From modular invariants to graphs: the modular splitting method

TL;DR

This paper introduces a systematic method to derive graphs and quantum symmetries from a given modular invariant in su(n)_k conformal field theories, providing explicit constructions for boundary conditions and defect lines.

Contribution

It presents a general approach to solve the modular splitting equation and determine associated graphs and symmetries without prior knowledge of the graph structure.

Findings

Explicit construction of boundary conditions and defect lines.

Determination of quantum symmetries for su(3)_k cases.

Application to exceptional cases at levels 5 and 9.

Abstract

We start with a given modular invariant M of a two dimensional su(n)_k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several su(3)_k exceptional cases at levels 5 and 9.