Alternative formulation for the operator algebra over the space of paths in a ADE $SU(3)$ graph

TL;DR

This paper introduces a new operator algebra framework for the path space over $SU(3)$ ADE graphs, incorporating novel collapsed triangular cells and alternative operators, advancing the understanding of modular invariant CFTs.

Contribution

It provides explicit collapsed triangular cell values and demonstrates their role in fulfilling Kuperberg relations, offering an alternative approach using closed triangular sequences.

Findings

Explicit values for collapsed triangular cells are provided.

Operators satisfying Temperley-Lieb algebra lead to alternative path descriptions.

Essential paths using closed triangles are equivalent to those with back-and-forth sequences.

Abstract

In this work we discuss the elements required for the construction of the operator algebra for the space of paths over a simply laced $SU(3)$ graph. These operators are an important step in the construction of the bialgebra required to find the partition functions of some modular invariant CFTs. We define the cup and cap operators associated with back-and-forth sequences and add them to the creation and annihilation operators in the operator algebra as they are required for the calculation of the full space of essential paths prescribed by the fusion algebra. These operators require collapsed triangular cells that had not been found in previous works; here we provide explicit values for these cells and show their importance in order for the cell system to fulfill the Kuperberg relations for $SU(3)$ tangles. We also find that demanding that our operators satisfy the Temperley-Lieb algebra leads one naturally to consider operators that create and annihilate closed triangular sequences, which in turn provides an alternative the cup and cap operators as they allow one to replace back-and-forth sequences with closed triangular ones. We finally show that the essential paths obtained by using closed triangles are equivalent to those obtained originally using back-and-forth sequences.