On Non-Abelian Duality

TL;DR

Addresses non-abelian duality in two-dimensional $\sigma$-models and conformal field theories. Shows a mixed gauge-gravitational anomaly can obstruct conformal invariance for non-semisimple isometry groups. Provides an exact non-abelian dual for WZW models with anomaly-free subgroups and extends to coset CFTs $(G/H)_k \times H_k$, and to general $\sigma$-models with left and right chiral currents, yielding a heterotic-like duality using an auxiliary field $\beta$. Demonstrates gravitational-instanton examples and clarifies the role of the auxiliary field in revealing the dual's global topology, with BRST consistency ensuring conformal data match. Outlines open challenges, including extending to non-chiral cases, invertibility, operator mappings, and potential discrete or supersymmetric generalizations.

Abstract

A general study of non-abelian duality is presented. We first identify a possible obstruction to the conformal invariance of the dual theory for non-semisimple groups. We construct the exact non-abelian dual for any Wess-Zumino-Witten (WZW) model for any anomaly free subgroup, and the corresponding extension for coset conformal field theories. We characterize the exact non-abelian dual for $σ$-models with chiral isometries and extend the standard notion of duality to anomalous subgroups of WZW models, thus giving a way of constructing dual transformations for different groups on the left and on the right. We also present some new examples of non-abelian duality for four-dimensional gravitational instantons.