Tilting Generator for the $T^*Gr(2,4)$ Coulomb Branch

TL;DR

This paper constructs an explicit tilting generator for the derived category of coherent sheaves on the cotangent bundle of the Grassmannian Gr(2,4), providing a geometric description in a specific symplectic resolution.

Contribution

It offers a concrete geometric construction of the tilting generator on T*Gr(2,4), extending Kaledin's abstract framework to an explicit case with natural bundles.

Findings

Explicit tilting generator as a sum of natural bundles

Connection to KLRW algebras in low-dimensional case

Concrete geometric description of the generator

Abstract

Remarkable work of Kaledin, based on earlier joint work with Bezrukavnikov, has constructed a tilting generator of the category of coherent sheaves on a very general class of symplectic resolutions of singularities. In this paper, we give a concrete construction of this tilting generator on the cotangent bundle of $Gr(2,4)$, the Grassmannian of 2-planes in $\mathbb{C}^4$. This construction builds on work of the second author describing these tilting bundles in terms of KLRW algebras, but in this low-dimensional case, we are able to describe our tilting generator as a sum of geometrically natural bundles on $T^*Gr(2,4)$: line bundles and their extensions, as well as the tautological bundle and its perpendicular.