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# Pure and Applied Mathematics Quarterly

## Volume 17 (2021)

### Number 1

### Graded tilting for gauged Landau–Ginzburg models and geometric applications

Pages: 185 – 235

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n1.a5

#### Authors

#### Abstract

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.

Received 19 June 2020

Accepted 20 June 2020

Published 11 April 2021