Hodge Duality Operation And Its Physical Applications On Supermanifolds

TL;DR

This paper defines a Hodge duality operation on a specific supermanifold and applies it to analyze symmetries in 2D and 4D free Abelian gauge theories within a superfield framework.

Contribution

It introduces a novel rule for the Hodge duality operation on a (2+2)-dimensional supermanifold and demonstrates its application to (anti-)co-BRST symmetries in gauge theories.

Findings

Defined a Hodge duality operation on (2+2)-dimensional supermanifold.

Applied the definition to analyze (anti-)co-BRST symmetries in 2D and 4D gauge theories.

Extended the application to supermanifolds with different dimensions.

Abstract

An appropriate definition of the Hodge duality $\star$ operation on any arbitrary dimensional supermanifold has been a long-standing problem. We define a working rule for the Hodge duality $\star$ operation on the $(2 + 2)$-dimensional supermanifold parametrized by a couple of even (bosonic) spacetime variables $x^\mu (\mu = 0, 1)$ and a couple of Grassmannian (odd) variables $\theta$ and $\bar\theta$ of the Grassmann algebra. The Minkowski spacetime manifold, hidden in the supermanifold and parametrized by $x^\mu (\mu = 0, 1)$, is chosen to be a flat manifold on which a two $(1 + 1)$-dimensional (2D) free Abelian gauge theory, taken as a prototype field theoretical model, is defined. We demonstrate the applications of the above definition (and its further generalization) for the discussion of the (anti-)co-BRST symmetries that exist for the field theoretical models of 2D- (and 4D) free Abelian gauge theories considered on the four $(2 + 2)$- (and six $(4 + 2)$)-dimensional supermanifolds, respectively.