Witten deformation using Lie groupoids

Abstract

We express Witten's deformation of Morse functions using deformation to the normal cone and C⁎-modules. This allows us to obtain asymptotics of the ‘large eigenvalues’. This is then applied to the case of Morse-Bott functions inspired by [2].Our methods extend to Morse functions along a foliation. We construct the Witten deformation using any generic function on an arbitrary foliation on a compact manifold and establish the compactness of its resolvent. When the foliation has a holonomy invariant transverse measure we show that our result implies Morse inequalities obtained by Connes and Fack [10] in a slightly more general situation.