Modular Operads and Batalin-Vilkovisky Geometry

We describe the noncommutative Batalin-Vilkovisky geometry associated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic geometry of the corresponding tree-level cyclic operad. We show, in particular, that the algebras over the Feynman tran...

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Bibliographic Details
Published in:International mathematics research notices 2007-01, Vol.2007
Main Author: Barannikov, S.
Format: Article
Language:English
Subjects:
Algebraic Geometry
Mathematics
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Summary:We describe the noncommutative Batalin-Vilkovisky geometry associated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic geometry of the corresponding tree-level cyclic operad. We show, in particular, that the algebras over the Feynman transform of a twisted modular operad P are in one-to-one correspondence with solutions to the quantum master equation of Batalin-Vilkovisky geometry on the affine P–manifolds. As an application we give a construction of characteristic classes with values in the homology of the quotient of Deligne-Mumford moduli spaces. These classes are associated naturally with solutions to the quantum master equation on affine S[t]–manifolds, where S[t] is the twisted modular Det–operad constructed from symmetric groups, which generalizes the cyclic operad of associative algebras.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnm075