## November 22, 2005

### Line-2-Bundles and Bundle Gerbes

#### Posted by urs

I’d be grateful for any comments on the following notes.

Line-2-Bundles and Bundle Gerbes

Abstract:

A line-2-bundle with 2-connection is defined to be a smooth functor from 2-paths to ${\mathrm{Vect}}_{1}$, where ${\mathrm{Vect}}_{1}$ is regarded as a 2-category with a single object. Pre-trivializations of a line-2-bundles are defined and shown to be in bijection with abelian bundle gerbes. The 2-category of pre-trivialized line-2-bundles should be equivalent to that of abelian bundle gerbes over a fixed fibration.

$\phantom{\rule{thinmathspace}{0ex}}$

Parallel transport of points in bundles is most naturally described in terms of parallel transport functors. In fact, bundles with connection can be entirely encoded in a functor from some path category to some transport category.

Here we are interested in an analogous statement for categorifed abelian gauge theory and the parallel transport of abelian strings.

Our main result is that pre-trivializations of smooth 2-functors

(1)$\mathrm{tra}:{P}_{2}\left(M\right)\to {\mathrm{Vect}}_{1}$

from 2-paths in a smooth space to a smooth sub-2-category of the the monoidal category of 1-dimensional vector space (regarded as a 2-category with a single object) are in bijection with abelian bundle gerbes with connective structure. We expect that this extends to an equivalence of the respective 2-categories.

Our constructions entirely follow those in [13,15], the only difference being that we are not dealing with principal2-bundles as defined in [8] but with something we call line-2-bundles.

The fiber of a principal 2-bundle is defined to be a 2-torsor for its structure 2-group. The 2-torsor condition on the fibers turns out to be too rigid to describe general gerbes globally. Our line-2-bundles with 2-connection locally look like principal ${G}_{2}$-2-bundles with 2-connection, for ${G}_{2}$ given by the crossed module ${G}_{2}=\left(U\left(1\right)\to 1\right)$.

Segal proposed [1] that ‘string connections’ of the kind we are interested in should be 1-functors from 2-dimensional cobordisms to $\mathrm{Vect}$. A description of parallel transport of string in abelian gerbes in terms of 1-functors on 2-cobordisms has been discussed in [10]. However, Stolz and Teichner observed [2] that 1-functors are too coarse a tool to capture all aspects of string connections. They instead pass to 2-functors from a sort of 2-paths into some 2-category, such that Segal’s picture is obtained as a special case when these 2-paths form cobordisms.

In this sense, following [13,15], our definition of parallel transport in line-2-bundles given below is more along the lines of Stolz and Teichner’s string connections, than Segal’s cobordism 1-functors.

In fact, there is a natural motivation of our definition of 2-connection in line-2-bundles obtained from imagining a string as a continuous family of objects in a 1-cobordism category. This is discussed in detail in section 2.2 after some technical preliminaries in section 2.1.

In section 2.3 we then state our definition of a line-2-bundle with 2-connection and discuss how 1-automorphisms of line-2-bundles with 2-connection are related to ordinary line bundles with connection.

In section 2.4 this is used to obtain the concept of an abelian bundle gerbe from a line-2-bundle. A ‘pre-trivialization’ of a line 2-bundle is defined to be an operation where the line-2-bundle is locally identified with trivial line-2-bundles which are related by transition 1-morphisms of line-2-bundles. As mentioned above, these 1-morphisms turn out to correspond to ordinary line bundles with connection and are in fact the bundles appearing in the notion of a bundle gerbe.

A full trivialization of a line-2-bundle would then be obtained by further trivializing these transition bundles. Fully trivializing a line-2-bundle yields a locally trivialized $\left(U\left(1\right)\to 1\right)$-2-bundle along the lines of [13,15].

Posted at November 22, 2005 7:59 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/686

### Re: Line-2-Bundles and Bundle Gerbes

Hi Urs,

I hope someone else comes back with some useful comments for you. After skimming the paper, the most I can offer is that I found it to be surprisingly readable. The figures (which I know must have taken some effort) almost fooled me into believing I understood more than I did :)

I had one question though, which I understand if you are too busy to respond to my peripheral questions. You said, “The most powerful tool to go from n-arrows to p-forms is probably synthetic
differential calculus as used in [3, 4].” Sorry if I am beginning to sound like a broken record, but would there be any value in going from an n-arrow to a discrete p-form? When I was studying synthetic geometry before, I had somehow managed to convince myself that it may be a kind of continuum limit of the discrete geometry. Maybe that is just way off. Sorry :|

Eric

Posted by: Eric on November 23, 2005 12:12 AM | Permalink | Reply to this

### Re: Line-2-Bundles and Bundle Gerbes

Hi Eric,

I found it to be surprisingly readable.

:-)

but would there be any value in going from an n-arrow to a discrete p-form? When I was studying synthetic geometry before, I had somehow managed to convince myself that it may be a kind of continuum limit of the discrete geometry.

That synthetic stuff to a large extent is a math-plugin that allows you to think discretely and takes care automatically of the continuum interpretation, in a sense. That’s why it’s so nicely compatible with functorial reasoning. Take a functor which goes from here to there, feed it into synthetic calculus and obtain the ‘differential’ of this functor.

I think one could easily do the following: Consider line-2-bundles like discussed in these notes, but internalized into some discrete setting. That is, replace the 2-category of smooth 2-paths with some sort of 2-category of 2-simplices or something like that. Nothing changes in the entire discussion, except that prop. 5 is replaced by saying something like that a trivial line 2-bundle is a functor from simplices to complex numbers. A discrete cochain.

Once this is under control, synthetic calculus would essentially tell you how to make these simplices ‘infinitesimally small’ and interpret that discrete cochain as a 2-form. That’s what I tried to discuss in these other notes that I linked to recently.

Posted by: Urs on November 23, 2005 6:18 PM | Permalink | Reply to this
Read the post NABGs from 2-Transport I: Synthetic Bibundles
Weblog: The String Coffee Table
Excerpt: Deriving nonabelian bundle gerbes from 2-functor 1-morphisms.
Tracked: December 7, 2005 7:19 PM
Read the post Topological Strings from 2-Transport
Weblog: The String Coffee Table
Excerpt: Fukuma-Hosono-Kawai TFT is obtained from 2-transport in Vect.
Tracked: December 14, 2005 5:11 PM
Read the post On Transport Theory
Weblog: The String Coffee Table
Excerpt: Overview of the theory of n-transport.
Tracked: February 17, 2006 3:34 PM