Equivariant index theorem

In differential geometry, the equivariant index theorem, of which there are several variants, computes the (graded) trace of an element of a compact Lie group acting in given setting in terms of the integral over the fixed points of the element.

Assume a compact Lie group G acts on both E and M so that

Let E be given a connection that is compatible with the action of G. Finally, let D be a Dirac operator on E associated to the given data.

In particular, D commutes with G and thus the kernel of D is a finite-dimensional representation of G. The equivariant index of E is a virtual character given by taking the supertrace:

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