Singularity categories of normal crossings surfaces, descent, and mirror symmetry

Abstract

Given a smooth 3-fold $Y$, a line bundle $L \to Y$, and a section $s$ of $L$ such that the vanishing locus of $s$ is a normal crossings surface $X$ with graph-like singular locus, we present a way to reconstruct the singularity category of $X$ as a homotopy limit of several copies of the category of matrix factorizations of $xyz : \mathbb{A}^{3} \to \mathbb{A}^{1}$ (the mirror to the Fukaya category of the pair of pants). This extends our previous result for the case where $L$ is trivialized. The key technique is the classification of non-two-periodic autoequivalences of the category of matrix factorizations. We also present a conjectural mirror for these singularity categories in terms of the Rabinowitz wrapped Fukaya categories of Ganatra-Gao-Venkatesh for certain symplectic four-manifolds, and relate this construction to work of Lekili-Ueda and Jeffs.