The Group of Quasisymmetric Homeomorphisms of the Circle and Quantization of the Universal Teichm\"uller Space

TL;DR

This paper explores the complex geometry of the universal Teichmüller space, focusing on its structure as a subset of holomorphic quadratic differentials, and introduces a novel quantization approach for its smooth part using Hilbert-Schmidt embeddings and Connes' method.

Contribution

It provides a detailed geometric description of the universal Teichmüller space and develops a new quantization technique for its smooth subset via Hilbert-Schmidt embeddings and Connes' approach.

Findings

Describes the complex geometry of the universal Teichmüller space.

Introduces a quantization method for the smooth part of the space.

Applies Connes' quantization approach to the entire space.

Abstract

In the first part of the paper we describe the complex geometry of the universal Teichm\"uller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo M\"obius transformations may be treated as a smooth part of $\mathcal T$. In the second part we consider the quantization of universal Teichm\"uller space $\mathcal T$. We explain first how to quantize the smooth part $\mathcal S$ by embedding it into a Hilbert-Schmidt Siegel disc. This quantization method, however, does not apply to the whole universal Teichm\"uller space $\mathcal T$, for its quantization we use an approach, due to Connes.