Divergences in the moduli space integral and accumulating handles in the infinite-genus limit

Abstract

Abstract. The symmetries associated with the closed bosonic string partition function are examined so that the integration region in Teichmuller space can be determined. The conditions on the period matrix defining the fundamental region can be translated to relations on the parameters of the uniformizing Schottky group. The growth of the lower bound for the regularized partition function is derived through integration over a subset of the fundamental region.* Present address IntroductionThe elimination of infrared divergences in scattering amplitudes of superstring theories promises a consistent quantum theory including gravity as part of the low-energy limit.An understanding of space-time at the most fundamental level could be achieved with the development of such a theory.This requires a complete formulation of the theory based on the sum over string histories arising in the path integral. Both the entire perturbative amplitude, and possibly nonperturbative effects, could be obtained in this approach.Interest has centered recently on a divergence in bosonic string theory that arises from summing the contributions at each loop, which have been rendered finite individually through a genus-independent regularization [1]. The cut-off introduced for the bosonic string excludes effectively closed surfaces of sufficiently large genus, but it will be shown that surfaces having an ideal boundary with positive linear measure do arise in the sum over all orders in the perturbation expansion. The source of the divergence can then be traced to these surfaces, which may be interpreted as representing a non-perturbative effect in string theory.The counting of these surfaces, and the effectively closed surfaces at higher genus, in the path integral could impact on the finiteness properties of the superstring path integral.The investigation of the divergence begins here with a study of the measure and the domain in the integral for the partition function. The measure is derived from the path integral weighting factor and a choice of coordinates on the the moduli space of metrics.The two most frequently used measures are those defined by the light-cone and Polyakov approaches, which require manifest unitarity or covariance of the string theory respectively.The light-cone diagram is constructed so that the momentum of the external string is proportional to the distance between the cuts at initial and final times, while the internal cuts correspond to the joining and splitting of strings. The conformal mapping from the string diagram to a planar domain with disks removed and punctures at the positions of the 1 vertex operators transforms paths from the boundary of the diagram to the internal cuts and the paths around the cuts to a-cycles and b-cycles respectively [2]. The planar domains with 2g disks removed are the Schottky covering surfaces for a Riemann surface of genus g, and by the retrosection theorem, every compact Riemann surface can be uniformized by a Schottky group generated by g Mobius transformations [3].The path in...