Hodge Dualities on Supermanifolds

TL;DR

This paper explores the cohomology of superforms and integral forms on supermanifolds using a new Hodge dual operator, linking superspace constraints with curvature and Ramond-Ramond fields, and introducing operators and non-abelian curvatures.

Contribution

It introduces a novel Hodge dual operator for supermanifolds, translating superspace constraints between superforms and integral forms, and constructs non-abelian curvatures within this framework.

Findings

Established a new Hodge dual operator for superforms and integral forms.

Connected superspace constraints with curvature and Ramond-Ramond fields.

Developed non-abelian curvatures for gauge connections in superform spaces.

Abstract

We discuss the cohomology of superforms and integral forms from a new perspective based on a recently proposed Hodge dual operator. We show how the superspace constraints (a.k.a. rheonomic parametrisation) are translated from the space of superforms $\Omega^{(p|0)}$ to the space of integral forms $\Omega^{(p|m)}$ where $0 \leq p \leq n$, $n$ is the bosonic dimension of the supermanifold and $m$ its fermionic dimension. We dwell on the relation between supermanifolds with non-trivial curvature and Ramond-Ramond fields, for which the Laplace-Beltrami differential, constructed with our Hodge dual, is an essential ingredient. We discuss the definition of Picture Lowering and Picture Raising Operators (acting on the space of superforms and on the space of integral forms) and their relation with the cohomology. We construct non-abelian curvatures for gauge connections in the space $\Omega^{(1|m)}$ and finally discuss Hodge dual fields within the present framework.