Quantum Mathematics: Charles Young (Hertfordshire)

Quantum Maths Seminar, external speaker.

Speaker: Charles Young (University of Hertfordshire)

Title: Homotopy Manin triples and higher current algebras

Abstract: The notion of a Manin triple of Lie algebras crops up in many contexts. After recalling the general definition, I will describe one important class of examples involving current algebras, i.e. certain Lie algebras associated to the punctured formal disc in complex dimension one. Studying these examples naturally leads one to recover the ideas of vertex algebras and rational conformal blocks, as I will try to describe.

Now, one would like to generalize all this to higher complex dimensions. (I will sketch one source of motivation, coming from quantum Gaudin models and integrable quantum field theory.) A possible approach to doing so starts with higher current algebras in the sense Faonte, Hennion and Kapranov. I will review the definition, which involves passing from Lie algebras to their differential graded (dg) analogs. In the dg setting, it is natural to relax the definition of a Manin triple, by requiring some statements to hold only up to homotopy. I will go on to describe explicit models for certain higher current algebras, built from polynomial differential forms on semisimplicial sets of flags. In terms of these, I will describe some examples of such homotopy Manin triples.

This talk is based on recent work https://arxiv.org/abs/2208.06009 joint with Luigi Alfonsi.

One-hour in-person talk