Witten instanton complex and Morse-Bott inequalities on stratified pseudomanifolds

TL;DR

This paper develops Witten instanton complexes on stratified pseudomanifolds with wedge metrics, generalizing Morse-Bott inequalities to singular spaces with non-isolated critical points, and provides tools for computing local cohomology and Morse polynomials.

Contribution

It introduces a novel construction of Witten instanton complexes on stratified pseudomanifolds, extending Morse-Bott theory to singular settings with non-isolated critical points.

Findings

Constructed Witten instanton complexes for all mezzo-perversities.

Proved Morse inequalities in the stratified pseudomanifold setting.

Provided methods for computing local cohomology groups and Morse polynomials.

Abstract

In this paper we construct Witten instanton complexes on stratified pseudomanifolds with wedge metrics, for all choices of mezzo-perversities which classify the self-adjoint extensions of the Hodge Dirac operator. In this singular setting we introduce a generalization of the Morse-Bott condition and in so doing can consider a class of functions with certain non-isolated critical point sets which arise naturally in many examples. This construction of the instanton complex extends the Morse polynomial to this setting from which we prove the corresponding Morse inequalities. This work proceeds by constructing Hilbert complexes and normal cohomology complexes, including those corresponding to the Witten deformed complexes for such critical point sets and all mezzo-perversities; these in turn are used to express local Morse polynomials as polynomial trace formulas over their cohomology groups. Under a technical assumption of `flatness' on our Morse-Bott functions we then construct the instanton complex by extending the local harmonic forms to global quasimodes. We also study the Poincar\'e dual complexes and in the case of self-dual complexes extract refined Morse inequalities generalizing those in the smooth setting. We end with a guide for computing local cohomology groups and Morse polynomials.