The String Coffee Table

Re: Categorified Gauge Theory

Posted by Urs Schreiber

Over on sci.physics.research Thomas Larsson is trying to understand how drastic the restriction of vanishing fake curvature in a 2-bundle/nonabelian gerbe really is. Does it imply that the nonabelian 2-bundle/gerbe can be ‘reduced’ in some sense to an abelian 2-bundle/gerbe?

I am not sure. Here is what I can say about this question:

To recall some notions, we need the following:

A crossed module $(G,H,t,\alpha)$, where $G$ and $H$ are Lie groups and $t: H \rightarrow G$ is a homomorphisms and $\alpha : G \to {Aut}(H)$ an action of $G$ on $H$, together with its differential version $(\mathfr{g},\mathfr{h},dt,d\alpha)$. Given a good cover $\{U_i\}_{i\in I}$ of the base space $B$ there are $\mathfr{g}$-valued 1-forms $A_i$ and $\mathfr{h}$-valued 2-forms $B_i$ on each $U_i$ which induce a local connection 1-form

on any path space $\mathcal{P}_s^t(U_i)$. Here $\oint_A(\omega_1, \dots, \omega_n)$ denotes the differential form on path space obtained by pulling back the target space forms $\omega_j$ to a given path and integrating them over a parameter $n$-simplex.

Given a bigon in $U_i$, i.e. a thin homotopy equivalence class of a smooth parametrized surface with two corners, there is a curve in path space mapping to that bigon and the surface holonomy of that bigon can be defined to be the ordinary holonomy of $\mathcal{A}$ along that curve. This notion of surface holonomy can be shown to compute 2-group holonomy and induce on the $A_i$ and $B_i$ the transformation laws of a nonabelian gerbe with connection and curving – but only if if the ‘fake curvature’ vanishes: $F_A + dt(B) = 0$. (This is for strict 2-groups and gets modified for coherent ones.)

Does this imply that we can compute the nonabelian surface holonomy of closed surfaces by integrating an abelian 3-form over a 3-volume?

In order to answer apply the nonabelian Stokes theorem on path space. The curvature of $\mathcal{A}$ can be shown to be

i.e. for vanishing fake curvature

This takes values in the abelian subalgebra ${ker}(dt) \subset \mathfr{h}$.

If the 2-bundle/nonabelian gerbe induced an ordinary bundle on path space this would imply that the structure group of this bundle could be reduced to an abelian one. But this is not the case. Maybe a similar reduction is still possible, but I do not see how it would work. To see the subtleties, we can derive the nonabelian volume integral that computes the nonabelian surface holonomy at the boundary of its integration domain:

For starters, restrict attention to the case that the surface in question is the boundary $\partial V$ of a 3-dimensional submanifold $V$ in a single $U_i$. The 3-fold $V$ comes from a surface $\Sigma$ in path space and the path space holonomy over the boundary of that surface is by the nonabelian Stokes theorem given by the integral

where $T_\mathcal{A}$ denotes the parallel transport of $\mathcal{F}$ along a curve of a foliation of $\Sigma$ (all in path space itself).

There is an implicit integral over the paths that are points in this integral. If you write that out the whole thing becomes roughly the integral of an abelian 3-form $H_i = \mathbf{d}_{A_i} B_i$ over $V$ but $H_i$ here at every point is parallel transported with $A_i$ to the $\sigma$-origin and with $\mathcal{A}$ to a $\tau$-origin, where $\mathcal{A}$ itself involves lots of $\sigma$-integrals.

In general, this does not seem to have any simple expression in terms of an ordinary integral. Of course, when the adjoint action of $\mathfr{h}$ on itself is trivial the whole thing simplifies a lot. And this action indeed is trivial for the crossed modules that I know of.

All that would remain in that case is the parallel transport with respect to $A_i$. Only if we also assume that the action of $G$ on ${ker}(dt) \subset \mathfr{h}$ is trivial does the whole integral reduce to the ordinary

Note that this assumed that we can work in a single patch in the first place. In the general case where surfaces in different patches have to be glued together using the 2-bundle/gerbe cocylce transition laws, things get more involved.

So obviously vanishing fake curvature requires the 2-bundle to be ‘close’ to being abelian, in a sense. How close this really is is not clear to me yet. It seems that the non-abelianness of the 2-group connection becomes relevant mostly for ‘global’ problems, like surfaces that wrap cycles, where we cannot in principle work in a single patch $U_i$.

In this context one is reminded of the fact that the membranes attached to 5-branes described by this formalism are required to wrap nontrivial cycles, too.