Categories of singularities of invertible polynomials

TL;DR

This paper investigates categories of singularities from invertible polynomials in Landau-Ginzburg models, providing computational methods, explicit constructions for low dimensions, and confirming Orlov's conjecture on block decompositions.

Contribution

It introduces an efficient method for computing morphism spaces and constructs explicit full strongly exceptional collections for small dimensions, advancing understanding of these categories.

Findings

Computed morphism spaces efficiently

Constructed explicit exceptional collections for n≤3

Proved Orlov's conjecture on block decompositions

Abstract

We study the categories of singularities coming from Landau-Ginzburg models given by the invertible polynomials. Such categories appear on the B-side of the Berglund-H\"ubsch mirror symmetry. We provide an efficient method of computing morphism spaces in these categories and explicitly construct full strongly exceptional collections in the cases of small dimensions ($n\le 3$). Finally, we use this construction in order to prove Orlov's conjecture stating that such collections can be chosen to have block decompositions of size one more than the number of variables.