# Witten deformation using Lie groupoids

## TL;DR

This paper develops a novel approach to Witten's deformation of Morse functions using Lie groupoids and $C^*$-modules, extending to foliations and providing asymptotic eigenvalue analysis and Morse inequalities.

## Contribution

It introduces a new framework for Witten deformation via deformation to the normal cone and Lie groupoids, generalizing previous results to foliations and invariant measures.

## Key findings

- Asymptotics of large eigenvalues derived
- Compactness of the resolvent established
- Extension of Morse inequalities to foliations with invariant measures

## Abstract

We express Witten's deformation of Morse functions using deformation to the normal cone and $C^*$-modules. This allows us to obtain asymptotics of the `large eigenvalues'. Our methods extend to Morse functions along a foliation. We construct the Witten deformation using any generic function on an arbitrary foliation on a compact manifold and establish the compactness of its resolvent. When the foliation has a holonomy invariant transverse measure we show that our result implies Morse inequalities obtained by Connes and Fack in a slightly more general situation.