G_∞-Formality Theorem in Terms of Graphs and Associated Chevalley–Eilenberg–Harrison Cohomology^#

^#Communicated by M. Rousso.

In 1997, M. Kontsevich proved the L _∞-formality conjecture (which implies the existence of star-products for any Poisson manifold) using graphs. A year later, D. Tamarkin gave another proof of a more general conjecture (for G _∞-structures) using operads and cohomological methods. In this article, we show how Tamarkin's construction can be written using graphs. For that, we introduce a generalization of Kontsevich graphs on which we define a “Chevalley–Eilenberg–Harrison” complex. We show that this complex on graphs is related to the “Chevalley–Eilenberg–Harrison” complex for maps on polyvector fields, which is trivial and give Tamarkin's formality theorem as a consequence. This formality reduces to an L _∞-formality.

Key Words:

• Deformation quantization
• Graphs
• Homological methods
• Star-product

Mathematics Subject Classification:

• Primary 16E40
• 53D55
• Secondary 18D50
• 16S80

Notes

^#Communicated by M. Rousso.