Abstract:  Skein modules capture vector spaces of line operators in 3D Chern-Simons, equivalently line operators constrained to a 3-dimensional boundary in the Kapustin-Witten twist of 4D N=4 gauge theory.  They have an elementary mathemical definition via representation theory of quantum groups.

 

In recent work with Iordanis Romaidis we proved that when the quantum parameter is generic, the skein module of a 3-manifold is finitely generated relative to the skein algebra of its boundary and that moreover the resulting singular support variety is Lagrangian, hence that skein modules are holonomic. Our results confirm and strengthen a conjecture of Detcherry, and imply a conjecture of Frohman, Gelca and LoFaro from 2002 (the latter independently established this year by Beletti and Detcherry using other methods).

 

In the talk I will give an outline of the key ingredients of the proof, which recreate elements of the classical theory of differential operators in the skein setting.