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).

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?

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

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…

]]>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.

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.

]]>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.

]]>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 currently writing an article as follows:

*Classical field theory via higher differential geometry*

AbstractWe discuss here how the refined formulation of classical mechanics/classical field theory (Hamiltonian mechanics, Lagrangian mechanics) that systematically takes all global effects properly into account – such as notably non-perturbative effects, classical anomalies and the definition of and the descent to reduced phase spaces – naturally is a formulation in “higher differential geometry”. This is the context where smooth manifolds are allowed to be generalized first to smooth orbifolds and then further to Lie groupoids, then further to smooth groupoids and smooth moduli stacks and finally to smooth higher groupoids forming a higher topos for higher differential geometry. We introduce and explain this higher differential geometry as we go along. At the same time as we go along, we explain how the classical concepts of classical mechanics all follow naturally from just the abstract theory of “correspondences in higher slice toposes”.This text is meant to serve the triple purpose of being an exposition of classical mechanics for homotopy type theorists, being an exposition of geometric homotopy theory for physicists, and finally to serve as the canonical example for and seamlessley lead over to the formulation of a local prequantum field theory which supports a localized quantization to local quantum field theory in the sense of the cobordism hypothesis.

This started out as a motivational subsection of *Local prequantum field theory (schreiber)* and as the nLab page *prequantized Lagrangian correspondences*, but for various reasons it seems worthwhile to have this as a standalone exposition and as a pdf file.

I am still working on it. Section 1 and two have already most of the content which they are supposed to have, need more polishing, but should be readable. Section 3 is currently just piecemeal, to be ignored for the moment.

]]>The Hinich-Pridham-Lurie theorem on formal moduli problems says that unbounded $L_\infty$-algebras over some field are equivalently the (“infinitesimally cohesive”) infinitesimal $\infty$-group objects over the derived site over that field.

May we say anything about an analogous statement for infinitesimal group objects in the tangent $\infty$-topos of the $\infty$-topos over the site of smooth manifolds?

There exists for instance the $L_\infty$-algebra whose CE-algebra is

$\{ d h_3 = 0, d \omega_{2p+2} = h_3 \wedge \omega_{2p} | p \in \mathbb{Z} \} \,.$This looks like it wants to correspond to the smooth parameterized spectrum whose base smooth stack is the smooth group 2-stack $\mathbf{B}^2 U(1)$, and whose smooth parameterized spectrum is the pullback along $\mathbf{B}^2 U(1) \to \mathbf{B}GL_1(\mathbf{KU})$ of the canonical smooth parameterized spectrum over $\mathbf{B}GL_1(\mathbf{KU})$, for $\mathbf{KU}$ a smooth sheaf of spectra representing multiplicative differential KU-theory.

So it looks like it wants to be this way. How much more can we say?

]]>started

and for inclusion under “Related concepts”

]]>Chenchang Zhu had been running a course titled “higher bundle theory” in Göttingen last semester. It ended up being mostly about Lie groupoids and stacks. She and her students used the relevant $n$Lab pages as lectures notes, and they added more stuff to these $n$Lab pages as they saw the need.

I just learned of this from Chenchang.

She had created an $n$Lab page

which lists the $n$Lab entries that were used and edited.

For instance the first one is *Lie groupoid* and Chenchang Zhu as well as some of her students added some stuff to that entry, such as this section *Morphisms of Lie groupoids*. Below that they added a section on Morphisms of Lie algebroids. (Maybe some of this could be reorganized a little now.)

I added the bare statement of the list of conditions to *Artin-Lurie representability theorem*, and then added the remark highlighting that the clause on “infinitesimal cohesion” implies that the Lie differentiation of any DM $n$-stack at any point is a formal moduli problem, hence equivalently an $L_\infty$-algebra. Made the corresponding remark more explicit also at *cohesive (∞,1)-presheaf on E-∞ rings*.

have started model structure for L-infinity algebras

]]>