symmetric monoidal (∞,1)-category of spectra
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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 $E_2$ (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
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
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)
Last revised on November 8, 2018 at 03:40:01. See the history of this page for a list of all contributions to it.