I finally realized that this ought to exist. And sure enough, it had been constructed already: the *4d supergravity Lie 2-algebra*-extension of the 4d $N = 2$ super-Poincaré super-Lie algebra. I have added a minimum of an Idea-section and pointers to the references.

have started something at *orthosymplectic super Lie algebra* and have added little bits and pieces to various related entries, such as first sketchy notes at *super Lie algebra – classification* and at *supersymmetry – Classification – superconformal symmetry*.

Nothing of this is done yet, but I need to call it quits now.

]]>wrote an entry *Deligne’s theorem on tensor categories* on the statement that every regular tensor category is equivalent to representations of a supergroup. Added brief paragraphs pointing to this to *superalgebra* and *supersymmetry*, added cross-links to *Tannaka duality*, *Doplicher-Roberts reconstruction* etc. Also created a disambiguation page *Deligne’s theorem*

started *topologically twisted D=4 super Yang-Mills theory*, in order to finally write a reply to that MO question we were talking about. But am being interrupted now…

have created an entry for *Bott periodicity*

I gave the brane scan table a genuine $n$Lab incarnation and included it at *Green-Schwarz action functional* and at *brane*.

I have split of *super 2-algebra* from super algebra. It’sa stub. Currently the only content is to provide the pointers into the video of Kapranov’s talk (minutes:seconds.)

I am trying to give more of the entries of the brane scan in low ambient dimension their proper names.

Next to the little string in $D = 6$ the brane scan says that there is a Green-Schwarz action functional for a 3-brane $\sigma$-model in $D = 6$. This has been first written down in

- James Hughes, Jun Liu, Joseph Polchinski,
*Supermembranes*, Physics Letters B Volume 180, Issue 4, 20 November 1986, Pages 370–374

but it seems to go by no specific name apart from “the 3-brane in 6d”. So I created a stub entry with that title, *3-brane in 6d*.

quick note at *spin structure* on the characterization *over Kähler manifolds*

started working on superalgebra. But have to interrupt now.

]]>have noted down the basic properties of the irreducible representations of the Lorentzian spin group, at *spin representation – Properties*.

I am slowly beginning to work on *geometry of physics – supergeometry*. So far there is nothing but the definition of super-Minkowski spacetimes and now a computation (here) that the 7-cocycle on the extension of 11d super Minkowski spacetime by its 4-cocycle descends along the rational Hopf fibration $S^7 \to S^4$ to a cocycle on 11 d super Minkowski spacetime itself with coeffcients in the rational 4-sphere.

I needed a table *exceptional spinors and division algebras – table*, and so I have created one and included it into the relevant entries

[First time posting here.. please correct me kindly if I have not followed any explicit or implicit rules. And please excuse my question if it’s vague.]

It’s exciting to see that HoTT can prove concrete homotopical results, like $\pi_1(S^1) \simeq \mathbb{Z}$. It seems that to go further into other subfields, we need modality and cohesion in the theory to “buff up” the local structures – be it topological, smooth, or super-smooth..

However, I had a hard time tapping into modality and cohesion. First of, I’ve tried to understand what a cohesive topos means. As a first example of a cohesive topos, I’ve read through geometry+of+physics+–+smooth+sets and found it exciting. However, it remains unclear how to build up a theory with local model “X” from scratch by myself.

To do that, I planned to read Shulman’s proof on Brouwer’s fixed point.. hoping to understand the process by looking at a classical theorem in topology. But it is unfortunately too advanced for me, I’m not even sure where to start asking if there’s someone to help. Not being too familiar with type theory, I reckon there must be something missing.

What would you suggest if I want to quickly tap into these kind of modalic math, understand at least the proof for Brouwer fixed point, and hopefully be able to understand Schreiber’s foundation of geometry and physics?

4th year math PhD student. I’m not afraid of reading formal papers.. as long as it’s as much self-sufficient as possible. In fact, I find there’s “too much” motivation out there.. I hope to get serious, and even start to think/research in this language. Put in another way using an analogy in the world of programming: I regard Def/Thm/Proof as the source code of a program, and a regular math paper as the literate code or the manual/documentation of the source code. Sometimes it’s nice to have docs to read.. but sometimes you’d rather dive into the source code to see what *it is*. My mood is at the latter stage.

I’m willing to provide more details if it’d be helpful. Thank you very much!

]]>created super infinity-groupoid

(to be distinguished from smooth super infinity-groupoid!)

currently the main achievement of the page is to list lots of literature in support of the claim that the site of superpoints is the correct site to consider here.

]]>added a bit more of substance to *torsion constraints in supergravity*.

I have finally given the previous stub-entry *2d (2,0)-superconformal QFT* some genuine content, also added more citations with brief comments on what they are about.

Also cross-linked with and added related brief paragraphs to *Calabi-Yau manifolds and supersymmetry*, *heterotic string*.

Over in another thread, David Roberts asks for explanation of a bunch of terms in QFT (here).

In further reaction I have started a minimum of explanation for one more item in the list: *split supersymmetry*.

had need to create an entry *signs in supergeometry*, along the lines of Deligne-Freed’s “Sign manifesto”.

One section of this I copy-and-pasted also to make a new entry *super Cartesian space*.

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…

]]>I have added discussion of how the “superfields” in the physics literature are generalized elements of internal homs in the topos over supermanifolds: here

]]>I finally gave *spectral super-scheme* an entry, briefly stating the idea.

This goes back to the observation highlighted in Rezk 09, section 2. There is some further support for the idea that a good definition of supergeometry in the spectrally derived/$E_\infty$ context is nothing but $E_\infty$-geometry over even periodic ring spectra. I might add some of them later.

Thanks to Charles Rezk for discussion (already a while back).

]]>I am working on the next chapter in *geometry of physics*:

This is not done yet, but it should already be readable. To some extent I am taking the talk notes *Super Lie n-algebra of Super p-branes (schreiber)* and expand them into fully fledged lecture notes.

Since I am editing in a separate window, at this point I ask that everyone who feels like touching this page, even if it just concerns tiny changes (typos) to please alert me.

]]>started *Fierz identity* to collect some references. Am still searching for the good reference for the general case…