Mellin transformed exponential volumes and probabilistic moduli spaces

Abstract

A decorated surface is a surface with boundary and a finite set of marked boundary points called cusps. We continue to study the exponential volumes E(S,K,L) of moduli spaces of ideal hyperbolic structures on a decorated surface S, with the given values of lengths of boundary geodesic circles L and boundary geodesic intervals K, introduced in our previous work: 1. We prove that the Mellin--Laplace transform of E(S,K,L) admits Barnes integral presentations, and satisfies recursions, generalising Mirzakhani's recursions. 2. We specify a subset I of the blue boundary intervals, and assign to them any collection of weights. We show that the Mellin transform of the function E(S,K,L) along the variables K at the blue intervals, evaluated at the weights values, calculates weighted exponential volumes of various moduli spaces. If all boundary intervals are blue and all weights are 1/2, we recover Chekhov's volumes of moduli spaces of hyperbolic surfaces with crowns, discretizing the Schwarzian action. We derive recursions for the weighted exponential volumes. 3. We introduce probabilistic Teichmuller spaces related to decorated surfaces S. They are infinite dimensional and carry canonical Wiener type measures. The definition of the measures builds on the works of Bauerschmidt - Losev - Wildemann and Losev. Their quotients by the action of the modular group of S are probabilistic moduli spaces. We show that their volumes are finite. We prove recursion formulas for these measures, and hence for the volumes. The emerging theory generalizes the Schwarzian field theory of Stanford - Witten from the disc to any decorated surfaces, i.e. to open string format. In particular, we get recursions for the Feynman integrals in Schwarzian field theory on decorated surfaces.