Rabinowitz Floer homology and mirror symmetry

We define a quantitative invariant of Liouville cobordisms with monotone filling through an action-completed symplectic cohomology theory. We illustrate the non-trivial nature of this invariant by computing it for annulus subbundles of the tautological bundle over $\mathbb{C} P^1$ and give further conjectural computations based on mirror symmetry. We prove a non-vanishing result in the presence of Lagrangian submanifolds with non-vanishing Floer homology.