On Modular Invariance of Quantum Affine W-Algebras

Abstract

Abstract We find modular transformations of normalized characters for the following W-algebras: (a) W k min ( g ) , where g = D n ( n ≥ 4 ) , or E 6 , E 7 , E 8 , and k is a negative integer ≥ - 2 , or ≥ - h ∨ 6 - 1 , respectively; (b) quantum Hamiltonian reduction of the g ^ -module L ( k Λ 0 ) , where g is a simple Lie algebra, f is its non-zero nilpotent element, and k is a principal admissible level with the denominator u > θ ( x ) , where 2x is the Dynkin characteristic of f, and θ is the highest root of g . We prove that these vertex algebras are modular invariant. A conformal vertex algebra V is called modular invariant if its character t r V q L 0 - c / 24 converges to a holomorphic modular function in the complex upper half-plane on a congruence subgroup. We find explicit formulas for their characters. Modular invariance of V is important since, in particular, conjecturally it implies that V is simple, and that V is rational, provided that it is lisse.