The Geometry of Supermanifolds and New Supersymmetric Actions

TL;DR

This paper develops a new Hodge dual for supermanifolds using Grassmannian Fourier transform, enabling the construction of supersymmetric actions with higher derivative terms and a novel representation of supersymmetry.

Contribution

It introduces a natural construction of the Hodge dual for superforms, linking integral forms to supersymmetry representations and enabling new supersymmetric actions with higher derivatives.

Findings

Hodge dual for superforms constructed via Grassmannian Fourier transform

Supersymmetric actions with higher derivative terms derived

New representation of supersymmetry using integral and superforms

Abstract

We construct the Hodge dual for supermanifolds by means of the Grassmannian Fourier transform of superforms. In the case of supermanifolds it is known that the superforms are not sufficient to construct a consistent integration theory and that the integral forms are needed. They are distribution-like forms which can be integrated on supermanifolds as a top form can be integrated on a conventional manifold. In our construction of the Hodge dual of superforms they arise naturally. The compatibility between Hodge duality and supersymmetry is exploited and applied to several examples. We define the irreducible representations of supersymmetry in terms of integral and superforms in a new way which can be easily generalised to several models in different dimensions. The construction of supersymmetric actions based on the Hodge duality is presented and new supersymmetric actions with higher derivative terms are found. These terms are required by the invertibility of the Hodge operator.