J. E. Nelson

Quantum Holonomies in (2+1)-Dimensional Gravity

We describe an approach to the quantisation 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. Representations are constructed and a group of transformations - a quasi-modular group - which preserves this structure, is presented.Abstract We describe an approach to the quantisation of (2+1)-dimensional gravity with topology R ×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. Representations are constructed and a group of transformations – a quasi-modular group – which preserves this structure, is presented.

Quantum Matrix Pairs

The notion of quantum matrix pairs is defined. These are pairs of matrices with noncommuting 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. Such matrices appear in an approach by the authors to quantizing gravity in 2 space and 1 time dimensions with negative cosmological constant on the torus. Explicit examples and transformations which generate new pairs from a given pair are presented.

Parametrization of the Moduli Space of Flat SL(2, R) Connections on the Torus

The moduli space of flat SL(2, R)-connections modulo gauge transformations on the torus may be described by ordered pairs of commuting SL(2, R) matrices modulo simultaneous conjugation by SL(2, R) matrices. Their spectral properties allow a classification of the equivalence classes, and a unique canonical form is given for each of these. In this way the moduli space becomes explicitly parametrized, and has a simple structure, resembling that of a cell complex, allowing it to be depicted. Finally, a Hausdorff topology based on this classification and parametrization is proposed.

A quantum Goldman bracket in 2+1 quantum gravity

In the context of quantum gravity for spacetimes of dimension (2 + 1), we describe progress in the construction of a quantum Goldman bracket for intersecting loops on surfaces. Using piecewise linear paths in (representing loops on the spatial manifold, i.e. the torus) and a quantum connection with noncommuting components, we review how holonomies and Wilson loops for two homotopic paths are related by phases in terms of the signed area between them. Paths rerouted at intersection points with other paths occur on the rhs of the Goldman bracket. To better understand their nature we introduce the concept of integer points inside the parallelogram spanned by two intersecting paths, and show that the rerouted paths must necessarily pass through these integer points.

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.

QUANTUM HOLONOMIES IN (2+1)-DIMENSIONAL GRAVITY

We describe an approach to the quantisation of (2+1)–dimensional gravity with topology IR×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. Representations are constructed and a group of transformations a quasi–modular group which preserves this structure, is presented. P.A.C.S.04.60Kz, 02.10.Tq, 02.20.-a email: nelson@to.infn.it email: picken@math.ist.utl.pt

QUANTUM GRAVITY AND QUANTUM GROUPS

First order gravity in 2 + 1 space-time dimensions with cosmological constant Λ and spatial Riemann surfaces is considered. The representations of the fundamental group satisfy an algebra related to the quantum group SU(2)q. The Poincare theory appears as the limit Λ → 0.