Infinite hierarchies of nonlocal symmetries of the Chen--Kontsevich--Schwarz type for the oriented associativity equations

Infinite hierarc hies of nonlo cal symme tries of the Chen–Kon tsevic h–Sc h w arz t yp e for the orien ted asso ciativit y equations A. Ser gyeyev Mathemat ical Institut e, Si lesian Universit y in Opav a, Na Rybn ´ ı ˇ cku 1, 7 46 01 Opa v a, Czech Republic E-mai l : Artu r.Ser gyeyev@ math.slu.cz June 14 , 2009 W e construct infi nite hierarchies of nonlo cal higher symmetries for the orien ted asso ciativit y equa- tions using solutions of asso ciated v ector and scalar sp ect ral problems. The symmetries in question generalize those found b y Chen, Kon tsevic h and Sc hw arz [40] for the WD VV equations. As a bypro d- uct, we obtain a Darb oux-t yp e transformation and a (conditional) B¨ ac klund transformation for the orien ted asso ciativit y equations. In tro duction The Witten–Dijkgraaf–V erlinde–V erlinde (WD VV) equations [1, 2], and the related geometric s tructures, in particular, the F ro b enius manifolds [3]–[8], hav e attracted considerable atten tion because of their man- ifold applications in ph ysics and mathematics. More recen tly , the oriented asso ciativit y equations, a generalization of the WD VV equations, and t he related geometric structures, F -manifolds, see e.g. [7]–[19], hav e also b ecome a sub ject of in tense researc h. These eq uations ha v e first a pp eared in [3, Prop osition 2.3] as the equations for the displacemen t vec tor. The or iented asso ciativit y equations describ e inter alia isoasso ciative quan tum deformations of commuta- tiv e asso ciative algebras [14 , 15], cf. also [16 ]–[19]. The orien ted asso ciativit y equations (8) admit the gradient reduction (43 ) to the “usual” asso ciativit y equations (44). Equations (44) and the so-called Hes sian reduction (see [20, 21, 22, 23, 24] and [14]) of the o rien ted asso ciativit y equations naturally arise in top ological 2D gr avit y [1, 25], singularity theory and complex geome try ( see e.g. [8, 9]), and in differen tial g eometry and theory of integrable systems, se e [8], [26]–[36] a nd references therein. There is a conside rable b o dy of work on the symmetry prop erties of the WDVV equations, see e.g. [37, 38, 39, 40, 41 ] for the po int symmetries of the WDVV and generalized ( in the sense of [42] and ref- erences therein) WD VV equations, [4, 43, 44, 45] and references therein for finite symmetries, B¨ ac klund 1 transformations a nd dualities, and [26, 46, 47, 48] and references t herein for the higher symmetries and (bi-)Hamiltonian structures for the WDVV equations, a nd also for (44), in three a nd four independen t v ar ia bles. Although the approac h of [26, 46, 47, 48] in princ iple could [26] b e generalized to the WD VV equations in more than fo ur indep enden t v ariables, this w a s not done y et. Nev ertheless, in [3 9, 40] infinite sets of nonlo c al higher symme tries fo r the WD VV equations w ere found. T o the b est of our kno wledge, higher (or generalized [49]) symmetries of the orien te d asso ciativity equations in arbitrary dimension w ere nev er f ully explored. The go al of the presen t pap er is to construct nonlo cal higher symmetries for t he o rien ted asso ciativity equations equations using, in ana lo gy with [40], the solutions of auxiliary sp ectral problems. W e sho w that t he very solutions of the v ector sp ectral problem (9), either p er se or m ultiplied b y a suitably c hosen solution of the scalar sp ectral problem (10) with the opp osite sign of the sp ectral parameter, indee d ar e (infinitesimal) symmetries for the orien ted asso ciativit y equations (8), see Theorem 1 b elo w for details. The fact that solutions of (9) are symmetries for (8) is quite un usual in itself, as symmetries t ypically turn out to b e quadratic [50] ra ther than linear in solutio ns of auxiliary linear problems. Up on p erforming the gradien t reduction to the “usual” asso ciativity equations (44) we repro duce the results of [40], see Corollaries 6 and 8 b elow. Ho w ev er, not all nonlo cal symmetries from Theorem 1 surviv e the gradient reduction a nd yield symmetries for (44), see Corollary 6 and the subsequen t discussion. Expanding solutions of the spectral problems into formal T a ylor series in the spectral parameter yields infinite hierarchies of nonlo cal higher symmetries for (8) and (44), see Corollaries 4 and 8 b elow. Finally , as a b ypro duct, w e obtain a Darb oux-t yp e transformation and some B¨ ac klund-t ype trans- formations relating the solutions of “usual” and orien ted associativity equations, see Prop osition 1 and Corollaries 11 and 12 f o r further details. These transformations, a s w ell as the nonlo cal symmetries discusse d a b o ve, could p ossibly yield new solutions f or the oriented and “usual” asso ciativit y equations. 1 Preliminaries Let the Greek indices α, β , γ , . . . (except for λ, µ, η , ζ , σ , τ , χ, φ, ψ ) r un from 1 to n , where n is a fixed nat- ural n um b er, and summation o v er the rep eated indices b e understo o d unles s otherwis e explic itly stated. In what follow s w e also assume tha t all functions under study are sufficien tly smoo th for all necess ary deriv atives to exist. Consider the oriente d asso ciativity e quations , see e.g. [1 0]–[14], for the structure “constan ts” c δ αβ ( x 1 , . . . , x n ) of a comm utativ e ( c α ν ρ = c α ρν ) alg ebra: c ν αρ c ρ β γ = c ν ργ c ρ αβ , (1) ∂ c α β γ ∂ x ρ = ∂ c α ργ ∂ x β . (2) 2 The condition (1) means that the algebra in question is asso ciativ e, and ( 2 ) means that we consider isoasso ciativ e [14] quantum deformations o f the algebra in question. The oriented asso ciativit y equations (1), (2 ) can b e written as compatibility conditions of the Gauss– Manin equations, see e.g. [4 , 7, 14], for a scalar function χ ( λ ) (for t he sake of brevit y we shall often omit b elo w the dep endence on x 1 , . . . , x n ): ∂ 2 χ ( λ ) ∂ x α ∂ x γ = λc ν αγ ∂ χ ( λ ) ∂ x ν . (3) Here λ is the sp ectral parameter. Thes e eq uations ha v e a v ery interes ting in terpretation, with χ pla ying the role of a w a ve function, in the context of quantum deformations of asso ciativ e a lg ebras [14]. W e also ha v e a zero-curv ature represen tation for (1), (2) of the f o rm (see e.g. [6, 42]) ∂ ψ α ( λ ) ∂ x β = λc α β γ ψ γ ( λ ) , (4) where we again omit, for the sak e of brevit y , t he dep endence of ψ α on x 1 , . . . , x n . In other words, Eqs.(1 ) , (2) are precisely the compatibility conditions for (4). The quan tities ψ α ( λ ) are nothing but the comp onen ts of a generic ve ctor field whic h is cov arian tly constan t (in other terminology , parallel or flat) w ith resp ect to the co v aria n t deriv ativ e asso ciated with the one-parametric family of flat connections − λc α ν ρ . Up on in tro ducing the quantities φ α ( λ ) = ∂ χ ( λ ) /∂ x α Eq.(3) can b e written in the first-order form as ∂ φ α ( λ ) /∂ x β = λc δ αβ φ δ ( λ ) . (5) Quite obvious ly , the sp ectral problem (5) is, up to the c hange of sign of λ , adjoint to (4). No w let χ α ( λ ), α = 1 , . . . , n , b e the solutions of (3) no rmalized by the condition χ α ( λ ) | λ =0 = x α . It is we ll kno wn (see e.g. [4 , 8, 14]) that χ α are nothing but flat co ordina t es fo r the one-par ameter f a mily λc α ν κ of flat connections (the flatness readily follows fro m (1) a nd (2)). F ollowing [3, 4], we can represen t χ α in the form χ α ( λ ) = x α + λK α + O ( λ 2 ) , (6) where K α = K α ( x 1 , . . . , x n ) is the so- called displacemen t ve ctor. Plugging (6) in to (3 ) and restricting our atten t io n to the terms linear in λ yields c α β γ = ∂ 2 K α ∂ x β ∂ x γ . (7) The ansatz (7 ) automatically solv es (2), and (1 ) b oils down to the ov erdetermined system ∂ 2 K ν ∂ x α ∂ x ρ ∂ 2 K ρ ∂ x β ∂ x γ = ∂ 2 K ρ ∂ x α ∂ x β ∂ 2 K ν ∂ x ρ ∂ x γ (8) 3 for K α . In what follow s we shall refer to this system a s to the orien ted asso ciativity equations just as w e referred t o (1), (2), as, in com bination with (7), Eq.(8 ) is equiv alen t to (1), (2) pro vided c α ν ρ = c α ρν . Of course, the equations obtained b y plugging (7) into (4), that is, ∂ ψ α ( λ ) ∂ x β = λ ∂ 2 K α ∂ x β ∂ x γ ψ γ ( λ ) , (9) pro vide a zero-curv ature represen ta tion for (8). Lik ewise, plugging (7) into (3) yields a scalar sp ectral problem for (8): ∂ 2 χ ( λ ) ∂ x α ∂ x γ = λ ∂ 2 K ν ∂ x α ∂ x γ ∂ χ ( λ ) ∂ x ν . (10) 2 Darb o ux-t yp e transformation for orien ted asso ciativit y equations It is w ell kno wn, see e.g. [4, App endix B], [37] and references t herein, tha t there exist c hanges of v ariables that lea v e the se cond deriv a tiv es of the prep otential unc hanged and map solutions of the (generalized) WD VV equations in to new solutions. Quite in terestingly , there is a c hange of v ariables of this kind that in volv es [37] a solution of the spectral problem (4), and th us can b e though t of as a Darb oux-t ype transformation. It turns out tha t this transformation is readily generalized to the o r ien ted a ssociat ivity equations. Namely , the f ollo wing assertion holds. Prop osition 1 L et K α satisfy (8) an d ψ α ( λ ) solve the sp e ctr al pr oblem (9). Supp os e that det ∂ ψ α ( λ ) /∂ x β 6≡ 0 , intr o duc e new i n dep endent variables ˜ x α = ψ α ( λ ) , (11) and de fi ne (lo c al ly) new dep endent va ri a bles ˜ K β by the formulas ∂ ˜ K β ∂ ˜ x γ = ∂ K β ∂ x γ . (12) Then ˜ c α β γ = ∂ 2 ˜ K α ∂ ˜ x β ∂ ˜ x γ , (13) wher e ˜ K α = ˜ K α ( ˜ x ) ar e determine d f r om (12), sa tisfy ˜ c α β γ ˜ c γ ρν = ˜ c α ν γ ˜ c γ ρβ . (14) 4 Pr o of. First of all, w e need to sho w that (12) is w ell-defined, i.e., that there exist, at least lo cally , the functions ˜ K α ( ˜ x ) suc h that (12) holds. Quite clearly , this amoun ts to proving that w e hav e ∂ 2 ˜ K α ∂ ˜ x β ∂ ˜ x γ = ∂ 2 ˜ K α ∂ ˜ x γ ∂ ˜ x β , (15) or equiv alen tly (b y virtue of (12)), ∂ 2 K α ∂ x β ∂ ˜ x γ = ∂ 2 K α ∂ x γ ∂ ˜ x β . (16) F ro m (12) w e hav e ∂ 2 K α ∂ x β ∂ ˜ x γ = ∂ 2 K α ∂ x β ∂ x ε ∂ x ε ∂ ˜ x γ = c α β ε ∂ x ε ∂ ψ γ . (17) Hence, ∂ 2 K α ∂ x β ∂ ˜ x γ − ∂ 2 K α ∂ x γ ∂ ˜ x β =  c α β ε ∂ x ε ∂ ψ γ − c α γ ε ∂ x ε ∂ ψ β  . (18) W e w an t t o sho w tha t the expression on the left-hand side of (18) v anishes. But this is equiv alen t to v anishing of the following quan t it y: B α ρν =  c α β ε ∂ x ε ∂ ψ γ − c α γ ε ∂ x ε ∂ ψ β  ∂ ψ γ ∂ x ρ ∂ ψ β ∂ x ν . Using t he obv ious iden tit y ∂ ψ γ ∂ x ρ ∂ x ρ ∂ ψ ν = δ γ ν , where δ γ ν is the Kronec ke r delta, w e find tha t B α ρν =  c α β ρ ∂ ψ β ∂ x ν − c α γ ν ∂ ψ γ ∂ x ρ  . No w using (4) we obtain B α ρν = λ  c α γ ρ c γ ν κ − c α γ ν c γ ρκ  ψ κ ( 1 ) = 0 , and thus ∂ 2 K α ∂ x β ∂ ˜ x γ − ∂ 2 K α ∂ x γ ∂ ˜ x β = 0 , so (15) indeed holds, i.e., ˜ c α β γ = ˜ c α γ β . (19) No w w e only need to prov e that (14) holds, or equiv a len tly ˜ c α β γ ˜ c γ ρν − ˜ c α ν γ ˜ c γ ρβ = 0 . (2 0 ) Using ( 1 9), (17) a nd (18) w e o btain ˜ c α β γ = ˜ c α γ β = c α γ ε ∂ x ε ∂ ψ β = ∂ x ε ∂ ψ β c α γ ε . (21) 5 Using (21) for ˜ c α β γ in the first t erm of (20) and for ˜ c α ν γ in the second term of (20), and the formu la tha t immediately follo ws from (17), ˜ c ε κρ = c ε κν ∂ x ν ∂ ψ ρ , for ˜ c γ ρν in the first term o f (20) a nd for ˜ c γ ρβ in the second term of (20), w e see that the ex pression o n the left-hand side of (20) b oils do wn to ˜ c α β γ ˜ c γ ρν − ˜ c α ν γ ˜ c γ ρβ = ∂ x π ∂ ψ β ∂ x κ ∂ ψ ν  c α π γ c γ ρκ − c α κγ c γ ρπ  ( 1 ) = 0 , (22) and thus (14) indeed holds.  3 Nonlo cal symmetries for orien ted asso ciativi t y equatio n s Recall (see e.g. [49, 51, 52, 53]) that an (infinitesim al higher) symmetry fo r the oriented ass o ciativit y equations (8 ) is an evolutionary v ector field X = G α ∂ / ∂ K α suc h that G α satisfy the linearized v ersion of (8), t ha t is, ∂ 2 G ν ∂ x α ∂ x ρ ∂ 2 K ρ ∂ x β ∂ x γ + ∂ 2 K ν ∂ x α ∂ x ρ ∂ 2 G ρ ∂ x β ∂ x γ = ∂ 2 G ρ ∂ x α ∂ x β ∂ 2 K ν ∂ x ρ ∂ x γ + ∂ 2 K ρ ∂ x α ∂ x β ∂ 2 G ν ∂ x ρ ∂ x γ , (23) mo dulo (8) and differential consequence s thereof (or , informally , on solutions of (8)). This is equiv alent to compatibilit y of (8) with the flo w asso ciated with X , tha t is, ∂ K α /∂ τ = G α . A straightforw ard but somewhat tedious computation pro v es the follo wing generalization of the re- sults of Chen, Kon tsevic h and Sc h warz [40] (see Corollary 6 b elo w for the latt er) to the case of orien ted asso ciativit y eq uations. Theorem 1 The evolutionary ve ctor fields ψ α ( λ ) ∂ ∂ K α and ψ α ( λ ) χ ( − λ ) ∂ ∂ K α , wher e ψ α ( λ ) s a tisfy (9) an d χ ( λ ) satisfies (10), ar e nonlo c al higher symmetries for the oriente d a sso ciativity e quations (8), i.e., the flows ∂ K α ∂ τ λ = ψ α ( λ ) , (24) ∂ K α ∂ σ λ = ψ α ( λ ) χ ( − λ ) , (25) ar e c omp atible with ( 8). 6 Informally , compatibilit y here means that the flows (24 ) and (25) map the set S of (smo oth) solutions of (8) in to itse lf, i.e., S is in v ariant under the flows (24) and (25); see e.g. [5 1, 52, 53, 54, 55, 56] and references therein for the general theory of nonlo cal symmetries. In a more analytic language, Theorem 1 states that G α = ψ α ( λ ) a nd ˜ G α = ψ α ( λ ) χ ( − λ ) satisfy (23) provided (8), (9) and (10) hold. An un usual feature of the symmetries ψ α ( λ ) ∂ /∂ K α from Theorem 1 is that they are linear (rather t ha n quadratic, as it is the case for man y other systems , cf. [50]) in the solutions of auxiliary linear problem. It is natural to a sk whe ther the flows (24) and (25) are in t egrable system s in an y reasonable sense. The fo llo wing r esult provides linear sp ectral pro blems for these flow s and th us suggests t heir inte grabilit y . Corollary 1 T he flows (24) and (25) c an b e (nonuniquely) extende d to the flows for the quantities ψ α ( µ ) and χ ( µ ) as fol lows: ∂ ψ α ( µ ) ∂ τ λ = λµc α ν κ ψ ν ( λ ) ψ κ ( µ ) , (26) ∂ χ ( µ ) ∂ τ λ = λµ λ + µ ψ ν ( λ ) ∂ χ ( µ ) ∂ x ν , (27) ∂ ψ α ( µ ) ∂ σ λ = λµc α ν κ ψ ν ( λ ) ψ κ ( µ ) χ ( − λ ) + λµ λ − µ ∂ χ ( − λ ) ∂ x β ψ β ( µ ) ψ α ( λ ) , (28) ∂ χ ( µ ) ∂ σ λ = λµ λ + µ ψ ν ( λ ) ∂ χ ( µ ) ∂ x ν χ ( − λ ) . (29) In particular, b y Coro lla ry 1 Eq.(26) together with the system ∂ ψ α ( µ ) ∂ x β = µ ∂ 2 K α ∂ x β ∂ x γ ψ γ ( µ ) , (30) pro vide (assuming that (9) holds ) a zero-curv ature represen tation for t he extended system (8), (24), and th us ensure in tegrabilit y thereof. Lik ewise, the flo w (25) is in tegrable b ecaus e Eq.(30) along with the system ∂ 2 χ ( µ ) ∂ x α ∂ x β = µ ∂ 2 K δ ∂ x α ∂ x β ∂ χ ( µ ) ∂ x δ (31) and (28), (29) prov ide (a ssuming that (9) and (10) hold) a linear sp ectral pro blem for the extended system (8), (25), i.e., (8) and (25) are precisely the compatibility conditions for (28)–(3 1), a nd integrabilit y of the extended system in question follows . Using the extended flows fro m Corollary 1 w e readily obtain the follow ing result: Corollary 2 Al l flows (24) and (25) c ommute for al l values of p ar ame ters λ and µ : ∂ 2 K α ∂ τ λ ∂ τ µ = ∂ 2 K α ∂ τ µ ∂ τ λ , ∂ 2 K α ∂ τ λ ∂ σ µ = ∂ 2 K α ∂ σ µ ∂ τ λ , ∂ 2 K α ∂ σ λ ∂ σ µ = ∂ 2 K α ∂ σ µ ∂ σ λ . It is imp o r t an t to stress that this result p er se do es not imply comm utativit y of the extende d flo ws from Coro lla ry 1. Ho we v er, a straightforw ard computation prov es the follo wing assertion. 7 Corollary 3 T he extende d flows (24), (2 6), (27) c ommute, i.e., for al l values of λ , µ and ζ we have ∂ 2 K α ∂ τ λ ∂ τ µ = ∂ 2 K α ∂ τ µ ∂ τ λ , ∂ 2 ψ α ( ζ ) ∂ τ λ ∂ τ µ = ∂ 2 ψ α ( ζ ) ∂ τ µ ∂ τ λ , ∂ 2 χ ( ζ ) ∂ τ λ ∂ τ µ = ∂ 2 χ ( ζ ) ∂ τ µ ∂ τ λ . (32) W e in tend to study the remaining comm utat io n relations for the extended flo ws in more detail elsewhere. 4 Expansion in the s p ec t ral parameter and non lo cal p oten t ials No w consider a fo rmal T aylor expansion for ψ α in λ , ψ α ( λ ) = ∞ X k =0 ψ α k λ k . (33) It is immediate fr o m Theorem 1 that ψ α k ∂ / ∂ K α are symmetries for (8 ) , i.e., the flows ∂ K α ∂ τ k = ψ α k , k = 0 , 1 , 2 , . . . , (34) are compatible with (8). W e readily find from (9) the fo llo wing recursion relation: ∂ ψ α k ∂ x β = ∂ 2 K α ∂ x β ∂ x γ ψ γ k − 1 , k = 1 , 2 , . . . . (35) F or k = 0 w e hav e ∂ ψ α 0 /∂ x β = 0 for all β = 1 , . . . , n . No w let ( w 0 ) α β = δ α β , where δ α β is the Kro nec k er delta, and ( w 1 ) α β = ∂ K α /∂ x β . Define recursiv ely the follo wing sequence of nonlo cal quantities : ∂ ( w k ) β γ ∂ x α = ∂ 2 K β ∂ x α ∂ x ρ ( w k − 1 ) ρ γ , k = 2 , 3 , . . . (36) W e ha v e ψ α k = k X j =0 h γ j ( w k − j ) α γ , k = 0 , 1 , 2 , . . . , (37) where h γ j are arbitr a ry constan ts. In analog y with (33), consider a formal T a ylor expansion for χ in λ , χ ( λ ) = ∞ X k =0 χ k λ k . (38) W e obtain from (10) the following recursion relation: ∂ 2 χ k ∂ x α ∂ x γ = ∂ 2 K ν ∂ x α ∂ x γ ∂ χ k − 1 ∂ x ν , k = 1 , 2 , . . . . (39) 8 Set v α 0 = x α , v α 1 = K α , and, in analogy with (6) and (36), define the following sequence of nonlo cal quan tities v β k , β = 1 , . . . , n : ∂ 2 v β k ∂ x α ∂ x γ = ∂ 2 K ν ∂ x α ∂ x γ ∂ v β k − 1 ∂ x ν , k = 2 , 3 , . . . (40) In terms o f geometric theory of PDEs, see e.g. [51, 52, 55, 56], the quan tities ( w k ) α β and v γ k , α, β , γ = 1 , . . . , n , k = 2 , 3 , . . . , define an infinite-dimensional Ab elian co v ering o v er (8). W e ha v e the fo llo wing coun terpart of (37): χ k = b k + k X j =0 d k − j,γ v γ j , k = 0 , 1 , 2 , . . . , (41) where b k and d j,γ are arbitr a ry constants. Using (33) and (38) w e readily find tha t ψ α ( λ ) χ ( − λ ) = ∞ X k =0 ρ α k λ k , ρ α k def = k X j =0 ( − 1) j χ j ψ α k − j . It is no w immediate from The orem 1 that ( w k ) α β ∂ / ∂ K α and ρ α k ∂ / ∂ K α are symmetries for (8), and for k ≥ 2 these symmetries are nonlo cal. What is more, using Corollary 1 w e readily obtain the follow ing result. Corollary 4 T he oriente d as s o c i a tivity e quations (8) ha ve infinitely ma n y symm etries of the form X k ,β = ( w k ) α β ∂ ∂ K α and Y β k ,γ = k X j =0 ( − 1) j v β j · ( w k − j ) α γ ∂ ∂ K α , and al l asso ci a te d flows , i.e., ∂ K α ∂ τ β k = ( w k ) α β , ∂ K α ∂ σ γ k ,β = k X j =0 ( − 1) j v β j · ( w k − j ) α γ , c ommute: ∂ 2 K α ∂ τ β k ∂ τ ν l = ∂ 2 K α ∂ τ ν l ∂ τ β k , ∂ 2 K α ∂ σ β k ,γ ∂ τ ρ l,ν = ∂ 2 K α ∂ τ ρ l,ν ∂ σ β k ,γ , ∂ 2 K α ∂ σ β k ,γ ∂ σ ρ l,ν = ∂ 2 K α ∂ σ ρ l,ν ∂ σ β k ,γ , k , l = 0 , 1 , 2 , . . . , α, β , γ , δ, ν, ρ = 1 , . . . , n. It is readily seen t ha t the symmetries X k ,β and Y β k ,γ and the asso ciated flo ws a re no nlo cal for k ≥ 2 . W e stress once more that the part of Corollary 4 ab out comm uta t ivity of the flo ws a ssociat ed with X k ,β and Y β k ,γ mak es substan tial use o f the extended flo ws from Corolla ry 1 which are not uniquely defined. Moreov er, Corollary 4 do es not imply comm utativit y of t he extende d flows asso ciated with the symmetries X k ,β and Y β k ,γ . How ev er, the flows asso ciated with the symmetries X k ,β do commute after a (suitable) ex tension to the v ariables ( w k ) α β . Na mely , using Corollar ies 1 and 3 w e readily arriv e a t the following assertion. 9 Corollary 5 T he flows ∂ K α ∂ τ β k = ( w k ) α β , ∂ ( w l ) α π ∂ τ β k = c α ν ρ ( w k − 1 ) ν β ( w l − 1 ) ρ π , l = 0 , 1 , 2 , . . . , (42) wher e ( w 0 ) α β = δ α β , ( w 1 ) α β = ∂ K α /∂ x β , and we set ( w − 1 ) ν β ≡ 0 for c onvenie nc e, comm ute for a l l β , γ = 1 , . . . , n and al l k, l = 0 , 1 , 2 , . . . . Equivalently, the evo l utionary ve ctor fields ¯ X s,β ≡ X s,β = ( w s ) α β ∂ ∂ K α , s = 0 , ¯ X k ,β = ( w k ) α β ∂ ∂ K α + ∞ X l =2 c α κρ ( w k − 1 ) κ β ( w l − 1 ) ρ π ∂ ∂ ( w l ) α π , k = 1 , 2 , 3 , . . . , comm ute , i.e., [ ¯ X k ,β , ¯ X l,γ ] = 0 (se e e.g. [51] for the defin ition of the br ack e t [ , ] ), for al l β , γ = 1 , . . . , n and al l k , l = 0 , 1 , 2 , . . . . Th us, the orien ted asso ciativity equations (8) p ossess an infinite hierarc hy of commuting flows whose existence reconfirms integrabilit y of (8). 5 Nonlo cal symmetries for the gradi e n t reduct ion of orien ted asso ciativit y equations F ollowing [14], consider the so- called gradien t reduction of (8). Namely , assume that there exist a nonde- generate symmetric constant matrix η αβ and a function F = F ( x 1 , . . . , x n ), kno wn as a prep o ten tial in 2D top ological field theories [1, 2 , 3], such that K α = η αβ ∂ F /∂ x β . (43) Then (8) b oils down t o the fa mo us asso ciativit y equations for F [1, 2, 3, 4]: ∂ 3 F ∂ x α ∂ x β ∂ x δ η δγ ∂ 3 F ∂ x γ ∂ x ν ∂ x ρ = ∂ 3 F ∂ x α ∂ x ν ∂ x δ η δγ ∂ 3 F ∂ x γ ∂ x β ∂ x ρ . (44) Note that in t he standard theory of the WD VV equations (see e.g. [1, 2, 3, 4]) it is further required that ∂ 3 F ∂ x α ∂ x β ∂ x 1 = η αβ , (45) where η αβ is a nondegenerate constan t matrix suc h that η αβ η β γ = δ γ α . Ho wev er, in w hat follows w e shall not imp ose (45) and the s o-called quasihomogeneit y condition (see e.g. [1, 2, 3, 4, 7] for the discussion o f these conditions). 10 Up on assuming (43) we find that the auxiliary linear problem (9) also admits a reduction ψ α = η αβ ∂ χ/ ∂ x β . (46) This, along with (43), turns (9) into the following o v erdetermined sy stem of the Gauss–Manin equations for χ : ∂ 2 χ ( λ ) ∂ x α ∂ x γ = λη ρν ∂ 3 F ∂ x α ∂ x γ ∂ x ρ ∂ χ ( λ ) ∂ x ν . (47) This is nothing but ( 1 0) after the substitution (4 3 ), and a g ain the asso ciativity equations ( 44) are nothing but the compatibilit y conditions for (47); see e.g. [3, 4, 30] for the discussion of geometric asp ects of (47). Using Theorem 1 in conjunction with (43) and (46) we reco v er the f ollo wing result fr o m [40]. Corollary 6 F or any so lution χ ( λ ) of (47 ) the quantities χ ( λ ) ∂ ∂ F and χ ( λ ) χ ( − λ ) ∂ ∂ F ar e nonlo c a l higher symmetries for the asso ciativity e quations (44), i.e. , the e quations ∂ F ∂ τ λ = χ ( λ ) , (48) ∂ F ∂ ζ λ = χ ( λ ) χ ( − λ ) (49) ar e c omp atible with ( 44). The ab ove flows c an b e ( n onuniquely) extende d as fol lows ∂ χ ( µ ) ∂ τ λ = λµ λ + µ η ν β ∂ χ ( λ ) ∂ x β ∂ χ ( µ ) ∂ x ν , (50) ∂ χ ( µ ) ∂ ζ λ = λµ λ + µ η ν β ∂ χ ( λ ) ∂ x β ∂ χ ( µ ) ∂ x ν χ ( − λ ) + λµ λ − µ η ν β ∂ χ ( − λ ) ∂ x β ∂ χ ( µ ) ∂ x ν χ ( λ ) . (51) In particular, this result means that χ ( λ ) and χ ( λ ) χ ( − λ ) satisfy the linearized v ersion o f (44 ) pro vided (44) and (47) hold. Using the extended flow s from Corollary 6 w e readily obtain Corollary 7 Al l flows (48) and (49) c ommute: for a l l values of p ar am eters λ and µ w e have ∂ 2 F ∂ τ λ ∂ τ µ = ∂ 2 F ∂ τ µ ∂ τ λ , ∂ 2 F ∂ τ λ ∂ ζ µ = ∂ 2 F ∂ ζ µ ∂ τ λ , ∂ 2 F ∂ ζ λ ∂ ζ µ = ∂ 2 F ∂ ζ µ ∂ ζ λ . P erhaps a bit surprisingly , the prop er coun terpar t of the flow (49) f o r the orien t ed associativity equa- tions (8) is no t (25) itself but a linear combination of t he flow s (25) with the opp osite v alues of λ : ∂ K α ∂ ζ λ = ψ α ( λ ) χ ( − λ ) + ψ α ( − λ ) χ ( λ ) . Consider now a formal T aylor expansion in λ for a solution χ ( λ ) of ( 4 7), χ ( λ ) = ∞ X k =0 χ k λ k . 11 Notice that using a slightly differen t expansion of χ ( λ ), in v olving also λ − 1 , enables one t o construct solutions o f the WDVV equations directly from χ ( λ ), see [58] and references therein. Eqs.(41) remain v alid when χ ( λ ) satisfies (47) instead of (10) if w e substitute η αβ ∂ F /∂ x β for K α in to the definitions of v α k and (41 ). Then expanding the symmetries from Corolla r y 6 in p ow ers of λ yields Corollary 8 T he asso ciativity e quations (44) have infinitely many symmetries of the form ˜ X β k = v β k ∂ ∂ F , a nd ˜ Z αβ k = k X j =0 ( − 1) j v α j v β k − j ∂ ∂ F , k = 0 , 1 , 2 , . . . , α , β = 1 , . . . , n, and al l asso ci a te d flows , i.e., ∂ F ∂ τ β ,k = v β k , ∂ F ∂ ζ αβ ,k = k X j =0 ( − 1) j v α j v β k − j , (52) c ommute: ∂ 2 F ∂ τ β ,k ∂ τ ν,l = ∂ 2 F ∂ τ ν,l ∂ τ β ,k , ∂ 2 F ∂ τ γ ,k ∂ ζ αβ ,l = ∂ 2 F ∂ ζ αβ ,l ∂ τ γ ,k , ∂ 2 F ∂ ζ ρν,k ∂ ζ αβ ,l = ∂ 2 F ∂ ζ αβ ,l ∂ ζ ρν,k , k , l = 0 , 1 , 2 , . . . , α, β , γ , δ , ν, ρ = 1 , . . . , n. Again, it is clear that the symmetries ˜ X β k and ˜ Z αβ k and the asso ciated flows are nonlo cal for k ≥ 2. Note that the ab ov e results do not imply comm utat ivity for al l of the extende d flows fro m Coro llary 6 and therefore do not contradict the results fro m [39, 40]. On the o ther hand, using Corolla r y 3 w e r eadily find that, in p erfect agreemen t with [40], the extended flo ws (48), (50) do comm ute, i.e., ∂ 2 F ∂ τ λ ∂ τ λ ′ = ∂ 2 F ∂ τ λ ′ ∂ τ λ , ∂ 2 χ ( µ ) ∂ τ λ ∂ τ λ ′ = ∂ 2 χ ( µ ) ∂ τ λ ′ ∂ τ λ , (5 3) for a ll v a lues of λ , λ ′ and µ . The quan t it ies χ k coincide, up to a ch oice o f normalization, with the densities of Hamiltonians of in tegra ble bihamiltonian h ydro dynamic-t yp e systems asso ciated to an y solution of t he WDVV equations, see Lecture 6 of [4] and e.g. [57], and r eferences therein. It was mentioned in [4] that it is natural to consider these hy dro dynamic-t ype systems as higher (Lie–B¨ ac klund) symmetries for the WD VV equations, b ecause using these systems one can construct [4] the B¨ a cklund transformat io n for the WD VV equations. W e hav e no w seen that χ k and v α k can also b e in terpreted as symm etries o f the asso ciativity equations (44) (a nd, up on imp osing necess ary restrictions on F and χ , of the WDVV equations) in a far mor e straightforw ard manner. 6 In termed iate integrals and B¨ ac klund- typ e t r ans formations The compatibility conditions ∂ 2 ( w 2 ) β γ ∂ x α ∂ x ν = ∂ 2 ( w 2 ) β γ ∂ x ν ∂ x α for ( 36) with k = 2 yield precisely Eqs.(8), and w e hav e 12 Corollary 9 I f the functions K α and G β γ satisfy the system ∂ 2 K β ∂ x α ∂ x ρ ∂ K ρ ∂ x γ = ∂ G β γ ∂ x α , (54) then K α automatic al ly satisfy the oriente d asso c i ativity e quations ( 8 ). In p a rticular, if under t he ab o ve assumptions K α = η αβ ∂ F /∂ x β , wher e η αβ is a symmetric nonde gen- er ate c onstant matrix, then F automatic al ly s a tisfies the asso ci a tivity e quations (44). Just lik e (5 4), the compatibility conditions ∂ 3 v ν 2 ∂ x α ∂ x β ∂ x γ = ∂ 3 v ν 2 ∂ x γ ∂ x β ∂ x α for ( 40) with k = 2 also yield nothing but Eqs.(8), and w e obtain Corollary 10 I f the functions K α and G β satisfy ∂ 2 K ν ∂ x α ∂ x γ ∂ K β ∂ x ν = ∂ 2 G β ∂ x α ∂ x γ , (55) then K α automatic al ly satisfy the oriente d asso c i ativity e quations ( 8 ). In p a rticular, if under t he ab o ve assumptions K α = η αβ ∂ F /∂ x β , wher e η αβ is a symmetric nonde gen- er ate c onstant matrix, then F automatic al ly s a tisfies the asso ci a tivity e quations (44). Th us, (54) and (55) yield a kind of in t ermediate in tegrals for (8) (or, if K α = η αβ ∂ F /∂ x β , for (44)). In the terminology of [51, 52, 5 5, 56], (54) and (55) a lso define co v erings ov er (8) (resp ectiv ely , if K α = η αβ ∂ F /∂ x β , o v er (44)): for an y solution K α of (8) (resp. for any solution F o f (44)) there exist, at least lo cally , the functions G α β and G α suc h that ( 5 4) and ( 5 5) hold. Moreo ver, w e hav e the following observ ation. Corollary 11 I f the functions K α and H α satisfy the system ∂ 2 K β ∂ x α ∂ x ρ ∂ K ρ ∂ x γ = ∂ 2 H β ∂ x α ∂ x γ , (56) then K α and ˜ K α = H α automatic al ly satisfy the oriente d asso c i ativity e quations ( 8 ). Hence, (56) provides a conditional B¨ ac klund transformation for (8): if K α satisfy (8) a nd the conditions ∂ 2 K β ∂ x α ∂ x ρ ∂ K ρ ∂ x γ = ∂ 2 K β ∂ x γ ∂ x ρ ∂ K ρ ∂ x α (57) whic h are necessary for (56) to hold, then there exist, at least lo cally , ˜ K α = H α suc h that (56) holds and, moreo ver, these ˜ K α also satisfy (8). Setting K α = η αβ ∂ F /∂ x β , where η αβ is a symmetric nondegenerate constan t matrix, in Corollary 11 yields 13 Corollary 12 L et the functions F and H α satisfy the system η β ν η ρκ ∂ 3 F ∂ x α ∂ x ρ ∂ x ν ∂ 2 F ∂ x γ ∂ x κ = ∂ 2 H β ∂ x α ∂ x γ . (58) Then F automatic al ly sa tisfi e s the asso cia tivity e q uations (44) and ˜ K α = H α automatic al ly satisfy the oriente d ass o c iativity e quations (8). In complete analo g y with the ab ov e, (58) prov ides a B¨ ac klund tra nsformation relating the asso ciativity equations (4 4 ) supplemen ted with the conditions η ρκ ∂ 3 F ∂ x α ∂ x ρ ∂ x ν ∂ 2 F ∂ x γ ∂ x κ = η ρκ ∂ 3 F ∂ x γ ∂ x ρ ∂ x ν ∂ 2 F ∂ x α ∂ x κ (59) whic h a re necessary for (58) to hold, and the o rien ted asso ciativit y equ ations (8) for ˜ K α = H α . Note that the system (59) w as originally fo und and studied in [20, 22, 23] (in [20] it w as referred to as a condition for compat ible p o ten tial deformation of a pair o f the F r ob enius algebras) b ecause this sys- tem pla ys an imp ortan t ro le in classification of compatible Hamiltonian structures of hy dro dynamic type. Moreo ver, in [59] (cf. also [29]) it w as pro ved that (59) is also equiv alen t to the condition of inv olutivit y for a certain set of functionals constructed from the function F with resp ect to the constan t homogeneous first-order Poiss on brac k et of hydrodynamic type asso ciated with the flat contra v ariant metric η αβ . 7 Conclus ions and op en proble ms In the presen t pap er w e hav e found infinite hierarc hies of nonlo cal higher sym metries for the orien ted and “usual” asso ciativit y eq uations (8) and (44). These symme tries can b e emplo y ed for pro ducing new solutions from the kno wn ones and for constructing in v a rian t solutions using the standard tec hniques as presen ted e.g. in [4 9, 51, 52]. Moreo ver, it is natura l to ask whether there exist nonlo cal symme tries and conserv ation laws for (8) (resp. for (44)) that dep end o n the nonlo cal v ariables (36), ( 4 0) (resp. (40) with K α = η αβ ∂ F /∂ x β ) in a more complicated f ashion that t he symmetries found in Coro lla ry 4 (resp. Corollary 8). F or instance, o ne could lo ok for p oten tial (in the sense of [53, 60] and references therein) symmetries and conserv at io n law s of (8) in v olving the nonlo cal v a r iables (36) and (40). The next steps to tak e include elucidating the relatio nship among the nonlo cal symmetries o f (44) from Corollary 6 a nd the sym metries found in [37] for the gene ralized (in t he sens e of [42]) WDVV equations. The relationship (if an y exists) of t he flo ws (52) to the flo ws (5.15) from [61] could be of in terest to o . Understanding the precise relationship of t he symmetries from Corollary 8 to the tau- function a nd the B¨ a c klund transforma t io ns for the WDVV equations from [4] is y et another c hallenge. It w ould b e also 14 in teresting to find recursion op era t ors or master symmetries fo r (8) and (44) that generate the hierarc hies from Coro lla ries 4 and 8. Finally , it remains to b e seen whether the Darb oux-type transfor ma t io n from section 2 and the results from section 6 could indeed yield new solutions f or the oriented a nd “usual” asso ciativit y equations. W e in tend to address some of the ab o v e issues in our future w ork. Ac kno wledgmen ts This researc h w as supp orted in part by the Ministry of Education, Y outh and Sp orts of the Czec h Republic (M ˇ SMT ˇ CR) under grant MSM 4781305 9 04, and by Silesian Univ ersity in Opav a under gra nt IGS 9/2008. 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