Bar codes of persistent cohomology and Arrhenius law for p-forms

TL;DR

This paper links persistent cohomology bar codes to the asymptotic behavior of small eigenvalues of the Witten Laplacian, showing stability and providing a method for precise eigenvalue estimates without Morse function assumptions.

Contribution

It establishes a connection between persistent cohomology bar codes and eigenvalues of the Witten Laplacian, extending previous results beyond Morse functions.

Findings

Small eigenvalues are determined by bar code lengths.

Quantities are stable under C0 perturbations.

Provides a method for subexponential correction calculations.

Abstract

This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the function f. In particular, this proves that these quantities are stable in the C 0 topology on the space of functions. Additionally, our analysis provides a general method for computing the subexponential corrections in a large number of cases.