Braid group actions on matrix factorizations

TL;DR

This paper constructs an action of the extended affine Braid group on the $G$-equivariant derived category of matrix factorizations on a specific geometric setting involving the Grothendieck variety and cotangent bundle, linking algebraic group actions with derived categories.

Contribution

It introduces a novel action of the extended affine Braid group on the $G$-equivariant derived category of matrix factorizations in a geometric context involving the Grothendieck variety and moment maps.

Findings

Established a new algebraic action on derived categories.

Connected Braid group actions with matrix factorizations in geometric representation theory.

Provided a framework for further exploration of symmetries in algebraic geometry.

Abstract

Let $X$ be a smooth scheme with an action of a reductive algebraic group $G$ over an algebraically closed field $k$ of characteristic zero. We construct an action of the extended affine Braid group on the $G$-equivariant absolute derived category of matrix factorizations on the Grothendieck variety times $T^*X$ with potential given by the Grothendieck-Springer resolution times the moment map composed with the natural pairing.