In this paper we focus on $G=SU(N)$.

These theories can also be geometrically engineered by compactifying type IIB string theory on isolated hypersurface singularities [61] $$\begin{gathered} {\mathcal{T}}_{3,N}:x^3+y^3+z^3+w^N=0, \\ {\mathcal{T}}_{4,N}:x^2+y^4+z^4+w^N=0, \\ {\mathcal{T}}_{6,N}:x^2+y^3+z^6+w^N=0. \end{gathered}$$(B7)

An elliptic function $f(\mathfrak{a})$ satisfies the double periodicity $f(\mathfrak{a}+1)=f(\mathfrak{a}+\tau )=f(\mathfrak{a})$.

Due to the ${\vartheta }_1(\mathfrak{u}{)}^{N-1}$ in the denominator in (42), such a limit must be taken carefully.

In rigorous mathematical language, “ordinary” requires semi-simplicity in the $L_0$ action. To simplify the narration in this paper, however, by “ordinary” we include those having logarithmic feature, but with nonsingular unflavored characters.

In general ${\varphi }_{2k}$ can have poles in the fundamental domain of the complex moduli $\tau$. In the absence of poles, the equation is called “monic.”

The removal of complex root is useful for analyzing the Cardy behavior. The indicial roots $\alpha =-\frac{c_{2\mathrm{D}}}{24}+h$ is tied to the conformal weight $h$ of a module, and the minimal value $h_{\min }$ is related to the 4d central charge $a_{4\mathrm{d}}=\frac{h_{\min }}{2}-\frac{5c_{2\mathrm{D}}}{48}$ [15]. To some extend, the presence of a complex indicial root renders the notion of minimal $h$ ill-defined.

This property is discussed in [37]. Here we present it using our notation which is more convenient for the subsequent modulairty analysis.

The exponent in the integral is also known as Weierstrass’ elliptic function, $$P_2(y,\tau )≔-\sum _{n=1}^{\infty }\frac{1}{2n}{E}_{2n}(\tau )y^{2n}.$$(B8)

The Levi subalgebra $\mathfrak{l}$ is given by the centralizer of ${\mathfrak{h}}^f≔\{h\in \mathfrak{h}|[h,f]=0\}$, which contains $f$. The roots system of $\mathfrak{l}$ is given by ${\mathrm{\Delta }}_{\mathfrak{l}}=\{\alpha \in \mathrm{\Delta }|\alpha {|}_{{\mathfrak{h}}^f}=0\}$. The roots ${\mathrm{\Delta }}_{\mathfrak{l}}$ determine a parabolic subalgebra $\mathfrak{p}\subset \mathfrak{j}$ with its Levi decomposition $\mathfrak{p}=\mathfrak{l}+\mathfrak{n}$.

The Weyl group of ${\mathbf{P}}^{\vee }$ is constructed using the finite part of the roots ${\widehat{\mathrm{\Delta }}}_{+}$.

The mismatch also appears in the next example. It would be interesting to further clarify the relationship between the Jordan blocks of the $T$ matrix and the dimension computation of the fixed varieties.

In the literature these functions are often called twisted Eisenstein series. In this paper we will suppress the word twisted.