Communications in Number Theory and Physics

Volume 5 (2011)

Number 3

The Kontsevich constants for the volume of the moduli of curves and topological recursion

Pages: 643 – 698



Kevin M. Chapman (Department of Mathematics, University of California at Los Angeles)

Motohico Mulase (Department of Mathematics, University of California at Davis)

Brad Safnuk (Department of Mathematics, Central Michigan University, Mount Pleasant)


We give an Eynard–Orantin-type topological recursion formula for the canonical Euclidean volume of the combinatorial moduli space of pointed smooth algebraic curves. The recursion comes from the edge removal operation on the space of ribbon graphs. As an application we obtain a new proof of the Kontsevich constants for the ratio of the Euclidean and the symplectic volumes of the moduli space of curves.

Published 16 December 2011