Zengo Tsuboi

Baxter's Q-operators for supersymmetric spin chains

Abstract We develop Yang–Baxter integrability structures connected with the quantum affine superalgebra U q ( sl ˆ ( 2 | 1 ) ) . Baxters Q-operators are explicitly constructed as super-traces 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.

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 ˆ ( 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 above-mentioned 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 Backlund flows proposed in [V. Kazakov, A. Sorin, A. Zabrodin, Nucl. Phys. B790 (2008) 345, arXiv:hep-th/0703147 ].

Classical tau-function for quantum spin chains

A bstractFor an arbitrary generalized quantum integrable spin chain we introduce a “master T -operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the master T -operator, which allows one to identify it with τ -function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rational GL(N)-invariant R-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic) R-matrices and to supersymmetric spin chains.

The master T-operator for the Gaudin model and the KP hierarchy

Abstract Following the approach of [1] , we construct the master T -operator for the quantum Gaudin model with twisted boundary conditions and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. We also characterize the class of solutions to the KP hierarchy that correspond to eigenvalues of the master T -operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum Gaudin model and the classical Calogero–Moser system of particles.

Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl (M|N)

Abstract In [1] (Z. Tsuboi, Nucl. Phys. B 826 (2010) 399, arXiv:0906.2039 ), we proposed Wronskian-like solutions of the T-system for [ M , N ] -hook of the general linear superalgebra gl ( M | N ) . We have generalized these Wronskian-like solutions to the ones for the general T-hook, which is a union of [ M 1 , N 1 ] -hook and [ M 2 , N 2 ] -hook ( M = M 1 + M 2 , N = N 1 + N 2 ). These solutions are related to Weyl-type supercharacter formulas of infinite dimensional unitarizable modules of gl ( M | N ) . Our solutions also include a Wronskian-like solution discussed in [2] (N. Gromov, V. Kazakov, S. Leurent, Z. Tsuboi, JHEP 1101 (2011) 155, arXiv:1010.2720 ) in relation to the AdS 5 / CFT 4 spectral problem.

Supersymmetric quantum spin chains and classical integrable systems

A bstractFor integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0-th time of the hierarchy. This implies a remarkable relation between the quantum supersymmetric spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, we obtain a system of algebraic equations for the spectrum of the spin chain Hamiltonians.

Asymptotic representations and q-oscillator solutions of the graded Yang-Baxter equation related to Baxter Q-operators

Abstract We consider a class of asymptotic representations of the Borel subalgebra of the quantum affine superalgebra U q ( g l ˆ ( M | N ) ) . This is characterized by Drinfeld rational fractions. In particular, we consider contractions of U q ( g l ( M | N ) ) in the FRT formulation and obtain explicit solutions of the graded Yang–Baxter equation in terms of q-oscillator superalgebras. These solutions correspond to L-operators for Baxter Q-operators. We also discuss an extension of these representations to the ones for contracted algebras of U q ( g l ˆ ( M | N ) ) by considering the action of renormalized generators of the other side of the Borel subalgebra. We define model independent universal Q-operators as the supertrace of the universal R-matrix and write universal T-operators in terms of these Q-operators based on shift operators on the supercharacters. These include our previous work on U q ( s l ˆ ( 2 | 1 ) ) case [1] in part, and also give a cue for the operator realization of our Wronskian-like formulas on T- and Q-functions in [2] , [3] .

Nonlinear integral equations and high temperature expansion for the Uq(slˆ(r+1|s+1)) Perk–Schultz model

We propose a system of nonlinear integral equations (NLIE) which gives the free energy of the

Nonlinear Integral Equations and high temperature expansion for the

U_{q}(widehat{sl}(r+1|s+1))

U_{q}(\hat{sl}(r+1|s+1))

Perk-Schultz model. In contrast with traditional thermodynamic Bethe ansatz equations, our NLIE contain only r+s+1 unknown functions. In deriving the NLIE, the quantum (supersymmetric) Jacobi-Trudi and Giambelli formula and a duality for an auxiliary function play important roles. By using our NLIE, we also calculate the high temperature expansion of the free energy. General formulae of the coefficients with respect to arbitrarily rank r+s+1, chemical potentials