Equivariant Lagrangian Floer theory on compact toric manifolds

Abstract

We define an equivariant Lagrangian Floer theory for Lagrangian torus fibers in a compact symplectic toric manifold equipped with a subtorus action. We show that the set of all Lagrangian torus fibers with weak bounding cochain data whose equivariant Lagrangian Floer cohomology is non-zero can be identified with a rigid analytic space. We prove that the set of these Lagrangian torus fibers is the tropicalization of the rigid analytic space. This provides a way to locate them in the moment polytope. Moreover, we prove that the dimension of such a rigid analytic space is equal to the dimension of the subtorus when the symplectic manifold is $\mathbb{CP}^n$ or has complex dimension less than or equal to 2. We also show that the Lagrangian submanifolds with non-trivial equivariant Floer cohomology are non-displaceable by $G$-equivariant Hamiltonian diffeomorphisms.