Towards Noncommutative Topological Quantum Field Theory: Tangential Hodge-Witten cohomology

TL;DR

This paper introduces a novel approach to develop a Hodge theory for the tangential cohomology of foliations by adapting Witten's Morse theory perturbation method to the Laplacian.

Contribution

It proposes a new method to extend Hodge theory to foliations using perturbations inspired by Witten's approach, bridging Morse theory and tangential cohomology.

Findings

A new perturbation technique for tangential Laplacian

Establishment of a Hodge theory framework for foliations

Potential applications to noncommutative topological quantum field theory

Abstract

We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian