On Zero Energy States in SUSY Quantum Mechanics on Manifolds

TL;DR

This paper investigates conditions under which a deformed Dirac operator on a closed manifold has no zero energy states, providing both sufficient criteria and explicit counterexamples.

Contribution

It establishes sufficient conditions for the positivity of the deformed Dirac operator and demonstrates that these are not necessary through explicit counterexamples.

Findings

Sufficient conditions for absence of zero modes are derived.

Counterexamples show conditions are not necessary.

Analysis of zero modes in SUSY quantum mechanics on manifolds.

Abstract

We study the zero modes of the operator $H_f=D^*_fD_f$, with a Dirac type operator $D_f$, acting on the spinor bundle over a closed even dimensional Riemannian manifold $M$. The operator $D_f=D+ifI$ is a deformation of the Dirac operator $D$ by a smooth function $f$. We obtain sufficient conditions on the deformation function that guarantee the positivity of the operator $H_f$, that is, the absence of zero modes. We also show that these conditions are not necessary and provide an explicit counterexample of a zero mode of the operator $H_f$.