where

$$A_{\underline{k}} = \prod_{a \in \mathcal{A}} \prod_{I=1}^{k_{\bar{a}}} A(x_{\bar{a},I}), \quad W_{\underline{n}|\underline{m}} = : \prod_{a \in \mathcal{A}} \frac{\prod_{\alpha=1}^{n_{\bar{a}}} W_{\bar{a}}(v_{\bar{a},\alpha})}{\prod_{\beta=1}^{m_{\bar{a}}} W_{\bar{a}}(u_{\bar{a},\beta})} :; \qquad (4.28a)$$

$$\langle A_{\underline{k}}^{-1} W_{\underline{n}|\underline{m}} \rangle = \prod_{a,b \in \mathcal{A}} \prod_{\alpha=1}^{n_{\bar{a}}} \prod_{I=1}^{k_{\bar{b}}} \nu_a \left( \frac{v_{\bar{a},\alpha}}{x_{\bar{b},I}} \right) \prod_{a,b \in \mathcal{A}} \prod_{\beta=1}^{m_{\bar{a}}} \prod_{J=1}^{k_{\bar{b}}} \nu_a \left( \frac{u_{\bar{a},\alpha}}{x_{\bar{b},J}} \right)^{-1} \prod_{(a,I) < (b,J)} A_{\mathcal{C}^4} \left( \frac{x_{\bar{a},I}}{x_{\bar{b},J}} \right)^{-1}. \qquad (4.28b)$$

We expect that the operators in the denominators of (4.26) correspond to $\overline{D6}$-branes similar to the situation of the magnificent four system. We leave a detailed analysis of the evaluation of this contour integral formula and its relation with the 7d supergroup gauge theory for future work. We will see in later sections, that after tuning the parameters {$u_{\bar{a},\beta}$} to special values, we can further reduce the system and obtain the contour integral formula of the spiked instanton system.

## 4.3 Spiked instanton

Let us next consider the spiked instanton system where D4-branes wrapping $\mathcal{C}_A^2 \times \mathbb{S}^1$, ($A \in \mathcal{A}$) appear. We introduce the following vertex operators for $A \in \mathcal{A}$:

$$X_A(x) = x_{A,0}(x) : \exp \left( \sum_{n \neq 0} x_{A,n} x^{-n} \right) :; \quad [x_{A,n}, x_{B,m}] = - \frac{1}{n} \frac{\mathbf{P}_{\mathcal{A}}^{[n]}}{\mathbf{P}_{\mathcal{A}}^{[-n]} \mathbf{P}_{\mathcal{B}}^{[n]}} \delta_{n+m,0}, \qquad (4.29)$$

where $x_{A,0}(x)$ is the zero mode. The relation with the A-operator is

$$x_{A,n} = \frac{a_n}{\mathbf{P}_{\mathcal{A}}^{[-n]}}, \quad [a_n, x_{A,m}] = - \frac{1}{n} \mathbf{P}_{\mathcal{A}}^{[n]} \delta_{n+m,0}. \qquad (4.30)$$

Explicitly, the contraction formulas are

$$\begin{align}
A(x)X_A(\nu) &= S_{\bar{A}}(\nu/x)^{-1} \overline{a_0(x)X_{A,0}(x')} : A(x)X_A(\nu) :; \tag{4.31} \\
X_A(\nu)A(x) &= S_{\bar{A}}(q_A x/\nu)^{-1} \overline{X_{A,0}(x')} \overline{a_0(x)} : X_A(\nu)A(x) :. \nonumber
\end{align}$$

We impose the following condition on the zero modes so that the operator product of the right-hand side is the same rational function after analytic continuation (3.69):

$$\overline{a_0(x)X_{A,0}(x')} = x_{A,0}(x')\overline{a_0(x)}. \qquad (4.32)$$

We will use $\overline{a_0(x)X_{A,0}(x')} = 1$ (see section 4.5 for explicit forms).

The vertex operators $X_A(x)$ we introduced here are just the Y-operators in [60] up to shift of variables

$$X_A(x) = Y_A(q_A^{-1}x). \qquad (4.33)$$

Explicitly, if we consider the affine quiver $\tilde{A}_0$ gauge theory on $\mathcal{C}_{12}^2 \times \mathbb{S}^1$, the Y-operator<fnref n="18" /> in [60] will be described as $X_{34}(q_{12}^{-1}x)$.

<footnote n="18">¹⁸The Y-operators are operators representing defects on the bulk gauge theory we are focusing on. For the gauge theory on the D4-brane wrapping $\mathcal{C}_{12}^2 \times \mathbb{S}^1$, the point defect is the transverse D4-brane wrapping $\mathcal{C}_{34}^2 \times \mathbb{S}^1$. Thus, the Y-operator is related with the operator $X_{34}(x)$.</footnote>
**Proposition 4.4.** The contour integral formula (3.136) of the spiked instanton configuration is rewritten as

$$ \underline{q}^{\underline{k}} \mathcal{Z}_{\underline{k}}^{\mathrm{D}4} = \frac{\underline{q}^{\underline{k}} \mathcal{G}^{\underline{k}}}{\underline{k}!} \oint \prod_{A \in \mathbf{6}} \prod_{I=1}^{k_A} \frac{dx_{A,I}}{2\pi i x_{A,I}} \langle \mathbf{A}_{\underline{k}}^{-1} \mathbf{X}_{\underline{n}} \rangle \quad (4.34) $$

where

$$ \underline{q}^{\underline{k}} = \prod_{A \in \mathbf{6}} \underline{q}^{k_A}, \quad \underline{G}^{\underline{k}} = \prod_{A \in \mathbf{6}} (\mathcal{G}_{\mathrm{sup}(\bar{A})})^{k_A}, \quad \mathbf{A}_{\underline{k}} = \prod_{A \in \mathbf{6}} \prod_{I=1}^{k_A} \mathbf{A}(x_{A,I}), \quad \mathbf{X}_{\underline{n}} := \prod_{A \in \mathbf{6}} \prod_{\alpha=1}^{n_A} \mathbf{X}_A(v_{A,\alpha}), \quad (4.35\mathrm{a}) $$

$$ \langle \mathbf{A}_{\underline{k}}^{-1} \mathbf{X}_{\underline{n}} \rangle = \prod_{A,B \in \mathbf{6}} \prod_{I=1}^{k_B} \prod_{\alpha=1}^{n_A} \mathfrak{S}_{\bar{A}} \left( \frac{v_{A,\alpha}}{x_{B,I}} \right) \prod_{(A,I) < (B,J)} \mathcal{A}_{\mathfrak{C}^4} \left( \frac{x_{A,I}}{x_{B,J}} \right)^{-1}. \quad (4.35\mathrm{b}) $$

The one-loop perturbative part in (3.147) is obtained by the OPE of the X operators:

$$ \mathbf{X}_B(v_{B,\beta})\mathbf{X}_A(v_{A,\alpha}) = \mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D}4-\mathrm{D}4}(v_{A,\alpha}, A | v_{B,\beta}, B) : \mathbf{X}_B(v_{B,\beta})\mathbf{X}_A(v_{A,\alpha}) : . \quad (4.36) $$

**Relation with tetrahedron instanton.** Similar to the situation in (4.22), where the D8-√D8 coupled system is reduced to the tetrahedron system, using (4.26) and specializing the parameters, we can obtain the contour integral formula of the spiked instanton system. This comes from the following relation:

$$ \mathbf{X}_{ab}(x) \approx : \frac{\mathfrak{W}_{abc}(x)}{\mathfrak{W}_{abc}(q_c x)} : \quad (4.37) $$

where ≈ means they are equivalent up to zero modes. In our notation of the zero modes, it will become an exact identity (see (4.66)). For example, we have

$$ : \prod_{\alpha=1}^{n_{12}} \mathbf{X}_{12}(v_{12,\alpha}) : := : \prod_{\alpha=1}^{n_{12}} \frac{\mathfrak{W}_{123}(v_{12,\alpha})}{\mathfrak{W}_{123}(q_3 v_{12,\alpha})} : \quad (4.38) $$

which means by considering the D6-√D6 system spanning C3123 × S1 and tuning the positions of them with the parameter q3, we can obtain the 5d theory on C212 × S1.

**Supergroup generalization.** As mentioned in section 3.5, to obtain supergroup analogs of the gauge origami system of the spiked instanton, we need to include negative D-branes D0-, D4- to the system. Due to the fact that the corresponding vertex operators of D0+ and D4+ are A(x)−1 and X_A(x) in the contour integral formula, it is natural to relate the D0- and D4- branes with A(x) and X_A(x)−1. Therefore, the contour integral formula for the supergroup analog of the spiked instanton system should be proportional to

$$ \oint \prod_{A \in \mathbf{6}} \prod_{I=1}^{k_A^+} \prod_{J=1}^{k_A^-} \frac{dx_{A,I}^+}{2\pi i x_{A,I}^+} \frac{dx_{A,J}^-}{2\pi i x_{A,J}^-} \langle \mathbf{A}_{\underline{k}_+}^{-1} \mathbf{A}_{\underline{k}_-} \mathbf{X}_{\underline{n}_+ | \underline{n}_-} \rangle \quad (4.39) $$

where

$$ \mathbf{A}_{\underline{k}_+}^{-1} = \prod_{A \in \mathbf{6}} \prod_{I=1}^{k_{A,+}} \mathbf{A}(x_{A,I}^{+})^{-1}, \quad \mathbf{A}_{\underline{k}_-} = \prod_{A \in \mathbf{6}} \prod_{J=1}^{k_{A,-}} \mathbf{A}(x_{A,J}^{-}), \quad \mathbf{X}_{\underline{n}_+ | \underline{n}_-} := \prod_{A \in \mathbf{6}} \frac{\prod_{\alpha=1}^{n_A^+} \mathbf{X}_A(v_{A,\alpha}^+)}{\prod_{\beta=1}^{n_A^-} \mathbf{X}_A(v_{A,\beta}^-)} :. \quad (4.40) $$
This complete formula indeed reproduces the contour integral formula in (D.3) for the affine quiver supergroup gauge theory case. By evaluating the poles properly, we expect this gives the supergroup analog of the gauge origami system.

## 4.4 Coupled vortex system

Similar to the previous cases, let us consider the D2 coupled vortex system in (3.155). We introduce an operator corresponding to the D2-brane wrapping $\mathcal{C}_a \times \mathbb{S}^1(a \in \mathbf{4})$ as

$$S_a(x) = s_{a,0}(x) : \exp \left( \sum_{n \neq 0} s_{a,n} x^{-n} \right) : , \quad [s_{a,n}, s_{b,m}] = -\frac{1}{n} \frac{\mathbf{P}_{\bar{a}}^{[n]}}{\mathbf{P}_a^{[-n]} \mathbf{P}_b^{[n]}} \delta_{n+m,0}, \quad (4.41)$$

where $s_{a,0}(x)$ is the zero modes and the relation with the A-operator is

$$[a_n, s_{a,m}] = -\frac{1}{n} \mathbf{P}_{\bar{a}}^{[n]} \delta_{n+m,0}, \quad s_{a,n} = \frac{a_n}{\mathbf{P}_a^{[-n]}}. \quad (4.42)$$

The operator product formulas are

$$\begin{align}
\overline{A(x)S_a(x')} &= \overline{a_0(x)s_{a,0}(x')}g_{\bar{a}}(x'/x)^{-1} : A(x)S_a(x') : , \tag{4.43} \\
S_a(x')\overline{A(x)} &= s_{a,0}(x')\overline{a_0(x)}g_{\bar{a}}(q_a x/x') : A(x)S_a(x') : , \nonumber
\end{align}$$

where we impose the zero modes condition as

$$\overline{a_0(x)s_{a,0}(x')} = \overline{s_{a,0}(x')a_0(x)}. \quad (4.44)$$

so that the rational function arising on the right-hand side will be the same after using (3.69). Explicitly, we have $\overline{a_0(x)s_{a,0}(x')} = 1$ (see section 4.5). Under this condition, the contour integral in (3.155) is written as follows.

**Proposition 4.5.** The contour integral formula for the coupled vortex system shown in (3.155) is given by the following correlation function of the vertex operators,

$$q^{\underline{k}} \mathcal{Z}_{\underline{k}}^{\mathrm{D}2} = \frac{q^{\underline{k}} \mathcal{G}^{\underline{k}}}{\underline{k}!} \oint \prod_{a \in \mathbf{4}} \prod_{I=1}^{k_a} \frac{dx_{a,I}}{2\pi i x_{a,I}} \langle \mathbf{A}_{\underline{k}}^{-1} \mathbf{S}_{\underline{n}} \rangle, \quad (4.45)$$

where

$$q^{\underline{k}} = \prod_{a \in \mathbf{4}} q^{k_a}, \quad A_{\underline{k}} = \prod_{a \in \mathbf{4}} \prod_{I=1}^{k_a} A(x_{a,I}), \quad S_{\underline{n}} := \prod_{a \in \mathbf{4}} \prod_{\alpha=1}^{n_a} S_a(v_{a,\alpha}) : , \quad (4.46a)$$

$$\langle \mathbf{A}_{\underline{k}}^{-1} \mathbf{S}_{\underline{n}} \rangle = \prod_{a,b \in \mathbf{4}} \prod_{\alpha=1}^{n_a} \prod_{I=1}^{k_b} g_{\bar{a}} \left( \frac{v_{a,\alpha}}{x_{b,I}} \right) \prod_{(a,I) < (b,J)} \mathcal{A}_{\mathbb{C}^4} \left( \frac{x_{a,I}}{x_{b,J}} \right)^{-1}. \quad (4.46b)$$

The one-loop perturbative part in (3.168) is obtained from the S-operators:

$$S_b(v_{b,\beta})S_a(x_{a,\alpha}) = \mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D}2-\mathrm{D}2}(v_{a,\alpha}, a | v_{b,\beta}, b) : S_b(v_{b,\beta})S_a(x_{a,\alpha}) : . \quad (4.47)$$
The D2-brane S-operators are related to the screening currents of quiver W-algebras [60].
Let us consider the two screening currents S₁(x) and S₂(x). Focusing on S₂(x), we have

$$
[\mathbf{s}_{2,n}, \mathbf{s}_{2,m}] = -\frac{1}{n} \frac{1 - q_1^n}{1 - q_2^{-n}} (1 - q_3^n)(1 - q_4^n) \delta_{n+m,0} \quad (4.48)
$$

which gives one of the screening currents of the affine quiver W-algebra [60, eq. (3.33)].
The screening current S₁(x) gives the other screening current. The other two screening
currents S₃(x), S₄(x) are introduced in a symmetric way using the quadrality. Thus, using
two screening currents Sₐ(x), Sₛ(x') (a ≠ b) we will obtain six affine quiver W-algebras.

Remark 4.6. Observing the operators (4.6), (4.15), (4.29), (4.41), one can see that the D2, D4, D6, D8 vertex operators S<sub>a</sub>(x), X<sub>A</sub>(x), W<sub>ā</sub>(x), Z(x) are all related with the D0-brane operator in a similar way as

$$
\mathcal{A}_n = \frac{\mathbf{a}_n}{\mathbf{P}_S^{[-n]}}, \quad \mathcal{A}_n = \mathbf{s}_{a,n}, \quad \mathbf{x}_{A,n}, \quad \mathbf{w}_{a\bar{a},n}, \quad \mathbf{z}_n \qquad (4.49)
$$

where *S* is a subset of **4** depending on the subspace the D-branes are spanning. For example,
we have the following formal expansions:

$$
\begin{align}
\mathbf{s}_{a,n} &= -q_a^n \sum_{m=0}^{\infty} q_a^{nm} \mathbf{a}_n, \quad |q_a| < 1, \nonumber \\
\mathbf{x}_{ab,n} &= q_{ab}^n \sum_{l,m=0}^{\infty} q_a^{ln} q_b^{mn} \mathbf{a}_n, \quad |q_{ab}| < 1, \tag{4.50} \\
\mathbf{w}_{abc,n} &= -q_{abc}^n \sum_{k,l,m=0}^{\infty} q_a^{kn} q_b^{ln} q_c^{mn} \mathbf{a}_n, \quad |q_{abc}| < 1, \nonumber
\end{align}
$$

where we used

$$
\frac{1}{1 - q_a} = \begin{cases} \sum_{l=0}^{\infty} q_a^l, & |q_a| < 1, \\ -q_a^{-1} \sum_{l=0}^{\infty} q_a^{-l}, & |q_a| > 1. \end{cases} \tag{4.51}
$$

Note that we need to be careful of the analytic region of the *q*-parameters when doing such
formal expansions. Due to the condition $q_1q_2q_3q_4 = 1$, we can only impose at most three
parameters as $|q_a| < 1$ and thus, the $z_n$ will be expanded as

$$
z_n = -q_{123}^n \sum_{j,k,l,m=0}^{\infty} q_1^{jn} q_2^{kn} q_3^{ln} q_4^{-mn} a_n, \quad |q_{1,2,3}| < 1, \quad |q_4| > 1. \quad (4.52)
$$

Other formal expansions in different analytic regions are obtained using (4.51).

4.5 Zero modes conditions

Let us impose some conditions on the zero modes **a**₀(x), **z**̃<sub>0</sub><sup>K</sup>(x), **w**<sub>ā,0</sub>(x), **x**<sub>A,0</sub>(x), **s**<sub>a,0</sub>(x) and determine the free field realizations of them. Using the observation (4.49), we can see that the operator product with operators associated with D-branes intersecting only at a point will
give rational functions. We impose the zero modes so that the operator product will be the same rational functions after the analytic continuation. For the cases, when the A-operator involves, the conditions are given in (4.5), (4.11), (4.18), (4.32), (4.44). For the cases when the S-operators involve, we impose the following conditions:

• D2_a-D2_b (a ≠ b):

$$S_a(x)S_b(x') = \overline{s_{a,0}(x)s_{b,0}(x')S_{\bar{a}\bar{b}}(q_a x'/x)} : S_a(x)S_b(x') : , \quad (4.53a)$$

$$S_b(x')S_a(x) = \overline{s_{b,0}(x')s_{a,0}(x)} S_{\bar{a}\bar{b}}(q_b x'/x) : S_a(x)S_b(x') : \quad (4.53b)$$

which gives the zero mode condition

$$s_{a,0}(x)s_{b,0}(x') = \overline{s_{b,0}(x')s_{a,0}(x)} , \quad a \neq b \quad (4.54)$$

• D4_A-D2_c (c, d ∈ A):

$$X_A(x)S_c(x') = \overline{x_{A,0}(x)s_{c,0}(x')} \frac{1 - q_A x'/x}{1 - q_A q_d x'/x} : X_A(x)S_c(x') : , \quad (4.55a)$$

$$S_c(x')X_A(x) = \overline{s_{c,0}(x')x_{A,0}(x)} \frac{1 - q_A^{-1} x/x'}{1 - q_A^{-1} q_d^{-1} x/x'} : X_A(x)S_c(x') : \quad (4.55b)$$

which gives the zero mode condition

$$\overline{x_{A,0}(x)s_{c,0}(x')} = q_A^{-1} q_c^{-1} \overline{s_{c,0}(x')} \overline{x_{A,0}(x)} \quad (4.56)$$

• D2_a-D6_a:

$$W_{\bar{a}}(x)S_a(x') = \overline{w_{\bar{a},0}(x)s_{a,0}(x')} \frac{1}{1 - q_a^{-1} x'/x} : W_{\bar{a}}(x)S_a(x') : , \quad (4.57a)$$

$$S_a(x')W_{\bar{a}}(x) = \overline{s_{a,0}(x')w_{\bar{a},0}(x)} \frac{1}{1 - q_a x/x'} : W_{\bar{a}}(x)S_a(x') : \quad (4.57b)$$

which gives the zero mode condition

$$\overline{w_{\bar{a},0}(x)s_{a,0}(x')} = \left(-\frac{x'}{q_a x}\right) \overline{s_{a,0}(x')w_{\bar{a},0}(x)} \quad (4.58)$$

We can do the same analysis for the D4-brane operators as

$$X_A(x)X_{\bar{A}}(x') = \overline{x_{A,0}(x)\overline{x_{\bar{A},0}(x')} \left(1 - q_A \frac{x'}{x}\right)} : X_A(x)X_{\bar{A}}(x') : , \quad (4.59a)$$

$$X_{\bar{A}}(x')X_A(x) = \overline{x_{\bar{A},0}(x')x_{A,0}(x)} \left(1 - q_{\bar{A}} \frac{x}{x'}\right) : X_A(x)X_{\bar{A}}(x') : \quad (4.59b)$$

and impose the condition

$$x_{A,0}(x)\overline{x_{\bar{A},0}(x')} = \left(-q_A^{-1} \frac{x}{x'}\right) \overline{x_{\bar{A},0}(x')} \overline{x_{A,0}(x)} \quad (4.60)$$
but to make the discussion simple, we do not impose this condition.<fnref n="19" />

Under these conditions, the free field realizations of the zero modes are given as

$$ \mathbf{a}_0(x) = e^{\mathbf{a}_0}, \quad \mathbf{s}_{a,0}(x) = x^{\mathbf{s}_{a,0}} e^{\tilde{\mathbf{s}}_{a,0}}, \quad \mathbf{w}_{\bar{a}}(x) = x^{\mathbf{w}_{\bar{a},0}} e^{\tilde{\mathbf{w}}_{\bar{a},0}} e^{\tilde{\mathbf{w}}_{\bar{a},0}}, \qquad (4.61a) $$

$$ \chi_{A,0}(x) = e^{\chi_{A,0}}, \quad \tilde{z}_0^K(x) = x^{z_0^K} e^{\tilde{z}_0^K} e^{\tilde{z}_0^K} \qquad (4.61b) $$

with

$$ \mathbf{a}_0 = \mathbf{t}_0, \quad \mathbf{s}_{a,0} = -(\log q_a)^{-1} \mathbf{t}_0, \quad \tilde{\mathbf{s}}_{a,0} = -(\log q_a)^{-1} \tilde{\mathbf{t}}_0, \qquad (4.62a) $$

$$ \mathbf{w}_{\bar{a},0} = -\log q_a \tilde{\mathbf{t}}_0, \quad \tilde{\mathbf{w}}_{\bar{a},0} = -\log q_a \log(-q_a) \tilde{\mathbf{t}}_0, \quad \tilde{\tilde{\mathbf{w}}}_{\bar{a},0} = -\log q_a \partial_t, \qquad (4.62b) $$

$$ \chi_{A,0} = \log q_c \log q_d \tilde{\mathbf{t}}_0, \quad (\bar{A} = cd), \qquad (4.62c) $$

$$ \mathbf{z}_0^K = -\log K \tilde{\mathbf{t}}_0, \quad \tilde{\mathbf{z}}_0^K = -\log K \log(-K) \tilde{\mathbf{t}}_0, \quad \tilde{\tilde{\mathbf{z}}}_0^K = -\log K \partial_t \qquad (4.62d) $$

where we introduced two independent sets of zero modes

$$ [\partial_t, \mathbf{t}_0] = [\tilde{\partial}_t, \tilde{\mathbf{t}}_0] = 1, \quad [\mathbf{t}_0, \tilde{\mathbf{t}}_0] = [\partial_t, \tilde{\partial}] = [\mathbf{t}_0, \tilde{\partial}_t] = [\tilde{\mathbf{t}}_0, \partial_t] = 0. \qquad (4.63) $$

The normal ordering is defined as

$$ : \partial_t \mathbf{t}_0 := \mathbf{t}_0 \partial_t, \quad : \tilde{\partial}_t \tilde{\mathbf{t}}_0 := \tilde{\mathbf{t}}_0 \tilde{\partial}_t. \qquad (4.64) $$

See Lem. C.1 for the proof that the above explicit form of zero modes obeys the zero modes conditions.

Under the above conditions, we actually have extra relations

$$ \mathbf{a}_0(x) := \frac{\mathbf{s}_{a,0}(x)}{\mathbf{s}_{a,0}(q_a x)}; \quad \mathbf{x}_{ab,0}(x) := \frac{\mathbf{w}_{abc,0}(x)}{\mathbf{w}_{abc,0}(q_c x)}; \quad \mathbf{w}_{\bar{a},0}(x) = \tilde{\mathbf{z}}_0^{q_a}(x) \qquad (4.65) $$

which imply

$$ A(x) := \frac{S_a(x)}{S_a(q_a x)}; \quad X_{ab}(x) := \frac{W_{abc}(x)}{W_{abc}(q_c x)}; \quad W_{\bar{a}}(x) := \frac{Z(x)}{Z(q_a x)} := \tilde{Z}^{q_a}(x). \qquad (4.66) $$

See Cor. C.2 for the derivation of these relations. Note that in our notation, the relation between the D2 and D4 operators are

$$ S_a(x) = s_{a,0}(x) : \frac{X_{ab}(x)}{X_{ab}(q_b x)} : \qquad (4.67) $$

where extra zero modes appear in front. Using these relations, we also have

$$ A(x) = \mathbf{a}_0(x) : \frac{X_{ab}(x)X_{ab}(q_a x)}{X_{ab}(q_a x)X_{ab}(q_b x)} :; \qquad (4.68a) $$

$$ = \mathbf{a}_0(x) : \frac{W_{abc}(x)W_{abc}(q_a x)W_{abc}(q_b x)W_{abc}(q_c x)}{W_{abc}(q_a x)W_{abc}(q_b x)W_{abc}(q_c x)W_{abc}(q_a x)} : \qquad (4.68b) $$

$$ = \mathbf{a}_0(x) : \frac{Z(x)^2 \prod_{a<b} Z(q_a x)}{\prod_{a \in \mathfrak{I}} Z(q_a x) \prod_{a \in \mathfrak{I}} Z(q_a^{-1} x)} : \qquad (4.68c) $$

The contraction formulas of the D-brane vertex operators obtained after using the explicit form of the zero modes are summarized in Prop. C.3

<footnote n="19">¹⁹This condition only affects the zero modes when we are considering the quadratic relations of the $\overline{q\bar{q}}$-characters which will be derived in section 6.5.</footnote>
## 4.6 Quiver structure and generalizations

The vertex operators $A(x), S_a(x), X_A(x), W_{\bar{a}}(x), Z(x)$ are all defined by the commutation relations with the root current $A(x)$. Let us show that we can understand the commutation relations using graded quivers [106–109] (see also the references therein) and that the commutation relations have a $q$-Cartan matrix understanding. We also briefly discuss how to generalize our construction to other complicated geometries.

### 4.6.1 $\mathbb{C}^4$ geometry

We first denote $P_a, P_A, P_{\bar{a}}, P_4$ ($a \in \mathfrak{q}, A \in \mathfrak{b}$) as $c_a, c_A, c_{\bar{a}}, c_{1234}$ and call them the full $q$-Cartan matrix:

$$ \begin{aligned} c_a &= 1 - q_a, & c_{ab} &= (1 - q_a)(1 - q_b), & c_{abc} &= (1 - q_a)(1 - q_b)(1 - q_c), \\ c_{1234} &= (1 - q_1)(1 - q_2)(1 - q_3)(1 - q_4). \end{aligned} \qquad (4.69) $$

The full $q$-Cartan matrix can be decomposed into the half $q$-Cartan matrix $c_{a,A,\bar{a},1234}^+$ as

$$ \begin{aligned} c_a &= c_a^+ + c_a^-, & c_a^+ &= 1, & c_a^- &= -q_a c_a^{+\vee}, \\ c_{ab} &= c_{ab}^+ + c_{ab}^-, & c_{ab}^+ &= 1 - q_a, & c_{ab}^- &= q_a b c_{ab}^{+\vee}, \\ c_{abc} &= c_{abc}^+ + c_{abc}^-, & c_{abc}^+ &= 1 - q_a - q_b - q_c, & c_{abc}^- &= -q_a q_b q_c c_{abc}^{+\vee}, \\ c_{1234} &= c_{1234}^+ + c_{1234}^-, & c_{1234}^+ &= 1 - \sum_{a \in \mathfrak{q}} q_a + q_4 \sum_{i=1}^3 q_i, & c_{1234}^- &= c_{1234}^{+\vee}. \end{aligned} \qquad (4.70) $$

We can write this decomposition in a compact form as

$$ c_S = P_S = c_S^+ + c_S^-, \quad c_S^- = \prod_{i \in S} (-q_i) c_S^{+\vee}, \quad S \subseteq \mathfrak{q}. \qquad (4.71) $$

Note that the decomposition is not unique and is related to how we choose the square root part of the index as what we did in section 3.2. Since each vertex operator corresponds to the D-brane wrapping subspaces $C_a, C_A^2, C_A^3, C^4$, the $q$-Cartan matrices should include the information of these geometries. Let us show that the $q$-Cartan matrices can be read from quivers dual to each of the geometry $C_a, C_A^2, C_A^3, C^4$. To show this, let us first introduce the concept of graded quivers following the convention of [108, 109, section 2].

**Graded quivers.** A quiver is denoted $Q = (Q_0, Q_1)$, where $Q_0$ is a set of nodes and $Q_1$ is a set of arrows. For each arrow $I: i \to j$, $(i, j \in Q_0)$, we have the source node $s(I) = i$ and the target node $t(I) = j$.

We fix $m$ to be a nonnegative integer and denote the graded quiver as $\bar{Q} = (\bar{Q}_0, \bar{Q}_1)$. The set of nodes does not change $\bar{Q}_0 = Q_0$ but the set of arrows changes to a set of graded arrows. A graded arrow $I: i \to j$ has a source node $s(I) = i \in Q_0$, a target node $t(I) = j \in Q_0$ and additionally a degree $|I|$ which is an integer in $\{0, 1, \dots, m\}$. For every graded arrow $I \in \bar{Q}_1$, we also introduce its opposite $I_{op}: j \to i$, where the source node and target node are reversed and the degree is given $|I_{op}| = m - |I|$. Namely, we will focus on a particular kind of graded quiver, such that every arrow has an opposite arrow.
For every node $i \in \bar{Q}_0$, additionally, there is a loop $\ell_i$ with $s(\ell_i) = i$, $t(\ell_i) = i$ and degree $-1$. We also introduce the opposite of them as $\bar{\ell}_i$ with degree $m+1$. These are the only arrows with degrees not in $\{0, 1, \dots, m\}$ and will be not drawn explicitly in the quiver diagram.

For later use, we denote an arrow from $i$ to $j$ with degree $c$ as

$$ \Phi_{ij}^{(c)} : i \to j \qquad (4.72) $$

and the opposite of it as

$$ (\Phi_{ij}^{(c)})_{\text{op}} \equiv \bar{\Phi}_{ji}^{(m-c)}. \qquad (4.73) $$

We then can pair the double arrows as $(\Phi_{ij}^{(c)}, \bar{\Phi}_{ji}^{(m-c)})$ and illustrate them as

<figure><img src="image_12.png" /></figure>

Following the terminology in [108], we refer to this pair of arrows as $(c, m-c)$ arrow. The different types of arrows are then labeled by $0 \le c \le m/2$.

From the physical viewpoint, the graded quiver contains information of the components included in a quiver gauge theory. The quiver nodes are identified with the gauge groups,<fnref n="20" /> the loops with degree $-1, m+1$ are identified with vector multiples, and the arrows with degrees $\{0, 1, \dots, m\}$ are matter fields. Different degrees of arrows represent different matter fields, and thus the graded quiver enables us to describe a larger class of quiver gauge theories in various dimensions. Each arrow connecting quiver nodes $i, j$ will change under the (anti-)fundamental representation of the source and target nodes. Note that a self-loop arrow $\Phi_{ii}^{(c)}$ will be a matter multiplet transforming in the adjoint representation.

We may choose an orientation of the double arrow by drawing either $\Phi_{ij}^{(c)}$ or $\bar{\Phi}_{ji}^{(m-c)}$. In the quiver diagram, only one of them will be explicitly drawn.<fnref n="21" /> When $m$ is even, we have a pair of degree $c = m/2$ arrows $(\Phi_{ij}^{(m/2)}, \bar{\Phi}_{ji}^{(m/2)})$. These arrows will be drawn in unoriented arrows. Note again that the self-loops $\ell_i$ are not explicitly drawn in the quiver diagram.<fnref n="22" />

For later use, let us briefly explain the connection between graded quivers and supersymmetric gauge theories. Graded quivers of degree $m$ correspond to $(6-2m)$-dimensional gauge theories with $2^{3-m}$ supercharges. Consider a Type IIB string theory where $D(5-2m)$-branes probe the CY $(m+2)$-folds. The Calabi-Yau manifold arises as the classical moduli spaces of

<footnote n="20">²⁰We may assign integers $N_i$ to each quiver node and then the gauge group is identified as $U(N_i)$.</footnote>

<footnote n="21">²¹For a detailed discussion of this choice see [108, section 2].</footnote>

<footnote n="22">²²In addition to the quiver diagram, we can assign *graded quiver superpotentials* which imposes relations on the path algebra obtained from the graded quiver. When relating the quiver arrows with the *q*-deformation parameters of the *q*-Cartan matrix, they will play important roles, but in this paper, we omit the discussion of it and postpone it for future work.</footnote>
the gauge theories [110–113]. Depending on $m$, the arising gauge theory is given as

<table><thead><tr><th>m</th><th>gauge theory</th><th>geometry</th></tr></thead><tbody><tr><td>0</td><td>6d N = (1, 0) theory</td><td>D5-branes probing CY<sub>2</sub></td></tr><tr><td>1</td><td>4d N = 1 theory</td><td>D3-branes probing CY<sub>3</sub></td></tr><tr><td>2</td><td>2d N = (0, 2) theory</td><td>D1-branes probing CY<sub>4</sub></td></tr></tbody></table>

* $m=2$ case: the $m=2$ graded quivers correspond to 2d $N = (0, 2)$ quiver gauge theories. The graded arrows are described in $(c, m-c)$ double arrows with $0 \le c \le m/2$. In this case, we have $(0, 2)$ and $(1, 1)$ double arrows. From the gauge theoretic viewpoint, we have vector superfields, chiral superfields, and the Fermi superfields. The vector superfields are associated with each node of the quiver. The chiral (Fermi) superfields correspond with the degree $c=0$ ($c=1$) arrows of the graded quiver. We denote the chiral superfields as $X_{ij}$ and they are identified with the arrows as $(X_{ij}, \bar{X}_{ij}) \leftrightarrow (0, 2)$ arrow. The Fermi superfields $\Lambda_{ij}$ are identified with the arrows as $(\Lambda_{ij}, \bar{\Lambda}_{ij}) \leftrightarrow (1, 1)$ arrow. The quiver diagram is then described as

<figure><img src="image_13.png" /></figure>

The left diagram is the quiver diagram with double arrows as (4.74), while the right diagram is the quiver diagram with single arrows. The Fermi superfields are drawn in unoriented arrows because the degrees are self-conjugate. For examples of quivers having multiple nodes, see [113, 114]. Quivers of these types are denoted as $\bar{Q} = (\bar{Q}_0, \bar{Q}_1)$ in later discussions.

* $m=1$ case: the $m=1$ graded quivers correspond to 4d $N=1$ quiver gauge theories. We only have one type of double arrows which is $(0, 1)$. They correspond to the chiral superfields of the gauge theory which we denote $X_{ij}$. The quiver diagram is illustrated as

<figure><img src="image_14.png" /></figure>
Some examples are

<figure><img src="image_15.png" /></figure>

Such kind of quivers are denoted as $Q = (Q_0, Q_1)$ in later sections.

* $m=0$ case: the $m=0$ graded quivers correspond to 6d $\mathcal{N} = (1,0)$ theories. The nodes correspond to the vector supermultiplets, while the (0,0) double arrow corresponds to the hypermultiplets. The discussion for the tensor multiplets is omitted. The quiver diagram is described as

<figure><img src="image_16.png" /></figure>

where we denoted the hypermultiplets using $X_{ij}$.

Some examples of these quivers are

<figure><img src="image_17.png" /></figure>

The quivers coming from the toric Calabi-Yau two-fold are the affine A-type Dynkin quivers. Note that we can also consider affine D and E-type quivers though they are not toric. Moreover, by decoupling one of the arrows of the quivers, we can also consider finite-type A, D, and E Dynkin quivers.

These types of quivers coming from $m=0$ are denoted as $\Gamma = (\Gamma_0, \Gamma_1)$ or $\Upsilon = (\Upsilon_0, \Upsilon_1)$ in later sections.
**Gauge origami viewpoint.** The gauge origami system can be understood a 2d $\mathcal{N} = (0, 2)$ quiver gauge theory with flavor branes. Consider the type IIB theory on $Z \times \mathcal{C}$ where $Z$ is a toric $\mathrm{CY}_4$ and $\mathcal{C} = \mathbb{C}, \mathbb{C}^\times, \mathbb{T}^2$. Consider a situation where there are multiple D1-branes$^{23}$ wrapping $\mathcal{C}$ probing $Z$. The arising theory is a 2d $\mathcal{N} = (0, 2)$ quiver gauge theory. We then consider D3, D5, D7, D9-branes wrapping non-compact cycles of $Z$ while preserving two supersymmetries. These branes will play the role of flavor branes from the D1-brane theory viewpoint. On the other hand, from the viewpoint of the theory inside $Z$, we have a generalized gauge theory where the D-branes inside $Z$ intersect with each other, and the D1-branes play the role of the instantons of the theory. Depending on $\mathcal{C}$, we can study the dimensional reduction (matrix models/supersymmetric quantum mechanics/field theory), and the gauge origami partition function is the index of the system. When $Z = \mathbb{C}^4$, this is obvious from the discussion in section 3.1. Taking T-duality of the setups in (3.4), (3.20), (3.30), (3.35), for all cases, we indeed have a D1-brane theory with flavor branes wrapping the subspaces of $\mathbb{C}^4$.

**$\mathbb{C}^4$-case.** Let us consider the case when $\mathrm{CY}_4 = \mathbb{C}_{1234}^4$. The dual gauge theory is just the maximally supersymmetric Yang-Mills in 2d which is the 2d $\mathcal{N} = (8, 8)$ SYM. This is because placing the D1-brane in the transverse direction of $\mathbb{C}^4$ will give a SYM theory with 16 supercharges. Note that it can be obtained by a dimensional reduction of the 4d $\mathcal{N} = 4$ SYM. In the 2d $\mathcal{N} = (0, 2)$ language, we have one vector superfield, four adjoint chiral superfields $X_1, X_2, X_3, X_4$, and three Fermi superfields $\Lambda_1, \Lambda_2, \Lambda_3$. The quiver diagram is then described as

$$ X_1, X_2, X_3, X_4 \qquad \Lambda_1, \Lambda_2, \Lambda_3 \qquad (4.81) $$

Note that the chiral superfields $X_1, X_2, X_3, X_4$ are identified with the four complex coordinates of the transverse $\mathbb{C}_{1234}^4$.

After identifying the superfields with the commutative parameters in the $q$-Cartan matrix, we can define the half $q$-Cartan matrix of this quiver as

$$ c_{1234}^{+} = 1 - (X_1 + X_2 + X_3 + X_4) + (\Lambda_1 + \Lambda_2 + \Lambda_3). \qquad (4.82) $$

The term 1 corresponds with the vector superfield, the terms $X_i$ ($i = 1, 2, 3, 4$) correspond with the chiral superfields, and the red term $\Lambda_i$ ($i = 1, 2, 3$) corresponds with the Fermi superfields.

To relate with (4.70), we need the identification

$$ \begin{gathered} X_1 \leftrightarrow q_1, \quad X_2 \leftrightarrow q_2, \quad X_3 \leftrightarrow q_3, \quad X_4 \leftrightarrow q_4, \\ \Lambda_1 \leftrightarrow q_1 q_4, \quad \Lambda_2 \leftrightarrow q_2 q_4, \quad \Lambda_3 \leftrightarrow q_3 q_4. \end{gathered} \qquad (4.83) $$

For the conjugated fields such as $\bar{\Lambda}_1$, the parameter $q_1^{-1}q_4^{-1}$ is assigned.

Actually, this identification can be understood from the superpotential of the theory. The potential of the 2d $\mathcal{N} = (0, 2)$ theories takes the form

$$ W = \sum_a \left( \Lambda_a J_a(X) + \bar{\Lambda}_a E_a(x) \right) \qquad (4.84) $$

<footnote>$^{23}$We may add fractional D1-branes to make the gauge group for each quiver node differ.</footnote>
where a runs all the Fermi fields. For the $\mathbb{C}^4$ case, the $J$ and $E$-terms are

$$ \begin{array}{l@{\qquad}l} J & E \\ \Lambda_1: X_2X_3 - X_3X_2 & X_4X_1 - X_1X_4 \\ \Lambda_2: X_3X_1 - X_1X_3 & X_4X_2 - X_2X_4 \\ \Lambda_3: X_1X_2 - X_2X_1 & X_4X_3 - X_3X_4. \end{array} \qquad (4.85) $$

The vacuum of this potential is obtained from the vanishing $J$ and $E$-terms which come from $\partial W/\partial \Lambda_a = 0$. This is called the F-term condition.

Physically, the parameters $q_i$ ($i = 1, 2, 3, 4$) are identified with the $U(1)^4 \subset \text{Spin}(8)$ rotational symmetries of the $\mathbb{C}^4$. For the vacuum defined from $J_a = E_a = 0$ to be invariant under the rotational symmetry, we need $q_iq_j = q_jq_i$. Each monomial term of the $W$ also needs to be invariant under this symmetry. For example, from $\Lambda_1 X_2 X_3$ we can see that the charge of $\Lambda_1$ should be $q_2^{-1}q_3^{-1}$. Then, the charge of $\bar{\Lambda}_1$ will be $q_2q_3$ and for the term $\bar{\Lambda}_1 X_4 X_1$ coming from the $E$-term to be invariant, we need $q_2q_3q_1q_4 = 1$. Other terms will give the same condition and this is the Calabi-Yau condition imposed on the $q$-deformation parameters.

The other half $q$-Cartan is obtained as

$$ c_{1234} = 1 - \sum_{a \in \mathcal{A}} q_a^{-1} + q_4^{-1} \sum_{i=1}^{3} q_i^{-1}. \qquad (4.86) $$

Obviously, looking at the parameters assigned, we can see that the second term corresponds with the conjugate of the chiral superfields, while the third term corresponds with the conjugate of the Fermi superfields. Therefore, we can say that the total $q$-Cartan matrix $c_{1234} = c_{1234}^{+} + c_{1234}^{-}$ is associated with the graded quiver illustrated in the double arrows notation as in (4.74).

**$\mathbb{C}^3$-case.** Let us move on to the $\mathbb{C}^3$-case. The dual gauge theory is the maximally supersymmetric Yang-Mills in 4d which is the 4d $\mathcal{N} = 4$ theory. In the 4d $\mathcal{N} = 1$ language, there are one vector superfield and three adjoint chiral superfields $X'_1, X'_2, X'_3$. The quiver diagram is described as

$$ X'_1, X'_2, X'_3 \qquad (4.87) $$

Given this quiver diagram, we can read the half $q$-Cartan matrix as

$$ c_{\mathbb{C}^3}^{+} = 1 - (X'_{1} + X'_{2} + X'_{3}). \qquad (4.88) $$

The three chiral superfields are identified with the three complex coordinates of the transverse $\mathbb{C}^3$. Depending on which subspace $\mathbb{C}_a^3$ ($a = 1, 2, 3, 4$) we are considering, the identification with the $q$-deformation parameters will be different:

$$ \begin{array}{l@{\qquad}l} \mathbb{C}^3 = \mathbb{C}_{123}^3: & c_{123}^{+} = 1 - (q_1 + q_2 + q_3), \\ \mathbb{C}^3 = \mathbb{C}_{124}^3: & c_{124}^{+} = 1 - (q_1 + q_2 + q_4), \\ \mathbb{C}^3 = \mathbb{C}_{134}^3: & c_{134}^{+} = 1 - (q_1 + q_3 + q_4), \\ \mathbb{C}^3 = \mathbb{C}_{234}^3: & c_{234}^{+} = 1 - (q_2 + q_3 + q_4). \end{array} \qquad (4.89) $$
Since, in the gauge origami system, we are not imposing the Calabi-Yau condition on $\mathbb{C}^3$ but only on $\mathbb{C}^4$, the three $q$-deformation parameters of the half $q$-Cartan matrix are independent. In other words, we start from $\mathbb{C}^4$ and impose the CY condition only on the 2d $\mathcal{N} = (0, 2)$ graded quiver. The 4d $\mathcal{N} = 1$ quivers are understood as a subquiver of the $\mathrm{CY}_4$ quiver and with $q$-parameters associated. We then read the $q$-Cartan matrix from it.

The other half $q$-Cartan matrix and the total $q$-Cartan matrix can be understood similarly but we need to be careful about how we assign the $U(1)$ charges since we are not imposing the CY condition on $\mathbb{C}^3$. We will not discuss it in this paper. A detailed discussion of this part will be done in [74].

Let us comment on what will happen when we impose the CY condition on $\mathbb{C}_{123}^3$. The superpotential of the theory is given by

$$W = X'_1[X'_2, X'_3]. \qquad (4.90)$$

After identifying the chiral superfields as $X'_i \leftrightarrow q_i$ ($i = 1, 2, 3$), a similar analysis as the 2d case shows that we need the conditions $q_iq_j = q_jq_i$ and $q_1q_2q_3 = 1$. Therefore, we only have two independent parameters. Actually, such kind of parameter assignment was done similarly in [32, 115] where the authors defined the quiver Yangian using melting crystal methods [57, 116–119] and brane tiling techniques [57, 110, 112, 120–122]. Discussion on the relationship with such algebras will be discussed in section 9.

**$\mathbb{C}^2$-case.** The dual gauge theory is the 6d $\mathcal{N} = (1, 0)$ SYM. The quiver is a single node with a single unoriented arrow. The node corresponds to the vector multiplet, while the unoriented arrow corresponds to the hypermultiplet denoted as $\Phi$:

$$\Phi \quad (4.91)$$

The hypermultiplet corresponds to either the two complex coordinates of the transverse $\mathbb{C}^2$. Let us consider the case that the transverse geometry is $\mathbb{C}_{12}^2$. We then can identify the half $q$-Cartan matrix as

$$c_{12}^{+} = 1 - q_{1}. \qquad (4.92)$$

Note for this case, since the arrow is unoriented, it is arbitrary to choose either $c_{12}^{+} = 1 - q_1$ or $c_{12}^{+} = 1 - q_2$ to be the half $q$-Cartan matrix.

Depending on which subspace $\mathbb{C}_A^2$ ($A \in \mathbf{6}$) we are considering, we have the following six types of $q$-Cartan matrices:

$$\mathbb{C}^2 = \mathbb{C}_{ab}^2 : \quad c_{ab}^{+} = 1 - q_a, \qquad (4.93)$$

where we chose one of the indices to define the $q$-Cartan matrix.

We also note that when we impose the Calabi-Yau condition on the $\mathbb{C}^2$ part, we have only one deformation parameter and the total $q$-Cartan matrix is $c = (1 - q)(1 - q^{-1})$. This corresponds to the unrefined limit in the context of 4d (5d) gauge theory.
The half $q$-Cartan matrix is then given as

$$c_{Z,ij}^{+} = \delta_{ij} - \sum_{I \in \{j \to i\}} q_{I}^{(0)} + \sum_{I \in \{j \to i\}} q_{I}^{(1)}. \qquad (4.97)$$

The total $q$-Cartan matrix will be

$$c_{Z,ij} = 2\delta_{ij} - \sum_{I \in \{j \to i\}} q_I^{(0)} + \sum_{I \in \{j \to i\}} q_I^{(1)} \\ - \sum_{I \in \{i \to j\}} q_I^{(0)-1} + \sum_{I \in \{i \to j\}} q_I^{(1)-1} \qquad (4.98)$$

which obeys $c_{Z,ij}^{\vee} = c_{Z,ji}$. We can also obtain the structure function as

$$A_{Z,ij}(x) = \mathbb{I}[-c_{Z,ij}^{\vee}x^{\vee}] = \frac{\prod_{I \in \{j \to i\}} (1 - q_I^{(0)}x) \prod_{I \in \{i \to j\}} (1 - q_I^{(0)-1}x)}{(1 - x)^{2\delta_{ij}} \prod_{I \in \{j \to i\}} (1 - q_I^{(1)}x) \prod_{I \in \{i \to j\}} (1 - q_I^{(1)-1}x)}. \qquad (4.99)$$

Let us move on to the operator formalism. When considering general toric $\mathrm{CY}_4$, we have multiple quiver nodes, and thus the operators will be labeled by the quiver nodes. Based on the discussion of the $\mathbb{C}^4$ case, the D0-brane operator (root current) is defined as

$$A_i(x) = a_{i,0}(x) : \exp \left( \sum_{n \neq 0} a_{i,n} x^{-n} \right) : , \quad [a_{i,n}, a_{j,m}] = -\frac{1}{n} \delta_{n+m,0} c_{Z,ij}^{[n]}, \qquad (4.100)$$

where $i \in \bar{Q}_0$. We can also introduce a D8-brane wrapping $\mathrm{CY}_4$ whose corresponding vertex operator is defined as

$$Z_i(x) = z_{i,0}(x) : \exp \left( \sum_{n \neq 0} z_{i,n} x^{-n} \right) : , \quad a_{i,n} = \sum_{j \in \bar{Q}_0} z_{j,n} c_{Z,ji}^{[-n]}. \qquad (4.101)$$

We omit the discussion of the zero-modes. A detailed discussion will be done in a future publication.

The operator product formulas are given as

$$\begin{align}
A_i(x)A_j(x') &= \overline{a_{i,0}(x)a_{j,0}(x')} A_{Z,ij} (x'/x)^{-1} : A_i(x)A_j(x') : , \nonumber \\
Z_i(x)A_j(x') &= \overline{z_{i,0}(x)a_{j,0}(x')} (1 - x'/x)^{\delta_{ij}} : Z_i(x)A_j(x') : , \tag{4.102} \\
A_j(x')Z_i(x) &= \overline{a_{j,0}(x')z_{i,0}(x')} (1 - x/x')^{\delta_{ij}} : Z_i(x)A_j(x') : . \nonumber
\end{align}$$

Using them, we can define a similar contour integral formula as the magnificent four system which we propose to be the generalization of the magnificent four system to general toric $\mathrm{CY}_4$.

**Conjecture 4.7** ([74]). Let $Z$ be a toric Calabi-Yau four-fold. We denote the corresponding graded quiver as $\bar{Q} = (\bar{Q}_0, \bar{Q}_1)$ and the associated $q$-deformation parameters as $\{q_I\}_{I \in \bar{Q}_1}$. We then have the following.
1. After imposing suitable conditions, the q-deformation parameters reduce up to three independent parameters.

2. The corresponding D0 and D8-brane operators are defined as in (4.100) and (4.101).

3. The partition function where there are D8-branes wrapping the entire Z is proportional to

$$ \frac{1}{\underline{k}!} \oint \prod_{i \in \bar{Q}_0} \prod_{I=1}^{k_i} \frac{dx_{i,I}}{2\pi i} \langle A_{\underline{k}}^{-1} Z_{\underline{n}} \rangle \qquad (4.103) $$

where

$$ A_{\underline{k}} = \prod_{i \in \bar{Q}_0} \prod_{I=1}^{k_i} A_i(x_{i,I}), \quad Z_{\underline{n}} := \prod_{i \in \bar{Q}_0} \prod_{\alpha=1}^{n_i} Z_i(v_{i,\alpha}) :. \qquad (4.104) $$

This gives the BPS/CFT correspondence of this setup.

4. The poles are classified by 4d analogs of the 3d BPS crystals [57].

Following the discussion of the $\mathbb{C}^4$-case, one would like to study the gauge origami system where lower dimensional D-branes appear. To do this practically, we need to study the subspaces of the toric $\mathrm{CY}_4$ case-by-case. However, for special cases when the total $\mathrm{CY}_4$ is decomposed into a product of smaller spaces, we can give a general discussion. These will be studied in the following subsections.

We note that there is another way to construct the magnificent four system where Z is general toric $\mathrm{CY}_4$. One can understand Z as a combination of $\mathbb{C}^4$ patches. The partition function of Z is understood by gluing the results of $\mathbb{C}^4$. Studies from this approach were done in [76, 80] where the authors introduced a vertex formalism. The relation with our construction here is not so clear for the moment.

**Remark 4.8.** We expect the above conjecture is also applicable to non-toric Calabi-Yau four-folds such as the non-Abelian orbifolds. Given the quiver and proper parameter association, we can define the q-Cartan matrix and the contour integral formula, but the evaluation of the poles might be non-trivial.

### 4.6.3 $\mathrm{CY}_3 \times \mathbb{C}$

Consider the Calabi-Yau four-fold with the form $Z = X \times \mathbb{C}$, where X is a toric CY three-fold (see also [54, 76, 127]). We denote the corresponding quiver$^{25}$ of X as $Q = (Q_0, Q_1)$. For each arrow $I \in Q_1$ of the quiver diagram, we assign a q-deformation parameter $q_I$. Given the geometry, the F-term condition of the superpotential determines the relations between the q-deformation parameters. After this condition, the number of independent parameters will be three. In this paper, we will not give the procedure to determine them. Practically, we need to study case by case.

We denote the corresponding q-Cartan matrix as

$$ c_{Z,ij} = c_{X,ij}(1 - q_4), \quad c_{X,ij} = c_{X,ij}^{+} + c_{X,ij}^{-}, \quad c_{X,ij}^{-} = -q_4^{-1} c_{X,ji}^{+\vee}, \qquad (4.105) $$

<footnote>$^{25}$Strictly speaking, the quiver here should be the quiver of the entire CY four-fold Z. The quiver of X is understood as a subquiver of Z. However, at the level of q-Cartan matrix, we only need to take the product of the q-Cartan matrix associated with X and $\mathbb{C}$, so we use the quiver of X which is associated with 4d $\mathcal{N} = 1$ theories.</footnote>
where

$$c_{X,ij}^{+} = \delta_{ij} - \sum_{I \in \{j \to i\}} q_{I}. \qquad (4.106)$$

We are assuming that the q-deformation parameters satisfy the F-term conditions. Note also that we have the property $c_{Z,ij} = c_{Z,ji}^{\vee}$ which is the reality condition of the q-Cartan matrix. The structure functions associated with Z and X are

$$ \begin{aligned} \mathcal{A}_{Z,ij}(x) &= \mathbb{I}[-c_{Z,ij}^{\vee}x^{\vee}] = \frac{(1-q_4x)^{\delta_{ij}}(1-q_4^{-1}x)^{\delta_{ij}} \prod_{I \in \{j \to i\}} (1-q_I x) \prod_{I \in \{i \to j\}} (1-q_I^{-1} x)}{(1-x)^{2\delta_{ij}} \prod_{I \in \{i \to j\}} (1-q_I^{-1} q_4^{-1} x) \prod_{I \in \{j \to i\}} (1-q_4 q_I x)}, \\ \varphi_{X,ij}(x) &= \mathbb{I}[-c_{X,ij}^{\vee}x^{\vee}] = \frac{(1-q_4^{-1}x)^{\delta_{ij}} \prod_{I \in \{j \to i\}} (1-q_I x)}{(1-x)^{\delta_{ij}} \prod_{I \in \{i \to j\}} (1-q_4^{-1} q_I^{-1} x)}. \end{aligned} \qquad (4.107) $$

We then can introduce the D0-brane operators (root currents) and the D6-brane operator associated with X. The operators are labeled by the quiver nodes $i \in Q_0$ as

$$ \mathcal{A}_i(x) = \mathcal{a}_{i,0}(x) : \exp \left( \sum_{n \neq 0} \mathcal{a}_{i,n} x^{-n} \right) : , \quad \mathcal{W}_i(x) = \mathcal{w}_{i,0}(x) : \exp \left( \sum_{n \neq 0} \mathcal{w}_{i,n} x^{-n} \right) : \quad (4.108) $$

where

$$ [\mathcal{a}_{i,n}, \mathcal{a}_{j,m}] = -\frac{1}{n} \delta_{n+m,0} c_{Z,ij}^{[n]}, \quad \mathcal{a}_{i,n} = \sum_{j \in Q_0} \mathcal{w}_{j,n} c_{X,ji}^{[-n]}. \qquad (4.109) $$

For later use, we also define the screening current corresponding to the C part of Z:

$$ \mathcal{S}_i(x) = \mathcal{s}_{i,0}(x) : \exp \left( \sum_{n \neq 0} \mathcal{s}_{i,n} x^{-n} \right) : , \quad \mathcal{s}_{i,n} = \frac{\mathcal{a}_{i,n}}{1 - q_4^{-n}}. \qquad (4.110) $$

The zero-modes need to obey similar conditions as what we did for the $\mathbb{C}^4$ case in section 4.5, but we do not discuss them in this paper. A detailed derivation will be done in [74].

The operator products of the operators are

$$ \begin{aligned} \mathcal{W}_i(x) \mathcal{S}_j(x') &= \overline{\mathcal{w}_{i,0}(x) \mathcal{s}_j(x')} \frac{1}{(1 - q_4^{-1} x' / x)^{\delta_{ij}}} : \mathcal{W}_i(x) \mathcal{S}_j(x') : , \\ \mathcal{S}_j(x') \mathcal{W}_i(x) &= \overline{\mathcal{s}_{j,0}(x') \mathcal{w}_{i,0}(x)} \frac{1}{(1 - q_4 x / x')^{\delta_{ij}}} : \mathcal{W}_i(x) \mathcal{S}_j(x') : , \\ \mathcal{W}_i(x) \mathcal{A}_j(x') &= \overline{\mathcal{w}_{i,0}(x) \mathcal{a}_{j,0}(x')} \left( \frac{1 - x' / x}{1 - q_4^{-1} x' / x} \right)^{\delta_{ij}} : \mathcal{W}_i(x) \mathcal{A}_j(x') : , \\ \mathcal{A}_j(x') \mathcal{W}_i(x) &= \overline{\mathcal{a}_{j,0}(x') \mathcal{w}_{i,0}(x)} \left( \frac{1 - x / x'}{1 - q_4 x / x'} \right)^{\delta_{ij}} : \mathcal{W}_i(x) \mathcal{A}_j(x') : , \\ \mathcal{A}_i(x) \mathcal{A}_j(x') &= \overline{\mathcal{a}_{i,0}(x') \mathcal{a}_{j,0}(x')} \mathcal{A}_{Z,ij} (x' / x)^{-1} : \mathcal{A}_i(x) \mathcal{A}_j(x') : , \\ \mathcal{A}_i(x) \mathcal{S}_j(x') &= \overline{\mathcal{a}_{i,0}(x') \mathcal{s}_{j,0}(x')} \varphi_{X,ij} (x' / x)^{-1} : \mathcal{A}_i(x) \mathcal{S}_j(x') : , \\ \mathcal{S}_j(x') \mathcal{A}_i(x) &= \overline{\mathcal{s}_{j,0}(x') \mathcal{a}_{i,0}(x')} \varphi_{X,ji} (q_4 x / x') : \mathcal{S}_j(x') \mathcal{A}_i(x) : . \end{aligned} \qquad (4.111) $$
Using this, we can define the contour integral formula that computes the partition function of the system where multiple D6-branes wrap the subspace X of the total $CY_4 Z$. Let us summarize what we have obtained as a conjecture.

**Conjecture 4.9** ([74]). Let $Z$ be the Calabi-Yau four-fold with the form $Z = X \times \mathbb{C}$, where $X$ is a toric $CY_3$. Denote the corresponding quiver as $Q = (Q_0, Q_1)$ and the associated $q$-deformation parameters as $\{q_I\}_{I \in Q_1}$. We then have the following.

1. After imposing suitable conditions, we can reparametrize the parameters and reduce them up to three independent $q$-deformation parameters.

2. The corresponding D0 and D6-brane operators are $A_i(x)$ and $W_i(x)$ defined in (4.108).

3. The partition function of the tetrahedron instanton system where multiple D6-branes wrap the X subspace is proportional to

$$ \frac{1}{\underline{k}!} \oint \prod_{i \in Q_0} \prod_{I=1}^{k_i} \frac{dx_{i,I}}{2\pi i x_{i,I}} \langle \mathbf{A}_{\underline{k}}^{-1} \mathbf{W}_{\underline{n}} \rangle \quad (4.112) $$

where

$$ \mathbf{A}_{\underline{k}} = \prod_{i \in Q_0} \prod_{I=1}^{k_i} \mathbf{A}_i(x_{i,I}), \quad \mathbf{W}_{\underline{n}} := \prod_{i \in Q_0} \prod_{\alpha=1}^{n_i} \mathbf{W}_i(v_{i,\alpha}) :. \quad (4.113) $$

Moreover, this gives the BPS/CFT correspondence of this system.

4. The poles of this contour integral formula are labeled by the 3d BPS crystals of [57, 126].

**Remark 4.10.** After taking the limit $q_4 \to 1$, assume the $q$-deformation parameters are transformed to $\{q_I\}_{I \in Q_1} \to \{q_I\}_{I \in Q_1}$. There will be only two independent parameters in $\{q_I\}_{I \in Q_1}$. Under this limit, we have

$$ \varphi_{X,ij}(x) \xrightarrow{q_4 \to 1} \varphi^{i \to j}(x) := \frac{\prod_{I \in \{j \to i\}} (1 - q_I x)}{\prod_{I \in \{i \to j\}} (1 - q_I^{-1} x)}, \quad (4.114) $$

where the right-hand side is the structure functions$^{26}$ of the toroidal quiver algebra/quiver quantum toroidal algebra introduced in [32–34, 115, 128, 129].

### 4.6.4 $CY_2 \times CY_2$

Let us next consider the case when the $CY_4 Z$ takes the form of $CY_2 \times CY_2$. The corresponding quiver structure for such geometry is decomposed into a product of two quivers: $\bar{Q} = \Upsilon \times \Gamma$. We denote the nodes and arrows of $\Gamma$, $\Upsilon$ as $\Gamma_{0,1}$, $\Upsilon_{0,1}$. Each node of $\bar{Q}$ is written as $(i, j) \in \Upsilon_0 \times \Gamma_0$. Each arrow of $\bar{Q}$ is decomposed into the following two types:

$$ I : (i, j) \to (i', j'), \quad (j = j'), \quad I : (i, j) \to (i', j'), \quad (i = i'). \quad (4.115) $$

The first arrow is identified with an element of $\Upsilon_1$ while the second arrow is identified with an element of $\Gamma_1$. Note that the quivers $\Upsilon, \Gamma$ are dual to 6d $\mathcal{N} = (1, 0)$ theories. We focus on the case when the quiver structures are affine A, D, E type and the $CY_4$ is denoted as $\mathbb{C}^2/\Upsilon \times \mathbb{C}^2/\Gamma$. See [12] for the formulas of the gauge origami partition function.

<footnote>$^{26}$The notation of the structure functions differs from the original papers. They come from how we define the explicit zero-modes of the operators.</footnote>
We denote the q-deformation parameters associated with the arrows of $\Upsilon_1, \Gamma_1$ as $\{q_I\}_{I \in \Upsilon_1 \cup \Gamma_1}$. Following the discussions of previous subsections, the q-Cartan matrix can be written as

$$ 
\begin{aligned}
c_{Z,ab} &= c_{\Upsilon,ii'}c_{\Gamma,jj'}, \quad a = (i,j), \quad b = (i',j'), \\
c_{\Upsilon,ii'} &= (1+q_{12})\delta_{ii'} - \sum_{I \in \Upsilon_1: i' \to i} q_I - \sum_{I \in \Upsilon_1: i \to i'} q_{12}q_I^{-1}, \\
c_{\Gamma,jj'} &= (1+q_{34})\delta_{jj'} - \sum_{I \in \Gamma_1: j' \to j} q_I - \sum_{I \in \Gamma_1: j \to j'} q_{34}q_I^{-1}
\end{aligned} 
\quad (4.116) $$

where the CY condition is imposed as $q_{12}q_{34} = 1$. Explicitly, we have

$$ 
\begin{aligned}
c_{Z,ab} ={}& (2+q_{12}+q_{34})\delta_{ii'}\delta_{jj'} \\
& + \sum_{\substack{I: i' \to i \\ J: j' \to j}} q_I q_J + \sum_{\substack{I: i' \to i \\ J: j' \to j'}} q_I q_J^{-1} q_{34} + \sum_{\substack{I: i \to i' \\ J: j \to j'}} q_{12} q_I^{-1} q_J + \sum_{\substack{I: i \to i' \\ J: j \to j'}} q_I^{-1} q_J^{-1} \\
& - \sum_{I: i' \to i} q_I (1+q_{34})\delta_{jj'} - \sum_{I: i \to i'} (1+q_{12})\delta_{jj'} - \sum_{I: j' \to j} q_I (1+q_{12})\delta_{ii'} - \sum_{I: j \to j'} (1+q_{34})\delta_{ii'}
\end{aligned} 
\quad (4.117) $$

Note that we have the property $c_{Z,ab}^{\vee} = c_{Z,ba}$. We also assume the other parameters $q_I$ obey the F-term conditions. The q-Cartan matrices arising here are the q-Cartan matrices of the double quiver gauge theory recently introduced in [77].

The structure functions are given as

$$ 
\begin{aligned}
\mathcal{A}_{Z,ab}(x) &= \mathbb{I}[-c_{Z,ab}^{\vee}x^{\vee}], \\
\mathcal{S}_{\Upsilon,ii'}(x) &= \mathbb{I}[-c_{\Upsilon,ii'}^{\vee}x^{\vee}] = \frac{\prod_{I: i' \to i} (1-q_I x) \prod_{I: i \to i'} (1-q_{12}q_I^{-1}x)}{(1-x)^{\delta_{ii'}} (1-q_{12}x)^{\delta_{ii'}}}, \\
\mathcal{S}_{\Gamma,jj'}(x) &= \mathbb{I}[-c_{\Gamma,jj'}^{\vee}x^{\vee}] = \frac{\prod_{I: j' \to j} (1-q_I x) \prod_{I: j \to j'} (1-q_{34}q_I^{-1}x)}{(1-x)^{\delta_{jj'}} (1-q_{34}x)^{\delta_{jj'}}}.
\end{aligned} 
\quad (4.118) $$

Let us introduce the D0 and D4-brane operators. This time, the operators will have a double index labeled by $\Upsilon_0$ and $\Gamma_0$. The D0-brane operators are introduced as

$$ \mathcal{A}_j^i(x) = \mathcal{A}_{j,0}^i(x) : \exp \left( \sum_{n \neq 0} \mathcal{A}_{j,n}^i x^{-n} \right) : , \quad [\mathcal{A}_{j,n}^i, \mathcal{A}_{j',m}^{i'}] = -\frac{1}{n} \delta_{n+m,0} c_{\Upsilon,ii'}^{[n]} c_{\Gamma,jj'}^{[n]} \quad (4.119) $$

for $i, i' \in \Upsilon_0, j, j' \in \Gamma_0$. For the D4-brane operators, we can introduce two types depending on whether they wrap either $\mathbb{C}^2/\Upsilon$ or $\mathbb{C}^2/\Gamma$:

$$ 
\begin{aligned}
X_{\Upsilon,j}^i(x) &= x_{\Upsilon,j,0}^i(x) : \exp \left( \sum_{n \neq 0} x_{\Upsilon,j,n}^i x^{-n} \right) : , \quad a_{i,n}^j = \sum_{k \in \Upsilon_0} x_{\Upsilon,i,n}^k c_{\Upsilon,kj}^{[-n]}, \\
X_{\Gamma,j}^i(x) &= x_{\Gamma,j,0}^i(x) : \exp \left( \sum_{n \neq 0} x_{\Gamma,j,n}^i x^{-n} \right) : , \quad a_{i,n}^j = \sum_{k \in \Gamma_0} x_{\Gamma,k,n}^j c_{\Gamma,ki}^{[-n]}.
\end{aligned} 
\quad (4.120) $$
The operator products are given as

$$
\begin{align}
A_j^i(x) A_{j'}^{i'}(x') &= \overline{a_{j,0}^i(x) a_{j',0}^{i'}(x')} \mathcal{A}_{Z,ab}(x'/x)^{-1} : A_j^i(x) A_{j'}^{i'}(x') : , \quad a = (i, j), b = (i', j'), \nonumber \\
A_j^i(x) X_{\Upsilon,j'}^{i'}(x') &= \overline{a_{j,0}^i(x) X_{\Upsilon,j',0}^{i'}(x')} \mathcal{S}_{\Gamma,jj'}(x'/x)^{-\delta_{ii'}} : A_j^i(x) X_{\Upsilon,j'}^{i'}(x') : , \nonumber \\
A_j^i(x) X_{\Gamma,j'}^{i'}(x') &= \overline{a_{j,0}^i(x) X_{\Gamma,j',0}^{i'}(x')} \mathcal{S}_{\Upsilon,ii'}(x'/x)^{-\delta_{jj'}} : A_j^i(x) X_{\Gamma,j'}^{i'}(x') : . \tag{4.121}
\end{align}
$$

The gauge origami system of this setup was studied for some examples in [77] and the free field realization of the contour integral formula is

$$
\frac{1}{\underline{k}!} \oint \prod_{j \in \Gamma_0} \prod_{i \in \Upsilon_0}^{k_j} \frac{dx_{j,I}^i}{2\pi i x_{j,I}^i} \langle \underline{A}_{\underline{k}}^{-1} \mathbf{X}_{\Upsilon, \underline{n}} \mathbf{X}_{\Gamma, \underline{n}'} \rangle, \quad (4.122)
$$

where

$$
A_{\underline{k}} = \prod_{i \in \Upsilon_0} \prod_{I=1}^{k_j} A_j^i(x_{j,I}^i), \quad X_{\Upsilon, \underline{n}} := \prod_{i \in \Upsilon_0} \prod_{\alpha=1}^{n_j} X_j^i(\nu_{j,\alpha}^i) : , \quad X_{\Gamma, \underline{n}'} := \prod_{i \in \Upsilon_0} \prod_{\alpha=1}^{n'_j} X_j^i(\nu'^{i}_{j,\alpha}) : . \quad (4.123)
$$

Remark 4.11. Denoting the above system as $\mathbb{C}_{12}^2/\Upsilon \times \mathbb{C}_{34}^2/\Gamma$, since we are assigning stacks of D-branes in a transverse way, the arising instantons are crossed instantons. Let us focus on the case $\Upsilon = \mathbb{Z}_p$ and $\Gamma = \mathbb{Z}_q$ whose orbifold action on the complex four coordinates are

$$
(z_1, z_2, z_3, z_4) \mapsto (\omega^{r_1} z_1, \omega^{r_2} z_2, \sigma^{s_1} z_3, \sigma^{s_2} z_4), \quad \omega = e^{2\pi i/p}, \quad \sigma = e^{2\pi i/q} \quad (4.124)
$$

where $r_1 + r_2 \equiv 0 \pmod{p}$ and $s_1 + s_2 \equiv 0 \pmod{q}$. When there are only D4-branes on the $\mathbb{C}_{12}^2/\mathbb{Z}_p$ surface, the arising instanton partition function is the $\mathbb{Z}_p$ invariant part of the partition function of the $\hat{A}_0$ quiver gauge theory. This is the same situation for the D4-branes wrapping $\mathbb{C}_{34}^2/\mathbb{Z}_q$, where the $\mathbb{Z}_q$ invariant part appears as the result. Generally, we can also consider the case where D4-branes wrap other complex two-dimensional surfaces such as $\mathbb{C}_2/\mathbb{Z}_p \times \mathbb{C}_3/\mathbb{Z}_q$, where the orbifold action mixes with each other. For example when $q = 1$ ($\mathbb{C}_2/\mathbb{Z}_p \times \mathbb{C}_3$), the arising partition function corresponds to instantons with surface operators [130]. Including such D-branes enables us to consider the most general spiked instanton setup of the entire $\mathbb{C}_{12}^2/\mathbb{Z}_p \times \mathbb{C}_{34}^2/\mathbb{Z}_q$. The D4 vertex operators appearing for this case can not be written as (4.120) but both the $q$-Cartan matrices $c_{\Upsilon,ij}$, $c_{\Gamma,ij}$ mix with each other. A detailed analysis of this will be done in [74]. We also note that starting from affine type $q$-Cartan matrices and taking a specific limit of the $q$-parameters, one can obtain the free field realizations of the contour integral formulas for finite type $q$-Cartan matrices (see [77] for examples and details).

5 D2-brane qq-characters

We introduce the screening charges and discuss the relation with the D2 *qq*-characters in section 5.1. We then use these D2 *qq*-characters to show that they reproduce the partition function of the gauge origami system of D2-branes in section 5.2.
## 5.1 D2 $\bar{q}q$-characters and screening charges

**Definition 5.1.** We define the screening charges as

$$ \Omega_a(x) = \sum_{k \in \mathbb{Z}} S_a(q_a^k x) \qquad (5.1) $$

for $a \in \mathbb{Z}$. We also call these screening charges the D2 $\bar{q}q$-characters whose motivation will be discussed later.

**Theorem 5.2.** The screening charges with different indices commute with each other:

$$ [\Omega_a(x), \Omega_b(x')] = 0, \quad a \neq b, \qquad (5.2) $$

for $a, b \in \mathbb{Z}$.

*Proof.* Let us focus on the commutation relation between $\Omega_1(x)$ and $\Omega_4(x)$. Other cases are obtained by using the quadrality. The operator products between $S_1(x)$ and $S_4(x)$ are

$$ \begin{aligned} S_4(x)S_1(x') &= S_{23}(q_4 x'/x) : S_4(x)S_1(x') : , \\ S_1(x')S_4(x) &= S_{23}(q_1 x/x') : S_1(x')S_4(x) : . \end{aligned} \qquad (5.3) $$

We then have

$$ \begin{aligned} [S_4(q_4^k x), S_1(x')] &= \frac{(1-q_2)(1-q_3)}{(1-q_2q_3)} \left( \delta\left(\frac{x'}{q_4^{k-1}x}\right) : S_4(q_4^k x)S_1(q_4^{k-1}x) : - \delta\left(\frac{x'}{q_1q_4^kx}\right) : S_4(q_4^k x)S_1(q_1q_4^kx) : \right) \\ &= \frac{(1-q_2)(1-q_3)}{(1-q_2q_3)} \left( \delta\left(\frac{x'}{q_4^{k-1}x}\right) : S_4(q_4^{k-1}x)A^{-1}(q_4^{k-1}x)S_1(q_4^{k-1}x) : - \delta\left(\frac{x'}{q_1q_4^kx}\right) : S_4(q_4^k x)S_1(q_1q_4^kx) : \right) \\ &= \frac{(1-q_2)(1-q_3)}{(1-q_2q_3)} \left( \delta\left(\frac{x'}{q_4^{k-1}x}\right) : S_4(q_4^{k-1}x)S_1(q_1q_4^{k-1}x) : - \delta\left(\frac{x'}{q_1q_4^kx}\right) : S_4(q_4^k x)S_1(q_1q_4^kx) : \right) \end{aligned} \qquad (5.4) $$

where we used

$$ A(x) := \frac{S_1(x)}{S_1(q_1 x)} := \frac{S_4(x)}{S_4(q_4 x)} :=. \qquad (5.5) $$

The commutation relation between the screening charge $\Omega_4(x)$ and $S_1(x')$ is

$$ [\Omega_4(x), S_1(x')] = \frac{(1-q_2)(1-q_3)}{(1-q_2q_3)} \sum_{k \in \mathbb{Z}} \left( \delta\left(\frac{x'}{q_4^k x}\right) - \delta\left(\frac{x'}{q_1 q_4^k x}\right) \right) : S_4(q_4^k x)S_1(q_1 q_4^k x) : \qquad (5.6) $$

which is a total difference. Thus, we finally have

$$ [\Omega_4(x), \Omega_1(x')] = 0. \qquad (5.7) $$

We call the screening charges D2-brane $\bar{q}q$-characters because each term is related by the iWeyl reflection (see (3.212) for the iWeyl reflection in terms of partition functions). We define the operator version of the iWeyl reflection in (3.212) of the D2-brane operator as the following.
**Definition 5.3.** The iWeyl reflection of the D2-brane operator is

$$S_a(x) \to : S_a(x) A^{-1}(x) : = S_a(q_a x) \qquad (5.8)$$

where we used (4.66) in the second identity.

Starting from the screening current $S_a(x)$ and using the iWeyl reflection sequentially, we get a sequence

$$S_a(x) \xrightarrow{A^{-1}(x)} S_a(q_a x) \xrightarrow{A^{-1}(q_a x)} \cdots S_a(q_a^k x) \xrightarrow{A^{-1}(q_a^k x)} \cdots \qquad (5.9)$$

which generates half of the screening charge:

$$T_a^+(x) = \sum_{k \ge 0} S_a(q_a^k x), \quad a \in \underline{\mathbb{Z}}. \qquad (5.10)$$

To define the screening charge, the above $T_a^+(x)$ is not enough and we need the other half part. It is generated by the iWeyl reflection in the opposite direction as

$$\dots \xrightarrow{A(q_a^{-l+1} x)} S_a(q_a^{-l} x) \dots \xrightarrow{A(q_a^{-2} x)} S_a(q_a^{-1} x) \xrightarrow{A(q_a x)} S_a(x) \qquad (5.11)$$

and gives the other half of the screening charge as

$$T_a^-(x) = \sum_{k<0} S_a(q_a^k x), \quad Q_a(x) = T_a^+(x) + T_a^-(x). \qquad (5.12)$$

We still call $T_a^+(x)$ and $T_a^-(x)$ D2 $qq$-characters. We may rescale the root currents as $A(x) \to q^{-1}A(x)$ and then obtain

$$Q_a(x) = \sum_{k \in \mathbb{Z}} q^k S_a(q_a^k x), \quad T_a^+(x) = \sum_{k \ge 0} q^k S_a(q_a^k x), \quad T_a^-(x) = \sum_{k < 0} q^k S_a(q_a^k x). \qquad (5.13)$$

Each term of the screening charge has a nice pictorial interpretation using the one-dimensional partition labeled by $k \in \mathbb{Z}$ in (2.2):

$$S_a(q_a^k x), \quad k \in \mathbb{Z} \iff \begin{array}{c} \cdots \\ \hline 1 \quad 2 \quad \cdots \quad k \end{array} \to q_a \qquad (5.14)$$

The screening charge $Q_a(x)$ is understood as a collection of possible one-dimensional partitions labeled by $k \in \mathbb{Z}$ with the $q$-coordinate of the origin $x$:

$$Q_a(x) \iff \left\{ \begin{array}{c} \cdots \\ \hline 1 \quad 2 \quad \cdots \quad k \end{array} \right. \to q_a \quad | \quad k \in \mathbb{Z} \qquad (5.15)$$

Using (4.66), we have

$$S_a(q_a^k x) := \prod_{i=-\infty}^{k} A^{-1}(q_a^{i-1} x) : \qquad (5.16)$$
which means we can interpret each box of the one-dimensional partition with the q-coordinates $xq_a^{i-1}$ as the operator $A^{-1}(xq_a^{i-1})$. The screening charge $\mathcal{Q}_a(x)$ is then understood as a collection of the possible one-dimensional partitions labeled by $k \in \mathbb{Z}$.

We can do the same discussion for the D2 $qq$-characters $T_a^{\pm}(x)$. We omit the discussion for $T_a^-(x)$ and focus on $T_a^+(x)$. In this case, each term of $T_a^+(x)$ is represented as a one-dimensional partition labeled by $k \in \mathbb{Z}_{\ge 0}$ in (2.1) and $T_a^+(x)$ is interpreted as a collection of the possible one-dimensional partitions labeled by $k \in \mathbb{Z}_{\ge 0}$ with the q-coordinate of the origin $x$:

$$ T_a^+(x) \iff \left\{ \begin{array}{c} \begin{array}{|c|c|c|c|c|} \hline x & & & & \\ & 1 & 2 & \cdots & k \\ \hline \end{array} \rightarrow q_a \quad \middle| \quad k \in \mathbb{Z}_{\ge 0} \end{array} \right\}. \quad (5.17) $$

In the context of quiver W-algebra, the iWeyl reflection is usually generated from the highest weight only in one direction. For the screening charge, actually, the $qq$-character is associated with the vector representation of quantum toroidal $gl_1$ (see section 9.2) which is not a highest weight representation, and thus, we need to generate the monomial terms in two directions.

We also note that the coefficient factors of the screening charges are 1, which is identical to the U(1) partition function of the D2 theory introduced in section 3.6 (see appendix B.5 and (B.49)): $\mathcal{Z}_a^{\mathrm{D2}}[k; q_i, q_j] = 1$.

**Remark 5.4.** Note that the D2 $qq$-characters $T_a^+(x)$ are the operator version of the D2 $qq$-characters introduced in (3.216):

$$ T_a^+(x) \leftrightarrow \hat{\mathcal{J}}^{\bar{a}}(x), \hat{\mathcal{J}}^{\bar{a}\vee}(x), \quad a \in \mathbf{4}. \quad (5.18) $$

The highest weight and the root current have the following correspondence:

$$ \begin{align} S_a(x) &\leftrightarrow \hat{W}^{\bar{a}}(q_{\bar{a}}x)^{-1}, \hat{W}^{\bar{a}\vee}(q_{\bar{a}}x)^{-1}, \quad a \in \mathbf{4} \tag{5.19} \\ A(x)^{-1} &\leftrightarrow \hat{A}(x)^{-1}, \hat{A}^{\vee}(x)^{-1}. \nonumber \end{align} $$

Note also that the $qq$-characters $T_a^{\pm}(x)$ ($a \in \mathbf{4}$) do not commute with the screening charges $\mathcal{Q}_a(x)$ nor themselves:

$$ [T_a^{\pm}(x), \mathcal{Q}_b(x')] \neq 0, \quad [T_a^{\pm}(x), T_b^{\pm}(x')] \neq 0, \quad a, b \in \mathbf{4}. \quad (5.20) $$

For the commutativity of the screening charges, we need the sum of both $T_a^{\pm}(x)$.

**Remark 5.5.** Although the above D2-brane $qq$-character (screening charge) is a sum of monomial terms labeled by an integer $k \in \mathbb{Z}$, using the results of section 7, we can introduce a D2 and $\overline{D6}$ coupled $qq$-character where the monomial terms are labeled by $k \in \mathbb{Z}_{\ge 0}$. In this case, the generating current is

$$ \frac{S_a(x)}{W_{\bar{b}}(q_b^{-1}q_a^{-1}x)}; \quad b \neq a \in \mathbf{4}. \quad (5.21) $$

Focusing on $a = 1, b = 4$, one can show that the $qq$-character that commutes with the screening charge $\mathcal{Q}_4(x')$ is

$$ T_{\overline{D2_1-D6_4}}(x) = \sum_{k \ge 0} \mathcal{Z}^{\overline{D2_1-D6_4}}[k] : \frac{S_1(q_1^k x)}{W_4(q_1^{-1}q_4^{-1}x)} : , \quad \mathcal{Z}^{\overline{D2_1-D6_4}}[k] = \prod_{l=1}^{k} \frac{1 - q_4^{-1}q_1^{-l}}{1 - q_1^{-l}}, \quad (5.22) $$
with the property

$$[\mathcal{T}_{\mathrm{D21-D6}\bar{4}}(x), \mathcal{Q}_4(x')] = 0. \qquad (5.23)$$

The coefficient factor here gives the partition function of a 3d $\mathcal{N} = 2^*$ theory with an adjoint matter whose mass is $q_4^{-1}$ described in [41, eq. (3.10)]. A detailed analysis of such systems is postponed for future work.

## 5.2 Coupled vortex system and D2 $qq$-characters

The contour integral formula in (3.155) can be expressed using vertex operators as in (4.45). Let us see that the expanded form of the partition function (3.166) can be expressed using the D2 $qq$-characters.

Let us consider finite products of the screening charges $S_a(v_{a,\alpha})$, ($a \in \underline{4}$, $\alpha = 1, \dots, n_a$). The operator product of two screening currents is

$$ \begin{aligned} S_b(v_{b,\beta} q_b^{k_b^{(\beta)}}) S_a(v_{a,\alpha} q_a^{k_a^{(\alpha)}}) &:= S_b(v_{b,\beta}) \prod_{\substack{\bullet \in k_b^{(\beta)} \\ \square \in k_a^{(\alpha)}}} A^{-1}(\chi_{b,v_{b,\beta}}(\square)) : S_a(v_{a,\alpha}) \prod_{\square \in k_a^{(\alpha)}} A^{-1}(\chi_{a,v_{a,\alpha}}(\square)) : \quad (5.24) \\ &= \mathcal{Z}_{1-\mathrm{loop}}^{\mathrm{D2-D2}}(v_{a,\alpha}, a | v_{b,\beta}, b) \mathcal{Z}_{a|b}^{\mathrm{D2-D2}}[v_{a,\alpha}, k_a^{(\alpha)} | v_{b,\beta}, k_b^{(\beta)}], \end{aligned} $$

where $a \neq b$. Generally, for finite products of screening currents, we have

$$ \prod_{a=1}^{4} \prod_{\alpha=1}^{n_a} S_a(v_{a,\alpha} q_a^{k_a^{(\alpha)}}) = \prod_{(b,\beta)>(a,\alpha)} \mathcal{Z}_{1-\mathrm{loop}}^{\mathrm{D2-D2}}(v_{a,\alpha}, a | v_{b,\beta}, b) \prod_{(b,\beta)>(a,\alpha)} \mathcal{Z}_{a|b}^{\mathrm{D2-D2}}(v_{a,\alpha}, k_a^{(\alpha)} | v_{b,\beta}, k_b^{(\beta)}) \\ \times : \prod_{a=1}^{4} \prod_{\alpha=1}^{n_a} S_a(v_{a,\alpha} q_a^{k_a^{(\alpha)}}) : \qquad (5.25) $$

where the left-hand side is an ordered operator product and the ordering is defined as

$$ \dots S_b(v_{b,\beta} q_b^{k_b^{(\beta)}}) \dots S_a(v_{a,\alpha} q_a^{k_a^{(\alpha)}}) \dots \iff (b, \beta) > (a, \alpha). \qquad (5.26) $$

Therefore, we have the following claim.

**Theorem 5.6.** The gauge origami partition function involving D2-branes is given by the correlation function of a finite number of screening currents,

$$ \mathcal{Z}_{1-\mathrm{loop}}^{\mathrm{D2}} \mathcal{Z}_{\mathrm{cpl.vort.}}^{\mathrm{D2}}[\vec{v}, \vec{k}] = \langle 0 | \prod_{a=1}^{4} \prod_{\alpha=1}^{n_a} S_a(v_{a,\alpha} q_a^{k_a^{(\alpha)}}) | 0 \rangle. \qquad (5.27) $$

**Corollary 5.7.** The total partition function can be written using half of the screening charge as

$$ \mathcal{Z}_{1-\mathrm{loop}}^{\mathrm{D2}} \mathcal{Z}_{\mathrm{vort.}}^{\mathrm{D2}} = \sum_{\vec{k}} q^{[\vec{k}]} \mathcal{Z}_{1-\mathrm{loop}}^{\mathrm{D2}} \mathcal{Z}_{\mathrm{cpl.vort.}}^{\mathrm{D2}}[\vec{v}, \vec{k}] = \langle 0 | \prod_{a \in \underline{4}} \prod_{\alpha=1}^{n_a} T_a^+(v_{a,\alpha}) | 0 \rangle. \qquad (5.28) $$

*Remark 5.8.* Realization of the 3d partition function on $\mathcal{C}_a \times \mathbb{S}^1$ with finite number insertion of the screening charges has been discussed in [131–133]. See also [103, 134] for the case with different types of the screening charges.
**Coulomb branch formula.** The vortex partition function discussed above is given by summation over infinitely many topologically distinct sectors (Higgs branch formula). There is another description of the 3d partition function, which is given by the contour integral over the Cartan torus of the gauge group (Coulomb branch formula) [102, 135]. Let us show that this Coulomb branch formula is also obtained from the vertex operator formalism.

The screening charges $Q_a(x)$ can be written using the Jackson integral with the base $x$ as

$$Q_a(x) = \oint_{C_{a,x}} d_{q_a} z S_a(z), \qquad (5.29)$$

where the contour integral is denoted $C_x$. Let us first consider the 3d theory on $C_a \times S^1$. Finite products of the screening charges are then given as

$$\begin{aligned} \mathbf{Z}_{a, \vec{x}}^{n_a} &:= \prod_{\alpha=1}^{n_a} \mathcal{Q}_a(x_\alpha) = \oint_{C_{a,x_n}} \cdots \oint_{C_{a,x_2}} \oint_{C_{a,x_1}} d_{q_a} z_{n_a} \cdots d_{q_a} z_1 S_a(z_{n_a}) \cdots S_a(z_1) \\ &= \oint_{C_{a,x_1}, \dots, C_{a,x_n}} \prod_{\alpha=1}^{n_a} d_{q_a} z_i \Delta_a(\vec{z}; q_a) : \prod_{\alpha=1}^{n_a} S_a(z_\alpha) : \end{aligned} \qquad (5.30)$$

where

$$\Delta_a(\vec{z}; q_a) = \prod_{i<j} \Delta_a(z_i/z_j), \quad \Delta_a(x'/x) = \frac{(x'/x; q_a)_{\infty} \prod_{i \neq a} (q_i q_a x'/x; q_a)_{\infty}}{(q_a x'/x; q_a)_{\infty} \prod_{i \neq a} (q_i^{-1} x'/x; q_a)_{\infty}}, \qquad (5.31)$$

and we assumed $|q_a| < 1$. Note that $\Delta_a(x'/x)$ is just $Z_{1-\text{loop}}^{\text{D2-D2}}(x', a | x, a)$.

One-loop contributions of 3d $\mathcal{N} = 2$ on $C_a \times S^1$ are given by

$$z_{\text{vec}} = \prod_{i \neq j} (x_i/x_j; q_a)_{\infty}, \quad z_{\text{adj}} = \prod_{i \neq j} (\mu_{\text{adj}} x_i/x_j; q_a)_{\infty}^{-1}, \qquad (5.32)$$

where $\mu_{\text{adj}} = e^{m_{\text{adj}}}$. Meanwhile, the OPE factor of the screening currents is rewritten after using the theta function in (A.12) as

$$\begin{aligned} \Delta_a(\vec{z}; q_a) &= \prod_{i<j} \frac{(z_i/z_j; q_a)_{\infty} \prod_{b \neq a} (q_b q_a z_i/z_j; q_a)_{\infty}}{(q_a z_i/z_j; q_a)_{\infty} \prod_{b \neq a} (q_b^{-1} z_i/z_j; q_a)_{\infty}} \\ &= \prod_{i<j} \frac{\prod_{b \neq a} \theta(q_b^{-1} z_j/z_i; q_a)}{\theta(z_j/z_i; q_a)} \prod_{i \neq j} \frac{(z_i/z_j; q_a)_{\infty}}{\prod_{b \neq a} (q_b^{-1} z_i/z_j; q_a)_{\infty}}. \end{aligned} \qquad (5.33)$$

which is thus identified with 1 vector multiplet and 3 adjoint chiral multiples with masses $q_b^{-1}$, ($b \neq a$).$^{27}$ Additional theta function factors are identified with the boundary contribution $\partial C_a \times S^1 = \mathbb{T}^2$. In addition to the adjoint matters, we can also add the fundamental matters using the flavor brane vertex operators (see section 10.3).

Note that due to the condition $q_1q_2q_3q_4 = 1$, we can only choose up to three of the $q$-parameters to be $|q_a| < 1$. For the analytic region of $|q_a| > 1$, the above formulas will be modified after using the reflection formula in (A.3) (see also (A.8)).

<footnote>$^{27}$Hence, this factor $\Delta_a(\vec{z}; q_a)$ agrees with (a trigonometric version of) the measure part of the vertex function associated with the Hilbert scheme of points on $\mathbb{C}^3$ [54, Prop. 7.12] up to the boundary contribution.</footnote>
Let us next consider the contribution between $C_a \times S^1$ and $C_b \times S^1$ where $a \neq b$. For simplicity, let us focus on $a = 1, b = 2$:

$$ \begin{aligned} Q_1(x)Q_2(x') &= \oint_{C_{1,x}} \oint_{C_{2,x'}} d_{q_1}z d_{q_2}z' S_{34} (q_1 z'/z) \\ &= \oint_{C_{1,x}} \oint_{C_{2,x'}} d_{q_1}z d_{q_2}z' \frac{(1-q_{13}z'/z)(1-q_{14}z'/z)}{(1-q_1z'/z)(1-q_2^{-1}z'/z)} \end{aligned} \quad (5.34) $$

The contribution coming from $S_{34}(z)$ is understood as the chiral and Fermi multiplet living on the 1d intersection $S^1$ [103, 134]. Therefore, we conclude that the partition function obtained by $\prod_{a \in \mathfrak{a}} \mathbf{Z}_{a, \tilde{a}_a}^{n_a}$ is an intersecting gauge theory. Choosing the contour integral properly, we expect that we can reproduce the coupled vortex partition function in (3.166).

**Conjecture 5.9.** By specifying the contour integrals properly, we have the following identity:

$$ Z_{\text{1-loop}}^{\text{D2}} Z_{\text{vort.}}^{\text{D2}} = \langle 0 | \prod_{a \in \mathfrak{a}} \mathbf{Z}_{a, \tilde{a}_a}^{n_a} | 0 \rangle. \quad (5.35) $$

Namely, the coupled vortex system is obtained by finite products of screening charges.

# 6 D4-brane $\bar{q}q$-characters

In this section, we introduce the operator version of the D4 $\bar{q}q$-characters introduced in section 3.8. These $\bar{q}q$-characters are not new $\bar{q}q$-characters but they were already introduced in [10] and the corresponding algebraic structure is known to be the affine quiver W-algebra [60]. We introduce six types of D4 $\bar{q}q$-characters corresponding to the six possible configurations of D4-branes in $\mathbb{C}^4$ in an equal footing in section 6.1. We then discuss the relation with the spiked instantons in section 6.2 and show that their compositions give the gauge origami partition function of the spiked instantons. We also show that the partition function can be rewritten using the screening currents in section 6.3. We then extend the D4 $\bar{q}q$-characters to general D4 $\bar{q}q$-characters which give a supergroup analog of the gauge origami partition function in section 6.4. The quadratic relations of the D4 $\bar{q}q$-characters are discussed in section 6.5.

## 6.1 D4 $\bar{q}q$-characters and affine quiver W-algebra

Let us consider the $\bar{q}q$-character generated by the D4 operators $X_A(x)$ ($A \in \mathfrak{a}$). The D4 $\bar{q}q$-character generated here is identified with the generator of the affine quiver W-algebra of $\Gamma = \hat{A}_0$ in [60]. Let us review the derivation of it.

Each term of the $\bar{q}q$-character is decomposed into two parts, the operator part and the coefficient part. The operator part is determined by the iWeyl reflection which is defined as the following (see also (3.202)).

**Definition 6.1.** The iWeyl reflection of the D4 vertex operator $X_A(x)$ ($A \in \mathfrak{a}$) is

$$ X_A(x) \to : X_A(x) A^{-1}(x) : , \quad A \in \mathfrak{a}. \quad (6.1) $$

Using the property in (4.68a), we have

$$ X_{ab}(x) \to : a_0(x)^{-1} \frac{X_{ab}(q_a x) X_{ab}(q_b x)}{X_{ab}(q_{ab}x)} : , \quad a \neq b \in \mathfrak{a}. \quad (6.2) $$
Proof. Let us focus on $A = 12$ and $a = 4$ and derive the D4 $qq$-character. Other cases are obtained using the quadrality. Using the formulas in (C.12), we have

$$ \Lambda_{12,\lambda}(x)S_4(x') = \left[ q_3^{-1} \frac{y_{\lambda,x}^{12}(q_{12}x')}{y_{\lambda,x}^{12}(q_{123}x')} \right]_{-}^{x'} : \Lambda_{12,\lambda}(x)S_4(x') : , \\ S_4(x')\Lambda_{12,\lambda}(x) = \left[ q_3^{-1} \frac{y_{\lambda,x}^{12}(q_{12}x')}{y_{\lambda,x}^{12}(q_{123}x')} \right]_{+}^{x'} : \Lambda_{12,\lambda}(x)S_4(x') : \qquad (6.9) $$

where $[f(x)]_{\pm}^x$ means expansions of $f(x)$ in $x^{\mp}$ respectively. Assume that the $qq$-character takes the form of

$$ T_{12}(x) = \sum_{\lambda \in \mathcal{P}} F_{12}^{\mathrm{D4}}[\lambda] \Lambda_{12,\lambda}(x), \quad F_{12}[\emptyset] = 1 \qquad (6.10) $$

where $F_{12}^{\mathrm{D4}}[\lambda]$ are some coefficients, then the commutation relation is

$$ [T_{12}(x), S_4(x')] = -q_3^{-1} \left[ \sum_{\lambda} F_{12}^{\mathrm{D4}}[\lambda] \sum_{\square \in A(\lambda)} \delta\left(\frac{\chi_{12,x}(\square)}{q_4^{-1}x'}\right) \mathrm{Res}_{x'=q_3^{-1}\chi_{12,x}(\square)} x'^{-1} \frac{y_{\lambda,x}^{12}(x')}{y_{\lambda,x}^{12}(q_3x')} : \Lambda_{12,\lambda}(x)S_4(x') : \right. \\ \left. + \sum_{\lambda} F_{12}^{\mathrm{D4}}[\lambda] \sum_{\square \in R(\lambda)} \delta\left(\frac{\chi_{12,x}(\square)}{x'}\right) \mathrm{Res}_{x'=q_{34}^{-1}\chi_{12,x}(\square)} x'^{-1} \frac{y_{\lambda,x}^{12}(x')}{y_{\lambda,x}^{12}(q_3x')} : \Lambda_{12,\lambda}(x)S_4(x') : \right]. \qquad (6.11) $$

Shifting the second term as $\lambda = \lambda' + \square$, it will be

$$ -q_3^{-1} \sum_{\lambda'} \sum_{\square \in A(\lambda')} F_{12}^{\mathrm{D4}}[\lambda' + \square] \delta\left(\frac{\chi_{12,x}(\square)}{x'}\right) \mathrm{Res}_{x'=q_{34}^{-1}\chi_{12,x}(\square)} x'^{-1} \frac{y_{\lambda'+\square,x}^{12}(x')}{y_{\lambda'+\square,x}^{12}(q_3x')} : \Lambda_{12,\lambda'}(x)A^{-1}(\chi_{12,x}(\square))S_4(x') : \qquad (6.12) $$

Using

$$ : \Lambda_{12,\lambda'}(x)A^{-1}(\chi_{12,x}(\square))S_4(\chi_{12,x}(\square)) : := : \Lambda_{12,\lambda'}(x)S_4(q_4\chi_{12,x}(\square)) : \qquad (6.13) $$

and imposing the condition (see the recursion formula in Thm. B.2)

$$ \frac{F_{12}^{\mathrm{D4}}[\lambda + \square]}{F_{12}^{\mathrm{D4}}[\lambda]} = - \frac{\mathrm{Res}_{x'=q_3^{-1}\chi_{12,x}(\square)} x'^{-1} \frac{y_{\lambda,x}^{12}(x')}{y_{\lambda,x}^{12}(q_3x')}}{\mathrm{Res}_{x'=q_{34}^{-1}\chi_{12,x}(\square)} x'^{-1} \frac{y_{\lambda'+\square,x}^{12}(x')}{y_{\lambda'+\square,x}^{12}(q_3x')}} = q_3^{-1} \frac{\mathcal{Z}_{12}^{\mathrm{D4}}[\lambda + \square; q_3]}{\mathcal{Z}_{12}^{\mathrm{D4}}[\lambda; q_3]} = \frac{\tilde{\mathcal{Z}}_{12}^{\mathrm{D4}}[\lambda + \square]}{\tilde{\mathcal{Z}}_{12}^{\mathrm{D4}}[\lambda]} \qquad (6.14) $$

we then obtain

$$ [T_{12}(x), S_4(x')] = -q_3^{-1} \sum_{\lambda \in \mathcal{P}} \sum_{\square \in A(\lambda)} F_{12}^{\mathrm{D4}}[\lambda] \mathrm{Res}_{x'=q_3^{-1}\chi_{12,x}(\square)} x'^{-1} \frac{y_{\lambda,x}^{12}(x')}{y_{\lambda,x}^{12}(q_3x')} : \Lambda_{12,\lambda}(x)S_4(q_4\chi_{12,x}(\square)) : \\ \times \left( \delta\left(\frac{q_4\chi_{12,x}(\square)}{x'}\right) - \delta\left(\frac{\chi_{12,x}(\square)}{x'}\right) \right) \qquad (6.15) $$

which is a total difference. Therefore, under the condition $F_{12}[\lambda] = \tilde{\mathcal{Z}}_{12}^{\mathrm{D4}}[\lambda]$, the $qq$-character commutes with the screening charge $\mathcal{Q}_4(x')$:

$$ [T_{12}(x), \mathcal{Q}_4(x')] = 0. \qquad (6.16) $$

<figure><img src="image_18.png" /></figure>
Remark 6.4. Note that these D4 *qq*-characters $\mathcal{T}_A(x)$ ($A \in \underline{\mathbf{6}}$) are the operator version of the D4 *qq*-characters introduced in (3.201):

$$ \mathcal{T}_{\bar{A}}(x) \leftrightarrow \hat{\mathcal{T}}^A(x), \hat{\mathcal{T}}^{A\vee}(x), \quad A \in \underline{\mathbf{6}}. \qquad (6.17) $$

The highest weight and the root current have the following correspondence:

$$ \begin{aligned} \mathcal{X}_A(x) &\leftrightarrow \hat{y}^A(q_A x), \hat{y}^{A\vee}(q_A x), \quad A \in \underline{\mathbf{6}} \\ \mathcal{A}(x)^{-1} &\leftrightarrow \hat{\mathcal{A}}(x)^{-1}, \hat{\mathcal{A}}^{\vee}(x)^{-1}. \end{aligned} \qquad (6.18) $$

## 6.2 Spiked instantons and D4 *qq*-characters

Similar to the integral formula, the expanded version of the spiked instanton partition function can be expressed using the D4 *qq*-characters.

**Lemma 6.5.** The operator product of $\{\Lambda_{A,\lambda}(x)\}_{A \in \underline{\mathbf{6}}}^{}$ is

$$ \Lambda_{B,\mu}(x')\Lambda_{A,\lambda}(x) = Z_{\text{l-loop}}^{\text{D4-D4}}(x, A|x', B)Z_{A|B}^{\text{D4-D4}}(x, \lambda|x', \mu) : \Lambda_{B,\mu}(x')\Lambda_{A,\lambda}(x) : . \qquad (6.19) $$

When $A = B$, it gives the vector and adjoint hypermultiplet contributions, while when $A \neq B$, it gives the bifundamental-like contributions connecting gauge theories defined on different subspaces.

**Theorem 6.6.** The gauge origami partition function of the spiked instantons is written using the D4 *qq*-characters:

$$ Z_{\text{l-loop}}^{\text{D4}} Z_{\text{inst.}}^{\text{D4}} = \sum_{\vec{\lambda}} q^{|\vec{\lambda}|} Z_{\text{l-loop}}^{\text{D4}} Z_{\text{spk.inst.}}^{\text{D4}} [\vec{v}, \vec{\lambda}] = \langle 0 | \prod_{A \in \underline{\mathbf{6}}} \prod_{\alpha=1}^{n_A} \mathcal{T}_A(v_{A,\alpha}) | 0 \rangle \qquad (6.20) $$

Depending on the value of $A, B \in \underline{\mathbf{6}}$, we get different instanton contributions:

* $A = B$: instantons in $\hat{A}_0$ quiver gauge theory

* $A \cap B \in \underline{\mathbf{4}}$: folded instantons

* $B = \bar{A}$: crossed instantons

To summarize, we have the following table of BPS/CFT correspondence.

<table><thead><tr><th>BPS</th><th>CFT</th></tr></thead><tbody><tr><td>5d N = 1* U(1) on C<sup>2</sup><sub>ab</sub> &times; S<sup>1</sup> (D4<sub>ab</sub> &times; 1)</td><td>&lang;0| T<sub>ab</sub>(v)|0&rang;</td></tr><tr><td>5d N = 1* U(n<sub>ab</sub>) on C<sup>2</sup><sub>ab</sub> &times; S<sup>1</sup> (D4<sub>ab</sub> &times; n<sub>ab</sub>)<br/>crossed instanton: D4<sub>12</sub>-D4<sub>34</sub>-D0<br/>folded instanton: D4<sub>12</sub>-D4<sub>13</sub>-D0<br/>gauge origami of spiked instantons</td><td>&lang;0| T<sub>ab</sub>(v<sub>n<sub>ab</sub></sub>) &middot;&middot;&middot; T<sub>ab</sub>(v<sub>2</sub>)T<sub>ab</sub>(v<sub>1</sub>)|0&rang;<br/>&lang;0| T<sub>12</sub>(v)T<sub>34</sub>(v')|0&rang;<br/>&lang;0| T<sub>12</sub>(v)T<sub>13</sub>(v')|0&rang;<br/>&lang;0| &prod;<sub>A&in;6</sub> &prod;<sub>&alpha;=1</sub><sup>n<sub>A</sub></sup> T<sub>A</sub>(v<sub>A,&alpha;</sub>)|0&rang;</td></tr></tbody></table>

Remark 6.7. It has been argued in [136] that the partition function of the 5d $N = 1^*$ theory on $\mathbb{C}^2 \times S^1$ is given by the *qq*-character correlation function, and the 6d theory partition function is given by the corresponding torus correlation function.
## 6.3 Spiked instantons and screening currents

Since the two-dimensional partition can be understood using the (1,1)-type description, it is natural to expect that the partition functions have an operator representation using the screening currents. Using the character form in (3.175), we omit the singular terms that might occur and extract the square root part of the total index by specifying an order:

$$V = \sum_{A,B \in \mathcal{B}} \frac{\mathbf{P}_{\bar{s}(A)}}{\mathbf{P}_{\bar{s}(A)}^{\vee} \mathbf{P}_{\bar{s}(B)}} \mathbf{X}_{A}^{\vee} \mathbf{X}_{B} \rightarrow \mathbf{v} = \sum_{\substack{(x,A) < (x',B) \\ x \in \mathcal{X}_A, x' \in \mathcal{X}_B}} \frac{\mathbf{P}_{\bar{s}(A)}}{\mathbf{P}_{\bar{s}(A)}^{\vee} \mathbf{P}_{\bar{s}(B)}} \left( \frac{x'}{x} \right). \quad (6.22)$$

Up to one-loop perturbative factors, $\mathbb{I}[\mathbf{v}]$ is identical to the partition function in (3.143), (3.144). Using (4.41), we then have the following BPS/CFT correspondence.

**Proposition 6.8.** The index of (6.22) is the vacuum expectation value of the screening currents and is equivalent to the partition function in (3.143) and (3.144) up to one-loop perturbative factors.

$$\mathbb{I}[\mathbf{v}] = \langle 0 | \prod_{A \in \mathcal{B}, x \in \mathcal{X}_A} S_{\bar{s}(A)}(x) | 0 \rangle \approx \mathcal{Z}_{\text{1-loop}}^{\text{D4}} \mathcal{Z}_{\text{spkinst.}}^{\text{D4}} [\vec{v}, \vec{\lambda}]. \quad (6.23)$$

Note that we are implicitly defining an ordering in the products of the screening currents as

$$\cdots S_{\bar{s}(B)}(x') \cdots S_{\bar{s}(A)}(x) \cdots \Leftrightarrow (x, A) < (x', B). \quad (6.24)$$

The symbol “$\simeq$” means that they are identical up to one-loop perturbative factors.

**Fusion of D2 $qq$-characters.** Since the gauge origami partition function can be written using both the screening currents and the D4 $qq$-characters, one would like to ask if we have any relation connecting both descriptions. An interesting fact is that up to one-loop perturbative factors, the D4 $qq$-characters are just infinite products of the screening charges.

**Theorem 6.9.** The D4 $qq$-characters are infinite products of D2 $qq$-characters (screening charges):

$$\mathcal{T}_{ab}(x) \approx \overrightarrow{\prod_{i=1}^{\infty} \Omega_b(x q_a^{i-1})}, \quad a \neq b \in \mathcal{B}, \quad (6.25)$$

where the product is $\overrightarrow{\prod_{i=1}^{\infty} f(x_i)} = f(x_1)f(x_2)\cdots$.

*Proof.* We only give a sketch of the proof (see [60] and [64, eq. (6.2.23)–(6.2.27)] for a similar discussion). The vertex operator $\Lambda_{ab,\lambda}(x)$ of the D4 $qq$-character satisfies

$$\Lambda_{ab,\lambda}(x) \sim \prod_{i=1}^{\infty} S_b(x q_a^{i-1} q_b^{\lambda_i}) : \quad (6.26)$$

where the equality is up to zero modes. By direct computation, one can show that up to one-loop perturbative factors, we have$^{28}$

$$\mathcal{T}_{ab}(x) \approx \sum_{\lambda \in \mathcal{P}} \prod_{i=1}^{\infty} S_b(x q_a^{i-1} q_b^{\lambda_i}). \quad (6.27)$$

<footnote>$^{28}$Strictly speaking, we have to be careful of the zero-modes appearing on both hand sides. Moreover, the infinite product of the screening currents needs to be regularized properly.</footnote>
Using the property that for $i \le j$ and $\lambda_i < \lambda_j$

$$S_b(xq_a^{i-1}q_b^{\lambda_i})S_b(xq_a^{j-1}q_b^{\lambda_j}) = 0, \qquad (6.28)$$

we obtain

$$Q_b(x)Q_b(q_a x) = \sum_{k \in \mathbb{Z}} S_b(xq_b^k) \sum_{l \in \mathbb{Z}} S_b(xq_a q_b^l) = \sum_{k \ge l} S_b(xq_b^k)S_b(xq_a q_b^l). \qquad (6.29)$$

By computing the nontrivial coefficients appearing after taking the contractions of the screening currents, one obtains the statement. □

**Corollary 6.10.** The total partition function is written using infinite products of screening charges

$$Z_{\text{1-loop}}^{\text{D4}} Z_{\text{inst.}}^{\text{D4}} = \sum_{\vec{\lambda}} q^{|\vec{\lambda}|} Z_{\text{1-loop}}^{\text{D4}} Z_{\text{spk.inst.}}^{\text{D4}} [\vec{v}, \vec{\lambda}] \approx \langle 0 | \prod_{A \in \mathcal{B}} \prod_{\alpha=1}^{n_A} \prod_{i=1}^{\vec{\lambda}_A} \Omega_{\bar{s}(A)}(v_{A,\alpha} q_{s(A)}^{j-1}) | 0 \rangle . \qquad (6.30)$$

*Remark 6.11.* We can visualize the above procedure using the correspondence in (5.14) and (5.15). The equation (6.27) is the consequence of the fact that the 2d partition has a (1, 1)-type description. Each term $S_b(xq_a^{i-1}q_b^{\lambda_i})$ is understood as a one-dimensional partition extending in the $q_b$ direction as mentioned in section 2. Piling the one-dimensional partitions in the orthogonal direction, we get the two-dimensional partition and the corresponding vertex operator $\Lambda_{ab,\lambda}(x)$. We call this the **fusion** process. Namely, fusions of D2 $qq$-characters (screening charges) give the D4 $qq$-character.

<figure><img src="image_19.png" /></figure>

## 6.4 General D4 $qq$-characters

The highest weight of the D4 $qq$-character discussed in the previous section was $X_A(x)$ ($A \in \mathcal{B}$) and the coefficients appearing are the U(1) partition function of the affine quiver gauge theory on $\mathbb{C}_A^2 \times \mathbb{S}^1$. Higher rank D4 $qq$-characters are obtained as

$$T_A^{(n)}(\underline{x}) := X_A(x_1) \cdots X_A(x_n) : + \cdots \qquad (6.32)$$

where $\underline{x} = (x_1, \dots, x_n)$ are generic parameters. Imposing the commutativity with the screening charges $Q_a(x')$ ($a \in \bar{A}$), the coefficients appearing will be the partition function of the 5d $\mathcal{N} = 1^*$ U(n) gauge theory on $\mathbb{C}_A^2 \times \mathbb{S}^1$.
We can also include negative highest weights as

$$ \mathcal{T}_A^{(n|m)}(\underline{x}|\underline{y}) := \frac{\mathcal{X}_A(x_1) \cdots \mathcal{X}_A(x_n)}{\mathcal{X}_A(y_1) \cdots \mathcal{X}_A(y_m)} + \cdots \qquad (6.33) $$

where $\underline{x} = (x_1, \dots, x_n)$ and $\underline{y} = (y_1, \dots, y_m)$ are all generic parameters. One can show that after imposing the commutativity with the screening charges $\mathcal{Q}_a(x')(a \in \bar{A})$, the coefficients will be the partition function of the $U(n|m)$ supergroup gauge theory. The derivation can be similarly done as the non-supergroup gauge theory (see appendix D).

**Definition 6.12.** The iWeyl reflection of the operator $\mathcal{X}_A(x)^{-1}$ is defined as

$$ \mathcal{X}_A(x)^{-1} \to : \mathcal{X}_A(x)^{-1} \mathcal{A}(q_A^{-1} x) : \qquad (6.34) $$

which we call the **negative iWeyl reflection**. The normal iWeyl reflection will be called positive iWeyl reflection in contrast.<fnref n="29" />

Doing the negative iWeyl reflection recursively, we obtain the monomial terms

$$ \bar{\Lambda}_{ab,\lambda}(x) := \mathcal{X}_{ab}(x)^{-1} \prod_{\square \in \lambda} \mathcal{A}(\bar{\chi}_{ab,x}(\square)) : , \quad \bar{\chi}_{ab,x}(\square) = x q_a^{-i} q_b^{-j}, \qquad (6.35) $$

which gives the following D4 $qq$-character.

**Theorem 6.13.** The D4 $qq$-character generated by $\mathcal{X}_A^{-1}(x)$ is

$$ \mathcal{T}_A^{(0|1)}(x) = \sum_{\lambda \in \mathcal{P}} \tilde{\mathcal{Z}}_A^{\mathrm{D4}^-}[\lambda] : \bar{\Lambda}_{A,\lambda}(x) : , \qquad (6.36) $$

where $\tilde{\mathcal{Z}}_A^{\mathrm{D4}^-}[\lambda]$ is the $U(0|1)$ partition function (see appendix D). It commutes with the screening charges $\mathcal{Q}_a(x')(a \in \bar{A})$:

$$ [\mathcal{T}_A^{(0|1)}(x), \mathcal{Q}_a(x')] = 0, \quad a \in \bar{A}. \qquad (6.37) $$

By rescaling the root current as $\mathcal{A}(x) \to q^{-1}\mathcal{A}(x)$, the $qq$-character is

$$ \mathcal{T}_A^{(0|1)}(x) = \sum_{\lambda \in \mathcal{P}} q^{-|\lambda|} \tilde{\mathcal{Z}}_A^{\mathrm{D4}^-}[\lambda] : \bar{\Lambda}_{A,\lambda}(x) : . \qquad (6.38) $$

Note that the higher rank D4 $qq$-characters are simply obtained by taking compositions of $\mathcal{T}_A^{(1|0)}(x)$ and $\mathcal{T}_A^{(0|1)}(x)$ up to one-loop perturbative factors:

$$ \mathcal{T}_A^{(n|m)}(\underline{x}|\underline{y}) \propto \prod_{i=1}^{n} \mathcal{T}_A^{(1|0)}(x_i) \prod_{j=1}^{m} \mathcal{T}_A^{(0|1)}(y_j). \qquad (6.39) $$

Obviously, for fixed $A \in \mathcal{B}$, arbitrary products of $\mathcal{T}_A^{(1|0)}(x)$ and $\mathcal{T}_A^{(0|1)}(x)$ commute with the screening charges $\mathcal{Q}_a(x)$ ($a \in \bar{A}$) because each of the $qq$-characters commute with the screening charges. For example, consider the product $\mathcal{T}_{12}(x)\mathcal{T}_{12}(x')$ and the commutation relation with $\mathcal{S}_4(x'')$. We then have

$$ [\mathcal{T}_{12}(x)\mathcal{T}_{12}(x'), \mathcal{Q}_4(x'')] = [\mathcal{T}_{12}(x), \mathcal{Q}_4(x'')]\mathcal{T}_{12}(x') + \mathcal{T}_{12}(x)[\mathcal{T}_{12}(x'), \mathcal{Q}_4(x'')] = 0. \qquad (6.40) $$

<footnote n="29">²⁹We omit the *positive* when obvious.</footnote>
Using

$$ T_{12}(x)T_{12}(x') = Z_{1-\text{loop}}^{\text{D4-D4}}(x, 12 | x', 12) (X_{12}(x)X_{12}(x'') + \dots) \quad (6.41) $$

we can see that $T_{12}(x)T_{12}(x')/Z_{1-\text{loop}}^{\text{D4-D4}}(x, 12|x', 12)$ is a $qq$-character whose highest weight is : $X_{12}(x)X_{12}(x')$ :.

More generally, we can introduce $qq$-characters starting from operators like : $X_{12}(x)X_{13}(x')$ : where D4 vertex operators associated with different $\mathbb{C}^2$ subspaces appear. The resulting $qq$-character is proportional to the product $T_{12}(x)T_{13}(x')$.

**Theorem 6.14.** Let $a, b, c, d \in \mathbb{Z}_4$ be the four elements in the set $\mathbb{Z}_4$. We choose one of the screening charges $Q_d(x)$. The highest weight of the D4 $qq$-character commuting with this screening charge is generally written as

$$ T_{ab:bc:ac}^{(\vec{n}|\vec{m})}(\vec{x} | \vec{y}) = \frac{\prod_{\alpha=1}^{n_{ab}} X_{ab}(x_{ab,\alpha}) \prod_{\beta=1}^{n_{bc}} X_{bc}(x_{bc,\beta}) \prod_{\gamma=1}^{n_{ac}} X_{ac}(x_{ac,\gamma})}{\prod_{\alpha=1}^{m_{ab}} X_{ab}(y_{ab,\alpha}) \prod_{\beta=1}^{m_{bc}} X_{bc}(y_{bc,\beta}) \prod_{\gamma=1}^{m_{ac}} X_{ac}(y_{ac,\gamma})} + \dots \quad (6.42) $$

where $\vec{n} = (n_{ab}, n_{bc}, n_{ac})$, $\vec{m} = (m_{ab}, m_{bc}, m_{ca})$, $\vec{x} = (x_{A,\alpha})_{A \in \{ab,bc,ac\}}^{\alpha=1,...,n_A}$, $\vec{y} = (y_{A,\alpha})_{A \in \{ab,bc,ac\}}^{\alpha=1,...,m_A}$. The monomial terms are obtained by doing the positive and negative iWeyl reflections respectively, while the coefficients are obtained by imposing the commutativity with the screening charge $Q_d(x)$:

$$ [T_{ab:bc:ac}^{(\vec{n}|\vec{m})}(\vec{x} | \vec{y}), Q_d(x')] = 0. \quad (6.43) $$

Moreover, the $qq$-character obtained here is proportional to the products of the D4 $qq$-characters associated with $A = ab, bc, ac$:

$$ T_{ab:bc:ac}^{(\vec{n}|\vec{m})}(\vec{x} | \vec{y}) \propto \prod_{A \in \{ab,bc,ac\}} T_A^{(n_A|m_A)}(\underline{x_A|y_A}) \quad (6.44) $$

where $\underline{x_A} = (x_{A,\alpha})_{\alpha=1,...,n_A}$, $\underline{y_A} = (y_{A,\alpha})_{\alpha=1,...,m_A}$. Note that the coefficients appearing are the supergroup analog of the gauge origami partition function of vector and folded instantons.

**Remark 6.15.** The above theorem claims that for example if we fix the screening charge $Q_4(x)$, finite products of D4 $qq$-characters of $A = 12, 23, 34$ will give higher rank $qq$-characters. Such kind of $qq$-characters have nice physical meanings because the appearing coefficients are related to the gauge origami partition function of intersecting D4-branes inside the $\mathbb{C}^3_{123}$ subspace. However, since the $qq$-characters such as $T_A(x)$ ($A = 14, 24, 34$) do not commute with $Q_4(x)$, we can not use $X_A(x)^{\pm 1}$ ($A = 14, 24, 34$) as highest weights. Thus, the $qq$-character whose coefficients are the most general gauge origami partition (e.g. crossed instantons) does not appear by just imposing the commutativity with one of the screening charges.

**Remark 6.16.** Using the fact that screening charges of different types commute with each other, $S_{1,2,3}(x)$ can also be the highest weight of the $qq$-characters commuting with $Q_4(x)$. Such kinds of $qq$-characters are expected to be physically related with a D2-D4 coupled system. We leave the analysis of these cases for future work.
## 6.5 Quadratic relations of crossed instantons

The $qq$-characters $\{T_A(x)\}_{A \in \mathfrak{g}}$ are expected to generate a larger algebra than the affine quiver W-algebra. As usual deformed W-algebras, studying the quadratic relations of the generators is another way to understand the algebraic structure. Since deriving the complete quadratic relations of the $\{T_A(x)\}_{A \in \mathfrak{g}}$ is beyond the scope of this paper, we just give a part of the quadratic relations which can be derived easily.

**Theorem 6.17.** The D4 $qq$-characters $T_A(x)$ and $T_{\bar{A}}(x)$ for $A \in \mathfrak{g}$ (anti-)commute with each other up to trivial zero-modes factors:

$$xT_A(x)T_{\bar{A}}(x') + q_A x'T_{\bar{A}}(x')T_A(x) = 0. \qquad (6.45)$$

*Proof.* Let us consider the commutation relations between $T_{12}(x)$ and $T_{34}(x')$:

$$T_{12}(x) = \sum_{\lambda} F_{12}^{\mathrm{D4}}[\lambda] : \Lambda_{12,\lambda}(x) : , \qquad (6.46)$$

$$T_{34}(x') = \sum_{\mu} F_{34}^{\mathrm{D4}}[\mu] : \Lambda_{34,\mu}(x') : . \qquad (6.47)$$

The contraction formulas give

$$\Lambda_{12,\lambda}(x)\Lambda_{34,\mu}(x') = [\mathcal{Z}_{34|12}^{\mathrm{tot}}(x', \mu | x, \lambda)]_{|x'/x|\ll 1} : \Lambda_{12,\lambda}(x)\Lambda_{34,\mu}(x') : , \qquad (6.48)$$

$$\Lambda_{34,\mu}(x')\Lambda_{12,\lambda}(x) = [\mathcal{Z}_{12|34}^{\mathrm{tot}}(x, \lambda | x', \mu)]_{|x/x'|\ll 1} : \Lambda_{12,\lambda}(x)\Lambda_{34,\mu}(x') : ,$$

where

$$\mathcal{Z}_{\bar{A}|A}^{\mathrm{tot.}}(x', \mu|x, \lambda) = \mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D4-D4}}(x', \bar{A}|x, A) \mathcal{Z}_{\bar{A}|A}^{\mathrm{D4-D4}}(x', \mu|x, \lambda). \qquad (6.49)$$

Noting that the one-loop factors of the crossed instantons are rational functions

$$\mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D4-D4}}(x', \bar{X}|x, X) = (1 - q_A x'/x), \qquad (6.50)$$

we then obtain

$$\begin{aligned} T_{12}(x)T_{34}(x') - \left(-q_{12} \frac{x'}{x}\right)T_{34}(x')T_{12}(x) \\ = \sum_{\lambda, \mu} F_{12}^{\mathrm{D4}}[\lambda]F_{34}^{\mathrm{D4}}[\mu] \left\{ \sum_{\substack{\bullet \in A(\mu) \\ \Box \in R(\lambda)}} \mathrm{Res}_{\substack{\chi_{12,x(\Box)} \\ = \chi_{34,x'(\bullet)}}} \left(\frac{x}{x'}\right)^{-1} \mathcal{Z}_{34|12}^{\mathrm{tot}}(x', \mu | x, \lambda) \delta\left(\frac{\chi_{12,x(\Box)}}{\chi_{34,x'(\bullet)}}\right) : \Lambda_{12,\lambda}(x)\Lambda_{34,\mu}(x') : \right. \\ \left. + \sum_{\substack{\bullet \in R(\mu) \\ \Box \in A(\lambda)}} \mathrm{Res}_{\substack{\chi_{12,x(\Box)} \\ = \chi_{34,x'(\bullet)}}} \left(\frac{x}{x'}\right)^{-1} \mathcal{Z}_{34|12}^{\mathrm{tot}}(x', \mu | x, \lambda) \delta\left(\frac{\chi_{12,x(\Box)}}{\chi_{34,x'(\bullet)}}\right) : \Lambda_{12,\lambda}(x)\Lambda_{34,\mu}(x') : \right\}. \end{aligned} \qquad (6.51)$$
Similar to the proof in Thm. 6.3, we redefine the sum of the first term as $\lambda \to \lambda = \lambda' + \square$, $\mu \to \mu = \mu' + \blacksquare$. After this, the first term will be

$$ \sum_{\substack{\lambda', \mu' \blacksquare \in R(\mu') \\ \square \in A(\lambda')}} \sum_{\substack{F_{12}^{D4}[\lambda' + \square] F_{34}^{D4}[\mu' - \blacksquare] \\ \text{Res}_{\substack{\chi_{12,x}(\square) \\ =\chi_{34,x'}(\blacksquare)}} \left(\frac{x}{x'}\right)^{-1} \mathcal{Z}_{34|12}^{\text{tot}}(x', \mu' - \blacksquare | x, \lambda' + \square) \\ \times \delta\left(\frac{\chi_{12,x}(\square)}{\chi_{34,x'}(\blacksquare)}\right) : \Lambda_{12,\lambda'+\square}(x) \Lambda_{34,\mu'-\blacksquare}(x') : .} \quad (6.52) $$

Using

$$ \frac{F_{12}^{D4}[\lambda + \square] F_{34}^{D4}[\mu - \blacksquare]}{F_{12}^{D4}[\lambda] F_{34}^{D4}[\mu]} = - \frac{\text{Res}_{\substack{\chi_{12,x}(\square) \\ =\chi_{34,x'}(\blacksquare)}} \left(\frac{x}{x'}\right)^{-1} \mathcal{Z}_{34|12}^{\text{tot}}(x', \mu | x, \lambda) \times \text{Res}_{\substack{\chi_{12,x}(\square) \\ =\chi_{34,x'}(\blacksquare)}} \left(\frac{x}{x'}\right)^{-1} \mathcal{Z}_{34|12}^{\text{tot}}(x', \mu - \blacksquare | x, \lambda + \square)}{\chi_{34,x'}(\blacksquare)} \quad (6.53) $$

and

$$ \delta\left(\frac{\chi_{12,x}(\square)}{\chi_{34,x'}(\blacksquare)}\right) : \Lambda_{12,\lambda+\square}(x) \Lambda_{34,\mu-\blacksquare}(x') := \delta\left(\frac{\chi_{12,x}(\square)}{\chi_{34,x'}(\blacksquare)}\right) : \Lambda_{12,\lambda}(x) \Lambda_{34,\mu}(x') : \quad (6.54) $$

we obtain the claim.

<figure><img src="image_20.png" /></figure>

Remark 6.18. The extra factors in front of the *qq*-characters come from the operator product of the zero modes $\chi_{A,0}(x)$. We may modify the zero modes so that they obey (4.60) (see also footnote 19) and then get an exact commuting relation $[T_A(x), T_{\bar{A}}(x')] = 0$. Instead of doing that, we simply relax the commutativity condition and say that operators $\mathcal{O}(x)$ and $\mathcal{O}'(x')$ commute when they satisfy

$$ \mathcal{O}(x) - f(x, x')\mathcal{O}'(x') = 0 \quad (6.55) $$

where $f(x, x')$ are zero modes factors.

## 7 D6-brane *qq*-characters

We introduce the operator version of the D6 *qq*-characters in section 7.1. We show that monomial terms of the *qq*-character with the highest weight $W_{\bar{a}}(x)$ are classified by plane partitions and that it commutes with the screening charge $Q_a(x')$. We then show that the D6 *qq*-characters reproduce the tetrahedron instanton partition function in section 7.2. The D6 *qq*-characters can be obtained by fusion of an infinite number of D4 *qq*-characters (see section 7.3). Description in lower dimensional *qq*-characters are discussed in section 7.4. Finally, in section 7.5, we introduce general D6 *qq*-characters where truncations of plane partitions appear. We also give a conjecture of the D6 *qq*-characters associated with toric CY$_4$ taking the form toric $CY_3 \times \mathbb{C}$. The *qq*-characters we introduce in this section imply the existence of a large class of *qq*-characters which we call BPS *qq*-characters.
## 7.1 D6 $qq$-characters and plane partition

Following the construction of the D4 $qq$-characters and the affine quiver W-algebra, let us construct the D6 $qq$-characters. The D6 $qq$-characters are $qq$-characters whose generating currents are the D6 vertex operators $W_{\bar{a}}(x)$ ($a \in \mathbf{4}$) in (4.15). Similar to the D4 case, each term of the D6 $qq$-character is decomposed into the vertex operator and the coefficient parts. The operator part is determined by the iWeyl reflection of the operator part defined as the following.

**Definition 7.1.** The iWeyl reflection of the D6 vertex operator $W_{\bar{a}}(x)$ ($a \in \mathbf{4}$) is

$$W_{\bar{a}}(x) \rightarrow : W_{\bar{a}}(x) A^{-1}(x) : , \quad a \in \mathbf{4}. \qquad (7.1)$$

Using the property in (4.68b), the root current is rewritten in the D6 vertex operators as

$$W_{\bar{a}}(x) \rightarrow : a_0(x)^{-1} W_{\bar{a}}(q_a^{-1} x) \frac{\prod_{i \in \mathbf{4} \setminus \{a\}} W_{\bar{a}}(q_i x)}{\prod_{i \in \mathbf{4} \setminus \{a\}} W_{\bar{a}}(q_i^{-1} q_a^{-1} x)} : \qquad (7.2)$$

The operator part is obtained by changing the root currents to the D6-operators and applying the iWeyl reflection to the numerators recursively. After iWeyl reflections, the operators are classified by plane partitions. Let us study some terms after applying the iWeyl reflection. The operators obtained after $n$-times of iWeyl reflections are called operators at level $n$. We focus on the case $W_{\bar{4}}(x)$ and then have

$$W_{\bar{4}}(x) \rightarrow : a_0(x)^{-1} \frac{W_{\bar{4}}(q_{123}x) \prod_{i=1}^{3} W_{\bar{4}}(q_i x)}{\prod_{1 \le i < j \le 3} W_{\bar{4}}(q_i q_j x)} : \qquad (7.3)$$

* Level 0: we only have one operator

$$W_{\bar{4}}(x). \qquad (7.4)$$

We can associate this operator with a vacuum configuration of the plane partition in the space (1,2,3) where there is no box:

$$W_{\bar{4}}(x) \leftrightarrow \begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,0) {};
\node (2) at (1,0) {};
\node (3) at (2,0) {};
\draw (1) -- (2) node right () {$3$};
\draw (2) -- (3) node right () {$1$};
\draw (1) -- (3) node left () {$2$};
\node at (0.5,0.2) {$W_{\bar{4}}(x)$};
\end{tikzpicture} \qquad (7.5)$$

The spectral parameter $x$ corresponds to the $q$-coordinates in the origin. The operator $W_{\bar{4}}(x)$ represents the addable box of this plane partition configuration which is the box in the origin.

* Level 1: after applying the iWeyl reflection once, we have

$$: W_{\bar{4}}(x) A^{-1}(x) : := : a_0(x)^{-1} \frac{W_{\bar{4}}(q_{123}x) \prod_{i=1}^{3} W_{\bar{4}}(q_i x)}{\prod_{i<j \le 3} W_{\bar{4}}(q_i q_j x)} : \qquad (7.6)$$
The level 1 current can be described as

$$ : a_0(x)^{-1} \frac{\mathbf{W}_{\bar{4}}(q_4^{-1}x) \prod_{i=1}^{3} \mathbf{W}_{\bar{4}}(q_i x)}{\prod_{i<j \le 3} \mathbf{W}_{\bar{4}}(q_i q_j x)} : \longleftrightarrow \quad (7.7) $$

An observation is that the red terms $W_{\bar{4}}(q_i x)$ ($i = 1, 2, 3$) correspond to the addable boxes of this plane partition configuration. The variables $q_i x$ ($i = 1, 2, 3$) correspond to the $q$-coordinates of the addable boxes. The blue term $W_{\bar{4}}(q_4^{-1}x)$ corresponds to the removable box of the configuration, which is the box in the origin with coordinate $x$.

* Level 2: since we have three numerators in the level 1 operator, we will have three possible level 2 operators depending on which term we do the iWeyl reflection.

$$ 1. \pi_{1,1} = 1, \pi_{2,1} = 1 $$

$$ : W_{\bar{4}}(x) A^{-1}(x) A^{-1}(q_1 x) : \propto \frac{(2)(3)(1^2)(14^{-1})}{(1^2 2)(1^2 3)} \longleftrightarrow \quad (7.8) $$

where we simply denote $W_{\bar{4}}(q_1^a q_2^b q_3^c q_4^d x)$ as $(1^a 2^b 3^c 4^d)$. The $q$-coordinates of the addable boxes of this configuration are $q_1^2 x, q_2 x, q_3 x$ and correspond to the red terms in the numerator. The $q$-coordinate of the removable box of this configuration is $xq_1$ and correspond to the blue term with variable $q_4^{-1}q_1 x$.

$$ 2. \pi_{1,1} = 1, \pi_{1,2} = 1 $$

$$ : W_{\bar{4}}(x) A^{-1}(x) A^{-1}(q_2 x) : \propto \frac{(1)(3)(2^2)(12^2 3)}{(1^2 2)(2^2 3)} \longleftrightarrow \quad (7.9) $$

Similarly, the red terms correspond to the addable boxes and the blue terms correspond to the removable boxes of the plane partition configuration.

$$ 3. \pi_{1,1} = 2 $$

$$ : W_{\bar{4}}(x) A^{-1}(x) A^{-1}(q_3 x) : \propto \frac{(1)(2)(3^2)(123^2)}{(1^3 2)(2^3 3)} \longleftrightarrow \quad (7.10) $$

Again, the red terms correspond to the addable boxes and the blue terms correspond to the removable boxes of the plane partition configuration.
We can do this procedure recursively and then obtain the following statement.

**Lemma 7.2.** The operators generated from the iWeyl reflection starting from $W_{\bar{a}}(x)$ are classified by plane partitions:

$$ \Lambda_{\bar{a},\pi}(x) := :W_{\bar{a}}(x) \prod_{\mathfrak{a} \in \pi} A^{-1}(\chi_{\bar{a},x}(\mathfrak{a})) :; \quad a \in \mathbf{4}. \qquad (7.11) $$

Converting the root currents into the D6-operators, we have

$$ W_{\bar{a}}(x) \prod_{\mathfrak{a} \in \pi} A^{-1}(\chi_{\bar{a},x}(\mathfrak{a})) \propto \frac{\prod_{\mathfrak{a} \in A(\pi)} W_{\bar{a}}(\chi_{\bar{a},x}(\mathfrak{a})) \prod_{\mathfrak{a} \in R(\pi)} W_{\bar{a}}(q_a^{-1}\chi_{\bar{a},x}(\mathfrak{a}))}{\#} \qquad (7.12) $$

where we have some extra denominators determined recursively.<fnref n="30" />

The D6 $qq$-character is a sum of the vertex operators $\Lambda_{\bar{a},\pi}(x)$ with some specific coefficients:

$$ \mathcal{T}_{\bar{a}}(x) = \sum_{\pi \in \mathcal{P}\mathcal{P}} F_{\bar{a}}^{\mathrm{D6}}[\pi] \Lambda_{\bar{a},\pi}(x). \qquad (7.13) $$

We impose the condition that this $qq$-character commutes with the screening charge $\Omega_a(x)$. After imposing this condition, the coefficient is determined uniquely.

**Definition 7.3.** We define the D6 $qq$-character for $a \in \mathbf{4}$ as

$$ \mathcal{T}_{\bar{a}}(x) = \sum_{\pi \in \mathcal{P}\mathcal{P}} \tilde{Z}_{\bar{a}}^{\mathrm{D6}}[\pi] \Lambda_{\bar{a},\pi}(x), \quad a \in \mathbf{4}, \qquad (7.14) $$

where the coefficients $\tilde{Z}_{\bar{a}}^{\mathrm{D6}}[\pi]$ are identified with the U(1) partition function of the 7d gauge theory on $\mathbb{C}_{\bar{a}}^3 \times \mathbb{S}^1$ in (3.117). Rescaling the zero-modes $A(x) \to q^{-1}A(x)$, we have

$$ \mathcal{T}_{\bar{a}}(x) = \sum_{\pi \in \mathcal{P}\mathcal{P}} q^{\lvert \pi \rvert} \tilde{Z}_{\bar{a}}^{\mathrm{D6}}[\pi] \Lambda_{\bar{a},\pi}(x). \qquad (7.15) $$

**Theorem 7.4.** The D6 $qq$-characters $\mathcal{T}_{\bar{a}}(x)$ ($a \in \mathbf{4}$) commutes with the screening charge $\Omega_a(x)$:

$$ [\mathcal{T}_{\bar{a}}(x), \Omega_a(x')] = 0. \qquad (7.16) $$

*Proof.* Let us focus on $\mathcal{T}_{\bar{a}}(x)$ and see how the commutativity appears. Using the formulas in Thm. (C.12) and the property in (4.66), we have

$$ \begin{gathered} [W_{\bar{4}}(x), S_4(x')] = q_4 x \delta(q_4 x/x') : W_{\bar{4}}(x) S_4(q_4 x) : \\ [:\, W_{\bar{4}}(x) A^{-1}(x) : , S_4(x')] = x \delta(x'/x) \frac{\prod_{i=1}^{3} (1 - q_i)}{\prod_{1 \le i < j \le 3} (1 - q_i q_j)} : W_{123}(x) S_4(q_4 x) : + \dots, \end{gathered} \qquad (7.17) $$

where for the second term, we only extracted the pole coming from $x' = x$. Thus, for the pole coming from $W_{\bar{4}}(x)$ to disappear up to a total difference, we need the combination

$$ W_{\bar{4}}(x) - q_4 \frac{\prod_{1 \le i < j \le 3} (1 - q_i q_j)}{\prod_{i=1}^{3} (1 - q_i)} : W_{\bar{4}}(x) A^{-1}(x) : \qquad (7.18) $$

<footnote n="30">³⁰The explicit form of the right-hand side is related to the shell formula of the plane partition [137]. In this paper, the information of the denominator is not necessary so we will not write the explicit form.</footnote>
where the coefficient is $\tilde{Z}_4^{\text{D6}}[\mathfrak{D}]$ (see appendix (B.20)). Generally, using

$$ 
\begin{aligned}
\Lambda_{\bar{a},\pi}(x)S_4(x') &= -q_4x \left[ \mathcal{W}_{\pi,x}^{\bar{4}}(q_4^{-1}x')^{-1} \right]_{-}^{x'} : \Lambda_{\bar{a},\pi}(x)S_4(x') : , \\
S_4(x')\Lambda_{\bar{a},\pi}(x) &= -q_4x \left[ \mathcal{W}_{\pi,x}^{\bar{4}}(q_4^{-1}x')^{-1} \right]_{+}^{x'} : \Lambda_{\bar{a},\pi}(x)S_4(x') : .
\end{aligned}
\quad (7.19) $$

and the property in (3.123), we obtain

$$ 
\begin{aligned}
[\mathcal{T}_{\bar{4}}(x), S_4(x')] \\
&= q_4x \sum_{\pi \in \mathcal{P}\mathcal{P}} \tilde{Z}_4^{\text{D6}}[\pi] \left( \sum_{\mathfrak{D} \in A(\pi)} \operatorname{Res}_{x'=q_4\chi_{\bar{4},x}(\mathfrak{D})} x'^{-1} \mathcal{W}_{\pi,x}^{\bar{4}}(q_4^{-1}x')^{-1} \delta\left(\frac{x'}{q_4\chi_{\bar{4},x}(\mathfrak{D})}\right) : \Lambda_{\bar{4},\pi}(x)S_4(q_4\chi_{\bar{4},x}(\mathfrak{D})) : \right. \\
&\qquad \left. + \sum_{\mathfrak{D} \in R(\pi)} \operatorname{Res}_{x'=\chi_{\bar{4},x}(\mathfrak{D})} x'^{-1} \mathcal{W}_{\pi,x}^{\bar{4}}(q_4^{-1}x')^{-1} \delta\left(\frac{x'}{\chi_{\bar{4},x}(\mathfrak{D})}\right) : \Lambda_{\bar{4},\pi}(x)S_4(\chi_{\bar{4},x}(\mathfrak{D})) : \right).
\end{aligned}
\quad (7.20) $$

Shifting the second term as $\pi' = \pi - \mathfrak{D}$, the second term will be rewritten as

$$ \sum_{\pi' \in \mathcal{P}\mathcal{P}} \tilde{Z}_4^{\text{D6}}[\pi' + \mathfrak{D}] \sum_{\mathfrak{D} \in A(\pi')} \operatorname{Res}_{x'=\chi_{\bar{4},x}(\mathfrak{D})} x'^{-1} \mathcal{W}_{\pi' + \mathfrak{D},x}(q_4^{-1}x')^{-1} \delta\left(\frac{x'}{\chi_{\bar{4},x}(\mathfrak{D})}\right) : \Lambda_{\bar{4},\pi' + \mathfrak{D}}(x)S_4(\chi_{\bar{4},x}(\mathfrak{D})) : . \quad (7.21) $$

Using Thm. B.1, we have

$$ 
\begin{aligned}
&: \Lambda_{\bar{4},\pi'+\mathfrak{D}}(x)S_4(\chi_{\bar{4},x}(\mathfrak{D})) :=: \Lambda_{\bar{4},\pi'}(x)S_4(q_4\chi_{\bar{4},x}(\mathfrak{D})) : , \\
&\frac{\tilde{Z}_4^{\text{D6}}[\pi + \mathfrak{D}]}{\tilde{Z}_4^{\text{D6}}[\pi]} = - \frac{\operatorname{Res}_{x'=q_4\chi_{\bar{4},x}(\mathfrak{D})} x'^{-1} \mathcal{W}_{\pi,x}^{\bar{4}}(q_4^{-1}x')^{-1}}{\operatorname{Res}_{x'=\chi_{\bar{4},x}(\mathfrak{D})} x'^{-1} \mathcal{W}_{\pi+\mathfrak{D},x}(q_4^{-1}x')^{-1}} \quad (7.22)
\end{aligned}
$$

and then we obtain

$$ 
\begin{aligned}
[\mathcal{T}_{123}(x), S_4(x')] \\
&= q_4x \sum_{\pi \in \mathcal{P}\mathcal{P}} \tilde{Z}_4^{\text{D6}}[\pi] \sum_{\mathfrak{D} \in A(\pi)} \operatorname{Res}_{x'=q_4\chi_{\bar{4},x}(\mathfrak{D})} x'^{-1} \mathcal{W}_{\pi,x}^{\bar{4}}(q_4^{-1}x')^{-1} : \mathcal{W}_{\bar{4}}(x) \prod_{\mathfrak{D} \in \pi} \mathcal{A}^{-1}(\chi_{\bar{4},x}(\mathfrak{D})) S_4(q_4\chi_{\bar{4},x}(\mathfrak{D})) \\
&\qquad \times \left( \delta\left(\frac{x'}{\chi_{\bar{4},x}(\mathfrak{D})}\right) - \delta\left(\frac{x'}{q_4\chi_{\bar{4},x}(\mathfrak{D})}\right) \right)
\end{aligned}
\quad (7.23) $$

which gives the claim.

<figure><img src="image_21.png" /></figure>

The Thm. 5.2, 6.3, 7.4 are summarized in the following theorem.

**Theorem 7.5.** Let $\mathcal{T}$ be the tetrahedron corresponding to the $\mathbb{C}^4$ geometry (see figure 6). We denote the set of vertices, edges, and faces of $\mathcal{T}$ as $\mathbf{v} = \{a | a \in \mathbf{4}\}$, $\mathbf{e} = \{A | A \in \mathbf{6}\}$, $\mathbf{f} = \{\bar{a} | a \in \mathbf{4}\}$ respectively. We also introduce the union of them as $\mathcal{S} = \mathbf{v} \cup \mathbf{e} \cup \mathbf{f}$. For each element of $i \in \mathcal{S}$, we can associate a $\mathbf{qq}$-character. If $i \in \mathbf{v}$, we associate the D2 $\mathbf{qq}$-character (screening charge), if $i \in \mathbf{e}$, we associate the D4 $\mathbf{qq}$-character, if $i \in \mathbf{f}$, we associate the D6
qq-character. The qq-character associated with the elements $i, j \in \mathcal{T}$ commute with each other up to trivial zero modes (see (6.55)) when $i$ and $j$ do not intersect in $\mathcal{T}$:

$$ \mathcal{T}_i(x)\mathcal{T}_j(x') - f_{ij}(x, x')\mathcal{T}_j(x')\mathcal{T}_i(x) = 0 \iff i \cap j = \emptyset, \quad (7.24) $$

where $f_{ij}(x, x')$ are zero modes.

<figure><img src="image_22.png" /></figure>

**Remark 7.6.** Note that the D6 qq-characters $\mathcal{T}_{\bar{a}}(x)$ ($a \in \underline{4}$) are the operator version of the D6 qq-characters introduced in (3.231):

$$ \mathcal{T}_{\bar{a}}(x) \leftrightarrow \hat{\mathcal{T}}^a(x), \hat{\mathcal{T}}^{a\vee}(x), \quad a \in \underline{4}. \quad (7.25) $$

The highest weight and the root current have the following correspondence:

$$ \begin{aligned} W_{\bar{a}}(x) & \leftrightarrow \hat{U}^a(q_a x)^{-1}, \hat{U}^{a\vee}(q_a x)^{-1}, \quad a \in \underline{4} \\ A(x)^{-1} & \leftrightarrow \hat{A}(x)^{-1}, \hat{A}^{\vee}(x)^{-1}. \end{aligned} \quad (7.26) $$

**Remark 7.7.** Choosing the screening charge $\mathcal{Q}_4(x)$, Thm. 7.5 claims that the kernel of $\mathcal{Q}_4(x)$ is generated by $\mathcal{Q}_{1,2,3}(x)$, $\mathcal{T}_{12,23,13}(x)$, $\mathcal{T}_{123}(x)$. We expect that this will give a larger algebra compared with the affine quiver W-algebra given by the kernel of two screening charges. We leave a detailed analysis of this for future work.

**Remark 7.8.** We have shown here that the D6 qq-character $\mathcal{T}_{\bar{a}}(x)$ is uniquely determined by the commutativity with the screening charge $\mathcal{Q}_a(x)$ and that the coefficient factor is the U(1) partition function of the 7d theory on $\mathbb{C}_a^3 \times \mathbb{S}^1$. Compared with the magnificent four system where we need to take care of the sign problem [15, 16, 80], our discussion here gives an algebraic proof showing that there will be no sign problem for the 7d case. This is indeed compatible with a mathematical proof given in [53].

## 7.2 Tetrahedron instantons and D6 qq-characters

The expanded version of the tetrahedron instanton partition function can be expressed using the D6 qq-characters.

**Lemma 7.9.** The operator product of $\{\Lambda_{\bar{a},\pi}(x)\}_{a \in \underline{4}}$ is

$$ \Lambda_{\bar{b}, \pi^{(2)}}(x_2) \Lambda_{\bar{a}, \pi^{(1)}}(x_1) = \mathcal{Z}_{1-\text{loop}}^{\text{D6-D6}}(x_1, \bar{a} | x_2, \bar{b}) \mathcal{Z}_{\bar{a}|\bar{b}}^{\text{D6-D6}}(x_1, \pi^{(1)} | x_2, \pi^{(2)}) : \Lambda_{\bar{a}, \pi^{(1)}}(x_1) \Lambda_{\bar{b}, \pi^{(2)}}(x_2) : . \quad (7.27) $$

When $a = b$, it gives the vector multiplet contributions, while when $a \neq b$, it gives the bifundamental contribution connecting gauge theories defined on different $\mathbb{C}^3$ subspaces.
**Theorem 7.10.** The gauge origami partition function of the tetrahedron instanton is written using the D6 *qq*-characters:

$$ Z_{\text{1-loop}}^{\text{D6}} Z_{\text{inst.}}^{\text{D6}} = \sum_{\vec{\pi}} q^{|\vec{\pi}|} Z_{\text{1-loop}}^{\text{D6}} Z_{\text{tet.inst.}}^{\text{D6}} [\vec{v}, \vec{\pi}] = \langle 0 | \prod_{a \in \mathfrak{D}} \prod_{\alpha=1}^{n_{\bar{a}}} \mathcal{T}_{\bar{a}}(v_{\bar{a},\alpha}) | 0 \rangle. \quad (7.28) $$

Explicitly, we have the following table of BPS/CFT correspondence.

<table><thead><tr><th>BPS</th><th>CFT</th></tr></thead><tbody><tr><td>7d U(1) theory on C<sup>3</sup><sub>abc</sub> &times; S<sup>1</sup> (D6<sub>abc</sub> &times; 1)</td><td>&lt;0| T<sub>abc</sub>(v)|0&gt;</td></tr><tr><td>7d U(n<sub>ab</sub>) theory on C<sup>3</sup><sub>abc</sub> &times; S<sup>1</sup> (D6<sub>abc</sub> &times; n<sub>ab</sub>)</td><td>&lt;0| T<sub>abc</sub>(v<sub>n<sub>ab</sub></sub>) &cdots; T<sub>abc</sub>(v<sub>2</sub>) T<sub>abc</sub>(v<sub>1</sub>)|0&gt;</td></tr><tr><td>generalized folded instantons: D6<sub>123</sub>-D6<sub>234</sub>-D0</td><td>&lt;0| T<sub>123</sub>(v) T<sub>234</sub>(v')|0&gt;</td></tr><tr><td>gauge origami of tetrahedron instantons</td><td>&lt;0| &prod;<sub>a&isin;D</sub> &prod;<sub>&alpha;=1</sub><sup>n<sub>a</sub></sup> T<sub>a</sub>(v<sub>a&alpha;</sub>)|0&gt;</td></tr></tbody></table>

## 7.3 Fusion of D4 *qq*-characters to D6 *qq*-characters

Let us see that the D6 *qq*-characters are obtained by fusion of the D4 *qq*-characters. By studying the zeros and pole structure, one can show the following lemma.

**Lemma 7.11.** Given two Young diagrams $\lambda^{(1)}, \lambda^{(2)}$, the contraction of the operators $\Lambda_{A,\lambda}(x)$ are given in Lemma 6.5 as

$$ \Lambda_{A,\lambda^{(2)}}(x_2)\Lambda_{A,\lambda^{(1)}}(x_1) = Z_{\text{1-loop}}^{\text{D4-D4}}(x_1, A|x_2, A)Z_{A|A}^{\text{D4-D4}}(x_1, \lambda^{(1)}|x_2, \lambda^{(2)}): \Lambda_{A,\lambda^{(2)}}(x_2)\Lambda_{A,\lambda^{(1)}}(x_1): \quad (7.30) $$

When the Young diagrams obey $\lambda^{(2)} \succ \lambda^{(1)}$ with the parameters as $x_2 = q_a x_1$ ($a \in \bar{A}$), we have

$$ Z_{A|A}^{\text{D4-D4}}(x_1, \lambda^{(1)}|q_a x_1, \lambda^{(2)}) = 0, \quad a \in \bar{A} \quad (7.31) $$

which gives

$$ \Lambda_{A,\lambda^{(2)}}(x_2)\Lambda_{A,\lambda^{(1)}}(x_1) = 0. \quad (7.32) $$

Using the above lemma, finite products of the D4 *qq*-characters are given as

$$ \mathcal{T}_A(q_a^{N-1}x) \cdots \mathcal{T}_A(q_a x) \mathcal{T}_A(x) = \sum_{\lambda^{(N)} \preceq \cdots \preceq \lambda^{(2)} \preceq \lambda^{(1)}} q^{\sum_{i=1}^N |\lambda^{(i)}|} \tilde{Z}_A^{\text{D4}}[\lambda^{(1)}] \cdots \tilde{Z}_A^{\text{D4}}[\lambda^{(N)}] \\ \times \Lambda_{A,\lambda^{(N)}}(q_a^{N-1}x) \cdots \Lambda_{A,\lambda^{(2)}}(q_a x) \Lambda_{A,\lambda^{(1)}}(x) \quad (7.33) $$

for $\forall a \in \bar{A}$. Using the operator product and extracting the one-loop perturbative factor, we define the renormalized N-fusion D4 *qq*-character as

$$ \bar{T}_{A;a}^{(N)}(x) = \sum_{\lambda^{(N)} \preceq \cdots \preceq \lambda^{(1)}} q^{\sum_{i=1}^{N} |\lambda^{(i)}|} \prod_{i=1}^{N} \tilde{Z}_{A}^{\text{D4}}[\lambda^{(i)}] \prod_{i<j} Z_{A|A}^{\text{D4-D4}}(q_a^{i-1}x, \lambda^{(i)}|q_a^{j-1}x, \lambda^{(j)}) : \prod_{i=1}^{N} \Lambda_{A,\lambda^{(i)}}(q_a^{i-1}x) : \quad (7.34) $$
Note that this is the rank $N$ $qq$-character in (6.32) with spectral parameters tuned as $\bar{x} = (x_i)_{i=1}^N$ and $x_i = q_a^{i-1}x$ for $a \in \bar{A}$. Taking the limit $N \to \infty$ and considering infinite products of the $qq$-characters, we can see that they are related to the D6 $qq$-character.

**Theorem 7.12.** Taking the limit $N \to \infty$ of the renormalized $N$-fusion D4 $qq$-characters $\bar{T}_{ab;c}^{(N)}(x)$ give the D6 $qq$-character $T_{abc}(x)$:

$$ \bar{T}_{ab;c}^{(N)}(x) \xrightarrow{N \to \infty} T_{abc}(x). \qquad (7.35) $$

Equivalently, we have

$$ \left. \prod_{i=1}^{\infty} T_{ab}(x q_c^{i-1}) \right\} \simeq T_{abc}(x) \qquad (7.36) $$

where the symbol “$\simeq$” means the equality is true up to one-loop perturbative factors.

*Proof.* Let us focus on the case $\bar{T}_{12;3}^{(N)}(x)$ and $T_{123}(x)$. Since the infinite products diverge, we need to regularize it properly (see for example [138]). Moreover, we need to take the inductive limit so that at large $N$, $\bar{T}_{12;3}^{(N)}(x) = \bar{T}_{12;3}^{(N+1)}(x)$ and we denote this $\bar{T}_{12;3}^{(\infty)}(x)$. We only give a sketch of the proof so see [138] for details. The operator $\bar{T}_{12;3}^{(\infty)}(x)$ is expanded as

$$ \bar{T}_{12;3}^{(\infty)}(x) = \sum_{\emptyset \subset \dots \subset \lambda^{(2)} \subset \lambda^{(1)}} q^{\sum_{i=1}^{\infty} |\lambda^{(i)}|} \prod_{i=1}^{\infty} \tilde{Z}_{12}^{D4}[\lambda^{(i)}] \prod_{i<j}^{\infty} Z_{12|12}^{D4-D4}(x_i, \lambda^{(i)} | x_j, \lambda^{(j)}) : \prod_{i=1}^{\infty} \Lambda_{12,\lambda^{(i)}}(q_3^{i-1}x): \qquad (7.37) $$

where $x_i = q_3^{i-1}x$. Since the right-hand side is expanded in all possible Young diagrams $\{\lambda^{(i)}\}$ obeying the condition $\lambda^{(i)} \simeq \lambda^{(i+1)}$, the right-hand side is expanded in all possible plane partitions (see the (1,2) description in section 2): $\pi = (\lambda^{(1)}, \lambda^{(2)}, \dots, \emptyset, \emptyset, \dots)$. Let us show that the coefficients and the operator part indeed can be written using the plane partition. The infinite product of the U(1) part is given by

$$ \prod_{i=1}^{\infty} \tilde{Z}_{12}^{D4}[\lambda^{(i)}] = \prod_{i=1}^{\infty} \prod_{\square \in \lambda^{(i)}} S_{34} \left( \frac{x_i}{\chi_{12, x_i}(\square)} \right) \prod_{\substack{\square \in \lambda^{(i)} \\ \bullet \in \lambda^{(i)}}} g_{\square} \left( \frac{\chi_{12, x_i}(\square)}{\chi_{12, x_i}(\bullet)} \right)^{-1}. \qquad (7.38) $$

We can rewrite the vector multiplet part as

$$ \prod_{i<j} Z_{12|12}^{D4-D4}(x_i, \lambda^{(i)} | x_j, \lambda^{(j)}) = \prod_{i<j} \prod_{\square \in \lambda^{(i)}} S_{34} \left( \frac{x_j}{\chi_{12, x_i}(\square)} \right) \prod_{\bullet \in \lambda^{(j)}} S_{34} \left( \frac{x_i}{\chi_{12, x_j}(\bullet)} \right) \prod_{\substack{\square \in \lambda^{(i)} \\ \bullet \in \lambda^{(j)}}} A_{\square} \left( \frac{\chi_{12, x_i}(\square)}{\chi_{12, x_j}(\bullet)} \right)^{-1}. \qquad (7.39) $$

Using

$$ \prod_{1 \le i < j \le N} \prod_{\substack{\square \in \lambda^{(i)} \\ \bullet \in \lambda^{(j)}}} A_{\square} \left( \frac{\chi_{12, x_i}(\square)}{\chi_{12, x_j}(\bullet)} \right)^{-1} = \prod_{i \ne j} \prod_{\substack{\square \in \lambda^{(i)} \\ \bullet \in \lambda^{(j)}}} g_{\square} \left( \frac{\chi_{12, x_i}(\square)}{\chi_{12, x_j}(\bullet)} \right)^{-1} \qquad (7.40) $$
and

$$ 
\begin{aligned}
& \prod_{1 \le i < j \le \infty} \prod_{\square \in \lambda^{(i)}} S_{34} \left( \frac{x_j}{\chi_{12, x_i}(\square)} \right) \prod_{\blacksquare \in \lambda^{(j)}} S_{34} \left( \frac{x_i}{\chi_{12, x_j}(\blacksquare)} \right) \\
& = \prod_{i=1}^{\infty} \prod_{\square \in \lambda^{(i)}} S_{34} \left( \frac{x_i}{\chi_{12, x_i}(\square)} \right)^{-1} \prod_{i=1}^{\infty} \prod_{\square \in \lambda^{(i)}} \frac{(1 - q_4 x / \chi_{12, x_i}(\square))}{(1 - x / \chi_{12, x_i}(\square))},
\end{aligned} 
\quad (7.41) $$

the coefficient part will be

$$ \prod_{i=1}^{\infty} \tilde{Z}_{12}^{D4}[\lambda^{(i)}] \prod_{i<j} Z_{12|12}^{D4-D4}(x_i, \lambda^{(i)}|x_j, \lambda^{(j)}) = \prod_{\square \in \pi} \frac{(1-q_4 x / \chi_{\bar{4},x}(\square))}{(1-x/\chi_{\bar{4},x}(\square))} \prod_{\substack{\square \in \pi \\ \bar{\square} \in \pi}} g_{\bar{4}} \left( \frac{\chi_{\bar{4},x}(\square)}{\chi_{\bar{4},x}(\bar{\square})} \right)^{-1} = \tilde{Z}_{4}^{D6}[\pi]. \quad (7.42) $$

The operator part is obtained from

$$ \prod_{i=1}^{\infty} \Lambda_{12, \lambda^{(i)}}(q_3^{i-1} x) := \prod_{i=1}^{\infty} \chi_{12}(q_3^{i-1} x) \prod_{i=1}^{\infty} \prod_{\square \in \lambda^{(i)}} A(\chi_{12, x_i}(\square))^{-1} := \prod_{\square \in \pi} \chi_{\bar{4}}(x) \prod_{\bar{\square} \in \pi} A(\chi_{\bar{4}, x}(\bar{\square}))^{-1} \quad (7.43) $$

where we used

$$ \prod_{i=1}^{\infty} \chi_{12}(q_3^{i-1} x) := \prod_{i=1}^{\infty} \frac{W_{\bar{4}}(q_3^{i-1} x)}{W_{\bar{4}}(q_3^i x)} := W_{\bar{4}}(x). \quad (7.44) $$

We then get the identity $\bar{T}_{12;3}^{(\infty)}(x) = T_{123}(x)$. $\square$

Remark 7.13. The fusion procedure can be visualized as the case of the fusion of D2 to D4. Using the correspondence in (6.4) (see also section 2), we have

<figure><img src="image_23.png" /></figure>

## 7.4 Description in lower dimensional qq-characters

The previous theorem already implies that the gauge origami partition function of tetrahedron instantons can be reproduced algebraically using infinite products of lower dimensional qq-characters. Let us explicitly see this procedure.

(1, 2)-type description and D4 qq-characters. Using the decomposition of the character in (3.181), the gauge origami partition function can be rewritten using the vertex operators $\Lambda_{A,\lambda}(x)$ as follows.
**Lemma 7.14.** We have the following identity up to one-loop perturbative factors:

$$Z_{\text{1-loop}}^{\text{D6}} Z_{\text{tetinst.}}^{\text{D6}}[\vec{v}, \vec{\pi}] \approx \langle 0 | \prod_{(abc) \in \mathbf{4}^{\vee}} \prod_{\alpha=1}^{n_{abc}} \prod_{k=1}^{\infty} \tilde{Z}_{ab}^{\text{D4}}[\lambda_{abc,\alpha}^{(k)}] \Lambda_{ab,\lambda_{abc,\alpha}^{(k)}}(xq_c^{k-1}) |0 \rangle, \quad (7.46)$$

where the plane partitions on the right-hand side are rewritten in the (1, 2)-type description as Young diagrams on the left-hand side.

Using the property (7.32), we also have the following theorem.

**Theorem 7.15.** The gauge origami partition function of the tetrahedron instanton system is written using infinite products of D4 *qq*-characters.

$$Z_{\text{1-loop}}^{\text{D6}} Z_{\text{inst.}}^{\text{D6}} = \sum_{\vec{\pi}} q^{|\vec{\pi}|} Z_{\text{1-loop}}^{\text{D6}} Z_{\text{tetinst.}}^{\text{D6}}[\vec{v}, \vec{\pi}] \approx \langle 0 | \prod_{(abc) \in \mathbf{4}^{\vee}} \prod_{\alpha=1}^{n_{abc}} \prod_{i=1}^{\infty} \mathcal{T}_{ab}(v_{\alpha} q_c^{i-1}) |0 \rangle \quad (7.47)$$

**Description in screening currents.** We can also represent the partition function of the tetrahedron instanton system using the screening currents. Using the character form in (3.179), we omit the singular terms that might occur and extract the square root part of the total index by specifying an order in the elements as

$$V = \sum_{a,b \in \mathbf{4}} \frac{\mathbf{P}_{\mathbf{4}}}{\mathbf{P}_{\mathbf{k}(a)}^{\vee} \mathbf{P}_{\mathbf{k}(b)}} \mathbf{X}_{\bar{a}}^{\vee} \mathbf{X}_{\bar{b}} \rightarrow v = \sum_{\substack{(x,a) < (x',b) \\ x \in \mathcal{X}_{\bar{a}}, x' \in \mathcal{X}_{\bar{b}}}} \frac{\mathbf{P}_{\mathbf{4}}}{\mathbf{P}_{\mathbf{k}(a)}^{\vee} \mathbf{P}_{\mathbf{k}(b)}} \left(\frac{x'}{x}\right). \quad (7.48)$$

Using the contraction formulas of the screening currents in (4.41), the gauge origami partition function of the tetrahedron instantons is rewritten using the screening currents.

**Theorem 7.16.** We have the following identity up to one-loop perturbative factors:

$$\mathbb{I}[\mathbf{v}] = \langle 0 | \prod_{\substack{a \in \mathbf{4} \\ x \in \mathcal{X}_{\bar{a}}}} \mathbf{S}_{\mathbf{k}(a)}(x) | 0 \rangle \approx Z_{\text{1-loop}}^{\text{D6}} Z_{\text{tetinst.}}^{\text{D6}}[\vec{v}, \vec{\pi}]. \quad (7.49)$$

Note that we are implicitly defining an order in the products of the screening currents as

$$\cdots \mathbf{S}_{\mathbf{k}(b)}(x') \cdots \mathbf{S}_{\mathbf{k}(a)}(x) \cdots \iff (x, a) < (x', b). \quad (7.50)$$

## 7.5 General D6 *qq*-characters

We can consider higher rank analogs of them similar to the D4 *qq*-character

$$\mathcal{T}_{\bar{a}}^{(n)}(\underline{x}) := \mathcal{W}_{\bar{a}}(x_1) \cdots \mathcal{W}_{\bar{a}}(x_n) : + \cdots \quad (7.51)$$

This time the coefficients appearing in the expansion of the right-hand side correspond to the partition function of the U(n) gauge theory on the D6-brane on $\mathbb{C}_{\bar{a}}^3 \times \mathbb{S}^1$.

Analogous to the D4 case, one would like to add negative weights as the highest weights. General D6 *qq*-characters including negative weights are written as

$$\mathcal{T}_{\bar{a}}^{(n|m)}(\underline{x} | \underline{y}) := \frac{\mathcal{W}_{\bar{a}}(x_1) \cdots \mathcal{W}_{\bar{a}}(x_n)}{\mathcal{W}_{\bar{a}}(y_1) \cdots \mathcal{W}_{\bar{a}}(y_m)} : + \cdots \quad (7.52)$$

The explicit coefficients are then obtained by imposing the commutativity with the screening charge $\mathcal{Q}_a(x')$.
Similar to the D4 case, one might think we need to introduce the negative iWeyl reflection introduced to reproduce the supergroup analog of D4 $qq$-characters. However, noticing that

$$W_{\bar{a}}(x)^{-1}S_a(x') = x'^{-1}(1 - q_a^{-1}x'/x) : W_{\bar{a}}(x)^{-1}S_a(x') : \quad (7.53)$$

gives no new poles, $W_a(x)^{-1}$ and $S_a(x')$ commute with each other. Thus, compared to the D4 case where we need to introduce negative iWeyl reflections to cancel the poles arising from the operators in the denominators, for the D6 case, we do not need to do such kind of procedure. The right-hand side of (7.52) is expanded by the plane partitions generated by $W_{\bar{a}}(x_i)$ ($i = 1, \dots, n$) and no plane partitions generated by $W_{\bar{a}}(y_j)^{-1}$ ($j = 1, \dots, m$). We expect the coefficients appearing on the right-hand side give the instanton partition function of the 7d $U(n|m)$ theory on $\mathbb{C}_a^3 \times \mathbb{S}^1$. We leave a detailed analysis of this for future work.

For later use, let us consider the case when there is only one positive and one negative weight:

$$T_{\bar{a}}^{(1|1)}(x | Kx) := \frac{W_{\bar{a}}(x)}{W_{\bar{a}}(Kx)} : + \dots \quad (7.54)$$

where $K$ is a generic parameter. The contraction $W_{\bar{a}}(Kx)^{-1}$ with the screening current $S_4(x')$ will give a pole free rational function but when $K$ is generic, no poles will be canceled and the coefficients are only modified:

$$S_a(x') \Lambda_{\bar{a}, \pi}^{K}(x) = K^{-1} \left[ \left( 1 - \frac{K q_a x}{x'} \right) W_{\pi, x}^{\bar{a}} (q_a^{-1} x')^{-1} \right]_+ : S_a(x') \Lambda_{\bar{a}, \pi}^{K}(x) : , \\ W_{\pi, x}^{\bar{a}}(x') \rightarrow \frac{1}{(1 - K x/x')} W_{\pi, x}^{\bar{a}, K}(x') := W_{\pi, x}^{\bar{a}, K}(x'), \quad \tilde{Z}_{\bar{a}}^{\mathrm{D}6}[\pi] \rightarrow \prod_{\mathfrak{a} \in \pi} \frac{1 - K x / \chi_{\bar{a}, x}(\mathfrak{a})}{1 - K q_a x / \chi_{\bar{a}, x}(\mathfrak{a})} \tilde{Z}_{\bar{a}}^{\mathrm{D}6}[\pi], \quad (7.55)$$

where

$$\Lambda_{\bar{a}, \pi}^{K}(x) := \frac{W_{\bar{a}}(x)}{W_{\bar{a}}(Kx)} \prod_{\mathfrak{a} \in \pi} A^{-1}(\chi_{\bar{a}, x}(\mathfrak{a})) :. \quad (7.56)$$

Thus, we will obtain

$$T_{\bar{a}}^{(1|1)}(x | Kx) = \sum_{\pi \in \mathcal{P}\mathcal{P}} \tilde{Z}_{\bar{a}}^{\mathrm{D}6}[K, \pi] \Lambda_{\bar{a}, \pi}^{K}(x), \\ \tilde{Z}_{\bar{a}}^{\mathrm{D}6}[K, \pi] = \prod_{\mathfrak{a} \in \pi} \frac{(1 - Kx/\chi_{\bar{a},x}(\mathfrak{a})) (1 - q_a x/\chi_{\bar{a},x}(\mathfrak{a}))}{(1 - Kq_a x/\chi_{\bar{a},x}(\mathfrak{a})) (1 - x/\chi_{\bar{a},x}(\mathfrak{a}))} \prod_{\substack{\mathfrak{a} \in \pi \\ \mathfrak{a} \in \pi}} g_{\bar{a}} \left( \frac{\chi_{\bar{a},x}(\mathfrak{a})}{\chi_{\bar{a},x}(\mathfrak{a})} \right)^{-1}. \quad (7.57)$$

Note that after rescaling $A(x) \rightarrow q^{-1}A(x)$ we can change the topological term to $q^{\lvert \pi \rvert}$. For later use, let us list some properties of these operators. The operator products of $\{\Lambda_{\bar{a},\pi}^{K}(x)\}$ are

$$\Lambda_{\bar{b}, \pi^{(2)}}^{K_2}(x_2) \Lambda_{\bar{a}, \pi^{(1)}}^{K_1}(x_1) = \mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D6-D6}}(x_1, \bar{a}, K_1 | x_2, \bar{b}, K_2) \mathcal{Z}_{\bar{a}; K_1 | \bar{b}; K_2}^{\mathrm{D6-D6}}(x_1, \pi^{(1)} | x_2, \pi^{(2)}) \times : \Lambda_{\bar{b}, \pi^{(2)}}^{K_2}(x_2) \Lambda_{\bar{a}, \pi^{(1)}}^{K_1}(x_1) : \quad (7.58)$$

where

$$\mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D6-D6}}(x_1, \bar{a}, K_1 | x_2, \bar{b}, K_2) = \frac{\mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D6-D6}}(x_1, \bar{a} | x_2, \bar{b}) \mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D6-D6}}(K_1 x_1, \bar{a} | K_2 x_2, \bar{b})}{\mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D6-D6}}(K_1 x_1, \bar{a} | x_2, \bar{b}) \mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D6-D6}}(x_1, \bar{a} | K_2 x_2, \bar{b})}, \quad (7.59a)$$
$$
\begin{equation}
\begin{split}
\mathcal{Z}_{\bar{a}; K_1|b; K_2}^{\text{D6-D6}}(x_1, \pi^{(1)} | x_2, \pi^{(2)}) ={}& \mathcal{Z}_{\bar{a}|b}^{\text{D6-D6}}(x_1, \pi^{(1)} | x_2, \pi^{(2)}) \\
& \times \prod_{\mathbb{F} \in \pi^{(1)}} \left( q_b^{-1} \mathcal{V}_b \left( \frac{\chi_{\bar{a},x_1}(\mathbb{F})}{q_b K_2 x_2} \right) \right) \prod_{\mathbb{F} \in \pi^{(2)}} \mathcal{V}_a \left( \frac{K_1 x_1}{\chi_{\bar{b},x_2}(\mathbb{F})} \right)^{-1}.
\end{split}
\tag{7.59b}
\end{equation}
$$

**Pit reduction of D6 qq-characters.** When the parameter $K$ is generic, the coefficients are modified slightly without changing the structure of the *qq*-character. However, when we tune $K$ to specific values, the zeros appearing will cancel the poles, the iWeyl reflection will be restricted, and the right-hand side will not be expanded with arbitrary plane partitions but only by specified plane partitions. This can be understood also from the extra factors in the coefficients. When $K$ is tuned, the coefficient $\mathcal{Z}_{\bar{a}}^{\text{D6}}[K, \pi]$ will be zero for some plane partition configurations and such terms will disappear from the *qq*-character. This procedure is well known in the literature of MacMahon representations and is called the *pit reduction* [137, 139]. A plane partition with a pit P is a plane partition that does not contain a box at the position P.

Let us focus on the case when $a = 4$. For example, when we tune the parameter $K$ as $K = q_3x$, we have

$$ \therefore \frac{W_{\bar{4}}(x)}{W_{\bar{4}}(q_3 x)} := X_{12}(x) \qquad (7.60) $$

and the D6 *qq*-character will reduce to the D4 *qq*-character. This process is just placing a pit in $q_3x$ and reducing the plane partition in (123) and restricting it to a Young diagram in the (12)-plane:

<figure><img src="image_24.png" /></figure>

Physically, this is interpreted as the Higgs mechanism and also the tachyon condensation as discussed in section 4.2.

When $K = \chi_{\bar{4},x}(\mathbb{F})/x = q_1^{L-1}q_2^{M-1}q_3^{N-1}$, we will get a pit reduction of the plane partition:

$$ \mathcal{T}_{\bar{4}}^{\mathbb{F}}(x) := \frac{W_{\bar{4}}(x)}{W_{\bar{4}}(\chi_{\bar{4},x}(\mathbb{F}))} + \dots = \sum_{\substack{\pi:\text{plane partitions} \\ \text{with a pit at } \mathbb{F}}} \dots \qquad (7.62) $$

The highest weight has the following structure

$$
\begin{align}
\frac{W_{\bar{4}}(x)}{W_{\bar{4}}(q_1^{L-1} q_2^{M-1} q_3^{N-1} x)} &:= \prod_{i=1}^{L} \frac{W_{\bar{4}}(q_1^{i-1} q_2^{M-1} q_3^{N-1} x)}{W_{\bar{4}}(q_1^i q_2^{M-1} q_3^{N-1} x)} \prod_{j=1}^{M} \frac{W_{\bar{4}}(q_2^{j-1} q_3^{N-1} x)}{W_{\bar{4}}(q_2^j q_3^{N-1} x)} \prod_{k=1}^{N} \frac{W_{\bar{4}}(x q_3^{k-1})}{W_{\bar{4}}(x q_3^k)} : \nonumber \\
&:= \prod_{i=1}^{L} X_{23}(q_1^{i-1} q_2^{M-1} q_3^{N-1} x) \prod_{j=1}^{M} X_{13}(q_2^{j-1} q_3^{N-1} x) \prod_{k=1}^{N} X_{12}(x q_3^{k-1}) : \tag{7.63}
\end{align}
$$
which implies the pit reduction of the plane partition is related to the general D4 $qq$-characters $T_{12:23:13}^{(\vec{n}|\vec{0})}(\vec{x})$ in Thm. 6.14 after specializing the parameters as

$$n_{23} = L, n_{13} = M, n_{12} = N, \vec{x}_{12} = (xq_3^{k-1})_{k=1}^N, \vec{x}_{13} = (q_2^{j-1}q_3^{N-1}x)_{j=1}^M, \vec{x}_{12} = (q_1^{i-1}q_2^{M-1}q_3^{N-1}x)_{i=1}^L. \quad (7.64)$$

This fact is nothing special from the plane partition viewpoint. This is because we can pile $L$ Young diagrams spanning the (23)-plane, $M$ Young diagrams spanning the (13)-plane, and $N$ Young diagrams spanning the (12)-plane on top of each other to obtain all possible plane partitions with a pit at $(L, M, N)$. Note also that this decomposition in Young diagrams is not unique and so we have multiple descriptions in the D4 $qq$-characters. Physically, the system corresponding to the D6 $qq$-character with a pit-reduced plane partition is just the gauge origami system with folded instantons where the Coulomb branch parameters are tuned in a specific way.

Generally, we may add another pit to the plane partition and this is called the double-constrained plane partition which was introduced in [140] (see also the references there) to discuss minimal models of W-algebras. Let $(L_1, M_1, N_1)$ and $(L_2, M_2, N_2)$ be the coordinates of the two pits. The parameter $K$ needs to obey the conditions of the two pits

$$K = q_1^{L_1-1} q_2^{M_1-1} q_3^{N_1-1} = q_1^{L_2-1} q_2^{M_2-1} q_3^{N_2-1}. \quad (7.65)$$

Imposing this condition causes the $q$-parameters to be not generic anymore. The physical meaning of these types of $qq$-characters and their relation with minimal models are still unclear for the moment. We note that the condition above is just the Burge condition [141, 142], and thus the BPS/CFT correspondence arising should be an analog of the AGT dual of minimal models of W-algebras. See also [143] where some examples of these truncations were studied from the $qq$-character viewpoint.

**Plane partitions with boundary conditions.** The $qq$-characters are uniquely determined from the highest weight after imposing the commutativity with the screening charges. We further can construct a $qq$-character where each term corresponds to plane partitions with asymptotic Young diagrams $\lambda, \mu, \nu$ in the three axes:

<figure><img src="image_25.png" /></figure>
The left figure is the vacuum configuration of the plane partition with boundary conditions and the right figure is the configuration with boxes added to the vacuum configuration. The highest weight reproducing this configuration is

$$ : W_{\bar{a}}(x) \prod_{\vec{\sigma} \in \mathcal{P}_{\lambda, \mu, \nu}} A^{-1}(\chi_{\bar{a},x}(\vec{\sigma})) :; \qquad (7.67) $$

where $\mathcal{P}_{\lambda,\mu,\nu}$ is the set of boxes in the vacuum configuration. We denote the $qq$-character obtained from this highest weight as

$$ T_{\bar{a},\lambda\mu\nu}(x) := W_{\bar{a}}(x) \prod_{\vec{\sigma} \in \mathcal{P}_{\lambda,\mu,\nu}} A^{-1}(\chi_{\bar{a},x}(\vec{\sigma})) : + \dots \qquad (7.68) $$

As an example, let us focus on $a = 4$ and consider the case where the boundary conditions are $(\lambda, \mu, \nu) = (\square, \emptyset, \emptyset)$. The highest weight is

$$ : W_{\bar{4}}(x) \prod_{i=1}^{\infty} A^{-1}(x q_1^{i-1}) := : W_{\bar{4}}(x) \prod_{i=1}^{\infty} \frac{S_1(q_1^i x)}{S_1(q_1^{i-1} x)} := : \frac{W_{\bar{4}}(x)}{S_1(x)} : \qquad (7.69) $$

The contraction is

$$ : \frac{W_{\bar{4}}(x)}{S_1(x)} : S_4(q_4 x') = x' \frac{(1 - q_1 q_4 x'/x)}{(1 - q_3^{-1} x'/x)(1 - q_2^{-1} x'/x)} : \frac{W_{\bar{4}}(x)}{S_1(x)} S_4(q_4 x') : \qquad (7.70) $$

which gives poles at $x' = q_2x, q_3x$. One can show that the terms that cancel the poles coming from these terms are

$$ : W_{\bar{4}}(x) A^{-1}(q_2 x) S_1(x)^{-1} :; \quad : W_{\bar{4}}(x) A^{-1}(q_3 x) S_1(x)^{-1} : \qquad (7.71) $$

which correspond to the following plane partitions:

<figure><img src="image_26.png" /></figure>

By doing this procedure recursively, one will see that each term of the expansion of the $qq$-character indeed corresponds with the elements in $\mathcal{P}_{\square, \emptyset, \emptyset}$.

**Web of $qq$-characters.** We may represent the $qq$-character $T_{\bar{a},\lambda\mu\nu}$ using a trivalent vertex as

<figure><img src="image_27.png" /></figure>
where the three legs correspond to the three axes of the plane partitions with boundary conditions $\lambda, \mu, \nu$. This reminds us of the topological vertex [117–119, 144–146]. In the context of the topological vertex and the plane partition, one can glue the legs with the same boundary conditions. We expect that we can do the same gluing procedure for the D6 $qq$-characters and obtain $qq$-characters associated with complicated webs of the trivalent vertices such as

<figure><img src="image_28.png" /></figure>

Note that the trivalent vertex in (7.73) and the left and right diagrams in (7.74) are the brane webs dual to the toric diagrams of the $\mathbb{C}^3$, $\mathbb{C}^2/\mathbb{Z}_4 \times \mathbb{C}$ and the resolved conifold, respectively. Since the $qq$-character associated with the trivalent vertex captures the partition function of the D-branes on the $\mathbb{C}^3$ subspace of $\mathbb{C}^4$, we expect that the glued $qq$-characters will capture the partition functions of D-branes on general toric $\mathrm{CY}_3$ manifolds included in the $\mathrm{CY}_3 \times \mathbb{C}$ geometry. Such kind of generalizations are left for future work. We may also add boundary Young diagrams to each of the edges generally. They are expected to be $qq$-characters obtained after gluing $T_{\tilde{a},\lambda\mu\nu}(x)$.

We can combine the two kinds of $qq$-characters $T_{\tilde{a}}^{\mathbb{C}}(x)$, $T_{\tilde{a},\lambda\mu\nu}(x)$ and construct $qq$-characters where plane partitions with boundary conditions and a pit occur. The $qq$-character is uniquely determined from the highest weight as

$$ T_{abc,\lambda\mu\nu}^{L,M,N}(x) := \frac{W_{abc}(x)}{W_{abc}(q_a^{L-1}q_b^{M-1}q_c^{N-1}x)} \prod_{\mathcal{S} \in \mathcal{P}_{\lambda,\mu,\nu}} A^{-1}(\chi_{abc,x}(\mathcal{S})) : + \dots \quad (7.75) $$

We represent this $qq$-character using a trivalent vertex as

<figure><img src="image_29.png" /></figure>

and gluing of these $qq$-characters should lead to a larger class of $qq$-characters. We call this large class of $qq$-characters, the **web of $qq$-characters**.<fnref n="31" /> For the moment, we do not know how to glue these $qq$-characters explicitly and we leave it for future work.

<footnote n="31">³¹There is a similar concept called *web of W-algebras* [28] where the authors associated brane webs including integers inserted in the faces surrounded by the legs with W-algebras (7.76). Each trivalent vertex corresponds to the corner VOA [27] and the glued algebra gives the web of W-algebras. See also [147] where examples for the gluing process were explicitly done in the trigonometric language.</footnote>
**BPS qq-characters.** Another way to construct *qq*-characters associated with the CY$_4$ with the form $CY3 \times \mathbb{C}$ is to use the vertex operators introduced in (4.108) (see the notations there). We can define the screening charge corresponding to the $\mathbb{C}$-part as

$$Q_i(x) = \sum_{k \in \mathbb{Z}} S_i(q_4^k x), \quad i \in Q_0. \qquad (7.77)$$

We then can introduce D6 *qq*-characters associated with the CY$_3$ part as

$$T_i(x) = W_i(x) + \dots \qquad (7.78)$$

satisfying

$$[T_i(x), Q_j(x')] = 0, \quad \forall i, j \in Q_0. \qquad (7.79)$$

An interesting property is that the monomial terms of this *qq*-character are classified by the 3d BPS crystals [57, 126]. We present this property as a conjecture which will be clarified in a future publication. Details will be explained in [74].

**Conjecture 7.17 ([74]).** Let $Z$ be a toric $CY_4$ which takes the form as $Z = X \times \mathbb{C}$ where $X$ is a toric $CY_3$. The corresponding quiver of $X$ is $Q = (Q_0, Q_1)$ with $q$-deformation parameters $\{q_I\}_{I \in Q_1}$. Given this quiver, we can construct a three-dimensional crystal called BPS crystals (see [57, 126] for details). The BPS crystals are sets of colored *atoms* where the colors are labeled by $Q_0$. Namely, given a 3d BPS crystal $\Lambda$, we have a color projection map $c: \Lambda \to Q_0$. The BPS crystal has an atom in the origin which we denote $\mathfrak{o}$. Each atom $\bar{\mathfrak{c}}$ of the BPS crystal is associated with a coordinate function$^{32}$ $\chi_{X,x}(\bar{\mathfrak{c}})$:

$$\chi_{X,x}(\bar{\mathfrak{c}}) = x \times \prod_{I \in \text{path}[\mathfrak{o} \to \bar{\mathfrak{c}}]} q_I. \qquad (7.80)$$

Given two *atoms* with color $i$ and $j$, the difference of the coordinates comes from the parameter $q_{I:i \to j}$ which is associated with the arrow connecting the two *atoms*.

Under this condition, the *qq*-characters are given

$$T_i(x) = \sum_{\Lambda^{(i)}} q_i^{\Lambda^{(i)}} Z_i^{D6}[\Lambda^{(i)}] : W_i(x) \prod_{\bar{\mathfrak{c}} \in \Lambda^{(i)}} A_{c(\bar{\mathfrak{c}})}^{-1}(\chi_{X,x}(\bar{\mathfrak{c}})) : , \qquad (7.81)$$

$$T_i^K(x) = \sum_{\Lambda^{(i)}} q_i^{\Lambda^{(i)}} Z_i^{D6}[K, \Lambda^{(i)}] : \frac{W_i(x)}{W_i(Kx)} \prod_{\bar{\mathfrak{c}} \in \Lambda^{(i)}} A_{c(\bar{\mathfrak{c}})}^{-1}(\chi_{X,x}(\bar{\mathfrak{c}})) : . \qquad (7.82)$$

where $\Lambda^{(i)}$ is the crystal whose atom at the origin is with color $i \in Q_0$. The coefficients $Z_i^{D6}[\Lambda^{(i)}], Z_i^{D6}[K, \Lambda^{(i)}]$ are determined from the commutativity with the screening charges:

$$[T_i(x), Q_j(x')] = 0, \quad [T_i^K(x), Q_j(x')] = 0 \qquad (7.82)$$

Similar to the $\mathbb{C}^4$-case, we can introduce pit reductions by adding extra $W_i(x)$ at the denominators or tuning the parameter $K$. The monomial terms will then be classified by the subcrystals of the 3d BPS crystals. Such types of subcrystal were studied in [128, 129]. Since the *qq*-characters obtained in this way have a strong relationship with BPS crystals, we call them **BPS qq-characters**. Later in section 9.5, we will see that such BPS *qq*-characters have a relation with the quiver quantum toroidal algebras/toroidal quiver BPS algebras [33, 34].

<footnote>$^{32}$This coordinate function is a one-parameter deformation of the coordinate function of [33, 34]. We will discuss the derivation of this in [74]. For this paper, we only note that after taking the limit $q_4 \to 1$, this coordinate function will become the coordinate functions defined in [33, 34].</footnote>
# 8 Towards D8-brane $\bar{q}q$-characters

We use the fusion process of the D6 $\bar{q}q$-characters to define the D8 $\bar{q}q$-characters in section 8.1. We then study the contractions of the D8 $\bar{q}q$-characters and show that they reproduce the instanton partition function of the magnificent four system up to sign factors in section 8.2.

## 8.1 Fusion of D6 $\bar{q}q$-characters to D8 $\bar{q}q$-characters

After constructing D2, D4, D6 $\bar{q}q$-characters, we would like to obtain D8 $\bar{q}q$-characters that reproduce the magnificent four partition function in (3.102). In the lower-dimensional cases, thanks to Thm. 7.5, we can define the $\bar{q}q$-characters by choosing the highest weight and imposing the commutativity with the screening charges. The highest weight and the corresponding screening charges were chosen so that the associated subspaces do not intersect in the $\mathbb{C}^4$ geometry. However, for the D8 operators $Z(x)$, $\tilde{Z}^K(x)$, the only operator that makes the operator products become rational functions is the root current $\mathcal{A}(x)$. In this sense, it is natural to construct a screening charge related to the D0 operator. For the moment, we do not know how to construct such kind of screening currents. Instead, we will use the fusion process discussed in Thm. 6.9, 7.12 to define the D8 $\bar{q}q$-characters. Since we are interested in studying the relation between the magnificent four system, where D8 and anti D8-branes appear, we use $\tilde{Z}^K(x)$ as the highest weight.

**Definition 8.1.** We define the D8 $\bar{q}q$-characters as

$$T_{\mathbf{4};a}^{K}(x) = \sum_{\rho \in \mathcal{SP}} \mathcal{Z}_{\mathbf{4};a}^{\mathrm{D8}}[\rho, K] \Lambda_{\mathbf{4},\rho}^{K}(x), \quad a \in \mathbf{4}, \qquad (8.1)$$

where

$$\Lambda_{\mathbf{4},\rho}^{K}(x) := \tilde{Z}^{K}(x) \prod_{\mathbf{a} \in \rho} \mathcal{A}^{-1}(\chi_{\mathbf{4},x}(\mathbf{a})) : \qquad (8.2)$$

Rescaling the root current as $\mathcal{A}(x) \to q^{-1}\mathcal{A}(x)$, we can modify the topological terms as

$$T_{\mathbf{4};a}^{K}(x) = \sum_{\rho \in \mathcal{SP}} q^{\lvert\rho\rvert} \mathcal{Z}_{\mathbf{4};a}^{\mathrm{D8}}[\rho, K] : \tilde{Z}^{K}(x) \prod_{\mathbf{a} \in \rho} \mathcal{A}^{-1}(\chi_{\mathbf{4},x}(\mathbf{a})) : , \quad a \in \mathbf{4}. \qquad (8.3)$$

Let us show that this is obtained by taking the infinite products of the D6 $\bar{q}q$-characters. Using Lem. 7.9, one can obtain the following property which is a D8 analog of Lem. 7.11.

**Lemma 8.2.** For $a \in \mathbf{4}$, we have

$$\Lambda_{\bar{a},\pi^{(2)}}(x_2)\Lambda_{\bar{a},\pi^{(1)}}(x_1) = 0 \qquad (8.4)$$

when $x_2 = q_a x_1$ and $\pi^{(2)} \succ \pi^{(1)}$.

*Proof.* Let us focus on the case $a = 4$. It is enough to study the zeros of $\mathcal{Z}_{\mathbf{4}|\mathbf{4}}^{\mathrm{D6-D6}}(x_1, \pi^{(1)}|x_2, \pi^{(2)})$. Using the formulas in (3.117), (3.123), we have$^{33}$

$$\mathcal{Z}_{\mathbf{4}|\mathbf{4}}^{\mathrm{D6-D6}}(x_1, \pi^{(1)}|x_2, \pi^{(2)}) \propto \prod_{\mathcal{a} \in \pi^{(1)}} \frac{(1 - q_4^{-1} \chi_{\bar{a},x_1}(\mathcal{a})/x_2)}{(1 - \chi_{\bar{a},x_1}(\mathcal{a})/x_2)} \prod_{\mathcal{a} \in \pi^{(2)}} \frac{\prod_{\mathcal{a} \in A(\pi^{(1)})} (1 - q_4 \chi_{\bar{a},x_1}(\mathcal{a})/\chi_{\bar{a},x_2}(\mathcal{a})) \prod_{\mathcal{a} \in R(\pi^{(1)})} (1 - \chi_{\bar{a},x_1}(\mathcal{a})/\chi_{\bar{a},x_2}(\mathcal{a}))}{\prod_{\mathcal{a} \in A(\pi^{(1)})} (1 - \chi_{\bar{a},x_1}(\mathcal{a})/\chi_{\bar{a},x_2}(\mathcal{a})) \prod_{\mathcal{a} \in R(\pi^{(1)})} (1 - q_4^{-1} \chi_{\bar{a},x_1}(\mathcal{a})/\chi_{\bar{a},x_2}(\mathcal{a}))} \qquad (8.5)$$

<footnote>$^{33}$Strictly speaking, we need the information of the denominators of $W_{\pi,v}^{\bar{a}}(x)$ in (3.123), which is omitted here.</footnote>
When $\pi^{(2)} > \pi^{(1)}$, there is a cube $\Box' = (i, j, k)$ that is addable to $\pi^{(1)}$ that is included in $\pi^{(2)}$:

$$ \exists \Box' = (i, j, k) \in A(\pi^{(1)}), \quad \Box' \in \pi^{(2)}. \qquad (8.6) $$

When $x_2 = q_4x_1$, the term $(1 - q_4\chi_{\bar{4},x_1}(\Box'))/\chi_{\bar{4},x_2}(\Box')) = (1 - q_4x_1/x_2)$ then gives the zero. Thus, we obtain the claim. □

The above discussion is also true for $\Lambda_{\bar{a},\pi}^K(x)$, where K is generic.

**Lemma 8.3.** For $a \in \mathfrak{A}$, after using (7.58) and (7.59), we have

$$ \Lambda_{\bar{a}, \pi^{(2)}}^{K_2}(x_2) \Lambda_{\bar{a}, \pi^{(1)}}^{K_1}(x_1) = 0, \qquad (8.7) $$

when $x_2 = q_a x_1$ and $\pi^{(2)} > \pi^{(a)}$ and $K_{1,2}$ are generic.

The finite products of the D6 $qq$-characters $\mathfrak{T}_{\bar{a}}^K(x) := \mathfrak{T}_{\bar{a}}^{(1|1)}(x|Kx)$ are then expanded as

$$ \mathfrak{T}_{\bar{a}}^{K}(x_N) \dots \mathfrak{T}_{\bar{a}}^{K}(x_1) = \sum_{\pi^{(N)} \preceq \dots \preceq \pi^{(1)}} q^{\sum_{i=1}^{N} |\pi^{(i)}|} \prod_{i=1}^{N} \tilde{Z}_{\bar{a}}^{\mathrm{D}6}[K, \pi^{(i)}] \Lambda_{\bar{a}, \pi^{(N)}}^{K}(x_N) \dots \Lambda_{\bar{a}, \pi^{(1)}}^{K}(x_1), \qquad (8.8) $$

where $x_i = q_a^{i-1}x$ ($i = 1, \dots, N$). Using the operator product and extracting the one-loop perturbative part, we define the normalized N-fusion D6 $qq$-character as

$$ \bar{T}_{\bar{a}}^{K(N)}(x) := \sum_{\pi^{(N)} \preceq \dots \preceq \pi^{(1)}} q^{\sum_{i=1}^{N} |\pi^{(i)}|} \prod_{i=1}^{N} \tilde{Z}_{\bar{a}}^{\mathrm{D}6}[K, \pi^{(i)}] \prod_{i<j} \tilde{Z}_{\bar{a};K|\bar{a};K}^{\mathrm{D}6-\mathrm{D}6}(x_i, \pi^{(i)}|x_j, \pi^{(j)}) : \prod_{i=1}^{N} \Lambda_{\bar{a}, \pi^{(i)}}^{K}(x_i) : . \qquad (8.9) $$

Taking the limit $N \to \infty$ and considering the infinite products, we can obtain the D8 $qq$-character defined in (8.1), (8.3).

**Theorem 8.4.** The normalized N-fusion D6 $qq$-characters $\bar{T}_{\bar{a}}^{K(N)}(x)$ ($a \in \mathfrak{A}$) give the $qq$-characters $\mathfrak{T}_{\mathfrak{A};a}^K(x)$:

$$ \bar{T}_{\bar{a}}^{K(N)}(x) \xrightarrow{N \to \infty} \mathfrak{T}_{\mathfrak{A};a}^{K}(x). \qquad (8.10) $$

Equivalently, we have

$$ \left. \prod_{i=1}^{\infty} \mathfrak{T}_{\bar{a}}^{K}(q_{a}^{i-1} x) \right\} \approx \mathfrak{T}_{\mathfrak{A};a}^{K}(x), \qquad (8.11) $$

where the symbol $\approx$ means the equality is up to one-loop perturbative factors.

*Proof.* Let us focus on the case $a = 4$. Similar to what we have done in the fusion of D4 $qq$-characters, we need to regularize properly the infinite products. We only give a sketch of the proof of how to take the infinite products:

$$ \bar{T}_{\mathfrak{A}}^{K(\infty)}(x) := \sum_{\emptyset \preceq \dots \preceq \pi^{(2)} \preceq \pi^{(1)}} q^{\sum_{i=1}^{\infty} |\pi^{(i)}|} \prod_{i=1}^{\infty} \tilde{Z}_{\mathfrak{A}}^{\mathrm{D}6}[K, \pi^{(i)}] \prod_{i<j} \tilde{Z}_{\mathfrak{A};K|\mathfrak{A};K}^{\mathrm{D}6-\mathrm{D}6}(x_i, \pi^{(i)}|x_j, \pi^{(j)}) : \prod_{i=1}^{\infty} \Lambda_{\mathfrak{A}, \pi^{(i)}}^{K}(x_i) : . \qquad (8.12) $$
Since the right-hand side is expanded in arbitrary plane partitions where the q-coordinates are tuned properly, we may understand the right-hand side as a sum of arbitrary solid partitions. This comes from the (1, 3)-type description of the solid partition: $\rho = (\pi^{(1)}, \pi^{(2)}, \dots, \emptyset, \dots)$. The topological term is then rewritten as $q^{\sum_{i=1}^{\infty} \pi^{(i)}} = q^{\lvert\rho\rvert}$. Let us consider the coefficient parts. The infinite product of the U(1|1) part is given

$$ \prod_{i=1}^{\infty} \tilde{Z}_{4}^{\text{D6}}[K, \pi^{(i)}] = \prod_{i=1}^{\infty} \prod_{\mathfrak{q} \in \pi^{(i)}} \frac{(1 - Kx_i/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}]))(1 - x_{i+1}/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))}{(1 - Kx_{i+1}/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))(1 - x_i/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))} \prod_{\substack{\mathfrak{q} \in \pi^{(i)} \\ \bar{\mathfrak{q}} \in \pi^{(i)}}} g_{\bar{4}} \left( \frac{\chi_{\bar{4},x_i}(\bar{\mathfrak{q}})}{\chi_{\bar{4},x_i}(\bar{\mathfrak{q}})} \right)^{-1}. \quad (8.13) $$

The other part is given by

$$ \prod_{i<j} Z_{4;K|4;K}^{\text{D6-D6}}(x_i, \pi^{(i)}|x_j, \pi^{(j)}) = \prod_{j>i} \prod_{\mathfrak{q} \in \pi^{(i)}} \frac{\mathcal{V}_4(x_j/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))}{\mathcal{V}_4(Kx_j/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))} \prod_{\mathfrak{q} \in \pi^{(j)}} \frac{\mathcal{V}_4(x_i/\chi_{\bar{4},x_j}(\bar{\mathfrak{q}}))}{\mathcal{V}_4(Kx_i/\chi_{\bar{4},x_j}(\bar{\mathfrak{q}}))} \\ \times \prod_{\substack{j>i \\ \mathfrak{q} \in \pi^{(i)}}} \prod_{\substack{\mathfrak{q} \in \pi^{(j)} \\ \mathfrak{q} \in \pi^{(j)}}} \mathcal{A}_{\mathbb{C}^4} \left( \frac{\chi_{\bar{4},x_i}(\bar{\mathfrak{q}})}{\chi_{\bar{4},x_j}(\bar{\mathfrak{q}})} \right)^{-1}. \quad (8.14) $$

Using

$$ \prod_{j>i} \prod_{\mathfrak{q} \in \pi^{(i)}} \mathcal{A}_{\mathbb{C}^4} \left( \frac{\chi_{\bar{4},x_i}(\bar{\mathfrak{q}})}{\chi_{\bar{4},x_j}(\bar{\mathfrak{q}})} \right)^{-1} = \prod_{i \neq j} \prod_{\substack{\mathfrak{q} \in \pi^{(i)} \\ \mathfrak{q} \in \pi^{(j)}}} g_{\bar{4}} \left( \frac{\chi_{\bar{4},x_i}(\bar{\mathfrak{q}})}{\chi_{\bar{4},x_j}(\bar{\mathfrak{q}})} \right)^{-1}, \\ \prod_{i=1}^{\infty} \prod_{\mathfrak{q} \in \pi^{(i)}} \prod_{j=i+1}^{\infty} \frac{\mathcal{V}_4(x_j/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))}{\mathcal{V}_4(Kx_j/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))} = \prod_{i=1}^{\infty} \prod_{\mathfrak{q} \in \pi^{(i)}} \frac{(1 - Kx_{i+1}/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))}{(1 - x_{i+1}/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))}, \\ \prod_{j=1}^{\infty} \prod_{\mathfrak{q} \in \pi^{(j)}} \prod_{i=1}^{j-1} \frac{\mathcal{V}_4(x_i/\chi_{\bar{4},x_j}(\bar{\mathfrak{q}}))}{\mathcal{V}_4(Kx_i/\chi_{\bar{4},x_j}(\bar{\mathfrak{q}}))} = \prod_{j=1}^{\infty} \prod_{\mathfrak{q} \in \pi^{(j)}} \frac{(1 - x_j/\chi_{\bar{4},x_j}(\bar{\mathfrak{q}}))(1 - Kx/\chi_{\bar{4},x_j}(\bar{\mathfrak{q}}))}{(1 - x/\chi_{\bar{4},x_j}(\bar{\mathfrak{q}}))(1 - Kx_j/\chi_{\bar{4},x_j}(\bar{\mathfrak{q}}))}, \quad (8.15) $$

we obtain

$$ \begin{aligned} & \prod_{i=1}^{\infty} \tilde{Z}_{4}^{\text{D6}}[K, \pi^{(i)}] \prod_{i<j} Z_{4;K|4;K}^{\text{D6-D6}}(x_i, \pi^{(i)}|x_j, \pi^{(j)}) \\ &= \prod_{i=1}^{\infty} \prod_{\mathfrak{q} \in \pi^{(i)}} \frac{(1 - Kx/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))}{(1 - x/\chi_{\bar{4},x_i}(\bar{\mathfrak{q}}))} \prod_{\substack{i,j \\ \mathfrak{q} \in \pi^{(i)}}} \prod_{\substack{\mathfrak{q} \in \pi^{(j)} \\ \mathfrak{q} \in \pi^{(j)}}} g_{\bar{4}} \left( \frac{\chi_{\bar{4},x_i}(\bar{\mathfrak{q}})}{\chi_{\bar{4},x_j}(\bar{\mathfrak{q}})} \right)^{-1} \quad (8.16) \\ &= \prod_{\mathfrak{q} \in \rho} \frac{(1 - Kx/\chi_{\mathbf{4},x}(\mathfrak{q}))}{(1 - x/\chi_{\mathbf{4},x}(\mathfrak{q}))} \prod_{\substack{\mathfrak{q}, \mathfrak{q}' \in \rho \\ \mathfrak{q}, \mathfrak{q}' \in \rho}} g_{\bar{4}} \left( \frac{\chi_{\mathbf{4},x}(\mathfrak{q})}{\chi_{\mathbf{4},x}(\mathfrak{q}')} \right)^{-1}, \end{aligned} $$

which is exactly $Z_{4;4}^{\text{D8}}[\rho; K]$. For the operator part, we can also show

$$ \therefore \prod_{i=1}^{\infty} \Lambda_{\bar{4}, \pi^{(i)}}^{K}(x_i) :=: \tilde{Z}^{K}(x) \prod_{\mathfrak{q} \in \rho} A^{-1}(\chi_{\mathbf{4},x}(\mathfrak{q})), \quad (8.17) $$
where we used

$$ \prod_{i=1}^{\infty} \frac{W_4(q_4^{i-1}x)}{W_4(K q_4^{i-1}x)} := \prod_{i=1}^{\infty} \frac{\tilde{Z}^K(q_4^{i-1}x)}{\tilde{Z}^K(q_4^i x)} := \tilde{Z}^K(x). \quad (8.18) $$

Therefore, we obtain the claim.

<figure><img src="image_30.png" /></figure>

Depending on which D6 $qq$-character $T_{\bar{a}}(x)$ we use to take the infinite products, we have four possibilities of the D8 $qq$-character. The operator part is invariant under the permutation of the deformation parameters $q_1, q_2, q_3, q_4$ but the coefficient part $Z_{4;a}^{D8}[\rho, K]$ is not invariant under the permutation. For example, for lower levels, we have

$$ \rho = \{\{\{1\}\}\}, \quad \frac{Z_{4;1}^{D8}[\rho, K]}{Z_{4;2}^{D8}[\rho, K]} = \frac{Z_{4;2}^{D8}[\rho, K]}{Z_{4;3}^{D8}[\rho, K]} = \frac{Z_{4;3}^{D8}[\rho, K]}{Z_{4;4}^{D8}[\rho, K]} = \frac{Z_{4;4}^{D8}[\rho, K]}{Z_{4;1}^{D8}[\rho, K]} = 1, \quad (8.19a) $$

$$ \rho = \{\{\{2\}\}\}, \quad \frac{Z_{4;1}^{D8}[\rho, K]}{Z_{4;2}^{D8}[\rho, K]} = \frac{Z_{4;2}^{D8}[\rho, K]}{Z_{4;3}^{D8}[\rho, K]} = \frac{Z_{4;3}^{D8}[\rho, K]}{Z_{4;4}^{D8}[\rho, K]} = \frac{Z_{4;4}^{D8}[\rho, K]}{Z_{4;1}^{D8}[\rho, K]} = 1, \quad (8.19b) $$

$$ \rho = \{\{\{1,1\}\}\}, \quad \frac{Z_{4;1}^{D8}[\rho, K]}{Z_{4;2}^{D8}[\rho, K]} = \frac{Z_{4;2}^{D8}[\rho, K]}{Z_{4;3}^{D8}[\rho, K]} = \frac{Z_{4;3}^{D8}[\rho, K]}{Z_{4;4}^{D8}[\rho, K]} = \frac{Z_{4;4}^{D8}[\rho, K]}{Z_{4;1}^{D8}[\rho, K]} = 1, \quad (8.19c) $$

$$ \rho = \{\{\{1\}, \{1\}\}\}, \quad -\frac{Z_{4;1}^{D8}[\rho, K]}{Z_{4;2}^{D8}[\rho, K]} = -\frac{Z_{4;2}^{D8}[\rho, K]}{Z_{4;3}^{D8}[\rho, K]} = \frac{Z_{4;3}^{D8}[\rho, K]}{Z_{4;4}^{D8}[\rho, K]} = \frac{Z_{4;4}^{D8}[\rho, K]}{Z_{4;1}^{D8}[\rho, K]} = 1, \quad (8.19d) $$

where we described the solid partition as

$$ \rho = \{\{\rho_{11}, \rho_{12}, \dots\}, \dots, \{\rho_{i1}, \rho_{i2}, \dots, \rho_{im_i}\}, \dots, \{\rho_{l1}, \dots, \rho_{lm_l}\}\}, \quad \rho_{ij} = (\rho_{ij1}, \dots, \rho_{ijn_i}) \quad (8.20) $$

and the $q$-coordinates of the hypercubes are

$$ v q_1^{a_1-1} q_2^{b_1-1} q_3^{c_1-1} q_4^{d_1-1}, \quad 1 \le a \le l, \ 1 \le b \le m_a, \ 1 \le c \le n_{ab}, \ 1 \le d \le \rho_{abc}. \quad (8.21) $$

As mentioned in section 3.3, the partition function of the magnificent four has a sign factor $(-1)^{\sigma_a(\rho)}$ depending on the solid partition. In our construction here, this did not appear and thus the appearing coefficient is equal to the U(1) magnificent four partition function only up to sign factors. One of the reasons this happened is because we naively took the infinite products. For the lower-dimensional $qq$-characters, the infinite products were controlled so that even after taking the infinite products, it still commutes with at least one of the screening charges. To obtain the D8 $qq$-character with the correct sign factors, a different procedure to derive the D8 $qq$-character is necessary.

## 8.2 Higher rank magnificent four and D8 $qq$-characters

We still can use the D8 $qq$-characters we constructed in the previous section to derive the partition functions of the rank $N$ magnificent four$^{34}$ [16] up to sign factors. The operator products of $\Lambda_{\underline{4},\rho}^K(x)$ are

$$ \Lambda_{\underline{4},\rho^{(2)}}^{K_2}(x_2) \Lambda_{\underline{4},\rho^{(1)}}^{K_1}(x_1) = Z_{1-\text{loop}}^{\text{D8-D8}}(x_1, K_1 | x_2, K_2) Z_{K_1|K_2}^{\text{D8-D8}}(x_1, \rho^{(1)} | x_2, \rho^{(2)}) : \Lambda_{\underline{4},\rho^{(2)}}^{K_2}(x_2) \Lambda_{\underline{4},\rho^{(1)}}^{K_1}(x_1) : . \quad (8.22) $$

The rank $N$ magnificent four partition function is then written using the vertex operators as the following theorem.

<footnote>$^{34}$Note that the higher rank version is called the magnificent four with color in the original paper.</footnote>
**Theorem 8.5.** The composition of the D8 $qq$-characters gives the partition function of higher rank magnificent four system up to sign factors:

$$ \langle 0 | T_{\mathbf{4};a_N}(x_N) \cdots T_{\mathbf{4};a_1}(x_1) | 0 \rangle = \sum_{\rho^{(1)}, \dots, \rho^{(N)}} q^{\left|\rho\right|} \prod_{i=1}^{N} \mathcal{Z}_{\mathbf{4};a_i}^{\mathrm{D8}}[\rho^{(i)}, K_i] \prod_{j>i} \mathcal{Z}_{\mathrm{1-loop}}^{\mathrm{D8-D8}}(x_i, K_i | x_j, K_j) \\ \times \prod_{j>i} \mathcal{Z}_{K_i|K_j}^{\mathrm{D8-D8}}(x_i, \rho^{(i)} | x_j, \rho^{(j)}). \qquad (8.23) $$

This is the BPS/CFT correspondence of the magnificent four.

*Remark 8.6.* One can also use the other (2, 2), (3, 1)-type descriptions and use infinite products of screening charges or D4 $qq$-characters to rewrite the partition functions. Note that the fusion process we used was essentially the (1, 3)-type description.

## 9 Quantum toroidal algebras and BPS $qq$-characters

In this section, we review the quantum toroidal $gl_1$ and point out observations regarding the $qq$-characters we introduced in the previous sections. We will show that the D2, D4, D6 $qq$-characters are related with the vertical representations of the quantum toroidal $gl_1$. At the end, we have the following correspondence:

<table><thead><tr><th>section</th><th>qq-characters</th><th>quantum toroidal gl<sub>1</sub></th></tr></thead><tbody><tr><td>9.2</td><td>D2 qq-character</td><td>vector representation</td></tr><tr><td>9.3</td><td>D4 qq-character</td><td>Fock representation</td></tr><tr><td>9.4</td><td>D6 qq-character</td><td>MacMahon representation</td></tr></tbody></table>

We then move on to the gauge origami system on toric $CY_4 Z = X \times \mathbb{C}$ where $X$ is a toric $CY_3$ and show that the BPS $qq$-characters introduced in Conj. 7.17 are related to the vertical representations of general quantum toroidal algebras associated with toric $CY_3$, which were recently introduced in [33, 34].

### 9.1 Quantum toroidal $gl_1$

The quantum toroidal $gl_1$ is an infinite-dimensional quantum algebra with two independent deformation parameters [24, 25, 137, 148, 149]. We follow the notations in [150] (see also [151, section 5.2, 5.3] for a review).

**Definition 9.1.** Let $q_1, q_2, q_3$ be the deformation parameters$^{35}$ with the condition $q_1q_2q_3 = 1$. The quantum toroidal $gl_1$, which is denoted $\mathcal{E}$, is generated by three Drinfeld currents

$$ E(z) = \sum_{m \in \mathbb{Z}} E_m z^{-m}, \quad F(z) = \sum_{m \in \mathbb{Z}} F_m z^{-m}, \quad K^{\pm}(z) = K^{\pm} \exp \left( \sum_{r>0} \mp \frac{\kappa_r}{r} H_{\pm r} z^{\mp r} \right) \quad (9.1) $$

<footnote>$^{35}$In the literature, the deformation parameters of the algebra are denoted $q_1, q_2, q_3$. We use a different notation to prevent confusion with the parameters $q_1, q_2, q_3, q_4$ introduced in this paper. As mentioned in footnote 8, the structure function $g(z)$ is related to the structure function $g_4(z)$ after taking the limit $q_4 \to 1$.</footnote>
and central elements

$$C, \quad K^{-} = (K^{+})^{-1}. \qquad (9.2)$$

The defining relations are

$$ 
\begin{align} 
E(z)E(w) &= g(z/w)E(w)E(z), \quad F(z)F(w) = g(z/w)^{-1}F(w)F(z), \nonumber \\ 
K^{\pm}(z)K^{\pm}(w) &= K^{\pm}(w)K^{\pm}(z), \quad K^{-}(z)K^{+}(w) = \frac{g(C^{-1}z/w)}{g(Cz/w)}K^{+}(w)K^{-}(z), \nonumber \\ 
K^{\pm}(C^{(1\mp 1)/2}z)E(w) &= g(z/w)E(w)K^{\pm}(C^{(1\mp 1)/2}z), \tag{9.3} \\ 
K^{\pm}(C^{(1\pm 1)/2}z)F(w) &= g(z/w)^{-1}F(w)K^{\pm}(C^{(1\pm 1)/2}z), \nonumber \\ 
[E(z), F(w)] &= \tilde{g}\left(\delta\left(\frac{Cw}{z}\right)K^{+}(z) - \delta\left(\frac{Cz}{w}\right)K^{-}(w)\right) \nonumber 
\end{align} 
$$

where

$$ g(z) = \frac{\prod_{i=1}^{3}(1 - q_i z)}{\prod_{i=1}^{3}(1 - q_i^{-1} z)}, \quad \kappa_r = \prod_{i=1}^{3}(q_i^{r/2} - q_i^{-r/2}), \qquad (9.4) $$

and $\tilde{g} = 1/\kappa_1$.

Additionally, one needs the so-called Serre relations which are cubic relations of $E(z)$, $F(z)$. We do not need them so we do not write the explicit form (see [150]). The function $g(z)$ is called the *structure function* of the quantum toroidal $gl_1$.

The quantum toroidal $gl_1$ has a Hopf algebraic structure. We only list down the coproduct structure:

$$ 
\begin{align} 
\Delta E(z) &= E(z) \otimes 1 + K^{-}(C_1 z) \otimes E(C_1 z), \nonumber \\ 
\Delta F(z) &= F(C_2 z) \otimes K^{+}(C_2 z) + 1 \otimes F(z), \nonumber \\ 
\Delta K^{+}(z) &= K^{+}(z) \otimes K^{+}(C_1^{-1} z), \tag{9.5} \\ 
\Delta K^{-}(z) &= K^{-}(C_2^{-1} z) \otimes K^{-}(z), \nonumber \\ 
\Delta(X) &= X \otimes X, \quad X = C, K^{-}, \nonumber 
\end{align} 
$$

where $C_1 = C \otimes 1$ and $C_2 = 1 \otimes C$. Using this coproduct, we can construct tensor product representations.

The representations of the quantum toroidal $gl_1$ are obtained by determining the values of the central elements $C, K^-$. We have two classes of representations called *vertical representations* and *horizontal representations*. Vertical representations are representations when the central element $C$ is trivial: $C = 1$. We have three types of them

* vector representation [25]: $(C, K^-) = (1, 1)$

* Fock representation [149]: $(C, K^-) = (1, q_c^{1/2}) (c = 1, 2, 3)$

* MacMahon representation [137]: $(C, K^-) = (1, K^{1/2}) (K \in \mathbb{C}^\times)$

Multi-dimensional partitions appear as the bases of the representation spaces of these representations. For the vector representation, 1d partitions labeled by integers appear. For
the Fock and MacMahon representations, 2d and 3d partitions appear respectively (see [150] and [151, section 5.3.1] for the derivations).

On the other hand, horizontal representations are representations where the central charges are $(C, K^{-}) = (\mathfrak{q}_c^{1/2}, 1)(c = 1, 2, 3)$ [24, 139, 152–155]. Drinfeld currents are represented in vertex operators for these representations. See for example [150] and [151, section 5.3.2] for the explicit derivation of these representations.

## 9.2 Vector representation and D2 $\bar{q}\bar{q}$-character

There are three types of vector representations with central charges $(C, K^{-}) = (1, 1)$ and the actions of the Drinfeld currents are

$$ 
\begin{align} 
K^{\pm}(z)[u]_{j}^{(c)} &= \left[ \Psi_{[u]_{j}^{(c)}}(z) \right]_{\pm}^{z} [u]_{j}^{(c)} := [S_{c}(u\mathfrak{q}_{c}^{j}/z)]_{\pm}[u]_{j}^{(c)}, \nonumber \\ 
E(z)[u]_{j}^{(c)} &= \mathcal{E}\delta(u\mathfrak{q}_{c}^{j}/z) [u]_{j+1}^{(c)}, \tag{9.6} \\ 
F(z)[u]_{j}^{(c)} &= \mathcal{F}\delta(u\mathfrak{q}_{c}^{j-1}/z) [u]_{j-1}^{(c)}, \quad c = 1, 2, 3, \quad j \in \mathbb{Z} \nonumber 
\end{align} 
$$

where

$$ \mathcal{E}\mathcal{F} = \tilde{g} \frac{(1 - \mathfrak{q}_{c+1}^{-1})(1 - \mathfrak{q}_{c-1}^{-1})}{(1 - \mathfrak{q}_c)} , \quad S_c(z) = \frac{(1 - \mathfrak{q}_{c-1}z)(1 - \mathfrak{q}_{c+1}z)}{(1 - z)(1 - \mathfrak{q}_{c-1}\mathfrak{q}_{c+1}z)} . \quad (9.7) $$

We denote these representations $\mathcal{V}_c(u)$, $(c = 1, 2, 3)$. The bases $\{[u]_j^{(c)}\}_{j \in \mathbb{Z}}$ are represented by 1d partitions (see also section 2.2):

$$ [u]_j^{(c)} = \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.7] 
% Coordinates for vertices
\coordinate (top) at (0,1); 
\coordinate (bottom) at (0,0); 
\coordinate (left) at (-1,0); 
\coordinate (right) at (1,0); 
\coordinate (left_q) at (-1,0.5); 
\coordinate (right_q) at (1,0.5); 
\coordinate (mid_q) at (0,0.3); 
\node[above] at (top) {$u$}; 
\node[below] at (bottom) {$\mathfrak{q}_c$}; 
\node[left] at (left) {$1$}; 
\node[right] at (right) {$j$}; 
\node[above] at (mid_q) {$\cdots$}; 
\node[below] at (mid_q) {$\cdots$}; 
\node[above] at (left_q) {$\cdots$}; 
\node[below] at (right_q) {$\cdots$}; 
\draw[->] (top) -- (bottom); 
\draw[->] (left) -- (right); 
\draw[->] (left_q) -- (right_q); 
\draw[->] (mid_q) -- (mid_q) node[above] {$\mathfrak{q}_c^{j-1}$}; 
\draw[->] (mid_q) -- (mid_q) node[below] {$\mathfrak{q}_c$}; 
\end{tikzpicture} \quad (9.8) $$

The operator $K^{\pm}(z)$ acts diagonally, $E(z)/F(z)$ adds/removes boxes to/from the configuration.

To relate the D2 $\bar{q}\bar{q}$-characters with the vector representations, we choose one specific direction, which is $\mathbb{C}_4$, in the gauge origami system. The motivation for choosing this direction will be explained in section 10. Let us study the relation of the $\bar{q}\bar{q}$-characters included in the $\mathbb{C}_{123}^3 \times \mathbb{S}^1$. The operator products of $S_c(z)$ ($c = 1, 2, 3$) with $S_4(q_4z)$ are

$$ S_c(q_c^j u)S_4(q_4z) = [S_{\overline{c4}}(uq_1^j/z)]_-^z : S_c(q_c^j u)S_4(q_4z) : \quad (9.9a) $$

$$ S_4(q_4z)S_c(uq_c^j) = [S_{\overline{c4}}(uq_c^j/z)]_+^z : S_c(q_c^j u)S_4(q_4z) : . \quad (9.9b) $$

After taking the limit $q_4 \to 1$, we can see that we have

$$ S_{\overline{c4}}(uq_c^j/z) \to S_c(uq_c^j/z), \quad q_1, q_2, q_3 \to q_1, q_2, q_3. \quad (9.10) $$
Similar to the D2 case, we choose $\mathbb{C}_4$ to be a specific direction in the gauge origami system. Focusing on $\mathcal{T}_{12}(x)$ and using (6.9), we obtain

$$ : \mathcal{X}_{12}(u) \prod_{\square \in \lambda} \mathcal{A}^{-1}(\chi_{12,u}(\square)) : \mathcal{S}_4(q_4 z) = \left[ q_3^{-1} \frac{y_{\lambda,u}^{12}(q_3^{-1} z)}{y_{\lambda,u}^{12}(z)} \right]_{-}^{z} : \mathcal{X}_{12}(u) \prod_{\square \in \lambda} \mathcal{A}^{-1}(\chi_{12,u}(\square)) \mathcal{S}_4(q_4 z) : , \qquad (9.16a) $$

$$ \mathcal{S}_4(q_4 z) : \mathcal{X}_{12}(u) \prod_{\square \in \lambda} \mathcal{A}^{-1}(\chi_{12,u}(\square)) := \left[ q_3^{-1} \frac{y_{\lambda,u}^{12}(q_3^{-1} z)}{y_{\lambda,u}^{12}(z)} \right]_{+}^{z} : \mathcal{X}_{12}(u) \prod_{\square \in \lambda} \mathcal{A}^{-1}(\chi_{12,u}(\square)) \mathcal{S}_4(q_4 z) : \qquad (9.16b) $$

and at the limit $q_4 \to 1$, we have

$$ y_{\lambda,u}^{12}(z) \to \mathcal{Y}_{\lambda,u}^{(3)}(z), \quad \chi_{12,u}(\square) \to \chi_{u}^{(3)}(\square). \qquad (9.17) $$

Thus, at the limit $q_4 \to 1$, we can relate the monomial terms of the D4 $\mathfrak{qq}$-characters with the bases of the Fock representation of quantum toroidal $\mathfrak{gl}_1$ as

$$ \mathcal{S}_4(q_4 z) \to K^{\pm}(z), \quad : \mathcal{X}_{12}(u) \prod_{\square \in \lambda} \mathcal{A}^{-1}(\chi_{12,u}(\square)) : \to |u, \lambda\rangle^{(3)}. \qquad (9.18) $$

In this sense, we can call the D4 $\mathfrak{qq}$-characters the **Fock $\mathfrak{qq}$-characters**.

**Remark 9.2.** Actually, the Fock representation can be derived by an infinite number of tensor products of the vector representation [25]. The coproduct structure enables us to consider the action of the Drinfeld currents on tensor product representations $\otimes_{i=1}^N \mathcal{V}_{c-1}(u_i)$. We tune the spectral parameter as $u_i = u q_{c+1}^{i-1}$ and take the limit $N \to \infty$. After proper regularization of the infinite products, we obtain $\otimes_{i=1}^{\infty} \mathcal{V}_{c-1}(u q_{c+1}^{i-1}) \simeq \mathcal{F}_c(u)$. This property corresponds with the fact that the D4 $\mathfrak{qq}$-character can be obtained by the fusion process of the D2 $\mathfrak{qq}$-characters as discussed in Thm. 6.9.

**Higher rank $\mathfrak{qq}$-characters.** Let us show that the higher rank D4 $\mathfrak{qq}$-characters correspond to the tensor product representations of the vertical Fock representations. We only focus on the $\mathfrak{qq}$-characters with no negative weights. The monomial terms appearing in $\mathcal{T}_{12:23:13}(\vec{x} | \vec{0})$ (see Thm. 6.14) are

$$ : \prod_{\alpha=1}^{n_{12}} \Lambda_{12,\lambda_{\alpha}}(x_{12,\alpha}) \prod_{\beta=1}^{n_{13}} \Lambda_{13,\mu_{\beta}}(x_{13,\beta}) \prod_{\gamma=1}^{n_{23}} \Lambda_{23,\nu_{\gamma}}(x_{23,\gamma}) : \qquad (9.19) $$

where $\lambda_\alpha, \mu_\beta, \nu_\gamma$ are Young diagrams. The coefficient appearing after taking the operator product with $\mathcal{S}_4(q_4 z)$ is

$$ \prod_{\alpha=1}^{n_{12}} q_3^{-1} \frac{y_{\lambda_{\alpha},x_{12,\alpha}}^{12}(q_3^{-1}z)}{y_{\lambda_{\alpha},x_{12,\alpha}}^{12}(z)} \prod_{\beta=1}^{n_{13}} q_2^{-1} \frac{y_{\mu_{\beta},x_{13,\beta}}^{13}(q_2^{-1}z)}{y_{\mu_{\beta},x_{13,\beta}}^{13}(z)} \prod_{\gamma=1}^{n_{23}} q_1^{-1} \frac{y_{\nu_{\gamma},x_{23,\gamma}}^{23}(q_1^{-1}z)}{y_{\nu_{\gamma},x_{23,\gamma}}^{23}(z)}. \qquad (9.20) $$

After taking the limit $q_4 \to 1$, it corresponds with the Cartan eigenvalue of the tensor product representation $\otimes_{\alpha=1}^{n_{12}} \mathcal{F}_3(x_{12,\alpha}) \otimes_{\beta=1}^{n_{13}} \mathcal{F}_2(x_{13,\beta}) \otimes \otimes_{\gamma=1}^{n_{23}} \mathcal{F}_1(x_{23,\gamma})$. Note that the Cartan eigenvalues of tensor product representations are simply the products of the Cartan eigenvalue
of each representation. This is because at $C = 1$, using the coproduct structure (9.5), we have $\Delta K^{\pm}(z) = K^{\pm}(z) \otimes K^{\pm}(z)$. Note also that the ordering of the tensor products does not matter because of the existence of the universal R-matrix$^{37}$ [24, 158, 159].

We can do the same analysis for the negative weights where supergroup analogs appear. The corresponding representations in the quantum toroidal $gl_1$ were constructed in [99, 159, 160], where vertical Fock representations with negative levels were introduced. One can show that the representations in [99] correspond with the D4 $qq$-character generated by $X_A(z)^{-1}$.

## 9.4 MacMahon representation and D6 $qq$-character

MacMahon representations are representations with central charges $(C, K^{-}) = (1, K^{1/2})$ where $K \in \mathbb{C}^{\times}$ is a generic parameter. The action of the Drinfeld currents is given as

$$ 
\begin{aligned}
K^{\pm}(z)|u, \pi\rangle &= [\Psi_{\pi,u}(z)]_{\pm}^{z} |u, \pi\rangle, \\
E(z)|u, \pi\rangle &= \sum_{\mathfrak{z} \in A(\pi)} \#\delta\left(\frac{z}{\chi_u(\mathfrak{z})}\right) \sqrt{\mathrm{Res}_{z=\chi_u(\mathfrak{z})} z^{-1} \Psi_{\pi,u}(z)} |u, \pi + \mathfrak{z}\rangle, \\
F(z)|u, \pi\rangle &= \sum_{\mathfrak{z} \in R(\pi)} \#\delta\left(\frac{z}{\chi_u(\mathfrak{z})}\right) \sqrt{\mathrm{Res}_{z=\chi_u(\mathfrak{z})} z^{-1} \Psi_{\pi,u}(z)} |u, \pi - \mathfrak{z}\rangle,
\end{aligned} 
\quad (9.21) $$

where $\#$ is some coefficient factor and

$$ \chi_u(\mathfrak{z}) = u q_1^{i_1-1} q_2^{j_1-1} q_3^{k_1-1}, \quad \Psi_{\pi,u}(z) = K^{-1/2} \frac{1-Ku/z}{1-u/z} \prod_{\mathfrak{z} \in \pi} g\left(\frac{z}{\chi_u(\mathfrak{z})}\right). \quad (9.22) $$

We denote this representation $M(u, K)$. The explicit coefficients of the right-hand side of $E(z)$, $F(z)$ are omitted. The $qq$-character $\mathrm{T}_{123}(x)$ and screening current $S_4(x')$ gives

$$ \begin{aligned} & \frac{W_{\bar{4}}(u)}{W_{\bar{4}}(Ku)} \prod_{\mathfrak{z} \in \pi} A^{-1}(\chi_{\bar{4},u}(\mathfrak{z})) : S_4(q_4 z) = -q_4 u \left[ W_{\pi,u}^{\bar{4},K}(z)^{-1} \right]_{-}^{z} : \frac{W_{\bar{4}}(u)}{W_{\bar{4}}(Ku)} \prod_{\mathfrak{z} \in \pi} A^{-1}(\chi_{\bar{4},u}(\mathfrak{z})) S_4(q_4 z) : , \\ & \mathrm{T}_{123}(x) \end{aligned} \quad (9.23a) $$

$$ S_4(q_4 z) : \frac{W_{\bar{4}}(u)}{W_{\bar{4}}(Ku)} \prod_{\mathfrak{z} \in \pi} A^{-1}(\chi_{\bar{4},u}(\mathfrak{z})) : = -q_4 u \left[ W_{\pi,x}^{\bar{4},K}(z)^{-1} \right]_{+}^{z} : \frac{W_{\bar{4}}(u)}{W_{\bar{4}}(Ku)} \prod_{\mathfrak{z} \in \pi} A^{-1}(\chi_{\bar{4},u}(\mathfrak{z})) S_4(q_4 z) : . \quad (9.23b) $$

Taking the limit $q_4 \to 1$, we have $\chi_{\bar{4},u}(\mathfrak{z}) \to \chi_u(\mathfrak{z})$ and

$$ g_{\bar{4}}(z) \to g(z), \quad W_{\pi,u}^{\bar{4},K}(z)^{-1} \to \frac{1-Ku/z}{1-u/z} \prod_{\mathfrak{z} \in \pi} g\left(\frac{z}{\chi_u(\mathfrak{z})}\right) \quad (9.24) $$

which gives the identification

$$ S_4(q_4 z) \to K^{\pm}(z), \quad : W_{\bar{4}}(u) \prod_{\mathfrak{z} \in \pi} A^{-1}(\chi_{\bar{4},u}(\mathfrak{z})) : \to |u, \pi\rangle. \quad (9.25) $$

Thus, at the limit $q_4 \to 1$, we can relate the monomial terms of the D6 $qq$-characters with the bases of the MacMahon representation of the quantum toroidal $gl_1$. In this sense, we

<footnote>$^{37}$Recently there have been attempts to construct the $qq$-character using the R-matrix of the quantum toroidal $gl_1$ [156, 157].</footnote>
call the D6 **qq**-characters the **MacMahon **qq**-characters**. Under this identification, we can
see that the distance between the D6 and D6 branes, denoted as K, appear as the central
charge of the MacMahon representation.

Remark 9.3. Instead of considering the U(1|1) theory of D6-branes, we can consider the U(1) theory by taking the limit $K \to 0, \infty$. After taking the limit $q_4 \to 1$, the eigenvalue $\Psi_{\pi,u}(z)|_{K \to 0, \infty}$ is no longer a rational function with the same degrees in the numerator and the denominator. Note that such kind of representations are not the representation of quantum toroidal $gl_1$ but of the shifted quantum toroidal $gl_1$ (see [128, 129, 161]).

Remark 9.4. Similar to the relation between the vector representations and the Fock repre-
sentations, the MacMahon representation also can be obtained by infinite tensor products of
the Fock representations [137]. Consider the tensor product $\bigotimes_{i=1}^N \mathcal{F}_c(u_i)$ and tune the spectral
parameters as $u_i = u q_c^{i-1}$. We then take the limit $N \to \infty$ and regularize it. After this
process, we get $\bigotimes_{i=1}^\infty \mathcal{F}_c(u q_c^{i-1}) \simeq \mathcal{M}(K, u)$. Note that the nontrivial parameter K appears
from the regularization process. This property corresponds to the fusion process of the D4
qq-characters to D6 qq-characters discussed in Thm. 7.12.

**General D6 **qq**-characters.** Similar to the D4 case, a higher rank version of D6 **qq**-characters corresponds to the tensor products of the MacMahon representations:

$$
\begin{equation}
\begin{aligned}
\mathcal{S}_4(q_4 x) : \prod_{i=1}^{N} \Lambda_{\frac{4}{c}, \pi^{(i)}}^{K_i}(x_i) : \mathcal{S}_4(q_4 x) &= \prod_{i=1}^{N} (-q_4 x_i) \prod_{i=1}^{N} [W_{\pi^{(i)}, x_i}^{4, K_i}(z)^{-1}]_{-}^{z} : \prod_{i=1}^{N} \Lambda_{\frac{4}{c}, \pi^{(i)}}^{K_i}(x_i) \mathcal{S}_4(q_4 x) : ,
\end{aligned}
\tag{9.26}
\end{equation}
$$

$$
\mathcal{S}_4(q_4 x) : \prod_{i=1}^{N} \Lambda_{\frac{4}{c}, \pi^{(i)}}^{K_i}(x_i) : = \prod_{i=1}^{N} (-q_4 x_i) \prod_{i=1}^{N} [W_{\pi^{(i)}, x_i}^{4, K_i}(z)^{-1}]_{+}^{z} : \prod_{i=1}^{N} \Lambda_{\frac{4}{c}, \pi^{(i)}}^{K_i}(x_i) \mathcal{S}_4(q_4 x) : .
$$

After taking the limit $q_4 \to 1$, one can see that it matches with the Cartan eigenvalue of
the tensor product $\bigotimes_{i=1}^N \mathcal{M}(x_i, K_i)$.

As discussed in section 7.5, after specifying the value K, we can obtain the pit reduction
of the D6 **qq**-characters, which eventually gives the D4 **qq**-character. This situation is the
same in the quantum toroidal **gl**₁. Setting K = **q**_c in (9.22), the Cartan eigenvalue will
be (9.14). The residue of Ψ<sub>π,u</sub>(z)|<sub>K=**q**_c</sub> at z = **q**_c**u** will vanish and thus the action of E(z)
will stop the growth of the plane partition in the direction **q**_c. We then obtain the **F**<sub>c</sub>(**u**)
representation.<fnref n="38" /> For a general pit located at (L, M, N) in the MacMahon representation,
the central charge is K = **q**<sub>1</sub><sup>L-1</sup>**q**<sub>2</sub><sup>M-1</sup>**q**<sub>3</sub><sup>N-1</sup>. A similar analysis can be done and we will see
that the MacMahon representation will be reduced to **F**<sub>1</sub><sup>⊗L</sup> ⊗ **F**<sub>2</sub><sup>⊗M</sup> ⊗ **F**<sub>3</sub><sup>⊗N</sup> with the spectral
parameters tuned properly (see for example [150, section 5.1.4]).

We can also do the same analysis for the case when the plane partitions appearing
have nontrivial boundary Young diagrams. For these cases, the eigenvalue of the vacuum
configuration is determined as

$$
\Psi_{\lambda, \mu, \nu, u}^{\text{vac}}(z) = K^{-1/2} \frac{1 - Ku/z}{1 - u/z} \prod_{\mathfrak{z} \in \mathcal{P}_{\lambda, \mu, \nu}} g\left(\frac{z}{\chi_u(\mathfrak{z})}\right). \qquad (9.27)
$$

<footnote n="38">³⁸The coefficients of the action of E(z), F(z) on the bases in the MacMahon representation after setting K = q_c is different from the coefficients in the Fock representation. This comes from the degree of freedom to rescale the bases.</footnote>
After taking the $q_4 \to 1$ limit, the coefficient appearing after taking the operator product of (7.67) and $S_4(q_4z)$ will become this vacuum Cartan eigenvalue.

## 9.5 BPS $qq$-characters and quiver quantum toroidal algebras

Up to the previous section, we managed to relate the $qq$-characters with the vertical representation of the quantum toroidal $gl_1$. To construct the explicit representation of $\mathcal{E}$, the essential part is how to determine the bases and the eigenvalue of $K^{\pm}(z)$. The action of the other Drinfeld currents is determined schematically as (9.21), where the coefficients are proportional to the residue of the eigenvalue. Moreover, the eigenvalue has a general structure

$$ \Psi_{\Lambda,u}(x) = \psi_{0,u}(x) \prod_{\mathfrak{z} \in \Lambda} g \left( \frac{z}{\chi_u(\mathfrak{z})} \right) \qquad (9.28) $$

where $\Lambda$ is either the 1d, 2d, 3d partitions or general truncations of the plane partition. The eigenvalue is a product of the vacuum function $\psi_{0,u}(z)$, and the contributions come from the boxes in the configurations. Actually, one can show that the zero and pole structure of the vacuum function $\psi_{0,u}(z)$ determines how the bases should be constructed [128, 137, 162].

This property is similar to the procedure to obtain $qq$-characters. To determine the $qq$-characters, we first started by choosing a screening current. We then choose a highest weight and impose the commutativity with the screening current. The $qq$-character is automatically determined in this way. The highest weight therefore has a one-to-one correspondence with the vacuum function of the vertical representations of $\mathcal{E}$:

$$ \begin{align} S_c(u) &\leftrightarrow \frac{(1 - q_{c-1}u/z)(1 - q_{c+1}u/z)}{(1 - u/z)(1 - q_{c-1}q_{c+1}u/z)}, \nonumber \\ X_{c4}(u) &\leftrightarrow q_c^{-1/2} \frac{1 - q_c u/z}{1 - u/z}, \tag{9.29} \\ \frac{W_{123}(u)}{W_{123}(Ku)} &\leftrightarrow K^{-1/2} \frac{1 - Ku/z}{1 - u/z}. \nonumber \end{align} $$

Note that this is just a rewriting of the conclusion obtained in sections 9.2, 9.3, 9.4.

Recently a large class of quantum toroidal algebras associated with toric Calabi-Yau three-folds<fnref n="39" /> were constructed [33, 34]. They are called quiver quantum toroidal algebras or toroidal quiver BPS algebras.<fnref n="40" /> We denote such algebras as $\mathcal{E}_{(Q,W)}$, where $Q$ is the corresponding quiver and $W$ is the superpotential. The above correspondence between the $qq$-characters and the vertical representations implies that we should have generalizations to such cases too. Let us show that the BPS $qq$-characters introduced in Conj. 7.17 are the $qq$-characters corresponding to the representations of such algebras.

### 9.5.1 Quiver quantum toroidal algebra

Let us introduce the definition of the algebra $\mathcal{E}_{(Q,W)}$ following [34]. Let $X$ be a toric Calabi-Yau three-fold. To $X$, one can associate a quiver $Q = (Q_0, Q_1)$ and a superpotential $W$. For

<footnote n="39">³⁹There are also quantum toroidal algebras associated with non-toric CYs. They are related to affine D and E-type algebras. The discussion here can also be generalized to those cases.</footnote>

<footnote n="40">⁴⁰We use the former terminology.</footnote>
each arrow of the quiver, we can associate parameters ${\mathfrak{q}_I}_{I \in Q_1}$. The superpotential imposes nontrivial conditions on them and the independent parameters are reduced up to two. We assume such conditions are imposed implicitly in the parameters.

**Definition 9.5** ([33, 34, 128, 129]). The algebra $\mathcal{E}_{(Q,W)}$ is generated by the Drinfeld currents:

$$E_i(z) = \sum_{m \in \mathbb{Z}} E_{i,m} z^{-m}, \quad F_i(z) = \sum_{m \in \mathbb{Z}} F_{i,m} z^{-m}, \quad K_i^{\pm}(z) = K_i^{\pm} z^{r_i^{\pm}} \exp \left( \pm \sum_{r=1}^{\infty} H_{i, \pm r} z^{\mp r} \right), \qquad (9.30)$$

for $i \in Q_0$, where $r_i^{\pm} \in \mathbb{Z}$. For each node $i \in Q_0$, we assign a $\mathbb{Z}_2$ grading:

$$|i| = \begin{cases} 0 & (\exists I \in Q_1 \text{ s.t. } I \text{ is a loop}), \\ 1 & (\text{otherwise}). \end{cases} \qquad (9.31)$$

We have one universal central element which is denoted as $C$. The defining relations are

$$\begin{align*}
K_i^{\pm}(z)K_j^{\pm}(w) &= K_j^{\pm}(w)K_i^{\pm}(z), \\
K_i^{-}(z)K_j^{+}(w) &= \frac{\tilde{\varphi}^{j \to i}(z, Cw)}{\tilde{\varphi}^{j \to i}(Cz, w)} K_j^{+}(w)K_i^{-}(z), \\
K_i^{\pm}(C^{\frac{1\mp 1}{2}}z)E_j(w) &= \tilde{\varphi}^{j \to i}(z, w)E_j(w)K_i^{\pm}(C^{\frac{1\mp 1}{2}}z), \\
K_i^{\pm}(C^{\frac{1\pm 1}{2}}z)F_j(w) &= \tilde{\varphi}^{j \to i}(z, w)^{-1}F_j(w)K_i^{\pm}(C^{\frac{1\pm 1}{2}}z), \tag{9.32} \\
[E_i(z), F_j(w)] &= \delta_{ij} \left( \delta\left(\frac{Cw}{z}\right) K_i^{+}(z) - \delta\left(\frac{Cz}{w}\right) K_i^{-}(w) \right), \\
E_i(z)E_j(w) &= (-1)^{|i||j|} \tilde{\varphi}^{j \to i}(z, w)E_j(w)E_i(z), \\
F_i(z)F_j(w) &= (-1)^{|i||j|} \tilde{\varphi}^{j \to i}(z, w)^{-1}F_jF_i(z),
\end{align*}$$

where the structure functions $\tilde{\varphi}^{i \to j}(z, w)$ are defined as

$$\begin{align}
\tilde{\varphi}^{i \to j}(z, w) &= (-1)^{\chi_{i \to j}} \frac{\prod_{I \in \{j \to i\}} (\mathfrak{q}_I^{1/2} z - \mathfrak{q}_I^{-1/2} w)}{\prod_{I \in \{i \to j\}} (\mathfrak{q}_I^{-1/2} z - \mathfrak{q}_I^{1/2} w)} \tag{9.33} \\
&= (-1)^{\chi_{i \to j}} \frac{\prod_{I \in \{j \to i\}} (-\mathfrak{q}_I^{-1/2} w)}{\prod_{I \in \{i \to j\}} (-\mathfrak{q}_I^{1/2} w)} \varphi^{i \to j}(z/w) \nonumber
\end{align}$$

where $\chi_{i \to j}$ is a factor determined by imposing the associativity condition

$$\tilde{\varphi}^{j \to i}(z, w) \tilde{\varphi}^{i \to j}(w, z) = 1 \qquad (9.34)$$

and $\varphi^{i \to j}(z)$ is defined in (4.114).

**Remark 9.6.** We add the factor $z^{r_i^{\pm}}$ to the mode expansion of $K_i^{\pm}(z)$ and relaxed the algebra. Such modifications will be important only when we consider the mode expansions of representations. Moreover, they are related to shifted quantum algebras. They are not important in this paper so we will not discuss them.
Setting $C = 1$, one can see that $K_i^{\pm}(z)$ commutes with each other and we have diagonal bases. Such diagonal bases are labeled by 3d BPS crystals [57] and we have a natural action of the algebra on them (see [32] for details). The 3d BPS crystals are a set of colored atoms where the colors are labeled by $Q_0$ (see Conj. 7.17 for the notations). The color projection map $c: \Lambda \to Q_0$ maps the color of the crystal to the quiver nodes. Similar to the discussion in Conj. 7.17, we can define a coordinate function:

$$ \chi_u(\bar{\mathfrak{a}}) := u \times \prod_{I \in \text{path}[\mathfrak{a} \to \bar{\mathfrak{a}}]} q_I. \qquad (9.35) $$

The difference with the coordinate function in Conj. 7.17 is that we use the $q$-deformation parameters $q_I$. There, we have an additional parameter $q_4$, while here we do not have such a parameter.

Under this situation, the representation is schematically given as

$$ \begin{aligned} K_i^{\pm}(z) |u, \Lambda\rangle &= [\Psi_{\Lambda,u}^{(i)}(z)]_{\pm}^{z} |u, \Lambda\rangle, \\ E_i(z) |u, \Lambda\rangle &= \sum_{\mathfrak{a} \in A(\Lambda)} \# \sqrt{\operatorname{Res}_{z=\chi_u(\mathfrak{a})} z^{-1} \Psi_{\Lambda,u}^{(i)}(z) \delta\left(\frac{z}{\chi_u(\mathfrak{a})}\right)} |u, \Lambda + \mathfrak{a}\rangle, \qquad (9.36) \\ F_i(z) |u, \Lambda\rangle &= \sum_{\mathfrak{a} \in R(\Lambda)} \# \sqrt{\operatorname{Res}_{z=\chi_u(\mathfrak{a})} z^{-1} \Psi_{\Lambda,u}^{(i)}(z) \delta\left(\frac{z}{\chi_u(\mathfrak{a})}\right)} |u, \Lambda - \mathfrak{a}\rangle \end{aligned} $$

where $\Lambda$ is the 3d BPS crystal configuration and $A(\Lambda)$, $R(\Lambda)$ are addable and removable atoms from the configuration and $\#$ is some coefficient factor. The eigenvalue $\Psi_{\Lambda,u}^{(i)}(z)$ schematically has the form

$$ \Psi_{\Lambda,u}^{(i)}(z) = \psi_{\emptyset,u}^{(i)}(z) \prod_{j \in Q_0} \prod_{\substack{\mathfrak{a} \in \Lambda, \\ c(\mathfrak{a}) = j}} \tilde{\varphi}^{j \to i}(z, \chi_u(\mathfrak{a})) \qquad (9.37) $$

where $\psi_{\emptyset,u}^{(i)}(z)$ is the vacuum function. Namely, the eigenvalue is a product of the vacuum function and contributions coming from all of the atoms in the crystal configuration. The contribution to $\Psi_{\Lambda,u}^{(i)}(z)$ coming from an atom at the coordinate $\chi_u(\mathfrak{a})$ with color $j \in Q_0$ comes from $\tilde{\varphi}^{j \to i}(z, \chi_u(\mathfrak{a}))$. The poles of the eigenvalue $\Psi_{\Lambda,u}^{(i)}(z)$ are the $q$-coordinates of the addable and removable atoms with color $i \in Q_0$ and thus the action of $E_i(z)$, $F_i(z)$ are determined by the residue of them.

The possible configuration is uniquely determined by the vacuum function $\psi_{\emptyset,u}^{(i)}(z)$ (see [128] for examples). We only focus on 3d BPS crystals with only one atom at the origin. For such cases, the vacuum function is given

$$ \psi_{\emptyset,u}^{(i)}(z) = \left( \frac{K^{-1/2}z - K^{1/2}u}{z - u} \right)^{\delta_{ia}}. \qquad (9.38) $$

This vacuum function gives the BPS crystals where the atom at the origin has color $a$. We denote such crystal configurations as $\Lambda^{(a)}$ for later use.
Remark 9.7. When $X = \mathbb{C}^2/\mathbb{Z}_n \times \mathbb{C}$, the corresponding quantum toroidal algebra is the quantum toroidal gl$_n$ [163–165], which we denote $\mathcal{E}_n$ for later use. Applications of these algebras to supersymmetric gauge theories were done in [166]. For general abelian orbifold cases such as $X = \mathbb{C}^3/\mathbb{Z}_n$ or $X = \mathbb{C}^3/(\mathbb{Z}_m \times \mathbb{Z}_n)$, the representations and some applications were studied in [72, 75, 129, 167].

### 9.5.2 BPS $qq$-characters

Let $Z$ be a toric $\mathrm{CY}_4$ with the form $Z = X \times \mathbb{C}$ where $X$ is a toric $\mathrm{CY}_3$. We denote the corresponding quiver of $X$ as $Q = (Q_0, Q_1)$ (see section 4.6.3 for the notations). As discussed in section 4.6.3 and Conj. 7.17, we have a screening current $S_i(x)$ and a screening charge corresponding to the $\mathbb{C}$ part of $Z$. The $qq$-character $\mathrm{T}_j^K(x)$ is obtained from the highest weight: $\frac{W_i(x)}{W_j(Kx)}$; and the monomial terms are expanded in the 3d BPS crystal $\Lambda^{(j)}$. Let us study the operator products of the monomial terms with the screening current:

$$ 
\begin{aligned}
& \frac{W_j(u)}{W_j(Ku)} \prod_{\mathfrak{t} \in \Lambda^{(j)}} A_{c(\mathfrak{t})}^{-1}(\chi_{X,u}(\mathfrak{t})) :S_i(q_4z) : \\
& = h_{ji}(u,z) \left(\frac{1-K^{-1}z/u}{1-z/u}\right)^{\delta_{ij}} \prod_{\mathfrak{t} \in \Lambda^{(j)}} \varphi_{X,c(\mathfrak{t})i} \left(\frac{q_4z}{\chi_{X,u}(\mathfrak{t})}\right) :\frac{W_j(u)}{W_j(Ku)} \prod_{\mathfrak{t} \in \Lambda^{(j)}} A_{c(\mathfrak{t})}^{-1}(\chi_{X,u}(\mathfrak{t}))S_i(q_4z) :; \\
& S_i(q_4z) :\frac{W_j(u)}{W_j(Ku)} \prod_{\mathfrak{t} \in \Lambda^{(j)}} A_{c(\mathfrak{t})}^{-1}(\chi_{X,u}(\mathfrak{t})) : \\
& = \tilde{h}_{ji}(u,z) \left(\frac{1-Ku/z}{1-u/z}\right)^{\delta_{ij}} \prod_{\mathfrak{t} \in \Lambda^{(j)}} \varphi_{X,i c(\mathfrak{t})} \left(\frac{\chi_{X,u}(\mathfrak{t})}{z}\right)^{-1} :\frac{W_j(u)}{W_j(Ku)} \prod_{\mathfrak{t} \in \Lambda^{(j)}} A_{c(\mathfrak{t})}^{-1}(\chi_{X,u}(\mathfrak{t}))S_i(q_4z) :;
\end{aligned} 
\quad (9.39) $$

where $h_{ji}(z, u)$, $\tilde{h}_{ji}(z, u)$ are extra factors coming from the zero-modes.

After taking the limit $q_4 \to 1$, one can see that the contraction becomes the eigenvalue in (9.37) up to extra factors which do not affect the zero and pole structure:

$$ \left( \frac{1-K^{-1}z/u}{1-z/u} \right)^{\delta_{ij}} \prod_{\mathfrak{t} \in \Lambda^{(j)}} \varphi_{X,c(\mathfrak{t})i} \left( \frac{q_4z}{\chi_{X,u}(\mathfrak{t})} \right) \xrightarrow{q_4 \to 1} \left( \frac{1-K^{-1}z/u}{1-z/u} \right)^{\delta_{ij}} \prod_{\mathfrak{t} \in \Lambda^{(j)}} \varphi^{c(\mathfrak{t})} \xrightarrow{i} \left( \frac{z}{\chi_{u}(\mathfrak{t})} \right) \quad (9.40) $$

where $\chi_{X,u}(\mathfrak{t}) \xrightarrow{q_4 \to 1} \chi_u(\mathfrak{t})$. We thus have the following identification

$$ S_i(q_4z) \to K_i^{\pm}(z), \quad : \frac{W_j(u)}{W_j(Ku)} \prod_{\mathfrak{t} \in \Lambda^{(j)}} A_{c(\mathfrak{t})}^{-1}(\chi_{X,u}(\mathfrak{t})) : \to |u, \Lambda^{(j)}\rangle. \quad (9.41) $$

Therefore, we conclude that the BPS $qq$-characters associated with $Z = X \times \mathbb{C}$ are related to the vertical representations, the BPS crystal representations, of quiver quantum toroidal algebras. Actually, the terminology BPS $qq$-characters came from the fact that they are related to these BPS crystals.

## 10 Semi-classical analysis and Bethe ansatz equation

We have discussed the gauge origami system on $\mathbb{C}^4$ with four associated equivariant parameters, $\epsilon_{1,2,3,4}$ obeying the $\mathrm{CY}_4$ condition, $\sum_{a \in \mathfrak{t}} \epsilon_a = 0$. In the multiplicative notation, this is written
as $q_1q_2q_3q_4 = 1$. It has been known that the asymptotic behavior of the partition function in the semi-classical limit, $\epsilon_a \to 0$, provides various physical information. For example, for 4d $\mathcal{N} = 2$ theory on $\mathbb{C}_{12}$, one can extract the Seiberg-Witten prepotential from the partition function in the limit $\epsilon_{1,2} \to 0$ [1]. Moreover, one can observe the relation between gauge theory and quantum integrable system in the limit, $\epsilon_1$ finite, $\epsilon_2 \to 0$, which is called the Nekrasov-Shatashvili limit (Bethe/Gauge correspondence) [168]. In particular, the saddle point equation obtained from the gauge theory partition function in the NS limit is identified with the Bethe ansatz equation (BAE) in this context. In this section, we explore the NS-type limit of the gauge origami system and examine the corresponding BAE.

In general, the partition function defined on $\times_{a \in S} \mathbb{C}_a$ ($S \subset \mathbf{4}$) behaves in the semi-classical limit as follows,

$$ \mathcal{Z}_S \xrightarrow{\epsilon_{\forall a} \to 0} \exp \left( \frac{1}{\epsilon_S} \mathcal{F}_S + \dots \right), \quad \epsilon_S = \prod_{a \in S} \epsilon_a, \qquad (10.1) $$

where $\mathcal{F}_S$ is identified with the “prepotential” associated with gauge theory on $\times_{a \in S} \mathbb{C}_a$. We may also consider a partial semi-classical limit to obtain the following,

$$ \mathcal{Z}_S \xrightarrow{\epsilon_a \to 0} \exp \left( \frac{1}{\epsilon_a} \tilde{W}_a + \dots \right), \qquad (10.2) $$

where $\tilde{W}_a(= \mathcal{F}_a)$ is the twisted superpotential associated with 2d (resp. 3d) gauge theory on $\mathbb{C}_a$ ($\mathbb{C}_a \times \mathbb{S}^1$). In the context of Bethe/Gauge correspondence, this twisted superpotential is identified with the Yang-Yang functional, and the corresponding critical point condition (twisted F-term condition) provides the BAE,

$$ \exp \left( \frac{\partial \tilde{W}_a}{\partial x} \right) = 1. \qquad (10.3) $$

In fact, this is immediately obtained from the behavior of the partition function under the instanton adding/removing operation presented in this paper. Applying this process to the gauge origami system on $\mathbb{C}^4$, we obtain the following BAE in the NS-type limit.

**Theorem 10.1.** Let $\mathbf{3} = \{1, 2, 3\}$. We denote $q_a \to q_a$ ($a \in \mathbf{3}$) in the limit $q_4 \to 1$ obeying $q_1q_2q_3 = 1$. Consider a gauge origami system having the support on the $\mathbb{C}_4$-plane, where D-branes wrap the subspace $\mathcal{S} \times \mathbb{S}^1$, where $\mathcal{S} = \tilde{\mathcal{S}} \times \mathbb{C}_4$ with $\tilde{\mathcal{S}} \subset \mathbf{3}$ is generally

$$ \mathcal{S} = n\mathbb{C}_4 + (L_1\mathbb{C}_{14}^2 + M_1\mathbb{C}_{24}^2 + N_1\mathbb{C}_{34}^2) + (L_2\mathbb{C}_{234}^3 + M_2\mathbb{C}_{134}^3 + N_2\mathbb{C}_{124}^3). \qquad (10.4) $$

The corresponding BAE of this gauge origami system obtained in the limit $q_4 \to 1$ is

$$ 1 = -q \frac{\mathcal{Q}_S(\mathfrak{q}_1^{-1}x)\mathcal{Q}_S(\mathfrak{q}_2^{-1}x)\mathcal{Q}_S(\mathfrak{q}_3^{-1}x)}{\mathcal{Q}_S(\mathfrak{q}_1x)\mathcal{Q}_S(\mathfrak{q}_2x)\mathcal{Q}_S(\mathfrak{q}_3x)}, \qquad (10.5) $$

where the corresponding Q-functions for the D2/D4/D6 system are given by (10.24), (10.8), (10.16), (10.40), (10.43).

We provide the proof of this Theorem for each system in the following.
## 10.1 D4 system

We start with the D4 system, which realizes 5d $N = 1$ gauge theory compactified on a circle with the adjoint matter, i.e., 5d $N = 1^*$ theory. The corresponding BAE was obtained in [168, 169]. We here rederive the BAE from our analysis presented in appendix B.

Consider 5d U(1) gauge theories defined on $\mathbb{C}_{a4}^2 \times \mathbb{S}^1$, where $a \in \mathbf{3}$. See section 10.4 for U(n) case. Namely, the gauge theories share a subspace $\mathbb{C}_4 \times \mathbb{S}^1$. Under generic $\Omega$-background, the structure functions $y_{\lambda,v}^{a4}(x)$ ($a \in \mathbf{3}$) can be written using the (1,1)-type description in section 3.7.2 as

$$y_{\lambda,v}^{a4}(x) = \prod_{i=1}^{\infty} \frac{1 - x_{v;i}^{(a)}/x}{1 - q_a x_{v;i}^{(a)}/x} = \prod_{i=1}^{\infty} \nu_a \left( \frac{x_{v;i}^{(a)}}{x} \right)^{-1} \qquad (10.6)$$

where

$$x_{v;i}^{(a)} = v q_a^{i-1} q_4^{\lambda_i}. \qquad (10.7)$$

Introducing the Q-functions [9] under generic instanton background

$$Q_{a4}(x) = \prod_{i=1}^{\infty} \left(1 - x_{v;i}^{(a)}/x\right) \qquad (10.8)$$

we may rewrite the $\mathcal{Y}$-functions as

$$y_{\lambda,v}^{a4}(x) = \frac{Q_{a4}(x)}{Q_{a4}(q_a^{-1}x)}. \qquad (10.9)$$

The D4 partition function behaves as Thm. B.2 and (B.42):

$$\begin{aligned} \frac{\mathcal{Q}^{|\lambda+|} \tilde{Z}_{a4}^{D4}[\lambda+\Box]}{\mathcal{Q}^{|\lambda|} \tilde{Z}_{a4}^{D4}[\lambda]} &= -\mathcal{Q} \frac{y_{\lambda,v}^{a4}(q_b^{-1}\chi_{a4,v}(\Box)) y_{\lambda+\Box,v}^{a4}(q_c^{-1}\chi_{a4,v}(\Box))}{y_{\lambda,v}^{a4}(\chi_{a4,v}(\Box)) y_{\lambda+\Box,v}^{a4}(q_{bc}^{-1}\chi_{a4,v}(\Box))} \\ &= -\mathcal{Q} \frac{S_{a4}(q_{c4}\chi_{a4,v}(\Box)/x) y_{\lambda,v}^{a4}(q_{ac}x) y_{\lambda,v}^{a4}(q_{ab}x)}{S_{a4}(q_{bc4}\chi_{a4,v}(\Box)/x) y_{\lambda,v}^{a4}(q_{abc}x) y_{\lambda,v}^{a4}(q_a x)} \end{aligned} \qquad (10.10)$$

where $a, b, c \in \mathbf{3}$ and $x = q_4\chi_{a4,v}(\Box)$. In the NS limit $q_4 \to 1$, the left hand side becomes $\exp(\partial\tilde{W}_4/\partial x)$ and $S_{a4}(x) \to 1$. Hence, the twisted F-term condition reads

$$1 = -\mathcal{Q} \frac{y_{\lambda,v}^{a4}(q_{ac}x) y_{\lambda,v}^{a4}(q_{ab}x)}{y_{\lambda,v}^{a4}(q_{abc}x) y_{\lambda,v}^{a4}(q_a x)}. \qquad (10.11)$$

Denoting the solution to this twisted F-term condition as $\lambda_*$ (critical configuration; limit shape), we can rewrite the $\mathcal{Y}$-functions in terms of the Q-functions, where the $x$-variables will take values $x_{v;i}^{(a)}|_{\lambda=\lambda_*}$. The BAE is then written as

$$1 = -\mathcal{Q} \frac{Q_{a4}(q_a^{-1}x) Q_{a4}(q_b^{-1}x) Q_{a4}(q_c^{-1}x)}{Q_{a4}(q_a x) Q_{a4}(q_b x) Q_{a4}(q_c x)}, \quad \{a, b, c\} = \mathbf{3} \qquad (10.12)$$

where $q_{a,b,c} \to q_{a,b,c}$ under $q_4 \to 1$ (see also section 9.1). Using the structure function of the quantum toroidal $\mathfrak{gl}_1$ in (9.4), we can rewrite this BAE as

$$1 = -\mathcal{Q} \prod_{i=1}^{\infty} \mathfrak{g} \left( x_{v;i}^{(a)} / x \right). \qquad (10.13)$$
Note that we have the condition $q_a q_b q_c = 1$. This is the BAE for the elliptic Calogero-Moser/Ruijsenaars-Schneider model, which is the integrable system associated with $\hat{A}_0$ quiver gauge theory [71, 170, 171]. Note that in the quantum integrable model side, the parameter $q_a$ corresponds to the Planck constant, while one of the parameters $q_b, q_c$ correspond to the mass parameter. The instanton counting parameter $q$ becomes the elliptic parameter of the potential function. The weak coupling limit corresponds to the trigonometric limit. See, for example, [172].

## 10.2 D6 system

We then consider the D6 system and derive the corresponding BAE. We consider 7d U(1) gauge theories defined on $\mathbb{C}_{ab4}^3 \times \mathbb{S}^1$ ($ab \in \{12, 23, 13\}$). Under generic $\Omega$-background, the structure functions $W_{\pi,v}^{ab4}(x)$ ($ab \in \{12, 13, 23\}$) associated with the plane partition configuration $\pi$ can be written using the (2, 1)-type description in section 3.7.3 as

$$W_{\pi,v}^{ab4}(x) = \prod_{i,j=1}^{\infty} \frac{(1 - x_{v;i,j}^{(ab)}/x)(1 - q_{ab}x_{v;i,j}^{(ab)}/x)}{(1 - q_a x_{v;i,j}^{(ab)}/x)(1 - q_b x_{v;i,j}^{(ab)}/x)} = \prod_{i,j=1}^{\infty} S_{ab} \left( \frac{x_{v;i,j}^{(ab)}}{x} \right)^{-1} \quad (10.14)$$

where we introduced

$$x_{v;i,j}^{(ab)} = v q_a^{i-1} q_b^{j-1} q_4^{\pi_{ij}}. \quad (10.15)$$

Similar to the D4-case, we introduce the Q-functions as

$$Q_{ab4}(x) = \prod_{i,j=1}^{\infty} \left(1 - x_{v;i,j}^{(ab)}/x\right) \quad (10.16)$$

and then the W-functions are rewritten as

$$W_{\pi,v}^{ab4}(x) = \frac{Q_{ab4}(x)Q_{ab4}(q_{ab}^{-1}x)}{Q_{ab4}(q_a^{-1}x)Q_{ab4}(q_b^{-1}x)}. \quad (10.17)$$

The recursion formula of the D6-partition function comes from (B.25) and Thm. B.1:

$$\frac{\mathfrak{q}^{|\pi+\bar{\mathfrak{\Gamma}}|} \tilde{\mathcal{Z}}_{ab4}^{\mathrm{D6}}[\pi+\bar{\mathfrak{\Gamma}}]}{\mathfrak{q}^{|\pi|} \tilde{\mathcal{Z}}_{ab4}^{\mathrm{D6}}[\pi]} = -\mathfrak{q} \frac{W_{\pi+\bar{\mathfrak{\Gamma}},v}^{ab4}(q_{ab4}\chi_{ab4,v}(\bar{\mathfrak{\Gamma}}))}{W_{\pi,v}^{ab4}(\chi_{ab4,v}(\bar{\mathfrak{\Gamma}}))} \\ = -\mathfrak{q} g_{ab4} \left( \frac{\chi_{ab4,v}(\bar{\mathfrak{\Gamma}})}{q_{ab}x} \right) \frac{W_{\pi,v}^{ab4}(q_{ab}x)}{W_{\pi,v}^{ab4}(q_4^{-1}x)} \quad (10.18)$$

where $x = q_4\chi_{ab4,v}(\bar{\mathfrak{\Gamma}})$. Taking the NS limit $q_4 \to 1$ and using $g_{ab4}(x) \xrightarrow{q_4 \to 1} 1$, we obtain the twisted F-term condition as

$$1 = -\mathfrak{q} \frac{W_{\pi,v}^{ab4}(q_{ab}x)}{W_{\pi,v}^{ab4}(q_4^{-1}x)}. \quad (10.19)$$

Denoting the critical configuration $\pi_*$ and using the Q-functions, we have the following BAE:

$$1 = -\mathfrak{q} \frac{Q_{ab4}(q_a^{-1}x)Q_{ab4}(q_b^{-1}x)Q_{ab4}(q_c^{-1}x)}{Q_{ab4}(q_a x)Q_{ab4}(q_b x)Q_{ab4}(q_c x)}, \quad a, b, c \in \mathbf{3} \quad (10.20)$$