  Given a Liouville manifold $M$, we introduce an invariant of $M$ that we call
the Heegaard Floer symplectic cohomology $SH^*_\kappa(M)$ for any $\kappa \ge
1$ that coincides with the symplectic cohomology for $\kappa=1$. Writing
$\hat{M}$ for the completion of $M$, the differential counts pseudoholomorphic
curves of arbitrary genus in $\mathbb{R} \times S^1 \times \hat{M}$ that are
required to be branched $\kappa$-sheeted covers when projected to the
$\mathbb{R} \times S^1$-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 $\hat{M}=T^*Q$ 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 $SH^*_\kappa(T^*Q)$ are
isomorphic to the cohomology groups of the free multiloop complex of $Q$.
