The $\hat{\Gamma}$-genus and a regularization of an $S^1$-equivariant Euler class

TL;DR

This paper introduces the $\hat{ ext{Gamma}}$-genus, a new multiplicative genus derived from Atiyah and Witten's work, with connections to multiple zeta values and algebraic regularization methods.

Contribution

It proposes the $\hat{ ext{Gamma}}$-genus as a novel generalization of existing genera, providing explicit calculations and exploring its algebraic and geometric properties.

Findings

Defined the $\hat{ ext{Gamma}}$-genus and computed examples.

Established a link between the genus and multiple zeta values.

Highlighted properties and potential applications of the genus.

Abstract

We show that a new multiplicative genus, in the sense of Hirzebruch, can be obtained by generalizing a calculation due to Atiyah and Witten. We introduce this as the $\hat{\Gamma}$-genus, compute its value for some examples and highlight some of its interesting properties. We also indicate a connection with the study of multiple zeta values, which gives an algebraic interpretation for our proposed regularization procedure.