Topological quantum field theory structure on symplectic cohomology

TL;DR

This paper develops a topological quantum field theory framework on symplectic cohomology and wrapped Floer cohomology, revealing new insights and applications in symplectic topology, including the Arnold chord conjecture and contact embeddings.

Contribution

It constructs a TQFT on symplectic and wrapped Floer cohomology, demonstrating its compatibility with restriction maps and cohomology, and applies it to solve problems in symplectic topology.

Findings

TQFT respects Viterbo restriction maps and canonical cohomology maps.

Vanishing of symplectic cohomology implies the Arnold chord conjecture.

The TQFT recovers string topology TQFT via cotangent bundle isomorphism.

Abstract

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic cohomology. These constructions yield 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 of M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian lying in the boundary. 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. Finally, we use the TQFT to prove that symplectic cohomology vanishes iff Rabinowitz Floer cohomology vanishes.