Hodge Duality Operation And Its Physical Applications On Supermanifolds

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 $x^\mu (\mu = 0, 1)$,ischosentobeaflatmanifoldonwhichatwo$(1 + 1)$-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.