On Superpotentials and Charge Algebras of Gauge Theories

TL;DR

This paper introduces a covariant Hamiltonian-inspired method to define gauge symmetry charges and superpotentials, simplifying calculations and clarifying charge algebra structures in various gauge theories, including Chern-Simons models.

Contribution

The paper presents a new ambiguity-free covariant formula for superpotentials and charges, applicable to a broad class of gauge theories, with explicit computations in Chern-Simons models.

Findings

Charge algebra with central extensions in 3D Chern-Simons.

No central charge for higher-dimensional non-abelian Chern-Simons.

Explicit superpotential calculations for various gauge theories.

Abstract

We propose a new "Hamiltonian inspired" covariant formula to define (without harmful ambiguities) the superpotential and the physical charges associated to a gauge symmetry. The criterion requires the variation of the Noether current not to contain any derivative terms in $\partial_{\mu}\delta \f$. The examples of Yang-Mills (in its first order formulation) and 3-dimensional Chern-Simons theories are revisited and the corresponding charge algebras (with their central extensions in the Chern-Simons case) are computed in a straightforward way. We then generalize the previous results to any (2n+1)-dimensional non-abelian Chern-Simons theory for a particular choice of boundary conditions. We compute explicitly the superpotential associated to the non-abelian gauge symmetry which is nothing but the Chern-Simons Lagrangian in (2n-1) dimensions. The corresponding charge algebra is also computed. However, no associated central charge is found for $n \geq 2$. Finally, we treat the abelian p-form Chern-Simons theory in a similar way.