[Paper Review] Modular Invariant of Quantum Tori

C. Castaño Bernard, T. M. Gendron|arXiv (Cornell University)|Sep 1, 2009

Advanced Mathematical Identities24 references3 citations

TL;DR

This paper introduces a discontinuous, multi-valued quantum modular invariant $j^{\rm qt}(\theta)$ for real numbers $\theta$ using the distance-to-the-nearest-integer function, showing it is invariant under $\mathrm{PGL}_2(\mathbb{Z})$ and finite for quadratic irrationals. It constructs a universal modular invariant ${}^{\diamond}j$ on an ultrasolenoid space, unifying the classical and quantum modular invariants as continuous quotients, with explicit formulas for the golden ratio case using Rogers-Ramanujan functions.

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.

Motivation & Objective

• To define a modular invariant for quantum tori that generalizes the classical modular invariant in the context of noncommutative geometry and real multiplication.
• To address the challenge of defining continuous invariants on spaces with dense orbits, such as quantum tori, by introducing a discontinuous, multi-valued map.
• To construct a universal modular invariant ${}^{\diamond}j$ on an ultrasolenoid that unifies both classical and quantum modular invariants as continuous, single-valued quotients.
• To provide explicit formulas for the quantum modular invariant at the golden mean $\varphi$, using weighted Rogers-Ramanujan functions.
• To present experimental evidence suggesting that $j^{\rm qt}(\theta)$ is finite for quadratic irrationals $\theta$, with cardinality growing linearly with the discriminant.

Proposed method

• Define the quantum modular invariant $j^{\rm qt}(\theta)$ as a multi-valued map using the distance-to-the-nearest-integer function $\|\cdot\|$, invariant under $\mathrm{PGL}_2(\mathbb{Z})$ action.
• For the golden mean $\varphi$, derive explicit formulas involving weighted Rogers-Ramanujan functions for the infimum and supremum of $j^{\rm qt}(\varphi)$.
• Construct the universal modular invariant ${}^{\diamond}j$ as a continuous, single-valued map on an ultrasolenoid ${}^{\diamond}\widehat{\sf Mod}$, using ultraproducts and ultratransversals.
• Realize the classical modular invariant $j^{\rm cl}$ as a quotient of ${}^{\diamond}j$ restricted to a subsolenoid ${}^{\diamond}{\sf Mod}^{\rm cl}$ fibering over the classical moduli space.
• Realize the quantum modular invariant $j^{\rm qt}$ as a quotient of ${}^{\diamond}j$ restricted to a subsolenoid ${}^{\diamond}{\sf Mod}^{\rm qt}$ fibering over the moduli space with Kronecker foliations.
• Use PARI/GP experiments to compute $j^{\rm qt}(\theta)$ for fundamental units in real quadratic fields, suggesting finite and bounded sets with cardinality $O(D)$.

Experimental results

Research questions

• RQ1 Is the quantum modular invariant $j^{\rm qt}(\theta)$ finite for quadratic irrationalities $\theta$?
• RQ2 Can a universal modular invariant be constructed that continuously unifies both classical and quantum modular invariants?
• RQ3 What explicit formulas describe the extremal values of $j^{\rm qt}(\theta)$ for the golden mean $\varphi$?
• RQ4 How does the cardinality of $j^{\rm qt}(\theta)$ scale with the discriminant $D$ of the real quadratic field $\mathbb{Q}(\sqrt{D})$?
• RQ5 What is the topological and geometric structure of the space $\widehat{\sf Mod}^{\rm qt}$, and how does it relate to the moduli space of elliptic curves with Kronecker foliations?

Key findings

• For $\theta \in \mathbb{Q}$, the quantum modular invariant satisfies $j^{\rm qt}(\theta) = \infty$, indicating a degenerate case.
• For the golden mean $\varphi$, the experimental supremum and infimum of $j^{\rm qt}(\varphi)$ are approximated by weighted Rogers-Ramanujan functions, yielding $j_{\rm best}^{\rm qt}(\varphi) \approx 9538.2496$.
• PARI/GP experiments suggest that for fundamental units $u$ in $\mathbb{Q}(\sqrt{D})$, the set $j^{\rm qt}(u)$ is finite and its cardinality is approximately $O(D)$.
• The universal modular invariant ${}^{\diamond}j$ is a continuous, single-valued map on the ultrasolenoid ${}^{\diamond}\widehat{\sf Mod}$, constructed via ultraproducts and ultratransversals.
• The classical modular invariant $j^{\rm cl}$ arises as a quotient of ${}^{\diamond}j$ restricted to the subsolenoid ${}^{\diamond}{\sf Mod}^{\rm cl}$, which fibers over the classical moduli space of elliptic curves.
• The quantum modular invariant $j^{\rm qt}$ arises as a quotient of ${}^{\diamond}j$ restricted to the subsolenoid ${}^{\diamond}{\sf Mod}^{\rm qt}$, which fibers over the moduli space of elliptic curves equipped with a Kronecker foliation.