Hypoellipticity of the Asymptotic Bismut Superconnection

We generalize the construction of the Bismut superconnection to the non-integrable setting of subRiemannian manifolds equipped with a transverse distribution. The non-integrability produces singularities within the superconnection, and it is shown that if one extracts the finite part then the resulting operator is hypoelliptic on two-step subRiemannian manifolds. Moreover, it is shown that a perturbation produces a hypoelliptic operator on any subRiemannian manifold. The explicit form of the operator on principal circle bundles is computed and its kernel is determined for principal circle bundles over the [Formula: see text]-sphere with conformal curvature. A discussion of possible directions for generalizations and open questions follows; in the direction of index theory we introduce matrix twisted Clifford relations in the transverse directions to produce asymptotic superconnections with non-vanishing Fredholm index. We conclude with a possible relationship between our hypoelliptic operator and a constrained supersymmetric sigma model.