Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

TL;DR

This paper constructs a categorification linking the flag Hilbert scheme with Soergel bimodules, providing a geometric framework for understanding Khovanov-Rozansky homology and categorified Jones-Wenzl projectors.

Contribution

It introduces a monoidal functor between coherent sheaves on the flag Hilbert scheme and Soergel bimodules, connecting geometric and algebraic categorifications of the Hecke algebra.

Findings

Matching Hochschild homology of braids with sheaf Euler characteristics

Identification of categorified Jones-Wenzl projectors with Koszul complexes

Conjecture relating endomorphism algebras to dg algebras on affine charts

Abstract

We construct a categorification of the maximal commutative subalgebra of the type $A$ Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials $d_{N}$ on these dg algebras and conjecture that their homology matches that of the $\mathfrak{gl}_N$ projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende.