Joyce invariants for K3 surfaces and mock theta functions

Jo yce in v arian ts for K3 surfaces and mo c k theta functions An ton Mellit and So Ok ada ∗ No v em b er 6, 2021 Abstract W e will discuss Joyce inv ariants of stability conditions for K3 surfaces and mo ck theta functions. 1 In tro duction Bridgeland in tro duced the notion of stabilit y conditions on triangula ted cate- gories [Br07], this notion extends standard stabilities suc h as Giesek er stabilities on the a b elian category of co herent sheav es of a v ariety X , denoted b y Coh X , to the b ounded derived categ ory o f Coh X , deno ted by D( X ). One wa y to think of the notion is that it is a to ol to mak e in teresting in- v ariant s of mo duli sta cks, as we hav e seen in the foundationa l work [HaNa], in which the notion of H ar der-Nar asimhan filtr ations , in today’s ter m, was given birth to discuss T amagawa numb ers that a re certain volumes o f mo duli spaces on curves. W e would lik e to recall that for D-branes in s upers tring theory , Douglas’s work [Do] on Π-stabilities motiv ated the notion of stability conditions. In a Calabi-Y au v ariet y X , our str ings form Riemann s urfaces whos e b oundaries restrict to subv a rieties ca lled B-br anes , that ar e kinds of D-branes. With Kont- sevich’s framework [Ko], in D( X ), the n otion of Π-stabilities discusses config ura- tion of B- branes a nd its deforma tion, which is lo cally parameterize d by c entr al char ges of B-branes . In this term, we ar e taking inv aria n ts out of B-br anes whose central charges a lig n in the complex plane. Now, we b egin to b e mo re sp ecific for our pap er, leaving formality a bit out for later se c tio ns. F or stability conditions o f triangula ted categorie s , Joyce started to extend Donalds o n-Thomas inv ar iant s so that wall-crossings of s tabil- it y conditions g ive differential equations ov er his inv ariants, which we ca ll Joyce inv ariants. A c omm utative Q algebr a Λ c on taining l a nd a motivic invariant I fr om the category o f Artin stacks of finite t yp e to Λ sa tisfy the following: for I ( C ) = l , we ∗ Emails: m ellit@ihes.fr and ok ada@ihes.fr, Addresses: IH ´ ES, Le Bois -Marie, 35, route de Chartres, F-91440, Bures- sur-Yvet te, F r ance. 1 hav e I (GL( n, C )) = l n 2 (1 − l − 1 ) · · · (1 − l − n ) inv ertible in Λ, fo r quasipro jective v arieties X and Y , we have I ( X × Y ) = I ( X ) I ( Y ), for a c lo sed quasipr o jective v ariet y Y in X , we hav e I ( X ) = I ( Y ) + I ( X/ Y ), and for a quotient stack [ X/G ] with a sp ecial algebr aic group G , which is a group embedded in so me GL( n, C ) with GL( n, C ) → GL( n, C ) /G having loca lly trivia l fibe rs, we hav e I ([ X/G ]) = I ( X ) /I ( G ). F or example, so me mo tivic inv a riant ex tends the r ing structures of Poincar´ e or Ho dge p olynomials on the catego r y of smo oth pro jective v arieties to our cate- gory . Generally , each motivic inv arian t factors through the ring of isomorphis m classes of ab ov e quo tien t stacks [ X/ G ] [Jo 07b]. F rom here , we will a ssume tha t X denotes an algebraic K3 sur face X , a nd, in the stabilit y manifold of stability conditions on D( X ), Sta b ∗ ( X ) denotes the connected comp onent co nstructed by Bridg eland [B r 08]. F or a stability condition σ o f Gieseker o n Coh X or of B r idgeland on D( X ) a nd Mukai ve ctors α in the Mukai lattic e o f X [Mu], which is a nondegener ate even integer la ttice, let M α ( σ ) b e mo duli stacks of semistable ob jects with r espe c t to σ ; now, J oyce inv ariants J α ( σ ) are defined with these mo duli stacks a nd motivic inv ariants. In [Jo 08], on Co h X , J o yce proved that his inv a r iant s e xist indep e ndently of the choice of Giesek er stabilit y conditions; then, on D( X ), he dis c us sed his inv ariants, s upposing that his inv ar iant s exis t indep e nden tly of the choice of stability conditions in Stab ∗ ( X ), which was prov ed by T o da [T o]. Now, with the notion of numeric al ly faithfulness ( fai thful for sho rt), which was introduced by the second author [Ok07 b], for each mo duli stack, the inde- pendenc e of the c hoice of stability conditio ns in Stab ∗ ( X ) for Jo yce inv ariants of D( X ) manifests itself as follows. Theorem 1 .1. F or e ach K3 surfac e X , Mukai ve ctor α of X , faithful stability c onditio ns σ, σ ′ ∈ Stab ∗ ( X ) , and motivic invariant I , we have I ( M α ( σ )) = I ( M α ( σ ′ )) . Here, by [Ok07 b ], faithful stability conditions exist as a dens e subset in Stab ∗ ( X ). So, in Stab ∗ ( X ), for a se t of s emistable ob jects with a b ounded mass, by wall structures examined in [Br0 8 ], for each Muk a i vector α a nd p olarization of X , we hav e some faithful stability condition σ ∈ Stab ∗ ( X ) s uc h that M α ( σ ) consists of Gieseker semistable coherent sheav es. One can chec k that Theo rem 1 .1 holds on any o ther known sta bilit y mani- folds fo r Calabi-Y au surfaces such as ab elian surfaces a nd minimal resolutio ns of surface singula rities (for references o f these stability ma nifolds, one can co ns ult with the Br idg eland’s survey [Br0 6]). Also, for so me mo duli s ta c ks o f stable ob jects and mo duli sta c ks of µ -semistable cohere nt sheaves on X , one can co m- pare Theorem 1.1 with a sequence of flops in [ArB eLi] and dimension co un ting in [Y o]. T o make explicit computation of motivic inv aria n ts, we first wan t to know our mo duli sta cks as mo duli spaces in some details, and then c ompute iso morphism groups of ob jects of mo duli stacks. F or example, for primitive Muk ai vectors of p ositive ranks , by [Y o], mo duli spa c es of Gieseker stable coherent sheav es 2 are defor ma tion equiv alent to Hilb ert schemes, and they hav e the trivial C ∗ isomorphism group for ea c h p oin t in the mo duli sta c ks. Going beyond ab ov e primitive cases is a challenge; reasons include that in general mo duli spaces ca n b e sing ular and co mputing isomo r phism groups of ob jects is demanding. T o explain what ha ppens in a situation, for ob jects E , F a nd their Muk ai vectors [ E ] , [ F ], let [ E ] . [ F ] = P i ( − 1) i dim E x t i ( E , F ) b e the Muk a i par ing of [ E ] and [ F ]. F or ob jects in the moduli stack of a Muk ai vector with non-p ositive self Muk ai paring, in the mo duli sta c k of a m ultiple of the Muk ai vector, their dir e c t pro ducts hav e a nontrivial fib er w ith so me isomorphism group for ea c h p oin t in the fib er. On the other hand, for a Muk ai vector with pos itiv e self Muk ai paring, by [Ok0 7b], Theo r em 1.1 b oils down to Co r ollary 1.2. Let us r ecall Muk ai vectors α are called spheric al , if their self-intersections a re tw o; in other words, α corresp ond to spheric al obje cts whic h not only give r ise to auto equiv alences of D( X ) [ST], but als o include structur e sheav es supp orted ov er rational curves on X , the structure sheaf of X , and their t wists b y line bundles. Notice that each Muk ai vector v with v .v > 0 is a multiple of a spher ic al Muk ai vector α . Corollary 1. 2. F or each s pher ical class α , faithful σ ∈ Stab ∗ ( X ), po sitiv e int eger n , and motivic inv a r iant I , we hav e I ( M nα ( σ )) = I ([1 / GL( n, C )]) = 1 l n 2 (1 − l − 1 ) ··· (1 − l − n ) . In o ther words, for faithful stability conditions, we a lw ays hav e a stable spherical ob ject for ea c h spher ical class . As p oin ted out to the authors by Bridgeland, the existence of a sta ble spherical o b ject of each spher ical cla ss in Corollar y 1.2 in particula r g iv es a nother wa y to pr o ve that Stab ∗ ( X ) is lo cally a bundle over the p erio d domain of X , which consists of complexified K¨ ahler classes of X w itho ut o nes tha t ar e o rthogonal to spherical classes. Once we know our mo duli stacks in these deta ils , then we are able to compute v arious inv arian ts. Indeed, a fter the seco nd a uthor discussed s ome part of the conten t o f this pap er such as Corolla ry 1.3 (in the original form of Joyce inv ari- ants for so me α ) at [O k07a] and whilst the author s were prepa ring this pap er, they got notified that for sta nda rd stabilities of coher en t sheaves of ra tional el- liptic surfaces, Y oshiok a–Nak a jima computed their inv a riants [NaY o]. Also, for stability conditions of Cala bi-Y au catego ries of dimension three (a .k.a. 3-Cala bi- Y au categorie s ), Kontsevic h–Soib elman discussed their inv aria n ts [KoSo]. Here we will stick to Joyce inv aria nts for K3 surfaces, but let us make some comments fo r o ur rea ders. Unlike inv aria n ts defined by Nak a jima–Y oshiok a, Joyce in v a riants inv olve not only arbitrar y motivic inv ariants, but a ls o correctio n terms of p ow ers of q ba sed on Lie algebras asso cia ted to each stability condition. The inv aria n ts discussed by Kon tsevich–Soib elman are (pr esumably) com- patible with Joyce inv a riants, and they put pr imary emphasis on nontrivial wall-crossing formulas o f their inv a riants for Calabi-Y au ca tegories o f dimensio n three. Now, let us go ba ck to o ur case; for the Joyce inv a riants in Corollary 1.2, we compute a s b elow. F or the conv enience of our formulas, we w ill us e q = l − 1 , switching b et ween T ate m otive and L efschetz motive . 3 Corollary 1.3. J nα ( σ ) = q n 2 n (1 − q n ) . Here we would like to mention that w e are slightly modifying the o riginal formulation of Joyce inv arian ts for K3 sur faces, a s s uggested to the author s by Za gier. Namely , in order to obtain more natural expressio ns, w e omit the factor ( q − 1 − 1) (this is ( l − 1) in [Jo0 8]). Reca ll that the factor ( q − 1 − 1) was inv olved so that we are a ble to get num ber s on mo duli stacks of stable ob jects by replacing q by o ne . Instead, we take residues at q = 1 to extend the notion of Euler characteris tics to mo duli s tac ks, which a r e not necessa rily only of stable ob jects. W e may rega rd Jo yce inv aria n ts as volumes for ea c h Muk ai vector by the following r e a son. By Theorem 1.1, for each Muk ai vector and g eneric c hoices of stability conditio ns , motivic inv ar ian ts igno re the difference of mo duli sta c ks, but unlike Joyce inv ar ian ts, o n defor mations of s tabilit y conditions on stability manifolds, we do not know whether mo tivic in v ariants deform o n mo duli stacks. So now, we would like to take the following generating functions of Joyce inv ariants: J k = X n> 0 J nα ( σ ) n k = X n> 0 q n 2 n k +1 (1 − q n ) . Let us p oint out that taking r esidues termwise at q = 1 gives − ζ ( k + 2). The generating function J k actually app ears in the following sum sugg ested by Jo yce [Jo0 7b]. Namely , on a stability manifold, we can consider the form P α 6 =0 J nα ( σ ) Z ( nα ) k , which is inv ar ian t under auto equiv alences. Also let us no te that we can take the smaller form P α.α =2 J nα ( σ ) Z ( nα ) k , which is a gain inv ariant under auto equiv alences. Here we would like to study its building piece J k . It is clear that case s of k b eing o dd give dege ner ated for ms; so we will concentrate o n cases when k is e ven. Let us a lso mention that by the work of Br idgeland–T oleda no -Laredo [BrT o] and Kontsevich–Soibelman [KoSo], it has b ecame clea r that inv ar iant s of the mo duli sta c ks whose images of cent ra l c harg es align make a building block to study Lie algebr as ass ocia ted to stability co nditions. As we hav e se en, gener ating functions coming out of physics have b een discussed with mo dular forms. Now, J k are already some quantum polylo ga- rhithms, as they ar e q -deforma tions of p olylo garhithms. Ho wev er, the prese nc e of q n 2 in the numerator do es no t ma k e in par ticular J 0 the well-kno wn quantum dilogarithm (for example, see [Za 07b]), but instead J k lo ok simila r to s ome of mo ck theta functions , whic h were intro duce d b y Ra man ujan [Ra00] [Ra 8 8] a nd carry tra nsformation laws similar to ones of theta functions. W e will pursue this view p oint. Let us take a quick r eview at mo ck theta functions. The explicit definition on these functions was not given by Rama nujan, and this issue had remained for a long time. How ever, quite recently , Zwegers in his thesis [Zw] provided a wa y to add corre ction ter ms to the Rama nujan’s mo ck theta functions to make them into harmonic we ak Maass forms of weig ht 1 2 [Za07a], which is expla ined as follows. 4 F or τ in q = e 2 π iτ , let D b e the differential op erator 1 2 π i d dτ , M k be the space o f meromorphic mo dular fo rms of weigh t k for k ∈ 1 2 Z with p oles only at the cusps, and τ = x + iy . Then, ha rmonic weak Maass forms of weigh t k are real analytic mo dular forms whose deriv atives with resp ect to D fall into the space M 2 − k y k ; here, these deriv atives for mo ck theta functions are called shadows [Za07a]. Since this under standing of mo c k theta functions surfaced, we hav e seen achiev ements such as [BrOn06], [BrO n07], a nd [BrOn]. Esp ecially , F ourier co- efficients o f harmonic weak Ma ass forms of weigh t 1 2 play ed a cent ral r ole, in particular, for solving the A ndr ews-Dr agonette Conje ctur e , that is to pr o ve an exact formula of F our ie r co efficients o f a mo c k theta function. Also, for an even int eger k > 2, the first autho r in his thesis [Me ] studied the so-called higher Gr e en ’s funct ions o f weight k , which ar e dire c tly rela ted to the harmonic weak Ma a ss forms of weight 2 − k by the Maass op er ators ( y 2 D ) k − 2 2 and ( D + 2 4 π iy ) · · · ( D + k − 2 4 π iy ) with the differential o per a tor D = 1 2 π i d dτ . Now, going back to our J k , with certain duality , we want to comp ensate our choices of po sitiv e integers k . This can b e done in terms of different ial equa tio ns, mo dular forms, a nd c ertain corr ection terms to J k . Her e, differential op erator D ma y cor resp ond to infinit esimal deriv ativ es of o ur volumes. Let E k and B k be the Eisens tein ser ie s and the Bernoulli num b e rs. Then, we hav e D k − 1 J k − 2 = B k 2 k (1 − E k ) − J − k + X n> 0 q n 2 n 1 − k . W e would like to hav e a dua lity formula which contains o nly mo dular forms as follows. Let us r e call that in the s pa ce of mo dular forms of a given degree, Eisenstein s eries make distinguis he d bas is of the subspa ce tha t is orthog onal to cusp forms. Now, we take the following. Definition 1.4. J k = B − k 2 k − 1 2 X n> 0 q n 2 n k +1 + J k = B − k 2 k + X n 6 =0 q n 2 n k +1 (1 − q n ) . Then, this time, for po sitiv e even int eger s k , we have D k − 1 J k − 2 + J − k = − B k 2 k E k . Now, we will take J k as granted, and study J − 2 in some detail. Here, J − 2 ( τ ) = − 1 24 − 1 2 X n> 0 nq n 2 + J − 2 , and P n> 0 nq n 2 is a half-theta function . Let θ 1 ( τ ) = P n ∈ Z e π in 2 τ and θ 3 ( τ ) = P n ∈ Z e π i ( n + 1 2 ) 2 τ be half-p erio d Ja cobi theta functions (at z = 0). Then we hav e the following. 5 Theorem 1. 5. F or SL(2 , Z ) , with b ounde d gr owth at the cusp, ther e is a unique r e al anal ytic mo dular form of weight two ˜ g ( τ ) s uch that the derivative of ˜ g ( τ ) with r esp e ct t o D is − θ 1 (2 τ ) θ 1 (2 τ )+ θ 3 (2 τ ) θ 3 (2 τ ) 64 π 2 y 3 2 . Now the holomorph ic p art of ˜ g ( τ ) c oincid es wi th J − 2 ( τ ) . Let us explain the words “ho lomorphic part” in Theo r em 1.5; for ho lomor- phic functions a ( τ ) a nd b ( τ ), ther e is a cano nical wa y to pro duce a function whose deriv ativ e with r espect to D is a function o f the form a ( τ ) b ( τ ) y k . It is given by the following integral (whenever the integral co n verges): R a ( τ ) b ( τ ) y k ; τ ! := 2 π ia ( τ ) Z τ i ∞ b ( z ) dz ( − i 2 ( z − ¯ τ )) k . Now, the diff erence ˜ g ( τ ) − R ( D ( ˜ g ( τ )) v a nishes b y D , and we call it the holo- morphic p art . The story of Raman ujan’s functions is pa rallel to Theo rem 1.5 , since they can b e o btained as the holomo rphic pa rts of cer tain harmonic weak Maass forms of weigh t 1 2 . Indeed, we prov e J − 2 ( τ ) is in the spa ce o f mo ck theta functions o f weigh t 3 2 tensored by the spa ce M 1 2 . Let us explain a bit mor e. Here, the s hadow is not in the space M 1 2 y 3 2 , but in the t wisted space M 1 2 ⊗ M 1 2 y 3 2 . Also, the holo morphic par t of ˜ g ( τ ) is not a mo c k theta function, but a sum of pro ducts of or dinary theta functions of weigh t 1 2 and mo ck theta functions of weigh t 3 2 , which will be derived in this pap er from the L er ch function in [Zw]. W e will then b e able to iden tify the F ourier co efficien ts of the sum to end the pro of of Theo rem 1 .5. Now, authors are aw are that w e are leaving ma n y questions op en. F o r ex- ample, we would want so me under standing of mo duli sta c ks of cases other than ones consider ed here a nd J k for k 6 = − 2 , but it is our impression that they rather p ose fundament al q uestions on isomorphis m gro ups of p oints in moduli stacks, algebra s on mo duli stacks, a nd mo ck theta functions. Y e t, here, we investigated o ur case s in some detail and thank the Dyson’s dream [Dy, Sec tio n 6 ], which at some po in t enco ur aged us to lo ok for mo ck symmetries in this context. 2 Definitions Let us recall fundamental notio ns from [Br0 7]. In this pap er, o ur tria ngulated category T is assumed to b e D( X ) for s ome K3 surface X . Let K ( T ) be the Gr othendie ck gr oup o f T ;i.e., K ( T ) is the ab elian gr o up g enerated by classes of ob jects of T such that for ob jects E , F , G in T , we have [ F ] = [ E ] + [ G ] in K ( T ) whenever we have an exa ct tria ngle E → F → G in T . 6 2.1 Stabilit y conditions A stability c ondition σ = ( Z, P ) o n T consists of a group ho momorphism Z from K ( T ) to the complex num b er C and a family P ( φ ) of full ab elian sub catego ries of T index ed by re a l num ber s φ . Each Z and P are called a c entr al char ge and a slicing . They need to satisfy the fo llo wing compatibilities. • If for some φ ∈ R , E is a nonzero o b ject in P ( φ ), then for so me p ositive real num ber m ( E ), called mass o f E , we hav e Z ( E ) = m ( E ) ex p( iπ φ ). • F or ea c h real num be r φ , we hav e P ( φ + 1 ) = P ( φ )[1]. • F or r eal n umbers φ 1 > φ 2 and ob jects A i ∈ P ( φ i ), we ha ve Hom T ( A 1 , A 2 ) = 0. • F or any nonzero o b ject E ∈ T , there exist real num bers φ 1 > · · · > φ n and o b jects A i ∈ P ( φ i ) such that ther e exis ts a sequence of exact tr iangles E i − 1 → E i → A i with E 0 = 0 and E . The sequence ab ov e is called the Har der-Nar asimha n filtr ation ( HN-filtr ation for short) of E . The HN-filtration o f any ob ject is unique up to iso morphisms. F o r each φ ∈ R , nonzero ob jects in P ( φ ) are ca lled semistable with phase φ . If mo re- ov er a semistable ob ject in P ( φ ) has only the triv ia l J ordan-H¨ older filtra tio n in P ( φ ), then it is called st able . W e will assume tha t our cent ra l charge Z fac to rs through the map [ E ] ∈ K ( T ) 7→ ch( E ) √ X , w hich is the Mukai ve ctor o f E in the Mukai lattic e of X . F or Muk ai vectors v , w , let v .w b e the Mukai p aring . A stability condition σ = ( Z , P ) is ca lled n umeric ally faithful [Ok07b, Def- inition 3.1] ( fai thful for short), if for each rea l num b er r , we ha ve a primitive Muk ai vector v s uc h that for ea c h semistable ob ject E o f the phase r , [ E ] is a sum of v . Here, by [B r08, Pro pos ition 8.3], the connected co mponent Stab ∗ ( X ) satisfies the assumption o f [Ok07b, Lemma 3.1]. So faithful stability c o nditions are dense in Stab ∗ ( X ). F or a re a l num ber r , let P ( r − 1 , r ] be the extension-c losed full s ubcate- gory co nsisting of semistable ob jects whos e phases are in the interv al ( r − 1 , r ], and C ( r ) b e the Muk ai vectors of the ob jects in P ( r − 1 , r ]. Then, for each stability condition σ = ( Z, P ) ∈ Stab ∗ ( X ), Muk ai vector α , and the r eal num- ber r such that Z ( α ) ∈ R > 0 e iπr , we define the Joyce inv arian t J α ( σ ) to b e P ∞ n =1 P α 1 + ··· + α n = α,α i ∈ C ( r ) q P j>i α j .α i ( − 1) n − 1 n Π i =1 I ( M α i ( σ )) [J o08, Definition 6.22], [T o, Definition 5.9] (let us r ecall that as explaine d in the introduction, we let q = l − 1 and omit ( l − 1) from their original definitions). 2.2 Mo dular forms Let us recall the definition and pro perties of the Dedekind eta function (we denote q = e 2 π iτ ), τ b elongs to the upp er ha lf plane. η ( τ ) = e πiτ 12 ∞ Y n =1 (1 − e 2 π inτ ) = q 1 / 24 (1 − q − q 2 + q 5 + q 7 · · · ) . 7 The eta functions tra nsforms like a mo dular form of weigh t 1 2 : η ( τ + 1) = e πi 12 η ( τ ) , η  − 1 τ  = e πi 4 √ τ η ( τ ) . W e will need the following identit y: η  τ 2  η  1 + τ 2  η (2 τ ) = e πi 24 η ( τ ) 3 . (1) Next we reca ll the half-per iod Jacobi theta functions (at z = 0), note that we slightly changed the indexing: θ 1 ( τ ) = θ 00 (0; τ ) = X n ∈ Z e π in 2 τ = 1 + 2 q 1 2 + 2 q 2 + 2 q 9 2 + 2 q 8 + · · · , θ 2 ( τ ) = θ 01 (0; τ ) = X n ∈ Z ( − 1) n e π in 2 τ = 1 − 2 q 1 2 + 2 q 2 − 2 q 9 2 + 2 q 8 + · · · , θ 3 ( τ ) = θ 10 (0; τ ) = X n ∈ Z e π i ( n + 1 2 ) 2 τ = 2 q 1 8 + 2 q 9 8 + 2 q 25 8 + 2 q 49 8 + · · · . The theta functions can b e expr essed in terms of the eta function in the following wa y: θ 1 ( τ ) = e − πi 12 η  1+ τ 2  2 η ( τ ) , θ 2 ( τ ) = η  τ 2  2 η ( τ ) , θ 3 ( τ ) = 2 η ( 2 τ ) 2 η ( τ ) . (2) W e know their tra nsformation pr o per ties: θ 1 ( τ + 1) = θ 2 ( τ ) , θ 1  − 1 τ  = e − πi 4 √ τ θ 1 ( τ ) , θ 2 ( τ + 1) = θ 1 ( τ ) , θ 2  − 1 τ  = e − πi 4 √ τ θ 3 ( τ ) , θ 3 ( τ + 1) = e πi 4 θ 3 ( τ ) , θ 3  − 1 τ  = e − πi 4 √ τ θ 2 ( τ ) . In particular , they ar e mo dular forms for the g roup Γ(2). W e also need the cla ssical Eisenstein ser ie s of weight 2 for SL(2 , Z ): E 2 ( τ ) = 2 4 η ′ ( τ ) η ( τ ) = 1 − 24 ∞ X k,n =1 k q nk = 1 − 2 4 q − 72 q 2 − 96 q 3 − · · · . The function E 2 is no t a mo dular for m, but is quasi- mo dular form. The fourth p ow ers of the theta functions are Eisenstein series for Γ(2) and we hav e 8 the following relatio ns : E 2  1 + τ 2  − 2 E 2 ( τ ) = θ 4 1 ( τ ) − 2 θ 2 ( τ ) 4 , (3) E 2  τ 2  − 2 E 2 ( τ ) = − 2 θ 4 1 ( τ ) + θ 2 ( τ ) 4 , (4) 4 E 2 (2 τ ) − 2 E 2 ( τ ) = θ 4 1 ( τ ) + θ 2 ( τ ) 4 , (5) θ 4 1 ( τ ) = θ 4 2 ( τ ) + θ 4 3 ( τ ) . (6) 2.3 The Lerch function Having intro duced some classical mo dular forms, we turn to the thesis of Zwegers [Zw]. In this thesis we find the following definition of the Le r c h function: µ ( u, v ; τ ) = e π iu θ ( v ; τ ) X n ∈ Z ( − 1) n e π i ( n 2 + n ) τ +2 π inv 1 − e 2 π inτ +2 π iu ( u, v ∈ C \ ( Z τ + Z )) . The definition of the theta function he use s is the fo llo wing one: θ ( z ; τ ) = X ν ∈ 1 2 + Z e π iν 2 τ +2 π iν ( z + 1 2 ) . Note the following symmetr y: θ ( z + 1 ; τ ) = θ ( − z ; τ ) = − θ ( z ; τ ) , θ ( z + τ ; τ ) = − e − π iτ − 2 π iz θ ( z ; τ ) . The theta functions θ 1 , θ 2 and θ 3 are related to θ in the following way: θ  τ 2 ; τ  = − ie − πiτ 4 θ 2 ( τ ) , θ  1 + τ 2 ; τ  = − e − πiτ 4 θ 1 ( τ ) , θ  1 2 ; τ  = − θ 3 ( τ ) . Moreov er we have θ (0 ; τ ) = 0 , d 2 π ids     s =0 θ ( s ; τ ) = iη 3 ( τ ) . Zwegers found a wa y to add a corr ection term to µ so that the new function e µ ha s g oo d transfo r mation pro p erties. Na mely , he defines e µ ( u, v ; τ ) = µ ( u, v ; τ ) + i 2 R ( u − v ; τ ) , where R ( u ; τ ) = X ν ∈ 1 2 + Z  sign( ν ) − E  ν + ℑ u y  p 2 y  ( − 1) ν − 1 2 e − π iν 2 τ − 2 π iν u . 9 Here y = ℑ τ and E is the function E ( z ) = 2 Z z 0 e − π t 2 dt = 1 − er fc( z √ π ) . The result of Zwegers is the following tra ns formation pro per ties of e µ : Theorem 2 .1. [Zw, The or em 1.11 ] The function e µ s atisfies e µ ( u, v ; τ ) = e µ ( v , u ; τ ) = e µ ( − u, − v ; τ ) , and e µ ( u, v ; τ + 1) = e − πi 4 e µ ( u, v ; τ ) , e µ  u τ , v τ ; − 1 τ  = − e − πi 4 − πi ( u − v ) 2 τ √ τ e µ ( u, v ; τ ) e µ ( u + 1 , v ; τ ) = − e µ ( u, v ; τ ) , e µ ( u + τ , v ; τ ) = − e 2 π i ( u − v )+ π iτ e µ ( u, v ) . Here is a list of prop erties that the functions R and µ satisfy separa tely: Prop osition 2 .2. [Zw, Pr op ositions 1.4 and 1.9] The funct ions µ and R s atisfy µ ( u, v ; τ ) = µ ( v , u ; τ ) = µ ( − u , − v ; τ ) , R ( − z ; τ ) = R ( z ; τ ) , and we have µ ( u + 1 , v ; τ ) = − µ ( u, v ; τ ) , R ( z + 1 ; τ ) = − R ( z ; τ ) . W e also mention o ne last prop erty which we will use: Prop osition 2.3. [Zw, Pr op osition 1.4 and The or em 1.11] Both the function µ and e µ ( if you plug it in plac e of µ ) satisfy µ ( u + z , v + z ; τ ) − µ ( u, v ; τ ) = iη 3 ( τ ) θ ( u + v + z ; τ ) θ ( z ; τ ) θ ( u ; τ ) θ ( v ; τ ) θ ( u + z ; τ ) θ ( v + z ; τ ) for u , v , u + z , v + z / ∈ Z + τ Z . 3 Pro ofs Let us prov e Theorem 1.1. Pr o of. F or faithful stability conditions σ , in terms o f Muk a i vectors, J α ( σ ) admit unique expressions. Since b y [T o, Theorem 1.5], we hav e J α ( σ ) = J α ( σ ′ ) for any α , esp ecially for pr imitiv e ones, the statement follows. In terms of faithful stability co nditions o ver integer lattices, for inv aria n ts of mo duli stacks o f alig ned central charges, Theo rem 1.1 is a g e neral feature of their deformation inv ar iance on stability manifolds. W e will pr o ve Corollar y 1.2. Now, an ob ject E ∈ T is called spheric al if Ext i ( E , E ) = C for i = 0 , 2 a nd Ext i ( E , E ) = 0 for els e; spheric a l class es ar e Muk ai vectors of spherical ob jects. 10 Pr o of. F or the ca se when α is with a nonzero rank, by [Y o, Theorem 0.1(1)] and [Br0 8, Prop ositio n 1 4 .2], for some faithful σ ∈ Stab ∗ ( X ), we have a stable spherical ob ject whose cla ss is α . So, b y [Ok07b, Pro positio n 4.9 ], the statement follows. F or other cases, by [F r , Lemma 25], the first Chern clas s of α is either effective or anti-effectiv e. So, by r eplacing α with − α , if neces sarily , one recalls that s ome co herent sheaf E with [ E ] = α is Gieseker semistable. Then, b y [T o, Theorem 6.6], the statement follows. Let us prove Cor ollary 1.3. Pr o of. Since α.α = 2, by choo sing σ to be faithful, we have that for p ositive inte- gers k i , J nα ( σ ) is equa l to P ∞ m =1 P k 1 + ··· + k m = n q P i>j 2 k i k j ( − 1) n − 1 n Π n i =1 1 I (GL( k i , C )) . Since P i>j 2 k i k j = ( P k i ) 2 − P k 2 i = n 2 − P k 2 i , we hav e that J nα ( σ ) is equal to q n 2 P ∞ m =1 P k 1 + ··· + k m = n ( − 1) n − 1 n Π n i =1 q − k 2 i I (GL( k i , C )) . Let F ( x ) = P m ≥ 0 q − m 2 I (GL ( m, C )) x m . Then we have F ( x ) − F ( q x ) = xF ( x ), and J nα is the n -th co efficient o f q n 2 P ( − 1) n − 1 n ( F ( x ) − 1) = q n 2 log F ( x ). Since log F ( x ) + log (1 − x ) = log F ( q x ), the n - th co efficien t of log F ( x ) is 1 n (1 − q n ) . So the statement follows. The rest of this se ction is devoted to the pro of of Theorem 1.5. The pla n is to s ee the existence of a function with the holomor phic part b eing J − 2 and g oo d transformatio n pro perties . Now, the first clue is to notice that J − 2 lo oks similar to µ , which is the holomor phic part o f ˜ µ , but to b e pre c ise, we will here derive several functions from µ a nd subsequently mo dify them with theta functions. Let us study b ehavior o f the functions µ, e µ, R at the “p oin ts o f order tw o” . The v alues at these p oints are not interesting s ince we hav e Prop osition 3.1. e µ  1 2 , τ 2 ; τ  = e µ  1 2 , 1 + τ 2 ; τ  = e µ  τ 2 , 1 + τ 2 ; τ  = 0 . Pr o of. W e simply take the definition of µ and R ab ov e and use the fo llo wing trick. F or e x ample, in the case of e µ  1 2 , τ 2  the trick is to write X n ∈ Z ( − 1) n e π i ( n 2 +2 n ) τ 1 + e 2 π inτ = 1 2 X n ∈ Z ( − 1) n e π i ( n 2 +2 n ) τ 1 + e 2 π inτ + X n ∈ Z ( − 1) n e π i ( n 2 − 2 n ) τ 1 + e − 2 π inτ ! = 1 2 X n ∈ Z ( − 1) n e π in 2 τ = θ 2 ( τ ) 2 . Therefore for e µ  1 2 , τ 2  we obtain µ  1 2 , τ 2 ; τ  = − e πiτ 4 2 . 11 A trick similar to the one used ab ov e g iv es R  1 − τ 2 ; τ  = − ie πiτ 4 . Thu s e µ  1 2 , τ 2 ; τ  = 0. The other cases are similar with µ  1 2 , 1 + τ 2 ; τ  = − ie πiτ 4 2 , µ  τ 2 , 1 + τ 2 ; τ  = 0 , R  − τ 2 ; τ  = e πiτ 4 , R  − 1 2 ; τ  = 0 . Because o f the last pro pos ition the der iv a tives of e µ at the p oints of o rder 2 should hav e nice transformatio n prop erties. Namely , we define µ 1 ( τ ) = d 2 π ids     s =0 e µ  1 2 , τ 2 + s ; τ  , µ ′ 1 ( τ ) = d 2 π ids     s =0 e µ  1 2 + s, τ 2 ; τ  , µ 2 ( τ ) = d 2 π ids     s =0 e µ  1 2 , 1 + τ 2 + s ; τ  , µ ′ 2 ( τ ) = d 2 π ids     s =0 e µ  1 2 + s, 1 + τ 2 ; τ  , µ 3 ( τ ) = d 2 π ids     s =0 e µ  τ 2 , 1 + τ 2 + s ; τ  , µ ′ 3 ( τ ) = d 2 π ids     s =0 e µ  τ 2 + s, 1 + τ 2 ; τ  . Prop osition 3.2. We ha ve µ 1 ( τ ) + µ ′ 1 ( τ ) = − e πiτ 4 θ 1 ( τ ) 3 4 , µ 2 ( τ ) + µ ′ 2 ( τ ) = − ie πiτ 4 θ 2 ( τ ) 3 4 , µ 3 ( τ ) + µ ′ 3 ( τ ) = − θ 3 ( τ ) 3 4 . Pr o of. Using Prop osition 2.3 we obta in µ 1 ( τ ) + µ ′ 1 ( τ ) = d 2 π ids     s =0  e µ  1 2 , τ 2 + s  − e µ  1 2 − s, τ 2  = d 2 π ids     s =0 iη ( τ ) 3 θ  1+ τ 2 ; τ  θ ( s ; τ ) θ  1 2 − s ; τ  θ  1 2 ; τ  θ  τ 2 − s ; τ  θ  τ 2 ; τ  = − η ( τ ) 6 θ  1+ τ 2 ; τ  θ  1 2 ; τ  2 θ  τ 2 ; τ  2 = − η 6 ( τ ) e πiτ 4 θ 1 ( τ ) θ 2 ( τ ) 2 θ 3 ( τ ) 2 . W e hav e (using (2) and (1)) θ 1 ( τ ) θ 2 ( τ ) θ 3 ( τ ) = 2 η ( τ ) 3 . 12 Therefore µ 1 ( τ ) + µ ′ 1 ( τ ) = − e πiτ 4 θ 1 ( τ ) 3 4 . Similarly , µ 2 ( τ ) + µ ′ 2 ( τ ) = − η ( τ ) 6 θ  τ 2 + 1; τ  θ  1 2 ; τ  2 θ  1+ τ 2 ; τ  2 = − iη 6 ( τ ) e πiτ 4 θ 1 ( τ ) θ 2 ( τ ) 2 θ 3 ( τ ) 2 = − ie πiτ 4 θ 2 ( τ ) 3 4 , µ 3 ( τ ) + µ ′ 3 ( τ ) = − η ( τ ) 6 θ  1 2 + τ ; τ  θ  τ 2 ; τ  2 θ  1+ τ 2 ; τ  2 = − η ( τ ) 6 θ 3 ( τ ) θ 1 ( τ ) 2 θ 2 ( τ ) 2 = − θ 3 ( τ ) 3 4 . Now we ar e ready to for m ulate the trans fo rmation pro perties o f µ i . Prop osition 3.3. We ha ve µ 1 (1 + τ ) = e − πi 4 µ 2 ( τ ) , µ 2 (1 + τ ) = − e − πi 4 µ 1 ( τ ) , µ 3 (1 + τ ) = e − πi 4  θ 3 ( τ ) 3 4 + µ 3 ( τ )  , µ 1  − 1 τ  = e πi 4 − π i 1+ τ 2 4 τ τ 3 2 e πiτ 4 θ 1 ( τ ) 3 4 + µ 1 ( τ ) ! , µ 2  − 1 τ  = e − πi 4 − πi 4 τ τ 3 2 µ 3 ( τ ) , µ 3  − 1 τ  = − e − πi 4 − πiτ 4 τ 3 2 µ 2 ( τ ) . Pr o of. The pro of of the first three equations go es by applying the op erator d 2 π ids   s =0 to bo th sides of the following equations obtained fro m Theor em 2.1. e µ  1 2 , 1 + τ 2 + s ; τ + 1  = e − πi 4 e µ  1 2 , 1 + τ 2 + s ; τ  , e µ  1 2 , τ 2 + 1 + s ; τ + 1  = − e − πi 4 e µ  1 2 , τ 2 + s ; τ  , e µ  1 + τ 2 , τ 2 + 1 + s ; τ + 1  = − e − πi 4 e µ  τ 2 + s, 1 + τ 2 ; τ  . Similarly , for the last three equations we use e µ  1 2 , − 1 2 τ + s ; − 1 τ  = − e − πi 4 − πi ( τ +1 − 2 sτ ) 2 4 τ √ τ e µ  τ 2 , − 1 2 + sτ ; τ  , e µ  1 2 , τ − 1 2 τ + s ; − 1 τ  = − e − πi 4 − πi (1 − 2 sτ ) 2 4 τ √ τ e µ  τ 2 , τ − 1 2 + sτ ; τ  , e µ  − 1 2 τ , τ − 1 2 τ + s ; − 1 τ  = − e − πi 4 − πi ( τ +2 sτ ) 2 4 τ √ τ e µ  − 1 2 , τ − 1 2 + sτ ; τ  . 13 Lo oking at the prop osition ab ov e it is clear that we should consider the following three functions e h 1 ( τ ) = e − πiτ 4 µ 1 ( τ ) + θ 1 ( τ ) 3 8 , e h 2 ( τ ) = − ie − πiτ 4 µ 2 ( τ ) + θ 2 ( τ ) 3 8 , e h 3 ( τ ) = − µ 3 ( τ ) − θ 3 ( τ ) 3 8 . Then we hav e e h 1 ( τ + 1 ) = e h 2 ( τ ) , e h 1  − 1 τ  = e πi 4 τ 3 2 e h 1 ( τ ) , e h 2 ( τ + 1 ) = e h 1 ( τ ) , e h 2  − 1 τ  = e πi 4 τ 3 2 e h 3 ( τ ) , e h 3 ( τ + 1 ) = e − πi 4 e h 3 ( τ ) , e h 3  − 1 τ  = e πi 4 τ 3 2 e h 2 ( τ ) . Next w e need to find the F ourie r expansions of e h i . W e would lik e to hav e them similar to the deco mpositio n e µ = µ + i 2 R . Therefore we compute Prop osition 3.4. d 2 π ids     s =0 R  1 − τ 2 − s ; τ  = − ie πiτ 4 X n ∈ Z | n | β (2 y n 2 ) e − π in 2 τ + 1 2 − θ 1 ( τ ) π √ 2 y ! , d 2 π ids     s =0 R  − τ 2 − s ; τ  = e πiτ 4 X n ∈ Z ( − 1) n | n | β (2 y n 2 ) e − π in 2 τ + 1 2 − θ 2 ( τ ) π √ 2 y ! , d 2 π ids     s =0 R  − 1 2 − s ; τ  = i   X ν ∈ Z + 1 2 | ν | β (2 y ν 2 ) e − π iν 2 τ − θ 3 ( τ ) π √ 2 y   . Pr o of. Differentiating term by ter m gives d 2 π ids     s =0 R ( u − s ; τ ) = X ν ∈ 1 2 + Z ν  sign( ν ) − E  ν + ℑ u y  p 2 y  ( − 1) ν − 1 2 e − π iν 2 τ − 2 π iν u − X ν ∈ 1 2 + Z 1 π √ 2 y ( − 1) ν − 1 2 e − 2 π y ( ν + ℑ u y ) 2 − π iν 2 τ − 2 π iν u . 14 Therefore for the first case we obtain − ie πiτ 4 X n ∈ Z  n + 1 2   sign  n + 1 2  − E ( n p 2 y )  e − π in 2 τ + ie πiτ 4 π √ 2 y X n ∈ Z e − π in 2 ¯ τ . The first summand can be transfor med into − ie πiτ 4 X n ∈ Z  n + 1 2  (sign( n ) − E ( n p 2 y )) e − π in 2 τ − ie πiτ 4 2 . Using the function β , β ( x ) = ∞ X x t − 1 2 e − π t dt = 1 − E ( √ x ) = erfc ( √ π x ) , we obtain d 2 π ids     s =0 R  1 − τ 2 − s ; τ  = − ie πiτ 4 X n ∈ Z  n + 1 2  sign( n ) β (2 y n 2 ) e − π in 2 τ − ie πiτ 4 2 + ie πiτ 4 π √ 2 y θ 1 ( τ ) , and the final r e sult eas ily fo llows fr o m this formula. Analogously , in the second case we obtain d 2 π ids     s =0 R  − τ 2 − s ; τ  = e πiτ 4 X n ∈ Z ( − 1) n  n + 1 2  sign( n ) β (2 y n 2 ) e − π in 2 τ + e πiτ 4 2 − e πiτ 4 π √ 2 y θ 2 ( τ ) , In the third c ase the result follows right fro m the following formula: d 2 π ids     s =0 R  − 1 2 − s ; τ  = i X ν ∈ Z + 1 2 | ν | β (2 y ν 2 ) e − π iν 2 τ − i X ν ∈ Z + 1 2 e − π iν 2 ¯ τ π √ 2 y . It remains to differentiate the function µ . 15 Prop osition 3.5. The c orr esp onding derivatives of µ ar e given by d 2 π ids     s =0 µ  1 2 , τ 2 + s ; τ  = − e πiτ 4 24 θ 1 ( τ ) − 2 + 6 θ 1 ( τ ) + 3 θ 4 1 ( τ ) − E 2 ( τ ) + 48 ∞ X n =1 e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 ! , d 2 π ids     s =0 µ  1 2 , 1 + τ 2 + s ; τ  = − i e πiτ 4 24 θ 2 ( τ ) − 2 + 6 θ 2 ( τ ) + 3 θ 4 2 ( τ ) − E 2 ( τ ) + 48 ∞ X n =1 ( − 1) n e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 ! , d 2 π ids     s =0 µ  τ 2 , 1 + τ 2 + s ; τ  = 1 24 θ 3 ( τ ) 1 − 3 θ 3 ( τ ) 4 − E 2 ( τ ) + 24 ∞ X n =1 e π i ( n 2 + n ) τ 1 + e 2 π inτ (1 − e 2 π inτ ) 2 ! . Pr o of. W e use the following decomp osition of µ : µ ( s, z ; τ ) = e π is θ ( z ; τ )(1 − e 2 π is ) + 1 θ ( z ; τ ) ∞ X n =1 ( − 1) n e π i ( n 2 + n ) τ  e 2 π inz + π is 1 − e 2 π inτ +2 π is − e − 2 π inz − π is 1 − e 2 π inτ − 2 π is  . W e compute the T aylor expansion with resp ect to 2 π is around s = 0 of the expression ab ov e for the following v alues of z : 1+ τ 2 , τ 2 , 1 2 . W e need o nly the co efficien t at 2 π i s . T his co efficient equals 1 24 θ ( z ; τ ) + 1 θ ( z ; τ ) ∞ X n =1 ( − 1) n e π i ( n 2 + n ) τ (1 + e 2 π inτ )( e 2 π inz + e − 2 π inz ) 2(1 − e 2 π inτ ) 2 . Therefore in the ca s e z = 1+ τ 2 we obtain − e πiτ 4 θ 1 ( τ ) 1 24 + ∞ X n =1 e π in 2 τ (1 + e 2 π inτ ) 2 2(1 − e 2 π inτ ) 2 ! = − e πiτ 4 θ 1 ( τ ) − 5 24 + θ 1 ( τ ) 4 + 2 ∞ X n =1 e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 ! , similarly , in the cas e z = τ 2 i e πiτ 4 θ 2 ( τ ) − 5 24 + θ 2 ( τ ) 4 + 2 ∞ X n =1 ( − 1) n e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 ! , 16 and finally , in the case z = 1 2 : − 1 θ 3 ( τ ) 1 24 + ∞ X n =1 e π i ( n 2 + n ) τ 1 + e 2 π inτ (1 − e 2 π inτ ) 2 ! . F or a pply ing Pro pos itio n 2.3 we a lso need to compute the co efficients at 2 π is of the following expr essions (we o mit τ from the arguments of θ ): iη ( τ ) 3 θ ( 1 2 + s ) θ ( τ 2 ) θ ( 1 − τ 2 ) θ ( 1 2 ) θ ( s ) θ ( τ 2 + s ) , iη ( τ ) 3 θ ( 1 2 + s ) θ ( 1+ τ 2 ) θ ( − τ 2 ) θ ( 1 2 ) θ ( s ) θ ( 1+ τ 2 + s ) , iη ( τ ) 3 θ ( τ 2 + s ) θ ( 1+ τ 2 ) θ ( − 1 2 ) θ ( τ 2 ) θ ( s ) θ ( 1+ τ 2 + s ) , F or this we need to c o mpute the T aylor expans io ns of θ ( s ), θ ( 1 2 + s ), θ ( τ 2 + s ) and θ ( 1+ τ 2 + s ) up to seco nd term with re s pect to 2 π is . W e hav e (denoting by ′ the op erator d 2 π idτ ): θ ( s ) = (2 π is ) iη ( τ ) 3  1 + (2 π is ) 2 η ′ ( τ ) η ( τ )  + · · · , θ ( 1 2 + s ) = − θ 3 ( τ )  1 + (2 π is ) 2 θ ′ 3 ( τ ) θ 3 ( τ )  + · · · , θ ( τ 2 + s ) = − i e − πiτ 4 θ 2 ( τ )  1 − 2 π is 2 + (2 π is ) 2  1 8 + θ ′ 2 ( τ ) θ 2 ( τ )  + · · · , θ ( 1+ τ 2 + s ) = − e − πiτ 4 θ 1 ( τ )  1 − 2 π is 2 + (2 π is ) 2  1 8 + θ ′ 1 ( τ ) θ 1 ( τ )  + · · · . Thu s the co efficients a t 2 π i s of the express ions in question ar e, corr e s pond- ingly , − e πiτ 4 θ 1 ( τ )  θ ′ 3 ( τ ) θ 3 ( τ ) − η ′ ( τ ) η ( τ ) + 1 8 − θ ′ 2 ( τ ) θ 2 ( τ )  , − i e πiτ 4 θ 2 ( τ )  θ ′ 3 ( τ ) θ 3 ( τ ) − η ′ ( τ ) η ( τ ) + 1 8 − θ ′ 1 ( τ ) θ 1 ( τ )  , 1 θ 3 ( τ )  θ ′ 2 ( τ ) θ 2 ( τ ) − η ′ ( τ ) η ( τ ) − θ ′ 1 ( τ ) θ 1 ( τ )  . Putting everything tog ether d 2 π ids     s =0 µ  1 2 , τ 2 + s ; τ  = d 2 π ids     s =0  µ  1 − τ 2 , s ; τ  + iη ( τ ) 3 θ ( 1 2 + s ) θ ( τ 2 ) θ ( 1 − τ 2 ) θ ( 1 2 ) θ ( s ) θ ( τ 2 + s )  = − e πiτ 4 θ 1 ( τ ) − 1 12 + θ 1 ( τ ) 4 + θ ′ 3 ( τ ) θ 3 ( τ ) − η ′ ( τ ) η ( τ ) − θ ′ 2 ( τ ) θ 2 ( τ ) + 2 ∞ X n =1 e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 ! , 17 d 2 π ids     s =0 µ  1 2 , 1 + τ 2 + s ; τ  = d 2 π ids     s =0  µ  − τ 2 , s ; τ  + iη ( τ ) 3 θ ( 1 2 + s ) θ ( 1+ τ 2 ) θ ( − τ 2 ) θ ( 1 2 ) θ ( s ) θ ( 1+ τ 2 + s )  = − i e πiτ 4 θ 2 ( τ ) − 1 12 + θ 2 ( τ ) 4 + θ ′ 3 ( τ ) θ 3 ( τ ) − η ′ ( τ ) η ( τ ) − θ ′ 1 ( τ ) θ 1 ( τ ) + 2 ∞ X n =1 ( − 1) n e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 ! , d 2 π ids     s =0 µ  τ 2 , 1 + τ 2 + s ; τ  = d 2 π ids     s =0  µ  − 1 2 , s ; τ  + iη ( τ ) 3 θ ( τ 2 + s ) θ ( 1+ τ 2 ) θ ( − 1 2 ) θ ( τ 2 ) θ ( s ) θ ( 1+ τ 2 + s )  = 1 θ 3 ( τ ) 1 24 + θ ′ 2 ( τ ) θ 2 ( τ ) − η ′ ( τ ) η ( τ ) − θ ′ 1 ( τ ) θ 1 ( τ ) + ∞ X n =1 e π i ( n 2 + n ) τ 1 + e 2 π inτ (1 − e 2 π inτ ) 2 ! . The statements we need to prov e follow from the following identities: θ ′ 3 ( τ ) θ 3 ( τ ) − η ′ ( τ ) η ( τ ) − θ ′ 2 ( τ ) θ 2 ( τ ) = θ 4 1 ( τ ) 8 − E 2 ( τ ) 24 , θ ′ 3 ( τ ) θ 3 ( τ ) − η ′ ( τ ) η ( τ ) − θ ′ 1 ( τ ) θ 1 ( τ ) = θ 4 2 ( τ ) 8 − E 2 ( τ ) 24 , θ ′ 2 ( τ ) θ 2 ( τ ) − η ′ ( τ ) η ( τ ) − θ ′ 1 ( τ ) θ 1 ( τ ) = − θ 4 3 ( τ ) 8 − E 2 ( τ ) 24 , Prop ositions 3.4 and 3.5 together give the F our ie r expa ns ions of e h i . Denote h 1 ( τ ) = 1 24 θ 1 ( τ ) 2 + E 2 ( τ ) − 48 ∞ X n =1 e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 ! , h 2 ( τ ) = 1 24 θ 2 ( τ ) 2 + E 2 ( τ ) − 48 ∞ X n =1 ( − 1) n e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 ! , h 3 ( τ ) = 1 24 θ 3 ( τ ) − 1 + E 2 ( τ ) − 24 ∞ X n =1 e π i ( n 2 + n ) τ 1 + e 2 π inτ (1 − e 2 π inτ ) 2 ! . The series h i are holomo r phic p ow er serie s c on verging on the upp er half plane. 18 Denote R 1 ( τ ) = 1 2 X n ∈ Z | n | β (2 y n 2 ) e − π in 2 τ − θ 1 ( τ ) 2 π √ 2 y , R 2 ( τ ) = 1 2 X n ∈ Z ( − 1) n | n | β (2 y n 2 ) e − π in 2 τ − θ 2 ( τ ) 2 π √ 2 y , R 3 ( τ ) = 1 2 X ν ∈ Z + 1 2 | ν | β (2 y ν 2 ) e − π iν 2 τ − θ 3 ( τ ) 2 π √ 2 y . Prop osition 3.6. F or i = 1 , 2 , 3 we have e h i ( τ ) = h i ( τ ) + R i ( τ ) . In his thesis Zwegers a lso r epresents R a s a certa in integral inv olving a theta function of w eight 3 2 . In our case, we also hav e such a representation but with theta functions of weight 1 2 . Prop osition 3.7. F or i = 1 , 2 , 3 R i ( τ ) = 1 4 π i Z i ∞ τ θ i ( z ) dz ( − i ( z − τ )) 3 2 . Pr o of. Note that the integral on the rig ht conv erg e s. W e prove the identit y termwise using the following for m ula for a rea l num b er a : 1 2 | a | β (2 y a 2 ) e − π ia 2 τ − e π ia 2 τ 2 π √ 2 y = − Z i ∞ τ a 2 ie π ia 2 z dz 2 p − i ( z − τ ) − e π ia 2 τ 2 π √ 2 y = 1 4 π i Z i ∞ τ e π ia 2 z dz ( − i ( z − τ )) 3 2 . This formula is obtained using the integral repr esen tation of β β (2 y a 2 ) = Z ∞ 2 y a 2 e − π t dt √ t after the substitution t = − i ( z − τ ) a 2 , and then integration b y parts. The c a se a = 0 sho uld b e consider ed separa tely . W e would lik e to compute the F o urier co efficient s of h i θ i explicitly . Since we know the F ourier co efficients o f E 2 , it remains to consider the following expressions : 19 Prop osition 3.8. We ha ve 2 ∞ X n =1 e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 = X m>n> 0 , m − n even me π imnτ − X n>m> 0 , m − n even me π imnτ 2 ∞ X n =1 ( − 1) n e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 = X m>n> 0 , m − n even m ( − 1) m e π imnτ − X n>m> 0 , m − n even m ( − 1) m e π imnτ ∞ X n =1 e π i ( n 2 + n ) τ 1 + e 2 π inτ (1 − e 2 π inτ ) 2 = X m>n> 0 , m − n o dd me π imnτ − X n>m> 0 , m − n o dd me π imnτ Pr o of. It is clear . Lo oking at the expansio ns we o bserve that 2 ∞ X n =1 e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 + 2 ∞ X n =1 ( − 1) n e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 = 4 X m>n> 0 me 4 π imnτ − X n>m> 0 me 4 π imnτ ! , (7) and 2 ∞ X n =1 e π i ( n 2 +2 n ) τ (1 − e 2 π inτ ) 2 + ∞ X n =1 e π i ( n 2 + n ) τ 1 + e 2 π inτ (1 − e 2 π inτ ) 2 = X m>n> 0 me π imnτ − X n>m> 0 me π imnτ (8) Therefore it is no t difficult to co mplete the pro of of the following sta temen t: Prop osition 3.9. We ha ve h 1 ( τ ) θ 1 ( τ ) + h 2 ( τ ) θ 2 ( τ ) − 4  h 1 (4 τ ) θ 1 (4 τ ) + h 3 (4 τ ) θ 3 (4 τ )  = − θ 1 (2 τ ) 4 4 . Using the co rresp onding identities b etw een the theta functions , namely θ 1 ( τ ) + θ 2 ( τ ) = 2 θ 1 (4 τ ) , θ 1 ( τ ) + θ 3 ( τ ) = θ 1 ( τ 4 ) , the integral representation of R i from Pro p osition 3.7 and the change of v a riables 4 π iR i (4 τ ) = Z i ∞ 4 τ θ i ( z ) dz ( − i ( z − 4 τ )) 3 2 = 1 2 Z i ∞ τ θ i (4 z ) dz ( − i ( z − τ )) 3 2 we obtain R 1 ( τ ) θ 1 ( τ ) + R 2 ( τ ) θ 2 ( τ ) − 4 ( R 1 (4 τ ) θ 1 (4 τ ) + R 3 (4 τ ) θ 3 (4 τ )) = 0 . Therefore we also have the following. 20 Prop osition 3.10. e h 1 ( τ ) θ 1 ( τ ) + e h 2 ( τ ) θ 2 ( τ ) − 4  e h 1 (4 τ ) θ 1 (4 τ ) + e h 3 (4 τ ) θ 3 (4 τ )  = − θ 1 (2 τ ) 4 4 . Now we are r eady to prov e Theo rem 1.5. This is done in a series of prop o- sitions. T ake e g ( τ ) = − e h 1 (2 τ ) θ 1 (2 τ ) + e h 3 (2 τ ) θ 3 (2 τ ) 2 + θ 1 ( τ ) 4 + θ 2 ( τ ) 4 96 , then the following holds. Prop osition 3.11. The function e g ( τ ) tr ansforms like a mo dular form of weight 2 : e g ( τ + 1) = e g ( τ ) , e g  − 1 τ  = τ 2 e g ( τ ) . The function e g ( τ ) decomp oses as e g ( τ ) = g ( τ ) + r ( τ ) , where g ( τ ) = − h 1 (2 τ ) θ 1 (2 τ ) + h 3 (2 τ ) θ 3 (2 τ ) 2 + θ 1 ( τ ) 4 + θ 2 ( τ ) 4 96 and r ( τ ) = − R 1 (2 τ ) θ 1 (2 τ ) + R 3 (2 τ ) θ 3 (2 τ ) 2 . It is not difficult to compute the F ourier e xpansion of g : Prop osition 3.12. W e have g ( τ ) = − E 2 ( τ ) 24 − 1 2 X n ∈ Z \{ 0 } nq n 2 1 − q n = − 1 24 + ∞ X n =1 σ ′ ( n ) q n , wher e σ ′ ( n ) denotes the sum of p ositive divisors of n which ar e gr e ater t han √ n , plus half √ n in the c ase if n is a p erfe ct squar e. Pr o of. Using the third identit y fro m (3) a nd (8) we find g ( τ ) = 1 48 − 1 − E 2 ( τ ) + 24 X m>n> 0 me 2 π imnτ − X n>m> 0 me 2 π imnτ !! = − 1 24 + 1 2 X m> 0 , n> 0 me 2 π imnτ + 1 2 X m>n> 0 me 2 π imnτ − 1 2 X n>m> 0 me 2 π imnτ = − 1 24 + X m> 0 , n> 0 me 2 π imnτ − X n ≥ m> 0 me 2 π imnτ + 1 2 X n> 0 ne 2 π in 2 τ . Then the statement follows. 21 Now, by Prop osition 3 .7, r ( τ ) = R ( D ( r ( τ ))) = R ( D ( ˜ g ( τ ))), where R is the op erator fr o m the introduction, hence g is the holomorphic part o f ˜ g . Propo - sition 3.12 g iv es the F ourier expansio n of g , which, a s one can ea sily verify , coincides with J − 2 . Having the transfo rmation pr oper ties of ˜ g prov ed in Prop o- sition 3.1 1, it remains to chec k only the uniq uene s s statement. This is obvious since there ar e no holomor phic mo dular for ms of weigh t 2 for SL(2 , Z ). 4 Ac kno wledgmen ts Authors thank Max- P lanck-Institut f ¨ ur Mathematik and Institut de s Hautes ´ Etudes Scientifiques. 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