On the Witten Rigidity Theorem for Odd Dimensional Manifolds

TL;DR

This paper proves new rigidity and vanishing theorems for twisted Toeplitz operators on odd-dimensional manifolds, highlighting the role of the fundamental group and employing modular techniques and odd K-theory analysis.

Contribution

It introduces novel Witten type rigidity and vanishing theorems for odd-dimensional manifolds using modular methods and odd K-theory, emphasizing the fundamental group's influence.

Findings

Established rigidity theorems for twisted Toeplitz operators on odd manifolds.

Identified the fundamental group's role in odd-dimensional rigidity.

Applied modular transgression and odd Chern class analysis.

Abstract

We establish several Witten type rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds. We obtain our results by combining the modular method, modular transgression and some careful analysis of odd Chern classes for cocycles in odd $K$-theory. Moreover we discover that in odd dimensions, the fundamental group of manifolds plays an important role in the rigidity.