First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)operad of CW–complexes whose constituent spaces form a homotopy associative version of the cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyclic version of Deligne’s conjecture. This chain model contains the minimal operad of Kontsevich and Soibelman as a suboperad and restriction of the action to this suboperad recovers the results of Kontsevich and Soibelman [Math. Phys. Stud. 21, Kluwer Acad. Publ., Dordrecht (2000) 255–307] and Kaufmann and Schwell [Adv. Math. 223 (2010) 2166–2199] in the unframed case. Additionally this proof recovers the work of Kaufmann in the case of a strict Frobenius algebra. We then extend our results to the context of cyclic categories, with an eye toward the homotopy BV structure present on the Hochschild cochains of the Fukaya category of a suitable symplectic manifold.
"Cyclic $A_\infty$ structures and Deligne's conjecture." Algebr. Geom. Topol. 12 (3) 1487 - 1551, 2012. https://doi.org/10.2140/agt.2012.12.1487