Seiberg-Witten curves and double-elliptic integrable systems

An old conjecture claims that commuting Hamiltonians of the double-elliptic integrable system are constructed from the theta-functions associated with Riemann surfaces from the Seiberg-Witten family, with moduli treated as dynamical variables and the Seiberg-Witten differential providing the pre-symplectic structure. We describe a number of theta-constant equations needed to prove this conjecture for the $N$-particle system. These equations provide an alternative method to derive the Seiberg-Witten prepotential and we illustrate this by calculating the perturbative contribution. We provide evidence that the solutions to the commutativity equations are exhausted by the double-elliptic system and its degenerations (Calogero and Ruijsenaars systems). Further, the theta-function identities that lie behind the Poisson commutativity of the three-particle Hamiltonians are proven.

. Note that at the level of AGT with one of the deformation parameters set zero (so corresponding to a quantum integrable system [32,33,34,35,36]), the p − q duality has quite an unexpected avatar: it is the spectral duality that describes a duality between the SL(N ) spin chain and the Gaudin model with N + 2 marked points (these two systems describe the two sides of the AGT correspondence) [37,38]. The spectral duality is lifted to 5d where it describes a duality of the SL(N ) spin chain on M + 2 sites and the SL(M ) spin chain on N + 2 sites [39,40]. Moreover, one can consider the full two-parametric AGT at this 5d level [41,42] where the spectral duality has a clear brane-picture interpretation.
However, an exact correspondence between p − q duality and spectral duality has yet to be established. Moreover, in contrast to the above cases, the probable generalizations of spectral duality to 6d systems are not known even at the classical level, while the p − q duality corresponding to 6d systems is known, as explained above, and is described by the self-duality of the double-elliptic systems.
In this paper we continue the recent study [10] of the double elliptic systems at the classical level. Our goal is to derive the theta-constant equations for the period matrix of the Seiberg-Witten curve of the double-elliptic integrable system. The main tool is the involutivity condition for the N -particle Hamiltonians constructed in [4,6,17] with respect to the Poisson bracket introduced in [10]. Initially the Hamiltonians for the integrable systems under consideration were defined in [4,6] in the form of the ratios of theta-functions on Abelian varieties associated with the Jacobians of the corresponding Seiberg-Witten curves (see below). The hypothesis of [6] was that the Hamiltonians are Poisson commuting with respect to the Seiberg-Witten symplectic structure. Following this idea a new approach, which deals with arbitrary Riemann surfaces instead of the Seiberg-Witten curves was proposed in [10]. Within this approach the concept of the Poisson bracket in terms of the coordinates on the Jacobian and the elements of the period matrix was introduced. An advantage of this approach is that it could lead to integrable systems not necessarily associated with the Seiberg-Witten curves. Indeed, it was demonstrated in [10] that the Poisson commutativity of the three-particle Hamiltonians is related just to some new theta-function identities of genus two, not making uses of the Seiberg-Witten curve. In this paper, we extend the approach to the case of N -particle systems with N > 3. When N = 4 we describe evidence that the four-particle Hamiltonians are still in involution for an arbitrary Riemann surface of genus g = 3. However, in the general case when N > 4 the involutivity condition places restrictions on the period matrix and we find these satisfied for the special period matrices corresponding to the Seiberg-Witten curve of the double-elliptic integrable system, in accordance with the original expectation in [6]. Thus, the involutivity condition can be used to determine the dependence of the period matrix on the Seiberg-Witten flat moduli, providing an independent method for calculating the Seiberg-Witten prepotentials (including their instanton corrections).
In Section 2 we recall the definition of the N -particle Hamiltonians [4,6,17] and introduce the involutivity condition with respect to the Poisson bracket from [10]. In Section 3 the involutivity condition is reformulated as a linear problem in terms of the vectors in a linear space of weight 3 theta functions. In Section 4 we prove the theta-function identities that lie behind the Poisson commutativity of the three-particle Hamiltonians. In Section 5 the method for calculating the Seiberg-Witten prepotentials of the double-elliptic integrable systems is presented. To illustrate the method we derive the Seiberg-Witten perturbative prepotential in the case of N = 5.

The Setting
Motivated by considerations of duality and the known rational, trigonometric and elliptic dependencies of the Calogero-Moser system the work of [4] sought to construct integrable systems with compact momentum dependence, extending the rational and hyperbolic dependence of momentum of the Calogero-Moser and Ruijsenaars-Schneider systems respectively. A 2-particle system that reproduced the Ruijsenaars-Schneider (and so also the Calogero-Moser) system as a limit was constructed in which the reduced momentum had elliptic dependence. That momentum dependence led to this class of models being called 'double-elliptic', though actually the momenta posited in the paper more generally took values in some abelian variety A (possibly a Jacobian) and the Hamiltonians were argued to be ratios of theta functions, θ a (z | Ω)/θ b (z | Ω) where z is the momenta and Ω the period matrix of A . The relevant theta functions were introduced by analogy to those of the N -particle elliptic Calogero-Moser system. There the genus N spectral curve C is an N -fold covering of an elliptic curve E and the Jacobian Jac(C) is isogenous to an Abelian variety of the form Jac(E) ⊕ A. If T is the period matrix of C the general theory of coverings enables us to write 1 where (z, z) T = Mẑ T for an appropriate M separating out the centre of mass mode z and M T M T = N τ 0 0 Ω .
Such decompositions are far from unique, for example the identity (here e(x) := e 2iπx ) lead to alternative expressions and we write these simply as specifying θ a := θ a (z| Ω) as required. Although [4] suggested the form of the Hamiltonians of the dual system, the exact nature of the θ a was left unspecified (for N > 2) and neither their Poisson commutativity nor their geometric setting was addressed at that time. The Seiberg-Witten picture tells us that Ω depends on the conjugate coordinates, but the dependence is left unspecified. Before turning to the issue of Poisson commutativity we recall that Braden and Hollowood have given a geometric setting for such Hamiltonians [17]. There the spectral curve is of genus N + 1 and lies in a (1, N )polarized abelian surface and is given by an equation of the form Here Θ is the theta function of the abelian surface and θ a are of the form (2.1). Degenerations of this give precisely the Ruijsenaars and elliptic Calogero-Moser systems.
Regarding the Poisson commutativity of the ratios of theta functions (Hamiltonians) associated with the dual models of [4], significant evidence for this was amassed in [6] using a perturbative (in instanton number) expansion. Here it was observed that resulting equations for commutativity gave relations between the various terms of the instanton expansion of the Seiberg-Witten prepotential associated with C. (A proof of the Poisson commutativity in the case of the elliptic Calogero-Moser system was later provided in [43] .) Following [44] we have that (for all a, b, c, d) where the actionâ i is conjugate to the variableẑ i and in obtaining the final equality we express the simplectic form as Here the Riemann θ-function with characteristics a, b ∈ Q and g × g period matrix T is noting that θ a is independent of the centre of mass. Geometrically the Seiberg-Witten curve is the fibre over the moduli space of actions: this means the dependence of the theta functions on the actionâ i is via the period matrix T of C and hence Ω. Upon using the heat equation, that (τ, a) T = M −1 TâT , and the prepotential F we have that thus the Poisson-commutativity reduces to showing that where P rst is totally symmetric (for this example P rst = M ri M sj M tk ∂ 3 F /∂â i ∂â j ∂â k ) and (2.8) As noted above, this Poisson commutativity has been established for systems arising as degenerations of the elliptic Calogero-Moser system. In the works just described we have utilised properties of the underlying spectral curve C to construct examples of double-elliptic systems. The recent paper [10] goes beyond this. Let us henceforth assume Hamiltonians of the form (a = 1, . . . , N − 1) They discovered that P rst were totally symmetric and their solutions were expressed in terms of theta-function identities. We reformulate and extend this as follows. Suppose one has a family of abelian varieties for which there is a symplectic structure on the total space with respect to which the abelian varieties are Lagrangian: we have in the above coordinates where the period matrix Ω = Ω(a) are some special functions of the Seiberg-Witten flat moduli a. We know from the work of Donagi and Markman [45] that (assuming holomorphicity) the differential of the period map at each point is the contraction of a cubic, and so P rst is totally symmetric. (This was a result of [10] that in the light of [45] we assume from the outset.) What can be said about the solutions to (2.7) for arbitrary g = N − 1? We will show that for g = 2 (2.7) holds for an arbitrary symmetric g × g period matrix Ω and are actually theta-function identities. In the case when g 4 the relations (2.7) define some special constraints on the elements of the period matrix Ω. We will show that these constraints describe the dependence of Ω on the flat moduli of the corresponding Seiberg-Witten curves. Thus, in the general case, the Poisson commutativity of the Hamiltonians (2.9) holds only for some special class of period matrices, as suggested in [4,6].
, where i k = 0, . . . , N − 1 and the elements from {i k } j satisfy j + N−1 k=1 i k ∈ N · Z may be expressed as

Poisson commutativity as a linear problem
The relations (2.7) are strongly connected with weight 3 theta functions and their corresponding linear spaces [46]. Recall that an entire function f (z | Ω) = f (z) on C g with fixed symmetric period matrix Ω is called a theta function of weight λ ∈ N and characteristic δ ǫ , if for all p, q ∈ Z g . Such functions form a linear space Θ Ω λ δ ǫ of dimension λ g with standard bases [47]: It is convenient to define the general and so (3.7) We emphasise that although none of the terms H abc rst individually possess this property the symmetrised sum is a third order theta function. This result has several important consequences. First, let {f α (z)} be any basis ; then we have an expansion and (2.7) becomes for each α 0 = g r,s,t=1 This important relation entails several things. First it expresses that the g (g + 1) (g + 2) /6 vectors C α {rst} (each one has 3 g coordinates) are linearly dependent. Second, that we have a linear problem to determine the P rst 's; and third, that the P rst 's will be expressible in terms of the constants C α {rst} if there is a nontrivial solution. Because the C α {rst} 's are given in terms of theta-constants, these are the theta-constant identities referred to earlier and generalise those obtained in [10]. In the next section we shall illustrate this general setting for the Hamiltonians (2.9). For ease of description in what follows we will describe C α {rst} as a 3 g × g (g + 1) (g + 2) /6 matrix.

An Example
We shall now apply the above considerations to the Hamiltonians (2.9) and prove a conjecture raised in [10]. The relevant characteristics are and we may obtain the expansion (3.8) as follows. The determinants H ab rst have the following Fourier decomposition: To obtain the decomposition in the basis (3.2) we change the summation variables in (4.1) as follows Then, using the relation (4.4), we obtain or in terms of theta constants (with θ ′  In terms of the basis elements we have Hence the only nonzero coefficients in the decomposition (4.6) are those C α {rst} for which e k l α l 3 = 1 (4.11) so leaving 3 g−1 possible nonzero coordinates in each column of C α {rst} . Moreover, the symmetry reduces the independent terms further to only (3 g−1 + 1)/2 different nonzero rows in the matrix C α {rst} . Now in the present setting we have g (g + 1) (g + 2) /6 > (3 g−1 + 1)/2 and so more variables P {rst} than equations, from which we deduce that for arbitrary Ω equation (3.9) holds for nontrivial P rst and the corresponding C α {rst} are linearly dependent. This establishes the theta-constant relations conjectured in [10].
We remark in passing the above argument actually shows that for each pair of nonzero vectors a, b ∈ Z g /3Z g with property a + b ≡ 0 mod 3 there exists a nontrivial set of quantities P rst for which the corresponding Hamiltonians commute for g ≤ 4. This is because z → z+a/3 is still a symmetry of H ab rst , now leading to nonzero C α {rst} when e ( α · a/3) = 1. Together with the restriction coming from parity, we have (g + 1) (g + 2) /6 > (3 g−1 + 1)/2 for g ≤ 4 and again there are corresponding theta-constant identities. When e ( α · a/3) = 1 we find the identities C α {rst} = 0.

Theta-constant representation for the Seiberg-Witten curves
In this section we shall use the constraint (2.7) to make various deductions about the prepotential. This work will focus on the perturbative prepotential to establish the method, leaving the instanton corrections to a later work.
It is helpful to isolate the assumptions being made. First we are assuming that there exists an underlying Seiberg-Witten curve C with period matrix T given by a prepotential F . Second, we shall assume that the curve C is of genus N and covers an elliptic curve, and so is related to the elliptic Calogero-Moser family. With a choice of M given by In passing we note that at this stage one could have chosen to use a different M (for example [44]) giving equivalent expansions, or chosen to have the Braden-Hollowood genus N + 1 curve. Together these assumptions provide us with an instanton expansion (for i = j) Here F (0) is the perturbative prepotential and the instanton corrections F (k) to the Seiberg-Witten prepotential F are only known to low order. We may obtain new information about this expansion as follows. With P rst = M ri M sj M tk ∂ 3 F /∂â i ∂â j ∂â k the constraints (3.9) with (4.8) and the instanton expansions of these constrain the P rst 's. Let us write the perturbative expansions (corresponding to the trigonometric limit Im τ → +∞) as The constraints (3.9) become relations between the coefficients of these expansions. Before illustrating this we note that although the dependence on the coordinates is not a priori known, we know that in the case of the GL (N ) systems under consideration (for example, from the Toda limits of [6]) the quantities F (0) ij from (5.3) are just functions of (â i −â j ), the difference of the Seiberg-Witten flat moduli. In general, the variables may be represented by the following series: In the previous section we considered the case when N = 3. When N = 4 we have 10 vectors C α {rst} in a 27 dimensional space. Consider the determinants H ab rst with a = 1 and b = 3. Then due to the symmetry we have only 14 different rows in the matrix consisting of vectors C α {rst} . The resulting 10 × 14 matrix has rank 9 due to some theta-constant identities, thus giving the relations (2.7).
We therefore consider the case N = 5. Here we have 20 vectors C α {rst} in an 81 dimensional space. Again, if we choose the determinants H ab rst with a = 1 and b = N − 1 = 4, we obtain 41 different rows in the matrix consisting of vectors C α {rst} . Then in order to satisfy the relations (2.7), the rank of this matrix must be at most equal to 19. The latter gives at least 22 equations on the elements of the period matrix Ω. The number 22 alone does not give useful information, as some of these equations may be equivalent. Now taking the constraints (3.9) in the case of N = 5 and the perturbative expansions above, we obtain the linear system for the first nonzero order of the elements p ijk : (the remaining p's appear at higher orders) and Thus, the first constraint is of the form det L = 0. Moreover, the constraint fixes the lowest term in the series (5.6) to be 1 or (â i −â j ) 2 up to some constant factor. These recurrences are satisfied by the three sets of functions q ij : • corresponding to the GL(N ) elliptic Calogero system • the GL(N ) elliptic Ruijsenaars model (5.14) • the GL(N ) double elliptic system (whereτ is the modulus of the second torus) (5.15) Let us elaborate on the parameter count for the last of these (the former two being obtained as scaling limits of this). As we remarked above, there is an overall scaling of the functions q ij left undetermined by the constraint (5.10). Further the scale √ µ of the moduliâ i , the mass m and the periodτ are also parameters. Setting q III ij = m 2 /µ 1 − m 2 /sn √ µ (â i −â j ) τ 2 −1 scales c 0 = −1 and the coefficients c 1,2,3 encode these three parameters; the recursions for c l (l 4) then express the remaining coefficients implicitly in terms of µ, m, τ .
The results just obtained suggest the following hypothesis. The constraints (3.9) are actually theta-constant equations of the period matrix T for the Seiberg-Witten curve associated with the N -particle double-elliptic integrable system for N ≥ 5. Thus, the equations can be used to determine the dependence of the period matrix on the Seiberg-Witten flat moduli, providing an independent method for calculating the Seiberg-Witten prepotentials (including the instanton corrections) of the GL (N ) elliptic Calogero, Ruijsenaars and doubleelliptic systems.