Equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted by a Lie group G is a polynomial map

$\alpha :{\mathfrak {g}}\to \Omega ^{*}(M)$

from the Lie algebra ${\mathfrak {g}}=\operatorname {Lie} (G)$ to the space of differential forms on M that is equivariant; i.e.,

$\alpha (\operatorname {Ad} (g)X)=g\alpha (X).$

In other words, an equivariant differential form is an invariant element of $\mathbb {C} [{\mathfrak {g}}]\otimes \Omega ^{*}(M).$

For an equivariant differential form $\alpha$, the equivariant exterior derivative $d_{\mathfrak {g}}\alpha$ of $\alpha$ is defined by

$(d_{\mathfrak {g}}\alpha )(X)=d(\alpha (X))-i_{X^{\#}}(\alpha (X))$

where d is the usual exterior derivative and $i_{X^{\#}}$ is the interior product by the fundamental vector field generated by X. It is easy to see $d_{\mathfrak {g}}\circ d_{\mathfrak {g}}=0$ (use the fact the Lie derivative of $\alpha (X)$ along $X^{\#}$ is zero) and this makes the space of equivariant differential forms a complex. One can then put

$H_{G}^{*}(X)=\operatorname {ker} d_{\mathfrak {g}}/\operatorname {im} d_{\mathfrak {g}}$,

which is called the equivariant cohomology of M (This coincides with the ordinary equivariant cohomology defined in terms of Borel construction.) The definition is due to H. Cartan. The notion has an application to the equivariant index theory.

$d_{\mathfrak {g}}$-closed or $d_{\mathfrak {g}}$-exact forms are often called equivariantly closed or equivariantly exact.

The integral of an equivariantly closed form may be evaluated from its fixed point subspaces by means of the localization formula.

References

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