W-algebras with Non-admissible Levels and the Deligne Exceptional Series. (23rd December 2016) - Kawasetsu, Kazuya
Abstract: The C_2 -cofinite rational simple 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 W -algebras are explicitly described. One of the main techniques developed in this article is to embed W -algebras and affine vertex algebras into abelian intertwining algebras, which are generalization of the vertex algebras. The Deligne exceptional series A_1A_2 G_2 D_4 F_4 E_6 E_7 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 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 W -algebras. As one of applications of the classification, modular invariance of the characters of the W -algebras appeared in this paper are proved.