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 $\Lambda/\langle s\rangle$:$D^b($\mathrm{c}\mathrm{o}\mathrm{h}$ Z(s))\simeq D^{$\mathrm{g}\mathrm{r}$}_{$\mathrm{s}\mathrm{g}$}(\Lambda/\langle s\rangle)$. Our geometric applications all come from homogeneous gauged linear sigma models. In this case $\mathrm{\Lambda }$ is a non-commutative resolution of the invariant ring which defines the ${\mathbb{C}}^{\ast }$-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.