Seidel Representation for Symplectic Orbifolds

TL;DR

This paper extends the Seidel representation to symplectic orbifolds by defining a homomorphism from the fundamental group of Hamiltonian diffeomorphisms to the invertible elements of the orbifold quantum cohomology ring.

Contribution

It introduces a new Seidel representation for symplectic orbifolds, linking their Hamiltonian diffeomorphism groups to orbifold quantum cohomology.

Findings

Defined the fundamental group of Hamiltonian diffeomorphisms for orbifolds

Constructed a homomorphism to invertible elements in orbifold quantum cohomology

Extended Seidel representation to the orbifold setting

Abstract

Let $(\X,\omega)$ be a compact symplectic orbifold. We define $\pi_1(Ham(\X, \omega))$, the fundamental group of the 2-group of Hamiltonian diffeomorphisms of $(\X, \omega)$, and construct a group homomorphism from $\pi_1(Ham(\X, \omega))$ to the group $QH_{orb}^*(\X,\Lambda)^{\times}$ of multiplicatively invertible elements in the orbifold quantum cohomology ring of $(\X, \omega)$. This extends the Seidel representation ([Se], [M]) to symplectic orbifolds.