### 2-Holonomy in Terms of 2-Torsors

#### Posted by Urs Schreiber

One thing that is sort of obvious, but which has not been written out in detail yet, is that the global 2-holonomy 2-functor defined in terms of local trivializations as discussed here more intrinsically is a 2-functor

from 2-paths (surfaces) in the base manifold $M$ to the 2-category of $G_2$-2-torsors.

A discussion of this can be found here:

**$p$-Functors from $p$-Paths to $p$-Torsors**

http://www-stud.uni-essen.de/~sb0264/Transport.pdf

as well as in section 12.4 of this.

Speaking of which, I would also like to mention here that no less than three of the early-morning talks at the Streetfest will be about aspects of categorified gauge theory.

Michael Kapranov will talk about Noncommutative Fourier-Transforms, Chen’s iterated integrals and Higher-Dimensional Holonomy. In the abstract he says:

[…] Then we explain how to extend this correspondence to represent higher-dimensional membranes by elements of a certain differential graded algebra $A$. This is related to the concept of holonomy of gerbes that attracted a lot of attention recently. We will also give the interpretation of higher gerbe holonomy in terms of Chen’s iterated integrals of forms of higher degree with coefficients in Lie-algebraic analogs of crossed modules and crossed complexes. If one views higher holonomy as a “pasting integral” then 2-dimensional associativities translate into vanishing of some brackets in the structure dg-Lie algebra which is automatic in the crossed module case but has to be imposed in general.

I am eager to learn more about the details he has in mind. This must be closely related to what I have been thinking about lately.

Then, Ezra Getzler will talk about Lie theory for $L_\infty$-algebras.

The Deligne groupoid is a homotopy functor from nilpotent differential graded Lie algebras concentrated in positive degree to groupoids. We generalize this to a functor from nilpotent $L_\infty$-algebras concentrated in degrees $(-n, \infty)$ to $n$-groupoids, or rather, to their nerves. On restriction to abelian differential graded Lie algebras, i.e. chain complexes, our functor is the Dold–Kan functor. The construction uses methods from rational homotopy theory, and gives a generalization of the Campbell-Hausdorff formula.

This is presumeably based on

E. Getzler

**Lie theory for nilpotent $L_\infty$-algebras**

math.AT/0404003

As far as I understand, this is about the problem of integrating weak (semistrict, really) Lie $p$-algebras to weak Lie $p$-groups. You might recall that I had recently made some elemetary comments on that here.

## Streetfest

Hi Urs

Yes, Mathematical Physics is one of the focuses of the Streetfest. Ross Street himself is publishing in physics journals now, with for instance

“Frobenius monads and pseudomonoids” J. Math. Phys. 45, 10 (2004) 3930

There is also a workshop following the Streetfest, with some exciting talks. For example, Tim Porter will speak about “Homotopy Quantum Field Theories with background a 2-type, formal maps and gerbes” and Robin Cockett will talk about “Differential and Smooth Categories”.