This paper concerns the compatibility between chain level \(S^{1}\)-actions arising in two different types of Floer theory on a symplectic manifold \(M\).

(1)

The first of these \(C_{-\ast} (S^{1})\)-actions is induced geometrically on the Hamiltonian Floer homology chain complex \(CF^{\ast} (M)\), formally a type of Morse complex for an action functorial on the free loop space, through rotating free loops. The homological action of \(\left[ S^{1}\right]\) is known as the BV operator \(\left[ \Delta\right]\), and the \(C_{-\ast} (S^{1})\)-action is to be used to define \(S^{1}\)-equivariant Floer homology theories [F. Bourgeois and A. Oancea, Int. Math. Res. Not. 2017, No. 13, 3849–3937 (2017; Zbl 1405.53123); P. Seidel, in: Current developments in mathematics, 2006. Somerville, MA: International Press. 211–253 (2008; Zbl 1165.57020)].

(2)

The second \(C_{-\ast} (S^{1})\)-action lies on the Fukaya category of \(M\), and has discrete or combinatorial origins, coming from the hierarchy of compatible \(\mathbb{Z}/k\mathbb{Z}\)-actions on cyclically composable chains of morphisms between Lagrangians. A. Connes [Publ. Math., Inst. Hautes Étud. Sci. 62, 41–144 (1985; Zbl 0592.46056)], B. L. Tsygan [Russ. Math. Surv. 38, No. 2, 198–199 (1983; Zbl 0526.17006)] and J.-L. Loday and D. Quillen [Comment. Math. Helv. 59, 565–591 (1984; Zbl 0565.17006)] observed that such a structure, which exists on any category \(\mathcal{C}\), can be packaged into a \(C_{-\ast} (S^{1})\)-action on the Hochschild homology chain complex \(\mathrm{CH}_{\ast}(\mathcal{C})\) of the category.

A relationship between the Hochschild homology of the Fukaya category \(\mathcal{F}\) and the Floer homology of \(M\) is provided by the so-called open-closed string map [M. Abouzaid, Publ. Math., Inst. Hautes Étud. Sci. 112, 191–240 (2010; Zbl 1215.53078)] \[ \mathcal{OC}:\mathrm{CH}_{\ast} (\mathcal{F})\rightarrow CF^{\ast+n} (M). \] The main result of this paper concerns the comparibility of \(\mathcal{OC}\) with \(C_{-\ast} (S^{1})\)-actions.
The synopsis of the paper is as follows.

§ 2

recalls a convenient model for the category of \(A_{\infty}\)-modules over \(C_{-\ast} (S^{1})\) and various equivariant homology functors from this category.

§ 3

reviews the (compact and wrapped) Fukaya category along with \(C_{-\ast} (S^{1})\)-action on its nonunital Hochschild chain complex.

§ 4

recalls the construction of the \(A_{\infty}C_{-\ast}(S^{1})\)-module structure on the (Hamiltonian) Floer chain complex after {F. Bourgeois} and {A. Oancea} [loc. cit.] and {P. Seidel} [loc. cit.].

§ 5

establishes the main result concerning the comparibility of \(\mathcal{OC}\) with \(C_{-\ast} (S^{1})\)-actions.

§ 6

applies the main result to construct proper and smooth Calabi-Yau structures, establishing the following two theorems.
Theorem. The Fukaya category of compact Lagrangians has, under some technical hypotheses, a canonical geometrically defined proper Calabi-Yau structure over any ground field.
Theorem. Under some techinical hypotheses, suppose further that the symplectic manifold \(M\) is nondegenerate in the sense of [S. Ganatra, “Symplectic cohomology and duality for the wrapped Fukaya category”, Preprint, arXiv:1304.7312]. Then its (compact or wrapped) Fukaya category possesses a canonical, geometrically defined strong smooth Calabi-Yau structure.

The appendix

is concerned with moduli spaces and operations.