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Modular operads. (English) Zbl 0894.18005
There has been recently a renewal of the theory of operads motivated by the study of moduli spaces of curves [V. Ginzburg and M. Kapranov, “Koszul duality for operads”, Duke Math. J. 76, No. 1, 203-272 (1994; Zbl 0855.18006)]. In fact, the operadic formalism is closely related to curves of genus 0. The authors introduce in this paper a higher genus analogue of the theory of operads. The resulting objects are called modular operads; their definition relies on the combinatorics of graphs in the same way as the definition of usual operads relies on the combinatorics of trees. Modular operads are then studied systematically and various examples coming from the theory of moduli spaces of curves are given.
The bar construction is one of the key tools in the theory of operads. An analogous functor, the Feynman transform, is constructed on the category of dg-modular operads, which generalizes Kontsevich’s graph complexes [M. Kontsevich, “Formal (non)-commutative symplectic geometry”, in: The Gelfand mathematics seminars, 1990-1992 , 173-187 (1993; Zbl 0821.58018)].
Using the theory of symmetric functions, the (Euler) characteristics of cyclic and modular operads are defined and studied [for what concerns cyclic operads, see E. Getzler and M. Kapranov, “Cyclic operads and cyclic homology”, in: Geometry, topology and physics for Raoul Bott, Conf. Proc. Lect. Notes Geom. Topol. 4, 167-201 (1995; Zbl 0883.18013)]. For such purposes, powerful analogues of the Legendre and Fourier transforms are introduced in the setting of symmetric functions. Euler characteristics of Feynman transforms are then calculated. This calculation is closely related to Wick’s theorem and to the computation by Harer-Zagier and Kontsevich of the Euler characteristics of moduli spaces [M. Kontsevich, “Intersection theory on moduli spaces of curves and the matrix Airy function”, Commun. Math. Phys. 147, No. 1, 1-23 (1992; Zbl 0756.35081)].