Current Algebras and QP Manifolds

TL;DR

This paper extends the concept of current algebras to higher dimensions using QP manifolds, revealing their underlying Leibniz/Loday algebroid structures and connections to homotopy Lie algebroids and Dirac structures.

Contribution

It generalizes current algebras to arbitrary dimensions via graded manifolds and QP structures, unifying various algebraic frameworks.

Findings

Current algebras in higher dimensions are described by QP manifolds.

Leibniz/Loday algebroid structures characterize these current algebras.

Anomaly cancellation conditions relate to generalized Dirac structures.

Abstract

Generalized current algebras introduced by Alekseev and Strobl in two dimensions are reconstructed by a graded manifold and a graded Poisson brackets. We generalize their current algebras to higher dimensions. QP manifolds provide the unified structures of current algebras in any dimension. Current algebras give rise to structures of Leibniz/Loday algebroids, which are characterized by QP structures. Especially, in three dimensions, a current algebra has a structure of a Lie algebroid up to homotopy introduced by Uchino and one of the authors which has a bracket of a generalization of the Courant-Dorfman bracket. Anomaly cancellation conditions are reinterpreted as generalizations of the Dirac structure.