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. If the element is neutral, then the theorem reduces to the usual index theorem. The classical formula such as the

Atiyah–Bott formula In algebraic geometry, the Atiyah–Bott formula says the cohomology ring :\operatorname^*(\operatorname_G(X), \mathbb_l) of the moduli stack of principal bundles is a free graded-commutative algebra on certain homogeneous generators. The orig ...

is a special case of the theorem.

Statement

Let

\pi: E \to M

be a

clifford module bundle In differential geometry, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a vector bundle whose fibers are Clifford modules, the representations of Clifford algebras. The canonical example is a spinor bundle. In ...

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

\pi

equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...

. Let ''E'' be given a connection that is compatible with the action of ''G''. Finally, let ''D'' be a

Dirac operator In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise forma ...

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 In the theory of superalgebras, if ''A'' is a commutative superalgebra, ''V'' is a free right ''A''- supermodule and ''T'' is an endomorphism from ''V'' to itself, then the supertrace of ''T'', str(''T'') is defined by the following trace diagram: ...

: :

\operatorname(g\mid\ker D) = \operatorname(g\mid\ker D^+) -  \operatorname(g\mid\ker D^-).

References

*Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag Differential geometry {{differential-geometry-stub

Statement

See also

References