Quantum Holonomies in (2+1)-Dimensional Gravity

Details Quantum Holonomies in (2+1)-Dimensional Gravity Quantum Holonomies in (2+1)-Dimensional Gravity

2004-08-25

arxiv.org/abs/gr-qc/0408082v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

5 pages, to appear in the proceedings of 10th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental

Scientific paper

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.

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