Abstract. One of the basic geometric objects in conformal field theory (CFT) is the the moduli space of Riemann surfaces whose n boundaries are "rigged" with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of n-punctured Riemann surfaces together with local biholomorphic coordinates at the punctures. We refer to both of these moduli spaces as the "rigged Riemann moduli space".By generalizing to quasisymmetric boundary parametrizations, and defining rigged Teichmüller spaces in both the border and puncture pictures, we prove the following results:(1) The Teichmüller space of a genus-g surface bordered by n closed curves covers the rigged Riemann and rigged Teichmüller moduli spaces of surfaces of the same type, and induces complex manifold structures on them. (2) With this complex structure the sewing operation is holomorphic. (3) The border and puncture pictures of the rigged moduli and rigged Teichmüller spaces are biholomorphically equivalent.These results are necessary in rigorously defining CFT (in the sense of G. Segal), as well as for the construction of CFT from vertex operator algebras.

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmüller space of such Riemann surfaces and demonstrate that in the case that 2g + 2 − n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller space (which is a Banach manifold) is holomorphic.We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of nonoverlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally we show that the rigged moduli space is the quotient of the refined Teichmüller space by a properly discontinuous group of biholomorphisms.2010 Mathematics Subject Classification. Primary 30F60 ; Secondary 30C55, 30C62, 32G15, 46E20, 81T40. Key words and phrases. Refined Teichmüller space, Hilbert manifold, quasiconformal maps, moduli space of rigged Riemann surfaces, conformal field theory.Eric Schippers is partially supported by the National Sciences and Engineering Research Council. He would like to thank Nina Zorboska for several helpful conversations.

Let Σ be a Riemann surface with n distinguished points p 1 , . . . , pn. We prove that the set of n-tuples (φ 1 , . . . , φn) of univalent mappings φ i from the unit disc D into Σ mapping 0 to p i , with non-overlapping images and quasiconformal extensions to a neighbourhood of D, carries a natural complex Banach manifold structure. This complex structure is locally modeled on the n-fold product of a two complex-dimensional extension of the universal Teichmüller space. Our results are motivated by Teichmüller theory and two-dimensional conformal field theory.

Abstract. We show that the infinite-dimensional Teichmüller space of a Riemann surface whose boundary consists of n closed curves is a holomorphic fiber space over the Teichmüller space of n-punctured surfaces. Each fiber is a complex Banach manifold modeled on a twodimensional extension of the universal Teichmüller space. The local model of the fiber, together with the coordinates from internal Schiffer variation, provides new holomorphic local coordinates for the infinite-dimensional Teichmüller space.

Consider a multiply-connected domain Σ in the sphere bounded by n nonintersecting quasicircles. We characterize the Dirichlet space of Σ as an isomorphic image of a direct sum of Dirichlet spaces of the disk under a generalized Faber operator. This Faber operator is constructed using a jump formula for quasicircles and certain spaces of boundary values.Thereafter, we define a Grunsky operator on direct sums of Dirichlet spaces of the disk, and give a second characterization of the Dirichlet space of Σ as the graph of the generalized Grunsky operator in direct sums of the space H 1/2 (S 1 ) on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.

The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal Teichmüller theory and geometric function theory.In particular we propose that the natural analytic setting for conformal field theory is the moduli space of Riemann surfaces with so-called Weil-Petersson class parametrizations. A collection of rigorous analytic results is advanced here as evidence. This class of parametrizations has the required regularity for CFT on one hand, and on the other hand are natural and of interest in their own right in geometric function theory.

Let Σ be a Riemann surface of genus g bordered by n curves homeomorphic to the circle S 1 , and assume that 2g + 2 − n > 0. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. Based on this, we present alternate models of the aforementioned Teichmüller space and show in particular that it is locally modelled on a Hilbert space of L 2 Beltrami differentials, which are holomorphic up to a power of the hyperbolic metric, and has a convergent Weil-Petersson metric.

This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of holomorphic functions in continuum domains as well as corresponding spaces of discrete holomorphic functions in lattice domains. We find distinguished sets of functions characterized by their singular behavior in the three infinite directions in the slit-strip domains. We prove convergence results of the distinguished discrete holomorphic functions to the continuum ones. In the subsequent articles, the discrete holomorphic functions will be used for the calculation of the Ising model fusion coefficients (as well as for the diagonalization of the Ising transfer matrix), and the convergence of the functions is used to prove the convergence of the fusion coefficients. It will also be shown that the vertex operator algebra of the boundary conformal field theory can be recovered from the limit of the fusion coefficients via geometric transformations involving the distinguished continuum functions.