Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces

TL;DR

This paper proves that homological mirror symmetry for very affine hypersurfaces aligns with natural symplectic operations, confirming conjectures and revealing dual mirrors related through non-geometric equivalences.

Contribution

It verifies Auroux's conjectures on the compatibility of mirror symmetry with symplectic operations and introduces new techniques for Liouville manifold presentations.

Findings

Mirror symmetry respects symplectic functors between Fukaya categories.

Complement of a hypersurface has two natural mirrors, one being a derived scheme.

Kn"orrer periodicity relates the two mirrors via a non-geometric equivalence.

Abstract

We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has in fact two natural mirrors, one of which is a derived scheme. These two mirrors are related via a non-geometric equivalence mediated by Kn\"orrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors.