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Bimodules and branes in deformation quantization

Published online by Cambridge University Press:  11 August 2010

Damien Calaque
Affiliation:
Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France (email: [email protected])
Giovanni Felder
Affiliation:
Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland (email: [email protected])
Andrea Ferrario
Affiliation:
Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland (email: [email protected])
Carlo A. Rossi
Affiliation:
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal (email: [email protected])
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Abstract

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We prove a version of Kontsevich’s formality theorem for two subspaces (branes) of a vector space X. The result implies, in particular, that the Kontsevich deformation quantizations of S(X*) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet’s recent paper on Koszul duality in deformation quantization.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Arnal, D., Manchon, D. and Masmoudi, M., Choix des signes pour la formalité de M. Kontsevich, Pacific J. Math. 203 (2002), 2366, in French with English summary; MR 1895924(2003k:53123).CrossRefGoogle Scholar
[2]Beilinson, A., Ginzburg, V. and Soergel, W., Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473527; MR 1322847(96k:17010).CrossRefGoogle Scholar
[3]Calaque, D. and Rossi, C. A., Lectures on Duflo isomorphisms in Lie algebras and complex geometry, Lecture notes, ETH Zürich (2008), available at http://math.univ-lyon1.fr/∼calaque/LectureNotes/LectETH.pdf.Google Scholar
[4]Calaque, D. and Rossi, C. A., Compatibility with cap-products in Tsygan’s formality and homological Duflo isomorphism, Preprint (2008), available at arXiv:0805.2409v2, Lett. Math. Phys., to appear.Google Scholar
[5]Cattaneo, A. S. and Felder, G., Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. Math. Phys. 69 (2004), 157175; MR 2104442(2005m:81285).CrossRefGoogle Scholar
[6]Cattaneo, A. S. and Felder, G., Relative formality theorem and quantisation of coisotropic submanifolds, Adv. Math. 208 (2007), 521548; MR 2304327(2008b:53119).CrossRefGoogle Scholar
[7]Cattaneo, A. S. and Torossian, C., Quantification pour les paires symétriques et diagrammes de Kontsevich, Ann. Sci. Éc. Norm. Super. (4) 41 (2008), 789854, in French with English and French summaries; MR 2504434.CrossRefGoogle Scholar
[8]Dolgushev, V., A formality theorem for Hochschild chains, Adv. Math. 200 (2006), 51101; MR 2199629(2006m:16010).CrossRefGoogle Scholar
[9]Ferrario, A., Poisson sigma model with branes and hyperelliptic Riemann surfaces, J. Math. Phys. 49 (2008), 092301; MR 2455835.CrossRefGoogle Scholar
[10]Getzler, E. and Jones, J. D. S., A∞-algebras and the cyclic bar complex, Illinois J. Math. 34 (1990), 256283; MR 1046565(91e:19001).CrossRefGoogle Scholar
[11]Keller, B., Introduction to A-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), 135; MR 1854636(2004a:18008a).CrossRefGoogle Scholar
[12]Keller, B., Derived invariance of higher structures on the Hochschild complex, Preprint (2003), available at http://people.math.jussieu.fr/∼keller/publ/dih.pdf.Google Scholar
[13]Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157216; MR 2062626(2005i:53122).CrossRefGoogle Scholar
[14]Lefèvre-Hasegawa, K., Sur les A -catégories, PhD thesis, Université Paris 7 (2003), available at http://people.math.jussieu.fr/∼keller/lefevre/TheseFinale/tel-00007761.pdf.Google Scholar
[15]Rinehart, G. S., Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195222; MR 0154906(27#4850).CrossRefGoogle Scholar
[16]Shoikhet, B., Koszul duality in deformation quantization and Tamarkin’s approach to Kontsevich formality, Preprint (2008), arXiv:0805.0174, Adv. Math., to appear.Google Scholar
[17]Tamarkin, D. E., Another proof of M. Kontsevich formality theorem, Preprint (1998), arXiv:math/9803025v4.Google Scholar
[18]Tamarkin, D. and Tsygan, B., Cyclic formality and index theorems, Lett. Math. Phys. 56 (2001), 8597, EuroConférence Moshé Flato 2000, Part II (Dijon); MR 1854129(2003e:19008).CrossRefGoogle Scholar
[19]Tradler, T., Infinity-inner-products on A-infinity-algebras, J. Homotopy Relat. Struct. 3 (2008), 245271; MR 2426181.Google Scholar
[20]Voronov, A. A. and Gerstenhaber, M., Higher-order operations on the Hochschild complex, Funktsional. Anal. i Prilozhen. 29 (1995), 16, 96, in Russian; English translation in Funct. Anal. Appl. 29 (1995), 1–5; MR 1328534(96g:18006).CrossRefGoogle Scholar
[21]Willwacher, T., A counterexample to the quantizability of modules, Lett. Math. Phys. 81 (2007), 265280; MR 2355492(2008j:53160).CrossRefGoogle Scholar