Heegaard Floer Symplectic homology and Viterbo's isomorphism theorem in the context of multiple particles

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