Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 060, 23 pages      arXiv:1305.7479      http://dx.doi.org/10.3842/SIGMA.2013.060
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Generalized Fuzzy Torus and its Modular Properties

Paul Schreivogl and Harold Steinacker
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria

Received June 19, 2013, in final form October 11, 2013; Published online October 17, 2013

Abstract
We consider a generalization of the basic fuzzy torus to a fuzzy torus with non-trivial modular parameter, based on a finite matrix algebra. We discuss the modular properties of this fuzzy torus, and compute the matrix Laplacian for a scalar field. In the semi-classical limit, the generalized fuzzy torus can be used to approximate a generic commutative torus represented by two generic vectors in the complex plane, with generic modular parameter τ. The effective classical geometry and the spectrum of the Laplacian are correctly reproduced in the limit. The spectrum of a matrix Dirac operator is also computed.

Key words: fuzzy spaces; noncommutative geometry; matrix models.

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