Limits of the quantum SO(3) representations for the one-holed torus

TL;DR

This paper investigates the asymptotic behavior of a sequence of SO(3)-TQFT representations of the mapping class group of the one-holed torus, demonstrating convergence and describing the limiting action of SL(2,Z).

Contribution

It proves the convergence of these quantum representations and characterizes their limits, confirming a conjecture by Andersen, Masbaum, and Ueno.

Findings

Representations converge as p tends to infinity.

Limits describe SL(2,Z) action on polynomial spaces.

Conjecture of Andersen, Masbaum, and Ueno verified.

Abstract

For $N \geq 2$, we study a certain sequence $(\rho_p^{(c_p)})$ of N-dimensional representations of the mapping class group of the one-holed torus arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and Ueno \cite{1} holds for these representations. This is done by proving that, in a certain basis and up to a rescaling, the matrices of these representations converge as $p$ tends to infinity. Moreover, the limits describe the action of $SL_{2}(\mathbb{Z})$ on the space of homogeneous polynomials of two variables of total degree $N-1$.