On modular invariance of quantum affine $W$-algebras

TL;DR

The paper establishes modular invariance for two families of quantum affine $W$-algebras: the minimal $W$-algebras $W_k^{min}(\frak{g})$ for $\frak{g}=D_n(n\ge4)$ or $E_6,E_7,E_8$ at specific negative levels, and the quantum Hamiltonian reductions $\widetilde{W}_k(\frak{g},f)=H_f(k\Lambda_0)$ at principal admissible levels with $u>\theta(x)$. It develops a unified approach via explicit algebraic formulas for the numerators $A_\Lambda^{[\alpha]}$ and their Jacobi-form reinterpretation, proving modular invariance by deriving explicit SL$_2(\mathbb{Z})$-transformation properties of the associated characters and Jacobi functions, including a detailed product expression in principal cases. The results yield holomorphic modular functions for the targeted characters, provide explicit character formulas, and connect modular invariance to potential simplicity and rationality under lisse assumptions, thereby advancing the conjectural picture for the rationality of these $W$-algebras. The work also supplies a corrected framework for earlier results to ensure consistency in the modular behavior of admissible-module characters and their reductions.

Abstract

We find modular transformations of normalized characters for the following $W$-algebras: (a) $W^{min}_k(\frak{g})$, where $\frak{g}=D_n \, (n \geq 4)$, or $E_6$, $E_7$, $E_8$, and $k$ is a negative integer $\geq -2$, or $\geq -\frac{h^{\vee}}{6}-1$, respectively; (b) quantum Hamiltonian reduction of the $\hat{\frak{g}}$-module $L(kΛ_0)$, where $\frak{g}$ 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 $\frak{g}$. We prove that these vertex algebras are modular invariant. A conformal vertex algebra is called modular invariant if its character $tr_V q^{L_0-c/24}$ 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.