The following numerical work was carried out:
a fast Fourier transform algorithm was developed to overcome the problem of critical slowing down near continuous phase transitions;
supersymmetric random surfaces (discretized versions of superstrings) were studied, with an extrinsic curvative term being added.
In the context of the double scaling limit of matrix models, attempts were made to define string theory and the energing matrix model field theories by means of a natural supersymmetric extensions, known as the Fokker-Planck Hamiltonian, in one dimension higher.
It has been shown that the one matrix model of hermitean matrices is constrained by differential operators satisfying the full W infinity algebra, which in this case is reducible to the usual Virasoro algebra.
In dealing with the problems of coupling field theories to gravity and solving quantum field theories on random triangulated graphs, it has been shown that, in 2-dimensions, Yang-Mills theory is completely independent of the metric. It was also shown that 2-dimensional lattice gauge theories may give rise to an infinite number of phase transitions if a Chern term is added.
In the context of random surfaces, it was shown that the spectrum of the Dirac operator on a 2-dimensional regular lattice with random inpurities behaves in many ways like the same system on a random lattice. The work on discretized random surfaces has been extended to the related question of gravity in a discretized formulation. Numerical simulations have demonstrated the possiblity that a nontrivial continuum theory of quantum gravity can be defined in 4-dimensions. 3-dimensional gravity coupled to matter fields was also considered.
First the problem of weak interaction cross sections at energies of the order of 1 TeV to 10 TeV in very large external magnetic fields was investigated. It was found that there was a dramatic increase in the cross sections of certain specific (bary on number violating) processes in large regions of strong magnetic fields. This is a truly nonperturbative phenomenon, induced by instanton configurations in the gauge fields.
Closely related to this work is the possiblity of experimentally observing signals for the new phase of the electroweak theory is very strong magnetic fields. The conclusion was reached that the formation of W and Z condensates many result in enhanced cross sections for processes involving these vector bosons.
Extensive computer simulations were performed of the bary on number violating processes in the context of very high temperatures. The resulting estimate for the rate of sphaleron transitions is in accord with pervious studies.
The problem of spontaneous symmetry breaking in the standard model of electroweak interactions was investigated. It was shown that the technique of chiral lagrangians can be used to detect even very small deviations from the predictions of the standard model in the Higgs sector.
A novel application of conformal field theory techniques to the study of deconfinement phase transitions in 3-dimensions was discovered. The question of deconfinement of higher representations sources was investigated and the strong coupling effective action of finite temperature lattice gauge theories was studied by means of a variational real space renormalization group.
Investigations were also carried out in the following areas:
spontaneous parity violation in a lattice regularized gauge theory in the high temperature limit;
the effect of instabilities in the renormalization group flow of general quantum field theories;
the Becchi, Rouet, Stora transformation (BRST) symmetry of field redefinitions;
a new appoach to 2-dimen sional bosonization.
Non-abelian gauge are analyzed non-perturbatively by the use of lattice regularization techniques. Special attention is paid to the implementation of fermionic degrees of freedom and to the study universality near the deconfinement phase transition between hadronic and quark-gluon matter. As a natural extension of this construction, quantum theories of strings shall be formulated as specific sums over discretized random surfaces and then treated by both numerical and analytical techniques.