A field theory for symplectic fibrations over surfaces

Lalonde, Francois

doi:10.2140/gt.2004.8.1189

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A field theory for symplectic fibrations over surfaces

François Lalonde

Geometry & Topology 8 (2004) 1189–1226

DOI: 10.2140/gt.2004.8.1189

arXiv: math.SG/0309335

Abstract

We introduce in this paper a field theory on symplectic manifolds that are fibered over a real surface with interior marked points and cylindrical ends. We assign to each such object a morphism between certain tensor products of quantum and Floer homologies that are canonically attached to the fibration. We prove a composition theorem in the spirit of QFT, and show that this field theory applies naturally to the problem of minimising geodesics in Hofer’s geometry. This work can be considered as a natural framework that incorporates both the Piunikhin–Salamon–Schwarz morphisms and the Seidel isomorphism.

Keywords

symplectic fibration, field theory, quantum cohomology, Floer homology, Hofer's geometry, commutator length

Mathematical Subject Classification 2000

Primary: 53D45

Secondary: 53D40, 81T40, 37J50

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Publication

Received: 20 September 2003

Revised: 22 August 2004

Accepted: 11 July 2004

Published: 10 September 2004

Proposed: Leonid Polterovich

Seconded: Yasha Eliashberg, Robion Kirby

Authors

François Lalonde
Department of Mathematics and Statistics University of Montreal Montreal H3C 3J7 Quebec Canada

Volume 8, issue 3 (2004)