Solutions of the T-system and Baxter equations for supersymmetric spin chains

Abstract

We propose Wronskian-like determinant formulae for the Baxter Q-functions and the eigenvalues of transfer matrices for spin chains related to the quantum affine superalgebra U q ( gl(M |N )). In contrast to the supersymmetric Bazhanov-Reshetikhin formula (the quantum supersymmetric Jacobi-Trudi formula) proposed in [Z. Tsuboi, J. Phys. A: Math. Gen. 30 (1997) 7975], the size of the matrices of these Wronskian-like formulae is less than or equal to M + N . Base on these formulae, we give new expressions of the solutions of the T -system (fusion relations for transfer matrices) for supersymmetric spin chains proposed in the abovementioned paper. Baxter equations also follow from the Wronskian-like formulae. They are finite order linear difference equations with respect to the Baxter Q-functions. Moreover, the Wronskian-like formulae also explicitly solve the functional relations for Bäcklund flows proposed in [V. Kazakov, A. Sorin, A. Zabrodin, Nucl. Phys. B790 (2008) 345 [arXiv:hep-th/0703147]].

Solving the AdS/CFT Y-system

“…The AdS 5 /CFT 4 Y-and T-systems with T-hook boundary conditions proposed in [7] and summarized in figure 1 were later shown to be equivalent, with certain analyticity requirements [11][12][13], to the TBA equations [14][15][16]. It was shown in [17,18] (see also [19,20]) that the T-system, and hence the Y-system, in T-hook can be formally solved in terms of Wronskian determinants of a finite number of Q-functions -a generalization of Baxter's Q-function. But the corresponding finite system of nonlinear integral equations (FiNLIE), a remote analogue of Destri-de Vega equations, was still missing.…”

“…Provided that the interpretation of [20] is correct and the T-functions of AdS/CFT are indeed the transfer matrix eigenvalues one should expect that the auxiliary Bethe roots (carrying no momentum and energy) can be found by requiring some good analytic properties for the physical transfer matrices. In this paper we restrict ourselves to the slp2q sector, which has no auxiliary roots, and thus the analyticity should be simpler.…”

“…For that we use the Plücker relations which follow from (4.14) and are explicitly given in [18,20,36]:…”

“…is divergent, however the divergence is just an infinite constant which can be regularized, as in (2.16), so that 20) where the kernel Ψ is defined in (2.16). Note that in principle one can add a constant to the r.h.s.…”

“…34 The value ofũ 1 is determined by the condition that T1 ,0 should have zeroes at position of the Bethe roots. The other q's can be found through the set of Plücker relations [20,36] q H q ij " qì qj´qj qí , (6.9)…”