Monodromic model for Khovanov-Rozansky homology

TL;DR

This paper introduces a geometric model for Hochschild cohomology of Soergel bimodules, providing a new perspective on Khovanov-Rozansky knot homology through monodromic Hecke categories and character sheaves.

Contribution

It presents a novel geometric framework for Hochschild cohomology and offers a different geometric description of Khovanov-Rozansky homology from previous models.

Findings

Identifies objects representing Hochschild cohomology groups in the monodromic Hecke category.

Relates these objects to explicit character sheaves of the Weyl group.

Provides a new geometric approach to Khovanov-Rozansky knot homology.

Abstract

We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild cohomology groups (for the zero and the top degree cohomology this reduces to an earlier result of Gorsky, Hogancamp, Mellit and Nakagane). These objects turn out to be closely related to explicit character sheaves corresponding to exterior powers of the reflection representation of the Weyl group. Applying the described functors to the images of braids in the Hecke category of type A we obtain a geometric description for Khovanov-Rozansky knot homology, essentially different from the one considered earlier by Webster and Williamson.