Gravitational Descendants in Symplectic Field Theory

Fabert, Oliver

doi:10.1007/s00220-010-1180-y

Cited by 7 publications

(15 citation statements)

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“…The corresponding invariants SFT(γ) for closed Reeb orbits γ were introduced by the author in [5,6] by counting holomorphic curves in the moduli spaces…”

Section: A Local Version Of Symplectic Field Theorymentioning

confidence: 99%

“…Instead of getting invariants for contact manifolds, we now get the invariants for closed Reeb orbits that were already studied in [5,6]. Note that for the orbit curves we used an infinitesimal energy estimate to show that multiple covers of orbit cylinders are isolated in the moduli space of holomorphic curves.…”

Section: Introductionmentioning

confidence: 99%

“…Counting multiple covers of fixed holomorphic curves, we show that immersed holomorphic curves with elliptic asymptotic orbits in fourdimensional symplectic cobordisms define morphisms between the local SFT invariants assigned to their asymptotic closed Reeb orbits, defined and computed in [5,6].…”

Section: Introductionmentioning

confidence: 99%

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Local Symplectic Field Theory

Fabert

2013

Int. J. Math.

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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 fourdimensional 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. Int. J. Math. 2013.24. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. O. Fabert for rational multiple covers u = v • ϕ in M v,d (Γ + , Γ − ), then every infinitesimal deformation of u as a holomorphic curve is again a multiple cover of v. Furthermore, the cokernels of the linearized Cauchy-Riemann operator∂ J fit together to a smooth obstruction bundle Coker v∂JUsing these obstruction bundles we can solve the transversality problem for multiple covers of immersed curves with elliptic orbits without employing the polyfold machinery from [7], see [9, Sec. 7.2] for the general approach. We emphasize that the following transversality result is a generalization of the well-known automatic transversality result in dimension four from [10]. Indeed, note that when the multiple cover is still immersed, then the unperturbed moduli space M is discrete and hence the obstruction bundle vanishes.which is extended (using parallel transport and cut-off functions, as described in [5,9,8]) to a section in the full Banach space bundle E → B. Then it holds:• If ν is a transversal section in Coker∂ J , then∂ ν J is a transversal section in E, i.e. M ν is regular. • The linearization of ν at every zero is a compact operator, so that the linearizations of∂ J and∂ ν J belong to the same class of Fredholm operators. 1350041-3 Int. J. Math. 2013.24. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. M 0 or M 0 × M − the cokernel bundle Coker∂ J = Coker v,d∂J (Γ + , Γ − ) is given by2 ) denote the cokernel bundle over the moduli space M 0 of multiple covers of the immersed curves with elliptic orbits and the moduli spaces M + , M − of multiple covers of cylinders over positive or negative asymptotic Reeb orbits γ ± of v, respectively. With this we can now give the analogue of the above definition of special sectionsν =ν v,d (Γ + , Γ − ) in obstruction bundles Coker∂ J = Coker v,d∂J (Γ + , Γ − ) 1350041-11 Int. J. Math. 2013.24. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. O. Fabert over moduli spaces M = M v,d (Γ + , Γ − ) of multiple covers of immersed curves with elliptic orbits v. Assume that we have already coherently chosen sections ν ± =ν γ ± ,d (Γ + , Γ − ) in the cokernel bundles Coker ±∂ J = Coker γ ± ,d∂J (Γ + , Γ − ) over all moduli spaces M ± = M γ,d (Γ + , Γ − ) of branched covers of cylinders over positive and negative asymptotic Reeb orbits γ ± of v. Definition 4.1. Assume that we have chosen sectionsν in the cokernel bundles Coker∂ J over all moduli spaces M of multiple covers of the immersed cu...

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“…The corresponding invariants SFT(γ) for closed Reeb orbits γ were introduced by the author in [5,6] by counting holomorphic curves in the moduli spaces…”

Section: A Local Version Of Symplectic Field Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local Symplectic Field Theory

Fabert

2013

Int. J. Math.

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“…Higher powers ψ l can be inductively defined as the zero sets of generic coherent sections of L ⊗l over the zero sets representing ψ l−1 , weighted by a factor of 1 l , since c 1 (L) = 1 l c 1 (L ⊗l ). For a detailed treatment of coherent sections, see [Fab10]. Note that coherent collections can always be constructed inductively.…”

Section: Introductionmentioning

confidence: 99%

Correlators and Descendants of Subcritical Stein Manifolds

2013

Int. J. Math.

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We determine contact homology algebra of a subcritical Stein-fillable contact manifold whose first Chern class vanishes. We also compute the genus-0 one point correlators and gravitational descendants of compactly supported closed forms of their subcritical Stein fillings. This is a step towards determining the full potential function of the filling as defined in [EGH00]. These invariants also give a canonical presentation of the cylindrical contact homology. With respect to this presentation, we determine the degree-2 differential in the Bourgeois-Oancea exact sequence of [BO09]. As a further application, we proved that if a Kähler manifold M 2n admits a subcritical polarization and c 1 vanishes in the subcritical complement, then M is uniruled.Theorem 1.1 ([Yau04]). Let (M 2n , ∂M), n ≥ 2, be a subcritical Stein domain of finite type, and ξ the maximal complex subbundle on ∂W . If c 1 (ξ) = 0, thenGiven a graded vector space V , let Λ(V ) denote the tensor algebra generated by V modulo the graded commuting relations, and V [n] the graded vectors space consisting of elements of V with a grading shift of positive n.Date: June, 2012.1 4 JIAN HE Reeb dynamics are described in more detail, essentially summarizing the previous work of Yau, and we prove Theorems 1.2 and 1.3 modulo the technical Proposition 3.15; in section 4 we apply Theorem 1.3 to subcritical polarizations; in section 5 we prove Proposition 3.15; and in the last section we discuss connections with the Bourgeois-Oancea exact sequence and prove Theorem 1.5. Symplectic Field Theory and DescendantsIn this section we will give an extremely brief overview of aspects of symplectic field theory and define the invariants we wish to compute. See [EGH00] for a more complete discussion. Throughout this section we assume the polyfold theory of Hofer, Zehnder and Wysocki, [HWZ06], [HWZ07], [HWZ08], which forms the analytical foundation of SFT. In other words, there exists a abstract perturbation scheme under which all moduli spaces are branched manifolds with boundaries and corners of the expected dimension.Let (V 2n−1 , ξ) be a contact manifold with a contact 1-form α, i.e., (dα) n−1 ∧ α is a volume form and ξ = Ker(α). The Reeb vector field is the unique vector field R such thatThe flow of the Reeb vector field preserves the contact structure ξ. A (possibly multiply covered) Reeb orbit γ is non-degenerate if the linearized Poincaré return map of the Reeb flow has no eigenvalue equal to 1. For a generic choice of α, there are countably many closed Reeb orbits, all of which are non-degenerate. Let κ γ denote the multiplicity of the orbit γ.Definition 2.1. A Reeb orbit is good if it is not an even multiple of another orbit γ such that the linearized Poincaré return map along γ has an odd total number of eigenvalues (counted with multiplicity) in the interval (−1, 0).Remark 2.2. All orbits appearing in this paper are easily seen to be good.If c 1 (ξ) = 0, then the Reeb orbits admit a consistent Z-grading. Define the index of a Reeb orbit γ to be µ(γ) = µ(γ) + (n − 3...

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“…This paper is organized as follows: While in section two we review the most important definitions and results about SFT with gravitational descendants and its relation with integrable systems in [F2,FR], in section three we first show, as a motivation for our main result, how the topological recursion relations in Gromov-Witten theory carry over to symplectic Floer theory. Since this example suggests that the localization theorem for gravitational descendants needs a non-equivariant version of cylindrical contact homology which, similar to symplectic Floer homology, is generated by parameterized instead of unparameterized closed Reeb orbits, we then recall the definition of non-equivariant cylindrical homology from [BO] and prove the topological recursion relations in the non-equivariant situation.…”

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confidence: 99%

Non-equivariant cylindrical contact homology

Fabert

Rossi

2013

Journal of Symplectic Geometry

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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.Contents * =ĤF * ⊕ȞF * , that the quantum cohomology acts on both factors separately and agrees with the action defined in [PSS].

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Gravitational Descendants in Symplectic Field Theory

Cited by 7 publications

References 16 publications

Local Symplectic Field Theory

Local Symplectic Field Theory

Correlators and Descendants of Subcritical Stein Manifolds

Non-equivariant cylindrical contact homology

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