Can fusion coefficients be calculated from the depth rule ?

TL;DR

This paper explores the possibility of calculating fusion coefficients from the depth rule, reformulates the rule precisely, and proposes an approximate method that yields exact results for  su(3) fusion coefficients, offering an efficient computational approach.

Contribution

The paper reformulates the depth rule for fusion coefficient calculation and introduces an approximate method that is exact for  su(3), providing a new computational technique.

Findings

Exact results for  su(3) fusion coefficients using the method.

The basis elements for tensor product coefficients lack a well-defined depth.

The approximate method provides accurate lower bounds for fusion levels.

Abstract

The depth rule is a level truncation of tensor product coefficients expected to be sufficient for the evaluation of fusion coefficients. We reformulate the depth rule in a precise way, and show how, in principle, it can be used to calculate fusion coefficients. However, we argue that the computation of the depth itself, in terms of which the constraints on tensor product coefficients is formulated, is problematic. Indeed, the elements of the basis of states convenient for calculating tensor product coefficients do not have a well-defined depth! We proceed by showing how one can calculate the depth in an `approximate' way and derive accurate lower bounds for the minimum level at which a coupling appears. It turns out that this method yields exact results for $\widehat{su}(3)$ and constitutes an efficient and simple algorithm for computing $\widehat{su}(3)$ fusion coefficients.