Homological mirror symmetry for Brieskorn-Pham singularities

TL;DR

This paper establishes a deep equivalence between symplectic and algebraic categories for Brieskorn-Pham singularities, advancing the understanding of homological mirror symmetry in singularity theory.

Contribution

It proves the homological mirror symmetry conjecture for Brieskorn-Pham singularities using symplectic Picard-Lefschetz theory.

Findings

Derived Fukaya category is equivalent to the triangulated category of singularities.

The equivalence involves a grading by an abelian group of rank one.

The proof relies on symplectic Picard-Lefschetz theory.

Abstract

We prove that the derived Fukaya category of the Lefschetz fibration defined by a Brieskorn-Pham polynomial is equivalent to the triangulated category of singularities associated with the same polynomial together with a grading by an abelian group of rank one. Symplectic Picard-Lefschetz theory developed by Seidel is an essential ingredient of the proof.