The W_N minimal model classification

TL;DR

This paper establishes a bijection between W_N minimal models and SU(N)xSU(N) WZW theories, leading to a complete classification of modular invariants for W_3 models and insights into their structure.

Contribution

It provides the first complete classification of modular invariants for W_3(p,q) minimal models and introduces a new correspondence simplifying the classification of W_N models.

Findings

Complete classification of W_3(p,q) modular invariants.

Identification of a new infinite series of exceptional SU(3)xSU(3) invariants.

W_3 invariants factorize into SU(3) invariants, unlike larger N.

Abstract

We first rigourously establish, for any N, that the toroidal modular invariant partition functions for the (not necessarily unitary) W_N(p,q) minimal models biject onto a well-defined subset of those of the SU(N)xSU(N) Wess-Zumino-Witten theories at level (p-N,q-N). This permits considerable simplifications to the proof of the Cappelli-Itzykson-Zuber classification of Virasoro minimal models. More important, we obtain from this the complete classification of all modular invariants for the W_3(p,q) minimal models. All should be realised by rational conformal field theories. Previously, only those for the unitary models, i.e. W_3(p,p+1), were classified. For all N our correspondence yields for free an extensive list of W_N(p,q) modular invariants. The W_3 modular invariants, like the Virasoro minimal models, all factorise into SU(3) modular invariants, but this fails in general for larger N. We also classify the SU(3)xSU(3) modular invariants, and find there a new infinite series of exceptionals.