On superpotentials and charge algebras of gauge theories

Abstract

We propose a new “Hamiltonian inspired” covariant formula to define (without harmful ambiguities) the superponential 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 ∂ μδ φ. The examples of Yang-Mills (in its first order formulation) and three-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 (2 n + 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 (2 n − 1) dimensions. The corresponding charge algebra is also computed. However, no associated central charge is found for n ⩾ 2. Finally, we treat the abelian p-form Chern-Simons theory in a similar way.