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November 23, 2006

The Baby Version of Freed-Hopkins-Teleman

Posted by Urs Schreiber

Recently I had discussed # one aspect of the paper

Simon Willerton
The twisted Drinfeld double of a finite group via gerbes and finite groupoids
math.QA/0503266 .

There are many nice insights in that work. One of them is a rather shockingly simple explanation of the nature of the celebrated Freed-Hopkins-Teleman result # - obtained by finding its analog for finite groups.

Here I will briefly say what Freed-Hopkins-Teleman have shown for Lie groups, and how Simon Willerton finds the analog of that for finite groups.

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

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

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

(1)Ω^ kG \hat \Omega_k G

of the loop group ΩG\Omega G of GG.

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

(2)Rep(Ω^ kG) \mathrm{Rep}(\hat \Omega_k G)

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 Rep(Ω^ kG)\mathrm{Rep}(\hat \Omega_k G) this is known as the Verlinde ring

(3)R k(Ω^G) R^k(\hat \Omega G)

of GG at level kk.

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

And there is another ring which we can canonically associate to GG and kk:

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

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

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

(4)K G k+r(G), K^{k+r}_G(G) \,,

where rr is some offset that I won’t describe.

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

(5)R k(Ω^G)K G k+r(G). R^k(\hat \Omega G) \simeq K_G^{k+r}(G) \,.

So there.

Find the details in

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

Now Willerton’s version of this theorem for GG 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

(6)par=Σ() \mathrm{par} = \Sigma(\mathbb{Z})

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

(7)P=Σ(G) P = \Sigma(G)

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


(8)conf:=ΛG:=Hom(par,P) \mathrm{conf} := \Lambda G := \mathrm{Hom}(\mathrm{par},P)

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 conf\mathrm{conf} does indeed behave like the loop group of a Lie group.


(9)BΛGΩBG. B\Lambda G \simeq \Omega B G \,.

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

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

(10)BΩGΩBG. B \Omega G \simeq \Omega B G \,.

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

(11)ΛG \Lambda G

plays the role of the loop group of GG.

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

Notice that ΛG\Lambda G is in fact a groupoid. Its objects are elements gg of GG. Its morphisms

(12)ghg g \stackrel{h}{\to} g'

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

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

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

(13)ρ:ΛGVect. \rho : \Lambda G \to \mathrm{Vect} \,.

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

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

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

That’s because for each gGg \in G we have a vector space ρ g\rho_g, which we may regard as the fiber of a vector bundle over gg. Furthermore, ρ(ghg)\rho(g \stackrel{h}{\to} g') is an isomorphism between the fibers over gg and gg' that are related under the adjoint action. This means we get an Ad G\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=0k=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 ΛG\Lambda G is the same as that of equivariant vector bundles over GG.

To get the twist, we need to know what the analogue of a gerbe over GG is in the case that GG 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 Σ(Σ(U(1)))\Sigma(\Sigma(U(1))) #.

If the domain is a groupoid, then, in turn, it is easily verified that a pseudofunctor from the groupoid to Σ(Σ(U(1)))\Sigma(\Sigma(U(1))) is nothing but a groupoid 2-cocycle (with values in U(1)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 ΛG\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 ΛG\Lambda G is the same as that of equivariant twisted vector bundles over GG.

Compare this to the statement of the FHT result above, and notice that the level kk 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 ΛG\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 ΛG\Lambda G is, in turn, nothing but the action groupoid of GG on itself.

Posted at November 23, 2006 7:06 PM UTC

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3 Comments & 7 Trackbacks

Re: The Baby Version of Freed-Hopkins-Teleman

Yes, its true - it seems so obvious in Simon’s paper! Its the reason I like the whole setup described in `The twisted Drinfeld double of a finite group via gerbes and finite groupoids’. Freed-Teleman-Hopkins is trivial! … for finite groups, at least :-)

Moreover, it seems you guys have already invented the technology to extend this stuff to Lie groups… Higher gauge theory and 2-bundles, for instance.

Posted by: Bruce Bartlett on November 23, 2006 10:12 PM | Permalink | Reply to this

Re: The Baby Version of Freed-Hopkins-Teleman

Yes, its true - it seems so obvious in Simon’s paper!

Yes, it’s great. The best thing that can happen: turn a deep statement into a triviality by adopting the right point of view.

Here, the change of viewpoint which makes this possible is to realize that we should be thinkong of the loop group in terms of the action groupoid, using the Parmesan Theorem.

I am wondering: what happens to the Parmesan Theorem as we remove the restriction that GG be a finite group?

it seems you guys have already invented the technology to extend this stuff to Lie groups…

At least I am trying to think about that.

I have a candidate # for what replaces the “target space” Σ(G)\Sigma(G) in this context: namely something like the 2-group coming from the crossed module # (ΩGPG)(\Omega G \to P G), or rather INN(ΩGPG)\mathrm{INN}(\Omega G \to P G).

Notice how this is nothing but a “fattened” version of Σ(G)\Sigma(G), since

(1)(ΩGPG)(1G). (\Omega G \to P G) \simeq (1 \to G) \,.

The point is that on this “fattened” realization of GG, we know how to implement the twist, by centrally extending the loop group.

I am guessing that what in the finite group setup is a groupoid 3-cocycle, namely a 3-functor (pseudo functor)

(2)tra:Σ(G)Σ 3(U(1)), \mathrm{tra} : \Sigma(G) \to \Sigma^3(U(1)) \,,

realizing a flat abelian 2-gerbe with connection on Σ(G)\Sigma(G), becomes in the Lie group case a 3-functor with values in the canonical 1-dimensional 3-vector space #, which assigns to each 2-cell (which now is a loop on GG) the 1-dimensional ordinary vector space that is associated to the central extension living over that loop.

Then one should probably redo what I did for 2-functors here, but now using that 3-functor.

That’s currently my line of attack.

Notice how in that approach, too (check out the latest version # of my notes), the loop groupoid of GG arises automatically.

In particular, I seem to find that on parameter space we do get a functor to Rep(ΛG)\mathrm{Rep}(\Lambda G)-module categories.

That’s what I was hoping for. For the CFT description on parameter space I know I need a functor to Rep(Ω^G)\mathrm{Rep}(\hat \Omega G)-module categories.

So this would be the right thing - iff something like the Parmesan Theorem also holds for the Lie group GG. At least if Rep(ΛG)\mathrm{Rep}(\Lambda G) would be equivalent to Rep(ΩG)\mathrm{Rep}(\Omega G).

So: what happens to the Parmesan Theorem for GG Lie?

Posted by: urs on November 24, 2006 9:27 AM | Permalink | Reply to this

Re: The Baby Version of Freed-Hopkins-Teleman

what happens to the Parmesan Theorem for GG Lie?

Actually, using Freed-Hopkins-Teleman, I can turn this question around:

I only care about groupoids up to Morita equivalence. Hence any two groupoids with the same representation category should be considered equivalent.

While FHS does not say that the category of reps of the loop group is equivalent to that of the action groupoid, it does say so after we decategorify.

Hm. Just a while ago we had a visitor who was thinking about if and how an abelian modular tensor category can be reconstructed (up to something) from its Grothendieck ring.

That’s what we’d need here.

That is, assuming that nothing goes wrong with identifying the (twisted) equivariant K-theory of GG with the Grothendieck ring of (twisted) reps of its action Lie groupoid. That should be right, shouldn’t it?

Or I can put it this way:

if I knew that the (twisted) representation category of the action Lie groupoid of the (simple, simply connected, compact) Lie group GG were a modular tensor category, then I would take that as the definition of the modular tensor category used to set up the 3DTFT/2DCFT following Turaev/FRS.


for GG a suitable Lie group, is

(1)Rep τ(G/G) \mathrm{Rep}^\tau(G/G)


Posted by: urs on November 24, 2006 11:20 AM | Permalink | Reply to this
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