Perturbations of Equivariant Dirac Operators

Let M be a compact Riemannian manifold with an action by isometries of a compact Lie group G. Suppose that this action could be lifted to an action by isometries on a Clifford bundle E over M. We use the method of the Witten deformation to compute the virtual representation-valued index of a transversally elliptic Dirac operator on E. We express the multiplicities of the associated representation in terms of the local action of G near the singular set of the deformation. A complete answer is obtained when G = S. 1. The Dirac Operator Here we give an idea of what the standard Dirac operator is. Recall that the Laplacian in R is the second order differential operator ∆ = − ∂ 2 ∂x1 − ∂ 2 ∂x2 − ∂ 2 ∂x3 . This is an extremely useful operator in physics. Examples include the heat equation ∂u ∂t + ∆u = 0 and the wave equation ∂u ∂t2 −∆u = 0. Dirac wanted to find a square root for the Laplacian, for reasons of physics. Well, let’s see. It should be a 1st order differential operator. Oh, let’s just try to guess what it should be: call it D (for Dirac). Let’s try