Operads, configuration spaces and quantization

TL;DR

This paper reviews and constructs operads related to configuration spaces, linking them to algebraic structures and developing quantization methods using Feynman graphs and de Rham theory.

Contribution

It introduces new operads of configuration spaces and develops a framework for their quantized representations via Feynman graphs and propagators.

Findings

Constructed new operads in smooth manifolds with corners.

Established quantized representations using Feynman sums over graphs.

Connected operadic structures to algebraic and homotopy theories.

Abstract

We review several well-known operads of compactified configuration spaces and construct several new such operads, C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A-infinity algebras and their homotopy morphisms, (ii) the 2-coloured operad of L-infinity algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on C - are introduced and used to construct quantized representations of the (fundamental) chain operad of C which are given by Feynman type sums over graphs and depend on choices of propagators.