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.

Notes on file formats

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

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