LOCAL SYMPLECTIC FIELD THEORY

Generalizing local Gromov–Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is four-dimensional and the underlying simple curve is regular by automatic transversality, we establish a transversality result for all its multiple covers and discuss the resulting algebraic structures.

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Local contact homology and applications

Journal of Topology and Analysis ◽

10.1142/s1793525315500119 ◽

2015 ◽

Vol 07 (02) ◽

pp. 167-238 ◽

Cited By ~ 11

Author(s):

Umberto L. Hryniewicz ◽

Leonardo Macarini

Keyword(s):

Field Theory ◽

Periodic Orbit ◽

Periodic Orbits ◽

Local Version ◽

Contact Homology ◽

Local Contact ◽

Symplectic Field Theory ◽

Uniformly Bounded

We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations. Several applications are obtained, including a generalization of Gromoll–Meyer's theorem on the existence of infinitely many simple periodic orbits, resonance relations and conditions for the existence of non-hyperbolic periodic orbits. Most of the results of this paper remain conjectural until the foundational issues of Symplectic Field Theory are resolved.

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Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations

Mathematische Annalen ◽

10.1007/s00208-009-0471-0 ◽

2010 ◽

Vol 348 (2) ◽

pp. 265-287 ◽

Cited By ~ 14

Author(s):

Paolo Rossi

Keyword(s):

Field Theory ◽

Witten Theory ◽

Symplectic Field Theory

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Higher algebraic structures in Hamiltonian Floer theory

Advances in Geometry ◽

10.1515/advgeom-2019-0017 ◽

2020 ◽

Vol 20 (2) ◽

pp. 179-215

Author(s):

Oliver Fabert

Keyword(s):

Algebraic Structures ◽

Floer Theory ◽

Manifold Structure ◽

Lie Bracket ◽

Witten Theory ◽

Mapping Tori ◽

Symplectic Field Theory ◽

The Rich ◽

Algebraic Formalism ◽

Hamiltonian Mapping

AbstractIn this paper we show how the rich algebraic formalism of Eliashberg–Givental–Hofer’s symplectic field theory (SFT) can be used to define higher algebraic structures in Hamiltonian Floer theory. Using the SFT of Hamiltonian mapping tori we define a homotopy extension of the well-known Lie bracket and discuss how it can be used to prove the existence of multiple closed Reeb orbits. Furthermore we define the analogue of rational Gromov–Witten theory in the Hamiltonian Floer theory of open symplectic manifolds. More precisely, we introduce a so-called cohomology F-manifold structure in Hamiltonian Floer theory and prove that it generalizes the well-known Frobenius manifold structure in rational Gromov–Witten theory.

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TETRAHEDRA AND POLYNOMIAL EQUATIONS IN TOPOLOGICAL FIELD THEORY

International Journal of Modern Physics A ◽

10.1142/s0217751x91001738 ◽

1991 ◽

Vol 06 (20) ◽

pp. 3571-3598 ◽

Cited By ~ 1

Author(s):

NOUREDDINE CHAIR ◽

CHUAN-JIE ZHU

Keyword(s):

Field Theory ◽

Topological Field Theory ◽

Fundamental Representation ◽

Conformal Blocks ◽

Conformal Field ◽

Witten Theory ◽

Topological Field ◽

General Program ◽

The One ◽

Chern Simons

Some tetrahedra in SUk(2) Chern-Simons-Witten theory are computed. The results can be used to compute an arbitrary tetrahedron inductively by fusing with the fundamental representation. The results obtained are in agreement with those of quantum groups. By associating a (finite) topological field theory (FTFT) to every rational conformal field theory (RCFT), we show that the pentagon and hexagon equations in RCFT follow directly from some skein relations in FTFT. By generalizing the operation of surgery on links in FTFT, we also derive an explicit expression for the modular transformation matrix S(k) of the one-point conformal blocks on a torus in RCFT and the equations satisfied by S(k), in agreement with those required in RCFT. The implication of our results on the general program of classifying RCFT is also discussed.

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