### Abstract

Original language | English |
---|---|

Pages (from-to) | 405-448 |

Journal | Journal of Symplectic Geometry |

Volume | 11 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 |

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### Cite this

*Journal of Symplectic Geometry*,

*11*(3), 405-448. https://doi.org/10.4310/jsg.2013.v11.n3.a5

}

*Journal of Symplectic Geometry*, vol. 11, no. 3, pp. 405-448. https://doi.org/10.4310/jsg.2013.v11.n3.a5

**Topological recursion relations in non-equivariant cylindrical contact homology.** / Fabert, O.; Rossi, P.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - Topological recursion relations in non-equivariant cylindrical contact homology

AU - Fabert, O.

AU - Rossi, P.

PY - 2013

Y1 - 2013

N2 - It was pointed out by Eliashberg in his ICM 2006 plenary talk that the integrable systems of rational Gromov-Witten theory very naturally appear in the rich algebraic formalism of symplectic field theory (SFT). Carefully generalizing the definition of gravitational descendants from Gromov-Witten theory to SFT, one can assign to every contact manifold a Hamiltonian system with symmetries on SFT homology and the question of its integrability arises. While we have shown how the well-known string, dilaton and divisor equations translate from Gromov-Witten theory to SFT, the next step is to show how genus-zero topological recursion translates to SFT. Compatible with the example of SFT of closed geodesics, it turns out that the corresponding localization theorem requires a non-equivariant version of SFT, which is generated by parameterized instead of unparameterized closed Reeb orbits. Since this non-equivariant version is so far only defined for cylindrical contact homology, we restrict ourselves to this special case. As an important result we show that, as in rational Gromov-Witten theory, all descendant invariants can be computed from primary invariants, i.e., without descendants.

AB - It was pointed out by Eliashberg in his ICM 2006 plenary talk that the integrable systems of rational Gromov-Witten theory very naturally appear in the rich algebraic formalism of symplectic field theory (SFT). Carefully generalizing the definition of gravitational descendants from Gromov-Witten theory to SFT, one can assign to every contact manifold a Hamiltonian system with symmetries on SFT homology and the question of its integrability arises. While we have shown how the well-known string, dilaton and divisor equations translate from Gromov-Witten theory to SFT, the next step is to show how genus-zero topological recursion translates to SFT. Compatible with the example of SFT of closed geodesics, it turns out that the corresponding localization theorem requires a non-equivariant version of SFT, which is generated by parameterized instead of unparameterized closed Reeb orbits. Since this non-equivariant version is so far only defined for cylindrical contact homology, we restrict ourselves to this special case. As an important result we show that, as in rational Gromov-Witten theory, all descendant invariants can be computed from primary invariants, i.e., without descendants.

U2 - 10.4310/jsg.2013.v11.n3.a5

DO - 10.4310/jsg.2013.v11.n3.a5

M3 - Article

VL - 11

SP - 405

EP - 448

JO - Journal of Symplectic Geometry

JF - Journal of Symplectic Geometry

SN - 1527-5256

IS - 3

ER -