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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.

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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