Workshop on Equivariant Gromov-Witten Theory and Symplectic Vortices; July 2009, Luminy, France

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Abstract Award: DMS-0835558 Principal Investigator: Christopher Woodward Algebraic geometry, symplectic geometry and low-dimensional topology have been revolutionized in recent years by the introduction of holomorphic curve techniques. This workshop, to be held July 6-10, 2009, at the CIRM (Luminy) conference center, concerns equivariant generalizations of invariants defined by holomorphic curves. In Gromov-Witten theory, equivariant invariants have been extensively studied by Givental and others, in the sense of integrals of equivariant cohomology classes over moduli spaces of stable maps. The focus of the workshop will be on the following more sophisticated version of equivariant Gromov-Witten theory: the study of moduli spaces of maps to the stack-theoretic quotient of, say, a smooth projective variety by a reductive group. By definition, such a map consists of a holomorphic principal bundle, together with a section of the associated bundle. From the symplectic point of view, suitable moduli spaces of such pairs are known as moduli spaces of {\em symplectic vortices}. The workshop is organized around this specific direction, with an aim to bring together researchers in algebraic and symplectic geometry who have had no previous interaction. On the other hand, the workshop will promote discussion of the interaction of holomorphic maps, gauge theory, and group actions more broadly. Applications of the theory are expected for instance, to the relation between Gromov-Witten theories of different varieties, and to the relation between the gauge-theoretic invariants such as Floer-Donaldson invariants and their symplectic analogs. We propose a workshop which will bring together gauge theory and holomorphic curves. Gauge theory is the study of local symmetry in mathematics and physics; the most famous example is electromagnetism. Holomorphic curve techniques aim, in algebraic geometry, to understand spaces by understanding the certain classes of curves in them, or in symplectic geometry, to understand the periodic orbits of dynamical systems. Symmetry of the target varieties may be used as a computational tool, and also combined with holomorphic curves to define new invariants. The workshop will focus on recent developments and emerging applications to algebraic geometry, symplectic geometry, and low-dimensional topology. The workshop will benefit a number of graduate students and postdoctoral fellows working in this area who have had little interaction with other groups working on related problems. The conference web-page controlled by the organizers is www.math.rutgers.edu/~ctw/Luminy, while that controlled by the conference center CIRM is www.cirm.univ-mrs.fr/web.ang/liste_rencontre/Rencontres2009/Renc346/Renc346.html.