  We introduce and study several new topological operads that should be
regarded as nonsymmetric analogues of the operads of little 2-disks, framed
little 2-disks, and Deligne-Mumford compactifications of moduli spaces of genus
zero curves with marked points. These operads exhibit all the remarkable
algebraic and geometric features that their classical analogues possess; in
particular, it is possible to define a noncommutative analogue of the notion of
cohomological field theory with similar Givental-type symmetries. This relies
on rich geometry of the analogues of the Deligne-Mumford spaces, coming from
the fact that they admit several equivalent interpretations: as the toric
varieties of Loday's realisations of the associahedra, as the brick manifolds
recently defined by Escobar, and as the De Concini-Procesi wonderful models for
certain subspace arrangements.
