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1 Equivariant cohomology as a relative Lie algebra cohomology
2 Weyl model

Equivariant cohomology

1 Equivariant cohomology as a relative Lie algebra cohomology

To define the equivariant cohomology, we need to remember that is actually a differential Lie algebra. It has a differentiation . In our definition of the cohomology, we have not yet used . The way to use it is to consider a larger algebra , which is obtained from by adding as an extra generator.

The Koszul dual of has an additional generator which we call . The differential on (which comes from the non-homogeneity of ) acts as follows:



Notice that is zero. We will simply put . Then the differential formally coincides with the given by Eq. (3). This is called Weyl algebra (a commutative differential superalgebra).

Definition 1 We define the equivariant cohomology of as the relative Lie algebra cohomology:

The ordinary (not relative) cohomology of with coefficients in is computed as follows:

with the differential acting as follows:

where and and .

This is the usual BRST-like operator defining Lie algebra cohomology. We have visually separated the terms into two groups: the first three terms account for the action of on , and the rest for the structure constants of .

To define the relative cohomology, we restrict to the subspace of cochains which are:


and -invariant:   

Now our cochains to not depend on and are -invariant; the differential becomes:

This is the Cartan model of equivariant cohomology.

2 Weyl model

To pass to the Weyl model, we act on cochains by . This does not change the condition of invariance, but changes the condition of horizonthality:


and gets replaced with: