Operads and moduli spaces of genus 0 Riemann surfaces.

*(English)*Zbl 0851.18005
Dijkgraaf, R. H. (ed.) et al., The moduli space of curves. Proceedings of the conference held on Texel Island, Netherlands during the last week of April 1994. Basel: Birkhäuser. Prog. Math. 129, 199-230 (1995).

The author studies two \(dg\)-operads, dual in the sense of Ginzburg-Kapranov, which are related to the moduli spaces of the title. The algebras described by these operads are, respectively, the author’s “gravity” algebras [Commun. Math. Phys. 163, No. 3, 473-489 (1994; Zbl 0806.53073)] and the algebras discovered by R. Dijkgraaf, H. Verlinde and E. Verlinde [Nuclear Phys. B 352, No. 1, 59-86 (1991)], which the author rechristens “polycommutative”. The latter have a sequence of operations \(A^{\otimes n} \to A\) satisfying an appropriate generalization of associativity. An important class of examples is provided by the quantum cohomology of compact Kähler manifolds.

As in much other recent work, a key role is played by the moduli spaces \({\mathcal M}_{0,n}\) of \(n\)-punctured Riemann spheres and the Knudsen-Deligne-Mumford compactification \(\overline {\mathcal M}_{0,n}\). The relevant technical tools include the spectral sequence of the natural stratification of the compactified moduli space and mixed Hodge theory which for genus 0 is pure. The author is building on much of his earlier work, especially that with Kapranov, cf. cyclic and modular operads. Graphs and trees play a significant part, although the given combinatorial definition of graph is far from perspicuous. He also obtains new formulas for the characters of the homology of \({\mathcal M}_{0, n}\) and of \(\overline {\mathcal M}_{0,n}\) as \({\mathcal S}_n\)-modules.

For the entire collection see [Zbl 0827.00037].

As in much other recent work, a key role is played by the moduli spaces \({\mathcal M}_{0,n}\) of \(n\)-punctured Riemann spheres and the Knudsen-Deligne-Mumford compactification \(\overline {\mathcal M}_{0,n}\). The relevant technical tools include the spectral sequence of the natural stratification of the compactified moduli space and mixed Hodge theory which for genus 0 is pure. The author is building on much of his earlier work, especially that with Kapranov, cf. cyclic and modular operads. Graphs and trees play a significant part, although the given combinatorial definition of graph is far from perspicuous. He also obtains new formulas for the characters of the homology of \({\mathcal M}_{0, n}\) and of \(\overline {\mathcal M}_{0,n}\) as \({\mathcal S}_n\)-modules.

For the entire collection see [Zbl 0827.00037].

Reviewer: J.Stasheff (MR 96k:18008)

##### MSC:

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

14H10 | Families, moduli of curves (algebraic) |

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

18G99 | Homological algebra in category theory, derived categories and functors |

53Z05 | Applications of differential geometry to physics |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

17A42 | Other \(n\)-ary compositions \((n \ge 3)\) |

81T70 | Quantization in field theory; cohomological methods |