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

## TL;DR

This paper develops a graded tilting theory for gauged Landau-Ginzburg models, linking derived categories of zero loci to graded singularity categories of non-commutative algebras, with applications to various geometric varieties.

## Contribution

It introduces a new graded tilting framework for gauged Landau-Ginzburg models and provides algebraic descriptions of derived categories for several classes of varieties.

## Key findings

- Derived category of zero locus is equivalent to graded singularity category of a non-commutative algebra.
- Provides algebraic models for derived categories of complete intersections and special Grassmannians.
- Connects geometric models to non-commutative resolutions via GIT quotients.

## 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 $\Lambda/\langle s\rangle$: $D^b(\mathrm{coh} Z(s))\simeq D^{\mathrm{gr}}_{\mathrm{sg}}(\Lambda/\langle s\rangle)$. Our geometric applications all come from homogeneous gauged linear sigma models. In this case $\Lambda$ is a non-commutative resolution of the invariant ring which defines the $\mathbb{C}^*$-equivariant affine GIT quotient of the model. We obtain purely algebraic descriptions of the derived categories of the following families of varieties: - Complete intersections. - Isotropic symplectic and orthogonal Grassmannians. - Beauville-Donagi IHS 4-folds.