Graded tilting for gauged Landau–Ginzburg models and geometric applications

In this paper we develop a graded tilting theory for gauged Landau–Ginzburg models of regular sections in vector bundles over projective varieties. Our main theoretical result describes—under certain conditions—the bounded derived category of the zero locus $Z(s)$ of such a section s as a graded singularity category of a non-commutative quotient algebra $\Lambda / {\langle s \rangle} : D^b (\operatorname{coh}Z(s)) \simeq D^\mathrm{gr}_\mathrm{sg} \Lambda / {\langle s \rangle}$. Our geometric applications all come from homogeneous gauged linear sigma models. In this case Λ is a graded noncommutative resolution of the invariant ring which defines the $\mathbb{C}^\ast$-equivariant affine GIT quotient of the model.

We obtain algebraic descriptions of the derived categories of the following families of varieties:

1. Complete intersections.

2. Isotropic symplectic and orthogonal Grassmannians.

3. Beauville–Donagi IHS 4-folds.

Keywords

derived category, derived singularity category, GIT quotient, gauged Landau–Ginzburg model, tilting, gauged linear sigma model, non-commutative resolutions, rank varieties

2010 Mathematics Subject Classification

Primary 14F05, 14L24, 14L30. Secondary 18E30.

Full Text (PDF format)

Received 19 June 2020

Accepted 20 June 2020

Published 11 April 2021