Cyclic A∞ structures and Deligne’s conjecture

Ward, Benjamin C¹

¹Department of Mathematics, Purdue University, 150 N University St, West Lafayette IN 47907, USA

Algebraic and Geometric Topology, Tome 12 (2012) no. 3, pp. 1487-1551

Résumé

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.

Détail

DOI : 10.2140/agt.2012.12.1487

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

Bibliographie

[1] A Abbondandolo, M Schwarz, On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006) 254

[2] A Abbondandolo, M Schwarz, Floer homology of cotangent bundles and the loop product, Geom. Topol. 14 (2010) 1569

[3] M A Batanin, Symmetrisation of $n$–operads and compactification of real configuration spaces, Adv. Math. 211 (2007) 684

[4] C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805

[5] R Bott, C Taubes, On the self-linking of knots, J. Math. Phys. 35 (1994) 5247

[6] M Chas, D Sullivan, String topology, to appear in Ann. of Math.

[7] K Cieliebak, A Floer, H Hofer, Symplectic homology II: A general construction, Math. Z. 218 (1995) 103

[8] R L Cohen, K Hess, A A Voronov, String topology and cyclic homology, Adv. Courses in Math. CRM Barcelona, Birkhäuser (2006)

[9] R L Cohen, J D S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773

[10] K Costello, Topological conformal field theories and Calabi–Yau categories, Adv. Math. 210 (2007) 165

[11] V A Dolgushev, D E Tamarkin, B L Tsygan, Proof of Swiss cheese version of Deligne's conjecture, Int. Math. Res. Not. 2011 (2011) 4666

[12] G Drummond-Cole, B Vallette, The minimal model for the Batalin–Vilkovisky operad

[13] Y Félix, L Menichi, J C Thomas, Gerstenhaber duality in Hochschild cohomology, J. Pure Appl. Algebra 199 (2005) 43

[14] I Galvez-Carrillo, A Tonks, B Vallette, Homotopy Batalin–Vilkovisky algebras

[15] M Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. 78 (1963) 267

[16] E Getzler, Cartan homotopy formulas and the Gauss–Manin connection in cyclic homology, from: "Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992)" (editors A Joseph, S Shnider), Israel Math. Conf. Proc. 7, Bar-Ilan Univ. (1993) 65

[17] E Getzler, Batalin–Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994) 265

[18] E Getzler, M M Kapranov, Cyclic operads and cyclic homology, from: "Geometry, topology, physics" (editor S T Yau), Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press (1995) 167

[19] J Giansiracusa, P Salvatore, Formality of the framed little $2$–discs operad and semidirect products, from: "Homotopy theory of function spaces and related topics" (editors Y Félix, G Lupton, S B Smith), Contemp. Math. 519, Amer. Math. Soc. (2010) 115

[20] V Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997) 3291

[21] T V Kadeishvili, The structure of the $A(\infty)$–algebra, and the Hochschild and Harrison cohomologies, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988) 19

[22] R M Kaufmann, On several varieties of cacti and their relations, Algebr. Geom. Topol. 5 (2005) 237

[23] R M Kaufmann, Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: Cell operads, J. Noncommut. Geom. 1 (2007) 333

[24] R M Kaufmann, Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: Correlators, J. Noncommut. Geom. 2 (2008) 283

[25] R M Kaufmann, A proof of a cyclic version of Deligne's conjecture via cacti, Math. Res. Lett. 15 (2008) 901

[26] R M Kaufmann, M Livernet, R C Penner, Arc operads and arc algebras, Geom. Topol. 7 (2003) 511

[27] R M Kaufmann, R Schwell, Associahedra, cyclohedra and a topological solution to the $A_\infty$ Deligne conjecture, Adv. Math. 223 (2010) 2166

[28] M Kontsevich, Feynman diagrams and low-dimensional topology, from: "First European Congress of Mathematics, Vol. II (Paris, 1992)" (editors A Joseph, F Mignot, F Murat, B Prum, R Rentschler), Progr. Math. 120, Birkhäuser (1994) 97

[29] M Kontsevich, Y Soibelman, Deformations of algebras over operads and the Deligne conjecture, from: "Conférence Moshé Flato 1999, Vol. I (Dijon)", Math. Phys. Stud. 21, Kluwer Acad. Publ. (2000) 255

[30] K Lefevre-Hasegama, Sur les $a_\infty$ categories

[31] J L Loday, Cyclic homology, Grundl. Math. Wissen. 301, Springer (1998)

[32] J L Loday, B Vallette, Algebraic operads version0.99, book preprint (2011)

[33] M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monogr. 96, Amer. Math. Soc. (2002)

[34] J E Mcclure, J H Smith, A solution of Deligne's Hochschild cohomology conjecture, from: "Recent progress in homotopy theory (Baltimore, MD, 2000)" (editor D M Davis), Contemp. Math. 293, Amer. Math. Soc. (2002) 153

[35] L Menichi, Batalin–Vilkovisky algebras and cyclic cohomology of Hopf algebras, $K$–Theory 32 (2004) 231

[36] P Seidel, Fukaya categories and deformations, from: "Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002)" (editor T Li), Higher Ed. Press (2002) 351

[37] P Seidel, Fukaya categories and Picard–Lefschetz theory, Zurich Lectures in Adv. Math., European Math. Soc. (2008)

[38] P Seidel, Symplectic homology as Hochschild homology, from: "Algebraic geometry—Seattle 2005. Part 1" (editors D Abramovich, A Bertram, L Katzarkov, R Pandharipande, M Thaddeus), Proc. Sympos. Pure Math., Amer. Math. Soc. (2009) 415

[39] P Ševera, Formality of the chain operad of framed little disks, Lett. Math. Phys. 93 (2010) 29

[40] D Tamarkin, B Tsygan, Noncommutative differential calculus, homotopy BV algebras and formality conjectures, Methods Funct. Anal. Topology 6 (2000) 85

[41] T Tradler, The Batalin–Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Ann. Inst. Fourier (Grenoble) 58 (2008) 2351

[42] T Tradler, M Zeinalian, On the cyclic Deligne conjecture, J. Pure Appl. Algebra 204 (2006) 280

[43] C Viterbo, Functors and computations in Floer homology with applications, I, Geom. Funct. Anal. 9 (1999) 985

[44] A A Voronov, Homotopy Gerstenhaber algebras, from: "Conférence Moshé Flato 1999, Vol. II (Dijon)" (editors G Dito, D Sternheimer), Math. Phys. Stud. 22, Kluwer Acad. Publ. (2000) 307

[45] A A Voronov, Notes on universal algebra, from: "Graphs and patterns in mathematics and theoretical physics" (editors M Lyubich, L Takhtajan), Proc. Sympos. Pure Math. 73, Amer. Math. Soc. (2005) 81

[46] N Wahl, C Westerland, Hochschild homology of structured algebras