Quantum Mechanical Anomalies and the De Witt Effective Action

TL;DR

This paper investigates the partition function of supersymmetric quantum mechanics on Riemannian manifolds with a chemical potential, revealing anomalies in odd dimensions that affect the De Witt effective action.

Contribution

It provides an exact evaluation of the fermionic path integral and uncovers dimension-dependent anomalies in the De Witt term.

Findings

In even dimensions, the De Witt term has a definite numerical factor.

In odd dimensions, a quantum anomaly causes ambiguity in the De Witt term.

The limit $ o  o  $ reveals dimension-dependent behavior.

Abstract

We study the partition function of N=1 supersymmetric De Rham quantum mechanics on a Riemannian manifold, with a nontrivial chemical potential $\mu$ for the fermions. General arguments suggest that when $\mu \to \infty$ we should get the partition function of a free point particle. We investigate this limit by exact evaluation of the fermionic path integral. In even dimensions we find the De Witt term with a definite numerical factor. However, in odd dimensions our result is pestered by a quantum mechanical anomaly and the numerical factor in the De Witt term remains ambiquous.