In my master thesis I have stumbled upon some issue which makes me go crazy. I want to state the problem but I should discuss it so that it would not mislead me.

There is a model at hand, a field theory which interacts with gravity non minimally via metric in the lagrangian. This field theory is nontrivial topologically and prequantizes. Nonetheless, it has a prequantization when we fix the metric but it seems that it has no canonical prequantization as a field theory on the bundle of fields + metrics (satisfying certain conditions). Moreover, there is an obstruction to prequantization of such a field theory, since if we compute metric energy-momentum tensor locally it shall not glue properly to a global one.

Initially I intended to find out how gravity interacts with nontrivial topologically field theories so as to conclude inconsistencies with gravity even at the level of prequantization.

It seems that given a prequantum field theory which has interaction with gravity (via energy-momentum tensor) we must deform it (canonically some way, so that it could be functorial, may be additive…) so as to extend the prequantization to the bundle of fields + metrics.

Why this idea? I guess the answer is that we already do this with even locally-defined field theories. An obstruction to a proper nonminimal interaction (via Energy-Momentum tensor) of a field theory with gravity is that the Energy-Momentum tensor must be divergence-free. When we deform a given field theory we usually fall to such field configurations that satisfy divergence-freeness or even fall on-shell of the theory.

So we had:

obstruction to nonminimal interaction - divergence-freeness of EMT (analytical condition). We deform a bundle of a theory but not lagrangians. This deformation is “canonical”.

Now we have in addition:

obstruction to nonminimal interaction - EMT should be a globally defined tensor on a manifold (topological condition). We deform a field theory but I guess here we change the Lagrangians so that they could prequantize on a bundle of fields + metrics.

What do you think I could do in this case? ]]>

added briefly the definition to *Einstein-Yang-Mills theory*

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

at *Aharonov-Bohm effect* I have polished/expanded the Idea-section and little and then I wrote out in some detail the standard 1-form vector potential that one uses to describe the effect.

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).