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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)

The following paper constructs a canonical chain of formality quasiisomorphisms for the operad of chains on framed little disks and the operad of chains on little disks. The construction is done in terms of logarithmic algebraic geometry and is remarkable for being rational (and indeed definable integrally) in de Rham cohomology:

• Dmitry Vaintrob, Formality of little disks and algebraic geometry, arXiv:2103.15054.