The n-Category Café

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 $D$-branes and Verlinde rings
arXiv:math-ph/0505040

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 $H$-flux does about this.

So an $H$-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 $U(1)_t$ (the base space will be given by context). That is, something like

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

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

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

$$E \times_{\pi, s} \Gamma_1 \to E$$

where the fibred product over $\Gamma_0$ 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 $G$.

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

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

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

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

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

Proposition 4.1 of [1] tells us that the pullback of $\mathcal{G}_k$ to $\mathcal{M}_\Sigma$ 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 $g$ Riemann surface with $k$ boundary components, $\Sigma_{g,n}$, where the moment map is now to $G^n$, and we pull pack the bundle gerbe $\mathcal{G}_k$ via the various projection maps $p_i:G^n \to G$ to get $\bigotimes_i p_i^* \mathcal{G}_k$ over $G^n$. The pullback to $\mathcal{M}_{\Sigma_{g,n}}$ is equivariantly trivialised as before.

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