Quasisymmetric Sewing in Rigged Teichmüller Space

One of the basic geometric objects in conformal field theory (CFT) 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. We generalize this picture to quasisymmetric boundary parametrizations. By using tools such as the extended λ-lemma and conformal welding we prove: (1) The universal Teichmüller space induces complex manifold structures on the Riemann and Teichmüller moduli spaces of rigged surfaces. (2) The border and puncture pictures of the rigged moduli and rigged Teichmüller spaces are biholomorphically equivalent. (3) The sewing operation is holomorphic. Because of the simplified picture we obtain it appears this is the natural setting for the geometric objects in CFT.