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*).

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

]]>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.

]]>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.

]]>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

]]>I am back to working on *geometry of physics*. I’ll be out-sourcing new paragraphs there to their own $n$Lab entries as much as possible (because the length of the page makes saving and hence previewing it take many minutes, so I need to work in smaller sub-entries and then copy-and-paste).

In this context I now started an entry *prequantum field theory*. To be further expanded.

This comes with a table of related concepts *extended prequantum field theory - table*:

**extended prequantum field theory**

$0 \leq k \leq n$ | transgression to dimension $k$ |
---|---|

$0$ | extended Lagrangian, universal characteristic map |

$k$ | (off-shell) prequantum (n-k)-bundle |

$n-1$ | (off-shell) prequantum circle bundle |

$n$ | action functional = prequantum 0-bundle |

Today at *Spaces for Mathematics and Physics*
Maxim Kontsevich gave a survey of what the $n$Lab calls *derived noncommutative geometry*. That reminded me that I had long wanted to extract some more of the essence in a clear way.

The discussions I have seen about this, and today was no exception, have a lot of dg-stuff in it, and fall back to presentations when possible, and generally seem to be more interested in describing examples than in developing abstract theory.

But at least verbally there is the indication that what should really be going on is the following, and in saying this I allow myself freely to strip away the dg-ism and just speak $\infty$-categorically right away. Then in my words the suggestion that I hear is being made is the following.

We recall that

We may think of $\infty$-sheaf $\infty$-toposes with geometric morphisms between them as geometric spaces.

By the $\infty$-Giraud theorem these are (accessible) left exact reflections of $\infty$-catgegories of $\infty$-presheaves with values in $\infty$-groupoids.

The “stable $\infty$-Giraud theorem” (also by Lurie, of course) says that, analogously, every locally presentable stable $\infty$-category is the left exact localization of an $\infty$-category of presheaves of spectra.

Given that, it is entirely reasonable to ask whether one gets a sensible notion of geometry from the category of locally presentable stable $\infty$-categories with some suitable kind of geometric morphisms between them.

This is Maxim Kontsevich’s proposal, modulo the preference for stable $\infty$-categories which come from $H k$-module spectra, these are equivalently the $k$-linear enhanced dg-categories (by stable Dold-Kan correspondence).

Indeed, this perspective via the “stable Giraud theorem” makes it natural *not* to consider monoidal stable $\infty$-categories, which is the key point that is being advertized as making this be about *non-commutative geometry* (because the categories of $A_\infty$-modules over $A_\infty$-rings are not in general monoidal, and these serve as the affine spaces in this context).

The $n$Lab entry *derived noncommutative geometry* more or less says this already, but it could say it more clearly still, it seems to me.

But is this actually said in a clear way anywhere in the literature? I mean the abstract story. Or any abstract story.

There are several choices of what one would want to call geometric morphisms between stable $\infty$-categories, and maybe some care would be justified to lay these out a bit.

Today was suggested the definition that the $n$Lab calls the *Grothendieck context* (except for the monoidal structure, which is being ignored here, as I said). That’s clearly good for some things, but elsewhere one would want what the $n$Lab calls the Wirthmüller context (again without the monoidal structure). In fact that choice would connect the whole idea just seamlessly into the linear homotopy type theory story. Indeed, there the lack of the tensor product is quite natural: it just means that we start in a very weak fragment of linear type theory.

I am saying this in part just as a reminder to myself that, viewed this way, the story of *Quantization via Linear homotopy types (schreiber)* connects nicely – indeed it gives some conceptualization of what happens as we linearize those slice $\infty$-toposes to stable $\infty$-categories: we quantize, and indeed that’s where all the motivation for the Kontsevich style stable geometry comes from, the example of interest are the Fukaya- and $DQCoh$-categories that reflect the quantization of Calabi-Yau geometries as seen by quantum tological strings propagating on these.

Okay, that was just me mumbling to myself. Back to turning this here into a conversation: is there any place in the literature where an abstract “stable $\infty$-geometry” in direct analogy to “geoemtry as $\infty$-toposes with geometric morphisms between them” is laid out, or else discussed far enough that one may directly extract what should be the abstract $\infty$-categorical formulation?

]]>It’s time that I think a bit more about the combination of smooth cohesion with Charles Rezk’s global equivariant cohesion. Here are some simple thoughts, nothing deep, just to warm up.

I’ll write $\infty Grpd_{Glob}$ for the global equivariant homotopy theory and by its smooth version I mean

$\mathbf{H} \coloneqq Sh_\infty(SmoothMfd, \infty Grpd_{Glob}) \,.$This sits now in a commuting square of geometric morphisms, each one of which exhibits cohesion over its codomain:

$\array{ Sh_\infty(SmoothMfd, \infty Grpd_{Glob}) &\stackrel{\Gamma_{smth}}{\longrightarrow}& \infty Grpd_{Glob} \\ \downarrow^{\mathrlap{\Gamma_{glob}}} &\searrow^{\mathrlap{\Gamma}}& \downarrow \\ Sh_\infty(SmoothMfd, \infty Grpd) &\longrightarrow& \infty Grpd } \,.$This provides a more refined perspective on smooth quotient spaces: for instance for $X$ a smooth manifold equipped with the action of a group $G$, then this defines the presheaf on manifolds

$X /_{glob}G : U \mapsto \delta_{C^\infty(U,G)} (C^\infty(U,X), C^\infty(U,G)) \in \infty Grpd_{glob} \,,$where we regard $(C^\infty(U), C^\infty(U,G))$ as a topological $C^\infty(U,G)$-space (which happens to be topologically discrete in this example) and $\delta_{C^\infty(U,G)}$ regards that as a presheaf over $Glob$.

Then

$\Gamma_{glob} (X/_{glob} G)$ is the smooth orbifold coresponding to the $G$-action on $X$

$\Pi_{glob} (X/_{glob} G)$ is the diffeological quotient space of $X$ by $G$.

I think this is going to be important for the application to singular $G_2$-compactifications of 11d supergravity. There one needs smooth spaces with conical singularities of ADE type, but the actual physical manifold is not supposed to be the ADE orbifold, but really the naive quotient with that singularity.

In fact what one really wants is that one considers the singular quotient in complex analytic cohesion and then blows up the singularity, replacing the singular point by a system of spheres that touch each other such as to form the corresponding ADE Dynkin diagram. I am wondering if there is any way to capture this abstractly.

]]>am starting *complex analytic infinity-groupoid* (in line with “smooth infinity-groupoid” etc.) and *higher complex analytic geometry*. Currently there is mainly a pointer to Larusson. To be expanded.

A thought.

On the one hand, one may see (discussed at *Structure Theory for Higher WZW Terms (schreiber)*) that 11d supergravity is the first-order integrable higher super Cartan geometry for the higher WZW terms of
the M2 and the M5, the latter twisted by the former.

On the other hand there have long been hints that 11d sugra should have a natural formalization in terms of exceptional generalized geometry.

Here is an observation of a possible connection:

The central result in the last section of

D’Auria, Fre, *Geometric Supergravity and its Hidden Supergroup* (pdf)

of which a more comprehensive analysis is in

Azcarraga et al, *On the underlying gauge group structure of D = 11 supergravity* (arXiv:hep-th/0406020)

is that the ordinary 1-Cartan geometry on which the M2-brane twist trivializes is locally modeled on the extension of super-Minkowski tangent spaces by the 2-forms $\wedge^2 T^\ast$ and by the 5-forms $\wedge^5 T^\ast$.

In fact these authors realize this extension as a super Lie algebra extension of the supersymmetry Lie algebra in 11d. But disregarding this super Lie structure for a moment, it is of course precisely the exceptional generalized tangent bundle considered originally in

- Chris Hull,
*Generalised Geometry for M-Theory*JHEP 0707:079 (2007) (arXiv:hep-th/0701203)

where it was observed that the E-series of the exceptional groups canonically acts on this extension, at least up do d = 7.

So this seemes to give a plausible path connecting the higher geometry of M-theory with its exceptional geometry. Since the exceptional geometry here may be thought of as cover of spacetime over which the M2 2-gerbe trivializes, there may be a chance that the exceptional geometry may be understood as well-adapted local data for the higher geometry in some sense.

]]>