The role of string topology in symplectic field theory
@article{Cieliebak2007TheRO, title={The role of string topology in symplectic field theory}, author={Kai Cieliebak and Janko Latschev}, journal={arXiv: Symplectic Geometry}, year={2007} }
We outline a program for incorporating holomorphic curves with Lagrangian boundary conditions into symplectic field theory, with an emphasis on ideas, geometric intuition, and a description of the resulting algebraic structures.
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References
SHOWING 1-10 OF 36 REFERENCES
Compactness results in Symplectic Field Theory
- Mathematics
- 2003
This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic…
Introduction to Symplectic Field Theory
- Mathematics
- 2000
We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds…
Symplectic field theory and its applications
- Mathematics
- 2006
Symplectic field theory (SFT) attempts to approach the theory of holomorphic curves
in symplectic manifolds (also called Gromov-Witten theory) in the spirit of a topological field
theory. This…
NONCOMMUTATIVE DIFFERENTIAL CALCULUS, HOMOTOPY BV ALGEBRAS AND FORMALITY CONJECTURES
- Mathematics
- 2000
We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these…
Coherent orientations in symplectic field theory
- Mathematics
- 2001
Abstract.We study the coherent orientations of the moduli spaces of holomorphic curves in Symplectic Field Theory, generalizing a construction due to Floer and Hofer. In particular we examine their…
Floer Homology and the Heat Flow
- Mathematics
- 2006
Abstract.We study the heat flow in the loop space of a closed Riemannian manifold M as an adiabatic limit of the Floer equations in the cotangent bundle. Our main application is a proof that the…
String Topology
- Mathematics
- 1999
There is a diffeomorphism invariant structure in the free loop space of a manifold defined (with Moira Chas) by considering transversal intersections in families of collections of closed curves. At…
Symplectic hypersurfaces and transversality in Gromov-Witten theory
- Mathematics
- 2007
We use Donaldson hypersurfaces to construct pseudo-cycles which define Gromov-Witten invariants for any symplectic manifold which agree with the invariants in the cases where transversality could be…
Closed String Operators in Topology Leading to Lie Bialgebras and Higher String Algebra
- Mathematics
- 2004
Imagine a collection of closed oriented curves depending on parameters in a smooth d-manifold M. Along a certain locus of configurations strands of the curves may intersect at certain sites in M. At…