A complex structure on the moduli space of rigged Riemann surfaces

TL;DR

This paper establishes that the moduli space of rigged Riemann surfaces with quasisymmetric boundary parametrizations has a complex structure, enabling holomorphic sewing operations crucial for conformal field theory constructions.

Contribution

The authors prove that the moduli space of rigged Riemann surfaces admits a complex structure and that sewing operations are holomorphic, extending previous results to quasisymmetric parametrizations.

Findings

Moduli space has a complex manifold structure.

Sewing operations are holomorphic.

Supports rigorous CFT constructions from vertex operator algebras.

Abstract

The study of Riemann surfaces with parametrized boundary components was initiated in conformal field theory (CFT). Motivated by general principles from Teichmueller theory, and applications to the construction of CFT from vertex operator algebras, we generalize the parametrizations to quasisymmetric maps. For a precise mathematical definition of CFT (in the sense of G. Segal), it is necessary that the moduli space of these Riemann surfaces be a complex manifold, and the sewing operation be holomorphic. We report on the recent proofs of these results by the authors.