The celebrated Kontsevich formality theorem [M. Kontsevich, Lett. Math. Phys. 66 (2003), no. 3, 157--216; MR2062626 (2005i:53122)] states that the differential graded Lie algebra gG of polydifferential operators on a smooth manifold M is formal, i.e., it is quasi-isomorphic to its cohomology which is, in turn, identified with the Schouten Lie algebra gS of polyvector fields on M.

Moreover, this quasi-isomorphism is realized by a certain L∞ map from gS to gG whose components are expressed through certain correlation functions of a topological field theory on the upper half-plane as shown by the present authors [A. S. Cattaneo and G. Felder, Comm. Math. Phys. 212 (2000), no. 3, 591--611; MR1779159 (2002b:53141)].

In the paper under review the authors consider the case of a manifold M endowed with a volume form and the differential graded Lie algebra gS[v], where v is a formal parameter and the differential is the divergence operator times v. Via the Kontsevich formality map the complex of negative cyclic chains of the algebra of smooth functions on M becomes an L∞-module over gS[v]. Consider also gS endowed with the divergence operator viewed as a differential and with the trivial action of gS[v]. The main result of the paper is the construction of an L∞-map between these L∞-modules.

The relevant quantum field theory is a BF theory on a disc (as opposed to the upper half-plane) which is treated in the framework of the Batalin-Vilkovisky quantization. The new feature is the presence of the zero modes of the action functional.

As an application the authors construct traces on algebras of functions with star-products associated with unimodular Poisson structures.

Cattaneo, A S; Felder, G (2011). *Effective Batalin-Vilkovisky theories, equivariant configuration spaces and cyclic chains.* In: Cattaneo, A S; Giaquinto, A; Xu, P. LinkHigher structures in geometry and physics : In honor of Murray Gerstenhaber and Jim Stasheff. Boston: Birkhäuser, 111-137.

## Abstract

The celebrated Kontsevich formality theorem [M. Kontsevich, Lett. Math. Phys. 66 (2003), no. 3, 157--216; MR2062626 (2005i:53122)] states that the differential graded Lie algebra gG of polydifferential operators on a smooth manifold M is formal, i.e., it is quasi-isomorphic to its cohomology which is, in turn, identified with the Schouten Lie algebra gS of polyvector fields on M.

Moreover, this quasi-isomorphism is realized by a certain L∞ map from gS to gG whose components are expressed through certain correlation functions of a topological field theory on the upper half-plane as shown by the present authors [A. S. Cattaneo and G. Felder, Comm. Math. Phys. 212 (2000), no. 3, 591--611; MR1779159 (2002b:53141)].

In the paper under review the authors consider the case of a manifold M endowed with a volume form and the differential graded Lie algebra gS[v], where v is a formal parameter and the differential is the divergence operator times v. Via the Kontsevich formality map the complex of negative cyclic chains of the algebra of smooth functions on M becomes an L∞-module over gS[v]. Consider also gS endowed with the divergence operator viewed as a differential and with the trivial action of gS[v]. The main result of the paper is the construction of an L∞-map between these L∞-modules.

The relevant quantum field theory is a BF theory on a disc (as opposed to the upper half-plane) which is treated in the framework of the Batalin-Vilkovisky quantization. The new feature is the presence of the zero modes of the action functional.

As an application the authors construct traces on algebras of functions with star-products associated with unimodular Poisson structures.

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## Additional indexing

Item Type: | Book Section, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2011 |

Deposited On: | 09 Sep 2011 06:48 |

Last Modified: | 05 Apr 2016 14:57 |

Publisher: | Birkhäuser |

Series Name: | Progress in Mathematics |

Number: | 287 |

ISBN: | 978-0-8176-4734-6 |

Publisher DOI: | 10.1007/978-0-8176-4735-3 |

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