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arXiv · 0909.0143

Modular Invariant of Quantum Tori

C. Castaño Bernard·T. M. Gendron
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Abstract

The quantum modular invariant of a real number is defined as a discontinuous, PGL(2,Z)-invariant multi-valued map using the distance-to-the-nearest-integer function. On the rationals, the quantum modular invariant is shown to be infinity and for quadratic irrationalities PARI/GP experiments suggest it is a finite set. In the case of the golden mean, we produce explicit formulas involving weighted versions of the Rogers-Ramanujan functions for the experimental supremum and infimum of its quantum modular invariant. We then define a universal modular invariant as a continuous and single valued map of ultrasolenoids, such that 1) the classical modular invariant is a quotient of its restriction to a subsolenoid fibering over the classical moduli space of elliptic curves and 2) the quantum modular invariant is a quotient of its restriction to a subsolenoid fibering over the moduli space of elliptic curves equipped with a Kronecker foliation.

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C. Castaño Bernard, T. M. Gendron. 2009-09-01. Modular Invariant of Quantum Tori. https://arxiv.org/abs/0909.0143

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