2009
DOI: 10.1007/s00220-009-0766-8
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Noncommutative Riemann Surfaces by Embeddings in $${\mathbb{R}^{3}}$$

Abstract: Abstract:We introduce C-Algebras of compact Riemann surfaces as non-commutative analogues of the Poisson algebra of smooth functions on . Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.

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Cited by 15 publications
(12 citation statements)
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“…The idea is to begin with a fuzzy sphere and to deform the matrices in a simple form to go from a sphere to a torus. The basic idea is to follow the construction of the giant torus as described in [26] (other examples of embeddings of Riemann surfaces in R 3 can be found in [27], and in [28] one can also find a different example that interpolates between sphere and tori). In the case of the giant torus, one is supposed to add strings with maximal angular momentum to a sphere until the geometry transitions to a torus.…”
Section: B From a Sphere To A Torusmentioning
confidence: 99%
“…The idea is to begin with a fuzzy sphere and to deform the matrices in a simple form to go from a sphere to a torus. The basic idea is to follow the construction of the giant torus as described in [26] (other examples of embeddings of Riemann surfaces in R 3 can be found in [27], and in [28] one can also find a different example that interpolates between sphere and tori). In the case of the giant torus, one is supposed to add strings with maximal angular momentum to a sphere until the geometry transitions to a torus.…”
Section: B From a Sphere To A Torusmentioning
confidence: 99%
“…The results can also be thought of as a concrete realization of the abstract idea in the classic work by Kontsevich, [12]. Related recent work includes [13,14], though our construction appears more general as it allows us to vary the local noncommutativity independent of the shape of the surface.…”
Section: Introductionmentioning
confidence: 77%
“…For example in [12] and [34] more general setups for metrics on the noncommutative torus are considered. And of course one could attempt to study Ricci flow and the resulting scalar curvature on other matrix geometries, like the fuzzy sphere [25], or even more general frameworks like noncommutative Riemann surfaces [2], [3]. It would also be possible to study the noncommutative Ricci flow equation numerically.…”
Section: Further Workmentioning
confidence: 99%