Witten Genus and Elliptic genera for proper actions

TL;DR

This paper extends the concepts of Witten and elliptic genera to noncompact manifolds with proper group actions, establishing new vanishing and rigidity theorems and computing examples.

Contribution

It introduces the first construction of Witten and elliptic genera for noncompact manifolds with proper cocompact group actions, generalizing known results.

Findings

Vanishing theorems for the constructed genera

Rigidity results under proper group actions

Explicit computations for specific examples

Abstract

In this paper, we construct for the first time, the Witten genus and elliptic genera on noncompact manifolds with a proper cocompact action by an almost connected Lie group and prove vanishing and rigidity results that generalise known results for compact group actions on compact manifolds. We also compute our genera for some interesting examples, and study the non-spinc case in an appendix.