Symplectic Morse Theory and Witten Deformation

TL;DR

This paper develops a Morse-type complex on symplectic manifolds, using Witten deformation to show its cohomology matches TTY cohomology, providing new inequalities relating critical points and symplectic structure.

Contribution

It introduces a novel Morse-type complex on symplectic manifolds and proves its cohomology is invariant and isomorphic to TTY cohomology, extending Morse theory to symplectic geometry.

Findings

Cohomology of the complex is independent of metric and Morse function.

The complex's cohomology is isomorphic to TTY cohomology.

Derived Morse-type inequalities relating critical points and symplectic structure.

Abstract

On symplectic manifolds, we introduce a Morse-type complex with elements generated by pairs of critical points of a Morse function. The differential of the complex consists of gradient flows and an integration of the symplectic structure over spaces of gradient flow lines. Using the Witten deformation method, we prove that the cohomology of this complex is independent of both the Riemannian metric and the Morse function used to define the complex and is in fact isomorphic to the cohomology of differential forms of Tsai, Tseng and Yau (TTY). We also obtain Morse-type inequalities that bound the dimensions of the TTY cohomologies by the number of Morse critical points and the interaction of symplectic structure with the critical points.