Noncommutative Riemann Surfaces

TL;DR

This paper constructs a noncommutative version of higher-genus Riemann surfaces using advanced algebraic and geometric methods, extending matrix theory concepts to complex topologies.

Contribution

It introduces a novel noncommutative Riemann surface framework via projective unitary representations and explores its algebraic properties and Morita equivalence.

Findings

Constructed a projective unitary representation of pi_1(Sigma) on L^2(H)

Defined a noncommutative Riemann surface as a C*-algebra

Analyzed Morita equivalence of the constructed algebra

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.