Symplectomorphisms of some Weinstein 4-manifolds

TL;DR

This paper introduces new symplectomorphisms of certain Weinstein 4-manifolds, demonstrating their mirror correspondence to line bundle tensors and automorphisms, and explores their role in the autoequivalence group of the wrapped Fukaya category.

Contribution

It defines two novel families of symplectomorphisms and establishes their mirror relationships, advancing understanding of symplectic automorphisms in mirror symmetry for Weinstein 4-manifolds.

Findings

Lagrangian translations mirror to line bundle tensors

Nodal slide recombinations mirror to automorphisms of (Y,D)

Symplectomorphisms, together with spherical twists, generate autoequivalences

Abstract

Let M be a Weinstein four-manifold mirror to Y\D for (Y,D) a log Calabi--Yau surface; intuitively, this is typically the Milnor fibre of a smoothing of a cusp singularity. We introduce two families of symplectomorphisms of M: Lagrangian translations, which we prove are mirror to tensors with line bundles; and nodal slide recombinations, which we prove are mirror to automorphisms of (Y,D). The proof uses a detailed compatibility between the homological and SYZ view-points on mirror symmetry. Together with spherical twists, these symplectomorphisms are expected to generate all autoequivalences of the wrapped Fukaya category of M which are compactly supported in a categorical sense. A range of applications is given.