Compositio Mathematica

Modular operads

E. GETZLER a1a2 and M. M. KAPRANOV a2
a1 Department of Mathematics, MIT, Cambridge, Massachusetts 02139 U.S.A. e-mail:
a2 Department of Mathematics, Northwestern University, Illinois 60208 U.S.A. e-mail:

Article author query
getzler e   [Google Scholar] 
kapranov mm   [Google Scholar] 


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 of pointed algebraic curves.

Key Words: operad; bar-construction; graph; symmetric functions; Feynman diagram..