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October 4, 2006

Wang at AustMS 2006

Posted by Guest

The following “guest post” is by David Roberts:

At the Australian Mathematical Society’s annual meeting in Sydney, Brian Wang gave the talk
Gerbes, D-branes and loop group representations. This is of particular interest to at least one blogger here, so I provide some details.

The talk wasn’t precisely on all the mentioned buzz-words, but really outlined a different method of thinking about the geometry in

[1] A.L. Carey and Bai-Ling Wang
Fusion of symmetric DD-branes and Verlinde rings

I will leave it to those who know better to explain the gist of that paper, and focus on the talk. All groupoids in what follows are Lie groupoids (and if necessary, Fréchet).

Essentially, we want to be able to define D-branes for spacetime represented by a groupoid, and think sensibly about what the HH-flux does about this.

So an HH-flux is just the 3-curvature of a bundle gerbe. We want to think of as an extension of groupoids, with kernel the trivial bundle of groups U(1) tU(1)_t (the base space will be given by context). That is, something like

U(1) tG X U(1)_t \to G_{\bullet} \to X_{\bullet}

for X X_{\bullet} “spacetime”, G G_{\bullet} the bundle gerbe (with G 0=X 0G_0 = X_0)and U(1) t=U(1)×X 0U(1)_t = U(1) \times X_0. Examples of X X_{\bullet} to keep in mind are: (1) the Cech groupoid of an open cover of a manifold MM, or more generally of a surjective submersion; (2) the action groupoid associated to a GG-manifold and (3) the suspension ΣG\Sigma G of a group.

Definition: Given any groupoid Γ \Gamma_{\bullet}, a geometric cycle for Γ \Gamma_{\bullet} is a vector bundle π:EΓ 0\pi:E \to \Gamma_0 with an action

E× π,sΓ 1EE \times_{\pi, s} \Gamma_1 \to E

where the fibred product over Γ 0\Gamma_0 is by the indicated maps. In the cases above a geometric cycle is (1) a vector bundle, (2) an equivariant vector bundle and (3) a representation of the group GG.

[A morphism of geometric cycles, would clearly be an equivariant map of vector bundles, or equivalently, a smooth functor between the action groupoids arising from the groupoid action on the vector bundle]

We can thus define a category Cycle Γ\mathbf{\mathrm{Cycle}}_{\Gamma} of geometric cycles for a groupoid Γ\Gamma (and, I imagine, for groupoids in general). It doesn’t take too much to guess what happens next: Take the Grothendieck group of Cycle Γ\mathbf{\mathrm{Cycle}}_{\Gamma}! Well maybe you didn’t see it coming, but let’s look at our examples: (1) gives us K 0(M)K^0(M), (2) K G 0(M)K^0_G(M) and (3) R(G)R(G), the representation ring of GG.

If we have a manifold QQ and a map ϕ:QΓ 0\phi:Q \to \Gamma_0, we can form the pullback groupoid ϕ *Γ\phi^*\Gamma (sometimes denoted Γ[Q]\Gamma[Q] and Γ| Q\Gamma|_Q) in the obvious way, as the source and target are submersions.

Definition A D-brane for the space X X_{\bullet} supporting the bundle gerbe G G_{\bullet} as above is a manifold QQ, a map ϕ:QX\phi: Q \to X and a geometric cycle EQE \to Q for ϕ *G \phi^*G_{\bullet}

The example to keep in mind: Let 1U(1)ΩG^ kΩG11 \to U(1) \to \widehat{\Omega G}_k \to \Omega G \to 1 be a level kk central extension of the based loop group of a simply connected Lie group GG. Denote by 𝒜 S 1Ω 1(S 1,𝔤)\mathcal{A}_{S^1} \simeq \Omega^1(S^1,\mathfrak{g}) the space of connections on the trivial GG-bundle over the (parameterised, pointed) circle. The holonomy map 𝒜 S 1G\mathcal{A}_{S^1} \to G gives us an ΩG\Omega G bundle (ΩG\Omega G is the space of based gauge transformations - those which are the identity at the basepoint) which is universal - recall that GBΩGG \sim B\Omega G. We can then form the mulitplicative lifting bundle gerbe 𝒢 k\mathcal{G}_k using the central extension above. A D-brane is supplied by a quasihamiltonian GG-space and its moment map.

And where does one buy a quasihamiltonian GG-space? I hear you ask. Well, given a Riemann surface Σ\Sigma with Σ=S 1\partial \Sigma = S^1, and a point on the boundary, the moduli space Σ\mathcal{M}_\Sigma of flat connections on Σ×G\Sigma \times G modulo the based gauge transformations (wrt the point on the boundary) is a quasihamiltonian GG-space and the holonomy around the boundary is the moment map.

Proposition 4.1 of [1] tells us that the pullback of 𝒢 k\mathcal{G}_k to Σ\mathcal{M}_\Sigma is then stably trivial (or, if you like, admits a module of rank 1) and even equivariantly so.

We can generalise this to a genus gg Riemann surface with kk boundary components, Σ g,n\Sigma_{g,n}, where the moment map is now to G nG^n, and we pull pack the bundle gerbe 𝒢 k\mathcal{G}_k via the various projection maps p i:G nGp_i:G^n \to G to get ip i *𝒢 k\bigotimes_i p_i^* \mathcal{G}_k over G nG^n. The pullback to Σ g,n\mathcal{M}_{\Sigma_{g,n}} is equivariantly trivialised as before.

The next step is to consider the sphere with 3 holes, Σ 0,3\Sigma_{0,3} - i.e. the pair of pants

Posted at October 4, 2006 8:23 AM UTC

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gerbe modules

Thanks a lot for the report!

Definition A D-brane for the space X X_\bullet supporting the bundle gerbe G G_\bullet as above is a manifold QQ, a map ϕ:QX\phi : Q \to X and a geometric cycle EQE \to Q for ϕ *G\phi^* G

Unless I am missing something, this geometric cycle is usually called a module for a gerbe (which can be thought of as a twisted bundle, e.g. for U(H)PU(H)U(H) \to P U(H) , for ordinary U(1)U(1)-gerbes).

I guess the point here is to emphasize that a gerbe module can be regarded as a special case of a more general concept - namely geometric cycles for groupoids? Is that right?

In fact, another way to realize a gerbe module as a special case of a more universal principle is to conceive it as a trivialization of a gerbe, but in an enlarged ambient category. (This is related to what we talked about here.)

The example to keep in mind: […]

The holonomy map classifies an ΩG\Omega G-bundle on the space 𝒜 S 1\mathcal{A}_{S^1} of connections on the trivial GG-bundle G×S 1S 1G \times S^1 \to S^1.

Lifting the structure group to ΩG^ k\widehat {\Omega G}_k produces a lifting gerbe G G_\bullet on that space.

We can pull this back to the space of flat connections Σ\mathcal{M}_\Sigma on a Riemann surface Σ\Sigma with an S 1S^1-boundary.

(1) Σp𝒜 Σ \mathcal{M}_\Sigma \stackrel{p}{\to} \mathcal{A}_\Sigma

The proposition tells us that G G_\bullet has a rank-1 module, hence that p *G p^* G_\bullet is trivial.

So I guess as a direct corollary we find that the pullback of the ΩG\Omega G-bundle from 𝒜 S 1\mathcal{A}_{S^1} to Σ\mathcal{M}_\Sigma does admit a lift to a ΩG^ k\widehat {\Omega G}_k-bundle (since the lifting gerbe is trivial there).

Okay, nice. Surely this is next applied to - something…

Posted by: urs on October 5, 2006 12:54 PM | Permalink | Reply to this

Re: gerbe modules

urs wrote:

I guess the point here is to emphasize that a gerbe module can be regarded as a special case of a more general concept - namely geometric cycles for groupoids? Is that right?

exactly. I should have mentioned this among the examples of geometric cycles.

Surely this is next applied to - something…

Indeed, but I wanted to get something up. The final aim is to relate the fusion ring of D-branes to the Verlinde ring of GG. The fusion coefficients apparently come from the pair of pants, but I’m still thinking about that. There is a bit of Chern-Simons in there as well, but what I have above is all of the talk. I’m also still trying to understand the paper!

Posted by: David Roberts on October 6, 2006 6:54 AM | Permalink | Reply to this

Re: Wang at AustMS 2006

I just noticed that slides of Wang’s talk in Vienna this summer # are available here.

That’s not exactly the talk that David reported about above, but closely related, and with some overlap.

Posted by: urs on November 24, 2006 10:48 AM | Permalink | Reply to this
Read the post D-Branes from Tin Cans, II
Weblog: The n-Category Café
Excerpt: Gerbe modules from 2-sections.
Tracked: November 28, 2006 9:44 PM
Read the post QFT of Charged n-particle: Chan-Paton Bundles
Weblog: The n-Category Café
Excerpt: Chan-Paton bundles from the pull-push quantization of the open 2-particle.
Tracked: February 7, 2007 9:59 PM

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