The n-Category Café

Giovanni Felder reported on recent work on studying the Poisson sigma-model in the presence of branes in the target.

It turns out that where in the absence of branes the string product of the sigma-model is famously just the ordnary algebra which is the deformation quantization of the Poisson manifold target space, in the presence of branes one sees genuine $A_\infty$-algebras and -modules. See the references here.

I suppose there should be a nice way to see the Poisson $\sigma$-model concretely as an example of a “TCFT” or something very similar. Does anyone have more details on this?

André Henriques gave a survey of some of the aspects and conjectured implications of his work with Anrthur Bartels and Chris Douglas on fermioninc conformal nets (see references there):

as we mentioned here various times before, the idea is that the 2d CFT $Ferm(1)$ given by one free fermion on the worldsheet is to the real Clifford algebra $Cl(1)$ as tmf is to real K-theory: as $Cl(1)$ is a period-8 generator of the symmetric monoidal bicategory of Clifford algebras, bimodules and transformations, $Ferm(1)$ is (semi-conjecturally) the period $24^2$-generator of a symmetric monoidal tricategory, which conjecturally allows a geometric construction of tmf in generalization of the way Clifford module bundles give geometric cocycles for K-theory.

Robbert Dijkgraaf talked about his work with Cumrum Vafa, Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems.

The nice underlying story is that the famously elusive nonabelian gerbe worldvolume theory on a pile (“stack”) of $N \in \mathbb{N}$ $M5$-branes of the topology $four-manifold \times Riemann-surface$ is supposed to give rise after integrating out either of the two factors to both a Yang-Mills theory on $X_4$ (whose $S$-duality invariance is the remnant of the conformal invariance of the Riemann surface) as well as a 2d-conformal field theory on that surface.

With Vafa he worked out various explict realizations of the correspondence of 4d gauge theories and 2d CFTs induced this way, and to their random matrix model reformulation, as described in the above article.

Alain Connes’ talk was titled Spacetime and $Spec(\mathbb{Z})$ – too elusive spaces , but both these gadgets appeared in his talk only if you read deep in between the lines.

What he did talk about was a wealth of ideas and results revolving around his work with Consani on hyperrings. See there for two simple examples central to his talk, and for the reference.

Thomas Schick gave an introductory talk to his work with Uli Bunke on multiplicative and differential geometric cocycle models for differential K-theory (see there for the references).

After the talk Sir Michael Atiyah voiced a complaint about the general notion of differential cohomology as traditionally understood, which I found noteworthy. At the risk of misrepresenting by paraphrasing from memory, maybe I may still formulate that complaint, which maybe deserves to be discussed independently of who may have said it or may not have said it:

paraphrase: “The usual definition of differential cohomology with its pairing of spectra from the world of stable homotopy theory on the one hand, and differential forms from the world of differential geometry on the other is ugly : since these two concepts don’t naturally live in the same context. There should instead be another definition, one that is all of a piece . “

Now, I guess one might argue that this pairing of two worlds is precisely the whole point of differential cohomology. This is true to some extent, and I imagine this statement would win a show-of-hands at any random conference on this topic. Which is why I found it noteworthy that somebody like Atiyah does voice a complaint after all.

Now, if I may recount that here Sir Atiyah also suggested a solution to his complaint: he suggested to look at models for differential cohomology in the world of von Neumann algebra factors. I’d like to see more details on that, unfortunately the beginning of the next talk put an end to this interesting discussion.

But I would dare to suggest another solution: there is a canonical fusion of the worlds of (stable) homotopy theory and of differential geometry: that’s the $(\infty,1)$-topos $\mathbf{H} =$ $\infty$LieGrpd of $\infty$-stacks on differential geometric test spaces. Every $(\infty,1)$-topos comes with its intrinsic notion of cohomology, and some, like this one, even come with their intrinsic notion of differential cohomology. And for suitable refinements of coefficient objects in $\infty Grpd \simeq Top$ to $\infty LieGrpd$, this intrinsic notion, all of a piece, does coincide with the one given by the usual extrinsic definition.