Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology.

*(English)*Zbl 1254.57024
Itenberg, Ilia (ed.) et al., Perspectives in analysis, geometry, and topology. On the occasion of the 60th birthday of Oleg Viro. Based on the Marcus Wallenberg symposium on perspectives in analysis, geometry, and topology, Stockholm, Sweden, May 19–25, 2008. Basel: Birkhäuser (ISBN 978-0-8176-8276-7/hbk; 978-0-8176-8277-4/ebook). Progress in Mathematics 296, 109-145 (2012).

The main objects of study of the paper under review are exact Lagrangian cobordisms, that is exact Lagrangian submanifolds in an exact symplectic cobordism such that, in the ends of the symplectic cobordism, the Lagrangian is a product of a Legendrian with a line. More specifically, the paper studies connected exact Lagrangian cobordisms with ends only in the convex ends of the symplectic cobordism. There are various algebraic invariants associated to such a cobordism and the paper studies three of these and the relations between them: rational symplectic field theory (SFT), linearised Legendrian contact homology and a version of Lagrangian Floer theory. For the purposes of this paper, these invariants are all defined over the field \(\mathbb{Z}_2\).

The symplectic cobordism is assumed to be simply-connected and to have vanishing first Chern class and the Lagrangian is assumed to be Maslov zero. Moreover there is a technical assumption that the ends of the symplectic cobordism are “good”, explained in Section 2.2.

The rational SFT is reviewed in Section 2. It was defined in [T. Ekholm, J. Eur. Math. Soc. (JEMS) 10, No. 3, 641–704 (2008; Zbl 1154.57020)] as the homology of a complex whose chain group is the vector space generated by the “admissible formal discs”, certain homotopy classes of maps from the disc into the symplectic cobordism where certain arcs in the boundary of the disc are mapped to the Lagrangian and the remaining arcs are mapped onto Reeb chords for the Legendrian at the end. The differential involves counts of 1-dimensional moduli spaces of holomorphic punctured discs.

The Legendrian contact cohomology (reviewed in Section 3) is the cohomology of a complex whose chain group is the algebra of Reeb chords on the Legendrian at the end. Using the Lagrangian cobordism one can define a linearisation of this cohomology, modifying the differential by a certain count of holomorphic punctured discs in the cobordism (an augmentation). This invariant originated in [Yu. Chekanov, Invent. Math. 150, No. 3, 441–483 (2002; Zbl 1029.57011)] and [Y. Eliashberg, A. Givental and H. Hofer, in: N. Alon (ed.) et al., GAFA 2000. Visions in mathematics – Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25-September 3, 1999. Part II. Basel: Birkhäuser, 560–673 (2000; Zbl 0989.81114)].

The main result of the paper is Theorem 1.1 which, under a monotonicity assumption, proves that there is an isomorphism between the Legendrian contact cohomology and the quotient of the rational SFT obtained by forgetting the homotopy class of the formal discs and remembering only the Reeb chords to which they are asymptotic.

Section 4 defines a version of Lagrangian intersection Floer cohomology for a pair of connected exact Lagrangian cobordisms with only convex ends. Its basic properties, like deformation invariance, are also proved. This theory is analogous to the wrapped Floer cohomology of M. Abouzaid and P. Seidel [Geom. Topol. 14, No. 2, 627–718 (2010; Zbl 1195.53106)].

The author makes a conjecture about the moduli space of holomorphic discs with boundary on a Lagrangian cobordism and a Hamiltonian pushoff, similar to the compact case considered in Theorem 3.6 of [T. Ekholm, J. B. Etnyre and J. M. Sabloff, Duke Math. J. 150, No. 1, 1–75 (2009; Zbl 1193.53179)] and shows that this would lead to an exact sequence relating the rational SFT, the Lagrangian Floer cohomology and the ordinary cohomology of the Lagrangian cobordism, analogous to the Bourgeois-Oancea exact sequence [F. Bourgeois and A. Oancea, Invent. Math. 175, No. 3, 611–680 (2009; Zbl 1167.53071)]. In particular, when the Lagrangian Floer cohomology vanishes (for instance when the cobordism is displaceable) this would imply that the rational SFT is isomorphic to the ordinary cohomology of the Lagrangian cobordism.

For the entire collection see [Zbl 1230.00045].

The symplectic cobordism is assumed to be simply-connected and to have vanishing first Chern class and the Lagrangian is assumed to be Maslov zero. Moreover there is a technical assumption that the ends of the symplectic cobordism are “good”, explained in Section 2.2.

The rational SFT is reviewed in Section 2. It was defined in [T. Ekholm, J. Eur. Math. Soc. (JEMS) 10, No. 3, 641–704 (2008; Zbl 1154.57020)] as the homology of a complex whose chain group is the vector space generated by the “admissible formal discs”, certain homotopy classes of maps from the disc into the symplectic cobordism where certain arcs in the boundary of the disc are mapped to the Lagrangian and the remaining arcs are mapped onto Reeb chords for the Legendrian at the end. The differential involves counts of 1-dimensional moduli spaces of holomorphic punctured discs.

The Legendrian contact cohomology (reviewed in Section 3) is the cohomology of a complex whose chain group is the algebra of Reeb chords on the Legendrian at the end. Using the Lagrangian cobordism one can define a linearisation of this cohomology, modifying the differential by a certain count of holomorphic punctured discs in the cobordism (an augmentation). This invariant originated in [Yu. Chekanov, Invent. Math. 150, No. 3, 441–483 (2002; Zbl 1029.57011)] and [Y. Eliashberg, A. Givental and H. Hofer, in: N. Alon (ed.) et al., GAFA 2000. Visions in mathematics – Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25-September 3, 1999. Part II. Basel: Birkhäuser, 560–673 (2000; Zbl 0989.81114)].

The main result of the paper is Theorem 1.1 which, under a monotonicity assumption, proves that there is an isomorphism between the Legendrian contact cohomology and the quotient of the rational SFT obtained by forgetting the homotopy class of the formal discs and remembering only the Reeb chords to which they are asymptotic.

Section 4 defines a version of Lagrangian intersection Floer cohomology for a pair of connected exact Lagrangian cobordisms with only convex ends. Its basic properties, like deformation invariance, are also proved. This theory is analogous to the wrapped Floer cohomology of M. Abouzaid and P. Seidel [Geom. Topol. 14, No. 2, 627–718 (2010; Zbl 1195.53106)].

The author makes a conjecture about the moduli space of holomorphic discs with boundary on a Lagrangian cobordism and a Hamiltonian pushoff, similar to the compact case considered in Theorem 3.6 of [T. Ekholm, J. B. Etnyre and J. M. Sabloff, Duke Math. J. 150, No. 1, 1–75 (2009; Zbl 1193.53179)] and shows that this would lead to an exact sequence relating the rational SFT, the Lagrangian Floer cohomology and the ordinary cohomology of the Lagrangian cobordism, analogous to the Bourgeois-Oancea exact sequence [F. Bourgeois and A. Oancea, Invent. Math. 175, No. 3, 611–680 (2009; Zbl 1167.53071)]. In particular, when the Lagrangian Floer cohomology vanishes (for instance when the cobordism is displaceable) this would imply that the rational SFT is isomorphic to the ordinary cohomology of the Lagrangian cobordism.

For the entire collection see [Zbl 1230.00045].

Reviewer: Jonathan D. Evans (Oxford)

##### MSC:

57R56 | Topological quantum field theories (aspects of differential topology) |

57R58 | Floer homology |

53D35 | Global theory of symplectic and contact manifolds |

53D40 | Symplectic aspects of Floer homology and cohomology |

53D42 | Symplectic field theory; contact homology |