de Rham cohomology of SO(n) and some related manifolds by supersymmetric quantum mechanics

TL;DR

This paper uses supersymmetric quantum mechanics to explicitly compute the de Rham cohomology of certain manifolds, confirming theoretical predictions through concrete analysis of vacuums and symmetries.

Contribution

It provides a detailed, concrete verification of Witten's Morse theory for specific manifolds using supersymmetric quantum mechanics, including symmetry-based vacuum selection.

Findings

Number of vacuums matches de Rham cohomology

Reflection symmetry effectively selects true vacuums

Confirmed instanton picture of cohomology

Abstract

We study supersymmetric quantum mechanics on RP_{n},SO(n),G_{2} and U(2) to examine Witten's Morse theory concretely. We confirm the simple instanton picture of the de Rham cohomology that has been given in a previous paper. We use a reflection symmetry of each theory to select the true vacuums. The number of selected vacuums agrees with the de Rham cohomology for each of the above manifolds.