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the little n-disk operad is formal

Contents

Context

Higher algebra

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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 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

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