Affine Lie Algebras in Massive Field Theory and Form-Factors from Vertex Operators

TL;DR

This paper explores the application of affine Lie algebras to massive 2D quantum field theory, specifically the sine-Gordon model at the free fermion point, leading to new methods for computing form-factors via vertex operators.

Contribution

It introduces a novel approach to relate affine Lie algebras with form-factor calculations in massive quantum field theory using vertex operators and radial quantization.

Findings

Decomposition of affine charges into two independent affine algebras

Organization of fields using level-1 highest weight representations

Development of a momentum space bosonization method for form-factors

Abstract

We present a new application of affine Lie algebras to massive quantum field theory in 2 dimensions, by investigating the $q\to 1$ limit of the q-deformed affine $\hat{sl(2)}$ symmetry of the sine-Gordon theory, this limit occurring at the free fermion point. Working in radial quantization leads to a quasi-chiral factorization of the space of fields. The conserved charges which generate the affine Lie algebra split into two independent affine algebras on this factorized space, each with level 1 in the anti-periodic sector, and level $0$ in the periodic sector. The space of fields in the anti-periodic sector can be organized using level-$1$ highest weight representations, if one supplements the $\slh$ algebra with the usual local integrals of motion. Introducing a particle-field duality leads to a new way of computing form-factors in radial quantization. Using the integrals of motion, a momentum space bosonization involving vertex operators is formulated. Form-factors are computed as vacuum expectation values in momentum space. (Based on talks given at the Berkeley Strings 93 conference, May 1993, and the III International Conference on Mathematical Physics, String Theory, and Quantum Gravity, Alushta, Ukraine, June 1993.)