From infinite-dimensional Teichmueller theory to conformal field theory and back

The mathematical definition (in the original sense of G. Segal) and construction of Conformal Field Theory (CFT) requires deep developments in algebra, analysis and geometry. The algebraic aspects involving vertex operator algebras have been well developed over the past twenty-five years, and the construction of CFT is nearing completion. However, many problems in analysis and geometry must be urgently addressed. These problems involve the infinite-dimensional moduli and Teichmuller spaces of Riemann surfaces with parametrized boundaries. Moreover, ideas from CFT are used to gain new insights and prove new theorems in Teichmuller theory.

Together with E. Schippers we have shown that by using quasisymmetric boundary parametrizations the rigged moduli space can, surprisingly, be obtained from the usual infinite-dimensional Teichmuller space of bordered surfaces. In the simplified picture we obtain, necessary results for CFT are proved. Also, we have given new holomorphic coordinates for this Teichmuller space, as well as a fiber structure. Many interesting problems remain, especially regarding determinant line bundles. An overview of these recent results will be presented.

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