# Rational symplectic field theory over ${\mathbb{Z}}^{2}$ for exact Lagrangian cobordisms

Journal of the European Mathematical Society (2008)

- Volume: 010, Issue: 3, page 641-704
- ISSN: 1435-9855

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topEkholm, Tobias. "Rational symplectic field theory over $\mathbb {Z}^2$ for exact Lagrangian cobordisms." Journal of the European Mathematical Society 010.3 (2008): 641-704. <http://eudml.org/doc/277183>.

@article{Ekholm2008,

abstract = {We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\mathbb \{Z\}$-graded chain complex of vector spaces over $\mathbb \{Z\}_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a negative end of a pair $(X^\{\prime \},L^\{\prime \})$ to a positive end of a pair $(X^\{\prime \prime \},L^\{\prime \prime \})$, then there are natural morphisms from the spectral sequences of $(X^\{\prime \},L^\{\prime \})$ and of $(X^\{\prime \prime \},L^\{\prime \prime \})$ to the spectral sequence of $(X,L)$. As an application, we show that if $\Lambda \subset Y$ is a Legendrian submanifold of a contact manifold then the spectral sequences associated to $(Y\times \mathbb \{R\},\Lambda _k^s\times \mathbb \{R\})$, where $Y\times \mathbb \{R\}$ is the symplectization of $Y$ and where $\Lambda _k^s\subset Y$ is the Legendrian submanifold consisting of $s$ parallel copies of $\Lambda $ subdivided into $k$ subsets, give Legendrian isotopy invariants of $\Lambda $.},

author = {Ekholm, Tobias},

journal = {Journal of the European Mathematical Society},

keywords = {holomorphic curve; Lagrangian submanifold; Legendrian submanifold; symplectic cobordism; symplectic field theory; holomorphic curve; Lagrangian submanifold; Legendrian submanifold; symplectic cobordism; symplectic field theory},

language = {eng},

number = {3},

pages = {641-704},

publisher = {European Mathematical Society Publishing House},

title = {Rational symplectic field theory over $\mathbb \{Z\}^2$ for exact Lagrangian cobordisms},

url = {http://eudml.org/doc/277183},

volume = {010},

year = {2008},

}

TY - JOUR

AU - Ekholm, Tobias

TI - Rational symplectic field theory over $\mathbb {Z}^2$ for exact Lagrangian cobordisms

JO - Journal of the European Mathematical Society

PY - 2008

PB - European Mathematical Society Publishing House

VL - 010

IS - 3

SP - 641

EP - 704

AB - We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\mathbb {Z}$-graded chain complex of vector spaces over $\mathbb {Z}_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a negative end of a pair $(X^{\prime },L^{\prime })$ to a positive end of a pair $(X^{\prime \prime },L^{\prime \prime })$, then there are natural morphisms from the spectral sequences of $(X^{\prime },L^{\prime })$ and of $(X^{\prime \prime },L^{\prime \prime })$ to the spectral sequence of $(X,L)$. As an application, we show that if $\Lambda \subset Y$ is a Legendrian submanifold of a contact manifold then the spectral sequences associated to $(Y\times \mathbb {R},\Lambda _k^s\times \mathbb {R})$, where $Y\times \mathbb {R}$ is the symplectization of $Y$ and where $\Lambda _k^s\subset Y$ is the Legendrian submanifold consisting of $s$ parallel copies of $\Lambda $ subdivided into $k$ subsets, give Legendrian isotopy invariants of $\Lambda $.

LA - eng

KW - holomorphic curve; Lagrangian submanifold; Legendrian submanifold; symplectic cobordism; symplectic field theory; holomorphic curve; Lagrangian submanifold; Legendrian submanifold; symplectic cobordism; symplectic field theory

UR - http://eudml.org/doc/277183

ER -

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