Wheeled PROPs, graph complexes and the master equation

Details Wheeled PROPs, graph complexes and the master equation Wheeled PROPs, graph complexes and the master equation

2006-10-23

arxiv.org/abs/math/0610683v3

J. Pure Appl. Algebra 213 (2009), no. 4, 496-535

Mathematics

Algebraic Geometry

LaTeX2e, 63 pages; Theorem 4.2.5 on bar-cobar construction is strengthened

Scientific paper

10.1016/j.jpaa.2008.08.007

We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and Ass as rather non-obvious extensions of Com_infty and Ass_infty, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich's complex of ribbon graphs.

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