On the equatorial Dehn twist of a Lagrangian nodal sphere

TL;DR

This paper proves that an equatorial Dehn twist on a Lagrangian nodal sphere cannot be extended to a Hamiltonian diffeomorphism in certain symplectic manifolds and explores related mirror symmetry predictions.

Contribution

It establishes the non-extendability of equatorial Dehn twists to Hamiltonian diffeomorphisms and confirms mirror symmetry predictions for their action on Floer theory.

Findings

Equatorial Dehn twist does not extend to Hamiltonian diffeomorphism.

Mirror symmetry predicts specific actions on Floer theory.

Analogues and conjectures for more singular Lagrangians.

Abstract

Let $(M^4,\omega)$ be a geometrically bounded symplectic manifold, and $L\subset M$ a Lagrangian nodal sphere such that $\omega\mid_{\pi_2(M,L)}=0$. We show that an equatorial Dehn twist of $L$ does not extend to a Hamiltonian diffeomorphism of $M$. We also confirm a mirror symmetry prediction about the action of a symplectomorphism extending an equatorial Dehn twist on the Floer theory of the nodal sphere. We present analogues of the equatorial Dehn twist for more singular Lagrangians, and make concrete conjectures about them.