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).
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:
| |||||||||
|
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: