The String Coffee Table

Over at sci.math.research there are some comments on categorification and its relation to string theory.

Of course categories, being the language in which god wrote math, show up all over the place once people start to seriously do some mathematical physics, and hence here and there in string theory, but apart from their ‘general purpose applications’ the notion of categorization has a rather special nature, since it always amounts to identifying some well known things as ‘points’ and generalizing them to linearly extended things: morphisms, arrows - strings.

The most obvious example is the categorification of the configuraton space of a single particle: This is just some sort of path space over the original config space (where the paths are morphisms between the point objects in the obvious way) which can generically be identified with the configuration space of a single string. In fact, such a space is known as a 2-space.

If our particle was a superparticle its configuration space was something like the exterior bundle over the original configuration space and similarly we expect the config space of the superstring to be the exterior bundle over a 2-space, which it is. The superparticle may be governed by supersymmetric quantum mechanics of the well-known form $e^{-W}d e^{W}|\psi\rangle = 0 = e^{W^\dagger}d^\dagger e^{-W^\dagger}|\psi\rangle$ and lifting that to path space indeed produces SCFTs describing string propagation in various backgrounds encoded in $W$.

I should take the time to formalize that a little better, but it seems indeed that one can justly say that strings are categorized point particles, in a sense.

My recent occupation with higher gauge theory has added considerably to my impression that this is much more than a general abstract nonsensical play of words. If strings on stacks of 5-branes really are to nonabelian gerbes as points are to fiber bundles and given the fact that these nonabelian gerbes are essentially categorified fiber bundles precisely along the space$\to$2-space line of reasoning sketched above, this shows that there is nontrivial string physics made visible by the strings-as-categorized-points point of view which is otherwise not or not as clearly tractable - it seems to me.

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’.)

Theorem: A $G$-2-bundle with (‘categorically discrete’) base 2-space, strict structure automorphism 2-group, connection and 2-holonomy defines a nonabelian gerbe with connection and inner curving .

Proof:

The following sketch of a proof heavily uses definitions and results from math.CT/0410328, hep-th/0409200 and hep-th/0309173. But a $G$-2-bundle with connection has not been defined yet, so here is the definition:

Definition: A G-2-bundle with connection is a 2-bundle with 2-cover $U$ together with a 2-map

(where $TU$ is the tangent 2-space to U) and with a natural transformation $\kappa$ between the 2-map given by $A_i$ and that given by $g_{ij}A_j g_{ij}^{-1}$ on double overlaps.:

(This is just the categorification of the transition law for a connection 1-form in an ordinary bundle.)

Now one can check the following:

(Let $G = (H,{Aut}(H),t={Ad})$ be the strict automorphism 2-group and $g$ be the transition 2-map.)

1) A 2-transition on the 2-bundle is a natural transformation that encodes functions $f_{ijk} \in \Omega^0 \otimes {Lie}(H)$ satisfying

on triple overlaps.

2) The coherence law for this natural transformation says that

on quadruple overlaps.

3) The natural transformation $\kappa$ encodes functions $a_{ij} \in \Omega^1 \otimes {Lie}(H)$ satisfying

on double overlaps.

4) The coherence law associated with $\kappa$ gives a further relation between $g_{ij}$, $A_i$, $a_{ij}$ and $f_{ijk}$.

5) The existence of a 2-holonomy in the 2-bundle implies locally the existence of 2-forms $B_i \in \Omega^2 \otimes {Lie}(H)$ satisfying

where $K_i$ is the curvature of $A_i$.

6) From 3) it follows that the transition law for $K_i$ is

where

7) From the fact that the 2-holonomy is globally defined (by assumption) it hence follows that the local $B_i$ are related by

on double overlaps.

The above list of equations characterizing properties of the 2-bundle with connection and holonomy can be checked to be the defining equations of a nonabelian gerbe with connection and curving characterized by the generalized cocycle

for the special case

${ad}(B_i) = -K_i$, $d_{ij} = 0$, ${ad}(H_i) = 0$.

Remark:

There are lots of of points where the above can be generalized: For one, it is possible to show how by allowing base 2-spaces whose arrow space is that of based loops one can get twisted nonabelian gerbes.

Then there is a strange dichotomy between generalizations possible on the 2-bundle side and those possible on the gerbe side: At the beginning of the above proof I severely restricted the possible properties of 2-bundles (e.g. they don’t need to have automorphism groups as structure 2-groups, as opposed to gerbes, and in fact they can have weak structure 2-groups), while at the end I restricted those of the nonabelian gerbe (by specializing to a very specific class of curving data).

But the latter restriction comes from the assumption that a 2-holonomy exists, as I have discussed a couple of times before here. It would be very interesting to better understand how this can be relaxed, if at all.