The n-Category Café

I think this is great. I don’t know if the 2-Yang Mills and 3-Yang Mills equations studied here are interesting in physics — they are, in any event, being studied on Euclidean $\mathbb{R}^n$ — but they might be interesting in differential geometry. And I’m very glad that Gastel read and tried to fix my old paper on higher Yang–Mills theory.

More generally, I think it’s great to get analysts involved in studying higher structures. Categorically-minded mathematicians and homotopy type theorists shouldn’t just soar up to the paradise of $\infty$-categories and leave their colleagues stranded on Earth in some mathematical equivalent of the Rapture.

There are interesting discoveries to be made by bringing other mathematicians into the conversation! For example, Gastel writes:

… we have a 2-Yang-Mills functional that really resembles Yang–Mills. Therefore, it is tempting to ask whether there is a higher form of Uhlenbeck’s gauge theorem, controlling norms like $\|A\|_{W^{2,2}}+\|B\|_{W^{1,2}}$ by $\|Z_{A,B}\|_{L^2}$ in dimensions $m \le 6$ once a suitable 2-gauge is fixed, maybe under a smallness condition for the latter norm. One of our results will be that this really works. But, surprisingly enough for the author, it turns out that the good 2-gauge exists without a smallness condition, and moreover the transformed 2-connection has a canonical form that has the potential to simplify the theory very much.

This kind of blend of ideas makes me happy: Sobolev spaces, differential geometry and higher categories!

Here $Z_{A,B}$ is a kind of ‘curvature’ of the 2-connection $(A,B)$, which consists of a Lie-algebra-valued 1-form $A$ and a Lie-algebra-valued 2-form $B$:

$$Z_{A,B} = d B + \underline{\alpha}(A) \wedge B$$

$A$ and $B$ take values in different Lie algebras, in general, because they come from a Lie 2-algebra. But the first of these Lie algebras acts as derivations of the second via $\underline{\alpha}$, and this lets us form the wedge product $\underline{\alpha}(A) \wedge B$.

The point, therefore, is that if the $L^2$ norm of the curvature is small, we can find a gauge in which $A$ and $B$ are small in a suitable sense. This is a generalization of a fundamental result by Uhlenbeck for ordinary gauge theories, proved in 1982.