# nLab the little n-disk operad is formal

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

cohomology

# Contents

## 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.

## References

The special case of formality of $E_2$ (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 $E_n$ for all $n$ 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

Last revised on November 8, 2018 at 03:40:01. See the history of this page for a list of all contributions to it.