Supersymmetric extension of qKZ-Ruijsenaars correspondence

We describe the correspondence of the Matsuo-Cherednik type between the quantum $n$-body Ruijsenaars-Schneider model and the quantum Knizhnik-Zamolodchikov equations related to supergroup $GL(N|M)$. The spectrum of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the ${\mathbb Z}_2$-grading for a fixed value of $N+M$, so that $N+M+1$ different qKZ systems of equations lead to the same $n$-body quantum problem. The obtained results can be viewed as a quantization of the previously described quantum-classical correspondence between the classical $n$-body Ruijsenaars-Schneider model and the supersymmetric $GL(N|M)$ quantum spin chains on $n$ sites.


Introduction
The KZ-Calogero and qKZ-Ruijsenaars correspondences are the Matsuo-Cherednik type constructions [12,10,18,19] for solutions of the Calogero-Moser-Sutherland [4] and Ruijsenaars-Schneider [14] quantum problems by means of solutions of the Knizhnik-Zamolodchikov (KZ) [8] and quantum Knizhnik-Zamolodchikov (qKZ) equations [11] respectively. Consider, for example, the qKZ equations 4 related to the Lie group GL(K): where g = diag(g 1 , . . . , g K ) is a diagonal K×K (twist) matrix, and g (i) acts by g multiplication in the i-th tensor component of the Hilbert space V = (C K ) ⊗n . The quantum R-matrices R ij are in the fundamental representation of GL(K). They act in the i-th and j-th tensor components of V and satisfy the quantum Yang-Baxter equation, which guarantees compatibility of equations (1.1). The twist matrix g is the symmetry of R ij : g (i) g (j) R ij = R ij g (i) g (j) . In the rational case we deal with the Yang's R-matrix [17]: where I is identity operator in End(V), and P ij is the permutation operator, which interchanges the i-th and j-th tensor components in V. ).
QC-duality. Using the asymptotics of solutions to the (q)KZ equations [15] it was also argued in [18,19] that the qKZ-Ruijsenaars correspondence can be viewed as a quantization of the quantum-classical duality [1,7,2] (see also [13,5]), which relates the generalized inhomogeneous quantum spin chains and the classical Ruijsenaars-Schneider model. Consider the classical Kbody Ruijsenaars-Schneider model, where the positions of particles {x i } are identified with the inhomogeneity parameters of the spin chain which is described by its transfer matrix The quantum spin chain Hamiltonians are defined as follows: (1.14) Therefore, Identify also the generalized velocities {ẋ i } with the eigenvalues of (1.14). Then the action variables {I i | i = 1, ..., K} of the classical model (eigenvalues of the Lax matrix) are given by the values of g 1 , ..., g K with multiplicities M 1 , ..., M K : (1. 16) See details in [7], where this statement was proved using the algebraic Bethe ansatz technique.
(1.17) More precisely, it was shown in [16] that the previous statement (1.16) is valid for all supersymmetric chains with supergroups (1.17).
The aim of this paper is to quantize the (supersymmetric) quantum-classical correspondence, that is to establish supersymmetric version of the qKZ-Ruijsenaars correspondence for the qKZ equations related to the supergroups GL(N|M). We construct generalizations of the vector Ω (1.8) and show that the quantum K-body Ruijsenaars-Schneider model follows from all K + 1 qKZ systems of equations related to the supergroups GL(N|M) with N + M = K (1.17). The skew-symmetric vectors Ω − with the property Ω − P ij = − Ω − (instead of symmetric vector (1.9)) are described as well. They lead to the Ruijsenaars-Schneider model with different sign of the coupling constant η and .
The paper is organized as follows. For simplicity we start with the rational KZ-Calogero correspondence. Then we proceed to the rational and trigonometric qKZ-Ruijsenaars relations. Most of notations are borrowed from [18,19,16]. We briefly describe the notations and definitions related to the graded Lie algebras (groups) in the Appendix.

SUSY KZ-Calogero correspondence
The rational Knizhnik-Zamolodchikov (KZ) equations [8] have the form form the commutative set of Gaudin Hamiltonians [6]. Similarly to non-supersymmetric case they also commute with the operators: where e a are basis vectors in V and the number of indices a k such that a k = a is equal to M a for all a = 1, . . . , N + M. A general solution to (2.1) can be written as where the coefficients Φ J are functions of all parameters entering (2.1).
To proceed further we need to find a (co)vector where in contrast to (1.9) the permutation operator P ij acts in the graded space (it has the form (A.7)). Having such a vector and taking into account the identities (A.11) and (A.12), we can repeat all the calculations from [18] without any changes. They lead to the eigenvalue equation for the second Calogero-Moser Hamiltonian: (2.9) Let us construct the vector Ω . Due to (A.9) the basis vector J entering Ω can not contain two identical fermions (vectors e a with p(a) = 1). Otherwise we get a contradiction with (2.6). Keeping this in mind choose a vector J with a 1 ≤ a 2 ≤ ... ≤ a n from V({M a }), and fix the coefficient Ω a 1 ≤a 2 ≤...≤an = 1 for this set. Next, generate the rest of vectors J by the rule that the permutation of two nearby indices multiplies the coefficient by the standard parity factor: Then Then (2.12) Then It is also worth noting that in order to change the sign of κ in the Hamiltonian (2.7) we need to construct vector Ω − , which is antisymmetric under the action of permutations: where the sign is opposite to the one in (2.6). Such a vector can not contain two identical bosons because the permutation of them contradicts assumption (2.14). In other situations it can be constructed. The example is given below. Then where the operators in the r.h.s are constructed by means of the quantum R-matrix R, which is a (unitary) solution of the graded Yang-Baxter equation. We start with the rational one where P ij is the graded permutation operator (A.7). Similarly to the non-supersymmetric case introduce the rescaled R-matrix: The transfer matrix of the corresponding supersymmetric spin chain provides non-local Hamiltonians as its residues: Explicitly, Alternatively, From comparison of expansions of the transfer matrix as x → ∞ in the forms (3.5) and (3.6) we obtain: where the property (A.12) was used. To obtain the correspondence we project the qKZ-equations on the vector Ω (2.6), constructed in the previous section: and repeat all calculations from [19]. This yields: is the eigenfunction and is the eigenvalue.
in (3.1) instead of (3.3). The R-matrix (3.14) is still unitary and acts identically on the antisymmetric vector Ω − (2.14) which is to be used instead of Ω .

Higher Hamiltonians
Following the construction in the non-supersymmetric case, it can be shown that the wave function Ψ = Ω Φ satisfies the equations The proof of this statement is the same as in [19]. One more point needed for the correspondence is the determinant identity det 1≤i,j≤n It was proven for the supersymmetric case in [16]. Therefore, the correspondence works in the supersymmetric case as well. Namely, given a solution |Φ of the qKZ equations the wave function of the rational Ruijsenaars-Schneider quantum problem is given by (3.12). The eigenvalues are the same symmetric polynomials as in the non-supersymmetric case (1.11).

SUSY qKZ-Ruijsenaars correspondence, trigonometric case
The trigonometric (hyperbolic) solution to the graded Yang-Baxter equation has the following form [3]: where q = e η . It can be rewritten as follows: where P 12 is the graded permutation operator (A.7), P q 12 -its q-deformation (the quantum permutation operator)  or The R-matrix entering the transfer matrix differs from (4.1) by a scalar factor: and the transfer matrix itself is defined similarly to (3.5). The Hamiltonians are introduced through the expansion They are related to the operators in the r.h.s of the qKZ-equations by the same formulae as in non-supersymmetric case:

Construction of q-symmetric vectors
Our strategy is as follows. Following the non-supersymmetric construction [19], we now need to find a vector Ω q with the property Let us show that this vector has the form: where Ω J are the same as in the rational case (2.7), (2.10), while ℓ(J) is defined to be the minimal number of elementary permutations required to get the multi-index J = (j 1 , j 2 , . . . , j n ) starting from the "minimal" one. The "minimal" order implies that the j k 's are ordered as 1 ≤ j 1 ≤ j 2 ≤ . . . ≤ j n ≤ N (see [19]). The proof is straightforward. First, by the construction we see that Ω q P q i,i−1 = Ω q . (4.11) In contrast to the non-supersymmetric case we have additional terms G + i,i−1 in R-matrices (4.2). However, they do not provide any effect when acting on Ω q : It happens because of the tensor structure (4.4). Indeed, so that only the same basis vectors e a i entering J may contribute. But we have already assumed that our vector Ω q does not contain two identical fermions, and for bosons G + a = 0. Finally, using (4.2) we arrive at (4.9).  Calculation of the eigenvalue Coming back to the proof of the correspondence we need the identity which follows from P i i−1 P q i i−2 = P q i−1 i−2 P i i−1 and an analogue of the identity for the supersymmetric case. It is as follows.  Proof: We will prove the first equality. The proof of the second one is similar. Let us first find the asymptotics of the R-matrix: The off-diagonal part does not contribute to the trace in (3.5). Therefore, and, finally,  Notice that although this expression depends on the choice of B and F the eigenvalue of the Ruijsenaars-Schneider Hamiltonian is independent of it:

Construction of q-antisymmetric vectors
In order to extend the correspondence to the case of the Hamiltonian with the opposite sign of η we should start with a different R-matrix: (4.24) It is an analog of (3.14) in the rational case. Expression (4.24) can be rewritten in the form  Similarly to the case of symmetric vector (and also similarly to (2.14)) it is easy to see that the vector Ω q with the property can not contain two or more identical bosonic vectors. On the other hand, G − 12 acts by zero on the pair of identical fermions. Thus (4.28) Repeating the steps from the previous paragraphs we obtain the following expressions for the asymptotics of the R-matrix at infinity: where It is easy to see that these asymptotics differ from the corresponding asymptotics in the qsymmetric case by non-diagonal part only, but the latter does not contribute to the trace in the transfer matrix. Therefore, the Hamiltonian with the opposite sign of η has the same eigenvalue:

Symmetry between q-(anti)symmetric vectors
In this paragraph we will show that the usage of q-antisymmetric vectors do not actually lead to any new wave functions of the Ruijsenaars-Schneider system. For this paragraph let us introduce more refined notations:  and where the index p stands for a fixed choice of grading.
Let us introduce the operator Q of the grading change: p(Qe a ) = p(e a ) + 1 (mod 2). (4.34) This operator simply changes all basis bosonic vectors e a to fermionic ones and vice versa. It is easy to see from this definition that the R-matrix has a symmetry where the index p + 1 means simultaneous shift of all grading parameters by 1 modulo 2 in (4.32). Therefore, For the special vectors (on which we project the solutions) we also reserve the following notation: By changing all bosons to fermions in these equations and vice versa, and taking into account that As a first step towards the explanation of the origin of the wavefunctions for Hamiltonians with signs of η and changed we will prove the following Proof: Consider the qKZ-equations: Changing signs of η and yields Using the symmetry (4.35) this could be rewritten in the form: It can be seen from here that the desired solution Φ p+1 + (x|η, ) is the following: Consider the space of all wavefunctions Ψ − (x|η, ) of the Ruijsenaars Hamiltonian with signs of η and changed: which could be obtained with our construction, i.e. they have the form    The proof follows from the previous proposition with Φ p+1 + (x|η, ) defined as in (4.40) and the remark (4.39).
This proposition actually means that for any wavefunction constructed with the help of the q-antisymmetric vector the existence of the corresponding solution of the qKZ equation is a simple consequence of the existence of such solution for the wavefunction with signs of η and changed, constructed with the help of the q-symmetric vector.

Appendix
Here we give a short summary of notations and definitions related to the Lie superalgebra gl(N|M). For any homogeneous (with a definite grading) operators {A i ∈ End(C N |M )} 4 i=1 and homogeneous vectors x , y ∈ C N |M we have: The graded permutation operator P 12 ∈ End(C N |M ⊗ C N |M ) is of the form: