On operadic open-closed maps in characteristic p

Consider a closed monotone symplectic manifold (M,ω). {Gan2} constructed a cyclic open-closed map, which goes from the cyclic homology of the Fukaya category of M to the S¹-equivariant quantum cohomology of M. In this paper, we show that with mod p coefficients, Ganatra's cyclic open-closed map is compatible with a certain Z/p-equivariant open-closed map under the natural Z/p-Gysin type comparison map for Hochschild homology. Along with the proof, this paper gives a new homotopy theoretic framework for studying open-closed maps in symplectic topology. These will be used in an upcoming work {Che} to study mod p equivariant enumerative invariants such as the Quantum Steenrod operations. The main insights of this paper are: 1) a Z/p-Gysin comparison result for (A∞-) cyclic objects, 2) a new construction of the open-closed map using operadic Floer theory of {AGV}, which gives rise to a new interpretation of its `S¹-equivariant' property, and 3) comparison of the new construction with its classical counterpart.