Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 5 (2009), 088, 10 pages T rigonomet ric Solutio ns of WD VV Equations and Generalize d Calogero –Moser–S utherland Systems Misha V. FEIGIN Dep artment of Mathematics, U niversity of Glasgow , G12 8QW, UK E-mail: M.F eigin@maths.gla.ac.uk URL: http://www.math s.gla.ac.uk/ ~ mf/ Received May 18, 2009, in f inal for m September 0 7, 200 9; Published o nline September 1 7 , 2 009 doi:10.38 42/SIGMA.20 09.088 Abstract. W e conside r trigonometric solutio ns of WDVV equations and derive geometric conditions when a co llection of vectors with multiplicities deter mines such a s olution. W e incorp ora te these conditions into the notio n of trigono metric V eselov system ( ∨ -system) and we determine a ll trigonometr ic ∨ -sys tems with up to f ive vectors. W e show that gener a li- zed Calo gero– Moser– Sutherland op er ator a dmits a factoriz e d eigenfunction if a nd only if it corres p o nds to the trig onometric ∨ -sy stem; this in verts a o ne-wa y implication o bserved by V eselov for the ratio na l so lutions. Key wor ds: Witten–Dijkgraaf–V erlinde–V erlinde equatio ns , ∨ -systems, Calogero–Moser – Sutherland s y stems 2000 Mathematics Subje ct Classific ation: 35Q40; 52C9 9 1 In tro duction In this note we are inte rested in th e sp ecial trilogarithmic s olutions of the generalized Witten– Dijkgraaf–V erlinde–V erlind e (WD VV) equ ations [1]. Suc h solutions are determined by a f inite collect ion A of co v ecto rs α with m ultiplicities c α . More sp ecif ically , the p rep otenti al satisfying the WD VV equations h as th e form F = X α ∈ A c α Li 3  e 2 iα ( x )  + cubic terms , (1) where L i 3 is the trilogarithm fu nction. Solution of th is t yp e for th e A n ro ot system app eared in [2] in relation with Seib erg–Witten theory . More systematically such solutions were studied b y Ho ev enaars and Martini in [3, 4] who determined solutions for all irreducible reduced ro ot systems [4]. More recen tly solutions of the form (1) w ere derive d from redu ctions of Egoro v h ydro dy n amic c hains in [5 ]. The rational versions of solutions (1) p la y an imp ortan t role in th e theory of F rob enius manifolds, a geometric fr amew ork for the WD VV equations [6]. Thus solutions corresp onding to the Coxet er ro ot systems are almost dual to the F rob enius structures on the orbit spaces of f inite Co xeter group s [7]. In the trigonometric case suc h a duality is verif ied for the af f ine A n case in [8, 9]. The study of general rational solutions of th e form F = X α ∈ A α ( x ) 2 log α ( x ) (2) w as initiate d by V eselo v in [10] where a geomet ric n otion of the ∨ -system equiv al en t to the WD VV equations for (2) w as in tro duced. It was sh o wn in [11] that any generalized C alogero– Moser–Sutherland (CMS) op erator admitting a factorized eigenfunction determin es a ∨ -system. 2 M.V. F eigin In this note we are interested in the solutions (1) where the cub ic terms in v olv e extra v ariable lik e in the w orks [3, 4] on the solutions for the ro ot systems. W e derive geometric and algebraic conditions for a system of ve ctors with multipliciti es so that the corresp on d ing f u nction (1) satisf ies the WDV V equations. These conditions should b e though t of as trigonometric analogue of the notion of th e ∨ -system. The conditions carry rather strong geometrica l restrictions on the collection of v ectors formulate d in terms of series of v ectors parallel to a c hosen one. W e illustrate this by determinin g all trigonometric ∨ -systems with up to f iv e v ectors in the plane. T r igonometric ansatz, in con trast to the rational one, allo w s to def ine the generalized CMS op erator corresp onding to th e solution (1). W e show that this op erator h as a factorized eigen- function. This s tatement in v erts the one for the rational ∨ -systems obtained in [11]. In fact our argumen ts follo w [11] v ery closely . W e also discuss additional condition needed to ha v e trigonometric s olution to the WD VV equations starting f r om a CMS op erator with factorized eigenfunction. 2 T rigonomet ric ∨ -systems Consider a function F of the form F = 1 3 y 3 + X α ∈ A c α α ( x ) 2 y + λ X α ∈ A c α f ( α ( x )) , (3) where A is a f inite collection of co v ectors on V ∼ = C n , x = ( x 1 , . . . , x n ), c α , λ are non-zero constan ts and fun ction f ( x ) satisf ies f ′′′ ( x ) = cot x . The last equation f ixes fun ction f ( x ) up to 2nd order terms whic h w ill not b e imp ortan t for the WD VV equ ations b elo w. W e ma y f ix a c hoice of f ( x ) by f ( x ) = 1 6 ix 3 + 1 4 Li 3  e − 2 ix  . The ansatz (3) in tro ducing extra v ariable y wa s prop osed in [4] in the case of ro ot systems A = R . The form (3) guaran tees that the matrix of third deriv ativ es in v olving y is constant, as w e will explore b elo w. We wil l assume thr oughout the p ap er that c ol le ction A of c ove c tors α b elongs to an n - dimen- sional lattic e, and that the biline ar form ( u, v ) := X α ∈ A c α α ( u ) α ( v ) (4) is non-de gener ate on V . The form ( · , · ) identif ies V and V ∗ and f ollo w ing [10] we will denote b y γ ∨ the ve ctor du al to the co v ecto r γ . W e w ill also d enote through ( · , · ) the corresp onding inner pr o duct on V ∗ . W e are inte rested in the conditions on { α, c α , λ } w hen function F satisf ies the WD VV equations F i F − 1 k F j = F j F − 1 k F i , (5) i, j, k = 0 , 1 , . . . , n . Here F i are ( n + 1) × ( n + 1) matrices of thir d deriv ati v es in the co ordinates ( x 0 = y , x 1 , . . . , x n ), ( F i ) ab = ∂ 3 F ∂ x i ∂ x a ∂ x b . It is suf f icient to f ix k = 0, th en F 0 = F y is the follo wing non-degenerate m atrix F y = 2 1 0 0 P α ∈ A c α α ⊗ α ! . T r igonometric Solutions of WD VV Equ ations and Generalized CMS S ystems 3 Similarly F i =    0 2 P α ∈ A c α α i α 2 P α ∈ A c α α i α λ P α ∈ A c α α i cot α ( x ) α ⊗ α    , where w e denoted by α b oth column and row ve ctors α = ( α 1 , . . . , α n ). The WD VV cond itions for a function F can b e reformulate d p artly using geo metry of th e system A . F or any α ∈ A let us collect all the co vecto rs from A non-parallel to α in to the disjoin t un ion of α α α -series Γ 1 α , . . . , Γ k α . These series are determined by the pr op ert y that for an y s = 1 , . . . , k and for any tw o co v ec tors γ 1 , γ 2 ∈ Γ s α one has either γ 1 − γ 2 = nα or γ 1 + γ 2 = nα for some inte ger n . W e also assume that the series are maximal, that is if β ∈ Γ i α then Γ i α m ust con tain all the vect ors of th e form ± β + nα ∈ A with n ∈ Z . W e n ote th at solution (3) is not af fected if some of the co v ectors α ∈ A are r ep laced w ith − α . By ap p ropriate choic e of s igns the vec tors can b e made to b elong to a h alf-space, w e will denote suc h systems as A + . Moreo v er, for an y α ∈ A one can choose a p ositiv e system A + ∋ α in such a wa y that α -series Γ s α will consist of vec tors of the form β s + n i α ∈ A + for appropriate intege r parameters n i with β s ∈ A + . Definition 1. Let A ⊂ V ∗ ∼ = C n b e a f in ite collection of co vecto rs α with multiplici ties c α suc h that the corresp onding form (4) is non-degenerate and th e co vect ors α b elong to an n - dimensional lattice. W e say that A is a trigo nometric ∨ -system if for an y α ∈ A an d for any α -series Γ s α one has X β ∈ Γ s α c β ( α, β ) α ∧ β = 0 . (6) Notice that α ∧ β 1 = ± α ∧ β 2 if β 1 , β 2 b elong to the same α -series Γ s α so identit ies (6) may b e simplif ied accordingly . Also r ep lacemen t of some of the co v ectors by their opp osite preserve s the class of trigonometric ∨ -systems. Note also that the non-degenerate linear transf ormations act naturally on the trigonometric ∨ -sys tems, and that the direct sum A 1 ⊕ A 2 of the trigonometric ∨ -systems A 1 ⊂ V ∗ 1 , A 2 ⊂ V ∗ 2 considered as a set of co v ec tors in V 1 ⊕ V 2 is again a trigonometric ∨ -system. The systems obtained in this wa y w ill b e called r e ducible . If such a decomp osition is not p ossible th en the (trigonometric ∨ -)system is called ir reducible. Theorem 1. The WDVV e quations F i F − 1 y F j = F j F − 1 y F i , i, j = 0 , 1 , . . . , n , f or the f unction (3) ar e e quivalent to the fol lowing two c onditions: 1) A is a trigonometric ∨ -system; 2) for a p ositive system A + and for any ve ctors a, b, c, d ∈ V X α,β ∈ A +  1 4 λ 2 ( α, β ) − 1  c α c β B α,β ( a, b ) B α,β ( c, d ) = 0 , (7) wher e B α,β ( a, b ) = α ∧ β ( a, b ) = α ( a ) β ( b ) − α ( b ) β ( a ) . Pro of . F or a vec tor a ∈ V we def in e F ∨ a = F − 1 y F a where F a = n P i =1 a i F i . Th e WD VV equations are equiv alent to the comm utativit y [ F ∨ a , F ∨ b ] = 0 for any a, b ∈ V . W e ha v e F ∨ i =    0 P α ∈ A c α α i α P α ∈ A c α α i α ∨ λ 2 P α ∈ A c α α i cot α ( x ) α ⊗ α ∨    , 4 M.V. F eigin where α ∨ is the (column) v ector dual to the (row) co v ect or α und er the bilinear f orm G = P α ∈ A c α α ⊗ α . Th erefore F ∨ a =    0 P α ∈ A c α α ( a ) α P α ∈ A c α α ( a ) α ∨ λ 2 P α ∈ A c α α ( a ) cot α ( x ) α ⊗ α ∨    for an y a ∈ C n . No w the pro du ct F ∨ a F ∨ b equals      P α,β ∈ A c α c β α ( a ) β ( b ) α ( β ∨ ) λ 2 P α,β ∈ A c α c β α ( a ) β ( b ) α ( β ∨ ) cot β ( x ) β λ 2 P α,β ∈ A c β c α α ( a ) β ( b ) α ( β ∨ ) cot α ( x ) α ∨ P α,β ∈ A c α c β α ( a ) β ( b ) α ∨ ⊗ β + λ 2 4 P α,β ∈ A c α c β α ( a ) β ( b ) α ( β ∨ ) cot α ( x ) cot β ( x ) β ⊗ α ∨      . Therefore [ F ∨ a , F ∨ b ] = 0 is equiv alent to the iden tities X α,β ∈ A c α c β B α,β ( a, b )( α, β ) cot α ( x ) α ∨ = 0 , (8) X α,β ∈ A  λ 2 4 c α c β ( α, β ) cot α ( x ) cot β ( x ) + c α c β  B α,β ( a, b ) α ∧ β = 0 . (9) T o cancel singularities in (9) one shou ld ha v e X β ∈ A β ≁ α c β ( α, β ) cot β ( x ) B α,β ( a, b ) α ∧ β = 0 when cot α ( x ) = 0. A linear com bination of functions cot β ( x ) | cot α ( x )=0 can v anish only if it v anishes for eac h α -series: X β ∈ Γ s α c β ( α, β ) cot β ( x ) B α,β ( a, b ) α ∧ β = 0 for all α -series Γ s α (see e.g. [14] for more detailed explanation). The last r elation can b e sim- plif ied as X β ∈ Γ s α c β ( α, β ) α ∧ β = 0 , (10) whic h means that A is a trigonometric ∨ -system. Iden tities (10) guaran tee that the left-hand side of (9) is non-singular. S ince all the v ectors fr om A b elong to an n -dimensional lattice with basis e 1 , . . . , e n , the left-hand side of (9) is a rational function in the exp onent ial co ord inates e e i ( x ) . This rational fun ction has degree zero and th erefore it is a constant. W e can assume that all co v ectors from A b elong to a half-space hence form a p ositiv e system A + , so in appropriate limit cot( α, x ) → i for all α ∈ A + . Thus prop ert y (9) is equiv al en t to (10) together with the condition X α,β ∈ A +  λ 2 4 c α c β ( α, β ) − c α c β  B α,β ( a, b ) α ∧ β = 0 . The remaining condition (8) is equiv alen t to the s et of prop erties X β ∈ A c β ( α, β ) B α,β ( a, b ) = 0 , (11) for any α ∈ A . I den tities (11) f ollo w from the ∨ -conditions (10), this completes the pro of of the theorem.  T r igonometric Solutions of WD VV Equ ations and Generalized CMS S ystems 5 Remark 1. Let trigonometric ∨ -systems A 1 ⊂ V ∗ 1 , A 2 ⊂ V ∗ 2 def ine the solutions (3) of the WD VV equations for some λ 1 , λ 2 . Th en the trigonometric ∨ -system A 1 ⊕ A 2 do es n ot def ine a solution. Indeed, let us tak e vect ors a, c ∈ V 1 and b, d ∈ V 2 . Then p rop erty (7) implies th at ( a, c ) 1 ( b, d ) 2 = 0 , (12) where ( · , · ) 1 , 2 are ∨ -forms (4) in the corresp ond ing spaces V 1 , 2 . Clearly , the relation (12) d o es not hold for general vec tors a , b , c , d . Remark 2. Not all the trigonometric solutions of the WD VV equations h a v e the form (3). It is s h o wn in [15] that trilogarithmic fu nctions ha v e to arise when ansatz for F is giv en by summation of g (( α, x )) o v er the ro ots of a ro ot system, x ∈ V . Remark 3. A slightl y more general ansatz f or the solutions F can b e considered w h en cub ic terms in x are added to F . Similarly to the pro of of Theorem 1 it follo ws that A still has to b e a trigonometric ∨ -system. The almost dual p otenti als corresp onding to the A n af f ine W eyl group orb it spaces h a v e su c h a f orm [9 ]. The corresp ondin g trigonometric ∨ -system A is the A n ro ot system in this case. Prop osition 1. L et A = { α, c α } b e a trigonometric ∨ -system. Then the set of ve ctors { √ c α α } is a ( r atio nal ) ∨ - system, that is F rat = P α ∈ A c α α ( x ) 2 log α ( x ) is a solution of the W DVV e quations in the sp ac e V . Pro of . By def inition of the trigonometric ∨ -system for any α ∈ A relations (6) hold. Consider t w o-dimensional plane π ⊂ V and sum up relations (6) ov er s so th at the α -series Γ s α b elong to the plane π ∋ α . W e arriv e at the r elations X β ∈ A ∩ π c β ( α, β ) α ∧ β = 0 , or, equiv al en tly , X β ∈ A ∩ π c β ( α, β ) β is prop ortional to α. (13) Relations (13) is a def inition of the (rational) ∨ -system for the set of co v ecto rs { √ c α α } (see [10] and [13] for the complex space). It is equiv ale n t to the p rop erty that F rat satisf ies WD VV equations in the sp ace V [10, 13]. Prop osition is pro v en.  Due to existence of extra v ariable y in the ansatz (3) the WD VV equations are n ontrivial al- ready when n = 2. Thus it is natural to study at f irst tw o-dimen s ional conf igur ations A d ef ining solutions of WD VV equations. When A consists of one v ector the corresp ondin g form (4) is d e- generate. If A consists of tw o non-collinear v ectors α , β then it follo ws that ( α, β ) = 0 therefore relation (6) holds and A is a trigonometric ∨ -system. How ever relation (7) cannot hold then for an y λ and therefore a p air of vec tors do es not def ine a solution to WD VV equations (see also Remark 1 ab o v e). The f ollo wing p r op ositions d eal with the next simplest cases w hen A consists of 3, 4 and 5 ve ctors r esp ectiv ely . In fact all irredu cible trigonometric ∨ -systems with up to 5 co v ectors hav e to b e tw o- dimensional. Prop osition 2. L et system A c onsist of thr e e ve ctors α , β , γ with nonzer o multiplicities c α , c β , c γ . Then A is an irr e ducible trigonometric ∨ -system iff α ± β ± γ = 0 for some c hoic e of signs. The non-de ge ne r acy c ondition for the form (4) is then g i ven by c α c β + c α c γ + c β c γ 6 = 0 . Any such system A defines the solution (3) of the WDVV e quations with λ = 2( c α c β + c α c γ + c β c γ )( c α c β c γ ) − 1 / 2 . 6 M.V. F eigin Pro of . It follo ws from relations (6) that γ = α + β u p to m ultiplicatio n of s ome of the ve ctors b y − 1. W e tak e a basis e 1 = α , e 2 = β in C 2 . The b ilinear form (4) takes the form G = c α x 2 1 + c β x 2 2 + c γ ( x 1 + x 2 ) 2 . This form is non-d egenerate if f c α c β + c α c γ + c β c γ 6 = 0. One can c hec k that e 1 ∨ = ( c β + c γ ) e 1 − c γ e 2 c α c β + c α c γ + c β c γ , e 2 ∨ = − c γ e 1 + ( c α + c γ ) e 2 c α c β + c α c γ + c β c γ , where e 1 , e 2 is dual basis to e 1 , e 2 , that is e i ( e j ) = δ i j . Relatio ns (6) lo ok as follo ws  e 1 ∨ , e 2 ∨  ( c β + c γ ) +  e 1 ∨ , e 1 ∨  c γ = 0 ,  e 1 ∨ , e 2 ∨  ( c α + c γ ) +  e 2 ∨ , e 2 ∨  c γ = 0 ,  e 1 ∨ , e 2 ∨  ( c α − c β ) +  e 1 ∨ , e 1 ∨  c α −  e 2 ∨ , e 2 ∨  c β = 0 , and they are automatically satisf ied. Relation (7) r esults to one scalar equation λ 2 4  c α c β ( α ∨ , β ∨ ) + c α c γ ( α ∨ , γ ∨ ) + c β c γ ( β ∨ , γ ∨ )  = c α c β + c α c γ + c β c γ , whic h has solution as stated in the formulat ion.  In the follo wing pr op osition w e study conf igurations consisting of four co v ec tors. Prop osition 3. L et system A c onsist of four ve ctors α , β , γ , δ with nonzer o multiplicities c α , c β , c γ , c δ . Then A is an irr e duci ble trigonometric ∨ -system iff the ve ctors in A + take the form e 1 , e 2 , e 1 ± e 2 in a suitable b asis, and the c orr esp onding multiplicities c 1 , c 2 , c ± satisfy c 1 = c 2 . This pr op erty is e qui valent to the ortho gonality ( e 1 + e 2 , e 1 − e 2 ) = 0 under the c orr esp onding ∨ -pr o duct. The non-de gener acy c ondition for the form (4) is then given by ∆ = ( c 1 + 2 c + )( c 1 + 2 c − ) 6 = 0 . These systems A define the solutions (3) of the WDVV e quations with λ = 2∆ c − 1 / 2 1 (4 c + c − + c 1 ( c + + c − )) − 1 / 2 onc e λ is finite. Pro of . It follo ws from the series relations (6) that there is a a vec tor α ∈ A such that all the remaining ve ctors β , γ , δ ∈ A b elong to single α -series Γ 1 α . Indeed, otherwise, up to renaming the co v ect ors and taking opp osite, w e ha v e δ = γ + nα , n ∈ N , ( α, β ) = ( γ , δ ) = 0. Then consideration of β -series giv es 2 γ + nα = mβ for some m ∈ Z . And consideration of γ -series giv es α + pγ = ± β for s ome p ∈ Z . Therefore 2 γ + n α = ± m ( α + pγ ), hence n = ± m and 2 = ± mp , th us m = ± 1 or p = ± 1. In the case m = ± 1 we h a v e n = 1 hence γ -series conta ins δ together with α and β . And in the case p = ± 1 the α -series con tains β together with γ and δ . So w e can assu m e th at there is only one α -series so that the remaining ve ctors take th e f orm γ = β + n 1 α , δ = β + n 2 α w ith integer n 2 > n 1 > 0. By considerin g β -series w e conclude that n 1 = 1. Consider no w the δ -series. I t is easy to see that cov ector β has to form a single series, therefore ( β , δ ) = 0 and the co v ecto rs β + α and α b elong to a δ -series. This is p ossible only if n 2 = 2. T aking no w the basis v ectors as e 1 = α , e 2 = β + α we conclude that the sys tem A consists of co v ect ors e 1 , e 2 , e 1 ± e 2 . The bilinear form (4) take s no w th e form G = c 1 x 2 1 + c 2 x 2 2 + c + ( x 1 + x 2 ) 2 + c − ( x 1 − x 2 ) 2 = ( c 1 + c + + c − ) x 2 1 + ( c 2 + c + + c − ) x 2 2 + 2( c + − c − ) x 1 x 2 , whic h has determinant ∆ = c 1 c 2 + ( c 1 + c 2 )( c + + c − ) + 4 c + c − . Therefore  e 1 , e 1  = ∆ − 1 ( c 2 + c + + c − ) ,  e 2 , e 2  = ∆ − 1 ( c 1 + c + + c − ) ,  e 1 , e 2  = ∆ − 1 ( c − − c + ) . (14) T r igonometric Solutions of WD VV Equ ations and Generalized CMS S ystems 7 No w w e analyze the series relations (6). The orthogonalit y ( e 1 − e 2 , e 1 + e 2 ) = 0 is clearly equiv alent to the condition c 1 = c 2 . Then the r emaining conditions (6) on ( e 1 ± e 2 )-series are automatica lly satisf ied. The condition (6) for the e 1 -series has the form c −  − e 1 + e 2 , e 1  + c 2  e 2 , e 1  + c +  e 1 + e 2 , e 1  = 0 , and it follo ws from the scalar p ro du cts (14). The condition on the e 2 -series is also satisf ied. It is easy to c hec k that relation (7) holds if f λ is as stated, h ence prop osition is p ro v en.  Prop osition 4. L et irr e ducible trigonometric ∨ -system A c onsist of five ve ctors with non-zer o multiplicities. Then in the appr opriate b asis A + takes the form e 1 , 2 e 1 , e 2 , e 1 ± e 2 and the c orr esp ond ing multiplicities c 1 , e c 1 , c 2 , c ± satisfy c + = c − (e quiv alently, ( e 1 , e 2 ) = 0 ) and 2 e c 1 c 2 = c + ( c 1 − c 2 ) . The f orm (4) is then non-de gener ate when ∆ = ( c 1 + 4 e c 1 + 2 c + )( c 2 + 2 c + ) 6 = 0 . The c orr e- sp onding solution of the W DVV e quations has the form (3) with λ = √ 2∆( c 2 + 2 c + ) − 1 / 2 ( c 1 + 4 e c 1 ) − 1 / 2 c − 1 / 2 + . Pro of is obtained b y simple analysis of the series conditions (6). One can f irstly establish that A is tw o- dimensional. Th en it is easy to see that A has to cont ain collinear v ectors, and the required form follo ws. T o conclude this section we present a few examples of trigonometric ∨ -systems on the plane with higher n um b er of vec tors. R ecall f irstly that the p ositiv e ro ots of th e r o ot sys tem G 2 can b e written as α , β , β + α , β + nα , β + ( n + 1) α , 2 β + ( n + 1) α w here n = 2. One can sho w that for integ er n > 2 the ab ov e vec tors never form a trigonometric ∨ -system, and th at for n = 2 the m ultiplicities hav e to satisfy c α = c β + α = c β + nα and c β = c 2 β +( n +1) α = c β +( n +1) α whic h is the case of the G 2 system. There are though some p ossibilities to extend the G 2 system. Firstly , one can sho w that G 2 ∪ A 2 where the system A 2 consists of d oubled sh ort ro ots of G 2 , is a trigonometric ∨ -system for appropriate multipliciti es. Secondly th e follo wing prop osition tak es place. Prop osition 5. L et A c ons ist of the ve ctors e 1 , e 2 , 2 e 2 , 1 2 ( e 1 ± e 2 ) , 1 2 ( e 1 ± 3 e 2 ) with the c orr esp ond ing nonzer o multiplicities c 1 , c 2 , e c 2 , a , b . Then A is a trigonometric ∨ -system iff the multiplicities satisfy the r elations a = 3 b , c 2 = a + 2 e c 2 , (2 c 1 + b ) c 2 = ( c 1 + 2 b ) a . Note that in the limiting case e c 2 = 0 w e reco v er the sys tem G 2 with sp ecial multiplicit ies. An example of trigonometric ∨ -system w ith yet h igher num b er of vecto rs is give n by vect ors e 1 , 2 e 1 , e 2 , 2 e 2 , e 1 ± e 2 , e 1 ± 2 e 2 , 2 e 1 ± e 2 where the m ultiplicities can b e c hosen approp r iately . 3 Relations with generalized Calogero–Moser–Sutherland systems Relation b et w een ∨ -systems and the pr op ert y of a Schr¨ odinger op erator of C MS typ e to hav e a factorized eigenfunction wa s observed by V eselo v in [11]. Namely , it w as sh o wn in [11 ] that if an op erator L = − ∆ + X α ∈ A + m α ( m α + 1)( α, α ) sin 2 ( α, x ) has a formal eigenfunction ψ = Y α ∈ A + sin − m α ( α, x ) , Lψ = µψ , 8 M.V. F eigin then F = P α ∈ A + m α ( α, x ) 2 log( α, x ) satisf ies the WD VV equ ations. T he follo wing theorem es- tablishes the con v erse statemen t in the case of trigonometric ∨ -systems. Theorem 2. L et A b e a trigonometric ∨ -system c onsisting of p airwise non-c ol line ar c ove ctors α with multiplicities c α . Then Schr¨ odinger op er ator L = − ∆ + X α ∈ A c α ( c α + 1)( α, α ) sin 2 α ( x ) (15) c onstructe d by the metric (4) has the formal eigenfunction ψ = Y α ∈ A sin − c α α ( x ) , L ψ = µψ . (16) Pro of . The pr op ert y Lψ = µψ is equiv alen t to the id entit y X α 6 = β c α c β ( α, β ) cot α ( x ) cot β ( x ) = const . (17) T o establish th e last identi t y it is su f f icien t to sh o w that the left-hand side of (17) is n on-singular. In other w ords, we need to sho w th at X β ,β 6 = α c β ( α, β ) cot β ( x ) = 0 (18) if cot α ( x ) = 0. T he last p rop erties are suf f icien t to c hec k when summation is tak en along arbitrary α -series, th en it is gu aranteed by relation (6). This prov es the theorem.  Corollary 1. Assume that function (3) c onstructe d by a set of p airw ise non-c ol line ar c ove c- tors α with multiplicities c α satisfies the WDVV e qu ations (5) . Then r elation (16) holds for the Schr¨ odinger op er ator (15) . Con v ersely , the p rop erty of a S c hr¨ odinger op erator to ha v e a factorized eigenfunction im p lies that th e corresp ondin g ve ctors √ c α α form a r ational ∨ -system [11]. T his prop erty is also suf f icien t to obtain the trigonometric ∨ -system, and the argum en ts are close to [11]. Theorem 3. Assume that the Schr¨ odinger op er ator (15) has an eigenfu nc tion (16) . Then the set A of v e ctors α with the multiplicities c α forms the trigonometric ∨ -system. Pro of . F r om equation (16), (15 ) it follo ws iden tit y (18) at cot α ( x ) = 0. Therefore for eac h α -series Γ s α w e ha v e X β ∈ Γ s α c β ( α, β ) α ∧ β = 0 . (19) Let β w denote a v ector dual to β with resp ect to the inner pro du ct ( · , · ) inv o lv ed in th e Sc hr¨ odinger equation. By summing identit ies (19) along all th e α -series w e conclude that X β ∈ A c β β ( α w ) β w is prop ortional to α w . (20) No w w e can d ecomp ose the sp ace V = V 1 ⊕ · · · ⊕ V k so that the op erator P β ∈ A c β β ⊗ β w is equal to constan t µ i on V i . W e can also assume that ( V i , V j ) = 0 if i 6 = j . It f ollo w s fr om (20) that G ( · , · ) | V i = µ i ( · , · ) | V i . Th erefore id en tities (19) imply X β ∈ Γ s α c β α ( β ∨ ) α ∧ β = 0 whic h are identit ies (6) fr om the def inition of the trigonometric ∨ -systems.  T r igonometric Solutions of WD VV Equ ations and Generalized CMS S ystems 9 Corollary 2. Assume that the Schr¨ odinger op er ator (15) has an eigenfunction (16) . Assume also that the system A is irr e ducible and that for some Λ and any a, b, c, d ∈ V the pr op erty X α,β ∈ A + (Λ( α, β ) − 1) c α c β B α,β ( a, b ) B α,β ( c, d ) = 0 (21) holds. Then the c orr esp onding function (3) with appr opriate λ satisfies the WDVV e quations (5) . Remark 4. T he previous corollary also holds for the r educible s y s tems A if we replace th e Sc hr¨ odinger equation metric ( α, β ) in (21) by the ∨ -pro du ct α ( β ∨ ). In this case λ = 2 √ Λ. 4 Concluding remarks T r igonometric ∨ -systems require fu rther inv estig ations. It w ould b e interesti ng to obtain almost dual p rep otent ials for the F rob enius manifolds of the af f ine W eyl groups as w ell as for their dis- criminan ts (cf. rational case [7, 12]). Comparison with a recen t work on th e elliptic s olutions [16] migh t also b e interesting. W e also h op e that the series cond itions would allo w understanding and eve n tually classif ication of the trigonometric ∨ -systems. 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