A note on coherent orientations for exact Lagrangian cobordisms

TL;DR

This paper proves that the Legendrian contact homology of exact Lagrangian cobordisms can be defined with integer coefficients, extending previous mod 2 results, by orienting moduli spaces of pseudo-holomorphic disks using capping operators.

Contribution

It establishes the functoriality of Legendrian contact homology with integer coefficients for exact Lagrangian cobordisms, using orientation techniques of moduli spaces.

Findings

Legendrian contact homology can be defined with integer coefficients.

The functorial morphism between DGA:s extends to integer coefficients.

Orientation of moduli spaces is achieved via capping operators at Reeb chords.

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:s 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:s change if we change the capping operators.