The n-Category Café

The FHT theorem says, roughly, that the Grothendieck ring of the category of highest weight representations of a loop group $\Omega G$ is isomorphic to the twisted equivariant K-theory of $G$.

Slightly more precisely: assume $G$ to be a simple and simply connected compact Lie group.

For any level $k \in H^3(G,\mathbb{Z}) \simeq \mathbb{Z}$, there is a canonical central extension

of the loop group $\Omega G$ of $G$.

There is a special sort of representations of this extended loop group - known as the “positive energy” representations. There is also a special notion of tensor product of such representations - known as fusion.

Restricting to positive energy representations and using the fusion product, we get the monoidal category

of representations. Abelian monoidal categories like this are in essence categorified rings. We can decategorify this and group complete the result to obtain an ordinary ring. For the case of $\mathrm{Rep}(\hat \Omega_k G)$ this is known as the Verlinde ring

of $G$ at level $k$.

This ring plays a major role in the conformal field theory of maps from Riemann surfaces into $G$.

And there is another ring which we can canonically associate to $G$ and $k$:

For every $k$, there is a canonical gerbe on $G$, whose Dixmier-Douady class is just that $k \in H^3(G,\mathbb{Z})$. In fact, from a certain point of view #, the central extension $\hat \Omega_k G$ is that gerbe on $G$.

Like a bundle is trivialized by a function. A gerbe may be trivialized by a bundle. Even if the gerbe is nontrivial, it can still be “trivialized” in a generalized sense - by a twisted bundle.

Like we have a notion of K-theory obtained by taking the group completion of the decategorification of the category of vector bundles on a space, we have a notion of twisted K-theory by doing the same with such twisted bundles.

Now, the group $G$ acts on itself by conjugation. We might hence be interested in the K-theory of the quotient $G/G$. This amounts to looking at the (twisted) K-theory of $G$-equivariant (twisted) bundles on $G$.

So that’s some ring, let’s denote it by

where $r$ is some offset that I won’t describe.

The Freed-Hopkins-Teleman theorem says that these two rings are the same

So there.

Find the details in

D. Freed, M. Hopkins, C. Teleman
Twisted K-theory and loop group representations
math.AT/0312155,

Now Willerton’s version of this theorem for $G$ a finite group.

First: what on earth is the loop group of a finite group?

Think not in terms of spaces, but in terms of categories and you’ll be enlightened.

Consider this: let

be the category with a single object, one morphism per natural number, with composition of morphisms being addition of natural numbers. That’s our “parameter space”, supposed to model the circle.

Moreover, let

be the category with a single object, and our finite group worth of morphisms. That’s our “target space”.

Let

be the “configuration space”, namely the category of functors from parameter space to target space.

It is crucial here that we do remember the morphisms (natural transformations) between these functors.

Now, Willerton proves a cool theorem, which essentially tells us that $\mathrm{conf}$ does indeed behave like the loop group of a Lie group.

Theorem:

Here $B$ denotes the operation of taking the classifying space, and $\Omega$ is taking the ordinary loop space.

For ordinary spaces, $B$ and $\Omega$ are like inverses up to homotopy. In particular, for topological groups we always have

The above theorem hence says that we can pull $\Lambda$, our would-be analog of the $\Omega$-operation, from the world of categories into the world of topological spaces, and there it then looks indeed like $\Omega$.

This nice theorem also has a nice name: that’s the parmesan theorem.

For our purposes, it is not all that crucial to actually understand this theorem. What is crucial is that this theorem suggests that the category

plays the role of the loop group of $G$.

Once we accept this, the finite-group version of Freed-Hopkins-Teleman becomes almost a triviality:

Notice that $\Lambda G$ is in fact a groupoid. Its objects are elements $g$ of $G$. Its morphisms

are elements $h \in G$ such that $g ' = h g h^{-1} = \mathrm{Ad}_h g$.

In other words: $\Lambda G$ is not just something like the loop group of $G$, it is also at the same time the action groupoid of the adjoint $G$-action on itself.

What would be a representation of $\Lambda G$? Well, a representation of any groupoid is nothing but a functor from that groupoid to vector spaces. Same here: a representation of $\Lambda G$ is a functor

But notice how this is now saying two things at the same time:

in as far as we regard $\Lambda G$ as the loop group of $G$, this says that $\rho$ is a representation of that loop group.

but in as far as $\Lambda G$ is regarded as the action groupoid of the adjoint action of $G$ on itself, this says that $\rho$ is – an equivariant vector bundle on $G$!

That’s because for each $g \in G$ we have a vector space $\rho_g$, which we may regard as the fiber of a vector bundle over $g$. Furthermore, $\rho(g \stackrel{h}{\to} g')$ is an isomorphism between the fibers over $g$ and $g'$ that are related under the adjoint action. This means we get an $\mathrm{Ad}_G$-equivariant structure on the vector bundle.

This statement actually holds also for vector bundles over topological or smooth spaces: an equivariant vector bundle is a (continuous, or smooth) functor from the action groupoid to vector spaces.

I haven’t introduced the twist yet, so what I said so far corresponds to vanishing level, $k=0$, in the FHT theorem. This will be remedied shortly. But for the moment, just note how the analogue of the FHT theorem has now emerged:

Fact: The category of representations of the $\Lambda G$ is the same as that of equivariant vector bundles over $G$.

To get the twist, we need to know what the analogue of a gerbe over $G$ is in the case that $G$ is a finite group. But this is what I talked about in my first posting on Willerton’s work:

Flat Sections and Twisted Groupoid Reps

For finite groups, all connections on bundles we might consider are flat, as are those for gerbes. But a flat gerbe with connection is nothing but a closed 2-form. Moreover, a closed 2-form is nothing but a (pseudo)functor to $\Sigma(\Sigma(U(1)))$ #.

If the domain is a groupoid, then, in turn, it is easily verified that a pseudofunctor from the groupoid to $\Sigma(\Sigma(U(1)))$ is nothing but a groupoid 2-cocycle (with values in $U(1)$).

This is how Simon Willerton defines the “twist” for finite groups.

Now, as discussed at the link given above (which in turn discusses Willerton’s account), for given such twist we can consider the corresponding twisted representations of the groupoid.

And, again, since our groupoid is $\Lambda G$, this means two things at the same time for us:

Fact (Willerton’s finite-group version of the FHT theorem): the category of twisted representations of $\Lambda G$ is the same as that of equivariant twisted vector bundles over $G$.

Compare this to the statement of the FHT result above, and notice that the level $k$ appearing there is nothing but the continuous version of the twist in the finite case.

I should maybe emphasize that the nontrivial theorem here is the “parmesan theorem” which shows that it is justified to think of $\Lambda G$ as the finite analog of the loop group.

The beautiful thing is that given this, the finite-group analog of FHT is essentially an elementary triviality, following from the observation that $\Lambda G$ is, in turn, nothing but the action groupoid of $G$ on itself.