A Weitzenb\"ock formula on Sasakian holomorphic bundles

Abstract

This work seeks to advance the understanding of the smooth structure of the moduli space of self-dual contact instantons (SDCI) on Sasakian 7-manifolds M. A neighborhood of a smooth point of M is locally modeled on the first cohomological group of an elliptic complex (1.4). There is a cohomological obstruction to the smoothness for the moduli space, in terms of a second basic cohomological group, in this paper we study conditions under which this obstruction disappears, by computing a Weitzenb\"ock formula and using a Bochner-type method to obtain a vanishing theorem. Given an SDCI on a Sasakian bundle E, we find sufficient conditions for the vanishing of the obstruction in the positivity of a couple of operators R and F depending on the curvatures of the connection and the Riemann curvature of the Sasakian metric g. In particular, we find that if M is transversely Ricci positive and F positive, the moduli space of SDCI must be smooth. However, in general, the operator F is not positive definite and we describe bundles over the Stiefel manifold for which it is the case. Finally, we show that when the energy of the curvature is less than the first non-zero eigenvalue of RicT the obstruction vanishes.