The String Coffee Table

Weak groups, hard computation

Posted by Urs Schreiber

2-bundles are great. They connect path space/loop space differential geometry and path space bundles with other stuff, like nonabelian gerbes. That’s nice for physics, because it allows to ‘see’ the string in the nonabelian background described by the gerbe: It’s configuration space is the arrow space of the base 2-space of the 2-bundle. Its constraints are gauge-covariant deRham operators on that space.

I have recently sketched a proof for how a 2-bundle with strict structure 2-group yields a (possibly twisted) nonabelian gerbe with curving and connection of a certain kind. In fact, it seems that except for one constraint the 2-bundle is more general. (For instance it turns out that the gerbe data encoded in the $d_{ij} \in {Lie}(H)\otimes \Omega^2(U_{ij})$ forms comes from infinitesimal loops in the arrow space of the 2-bundle’s base 2-space and are enriched for larger loops.)

That one constraint is the infamous $dt(B_i) + F_{A_i} = 0$, which comes from the nature of the strict structure 2-group.

But the most general 2-bundle has a coherent structure 2-group instead, and I have now worked out some facts related to surface holonomy using coherent 2-connections. There the above constraint is indeed alleviated! This might be interesting, since at the same time the data which makes a strict 2-group coherent is encoded in an object with three group indices, which might be a candidate carrier of the respective $\sim n^3$ degrees of freedom seen on 5-branes.

I don’t know if it is, but I know how to construct a generalization of a nonabelian gerbe with consistent surface holonomy which depends on a couple of algebra-valued $p$-forms plus an element of $H^3(G,K)$, where $K \subset H$ is an abelian group inside an non-associative algebra $H$. The key point is that in going from strict to coherent structure 2-groups one finds that up to “weakening” the essential equations remain intact:

Where the strict structure 2-group is described by a crossed module which involves the semidirect product of two groups, the coherent structure 2-group is described by what I tend to call a weak crossed module where the well-known relations hold only up to generalized similarity transformations which are determined by that element of $H^3(G,K)$. This leads to a weakened form of path space connection and hence to a new notion of surface holonomy.

Arriving at this point involved a lot of work though. The results that I have managed to extract so far are summarized in this set of notes . (Look for the subsection ‘Coherent 2-Groups’.)