Cyclic A_\infty Structures and Deligne's Conjecture

TL;DR

This paper introduces cyclic A_infinity algebras and constructs a chain model for the framed little disks operad, proving a cyclic version of Deligne's conjecture and connecting to homotopy BV structures.

Contribution

It defines cyclic A_infinity algebras via cyclic operads, constructs a new combinatorial operad acting on Hochschild cochains, and proves a cyclic Deligne's conjecture using topological operads.

Findings

Constructed a chain model for the framed little disks operad.

Proved a cyclic A_infinity version of Deligne's conjecture.

Connected the operad to homotopy BV structures on Hochschild cochains.

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 their results 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 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.