## November 3, 2004

### 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

(1)$A:\mathrm{TU}⟶\mathrm{Lie}\left(G\right)$

(where $\mathrm{TU}$ 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 ${g}_{\mathrm{ij}}{A}_{j}{g}_{\mathrm{ij}}^{-1}$ on double overlaps.:

(2)$\kappa :{A}_{i}\to {g}_{\mathrm{ij}}\left(d+{A}_{j}\right){g}_{\mathrm{ij}}^{-1}\phantom{\rule{thinmathspace}{0ex}}.$

(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 $G=\left(H,\mathrm{Aut}\left(H\right),t=\mathrm{Ad}\right)$ 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 ${f}_{\mathrm{ijk}}\in {\Omega }^{0}\otimes \mathrm{Lie}\left(H\right)$ satisfying

(3)${g}_{\mathrm{ij}}{g}_{\mathrm{jk}}={\mathrm{Ad}}_{{f}_{\mathrm{ijk}}}\phantom{\rule{thinmathspace}{0ex}}{g}_{\mathrm{ik}}$

on triple overlaps.

2) The coherence law for this natural transformation says that

(4)${f}_{\mathrm{ikl}}^{-1}{f}_{\mathrm{ijk}}^{-1}{g}_{\mathrm{ij}}\left({f}_{\mathrm{jkl}}\right){f}_{\mathrm{ijl}}=1$

on quadruple overlaps.

3) The natural transformation $\kappa$ encodes functions ${a}_{\mathrm{ij}}\in {\Omega }^{1}\otimes \mathrm{Lie}\left(H\right)$ satisfying

(5)${A}_{i}+\mathrm{ad}\left({a}_{\mathrm{ij}}\right)={g}_{\mathrm{ij}}\left(d+{A}_{j}\right){g}_{\mathrm{ij}}^{-1}$

on double overlaps.

4) The coherence law associated with $\kappa$ gives a further relation between ${g}_{\mathrm{ij}}$, ${A}_{i}$, ${a}_{\mathrm{ij}}$ and ${f}_{\mathrm{ijk}}$.

5) The existence of a 2-holonomy in the 2-bundle implies locally the existence of 2-forms ${B}_{i}\in {\Omega }^{2}\otimes \mathrm{Lie}\left(H\right)$ satisfying

(6)${K}_{i}+\mathrm{ad}\left({B}_{i}\right)=0\phantom{\rule{thinmathspace}{0ex}},$

where ${K}_{i}$ is the curvature of ${A}_{i}$.

6) From 3) it follows that the transition law for ${K}_{i}$ is

(7)${K}_{i}+\mathrm{ad}\left({k}_{\mathrm{ij}}\right)={g}_{\mathrm{ij}}\left(d+{A}_{j}\right){g}_{\mathrm{ij}}^{-1}\phantom{\rule{thinmathspace}{0ex}},$

where

(8)${k}_{\mathrm{ij}}=d{a}_{\mathrm{ij}}+{a}_{\mathrm{ij}}\wedge {a}_{\mathrm{ij}}+{A}_{i}\left({a}_{\mathrm{ij}}\right)\phantom{\rule{thinmathspace}{0ex}}.$

7) From the fact that the 2-holonomy is globally defined (by assumption) it hence follows that the local ${B}_{i}$ are related by

(9)${B}_{i}={g}_{\mathrm{ij}}\left({B}_{j}\right)+{k}_{\mathrm{ij}}$

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

(10)$\left({f}_{\mathrm{ijk}},{g}_{\mathrm{ij}},{a}_{\mathrm{ij}},{A}_{i},{B}_{i},{d}_{\mathrm{ij}},{H}_{i}\right)$

for the special case

$\mathrm{ad}\left({B}_{i}\right)=-{K}_{i}$, ${d}_{\mathrm{ij}}=0$, $\mathrm{ad}\left({H}_{i}\right)=0$.

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.

Posted at November 3, 2004 2:37 PM UTC

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