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RESEARCH PRODUCT

Kontsevich formality and cohomologies for graphs

Didier Arnal

subject

AlgebraPure mathematicsMathematics::K-Theory and HomologyMathematics::Quantum AlgebraComplex systemStatistical and Nonlinear PhysicsQuasi-isomorphismFormalitySpace (mathematics)Mathematical PhysicsCohomologyManifoldMathematics

description

A formality on a manifold M is a quasi isomorphism between the space of polyvector fields (Tpoly(M)) and the space of multidifferential operators (Dpoly(M)). In the case M=R d , such a mapping was explicitly built by Kontsevich, using graphs drawn in configuration spaces. Looking for such a construction step by step, we have to consider several cohomologies (Hochschild, Chevalley, and Harrison and Chevalley) for mappings defined on Tpoly. Restricting ourselves to the case of mappings defined with graphs, we determine the corresponding coboundary operators directly on the spaces of graphs. The last cohomology vanishes.

yearjournalcountryeditionlanguage
2004-07-01Letters in Mathematical Physics
https://doi.org/10.1007/s11005-004-1220-7