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. These share the same representation theory as quantum groups at roots of unity and physically correspond to the Wess-Zumino-Witten theories. One would like to identify the full (bulk) conformal field theories whose chiral halves are those VOAs. Mathematically, these correspond to the module categories for their representation theory. 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). Our paper focuses on the extensions. We prove that, for any simple g, there is a bound K(g) growing like the rank-cubed, such that for any level k>K(g), the only extensions are generic (simple-current ones). We use that bound to find all extensions (hence module categories) for g=sl(4) and sl(5). In the sequel to this paper, we find all extensions for all simple g of rank less than or equal to 6.