Universal Teichm\"uller space in geometry and physics

TL;DR

This paper surveys the geometry of the universal Teichmüller space, highlighting its mathematical significance in Teichmüller theory and its emerging role in physics, particularly in string theory.

Contribution

It provides a comprehensive overview of the current understanding and conjectures regarding the geometry and physical implications of the universal Teichmüller space.

Findings

Universal Teichmüller space embeds all Fuchsian group Teichmüller spaces.

It is a promising geometric framework for non-perturbative bosonic string theory.

Current knowledge includes geometric properties and conjectured physical interpretations.

Abstract

Lipman Bers' universal Teichm\"uller space, classically denoted by $T(1)$, plays a significant role in Teichm\"uller theory, because all the Teichm\"uller spaces $T(G)$ of Fuchsian groups $G$ can be embedded into it as complex submanifolds. Recently, $T(1)$ has also become an object of intensive study in physics, because it is a promising geometric environment for a non-perturbative version of bosonic string theory. We provide a non-technical survey of what is currently known about the geometry of $T(1)$ and what is conjectured about its physical meaning.