The Concept of a Noncommutative Riemann Surface

We consider the compactification M(atrix) theory on a Riemann surface Sigma of genus g>1. A natural generalization of the case of the torus leads to construct a projective unitary representation of pi_1(\Sigma), realized on the Hilbert space of square integrable functions on the upper half--plane. A uniquely determined gauge connection, which in turn defines a gauged sl_2(R) algebra, provides the central extension. This has a geometric interpretation as the gauge length of a geodesic triangle, and corresponds to a 2-cocycle of the 2nd Hochschild cohomology group of the Fuchsian group uniformizing Sigma. Our construction can be seen as a suitable double-scaling limit N\to\infty, k\to-\infty of a U(N) representation of pi_1(Sigma), where k is the degree of the associated holomorphic vector bundle, which can be seen as the higher-genus analog of 't Hooft's clock and shift matrices of QCD. We compare the above mentioned uniqueness of the connection with the one considered in the differential-geometric approach to the Narasimhan-Seshadri theorem provided by Donaldson. We then use our infinite dimensional representation to construct a C^\star-algebra which can be interpreted as a noncommutative Riemann surface Sigma_\theta. Finally, we comment on the extension to higher genus of the concept of Morita equivalence.