Central charges without finite-size effects

TL;DR

This paper presents a method to determine the ultraviolet central charge directly from the exact S-matrix for various models with U(1) symmetry by coupling to a background field, bypassing finite-size effects.

Contribution

It generalizes the calculation of the central charge from N=2 superconformal theories to a broad class of U(1)-invariant models using quantum-group symmetry deformation.

Findings

Central charge obtained from S-matrix matches known values.

Method applies to sine-Gordon and SU(2) WZW models.

Provides a consistency check independent of thermodynamic Bethe ansatz.

Abstract

We show how to obtain the ultraviolet central charge from the exact S-matrix for a wide variety of models with a $U(1)$ symmetry. This is done by coupling the $U(1)$ current $J$ to a background field. In an $N$=2 superconformal theory with $J$ the fermion number current, the OPE of $J$ with itself and hence the free energy are proportional to $c$. By deforming the supersymmetry into affine $\widehat SU(2)_q$ quantum-group symmetry, this result can be generalized to many $U(1)$-invariant theories, including the N=0 and N=1 sine-Gordon models and the $SU(2)_k$ WZW models. This provides a consistency check on a conjectured S-matrix completely independent of the finite-size effects expressed in terms of dilogarithms resulting from the thermodynamic Bethe ansatz.