QKZ-Ruijsenaars correspondence revisited

We discuss the Matsuo-Cherednik type correspondence between the quantum Knizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle quantum Ruijsenaars model, with $n$ being not necessarily equal to $N$. The quasiclassical limit of this construction yields the quantum-classical correspondence between the quantum spin chains and the classical Ruijsenaars models.


Introduction
The quantum Knizhnik-Zamolodchikov (qKZ) equations [8] is a system of holonomic difference equations e ηh∂x i Φ = K i in the right hand side is constructed as a chain product of quantum R-matrices: Here R ij (x) is the R-matrix acting in the i-th and j-th tensor factors (it has to satisfy the unitarity condition R ij (x)R ji (−x) = id), g = diag(g 1 , . . . , g N ) is a diagonal N ×N matrix and g (i) is the operator in V acting as g on the i-th factor (and identically on all other factors). For example, the rational R-matrix is of the form where x is the spectral parameter, I is the identity operator and P ij is the permutation of the i-th and j-th tensor factors. Compatibility of the qKZ equations follows from the Yang-Baxter equation for the R-matrix and from the commutativity [g ⊗ g, R(x)] = 0.
The remarkable correspondence of the qKZ equations with the Macdonald type difference operatorĤ was discussed in [5,11,14,17] in the case N = n. In this case solutions to (1)  Ce a and S n is the symmetric group.
If such Φ is a solution to the qKZ equations, then the function Ψ = σ∈Sn Φ σ solves the spectral problem for the operatorĤ: This is referred to as the Matsuo-Cherednik type correspondence [13,4]. A similar correspondence holds true for qKZ equations with trigonometric R-matrices.
The operatorĤ is essentially the Hamiltonian of the quantum Ruijsenaars model [15] which is a relativistic version of the Calogero model (more precisely, the operatorĤ and the Ruijsenaars Hamiltonian are connected by a similarity transformation, see [10]). The parameter η plays the role of the inverse velocity of light.
We will give a simple proof of the correspondence valid also in the case when n is not necessarily equal to N = dim V . In this form it looks like a quantum deformation of the quantum-classical correspondence [1,2,9,21,3] between the quantum XXX or XXZ spin chains and the classical Ruijsenaars models.
The spectral problem for the spin chain appears as a "quasiclassical" limit of qKZ as h → 0. Indeed, ash → 0 the qKZ solutions have the asymptotic form [18,19] where S is some scalar function. Upon substitution to the qKZ equations (1), this leads, in the leading order, to the joint eigenvalue problems for the commuting operators K i . They are Hamiltonians of the inhomogeneous quantum spin chain. They can be diagonalized using the algebraic Bethe ansatz. In the quantum-classical correspondence, the p i 's are identified with momenta of the Ruijsenaars particles. The parameterh becomes the true Planck constant after quantization of the corresponding Ruijsenaars model. The paper is organized as follows. In section 2, we review the inhomogeneous XXX spin chain and the associated qKZ equations. In section 3 the qKZ-Ruijsenaars correspondence is established by a simple direct calculation. In section 4 we extend the result to higher Ruijsenaars Hamiltonians. Section 5 is devoted to the qKZ equations with trigonometric R-matrices. Finally, in section 6 we discuss the interpretation of the results as a "quantum" deformation of the quantum-classical correspondence. The link to spin chains and their solutions by means of the algebraic Bethe ansatz appears naturally because of the usage of R-matrices in fundamental representation. It is alternative to the approach based on the affine Hecke algebras [4,5,6], where similar results were originally presented. Explicit relationship between these two approaches deserves further consideration.

The XXX spin chain and qKZ equations
Let e ab be the standard basis in the space of N×N matrices: the matrix e ab has only one non-zero element (equal to 1) at the place ab: (e ab ) a ′ b ′ = δ aa ′ δ bb ′ . Note that I = a e aa is the unity operator and P = ab e ab ⊗ e ba is the permutation operator in the space C N ⊗ C N . We embed e ab into End (V ⊗n ) in the usual way: e (i) a ′ b ′ commute for any i = j because they act non-trivially in different spaces. Similarly, for any matrix g ∈ End(C N ) we define g (i) acting in the tensor product V ⊗n : g (i) = I ⊗(i−1) ⊗ g ⊗ I ⊗(n−i) ∈ End(V). In this notation, the permutation operator of the i-th and j-th tensor factors in Let g ∈ GL(N) be a diagonal matrix g = diag (g 1 , g 2 , . . . , g N ) = N a=1 g a e aa . We call it the twist matrix with twist parameters g a . It is used for the construction of an integrable spin chain with twisted boundary conditions. Together with the R-matrix (3), we introduce another rational R-matrix which differs from the R(x) by a scalar factor: It obeys the same Yang-Baxter equation and commutes with g ⊗ g. The transfer matrix of the inhomogeneous spin chain (or, equivalently, of the associated statistical vertex model on the 2D lattice) is defined in the standard way as a trace of the chain product of R-matrices in the auxiliary space V = C N with index 0: where x 1 , x 2 , . . . , x n are inhomogeneity parameters. (We assume that they are in general position meaning that x i = x j and x i = x j ±η for all i = j.) As is known, the Yang-Baxter equation for the R-matrix implies that the transfer matrices with fixed inhomogeneity and twist parameters commute: The dynamical variables of the model (we call them "spins" in analogy with the rank 1 case) are vectors in the vector representation of GL(N) realized in the spaces V = C N attached to each site. Non-local commuting Hamiltonians H j are defined as residues of T(x) at x = x j : From (8) it follows that their explicit form is We obviously have where Let us introduce the operators They commute among themselves and with the Hamiltonians: We call them weight operators. Clearly, a M a = nI. Comparing the expansion of (8) as x → ∞, with that of (9), we conclude that so the system has n + N − 2 independent integrals of motion. The joint spectral problem is The common eigenstates of the Hamiltonians can be classified according to eigenvalues of the operators M a . Let be the weight decomposition of the Hilbert space of the spin chain, V, into the direct sum of eigenspaces for the operators M a with the eigenvalues M a ∈ Z ≥0 , a = 1, . . . , N (recall that M 1 + . . . + M N = n). The common eigenstates of the operators M a and H i belong to the spaces V({M a }).

The basis vectors in
where the number of indices j k such that j k = a is equal to M a for all a = 1, . . . , N. We also introduce dual vectors Associated with the inhomogeneous XXX spin chain is the system of qKZ equations i given by (2). The compatibility condition follows from the Yang-Baxter equation for the R-matrix. The operators K (h) i respect the weight decomposition (14), hence the solutions to the qKZ system belong to the weight subspaces V({M a }).

The qKZ-Ruijsenaars correspondence in the rational case
Let Φ = J Φ J J be any solution of the qKZ equations belonging to the weight sub- . We claim that the function is an eigenfunction of the Macdonald operatorĤ with the eigenvalue E = N a=1 M a g a : For the proof we consider the covector Ω equal to the sum of all basis (dual) vectors: then Ψ = Ω Φ . It is important to note that Ω P ij = Ω and, therefore, Ω R ij (x) = Ω . It then follows that Ω K i , so the projection of the i-th qKZ equation onto the covector Ω reads Therefore, multiplying by n j =i and summing over i, we get: where we have used (11) and (13).
In the next section we show that Ψ is the common eigenfunction for all higher Ruijsenaars HamiltoniansĤ d with the eigenvalues e d (g 1 , . . . , g 1

Higher Hamiltonians
Here we show that Ψ (15) is an eigenfunction of the higher rational Macdonald-Ruijsenaars HamiltoniansĤ d defined bŷ For example, for d = 2 For the proof we introduce the notation R ab = R ab (x a − x b ) and Consider first the case d = 2. For i < j we have The "underbraced" expressions in (21) consist of commuting products of R-matrices thanks to i < j. Therefore, these expressions can be permuted. Using then the property Ω R kl (x) = Ω we conclude that all shifts by ηh in the R-matrix arguments can be removed. Finally, we can permute the products of R-matrices coming back to the initial order (but with non-shifted arguments).
For d = 3 and i < j < k we have Again, since i < j < k we see that "underbraced" expressions commute as well as the "overbraced" expressions. Permute first the "underbraced" expressions. Then we get Now the "overbraced" expression from the second line of (23) commutes with the whole third line. By permuting them we get the product where all R-matrices with shifted arguments (R + ) are to the left from twist matrices and act on Ω (to the left). Then we can apply the previously used reasoning and remove all the shifts of arguments.
It is easy to see that the same proof holds true for arbitrary d. The choice of ordering i 1 < ... < i d is convenient but the final answer is independent of it since [K Multiplying both parts of (20) by the products transforming R(x) to R(x) (7), (11), we get d k=1 n r =i k A comparison between (18) and (24) shows that We are now in a position to use the following operator relation (see [20, eq.
The both sides are polynomials in z. Equating the coefficients in front of the powers of z, we have where e d ({P j }) are elementary symmetric functions defined by the generating function as and P k = a M a g k a . In particular, for k = 1, 2, 3 The vector Φ is an eigenvector for these operators with the eigenvalues given by the same formulas with M a → M a . Therefore, plugging (27) into (25) we It is easy to see that the right hand side is the elementary symmetric polynomial of n variables e d (g 1 , . . . , g 1 (30) (In this section we use the same notation for analogous objects in the trigonometric and rational cases.) After some algebra it can be represented in the form where P q 12 = N a=1 e aa ⊗ e aa + q n a>b e ab ⊗ e ba + q −1 n a<b e ab ⊗ e ba , q = e η , is the q-permutation operator acting as follows: Note that P q ij = P 1/q j i . The unitarity condition can be easily checked. We also introduce the R-matrix and the transfer matrix T(x) = tr 0 R 0n (x−x n ) . . . R 01 (x−x 1 ) g (0) of the inhomogeneous XXZ spin chain with twisted boundary conditions. Similarly to the rational case, the commuting Hamiltonians are defined by the pole expansion They are expressed through the R-matrices by the same formula (10). We have (see [2]): g a e ±ηMa , where C is some operator and the weight operators M a are defined in the same way as before. Hence Similarly to (11), we have: As in the rational case, the trigonometric operators K (h) i respect the weight decomposition (14). Let Φ = J Φ J J be any solution of the trigonometric qKZ equations belonging to the weight subspace V({M a }). Let ℓ(J) be the minimal number of elementary permutations σ i i+1 ∈ S n which are required to get the multi-index J = (j 1 , j 2 , . . . , j n ) from the "minimal" one 1 , where the j k 's are ordered as 1 ≤ j 1 ≤ j 2 ≤ . . . ≤ j n ≤ N. In the case n = N, M 1 = M 2 = . . . = M N = 1 the ℓ(J) is what is called length of the permutation (12 . . . N) → (j 1 j 2 . . . j N ). We claim that the function is an eigenfunction of the trigonometric Macdonald operatorĤ with the eigenvalue The idea of the proof is the same as in the rational case. We consider the covector (the second term in (31) disappears). Using the relation P i i−1 P q i i−2 = P q i−1 i−2 P i i−1 , one can show by induction that for all i = 2, . . . , n. Since the right hand side does not depend on the spectral parameters, we can substitute each R-matrix in the left hand side by the same one withh = 0 and conclude that Ω q K i . The projection of the i-th qKZ equation onto the covector Ω q reads where J (i) = e j (i) 1 ⊗ ... ⊗ e j (i) n and each time j ki+1 . 2 Indeed, from (32) and construction of J from J min one gets P q i−1,i Ω q = Ω q . The matrix transposition of this equality gives Ω q = Ω q P q i−1,i In the same way as in the rational case, we multiply this by sum over i to get: Here we have used (34) and (35).
Similarly to the rational case, the correspondence can be extended to the higher trigonometric Macdonald-Ruijsenaars operators.

Conclusion and discussion
We have established the correspondence between solutions to the qKZ equations (1) in different weight subspaces of V ⊗n and solutions to the spectral problem for the n-body Ruijsenaars model with Planck's constanth, with V being the space of N-dimensional vector representation of GL(N). The proof appears to be even simpler than in the case of the correspondence between the differential KZ equations and the quantum Calogero model (see [13,4,7] and [22]). In the limith → 0 we obtain the quantum-classsical correspondence [1,9,20] between the quantum spin chain (XXX or XXZ) and the classical Ruijsenaars system of particles [16] (rational or trigonometric).
The Hamiltonian of the classical Ruijsenaars system has the form with the usual Poisson brackets {p i , x j } = δ ij (for simplicity we consider the rational case). The model is integrable, with the Lax matrix is the velocity of the i-th particle. The higher Hamiltonians in involution are coefficients of the characteristic polynomial of the Lax matrix:  (g 1 , . . . , g N ). In fact the admissible values ofẋ i 's obey a system of algebraic equations (see [20]). Different solutions of this system correspond to different eigenstates of the spin chain Hamiltonians.
In the trigonometric case eigenvalues of the Lax matrix L trig ij =ẋ j sinh(x i − x j + η) should form "multiplicative strings" of lengths M a centered at g a : Comparing this with (6) and taking into account (11), (40), we see that in the limit h → 0 of the qKZ system (which is the spectral problem for H i 's or K (0) i 's) the other side of the correspondence becomes the classical Ruijsenaars model. In other words, we can say that the quantization of the classical Ruijsenaars system of particles with the Planck constanth (p i →h∂ x i ) corresponds to the passage from the spectral problem for the spin chain to the system of qKZ equations with the step parameter ηh.