Distributions and Integration in superspace

TL;DR

This paper extends the theory of distributions in superspace to define integrals over more general domains and surfaces, providing new formulas and examples that unify previous approaches in supersymmetric analysis.

Contribution

It introduces a generalized distributional framework for integration over arbitrary domains and surfaces in superspace, independent of phase function choice, and presents a new unified Cauchy-Pompeiu formula.

Findings

Defined domain and surface integrals via superdistributions

Proved independence from phase function choice

Derived a new distributional Cauchy-Pompeiu formula

Abstract

Distributions in superspace constitute a very useful tool for establishing an integration theory. In particular, distributions have been used to obtain a suitable extension of the Cauchy formula to superspace and to define integration over the superball and the supersphere through the Heaviside and Dirac distributions, respectively. In this paper, we extend the distributional approach to integration over more general domains and surfaces in superspace. The notions of domain and surface in superspace are defined by smooth bosonic phase functions $g$. This allows to define domain integrals and oriented (as well as non-oriented) surface integrals in terms of the Heaviside and Dirac distributions of the superfunction $g$. It will be shown that the presented definition for the integrals does not depend on the choice of the phase function $g$ defining the corresponding domain or surface. In addition, some examples of integration over a super-paraboloid and a super-hyperboloid will be presented. Finally, a new distributional Cauchy-Pompeiu formula will be obtained, which generalizes and unifies the previously known approaches.