Solutions of B C n Type of WDVV Equations
aa r X i v : . [ m a t h - ph ] F e b SOLUTIONS OF BC n TYPE OF WDVV EQUATIONS
MAALI ALKADHEM, GEORGIOS ANTONIOU, AND MISHA FEIGIN
To the memory of Boris Dubrovin
Abstract.
We give a family of solutions of Witten–Dijkgraaf–Verlinde–Verlindeequations in n -dimensional space. It is defined in terms of BC n root system and n + 2 independent multiplicity parameters. We also apply these solutions to definesome N = 4 supersymmetric mechanical systems. Introduction
In this note we are interested in trigonometric solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Let us recall these equations, their specialsolutions and where they arise.Let F = F ( x , . . . , x n ) be a function in V ∼ = C n . Consider a vector field e = n X i =1 A i ( x ) ∂ x i , where A i ( x ) = A i ( x , . . . , x n ) are some functions. Define n × n matrix B = ( B ij ) ni,j =1 by B ij = e ( F ij ) = n X k =1 A k ( x ) F ijk , i, j = 1 , . . . , n, (1.1)where F ijk = ∂ F∂x i ∂x j ∂x k . Then WDVV equations are the following equations for the function F : F i B − F j = F j B − F i , i, j = 1 , ..., n, (1.2)where F i is the n × n matrix constructed from the third order partial derivatives:( F i ) jk = F ijk and B − is the inverse of B. It was noted in [18] (see also [15]) thatequations (1.2) are equivalent to generalized WDVV equations F i F − k F j = F j F − k F i , (1.3) The work of M.A. was funded by Imam Abdulrahman Bin Faisal University, Kingdom of SaudiArabia.The work of G.A. was funded by EPSRC doctoral training partnership grants EP/M506539/1,EP/M508056/1, EP/N509668/1. where i, j, k = 1 , ..., n, provided that all the matrices F k and B are invertible.Rational solutions of equations (1.3) have the form F rat = X α ∈A ( α, x ) log( α, x ) , (1.4)where A is a configuration of vectors in V . Solutions of the form (1.4) for theroot systems A appear in Frobenius manifolds theory as almost dual prepotentialsfor the Coxeter orbit spaces polynomial Frobenius manifolds [7]. They also appearin Seiberg-Witten theory as perturbative parts of prepotentials [18]. For general A they were interpreted geometrically in [21] through the introduced notion of a ∨ -system. Multiparameter deformations related to A n and B n root systems wereconstructed in [6]. The class of ∨ -systems and the corresponding rational solutions(1.4) are closed under the natural operation of taking restrictions and subsystems[12], [13].Trigonometric generalisation of solutions (1.4) also arise in theory of Frobeniusmanifolds. These solutions have the form F trig = X α ∈A c α f (( α, x )) + Q, (1.5)where c α ∈ C are some multiplicity parameters and Q is a cubic polynomial in x = ( x , . . . , x n ) which also often depends on additional variable y, and f is thefunction of a single variable z given by f ( z ) = 16 z −
14 Li ( e − z ) , (1.6)so that f ′′′ ( z ) = coth z . Such solutions appear as almost dual prepotentials for theorbit spaces of affine Weyl groups Frobenius manifolds ([9], [10], [8]). They alsoappear in the study of quantum cohomology of resolutions of simple singularities[5]. Trigonometric version of a ∨ -system type geometrical conditions for solutions(1.5) was proposed in [14]. Trigonometric solutions and ∨ -systems allow operationsof taking restrictions and subsystems respectively [1].In this work we are interested in the case when the cubic corrections are absent,that is Q = 0 . The corresponding solution (1.5) does not exist in general even forthe case of root system A and invariant multiplicities c α . However, it is known toexist for the root system B n and specific choice of invariant multiplicities [17]. Inthis note we generalize this solution so that it is included in ( n + 2)-parametricfamily. The underlying configuration A is the positive half of BC n root system, andmultiplicities are chosen in a specific way. In order to get such a solution we findfirstly a two-parameter family of solutions where the configuration A is the positivehalf of BC n and multiplicities are Weyl-invariant. We obtain solutions with manyparameters by taking special restrictions of these solutions (cf. [13], [1]).We also apply these solutions in order to construct N = 4 supersymmetric me-chanical systems. Relations of such mechanical systems with WDVV equations wereknown since [22] and [3]. Trigonometric solutions of WDVV equations were used OLUTIONS OF BC n TYPE OF WDVV EQUATIONS 3 to construct N = 4 supersymmetric Hamiltonians in [2]. This gave, in particu-lar, supersymmetric version of quantum Calogero–Moser–Sutherland system of type BC n with two independent coupling parameters. Thus we extend this Hamiltonianinto multiparameter family. Other N = 4 supersymmetric systems related withCalogero–Moser–Sutherland system with extra spin variables were obtained in [11]using the superfield approach.It would be interesting to see whether constructed solutions of WDVV equationsadmit extra Frobenius manifolds structures or further links to geometry as it hap-pens for root systems solutions with non-zero cubic terms Q depending on an extravariable. Possible elliptic generalizations may also be interesting (see [19], [4] and[20] for elliptic solutions of WDVV equations).2. Metric for a family of BC n type configurations We are going to present solution F to the equations (1.2) for a suitable vector field e. This solution is related to BC n root system with prescribed multiplicities of theroot vectors. Let m = ( m , . . . , m n ) ∈ C n and let r, s, q ∈ C . Let BC n ( r, s, q ; m ) ⊂ C n be the following configuration of vectors α with corresponding multiplicities c α : e i , with multiplicities rm i , ≤ i ≤ n, e i , with multiplicities sm i + 12 qm i ( m i − , ≤ i ≤ n,e i ± e j , with multiplicities qm i m j , ≤ i < j ≤ n, where e , . . . , e n is the standard basis in C n . If all the multiplicities m i = 1 thenthe configuration reduces to the configuration BC n ( r, s, q ) which is a positive halfof the root system BC n with an invariant collection of multiplicities r, s, q. Let us consider the function F given by F = X α ∈ BC n ( r,s,q ; m ) c α f (( α, x )) , (2.1)where f is given by (1.6). More explicitly the function F can be written as follows: F = n X i =1 rm i f ( x i ) + n X i =1 ( sm i + 12 qm i ( m i − f (2 x i ) + n X i We have the following expression for the third order derivatives of F : F klt = rm k b k δ kl δ lt + 4(2 sm k + qm k ( m k − e b k δ kl δ lt + qδ kl δ lt n X j =1 j = k m j m k b kj (2.4)+ qm t m k b tk δ kl + qm l m k b lk δ kt + qm k m l b kl δ lt , where k, l, t = 1 , ..., n, and δ is the Kronecker symbol.Proof. We note that the first two terms in (2.4) are obtained from the first two termsin formula (2.2). The last term in (2.2) contributes the following sum in F klt : q n X i The sum of the terms (2.8) and (2.9) equals to qm t m k b tk δ kl . Similarly, the sum ofthe terms q n X i We have the following identities: A k b kj + A j b jk = 2(cosh 2 x k + cosh 2 x j ) , ≤ j = k ≤ n, (2.10) and A k b jk + A j b kj = 0 , j, k = 1 , . . . , n, (2.11) where A k is given by (2.3).Proof. We have A k b kj + A j b jk = A k sinh 2 x k − A j sinh 2 x j sinh( x k + x j ) sinh( x k − x j ) = cosh 4 x k − cosh 4 x j cosh 2 x k − cosh 2 x j , which implies the first formula (2.10). Identity (2.11) follows similarly. (cid:3) Now we show that the matrix B is diagonal. Moreover, it is proportional to aconstant diagonal matrix under a particular restriction on the parameters r, q, s . Proposition 2.3. The matrix B = B ( x ) with the matrix entries B lt = n X k =1 A k F klt , l, t = 1 , . . . , n is diagonal. Furthermore, if the multiplicities r , q , s and m satisfy the relation r = − s − q ( N − , (2.12) where N = P nk =1 m k , then the matrix B takes the form B lt = m l h ( x ) δ lt , (2.13) MAALI ALKADHEM, GEORGIOS ANTONIOU, AND MISHA FEIGIN where h ( x ) = 2 q P nk =1 m k cosh 2 x k + r .Proof. It follows by Lemma 2.1 that for l = tB lt = qm l m t ( A l b tl + A t b lt ) , which is equal to zero by Lemma 2.2.Let us now consider the diagonal entries of B . We have by Lemmas 2.1, 2.2 B ll = rm l A l b l + 4(2 sm l + qm l ( m l − A l ˜ b l + q n X k =1 k = l m k m l ( A l b lk + A k b kl )= 2 rm l cosh x l + 4(2 sm l + qm l ( m l − x l + 2 q n X k =1 k = l m k m l (cosh 2 x k + cosh 2 x l ) . Then B ll = 2 rm l cosh x l + 4(2 sm l + qm l ( m l − x l + 2 q ( N − m l ) m l cosh 2 x l + 2 qm l n X k =1 m k cosh 2 x k = m l (cid:16)(cid:0) r + 8 s + 2 q ( N − (cid:1) cosh 2 x l + 2 q n X k =1 m k cosh 2 x k + r (cid:17) , which implies the statement. (cid:3) Below summation over repeated indices will be assumed. Let us now assume thatmultiplicities m i = 1 for all i = 1 , ..., n. For any vector v = ( v , . . . , v n ) ∈ V let usintroduce the vector field ∂ v = v i ∂ x i ∈ T V . For any u = ( u , . . . , u n ) ∈ V we definethe following multiplication on the tangent plane T x V for generic x ∈ V : ∂ u ∗ ∂ v = F ijk u i v j ∂ x k . (2.14)Note that multiplication (2.14) defines a commutative algebra on T x V . The followingtheorem takes place. Theorem 2.4. Suppose that parameters r , s and q satisfy the linear relation (2.12).Then function F = r n X i =1 f ( x i ) + s n X i =1 f (2 x i ) + q X ≤ i Proof. It has been shown in [2] that the function (2.15) satisfies the following systemof equations if the linear relation (2.12) holds: F i F j = F j F i , (2.16)for all i, j = 1 , . . . , n . It then follows from Proposition 2.3 that conditions (2.16)are equivalent to WDVV equations (1.2) since the matirx B is proportional to theidentity matrix. Also it is easy to see that associativity of the multiplication (2.14)is equivalent to the relation (2.16). (cid:3) In the remaining part of the paper we prove generalization of Theorem 2.4 to theconfiguration BC n ( q, r, s ; m ) , that is to the case of arbitrary multiplicities m i . Thisgeneralization can be formulated as follows. Theorem 2.5. Suppose parameters r, s, q and m satisfy the relation r = − s − q ( N − , (2.17) where N = P ni =1 m i . Then prepotential (2.2) satisfies WDVV equations (1.2) where B = P ni =1 sinh 2 x i F i . Remark . Theorem 2.5 generalizes Theorem 2.3 from [17]. In this case we haveall m i = 1 and s = 0 . Then putting q = 1 we get the standard B N root system andthe condition (2.17) reduces to r = − N − 2) which is the multiplicity of the shortroot of B N root system considered in [17]. Remark . In the rational limit solutions (2.2) of WDVV equations reduce to B n family of ∨ -system found in [6].3. Proof through restrictions Let A be the configuration A = BC N ( r, s, q ) ⊂ W ∼ = C N , N ∈ N . Let e , . . . , e N be the standard basis of W. Let ( · , · ) be the standard inner product which is definedby ( x, y ) = N X i =1 x i y i , where x = ( x , . . . , x N ) , y = ( y , . . . , y N ) ∈ W. Let n ∈ N and m = ( m , . . . , m n )with m i ∈ N such that P ni =1 m i = N . Let us consider subsystem B ⊂ A as follows: B = { e P i − j =1 m j + k − e P i − j =1 m j + l , ≤ k < l ≤ m i , i = 1 , . . . , n } . Now let us consider the corresponding subspace of W of dimension n given by W B = { x ∈ W : ( β, x ) = 0 , ∀ β ∈ B} . MAALI ALKADHEM, GEORGIOS ANTONIOU, AND MISHA FEIGIN More explicitly, vectors x = ( x , . . . , x N ) ∈ W B satisfy conditions: x = · · · = x m ,x m +1 = · · · = x m + m , ... x m + m + ··· + m n − +1 = · · · = x N . (3.1)For any vector v = ( v , . . . , v N ) ∈ W let us define the vector field ∂ v = v i ∂ x i ∈ T W .For any u = ( u , . . . , u N ) ∈ W we define the following multiplication on the tangentplane T x W for generic x ∈ W : ∂ u ∗ ∂ v = u i v j F ijk ∂ x k , (3.2)where the function F is given by F = X α ∈A c α f (( α, x )) . (3.3)Assume that parameters r, s, q and m satisfy the relation r = − s − q ( N − . Thenmultiplication (3.2) is associative by Theorem 2.4 (applied with n = N ). Note thatfunction (3.3) satisfies F ijk = X α ∈A c α α i α j α k coth( α, x ) , hence multiplication (3.2) can be expressed as follows: ∂ u ∗ ∂ v = X α ∈A c α ( α, u )( α, v ) coth( α, x ) ∂ α . (3.4)If we identify W with T x W ∼ = W, then multiplication (3.4) takes the form u ∗ v = X α ∈A c α ( α, u )( α, v ) coth( α, x ) α. (3.5)Define M B = W B \ S α ∈A\B Π α , where Π α = { x ∈ W : ( α, x ) = 0 } . Consider now apoint x ∈ M B and two tangent vectors u , v ∈ T x M B . We extend vectors u and v to two local analytic vector fields u ( x ) , v ( x ) in the neighbourhood U of x thatare tangent to the subspace W B at any point x ∈ M B ∩ U such that u = u ( x )and v = v ( x ). Now we want to study the limit of u ( x ) ∗ v ( x ) when x tends to x . The limit may have singularities at x ∈ W B as coth( α, x ) with α ∈ B is not definedfor such x. Also we note that outside W B we have a well-defined multiplication u ( x ) ∗ v ( x ) . Similarly to the rational case considered in [12] and trigonometric casewith extra variable [1] the following lemma holds. Lemma 3.1. The product u ( x ) ∗ v ( x ) has a limit when x tends to x ∈ M B given by u ∗ v = X α ∈A\B c α ( α, u )( α, v ) coth( α, x ) α. (3.6) In particular u ∗ v is determined by u and v only. OLUTIONS OF BC n TYPE OF WDVV EQUATIONS 9 Now we are going to show that the product u ∗ v belongs to T x M B . We willneed the following lemma (cf. [12], [1]). Lemma 3.2. Let α ∈ A . Let x ∈ Π α be generic. Then the identity X β ∈A β ≁ α c β ( α, β ) coth( β, x ) B α,β ( a, b ) α ∧ β = 0 (3.7) holds for all a, b ∈ V provided that ( α, x ) = 0 , where B α,β ( a, b ) = ( α, a )( β, b ) − ( α, b )( β, a ) and α ∧ β = α ⊗ β − β ⊗ α. Proof. For any β ∈ A such that β ≁ α let γ = s α β , where s α is the orthogonalreflection about Π α . Note that coth( γ, x ) = coth( β, x ) at ( α, x ) = 0 . Also note that( α, γ ) = − ( α, β ) , B α,γ ( a, b ) = B α,β ( a, b ) , α ∧ γ = α ∧ β. We have that either γ or − γ is an element of A . Suppose firstly that γ ∈ A . Then c β ( α, β ) coth( β, x ) B α,β ( a, b ) α ∧ β + c γ ( α, γ ) coth( γ, x ) B α,γ ( a, b ) α ∧ γ = 0 (3.8)at ( α, x ) = 0 since multiplicities are B N -invariant. If one replaces γ with − γ then(3.8) holds as well. (cid:3) Proposition 3.3. Let u, v ∈ T x M B where x ∈ M B . Then u ∗ v ∈ T x M B . Proof. By Lemma 3.1 it is enough to show that X β ∈A\B c β ( β, u )( β, v )( α, β ) coth( β, x ) = 0 (3.9)for all α ∈ B . Assume firstly that W B has codimension 1 . By Lemma 3.2 we get X β ∈A β ≁ α c β ( α, β ) coth( β, x ) (cid:18) ( α, a )( β, b ) − ( α, b )( β, a ) (cid:19)(cid:18) ( α, y )( β, z ) − ( α, z )( β, y ) (cid:19) = 0(3.10)for any a, b, y, z ∈ V and generic x ∈ Π α . Assume that a, y / ∈ Π α and let b = u ∈ Π α and z = v ∈ Π α . Then ( α, b ) = ( α, z ) = 0 and relation (3.10) implies that X β ∈A β ≁ α c β ( α, β )( β, u )( β, v ) coth( β, x ) = 0 . As W B has codimension 1 the relation β ≁ α is equivalent to β ∈ A \ B and lemmafollows.Let us now suppose that W B has codimension 2 . Let α, γ ∈ B , α ≁ γ. By the abovearguments for generic x ∈ Π α and u, v ∈ T x Π α , we have u ∗ v ∈ T x Π α . Similarly if x ∈ Π γ is generic and u, v ∈ T x Π γ , then u ∗ v ∈ T x Π γ . By Lemma 3.1, u ∗ v existsfor x ∈ M B and u, v ∈ T x M B . It follows that for any x ∈ M B we have u ∗ v ∈ T x M B , which proves the statement for the case when W B has codimension 2 . General W B is dealt with similarly. (cid:3) Consider now the orthogonal decomposition W = W B ⊕ W ⊥B (3.11)with respect to the standard inner product. Any α ∈ W can be written as α = e α + w, (3.12)where e α ∈ W B is the orthogonal projection of vector α to W B and w ∈ W ⊥B . For any x ∈ M B and u, v ∈ T x M B one can represent product u ∗ v as u ∗ v = X α ∈A\B c α ( α, u )( α, v ) coth( α, x ) e α (3.13)by Proposition 3.3. Hence, we have ∂ u ∗ ∂ v = X α ∈A\B c α ( α, u )( α, v ) coth( α, x ) ∂ e α . (3.14)Let us define vectors f i , ≤ i ≤ n by f i = m i X j =1 e P i − s =1 m s + j . (3.15)These vectors form a basis for W B . The following lemma gives the general formula for the orthogonal projection ofany vector u ∈ W to W B . Lemma 3.4. Let u = P Ni =1 u i e i ∈ W. Then the projection e u has the form e u = m m X i =1 u i , . . . , m m X i =1 u i | {z } m , . . . , m n m n X i =1 u P n − s =1 m s + i , . . . , m n m n X i =1 u P n − s =1 m s + i | {z } m n ! . (3.16)Let us now project A to the subspace W B . Notice that by Lemma 3.4 e e P i − s =1 m s +1 = · · · = e e P is =1 m s = m − i f i , i = 1 , . . . , n. Let us denote the projected system as e A = BC n ( q, r, s ; m ) ⊂ C n . It consists ofvectors α with the corresponding multiplicities c α given as follows: b f i = m − i f i , with multiplicities rm i , ≤ i ≤ n, b f i = 2 m − i f i , with multiplicities sm i + 12 qm i ( m i − , ≤ i ≤ n, b f i ± b f j = m − i f i ± m − j f j , with multiplicities qm i m j , ≤ i < j ≤ n. OLUTIONS OF BC n TYPE OF WDVV EQUATIONS 11 By Lemma 3.4, for any α ∈ W, its orthogonal projection has the form e α = n X k =1 e α k f k , where the basis f k is given by (3.15) and e α k = ( e α, f k )( f k , f k ) = ( α, f k ) m k . (3.17)Let us define e F ( e x ) = X γ ∈ e A c γ f (( γ, e x )) , (3.18)where e x = n X i =1 e x i f i ∈ W B . (3.19)Note that function (3.18) can also be represented as e F ( e x ) = X α ∈A\B c α f (( α, e x )) . Let e F i be the n × n matrix constructed from the third-order partial derivatives ofthe function e F , that is ( e F i ) jk = e F ijk = ∂ e F∂ e x i ∂ e x j ∂ e x k ,i, j, k = 1 , . . . , n. The following lemma gives another way to represent multiplication (3.14) on W B . Lemma 3.5. The multiplication (3.14) takes the form ∂ f i ∗ ∂ f j = n X k =1 m − k e F ijk ∂ f k , i, j = 1 , ..., n. Proof. We rearrange ∂ e α in the right hand side of (3.14) as ∂ e α = n X k =1 e α k ∂ f k = n X k =1 m − k ( α, f k ) ∂ f k by (3.17). Therefore the multiplication (3.14) can be rewritten as ∂ f i ∗ ∂ f j = X α ∈A\B n X k =1 c α m − k ( α, f i )( α, f j )( α, f k ) coth( α, e x ) ∂ f k = n X k =1 m − k e F ijk ∂ f k , i, j = 1 , ..., n, as required. (cid:3) Let H B be the matrix of the restriction of the standard inner product on W B inthe basis f , . . . , f n . That is( H B ) lt = ( f l , f t ) = m l δ lt , l, t = 1 , . . . , n. (3.20) Lemma 3.6. The matrix H B can be written as a linear combination H B = n X i =1 a i e F i , where functions a i are given by a i = h ( e x ) − sinh 2 e x i , and h ( e x ) = 2 q P nk =1 m k cosh 2 e x k + r. Proof. By Proposition 2.3 and (3.20), we have H B = h ( e x ) − B ( e x ) , where B ( e x ) = n X i =1 (sinh 2 e x i ) e F i . This implies the statement. (cid:3) The previous considerations allow us to prove the following theorem, which isa version of stated earlier Theorem 2.5 where multiplicities m i do not have to beinteger. Theorem 3.7. Let e A = BC n ( q, r, s ; m ) ⊂ C n . Suppose parameters r, s, q and m satisfy the relation r = − s − q ( N − , (3.21) where N = P ni =1 m i . Then the prepotential e F = X α ∈ e A c α f (( α, e x )) , e x ∈ W B , (3.22) satisfies the WDVV equations e F i B − e F j = e F j B − e F i , i, j = 1 , ..., n, (3.23) where B = P ni =1 sinh 2 e x i e F i . The corresponding associative multiplication has the form u ∗ v = X e α ∈ e A c e α ( e α, u )( e α, v ) coth( e α, e x ) e α, (3.24) for any u, v ∈ T e x M B , e x ∈ M B , where e α is the orthogonal projection of α to W B . Proof. Let us assume firstly that m i ∈ N for any i. Consider the multiplication u ∗ v = X α ∈ BC N ( r,s,q ) c α ( α, u )( α, v ) coth( α, x ) α (3.25)on the tangent space T x W for x ∈ W. By Theorem 2.4 the multiplication (3.25) isassociative. Now as x tends to e x ∈ M B , by Lemmas 3.1, 3.5 and Proposition 3.3 OLUTIONS OF BC n TYPE OF WDVV EQUATIONS 13 this product restricts to an associative product on the tangent space T e x M B whichhas the form ∂ f i ∗ ∂ f j = n X l =1 m − l e F ijl ∂ f l . (3.26)The associativity condition( ∂ f i ∗ ∂ f j ) ∗ ∂ f k = ∂ f i ∗ ( ∂ f j ∗ ∂ f k ) , for any i, j, k = 1 , . . . , n, can be rearranged as n X l =1 m − l e F ijl ∂ f l ∗ ∂ f k = n X l =1 m − l e F jkl ∂ f i ∗ ∂ f l . Hence, we have n X l =1 m − l e F ijl e F lkp = n X l =1 m − l e F jkl e F ilp , (3.27)for any i, j, k, p. In the matrix form we have e F i H − B e F k = e F k H − B e F i . By Lemma 3.6 we obtain relation (3.23) where B = P ni =1 sinh 2 e x i e F i as required.This proves the theorem for the case when m i ∈ N . Since m i can take arbitraryinteger values the statement follows for general m i as well. (cid:3) Application to supersymmetric mechanics Let us define coordinates b x = ( b x , . . . , b x n ) ∈ C n by b x i = m / i x i (1 ≤ i ≤ n ). LetΛ be the n × n diagonal matrix Λ = ( m / i δ ij ) ni,j =1 . Let F be given by formula (2.2)with relation (2.12) on parameters r, q, s . By Proposition 2.3 the matrix B can berepresented as B = h ( x )Λ . Let us define a function b F ( b x ) such that b F ( b x ) = F ( x ( b x )).Consider the n × n matrices b F k with entries( b F k ) lt = b F klt = ∂ b F∂ b x k ∂ b x l ∂ b x t , (4.1) k, l, t = 1 , . . . , n. Note that b F k = m − / k Λ − F k Λ − , where F k is the matrix withentries ( F k ) lt = F klt . Let b B be the n × n matrix with entries b B ij = hδ ij , where h = h ( x ( b x )) is given in Proposition 2.3. Proposition 4.1. The metric b B can be represented as b B = n X k =1 m / k sinh(2 m − / k b x k ) b F k , and the function b F satisfies generalised WDVV equations of the form b F i b F j = b F j b F i , (4.2) for all i, j = 1 , . . . , n .Proof. The first part of the statement is immediate by Proposition 2.3. Considerthe left-hand-side of (4.2). We have b F i b F j = ( m i m j ) − / Λ − F i Λ − F j Λ − = h ( m i m j ) − / Λ − F i B − F j Λ − . (4.3)It follows by Theorem 2.5 that the right-hand-side of (4.3) is symmetric under theswap of i and j , hence the statement follows. (cid:3) Two different representations of N = 4 supersymmetry algebra were constructedin [2] (see also references therein). The corresponding supercharges depend on aprepotential of the form (1.5) (with Q = 0). This prepotential is assumed to satisfyequations of the form (2.16). It follows by Proposition 4.1 that the function b F satisfies such type of equations, hence we obtain in this way two supersymmetricHamiltonians H i , i = 1 , BC n type configurations. We give theseHamiltonians in detail. Let us consider the following configuration b A ⊂ C n ofvectors α with multiplicities c α : m − / i e i , with multiplicities rm i , ≤ i ≤ n, m − / i e i , with multiplicities sm i + 12 qm i ( m i − , ≤ i ≤ n,m − / i e i ± m − / j e j , with multiplicities qm i m j , ≤ i < j ≤ n. Consider n (quantum) particles on a line with coordinates b x j and momenta p j = − i∂ b x j , j = 1 , . . . , n to each of which we associate four fermionic variables ψ aj , ¯ ψ ja , a =1 , . These variables may be thought of as operators acting on wavefunctions whichdepend on bosonic and fermionic variables. Let ǫ ab be the fully anti-symmetrictensors in two dimensions such that ǫ = − ǫ = 1.Fermionic variables are assumed to satisfy the following (anti)-commutation rela-tions ( a, b = 1 , j, k = 1 , . . . , n ): { ψ aj , ¯ ψ kb } = − δ jk δ ab , { ψ aj , ψ bk } = { ¯ ψ ja , ¯ ψ kb } = 0 . We consider supercharges of the form Q a = − i ∂∂ b x r ψ ar + i b F rjk ( ǫ bc ǫ da ψ br ψ cj ¯ ψ kd − ψ ar δ jk ) , ¯ Q a = − i ∂∂ b x l ¯ ψ la + i b F lmn ( ǫ bd ǫ ac ¯ ψ ld ¯ ψ mb ψ cn − 12 ¯ ψ la δ nm ) . where a = 1 , b F ijk is defined in (4.1), and we assume summation over repeatedindecies. Let ∆ = P nj =1 ∂ b x j be the Laplacian in C n . We have the following statementon supersymmetry algebra which follows from [2]. OLUTIONS OF BC n TYPE OF WDVV EQUATIONS 15 Proposition 4.2. For all a, b = 1 , the supercharges Q a , ¯ Q b generate N = 4 supersymmetry algebra of the form { Q a , Q b } = { ¯ Q a , ¯ Q b } = 0 , and { Q a , ¯ Q b } = − H δ ab , where the Hamiltonian H is given by H = − ∆ + 12 X α ∈ b A c α ( α, α ) sinh ( α, b x ) + 14 X α,β ∈ b A c α c β ( α, α )( β, β )( α, β ) coth( α, b x ) coth( β, b x ) + Φ , with the fermionic term Φ = X α ∈ b A c α α i α j sinh ( α, b x ) (cid:0) α l α k ǫ bc ǫ ad ψ bi ψ cj ¯ ψ ld ¯ ψ ka + ( α, α ) ψ ai ¯ ψ ja (cid:1) . (4.4)The Hamiltonian H is formally self-adjoint. Similar considerations (see [2]) yieldnot self-adjoint Hamiltonian of the form H = − ∆ + X α ∈ b A c α ( α, α ) coth( α, b x ) ∂ α + Φwith the same fermionic term Φ . In fact Hamiltonians H , H satisfy gauge relation H = gH g − , where g = Q α ∈ b A sinh cα ( α,α ) ( α, b x ). References [1] Alkadhem M.A. and Feigin M., On trigonometric solutions of WDVV equations , in prepara-tion.[2] Antoniou G. and Feigin M., Supersymmetric V -systems , JHEP. 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