3D TQFT, Soergel bimodules and knot homology

Geometry, Symmetry and Physics
Event time: 
Monday, October 14, 2019 - 4:30pm
Alexei Oblomkov
Speaker affiliation: 
UMass Amherst
Event description: 

Talk is based on joint work with Lev Rozansky. I will explain a mathematical
construction of N=4 gauge 3D TQFT, known as Kapustin-Saulina-Rozansky theory.
The defects in this theory encode the braids and that allows us to categorify
Ocneanu-Jones trace and obtain a triply-graded knot homology. We show the
homology coincide with the Khovanov-Rozansky trace on the Rouquier complexes
on Soergel bimodules. Because of the geometric nature of our TQFT
construction we obtain an interpretation of the homology as space  of   global sections of a sheaf on the Hilbert scheme of point on the plane. I will also explain the relation between our result and the conjectures of Gorsky-Negut-Rasmussen.