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Relative formality theorem and quantisation of coisotropic submanifolds


Cattaneo, A S; Felder, G (2007). Relative formality theorem and quantisation of coisotropic submanifolds. Advances in Mathematics, 208(2):521-548.

Abstract

We prove a relative version of Kontsevich's formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich's theorem if C=M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L∞-quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to the deformation quantisation of coisotropic submanifolds are given. The proof uses a duality transformation to reduce the theorem to a version of Kontsevich's theorem for supermanifolds, which we also discuss. In physical language, the result states that there is a duality between the Poisson sigma model on a manifold with a D-brane and the Poisson sigma model on a supermanifold without branes (or, more properly, with a brane which extends over the whole supermanifold).

Abstract

We prove a relative version of Kontsevich's formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich's theorem if C=M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L∞-quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to the deformation quantisation of coisotropic submanifolds are given. The proof uses a duality transformation to reduce the theorem to a version of Kontsevich's theorem for supermanifolds, which we also discuss. In physical language, the result states that there is a duality between the Poisson sigma model on a manifold with a D-brane and the Poisson sigma model on a supermanifold without branes (or, more properly, with a brane which extends over the whole supermanifold).

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2007
Deposited On:07 Dec 2009 14:25
Last Modified:18 Feb 2018 12:51
Publisher:Elsevier
ISSN:0001-8708
OA Status:Hybrid
Publisher DOI:https://doi.org/10.1016/j.aim.2006.03.010
Related URLs:http://arxiv.org/abs/math/0501540

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Download PDF  'Relative formality theorem and quantisation of coisotropic submanifolds'.
Preview
Content: Accepted Version
Filetype: PDF (Accepted manuscript, Version 1)
Size: 340kB