Alexei B. Zamolodchikov

Factorized S-Matrices in Two Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Theory Models

Abstract The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N ⩾ 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N ⩾ 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N ⩾ 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.

CONFORMAL BOOTSTRAP IN LIOUVILLE FIELD THEORY

Attention to the two-dimensional Liouville Field Theory (LFT) is drawn basically for two reasons. First it was recognized [1] as an effective field theory of the 2d. quantum gravity. In particular it is very relevant in the string theory [1–4]. Second, it is an example of non-rational conformai field theory (CFT) which is very likely exactly solvable (e.g., the classical equations of motion are integrable).The interest to LFT was renewed recently with the development of the matrix model approach to 2d gravity [5, 6]. It was shown that LFT is able to reproduce some of the predictions of the matrix model approach, in particular the scaling behavior [7–9], the genus one partition functions [10] and some of the integrated correlation functions [11–15]. It is very plausible therefore that LFT describes the same 2d quantum gravity as the matrix models do (at least in the “weak coupling region” C L > 25).An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere. In the classical limit it coincides with what the classical Liouville theory predicts. Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative. We consider also the Liouville reflection amplitude which follows explicitly from the structure constants.

On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories

Abstract We remark about an amusing universality in the thermodynamic Bethe ansatz equations for ADE-related diagonal scattering theories.

Massless factorized scattering and sigma models with topological terms

Abstract We propose two massless non-diagonal scattering theories with SU(2) isotopic symmetry. While being identical in the right- and left-moving sectors, they differ in the right-left scattering. The first theory has reflectionless resonance right-left amplitudes and exhibits two independent (right and left) SU(2) symmetries. It is suggested as a scattering theory of the SU(2) × SU(2) principal chiral model with the Wess-Zumino term of level k = 1. In the second scattering theory only one SU(2) group survives at finite distances. We interpret this one as corresponding to the O(3) sigma model with the θ = π topological term. In both the cases the thermodynamic Bethe ansatz equations are derived.

Expectation values of local fields in the Bullough-Dodd model and integrable perturbed conformal field theories

Exact expectation values of the fields eaϕ in the Bullough-Dodd model are derived by adopting the “reflection relations” which involve the reflection S-matrix of the Liouville theory, as well as a special analyticity assumption. Using this result we propose explicit expressions for expectation values of all primary operators in the c < 1 minimal CFr perturbed by the operator φ1,2 or φ2,1. Some results concerning the φ1,5 perturbed minimal models are also presented.

EXPECTATION VALUES OF DESCENDENT FIELDS IN THE SINE-GORDON MODEL

We obtain exactly the vacuum expectation values

Expectation values of boundary fields in the boundary sine-Gordon model

<(\partial\phi)^2 ({\bar\partial}\phi) e^{i\alpha\phi}>

Moduli integrals and ground ring in minimal Liouville gravity

in the sine-Gordon model and

Perturbed Conformal Field Theory on A Sphere

in