Current Algebras and Differential Geometry

TL;DR

The paper addresses how to describe symmetries and gauge symmetries of a broad class of 2D sigma-models using a new current-algebra formalism. It introduces currents J_{(v,α)} labeled by a pair (v,α) on the target manifold M and proves their Poisson brackets realize the Courant bracket, with an anomaly controlled by the symmetric pairing ⟨(u,α),(v,β)⟩_+; anomaly-free subalgebras arise from Dirac structures and generalized complex structures correspond to a splitting into two such anomaly-free subalgebras. The authors provide explicit formulas for the current algebra, connect it to the extended Courant bracket on an enlarged bundle, and illustrate how familiar models like WZW, Poisson and WZ-Poisson sigma-models fit into this framework, including boundary considerations for D-branes. This geometric viewpoint unifies diverse physical models under the umbrella of generalized geometry, clarifies anomaly cancellation conditions, and suggests avenues for supersymmetric extensions and higher-derivative current algebras with broader mathematical significance.

Abstract

We show that symmetries and gauge symmetries of a large class of 2-dimensional sigma models are described by a new type of a current algebra. The currents are labeled by pairs of a vector field and a 1-form on the target space of the sigma model. We compute the current-current commutator and analyse the anomaly cancellation condition, which can be interpreted geometrically in terms of Dirac structures, previously studied in the mathematical literature. Generalized complex structures correspond to decompositions of the current algebra into pairs of anomaly free subalgebras. Sigma models that we can treat with our method include both physical and topological examples, with and without Wess-Zumino type terms.