Topological quantum field theory structure on symplectic cohomology

We construct the topological quantum field theory (TQFT) on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and we prove that Viterbo restriction preserves the TQFT. This yields new applications in symplectic topology relating to the Arnol'd chord conjecture and to exact contact embeddings. We prove that if a Liouville domain M admits an exact embedding into an exact convex symplectic manifold X, and the boundary ∂ M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian in ∂ M. The TQFT respects the isomorphism between the symplectic cohomology of a cotangent bundle and the homology of the free loop space, so it recovers the TQFT of string topology.