On modular invariance of quantum affine $W$-algebras

Abstract

We find modular transformations of normalized characters for the following $W$-algebras: (a) $W_k^{min}(\mathfrak{g})$, where $\mathfrak{g}=D_n(n\ge 4)$, or $E_6$, $E_7$, $E_8$, and $k$ is a negative integer $\ge -2$, or $\ge -\frac{h^{\vee }}{6}-1$, respectively; (b) quantum Hamiltonian reduction of the $\hat{\mathfrak{g}}$-module $L(k{\Lambda }_0)$, where $\mathfrak{g}$ is a simple Lie algebra, $f$ is its non-zero nilpotent element, and $k$ is a principal admissible level with the denominator $u>\theta (x)$, where $2x$ is the Dynkin characteristic of $f$ and $\theta$ is the highest root of $\mathfrak{g}$. We prove that these vertex algebras are modular invariant. A conformal vertex algebra 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.