Witten Deformation And Some Topics Relating To It

TL;DR

This paper provides an accessible overview of Witten deformation, its mathematical foundations, and various applications including proofs of classical theorems like Poincaré-Hopf, Morse inequalities, and Atiyah vanishing theorem.

Contribution

It offers a simplified exposition of Witten deformation and demonstrates its use in proving several fundamental theorems in differential geometry and topology.

Findings

Analytic proof of Poincaré-Hopf index theorem

Proof that Thom-Smale complex is quasi-isomorphic to de Rham complex

Analytic proof of Atiyah vanishing theorem

Abstract

This is a simple reading report of professor Weiping Zhang's lectures. In this article we will mainly introduce the basic ideas of Witten deformation, which were first introduced by Edward Witten on, and some applications of it. The first part of this article mainly focuses on deformation of Dirac operators and some important analytic facts about the deformed Dirac operators. In the second part of this article some applications of Witten deformation will be given, to be more specific, an analytic proof of Poincar$\acute{e}$-Hopf index theorem and Real Morse Inequilities will be given. Also we will use Witten deformation to prove that the Thom Smale complex is quasi-isomorphism to the de-Rham complex (Witten suggested that Thom Smale complex can be recovered from his deformation and his suggestion was first realized by Helffer and Sj$\ddot{o}$strand, the proof in this article is given by Bismut and Zhang). And in the last part an analytic proof of Atiyah vanishing theorem via Witten deformation will be given.