Boundary Conformal Field Theory and Fusion Ring Representations

TL;DR

This paper explores the relationship between boundary conformal field theories, fusion ring representations, and modular invariants, revealing discrepancies and proposing questions about their classification and physical relevance.

Contribution

It develops theory and classifications of NIM-reps, compares them with modular invariants, and highlights the existence of modular invariants without corresponding NIM-reps.

Findings

Many WZW modular invariants lack NIM-rep counterparts

Some modular invariants may be physically inconsistent

Classification of modular invariants is complex and context-dependent

Abstract

To an RCFT corresponds two combinatorial structures: the amplitude of a torus (the 1-loop partition function of a closed string, sometimes called a modular invariant), and a representation of the fusion ring (called a NIM-rep or equivalently a fusion graph, and closely related to the 1-loop partition function of an open string). In this paper we develop some basic theory of NIM-reps, obtain several new NIM-rep classifications, and compare them with the corresponding modular invariant classifications. Among other things, we make the following fairly disturbing observation: there are infinitely many (WZW) modular invariants which do not correspond to any NIM-rep. The resolution could be that those modular invariants are physically sick. Is classifying modular invariants really the right thing to do? For current algebras, the answer seems to be: Usually but not always. For finite groups a la Dijkgraaf-Vafa-Verlinde-Verlinde, the answer seems to be: Rarely.