Integration of Superforms and Super-Thom Class

TL;DR

This paper develops a method to construct the Thom class for supermanifolds, introduces new examples of supermanifolds, and discusses regularization techniques for singular fermionic Thom classes, with applications to Calabi-Yau spaces.

Contribution

It extends the Thom class construction to supermanifolds, provides explicit examples, and introduces regularization methods for singular cases.

Findings

Explicit computation of Thom class for CP^{(1|2)} with regularization

Introduction of superforms transforming as Berezinian

Discussion on embedding super-Calabi-Yau spaces

Abstract

We address the basic problem of constructing the Thom class for a supermanifold. Given a cohomological class of a supermanifold and the restriction of the supermanifold to its bosonic submanifold, the Thom class gives a prescription to define the integral over the bosonic submanifold in terms of the integral over the entire supermanifold. In addition, we provide some new interesting examples of supermanifolds obtained by extending a given bosonic manifold, we discuss the construction of superforms of special type (which transform as Berezinian under change of supercoordinates) and we define the de Rham cohomology. We review the construction of the Thom class in the conventional geometry and we discuss the extension to the supermanifolds. Then, we compute explicitly the Thom class for the case of CP^{(1|2)} and, as expected, the result is singular. We provide a regularization technique to handle the fermionic Thom class in practical applications. We conclude with some remarks about Calabi-Yau spaces and their embedding into super-CY.