$\boldsymbol{\mathcal{W}}$-algebras with Non-admissible Levels and the Deligne Exceptional Series. (23rd December 2016) - Kawasetsu, Kazuya
Abstract: The $C_2$ -cofinite rational simple $\mathcal{W}$ -algebras which are associated with minimal nilpotent elements, and related with simple current extensions of vertex operator algebras are classified. A simple current $M$ is a module over a vertex operator algebra $V$ such that the fusion product of $M$ and any irreducible module is also irreducible. The branching rules of such $\mathcal{W}$ -algebras are explicitly described. One of the main techniques developed in this article is to embed $\mathcal{W}$ -algebras and affine vertex algebras into abelian intertwining algebras, which are generalization of the vertex algebras. The Deligne exceptional series $ A_1\subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8 $ appears in the classification as in the Mathur-Mukhi-Sen classification of the rational conformal field theories with at most two linearly independent irreducible characters. Several $\mathcal{W}$ -algebras associated with the Lie algebras $D_4, E_6, E_7$ and $E_8$ given have the non-admissible levels $-1, -2, -3$ and $-5$, and they are new examples of $C_2$ -cofinite rational $\mathcal{W}$ -algebras. As one of applications of the classification, modular invariance of the characters of the $\mathcal{W}$ -algebras appeared in this paper are proved.