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 beginning at geometric Langlands correspondence

]]>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.

]]>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…

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

added to *Haefliger groupoid* some of the pertinent facts proven in Carchedi 12.

Inspired by Matthew Ando’s talk at the Conference on twisted cohomology that I am currently attending, I finally typed up a note on

]]>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…

]]>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*).

prompted by this MO question I have started to compile a list of references at *higher differential geometry*.

Of course this has room to be much expanded.

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

]]>You may recall that we have been thinking about extended geometric *pre*-quantization a good bit. So far we haven’t been sure how exactly to generalize the notion of polarization to n-plectic moduli $\infty$-stacks in general and hence only had some tentative steps towards genuine extended geometric quantization. (Chris Rogers has a discussion of 2-plectic geometric quantization of the topological WZW term in preparation, that certainly goes in the right direction).

Now, it occurred to me that for the extended geometric quantization of 2d Chern-Simons theory, the answer is actually already in the literature – in disguise and hence not fully recognized. Exploring this example should help to understand how the general case will work.

So I started making some notes that contain the story to the degree that I currently understand it, in the hope that this will be further developed. This is now at

*extended geometric quantization of 2d Chern-Simons theory*.

There is an introduction and overview there, which should give the main idea.

]]>We are in the process of finalizing an article on prequantum theory in higher geometry. An early version of our writeup I have now uploaded. It needs a few more cycles of polishing, but I thought I’ll provide this here on the nForum right now as a kind of explanation for the sheer drop in the amount of noise that I have been making around the nLab as of lately ;-:

Domenico Fiorenza, Chris Rogers, Urs Schreiber,

Regulars here will recognize various things that I/we have been talking about for a good while now. Finally they are materializing in a more pdf-kind of incarnation…

]]>am working on putting some genuine detailed content into *smooth groupoid*. So far there is now discussion of the groupoid-enriched category of groupoid-valued presheaves, Cech nerves, and the stack condition.

Then it breaks off and some rough old material kicks in which needs to be harmonized. Will continue later, need to go offline now for a little.

]]>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.

]]>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.

]]>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.
=–