3 Corrected (I hope!) comments on open closed string maps.
(Edited: my first version was not correct.) There are actually two versions of Seidel's open-closed string map, derived from the same moduli spaces:One expects that both are isomorphisms when Here $F(M)$ is the Fukaya category of exact, compact Lagrangians. Moreover, there are "enough" Lagrangian submanifolds. The absence extensions of a dualisation looks strange, but the point is that these maps to the wrapped Fukaya category $F(M)$ is expected to have a structure W(M)$. The absence of$n$-dimensional Calabi-Yau$A_\infty$-category (Fukaya has proved a part of this statement), dualisation - hence its (shifted) the connection of symplectic cohomology to both Hochschild homology and cohomology should be isomorphic- looks strange. Mohammed Abouzaid has shown points out in his comment below that this is a manifestation of a self-duality property for the wrapped category. He shows in http://arxiv.org/abs/1003.4449 that for$M=T^\ast L$and the wrapped category, the map$\lambda$is an isomorphism. So far as I know,$\kappa$for$T^\ast L$is not explicitly studied in the literature; but it's anyway less natural for string topology. For simply connected, spin cotangent bundles$T^\ast L$of simply connected, spin manifolds, one has$SH^\ast_{S^1}(M) \cong H_{n-*}^{S^1}(\mathcal{L}L)$. 2 correction about grading of symplectic cohomology Some blah on symplectic homology vs. cohomology. There's an invariant$SH(M)$of Liouville domains$M$which some people call symplectic homology and some symplectic cohomology. This is the direct limit of Hamiltonian Floer groups associated with functions of increasing eventual slope. The dual theory has two rather unpleasant features: it involves inverse limits, hence one must worry about$\varprojlim^1$-terms; and in general it's not countably generated. It's not often used. Why the confusion about terminology? Well, depending on your convention for the sign of the symplectic action functional, you may regard this as Morse homology or compactly supported Morse cohomology of this function. From the perspective of Lagrangian Floer cohomology, consistency demands that one calls it symplectic cohomology; I do. However, symplectic field theorists (including Bourgeois-Oancea, I think) prefer the contrary convention. Blah about grading. The integer grading on$SH^\ast(M)$is defined canonically for exact symplectic manifolds with when$c_1=0$. c_1=0$, and is canonical when $H^1(M)=0$. One has Viterbo's map $H^\ast(M)\to SH^\ast(M)$, and one convention makes this preserve degree (I'll take that option), while another makes it shift degree by the complex dimension $n$ of $M$.

Seidel's map for cotangent bundles. There are actually two versions of Seidel's open-closed string map, derived from the same moduli spaces: $$\kappa: SH^\ast(M) \to HH^\ast(F(M),F(M))$$ and $$\lambda : HH_\ast(F(M),F(M)) \to SH^{n+\ast}(M).$$ One expects that both are isomorphisms when there are "enough" Lagrangian submanifolds. The absence of a dualisation looks strange, but the point is that the Fukaya category $F(M)$ is expected to have a structure of $n$-dimensional Calabi-Yau $A_\infty$-category (Fukaya has proved a part of this statement), hence its (shifted) Hochschild homology and cohomology should be isomorphic.

Abouzaid has shown in http://arxiv.org/abs/1003.4449 that for $M=T^\ast L$, the map $\lambda$ is an isomorphism. So far as I know, $\kappa$ for $T^\ast L$ is not explicitly studied in the literature; but it's anyway less natural for string topology.

Cyclic version. My expectations are slightly different from those stated in the question. I'd guess that $\lambda$ extends to a map from cyclic homology to circle-equivariant symplectic cohomology, $$HC_\ast(F(M)) \to SH^{n+\ast}_{S^1}(M)$$ and that this should be an isomorphism when $\lambda$ is.

For simply connected, spin cotangent bundles $T^\ast L$, one has $SH^\ast_{S^1}(M) \cong H_{n-*}^{S^1}(\mathcal{L}L)$.

Linearised contact (co)homology is, according to Bourgeois-Oancea (if I have it right), the mapping cone of the (cochain level) Viterbo map $H^\ast(M; H^\ast_{S^1}(pt.)) \to SH^\ast_{S^1}(M)$. For cotangent bundles as before, Viterbo's map should be identifiable with the map induced by the equivariant inclusion of constant loops: $H^\ast(T^\ast L)[u] = H^\ast (L)[u] = H_{n-\ast}(L)[u] \to H_{n-\ast}^{S^1}(\mathcal{L}L)$.

1

Some blah on symplectic homology vs. cohomology. There's an invariant $SH(M)$ of Liouville domains $M$ which some people call symplectic homology and some symplectic cohomology. This is the direct limit of Hamiltonian Floer groups associated with functions of increasing eventual slope. The dual theory has two rather unpleasant features: it involves inverse limits, hence one must worry about $\varprojlim^1$-terms; and in general it's not countably generated. It's not often used.

Why the confusion about terminology? Well, depending on your convention for the sign of the symplectic action functional, you may regard this as Morse homology or compactly supported Morse cohomology of this function. From the perspective of Lagrangian Floer cohomology, consistency demands that one calls it symplectic cohomology; I do. However, symplectic field theorists (including Bourgeois-Oancea, I think) prefer the contrary convention.

Blah about grading. The grading on $SH^\ast(M)$ is defined canonically for exact symplectic manifolds with $c_1=0$. One has Viterbo's map $H^\ast(M)\to SH^\ast(M)$, and one convention makes this preserve degree (I'll take that option), while another makes it shift degree by the complex dimension $n$ of $M$.

Seidel's map for cotangent bundles. There are actually two versions of Seidel's open-closed string map, derived from the same moduli spaces: $$\kappa: SH^\ast(M) \to HH^\ast(F(M),F(M))$$ and $$\lambda : HH_\ast(F(M),F(M)) \to SH^{n+\ast}(M).$$ One expects that both are isomorphisms when there are "enough" Lagrangian submanifolds. The absence of a dualisation looks strange, but the point is that the Fukaya category $F(M)$ is expected to have a structure of $n$-dimensional Calabi-Yau $A_\infty$-category (Fukaya has proved a part of this statement), hence its (shifted) Hochschild homology and cohomology should be isomorphic.

Abouzaid has shown in http://arxiv.org/abs/1003.4449 that for $M=T^\ast L$, the map $\lambda$ is an isomorphism. So far as I know, $\kappa$ for $T^\ast L$ is not explicitly studied in the literature; but it's anyway less natural for string topology.

Cyclic version. My expectations are slightly different from those stated in the question. I'd guess that $\lambda$ extends to a map from cyclic homology to circle-equivariant symplectic cohomology, $$HC_\ast(F(M)) \to SH^{n+\ast}_{S^1}(M)$$ and that this should be an isomorphism when $\lambda$ is.

For simply connected, spin cotangent bundles $T^\ast L$, one has $SH^\ast_{S^1}(M) \cong H_{n-*}^{S^1}(\mathcal{L}L)$.

Linearised contact (co)homology is, according to Bourgeois-Oancea (if I have it right), the mapping cone of the (cochain level) Viterbo map $H^\ast(M; H^\ast_{S^1}(pt.)) \to SH^\ast_{S^1}(M)$. For cotangent bundles as before, Viterbo's map should be identifiable with the map induced by the equivariant inclusion of constant loops: $H^\ast(T^\ast L)[u] = H^\ast (L)[u] = H_{n-\ast}(L)[u] \to H_{n-\ast}^{S^1}(\mathcal{L}L)$.