Volume 8, issue 3 (2004)

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A field theory for symplectic fibrations over surfaces

François Lalonde

Geometry & Topology 8 (2004) 1189–1226

arXiv: math.SG/0309335


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.

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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Received: 20 September 2003
Revised: 22 August 2004
Accepted: 11 July 2004
Published: 10 September 2004
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Robion Kirby
François Lalonde
Department of Mathematics and Statistics
University of Montreal
Montreal H3C 3J7