Defects in skein theory

Abstract: I will give an overview of some recent progress constructing skein module invariants of 3-manifolds with defects. The defects to be discussed come in three different classes detailed below. Warning: with all these examples to cover, there might be precious few proofs!

I will explain how "electric-magnetic 1-form symmetry" arises in skein theory as invertible line defects, and how it enters into (conjectural) electric-magnetic/Langlands/S-duality, following joint work with Gunningham and Safronov.

I will then discuss a recent work of Jennifer Brown and myself; independent works of Juan Ramon Gomez; separate independent works of Julia Bierent and Matthias Vancraeynest, which ares all around constructing defects in skein theory modelling parabolic induction and restriction (by non-invertible plane defects), as well as Weyl group twists (by invertible line defects), with applications to the quantum A-polynomial, the irregular Deligne--Simpson problem and the abelianisation program of Gaiotto--Moore-Neitzke.

Finally, I will touch on some nascent joint work with Eric Chen and Iordanis Romaidis aimed at constructing and analysing line defects associated to quantum symmetric pairs, with a view towards exploring the relative Langlands program of Ben-Zvi--Sakellaridis--Venkatesh.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home