Volume 12, issue 3 (2012)

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Cyclic $A_\infty$ structures and Deligne's conjecture

Benjamin C Ward

Algebraic & Geometric Topology 12 (2012) 1487–1551
Abstract

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 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 ${A}_{\infty }$ categories, with an eye toward the homotopy BV structure present on the Hochschild cochains of the Fukaya category of a suitable symplectic manifold.

Keywords
operad, cactus, BV algebra, cyclic A infinity algebra, Hochschild cohomology, Fukaya category
Mathematical Subject Classification 2010
Primary: 16E40, 18D50