double quantum torus R Γ ⊕ Γ ∨ ,q ≃ D ( R Q,q ). This means that it is an auto - morphism of the ( well-deﬁ ned) sk ew ﬁeld K Γ ⊕ Γ ∨ ,q of fractions of this quantu m torus. In t he equiv a lent language of quiv ers it suﬃces to require that Q ∈ P . Let us denote by Φ E the automorphism of K Γ ⊕ Γ ∨ ,q giv en b y Φ E ( x ) = (Ad − 1 A E ,q ◦ τ )( x ) , where τ is the inv olution induced by the an tip o dal in v olution γ 7→ − γ of Γ ⊕ Γ ∨ . Prop osition 17 If E ′ = ( E ′ i ) i ∈ I is the cluster c ol le ction obtaine d by the mu- tation at E 0 then Ad − 1 E ( ˆ e cl k ( E 0 ) ) ◦ Φ E ◦ Ad E ( ˆ e cl k ( E 0 ) ) = Φ E ′ . Pr o of. F rom the kno wn iden tit y Ad − 1 E ( ˆ e cl k ( E 0 ) ) ◦ Ad A E ,q = Ad A E ′ ,q ◦ Ad − 1 E ( ˆ e − cl k ( E 0 ) ) w e obtain the desired one b y m ultiplying it from the righ t b y τ ◦ Ad E ( ˆ e cl k ( E 0 ) ) .  No w w e can state a similar result for a quiv er Q whic h satisﬁes the condi- tion that Ad E Q is a birational transformation of the sk ew ﬁeld K Q of fractions of the double quan tum to rus D ( R Q,q ). L et us deﬁne Φ Q := Ad − 1 E Q ◦ τ where τ is the obv ious in v olution: τ ( ˆ e i ) = ˆ e − 1 i , τ ( ˆ e ∨ i ) = ( ˆ e ∨ i ) − 1 . Let Q ′ b e the quiv er o btained as a mutation o f Q at the v ertex 0 ∈ I . Then w e hav e t he follo wing corollary of the ab o v e Prop osition. Corollary 4 L et us deﬁne the m a p C Q, 0 : K Q,q → K Q ′ ,q as the c omp osition K Q,q → K Γ ⊕ Γ ∨ ,q → K Γ ⊕ Γ ∨ ,q → K Q ′ ,q , wher e the midd le arr ow is the automorphi s m Ad − 1 E ( ˆ e cl k ( E 0 ) ) while the other m aps ar e o b v ious isomorph isms of skew ﬁelds. Then C Q, 0 ◦ Φ Q = Φ Q ′ ◦ C Q, 0 . 141 Pr o of. It is just a reform ulation of the previous Prop osition in the lan- guage of quiv ers.  Let us compute C Q, 0 ( ˆ e i ), where ˆ e i = ˆ e cl k ( E i ) , i ∈ I , as we ll as C Q, 0 ( ˆ e ∨ i ) , i ∈ I . W e hav e to compute the action of Ad − 1 E ( ˆ e cl k ( E 0 ) ) on t hese generators. Th us w e o btain ˆ e 0 7→ ( ˆ e ′ 0 ) − 1 , ˆ e i 7→ ˆ e ′ i · Q 0 6 n 6 a i 0 − 1 (1 + q n +1 / 2 ( ˆ e ′ 0 ) − 1 ) − 1 , i < 0 , ˆ e i 7→ ˆ e ′ i · Q 0 6 n 6 a i 0 − 1 (1 + q n +1 / 2 ˆ e ′ 0 ) , i > 0 . Similarly w e obtain that ˆ e ∨ i 7→ ˆ e ∨ i ′ , i 6 = 0 , ˆ e ∨ 0 7→ ( ˆ e ∨ 0 ′ ) − 1 · Q i> 0 ( ˆ e ∨ i ′ ) a 0 i · (1 + q 1 / 2 ( ˆ e ′ 0 ) − 1 ) − 1 . Under quasi-classical limit the generators ˆ e i , i ∈ I go to the co ordinates y i , i ∈ I and ˆ e ∨ i go to the co ordinates x i , i ∈ I o f the symplectic double tor us (see Section 2.6). Then in those co o rdinates w e obtain y i 7→ y ′ i (1 − 1 /y ′ 0 ) a i 0 , i < 0 , y 0 7→ ( y ′ 0 ) − 1 , y i 7→ y ′ i (1 − y ′ 0 ) a 0 i , i > 0 . F or the dual coor dinates w e ha v e: x i 7→ x ′ i , i 6 = 0 , x 0 7→ ( x ′ 0 ) − 1 · Q i> 0 ( x ′ i ) a 0 i · (1 − 1 /y ′ 0 ) − 1 . Up to a ch ange of sign these are cluster transformations. Namely , if w e set X i = − y i , X ′ i = − y ′ i , A i = 1 /x i , i ∈ I then our formulas b ecome form ulas (17) and (18) from [23] (in the notation from lo c. cit). Remark 31 L et us r e c al l the variety N fr om Se c tion 2.6 d e ﬁne d by the e qua- tions N = { y i = − Q j ∈ I x a ij j , i ∈ I } , a nd let N ′ b e a sim ilar variety deﬁne d for the tr a n sforme d c o or dinates x ′ i , y ′ i , i ∈ I . O ne c an che ck that the quasi- classic al limit of C Q, 0 tr ansfo rms N into N ′ . F urthermor e, the quasi-classic al limit of the a utomorp h ism Φ Q pr e s e rves N . Remark 32 1) L et us assume that Ad A E ,q is bir ational (e.g. for Q ∈ P ).The ab ove c onsider ations show that the c onjugacy cla s s of the element Φ Q is a n 142 invariant of the q uiver Q unde r mutations. Passi n g to quasi-classic al limi t we obtain an invariant of a quiver (under mutations) which is a c onjugacy class in the gr oup of bir ational tr ansform a tions of the classic al d o uble torus. 2) The c ate goric al vers i o n of the ab ove r emark hold s in a g r e ater gener ality. 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