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arXiv · 1510.03261

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

Vladimir Dotsenko·Sergey Shadrin·Bruno Vallette
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Abstract

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.

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Vladimir Dotsenko, Sergey Shadrin, Bruno Vallette. 2015-10-12. Toric varieties of Loday's associahedra and noncommutative cohomological field theories. https://doi.org/10.1112/topo.12091

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