The n-Category Café

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.