Description
I continue the series of my not-so-regular talks devoted to problems of non-abelian integration over surfaces. I will start with a broad-brush introduction to the subject, covering in particular Eckmann-Hilton argument, a sort of no-go theorem essentially forbidding two-dimensional non-abelian constructions relying on associative multiplication. We will hopefully see how this argument affects a version of non-abelian 2-form fields relying on the crossed module theory, essentially bounding their 3-curvatures to abelian sector. After that I will try and explain my own work on the subject, which uses contraction of three-index tensors instead of associative multiplication as a base operation. The basic example I hope to understand completely soon enough is the restoration of (infinitely-indexed) multiplication tensor from structure constants in Lie algebras, given by continuous-limit BCH formula. One of the key understanding checks for non-abelian surface integration theory would be giving a loop-space differential equation-based derivation for this formula. The prevoznemoganie techniques introduced and developed by Profs. Zaigraev and Kolganov are essential in this process, but the fast-working methods have been surprisingly fruitless so far.
