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Modular operads

Published online by Cambridge University Press:  04 December 2007

E. GETZLER  and
M. M. KAPRANOV
E. GETZLER
Affiliation:
Department of Mathematics, MIT, Cambridge, Massachusetts 02139 U.S.A. e-mail: getzler@math.math.nwu.edu Department of Mathematics, Northwestern University, Illinois 60208 U.S.A. e-mail: kapranov@math.nwu.edu
M. M. KAPRANOV
Affiliation:
Department of Mathematics, Northwestern University, Illinois 60208 U.S.A. e-mail: kapranov@math.nwu.edu
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Abstract

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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.

Keywords

operadbar-constructiongraphsymmetric functionsFeynman diagram.
Type
Research Article
Information
Compositio Mathematica , Volume 110 , Issue 1 , January 1998 , pp. 65 - 125
Copyright
© 1998 Kluwer Academic Publishers