Invariants for Hamiltonian Spaces

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AbstractAward: DMS-1104670Principal Investigator: Eduardo GonzalezThe principal investigator will investigate, using symplecticgeometry techniques, invariants associated to Hamiltonian groupactions. The first project will study (joint with A. Ott,C. Woodward and F. Ziltener) the construction of gaugedGromov-Witten invariants on punctured surfaces with fixedholonomies. The PI will also investigate the particular case oftoric actions on affine complex spaces. Our second project is onthe study of the invariance of gauged Gromov-Witten potentialsfor toric actions on affine spaces. We will use establishedformulae to show that for Calabi-Yau wall-crossing the gaugedGromov-Witten potentials do not change. After this project isfinished we will study the behaviour of the potential in thenon-toric case. This will be done jointly with C. Woodward. Ourthird proposed activity, joint with H. Iritani, will explain therelation between Seidel elements and Mirror Transformations fornef toric non-singular projective varieties. We will then try toapply our methods to the non-nef case. Our last proposed projectwith A. Cotton-Clay will attempt to resolve a conjecture due toM. Thaddeus relating Floer cohomology for symplectomorphisms andorbifold (stacky) cohomology for global quotients by finitegroups.Our research will advance the interplay of several subjects inmathematics and physics where symmetries play a relevantrole. The first project in this grant will advance theunderstanding and applications of symplectic geometry and gaugetheory (the modern language of elementary particle physics) intophysical theories called "gauged sigma models". Our otherprojects will provide a different approach to the Mirror Symmetryphenomena using symplectic geometry invariants.