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Rational symplectic field theory over $$\mathbb Z_{2}$$ for exact Lagrangian cobordisms. (English) Zbl 1154.57020
Let $$X$$ be a symplectic $$2n$$-manifold with a cylindrical ends and $$L$$ a Lagrangian $$n$$-submanifold such that outside a compact subset, the pair $$(X, L)$$ is symplectomorphic to the disjoint union of $$(Y^+\times\mathbb R_+, \Lambda^+\times\mathbb R_+)$$ and $$(Y^-\times\mathbb R_-, \Lambda^-\times\mathbb R_-)$$, where $$\Lambda^{\pm}\subset Y^{\pm}$$ is a Legendrian $$(n-1)$$-submanifold. Let $$(\overline X, \overline L)$$ denote the compact part of $$(X, L)$$. The author calls $$(X, L)$$ an exact cobordism if the symplectic form $$\omega$$ on $$X$$ satisfies $$\omega=d\beta$$ for some $$1$$-form $$\beta$$ and $$\beta| _L=df$$ for some function $$f$$. The author considers a formal disk in $$(X, L)$$, that is a homotopy class of maps of the 2-disk $$D$$ with some marked disjoint closed subintervals in $$\partial D$$ into $$\overline X$$, where the marked intervals are required to map in orientation preserving (reversing) manner to Reeb chords of $$\partial\overline L$$ in the $$(+\infty)$$-boundary (in the $$(-\infty)$$-boundary) and where the remaining parts of the boundary $$\partial D$$ map to $$\overline L$$. The action of a Reeb chord is the integral of the contact form $$\beta$$ in the $$(\pm\infty)$$-boundary along it and the $$(+)$$-action of the formal disk is the sum of the actions of its Reeb chords in the $$(+\infty)$$-boundary. The author introduces the notion of an admissible formal disk in $$(X, L)$$ so that if a formal disk determined by a $$J$$-holomorphic disk in $$X$$ with boundary on $$L$$ is admissible then bubbling is impossible for topological reasons.
Let $$\mathbf V(X, L)$$ denote the graded vector space over $$\mathbb Z_2$$ consisting of all formal sums of admissible disks of $$(X, L)$$ which contain only a finite number of summands of the $$(+)$$-action below any given number and with grading such that the degree of a formal disk is the formal dimension of the moduli space of $$J$$-holomorphic disks homotopic to the formal disk. The filtration $$0\subset F^k\mathbf V(X, L)\subset \cdots\subset F^2\mathbf V(X, L) \subset F^1\mathbf V(X, L)=\mathbf V(X, L)$$ with respect to the number $$k$$ of pieces of $$L$$ is defined where the filtration level is determined by the number of Reeb chords of a formal disk in the $$(+\infty)$$-boundary of $$(X, L)$$. The author defines a differential $$d: \mathbf V(X, L)\to\mathbf V(X, L)$$ which counts all admissibe formal disks which are built from the following disks by gluing at Reeb chords: one admissible $$J$$-holomorphic disk in the symplectization of the $$(\pm\infty)$$-boundary of degree 1, one copy of the admissible formal disks and any number of admissible $$J$$-holomorphic disk in $$(X, L)$$ of degree 0. Thus $$d$$ increases grading by 1, respects the filtration and is $$(+)$$-action non-decreasing.
For $$\alpha>0$$, the finite $$(+)$$-action subspace $$\mathbf V_{[\alpha]}(X, L) \subset\mathbf V(X, L)$$ is defined by the subspace of formal sums of admissible disks of $$(+)$$-action at most $$\alpha$$. The differential $$d$$ induces filtration preserving differentials $$d_{\alpha}$$ in $$\mathbf V_{[\alpha]}(X, L)$$ and the natural projection map $$\pi^{\alpha}_{\beta}: \mathbf V_{[\alpha]}(X, L)\to\mathbf V_{[\beta]}(X, L), \alpha>\beta$$ is a filtration preserving chain map. The author obtains the rational admissible SFT spectral sequence $$\{E^{p,q}_{r, [\alpha]}(X, L)\}^k_{r=1}$$ as the spectral sequence induced by $$d_{\alpha}$$. By restricting the contact form of the contact manifolds at $$(\pm\infty)$$-boundary of cobordism, the spectral sequence is invariant under compactly supported deformations (Theorem 1.1). If $$\Lambda\subset Y$$ is a Legendrian submanifold of a contact manifold $$Y$$ and if $$\Lambda$$ is subdivided into pieces $$\Lambda=\Lambda_1\cup\cdots\cup \Lambda_k\subset Y$$, then the spectral sequence $$\{E^{p,q}_r(Y, L)\}^k_{r=1}=\underset\alpha{\underleftarrow {\lim}} \{E^{p,q}_{r, [\alpha]}(Y\times\mathbb R, \Lambda\times\mathbb R)\}^k_{r=1}$$ gives Legendrian isotopy invariants of $$\Lambda$$ (Theorem 1.2).

##### MSC:
 57R17 Symplectic and contact topology in high or arbitrary dimension 57R58 Floer homology 53D40 Symplectic aspects of Floer homology and cohomology 53D12 Lagrangian submanifolds; Maslov index
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