gl(N,N) Current Algebras and Topological Field Theories

TL;DR

This paper explores the structure of the $gl(N,N)$ affine Lie superalgebra in 2D conformal field theory, revealing a topological symmetry with a novel algebraic structure and connections to coset models and topological sigma models.

Contribution

It constructs the energy-momentum tensor with vanishing Virasoro anomaly and identifies a new topological algebra structure in the $gl(N,N)$ model, including explicit free-field realizations.

Findings

The energy-momentum tensor has vanishing Virasoro anomaly.

The topological algebra differs from twisted $N=2$ models and closes with $gl(N)$ currents.

The $GL(N,N)$ Wess-Zumino-Witten model provides explicit realizations and decompositions into $sl(N)$ and $U(1)$ components.

Abstract

The conformal field theory for the $gl(N,N)$ affine Lie superalgebra in two space-time dimensions is studied. The energy-momentum tensor of the model, with vanishing Virasoro anomaly, is constructed. This theory has a topological symmetry generated by operators of dimensions 1, 2 and 3, which are represented as normal-ordered products of $gl(N,N)$ currents. The topological algebra they satisfy is linear and differs from the one obtained by twisting the $N=2$ superconformal models. It closes with a set of $gl(N)$ bosonic and fermionic currents. The Wess-Zumino-Witten model for the supergroup $GL(N,N)$ provides an explicit realization of this symmetry and can be used to obtain a free-field representation of the different generators. In this free-field representation, the theory decomposes into two uncoupled components with $sl(N)$ and $U(1)$ symmetries. The non-abelian component is responsible for the extended character of the topological algebra, and it is shown to be equivalent to an $SL(N)/ SL(N)$ coset model. In the light of these results, the $G/ G$ coset models are interpreted as topological sigma models for the group manifold of $G$