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

Abstract

Given a smooth 3-fold Y , a line bundle L → 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 : A 3 → 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.