Product structures in Floer theory for Lagrangian cobordisms

TL;DR

This paper develops an algebraic product structure on Floer complexes for Lagrangian cobordisms, extending to an $A_$-structure and establishing a ring isomorphism via an $A_$-morphism.

Contribution

It introduces a new product on Floer complexes for Lagrangian cobordisms and extends the Ekholm-Seidel isomorphism to an $A_$-morphism, enhancing the algebraic framework.

Findings

Defined a product $$ on Floer complexes using pseudo-holomorphic disks.

Established that the family of maps forms an $A_$-structure.

Proved the Ekholm-Seidel isomorphism extends to an $A_$-morphism, preserving ring structures.

Abstract

We construct a product on the Floer complex associated to a pair of Lagrangian cobordisms. More precisely, given three exact transverse Lagrangian cobordisms in the symplectization of a contact manifold, we define a map $\mathfrak{m}_2$ by a count of rigid pseudo-holomorphic disks with boundary on the cobordisms and having punctures asymptotic to intersection points and Reeb chords of the negative Legendrian ends of the cobordisms. More generally, to a $(d+1)$-tuple of exact transverse Lagrangian cobordisms we associate a map $\mathfrak{m}_d$ such that the family $(\mathfrak{m}_d)_{d\geq1}$ are $A_\infty$-maps. Finally, we extend the Ekholm-Seidel isomorphism to an $A_\infty$-morphism, giving in particular that it is a ring isomorphism.