One-Form Abelian Gauge Theory As The Hodge Theory

TL;DR

This paper shows that a 2D free Abelian gauge theory models Hodge theory, with symmetries and algebraic structures mirroring de Rham cohomology, and introduces a novel superfield derivation of equations of motion.

Contribution

It establishes a new superfield approach to derive equations of motion in the BRST formalism for Hodge theory models.

Findings

Symmetries correspond to de Rham cohomological operators

Conserved charges form an algebra similar to de Rham algebra

Super Laplacian operator yields equations of motion

Abstract

We demonstrate that the two (1 + 1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical ingredients that are required in the definition of the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, constitute an algebra that is reminiscent of the algebra obeyed by the de Rham cohomological operators. The topological features of the above theory are discussed in terms of the BRST and co-BRST operators. The super de Rham cohomological operators are exploited in the derivation of the nilpotent (anti-)BRST, (anti-)co-BRST symmetry transformations and the equations of motion for all the fields of the theory, within the framework of the superfield formulation. The derivation of the equations of motion, by exploiting the super Laplacian operator, is a completely new result in the framework of the superfield approach to BRST formalism. In an Appendix, the interacting 2D Abelian gauge theory (where there is a coupling between the U(1) gauge field and the Dirac fields) is also shown to provide a tractable field theoretical model for the Hodge theory.