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October 18, 2018

Analysis in Higher Gauge Theory

Posted by John Baez

Higher gauge theory has the potential to describe the behavior of 1-dimensional objects and higher-dimensional membranes much as ordinary gauge theory describes the behavior of point particles. But ordinary gauge theory is also a source of fascinating differential equations, which yield interesting results about topology if one uses enough analysis to prove rigorous results about their solutions. What about higher gauge theory?

Andreas Gastel has a new paper studying higher gauge theory using some techniques of analysis that are commonly used in ordinary gauge theory. He’s finding some interesting similarities but also some ways in which higher gauge theory is simpler:

Abstract. We study the problem of finding good gauges for connections in higher gauge theories. We find that, for 2-connections in strict 2-gauge theory and 3-connections in 3-gauge theory, there are local “Coulomb gauges” that are more canonical than in classical gauge theory. In particular, they are essentially unique, and no smallness of curvature is needed in the critical dimensions. We give natural definitions of 2-Yang-Mills and 3-Yang-Mills theory and find that the choice of good gauges makes them essentially linear. As an application, (anti-)selfdual 2-connections over B 6B^6 are always 2-Yang-Mills, and (anti-)selfdual 3-connections over B 8B^8 are always 3-Yang-Mills.

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 n\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\|A\|_{W^{2,2}}+\|B\|_{W^{1,2}} by Z A,B L 2\|Z_{A,B}\|_{L^2} in dimensions m6m \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,BZ_{A,B} is a kind of ‘curvature’ of the 2-connection (A,B)(A,B), which consists of a Lie-algebra-valued 1-form AA and a Lie-algebra-valued 2-form BB:

Z A,B=dB+α̲(A)B Z_{A,B} = d B + \underline{\alpha}(A) \wedge B

AA and BB 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 α̲\underline{\alpha}, and this lets us form the wedge product α̲(A)B\underline{\alpha}(A) \wedge B.

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

Posted at October 18, 2018 9:14 PM UTC

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