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 then discuss the explicit form of the operator on principal $\bS^1$-bundles and discuss possible directions for generalizations and open questions; in particular, a possible relationship between our hypoelliptic operator and a constrained supersymmetric sigma model.