Minicourse starting July 13 at 3:00 (note special time!):

Lev Rozansky

2-categories of affine Nakajima quivers, categorical representations of braid groups and a categorification of HOMFLY-PT polynomial

Abstract:

This is a joint work with Alexei Oblomkov. I will try to explain the following:

1. To a symplectic variety X one associates a special 2-category B(X). Its objects are fibrations whose bases are Lagrangian subvarieties of X. The morphisms between objects are defined as categories of matrix factorizations. Generally, this construction is work in progress. When X is a cotangent bundle: X = T*Y, the 2-category B(X) is (presumably) equivalent to the 2-category of Y appearing in `derived algebraic geometry'.

2. The definition of B(X) is rigorous when X is (an open subvariety of) a cotangent bundle with the Hamiltonian action of a group G. In particular, B(X) can be defined for Nakajima quiver (and Nakajima-Cherkis quiver-bow) varieties.

3. Consider a special case when X is the variety of an affine quiver corresponding to the moduli space of n U(r) instantons on C^2. For a special object FL in B(X) (related to a flag variety) we define a homomorphism from the (affine) braid group on n strands to the monoidal category End(FL). When r=0 (`commuting variety') it categorifies the affine Hecke algebra (equivalent to Bezrukavnikov-Riche), when r=1 (Hilbert scheme of points on C^2) it categorifies the ordinary Hecke algebra (equivalent to Soergel-Rouquier) and for r>1 it categorifies the cyclotomic Hecke algebra.

4. In case of r=1 the space of morphisms between the images of a braid and the identity braid categorifies the HOMFLY-PT polynomial of the braid closure. Then the categorical version of the Riemann-Roch theorem says that to a closed braid one associates a complex of sheaves on the Hilbert scheme whose homology coincides with the link homology.

5. The object FL comes from the `quiver-in-quiver' construction related to a 3d TQFT with defects. In Nakajima quiver terms, we interpret a circle with number n as 2-category of a moduli space of n instantons and an edge connecting two such circles as a functor of a Lagrangian correspondence between two instanton moduli spaces. Then an A-type quiver 0-1-2- … - n - … - 2-1-0 becomes the monoidal category End(FL) used for braid representations. From the 3d TQFT perspective, we consider a surface split into regions by intersecting curves: the regions are `colored’ with instanton moduli spaces, the curves are colored by Nakajima-type Lagrangian correspondences and the intersection points are colored by matrix factorizations corresponding to braid crossings.