Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Zero Energy States in SUSY Quantum Mechanics on Manifolds

Ivan G. Avramidi

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 $H_f$.

On Zero Energy States in SUSY Quantum Mechanics on Manifolds

Abstract

We study the zero modes of the operator , with a Dirac type operator , acting on the spinor bundle over a closed even dimensional Riemannian manifold . The operator is a deformation of the Dirac operator by a smooth function . We obtain sufficient conditions on the deformation function that guarantee the positivity of the operator , 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 .

Paper Structure

This paper contains 9 sections, 8 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Supersymmetric Quantum Mechanics
  3. Dirac Operator
  4. Deformed Dirac Operator $D_f{}{}$
  5. Two-dimensional Manifolds
  6. Sufficient Condition for Positivity
  7. Properties of the Zero Mode
  8. Example of a Zero Mode
  9. Conclusion

Key Result

Proposition 1

If one of the following conditions is valid for any $\varphi$ then there are no zero modes of the operator $H_f{}{}$ and the functional $S_f{}(\varphi)$ is strictly positive.

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 1
  • Proposition 5
  • Proposition 6
  • Proposition 7