Noncommutative Riemann Surfaces
Abstract
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.
Cite
@article{arxiv.hep-th/0003131,
title = {Noncommutative Riemann Surfaces},
author = {G. Bertoldi and J. M. Isidro and M. Matone and P. Pasti},
journal= {arXiv preprint arXiv:hep-th/0003131},
year = {2018}
}
Comments
LaTeX, 1+14 pages. Contribution to the TMR meeting ``Quantum aspects of gauge theories, supersymmetry and unification'', Paris 1-7 September 1999