Towards a Classification of su(2)$⨁\cdots ⨁$su(2) Modular Invariant Partition Functions

Abstract

The complete classification of WZNW modular invariant partition functions is known for very few affine algebras and levels, the most significant being all levels of $A_1$ and $A_2$ and level 1 of all simple algebras. Here, we address the classification problem for the nicest high rank semi-simple affine algebras: $(A_1^{(1)}{)}^{{\oplus }_r}$. Among other things, we explicitly find all automorphism invariants, for all levels $k=(k_1,\dots ,k_r)$, and complete the classification for $A_1^{(1)}\oplus A_1^{(1)}$, for all levels $k_1,k_2$. We also solve the classification problem for $(A_1^{(1)}{)}^{{\oplus }_r}$, for any levels $k_i$ with the property that for $i\ne j$ each $gcd(k_i+2,k_j+2)\le 3$. In addition, we find some physical invariants which seem to be new. Together with some recent work by Stanev, the classification for all $(A_1^{(1)}{)}_k^{{\oplus }_r}$ could now be within sight.

38 pp, (plain tex)

References in corpus (9)

• Simple currents versus orbifolds with discrete torsion -- a complete classification
• The Classification of Affine SU(3) Modular Invariant Partition Functions
• WZW Commutants, Lattices, and Level 1 Partition Functions
• Implications of an arithmetical symmetry of the commutant for modular invariants
• Partition Functions for Heterotic WZW Conformal Field Theories
• On the Classification of Diagonal Coset Modular Invariants
• The Rank four Heterotic Modular Invariant Partition Functions
• The Low Level Modular Invariant Partition Functions of Rank-Two Algebras
• On Modular Invariant Partition Functions for Tensor Products of Conformal Field Theories

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