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

TL;DR

This paper develops a method to reconstruct the singularity category of certain normal crossings surfaces using matrix factorizations, extending previous results and proposing a conjectural mirror symmetry framework involving wrapped Fukaya categories.

Contribution

It introduces a new approach to reconstruct singularity categories via homotopy limits of matrix factorizations, generalizing earlier trivial line bundle cases and classifying autoequivalences.

Findings

Reconstruction of singularity categories as homotopy limits.

Extension of previous results to non-trivial line bundles.

Conjectural mirror symmetry involving wrapped Fukaya categories.

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.