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Cyclic \(A_\infty\) structures and Deligne’s conjecture. (English) Zbl 1277.18013

Summary: First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic \(A_\infty\) 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 \(A_\infty\) 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 M. Kontsevich and Y. Soibelman [Conférence Moshé Flato 1999: Quantization, deformation, and symmetries, Dijon, France, September 5–8, 1999. Volume I. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 21, 255–307 (2000; Zbl 0972.18005)] and R. M. Kaufmann and R. Schwell [Adv. Math. 223, No. 6, 2166–2199 (2010; Zbl 1194.55014)] 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 \(A_\infty\) categories, with an eye toward the homotopy BV structure present on the Hochschild cochains of the Fukaya category of a suitable symplectic manifold.

MSC:

18D50 Operads (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55P48 Loop space machines and operads in algebraic topology
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