Noncommutative Riemann Surfaces

Details Noncommutative Riemann Surfaces Noncommutative Riemann Surfaces

2000-03-15

arxiv.org/abs/hep-th/0003131v1

Physics

High Energy Physics

High Energy Physics - Theory

LaTeX, 1+14 pages. Contribution to the TMR meeting ``Quantum aspects of gauge theories, supersymmetry and unification'', Paris

Scientific paper

We compactify M(atrix) theory on Riemann surfaces Sigma with genus g>1. Following [1], we construct a projective unitary representation of pi_1(Sigma) realized on L^2(H), with H the upper half-plane. As a first step we introduce a suitably gauged sl_2(R) algebra. Then a uniquely determined gauge connection provides the central extension which is a 2-cocycle of the 2nd Hochschild cohomology group. Our construction is the double-scaling limit N\to\infty, k\to-\infty of the representation considered in the Narasimhan-Seshadri theorem, which represents the higher-genus analog of 't Hooft's clock and shift matrices of QCD. The concept of a noncommutative Riemann surface Sigma_\theta is introduced as a certain C^\star-algebra. Finally we investigate the Morita equivalence.

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