# David Radnell

## Quasisymmetric Sewing in Rigged Moduli Space

One of the basic geometric objects in conformal field theory is the moduli space of Riemann surfaces whose boundaries are ``rigged'' with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations. By using conformal welding we generalize this picture to quasisymmetric boundary parametrizations. Because of the simplified picture we obtain it appears this is the natural setting for the geometric objects in conformal field theory.

We show how complex manifold structures on the Riemann and
Teichmueller moduli spaces of rigged surfaces are inherited from the universal Teichmueller space. A straightforward proof of the holomorphicity of the sewing operation will be given. These results are necessary in rigorously defining higher-genus weakly conformal field theories in the sense of G. Segal. The definition of conformal field theory and appropriate background in Teichmueller theory will be
explained.

This is joint work with E. Schippers.