Weyl groups and Elliptic Solutions of the WDVV equations

WEYL GR OUPS AND ELLIPTIC SO LUTIONS OF THE WD VV EQUA TIONS IAN A. B. STRA CHAN Abstract. A funct ional ansatz is dev eloped which give s certain elli ptic solu- tions of the Witten-Dijkgraaf-V erlinde-V erlinde (or WDVV) equ ation. This is based on the elliptic tri logarithm function introduced b y Beili nson and Levin. F or this to b e a solution results in a num ber of purel y algebraic conditions on the set of v ectors that app ear in the ansatz, this providing an elliptic version of the i dea, in tro duced b y V eselov, of a ∨ -system. Rational and trigonometric l imits are studied together wi th examples of elliptic ∨ -syste ms base d on v arious W eyl gr oups. Jacobi group orbit spaces are studied: these carry the structure of a F r obenius manifold. The corresp onding ‘almost dual’ structure is sho wn, in the A N and B N and conjecturally for an arbitrary W eyl group, to corresp ond to the elli ptic solutions of the WDVV equations. T r ansformation properties, under the Jacobi group, of the elli ptic triloga- rithm are derived together with v arious f unctional iden tities whi c h generalize the classical F rob enius-Stick elburger relations. Contents 1. Int ro duction 2 1.1. F rob enius Manifolds and almo st-duality 2 1.2. Examples 4 2. The Elliptic P olyloga rithm and its Pro p er ties 7 2.1. Notation 7 2.2. The elliptic polylo garithm 8 3. T rans formation prop erties of the WDVV equations 11 3.1. Analysis of the WDV V equations 12 3.2. Modula r transfo rmations of the structure functions 12 3.3. Periodicity prop erties of the structure functions 14 4. Singularity pro pe r ties 15 5. The Main Theorem 18 5.1. Rational and T rigono metric Limits 19 6. Examples of elliptic ∨ -systems 20 6.1. The case U = R W 21 6.2. The case U = R W ∪ R irr eg W 22 7. F rob enius- Stickelberger Identities 26 8. Jacobi Group O rbit Spa ces 27 8.1. Jacobi gr oups and Jaco bi forms 28 Date : July 21, 2018. 1991 Mathematics Subje ct Classific ation. 11F55, 53B50, 53D45. Key wor ds and phr ases. F rob enius manifolds, WD VV equations, Jacobi groups, elliptic func- tions, ell iptic p olylogarithms. 1 2 IAN A. B. STRAC HAN 8.2. Hurwitz spaces 30 9. Comments 33 Ac knowledgmen ts 34 References 34 1. Introduction One recurrent theme in the theor y of integrable systems is the tow er of general- izations rational − → trig onometric − → elliptic , the para digm b eing pro vided by the Calo gero-Mo ser system, where the original rational interaction term may b e g eneralized 1 z 2 − → 1 sin 2 z − → ℘ ( z ) whilst retaining in tegrability . A sec o nd recurrent theme is the app ea r ance of ro ot systems, the paradig m b e ing again provided by the Calo gero- Mo ser system where the interaction term X i 6 = j 1 ( z i − z j ) 2 can, on fixing the centre of mas s, b e wr itten as X α ∈R A N 1 ( α, z ) 2 , where the s um is taken ov e r the r o ots R A N of the A N Coxeter group [24]. The int egrability o f the system is prese r ved if o ther ro o t systems ar e used. These tw o themes o ccur in ma ny other in tegrable structures ; R ma tr ices, quan- tum gr oups, Dunkl op era tors, K Z-equations a ll admit (to a greater o r lesser extent) rational, trigo nometric a nd elliptic versions and g eneralizatio ns to arbitrar y ro ot systems (see for example [8] and the references therein). In this pap er elliptic solutions of the Witten-Dijkgr aaf-V er linde-V erlinde (or WD VV) equations will be studied for ar bitrary W ey l gr oups, these sitting at the rig ht of the following to wer of genera lizations: C N /W − → C N +1 / f W − → Ω /J ( g ) .  Coxeter gr oup orbit space  − →  Extended a ffine W eyl orbit space  − →  Jacobi gro up orbit space  W e begin by defining a F rob enius manifold. 1.1. F rob enius M anifolds and al m ost-duality. Definition 1. An algebr a ( A , ◦ , η , e ) over C is a F r ob enius algebr a if: • the algebr a {A , ◦ } is c ommutative, asso ciative with u nity e ; • the multiplic ation is c omp atible with a C -value d biline ar, symmetric, non- de gener ate inner pr o duct η : A × A → C WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 3 in the sense that η ( a ◦ b, c ) = η ( a, b ◦ c ) for al l a, b, c ∈ A . With this structure one may define a F rob enius manifold [9]: Definition 2. ( M , ◦ , e, η , E ) is a F r ob enius manifold if e ach tangent sp ac e T p M is a F r ob enius algebr a varying smo othly over M with the additional pr op erties: • the inner pr o duct is a flat metric on M (the term ‘metric’ wil l denote a c omplex-value d quadr atic form on M ). • ∇ e = 0 , wher e ∇ is the L evi-Civita c onne ction of the metric; • the tensor ( ∇ W ◦ )( X , Y , Z ) is t otal ly symmetric for al l ve ctors W, X , Y , Z ∈ T M ; • the ve ctor field E (the Euler ve ctor field) has the pr op erties ∇ ( ∇ E ) = 0 and the c orr esp onding one-p ar ameter gr oup of diffe omorphi sms acts by c on- formal tr ansformations of the m et ric and by r esc alings on the F r ob enius algebr as T p M . Since the metric η is fla t there exists a distinguished co or dinate sys tem (defined up to linear tra ns formations) of so-called flat co ordinates 1 { t α , α = 0 , . . . , N + 1 } in which the co mpo nent s of the metric are co nstant. F r om the v ario us symmetry prop erties o f tensors ◦ and ∇◦ it then follows tha t there exis ts a function F , the prep otential, s uch that in the fla t co ordinate system, c αβ γ = η  ∂ ∂ t α ◦ ∂ ∂ t β , ∂ ∂ t γ  , = ∂ 3 F ∂ t α ∂ t β ∂ t γ , and the a sso ciativity condition then implies that the pair ( F, η ) satisfy the WD VV- equations ∂ 3 F ∂ t α ∂ t β ∂ t λ η λµ ∂ 3 F ∂ t µ ∂ t γ ∂ t δ − ∂ 3 F ∂ t δ ∂ t β ∂ t λ η λµ ∂ 3 F ∂ t µ ∂ t γ ∂ t α = 0 , where α , β , γ , δ = 0 . . . , N + 1 . Consider the vector field E − 1 defined by the condition E − 1 ◦ E = e . This is defined o n M ⋆ = M \ Σ , where Σ is the discriminant s ubmanifold on which E − 1 is undefined. With this field one may define a new ‘dual’ multiplication ⋆ : T M ⋆ × T M ⋆ → T M ⋆ by X ⋆ Y = E − 1 ◦ X ◦ Y , ∀ X , Y ∈ T M ⋆ . This new multiplication is clea rly commutativ e and as so ciative, with the Euler vector field being the unity field for the new multiplication. F urthermor e, this new multiplication is compatible with the int ersection form g on the F rob enius manifold, i.e. g ( X ⋆ Y , Z ) = g ( X, Y ⋆ Z ) , ∀ X , Y , Z ∈ T M ⋆ . 1 This labeling is for future notational con v enience. 4 IAN A. B. STRAC HAN Here g is defined by the equa tion g ( X , Y ) = η ( X ◦ Y , E − 1 ) , ∀ X , Y ∈ T M ⋆ (and hence is well-defined o n M ⋆ ). Alternatively one may use the metric η to extend the o riginal multiplication to the cotangent bundle and define g − 1 ( x, y ) = ι E ( x ◦ y ) , ∀ x , y ∈ T ⋆ M ⋆ . The in tersection form has the imp ortant pr op erty that it is flat, and hence there exists a distinguished co ordinate system { p } in which the comp onents of the in- tersection form a re constant. It turns out that ther e exists a dual prep otential F ⋆ such that its third deriv a tives give the structure functions c ⋆ ij k for the dual m ultiplication. More pre cisely [10]: Theorem 3. Given a F r ob enius m anifold M , ther e ex ist s a fu nction F ⋆ define d on M ⋆ such that: c ⋆ ij k = g  ∂ ∂ p i ⋆ ∂ ∂ p j , ∂ ∂ p k  , = ∂ 3 F ⋆ ∂ p i ∂ p j ∂ p k . Mor e over, the p air ( F ⋆ , g ) satisfy the WDVV-e quations in the flat c o or dinates { p } of the metric g . Thu s given a spe c ific F rob enius manifold one may construct a ‘dua l’ solution to the WDVV-equations by constructing the fla t-co ordinates o f the intersection form and using the a b ove result to find the tensor c ⋆ ij k from which the dual prep otential may b e co nstructed. 1.2. E xamples. The simplest c la ss of F ro b enius manifo lds is given by the so-called Saito construction on the spa ce of orbits of a Coxeter group. Let W b e an irreducible Coxeter gro up acting on a real vector space V of dimens io n N . The ac tion extends to the co mplexified space V ⊗ C . The orbit space V ⊗ C /W ∼ = C N /W has a par ticularly nice str ucture, this following from Chev alley’s theorem on the ring of W - inv ar ia nt po lynomials: Theorem 4. Ther e ex ists a set of W -invariant p olynomial s i ( z ) , i = 1 , . . . , N such that C [ z 1 , . . . , z N ] W ∼ = C [ s 1 , . . . , s N ] . On this orbit space one ma y define a metric (a complex-v a lued qua dratic for m) by taking the Lie-deriv ative of the W -inv ar iant Euclidean metr ic g on V ⊗ C η − 1 = L e g − 1 where e is a v ector field constructed from the hig hest degree inv ariant p olynomial. It was proved b y K. Saito that this metric is non-deg e nerate a nd flat [28]. One therefore obtains a flat p encil of metrics from whic h one may construct a p olyno- mial solution - p olyno mial in the fla t co ordina tes of the metr ic η - to the WDVV equations. WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 5 The dua l prep otential for this class of F rob enius manifolds is pa rticularly simple: (1) F = 1 4 X α ∈R W ( α, z ) 2 log( α, z ) 2 where the sum is taken ov er the r o ots of the Coxeter gro up W [20, 21, 22]. How ever, the space of solutions of the same functional for m is far larg er. V eselov [32] derived the algebr aic c o nditions, known as ∨ -co nditions, on the set o f vectors U that are required for the prep otential F = 1 4 X α ∈ U ( α, z ) 2 log( α, z ) 2 to satisfy the WD VV equa tio ns (we assume throughout this pa p er that if α ∈ U then − α ∈ U automa tica lly). What is required here is a refinement of this idea, namely that of a complex Euclidean ∨ -sys tem [15] Definition 5. L et h b e a c omplex ve ctor sp ac e with non-de gener ate biline ar form ( , ) and let U b e a c ol le ction of ve ctors in h . A c omplex Eu clide an ∨ -system U satisfies the fol lowing c onditions: • U is wel l distribute d, i.e. P α ∈ U h α ( α, u )( α, v ) = 2 h ∨ U ( u , v ) for some λ ; • on any 2-dimensional plane Π the set Π ∩ U is either wel l distribute d or r e du cible (i.e. the u n ion of two non-empty ortho gonal subsystems). Note the following: • the constants h α could b e a bsorb ed in to the α . In applications these c o n- stants will b e b oth p ositive and neg a tive. Hence the r equirement o f a complex v ector space. • the co nstant h ∨ U can b e zero in certa in spaces. One further comment has to b e made in the case when h ∨ U = 0 . W e requir e her e that the in verse metric used in the WDVV equations is the non-deg e nerate bilinea r form ( , ) o n h rather than one - p o ssibly degenerate - constructed from the sum of deriv atives o f F as us ed in [14] . T rigo nometric s olutions were studied in [11], corresp o nding to extended affine W eyl groups. As in the Co xeter ca s e one has a Chev alley- t yp e theo rem and a well defined orbit s pace on which one may define, following the Sa ito -construction, a flat metric and hence a solution to the WD VV e quations. It is to be exp ected, thoug h a full pro of for arbitr a ry W eyl groups is currently lac king, that the corr esp onding dual solutions will take the following functional form (2) F = cubic terms + X α ∈R W h α Li 3  e i ( α, x )  where Li 3 ( x ) is the triloga rithm and h α are W ey l- inv ar iant sets of constants. Solu- tions of the WD VV equations of this t y pe hav e b een studied b y a num b er of author s [21, 23] but are only known to b e almost dual solutions to the extended a ffine W eyl F rob enius ma nifolds in certain sp ecial cas es (e.g. W = A ( k ) N ) [26]. T rigonometr ic ∨ -conditions, conditio ns on the v ectors α that ensure that the pr epo tential F = cubic terms + X α ∈ U h α Li 3  e i ( α, x )  6 IAN A. B. STRAC HAN satisfies the WDVV equatio ns, hav e a lso b een studied recently [13]. Elliptic so lutions were s tudied in [2], b eing defined o n the Ja cobi group orbit space Ω /J ( g ). F urther details and definitions will b e given in Section 8, following [2],[12] a nd [33]. The Jacobi g r oup J ( g ) (where g is a co mplex finite dimensiona l simple Lie algebra of ra nk N with W ey l gr oup W ) acts on the space Ω = C ⊕ h ⊕ H where h is the complex Car tan subalgebr a of g and H is the upper- half-plane, and this leads to the study of in v ariant functions - the Ja cobi forms. Ana lo gous to the Coxeter cas e, the o rbit space Ω /J ( g ) is a manifold and ca rries the structur e of a F rob enius manifold. In [27] , using the Hurwitz space des cription (see Section 8.2) Ω /J ( A N ) ∼ = H 1 ,N +1 ( N + 1 ) , the dual pr ep otential was constr uc ted. Theorem 6. [27] The int erse ction form on the sp ac e Ω /J ( A N ) is given by the formula g = 2 du dτ − N X i =0 ( dz i ) 2      P N j =0 z j =0 (wher e u ∈ C , z ∈ h and τ ∈ H ). The dual pr ep otential is given by the formula 2 F ⋆ ( u , z , τ ) = 1 2 τ u 2 − 1 2 u N X i =0 ( z i ) 2 + 1 2 X i 6 = j ′ 1 (2 π i ) 3 n L i 3 ( e 2 i ( z i − z j ) , e 2 π iτ ) − L i 3 (1 , e 2 π iτ ) o − ( N + 1) X j ′ 1 (2 π i ) 3 n L i 3 ( e 2 iz j , e 2 π iτ ) − L i 3 (1 , e 2 π iτ ) o . wher e this function is evaluate d on the plane P N j =0 z j = 0 . The precise definitio ns of the v arious terms in these formulae will b e given b elow, but for now w e no te that this dual prep otential is given in terms of the elliptic trilogar ithm L i 3 ( z , q ) introduced by Beilinson and Levin [1, 19]. This function has app eared a lready in the theory o f F rob enius ma nifolds in the enumeration of curves [18]. This r esult is curious - as well a s the A N ro ot vectors app earing in the solution certain extra vectors (in fa c t weigh t vectors) app ear : these do not app ear in the corres p o nding rationa l and trigonometr ic so lutions. This work rais ed a num b er o f questions: • Is ther e a direct verification that the function that app ear s in Theorem 6 s atisfies the WDVV equations? Recall that its construction w as via a Hurwitz spa ce construction in ter ms of cer ta in holomor phic ma ps b etw een the complex torus and the Riemann sphere . • What is the or igin of the ‘extra’ vectors in the s olution? 2 Note, P j ′ includes the term j = 0 . WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 7 • Can one construct so lutions for o ther W eyl groups? The purp ose of this pap er is to study so lutions of the WDVV equations which take the functional fo rm F ( u, z , τ ) = 1 2 u 2 τ − 1 2 u ( z , z ) + X α ∈ U h α f ( z α , τ ) , where f ( z , τ ) = 1 (2 π i ) 3  L i 3 ( e 2 π iz , e 2 π iτ ) − L i 3 (1 , e 2 π iτ )  , deriving a s e t o f elliptic ∨ -conditions on the ‘ro ots’ contained in the set U . Thus the ab ove questions can all b e answered affirmatively . This leav es the following question: • F or which elliptic ∨ -systems is the solutio n the almos t-dual so lution to the Jacobi g roup orbit spa ce Ω /J ( g ) ? This ques tion has b een ans wered alrea dy in the A N case [27] and in this pap er we extend the results to the B N case. F o r other W eyl groups it r e ma ins an op en problem. 2. The Elliptic P ol ylogarithm and its Proper ties The functional form of the a b ov e prep o ten tial uses the elliptic p oly logarithm. In this se c tion this is defined and its transformatio n pr op erties under shifts and mo dular tra nsformations ar e studied. Befor e this we define v a rious sp ecial functions and the no tation that will b e used throughout the rest of this pap er. 2.1. N otation. There ar e, unfortunately , ma ny different definitions and normal- izations for elliptic, num b er- theoretic and other sp ecial functions. Here we list the definitions used in this paper. Let q = e 2 π iτ , where τ ∈ H . • ϑ 1 -function: ϑ 1 ( z | τ ) = − i  e π iz − e − π iz  q 1 8 ∞ Y n =1 (1 − q n )  1 − q n e 2 π iz   1 − q n e − 2 π iz  . The fundamental la ttice is ge nerated by z 7→ z + 1 , z 7→ z + τ , and the function itself satisfies the complex heat equation ∂ 2 ϑ 1 ∂ z 2 = 4 π i ∂ ϑ 1 ∂ τ . • Bernoulli num b ers and Bernoulli p olyno mials: x e x − 1 = ∞ X n =0 B n x n n ! , B n ( z ) = n X k =0  n k  B k z n − k . • Eisenstein se ries: E k ( τ ) = 1 − 2 k B k ∞ X n =1 σ k − 1 ( n ) q n , k ∈ 2 N where σ k ( n ) = P d | n d k . 8 IAN A. B. STRAC HAN • Dedekind η -function: η ( τ ) = q 1 24 ∞ Y n =1 (1 − q n ) . • Polylogarithm function: Li N ( z ) = ∞ X n =1 z n n N , | z | < 1 . Note that ϑ 1 , E 2 and η are rela ted: η ′ ( τ ) η ( τ ) = 2 π i 24 E 2 ( τ ) = 1 12 π i ϑ ′′′ 1 (0 , τ ) ϑ ′ 1 (0 , τ ) . These hav e the following prop erties under inv er sion of the indep endent v ariable: τ − n E n  − 1 τ  = E n ( τ ) , n ≥ 4 ; τ − 2 E 2  − 1 τ  = E 2 ( τ ) + 12 2 π iτ ; η  − 1 τ  = r τ i η ( τ ) where in the last formula the square- r o ot is taken to hav e non-negative rea l part. The poly logarithm has the inv ersion prop er t y (for n ∈ N - other mor e complicated versions hold for other v alues): (3) Li n  e 2 π iz  + ( − 1) n Li n  e − 2 π iz  = − (2 π i ) n n ! B n ( z ) . This inv ersion formula ho lds: if ℑ ( z ) ≥ 0 for 0 ≤ ℜ ( z ) < 1 , a nd if ℑ ( z ) < 0 for 0 < ℜ ( z ) ≤ 1 . This, and other simila r formulae, may b e us e d to a nalytically contin ue the function outside the unit disc to a multi-v a lued holomor phic function on C \{ 0 , 1 } . F or a dis c ussion of the monodr omy of the polylo garithm function s ee [25]. This m ultiv aluedness will also oc c ur in the so lutio n of the WD VV e q uations. How ever this multiv aluedness o ccur s in the quadr atic terms o nly and hence any ph ysical quantities are sing le-v alued. 2.2. The ellipti c p olylog arithm . An ‘obvious’ elliptic generalizatio n o f the p o ly - logarithm function is L i r ( ζ , q ) = ∞ X n = −∞ Li r ( q n ζ ) . How ever this series div erges, but by using the inv ersion formula (3) and ζ -function regular iz ation one can arr ive at the following definition of the elliptic p olylo garithm function [1, 19]: L i r ( ζ , q ) = ∞ X n =0 Li r ( q n ζ ) + ∞ X n =1 Li r ( q n ζ − 1 ) − χ r ( ζ , q ) , r o dd , WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 9 where χ r ( ζ , q ) = r X j =0 B j +1 ( r − j )!( j + 1 )! (log ζ ) ( r − j ) (log q ) j . A rea l-v alued version of this function had previously been studied by Zagier [35]. With this the function f may b e defined. Definition 7. The function f ( z , τ ) , wher e z ∈ C , τ ∈ H , is define d t o b e: f ( z , τ ) = 1 (2 π i ) 3  L i 3 ( e 2 π iz , q ) − L i 3 (1 , q )  . It follows from the definitions that (4)  d dτ  3 1 (2 π i ) 3 L i 3 (1 , q ) = 1 120 E 4 ( τ ) and (5)  ∂ ∂ z  2 f ( z , τ ) = − 1 2 π i log  ϑ 1 ( z , τ ) η ( τ )  . Thu s the elliptic-triloga rithm may be thought of as a classical function (or, at lea st, a neo classical function) as it may b e o btained from classica l functions via nes ted int egratio n a nd other standard pro cedures. It does , howev er , provide a s ystematic wa y to deal w ith the arbitra ry functions that would appe a r this wa y . The notation F ≃ G will b e used if the functions F and G differ b y a qua dratic function in the v ariables { u, z , τ } (recall that any prep otential satisfying the WD VV equations is only defined up to quadr atic terms in the flat-c o ordint es). Prop ositi o n 8. The function f has the fol lowing tr ans formation pr op erties: f ( z + 1 , τ ) ≃ f ( z , τ ) ; f ( z , τ + 1) ≃ f ( z , τ ) ; f ( z + τ , τ ) ≃ f ( z , τ ) +  1 6 z 3 + 1 4 z 2 τ + 1 6 z τ 2 + 1 24 τ 3  ; f ( − z , τ ) ≃ f ( z , τ ) . The function also has the alternative exp ansions: (6) f ( z , τ ) ≃ − 1 (2 π i )  1 2 z 2 log z + z 2 log η ( τ )  + 1 (2 π i ) 3 ∞ X n =1 ( − 1) n E 2 n ( τ ) B 2 n (2 n + 2)!(2 n ) (2 π z ) 2 n +2 and (7) f ( z , τ ) ≃ 1 (2 π i ) 3 Li 3  e 2 π iz  + 1 12 z 3 − 1 24 z 2 τ − 4 (2 π i ) 3 ∞ X r =1  q r (1 − q r )  sin 2 ( π rz ) r 3 F u rthermor e, f  z τ , − 1 τ  ≃ 1 τ 2 f ( z , τ ) − 1 τ 3 z 4 4! . 10 IAN A. B. STRAC HAN Pro of The first t wo r elations follow from the definition. The third and fourth use the inversion for m ula for polylo garithms (3). The pro of o f (6) and (7) just inv o lves some care ful resumming. Consider the first tw o terms in the definition o f f : ∞ X n =0 Li 3 ( q n e 2 π iz ) + ∞ X n =1 Li 3 ( q n e − 2 π iz ) = Li 3 ( e 2 π iz ) + 2 ∞ X s =0 ( − 1) s (2 s )! ( ∞ X n,r =1 q nr r 2 s − 3 ) (2 π z ) 2 s . F rom this series (7) follows immediately . T o obtain (6 ) one re arrang es the terms. The s = 0 term cancels in the final expre ssion and the remaining terms may be re-expres sed in terms of Eise nstein series (for s > 1) or the Dedekind function (for s = 1). Finally , using the result 1 (2 π i ) 3 d 3 dz 3 Li 3  e 2 π iz  = − 1 2 [1 + coth( πi z )] , = − " 1 2 + 1 (2 π z ) + ∞ X n =1 B 2 n (2 n )! (2 π iz ) 2 n − 1 # one may obtain a series for Li 3  e 2 π iz  . Putting all these parts together gives the series (6).  Theorem 9 . The function h ( z , τ ) = f (2 z , τ ) − 4 f ( z , τ ) satisfies the p artial differ ential e quation h (3 , 0) ( z , τ ) h (1 , 2) ( z , τ ) − h h (2 , 1) ( z , τ ) i 2 + 4 h (0 , 3) ( z , τ ) = 0 wher e h ( m,n ) ( z , τ ) = ∂ n + m h ∂ z m ∂ τ n . Pro of Le t ∆( z , τ ) = h (3 , 0) ( z , τ ) h (1 , 2) ( z , τ ) − h h (2 , 1) ( z , τ ) i 2 + 4 h (0 , 3) ( z , τ ) . Using the transfo r mation pro pe rties in Prop o s ition 8 one ma y der ive the transfor - mation prop er ties of the deriv atives and he nce for the co mbin ation ∆ . While the individual terms have quite complicated tra nsformation prop er ties , those for ∆ are very simple: ∆( z + 1 , τ ) = ∆( z , τ ) , ∆( z + τ , τ ) = ∆( z , τ ) , ∆  z τ , − 1 τ  = τ 4 ∆( z , τ ) . The first tw o of these equations imply that the function ∆ is doubly p er io dic. F ro m the series expansio n in Prop os itio n 8 it follo ws that ∆ has no p oles : the only term which has a p ole is f (3 , 0) and this cancels with the zero in f (1 , 2) . Thus ∆ is doubly WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 11 per io dic with no p oles and hence must be a function of τ -alone (i.e. a theta- constant). The r emaining transforma tion prop erty implies that ∆ is a mo dular function o f deg ree 4 . The q - series representation of the function f in P rop osition 8 implies that ∆ is actually a cusp-form. But the space of cus p-forms of de g ree 4 is empt y and hence ∆ = 0 .  Corollary 10. The p air F ( u, z , τ ) = 1 2 u 2 τ − uz 2 + h ( z , τ ) , = 1 2 u 2 τ − uz 2 + f (2 z , τ ) − 4 f ( z , τ ) and g = 2 du dτ − 2 dz 2 satisfy the WDVV e qu ations. Pro of The WD VV equations for the a b ove prep otential reduce to the single equation ∆ = 0 so the result follows immediately from the a bove Theo rem.  Using the same metho ds it is str a ightforw ard to show that the function (8) ˜ h ( z , τ ) = 3 4(2 π i ) 3  L i 3 ( e 2 π iz , q ) + 3 2 L i 3 (1 , q )  also satisfies the equatio n ∆ = 0 . It is, how ever, the function h which de fines the dual prep otential to the A 1 -Jacobi gr oup orbit s pace - s ee Theorem 6. 3. Transforma tion proper ties of the WDVV equa tions Recall that w e see k a solution o f the WDVV of the form (9) F ( u, z , τ ) = 1 2 u 2 τ − 1 2 u ( z , z ) + X α ∈ U h α f ( z α , τ ) with f ( z , τ ) b eing given by Definition 7 . Sometimes the notation z α will be us ed to denote ( z , α ) , esp ecia lly for terms inv olving the function f . T hus f (( z , α ) , τ ) will be written f ( z α , τ ) or even f ( z α ) . The co ordinates { t α , α = 0 , 1 , . . . N , N + 1 } a re defined to b e t 0 = u , t i = z i , i = 1 , . . . N , t N +1 = τ . Latin indices will range fro m 1 to N a nd Greek fro m 0 to N + 1 so the dimension of the manifo ld is N + 2 , with N ≥ 1 . In addition u ∈ C , z ∈ h ∼ = C N , τ ∈ H , so ( u, z , τ ) ∈ Ω . Later , h will b e the co mplex Car ta n subalge br a of a simple Lie algebra g of ra nk N with W eyl group W , but for now it may b e thoug ht of a just C N . Also, ( , ) denotes the standa rd Euc lidea n inner pro duct on h . It follows fro m the functional dep endence on t 0 = u that ∂ u is the unity vector field and he nc e the metric on Ω is g = dτ du + du dτ − ( d z , d z ) . One of the main ideas of this paper is to extend Theorem 9 to hig her dimension, using do ubly-p erio dicity and mo dular arg umen ts to pr ov e that the WDVV e qua- tions ar e satisfied. T o b e gin we require a detailed analys is of the WDV V equations themselves. 12 IAN A. B. STRAC HAN 3.1. Analysi s of the WD VV equations. The WDVV equations are the condi- tions for a commutativ e algebra to b e asso cia tive. Thus they may b e written in terms of the v anishing of the ass o ciator ∆[ X , Y , Z ] = ( X ◦ Y ) ◦ Z − X ◦ ( Y ◦ Z ) . Since in the case b eing consider ed we hav e a unity ele ment these simplify further: if a ny of the vector field is equal to the unity field then ∆ v anishes identically . Since the vector field ∂ τ ∈ T H is sp ecial (for example, it b ehaves differently to the other v ariables under mo dula rity trans fo rmation), we decomp ose these e quation further, taking the inner pro duct with a r bitrary v ector fields to obtain scalar - v alued equations. Prop ositi o n 11. The WDVV e quations for a mu ltiplic ation with un ity field ar e e quivalent to the vanishing of the fol lowing functions: ∆ (1) ( u , v ) = g ( ∂ τ ◦ ∂ τ , u ◦ v ) − g ( ∂ τ ◦ u , ∂ τ ◦ v ) , ∆ (2) ( u , v , w ) = g ( ∂ τ ◦ u , v ◦ w ) − g ( ∂ τ ◦ w , u ◦ v ) , ∆ (3) ( u , v , w , x ) = g ( u ◦ v , w ◦ x ) − g ( u ◦ x , v ◦ w ) for al l u , v , w , x ∈ T h . In terms of co ordinate vector fields these conditions ar e: ∆ (1) ij = g ij c τ τ τ + g pq { c τ τ p c ij q − c τ ip c τ j q } , ∆ (2) ij k = { g j k c τ τ i − g ij c τ τ k } + g pq { c τ ip c j kq − c τ k p c ij q } , ∆ (3) ij r s = { g ij c τ r s + g r s c τ ij − g is c τ r j − g r j c τ is } + g pq { c ij p c r sq − c isp c r j q } where g ij = − ( ∂ i , ∂ j ) . The function ∆ in theorem (9 ) is, s ince dim C h = 1, pr op or- tional to ∆ (1) ( x , x ) . 3.2. M o dular transformations of the structure functions. Lemma 12 . L et (10) ˆ u = u − ( z , z ) 2 τ , ˆ z = z τ , ˆ τ = − 1 τ . Then F ( ˆ u, ˆ z , ˆ τ ) = 1 τ 2  F ( u, z , τ ) − 1 2 u (2 uτ − ( z , z ))  if and only if (11) X α ∈ U h α ( α, z ) 4 = 3( z , z ) 2 . Pro of This follows immediately from Pr op osition 8.  WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 13 The origin of the transformation (10) comes from the study of symmetries of the WD VV equations (see [9] Appendix B). A sy mmetr y is a transforma tion t α 7→ ˆ t α , g αβ 7→ ˆ g αβ , F 7→ ˆ F that acts on the solution spa ce of the WD VV equa tions. In particular, (10) is just the trans formation, denoted I in [9], ˆ t 0 = 1 2 t σ t σ t N +1 , ˆ t i = t i t N +1 , i = 1 , . . . , N , ˆ t N +1 = − 1 t N +1 , ˆ g αβ = g αβ , ˆ F ( ˆ t ) =  t N +1  − 2  F ( t ) − 1 2 t 0 ( t σ t σ )  which induces a symmetry o f the WDVV equations. Up to a simple equiv alence, I 2 = I . It follows from this a nd Lemma 12 that we are at the fixed p oint of this inv olution and that, rather than telling one how to constr uc t a new solution from a seed so lutio n, it gives the transfor mation prop erty of the v arious functions under the mo dular transfor mations. A simple mo dification of Lemma B.1 [9] immediately gives: Prop ositi o n 1 3. Su pp ose F is given by e quation (9) wher e c ondition (11 ) is as- sume d to hold. L et c αβ γ = ∂ α ∂ β ∂ γ F ( t ) . Then c αβ γ ( z , τ + 1) = c αβ γ ( z , τ ) and c ij k  z τ , − 1 τ  = τ c ij k ( z , τ ) − g ij z k − g j k z i − g ki z j , c τ ij  z τ , − 1 τ  = τ c ij α ( z , τ ) t α − 1 2 g ij ( t σ t σ ) − z i z j , c τ τ i  z τ , − 1 τ  = τ c iαβ ( z , τ ) t α t β − z i ( t σ t σ ) , c τ τ τ  z τ , − 1 τ  = τ c αβ γ ( z , τ ) t α t β t γ − 3 4 ( t σ t σ ) 2 . With these, the tr ansformation pr op erties of the functions ∆ ( i ) ar e ∆ ( i ) ( z , τ + 1) = ∆ ( i ) ( z , τ ) , i = 1 , 2 , 3 14 IAN A. B. STRAC HAN and ∆ (3) ij r s  z τ , − 1 τ  = τ 2 ∆ (3) ij r s ( z , τ ) , ∆ (2) ij k  z τ , − 1 τ  = τ 3 ∆ (2) ij k ( z , τ ) + τ 2 z r ∆ (3) irk j ( z , τ ) , ∆ (1) ij  z τ , − 1 τ  = τ 4 ∆ (1) ij ( z , τ ) − τ 3 z r n ∆ (2) ij r ( z , τ ) + ∆ (2) j ir ( z , τ ) o + τ 2 z a z b ∆ (3) abij ( z , τ ) . Pro of The pro o f is straightforw ard a nd uses the transforma tion pr op erties of f derived in Prop osition 8.  It is imp ortant to note that these ∆ ( i ) are p ow er s series, not La ur ent series, in the q -v ar iable. Again, this follows from the q - expansion of f given in pr op osition 8. 3.3. Perio dicit y prop e rti es of the structure functions. W e assume that there exists a vector p ∈ h s uch that ( α, p ) ∈ Z for all α ∈ U . Later we will requir e the existence of a full N -dimensional la ttice (the ‘w eight lattice’ asso cia ted to the ‘ro ots’ in U ), but for now we just require a single such vector. F rom Prop osition 8 it follows that f (( α, z + p ) , τ ) ≃ f (( α, z ) , τ ) and hence F ( u, z + p , τ ) ≃ F ( u, z , τ ) . Th us ∆ ( i ) ( z + p , τ ) = ∆ ( i ) ( z , τ ) , i = 1 , 2 , 3 . The calcula tion of the transforma tio ns under shifts z 7→ z + p τ requires more c are. Prop ositi o n 14. Assume that the fol lowing c onditions hold: X α ∈ U h α ( α, z ) 4 = 3( z , z ) 2 , and ( α, p ) ∈ Z for al l α ∈ U . Then h ( z + p τ , τ ) ≃ h ( z , τ ) + 1 8    4( p , z )( z , z ) + τ  4( p , z ) 2 + 2( p , p )( z , z )  +4 τ 2 ( p , z )( p , p ) + τ 3 ( p , p ) 2    wher e h ( z , τ ) = X α ∈ U h α f ( z α , τ ) . Pro of F rom Pr op osition 8 it follows by induction, for n ∈ Z , that f ( z + nτ , τ ) ≃ f ( z , τ ) + 1 24  4 nz 3 + 6 n 2 τ z 2 + 4 n 3 τ 2 z + n 4 τ 3  and since, b y assumption, ( α, p ) ∈ Z , it follows immedia tely that f (( α, z ) + ( α, p ) τ , τ ) ≃ f (( α, z ) , τ ) + 1 24    4( α, p )( α, z ) 3 + 6 τ ( α, p ) 2 ( α, z ) 2 + 4 τ 2 ( α, p ) 3 ( α, z ) + τ 3 ( α, p ) 4    . WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 15 Hence, on summing ov er α , h ( z + p τ , τ ) ≃ h ( z , τ ) + 1 24    4 P α h α ( α, p )( α, z ) 3 + 6 τ P α h α ( α, p ) 2 ( α, z ) 2 + 4 τ 2 P α h α ( α, p ) 3 ( α, z ) + τ 3 P α h α ( α, p ) 4    . Hence, using the first co ndition (and its p olarized v ersion), the r esult follows.  With this the follo wing Prop os ition may b e proved: the first part is immediate and the s econd part follo ws from routine but tedious calculations. Prop ositi o n 15. Under the c onditions of the ab ove pr op osition, the stru ctur e func- tions have the fol lowing tra nsformation pr op ert ies: c ij k ( z + p τ , τ ) = c ij k ( z , τ ) + p i g j k + p j g ki + p k g ij , c τ ij ( z + p τ , τ ) = c τ ij ( z , τ ) − p a c ij a ( z , τ ) −  p i p j + 1 2 ( p , p ) g ij  , c τ τ i ( z + p τ , τ ) = c τ τ i ( z , τ ) − 2 p a c τ ai ( z , τ ) + p a p b c abi ( z , τ ) + ( p , p ) p i , c τ τ τ ( z + p τ , τ ) = c τ τ τ ( z , τ ) − 3 p a c τ τ a ( z , τ ) + 3 p a p b c τ ab ( z , τ ) − p a p b p c c abc ( z , τ ) − 3 4 ( p , p ) 2 . The ∆ ( i ) have the fol lowing tr ansformation pr op erties: ∆ (3) ij r s ( z + p τ , τ ) = ∆ (3) ij r s ( z , τ ) , ∆ (2) ij k ( z + p τ , τ ) = ∆ (2) ij k ( z , τ ) + p a ∆ (3) ij k a ( z , τ ) , ∆ (1) ij ( z + p τ , τ ) = ∆ (1) ij ( z , τ ) + p a n ∆ (2) ij a ( z , τ ) + ∆ (2) j ia ( z , τ ) o + p a p b ∆ (3) ij ab ( z , τ ) , W e a re now in the p os ition to rehear se the main theorem. If we hav e a full N -dimensiona l weight lattice, then ∆ (3) is doubly p erio dic in all z v ariables and if we can show it has no p ole s then it m ust b e a function o f τ alone. The mo du- larity prop er ties of ∆ (3) then imply that it m ust b e zero. Rep eating the argument sequentially for ∆ (2) and then ∆ (1) will give the desired result. T o pro ceed further requires the examination of the singularity pro p e rties of the ∆ ( i ) . 4. Singularity pr oper ties T o study the singula rity prop erties of the ∆ ( i ) we require a more detailed a nalysis of these functions. Using equation (9) and Pro p osition 11 one o btains: 16 IAN A. B. STRAC HAN ∆ (1) = ∆ (1) ( u , v ) = − ( u , v ) X α ∈ U h α f (0 , 3) ( z α , τ ) + X α,β ∈ U h α h β ( α, β )( α, v )  +( β , u ) f (2 , 1) ( z β , τ ) f (2 , 1) ( z α , τ ) − ( α, u ) f (1 , 2) ( z β , τ ) f (3 , 0) ( z α , τ )  ∆ (2) = ∆ (2) ( u , v , w ) = X α ∈ U h α [( u , v )( α, w ) − ( w , v )( α, u )] f (1 , 2) ( z α , τ ) + X α,β ∈ U h α h β ( α, β )( α, v ) [ ( α ∧ β )( u , w )] f (2 , 1) ( z β , τ ) f (3 , 0) ( z α , τ ) ∆ (3) = ∆ (3) ( u , v , w , x ) = X α ∈ U h α  +( α, v )( α, w )( u , x ) − ( α, x )( α, w )( u , v ) +( α, u )( α, x )( v , w ) − ( α, u )( α, v )( w , x )  f (2 , 1) ( z α , τ ) − 1 2 X α,β ∈ U h α h β ( α, β )[( α ∧ β )( u , w )][( α ∧ β )( v , x )] f (3 , 0) ( z α , τ ) f (3 , 0) ( z β , τ ) , where ( α ∧ β )( u , v ) = ( α, u )( β , v ) − ( α, v )( β , u ) . The only deriv a tive o f f that g ives rise to a p ole is the f (3 , 0) deriv ative; all other deriv atives are analytic in z - this following from (6). Therefor e the only parts of the ∆ ( i ) that could con tribute to a singularity a re those whic h contain this deriv a tive. Prop ositi o n 16. L et Π α denote a plane thr ough the origin c ontaining the ve ct or α and α ⊥ a ve ctor in Π α p erp endicular to α . Then, at ( α, z ) = 0 : • ∆ (1) ( u , v ) has no p ole if the sc alar e qu ation (12) X β ∈ Π α ∩ U h β ( α, β )( β , α ⊥ ) 2 n +1 = 0 , n = 1 , 2 , . . . , holds. • ∆ (2) ( u , v , w ) has no p ole if the biline ar form e quation (13) X β ∈ Π α ∩ U h β ( α, β )( α ∧ β )( β , α ⊥ ) 2 n = 0 , n = 1 , 2 , . . . , holds. • ∆ (3) ( u , v , w , x ) has no p ole if t he 4-line ar form e qu ation (14) X β ∈ Π α ∩ U h β ( α, β )( α ∧ β ) ⊗ ( α ∧ β )( β , α ⊥ ) 2 n +1 = 0 , n = 1 , 2 , . . . , holds. WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 17 Her e ( α ∧ β )( u , v ) = ( α, u )( β , v ) − ( α, v )( β , u ) and ( α ∧ β ) ⊗ ( α ∧ β )( u , v , w , z ) = [( α ∧ β )( u , v )] [( α ∧ β )( w , x )] . Pro of The only third deriv ative of f which contains a po le is the f (3 , 0) -deriv ative and hence the only par t of ∆ (1) that could contain p o le s is the term X α,β ∈ U h α h β ( α, β )( α, v )( α, u ) f (1 , 2) ( z β ) f (3 , 0) ( z α ) . Since f (3 , 0) only has a simple po le the ter m inv olving the p o les is, up to a non- z e ro constant, X α ∈ U h α ( α, u )( α, v ) ( z , α ) X β ∈ U h β ( α, β ) f (1 , 2) ( z β ) . The function f (1 , 2) ( z β ) is o dd and hence may be written as P ∞ n =0 A n ( τ )( z , β ) 2 n +1 (the explicit expressions for the no n- zero functions A n are not required - they may be derived from Prop ositio n 8). Thus a sufficient co nditio n fo r the absence o f p oles is, for a rbitrary α ∈ U : ( α, z ) divides X α fixe d , β ∈ U h β ( α, β )( β , z ) 2 n +1 , n = 0 , 1 , . . . . Note that this is a utomatically satisfie d if n = 0 by the fir st condition in Definition 5. T his sum may b e re wr itten as sums over vectors in 2 -planes Π α containing α , and hence a sufficient condition for the absence of p o les is , for arbitra ry α ∈ U : ( α, z ) divides X β ∈ Π α ∩ U h β ( α, β )( β , z ) 2 n +1 , n = 1 , 2 , . . . . On decomp osing each β in the plane Π α as β = µα + ν α ⊥ (so ν = ( β , α ⊥ ) / ( α ⊥ , α ⊥ )) one finds that a ll terms in the binomia l expansion of ( β , z ) 2 n +1 contain a ( α, z )- term except the fina l [ ν ( z , α ⊥ )] 2 n +1 -term. T hus a sufficient condition co ndition for the abse nce o f po les in ∆ (1) is X β ∈ Π α ∩ U h β ( α, β )( β , α ⊥ ) 2 n +1 = 0 , n = 1 , 2 , . . . . The pro of o f the ∆ (2) condition is identical: f (2 , 1) is an even function, and the low est term v anishes o n using the firs t condition in Definition 5 . The function ∆ (3) contains a term X α ,β ∈ U h α h β ( α, β ) [ ( α ∧ β )( u , w )] [( α ∧ β )( v , x )] 1 ( α, z ) 1 ( β , z ) . This v anishes by definition of a complex Euclidean ∨ -sys tem [15] . The pro of of the remaining ∆ (3) condition is ide ntical to the ab ove: f (3 , 0) is an o dd function, a nd the low est term v anishes on us ing a p ola rized version o f condition (11) .  18 IAN A. B. STRAC HAN 5. The Main Theorem W e can now draw the v arious comp onents together, but first w e define an elliptic ∨ -system. Definition 17 . L et U b e a c omplex Euclide an ∨ -system. An el liptic ∨ -system is a c omplex Euclide an ∨ -syst em with the fol lowing additio nal c onditions: • P α ∈ U h α ( α, z ) 4 = 3( z , z ) 2 ; • The thr e e c onditions in Pr op osition 16 hold; • Ther e ex ists a ful l N -dimensional weight lattic e of ve ctors p such t hat ( p , α ) ∈ Z for al l α ∈ U . Examples o f elliptic ∨ -systems w ill b e constructed in the next sectio n. With this definition in pla ce o ne arrives a t the main theorem. Theorem 1 8 . L et U b e an el liptic ∨ -system. If h ∨ U = 0 then t he function (15) F ( u, z , τ ) = 1 2 u 2 τ − 1 2 u ( z , z ) + X α ∈ U h α f ( z α , τ ) satisfies the WDVV e qu ations. If h ∨ U 6 = 0 t hen the mo difie d pr ep otential (16) F − → F + 10 ( h ∨ U ) 2 3(2 π i ) 3 L i 3 (1 , q ) satisfies the WDVV e qu ations. Pro of F rom the co nditions in the definition of an elliptic ∨ -system and P rop o- sition 1 5 it fo llows that ∆ (3) is doubly p erio dic in a ll z -v a riables, and from the conditions in Pro p o sition 16 it follows that it has no po les. It therefore m ust b e a function of τ alone. F rom P rop osition 1 3 it follows that ∆ (3) is a mo dular function of degr ee 2 a nd fro m Pr op osition 13 it follows that it co ntains only p ositive p owers in its q -expansion. Hence it is a mo dular form of degree 2 and hence must b e z ero. This argument can now be rep eated for ∆ (2) (a modula r function of degree 3 with only p ositive p ow ers in its q -expansion and hence a mo dula r form of degree 3 and so must b e zero). Finally , the same arg ument s implies that ∆ (1) is a mo dular for m o f deg r ee 4, and hence it must b e a multiple of the modular form E 4 ( τ ) . Th us ∆ (1) ( u , v ) = m ( u , v ) E 4 ( τ ) . T o find m ( u , v ) one just requires the O (1)-ter ms in the q - expansion of ∆ (1) . On using equatio n (7) one finds that m ( u , v ) = X α,β ∈ U h α h β ( α, β )( α, v )( β , u )  − 1 12  2 , = 1 36 ( h ∨ U ) 2 ( u , v ) . Hence ∆ (1) = 1 36 ( h ∨ U ) 2 E 4 ( τ ) ( u , v ) . WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 19 Thu s if h ∨ U = 0 then ∆ ( i ) = 0 for i = 1 , 2 , 3 and hence (15) s a tisfies the WD VV equations. If h ∨ U 6 = 0 one has to mo dify the ansatz for F : F − → F + µ 1 (2 π i ) 3 L i 3 (1 , q ) . This change only effects c τ τ τ and hence the ab ov e a rgument on the v a nishing of ∆ (3) and ∆ (2) is unchanged. With this new ansa tz ∆ (1) undergo es a slight change: ∆ (1) − → ∆ (1) − µ 1 120 E 4 ( τ ) ( u , v ) , on using (4 ). Thus if µ = 10 3 ( h ∨ U ) 2 then the modified ∆ (1) is zero and hence (16) satisfies the WD VV equations.  5.1. R ational and T rigonometric Limits . F rom the lea ding order b ehaviour, obtained from P rop ositio n 8 , f = − 1 (4 π i ) z 2 log z as z → 0 , and f ≃ 1 (2 π i ) 3 Li 3  e 2 π iz  + 1 12 z 3 as q → 0 one may o btain r ational a nd tr igonometric solutions, of low er dimension, of the WD VV equations . Prop ositi o n 19. Given an el liptic ∨ -s yst em U the fol lowing ar e solutions of the WDVV e quations: R ational limit F r ational = X α ∈ U h α ( α, z ) 2 log( α, z ) . The metric in this c ase is the standar d Euclide an inner pr o duct on h . T rigonometric limit I If h ∨ U = 0 then F tr ig = X α ∈ U h α Li 3  e 2 π i ( z ,α )  . The metric in this c ase is the standar d Euclide an inner pr o duct on h . T rigonometric limit II If h ∨ U 6 = 0 then F tr ig = 1 6 u 3 − 1 2 u ( z , z ) + 1 (2 π i ) 3  3 h ∨ U  1 2 X α ∈ U h α Li 3  e 2 π i ( z ,α )  . In this c ase one has a c ovariantly c onstant unity ve ct or field ∂ u and henc e the m et ric in this c ase g = du 2 − ( d z , d z ) . The pro of just inv olves the examination o f the asso cia tor ∆ (3) under the a b ov e men tioned limits. 20 IAN A. B. STRAC HAN 6. Examples of el liptic ∨ -systems In this section we construct e xamples of elliptic ∨ -s ystems based o n a W eyl group W . Recall that by as sumption, if α ∈ U then − α ∈ U . W e now also ass ume that the constants h α are W eyl inv a riant, i.e. h w ( α ) = h α for w ∈ W . W e denote the nu mber of vectors in U by | U | . The calc ula tions for sp ecific gro ups will b e done using the standard notion fo r ro ots and w eights, see for example [1 6]. Two classes of exa mples will b e given, the first whe r e U = R W (where R W is the ro ot system of W ) and the second whe r e U = R W ∪ R irr eg W , where R irr eg W contains a set of W -inv a riant vectors that form an irr egular orbit under the actio n of W . W e first co nstruct W -inv ariant sets o f vectors (and constants h α ) sa tisfying the tw o conditions X α ∈ U h α ( α, z ) 4 = 3( z , z ) 2 , (17) X α ∈ U h α ( α, u )( α, v ) = 2 h ∨ U ( u , v ) (18) and then chec k that the co nditio ns in Prop ositio n 16 are satisfied, which will be done with the help of the following lemma. Recall that these conditio ns inv olve summing ov er vectors in the plane Π α ∩ U . In the c a ses to be discussed here these vectors o ccur in pairs , related by certain re fle c tio ns, and the corres p o nding terms in the s um c a ncel. Let σ α β denote the re flec tion of the vector β in the line with normal vector α . The pairs in the set of vectors Π α ∩ U will o cc ur in tw o types: ✟ ✟ ✟ ✟ ✯ ❍ ❍ ❍ ❍ ❨ ✲ σ α β β α Type A: α ∈ R W ✟ ✟ ✟ ✟ ✯ ❍ ❍ ❍ ❍ ❥ ✲ σ β ⊥ α α β = α + σ β ⊥ α Type B Type A pa irs ar e very familia r: they o ccur in W eyl group (indeed, Co xeter gro up) ro ot sys tems (with cer tain sp ecia l a ng les). Type B pairs will o ccur when an extra set of W e yl inv ar iant vectors is a pp ended to the ro ot system - see Section 6.2. Both these types of co nfiguration app ear in ∨ -s ystems a nd defor med ro ot systems [7, 14, 15, 32]. Lemma 20. L et α ∈ U and supp ose t hat the terms in t he sums P β ∈ Π α ∩ U o c cur in p airs of T yp e A or T yp e B. Then the c onditions in Pr op osition 16 ar e s atisfie d: (a) for typ e A c onfigura tions if and only if h β = h σ α β ; (b) for typ e B c onfigur ations if and only if ( α, β ) h β = ( α, α − β ) h σ β ⊥ α . Pro of Consider the firs t co nditio n in P rop ositio n 16, namely e quation (12), and consider the par tial sums: WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 21 Type A Ξ A = h β ( α, β )( α ⊥ , β ) n + h σ α β ( α, σ α β )( α ⊥ , σ α β ) n ; Type B Ξ B = h β ( α, β )( α ⊥ , β ) n + h σ β ⊥ α ( α, σ β ⊥ α )( α ⊥ , σ β ⊥ α ) n . It is easy to show that Ξ A = 0 if and only if h β = h σ α β . Similarly o ne may show (and here the condition tha t β = α + σ β ⊥ α is used) that Ξ B = 0 if and o nly if (19) ( α, β ) h β = ( α, α − β ) h σ β ⊥ α The full s um is made up of sums of such paired-ter ms, and hence is z e ro. Rep eating the ar gument fo r the terms that appea r in equa tio ns (13) and (14) y ields no further conditions.  Note that we hav e assumed that the h α are W e yl inv ariant and hence for type A configuratio ns the conditions in P rop ositio n 16 are automa tically satisfied with no extra conditions. T o illustrate this we b egin with the simplest ca se, where W = A 1 , which will repro duce the examples constructed e arlier. Example 21. W = A 1 • | U | = 2 L et U = R A 1 = {± α } (normalize d so that ( α, α ) = 2 ). Then c onditions (17) and (18) imply that h α = 3 8 , h ∨ A 1 = 3 4 (note t heir r atio is 2, which is the (dual) Coxeter numb er of A 1 ). The p ole c onditions ar e vacuous in this c ase. This gives solution (8). • | U | = 4 L et U = {± α, ± ˜ α } with ( α, α ) = 2 , ( ˜ α, ˜ α ) = ν . Then c onditions (17) and (18) imply that 8 h α + 2 ν 2 h ˜ α = 3 , 2 h α + ν h ˜ α = h ∨ U . Without loss of gener ality let h α = 1 2 . Then h ˜ α = − 1 2 ν 2 , h ∨ U = 1 − 1 2 ν . A gain t he p ole c onditions ar e vacuous. The choic e ν = 1 2 is sp e cial ( h ∨ U = 0) and le ads to the solution obtaine d in Cor ol lary 10. 6.1. The case U = R W . In this ca se it follows from g e ne r al theory that (18) is sa tisfied for a ll W eyl groups (if h α = 1 for all ro ots then h ∨ U is just the dual Coxeter num b er of W ). Since the quartic express ion P h α ( α, z ) 4 is W eyl inv ar iant, by Chevella y’s Theorem (Theorem 4) it may b e written in terms of fundamental inv ariant p o lynomials of deg ree 2 and degree 4, i.e. X h α ( α, z ) 4 = A [ s 2 ( z )] 2 + B s 4 ( z ) 22 IAN A. B. STRAC HAN if s uch p olyno mials exist. The quadr atic p olyno mia l s 2 exists for a ll g roups W ; one may take s 2 ( z ) = ( z , z ) . Inv ariant po lynomials o f degr ee 4 do not exist for W = A 2 , E 6 , 7 , 8 , F 4 , G 2 . Th us for these gro ups it follows immediately that (17) is satisfied. By dir ect ca lculation one ma y show that for the rema ining W eyl groups, W = A N ≥ 3 , B N , D N (where such an inv ar iant p olynomial do es e x ist) conditio n (17) fails, e x cept for B 2 where it holds if a sp ecific r elationship b etw e en h ( long ) and h ( shor t ) exists. Thus, in general, for the thr ee infinite families of gro ups, condition (17) fails and one has to app end an extra set of W ey l- group in v aria nt vectors in order to sa tisfy this condition: this will b e done in the next se ction. Since the constants h α are W eyl inv ar iant the analysis decomp os es in to c a ses lab eled by the num ber of indep endent W eyl orbits: • F or W = A 2 , E 6 , E 7 , E 8 one has a single W eyl or bit, so the constants h α are all identical. The v alues of this consta nt, and the cons ta nt h ∨ U are tabulated b elow: W eyl group A 2 E 6 E 7 E 8 h α 1 3 1 6 1 8 1 12 h ∨ U 1 2 9 4 5 2 (note, h ∨ U /h α =(dual) Coxeter n umber, a s requir ed). • F or W = B 2 , G 2 , F 4 one has tw o W eyl orbits, lab eled by short a nd lo ng ro ots. By dire c t c omputation o ne finds that conditions (17) and (18) a re satisfied with the following data: W eyl group B 2 G 2 F 4 h ( long ) 1 4 1 − h 18 3 − h 6 h ( shor t ) 1 3 h − 1 6 2 h − 3 3 h ∨ U 3 2 h h It re ma ins now to chec k the co nditions a pp earing in Pro po sition 16. It is w ell known that for a ro ot system R W the configuratio ns Π α ∩ R W are tw o dimensiona l ro ot systems, na mely one o f R A 1 × A 1 , R A 2 , R B 2 or R G 2 , and all of these configuratio ns are of type A. Hence by Lemma 20 these are elliptic ∨ -s ystems and hence provide solutions of the WD VV equations. 6.2. The case U = R W ∪ R irr eg W . W e now turn our a ttention to the three infinite families, where o ne has to app end a n extra set of v ectors to the standar d ro ots in order to satisfy co ndition (18). Note the the W eyl gro ups A 1 , A 2 and B 2 app ear to b e s pec ial in the sense that there are solutions with b oth U = R W and U = R W ∪ R irr eg W . F or A N ≥ 3 and B N ≥ 3 condition (1 8) fails for U = R W . 6.2.1. The c ase W = A N ≥ 2 . WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 23 Let z = P N +1 i =1 z i e i , with P N +1 i =1 z i = 0 . With the later co ndition the following ident ities immedia tely follow: 1 2 X i 6 = j ( z i − z j ) 2 − ( N + 1 ) 2 X i  ( z i ) 2 + ( − z i ) 2  = 0 , 1 2 X i 6 = j ( z i − z j ) 4 − ( N + 1 ) 2 X i  ( z i ) 4 + ( − z i ) 4  = 3 X i ( z i ) 2 ! 2 . F rom these one may obtain the α a nd h α satisfying c onditions (17) and (18) on using the standard Euclidean inner pro duct. Let α ( ij ) = e i − e j , β ( i ) = 1 ( N + 1)   N e i − X j 6 = i e j   (so ( α ( ij ) , z ) = z i − z j and ( β ( i ) , z ) = z i ). No te b o th these vectors lie on the hyperplane P N +1 i =1 z i = 0 . With these it follows that U = R A N ∪ R irr eg A N where: R A N = { α ( ij ) , i 6 = j } , h α = 1 / 2 if α ∈ R A N ; R irr eg A N = {± β ( i ) , , i = 1 , . . . N + 1 } , h α = − ( N + 1) / 2 if α ∈ R irr eg A N . Note that R A N is just the r o ot system for A N . The ge o metry o f this config uration will now b e discussed. Let σ α β denote the r eflection of β in the plane per p endicular to α . Then σ α ( ij ) β ( i ) = β ( i ) − 2( α ( ij ) , β ( i ) ) ( α ( ij ) , α ( ij ) ) α ( ij ) , = β ( i ) − α ( ij ) , = β ( j ) , σ α ( ij ) β ( k ) = β ( k ) , i, j, k distinct . Thu s the se t R irr eg A N is inv ariant under the action of W (which is g e nerated by reflections defined by the vectors in R A N ). Thus for N ≥ 3 the W eyl o rbit of an element of R irr eg A N is sma ller (since |R irr eg A N | = 2( N + 1)) than the size of the or bit of a g eneric v ector (which would b e | A N | = ( N + 1)!) . Thus R irr eg A N ≥ 2 consists of the union of t wo irre g ular orbits R irr eg A N ≥ 2 ∼ = { + β ( i ) | i = 1 , . . . , N + 1 } ∪ { − β ( i ) | i = 1 , . . . , N + 1 } . There ar e cer tain degeneracies if N = 1 or 2 : if N = 1 then β (1) = − β (2) and hence the set {± β ( i ) } double counts the vectors (this degeneracy was remov ed in the earlier discussion o f the A 1 solution); if N = 2 then |R irr eg A N | = |R A N | = ( N + 1)! . In fact this c a se coincides with the G 2 example a bove, i.e. R G 2 ∼ = R A 2 ∪ R irr eg A 2 for a sp ecific v alue of the constant h , namely h = 0 . Note that since ( β ( i ) , α ( j k ) ) = 0 and ( β ( i ) , α ( ij ) ) = 1 the s e t R irr eg A N consists of vectors fro m the weight lattice o f A N . In terms of fundamental weigh ts ∆ ( i ) = i X r =1 e r − i ( N + 1 ) N +1 X r =1 e r 24 IAN A. B. STRAC HAN θ ❍ ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✟ ✙ ✟ ✟ ✟ ✟ ✯ ❍ ❍ ❍ ❍ ❨ ✲ ✛ − β ( i ) + β ( j ) − β ( j ) + β ( i ) α ( ij ) = β ( i ) − β ( j ) α ( j i ) = β ( j ) − β ( i ) cos θ = − 1 N Figure 1. The configura tion U A N ∩ Span { β ( i ) , β ( j ) } one has R irr eg A N ∼ =  ± w (∆ ( N ) ) : w ∈ W  (note that ± ∆ (1) also lie in these tw o orbits). The orbits of other fundamental weigh ts form other irr egular orbits. F urthermor e, if N ≥ 3 one obtains the configurations U ∩ Span { α ( ij ) , α ( r s ) } = R A 1 × A 1 , { i, j } ∩ { r , s } 6 = ∅ , U ∩ Span { α ( ij ) , α ( ik ) } = R A 2 , i, j, k distinct together with the new co nfig uration U ∩ Span { β ( i ) , β ( j ) } = { ± β ( i ) , ± β ( j ) , ± α ( ij ) } . The g eometry of this new configuratio n is shown in Figur e 1. This is precisely a type B configuration, a nd the c o ndition (19) is satisfied, since α = β ( i ) , h α = − ( N + 1) / 2 and β = α ( ij ) , h β = 1 / 2 . Hence b y Lemma 20 we hav e an elliptic ∨ -sy stem a nd hence a so lution to the WD VV equations. 6.2.2. The c ase W = B N . The dua l prep otential for the Jacobi group or bit space Ω /J ( B N ) may b e calculated in the same was as the Ω /J ( A N ) dual prep otential was der ived in Theore m 6 (see also Example 27), and from this the set U and the constants h α may b e extrac ted. Given this origin of the set one might exp ect that it should b e re la ted to the ro ot system R B N . It turns out that one may des crib e this set in t wo wa y s: either in terms of the ro ot system R B C N or in terms of the ro ot system R C N (whic h is, of course , dual to the ro ot system R B N ) toge ther with an ir regular orbit R irr eg C N . Consider the following identities 3 : X i 6 = j ( z i − z j ) 2 + ( z i + z j ) 2 + N X i =1 (2 z i ) 2 − 4 N N X i =1 (2 z i ) 2 = 0 , X i 6 = j ( z i − z j ) 4 + ( z i + z j ) 4 + N X i =1 (2 z i ) 4 − 4 N N X i =1 (2 z i ) 4 = 12 X i ( z i ) 2 ! 2 . 3 Note the condition P z i = 0 used in the l ast section is not used in this section. WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 25 ✲ ✛ ✻ ❄   ✒   ✠ ❅ ❅ ■ ❅ ❅ ❘ ✲ ✛ ✻ ❄ Figure 2. The configuration U B N ∩ Span { e i , e j } On defining the inner pro duct to b e twice the sta ndard Euclidean pro duct (that is, ( z , z ) = 2 P i ( z i ) 2 ) one may obtain the α and h α satisfying co nditions (17) and (18). In terms of the r o ot sys tem R B C N , one has U B N = R B C N where R B C N =  1 2 ( ± e i ± e j ) , i 6 = j  ∪ {± e i } ∪  ± 1 2 e i  and h α =    1 2 if α is a long ro o t , 1 if α is a middle ro o t , − 2 N if α is a short ro ot . Alternatively (and this pr ovides a descr iption that is closer to the A N configuratio n ab ov e ) U = R C N ∪ R irr eg C N , = R ∨ B N ∪ R ∨ ir r eg B N where: R C N = { 1 2 ( ± e i ± e j ) , i 6 = j } ∪ {± e i } , h α =  1 if α short 1 / 2 if α long  if α ∈ R C N ; R irr eg C N = {± 1 2 e i } , h α = − 2 N if α ∈ R irr eg C N . As in the A N case, R irr eg C N is an irregular orbit (a single or bit in this case): R irr eg C N = { w (∆ ( N ) ) | w ∈ W } for a c e rtain fundamental weigh t ∆ ( N ) . In either case, the only new tw o dimensio na l configuration on v ectors is U C N ∩ Span { e i , e j } . This is shown in Figure 2, where the vectors of R irr eg C N hav e b een displaced slightly for visua l rea sons (this is actually the B C 2 system). The pro of that this is an elliptic ∨ system follows the A N case and will b e omitted. It als o follows from the Hurwitz spa ce description that will be given in Section 8.2 . 26 IAN A. B. STRAC HAN 6.2.3. The c ase W = D N . The D N configuratio ns are co mbinatorially quite complicated, as, even at N = 5 several W eyl orbits of fundamen tal w eights have to be app ended to the ba sic r o ot system R D N in order to satisfy (17) and (18 ). Some o f the resulting co nfigurations Π α ∩ U are not of Type A and Type B. This do es not mean that the conditions in Prop ositio n 16 must b e fals e - there may be other r easons wh y the v ario us terms could v anish. In the N = 4 cas e U = R D 4 ∪ R irr eg D 4 which coincides with the F 4 example considered ab ove, with the long ro ots o f F 4 being the ro o ts of D 4 and the short ro ots b e ing interpreted as the irregula r o rbits of the fundamental weights of D 4 . Clearly more w ork is required to co nstruct an exa mple of a D N elliptic ∨ -sys tem. The results in this section hav e b een obta ine d on a case - by-case basis. It would b e nice if there was a more abstrac t deriv ation of the results. 7. Frobenius-Stickelber ger Id entities Hidden within the v anis hing of the ∆ ( i ) are a num b er of in teresting functional ident ities satisfied by the v arious third deriv atives of the elliptic trilog arithm, the simplest of these reducing to 1 9 th centu ry ϑ -function identities. W e build up to these by first c o nsidering the ratio nal a nd trigonometr ic versions. Given no n-zero a , b , c ∈ C suc h that a + b + c = 0 then 1 a . 1 b + 1 b . 1 c + 1 c . 1 a = 0 and cot( a ) cot( b ) + cot( b ) cot( c ) + co t( c ) cot( a ) = 1 . Such iden tities are used in the direct verification that the ratio nal (1) a nd trigono- metric (2) prep otentials sa tisfy the WD VV equations. The elliptic version (where the dep endence on τ has b een suppressed for notational convenience) is            f (3 , 0) ( a ) f (3 , 0) ( b ) + f (3 , 0) ( b ) f (3 , 0) ( c ) + f (3 , 0) ( c ) f (3 , 0) ( a )            − n f (2 , 1) ( a ) + f (2 , 1) ( b ) + f (2 , 1) ( c ) o = 0 Using (5) this may be written in ter ms of ϑ -functions 4 : ϑ ′ 1 ( a ) ϑ 1 ( a ) ϑ ′ 1 ( b ) ϑ 1 ( b ) + ϑ ′ 1 ( b ) ϑ 1 ( b ) ϑ ′ 1 ( c ) ϑ 1 ( c ) + ϑ ′ 1 ( c ) ϑ 1 ( c ) ϑ ′ 1 ( a ) ϑ 1 ( a ) + 1 2  ϑ ′′ 1 ( a ) ϑ 1 ( a ) + ϑ ′′ 1 ( b ) ϑ 1 ( b ) + ϑ ′′ 1 ( c ) ϑ 1 ( c )  = 1 2 ϑ ′′′ 1 (0) ϑ ′ 1 (0) where a + b + c = 0 . With the identifi cation a = ( α, z ) , b = ( β , z ) , c = − ( α + β , z ) these identities may b e s een as identities connected to the A 2 Coxeter gr oup, with α and β b eing the p ositive ro ots. This immediately motiv ates the following: 4 This formula was found by the author during the researches that led to [ 27] and it has also appeared recen tly , with proof, in the work of Calaque, Enriques and Etingof [6]. How ev er it is a classical formula; in terms of W eiers trass functions it is just the well kno wn F robenius- Stick elberger equation [34] ( ζ ( a ) + ζ ( b ) + ζ ( c )) 2 = ℘ ( a ) + ℘ ( b ) + ℘ ( c ) , ( a + b + c = 0) re-written i n terms of ϑ -functions, an observ ation due to Prof. H. W.Braden. WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 27 Lemma 22. L et R b e the ro ot system for the 2-dimensional Coxeter gr oups A 2 , B 2 or G 2 , with the standar d n ormalization for α , β p ositive simple r o ots: A 2 : ( α, α ) = ( β , β ) = 2 , ( α, β ) = − 1 , B 2 : ( α, α ) = 2 , ( β , β ) = 1 , ( α, β ) = − 1 , G 2 : ( α, α ) = 6 , ( β , β ) = 2 , ( α, β ) = − 3 . Then X α 6 = β ∈R + ( α, β ) f (3 , 0) ( z α , τ ) . f (3 , 0) ( z β , τ ) + X α ∈R + k α f (2 , 1) ( z α , τ ) = 0 , wher e: • A 2 : k α = 1 for al l r o ots; • B 2 : k shor t = 2 , k long = 1 ; • G 2 : k shor t = 10 , k long = 6 . The pro of is entirely standa rd a nd is omitted. Many other functional iden tities may b e derived using the same idea s. Ra ther than give a full list we present tw o of the A 2 ident ities:        f (3 , 0) ( x + y ) h f (2 , 1) ( x ) − f (2 , 1) ( y ) i + f (3 , 0) ( y ) h f (2 , 1) ( x + y ) − f (2 , 1) ( x ) i        + f (1 , 2) ( x ) − 1 2 f (1 , 2) ( y )+ 1 2 f (1 , 2) ( x + y ) = 0 and                  f (3 , 0) ( x ) h f (1 , 2) ( x + y ) − f (1 , 2) ( y ) i + f (3 , 0) ( y ) h f (1 , 2) ( x + y ) − f (1 , 2) ( x ) i − 2 3 f (3 , 0) ( x + y ) h f (1 , 2) ( x ) + f (1 , 2) ( y ) i                  +                    2 3 f (2 , 1) ( x + y ) f (2 , 1) ( x ) + 2 3 f (2 , 1) ( x + y ) f (2 , 1) ( y ) − 8 3 f (2 , 1) ( x ) f (2 , 1) ( y )                    + 10 9 f (0 , 3) ( x + y ) = − 1 108 E 4 ( τ ) . Clearly ther e is m uch scop e to inv estigate such neo-classic a l functional identities. More identities of these t y p e may b e found in [31] . 8. Jacobi Group Orbit Sp aces Ment ion has b een made a num b er of times to J a cobi g roups and their or bit spaces, but so far these hav e no t b een defined. In this section this is rectified and in addition the construction of the F r ob enius manifold structure on such or bit spaces will b e outlined. In particular, using a n a lternative des cription of such spaces as sp ecific Hurwitz spaces we cons truct the dual prep otentials for the W eyl groups 28 IAN A. B. STRAC HAN A N and B N , thus proving that the ex amples of elliptic ∨ -systems constructed earlier corres po nd to J acobi gro up orbit space s . This then motiv ates a conjectur e for ar bitr ary W eyl group. 8.1. Jacobi groups and Jacobi forms. The material in this section will clos ely follow [2], which in turn relies heavily on the fundamental pap ers of Wirthm¨ uller [33] and Eichler and Zag ier [12]. W e b egin by the definition of a J a cobi fo rm. These play the sa me ro le in the construction of the orbit s pace a s the sy mmetr ic po lynomials do in the or iginal Saito construction - they provide co ordinates on the orbit space . Definition 23. L et W b e a finite Weyl gr oup with r o ot lattic e Q and let g b e the c orr esp onding Lie algebr a with Cartan sub algebr a h . A Jac obi form of weight k ∈ Z and index m ∈ Z i s a holomorphic function φ : h ⊕ H → C with t he fol lowing pr op erties: φ ( z + q , τ ) = φ ( z , τ ) , φ ( z + q τ , τ ) = e − 2 π im ( q , z ) − π im ( q , q ) τ . φ ( z , τ ) , f or a l l q ∈ Q , φ  z cτ + d , aτ + b cτ + d  = ( cτ + d ) k . e cπ im ( z , z ) / ( cτ + d ) . φ ( z , τ ) , φ ( w. z , τ ) = φ ( z , τ ) , f or al l w ∈ W and φ ( z , τ ) is a l ocal l y b ounded f uncti on as ℑ m ( τ ) → + ∞ . Such forms ar e the elliptic analogues of the W -inv ariant polyno mials and they to o satisfy a Chev alley-type theorem. F ollowing Berto la [2] , o ne ca n defined a new function φ ( u, z , τ ) = e mu φ ( z , τ ) defined on the Tits cone Ω ∼ = C ⊕ h ⊕ H . This is also r eferred to as a Jacobi form and the spa ce of Ja c obi forms will b e denoted J W k,m . The J a cobi group J ( g ) itself generates the ab ov e transfor mations. The full deta ils a r e not required her e: J ( g ) is the semi-direc t pro duct W ⋊ S L (2 , Z ) wher e W = W ⋊ H R where W is a W ey l group and H R the Heisen b erg group obtained from the ro o t space R of W . The precise definitions of the v arious actions ma y b e found in [2, 33]. It is well k nown that the ring of mo dular for ms is a free graded algebr a over C genera ted by the E is enstein series E 4 and E 6 , i.e. M • = L k M k , where the subspace of modular forms of w eight k is M k = C [ E a 4 E b 6 , ∀ a , b ∈ N such tha t 4 a + 6 b = k ] . The ring of J acobi forms is particular ly nice; it sa tisfies an analogue of Chev alley’s Theorem (Theorem 4 ) : Theorem 24. [33] Given the Jac obi gr oup asso ciate d to any fi nite dimensional simple Lie algebr a g of r ank N (exc ept for p ossibly E 8 ): • the bi-gr ade d algebr a of Jac obi forms J W • , • = L k,m J W km is fr e ely gener ate d by N + 1 fun damental Jac obi forms { φ 0 , . . . , φ N } over the gr ade d ring of mo dular forms M • , J W • , • = M • [ φ 0 , . . . , φ N ] ; WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 29 • e ach φ j ∈ J W − k ( j ) ,m ( j ) wher e − k ( j ) ≤ 0 , m ( j ) > 0 ar e define d as fol lows: – k (0) = 1 and k ( j ) , j > 0 ar e the de gr e es of the gener ators of the invariant p olynomials in (4) ; – m (0) = 1 and m ( j ) , j > 0 ar e the c o efficients in the exp ansion of t he highest c or o ot ˜ α ∨ , ˜ α ∨ = N X j =1 m ( j ) α ∨ j wher e ˜ α is the highest r o ot and α ∨ j a b asis for R ∨ . Note that J W • , 0 ∼ = M • . It will also b e useful to define φ − 1 = τ , even though it is not a Jacobi form. These J acobi forms b ecome the co ordinates on the o rbit space Ω /J ( g ) . Before turning to the explicit cons tr uction o f such forms we prove the following simple result on the Jacobia n of the transforma tio n b etw een the t wo co ordinate systems { u, z , τ } and { φ − 1 , φ 0 , . . . , φ N } . Prop ositi o n 25. L et J ac ( u, z , τ ) = ∂ { φ − 1 , φ 0 , . . . , φ N } ∂ { u, z , τ } . Then J ac has the fol lowing tra nsformation pr op erties: 1 2 π i ∂ ∂ u J ac ( u, z , τ ) = h ∨ J ac ( u, z , τ ) , J ac ( u, z + q , τ ) = J ac ( u, z , τ ) , J ac ( u, z + q τ , τ ) = e − 2 π ih ∨ ( q , z ) − π ih ∨ ( q , q ) . J ac ( u, z , τ ) , J ac ( u, z , τ + 1) = J ac ( u, z , τ ) , J ac  u, z τ , − 1 τ  = τ −|R + W | e π ih ∨ ( z , z ) /τ . J ac ( u, z , τ ) , J ac ( u, w. z , τ ) = det( w ) . J ac ( u, z , τ ) , wher e h ∨ is t he dual Coxeter numb er and |R + W | t he nu mb er of p ositive r o ots . Mor e- over, up to an over al l c onstant, (20) J ac ( u, z , τ ) = e 2 π ih ∨ u Y α ∈R + W ϑ 1 ( z α , τ ) ϑ ′ 1 (0 , τ ) . These transforma tion prop erties may be elev ated to a definition o f an a nt i-inv a riant Jacobi for m. This result is the elliptic version of the well known result J ac ( z ) = Q α ∈R + W z α for Coxeter groups. Pro of By definition, the J a cobian is a determina nt , so by using pro per ties of the determinant, to gether with the tra nsformation prop erties of the individual Jacobi forms given in Pro p osition 23 the result follows. V ario us Lie-theory results are used, such a s h ∨ = N X i =0 m ( i ) , |R + W | = N X i =1 ( k ( i ) − 1) , pro ofs of whic h may b e found in Kac [17]. 30 IAN A. B. STRAC HAN T o prov e (20) one first prov es that the right-hand-side has the same transfor ma- tion prop er ties as J ac . Ther efore their r atio tr a nsformations like a J W 0 , 0 -Jacobi for m (the analytic pro per ties following from those of the ϑ 1 -function, such as its entire prop erty). B ut J W 0 , 0 ∼ = M 0 and there are no no n-trivial degr ee 0 mo dula r forms and hence the r atio must be a co nstant.  F urther prop erties of the for ms may b e found in [2, 3 3]. F or the A N and B N cases there is a v ery compact way to study the forms by co mbin ing them into a generating function. The in v ariant po lynomials for the A N -Coxeter group may b e obtained via a generating function (a res ult due to Vi ` ete) N Y i =0 ( v − z i )      P z i =0 = v N +1 + N − 1 X r =0 ( − 1) N +1 − r s r +1 ( z ) v r . Similarly , the A N Jacobi forms ma y b e o btained [2] from a similar expansion o f (21) λ A N ( v ) = e 2 π iu Q N i =0 ϑ 1 ( v − z i , τ ) ϑ 1 ( v , τ ) N +1      P z i =0 as a sum of W eier strass ℘ functions and their deriv atives, their co efficients being the A N -Jacobi forms. Using the embedding B N ⊂ A 2 N − 1 one may o btain a g e ne r ating function for the B N -Jacobi forms: (22) λ B N ( v ) = e 2 π iu Q N i =1 ϑ 1 ( v − z i , τ ) ϑ 1 ( v + z i , τ ) ϑ 1 ( v , τ ) 2 N These ge ne r ating functions are no t just fo r mal ob jects, they ar e holomor phic maps from the complex tor us to the Riema nn sphere. This means one can use a Hur w itz space construction to calculate the dual prep otential. 8.2. H urwitz spaces. Let H g,N ( k 1 , . . . , k l ) be the Hurwitz space 5 of equiv alence classes [ λ : L → P 1 ] of N -fold bra nched cov er ings λ : L → P 1 , where L is a co mpact Riemann surface of ge nus g a nd the holo morphic map λ of degree N is sub ject to the following conditions: • it has M simple r amification p oints P 1 , . . . , P M ∈ L with distinct finite images l 1 , . . . , l M ∈ C ⊂ P 1 ; • the pr eimage λ − 1 ( ∞ ) consists of l p oints: λ − 1 ( ∞ ) = { ∞ 1 , . . . , ∞ l } , and the ra mification index o f the map λ a t the p oint ∞ j is k j (1 ≤ k j ≤ N ). (W e define the r amification index at a po int as the num be r o f shee ts of the cov e ring which are glued tog ether at this po int . A po int ∞ j is a ramification p oint if and only if k j > 1 . A ra mification p oint is simple if the corresp o nding ramificatio n index equa ls 2.) The Riemann- Hurwitz for mu la implies that the dimension of this space is M = 2 g + l + N − 2 . One ha s also the eq uality k 1 + · · · + k l = N . Two branched cov er ings λ 1 : L 1 → P 1 and λ 2 : L 2 → P 1 are s aid to b e equiv alent if there exists a biholomorphic ma p f : L 1 → L 2 such that λ 2 f = λ 1 . W e a lso int ro duce the covering ˆ H g,N ( k 1 , . . . , k l ) of the space H g,N ( k 1 , . . . , k l ) consisting of pairs < [ λ : L → P 1 ] ∈ H g,N ( k 1 , . . . , k l ) , { a α , b α } g α =1 >, 5 Dubrov in [9] uses a sli gh tly different notation. In his notation the H urwitz space i s H g ; k 1 − 1 ,... ,k l − 1 . WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 31 where { a α , b α } g α =1 is a canonic a l basis of cycle s on the Riemann surface L . The spaces ˆ H g,N ( k 1 , . . . , k l ) a nd H g,N ( k 1 , . . . , k l ) a re co nnected c omplex manifolds and the lo cal co ordinates on these ma nifolds a re given by the finite critical v alues of the map λ . F or g = 0 the spaces ˆ H g,N ( k 1 , . . . , k l ) and H g,N ( k 1 , . . . , k l ) coincide. The v arious metric and multiplication tensors are given in terms of this holo- morphic map λ : L → P 1 (also known as the s uper p otential) b y the follo wing: Theorem 2 6. The interse ction form and dual multiplic ation on the Hurwitz sp ac e H g,N ( k 1 , . . . , k l ) ar e given by the fol lowing r esidue formulae: g ( ∂ ′ , ∂ ′′ ) = X res dλ =0 ∂ ′ (log λ ( v ) dv ) ∂ ′′ (log λ ( v ) dv ) d log λ ( v ) , c ⋆ ( ∂ ′ , ∂ ′′ , ∂ ′′′ ) = 1 2 π i X res dλ =0 ∂ ′ (log λ ( v ) dv ) ∂ ′′ (log λ ( v ) dv ) ∂ ′′′ (log λ ( v ) dv ) d log λ ( v ) . Her e ∂ , ∂ ′ and ∂ ′′ ar e arbitr ary ve ctor fields on the Hurwitz sp ac e H g,N ( k 1 , . . . , k l ) . The for mula for g app ea r ed in [9] while the fo rmula for c ⋆ follows immediately from the results in [10]. Note that with the s pe c ific dep endence of u in the super po tent ials (21) and (22) we hav e norma lized g and c ⋆ so that ∂ u is the unity vector field (ra ther than 1 2 π i ∂ u ). Thus c ⋆ ( ∂ u , ∂ ′ , ∂ ′′ ) = g ( ∂ ′ , ∂ ′′ ) . This also av oids a prolifer a tion of (2 π i )-factor s in the final result. Certain Hurwitz spaces ar e isomorphic to certain orbit s paces [9]. F o r example, C N / A N ∼ = H 0 ,N +1 ( N + 1) , C N +1 / ˜ A ( k ) N ∼ = H 0 ,N +1 ( k , N − k ) , Ω /J ( A N ) ∼ = H 1 ,N +1 ( N + 1) . Thu s the tow er of ge neralizations mentioned in the introduction has a unified de- scription, at least for the A N -cases, in terms of the theory of Hurwitz spaces. This also leads to a wa y to expa nd the tow er further via higher genus Hurwitz s paces, the mos t na tur al being the space H g,N +1 ( N + 1) . The B N examples come from int ro ducing a Z 2 grading onto the Huwitz space (e.g. the supe r p otentials ab ov e hav e a z ↔ − z symmetry). Example 27. (a) U sing the sup erp otential (21) one obtains the interse ction form and (dual) pr ep otential for the A N -Jac obi gr oup orbit sp ac e in The or em 6 ab ove [2 7] : g = 2 du dτ − N X i =0 ( dz i ) 2      P N j =0 z j =0 F ⋆ ( u , z , τ ) = 1 2 τ u 2 − 1 2 u N X i =0 ( z i ) 2      P N j =0 z j =0 + 1 2 X i 6 = j f ( z i − z j , τ ) − ( N + 1 ) X i f ( z i , τ ) wher e this funct ion is evaluate d on the plane P N i =0 z j = 0 . 32 IAN A. B. STRAC HAN (b) Using the sup erp otential (22) one obtains the interse ction form and (du al) pr ep otential for the B N -Jac obi gr oup orbit sp ac e: g = 2 du dτ − 2 N X i =1 ( dz i ) 2 , F ⋆ ( u , z , τ ) = 1 2 τ u 2 − 2 u N X i =0 ( z i ) 2 + X i 6 = j  f ( z i + z j , τ ) + f ( z i − z j , τ )  + X i f (2 z i , τ ) − 2 N X i f ( z i , τ ) . Combining this with the earlier results o n elliptic ∨ -systems gives: Theorem 28. The el liptic ∨ -systems given in se ctions (6.2.1) and (6.2.2) define pr ep otentials that ar e the almost-dual pr ep otentials asso ciate d t o the A N and B N Jac obi gr oup orbit sp ac es. The form o f this res ult, coupled with the examples of elliptic ∨ -sys tems leads to the following conjecture: Conjecture 29. L et W b e a Weyl gr oup. F or the Jac obi gr oup orbit sp ac e Ω /J ( W ) the dual pr ep otential t akes the form (15) with h ∨ U = 0 . F urthermor e, U = R W ∪ R irr eg W (or its dual) and wher e R irr eg W = { w (∆) | w ∈ W } or R irr eg W = {± w (∆) | w ∈ W } for some weight ve ctor ∆ . The conjecture seems plausible. I t is true for the A N examples a nd the B N examples (if one uses the dual ro ot system) and if all o rbit spaces ar e to b ehave in the same generic wa y in the trigonometric limit then o ne must have h ∨ U = 0 fr om the results of section 5 .1. One p ossible approa ch to proving this conjecture would b e to show that if a prep otential F lies at the fixed p oint of the inv o lutive symmetry then so do es the corres p o nding almost dual prep otential. Since this is true for the J a cobi gro up examples this would then prove the first part o f the conjecture, but not the second part on the structure of the set U . The Saito co nstruction of J acobi gro ups has recently b een studied in detail [29]. Perhaps a formulation of a dua l version of the result would provide a pro of of the conjecture. Within the cla ss of elliptic ∨ -systems there remains the problem of co nstructing examples with h ∨ U = 0 . F or W = G 2 , F 4 one ma y set h = 0 , but for E 6 , 7 , 8 one w ould hav e to app end an R irr eg W set o f vectors. These cases also r emain pro blematical. If h = 0 in the G 2 case one obtains a dual prep otential that is actually the dual prep otential for the A 2 Jacobi gro up orbit space, leaving a problem as to w ha t the correct G 2 solution would b e. This case lies in the so-called co-dimension one case and de s erves clo ser study . The G 2 Jacobi forms have also been constr ucted explicitly [2] so it may b e p ossible to find the dual prep otential in this case by direct calculation. It is a lso p ossible the the dua l prep otential is the s ame in these tw o cases: the reconstruction of the F rob enius manifold from the dua l picture requires additional data besides the almost dua l pre p o ten tial. WEYL GR OUPS AND ELLIPTIC S OLUTIONS OF THE WDVV EQUA TIONS 33 9. Comments The idea of a n elliptic ∨ -system may c le arly b e studied further. As well as the obvious question on the relationship betw e e n the functional ansatz and Ja cobi group orbit spaces summarize d in Conjecture 29, there are many other questio ns and problems that co uld b e addressed. Given a co mplex E uclidean ∨ -system one may study their res tr iction to low er dimensions and the conditions required for the restricted system to also b e a c omplex E uc lide a n ∨ -system. Clear ly the sa me q ues- tion can b e as ked for elliptic ∨ -systems. Examples along this line may b e obta ined from the restriction of the A N and B N Jacobi-g roup spaces to discriminants. This is achiev ed b y intro ducing multiplicities into the A N sup e rp otential (21), λ ( p ) = e 2 π iu m Y i =0  ϑ 1 ( v − z i , τ ) ϑ 1 ( v , τ )  k i , where P m i =0 k i = N + 1 , P m i =0 k i z i = 0 , or on mo re gener al Hurwitz spaces H 1 ,N ( n 1 , . . . , n m ) and their discriminants. Partial res ults have b een obtained in [26], and these pr ovide further examples of elliptic ∨ -systems. In fact, interest- ing examples of ∨ -system can b e found by lo oking on the induced structur e s on disciminants [14, 30] and c le a rly the s a me ideas c o uld b e applied here. Possible applica tions of these s olutions should come from Seib erg- Witten theory and the p erturbative limits of such theories. This link is well known for rational and trig onometric solutions, and the in terpretation of the elliptic so lutio ns found in [27] in ter ms of a 6 -dimensional field theor y has b een g iven in [3], and one would exp ect similar results for the more gener a l solutions co nstructed her e (though [3] do es use the existence of a sup erp otential which is la cking for general solutions constructed her e ). The tower of gener alizations mentioned in the introduction clear ly do es not hav e to stop at elliptic solutions. An arbitra r y Hur witz spac e H G,N ( k 1 , . . . , k l ) carries the s tructure o f a F rob enius manifold a nd hence an almost-dua l structur e . An int eresting question is whether or not there is an orbit space constructio n fo r these more genera l spaces: H 0 ,N ( N ) − → H 0 ,N ( k , N − k ) − → H 1 ,N ( N ) − → . . . − → H g,N ( N ) l l l l C N / A N − → C N +1 / e A ( k ) N − → Ω /J ( A N ) − → . . . − → orbit space structure? It seems sensible to co njecture that such an orbit space exists. One w ould exp ect Siegel modula r forms to play a role instead of the mo dular forms used here. Higher genus Jacobi forms certain hav e been studied, but their use has yet to p erc olate into the theory of integrable systems. The developmen t, a nd applica tions o f, the neo - classical ϑ -function iden tities studied in Section 7 remains to b e do ne systematica lly . Certain higher g e n us analogues o f these identities certainly exist, since there exist almost-dual prep otentials on these Hurwitz spaces whic h, by construction, satisfy the WDVV equatio ns. In the genus 0 and genus 1 cases, the pr ep otential is very closely r elated to the prime form o n the Riemann surface. This may be the starting po int for the developmen t of a functiona l ansa tz for the hig her genus ca ses. Central 34 IAN A. B. STRAC HAN to the r esults presented here ar e the quasi- p e r io dicity and mo dula rity prop erties of the elliptic p oly lo garithm, and these w ere obtained from the analytic prop erties of this function; the o nly role the ana ly tic pr op erties play were in the developmen t of these trans fo rmation prop erties. It would b e attrac tive if one c o uld obtain these directly from the geo metric prop er ties of the prime form. This appro a ch could then b e used in the higher genus case wher e the a nalytic pr op erties ar e likely to b e considerably mor e complicated. Ment ion has b een made already o f the b eautiful pap er [6]. It would b e in teresting to see if the idea s develop ed here co uld b e use d in the study of K Z and Dunkl-type systems. The idea w ould b e to study ob jects such as X α,β ∈ U h f (3 , 0) ( z α ) s α , f (3 , 0) ( z β ) s β i where s α and s β are shift o p erators . Co njecturally this quadratic term would be related to linea r terms in the function f (2 , 1) . The ra tional limit would then coincide with the classica l work of Dunkl [5]. Such a development would b e differen t to the elliptic Dunkl op era tors in the pioneering work o f Buchstaber et al. [ 4 ]. F or a preliminary discussio n of these ideas, see [3 1] . Finally , one thing that has bee n learnt from this work is that on going from rational and trigonometric structure related to a W eyl group W v ia the r o ot system R W to elliptic structure s, gener alizations based entirely on the use of the ro ot system R W alone may no t suffice. Ackno wledgments I would like to thank Harr y Br aden, Misha F eigin a nd Andrew Riley for their comments on this pap er. I am also very grateful to the referee for p o int ing o ut cer- tain error s in the orig inal v ersion, and for his/her car eful reading of the manuscript - this has resulted in m uch improved final version. References [1] Beilinson, A. and Levin, A., The Elliptic Polylogarithm , i n M otiv es (ed. Jannsen, U., K leiman, S,. Serre, J.- P .), Proc. Symp. Pure Math. vo l 55 , Amer. Math. So c., (1994), Part 2, 123-190. [2] Bertola, M. F r obenius manif old structure on orbit space of Jacobi groups; Pa rts I and I I , Diff. Geom. Appl. 13 , (2000), 19-41 and 13 , (2000) , 213-23. 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