Gauged Gromov-Witten theory and holomorphic quilts

Project Details

Description

Abstract

Award: DMS-0904358

Principal Investigator: Christopher Woodward

The PI will carry out projects on functoriality for Lagrangian

correspondences in Fukaya-Floer theory and functoriality for quotients

in Gromov-Witten theory. The first group of projects will have

applications in Gromov-Witten theory and ``cohomological'' mirror

symmetry, that is, in the sense of Givental etc. With F. Ziltener and

his former postdoctoral advisee E. Gonzalez he will investigate

functoriality for Gromov-Witten invariants under the symplectic

quotient construction. Potential applications include invariance of

Gromov-Witten invariants under symplectic birational equivalence, to

cohomological mirror symmetry for complete intersections of general

type. With K. Wehrheim and his former student S. Mau the PI will

study functoriality of Lagrangian correspondences in Floer-Fukaya

theory. Applications include symplectic definitions of non-abelian

Floer homology for tangles and arbitrary three-manifolds, possible

generalizations of Khovanov homology and categorification of quantum

groups. These projects will have applications in homological mirror

symmetry and low-dimensional topology. Some of the projects have a

graduate education component, and the PI also proposes several

undergraduate research projects.

Overall the research carried out under this grant will advance the

understanding of symplectic geometry, which is the mathematical

language for classical dynamical systems, and the relationship between

gauge theory, representation theory, and quantum physics. Gauge

theories arise naturally in a number of physical settings, such as

electromagnetism. The first part of the project concerns certain

gauge theories with an addition 'non-linear' field taking values in a

classical phase space, which have been substantially studied in the

physics literature in the linear case under the name 'gauged sigma

models'. The second part of the project concerns the structural

properties of 'Floer-theoretical' invariants which have been

extensively studied in relation to dynamical systems in recent years.

StatusFinished
Effective start/end date9/15/098/31/13

Funding

  • National Science Foundation: $401,899.00

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