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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 .


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 κ between the 2-map given by A i and that given by g ijA jg ij 1 on double overlaps.:

(2)κ:A ig ij(d+A j)g ij 1 .

(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 ijkΩ 0 Lie(H) satisfying

(3)g ijg jk=Ad f ijkg ik

on triple overlaps.

2) The coherence law for this natural transformation says that

(4)f ikl 1 f ijk 1 g ij(f jkl)f ijl=1

on quadruple overlaps.

3) The natural transformation κ encodes functions a ijΩ 1 Lie(H) satisfying

(5)A i+ad(a ij)=g ij(d+A j)g ij 1

on double overlaps.

4) The coherence law associated with κ 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 iΩ 2 Lie(H) satisfying

(6)K i+ad(B i)=0 ,

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+ad(k ij)=g ij(d+A j)g ij 1 ,


(8)k ij=da ij+a ija ij+A i(a ij).

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 ij(B j)+k 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)(f ijk,g ij,a ij,A i,B i,d ij,H i)

for the special case

ad(B i)=K i, d ij=0 , ad(H i)=0 .


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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