The Oriented Graph Complexes

By Thomas Willwacher. 2013

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The oriented graph complexes $${GC^{or}_n}$$GCnor are complexes of directed graphs without directed cycles. They govern, for example, the quantization of Lie bialgebras and infinite dimensional deformation quantization. Similar to the ordinary graph complexes GCn introduced by Kontsevich they come in two essentially different versions, depending on the parity of n. It is shown that, surprisingly, the oriented graph complex $${GC^{or}_n}$$GCnor is quasi-isomorphic to the ordinary commutative graph complex of opposite parity GCn-1, up to some known classes. This yields in particular a combinatorial description of the action of $${\mathfrak{grt}_{1} \cong H^0({\rm GC}_2)}$$grt1≅H0(GC2) on Lie bialgebras, and shows that a cycle-free formality morphism in the sense of Shoikhet can be constructed rationally without reference to configuration space integrals. Curiously, the obstruction class in the oriented graph complex found by Shoikhet corresponds to the well known theta graph in the ordinary graph complex.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d855bc4db714c736ca5801d64c20bffd47d16870