Construction of Four-Dimensional String Models: Induced Statistics in the Nonlinear Sigma Model

In the first part we describe a way to construct a very large number of classical vacua in string theory, in the context of the spin-structure construction. We present a review of the spin-structure construction of four-dimensional heterotic string models, and describe in detail the content of a generic massless spectrum. We explain how the massless spectrum can be quickly obtained from the spin-structure, and how the task of analyzing the content of a string vacuum can be computerized. We also discuss the problem of generating consistent spin structures, both systematically and at random, with the purpose of constructing as many classical ground states of the heterotic string as possible, and of obtaining a lower limit on their total number. At present, the number of classical vacua (nonsupersymmetric or with N = 1 supersymmetry) constructed in this way is about 180,000. Storage considerations were the only practical limits on this number. We discuss the conditions that the massless spectra of classical vacua must obey in order to be minimally acceptable from the phenomenological point of view. In the second part, we study the statistics of solitons in the O(3) nonlinear sigma model in (2 + 1) dimensions coupled to fermions with arbitrary dispersion relation. We show that P and T violation is not a sufficient condition for the appearance of fractional statistics in the spectrum. We give a simple formula for the coefficient of the Hopf term induced by quantum effects, and formulae for the induced Chern-Simons term and fermion number, where appropriate. The results depend only on the high and low momentum behavior of the fermion dispersion relation, and are stable against perturbations (such as anisotropies) of the dispersion relation which preserve its asymptotic behavior. We apply our results to models of high-T_{c } materials, and find that, even with P and T violation, fractional statistics in general do not emerge. In a relativistic model, the soliton is always a fermion. We discuss a hypothetical model in which fractional statistics is generated.