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We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2‐discs, framed little 2‐discs, 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.

Toric varieties of Loday's associahedra and noncommutative cohomological field theories