# Effective Batalin-Vilkovisky theories, equivariant configuration spaces and cyclic chains

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.

## 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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