Operads, configuration spaces and quantization (Q420563)

The paper studies various operads associated with configuration spaces and manifolds with corners. The paper start with a recollection on Stasheff's associahedra and their relation to a compactification of configurations of points on the line. It also discusses a similar relation for multiplihedra and introduces a new compactification of the configuration of points on the line which is a one-dimensional analog of the two-dimensional geometric model of the two-coloured operad \({\text{Mor}}(L_{\infty})\). After a recollection of Kontsevich's open-closed homotopy algebra the author introduces two configuration space models for the operad \(\text{Mor}(OC_{\infty})\) of morphisms of open-closed homotopy algebras. Next, the operad of Feynman graphs and their representations are investigated and the author introduces de Rham field theories on operads. Finally, some concrete examples are discussed. The paper is very detailed with many helpful diagrams and, although it assumes the knowledge of coloured operads, it provides an appendix with basics on coloured operads with definitions and examples.