The String Coffee Table

Nonabelian gerbes with connection and curving from 2-bundles with 2-holonomy

Posted by Urs Schreiber

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.