The n-Category Café

On n-Transport: 2-Vector Transport and Line Bundle Gerbes

Posted by Urs Schreiber

My first examples for the general concept of $n$-transport # had been local transitions of principal 2-transport ##, taking values in some Lie 2-group.

For many applications in physics, one needs to have associated $n$-transport, taking values in $n$-vector spaces on which some $n$-group is represented.

Here I present some notes which contain

• a working definition of associated $n$-transport;
• an identification of a class of 2-representations of Lie 2-groups on bimodules # #;
• a description of (torsion) $U(1)$-gerbes with connection as associated 2-transport with respect to the above representation of the 2-group $\Sigma(\Sigma(U(1)))$;
• a demonstration that the transition data of such a 2-transport is precisely the data of a line bundle gerbe with connection and curving (and that in fact the 2-category of these transiton tetrahedra is equivalent to that of line bundle gerbes with connection).

Line Bundle Gerbes from 2-Transport

Apart from being interesting in its own right, this is supposed to be a warm-up for describing more sophisticated 2-vector transport.

In particular, one can see that by replacing the 2-group $\Sigma(\Sigma(U(1)))$ in the above setup with the strict 2-group $\mathrm{String}_G$ #, one obtains – up to technical subteties related to the fact that $\mathrm{String}_G$ is infinite-dimensional – something very similar (maybe identical) to the the notion of string-connection that Stolz&Teichner defined #.

I conjecture that

A string connection as defined by Stolz&Teichner is the 2-vector 2-transport associated to a principal $\mathrm{String}_G$-2-transport by way of the 2-representation defined in the above notes.

I think that in as far as this conjecture is false it is just due to technicalities that can be fixed – like the fact that in the above notes I use the ordinary tensor product of bimodules, while for the $\mathrm{String}_G$-application we need “Connes fusion” #.

(I was waiting with posting this entry until I had tied up some lose ends in the above notes. But since we are now discussing this already in the comment sections #, I thought I’d just ahead and post it.)