Cattaneo
Fakultäten » Mathematisch-naturwissenschaftliche Fakultät » Mathematik, Institut für » Reine Mathematik » Prof. Dr. Alberto Cattaneo » Cattaneo
Current research project
| Title / Titel | Integral Invariants of Knots and Manifolds. Loop and Path Spaces and Topological BF Theories. Deformation Quantization and Poisson Sigma Models | ||||||
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| Summary / Zusammenfassung | Deformation quantization means finding an associative deformation of the Poisson algebra of functions on a Poisson manifold. The existence and classification of deformation quantizations in the general case was proved by Kontsevich who exhibited an explicit formula. This formula also appears as the perturbative expansion of a topological open string theory called the Poisson sigma model. The reduced phase space of this topological theory is (a singular version of) the symplectic groupoid of the Poisson manifold. Our research concentrates on studying the relation between Kontsevich's formula and the quantization of the symplectic groupoid, on other applications of the Poisson sigma model, on the problem of the quantization of Poisson morphisms, on the quantization of Lie bialgebroids, and on the integration of twisted Poisson structures. Weitere Informationen |
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| Publications / Publikationen | A. S. Cattaneo, G. Felder and T. Willwacher, “The character map in deformation quantization,” Adv. Math. 228, 1966–1989 (2011).A. S. Cattaneo, B. Dherin and A. Weinstein, “Symplectic micro- geometry II: generating functions,” Bull. Brazilian Math. Soc. 42, 507–536 (2011).A. S. Cattaneo and F. Schätz, “Introduction to supergeometry,” Rev. Math. Phys. 23, 669–690 (2011).A. S. Cattaneo and G. Felder, “Effective Batalin–Vilkovisky theories, equivariant configuration spaces and cyclic chains,” Progress in Mathematics 287, 111–137 (2011).A. S. Cattaneo, B. Dherin and G. Felder, “Formal Lagrangian op- erad,” International Journal of Mathematics and Mathematical Sci- ences, vol. 2010, Article ID 643605, 36 pages, 2010. doi:10.1155/2010/643605.A. S. Cattaneo, J. Qiu and M. Zabzine, “2D and 3D topological field theories for generalized complex geometry,” Adv. Theor. Math. Phys. 14, 695–725 (2010).A. S. Cattaneo, B. Dherin and A. Weinstein, “Symplectic microge- ometry, I: Micromorphisms,” J. Sympl. Geom. 8, 205–223 (2010).A. Alekseev, A. S. Cattaneo, Y. Kosmann-Schwarzbach, T. S. Ratiu Special Volume on Poisson Geometry (guest editors) Lett. Math. Phys. 90, Nos. 1-3 (2009)A. S. Cattaneo, G. Dito, M. Kontsevich, D. Sternheimer Special Issue on Deformation Quantization (guest editors) in SIGMA 4 (2008) and 5 (2009)A. S. Cattaneo, G. Felder, Th. Willwacher On L_\infty-morphisms of cyclic chains Preprint No. 06-2009; Lett. Math. Phys. 90, 85-101 (2009)A. S. Cattaneo, M. Zambon Graded geometry and Poisson reduction AIP Conf. Proc. 1093, 48-56 (2009)A. Cattaneo, F. Schätz Equivalences of higher derived brackets J. Pure and Applied Algebra 212, 2450-2460 (2008)A. Cattaneo, M. Zambon Coisotropic Embeddings in Poisson Manifolds to appear in Trans. AMSA. Cattaneo Deformation Quantization and Reduction Cont. Math. 450, 79-101 (2008)A. Cattaneo, M. Zambon Pre-Poisson submanifolds Travaux mathématiques 17, 61-74 (2007)A. Cattaneo, C. Torossian Quantification pour les paires symétriques et diagrammes de Kontsevich Ann. Sci. École Norm. Sup. 41, 787-852 (2008)A. Bruguières, A. Cattaneo, B. Keller, C. Torossian, Déformation, Quantification, Théorie de Lie, Panoramas et Synthése 20 (2005), viii+186 pages.A. Cattaneo, G. Felder, Relative formality theorem and quantisation of coisotropic submanifolds, Adv. Math. 208, 521-548 (2007).A. Cattaneo, From Topological Field Theory to Deformation Quantization and Reduction, Proceedings of ICM 2006, Vol. III, 339-365 (European Mathematical Society, 2006).A. Cattaneo, D. Fiorenza, R. Longoni, On the Hochschild-Kostant-Rosenberg Map for Graded Manifolds, IMRN 62, 3899-3918 (2005).F. Bonechi, A. Cattaneo, M. Zabzine, Geometric quantization and non-perturbative Poisson sigma model, Adv. Theor. Math. Phys. 10, 683-712 (2006).A. Cattaneo, D. Fiorenza, R. Longoni, Graded Poisson Algebras, Encyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T. , vol. 2, p. 560-567 (Oxford: Elsevier, 2006).A. Cattaneo, D. Indelicato, Formality and Star Products, in Poisson Geometry, Deformation Quantisation and Group Representations, (ed. S. Gutt, J. Rawnsley, D. Sternheimer), London Mathematical Society Lecture Note Series 323, 79-144 (Cambridge University Press, 2005).A. Cattaneo, B. Dherin, G. Felder, Formal symplectic groupoid, Commun. Math. Phys. 253, 645-674 (2005).A. Cattaneo, G. Felder, Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. Math. Phys. 69; 157-175 (2004).A. Cattaneo, On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds, Lett. Math. Phys. 67, 33-48 (2004).A. Cattaneo, P. Xu, Integration of twisted Poisson structures, J. Geom. Physics 49, 187-196 (2004).A. Cattaneo, G. Felder, L. Tomassini, From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115, 329-352 (2002).A. Cattaneo, G. Felder, On the globalization of Kontsevich's star product and the perturbative Poisson sigma model, Prog. Theor. Phys. Suppl. 144, 38-53 (2001).A. Cattaneo, G. Felder, L. Tomassini, Fedosov connections on jet bundles and deformation quantization, in: Deformation Quantization (ed. G. Halbout), IRMA Lectures in Mathematics and Theoretical Physics (ed. V. Turaev), 191-202 (de Gruyter, Berlin, 2002).A. Cattaneo, G. Felder, Poisson sigma models and symplectic groupoids, (ed. N. P. Landsman, M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198, 61-93 (Birkhäuser, 2001).A. Cattaneo, G. Felder, On the AKSZ formulation of the Poisson sigma model, Lett. Math. Phys. 56, 163-179 (2001) .A. Cattaneo, G. Felder, Poisson sigma models and deformation quantization, Mod. Phys. Lett. A 16, 179-190 (2001).A. Cattaneo, G. Felder, A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212, 591-611 (2000).Weitere Informationen |
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| Project leadership and contacts / Projektleitung und Kontakte |
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SNF (Personen- und Projektförderung) |
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| Duration of Project / Projektdauer | Oct 2010 to Sep 2013 |