Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

TL;DR

This paper examines the non-unitarity issues in quantum affine Toda theories, specifically a2(1) and a2(2), revealing fundamental problems in their scattering matrices and the impact of RSOS restrictions on these issues.

Contribution

It provides a detailed analysis of the scattering problems in quantum affine Toda theories and discusses how RSOS restrictions and regradations affect unitarity and consistency.

Findings

a2(1) soliton and breather scattering is flawed in both classical and quantum theories

a2(2) has issues with soliton-excited soliton scattering in unrestricted form

RSOS restriction can sometimes fix or leave unresolved the scattering problems depending on gradation

Abstract

There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue - a2(1) and a2(2). By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory, (with or without RSOS restriction), indeed has some fundamental problems, but that these problems are of two different sorts. For a2(1), the scattering of solitons and breathers is flawed in both classical and quantum theories, and RSOS restriction cannot solve this problem. For a2(2) however, while there are no problems with breather-soliton scattering there are instead difficulties with soliton-excited soliton scattering in the unrestricted theory. After RSOS restriction, the problems with kink-excited kink may be cured or may remain, depending in part on the choice of gradation, as we found in [12]. We comment on the importance of regradations, and also on the survival of R-matrix unitarity and the S-matrix bootstrap in these circumstances.