The braiding for representations of q-deformed affine $sl_2$

TL;DR

This paper calculates the braiding structure for a specific quantum affine algebra at complex deformation parameters, revealing its dependence on number-theoretic properties and providing a rigorous, assumption-free derivation.

Contribution

It provides a first-principles derivation of the braiding for the principal gradation of $U_q(\hat{sl}_2)$ at $|q|=1$, without relying on crossing or unitarity assumptions.

Findings

Demonstrates the uniqueness of the braiding normalization under analyticity assumptions.

Shows the convergence of the braiding depends on the number-theoretic properties of $\tau$.

Analyzes convergence probabilistically assuming a uniform distribution of $q$ on the unit circle.

Abstract

We compute the braiding for the `principal gradation' of $U_q(\hat{{\it sl}_2})$ for $|q|=1$ from first principles, starting from the idea of a rigid braided tensor category. It is not necessary to assume either the crossing or the unitarity condition from S-matrix theory. We demonstrate the uniqueness of the normalisation of the braiding under certain analyticity assumptions, and show that its convergence is critically dependent on the number-theoretic properties of the number $\tau$ in the deformation parameter $q=e^{2\pi i\tau}$. We also examine the convergence using probability, assuming a uniform distribution for $q$ on the unit circle.