nLab the little n-disk operad is formal



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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.


The special case of formality of E 2E_2 (see also Kontsevich formality) was finally proven in

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

A general proof of formality of E nE_n for all nn was sketched in

and fully spelled out in

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

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:

Last revised on March 30, 2021 at 13:48:06. See the history of this page for a list of all contributions to it.