Hypoellipticity of the Asymptotic Bismut Superconnection on Contact Manifolds

TL;DR

This paper extends the Bismut superconnection to contact and sub-Riemannian manifolds, demonstrating that a finite part of the resulting operator is hypoelliptic, with implications for index theory and potential links to supersymmetric models.

Contribution

It generalizes the Bismut superconnection to non-integrable contact and sub-Riemannian manifolds, establishing hypoellipticity of the finite part of the operator and exploring its index theory.

Findings

Finite part of the superconnection operator is hypoelliptic.

Hypoellipticity holds for two-step sub-Riemannian manifolds.

Explicit form of the operator on principal S^1-bundles is provided.

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.