Non-perturbative string theories and singular surfaces

Abstract

Singular surfaces are shown to be dense in the Teichmüller space of all Riemann surfaces and in the grassmannian. This happens because a regular surface of genus h, obtained identifying 2 h in pairs, can be approximated by a very large genus singular surface with punctures dense in the 2 h disks. A scale ε is introduced and the approximate genus is defined as half the number of connected regions covered by punctures of radius ε. The non-perturbative partition function is proposed to be a scaling limit of the partition function on such infinite genus singular surfaces with a weight which is the coupling constant g raised to the approximate genus. For a gaussian model in any space-time dimension the regularized partition function on singular surfaces of infinite genus is the partition function of a two-dimensional lattice gas of charges and monopoles. It is shown that modular invariance of the partition function implies a version of the Dirac quantization condition for the values of the e m charges. Before the scaling limit the phases of the lattice gas may be classified according to the 't Hooft criteria for the condensation of e m operators.