Remarks on elementary integral calculus for supersmooth functions on superspace ${\mathfrak{R}}^{m|n}$

TL;DR

This paper develops an elementary calculus for supersmooth functions on superspace, extending Berezin integral and addressing discrepancies in change of variables crucial for supermanifold analysis.

Contribution

It introduces a modified contour integral for supersmooth functions on superspace with infinite Grassmann generators, resolving issues in change of variables.

Findings

Extended Berezin integral to superspace with infinite Grassmann generators

Modified contour integral to correct change of variables discrepancies

Facilitated analysis on supermanifolds and PDE applications

Abstract

After introducing Berezin integral for polynomials of odd variables, we develop the elementary integral calculus based on supersmooth functions on the superspace ${\mathfrak{R}}^{m|n}$. Here, ${\mathfrak{R}}$ is the Fr\'echet-Grassmann algebra with countably infinite Grassmann generators, which plays the role of real number field ${\mathbb{R}}$. As is well-known that the formula of change of variables under integral sign is indispensable not only to treat PDE applying funtional analytic method but also to introduce analysis on supermanifolds. But, if we define naively the integral for supersmooth functions, there exists discrepancy which should be ameliorated. Here, we extend the contour integral modifying the parameter space introduced basically by de Witt, Rogers and Vladimirov and Volovich