Partition functions and Jacobi fields in the Morse theory

TL;DR

This paper explores the semiclassical partition function within Morse theory, linking phase factors to the eta invariant and utilizing Jacobi fields and geodesics on curved manifolds to deepen understanding of quantum systems.

Contribution

It introduces a novel approach connecting Morse theory, Jacobi fields, and the eta invariant to analyze the phase factors of partition functions in quantum systems.

Findings

Clarifies the phase factor of the partition function.

Relates the eta invariant to physical systems via Morse theory.

Uses geodesic and Hamilton-Jacobi frameworks to analyze eigenvalues.

Abstract

We study the semiclassical partition function in the frame work of the Morse theory, to clarify the phase factor of the partition function and to relate it to the eta invariant of Atiyah. Converting physical system with potential into a curved manifold, we exploit the Jacobi fields and their corresponding eigenvalue equations to be associated with geodesics on the curved manifold and the Hamilton-Jacobi theory.