Noncommutative Riemann Surfaces

We compactify M(atrix) theory on Riemann surfaces Σ with genus g > 1. Following [1], we construct a projective unitary representation of π1(Σ) realized on L (H), with H the upper half–plane. As a first step we introduce a suitably gauged sl2(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 → ∞, k → −∞ 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 Σθ is introduced as a certain C –algebra. Finally we investigate the Morita equivalence. Contribution to the TMR meeting ”Quantum Aspects of Gauge Theories, Supersymmetry and Unification”, Paris, September 1–7, 1997.