Quasiclassical expansion for Tr{(-1)^F exp(-\beta H)}

TL;DR

This paper develops a method to compute small-eta corrections to the fermion-graded partition function in supersymmetric quantum systems, revealing unexpected results like vanishing corrections and breakdown of quasiclassical expansion.

Contribution

It introduces a systematic approach for calculating quasiclassical corrections to the fermion-graded partition function in supersymmetric quantum mechanics, including specific analyses of supersymmetric Yang-Mills and sigma models.

Findings

Correction of order   vanishes in supersymmetric Yang-Mills quantum mechanics.

Quasiclassical expansion breaks down for certain conformally flat sigma models on S^3.

Method provides insights into quantum corrections in supersymmetric systems.

Abstract

We start with some methodic remarks referring to purely bosonic quantum systems and then explain how corrections to the leading--order quasiclassical result for the fermion--graded partition function Tr{(-1)^F exp(-\beta H)} can be calculated at small \beta. We perform such calculation for certain supersymmetric quantum mechanical systems where such corrections are expected to appear. We consider in particular supersymmetric Yang-Mills theory reduced to (0+1) dimensions and were surprised to find that the correction of order \beta^2 vanishes in this case. We discuss also a nonstandard N =2 supersymmetric sigma model defined on S^3 and other 3--dimensional conformally flat manifolds and show that the quasiclassical expansion breaks down for this system.