### Equivariant deRham Cohomology and Action Lie Algebroids

#### Posted by Urs Schreiber

The trimester program Geometry and Physics at the Hausdorff Institute in Bonnn is over. After a concentration of official activity around July it was getting a bit more relaxed again in August and we could pick up again our “internal seminar” among the program participants. Maybe a bit too relaxed: in the last weeks this internal seminar was attended just by Alejandro Cabrera and myself. But a 2-body seminar can be very useful.

I had mentioned a seminar talk by Alejandro on BV Poisson reduction recently here. Now in the internal seminar Alejandro was looking into the general mechanism of *localization* of integrals over graded spaces following

Richard Szabo
*Equivariant Localization of Path Integrals*

arXiv:hep-th/9608068

We talked about the relation of this to BV-BRST quantization. Maybe some of the things we discussed there are too top secret to be shared here for the moment, but I can share the following useful literature that Alejandro tracked down, on the relation between equivariant deRham cohomology and action Lie algebroids, and an interpretation of this in the context of rational approximations to universal $\infty$-bundles in the sense of *$L_\infty$-connections* (blog, arXiv).

Given the action of a compact connected Lie group $G$ on a manifold $X$, there are several closely related but different differential algebra models for the *$G$-equivariant deRham cohomology* of $X$: notably there is the ancient Cartan-Weil model and then more recently there is what is called the *BRST model*.

It was shown in

V. Mathai, D. Quillen
*Superconnections, Thom classes, and equivariant differential forms*

Topology, 25(1):85-110, 1986

that the underlying complexes of these two models are isomorphic. Now, both these complexes are obtained as sub-complexes of bigger complexes. In

J. Kallkman
*BRST model for equivariant cohomology and representatives for the equivariant Thom class*

Comm. Math. Phys., 153(3):447-463, 1993

it is shown that also these bigger complexes are isomorphic.

When Alejandro showed me this bigger complex I said: “hey, this looks like the Weil algebra of the action Lie algebroid of the action of $G$ on $X$”.

Alejandro quickly demonstrated that this is indeed the case and shortly afterwards also came up with a reference where, up to some inessential difference in language, precisely this statement is demonstrated:

Rajan Mehta
*$Q$-Algebroids and their cohomology*

arXiv:math/0703234

Recall what this statement about the Weil algebra of the action Lie algebroid means (I am following the notation and terminology from $L_\infty$-connections :

an $L_\infty$-algebroid with space of objects the smooth space $X$ is a non-positively graded cochain complex $g$ of $(A := C^\infty(X))$-modules equipped with a degree +1 derivation $d_g : \wedge^\bullet_A g^* \to \wedge^\bullet_A g^*$ covering the differential on $g^*$ and squaring to zero.

The “quasi free” (namely free just as a graded commutative algebra over $A$) differential graded commutative algebra
$CE_A(g) := (\wedge^\bullet_A g^*, d_g)$ is the *Chevalley-Eilenberg algebra* of the $L_\infty$-algebroid $(A,g)$.

The *Weil algebra* of the $L_\infty$-algebroid, obtained by throwing in another but shifted copy of $g$ as well as the “locally shifted copy” $\Gamma(T X)[-1]$ of $C^ \infty(X)$ is
$W(g) := (\wedge^\bullet ( \Gamma(T X)^* \oplus g^* \oplus g^*[1] ),
\;
d_{W(g)} =
\left(
\array{
d_g & 0
\\
\sigma & -\sigma \circ d_g \circ \sigma
}
\right)
)
\,,$
where the matrix on the right is supposed to indicate how the differential $d_{W(g)}$ acts on the original generators in $C^\infty(X) \oplus g^*[1]$ and their shifted copies $\Gamma(T X)^* \oplus g^*[2]$. Here $\sigma$ denotes the degree +1 derivation which restricts to the canonical isomorphism $g^*[1] \to g^*[2]$ on $g^*[1]$ and to the deRham differential on $C^\infty(X)$ and to zero on the shifted generators.

Action Lie algebroids provide examples for this: for $g$ an ordinary Lie algebra acting on an ordinary manifold $X$ by means of a Lie algebra morphism $\rho : g \to \Gamma(T X)$, the action Lie algebroid $Lie(X//G)$ comes from the free $C^\infty(X)$-module generated by $g$ concentrated in degree 0, with the differential being the dual of $\rho$ on $C^\infty(X)$ and the ordinary Chevalley-Eilenberg differential on $g^*$.

As Alejandro saw, it is a quick computation using Cartan’s formula for the Lie derivative on differential forms to show that the Weil algebra $W(Lie(X//G))$ in the above sense is precisely the Kalkman/BRST differential that appears as equation (21) in Mehta’s article.

Notice that Mehta follows common fashion and thinks of all things $L_\infty$ in terms of $\mathbb{Z}$-graded supergeometry. I tend to want to not do that, as the Lie-theoretic imagery seems to me to be paramount. But in the end it is just an inessential matter of language and a straightforward exercise to check that Mehta’s shifted tangent $Q$-algebroid $[-1]T g$ in his section 5.2 corresponds to the $W(g)$ above.

Finally a word on the original issue of models for equivariant deRham cohomology and the relavent sub-complexes of the Weil algebra complex:

as recalled and discussed in $L_\infty$-connections

the Chevalley-Eilenberg algebra $CE(g)$ plays the role of differential forms on the $\infty$-group $G$ integrating $g$

the Weil algebra $W(g)$ plays the role of differential forms on the universal $G$-bundle

the canonical morphism $CE(g) \stackrel{i^*}{\leftarrow} W(g)$ plays the role of the dual to the fiber injection $G \stackrel{i}{\to} E G \to B G$.

This defines yet another qDGCA, namely $W(g)_basic =: inv(g) \subset W(g)$, the algebra of *basic forms* or *invariant polynomials* on $g$ as the subalgebra of $W(g)$ which is invariant under those inner derivations on $W(g)$ that cover inner derivations on $CE(g)$.

the algebra $W(g)_{basic} = inv(g) \subset W(g)$ plays the role of differential forms on $B G$.

And $inv(Lie(X//G))$ is the Kalkman/BRST model of $G$-equivariant deRham cohomology on $X$.

To see this, notice that the only inner derivations $[d_g, \iota_t]$ on the CE-algebra $\mathrm{CE}(Lie(X//G))$ of the action Lie algebroid $Lie(X//G)$ are those coming from contractions $\iota_t$ with elements $t$ in the Lie algebra $g$. Hence so are the inner derivations $[d_{W(g)}, \iota_t]$ of the Weil algebra $W(Lie(X//G))$ covering these. So an element of $W(Lie(X//G))$ is in $inv(Lie(X//G))$ precisely if it is annihilated by all the contractions $\iota_t$ and all the inner derivations $[d_{W(g)}, \iota_t]$ for all $t \in g$. This subalgebra is usually denoted $inv(Lie(X//G)) = (\Omega(X) \otimes S g^*)^G$ and with the induced differential this is indeed the Cartan model for equivariant deRham cohomology. See for instance page 19 of Mehta’s article.

Finally, I should add a more general word on Mehta’s article. As mentioned here he is interested in groupoids internal to “$Q$-manifolds”. In the language I am using a “$Q$-manifold” is the geometric interpretation of the dual of the Chevalley-Eilenberg algebra of an $L_\infty$-algebroid. Hence these are groupoids internal to $L_\infty$-algebroids.

Such internal groupoids correspond of course to Lie algebroids internal to $L_\infty$-algebroids. Mehta calls these “$Q$-algebroids” and this is what his article is concerned with. The Weil algebra $W(g)$ of an $L_\infty$-algebroid is itself a qDGCA and hence itself a Chevalley-Eilenberg algebra of an $L_\infty$-algebroid which in $L_\infty$-connections is denoted $inn(g)$: the $L_\infty$-algebroid of inner derivations of $g$. From Mehta’s point of view this is a special case of a $Q$-algebroid, corresponding to the *shifted tangent bundle* of the corresponding “$Q$-manifold”.

$n$-Café regulars will recall our extensive discussion of the relation between tangents, inner derivations and universal bundles, for instance from More on tangent categories.

## Re: Equivariant deRham Cohomology and Action Lie Algebroids

Just 2 quick questions (probably obvious for those in the know):

What is BV reduction?

Berline-Vergne?

Any relation of this with Nekrasov’s approach to instanton counting?