Dual Teichm\" uller spaces

TL;DR

The paper develops explicit global coordinate systems for Teichmüller spaces of surfaces with holes and for lamination spaces (bounded/compact and unbounded/closed support), and demonstrates their asymptotic equivalence. It unifies Thurston’s lamination framework with conformal field theory concepts, and formalises a quantisation of the Weil–Petersson–Poisson structure, including a hbar <-> 1/hbar duality and projective mapping class group representations. It also provides explicit transformation rules under graph flips, analyzes length and intersection functions, and outlines rich applications such as Markov-number relations, dualities between Teichmüller spaces, and connections to Virasoro orbits. The work lays a concrete, constructive bridge between hyperbolic geometry, lamination theory, and quantum topology, with clear avenues for further rigorous development and quantisation-related applications.

Abstract

We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations with compact and closed support respectively. We show explicitly that the latter spaces are asymptotically isomorphic to the former. We discuss briefly quantisation of Teichm\" uller spaces and some other application of the constructed approach. The paper does not require any preliminary knowledge of the subject above the Poincar\' e uniformisation theorem.