Boundary Conformal Field Theory and Fusion Ring Representations

Details Boundary Conformal Field Theory and Fusion Ring Representations Boundary Conformal Field Theory and Fusion Ring Representations

2001-06-13

arxiv.org/abs/hep-th/0106105v2

Nucl.Phys. B627 (2002) 506-564

Physics

High Energy Physics

High Energy Physics - Theory

58 pages, 4 figures

Scientific paper

10.1016/S0550-3213(01)00632-0

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.

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