I have added to *Teichmüller theory* a mini-paragraph Complex structure on Teichmüller space with a minimum of pointers to the issue of constructing a complex structure on Teichmüller space itself.

Maybe somebody has an idea on the following: The Teichmüller orbifold itself should have a neat general abstract construction as the full subobject on the étale maps of the mapping stack formed in smooth $\infty$-groupoids/smooth $\infty$-stacks into the Haefliger stack for complex manifolds : via Carchedi 12, pages 37-38.

Might we have a refinement of this kind of construction that would produce the Teichmüller orbifold directly as on objects in $\infty$-stacks over the complex-analytic site?

]]>On the occasion of Brandenburg 14 I have – finally – created an entry *2-algebraic geometry*.

We had *almost* created that a few times before, only that we never did. Maybe the closest we came in the section *Derived algebraic geometry – Relation to noncommutative geometry*.

I have tried to do some minimum cross-linking, with 2-ring, etc. But one might want to do more.

]]>I gave the stub-entry *Hopf algebroid* a paragraph in the Idea-section that points out that already in commutative geometry there are two different kinds of Hopf algebroids associated with a groupoid (just as there are two versions of Hopf algebras associated with a group):

The commutative but non-co-commutative structure obtained by forming ordinary function algebras on objects and morphisms;

The non-commutative but co-commutative structure obtained by forming the groupoid convolution algebra.

For the moment I left the rest of the entry (which vaguely mentions commutative and non-commutative versions without putting them in relation) untouched, but I labelled the whole entry “under constructions”, since I think this issue needs to be discussed more for the entry not to be misleading.

I may find time to get back to this later…

]]>Since it touches on several of the threads that we happen to have here, hopefully I may be excused for making this somewhat selfish post here.

For various reasons I need to finally upload my notes on “differential cohomology in a cohesive ∞-topos” to the arXiv. Soon. Maybe by next week or so.

It’s not fully finalized, clearly, I could spend ages further polishing this – but then it will probably never be fully finalized, as so many other things.

Anyway, in case anyone here might enjoy eyeballing pieces of it (again), I am keeping the latest version here

]]>added some lines to *differential algebraic K-theory*

also a stub *Beilinson regulator*

For discussion at *geometry of physics* I need a way to point to the concept of “locality” in QFT, so I gave it a small entry: *local quantum field theory*.

Today I was asked for what I know about the development of the theory of Kan-fibrant simplicial manifolds. I realized that the nLab does not discuss this, so I have started a page now with the facts that come to mind right away. (Likely I forgot various things that should still be added.)

]]>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.

]]>have started model structure for L-infinity algebras

]]>created a stub for *twisted differential cohomology* and cross-linked a bit.

This for the moment just to record the existence of

- Ulrich Bunke, Thomas Nikolaus,
*Twisted differential cohomology*(arXiv:1406.3231)

No time right now for more. But later.

]]>at *Atiyah Lie groupoid* was this old query box discussion, which hereby I am moving from there to here:

+– {: .query} What is all of this $diag$ stuff? I don't understand either $(P \times P)/_{diag} G$ or $(P_x \times P_x)_{diag} G$. —Toby

David Roberts: It’s to do with the diagonal action of $G$ on $P\times P$ as opposed to the antidiagonal (if $G$ is abelian) or the action on only one factor. I agree that it’s a bad notation.

*Toby*: How well do you think it works now, with the notation suppressed and a note added in words? (For what it's worth, the diagonal action seems to me the only obvious thing to do here, although admittedly the others that you mention do exist.)

*Todd*: I personally believe it works well. A small note is that this construction can also be regarded as a tensor product, regarding the first factor $P$ as a right $G$-module and the second a left module, where the actions are related by $g p = p g^{-1}$.

*Toby*: H'm, maybe we should write diagonal action if there's something interesting to say about it.
=–

I have expanded the Idea-section at *moduli stack of elliptic curves*, have tried to add more pertinent references, and have touched the subsection on “Over general rings” and on the derived version.

In the course of this I started to split off some entries such as *nodal cubic curve* (which now has a little bit of content) and *cuspidal cubic curve* (which does not yet).

edited at *orbispace* in order to express Charles Rezk’s statement here more accurately.

I tried to polish the "Idea" and the "References" section at [[Courant algebroid]] to something more comprehensive.

]]>a beginning at geometric Langlands correspondence

]]>added a little bit to *foliation*: a brief list of equivalent alternative definitions and and Idea-section with some general remarks.

Thomas Holder has been working on *Aufhebung*. I have edited the formatting a little (added hyperlinks and more Definition-environments, added another subsection header and some more cross-references, cross-linked with *duality of opposites*).

We are in the process of finalizing this article here:

Domenico Fiorenza, Hisham Sati, Urs Schreiber

*Super Lie $n$-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields*

**Abstract.** We formalize higher dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the “FDA”-language used in the supergravity literature).
We show generally how the intersection laws
for such higher WZW-type sigma-model branes (open brane ending on background brane) are encoded precisely in (super-)$L_\infty$-extension theory and how the resulting “extended (super-)spacetimes” formalize spacetimes containing $\sigma$-model brane condensates. As an application we prove in Lie $n$-algebra homotopy theory that the complete super $p$-brane spectrum of superstring/M-theory is realized this way,
including the pure sigma-model branes (the “old brane scan”) but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional spacetime with an M2-brane condensate turns out to be the “M-theory super Lie algebra”. We also observe that in this
formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11-dimensional sugra/M-theory spacetime, and of (prequantum) S-duality for type II string theory. Finally we give the non-perturbative description of all this by higher WZW-type $\sigma$-models on higher super-orbispaces with higher WZW terms in higher differential geometry.

am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.

started a stubby *double dimensional reduction* in this context and added some first further pointers and references to *M5-brane*, to *D=5 super Yang-Mills theory* and maybe elsewhere.

But this still needs more details to be satisfactory, clearly.

]]>I’ve been thinking about generalizing the Cech-Delign double complex to the case where $U(1)$ is replaced by some Lie group, $G$, and $\mathbb{R}$ is replaced with $\mathfrak{g}$. I came across this post on Nonabelian Weak Deligne Hypercohomology by Urs a while back and was wondering if his musing was ever fully considered/resolved?

For full context of why I’m considering this: I was working on a project during my PhD that I’d like to eventually publish, but I constructed an element in a (Hochschild-like) curved dga with a Chen map to holonomy (path or surface) which is a chain map and map of algebras. It was suggested that this is not enough of a “result” unless I could find the right notion of equivalence to fully flesh out this map. I was hoping that some resolution of the referenced article above could allow me to put my work in that context.

P.S. This is my first post here so my apologies if I made a cultural error.

]]>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…

I have been adding and editing a bit at *axion* in the section *In string theory*.

The axion fields in string theory form a curious confluence point relating

- abstract concepts related to higher gauge theory

with

- fundamental questions in particle physics/cosmology phenomenology

as indicated schematically in this table (now also in the entry):

$\,$

$\array{ \mathbf{\text{higher gauge theory}} && && \mathbf{\text{particle physics/cosmology phenomenology}} \\ \\ \left. \array{ \text{higher gauge fields} \\ \text{higher characteristic classes} \\ \updownarrow \\ \text{non-perturbative QFT/string effects} \\ \text{in HET: Green-Schwarz anomaly cancellation} \\ \text{in IIA/B: higher WZW term for Green-Scharz D-branes} } \right\} &\longrightarrow& \array{ \text{axion fields} \\ \text{in the string spectrum} } &\longrightarrow& \left\{ \array{ \text{solve strong CP-problem as with P-Q robustly} \\ \text{solve dark matter problem by FDM} } \right. }$$\,$

I will be trying to expand on this a little more.

]]>Started something at *local BRST complex*.

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.

]]>