Rational symplectic field theory over Z_2 for exact Lagrangian cobordisms
2008 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 10, no 3, 641-704 p.Article in journal (Refereed) Published
We construct a version of rational symplectic ﬁeld theory for pairs (X, L), where X is an exact symplectic manifold, where L ⊂ 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 Z-graded chain complex of vector spaces over 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, L) to a positive end of a pair (X, L), then there are natural morphisms from the spectral sequences of (X, L) and of (X ,L) to the spectral sequence of (X, L). As an application, we show that if \Lambda ⊂ Y is a Legendrian submanifold of a contact manifold then the spectral sequences associated to (Y × R, \Lambda_s × R), where Y × R is the symplectization of Y and where \Lambda_s ⊂ Y is the Legendrian submanifold consisting of s parallel copies of \Lambda subdivided into k subsets, give Legendrian isotopy invariants of \Lambda.
Place, publisher, year, edition, pages
2008. Vol. 10, no 3, 641-704 p.
Holomorphic curve, Lagrangian submanifold, Legendrian submanifold, symplectic cobordism, symplectic field theory
IdentifiersURN: urn:nbn:se:uu:diva-17673DOI: 10.4171/JEMS/126ISI: 000257869200004OAI: oai:DiVA.org:uu-17673DiVA: diva2:45444