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Hypoellipticity of the Asymptotic Bismut Superconnection on Contact Manifolds

Jesus Sanchez, Andres Franco Valiente

Abstract

Given a contact sub-Riemannian manifold one obtains a non-integrable splitting of the tangent bundle into the directions along the contact distribution and the Reeb field. We generalize the construction of the Bismut superconnection to this non-integrable setting and show that although singularities appear within the superconnection, if one extracts the finite part then the resulting operator is hypoelliptic. We find that the hypoellipticity also holds in the setting of two-step subRiemannian manifolds and produce a modification for arbitrary subRiemannian manifolds which always gives a hypoelliptic operator. A discussion of the explicit form of the operator on principal $\bS^1$-bundles is provided. The index theory is worked out on contact manifolds and a matrix twisting of the Clifford relations produces operators with non-trivial Fredholm index. We conclude with a possible relationship between our hypoelliptic operator and a constrained supersymmetric sigma model.

Hypoellipticity of the Asymptotic Bismut Superconnection on Contact Manifolds

Abstract

Given a contact sub-Riemannian manifold one obtains a non-integrable splitting of the tangent bundle into the directions along the contact distribution and the Reeb field. We generalize the construction of the Bismut superconnection to this non-integrable setting and show that although singularities appear within the superconnection, if one extracts the finite part then the resulting operator is hypoelliptic. We find that the hypoellipticity also holds in the setting of two-step subRiemannian manifolds and produce a modification for arbitrary subRiemannian manifolds which always gives a hypoelliptic operator. A discussion of the explicit form of the operator on principal -bundles is provided. The index theory is worked out on contact manifolds and a matrix twisting of the Clifford relations produces operators with non-trivial Fredholm index. We conclude with a possible relationship between our hypoelliptic operator and a constrained supersymmetric sigma model.

Paper Structure

This paper contains 18 sections, 30 theorems, 372 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Structure of the paper
  4. Construction of the Asymptotic Bismut Superconnection
  5. The Asymptotic Bismut Superconnection
  6. The Clifford Contracted Curvature
  7. The Case of a Contact Manifold
  8. The Equivariant Setting
  9. The Heisenberg Calculus and Hypoellipticity
  10. The Rockland Condition
  11. Hypoellipticity on Contact Manifolds
  12. A subtlety within the choice of connection
  13. The Graded Rockland Condition
  14. The General Case
  15. An Example: Principal $\mathbb{S}^1$-bundles With Non-Vanishing Curvature
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that $\left( M,\theta,g_{\mathcal{F}}\right)$ is a contact sub-Riemannian manifold. Then for an appropriate choice of $\mathbb{Z}_2$-graded covariant Hermitian Clifford module the operator is hypoelliptic. If $M$ is closed, then $\textnormal{Ker}\mathbb{D}^{E}$ is a finite dimensional space of smooth sections.

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 1.1
  • proof
  • Proposition 1.2
  • Remark 1.3
  • ...and 40 more