On Superpotentials and Charge Algebras of Gauge Theories

TL;DR

The paper introduces a covariant, Hamiltonian-inspired formula to define superpotentials and conserved charges for gauge symmetries, anchored by the localizability condition that the variation of the Noether current contains no derivatives of the field variation. It demonstrates the method on Yang-Mills in first-order form and on Chern-Simons theories across dimensions, deriving explicit superpotentials and charge algebras, including central extensions in 3D CS and their absence in higher dimensions under chosen boundary conditions. The non-abelian $(2n+1)$-dimensional CS theory yields a superpotential equivalent to a $(2n-1)$-dimensional CS Lagrangian, with charge algebras that lack central charges for $n\ge2$, while abelian p-form CS theories show a similar pattern. The framework unifies covariant Noether theory with Hamiltonian boundary terms, clarifies the role of boundary conditions, and provides a practical route to compute gauge and diffeomorphism charges in diverse CS/YM settings.

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 $\partial_μδ\f$. 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.