III. Algebraic Geometrical Aspects of Quantum InformationNo Access

MONODROMY ARISING FROM THE MARKOFF THEORY

SERGE PERRINE

France Telecom, Research and Developments, 38-40 rue du General Leclerc, Issy les Moulineaux Cedex 9, 92794, France

https://doi.org/10.1142/S0217979206034339Cited by:1 (Source: Crossref)

Abstract

In a former work, recalling what the Markoff theory is, we summarized some existing links with the group GL(2, ℤ) of 2 × 2 matrices. We also quoted the relation with conformal punctured toruses. The monodromy representation of the Poincaré group of such a torus was considered. Here we explicit the corresponding solution of the associated Riemann-Hilbert problem, and the resulting Fuchs differential equation. We precisely describe how the calculus runs. The main result is the description of a complete family of Fuchs differential equations with, as the monodromy group, the free group with two generators. We also identify a link with some eigenvalues of a Laplacian. The introduction explains the links that we see with information and computation theory (classical or quantum).

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