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Hypercommutative operad as a homotopy quotient of BV. (English) Zbl 1281.55011
The hypercommutative operad considered in the title of this article is the operad in graded modules \(\mathsf{Hycomm}\) defined by the homology of the moduli spaces of stable point curves \(\overline{\mathcal{M}}_{n+1}\), \(n\geq 1\). This operad admits a simple presentation with a generating operation \(m_n = m_n(x_1,\dots,x_n)\) corresponding to the fundamental class of the moduli spaces \([\overline{\mathcal{M}}_{n+1}]\) in each arity \(n\geq 2\) together with relations modelling higher associativity and commutativity constraints. The second operad considered in the article, the Batalin-Vilkovsky operad (the BV operad for short) denoted by \(\mathsf{BV}\), governs structures associated with double loop spaces endowed with a circle action. This operad is generated by a two-ary associative and commutative multiplication operation \(\mu = \mu(x_1,x_2)\) together with a one-ary operator \(\Delta\), referred to as the BV-operator, which sastisfies the relation of a differential operator of order two with respect to the multiplication.
The BV-operad is also identified with the homology of an operad in topological spaces, the operad of framed little \(2\)-discs. Recently, G. C. Drummond-Cole [Homotopically trivializing the circle in the framed little disks, preprint arXiv:1112.1129, to appear in J. Topol.] proved that the topological operad of the moduli spaces of stable point curves is weakly-equivalent to the operad obtained by homotopically trivializing the action of the circle on framed little \(2\)-discs. The authors of the paper under review study a homological counterpart of this weak-equivalence of topological operads. They consider an operad in chain complexes \(\mathsf{BV}/\Delta\) governing BV-algebras equipped with linear endomorphisms that trivialize the action of the BV-operator in homology. The main result of the paper is an explicit definition of a quasi-isomorphism from the hypercommutative operad \(\mathsf{Hycomm}\) towards the operad \(\mathsf{BV}/\Delta\). The authors actually give two approaches for the construction of this quasi-isomorphism. The first approach relies on a diagram of operadic resolutions. The second approach relies on an operadic interpretation of the Givental loop group action.

MSC:
55P48 Loop space machines and operads in algebraic topology
22E67 Loop groups and related constructions, group-theoretic treatment
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T70 Quantization in field theory; cohomological methods
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