Non-Compact WZW Conformal Field Theories

TL;DR

This paper explores non-compact WZW conformal field theories, highlighting their unique features like continuous spectra and diverging partition functions, and presents new constructions of related sigma models and coset theories.

Contribution

It introduces non-compact WZW models with symmetric space targets, analyzes their distinctive properties, and constructs stable, explicit sigma models for non-rational conformal theories.

Findings

Continuous spectrum of conformal weights identified

Partition functions diverge, indicating non-rationality

Explicit stable sigma model for Euclidean 2D black hole constructed

Abstract

We discuss non-compact WZW sigma models, especially the ones with symmetric space $H^{\bf C}/H$ as the target, for $H$ a compact Lie group. They offer examples of non-rational conformal field theories. We remind their relation to the compact WZW models but stress their distinctive features like the continuous spectrum of conformal weights, diverging partition functions and the presence of two types of operators analogous to the local and non-local insertions recently discussed in the Liouville theory. Gauging non-compact abelian subgroups of $H^{\bf C}$ leads to non-rational coset theories. In particular, gauging one-parameter boosts in the $SL(2,\bC)/SU(2)$ model gives an alternative, explicitly stable construction of a conformal sigma model with the euclidean 2D black hole target. We compute the (regularized) toroidal partition function and discuss the spectrum of the theory. A comparison is made with more standard approach based on the $U(1)$ coset of the $SU(1,1)$ WZW theory where stability is not evident but where unitarity becomes more transparent.