Monodromy groups of Lagrangian tori in the symplectic 4-space

TL;DR

This paper characterizes the monodromy groups of Clifford tori in symplectic 4-space, revealing their algebraic structure and providing explicit formulas, with implications for isotopy classes of Lagrangian tori.

Contribution

It explicitly determines the Lagrangian and smooth monodromy groups of Clifford tori, including their algebraic structures and generating elements, and discusses isotopy properties.

Findings

L(T) is isomorphic to the infinite dihedral group

S(T) is generated by three reflections

Smooth isotopy to Clifford torus can be Lagrangian outside a disc

Abstract

We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three reflections. We give explicit formulas for both groups. We also show that if a Lagrangian torus is smoothly isotopic to a Clifford torus then the smooth isotopy can be chosen to be Lagrangian outside of a disc.