Colliding holes in Riemann surfaces and quantum cluster algebras

TL;DR

This paper introduces a new surgery on non-compact Riemann surfaces leading to bordered cusps, develops a Poisson and quantum structure for the associated Teichmüller space, and links it to quantum cluster algebras and string moduli spaces.

Contribution

It defines bordered cusped Teichmüller space, describes its combinatorial and Poisson structures, and connects mapping class group actions to cluster algebra mutations.

Findings

Explicit Poisson and quantum structures for bordered cusped Teichmüller space.

Complete combinatorial description via maximal cusped lamination.

Mapping class group acts as generalized cluster algebra mutation.

Abstract

In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appear when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincar\'e uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichm\"uller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichm\"uller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesics arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as $\lambda$-lengths in Thurston--Penner terminology. We compute the Goldman bracket explicitly in terms of these $\lambda$-lengths and show that the Mapping Class Group acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.