Random Surfaces in Lattice QCD and String Theory.
Abstract
We derive an expression for the free energy of pure gauge U(N) lattice gauge theory in the largeN, strong coupling limit, as a sum over closed, connected, oriented surfaces, formed by plaquettes on a hypercubic Euclidean space time lattice. The partition function for the pure gauge Wilson action is expanded in powers of the inverse strong coupling, each term of the expansion corresponding to a collection of plaquettes on the lattice. The one link integrals are expanded in powers of (1/N), each term of this expansion corresponds to a specific kind of joining of plaquettes on a given link. The surfaces formed by the joinings are classified topologically, and the power of (1/N) associated with the term is shown to be equal to the genus of the corresponding surface. Our major result for this section is the explicit calculation of all coefficients associated with the lowest order (genus = 0) surfaces. We generate, by a Monte Carlo technique, an ensemble of random surfaces, weighted by a discrete version of the Polyakov bosonic string action. By measuring the ensemble average of the mean square extent of the surface as a function of the area of the surface, we calculate the Hausdorff dimension. Our result is that the Hausdorff dimension is infinite. Above spacetime dimension about D = 26, the surface collapses to a new phase in which most of the surface is crushed close to the origin, but the surface develops long spikes. We apply the same technique to the Nambu string with a term proportional to the extrinsic curvature of the surface. Our result for this model is that the Hausdorff dimension is finite for nonvanishing coupling in the curvature, and decreases monotonically as the coupling is increased. The result was independent of the embedding dimension for all embeddings studied. Finally, we consider the modular invariance apparent in the continuum sum over surfaces. We show this modular invariance is maintained in the thermopartition function when finite temperature is introduced. We investigate the implications of this invariance for the 'limiting temperature' and simple compactification schemes.
 Publication:

Ph.D. Thesis
 Pub Date:
 1987
 Bibcode:
 1987PhDT........17O
 Keywords:

 Physics: Elementary Particles and High Energy