Geometry & Topology, Vol. 8 (2004) Paper no. 32, pages 1189--1226.

A field theory for symplectic fibrations over surfaces

Francois Lalonde

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

AMS subject classification. Primary: 53D45. Secondary: 53D40, 81T40, 37J50.

DOI: 10.2140/gt.2004.8.1189

E-print: arXiv:math.SG/0309335

Submitted to GT on 20 September 2003. (Revised 22 August 2004.) Paper accepted 11 July 2004. Paper published 10 September 2004.

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Francois Lalonde
Department of Mathematics and Statistics, University of Montreal
Montreal H3C 3J7, Quebec, Canada
Email: lalonde@dms.umontreal.ca

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