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In [[Differential geometry|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 point (mathematics)|fixed points]] of the element. If the element is neutral, then the theorem reduces to the usual index theorem.
The classical formula such as the [[Atiyah–Bott formula]] is a special case of the theorem.

== Statement ==
Let <math>\pi: E \to M</math> be a [[Clifford module bundle|clifford module bundle]]. Assume a compact Lie group ''G'' acts on both ''E'' and ''M'' so that <math>\pi</math> is equivariant. Let ''E'' be given a connection that is compatible with the action of ''G''. Finally, let ''D'' be a [[Physics:Dirac operator|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 [[Brauer's theorem on induced characters|virtual character]] given by taking the [[Supertrace|supertrace]]:
:<math>\operatorname{str}(g\mid\ker D) = \operatorname{tr}(g\mid\ker D^+) -  \operatorname{tr}(g\mid\ker D^-).</math>
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== See also ==
*Equivariant topological K-theory
*[[Kawasaki's Riemann–Roch formula]]

== References ==
*Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag

[[Category:Differential geometry]]



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