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2012 Cyclic $A_\infty$ structures and Deligne's conjecture
Benjamin C Ward
Algebr. Geom. Topol. 12(3): 1487-1551 (2012). DOI: 10.2140/agt.2012.12.1487


First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic A 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 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 categories, with an eye toward the homotopy BV structure present on the Hochschild cochains of the Fukaya category of a suitable symplectic manifold.


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Benjamin C Ward. "Cyclic $A_\infty$ structures and Deligne's conjecture." Algebr. Geom. Topol. 12 (3) 1487 - 1551, 2012.


Received: 24 February 2012; Revised: 24 April 2012; Accepted: 25 April 2012; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1277.18013
MathSciNet: MR2966694
Digital Object Identifier: 10.2140/agt.2012.12.1487

Primary: 16E40 , 18D50

Keywords: BV algebra , cactus , cyclic A infinity algebra , Fukaya category , Hochschild cohomology , operad

Rights: Copyright © 2012 Mathematical Sciences Publishers

Vol.12 • No. 3 • 2012
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