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the little n-disk operad is formal

Statement

The little n-disk operad is formal in the sense that for each of its component topological spaces there is a zig-zag of quasi-isomorphisms between their de Rham cohomology and their de Rham complex, and such that these morphisms are compatible with the induced cooperad-structure on both sides.

Concretely, the zig-zags may be taken to consist of one span of quasi-isomorphisms out of a suitable graph complex to the de Rham cohomology/de Rham complex of the Fulton-MacPherson operad, which in turn is weakly equivalent to the little n-disk operad (this Prop.).

Here the morphism from the graph complex to the de Rham complex of the Fulton-MacPherson operad regards the latter as the compactification of a configuration space of points, regards functions/differential forms on configuration spaces of points as n-point functions of a topological quantum field theory, regards suitable graphs as Feynman diagrams and proceeds by sending each such graph/Feynman diagram to a corresponding Feynman amplitude.

This idea of a proof was sketched in Kontsevich 99, a full account is due to Lambrechts-Volic 14.

Related concepts

• Kontsevich formality

References

The special case of formality of $E_2$ (see also Kontsevich formality) was finally proven in

• Dmitry Tamarkin, Another proof of M. Kontsevich formality theorem (arXiv:math/9803025)

• Dmitry Tamarkin, Formality of Chain Operad of Small Squares

See Kontsevich 99, p. 15 for the history of this result.

A general proof of formality of $E_n$ for all $n$ was sketched in

• Maxim Kontsevich, section 3, around Def. 15 and Lemma 3 in Operads and Motives in Deformation Quantization, Lett.Math.Phys.48:35-72,1999 (arXiv:math/9904055)

and fully spelled out in

• Pascal Lambrechts, Ismar Volić, sections 6 and 7 of Formality of the little N-disks operad, Memoirs of the American Mathematical Society ; no. 1079, 2014 (arXiv:0808.0457, doi:10.1090/memo/1079)

Further discussion of the graph complex as a model for the de Rham cohomology of configuration spaces of points is in

• Ricardo Campos, Thomas Willwacher, A model for configuration spaces of points (arXiv:1604.02043)

• Ricardo Campos, Najib Idrissi, Pascal Lambrechts, Thomas Willwacher, Configuration Spaces of Manifolds with Boundary (arXiv:1802.00716)

• Ricardo Campos, Julien Ducoulombier, Najib Idrissi, Thomas Willwacher, A model for framed configuration spaces of points (arXiv:1807.08319)