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Towards a classification of \(su(2)\oplus\cdots\oplus su(2)\) modular invariant partition functions. (English) Zbl 0833.17025

Summary: The complete classification of Wess-Zumino-Novikov-Witten 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, the classification problem is addressed for the nicest high rank semisimple affine algebras: \((A_1^{ (1)} )^{\oplus_r}\). Among other things, all automorphism invariants are found explicitly for all levels \(k= (k_1, \dots, k_r)\), and the classification for \(A_1^{ (1)} \oplus A_1^{ (1)}\) is completed for all levels \(k_1\), \(k_2\). The classification problem for \((A_1^{ (1)} )^{\oplus_r}\) is also solved for any levels \(k_i\) with the property that for \(i\neq j\) each \(\text{gcd} (k_i+2, k_j+ 2)\leq 3\). In addition, some physical invariants are found which seem to be new. Together with some recent work by Y. S. Stanev (see below), the classification for all \((A_1^{ (1)} )_k^{\oplus_r}\) could now be within sight.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
11F22 Relationship to Lie algebras and finite simple groups
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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