Irreducible Metric Maps and Weil–Petersson Volumes

Abstract

We consider maps on a surface of genus g with all vertices of degree at least three and positive real lengths assigned to the edges. In particular, we study the family of such metric maps with fixed genus g and fixed number n of faces with circumferences alpha _1,ldots ,alpha _n and a beta -irreducibility constraint, which roughly requires that all contractible cycles have length at least beta . Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume V_{g,n}^{(beta )}(alpha _1,ldots ,alpha _n) of this family of maps that arises naturally from the Lebesgue measure on the edge lengths. It is shown to be a homogeneous polynomial in beta , alpha _1,ldots , alpha _n of degree 6g-6+2n and to satisfy string and dilaton equations. Surprisingly, for g=0,1 and beta =2pi the volume V_{g,n}^{(2pi )} is identical, up to powers of two, to the Weil–Petersson volume V_{g,n}^{mathrm {WP}} of hyperbolic surfaces of genus g and n geodesic boundary components of length L_i = sqrt{alpha _i^2 - 4pi ^2}, i=1,ldots ,n. For genus gge 2 the identity between the volumes fails, but we provide explicit generating functions for both types of volumes, demonstrating that they are closely related. Finally we discuss the possibility of bijective interpretations via hyperbolic polyhedra.