A note on coherent orientations for exact Lagrangian cobordisms

Abstract

Let $L\subset \mathbb{R}\times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold $M$. Assume that $L$ has cylindrical Legendrian ends ${\Lambda }_{\pm }\subset J^1(M)$. It is well known that the Legendrian contact homology of ${\Lambda }_{\pm }$ can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent bundle of $M$. It is also known that this count can be lifted to a mod 2 count of pseudo-holomorphic disks in the symplectization $\mathbb{R}\times J^1(M)$, and that $L$ induces a morphism between the ${\mathbb{Z}}_2$-valued DGA

of the ends ${\Lambda }_{\pm }$ in a functorial way. We prove that this hold with integer coefficients as well.

The proofs are built on the technique of orienting the moduli spaces of pseudo-holomorphic disks using capping operators at the Reeb chords. We give an expression for how the DGA

change if we change the capping operators.