Exotic quantum subgroups and extensions of affine Lie algebra VOAs -- part I

Abstract

Prototypical rational vertex operator algebras are associated to affine Lie algebras at positive integer level k. They correspond physically to the Wess-Zumino-Witten theories, and their representation theory can be captured by quantum groups at roots of unity. One would like to identify the full (bulk) conformal field theories whose chiral halves are those VOAs. Mathematically, these correspond to their module categories. Until now, this has been done only for sl(2) (famously, the A-D-E classification of Cappelli-Itzykson-Zuber) and sl(3). The problem reduces to knowing the possible extensions of those VOAs, and the tensor equivalences between those extensions. Recent progress puts the tensor equivalences in good control, especially for sl(n). This paper focuses on the extensions. We prove that, for any simple g, there is a bound K(g) growing like rank$(g)^3$, such that for any level k>K(g), the only extensions are generic (i.e. simple-current ones). We use that bound to find all extensions for g=sl(4) and sl(5), at all levels, as well as all g=sl(n) at levels $k\le 5$ (only those for k=1 had been classified before). In the sequel to this paper, we find all extensions for all simple g of rank <7 (and the corresponding low level classifications).