# pullback of Lie algebra cocycles along Cartan connections

For $H \hookrightarrow G$ an inclusion of Lie groups, with $\mathfrak{h} \hookrightarrow \mathfrak{g}$ the corresponding inclusion of Lie algebras, an $(H\hookrightarrow G)$-Cartan connection on a smooth manifold $X$ is a $G$-principal connection $\nabla$ on $X$ equipped with a reduction of its structure group to $H$ and such that at each point $x \in X$ the composite (for any local trivialization)

$$\phi_x : T_x X \stackrel{\nabla}{\longrightarrow} \mathfrak{g} \longrightarrow \mathfrak{g}/\mathfrak{h}$$

is a linear isomorphism.

Now suppose that the inclusion is normal so that $\mathfrak{k} := \mathfrak{g}/\mathfrak{h}$ is a Lie algebra. Then for $\mu \in \mathrm{CE}^{n+1}(\mathfrak{k})$ an $H$-invariant Lie algebra $(n+1)$-cocycle on $\mathfrak{k}$ (hence an $L_\infty$-map $\mu : \mathfrak{k} \to \mathbb{R}[n]$), pullback of $\mu$ along $\phi$ produces a differential $(n+1)$-form $\phi^\ast \mu \in \Omega^{n+1}(X)$.

In the special case that $X = G/H$ and $\nabla$ is the Maurer-Cartan form, then this differential form is just the left-invariant extension of the cocycle over the Lie group, and is closed.

For general $X$ the form $\phi^\ast \mu$ is not in general closed. Asking it to be closed is akin to famous integrability conditions for differential form data, say in the definition of $G_2$-manifolds.

I would like to know if this concept of pulling back Lie algebra cocycles along Cartan connections this way and asking the result to be closed has been considered in generality, and under which name.

Moreover, what I am really interested in is the generalization of this condition to the case that $\mathfrak{g}$ is an $L_\infty$-algebra, $\nabla$ a higher principal connection and $X$ a suitable smooth higher (derived) stack. For this case, has anything vaguely similar to the above general condition been considered and given a name?

• I don't think that this has been studied systematically, but there is a lot of invariant calculus on Cartan geometries avaialble to study the question. Are you aware of intereting examples in which $\mathfrak g/\mathfrak h$ is canonically a Lie algebra (i.e. with $\mathfrak h$-equivariant bracket)? I don't know a nice example of this sitatuion. – Andreas Cap Dec 26 '14 at 10:18
• Thanks for your comment. My main motivating class of examples is that where $\mathfrak{g}$ is an extended super-Poincare Lie algebra and $\mathfrak{h}$ is the Lie algebra of the Lorentz group. Then $\mathfrak{g}/\mathfrak{h}$ is extended Minkowski-spacetime regarded as a super-translation Lie algebra. There are a finite number of exceptional super-Lie algebra cocycles on these, and in supergravity theory it is of key interest to prolong these to closed forms over a curved superspacetime, hence over an $(\mathfrak{h} \hookrightarrow \mathfrak{g})$-Cartan geometry. – Urs Schreiber Dec 27 '14 at 14:03
• I am not really familiar with the super-setting, but this sounds like a super-version of conformal geometry (which I am very familiar with). It seems to me that to do what you intend, you would need an action of $\mathfrak h$ on the space $V$ in which the cocycle has values in such a way that the action of $\mathfrak g/\mathfrak h\times V\to V$ is $\mathfrak h$-equivariant. I am not aware of examples of this in the usual conformal setting, but would be very intersted to get references for the cases you mention. – Andreas Cap Dec 28 '14 at 11:07
• Literature on the cocycles that I am thinking of is listed here: ncatlab.org/nlab/show/… Discussion of the need to prolong these to closed forms on curved superspace is listed here: ncatlab.org/nlab/show/… I am preparing some notes on this extension problem. Will send you more once its ready. – Urs Schreiber Dec 28 '14 at 17:47

The closest to this I have ever come is the work of Calderbank and Diemer (http://arxiv.org/abs/math/0001158) from the realm of Cartan geometries modelled on a pair $(G,P)$ for a parabolic subgroup $P$. They start with finite-dimensional irreducible $\mathfrak{g}$-representation $\mathbb{V}$ and construct a homotopy transfer from the twisted deRham sequence (it is in general no longer a complex: $(d^\nabla)^2 s= R\cdot s$) on exterior forms with values in the bundle associated to $\mathbb{V}$ to bundles associated to Lie algebra homology of the nilradical of $\mathfrak{p}$ with values in $\mathbb{V}$. In this way some curved infinity structures arise on the sections of homology bundles, whose operators are actually multi-differential operators. These structures have not been much exploited yet. The good thing is that one always has some control over the kernel of these operators between sections of homology bundles. E.g., there is an injection from the $\nabla$ parallel sections into the kernel of the operator acting on sections of the zeroth homology bundle with values in sections of the first homology bundle. In general, forms with values in $\mathcal{V}$ that are closed give rise to some special solutions of "geometric" differential equations.