A note on coherent orientations for exact Lagrangian cobordisms

Abstract

Let L⊂\R×J1(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 Λ±⊂J1(M). It is well known that the Legendrian contact homology of Λ± 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 \R×J1(M), and that L induces a morphism between the \Z2-valued DGA:s of the ends Λ± 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.