The String Coffee Table

New paper: 2-Connections on 2-Bundles

Posted by Urs Schreiber

I am glad to announce that the paper

John Baez & Urs Schreiber: Higher Gauge Theory: 2-Connections on 2-Bundles (2004)

is available now on the preprint server as

hep-th/0412325.

(This link will be generally accessible next Sunday or Monday. If it does not work yet a copy can be found here.)

Here is the abstract:

Connections and curvings on gerbes are beginning to play a vital role in differential geometry and mathematical physics — first abelian gerbes, and more recently nonabelian gerbes. These concepts can be elegantly understood using the concept of ‘2-bundle’ recently introduced by Bartels. A 2-bundle is a generalization of a bundle in which the fibers are categories rather than sets. Here we introduce the concept of a ‘2-connection’ on a principal 2-bundle. We describe principal 2-bundles with connection in terms of local data, and show that under certain conditions this reduces to the cocycle data for nonabelian gerbes with connection and curving subject to a certain constraint — namely, the vanishing of the ‘fake curvature’, as defined by Breen and Messing. This constraint also turns out to guarantee the existence of ‘2-holonomies’: that is, parallel transport over both curves and surfaces, fitting together to define a 2-functor from the `path 2-groupoid’ of the base space to the structure 2-group. We give a general theory of 2-holonomies and show how they are related to ordinary parallel transport on the path space of the base manifold.

Readers of this blog will recall many posts concerned with these issues.

The paper builds a bridge from work on local nonabelian surface holonomy as presented in

Alvarez, Ferreira & Sánchez Guillén: A new approach to integrable theories in any dimension (1997)

H. Pfeiffer Higher gauge theory and a non-Abelian generalization of 2-form electromagnetism (2003)

F. Girelli & H. Pfeiffer Higher gauge theory - differential versus integral formulation (2003)

U. Schreiber Nonabelian 2-forms and loop space connections from 2d SCFT deformations (2004)

to the cocycle description of nonabelian gerbes as described in

L. Breen & W. Messing: Differential geometry of gerbes (2001)

P. Aschieri, L. Cantini & B. Jurčo: Nonabelian bundle gerbes, their differential geometry and gauge theory (2003)

using the categorification approach of

J. Baez: Higher Yang-Mills theory (2002)

T. Bartels: Categorified gauge theory: 2-bundles (2004)

by demonstrating that a 2-bundle with 2-connection under certain conditions defines a nonabelian gerbe with connection and curving together with a notion of globally defined nonabelian 2-holonomy (surface holonomy) for that gerbe. (This uses strict structure 2-groups which implies that the ‘fake curvature’ $\mathrm{dt}(B)+F_A$ has to vanish.)

The basic tools here are path space differential calculus on the one hand (‘calculus of string’) and category theory (‘stringification’) on the other. Roughly, an ordinary connection $𝒜$ on path space can be used to construct a 2-functor $\mathrm{hol}$ from the 2-groupoid ${𝒫}_2(M)$ of ‘bigons’ (surface elements, roughly) to the structure 2-group $𝒢$

Having a notion of nonabelian surface holonomy for nonabelian gerbes is necessary for writing down actions for membranes ending on stacks of 5-branes and should hence also be required for nonabelian 6-dimensional conformal field theories and their relation to Yang-Mills theory in four dimensions.

As discussed in the last section of the above paper, the generalization to coherent structure 2-groups still needs to be properly done and is not treated in that paper. But hopefully in a followup.

Happy New Year!

Posted at December 30, 2004 11:28 AM UTC