JournalsjemsVol. 10, No. 3pp. 641–704

Rational symplectic field theory over ℤ<sub>2</sub> for exact Lagrangian cobordisms

  • Tobias Ekholm

    Uppsala Universitet, Sweden
Rational symplectic field theory over ℤ<sub>2</sub> for exact Lagrangian cobordisms cover
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Abstract

We construct a version of rational Symplectic Field Theory for pairs (X,L)(X,L), where XX is an exact symplectic manifold, where LXL\subset X is an exact Lagrangian submanifold with components subdivided into kk subsets, and where both XX and LL have cylindrical ends. The theory associates to (X,L)(X,L) a Z\Z-graded chain complex of vector spaces over Z2\Z_2, filtered with kk filtration levels. The corresponding kk-level spectral sequence is invariant under deformations of (X,L)(X,L) and has the following property: if (X,L)(X,L) is obtained by joining a negative end of a pair (X,L)(X',L') to a positive end of a pair (X,L)(X'',L''), then there are natural morphisms from the spectral sequences of (X,L)(X',L') and of (X,L)(X'',L'') to the spectral sequence of (X,L)(X,L). As an application, we show that if ΛY\Lambda\subset Y is a Legendrian submanifold of a contact manifold then the spectral sequences associated to (Y×R,Λks×R)(Y\times\R,\Lambda_k^s\times\R), where Y×RY\times\R is the symplectization of YY and where ΛksY\Lambda_k^s\subset Y is the Legendrian submanifold consisting of ss parallel copies of Λ\Lambda subdivided into kk subsets, give Legendrian isotopy invariants of Λ\Lambda.

Cite this article

Tobias Ekholm, Rational symplectic field theory over ℤ<sub>2</sub> for exact Lagrangian cobordisms. J. Eur. Math. Soc. 10 (2008), no. 3, pp. 641–704

DOI 10.4171/JEMS/126