Spectrum Generating Affine Lie Algebras in Massive Field Theory

TL;DR

This paper explores the application of affine Lie algebras to massive 2D quantum field theory, revealing a new structure at the free fermion point and proposing a novel method for computing form-factors.

Contribution

It introduces a new affine Lie algebra framework for massive field theory and develops a momentum space bosonization approach for form-factor calculations.

Findings

Affine Lie algebra symmetry emerges at the free fermion point.

Field space factorizes into two independent affine algebras.

A new momentum space form-factor computation method is proposed.

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 occuring at the free fermion point. We describe how 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. 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. Using the integrals of motion, a momentum space bosonization involving vertex operators is formulated. This leads to a new way of computing form-factors, as vacuum expectation values in momentum space.