Abstract
An appropriate definition of the Hodge duality ⋆ operation on any arbitrary dimensional supermanifold has been a long-standing problem. We define a working rule for the Hodge duality ⋆ operation on the (2+2)-dimensional supermanifold parametrized by a couple of even (bosonic) space–time variables xμ(μ = 0, 1) and a couple of odd (fermionic) variables θ and [Formula: see text] of the Grassmann algebra. The Minkowski space–time manifold, hidden in the supermanifold and parametrized by xμ(μ = 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.
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