Local symplectic field theory and stable hypersurfaces in symplectic blow-ups

Details Local symplectic field theory and stable hypersurfaces in symplectic blow-ups Local symplectic field theory and stable hypersurfaces in symplectic blow-ups

2011-04-18

arxiv.org/abs/1104.3504v1

Mathematics

Symplectic Geometry

26 pages

Scientific paper

In this paper we define a local version of symplectic field theory which generalizes local Gromov-Witten theory in the same way as standard symplectic field theory generalizes standard Gromov-Witten theory. While local symplectic field theory assigns invariants to closed Reeb orbits in contact manifolds, we show that nicely-embedded curves in four-dimensional symplectic cobordisms (in the sense of Wendl) define morphisms between the invariants assigned to their asymptotic closed Reeb orbits. After introducing gravitational descendants, we prove that a stable hypersurface intersecting an exceptional sphere (in a homologically nontrivial way) in a closed four-dimensional symplectic manifold must carry an elliptic orbit. For this we relate the local Gromov-Witten potential of the exceptional sphere with the local SFT Hamiltonian of the breaking orbits (obtained after neck-stretching along the hypersurface) as in the Hamilton-Jacobi equation of standard SFT.

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