Current Algebras and Differential Geometry

TL;DR

This paper introduces a novel current algebra framework for symmetries in 2D sigma models, linking algebraic structures to differential geometry concepts like Dirac and generalized complex structures.

Contribution

It develops a new type of current algebra labeled by vector fields and 1-forms, connecting symmetry analysis to geometric structures such as Dirac and generalized complex structures.

Findings

Derived the current-current commutator for the new algebra

Analyzed anomaly cancellation conditions geometrically

Applied framework to both physical and topological sigma models

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.