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 $String_{Fr} \in Grp(Frechet)$ on the topological string group $String_{Top} \in Grp(Top)$ (based on a model by Stefan Stolz) and

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

$$\pi_0(\widehat{Gau} \to String_{Fr}) \simeq Spin$$

$$\pi_1(\widehat{Gau} \to String_{Fr}) \simeq U(1) \,.$$

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

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

the classifying space $B String_{top}$ is the homotopy fiber of the first fractional Pontryagin class

$$\frac{1}{2}p_1 : B Spin \to B^3 U(1)$$

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

$$\frac{1}{2} \mathbf{p}_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1) \,,$$

where $\mathbf{B} Spin$ is the smooth moduli stack for smooth $Spin$-principal bundles, and $\mathbf{B}^3 U(1)$ 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

$$String_{smooth} := \Omega hofib(\frac{1}{2}\mathbf{p}_1) \,.$$

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

$$\pi_0 String_{smooth} \simeq Spin,\;\;\; \pi_1 String_{smooth} \simeq U(1)$$

in $SmoothSpaces = Smooth \infty Grpd_{\leq 0}$.

One can show that both the strict and the weak Lie integration of the $\mathfrak{string}$-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 $String_{smooth}$ has the properties that originally motivated the search for smooth models of $String_{top}$ (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 $(\widehat{Gau P} \to String_{Fr})$ represent $String_{smooth}$ in $Smooth \infty Grpd$?

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

$$(U(1) \to ) \to (\widehat{Gau P} \to String_{Fr}) \to (1 \to Spin)$$

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

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