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MiS Preprint

Topological recursion relations in non-equivariant cylindrical contact homology

Oliver Fabert and Paolo Rossi


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 parametrized instead of unparametrized 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.

MSC Codes:
53D42, 53D45, 53D40
symplectic field theory, integrable system, Gromov-Witten theory

Related publications

2013 Repository Open Access
Oliver Fabert and Paolo Rossi

Topological recursion relations in non-equivariant cylindrical contact homology

In: Journal of symplectic geometry, 11 (2013) 3, pp. 405-448