Localization and stationary phase approximation on supermanifolds

Given an odd vector field Q on a supermanifold M and a Q-invariant density mu on M, under certain compactness conditions on Q, the value of the integral integral(M) mu is determined by the value of mu on any neighborhood of the vanishing locus N of Q. We present a formula for the integral in the case where N is a subsupermanifold which is appropriately non-degenerate with respect to Q. In the process, we discuss the linear algebra necessary to express our result in a coordinate independentway. We also extend the stationary phase approximation and the Morse-Bott lemma to supermanifolds. Published by AIP Publishing.