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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].

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