Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Fabert, Oliver

doi:10.1007/s00209-010-0730-5

Cited by 6 publications

(39 citation statements)

References 19 publications

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“…If F is not a cylinder over , then F must have positive d˛1 -area, implying the strict inequality in (27). (Branched covers of trivial cylinders do not contribute to the differential by Fabert [13]. )…”

Section: Continuation Mapsmentioning

confidence: 99%

Sutures and contact homology I

et al. 2011

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Section: Continuation Mapsmentioning

confidence: 99%

Sutures and contact homology I

et al. 2011

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“…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:…”

mentioning

confidence: 91%

“…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%

“…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:…”

mentioning

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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Higher algebraic structures in Hamiltonian Floer theory

Fabert

2019

Advances in Geometry

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This is the second of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on symplectic cohomology. Using the SFT of Hamiltonian mapping tori we show how to define the analogue of rational Gromov-Witten theory for open symplectic manifolds. More precisely, we show that their symplectic cohomology can be equipped with the structure of a so-called cohomology F-manifold. After discussing applications to the quantum cohomology ring of the quintic, we outline why the the classical (closedstring) mirror symmetry conjecture for open Calabi-Yau manifolds shall be formulated as an isomorphism of cohomology F-manifolds. O.ANote that here we view the small quantum product as a product on the tangent space to Q at q = 0,Algebraically the big quantum (cup) product can equivalently be viewed as product on the space of vector fields T (1,0) Q on Q,In other words, it is an element in the space T (1,2) Q of (1, 2)-tensor fields on Q. It is the key ingredient of the Frobenius manifold structure on Q ∼ = QH * (M ) defined by Dubrovin, see [22], [11].Generalizing from closed to open symplectic manifolds M , it is known that quantum cohomology needs to be replaced by the symplectic cohomology SH * (M ) of M , defined using Floer theory. While it is well-known that the small quantum product on QH * (M ) generalizes to the pair-of-pants product, it is the main goal of this paper to define the generalization of the big quantum product and the resulting Frobenius manifold structure. After defining a big pair-of-pants product on symplectic cohomology, we will prove Theorem 0.1. Generalizing Dubrovin's definition of Frobenius manifolds for closed symplectic manifolds, the symplectic cohomology of a Liouville manifold, Calabi-Yau or not, can be equipped with the structure of a cohomology F-manifold in such a way that, on the tangent space at zero, we recover the ring structure given by the pair-of-pants product.Cohomology F-manifolds are generalizations of Frobenius manifolds defined by Merkulov in [20], [21]. Among other things, the vector field product ⋆ now just lives on a differential graded manifold instead of a graded linear space and one drops the requirement for an underlying potential. In other words, a cohomology F-manifold is a differential graded manifold (Q, X) equipped with a graded commutative and associative product for vector fields ⋆ : T (1,0) Q X ⊗ T (1,0) Q X → T (1,0) Q X .Following [8] and [19] 1 , a differential graded manifold is given by a pair of a graded linear space Q (more generally, a formal pointed graded manifold

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Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Cited by 6 publications

References 19 publications

Sutures and contact homology I

Sutures and contact homology I

Local Symplectic Field Theory

Higher algebraic structures in Hamiltonian Floer theory

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