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November 24, 2006

2-Monoid of Observables on String-G

Posted by Urs Schreiber

The baby version of the Freed-Hopkins-Teleman result, as explained by Simon Willerton, suggests that we should be thinking of the modular tensor category

(1)CRep(Ω^ kG) C \simeq \mathrm{Rep}(\hat \Omega_k G)

that govers GG-Chern Simons theory and GG Wess-Zumino-Witten theory rather in terms of the representation category

(2)Rep k(G/G) \mathrm{Rep}_k(G/G)

of a central extension of the action groupoid

(3)G/GΛG G/G \simeq \Lambda G

of the adjoint action of GG on itself.

This monoidal category should arise as the 2-monoid of observables # that acts on the 2-space of states over a point as we consider the 3-particle propagating on a target space that resembles BGB G.

In turn, this 2-monoid of observables should arise # as the endomorphisms of the trivial transport on target space

(4)𝒜=End(1 *). \mathcal{A} = \mathrm{End}(1_*) \,.

Here I would like to show that when we model target space as

(5)P=Σ(String G), P = \Sigma(\mathrm{String}_G) \,,

where String G\mathrm{String}_G is the strict string 2-group #, coming from the crossed module # Ω^ kGPG\hat \Omega_k G \to P G, then sections on configuration space of the boundary of the 3-particle form a module category for

(6)ΛRep k(ΛG). \Lambda \mathrm{Rep}_k(\Lambda G) \,.

As discussed elsewhere (currently at the end of these notes), this should imply that states over a point are a module for

(7)Rep k(G/G). \mathrm{Rep}_k(G/G) \,.

This proof is a slight variation of that of prop. 4 in the notes on sections of 2-reps. The difference is that there I considered just the 2-particle (“string”), whereas now I am looking at the boundary of the 3-particle (“membrane”).

This may sound like almost the same thing, but the extended configuration spaces differ in both situations.

For the closed 2-particle propagating on Σ(G 2)\Sigma(G_2), which we model by

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

the configuration space is the sub-2-category

(2)conf[par,Σ(G)] \mathrm{conf} \subset [\mathrm{par},\Sigma(G)]

whose morphisms are restricted to describe only rotations of the string, but no true propagation.

On the other hand, what is propagation for the string is, in this sense, just reparameterization of the membrane. Hence for par\mathrm{par} considered as the boundary of the 3-particle, we take all of

(3)conf=[par,Σ(G 2)]. \mathrm{conf} = [\mathrm{par},\Sigma(G_2)] \,.

Or so I propose. I am trying to check this proposed formalism against examples, that’s what this here is about.

So I think I checked the following two statements:

proposition: For 1:Σ(G 2)Bim 1 : \Sigma(G_2) \to \mathrm{Bim} the trivial 2-rep of the strict 2-group G 2G_2, and for 1 *:conf[par,Bim] 1_* : \mathrm{conf} \to [\mathrm{par},\mathrm{Bim}] its transgression to configuration space, we get End(1 *)ΛRep(ΛG 2) \mathrm{End}(1_*) \simeq \Lambda \mathrm{Rep}(\Lambda G_2) as an equivalence of monoidal categories.

Here ΛG 2\Lambda G_2 is the loop 1-groupoid of the 2-group G 2G_2. This is a slight generalization of the loop groupoid ΛG\Lambda G of an ordinary group as used by Simon Willerton. ΛG 2\Lambda G_2 is obtained from conf=[Σ(),Σ(G 2)]\mathrm{conf} = [\Sigma(\mathbb{Z}),\Sigma(G_2)] by identifying isomorphic 1-morphisms.

To make use of that, we need to figure out how the loop groupoid of the String 2-group is like.

proposition The loop groupoid ΛString G\Lambda \mathrm{String}_G of the String 2-group is a central extension of the loop groupoid of GG itself

(4)U(1)ΛString GΛG. U(1) \to \Lambda \mathrm{String}_G \to \Lambda G \,.

I didn’t try to determine exactly which central extension it is. But there is only one plausible candidate: it should be the central extension obtained from the 3-cocycle on GG at level kk - that which governs Ω^G\hat \Omega G and hence String G\mathrm{String}_G - transgressed from GG to ΛG\Lambda G, as described in Simon Willerton’s paper.

You can find the details here:

\;\;\; 2-Monoid of Observables of String G\mathrm{String}_G.

This is in a sense just an addendum to

\;\;\; sections of 2-reps.

I’d consider this result to be encouraging. In as far as it is meaningful, the next thing one would like to do is to pass from the configuration space of just the boundary of the 3-particle to the entire 3-particle. This should go along with passing from target space a (suspended) 2-group to a full 3-group - as discussed here and here.

Posted at November 24, 2006 4:26 PM UTC

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Rep(Lambda String_G) and K-theory

I am beginning to suspect that the representation ring of the groupoid that I called

(1)ΛString G=[Σ(),Σ(String G)] / \Lambda \mathrm{String}_G = [\Sigma(\mathbb{Z}), \Sigma(\mathrm{String}_G)]_{/\sim}

is indeed the twisted equivariant K-theory of GG.

I have included a discussion of that at the end of my above notes.

It amounts to observing that the groupoid ΛString G\Lambda \mathrm{String}_G is generated from String G\mathrm{String}_G (regarded as just a groupoid) together with the action groupoid of GG acting on PGP G.

But notice that, as a groupoid, String G\mathrm{String}_G is nothing but the canonical bundle gerbe # on GG, and that a groupoid representation of String G\mathrm{String}_G (i.e. suitable functor String GVect\mathrm{String}_G \to \mathrm{Vect}) is nothing but a gerbe module #, otherwise known as a twisted bundle.

So from the fact that ΛString G\Lambda\mathrm{String}_G is generated from String G\mathrm{String}_G and the action groupoid of the adjoint action, it follows that a representation of ΛString G\Lambda \mathrm{String}_G is in particular a gerbe module - and one equipped with the additional structure that makes it also a representation of the action groupoid. But means nothing but that it must be an equivariant gerbe module.

If follows that

(2)Rep(ΛString G) \mathrm{Rep}(\Lambda \mathrm{String}_G)

is a category of equivariant twisted bundles on GG.

I still need to check some literature, but this should say that the representation ring coming from Rep(ΛString G)\mathrm{Rep}(\Lambda \mathrm{String}_G) is nothing but the twisted equivariant K-theory of GG.

If there is anyone reading this who could confirm this, or else point out where I went wrong, I’d greatly appreciate it.

Posted by: urs on November 27, 2006 10:09 PM | Permalink | Reply to this
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