## March 11, 2006

### Remarks on String(n)

#### Posted by Urs Schreiber Let $P\mathrm{Spin}\left(n\right)$ be the 2-group ($\to$) whose nerve is the group $\mathrm{String}\left(n\right)$ ($\to$).

I would like to understand if Stolz/Teichner’s conception ($\to$) of $\mathrm{String}\left(n\right)$-connections can be understood in terms of 2-connections in 2-bundles ($\to$) which are associated by way of a 2-representation ($\to$) to a principal $P\mathrm{Spin}\left(n\right)$-2-bundle ($\to$).

Here are some remarks.

Let me fix some notation.

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

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

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

(1)$\Sigma \left(\mathrm{Aut}\left(C\right)\right)={\mathrm{Aut}}_{\mathrm{Cat}}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

A representation of a 2-group ${G}_{2}$ is a 2-functor on $\Sigma \left({G}_{2}\right)$.

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 $\mathrm{Aut}\left(K\right)$.

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

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

(2)$P\mathrm{Spin}\left(n\right)\to \mathrm{Aut}\left(\stackrel{̂}{\Omega }\mathrm{Spin}\left(n\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

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

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

So let

(3)$\rho :\Sigma \left(K\right)\to {\mathrm{Aut}}_{\mathrm{Vect}}\left({ℂ}^{n}\right)\subset Vect$

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

(4)$\rho :\Sigma \left(K\right)\to {\mathrm{Aut}}_{\mathrm{Hilb}}\left(H\right)\subset Hilb$

on Hilbert spaces.

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

(5)${\rho }_{2}:\Sigma \left(\mathrm{Aut}\left(K\right)\right)\to \mathrm{Aut}\left(\mathrm{Im}\left(\rho \right)\right)\subset {\mathrm{Aut}}_{\mathrm{Cat}}\left(\mathrm{Hilb}\right)$

as follows.

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

(6)${\rho }_{2}\left(•\right)=\mathrm{Im}\left(\rho \right)=\left\{H\stackrel{\rho \left(k\right)}{\to }H\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}k\in K\right\}\phantom{\rule{thinmathspace}{0ex}}.$

(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 $g:\mathrm{Im}\left(\rho \right)\to \mathrm{Im}\left(\rho \right)$ acting as

(7)$\begin{array}{c}H\\ \rho \left(k\right)↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ H\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{c}H\\ \rho \left(g\left(k\right)\right)↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ H\end{array}$

iii) the image of a 2-morphism

(8)$\begin{array}{c}•\stackrel{g}{\to }•\\ k\prime ⇓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ •\stackrel{g\prime }{\to }•\end{array}$

is the natural ismorphism given by these naturality squares

(9)$\begin{array}{ccc}H& \stackrel{\rho \left(k\prime \right)}{\to }& H\\ \rho \left(g\left(k\right)\right)↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\rho \left(g\prime \left(k\right)\right)\\ H& \stackrel{\rho \left(k\prime \right)}{\to }& H\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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

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. $P\mathrm{Spin}\left(n\right)$ can be regarded as a sub-2-group of $\mathrm{Aut}\left(\stackrel{̂}{\Omega }\mathrm{Spin}\left(n\right)\right)$.

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 $\Omega \mathrm{Spin}\left(n\right)$ on some Hilbert space $H$

(10)$\rho :\Omega G\to \mathrm{PU}\left(H\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(11)$1\to T\to U\left(H\right)\to \mathrm{PU}\left(H\right)\to 1\phantom{\rule{thinmathspace}{0ex}}.$

Let’s draw $\rho$ into this diagram:

(12)$\begin{array}{ccccccccc}1& \to & T& \to & U\left(H\right)& \to & \mathrm{PU}\left(H\right)& \to & 1\\ & & & & & & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↑\rho \\ & & & & & & \Omega G\end{array}$

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

(13)$\begin{array}{ccccccccc}1& \to & T& \to & U\left(H\right)& \to & \mathrm{PU}\left(H\right)& \to & 1\\ & & \mathrm{Id}↑\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& & \stackrel{̂}{\rho }↑\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↑\rho \\ & & T& \to & \stackrel{̂}{\Omega }G& \to & \Omega G\end{array}$

thus obtaining a unitary representation

(14)$\stackrel{̂}{\rho }:\stackrel{̂}{\Omega }G\to U\left(H\right)$

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 $\stackrel{̂}{\rho }$ of all loops that take values different from 1 only on the upper half circle $I\subset {𝕊}^{1}$

(15)${A}_{\rho }:=\stackrel{̂}{\rho }\left({\stackrel{̂}{\Omega }}_{I}G\right)″\phantom{\rule{thinmathspace}{0ex}}.$

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

So ${A}_{\rho }$ plays the role of the representation of $\stackrel{̂}{\Omega }\mathrm{Spin}\left(n\right)$.

The objects in $P\mathrm{Spin}\left(n\right)$ are based paths in $\mathrm{Spin}\left(n\right)$. 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 $\mathrm{Spin}\left(n\right)$ on ${A}_{\rho }$. Indeed, this is the case (Stolz/Teichner, p. 83).

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

(16)${A}_{\rho }\ni a↦\stackrel{̂}{\rho }\left(\stackrel{̂}{\gamma }\right)a\stackrel{̂}{\rho }\left({\stackrel{̂}{\gamma }}^{-1}\right)$

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

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

Hence, unless I am overlooking something, for every projective unitary representation of the loop group of $\mathrm{Spin}\left(n\right)$ we get a 2-representation of the 2-group $P\mathrm{Spin}\left(n\right)$ on $\mathrm{Aut}\left(\Sigma \left({A}_{\rho }\right)\right)$, 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 $\mathrm{Spin}\left(n\right)$ and let $\stackrel{̂}{\ell }$ be a centrally extended based loop in $\mathrm{Spin}\left(n\right)$ such that $\gamma \prime =\ell \cdot \gamma$. Then

(17)$\begin{array}{c}\gamma \\ \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\stackrel{̂}{\ell }\\ \gamma \prime \end{array}$

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

(18)$\begin{array}{c}H\\ a↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ H\end{array}$

by the naturality square

(19)$\begin{array}{ccc}H& \stackrel{\stackrel{̂}{\rho }\left(\stackrel{̂}{\ell }\right)}{\to }& H\\ \stackrel{̂}{\rho }\left(\stackrel{̂}{\gamma }\right)a\stackrel{̂}{\rho }\left({\stackrel{̂}{\gamma }}^{-1}\right)↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\stackrel{̂}{\rho }\left(\stackrel{̂}{\gamma }\prime \right)a\stackrel{̂}{\rho }\left({\stackrel{̂}{\gamma }}^{\prime -1}\right)\\ H& \stackrel{\to }{\stackrel{̂}{\rho }\left(\stackrel{̂}{\ell }\right)}& H\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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

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

(20)$\begin{array}{ccc}\mathrm{PU}\left(H\right)& \to & \mathrm{Inn}\left({A}_{\rho }\right)\\ ↓& & ↓\\ \mathrm{String}\left(n\right)& \to & \mathrm{Aut}\left({A}_{\rho }\right)\\ ↓& & ↓\\ \mathrm{Spin}\left(n\right)& \to & \mathrm{Out}\left({A}_{\rho }\right)\end{array}$

we have an action of $\mathrm{String}\left(n\right)$ 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 $P\mathrm{Spin}\left(n\right)$ 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).

Posted at March 11, 2006 1:28 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/765