Gauged Gromov-Witten theory and holomorphic quilts

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AbstractAward: DMS-0904358Principal Investigator: Christopher WoodwardThe PI will carry out projects on functoriality for Lagrangiancorrespondences in Fukaya-Floer theory and functoriality for quotientsin Gromov-Witten theory. The first group of projects will haveapplications in Gromov-Witten theory and ``cohomological'' mirrorsymmetry, that is, in the sense of Givental etc. With F. Ziltener andhis former postdoctoral advisee E. Gonzalez he will investigatefunctoriality for Gromov-Witten invariants under the symplecticquotient construction. Potential applications include invariance ofGromov-Witten invariants under symplectic birational equivalence, tocohomological mirror symmetry for complete intersections of generaltype. With K. Wehrheim and his former student S. Mau the PI willstudy functoriality of Lagrangian correspondences in Floer-Fukayatheory. Applications include symplectic definitions of non-abelianFloer homology for tangles and arbitrary three-manifolds, possiblegeneralizations of Khovanov homology and categorification of quantumgroups. These projects will have applications in homological mirrorsymmetry and low-dimensional topology. Some of the projects have agraduate education component, and the PI also proposes severalundergraduate research projects.Overall the research carried out under this grant will advance theunderstanding of symplectic geometry, which is the mathematicallanguage for classical dynamical systems, and the relationship betweengauge theory, representation theory, and quantum physics. Gaugetheories arise naturally in a number of physical settings, such aselectromagnetism. The first part of the project concerns certaingauge theories with an addition "non-linear" field taking values in aclassical phase space, which have been substantially studied in thephysics literature in the linear case under the name "gauged sigmamodels". The second part of the project concerns the structuralproperties of "Floer-theoretical" invariants which have beenextensively studied in relation to dynamical systems in recent years.