Ectoplasm with an Edge

TL;DR

This paper develops a method for constructing supersymmetric actions on manifolds with boundaries using the ectoplasm formalism, employing relative cohomology and relating it to superembedding, with examples in 3D, 4D, and 5D.

Contribution

It introduces a cohomological approach to supersymmetric invariants on manifolds with boundaries, extending the ectoplasm formalism and connecting it to superembedding techniques.

Findings

Constructed supersymmetric invariants using relative cohomology.

Related ectoplasm formalism to superembedding for brane actions.

Provided explicit examples in 3D, 4D, and 5D spacetimes.

Abstract

The construction of supersymmetric invariant actions on a spacetime manifold with a boundary is carried out using the "ectoplasm" formalism for the construction of closed forms in superspace. Non-trivial actions are obtained from the pull-backs to the bosonic bodies of closed but non-exact forms in superspace; finding supersymmetric invariants thus becomes a cohomology problem. For a spacetime with a boundary, the appropriate mathematical language changes to relative cohomology, which we use to give a general formulation of off-shell supersymmetric invariants in the presence of boundaries. We also relate this construction to the superembedding formalism for the construction of brane actions, and we give examples with bulk spacetimes of dimension 3, 4 and 5. The closed superform in the 5D example needs to be constructed as a Chern-Simons type of invariant, obtained from a closed 6-form displaying Weil triviality.