Yesterday I had filled in some minimum regarding the idea of rheonomy at *D’Auria-Fre formulation of supergravity*. (I thought I had done this long ago, but found the section empty now. )

There is something immensely curious going on here, which I was wanting to formalize ever since I started looking into this. Now maybe I am getting closer, but I am still stuck with something:

So the simple beautiful idea of rheonomy is that differential forms on a supermanifold $X$ are constrained to be something like “holomorphic” in the super-direction, in that they are fully determined by their restriction along the inclusion $\Re X \longrightarrow X$ of the underlying ordinary manifold, in (vague? or good?) analogy with how holomorphic functions on $\mathbb{C}$ are determined by their restriction along $\mathbb{R} \hookrightarrow \mathbb{C}$. This rheonomy constraint turns out to be equivalent to the more popular “superspace constraints” that are used elsewhere in the SuGra literature, but is evidently conceptually a much nicer perspective.

The striking claim then is that the equations of motion of supergravity theories enode precisely nothing but the constraint on a higher super-Cartan geometry on $X$, modeled on a given extended super-Minkowski spacetime, to have higher super-vielbein fields $E$ which, as super-differential forms on $X$, are rheonomic.

So the statement is something like that a solution to supergravity is nothing but a certain $G$-structure satisfying a “holomorphicity”-like constraint.

Apart from being beautiful and remarkable in itself, this smells like it has a good chance of having an “elementary” formalization in differential cohesion. That’s what I am after. I know how to naturally say “higher super-Cartan geometry” axiomatically in differential cohesion, but I don’t know yet how to say “rheonomy” in this way.

In fact I am pretty much in the dark about it at the moment, but from the above there are some evident guesses as to what one has to consider.

So given some manifold $X$ modeled on a framed space $V$, such as some extended super-Minkowski spacetime, then we are simply looking at an orthogonal structure exhibited by a diagram of the form

$\array{ X && \longrightarrow && \mathbf{B} O(V) \\ & \searrow &\swArrow_{E}& \swarrow \\ && \mathbf{B}GL(V) }$where the homotopy $E$ is (for some value of “is”) the vielbein, i.e. in the running example it is the super-vielbein together with the relevant higher form fields.

Now we may restrict this Cartan geometry to the underlying ordinary (reduced) manifold, simply by precomposing with the unit of the reduction modality

$\left( \array{ \Re X && \longrightarrow && \mathbf{B} O(V) \\ & \searrow &\swArrow_{\Re E}& \swarrow \\ && \mathbf{B}GL(V) } \right) \;\;\;\; \coloneqq \;\;\;\; \left( \array{ \Re X &\longrightarrow& X && \longrightarrow && \mathbf{B} O(V) \\ && & \searrow &\swArrow_{E}& \swarrow \\ && && \mathbf{B}GL(V) } \right)$So rheonomy is supposed to be some constraint on $E$ that makes it be fully determined by its restriction $\Re E$.

When one expresses $E$ in terms of actual differential forms with values in a super-$L_\infty$-algebra, then the constraint simply says that the curvature forms of this super $L_\infty$-algebra valued differential form are such that their components with incdices in directions perpendicular to $\Re X$ in $X$ are linear combinations of the components with all indices parallel to $X$.

So if I allowed myself to speak of components of $L_\infty$-algebra valued differential forms I’d be done. But I am suspecting that there is a more fundamental way to express what’s going on here, in terms of some general abstract differential cohesion yoga applied to the above diagrams.

And it looks like some kind of formal étaleness condition, or maybe formal smoothness condition on $E$. Hm…

]]>Am working on the entry *higher Cartan geometry*. Started writing a *Motivation* section.

This is just the first go, need to quit now, will polish tomorrow.

]]>A thought:

With $\mathbf{L}_{WZW}$ a higher connection on some group $V$, then there is the elementary concept (in differential cohesive HoTT) of a first-order integrable definite globalization $\mathbf{L}_{WZW}^X$ of $\mathbf{L}_{WZW}$ over some $V$-manifold $X$. This comes with an $\infty$-functor from such definite globalizations to $\mathrm{Stab}(\mathbf{L}_{WZW}^{inf})$-structures, on $X$, for $\mathbf{L}_{\mathrm{WZW}}^{inf}$ the restriction to the infinitesimal disk $\mathbb{D}^V \to V$.

When realized in the supergeometric model and with $V = \mathbb{R}^{10,1|N=1}$ and $\mathbf{L}_{\mathrm{WZW}} = \mathbf{L}_{M2}$ the GS-WZW term of the M2-brane, then this $\infty$-functor lands in the solutions of 11d supergravity with vanishing gravitino field strength and equipped with a genuine globalization of the M2 WZW term. So this means we may take the space of these definite globalizations as being the actual phase space of 11d SuGra. (Details are still here.)

An interesting question to ask then is which of the symmetries of $\mathbf{L}_{WZW}^{inf}$ covering $V$ carry over to $\mathbf{L}_{WZW}^X$, i.e. which common subgroups $Q$ there are of $Heis(\mathbf{L}_{WZW}) \coloneqq Stab_{V}(\mathbf{L}_{WZW})$ and $QuantMorph(\mathbf{L}_{\mathrm{WZW}}^X) \coloneqq Stab_{Aut(X)}(\mathbf{L}_{WZW}^X)$. In the above situation this means asking for *BPS states* of 11-d supergravity, namely spacetimes $X$ equipped with field configurations satisfying the SuGra equations of motion such that there are super-Killing vectors.

added to *Poisson bracket Lie n-algebra* the two definitions we have and the statement of their equivalence.

(I am about to edit at *conserved current* and need to point to these ingredients from there)

started a topic cluster table of contents *higher spin geometry - contents* and included it as a “floating table of contents” into relevant entries