The n-Category Café

Congratulations, Thomas!

Here a few more words and links on the content of the thesis:

Chapter 1 is about the notion of connection on a 2-bundle for the case that the structure 2-group is the circle 2-group ($\simeq$ ordinary bundle gerbes) or the (automorphism 2-group) of the ordinary circle group ($\simeq$ the case of “Jand gerbes”: orientifolds). It discusses the higher parallel transport/surface holonomy of these objects over surfaces with boundaries and with defects.

Chapter 2 is about stacks on smooth groupoids, hence (nonabelian) equivariant cohomology, then applied to the previous case of bundle gerbes.

The third chapter discusses four equivalent incarnations of the notion of principal 2-bundle: 1. in terms of nonabelian Cech cohomology, 2. by topological classifying maps 3. as bundle gerbes, 4. explicitly as total “spaces” with principal action.

The fourth chapter discusses a smooth Frechet manifold structure on the topological string group and a smooth 2-group obtained from it.

The fifth chapter discusses aspects of equivariant Dijkgraaf-Witten theory.

And I have to run now to catch a train.

@Tom: Thanks ;)

Also thanks to Urs for this very good summary. If someone is interested I could expand on that or provide more details of the respective chapters. Also if someone finds some errors or has some comments - please let me know.

I should just say that the chapters of the thesis are based on papers with the same name, but with some coauthors, which should be mentioned here:

Chapter 1: “Bundle gerbes and surface holonomy” with J. Fuchs, C. Schweigert, and K. Waldorf

Chapter 2: “Equivariance in higher geometry” with C. Schweigert

Chapter 3: “Four Equivalent Versions of Non-Abelian Gerbes” with K. Waldorf

Chapter 4: “A smooth model for the string group” with C. Sachse, and C. Wockel

Chapter 5: “Equivariant modular categories via Dijkgraaf-Witten theory” with J. Maier and C. Schweigert

I’ll briefly highlight one aspect of Thomas’s work. Then I make a lengthy comment. Finally I have a question.

The point of section 4 in Thomas’ thesis, the one on the string group, is

1. to state that there is indeed the structure of a Fréchet Lie group ${\mathrm{String}}_{\mathrm{Fr}}\in \mathrm{Grp}(\mathrm{Frechet})$ on the topological string group ${\mathrm{String}}_{\mathrm{Top}}\in \mathrm{Grp}(\mathrm{Top})$ (based on a model by Stefan Stolz) and

2. to observe that there is then a Fréchet 2-group coming from a crossed module $(\widehat{\mathrm{Gau}}\to {\mathrm{String}}_{\mathrm{Fr}})$ whose homotopy sheaves are

The second statement is important: naïvely one might be led to believe that ${\mathrm{String}}_{\mathrm{Fr}}$ is already a decent smooth model of $\mathrm{String}$. But it is not. Not if one means to have differential cohomology of smooth $\mathrm{String}$-principal bundles to come out right.

The way I like to think about this subtlety, in terms of cohesive $\mathrm{\infty }$-toposes, is this:

the classifying space $B{\mathrm{String}}_{\mathrm{top}}$ is the homotopy fiber of the first fractional Pontryagin class

in $\mathrm{Discrete}\mathrm{\infty }\mathrm{Grpd}:=\mathrm{\infty }\mathrm{Grpd}\simeq \mathrm{Top}$. One can show that this has, up to equivalence, a unique lift through geometric realization $\Pi :\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}\to \mathrm{Discrete}\mathrm{\infty }\mathrm{Grpd}$ to a smooth characteristic map of the form

where $B\mathrm{Spin}$ is the smooth moduli stack for smooth $\mathrm{Spin}$-principal bundles, and $B^{3}U(1)$ is the smooth moduli 2-stack for smooth circle 3-bundles (aka bundle 2-gerbes).

The “correct” smooth string 2-group is the delooping of the homotopy fiber of this smooth characteristic map

From this definition it follows immediately by the long exact sequence of geoemtric homotopy groups that

in $\mathrm{SmoothSpaces}=\mathrm{Smooth}\mathrm{\infty }{\mathrm{Grpd}}_{\le 0}$.

One can show that both the strict and the weak Lie integration of the $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}$-Lie 2-algebra that are in the literature are indeed models for this abstractly defined String Lie 2-group, and that nonabelian differential cohomology with coeffcients in ${\mathrm{String}}_{\mathrm{smooth}}$ has the properties that originally motivated the search for smooth models of ${\mathrm{String}}_{\mathrm{top}}$ (details on all this are in section 4.1 of differential cohomology in a cohesive topos ).

So from this prespective one would like to know (or at least I would like to know):

does the Sachse-Nikolaus-Wockel Fréchet 2-group $(\widehat{\mathrm{Gau}P}\to {\mathrm{String}}_{\mathrm{Fr}})$ represent ${\mathrm{String}}_{\mathrm{smooth}}$ in $\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$?

A sufficient condition for this to be true is that the exact sequence of Fréchet crossed modules

which they discuss presents a fiber sequence in $\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$.

Last time that I thought and talked about this was when I met Christoph Wockel in Cardiff. I was thinking: present $\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ by the structure of a Brown-“category of fibrant objects” on $\mathrm{CartSp}$ whose fibrations are stalkwise fibrations. Then it should be clear that $(\widehat{\mathrm{Gau}P}\to {\mathrm{String}}_{\mathrm{Fr}})\to (1\to \mathrm{Spin})$ is a fibration. And thus it follows that its ordinary fiber is also its homotopy fiber.

So finally my question: have you, Thomas, further thought about this? If not, we should try to nail it down.