Quantum Holonomies in (2+1)-Dimensional Gravity

We describe an approach to the quantization of (2+1)--dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q--commutation relation. Solutions of diagonal and upper--triangular form are constructed, which in the latter case exhibit additional, non--trivial internal relations for each holonomy matrix. This leads to the notion of quantum matrix pairs. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relations as the original pair. This has implications for the classical moduli space, described by ordered pairs of commuting SL(2,R) matrices modulo simultaneous conjugation by SL(2,R) matrices.Comment: 5 pages, to appear in the proceedings of 10th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG X MMIII), Rio de Janeiro, Brazil, 20-26 Jul 200

The Quantum Modular Group in (2+1)-Dimensional Gravity

The role of the modular group in the holonomy representation of (2+1)-dimensional quantum gravity is studied. This representation can be viewed as a "Heisenberg picture", and for simple topologies, the transformation to the ADM "Schr{\"o}dinger picture" may be found. For spacetimes with the spatial topology of a torus, this transformation and an explicit operator representation of the mapping class group are constructed. It is shown that the quantum modular group splits the holonomy representation Hilbert space into physically equivalent orthogonal ``fundamental regions'' that are interchanged by modular transformations.Comment: 23 pages, LaTeX, no figures; minor changes and clarifications in response to referee (basic argument and conclusions unaffected

Optical links in the angle-data assembly of the 70-meter antennas

In the precision-pointing mode the 70 meter antennas utilize an optical link provided by an autocollimator. In an effort to improve reliability and performance, commercial instruments were evaluated as replacement candidates, and upgraded versions of the existing instruments were designed and tested. The latter were selected for the Neptune encounter, but commercial instruments with digital output show promise of significant performance improvement for the post-encounter period

Entropy of Folding of the Triangular Lattice

The problem of counting the different ways of folding the planar triangular lattice is shown to be equivalent to that of counting the possible 3-colorings of its bonds, a dual version of the 3-coloring problem of the hexagonal lattice solved by Baxter. The folding entropy Log q per triangle is thus given by Baxter's formula q=sqrt(3)(Gamma[1/3])^(3/2)/2pi =1.2087...Comment: 9 pages, harvmac, epsf, uuencoded, 5 figures included, Saclay preprint T/9401

Quantum geometry from 2+1 AdS quantum gravity on the torus

Wilson observables for 2+1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.Comment: 19 pages, 11 figures, revised with more explanations, improved figures and extra figures. To appear GER

Anomalous coupling between topological defects and curvature

We investigate a counterintuitive geometric interaction between defects and curvature in thin layers of superfluids, superconductors and liquid crystals deposited on curved surfaces. Each defect feels a geometric potential whose functional form is determined only by the shape of the surface, but whose sign and strength depend on the transformation properties of the order parameter. For superfluids and superconductors, the strength of this interaction is proportional to the square of the charge and causes all defects to be repelled (attracted) by regions of positive (negative) Gaussian curvature. For liquid crystals in the one elastic constant approximation, charges between 0 and $4\pi$ are attracted by regions of positive curvature while all other charges are repelled.Comment: 5 pages, 4 figures, minor changes, accepted for publication in Phys. Rev. Let