The W_N minimal model classification

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.Comment: 25 page

Symmetries of the Kac-Peterson Modular Matrices of Affine Algebras

The characters $\chi_\mu$ of nontwisted affine algebras at fixed level define in a natural way a representation $R$ of the modular group $SL_2(Z)$. The matrices in the image $R(SL_2(Z))$ are called the Kac-Peterson modular matrices, and describe the modular behaviour of the characters. In this paper we consider all levels of $(A_{r_1}\oplus\cdots\oplus A_{r_s})^{(1)}$, and for each of these find all permutations of the highest weights which commute with the corresponding Kac-Peterson matrices. This problem is equivalent to the classification of automorphism invariants of conformal field theories, and its solution, especially considering its simplicity, is a major step toward the classification of all Wess-Zumino-Witten conformal field theories.Comment: 16 pp, plain te

Automorphism Modular Invariants of Current Algebras

We consider those two-dimensional rational conformal field theories (RCFTs) whose chiral algebras, when maximally extended, are isomorphic to the current algebra formed from some affine non-twisted Kac--Moody algebra at fixed level. In this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the Kac--Peterson modular matrices. We classify all such partition functions when the underlying finite-dimensional Lie algebra is simple. This gives all possible spectra for this class of RCFTs. While accomplishing this, we also find the primary fields with second smallest quantum dimension.Comment: 32 pages, plain Te

Modular differential equations for torus one-point functions

It is shown that in a rational conformal field theory every torus one-point function of a given highest weight state satisfies a modular differential equation. We derive and solve these differential equations explicitly for some Virasoro minimal models. In general, however, the resulting amplitudes do not seem to be expressible in terms of standard transcendental functions.Comment: 19 pages, LaTeX; reference adde

On the complete classification of the unitary N=2 minimal superconformal field theories

Aiming at a complete classification of unitary N=2 minimal models (where the assumption of space-time supersymmetry has been dropped), it is shown that each modular invariant candidate of a partition function for such a theory is indeed the partition function of a minimal model. A family of models constructed via orbifoldings of either the diagonal model or of the space-time supersymmetric exceptional models demonstrates that there exists a unitary N=2 minimal model for every one of the allowed partition functions in the list obtained from Gannon's work. Kreuzer and Schellekens' conjecture that all simple current invariants can be obtained as orbifolds of the diagonal model, even when the extra assumption of higher-genus modular invariance is dropped, is confirmed in the case of the unitary N=2 minimal models by simple counting arguments.Comment: 53 pages; Latex; minor changes in v2: intro expanded, references added, typos corrected, footnote added on p31; renumbering of sections; main theorem reformulated for clarity, but contents unchanged. Minor revisions in v3: typos corrected, footnotes 5, 6 added, lemma 1 and section 3.3.2 rewritten for greater generality, section 3.3 review removed. To appear in Comm. Math. Phy