The NilHecke schober

TL;DR

This paper constructs an explicit algebraic perverse schober to model the Fukaya category of a specific Hilbert scheme, advancing the categorical understanding in knot invariant categorification.

Contribution

It provides the first explicit algebraic construction of a perverse schober for the Fukaya category of a horizontal Hilbert scheme, based on $A_n$-schobers.

Findings

Construction verified via diagrammatic calculus.

Schober axioms are satisfied in the algebraic framework.

Connects perverse schobers with knot invariant categorification.

Abstract

Perverse schobers can be used to describe Fukaya categories but are hard to axiomatize and construct. In this paper, we give an explicit construction of a perverse schober intended to accurately describe the Fukaya category of the horizontal Hilbert scheme considered by Aganagic et al. in the frame of the knot invariant categorification program. The formalism is based on the notion of $ A_n $-schobers of Dyckerhoff and Wedrich. The construction is entirely algebraic and we check the schober axioms with the help of diagrammatic calculus.