Off-shell Bethe Ansatz Research Articles

A study of form factors in relativistic mixed-flux AdS3

We study the two-particle form-factors for the relativistic limit of the integrable S-matrix of the mixed-flux AdS3 × S3 × T4 string theory. The S-matrix theory was formally constructed in two distinct ways by two different teams. We focus on the massive theory built up by Frolov, Polvara and Sfondrini, and derive expressions for the minimal solutions to the axioms, in both integral and manifestly meromorphic form, and then proceed to apply the off-shell Bethe ansatz method of Babujian et al. We obtain the integral formulas for the two-particle complete form-factors and check the axioms at this particle number.

A study of integrable form factors in massless relativistic AdS 3

We show that the massless integrable sector of the AdS 3 × S 3 × T 4 superstring theory, which admits a non-trivial relativistic limit, provides a setting where it is possible to determine exact minimal solutions to the form factor axioms, in integral form, based on analyticity considerations, along the same lines of ordinary relativistic integrable models. We construct in full detail the formulas for the two- and three-particle case, and show the similarities as well as the differences with respect to the off-shell Bethe ansatz procedure of Babujian et al. We show that our expressions pass a series of non-trivial consistency checks which are substantially more involved than in the traditional case. We speculate on the problems concerned in a possible generalisation to an arbitrary number of particles, and on a possible connection with the hexagon programme.

Generalized Knizhnik–Zamolodchikov equations, off-shell Bethe ansatz and non-skew-symmetric classical r-matrices

Using non-skew-symmetric sl ( 2 ) -valued classical r-matrices with spectral parameters we construct a generalization of the Knizhnik–Zamolodchikov (KZ) equations. We obtain integral solutions of the constructed KZ-type equations using the “off-shell” Bethe ansatz technique. We consider several examples of the obtained generalized KZ equations and their integral solutions that correspond to the “K-twisted” non-skew-symmetric classical r-matrices parametrized by arbitrary complex parameter ξ.

The nested SU(N) off-shell Bethe ansatz and exact form factors

The form factor equations are solved for a SU(N) invariant S-matrix under the assumption that the anti-particle is identified with the bound state of N − 1 particles. The solution is obtained explicitly in terms of the nested off-shell Bethe ansatz where the contribution from each level is written in terms of multiple contour integrals. The general solution is illustrated for some operators, such as the energy–momentum tensor, the fields and the current.

The form factor program: A review and new results, the nested SU(N) off-shell Bethe ansatz and the 1/N expansion

The purpose of the “bootstrap program” for integrable quantum field theories in 1+1 dimensions is to construct a model explicitly in terms of its Wightman functions. We illustrate this program here mainly in terms of the SU(N) Gross-Neveu model. We construct the nested off-shell Bethe ansatz for an SU(N) factoring S-matrix and consider the problem of how to sum over intermediate states in the short-distance limit of the two-point Wightman function for the sinh-Gordon model.

Osp(1|2) off-shell Bethe ansatz equation with boundary terms

This work is concerned with the quasi-classical limit of the boundary quantum inverse scattering method for theosp(1|2) vertex modelwith diagonal K matrices. In this limit Gaudin’s Hamiltonians with boundary terms are presentedand diagonalized. Moreover, integral representations for correlation functions arerealized to be solutions of the trigonometric Knizhnik–Zamolodchikov equations.

The Form Factor Program: a Review and New Results - the Nested SU(N) Off-Shell Bethe Ansatz

The purpose of the bootstrap for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions. In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU(N) Gross-Neveu model. The nested off-shell Bethe ansatz for an SU(N) factorizing S-matrix is constructed. We review some previous results on sinh-Gordon form factors and the quantum operator field equation. The problem of how to sum over intermediate states is considered in the short distance limit of the two point Wightman function for the sinh-Gordon model.

TheA(2)2Gaudin model and its associated Knizhnik–Zamolodchikov equation

The semiclassical limit of the algebraic Bethe Ansatz for the Izergin-Korepin 19-vertex model is used to solve the theory of Gaudin models associated with the twisted $A_{2}^{(2)}$ R-matrix. We find the spectra and eigenvectors of the $N-1$ independents Gaudin Hamiltonians. We also use the off-shell Bethe Ansatz method to show how the off-shell Gaudin equation solves the associated trigonometric system of Knizhnik-Zamolodchikov equations.

Formula omitted] Gaudin magnet and its associated Knizhnik–Zamolodchikov equation

The semiclassical limit of the algebraic Bethe Ansatz method is used to solve the theory of Gaudin models for the s l ( 2 | 1 ) ( 2 ) R-matrix. We find the spectra and eigenvectors of the N − 1 independent Gaudin Hamiltonians. We also use the off-shell Bethe Ansatz method to show how the off-shell Gaudin equation solves the associated trigonometric system of Knizhnik–Zamolodchikov equations.

Supersymmetrict JGaudin models and KZ equations

Supersymmetric t-J Gaudin models with both periodic and open boundary conditions are constructed and diagonalized by means of the algebraic Bethe ansatz method. Off-shell Bethe ansatz equations of the Gaudin systems are derived, and used to construct and solve the KZ equations associated with $sl(2|1)^{(1)}$ superalgebra.

Exact form factors in integrable quantum field theories: the sine-Gordon model (II)

A general model independent approach using the ‘off-shell Bethe Ansatz’ is presented to obtain an integral representation of generalized form factors. The general techniques are applied to the quantum sine-Gordon model alias the massive Thirring model. Exact expressions of all matrix elements are obtained for several local operators. In particular soliton form factors of charge-less operators as for example all higher currents are investigated. It turns out that the various local operators correspond to specific scalar functions called p-functions. The identification of the local operators is performed. In particular the exact results are checked with Feynman graph expansion and full agreement is found. Furthermore all eigenvalues of the infinitely many conserved charges are calculated and the results agree with what is expected from the classical case. Within the frame work of integrable quantum field theories a general model independent ‘crossing’ formula is derived. Furthermore the ‘bound state intertwiners’ are introduced and the bound state form factors are investigated. The general results are again applied to the sine-Gordon model. The integrations are performed and in particular for the lowest breathers a simple formula for generalized form factors is obtained.

The BCS model and the off-shell Bethe ansatz for vertex models

We study the connection between the BCS pairing model and the inhomogeneous vertex model. The two spectral problems coincide in the quasi-classical limit of the off-shell Bethe Ansatz of the disordered six vertex model. The latter problem is transformed into an auxiliary spectral problem which corresponds to the diagonalization of the integrals of motion of the BCS model. A generating functional whose quasi classical expansion leads to the constants of motion of the BCS model and in particular the Hamiltonian, is identified.

Osp (1|2) off-shell Bethe ansatz equations

The semiclassical limit of the algebraic quantum inverse scattering method is used to solve the theory of the Gaudin model. Via Off-shell Bethe ansatz equations of an integrable representation of the graded osp(1|2) vertex model we find the spectrum of the N - 1 independents Hamiltonians of Gaudin. Integral representations of the N-point correlators are presented as solutions of the Knizhnik-Zamolodchikov equation. These results are extended for highest representations of the osp (1|2) Gaudin algebra.

KNIZHNIK–ZAMOLODCHIKOV–BERNARD EQUATIONS CONNECTED WITH THE EIGHT-VERTEX MODEL

By the use of the quasiclassical limit of Baxter's eight-vertex R matrix, an elliptic generalization of the Knizhnik–Zamolodchikov equation is constructed. An integral representation for the solution of this equation is obtained via the off-shell Bethe ansatz. It is shown that there exists a gauge transformation connecting this equation with the Knizhnik–Zamolodchikov–Bernard equation for the SU(2) WZNW model on a torus.

Generalized Knizhnik–Zamolodchikov equations and isomonodromy quantization of the equations integrable via the Inverse Scattering Transform: Maxwell–Bloch system with pumping

Canonical quantization of the isomonodromy solutions of equations integrable via the Inverse Scattering Transform leads to generalized Knizhnik–Zamolodchikov equations. One can solve these equations by the off-shell Bethe ansatz method provided the Knizhnik–Zamolodchikov equations are related with the highest weight representations of the corresponding Lie algebras: These solutions can be written in terms of multi-variable generalizations of special functions of the hypergeometric type. In this work, we consider a realization of the above scheme for the Maxwell–Bloch system with pumping: quantum states for this system are found in terms of the multi-variable confluent hypergeometric function.

Gaudin magnet and off-mass-shell Bethe wave functions

The general Bethe equation (or off-shell Bethe ansatz equation) is proved for the Gaudin magnet for a broad class of simple complex Lie algebras.

Off-shell Bethe ansatz for the XXZ vertex model and solution of the trigonometric Knizhnik-Zamolodchikov equations

We prove that the wavevector of the off-shell Bethe ansatz equation for the inhomogeneous XXZ vertex model in the quasiclassical limit gives the solution of the trigonometric Knizhnik-Zamolodchikov equation. We also observe that this solution is the quasiclassical limit of the solution of the quantum Knizhnik-Zamolodchikov equation corresponding to Uq(sll2).

Integral Solution of Trigonometric Knizhnik-Zamolodchikov Equation; Off-Shell Bethe Ansatz Method

We construct the solutions of the s u (2) trigonometric Knizhnik-Zamolodchikov (KZ) equation based on the Babujian's ”off-shell Bethe ansatz” method. We show that the quasi-classical limit of the Bethe ansatz equation including the ”unwanted term” for the inhomogeneous XXZ lattice reduces to the trigonometric KZ equation.

OFF-SHELL BETHE ANSATZ EQUATION FOR GAUDIN MAGNETS AND SOLUTIONS OF KNIZHNIK-ZAMOLODCHIKOV EQUATIONS

We generalize the previously established connection between the off-shell Bethe ansatz equation for inhomogeneous SU(2) lattice vertex models in the quasiclassical limit and the solutions of the SU(2) Knizhnik-Zamolodchikov equations to the case of simple Lie algebras of higher rank.

Off-shell Bethe ansatz equations and N-point correlators in the SU(2) WZNW theory

We prove that the wave vectors of the off-shell Bethe Ansatz equation for the inhomogeneous SU(2) lattice vertex model render in the quasiclassical limit the solution of the Knizhnik-Zamolodchikov equation.