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-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. Contents

Homotopy invariants for $\overline{\mathcal{M}}_{0,n}$ via Koszul duality

“…In [4, Remark 2.17], a question is raised of a geometric characterisation of the class of Kähler manifolds with Koszul cohomology algebras; such manifolds are of interest since they are coformal (over rational numbers), and their rational homotopy Lie algebras can be described in a very explicit way. We show that a smooth projective toric variety belongs to that class if and only if its fan is a flag complex (Theorem 5.1); in particular, this includes the Losev-Manin spaces [30] and the noncommutative Deligne-Mumford spaces [14]. We conjecture that certain types of De Concini-Procesi wonderful models of hyperplane arrangements [11] also belong to that class; another conjectural class of Koszul algebras of the same flavour is given by matroid Chow rings [1].…”

“…As a corollary to Theorem 5.1, we shall examine the noncommutative analogues ncM 0,n+1 defined in [14]. One of the geometric definitions of those varieties identifies them as (smooth projective) toric varieties whose fans are dual to Loday's realisations of associahedra, implying the following result.…”

The twisting procedure

“…We conclude this study with new computations of the cohomology groups of twisted operads: we treat the case of the classical operads encoding respectively Gerstenhaber and Batalin-Vilkovisky algebras and the case of their nonsymmetric analogous operads introduced recently in [15]. This latter cases give rise to interesting non-commutative graph complexes.…”

“…The fact that the twisted operator ad µ is a derivation with respect to the partial composition products • i is equivalent to the equations EXAMPLE 1.3. One can also twist the ns operad ncBV of non-commutative Batalin-Vilkovisky algebras [15] by its square-zero element ∆, under cohomological degree convention.…”

“…Each edge carries cohomological degree 1 and the total cohomological degree is thus equal to the number of edges. (These elements corresponds to right-combs of binary generators with the presentation given in [15,Section 3.1].) It is convenient to think that the edges are ordered and reordering generates the sign; for a given bamboo, we assume, by default, the ordering of the edges from left to right.…”