Duality in elliptic Ruijsenaars system and elliptic symmetric functions

We demonstrate that the symmetric elliptic polynomials $E_\lambda(x)$ originally discovered in the study of generalized Noumi-Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians that act on the mother function variable $y_i$ (substitute of the Young-diagram variable $\lambda$). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $P_R(x)$ are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov (KS) Hamiltonians. This means that these latter are related to the dual eRS Hamiltonians by a somewhat mysterious orthogonality transformation, which is well defined only on the full space of time variables, while the coordinates $x_i$ appear only after the Miwa transform. This observation explains the difficulties with getting the apparently self-dual Hamiltonians from the double elliptic version of the KS Hamiltonians.


Introduction
The Calogero system [1] and its various trigonometric [2,3] and elliptic [4,3] generalizations is one of the basic examples in the theory of integrable systems, which, after a discovery of integrable properties of QFT effective actions [5,6], are at one of the focuses of modern theoretical studies. The whole Calogero-Sutherland-Ruijsenaars-Schneider family is rich enough for study the P Q dualities [7,8,9,10] (see also the spectral duality version in [11]), and one may hope this would be the first case where the self-dual double-elliptic (dell) system will be clearly formulated and investigated [12,13,14,15]. At the moment, we are still stack at the previous stage: an explicit formulation of elliptic-trigonometric duality. Indeed, though the P Q duality has been explicitly formulated both at the classical [7,8,9,10] and quantum [16] levels for the rational and trigonometric systems, and, in principle, it is basically clear how it should be lifted to elliptic systems, any explicit formulas has been lacking so far in the multi-particle case. This paper is a big step towards a resolution of this problem.
We will discuss solely the quantum duality, where the task is to find a dual pair of functions (called mother functions) of two sets of variables, x and y, and the two sets of Hamiltonians for which they are the eigenfunctions. Within the Calogero-Sutherland-Ruijsenaars-Schneider family, one of the sets of variables is usually considered as the Miwa transform of the "time variables" {p k }, while the other set corresponds to an analytical continuation from lengths of the Young diagram λ. The mother function functions themselves are then related to various symmetric polynomials P λ {p} from the Schur-Macdonald family. The Hamiltonians are well known up to the eRS case, though their dual versions are not known. However, using the P Q duality, we are still able to find the eigenfunctions E λ {p k } of these latter. In this paper, we demonstrate that they are actually expressed through the elliptic Macdonald functions constructed recently in [17]. A candidate for the dual Hamiltonian was proposed by P. Koroteev and Sh. Shakirov [18] 4 (in fact, they proposed a dell version of the Hamiltonians, which is, however, not self-dual) but, as we explain in this paper, their eigenfunctions are not exactly the same elliptic Macdonald The elliptic Pochhammer symbol is defined as Θ(z; q, ω) n := Γ(q n z; q, ω) Γ(z; q, ω) We also use the notation In particular, ψ(x/q) = ψ(x −1 ) and ψ(tz) = ζ(z) q↔t .
Throughout the paper, we denote symmetric functions of variables x i as M λ (x i ), P λ (x i ), E λ (x i ), while these polynomials as functions of power sums p k : The elementary symmetric polynomials [23] are denoted through e k , and monomial symmetric polynomials [23], through m λ .

Duality
The notion of quantum duality. The idea of P Q-duality of integrable many-body systems was first proposed by S. Ruijsenaars [7] and was later discussed in [8,9,10] at the classical level and in [16] at the quantum level. While the classical P Q-duality is realized just in terms of Hamiltonians and their canonical transformations [10], the quantum duality requires the eigenvalue problem, i.e. the Hamiltonians are accompanied by the eigenfunctions from the very beginning. That is, if the eigenvalue problem for the HamiltonianĤ x , which is an operator acting on the variable x, looks likê then the dual Hamiltonian acts on the variable λ: so that Ψ (D) (λ; x) = Ψ(x; λ). Here E and E (D) are some fixed functions of the variables x and λ accordingly. In the case of many-body integrable system there are several coordinates x i , i = 1, ..., N and the corresponding λ i are associated with the separated variables. Integrability implies that, in this case, there are N commuting Hamiltonians and N dual Hamiltonians. In this context, one naturally considers the eigenfunction Ψ λ (x) as a function of the two continuous variables x and λ. In the case of Hamiltonians from the Calogero-Sutherland-Ruijsenaars-Schneider family, the most informative are the Hamiltonians of the Dell system, which are elliptic both in the coordinates and in momenta, and are self-dual, i.e.Ĥ k =Ĥ Duality in the trigonometric Ruijsenaars system. The simplest example is provided by the trigonometric Ruijsenaars system, which is self-dual. Its Hamiltonians are explicitly given by The eigenfunctions of these Ruijsenaars Hamiltonians are the Macdonald polynomials: where the eigenvalues are given by Since this system is self-dual, i.e. the eigenfunction coincides with its dual, one just needs the property Ψ(λ; x) = Ψ(x; λ). This is guaranteed by the duality relation of the Macdonald polynomials: In fact, the normalization coefficient M λ (t −i ) can be chosen in a different form, which we will need in further elliptic generalization.

Elliptic Macdonald polynomials
In this section, we define the two conjugate systems of symmetric polynomials that are the main players in the paper. We call them elliptic Macdonald polynomials. They come as a particular case of the generalized Noumi-Shiraishi (GNS) polynomials [17] defined as a certain truncation of an explicit series expression. These polynomials form a basis in the space of symmetric functions. An essential property of both these systems of polynomials is that their coefficients are expressed not though separate θ-functions, but through their peculiar combinations ζ(z) and ψ(z) differing by the permutation of q and t.

P λ {p} polynomials
The basic systems of polynomials, P (q,t,ω) λ (x i ) was defined in [24,25,17] (in the latter reference, this system is obtained as a particular case of the GNS polynomials when ξ(z) = θ ω (z)), and can be described in the following way generalizing the Noumi-Shiraishi representation of the Macdonald polynomials [26]: where m ij = 0 for i ≥ j, m ij ∈ Z ≥0 , the number of lines in the Young diagram λ does not exceed N , and As it was explained in [25, sec.3], P (q,t,ω) λ (x i ) are symmetric polynomials, which is a consequence of a series of non-trivial θ-function relations. They form a ring with the properties described in [17]. The first few of these polynomials are P (q,t,ω) [1] More examples can be found in [17].

E λ {p} polynomials
Orthogonality and E polynomials. Let us define a conjugate system of polynomials in the following way. Denote χ λ∆ the coefficients of the p-expansion of the Then, the set of polynomials with χ −1 being the inverse matrix, p ∆ := i=1 ∆ i and z ∆ being the standard symmetric factor of the Young diagram (order of the automorphism) [27], is orthogonal, w.r.t. to the measure Note that, in the Macdonald limit ω → 0, This suggests to define a system of symmetric polynomials The first few of E (q,t,ω) λ (p n ) polynomials are given by Ring structure. The E (q,t,ω) λ {p k } polynomials form a commutative ring (isomorphic to the ring of symmetric polynomials): The coefficients N ν λµ (q, t, ω) have some nice properties: 1. They vanish whenever the corresponding Littlewood-Richardson coefficient vanishes.
2. They are elliptizations of the (q, t)-Littlewood-Richardson coefficients for Macdonald polynomials, and, when the latter are factorized, they also factorize into products of theta functions.
It is not hard to guess a formula for some classes of ring coefficients. The important example is the Pieri rule: where µ + 1 i denotes a diagram obtained from the diagram µ by adding one box in column i. If µ + 1 i is not a Young diagram, the coefficient in front of E (q,t,ω) µ {p k } vanishes automatically.

E λ {p} as solutions to dual eRS system
The eRS Hamiltonians are manifestly given by Their eigenfunctions Ψ λ (y i ) eRS were conjectured in [28,Eq.] (see also (66) below). In this section, we are going to construct the dual of these eigenfunctions following the pattern of sec.34. That is, we use the duality relation where the dual functions Ψ λ (x i ) are eigenfunctions of the dual eRS Hamiltonians (yet to be evaluated). We demonstrate below that by checking that, upon a proper choice of the normalization factor, it satisfies the eigenvalue equations with the Hamiltonians (33) w.r.t. to the variables y i = q µi t −i . Here p n = i x n i .
eRS eigenfunctions. The first eRS Hamiltonian is given by where N is the number of particles. The eRS Hamiltonian (36) is related to the Pieri's rule (32). One can see this by conjugating H (q,tω) 1 with the function where β = ln t ln q and Υ (q,t,ω) (y) = n,m≥0 Up to a constant factor, it is an immediate elliptization of (15). The function Υ satisfies a simple difference equation and thus From Eq. (40) we see that after conjugation half of the factors in i =j in H (q,t,ω) 1 cancel, while the rest are "doubled". Finally, using the property of the Jacobi theta function θ ω (y −1 ) = −y −1 θ ω (y) on the factors in the first square brackets in Eq. (40), we get the conjugated Hamiltonian, which is expressible through the function ψ(y) from Eq. (7): If we send N → ∞ and set the variables y i to discrete values for some Young diagram µ we notice that the action of the conjugated Hamiltonian (41) coincides with the Pieri rule (32). Therefore, with proper normalizing constant, elliptic Macdonald polynomials are eigenfunctions of the eRS model. More specifically, the function Ψ (q,t,ω) (p n |y i ), satisfies H (q,t,ω) 1 Ψ (q,t,ω) (p k |y i ) = p 1 Ψ (q,t,ω) (p k |y i ).
The variables y k are the mother-function arguments, which substitute the Young diagram index µ in E µ {p} and E µ (x).
Higher eRS Hamiltonians and more Pieri rules. The eigenfunction Ψ (q,t,ω) (p n |y i ) is in fact an eigenfunction of a whole infinite set of eRS Hamiltonians, of which H (q,t,ω) 1 is only the first member. Conjugating this Hamiltonian with the function F (q,t,ω) ( y) we get These Hamiltonians can also be understood as further Pieri rules for elliptic Macdonald polynomials E The KS Hamiltonians are elliptic both in momenta and coordinates. Here we consider the case when the coordinate torus is degenerate so that the dependence on coordinates is trigonometric, we call this as ell-trig case. If one considers the trig-ell case instead, the KS Hamiltonians become the eRS Hamiltonians. Since the KS Hamiltonians are not self-dual, one should not expect that degenerating the coordinate torus would lead to the dual eRS system. Indeed, how we explain in this section, the corresponding eigenfunctions are the P λ (x) polynomials, which are conjugate to the wave functions of the dual eRS system, E (q,t,ω) λ w.r.t. to the Schur scalar product. As it follows from this scalar product, the conjugation can be realized with the substitutions , which is not easy to realize on the subspace of finite number of the Miwa variables x i . P λ (x) as eigenfunctions of the ell-trig KS Hamiltonians. One of the possibilities to proceed with the ell-trig KS Hamiltonians is to notice directly that their wave functions constructed in [21,Eqs.(72)-(73)] are nothing but the P λ (x) polynomials. In order to see this, one has to use rather tricky relations between the odd and even θ-functions, the simplest of which is The formulas in [17] actually used brute force calculations. A smarter approach is to use the generating function of the KS Hamiltonians in the determinant form [20], which can be written, in the ell-trig case, as This matrix is triangular in the basis of monomial symmetric polynomials m λ ( x). The current O trig (u) can be diagonalized with eigenvalues However, since O trig (u) for different values of u do not commute, the eigenfunctions in general depend on u. The exceptions here are the eigenfunctions correspond to λ = [ [1,1] m [2] (51) To get commuting Hamiltonians one should take the ratio of the generating functions O trig (u) at two different values of u: Let us compute the first nontrivial eigenfunction of H(v, u), which should not depend on u and v. The matrix of the operator H(v, u) in the basis of m λ on the second level reads We need left eigenfunctions, at the second level they are The eigenfunctions indeed are independent of u and v. This depends on the following identity: We notice that the eigenfunctions (54), (54) are precisely the P λ {p k } polynomials. Likewise, at the next level the left eigenfunctions are and which are exactly the P λ {p k } polynomials. We conjecture that this statement is true at all levels so that Note that counterparts of (55) behind (57) are more involved, the simplest one being The reason is that an elliptic function with two given poles is fully defined by one of its zeroes and the overall scale. This is also behind the identities like (48). There are plenty of other relations associated with multiple poles.
6 ELS-functions [25], dualities and conjugation In [25], there was introduced and discussed the ELS-function defined in the following way: where and p is another elliptic parameter. This function is a lift of the P (q,t,ω) λ ( x) polynomial with the Shiraishi functor [17] and is conjectured to play an essential role in description of the double elliptic systems. Here we note that various trig-ell and ell-trig limits of the ELS-function are related by dualities. Indeed, the P (q,t,ω) λ ( x) polynomial is obtained from P N (p N −i x i ; p|s N −i y i ; s|q, t, ω) in the p → 0 limit. Indeed, consider the limit Then, At the same time, this is related to the ω → 0 limit of the ELS-function, [28] E N (x i ; p|y i ; s|q, t) := lim ω→0 P N (x i ; p|y i ; s|q, t, ω) via the formula [24] with the normalization constant Moreover, this mother function E n (x i ; p|y i ; s|q, t) allows one to construct also the eigenfunctions of the eRS Hamiltonians: as was conjectured in [28], the function with α(p|y i ; s|q, t) being the constant term in the Shiraishi function E n (x i ; p|y i ; s|q, t), which does not depend on x i (this normalization constant is necessary, since otherwise the limit of s → 1 is singular), is an eigenfunction of the eRS Hamiltonian: More discussion of this equation can be found in [29]. These formulas imply the conjecture that the eigenfunctions of the full KS Hamiltonians can be obtained from the limit s → 1 of the ELS-functions P N (x i ; p|y i ; s|q, t, ω): with some normalization constant α f (p|y i ; s|q, q t , ω) that makes the expression non-singular in the s → 1 limit. This should be the case, since the ELS-function describes the Nekrasov function in the full Ω-background, while s = e −2π 2 → 1 [25, Eq.(95)] describes its Nekrasov-Shatashvili limit, and this the Nekrasov-Shatashvili limit that describes the quantum integrable system. As usual, this limit is singular and requires some accurate normalization. We have checked this conjecture in the first terms of expansion in the elliptic parameters p and ω.
Hence, we finally come to the diagram The checked line was the point of our interest in this paper. Its main drawback is the mysterious orthogonality relation between the first two eigenfunctions and the lack of explicit relation between the last two apart from (34). The top of the table (the ELS-functions) is conjectured to provide solutions to the full KS Hamiltonian eigenproblem via (68), which implies the next, most interesting step to be done: since (68) looks providing eigenfunctions of the non-self-dual KS Hamiltonians, what are appropriate self-dual functions?

Elliptic DIM algebra and ell-trig KS Hamiltonians
New view on the vertical Fock representation of the elliptic DIM algebra. Elliptic DIM algebra (eDIM) [30] is generated by currents x ± (z) and ψ ± (z) satisfying commutation relations with an elliptic structure function. To keep the presentation brief, we do not give here these relations, which can be found e.g. in [31]. It was also found in [31] that the eDIM algebra can be rewritten as a direct sum of the trigonometric DIM algebra and an additional Heisenberg subalgebra. We adopt this view in this exposition.
The eDIM algebra is bi-graded with generators lying in a Z 2 lattice, as shown in Fig. 1. Every node contains a single generator 6 e (n,m) , except for the nodes on the vertical line which contain two e ± (0,m) , which is marked by additional small circles. The extra generators e + (0,m) on the vertical axis form a decoupled Heisenberg subalgebra commuting with the rest of the DIM algebra.
There are two central charges C 1,2 in the algebra, C 2 associated with the "elliptic" direction (vertical in Fig. 1), and C 1 associated with the "trigonometric" one (horizontal in Fig. 1). The central charges control the commutation relations in the Heisenberg subalgebras of the eDIM algebra which correspond to lines of rational slopes in the Z 2 lattice. The Heisenberg subalgebra corresponding to the slope a b is spanned by the generators e (na,nb) (with an additional ± index in case of the vertical subalgebra) [e (na,nb) , e (ma,mb) ] ∼ (C −nb Therefore, if, for a particular representation, the ratio ln C1 ln C2 is rational, then there exists a commuting Heisenberg subalgebra inside the eDIM algebra of slope ln C1 ln C2 . e (3,2) e (−3,−2) slope 3 2 x + n = e (1,n) x − n = e (−1,n) H ± n = e ± (0,n) Figure 1: The grading lattice of eDIM algebra. The generator at vertex (n, m) is denoted by e (n,m) . The generating currents are framed in blue, x ± (z) = n∈Z e (±1,n) z −n and ψ ± (z) = exp n =0 H ± n z −n = exp n =0 e ± (0,n) z −n . Notice that the vertices on the vertical line unlike the others contain two generators e ± (0,n) . The red line denotes an example of a Heisenberg subalgebra of slope 3 2 .
We are particularly interested in the vertical Fock representation F (0,1) q,t −1 (u) of the eDIM algebra with C 1 = 1, C 2 = t/q. In this representation, the both vertical subalgebras spanned by e ± (0,n) are commutative. The e + (0,n) generators turn out to be irrelevant in this representation: they commute both between themselves and with the rest of the algebra. We set them to zero. The rest of the generators can be expressed through the horizontal Heisenberg subalgebra e (n,0) , which acts freely on the Fock space. The states of the representation are labelled by Young diagrams, e.g. |µ, u .
In the vertical representation, the generating currents act as follows [32,33]