Higher Gauge Theory and Elliptic Cohomology
Posted by John Baez
After some fun in Greece, I’ve been holed up in Greenwich the last two days preparing my talk for the 2007 Abel Symposium. This is an annual gettogether sponsored by the folks who put out the Abel prize, a belated attempt to create something like a Nobel prize for mathematicians.
One of the themes of this year’s symposium is “elliptic objects and quantum field theory”. So, while my true love is higher gauge theory, my talk will emphasize its relation to elliptic cohomology and related areas of math:

John Baez, Higher Gauge Theory and Elliptic Cohomology.
Abstract: The concept of elliptic object suggests a relation between elliptic cohomology and "higher gauge theory", a generalization of gauge theory describing the parallel transport of strings. In higher gauge theory, we categorify familiar notions from gauge theory and consider "principal 2bundles" with a given "structure 2group". These are a slight generalization of nonabelian gerbes. After a quick introduction to these ideas, we focus on the 2groups $String_k(G)$ associated to any compact simple Lie group $G$. We describe how these 2groups are built using central extensions of the loop group $\Omega G$ and how the classifying space for $String_k(G)$2bundles is related to the "string group" familiar in elliptic cohomology. If there is time, we shall also describe a vector 2bundle canonically associated to any principal 2bundle, and how this relates to the von Neumann algebra construction of Stolz and Teichner.
I hope you folks — especially Urs — can look this over and find typos and other problems before I take off for Oslo on Sunday.
Urs will be pleased to see that I’ve caved in and started using his notation for the string 2group.
He’ll also note that I’m deliberately not explaining his construction of a 2vector bundle from a principal 2bundle for the string 2group — the construction that has very strong ties to Stolz and Teichner’s work on elliptic objects. I thought it would take too long at the end of a very long talk. Instead, while I provide links to his work, I’ll explained a similar construction which is simpler and ‘more canonical’ — it works easily for any 2group. This is Alissa Crans and Danny Stevenson and I were working on this spring.
I would like to better understand the relation of the two constructions!
Re: Higher Gauge Theory and Elliptic Cohomology
Nice. Interesting construction of the 2rep.
I’ll think about this. But not right now, I have a guest here.
One point which might be worth mentioning:
Mere 2functors with values in $\Sigma \mathrm{String}_k(G)$ will only describe String2bundles which come from lifting flat $G$bundles.
One way to see this is that the fakeflatness condition for 2functors $d A + A \wedge A + \partial B = 0$ with $A$ and $B$ taking values in some path algebra, translates here into a true flatness condition at the end of these paths, since $B(2\pi) = 0$.
But it’s $A(2\pi)$ which is the original connection on the $G$bundle of which the $\mathrm{String}_k(G)$bundle is a lift.
So we do need to pass to flat $\mathrm{INN}(\mathrm{String}_k(G))$3transport to get the general String2bundles.
But that’s not a bug, but a feature. For one $\mathrm{Lie}(\mathrm{INN}(\mathrm{String}_k(G))$ is isomorphic to the ChernSimons Lie 3algebra $\mathrm{cs}_k(g) \,.$ I didn’t explicitly mention it there, but one should take everything I described here and apply one more $\mathrm{inn}(\cdot)$ to it, to really get the codomains of the transport.