Hodge Dualities on Supermanifolds

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 $0\le p\le n$, $n$ 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.

35 pages

References in corpus (5)

• Supergravity Actions with Integral Forms
• The Geometry of Supermanifolds and New Supersymmetric Actions
• Super-Chern-Simons Theory as Superstring Theory
• The Structure of Superforms
• Integration of Superforms and Super-Thom Class

Cited by in corpus (16)

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• Pictures from Super Chern-Simons Theory
• $A_{\infty }$-Algebra from Supermanifolds
• Self-Dual Forms in Supergeometry I: The Chiral Boson
• Super Chern-Simons Theory: BV-formalism and $A_{\infty }$-algebras
• Super Quantum Mechanics in the Integral Form Formalism
• Supergravities and Branes from Hilbert-Poincaré Series
• Wess-Zumino and Super Yang-Mills Theories in D=4 Integral Superspace
• Surface Operators in Superspace
• Integral representations in supermanifolds: super Hodge duals, PCOs and Liouville forms
• The Hodge Operator Revisited
• Embedding of the Georgi-Glashow $SU(5)$ model in the superconformal algebra