The $\hat{A}$ genus as a projective volume form on the derived loop space

TL;DR

This paper extends previous work by representing the $$ genus as a projective volume form on the derived loop space, linking it to BV operators in a one-dimensional sigma model quantization.

Contribution

It introduces a novel interpretation of the $$ genus as a projective volume form within the formalism of $L_$ spaces, applicable to both smooth and complex geometries.

Findings

Realization of $$ genus as a projective volume form

Development of an associated integration/expectation map

Application to both smooth and complex geometric contexts

Abstract

In the present work, we extend our previous work with Gwilliam by realizing $\hat{A}(X)$ as the projective volume form associated to the BV operator in our quanitization of a one-dimensional sigma model. We also discuss the associated integration/expectation map. We work in the formalism of $L_\infty$ spaces, objects of which are computationally convenient presentations for derived stacks. Both smooth and complex geometry embed into $L_\infty$ spaces and we specialize our results in both of these cases.