| In this dissertation, we define Seidel representation for compact symplectic orbifolds, and then apply it to compute the quantum cohomology of toric orbifolds.;Let (X, o) be a compact symplectic orbifold and let Ham(X,o) be the 2-group of Hamiltonian diffeomorphisms of (X, o). We develop a suitable notion of loops of Hamiltonian diffeomorphisms of (X, o), which is natural from a categorical view and necessary for application. Then we consider the set pi1(Ham(X,o)) of homotopy classes of (based) loops of Hamiltonian diffeomorphisms of ( X,o). Composition of loops gives pi1( Ham(X,o)) a natural group structure. We also generalize the notion of Hamiltonian fibration to the orbifold case, which is called Hamiltonian orbifibration. Each element in pi 1( Ham(X,o)) can be associated with an isomorphic class of Hamiltonian orbifibration over sphere. By counting orbicurves in the Hamiltonian fibration, we define a map from pi1(Ham(X,o)) to the quantum cohomology QH*(X,o) of (X,o). Then we prove the map respects the product in pi 1(Ham(X,o)) and the quantum product. This map is called Seidel representation map or Seidel morphism.;If X is compact toric, Seidel elements are defined as the images under Seidel morphism for the elements in pi1( Ham(X,o)) coming from Hamiltonian circle actions. We give a representation of QH*(X,o) by computing Seidel elements and showing that the relation between Seidel elements are enough to capture the relations in QH*( X,o). |
