Schiffer variation in Teichmuller space, determinant line bundles and modular functors

The pure mathematical incarnation of conformal field theory was introduced by Segal [48] and Kontsevich around 1987. Recently, Hu and Kriz [26] further rigorized Segal's definition. Conformal field theory is intimately connected to vertex operator algebras and the complex geometry of Riemann surfaces with analytically parametrized boundaries. This thesis is centered on the analytic and geometric aspects of this theory.; An explicit description of the complex structure of the infinite-dimensional moduli space of Riemann surfaces with analytically parametrized boundary components is given and the holomorphicity of the sewing operation is proved. The determinant line bundle is shown to be a holomorphic bundle over this moduli space and the sewing operation is proved to be holomorphic on these bundles. Applications to modular functors, which are high-rank generalizations of the determinant line bundle, are discussed.; All these results are needed in order to have rigorous definitions of a holomorphic (or chiral) conformal field theory and a holomorphic modular functor. So certainly this is a necessary step in the on-going project to construct higher-genus conformal field theories from vertex operator algebras.; The formulation and proofs of these results rely on deep aspects of analytic Teichmuller theory and quasiconformal mappings, the uniformization of higher-genus Riemann surfaces, and Schiffer variation. To my knowledge these techniques have not previously been applied in this context and will have continued applicability.