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

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.

70 pages, main changes concern title/abstract/introduction, a construction of two maps of "forgetting points", and a new formality result

References in corpus (7)

• Pre-Lie deformation theory
• Homological algebra related to surfaces with boundary
• The Frobenius properad is Koszul
• Mixed Hodge structures and formality of symmetric monoidal functors
• Algebras over Cobar(coFrob)
• Homotopy DG algebras induce homotopy BV algebras
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Cited by in corpus (9)

• The controlling $L_{\infty }$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples
• Celebrating Loday's Associahedron
• The twisting procedure
• Generalized cohomological field theories in the higher order formalism
• Gröbner bases and dimension formulas for ternary partially associative operads
• Reconnectads
• Homotopy invariants for ${\overline{\mathcal{M}}}_{0,n}$ via Koszul duality
• A generalization of operads based on subgraph contractions
• Real toric manifolds associated with chordal nestohedra