Baxter's Q-operators for supersymmetric spin chains

Abstract

We develop Yang-Baxter integrability structures connected with the quantum affine superalgebra U q ( sl(2|1)). Baxter's Q-operators are explicitly constructed as supertraces of certain monodromy matrices associated with (q-deformed) bosonic and fermionic oscillator algebras. There are six different Q-operators in this case, obeying a few fundamental fusion relations, which imply all functional relations between various commuting transfer matrices. The results are universal in the sense that they do not depend on the quantum space of states and apply both to lattice models and to continuous quantum field theory models as well.

Solving the AdS/CFT Y-system

“…To produce this function we should simply systematically replace all G andḠ in (3.10) byĜ: 16 only for s ě a, we can formally defineT a,s for any´8 ă s ă 8 by (3.17). Interestingly, for s " 1 we get then 19) which is consistent withT…”

“…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.…”

“…(D. 19) We see that o is some universal function which should be the same for any pair of indexes, since q ij is an antisymmetric tensor. Consider now…”

Classical tau-function for quantum spin chains

“…[45,[52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67]). In this section, we explain how the Baxter Q-operators emerge from the master T -operator using the approach of [12].…”

Review of AdS/CFT Integrability, Chapter III.7: Hirota Dynamics for Quantum Integrability

“…The analytic Bethe ansatz approach based on the Y-and T-systems for the fusion of transfer-matrices in various representations was successfully applied to various spin chains and 2D QFT's [6,7] and is especially efficient for the supersymmetric systems [8][9][10]. Being integrable, Hirota equation with specific boundary conditions stemming from the symmetry of the problem, can be often solved explicitly, either by the Bäcklund method [6,9] or in the determinant (Wronskian) form [6,11]. Of course to specify completely the physical solutions we have to precise the functional space for the functions of spectral parameter entering Hirota equation, or in other words, we also need to impose certain analyticity conditions on these solutions which is usually the hardest part of the problem.…”