Invariants for Hamiltonian Spaces

University Of Massachusetts Boston, Dorchester MA

Investigators

Abstract

Abstract Award: DMS-1104670 Principal Investigator: Eduardo Gonzalez The principal investigator will investigate, using symplectic geometry techniques, invariants associated to Hamiltonian group actions. The first project will study (joint with A. Ott, C. Woodward and F. Ziltener) the construction of gauged Gromov-Witten invariants on punctured surfaces with fixed holonomies. The PI will also investigate the particular case of toric actions on affine complex spaces. Our second project is on the study of the invariance of gauged Gromov-Witten potentials for toric actions on affine spaces. We will use established formulae to show that for Calabi-Yau wall-crossing the gauged Gromov-Witten potentials do not change. After this project is finished we will study the behaviour of the potential in the non-toric case. This will be done jointly with C. Woodward. Our third proposed activity, joint with H. Iritani, will explain the relation between Seidel elements and Mirror Transformations for nef toric non-singular projective varieties. We will then try to apply our methods to the non-nef case. Our last proposed project with A. Cotton-Clay will attempt to resolve a conjecture due to M. Thaddeus relating Floer cohomology for symplectomorphisms and orbifold (stacky) cohomology for global quotients by finite groups. Our research will advance the interplay of several subjects in mathematics and physics where symmetries play a relevant role. The first project in this grant will advance the understanding and applications of symplectic geometry and gauge theory (the modern language of elementary particle physics) into physical theories called "gauged sigma models". Our other projects will provide a different approach to the Mirror Symmetry phenomena using symplectic geometry invariants.