Modular operads

  title={Modular operads},
  author={Ezra Getzler and M. M. Kapranov},
  journal={Compositio Mathematica},
We develop a ‘higher genus’ analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich’s graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick’s theorem. We give applications to the theory of moduli spaces… 
Open-closed modular operads, the Cardy condition and string field theory
We prove that the modular operad of diffeomorphism classes of Riemann surfaces with both `open' and `closed' boundary components, in the sense of string field theory, is the modular completion of its
Topology of moduli spaces and operads
We give a new proof of formality of the operad of little disks. The proof makes use of an operadic version of a simple formality criterion for commutative differential graded algebras due to
We give a biased definition of a properad and an explicit example of a closed Frobenius properad. We recall the construction of the cobar complex and algebra over it. We give an equivalent
Cusp form motives and admissible G-covers
This thesis contains three articles related to operads and moduli spaces of admissible covers of curves. In Paper A we isolate cohomology classes coming from modular forms inside a certain space of
Properads and Homotopy Algebras Related to Surfaces
Starting from a biased definition of a properad, we describe explicitly algebras over the cobar construction of a properad. Equivalent description in terms of solutions of generalized master


Operads and Moduli Spaces of Genus 0 Riemann Surfaces
In this paper, we study two dg (differential graded) operads related to the homology of moduli spaces of pointed algebraic curves of genus 0. These two operads are dual to each other, in the sense of
Gromov-Witten classes, quantum cohomology, and enumerative geometry
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic
Models for operads
We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps including a homology isomorphism. ...
Koszul duality for Operads
(0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] and
The geometry of iterated loop spaces
Operads and -spaces.- Operads and monads.- A? and E? operads.- The little cubes operads .- Iterated loop spaces and the .- The approximation theorem.- Cofibrations and quasi-fibrations.- The smash
Batalin-Vilkovisky algebras and two-dimensional topological field theories
By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator Δ:A⊙→A⊙+1 such that Δ2 = 0, and [Δ,a]−Δa is a graded derivation ofA for alla∈A. In this article, we
Homotopy algebra and iterated integrals for double loop spaces
This paper provides some background to the theory of operads, used in the first author's papers on 2d topological field theory (hep-th/921204, CMP 159 (1994), 265-285; hep-th/9305013). It is intended
The decorated Teichmüller space of punctured surfaces
A principal ℝ+5-bundle over the usual Teichmüller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several
Degeneration of Abelian varieties
I. Preliminaries.- II. Degeneration of Polarized Abelian Varieties.- III. Mumford's Construction.- IV. Toroidal Compactification of Ag.- V. Modular Forms and the Minimal Compactification.- VI.