Symmetries of Lagrangian fibrations (Q2637913)

For a large class of symplectic manifolds with Lagrangian torus fibrations the authors construct fiber-preserving anti-symplectic involutions. In particular, they treat a K3 surface and a quintic Calabi-Yau threefold studied by the first two authors independently (in preparation). The results are interpreted as corroboration of the view that in homological mirror symmetry an anti-symplectic involution is the mirror of duality. In the same setting, the authors construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.