Short answer.

• For a Lefschetz Landau–Ginzburg model (Y, ω, W) with only Morse critical points (the setting in which the Fukaya–Seidel category FS(Y, W) is defined), the “closed string” A-model state space is the relative cohomology H*(Y, Y−∞; C), and there are natural open–closed and closed–open maps OC: HH∗(FS(Y, W)) → H∗+n(Y, Y−∞), CO: H∗(Y, Y−∞) → HH∗(FS(Y, W)), where n = dimC Y.

• The open–closed map is an isomorphism under the standard generation hypotheses (vanishing cycles generate FS), so HH∗(FS(Y, W)) ≅ H∗+n(Y, Y−∞) (up to the usual grading shift by n). This is the correct (and sharp) A-side identification for Hochschild homology.

• For Hochschild cohomology, there is in general no simple cohomological description like the one for homology. Instead, HH∗(FS(Y, W)) sits in a long exact sequence together with the relative cohomology H∗(Y, Y−∞) and the fixed-point Floer cohomology of the global monodromy φ of W on a regular fiber F:

  … → HF∗+n(φ) → H∗(Y, Y−∞) —CO→ HH∗(FS(Y, W)) → HF∗+n+1(φ) → H∗+1(Y, Y−∞) → …

Consequences and interpretations.

• The map CO is a unital ring map; its image consists of the “bulk” (closed-string) deformations. In general CO is not surjective: Hochschild cohomology is typically larger than the closed-string ring. The failure of surjectivity is precisely measured by fixed-point Floer cohomology of the monodromy.

• In the favorable situation that HF∗(φ) = 0 (e.g. in certain exact or “negatively monotone” situations where the monodromy has no Floer fixed points), CO is an isomorphism and then HH∗(FS(Y, W)) ≅ H∗(Y, Y−∞) as graded rings (up to the usual shift). In general, however, HH∗(FS) is strictly bigger (already for the mirror of CP^1 one finds dim HH∗ = 4 whereas dim H∗(Y, Y−∞) = 2).

• Under homological mirror symmetry, when FS(Y, W) ≅ DbCoh(X) for a smooth projective Fano X, one recovers the Hochschild–Kostant–Rosenberg description HH∗(FS(Y, W)) ≅ ⊕p+q=∗ Hp(X, ∧qTX), and the long exact sequence above mirrors the fact that the map from the quantum/closed A-model cohomology to HH∗(DbCoh(X)) need not be surjective.

Deformations (the HH2-piece).

• First-order deformations of the A∞-structure on FS(Y, W) are controlled by HH^2(FS(Y, W)). From the long exact sequence one obtains the exact piece HFn+2(φ) → H^2(Y, Y−∞) —CO→ HH^2(FS(Y, W)) → HFn+3(φ), which shows:
  • Image(CO) ⊂ HH^2 gives the “bulk” deformations coming from closed-string classes (varying ω and, in an LG setting, deforming W through admissible functions).
  • The kernel/cokernel are governed by fixed-point Floer cohomology of the monodromy. These correspond to additional, typically non-geometric, deformations (mirror to complex-structure/polyvector-field deformations on the B-side).

References.

• P. Seidel, Fukaya categories and Picard–Lefschetz theory, esp. Chapters 18–19: construction of FS(Y, W), open–closed/closed–open maps, and the identification HH∗(FS) ≅ H∗+n(Y, Y−∞) for homology; also the long exact sequence relating HH∗, relative cohomology, and HF∗(φ).
• M. Abouzaid – P. Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010): The long exact sequence quoted above (Theorem 1.4) and its consequences.
• S. Ganatra, Symplectic cohomology and duality for the wrapped Fukaya category, and S. Ganatra – J. Pardon – V. Shende (a series on Liouville sectors and wrapped Fukaya categories): general frameworks in which OC is an isomorphism and CO is a ring map; recovers HH∗(FS) ≅ H∗+n(Y, Y−∞) for homology and the LES for cohomology.

Summary.

• Hochschild homology of FS(Y, W) is canonically isomorphic (up to shift) to the relative cohomology H∗(Y, Y−∞).
• Hochschild cohomology is not, in general, equal to that relative cohomology. It fits into a natural long exact sequence with H∗(Y, Y−∞) and the fixed-point Floer cohomology HF∗(φ) of the global monodromy. In particular cases (HF∗(φ) = 0), one does get HH∗(FS) ≅ H∗(Y, Y−∞), but in general HH∗ is strictly larger; this matches the mirror-side HKR description when FS(Y, W) ≅ DbCoh(X).