Irreducible Metric Maps and Weil–Petersson Volumes

Abstract

AbstractWe 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 $$\alpha _1,\ldots ,\alpha _n$$ α 1 , … , α n and a $$\beta $$ β -irreducibility constraint, which roughly requires that all contractible cycles have length at least $$\beta $$ β . Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume $$V_{g,n}^{(\beta )}(\alpha _1,\ldots ,\alpha _n)$$ V g , n ( β ) ( α 1 , … , α 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 $$\beta , \alpha _1,\ldots , \alpha _n$$ β , α 1 , … , α n of degree $$6g-6+2n$$ 6 g - 6 + 2 n and to satisfy string and dilaton equations. Surprisingly, for $$g=0,1$$ g = 0 , 1 and $$\beta =2\pi $$ β = 2 π the volume $$V_{g,n}^{(2\pi )}$$ V g , n ( 2 π ) is identical, up to powers of two, to the Weil–Petersson volume $$V_{g,n}^{\mathrm {WP}}$$ V g , n WP of hyperbolic surfaces of genus g and n geodesic boundary components of length $$L_i = \sqrt{\alpha _i^2 - 4\pi ^2}$$ L i = α i 2 - 4 π 2 , $$i=1,\ldots ,n$$ i = 1 , … , n . For genus $$g\ge 2$$ g ≥ 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.