Knot Homologies from Landau-Ginsburg Models

Date: 2022-07-27

Time: 14:30 - 15:30


Miroslav Rapcak


In her recent work, Mina Aganagic introduced novel ways to construct knot homologies for any simple Lie algebra. One of her proposals relies on finding intersection points between Lagrangians in Landau-Ginsburg models on symmetric powers of Riemann surfaces. More concretely, she cuts the knot into halves and associates a Lagrangian to each of the two parts. In principle, such Lagrangians admit a description in terms of a complex of special “thimble” generators. Morphisms between the Lagrangians are then given by morphisms between such complexes in the homotopy category. The difficulty lies in the constructions of the complexes themselves.

In our work with Mina Aganagic and Elise LePage, we propose an explicit algorithm to find such complexes in two steps:
1. The system of thimbles together with an approximate differential can be read off directly from the knowledge of the support of a Lagrangian.
2. The full differential $delta$ can be bootstrapped by identifying an ansatz for correction terms and solving for $delta^2 =0$.
In my talk, I will illustrate this construction in various examples. I will also briefly comment on how to identify reduced homological invariants, $mathbb{Z}_2$ factors in homologies and how to extend the story to Lie super-algebras.