Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds

TL;DR

This paper develops Floer theory for Lagrangian torus fibers in symplectic toric orbifolds, classifies holomorphic orbi-discs, and introduces bulk-deformed potentials to explore orbifold Hamiltonian geometry.

Contribution

It introduces a classification of holomorphic orbi-discs and develops a bulk-deformed Floer theory for toric orbifolds, extending previous manifold results.

Findings

Classification of holomorphic orbi-discs with boundary on Lagrangian fibers

Existence of a bulk-deformed Floer theory incorporating twisted sectors

Development of potential and leading order potential for orbifolds

Abstract

We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk-deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk-deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential. We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta and Ono.