From conformal embeddings to quantum symmetries: an exceptional SU(4) example

TL;DR

This paper explores algebraic tools used to understand quantum symmetries in Boundary Conformal Field Theories, focusing on a specific SU(4) conformal embedding and its associated exceptional quantum graph.

Contribution

It introduces a detailed analysis of the algebraic structures linking conformal embeddings, quantum graphs, and quantum symmetries, exemplified by the SU(4) case.

Findings

Identification of the quantum graph E4(SU(4))

Connection between modular invariants and quantum symmetries

Illustration of algebraic tools in conformal field theory

Abstract

We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a quantum graph. For known examples, the corresponding modular invariant partition function, which is sometimes associated with a conformal embedding, provides enough information to recover the whole structure. We illustrate these notions with the example of the conformal embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the exceptional quantum graph E4(SU(4)).