Symmetries of Lagrangian fibrations

TL;DR

This paper constructs anti-symplectic involutions and symplectomorphisms for symplectic manifolds with Lagrangian torus fibrations, providing insights into mirror symmetry and dualities.

Contribution

It introduces fiber-preserving anti-symplectic involutions and symplectomorphisms for a broad class of symplectic manifolds, including K3 surfaces and quintic threefolds, linking them to mirror symmetry concepts.

Findings

Construction of anti-symplectic involutions for K3 and quintic threefolds

Interpretation of involutions as mirror dualities in homological mirror symmetry

Development of symplectomorphisms as mirrors to line bundle twistings

Abstract

We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.