The String Coffee Table

Let me fix some notation.

For $K$ a group, let $\Sigma (K)$ (the “suspension of $K$”) be the category with a single object $•$ and $K=\mathrm{Hom}(•,•)$.

Similarly, for $G_2$ a 2-group, let $\Sigma (G_2)$ be the 2-category with a single object $•$ and $G_2=\mathrm{Hom}(•,•)$.

For any category $C$, let $\mathrm{Aut}(C)$ be the 2-group whose objects are automorphisms of $C$ and whose morphisms are natural isomorphisms of these automorphisms. In other words

A representation of a group $K$ is a functor on $\Sigma (K)$.

A representation of a 2-group $G_2$ is a 2-functor on $\Sigma (G_2)$.

The main point of my remarks here is the simple but useful observation that every representation of a group $K$ induces a representation of the 2-group $\mathrm{Aut}(K)$.

I’ll spell that out in detail in a moment. The relevance for the case at hand is the following.

The 2-group $P\mathrm{Spin}(n)$ sits inside $\mathrm{Aut}(\hat{\Omega }\mathrm{Spin}(n))$, where $\hat{\Omega }\mathrm{Spin}(n)$ is a Kac-Moody central extension of the loop group of $\mathrm{Spin}(n)$.

Therefore, the above statement implies that every representation of $\hat{\Omega }\mathrm{Spin}(n)$ induces a 2-representation of $P\mathrm{Spin}(n)$.

But $\hat{\Omega }\mathrm{Spin}(n)$ is represented in terms of unitary elements in von Neumann algebras (“positive energy reps of loop groups”). The induced 2-representation of $P\mathrm{Spin}(n)$ is, implicitly, pretty much what Stolz/Teichner are considering.

Here is what I mean by the 2-representation of $\mathrm{Aut}(K)$ induced by a representation of $K$. It is a simple variation on the theme of the construction of $\mathrm{Aut}(K)$ itself.

So let

be a representation of $H$. Alternatively, think of a representation

on Hilbert spaces.

Denote by $\mathrm{Im}(\rho )$ the image category of $\rho$. Then we get a 2-representation

as follows.

i) the image of the single object $•$ is the image of $\rho$

(Here $H$ denotes the vector space/Hilbert space on which $K$ is represented by $\rho$.)

ii) the image of an automorphism $•\stackrel{g}{\to }•$ of $K$ is the functor $g:\mathrm{Im}(\rho )\to \mathrm{Im}(\rho )$ acting as

iii) the image of a 2-morphism

is the natural ismorphism given by these naturality squares

(Notice that this is nothing but the image under $\rho$ of the corresponding naturality squares in $\mathrm{Aut}(K)$).

While this makes manifest sense only if $\rho$ is faithful, one can check that it is also well defined in the general case.

Consider the example which is of interest here. $P\mathrm{Spin}(n)$ can be regarded as a sub-2-group of $\mathrm{Aut}(\hat{\Omega }\mathrm{Spin}(n))$.

Recall how Antony Wassermann explains that we get representations of these centrally extended loop groups (section 5.4 of Stolz/Teichner).

Let $\rho$ be a projective unitary representation of $\Omega \mathrm{Spin}(n)$ on some Hilbert space $H$

The projective unitary group is $\mathrm{PU}(H)=U(H)/T$ (T is the circle group), hence fits into the short exact sequence

Let’s draw $\rho$ into this diagram:

and consider the pullback of $\rho$ to the left

thus obtaining a unitary representation

of the Kac-Moody loop group on $H$.

Now think of the loops in $\Omega G$ as maps from the circle ${𝕊}^1$ into $G$ Then we get a vonNeumann algebra $A_{\rho }$ from this representation by looking at the closure of the image under $\hat{\rho }$ of all loops that take values different from 1 only on the upper half circle $I\subset {𝕊}^1$

(The two primes indicate taking the double commutant, which amouts to closing the image of $\hat{\rho }$ in weak operator norm.)

So $A_{\rho }$ plays the role of the representation of $\hat{\Omega }\mathrm{Spin}(n)$.

The objects in $P\mathrm{Spin}(n)$ are based paths in $\mathrm{Spin}(n)$. Hence, in order to be able to follow the above prescription for the construction of representations of a 2-group, we need to check that the technical subtleties involved in the construction of $A_{\rho }$ still allow us to act with based paths in $\mathrm{Spin}(n)$ on $A_{\rho }$. Indeed, this is the case (Stolz/Teichner, p. 83).

The trick is to regard an open path in $\mathrm{Spin}(n)$ as a map $\gamma :I\to \mathrm{Spin}(n)$, where, recall, $I$ was the upper half circle. Any such map may be extended to a map $\hat{\gamma }:{𝕊}^1\to \mathrm{Spin}(n)$ on all of ${𝕊}^1$ and then sent it with $\hat{\rho }$ to $U(H)$. By definition of $A_{\rho }$, conjugating any element with such a lift

is independent of the choice of lift $\hat{\gamma }$ of $\gamma$ and hence well defined.

(What Stolz/Teichner discuss is that this construction even works on $\mathrm{PU}(A_{\rho })$.)

Hence, unless I am overlooking something, for every projective unitary representation of the loop group of $\mathrm{Spin}(n)$ we get a 2-representation of the 2-group $P\mathrm{Spin}(n)$ on $\mathrm{Aut}(\Sigma (A_{\rho }))$, i.e. on the 2-category whose 1-morphisms are automorphisms of the category with single object $H$ and vonNeumann operators $A_{\rho }$ as morphisms, and whose 2-morphisms are natural isomorphisms between these.

More precisely, let $\gamma$ and $\gamma \prime$ be based paths in $\mathrm{Spin}(n)$ and let $\hat{\ell }$ be a centrally extended based loop in $\mathrm{Spin}(n)$ such that $\gamma \prime =\ell \cdot \gamma$. Then

is a morphism in $P\mathrm{Spin}(n)$ and it is sent by our 2-representation induced by $\rho$ to the natural isomorphism given for each morphism

by the naturality square

This should mean that there should be a 2-connection on a $\mathrm{String}(n)$-bundle obtained as follows.

From a principal $P\mathrm{Spin}(n)$-2-bundle we obtain a $\mathrm{String}(n)$-bundle using Branislav Jurčo’s construction ($\to$). Following Stolz/Teichner, who observed that (p. 85)

we have an action of $\mathrm{String}(n)$ on $A_{\rho }$ and hence can form an associated vonNeumann algebra bundle. A 2-connection taking values in the above representation of the 2-group $P\mathrm{Spin}(n)$ would give the local surface transport in a locally trivialized version of this.

I won’t go into the details right now. I’ll just note that the “fake flatness” condition ($\to$) on that 2-connection ensures that there is precisely a circle worth of surface transport between given fixed boundaries. This is precisely as in Stolz/Teichner’s formulation (paragraph below digram on p. 70).