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 have tried to clarify a bit more at *Kaluza-Klein monopole*, and at *D6-brane*

*[[level (Chern-Simons theory)]]*

created an entry for *Koszul-Malgrange theorem*

I wanted the links to *weak nuclear force* and *strong nuclear force* in various entries to cease appearing grayish and ugly. So I created a minimal entry *nuclear force*.

added some lines to *differential algebraic K-theory*

also a stub *Beilinson regulator*

needed to be able to point to *duality in physics*, so I created an entry. For the moment just a glorified redirect.

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.

]]>I have spent some minutes starting to put some actual expository content into the Idea-section on *higher gauge theory*. Needs to be much expanded, still, but that’s it for the moment.

started *gauged WZW model*, but no content yet, am just recording some references…

created *geometric quantization by push-forward*, collecting a bunch of references. Thanks to Chris Rogers for pointers.

Reading [[infinity-Chern-Weil theory introduction]] I see (Remark 3.33) that at some point there is a 'partition of unity' argument to show that every pseudo-connection can be replaced by some equivalent true connection. Is it known if this is still true in the complex case, and if not, how it changes the theory? In general, are there any references that build up complex Deligne cohomology in a similar way to this approach to smooth Deligne cohomology? ]]>

Hi all –

I have some basic questions about the cup product on the smooth Deligne complexes as defined at Beilinson–Deligne cup product:

the product is a bit odd in the sense that it’s written $\mathbb{Z}[i]_D^\bullet \otimes\mathbb{Z}[j]_D^\bullet \to \mathbb{Z}[i+j]_D^\bullet$, i.e. it looks like a graded algebra in the category of chain complexes (of sheaves). Is this the best way to think about it?

where does this multiplication come from, abstractly? For instance we can produce an $E_\infty$-algebra if we take a suitable homotopy construction on commutative algebras – can we write the Deligne complexes as such? Beilinson, in the 1985 paper linked on the page above, makes a remark (Remark 1.2.6) about the cup product coming from such an inverse homotopy limit but that seems to be in the wrong direction, i.e. along a given $k$th Deligne complex as opposed to the collection of complexes… what am I missing? Or maybe more generally can we show that this graded algebra in chain complexes is some sort of unit in a symmetric monoidal $\infty$-category?

Thanks! Nilay

]]>stub for *moduli space of connections*, started to collect some references

Together with Dennis Borisov I’ll be running an informal seminar (working group) on *F-theory* at the MPI in Bonn. I’ll be showing the schedule and developing notes here:

I have (finally) added some pointers to the result of Freed-Hopkins 13 to relevant $n$Lab entries.

Mostly at *Weil algebra – characterization in the smooth infinity-topos*

also at *invariant polynomials – As differential forms on the moduli stack of connections*

pointing out that this adds further rationalization to the construction of connections on principal infinity-bundles – via Lie integration.

In making these edits, I have created and then used a little table-for-inclusion

Presently this displays as follows:

**Chevalley-Eilenberg algebra CE $\leftarrow$ Weil algebra W $\leftarrow$ invariant polynomials inv**

differential forms on moduli stack $\mathbf{B}G_{conn}$ of principal connections (Freed-Hopkins 13):

$\array{ CE(\mathfrak{g}) &\simeq& \Omega^\bullet_{li \atop cl}(G) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\simeq & \Omega^\bullet(\mathbf{E}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\simeq& \Omega^\bullet(\mathbf{B}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})/G) }$ ]]>I have extracted one of the key statements from

- Ulrich Bunke,
*A regulator for smooth manifolds and an index theorem*(arXiv:1407.1379)

to an entry *algebraic K-theory of smooth manifolds*.

started some minimum at *odd Chern character* and cross-linked a little

I am working on a further chapter of *geometry of physics* titled *geometry of physics – WZW terms*.

So far there is just the introduction.

As usual, in the course of this I will be touching related entries. Right now I have copied the bulk of that introduction also to the entry *WZW model* in the section *Topological term – WZW term – For the 2d WZW model*, replacing the material that was there before (which I had had written, too, but the new version is better).

gave *torsion of a Cartan connection* its own entry, and cross-linked a bit.

I am writing some notes for a talk that I will give tomorrow:

I thought this might serve also as an exposition for a certain topic cluster of $n$Lab entries, so I ended up typing it right into the $n$Lab.

Notice that this is presently a super-rough version. At the moment this is mostly just personal jotted notes for myself. There will be an abundance of typos at the moment and at several points there are still certain jumps that in a more polished entry would be expanded on with more text.

So don’t look at this just yet if you have energy only for passive reading.

]]>started some bare minimum at *Spin Chern-Simons theory*

I am beginning to think that all these things are secretly the same:

Langlands’ conjecture 3;

the modular functor, hence quantization of 3dCS/2dWZW.

Read number-theoretic Langlands from the CS/WZW-theoretic perspective (using this and this) then one has: on the collection of flat connections (Galois representations) we assign the theta functions which are the partition functions (essentially: L-functions) with “source fields” these flat connections. That’s the definition of the conformal blocks.

But that’s also just the interpretation of equivariant elliptic cohomology in QFT language, which is a refined picture of the modular functor.

From this perspective Langlands’s “functoriality” is Lurie’s “global equivariance”, the fact that the construction is natural in the gauge group.

In this picture the automorphic forms don’t necessarily appear prominently, they rather seem a technical means to express the theta functions (their L-functions), analogous to how for writing down the standard modular functor it may be useful, but not necessary, to express the conformal blocks in certain preferred coordinates.

]]>started some minimum at *Donaldson-Uhlenbeck-Yau theorem*