Given a Liouville manifold M, we introduce an invariant of M that we call
the Heegaard Floer symplectic cohomology SHκ∗​(M) for any κ≥1 that coincides with the symplectic cohomology for κ=1. Writing
M^ for the completion of M, the differential counts pseudoholomorphic
curves of arbitrary genus in R×S1×M^ that are
required to be branched κ-sheeted covers when projected to the
R×S1-direction; this resembles the cylindrical reformulation
of Heegaard Floer homology by Lipshitz. These cohomology groups provide a
closed-string analogue of higher-dimensional Heegaard Floer homology introduced
by Colin, Honda, and Tian.
When M^=T∗Q with Q an orientable manifold, we introduce a
Morse-theoretic analogue of Heegaard Floer symplectic cohomology, which we call
the free multiloop complex of Q. When Q has vanishing relative second
Stiefel-Whitney class, we prove a generalized version of Viterbo's isomorphism
theorem by showing that the cohomology groups SHκ∗​(T∗Q) are
isomorphic to the cohomology groups of the free multiloop complex of Q.Comment: 78 pages, 14 figure