1 Introduction↩︎

In [1] and [2] we have explored various aspects of the non-stationary difference equation; \[\label{Shakirov-eq} {\mathcal{H}_{\mathrm S}}T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1} \cdot \Psi(\Lambda,x)=\Psi(\Lambda,x), \qquad \Psi(\Lambda,x) = \sum_{m,n \geq 0} c_{m,n} x^m (\Lambda/x)^n, \quad (c_{0,0}=1),\tag{1}\] which was first introduced in [3]. The Hamiltonian has mass parameters \(d_i\) and is given by \[\begin{align} \label{Shakirov-Hamiltonian} {\mathcal{H}_{\mathrm S}}=& \frac{1}{\varphi(qx)\varphi(\Lambda/x)} \cdot \mathcal{B}\cdot \frac{\varphi(\Lambda)\varphi(q^{-1} d_1d_2d_3d_4\Lambda)}{\varphi(-d_1x)\varphi(-d_2x)\varphi(-d_3 \Lambda/x)\varphi(-d_4 \Lambda/x)} \nonumber \\ &~~\cdot \mathcal{B}\cdot \frac{1}{\varphi(q^{-1}d_1d_2x)\varphi(d_3d_4 \Lambda/x)}, \end{align}\tag{2}\] where \(\varphi(z) := (z;q)_\infty\), \(\mathcal{B}\) is the \(q\)-Borel transformation and \(T_{\alpha,z}\) denotes the shift operator \(z \to \alpha z\). Other notations used throughout the paper are summarized in subsection 1.6 at the end of the introduction. The non-stationary difference equation 1 is related to the quantized discrete Painlevé VI equation [1]. Namely, the Hamiltonian 2 is equivalent to the Hamiltonian of the discrete Painlevé VI equation given by [4], in the sense that they have the same adjoint action on the canonical variables \((F,G)\) with \(FG=q^{-1}GF\). On the other hand, if we tune two of the mass parameters, say \(d_2=q^{-m},d_3=q^{-n},~m,n \in \mathbb{Z}_{\geq 0}\), the equation 1 can be also identified with the quantum Knizhnik-Zamolodchikov (\(q\)-KZ) equation for \(U_q(\widehat{\mathfrak{gl}}_2)\) with generic spins. Based on this fact we can prove that the \(K\)-theoretic Nekrasov partition function¹ coming from the affine Laumon space provides a solution to the equation 1.

In this paper we propose a \({\widehat{\mathfrak{gl}}_N}\) generalization of the non-stationary Hamiltonian 2 . For explicit expressions see Definitions 1 – 4 below. One of the significant differences from the \(N=2\) case is that the arguments of \(\varphi\) become \(q\)-commutative. Let us introduce two sets of \(q\)-commutative variables \((\hat{\mathsf{u}}_i,\check{\mathsf{u}}_i)~(i \in \mathbb{Z}/ N \mathbb{Z})\) with the following commutation relations; \[\label{q-com-1} \hat{\mathsf{u}}_i \hat{\mathsf{u}}_{j} = q^{\delta_{i, j-1}- \delta_{i-1,j}} \hat{\mathsf{u}}_{j} \hat{\mathsf{u}}_{i}, \qquad \check{\mathsf{u}}_i \check{\mathsf{u}}_{j} = q^{\delta_{i-1,j}-\delta_{i, j-1}}\check{\mathsf{u}}_{j} \check{\mathsf{u}}_{i},\tag{3}\] and \[\label{q-com-2} \hat{\mathsf{u}}_i \check{\mathsf{u}}_j = q^{2 \delta_{i,j} - \delta_{i,j+1} - \delta_{i, j-1}} \check{\mathsf{u}}_j \hat{\mathsf{u}}_i,\tag{4}\] where \(\delta_{i,j}\) is the Kronecker delta modulo \(N\). Note that the matrix which appears in the power of \(q\) is the Cartan matrix of \({A}_{N-1}^{(1)}\). To write down the Hamiltonian of the non-stationary \({\widehat{\mathfrak{gl}}_N}\) difference equation with \(N\) commutative variables \(x_i~(i \in \mathbb{Z}/ N\mathbb{Z})\), we employ the following representation of the algebra generated by \((\hat{\mathsf{u}}_i,\check{\mathsf{u}}_i)\); \[\label{x-rep} \hat{x}_i := \alpha_i x_i q^{\vartheta_i - \vartheta_{i-1}}, \qquad \check{x}_i := \beta_i x_i q^{-\vartheta_i + \vartheta_{i-1}},\tag{5}\] where \(\vartheta_i := x_i \frac{\partial}{\partial x_i}\) and \(\alpha_i, \beta_i\) are arbitrary scaling parameters. Since the index of \(x_i\) is in \(\mathbb{Z}/ N\mathbb{Z}\), we will identify \(x_0\) with \(x_N\) throughout the paper. From \(p_i x_j = q^{\delta_{i,j}} x_j p_i\) with \(p_i := q^{\vartheta_i}\), we see that \(\hat{x}_i\) and \(\check{x}_i\) satisfy the commutation relations 3 and 4 . The non-stationary \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian has \(3N\) parameters \(b_i, d_i, \overline{d}_i~(i \in \mathbb{Z}/ N\mathbb{Z})\). It also involves the quantum deformation parameter \(q\) and the shift parameter \(t^{-1}= \kappa^{N}\). In the supersymmetric gauge theory, \(x_i\) are instanton expansion parameters, \(b_i\) are Coulomb moduli and \((d_i,\overline{d}_i)\) are mass parameters. The equivariant parameters \((q,t)\) come from the torus action \((z_1, z_2) \to (q z_1, \kappa z_2)\) on \(\mathbb{C}^2\).

There are three equivalent definitions of \(\mathcal{A}_L= \mathcal{A}_L^{(i)}\) and \(\mathcal{A}_R=\mathcal{A}_R^{(i)}\) with \(i = \mathrm{s}, \mathrm{h}, \mathrm{n}\), which is one of the remarkable consequences of the fact that \(\mathcal{A}_L\) and \(\mathcal{A}_R\) involve the \(q\)-commutative variables \(\hat{x}_i\) and \(\check{x}_i\), respectively. To define \(\mathcal{A}_L\) and \(\mathcal{A}_R\), we choose the scaling parameters of \(\hat{x}_i\) and \(\check{x}_i\) as \(\alpha_i = d_i \overline{d}_i\) and \(\beta_i =1\) and set \(D_N := {\displaystyle\prod_{k=1}^{N}}~d_k \overline{d}_k\).

In section 2 we will prove the equivalence of three forms of the Hamiltonian. In subsection 2.1 we show the pentagon identity implies the equivalence of two factorized forms of the Hamiltonian; \(\mathcal{A}_L^{(\mathrm{s})}=\mathcal{A}_L^{(\mathrm{h})}\) and \(\mathcal{A}_R^{(\mathrm{s})}=\mathcal{A}_R^{(\mathrm{h})}\). On the other hand in subsection 2.2 we employ the \(q\)-binomial theorem to prove the equivalence to the normal ordered Hamiltonian; \(\mathcal{A}_L^{(\mathrm{h})}=\mathcal{A}_L^{(\mathrm{n})}\) and \(\mathcal{A}_R^{(\mathrm{h})}=\mathcal{A}_R^{(\mathrm{n})}\). These equalities among the Hamiltonians mean the identities as linear operators on the space of formal power series in \(x_i\).

1.1 Several Remarks↩︎

1.1.1 ↩︎

The arguments of the middle block \(\mathcal{A}_C\) are commutative variables \(x_i\). We note that \(\hat{x}_1 \cdots \hat{x}_N = q^{-1} D_N \Lambda,~\check{x}_1 \cdots \check{x}_N = q\Lambda,~ \hat{x}_N \cdots \hat{x}_1 = q^{1-N} D_N \Lambda\) and \(\check{x}_N \cdots \check{x}_1 = q^{N-1} \Lambda\) are central elements in the algebra.

1.1.2 ↩︎

Compared with the \(\widehat{\mathfrak{gl}}_2\) Hamiltonian 2 , the \(q\)-Borel transformation \(q^{\frac{1}{2}\Delta}\) is moved to both of the end positions. See Appendix 6 for the agreement of the \(N=2\) case of ?? and 2 . In \({\widehat{\mathfrak{gl}}_N}\) case if \(q^{\frac{1}{2}\Delta}\) is put between the blocks \(\mathcal{A}_i\), the formulas for \(\mathcal{A}_i\) will become more involved (see Proposition 19).

1.1.3 ↩︎

In terms of the Hamiltonian of normal ordered form, the Schrödinger equation for the wave function \(\psi\) can be written the following way; \[\begin{align} (\mathcal{A}_L^{(\mathrm{n})})^{-1} q^{-\frac{1}{2}\Delta} \psi &= \mathcal{A}_C \cdot \mathcal{A}_R^{(\mathrm{n})} \cdot q^{\frac{1}{2}\Delta} \mathsf{T} \psi, \\ \psi &= \sum_{\theta_1, \ldots, \theta_N =0}^\infty c_{\theta_1, \ldots, \theta_N} x_1^{\theta_1} \cdots x_N^{\theta_N}, \quad (c_{0,\ldots,0}=1). \end{align}\] By the gauge transformation of the form \(\psi \to \prod_{i} x_i^{\beta_i}\cdot \psi\) with an appropriate scaling of \(x_i\) we can eliminate the parameters \(b_i\) in the shift operator \(\mathsf{T}\) so that the dependence on \(b_i\) only appears in the wave function \(\psi\). See Remark 2.9 in [1] for an explicit example in the case \(N=2\).

1.1.4 ↩︎

In Definition 2 of simple root type the arguments \(x_0, x_1, \ldots, x_{N-1}\) of the function \(\varphi(z)\) correspond to the simple roots of the affine algebra \(A_{N-1}^{(1)}\) and \(\Lambda := x_0x_1 \cdots x_{N-1}\) corresponds to the null root. On the other hand in Definition 3 of higher root type, the Hamiltonian involves the \(q\)-exponential factors corresponding to the higher roots. Note that \(\mathcal{A}_L^{(\mathrm{h})}\) and \(\mathcal{A}_R^{(\mathrm{h})}\) involve \(N-1\) \(q\)-exponentials which correspond to the simple roots of \(A_{N-1}\) and \(N-1\) \(q\)-exponentials with variables for higher roots of the affine algebra. The factorized Hamiltonian of higher root type is more convenient to see the relation to the universal \(R\) matrix of \(U_q(A_{N-1}^{(1)})\) [6].

1.1.5 ↩︎

An interesting feature of the Hamiltonian of simple root type is that the factor corresponding to the last variable \(x_N=x_0\) is twisted by the adjoint action of \(G_L(\check{x}_i)\) or \(G_R(\hat{x}_i)\), which is the product of \(\varphi\) with variables \(x_1, \ldots, x_{N-2}\).² In contrast to the non-affine \(\mathfrak{gl}_N\) case, the cyclic symmetry of the Hamiltonian in \(x_1, \ldots, x_N\) is required for the affine \({\widehat{\mathfrak{gl}}_N}\) case, which is non-trivial, since the Hamiltonian involves the \(q\)-commuting variables. We note that the twisting guarantees the desired cyclic symmetry of the Hamiltonian. In subsection 2.3, we give a classical analogue of the twisting in \(\mathcal{A}_L^{(\mathrm{s})}\) and \(\mathcal{A}_R^{(\mathrm{s})}\). More generally, due to the pentagon identity for \(\varphi(z)^{-1}\) with \(q\)-commutative variables, the Hamiltonian \(\mathcal{H}^{{\widehat{\mathfrak{gl}}_N}}\) is invariant under the automorphism of the Dynkin diagram of \(A_{N-1}^{(1)}\), (See Corollary 14).

1.1.6 ↩︎

The \(q\)-commuting variables \(\hat{x}_i\) and \(\check{x}_i\) appear in the arguments of the \(q\)-exponential function \(\varphi(z)= e_q(z)^{-1}\). By using \[\operatorname{Ad} ( q^{\frac{1}{2}(\vartheta_i - \vartheta_{i-1})^2}) (\alpha_i x_i)^n = q^{\frac{n}{2}} (\hat{x}_i)^n, \qquad \operatorname{Ad} ( q^{-\frac{1}{2}(\vartheta_i - \vartheta_{i-1})^2}) (\beta_i x_i)^n = q^{-\frac{n}{2}} (\check{x}_i)^n,\] (see 62 ), we can replace the \(q\)-exponential functions with \(q\)-commuting variables by \(\varphi(x_i)= e_q(x_i)^{-1}\) with commuting variables \(x_i\). For example when \(N=3\) the left block \(\mathcal{A}_L^{(\mathrm{s})}\) can be written as follows;³ \[\begin{align} \label{eq:A95Lcom} & q^{\frac{1}{2}\Delta} \cdot \mathcal{A}_L^{(\mathrm{s})} \cdot \varphi(\Lambda)^{-1} \nonumber \\ &= q^{\frac{1}{2}(-\vartheta_1 \vartheta_2 + \vartheta_1 \vartheta_3 + \vartheta_2 \vartheta_3)} \varphi(q^{\frac{1}{2}}x_1)^{-1} q^{-\vartheta_1 \vartheta_3} \varphi(q^{\frac{1}{2}}x_3)^{-1} q^{\vartheta_1 \vartheta_3 -\vartheta_2 \vartheta_3} \varphi(q^{\frac{1}{2}}x_1) \nonumber \\ & \qquad q^{\vartheta_1\vartheta_2} \varphi(q^{\frac{1}{2}}x_2)^{-1} q^{-\vartheta_1 \vartheta_2} \varphi(q^{\frac{1}{2}}x_1)^{-1} q^{\frac{1}{2}(-\vartheta_1 \vartheta_3 + \vartheta_1 \vartheta_2 + \vartheta_2 \vartheta_3)} \cdot q^{\frac{1}{2}\Delta}. \end{align}\tag{6}\] Thus, the expense of eliminating \(q\)-commuting variables from the arguments of \(\varphi(z)\) is the scattered insertion of the operators of the form \(q^{\tiny quadratic in~\vartheta_i}\) between the \(q\)-exponential functions. Note that the position of the \(q\)-Borel transformation \(q^{\frac{1}{2}\Delta}\) is changed from the left of \(\varphi(\hat{x}_i)\) to the right of \(\varphi(q^{\frac{1}{2}}x_i)\) with commuting variables \(x_i\). For general \({\widehat{\mathfrak{gl}}_N}\) case, see subsection 2.4.

1.2 Affine Laumon partition function: Conjecture↩︎

In [2] we proved that the affine Laumon partition function of type \(A_1^{(1)}\) provides a solution to the non-stationary difference equation 1 . In general the affine Laumon partition function of type \(A_{N-1}^{(1)}\) is defined as follows;

Now we are ready to present our main claim in this paper.

Firstly, let us make a remark on the cases \(N=1,2\). The \({\widehat{\mathfrak{gl}}_2}\) case of the conjecture was proved in [2]. We can see it is also valid for \(N=1\) as follows; Dropping the indices, we simply write \(x=x_1,d=d_1,\overline{d}=\overline{d}_1\), etc. The Hamiltonian is simplified to \[\mathcal{H}^{\widehat{\mathfrak{gl}}_1}= { \varphi(x)\varphi(d\overline{d}x)\over \varphi(d x)\varphi(\overline{d}x)}T_{\kappa,x} =\exp\left(- \sum_{n=1}^\infty {1\over n} {(1-d^n)(1-\overline{d}^n) \over (1-q^n)} x^n\right)T_{\kappa,x},\] since \(\Delta=0\), \(\mathsf{T}= T_{\kappa,x}\), \(\mathcal{A}_C={1\over \varphi(dx)\varphi(\overline{d}x)}\), \(\mathcal{A}_L=\varphi(x)\), and \(\mathcal{A}_R=\varphi(d\overline{d}x)\). Hence, it is easy to see that the equation and the solution read \[\label{gl1case} \mathcal{H}^{\widehat{\mathfrak{gl}}_1} \psi=\psi,\qquad \psi=\exp\left(- \sum_{n=1}^\infty {1\over n} {(1-d^n)(1-\overline{d}^n) \over (1-q^n)(1-\kappa^n)} x^n\right).\tag{7}\] On the other hand, we have an impressive (double infinite product) expression for the \({\widehat{\mathfrak{gl}}_1}\) affine Laumon partition function \[\label{gl1partition-function} \mathcal{Z}^{\widehat{\mathfrak{gl}}_1}_{\rm AL} \left( \begin{array}{c} a \\ b\\c \end{array} \Biggl|\,\,\mathsf{x}\,\,\Biggl|q,\kappa\right)= \exp\left( \sum_{n=1}^\infty {1\over n} {[b^n/c^n][a^n/q^n\kappa^n b^n] \over [q^n][\kappa^n]}\mathsf{x}^n\right),\tag{8}\] where we used the symbol \([x]=x^{-1/2}-x^{1/2}\). As for a proof of the identity 8 , see e.g. Proposition 4.17 in [5]. Note that in idem., the Nekrasov partition function is defined by the ordinary Pochhammer symbol \((a;q)_n\) (as eq.(72) in idem.), instead of the shifted product of hyperbolic sine functions \([a;q]_n=[a][qa]\cdots [q^{n-1}a]\) defined in Definition 1.5. Comparison of these is achieved by applying Proposition B.1. We conclude that the solution \(\psi\) to the equation \(\mathcal{H}^{\widehat{\mathfrak{gl}}_1} \psi=\psi\) is given by the affine Laumon function as \[\mathcal{Z}^{\widehat{\mathfrak{gl}}_1}_{\rm AL} \left( \begin{array}{c} a\kappa b/d \\ b\\b/\overline{d}\end{array} \Biggl|\,\, \sqrt{d\overline{d}\over q\kappa} x\,\,\Biggl|q,\kappa\right)= \exp\left(- \sum_{n=1}^\infty {1\over n} {(1-d^n)(1-\overline{d}^n) \over (1-q^n)(1-\kappa^n)} x^n\right)=\psi.\]

Secondly, some remarks are in order concerning the verification of Conjecture 6 for \(N \geq 3\). For simplicity, we set \[{\mathcal{A}}_L^{-1} q^{-{1\over 2}\Delta}\psi- {\mathcal{A}}_C {\mathcal{A}}_R q^{{1\over 2}\Delta}\mathsf{T}\psi =\sum_{\theta_1,\ldots,\theta_N =0}^{\infty} e_{\theta_1,\ldots,\theta_N}x_1^{\theta_1} \cdots x_N^{\theta_N}.\] By straightforward brute-force calculations using Mathematica, we can show that \(e_{\theta_1,\ldots,\theta_N}=0\) at least for small \(N\) and lower total degree \(\mathsf{d}={\displaystyle\sum_{i=1}^{N}} \theta_i\). For \(N=3\), we have checked that \(e_{\theta_1,\theta_2,\theta_3}=0\) for \(0\leq \mathsf{d}\leq 5\), treating all eleven parameters \(d_i, \overline{d}_i ,b_i ~(i=1,2,3),q\) and \(\kappa\) as indeterminates. For \(6 \leq \mathsf{d}\), however, such a task becomes more computationally demanding and practically intractable. Instead, substituting random integers for all eleven parameters, we observe that \(e_{\theta_1,\theta_2,\theta_3}=0\) for \(6\leq \mathsf{d}\leq 15\), which means up to instanton number five. Similarly, for \(N=4\), we find that \(e_{\theta_1,\theta_2,\theta_3,\theta_4}=0\) for \(0 \leq \mathsf{d}\leq 9\), when we specialize the fourteen parameters in the same manner; for \(N=5\) the same method works for \(0 \leq \mathsf{d}\leq 7\).

1.3 Mass truncation and relation to the \(R\) matrix↩︎

In \({\widehat{\mathfrak{gl}}_2}\) case [2], we introduced the mass parameter truncation where half of the mass parameters are set to the form \(q^{-n}~(n \in \mathbb{Z}_{\geq 0})\). After the mass parameter truncation the non-stationary difference equation 1 is identified with the quantum KZ equation. Namely if we remove the shift operator \(\mathsf{T}\) from the Hamiltonian, it gives the (finite dimensional) \(R\) matrix of \(U_q(A_1^{(1)})\) with generic spins. Based on the normal ordered Hamiltonian [def:normal-ordered], we can show the same story for \({\widehat{\mathfrak{gl}}_N}\) case. It is quite remarkable the resulting finite dimensional \(R\) matrix of \(U_q(A_{N-1}^{(1)})\) is related the three dimensional (tetrahedron) \(R\) matrix [7]. In the formula of the components of the three dimensional \(R\) matrix there appears a basic building block \(\Phi_q\) defined by 27 (see [7], §13.5). We find the same function in our formula of the components of the \(R\) matrix (see Corollary 26).

1.4 Four dimensional limit↩︎

The four dimensional (cohomological) version of the affine Laumon partition function of type \({\widehat{\mathfrak{gl}}_2}\) satisfies a quantization of the differential Painlevé equation \(P_{\rm VI}\) (see e.g.[8] and references therein). In [1] we have seen how the non-stationary difference equation 1 provides a way to up grade the story to five dimensional/\(q\)-difference version. In four dimensional/differential situation, the generalization to \({\widehat{\mathfrak{gl}}_N}\) case was also considered in [9] where a quantization of a particular kind of higher rank generalization of \(P_{\rm VI}\) (called Fuji-Suzuki-Tsuda system) was studied as the relevant equation. One can check that the four dimensional limit of Conjecture 6 is consistent with the result in [9].

1.5 Organization of the paper↩︎

The present paper is organized as follows; In section 2, by using the pentagon identity for the \(q\)-exponential function \(e_q(z) = \varphi(z)^{-1}\) and the \(q\)-binomial theorem we prove that there are three equivalence forms of the \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian ?? ; two kinds of the factorized form and the normal ordered form. Each form has its own advantage. We also show that the Hamiltonian is invariant under the action of the Dynkin automorphisms of \(A_{N-1}^{(1)}\). We consider the mass truncation in section 3. Namely we tune half of the mass parameters in the Hamiltonian so that we can extract finite dimensional blocks of the \(R\)-matrix. We find an interesting relation to the tetrahedron (3D) \(R\)-matrix. In [2] Conjecture 6 for \(N=2\) was proved by rewriting the affine Laumon partition function in the form of the Jackson integral. In section 4 we show that it is straightforward to apply the same method to \(N \geq 3\), though it does not lead to a proof of Conjecture 6 at the moment. Finally in section 5, we show that a four dimensional limit of our system is nothing but the Fuji-Suzuki-Tsuda system, which is consistent with the conjecture. Some of technical details and miscellaneous topics are collected in appendices.

1.6 Notations and convention↩︎

We will use the following notations throughout the paper [10]; \[\label{inf-prod} \varphi(x):= (x;q)_\infty = \prod_{n=0}^\infty (1-x q^n) =\exp\left(-\sum_{n=1}^\infty \frac{1}{n} \frac{1}{1-q^n} x^n \right), \quad |x| < 1, \quad |q|<1.\tag{9}\] The \(q\)-shifted factorial is defined by \[\label{q-factorial} (x;q)_n = \frac{(x;q)_\infty}{(xq^n ; q)_\infty}.\tag{10}\] The following formula is useful; \[\label{q-factorial-inversion} (x;q)_n=(-x)^n q^{\frac{1}{2}n(n-1)}\frac{1}{(qx^{-1};q)_{-n}}, \qquad n \in \mathbb{Z}.\tag{11}\]

We employ the formulas of two \(q\)-exponential functions [10]; \[\begin{align} \tag{12} e_q(z) &= \sum_{n=0}^\infty \frac{z^n}{(q;q)_n} = \varphi(z)^{-1}, \quad |z| <1, \quad |q| <1, \\ \tag{13} E_q(z) &= \sum_{n=0}^\infty \frac{q^{\frac{1}{2}n(n-1)}z^n}{(q;q)_n} = \varphi(-z), \quad |z| <1, \quad |q| <1. \end{align}\] Finally, the \(q\)-binomial coefficients are defined by \[\label{q-binom} \Big[\begin{array}{c}n\\k\end{array}\Big]_{q}:= \frac{(q;q)_n}{(q;q)_{k} (q;q)_{n-k}}.\tag{14}\] We have \[(x;q)_n = \sum_{k=0}^n \Big[\begin{array}{c}n\\k\end{array}\Big]_{q} (-x)^k q^{\frac{1}{2}k(k-1)}.\]

The partition function on the gauge theory side is computed by the localization for the torus action. On the four dimensional space-time the action is \(\mathbb{R}^4 \simeq \mathbb{C}^2 \ni (z_1, z_2) \longrightarrow (q_1z_1, q_2z_2)\). In this paper we regard the equivariant parameters⁴ \[q_1 := e^{\epsilon_1}, \qquad q_2 := \kappa = t^{-\frac{1}{N}} = e^{\frac{\epsilon_2}{N}},\] as the canonical parameters of the theory. They are natural parameters of the quantum toroidal algebras. We simply denote \(q=q_1\) unless otherwise mentioned.

2 Non-stationary \({\widehat{\mathfrak{gl}}_N}\) difference equation↩︎

2.1 Pentagon identity and Dynkin automorphisms of \(A_{N-1}^{(1)}\)↩︎

By using the pentagon identity for the \(q\)-exponential function \(e_q(z)= \varphi(z)^{-1}\), we can recast the blocks \(\mathcal{A}_L^{(\mathrm{s})}\) and \(\mathcal{A}_R^{(\mathrm{s})}\) of the Hamiltonian ?? of factorized form of simple root type so that the correspondence to the factorization of the universal \(R\)-matrix becomes clear [6]. The pentagon identity also allows us to see that the \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian is actually symmetric in variables \(x_i\).

Since the \(q\)-commutative variables \(\hat{x}_i\) and \(\check{x}_i\) satisfy \(\hat{x}_i \hat{x}_j = q^{\delta_{i,j-1}-\delta_{i-1,j}}\hat{x}_j \hat{x}_i\) and \(\check{x}_i \check{x}_j = q^{\delta_{i-1,j}-\delta_{i,j-1}}\check{x}_j \check{x}_i\), we obtain \[\begin{align} \tag{15} e_q(-\hat{x}_i) e_q(-\hat{x}_{i+1}) &= e_q(-\hat{x}_{i+1}) e_q(-\hat{x}_{i+1}\hat{x}_i) e_q(-\hat{x}_i), \\ \tag{16} e_q(-\check{x}_{i+1}) e_q(-\check{x}_i) &= e_q(-\check{x}_i) e_q(-\check{x}_i\check{x}_{i+1}) e_q(-\check{x}_{i+1}). \end{align}\]

By Lemma 8 we can reduce the left block \(\mathcal{A}_L^{\mathrm{s}}\) and the right block \(\mathcal{A}_R^{\mathrm{s}}\) of the non-stationary \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian ?? into the factorized form of higher root type;

Let us introduce the cyclic shift \(\pi(\check{x}_i)=\check{x}_{i+1}~(i \in \mathbb{Z}/ N\mathbb{Z})\) and similarly for \(\hat{x}_i\). \(\pi\) is an automorphism of the algebra. Using the pentagon identity ?? , we can show the Hamiltonian enjoys the cyclic symmetry \(\pi(\mathcal{H}^{{\widehat{\mathfrak{gl}}_N}}) = \mathcal{H}^{{\widehat{\mathfrak{gl}}_N}}\). It is enough to prove \(\pi(\mathcal{A}_L) = \mathcal{A}_L\) and \(\pi(\mathcal{A}_R) = \mathcal{A}_R\), since other parts of \(\mathcal{H}^{{\widehat{\mathfrak{gl}}_N}}\) are manifestly symmetric under the cyclic permutation.

We have proved that the \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian is invariant under the shift \(\pi(\hat{x}_i) := \hat{x}_{i+1}\). By the pentagon identity we can also check the invariance under the automorphisms of the Dynkin diagram of \(A_{N-1}^{(1)}\) for \(N \geq 3\). Let \(\pi\) and \(s_{j}\) be the automorphism \(\pi(ab)=\pi(a)\pi(b)\) and the anti-automorphisms \(s_{j}(ab)=s_{j}(b)s_{j}(a)\), respectively, such that \[\pi(\hat{x}_i) := \hat{x}_{i+1},\qquad s_{j}(\hat{x}_i) := \hat{x}_{2j-i}.\] Since \(s_{\frac{n}{2}} = \pi^{n}\circ s_{0}\) for any \(n\in{\mathbb{Z}}\), the group generated by \(\pi\) and \(s_{\frac{n}{2}}\)’s \((n\in{\mathbb{Z}})\) is generated by \(\pi\) and \(s_{0}\), i.e., \(\langle \pi, s_{\frac{n}{2}} \rangle_{n\in{\mathbb{Z}}}=\langle \pi, s_{0} \rangle\). Note that the automorphism of the Dynkin diagram of \(A^{(1)}_{N-1}\), which is isomorphic to the dihedral group, is generated by \(\pi\) and \(s_{0}\) with \[\pi^N = s_{0}^2={\rm id}, \qquad s_{0}=\pi\circ s_{0}\circ\pi.\]

For \(i\in{\mathbb{Z}}/N{\mathbb{Z}}\), let us look at the following quantities \[\begin{align} { \mathcal{A}^{i-1}_{i}} := e_q(-\hat{x}_{i+1}) \cdots e_q(-\hat{x}_{i+N-2}) \cdot e_q(-\hat{x}_{i+N-1}) &\cdot e_q(-\hat{x}_{i+N-2})^{-1}\cdots e_q(-\hat{x}_{i+1})^{-1} \cr \cdot\; e_q(-\hat{x}_{i}) &\cdot e_q(-\hat{x}_{i+1}) \cdots e_q(-\hat{x}_{i+N-2}) \end{align}\] and \[\begin{align} {\mathcal{B}_{i}} :=& e_q(-\hat{x}_{i+1}) \cdots e_q(-\hat{x}_{i+N-2}) e_q(-\hat{x}_{i+N-1}) \cr \cdot\, & e_q(-\hat{x}_{i+N-2} \cdots \hat{x}_{i+1}\hat{x}_{i}) \cdots e_q(-\hat{x}_{i+1}\hat{x}_{i}) e_q(-\hat{x}_{i}), \cr {\mathcal{B}^{i}} :=& e_q(-\hat{x}_{i+N}) e_q(-\hat{x}_{i+N}\hat{x}_{i+N-1}) \cdots e_q(-\hat{x}_{i+N} \hat{x}_{i+N-1}\cdots \hat{x}_{i+2}) \cr \cdot\, & e_q(-\hat{x}_{i+1}) e_q(-\hat{x}_{i+2}) \cdots e_q(-\hat{x}_{i+N-1}). \end{align}\] Then \[\begin{align} \pi({ \mathcal{A}^{i-1}_{i}}) &= {\mathcal{A}^{i}_{i+1}}, \qquad s_{j}({ \mathcal{A}^{i-1}_{i}}) = {\mathcal{A}^{2j-i}_{2j-i+1}}, \cr \pi({\mathcal{B}_{i}}) &= {\mathcal{B}_{i+1}}, \qquad\quad s_{j}({\mathcal{B}_{i}}) = {\mathcal{B}^{2j-i}}, \cr \pi({\mathcal{B}^{i}}) &= {\mathcal{B}^{i+1}}, \qquad\quad s_{j}({\mathcal{B}^{i}}) = {\mathcal{B}_{2j-i}}. \end{align}\] Therefore, \(\{ { \mathcal{A}^{i-1}_{i}} \}_{i\in{\mathbb{Z}}/N{\mathbb{Z}}}\) and \(\{ {\mathcal{B}_{i}}, {\mathcal{B}^{i}} \}_{i\in{\mathbb{Z}}/N{\mathbb{Z}}}\) are invariant and transitive under the Dynkin automorphism group \(\langle \pi, s_{0} \rangle\).

By iteratively using the pentagon identity 16 we have (See the proof of Lemma 8)

Then, the relations ?? and ?? imply the following result, which generalizes Proposition 10;

By this lemma, \({\mathcal{B}_{i+1}}={\mathcal{A}^{i}_{i+1}}={ \mathcal{A}^{j}_{j+1}}={\mathcal{B}^{j}}\) for any \(i,j\in{\mathbb{Z}}/N{\mathbb{Z}}\). Thus, \({ \mathcal{A}^{i-1}_{i}}={\mathcal{B}_{j}}={\mathcal{B}^{k}}\) for any \(i,j,k\in{\mathbb{Z}}/N{\mathbb{Z}}\). Therefore, we finally obtain

The original definition of the right block \(\mathcal{A}_R\) of \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian employs \(\mathcal{A}_0^{N-1}\) in Proposition 13 (see Definition 1). On the other hand it is \[\label{another95expression95of95A95R} \mathcal{B}^{0}= e_q(-\hat{x}_0)e_q(-\hat{x}_0\hat{x}_{N-1})\cdots e_q(-\hat{x}_0\hat{x}_{N-1}\cdots\hat{x}_2) \cdot e_q(-\hat{x}_1)\cdots e_q(-\hat{x}_{N-1})\tag{18}\] that naturally appears in the Hamiltonian constructed from the universal \(R\) matrix of \(U_q (A_{N-1}^{(1)})\) [6]. We have focused on the right block \(\mathcal{A}_R\) of the non-stationary Hamiltonian. Similarly we can confirm the invariance under the Dynkin automorphism of the left block \(\mathcal{A}_L\) with \(q\)-commutative variables \(\check{x}_i\). The dihedral group invariance of the remaining parts of the Hamiltonian is trivial. Hence, we obtain Corollary 14.

2.2 Normal ordered form of the Hamiltonian↩︎

In this subsection we prove the equivalence of the Hamiltonian of factorized form and of normal ordered form (see Definitions 2, 3 and 4);

Recall that \(\hat{x}_i = d_i \overline{d}_i x_i q^{\vartheta_i - \vartheta_{i-1}}\) and \(\check{x}_i := x_i q^{-\vartheta_i + \vartheta_{i-1}}\).⁶ We are going to show the agreement of the factorized form of the building block \(\mathcal{A}_R^{(\mathrm{h})}\) of higher root type and the corresponding normal ordered form \(\mathcal{A}_R^{(\mathrm{n})}\). The agreement of \(\mathcal{A}_L^{(\mathrm{h})}\) and \(\mathcal{A}_L^{(\mathrm{n})}\) is proved similarly. To prove Proposition 15 for \(\mathcal{A}_R\) we need Proposition 17 below, which relies on the \(q\)-multinomial formula;

When \(m=2\), ?? is given as Exercise 1.35 in [10] (see also references there). The general case is proved by induction on \(m\) as follows;

Since Proposition 9 implies that the left hand side of ?? is equal to \(\mathcal{A}_R^{(\mathrm{s})} = \mathcal{A}_R^{(\mathrm{h})}\), Proposition 15 for \(\mathcal{A}_R\) follows from Proposition 17. Note that \(\hat{x}_{N-1}\cdots \hat{x}_1 \hat{x}_0 = q^{1-N} D_N \Lambda\) is central.

2.3 Classical analogue of \(\mathcal{A}_L^{(\mathrm{s})}\) and \(\mathcal{A}_R^{(\mathrm{s})}\)↩︎

This subsection is an interesting detour. Logically it is not necessary for the following sections and may be skipped. But we would like to make a remark on the factorization of the classical cyclic matrix, which is instructive for understanding \(\mathcal{A}_L^{(\mathrm{s})}\) and \(\mathcal{A}_R^{(\mathrm{s})}\) in the \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian. For \(0\leq i\leq n-1\) and \(x \in {\mathbb{C}}\), let \(J_i(x)\) be the \(n \times n\) elementary Jacobi matrix defined as \[\begin{align} &J_i(x)=\exp(x e_i)={\boldsymbol{1}}+x e_i, \nonumber \\ &e_i=E_{i,i+1}, (1\leq i\leq n-1), \quad e_0=z E_{n,1}, \end{align}\] where \({\boldsymbol{1}}={\boldsymbol{1}}_n\) is the identity matrix and \(E_{i,j}\) is the matrix unit: \((E_{i,j})_{k,l}=\delta_{i,k}\delta_{j,l}\). We define the matrix \(X\) by \[X={\boldsymbol{1}}+\sum_{i=0}^{n-1} x_i e_i=\left[ \begin{array}{cccccc} 1 & x_1&&& \\ & 1 & x_2 \\ &&\ddots&\ddots\\ &&& 1 & x_{n-1} \\ x_0 z && & & 1 \\ \end{array} \right],\] which is manifestly cyclic and plays fundamental role in tropical/geometric crystal and discrete integrable systems.

We have the following factorization of the cyclic matrix \(X\), where \(X^{\pm 1}\) may be viewed as the classical analog of \({\mathcal{A}}_L^{(\mathrm{s})}\) and \({\mathcal{A}}_R^{(\mathrm{s})}\) which enjoy the cyclic symmetry.

Note that the first factor in ?? can be written various ways as \[\begin{align} &g J_{0}(x_0) d_{n}(v z) g^{-1} \nonumber \\ =&\prod_{j=0}^{n-1}({\boldsymbol{1}}+ (-1)^{j} x_0 x_1 \cdots x_{j} z E_{n,j+1} )\nonumber \\ =&{\boldsymbol{1}}+\sum_{j=0}^{n-1} (-1)^j x_0 x_1 \cdots x_{j} z E_{n,j+1} \nonumber \\ =&\left[ \begin{array}{cccc|c} &&{\boldsymbol{1}}_{n-1} &&{\boldsymbol{0}} \\ \hline x_0 z &-x_0x_1z & \cdots & (-1)^{n-2}x_0 \cdots x_{n-2}z& vz \end{array} \right]. \end{align}\]

2.4 Other forms of \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian↩︎

The \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian involves the \(q\)-exponential function with \(q\)-commutative variables \(\hat{x}_i\) and \(\check{x}_i\). We can recast it in such a form that the arguments of the \(q\)-exponential function are commutative variables \(x_i\) by moving the position of the \(q\)-Borel transformation.

Let \(\vartheta:= x\frac{\partial}{\partial x}\). Since \(\vartheta^k x = x (1+\vartheta)^k\) for any natural number \(k\in{\mathbb{N}}\), we have \(q^{\vartheta} x = x q^{1+\vartheta}\) and \(q^{{\frac{1}{2}}\vartheta(\vartheta-1)} x = x q^{{\frac{1}{2}}(1+\vartheta)\vartheta} = x q^{\vartheta+{\frac{1}{2}}\vartheta(\vartheta-1)}\). Therefore, \(q^{c\vartheta} x^n =(q^c x)^n q^{c\vartheta}\) and \(q^{{\frac{1}{2}}\vartheta(\vartheta-1)} x^n = (x q^\vartheta)^n q^{{\frac{1}{2}}\vartheta(\vartheta-1)}\) for any integer \(n\in{\mathbb{Z}}\) and \(c\in{\mathbb{C}}\).

Assume \(N\geq 3\). For \(N=3\), \(f_{N-2}\cdots f_2 f_1\) stands for \(f_1\). Let us set \(\alpha_i=1\) for simplicity, so that \(\hat{x}_i = x_i q^{\vartheta_i - \vartheta_{i-1}}\).

Since \[\begin{align} q^{\vartheta_i(\vartheta_i-\vartheta_{i-1}-\vartheta_{i+1}-1)} x_i &= x_i q^{(1+\vartheta_i)(\vartheta_i-\vartheta_{i-1}-\vartheta_{i+1})} \cr &= x_i q^{(\vartheta_i-\vartheta_{i-1})+(\vartheta_i-\vartheta_{i+1})} q^{\vartheta_i(\vartheta_i-\vartheta_{i-1}-\vartheta_{i+1}-1)} , \end{align}\] we obtain \[q^{\sum_{j=1}^N\vartheta_j(\vartheta_j-\vartheta_{j-1}-1)} x_i^n = \left(x_i q^{(\vartheta_i-\vartheta_{i-1})+(\vartheta_i-\vartheta_{i+1})}\right)^n q^{\sum_{j=1}^N\vartheta_j(\vartheta_j-\vartheta_{j-1}-1)} .\] Therefore, we have \[q^{{\frac{1}{2}}\sum_{j=1}^N\vartheta_j(\vartheta_j-\vartheta_{j-1}-1)} \begingroup \let\macc@kerna\z@ \let\macc@kernb\z@ \let\macc@nucleus\@empty \hat{\mathchoice {\raisebox{.3ex}{\vphantom{\displaystyle x}}} {\raisebox{.3ex}{\vphantom{\textstyle x}}} {\raisebox{.16ex}{\vphantom{\scriptstyle x}}} {\raisebox{.14ex}{\vphantom{\scriptscriptstyle x}}} \smash{\hat{x}}}\endgroup_{i}^n = \hat{x}_{i}^n q^{{\frac{1}{2}}\sum_{j=1}^N\vartheta_j(\vartheta_j-\vartheta_{j-1}-1)} ,\] which also yields the equality of \((\ref{eq:lemmahat})\) and \((\ref{eq:lemmahathat})\).

3 Mass truncation and finite dimensional \(R\) matrix↩︎

In this section we study the \({\widehat{\mathfrak{gl}}_N}\) equation ?? by imposing a truncation condition 22 on mass parameters. We show that the normal ordered Hamiltonian [def:normal-ordered] gives rise to the finite dimensional \(R\)-matrix for the symmetric representation of \(U_q(A_{N-1}^{(1)})\). It is quite remarkable that the resulting \(R\) matrix is also related the three dimensional (tetrahedron) \(R\) matrix [7]. Namely, in the formula of the components of the three dimensional \(R\) matrix there appears a basic building block \(\Phi_q\) defined by 27 (see [7], §13.5). Exactly the same function is obtained in our formula of the components of the \(R\) matrix (see Corollary 26). We note that the mechanism for deriving these formulas is common to both cases.

3.1 The mass truncation↩︎

We can recast the normal ordered form of the non-stationary \({\widehat{\mathfrak{gl}}_N}\) equation as follows; \[:\!\prod_{i=1}^N \frac{\varphi(\hat{x}_i)}{\varphi(d_i x_i)}\!:~q^{\frac{1}{2}\Delta}\cdot \mathsf{T} \psi =~:\!\prod_{i=1}^N \frac{\varphi(\overline{d}_i x_i)}{\varphi(\check{x}_i)}\!:~q^{-\frac{1}{2} \Delta} \cdot \psi.\] Here the normal ordering \(: \quad :\) is defined as \[:F(x,\vartheta): x^\nu=F(x,\nu)x^\nu,\] for any commutative function \(F(x,\vartheta)=F(\{x_a\}, \{\vartheta_a\})\) and monomial \(x^{\nu}=\prod_{a=1}^N x_a^{\nu_a}\). By the \(q\)-binomial theorem \[\label{q-binomial} \sum_{n=0}^\infty \frac{(a;q)_n}{(q;q)_n} z^n = \frac{\varphi(az)}{\varphi(z)},\tag{20}\] we have \[\begin{align} \label{normal-ordered-form} & \sum_{\alpha_1, \ldots, \alpha_N \geq 0} : \prod_{i=1}^N (d_i x_i)^{\alpha_i} \frac{(\overline{d}_i q^{\vartheta_i - \vartheta_{i-1}};q)_{\alpha_i}}{(q;q)_{\alpha_i}} : q^{\frac{1}{2}\Delta}\cdot \mathsf{T} \psi \nonumber \\ & \qquad = \sum_{\alpha_1, \ldots, \alpha_N \geq 0} : \prod_{i=1}^N x_i^{\alpha_i} \frac{(\overline{d}_i q^{\vartheta_i - \vartheta_{i-1}};q)_{\alpha_i}}{(q;q)_{\alpha_i} (q^{\vartheta_i - \vartheta_{i-1}})^{\alpha_{i}}} : q^{-\frac{1}{2}\Delta}\cdot \psi. \end{align}\tag{21}\]

Let us impose the mass truncation condition, \[\label{mass-truncation} \overline{d}_i = q^{-m_i}, \quad m_i \in \mathbb{Z}_{\geq 0}, \qquad 1 \leq i \leq N.\tag{22}\] Set \(M:= m_1 + m_2 + \cdots +m_N\). Note that \(\overline{d}_1 \overline{d}_2 \cdots \overline{d}_N =q^{-M}\). Under the condition 22 , the coefficient for \(x_i^{\alpha_i}\) in 21 vanishes for \(\alpha_i \geq 1+m_i-\vartheta_i + \vartheta_{i-1}\). By using \[\sum_{\alpha=0}^{n} \frac{(q^{-n};q)_\alpha}{(q;q)_\alpha} z^{\alpha} = (q^{-n}z;q)_n,\] we obtain

Recall that we identify \(\Lambda = x_1x_2 \cdots x_N\) as the parameter of the instanton expansion. If we fix \(\theta_N \geq 0\) and regard it as the instanton number, the number of terms in the terminated expansion ?? is finite. They are labeled by the set \(S(m) \subset \mathbb{Z}^{N-1}\) defined by \[S(m) := \{ (\theta_1, \ldots, \theta_{N-1}) \mid \theta_1 \leq m_1, \theta_2 - \theta_1 \leq m_2, \ldots, \theta_{N-1}- \theta_{N-2} \leq m_{N-1}, -\theta_{N-1} \leq m_N \}.\] For example, when \(N=3\), \[S(m) = \{ (\theta_1, \theta_2) \in \mathbb{Z}^2 \mid \theta_1 \leq m_1, \theta_2 - \theta_1 \leq m_2, - \theta_2 \leq m_3 \},\] and the allowed \((\theta_1, \theta_2)\) in \(\mathbb{Z}^2\)-lattice is bounded by the triangle (See Figure 3 in Appendix 8). For general \(N\), we have \(\vert S(m) \vert = \binom{M+N-1}{N-1} = \binom{M+N-1}{M}\).

Let us make a shift \(\widetilde{\theta}_i := \theta_i + m_{i+1}+ \cdots + m_N\) so that the defining conditions for \(S(m)\) can be written by \(0 \leq \widetilde{\theta}_{N-1} \leq \widetilde{\theta}_{N-2} \leq \cdots \leq \widetilde{\theta}_2 \leq \widetilde{\theta}_1 \leq M\). See also Appendix 8 for the meaning of such a shift. Then define \(i_1 := M -\widetilde{\theta}_1, i_2 := \widetilde{\theta}_1 - \widetilde{\theta}_2, \cdots, i_{N-1}:= \widetilde{\theta}_{N-2} - \widetilde{\theta}_{N-1}, i_N := \widetilde{\theta}_{N-1}\), then \(\mathbf{i}=(i_1, i_2, \ldots, i_N)\) belongs to the set \[\label{IM-def} I_M :=\{ {\boldsymbol{i}}=(i_1,i_2, \ldots, i_N) \in \mathbb{Z}_{\geq 0}^N \mid i_1+i_2+\cdots +i_N=M \}.\tag{23}\] In fact we can define a bijection between \(S(m)\) and \(I_M\) as follows; let us define \(z_k := x_1 x_2 \cdots x_k,~k=1, \ldots, N-1\) and set \[z_1^{\alpha_1} z_2^{\alpha_2} \cdots z_{N-1}^{\alpha_{N-1}} = x_1^{\widetilde{\theta}_1} \cdots x_{N-1}^{\widetilde{\theta}_{N-1}}.\] Then we have \(\alpha_k = i_{k+1} ,k=1, \ldots, N-1\) and \(\alpha_1 + \alpha_2 + \cdots +\alpha_{N-1} \leq M\). Introducing \(\alpha_N := M - (\alpha_1 + \alpha_2 + \cdots +\alpha_{N-1}) = i_1\), we see that the elements of \(S(m)\) are in one to one correspondence with the monomials \(z_1^{\alpha_1} \cdots z_{N-1}^{\alpha_{N-1}} z_N^{\alpha_N},~ z_N = \Lambda\) with homogeneous degree \(M\).

3.2 \(R\) matrix as a connection matrix↩︎

We study the finite dimensional matrix \(R\) arising from the \({\widehat{\mathfrak{gl}}_N}\) Hamiltonian by the mass truncation. For \(N, M \in \mathbb{Z}_{\geq 0}\) we have introduced \(I^N_M = I_M\) in the last subsection (see 23 ). Note that \(|I_{M}|=\binom{N+M-1}{M}\). For variables \(\mathbf{z}=(z_1, z_2, \ldots, z_{N})\) and parameters⁸ \((\mu_1, \ldots, \mu_N)\), we define the polynomials \(B_{k, {\boldsymbol{i}}}\) (\(k=1,2\)) as \[\begin{align} \label{eq:z-basis} &B_{1,{\boldsymbol{i}}}(\mathbf{z},\Lambda)=\prod_{a=1}^N \left(\mu_a \frac{z_{a+1}}{z_a};q \right)_{i_a} z_a^{i_a}, \quad B_{2,{\boldsymbol{i}}}(\mathbf{z},\Lambda)=\prod_{a=1}^N \left(\frac{z_{a}}{z_{a+1}};q \right)_{i_a} z_{a+1}^{i_a}. \end{align}\tag{24}\] Here and in the followings, we always put \(z_{N+1}=\Lambda z_1\) regarding \(\Lambda\) as a free parameter.⁹ Note that we can change the normalization of the base polynomials 24 freely keeping the main structure of the matrix \(R\). See the remark at the end of the section. For generic \(\Lambda\), both \(\{B_{1,{\boldsymbol{i}}}(\mathbf{z},\Lambda) \mid {\boldsymbol{i}} \in I_M\}\) and \(\{B_{2,{\boldsymbol{i}}}(\mathbf{z},\Lambda) \mid {\boldsymbol{i}} \in I_M\}\) form a basis of the homogeneous polynomials of degree \(M\) in \(\mathbb{C}[z_1, \ldots, z_N]\). Hence we have a relation \[\label{eq:BRB} B_{1,{\boldsymbol{i}}}(\mathbf{z},\Lambda)=\sum_{{\boldsymbol{j}} \in I_M} R_{{\boldsymbol{i}},{\boldsymbol{j}}}(\Lambda) B_{2,{\boldsymbol{j}}}(\mathbf{z},\Lambda).\tag{25}\] The coefficients \(R(\Lambda)_{{\boldsymbol{i}},{\boldsymbol{j}}}\) are polynomial in \(\mu_a\) and rational in \(\Lambda\) and \(q\).

Since the size of \(R\)-matrix is \(|I_M|\), one can determine \(R_{{\boldsymbol{i}},{\boldsymbol{j}}}(\Lambda)\) by specializing 25 at \(|I_M|\) points. It is convenient to choose such \(|I_M|\) reference points \(z_{{\boldsymbol{k}}}\) as follows; \[\label{eq:zjsp} z_{{\boldsymbol{k}},1}=1, \quad z_{{\boldsymbol{k}},a}=q^{k_1+\cdots+k_{a-1}} \quad (1<a\leq N),\tag{26}\] with \({\boldsymbol{k}} = (k_1, k_2, \ldots, k_N) \in I_M\). We will solve the matrix equation \(B_{1,{\boldsymbol{i}}}(z_{{\boldsymbol{k}}},\Lambda)=\sum_{{\boldsymbol{j}} \in I_M} R_{{\boldsymbol{i}},{\boldsymbol{j}}}(\Lambda) B_{2,{\boldsymbol{j}}}(z_{{\boldsymbol{k}}}, \Lambda)\). As we will see in Proposition 25, the inverse of the matrix \(\big(B_{2,{\boldsymbol{i}}}(z_{{\boldsymbol{k}}},\Lambda)\big)_{{\boldsymbol{i}},{\boldsymbol{k}}}\) is obtained explicitly, hence one can derive an explicit formulae of \(R_{{\boldsymbol{i}},{\boldsymbol{j}}}(\Lambda)\).

To describe the inversion formulae, we prepare some notations. Let \(n \geq 1\). For any sequences of integers \({\boldsymbol{i}}=(i_1, i_2, \ldots, i_n),~{\boldsymbol{j}}=(j_1, j_2, \ldots, j_n)\in \mathbb{Z}^n\) of length \(n\), we put \[\begin{align} |{\boldsymbol{i}}|=\sum_{a=1}^n i_a, \qquad \overline{\boldsymbol{i}}=(i_1, \ldots, i_{n-1}), \qquad \langle {\boldsymbol{i}}, {\boldsymbol{j}} \rangle=\sum_{1\leq a<b\leq n} i_a j_b. \end{align}\] For \(\boldsymbol{\beta}, \boldsymbol{\gamma}\in \mathbb{Z}^{n}\) and \(\lambda, \mu \in \mathbb{C}\), we define¹⁰ \[\begin{align} \label{def:Phi95q} &\Phi_q({\boldsymbol{\gamma}}|{\boldsymbol{\beta}}; \lambda, \mu) =q^{\langle \boldsymbol{\beta}-\boldsymbol{\gamma}, \boldsymbol{\gamma}\rangle} \Big(\frac{\mu}{\lambda}\Big)^{|\boldsymbol{\gamma}|} \frac{(\lambda;q)_{|\boldsymbol{\gamma}|}(\frac{\mu}{\lambda};q)_{|\boldsymbol{\beta}|-|\boldsymbol{\gamma}|}}{(\mu;q)_{|\boldsymbol{\beta}|}} \prod_{a=1}^{n} \left[{ \genfrac{}{}{0pt}{}{\beta_a}{\gamma_a} }\right]_q, \end{align}\tag{27}\] where the \(q\)-binomial coefficients are define by 14 .

Note that \(\Phi_q({\boldsymbol{\gamma}}|{\boldsymbol{\beta}}; \lambda, \mu)=0\) unless \(\boldsymbol{\gamma}\leq \boldsymbol{\beta}\) (i.e. \(\forall a: \gamma_a\leq \beta_a\)). The function \(\Phi_q\) and the function \(A_{{\boldsymbol{i}},{\boldsymbol{j}}}^{{\boldsymbol{a}}, {\boldsymbol{b}}}\) defined below (see ?? ), which is quadratic in \(\Phi_q\), originate in the study of the three dimensional \(R\) matrix, where it was shown that the trace reduction of the three dimensional \(R\) matrix gives fundamental examples of the quantum \(R\) matrix of \(U_q(A_{N-1}^{(1)})\) with higher ‘’spin’’representations. See Chap.13 of [7] and references therein.

From ?? , we see \(B_{2, {\boldsymbol{i}}}(z_{\boldsymbol{j}}, \Lambda)=0\) unless \(\overline{\boldsymbol{i}} \leq \overline{\boldsymbol{j}}\). Moreover, the transition property ?? with \(a=c\) implies;

Thanks to ?? and the inversion formulae ?? , we have

In [7] the same function \(A_{{\boldsymbol{i}},{\boldsymbol{j}}}^{{\boldsymbol{a}}, {\boldsymbol{b}}}\) was introduced in the formula of a trace reduction of the tetrahedron (3D) \(R\) matrix \(R=R_{i,j,k}^{a,b,c}\); \[R^{\rm{tr}_3}(z)_{{\boldsymbol{i}}, {\boldsymbol{j}}}^{{\boldsymbol{a}}, {\boldsymbol{b}}} := \sum_{c_1, \ldots, c_N \geq 0} z^{c_1} R_{i_1,j_1,c_2}^{a_1,b_1,c_1} R_{i_2,j_2,c_3}^{a_2,b_2,c_2}\cdots R_{i_N,j_N,c_1}^{a_N,b_N,c_N},\] where \({\boldsymbol{i}}, {\boldsymbol{j}}, {\boldsymbol{a}}, {\boldsymbol{b}} \in I_{M}\). By the weight conservation \({\boldsymbol{c}}:={\boldsymbol{i}}+{\boldsymbol{j}}={\boldsymbol{a}}+{\boldsymbol{b}}\), one can regard \(R^{{\rm tr}_3}(z)_{{\boldsymbol{i}}, {\boldsymbol{j}}}^{{\boldsymbol{a}}, {\boldsymbol{b}}}\) as a function \(R(z; {\boldsymbol{c}})_{\boldsymbol{j},\boldsymbol{b}}\) of \({\boldsymbol{j}}, {\boldsymbol{b}} \in I_M\) and \({\boldsymbol{c}} \in {\mathbb{Z}_{\geq 0}}^N\). The dependence on the parameter \({\boldsymbol{c}}\) seems to be analytically continued to polynomials in \(q^{-c_i}\).

The following two results are relevant to our current problem.

Combining Corollary 26 and these two theorems, we see that \(R_{{\boldsymbol{i}}, {\boldsymbol{j}}}(\Lambda)\) is nothing but the \(R\)-matrix of \(U_q(A_{N-1}^{(1)})\) for the symmetric representations presented in [13].

The coefficients \(R_{{\boldsymbol{i}},{\boldsymbol{j}}}(\Lambda)\) explicitly given by ?? are to be related to the Hamiltonian without the shift operator \(\mathsf{T}\) of the mass truncated \({\widehat{\mathfrak{gl}}_N}\) equation in Proposition 22. Actually they are related by a gauge transformation of the form \(R \longrightarrow K^{-1} R L\), where \(K\) and \(L\) are diagonal matrices, which corresponds to a change of the normalization of a basis of homogeneous polynomials. The diagonal components of \(K\) and \(L\) come from the \(q\)-Borel transformation on monomials in \(\mathbf{z}\) and the inversion of the \(q\)-factorial, hence they are of the form \(K= \mathrm{diag.} (q^{\gamma_i})\) and \(L= \mathrm{diag.} (q^{\delta_i})\), where \(\gamma_i\) and \(\delta_i\) are at most quadratic in the powers \(\mathbf{i} \in I_M\) of \(\mathbf{z}\).

4 Affine Laumon partition function as Jackson integral↩︎

When we impose the mass truncation, the affine Laumon partition function is represented as a Jackson integral, which was a key to the relation to the \(q\)-KZ equation. See [14] for the general method of representing Nekrasov partition functions as Jackson integrals with the help of the truncation of Young diagrams to finite length. In this section by recasting the affine Laumon partition function as a Jackson integral we will show that it is identified with an \(N+2\) point correlation function, which provides a solution to the \(q\)-KZ equation of \(U_q(\widehat{\mathfrak{sl}}_2)\) [15], [16]. In the last section we have seen that after the mass parameter truncation the Hamiltonian of the non-stationary \({\widehat{\mathfrak{gl}}_N}\) difference equation is related to the finite dimensional \(R\)-matrix of \(U_q(\widehat{\mathfrak{sl}}_N)\) (See Corollary 26 and Theorems 27, 28). On the other hand what we will see in this section is the affine Laumon partition function, which is conjecture to be a solution to the non-stationary \({\widehat{\mathfrak{gl}}_N}\) difference equation, actually gives a solution to the \(q\)-KZ equation of \(U_q(\widehat{\mathfrak{sl}}_2)\) (See Proposition 31 and [17], [18]). At the moment, unfortunately, we are not able to fill the gap between \(U_q(\widehat{\mathfrak{sl}}_N)\) on the Hamiltonian side and \(U_q(\widehat{\mathfrak{sl}}_2)\) on the wave function side. It is natural to expect a duality of \(q\)-KZ equations between \(U_q(\widehat{\mathfrak{sl}}_2)\) and \(U_q(\widehat{\mathfrak{sl}}_N)\). Towards a proof of our conjecture such a duality may play an important role.

4.1 Jackson integral of symmetric Selberg type↩︎

The Jackson integral of symmetric Selberg type is¹⁴ \[\begin{align} \label{Jackson-Selberg} \langle \phi, \xi \rangle &:= \int_{0}^{\xi \infty} \phi(z) \Phi_{M,N}(z) \Delta(z) \frac{d_tz_1}{z_1} \wedge \cdots \wedge \frac{d_tz_M}{z_M} \nonumber \\ &=(1-t)^M \sum_{\nu \in \mathbb{Z}^M} [ \phi(z)\Phi_{M,N}(z) \Delta(z)]_{z_i=\xi_i t^{\nu_i}}, \end{align}\tag{28}\] where \[\label{weight-MN} \Phi_{M,N}(z) := \prod_{i=1}^{M} \left[ z_i^\alpha \prod_{r=1}^{N} \frac{(t \mathsf{a}_r^{-1}z_i;t)_\infty}{(\mathsf{b}_rz_i;t)_\infty} \right] \prod_{1 \leq j < k \leq M} z_j^{2\tau -1} \frac{(tq^{-1} z_k/z_j;t)_\infty}{(q z_k/z_j; t)_\infty},\tag{29}\] with \(q=t^{\tau}\), \[\Delta(z) := \prod_{1 \leq i < j \leq M} (z_i-z_j),\] and \(\phi(z)\) is an arbitrary symmetric function on \((\mathbb{C}^\times)^{M}\) [18]. For later convenience we also define \[\Delta(q,z) := \prod_{1 \leq i < j \leq M} (z_i- q^{-1} z_j), \qquad \Delta(z) = \Delta(1,z).\] Here as in [2] we have exchanged \(q\) and \(t\), because the shift parameter of the quantum Knizhnik-Zamolodchikov (\(q\)-KZ) equation of our interest is \(t\), while it was \(q\) in [18]. In the context of two dimensional conformal field theory, \(N\) is the number of the vertex operators in the correlation function and \((\mathsf{a}_r, \mathsf{b}_r)\) are evaluation parameters of each vertex operator of \(U_q\bigl(A_1^{(1)}\bigr)\). On the other hand, each integration variable \(z_i\) is associated with the screening operator inserted in the correlation function. Hence, the Jackson integral 28 corresponds to \(N\)-point correlation function (or \(N+2\) point vacuum expectation value) with the insertion of \(M\) screening operators.

Aomoto and Kato [19],[20] showed that the Jackson integral 28 satisfies a \(q\)-difference system of rank \(\binom{N+M-1}{N-1}\). In [18] the case \(N=2\) was studied and the \(R\) matrix appearing in the \(q\)-difference system of rank \(M+1\) is identified with the \(R\) matrix for the symmetric tensor representation of \(U_q(A_1^{(1)})\) obtained in [13] and [12]. Later in subsection 4.3, we will see that up to a pseudo constant the weight function \(\Phi_{M,N}(z)\) arises from the affine Laumon partition function after the mass parameter truncation (See 49 ). Moreover, the rank of a basis (See 50 and 51 ) of the cocycle function \(\phi(z)\) is exactly the same as that of the \(q\)-difference system of Aomoto-Kato.

4.2 Affine Laumon partition function↩︎

The affine Laumon partition function of type \(A_{N-1}^{(1)}\) is expressed as a summation over \(N\)-tuples of partitions \(\vec{\lambda} = (\lambda^{(1)}, \ldots, \lambda^{(N)})\); \[\begin{align} \label{glnAL} &\mathcal{Z}_{\mathrm{AL}}^{{\widehat{\mathfrak{gl}}_N}} \left( \left.\left.\begin{array}{ccc}a_1,\ldots, a_N \\b_1, \ldots ,b_N\\c_1,\ldots, c_N\end{array} \right| \mathsf{x}_1, \ldots, \mathsf{x}_N \right|q, \kappa \right) \nonumber \\ &= \sum_{\vec{\lambda}} \prod_{i,j=1}^N \frac{{\mathsf N}^{(j-i\vert N)}_{\emptyset,\lambda^{(j)}}(a_i/b_j|q, \kappa) {\mathsf N}^{(j-i\vert N)}_{\lambda^{(i)},\emptyset}(b_i/c_j|q, \kappa)}{{\mathsf N}^{(j-i\vert N)}_{\lambda^{(i)},\lambda^{(j)}}(b_i/b_j|q, \kappa)} \cdot \mathsf{x}_1^{k_1(\vec{\lambda})} \cdots \mathsf{x}_N^{k_N(\vec{\lambda})}, \end{align}\tag{30}\] where \(\mathsf{N}_{\lambda, \mu}^{(k\vert N)} (u \vert q, \kappa)\) is the orbifolded Nekrasov factor (see Definition 5). The powers of the expansion parameters \(\mathsf{x}_i\) are given by the number of boxes with a fixed color (see ?? ).

4.2.1 Exchange symmetry of the orbifolded Nekrasov factor↩︎

The orbifolded Nekrasov factor \(\mathsf{N}_{\lambda, \mu}^{(k\vert N)} (u \vert q,t)\) has the exchange symmetry. We employ the infinite product form of the orbifolded Nekrasov factor obtained in Appendix F to [2]¹⁵ \[\label{inf-prod-form} \mathsf{N}_{\lambda, \mu}^{(k\vert N)} (u \vert q, t^{-\frac{1}{N}}) = \prod_{i,j=1}^\infty \frac{[uq^{j-i} t^{1 -\frac{k}{N} + \lfloor\frac{\mu_{i}^\vee + k -\lambda_j^\vee}{N}\rfloor} ; t]_\infty}{[uq^{j-i-1} t^{1 -\frac{k}{N} + \lfloor\frac{\mu_{i}^\vee + k -\lambda_j^\vee}{N}\rfloor} ; t]_\infty} \frac{[uq^{j-i-1} t^{1 -\frac{k}{N}}; t]_\infty}{[uq^{j-i} t^{1 -\frac{k}{N}}; t]_\infty},\tag{31}\] where \([u;t]_\infty\) is a regularized version of \([u;q]_n\) as \(n \to \infty\), which is defined by \[\label{sinh-reg} [u;t]_\infty := \frac{(u;t)_\infty}{\vartheta_{t^{1/2}}(- u^{1/2})} = \frac{(u^{1/2}; t^{1/2})_\infty}{(-t^{1/2}u^{-1/2}; t^{1/2})_\infty}.\tag{32}\] with \(\vartheta_p(z):= (z;p)_\infty (pz^{-1};p)_\infty\). One can check \[\begin{align} \label{sinh-ratio} \frac{[u;q]_\infty}{[q^n u ; q]_\infty} &= \frac{(u^{1/2} ; q^{1/2})_\infty}{(-q^{1/2} u^{-1/2}; q^{1/2})_\infty} \frac{(-q^{(1-n)/2} u^{-1/2}; q^{1/2})_\infty}{(q^{n/2} u^{1/2} ; q^{1/2})_\infty} \nonumber \\ &= (u^{1/2}; q^{1/2})_n (-q^{(1-n)/2} u^{-1/2}; q^{1/2})_n = [u; q]_n. \end{align}\tag{33}\]

The exchange symmetry of the orbifolded Nekrasov factor implies that the matter contribution to the affine Laumon partition function is symmetric under \(d_i \leftrightarrow \overline{d}_{i}\). For matter contribution one of the partitions in the Nekrasov factor is empty and we have general formulas of finite product form (see eqs. (F.29) and (F.30) in [2]); \[\begin{align} \mathsf{N}_{\lambda, \varnothing}^{(k\vert N)} (u \vert q, \kappa) &= \prod_{i \geq 1} [uq^{i-1}\kappa^{k}; \kappa^N] _{\lfloor\frac{\lambda_{i}^\vee + n-1 -k}{N}\rfloor}, \\ \mathsf{N}_{\varnothing, \lambda}^{(\ell\vert N)} (u \vert q, \kappa) &= \prod_{i \geq 1}[uq^{-i}\kappa^{\ell -N\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor} ; \kappa^N] _{\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor}. \end{align}\]We compute \[\begin{align} \label{d-inversion} \mathsf{N}_{\varnothing, \lambda}^{(\ell\vert N)} (\frac{q\kappa}{d} \vert q, \kappa) &= \prod_{i \geq 1}[d^{-1}q^{1-i}\kappa^{1+ \ell -N\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor} ; \kappa^N] _{\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor}. \nonumber \\ &= \prod_{i \geq 1} (-1)^{\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor} [d q^{i-1}\kappa^{-1- \ell +N\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor} ; \kappa^{-N}] _{\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor} \nonumber \\ &= \prod_{i \geq 1} (-1)^{\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor} [d q^{i-1}\kappa^{N -1- \ell} ; \kappa^{N}] _{\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor} \nonumber \\ &= \prod_{i \geq 1} (-1)^{\lfloor\frac{\lambda_i^\vee + \ell }{N}\rfloor} \cdot \mathsf{N}_{\lambda, \varnothing}^{(N-1-\ell\vert N)} (d \vert q, \kappa), \end{align}\tag{38}\] where we have used 37 .

In general, if we take the specialization \[\label{mass-specialization} a_i = \frac{q \kappa b_{i-1}}{d_{i-1}}, \qquad (0\equiv N), \qquad c_j = \frac{b_j}{\overline{d}_{j}},\tag{39}\] the anti-fundamental factor is \[\begin{align} \mathsf{N}_{\varnothing, \lambda^{(j)}}^{(j-i\vert N)} (b_{i-1,j}\frac{q\kappa}{d_{i-1}} \vert q, \kappa) &\sim \mathsf{N}_{\lambda^{(j)}, \varnothing}^{(N+(i-1)-j\vert N)} (b_{j,i-1}d_{i-1} \vert q, \kappa) \nonumber \\ &= \mathsf{N}_{\lambda^{(j)}, \varnothing}^{(k-j\vert N)} (b_{jk}d_{k} \vert q, \kappa), \end{align}\] where \(b_{ij} := b_i/b_j\). On the other hand the fundamental factor is \[\mathsf{N}_{\lambda^{(i)}, \varnothing}^{(j-i\vert N)} (b_{ij}\overline{d}_{j} \vert q, \kappa).\] Hence, up to sign, they are the same under the exchange of mass parameters.

4.2.2 Vector multiplet↩︎

Let us parametrize the lengths of the columns of an \(N\)-tuple of Young diagrams as follows; \[\label{column-length} (\lambda^{(i)})^\vee= (\ell_1^{(i)}, \ell_2^{(i)}, \ldots), \qquad 1 \leq i \leq N.\tag{40}\] For \(i \leq j\) the (inverse of) vector multiplet contribution is \[\label{vector-general} {\mathsf N}^{(j-i \vert N)}_{\lambda^{(i)}, \lambda^{(j)}} (b_i/b_j \vert q, t^{-\frac{1}{N}}) = \prod_{k, m=1}^\infty \frac{[(b_i/b_j) q^{m -k} t^{1 -\frac{j-i}{N} + \lfloor\frac{\ell^{(j)}_{k} + j-i - \ell^{(i)}_{m})}{N}\rfloor} ; t]_\infty}{[(b_i/b_j) q^{m-k-1} t^{1 -\frac{j-i}{N} + \lfloor\frac{\ell^{(j)}_{k}+ j-i - \ell^{(i)}_m)}{N}\rfloor} ; t]_\infty},\tag{41}\] where we have deleted the normalization factor which is independent of \(\ell_k^{(i)}\). When \(i > j\), \(j-i <0\) in Eq.@eq:vector-general should be replaced by \(N+j-i\). But, since \[t^{1 -\frac{N+j-i}{N} + \lfloor\frac{\ell^{(j)}_{k} +(n + j-i) - \ell^{(i)}_{m})}{N}\rfloor} = t^{1 -\frac{j-i}{N} + \lfloor\frac{\ell^{(j)}_{k} + j-i - \ell^{(i)}_{m})}{N}\rfloor},\] we may use Eq.@eq:vector-general for any \(1 \leq i,j \leq N\).

Let us assume that the width of \(\lambda^{(i)}\) is at most \(L_i\) and set \(L=L_1 + L_2 + \cdots + L_N\). We introduce a disjoint decomposition of the index set \(\{ 1,2, \cdots L\}\) by \(S_1, S_2, \ldots, S_N\), where \[S_i = \{ L_1 + \cdots + L_{i-1} +1, \ldots, L_1 + \cdots + L_{i} \}, \qquad \vert S_i \vert = L_i.\] When \(I = L_1 + \cdots + L_{i-1} + m \in S_i,~ 1 \leq m \leq L_i\), we denote \(I=(i,m)\). Then we define the variables \[\label{Jackson-variables} z_{(i,m)}= b_i^{-1} q^{1-m} \kappa^{i-1} t^{\lfloor\frac{\ell^{(i)}_m +i-1}{N}\rfloor}, \qquad \kappa=t^{-1/N}.\tag{42}\] We can order \(L\) variables \(z_I\) in the lexicographic manner. In [2] in order to write down the weight function \(W_{m+n}(z)\) in a symmetric way, we defined (see below eq.(4.18)); \[\label{n612} z_{N_1 +j} = Q^{-1} q^{1-j} t^{\lfloor\frac{k_j-1}{2}\rfloor}, \qquad k_j = (\lambda^{(2)})^\vee_j.\tag{43}\] When \(N=2\) the definition 42 implies \[z_{(1,i)} = b_1^{-1} q^{1-i} t^{\lfloor\frac{\ell^{(1)}_i}{2}\rfloor}, \qquad z_{(2,j)} = b_2^{-1} q^{1-j} \kappa t^{\lfloor\frac{\ell^{(2)}_j+1}{2}\rfloor} = b_2^{-1} q^{1-j} \kappa^{-1} t^{\lfloor\frac{\ell^{(2)}_j-1}{2}\rfloor}.\] After the scaling by \(b_1\), this agrees with 43 with \(Q=\kappa(b_2/b_1)\).

Now we note¹⁶ \[\begin{align} \label{floor-difference} \lfloor\frac{\ell - m}{N}\rfloor & = \lfloor\frac{\ell}{N}\rfloor - \lfloor\frac{m}{N}\rfloor - \theta([\ell]_N, [m]_N), \end{align}\tag{44}\] where for a pair \((r,s)\) of the residue classes modulo \(N\) we define \[\begin{align} \theta(r,s) &= \begin{cases} 1 \qquad 0 \leq r < s \leq N-1 \\ 0 \qquad otherwise \end{cases}. \end{align}\] By making use of the formula 44 we can recast the (double product part of)¹⁷ vector multiplet contribution as follows; \[{\mathsf N}^{(j-i \vert N)}_{\lambda^{(i)}, \lambda^{(j)}} (b_i/b_j \vert q, t^{-\frac{1}{N}}) = \prod_{I =(i,m) \in S_i} \prod_{J =(j,k)\in S_j} \frac{[(tz_J/z_I)t^{- \theta(\ell^{(j)}_k + j-1, \ell^{(i)}_m + i-1)} ; t]_\infty}{[(t z_J/q z_I)t^{- \theta(\ell^{(j)}_k + j-1, \ell^{(i)}_m + i-1)} ; t]_\infty},\] where for \(i=j\), we should remove the factor with \(m=k\). Hence \[\prod_{i,j=1}^N {\mathsf N}^{(j-i \vert N)}_{\lambda^{(i)}, \lambda^{(j)}} (b_i/b_j \vert q, t^{-\frac{1}{N}}) = \prod_{I \neq J =1}^L \frac{[(tz_J/z_I)t^{- \theta(\ell^{(j)}_k + j-1, \ell^{(i)}_m + i-1)} ; t]_\infty}{[(t z_J/q z_I)t^{- \theta(\ell^{(j)}_k + j-1, \ell^{(i)}_m + i-1)} ; t]_\infty}.\] The additional factor \(t^{- \theta(\ell^{(j)}_k + j-1, \ell^{(i)}_m + i-1)}\) is a generalization of \(t^{\{(\ell_I)-1\}\cdot(\ell_J)}\) appearing in eq.(4.19) of [2], where \((\ell_I):=[\ell_I]_2\) is nothing but the parity of \(\ell_I\).

For each pair \((i,j),~(1 \leq i,j \leq N)\), the boundary contribution of the mass truncation comes from \[{(I)} \quad 1 \leq m \leq L_i,~L_j+1 \leq k < \infty \qquad {and} \qquad {(II)} \quad L_i +1 \leq m < \infty,~1 \leq k \leq L_j,\] where we can assume either \(\ell_k^{(j)}=0\) or \(\ell_m^{(i)}=0\). With the same integration variables as above, we compute the boundary contribution as follows;

Taking the product of the inverses of all the boundary contributions, we obtain \[\prod_{i=1}^N \prod_{m=1}^{L_i} \prod_{j=1}^N \frac{[z_{(i,m)} b_j q^{L_j-1} \kappa^{1-j} t^{{1- \theta(\ell_m^{(i)} + i-1, j-1)}};t]_\infty}{[z_{(i,m)}^{-1} b_j^{-1} q^{-L_j} \kappa^{j-1} t^{{1- \theta(j-1, \ell_m^{(i)} + i-1)}};t]_\infty}.\]

4.2.3 Matter multiplet↩︎

Substituting the specialization 39 , we obtain the matter multiplet contributions as follows;

1. Fundamental matter \[\begin{align} {\mathsf N}^{(j-i \vert N)}_{\lambda^{(i)}, \varnothing} &=\prod_{m=1}^\infty [ b_{ij} \overline{d}_{j} q^{m-1} t^{1 - \frac{j-i}{N} + \lfloor\frac{(j-1) - (\ell^{(i)}_m + i-1)}{N}\rfloor} ; t]_\infty \nonumber \\ &= \prod_{m=1}^\infty [b_j^{-1} \overline{d}_{j} z_{(i,m)}^{-1} \kappa^{j-1} t^{1 - \theta(j-1, \ell_m^{(i)} + i -1)}; t]_\infty, \end{align}\]

2. Anti-fundamental matter \[\begin{align} \label{anti-fund-gln} {\mathsf N}^{(j-i \vert N)}_{\varnothing, \lambda^{(j)}} &= \prod_{k=1}^\infty \frac{1}{[b_{i-1, j} d_{i-1}^{-1} q^{1-k} t^{1 - \frac{j-i+1}{N} + \lfloor\frac{\ell^{(j)}_k}{N}+j -i\rfloor}; t]_\infty} \nonumber \\ &= \prod_{k=1}^\infty \frac{1}{[b_{i-1} d_{i-1}^{-1} z_{(j,k)} \kappa^{2-i} t^{1 - \theta(\ell_k^{(j)} + j-1, i-1)} ]_\infty}, \end{align}\tag{45}\]

where the normalization factor is omitted in the same manner as 41 .

4.3 Mass parameter truncation and a basis of the cocycle↩︎

None

Let us impose the mass truncation condition \(\overline{d}_k = q^{-m_k},~m_k \in \mathbb{Z}_{\geq 0}\). Then by identifying \(\ell_k\) in the previous section with \(m_k\), we see that the fundamental matter contribution cancels half of the boundary part of the vector multiplet contribution. After the mass parameter truncation, the number of the columns of \(\lambda^{(i)}\) is at most \(m_i\), hence we have \(M := m_1 + \cdots + m_N\) columns in total. It is convenient to introduce the index \(I\), \(1 \leq I \leq M\) which is identified with the multi-index \((i,m)\): the \(m\)-th column of the \(i\)-th Young diagram through a bijection \[\label{bijection} \{1,2,\ldots, M\} \to\{(1,1),\ldots,(1,m_1), (2,1),\ldots,(2,m_2), \ldots, (N,1), \ldots, (N,m_N)\}.\tag{46}\] Using this identification, we define the shifted residue \(\mathsf{R}(\ell_I)\) as the residue class of \(\ell_m^{(i)} + i-1\) modulo \(N\). The shifted residue \(\mathsf{R}(\ell_I)\) tells the color of the end box in the corresponding column; See Fig.1. Then the affine Laumon partition function is obtained from the following weight function; \[W_M^{{\widehat{\mathfrak{gl}}_N}}(z) = \prod_{I=1}^M \prod_{k=1}^N \frac{[z_I b_k q^{m_k-1} \kappa^{1-k} t^{{1- \theta(\mathsf{R}(\ell_I), k-1)}} ;t]_\infty}{[z_I b_{k-1} d_{k-1}^{-1} \kappa^{2-k} t^{1 - \theta(\mathsf{R}(\ell_I), k-1)} ;t]_\infty} \cdot \prod_{I \neq J =1}^M \frac{[(t z_J/q z_I)t^{- \theta(\mathsf{R}(\ell_J), \mathsf{R}(\ell_I))} ; t]_\infty}{[(tz_J/z_I)t^{- \theta(\mathsf{R}(\ell_J), \mathsf{R}(\ell_I))} ; t]_\infty},\] with the cycle of the Jackson integral that is chosen according to the corresponding lattice truncation. Note that the weight function \(W_M^{{\widehat{\mathfrak{gl}}_N}}(z)\) depends on the fixed point data \(\{ \lambda^{(i)} \}\) only through the shifted residues \(\mathsf{R}(\ell_I)\).

When \(N=2\), \(\theta(X,Y)\) is non-vanishing only when \(X\) is even and \(Y\) is odd. Namely we have \(\theta(X,Y)= (1-(X))\cdot(Y)\). Hence, \[W_M^{\widehat{\mathfrak{gl}}_2}(z) = \prod_{I=1}^M \frac{[z_I q^{m-1} t ;t]_\infty}{[z_I Q d_{0}^{-1} t ;t]_\infty} \frac{[z_I Q q^{n-1} t^{1+\mathsf{R}(\ell_I)} ;t]_\infty}{[z_I d_{1}^{-1}t^{\mathsf{R}(\ell_I)} ;t]_\infty} \cdot \prod_{I \neq J =1}^M \frac{[(t z_J/q z_I)t^{(\mathsf{R}(\ell_J)-1)\cdot \mathsf{R}(\ell_I))} ; t]_\infty}{[(tz_J/z_I)t^{(\mathsf{R}(\ell_J)-1)\cdot \mathsf{R}(\ell_I))} ; t]_\infty},\] where we have substituted \(m_1=m, m_2=n\) and \(b_1=1, b_0=b_2 = Q/\kappa\). We see that this is exactly Eq.(4.19) in [2] with \(d_0 \leftrightarrow d_4\).

For a matching with the formulas in [18], we define \[\label{Jackson-parameters} \mathsf{a}_k := b_k^{-1} q^{1-m_k} \kappa^{k-1}, \qquad \mathsf{b}_k := b_{k} d_{k}^{-1} \kappa^{1-k},\tag{47}\] with \(\mathsf{b}_N = t \mathsf{b}_0\). Note that \(\mathsf{a}_k \mathsf{b}_k =q^{1-m_k}d_k^{-1}\).¹⁸ We obtain \[W_M^{{\widehat{\mathfrak{gl}}_N}}(z) = \prod_{I=1}^M \prod_{k=1}^N \frac{[\mathsf{a}_k^{-1} z_I t^{1- \theta(\mathsf{R}(\ell_I), k-1)} ;t]_\infty}{[ \mathsf{b}_{k-1} z_I t^{1- \theta(\mathsf{R}(\ell_I), k-1)} ;t]_\infty} \cdot \prod_{I \neq J =1}^M \frac{[(t z_J/q z_I)t^{- \theta(\mathsf{R}(\ell_J), \mathsf{R}(\ell_I))} ; t]_\infty}{[(tz_J/z_I)t^{- \theta(\mathsf{R}(\ell_J), \mathsf{R}(\ell_I))} ; t]_\infty},\] which should be compared with the weight function \(\Phi_{M,N}(z)\) (see 29 and also Eq.(4.27) of [2] in the case of \(N=2\)).

We may note that the sum in the Jackson integral 28 is bilateral, namely it is taken over \(\{\nu_i \}\in \mathbb{Z}^M\). On the other hand, the affine Laumon partition function involves a sum over Young diagrams. After the mass parameter truncation, the discrepancy is remedied by an appropriate choice of the cycle \(\xi=(\xi_1, \dots, \xi_M)\). In fact by taking the following cycle¹⁹ \[\label{cycle-choice} \xi=\xi_{m_i} =\bigl(\underbrace{\mathsf{a}_1, \mathsf{a}_1 q, \dots, \mathsf{a}_1 q^{m_1-1}}_{m_1}, \ldots\ldots, \underbrace{\mathsf{a}_N, \mathsf{a}_N q, \dots, \mathsf{a}_N q^{m_N-1}}_{m_N}\bigr),\tag{48}\] the lattice summation over \(\bigl\{z_i=\xi_i t^{\nu_i}\bigr\}\) is truncated to a cone ([17] section 3.2); \[\begin{align} &0\leq \nu_1\leq \nu_2 \leq \dots \leq \nu_{n-1} \leq \nu_{m_1}, \nonumber \\ &0\leq \nu_{m_1+1} \leq \nu_{m_1+2} \leq \dots \leq \nu_{n-1} \leq \nu_{m_1+m_2}, \nonumber \\ & \vdots \qquad \vdots \qquad \vdots \nonumber \\ &0 \leq \nu_{M- m_N+1} \leq \nu_{M -m_N +2}\leq \dots \leq \nu_{M-1} \leq \nu_{M}. \end{align}\]

In order to identify the cocycle factor \(\phi(z)\) in the Jackson integral 28 , let us introduce a disjoint decomposition of \(\{ 1,2, \ldots, M \} = R_0 \sqcup \cdots \sqcup R_{N-1}\) by \(R_k := \{ I \vert \mathsf{R}(\ell_I) =k \}\). Then we can decompose the weight function as \[W_M^{{\widehat{\mathfrak{gl}}_N}}(z) = W_L^{{\widehat{\mathfrak{gl}}_N}, (0)}(z) \cdot P_{\{R_0 \sqcup \cdots \sqcup R_{N-1}\}}(z),\] where \[\label{weight-function} W_M^{{\widehat{\mathfrak{gl}}_N}, (0)}(z) = \prod_{I=1}^M \prod_{k=1}^N \frac{[t \mathsf{a}_k^{-1} z_I ;t]_\infty}{[ \mathsf{b}_{k-1} z_I ;t]_\infty} \cdot \prod_{I \neq J =1}^M \frac{[(t z_J/q z_I) ; t]_\infty}{[(tz_J/z_I); t]_\infty},\tag{49}\] and \[\begin{align} P_{\{R_0 \sqcup \cdots \sqcup R_{N-1}\}}(z) \sim& \left(\prod_{k=2}^N \prod_{\ell < k -1} \prod_{J \in R_\ell} (1- \mathsf{a}_k^{-1} z_J)\right) \left(\prod_{k=1}^{N-1} \prod_{k \leq \ell } \prod_{J \in R_\ell} (1- \mathsf{b}_{k} z_J) \right) \nonumber \\ & \quad \times \left(\prod_{0 \leq k < \ell \leq N-1} \prod_{I \in R_k} \prod_{J \in R_\ell} \frac{z_J - q^{-1}z_I}{z_J - z_I}\right). \end{align}\] By Lemma 3.1 in [2] the second factor of \(W_M^{{\widehat{\mathfrak{gl}}_N}, (0)}(z)\) is \[\prod_{I \neq J =1}^M \frac{[(t z_J/q z_I) ; t]_\infty}{[(tz_J/z_I); t]_\infty} = C(z) \Delta(z) \prod_{I=1}^M z_I^{-\tau(M-1)} \prod_{1 \leq I<J \leq M} z_I^{2\tau -1} \frac{[(t z_J/q z_I) ; t]_\infty}{[(qz_J/z_I); t]_\infty},\] where \(\tau = \log_t q\) and \(C(z)\) is a pseudo constant that is invariant under \(z_I \to tz_I\) for each variable \(z_I\).

The last factor of \(P_{\{R_0 \cup \cdots \cup R_{N-1}\}}(z)\) is exactly such that we can apply Proposition G.1 in [2]. To apply the proposition we recast the remaining factors as follows; \[\label{cocycle-base} \prod_{\ell=0}^{N-1} \prod_{J \in R_{\ell}} f_\ell(z_J), \qquad f_{\ell}(z) := \prod_{\ell+1 < k} (1- \mathsf{a}_k^{-1} z) \cdot \prod_{k \leq \ell} (1- \mathsf{b}_{k} z).\tag{50}\] For example, when \(N=3\) \[\begin{align} f_0(z) &= (1- \mathsf{a}_2^{-1}z)(1- \mathsf{a}_3^{-1}z), \\ f_1(z) &= (1- \mathsf{a}_3^{-1}z)(1- \mathsf{b}_1 z), \\ f_2(z) &= (1- \mathsf{b}_1 z)(1- \mathsf{b}_2 z). \end{align}\]

Now let consider all the partitions \(R_0 \sqcup \cdots \sqcup R_{N-1}\) of \(\{1, \ldots, M\}\) with fixed \(|R_k| = r_k\). Then Proposition G.1 in [2] tells \[\begin{align} \label{symmetrization} &\phi_{(r_0,r_1, \ldots, r_{N-1})}(z) :=\sum_{R_0 \sqcup \cdots \sqcup R_{N-1}} P_{\{R_0 \sqcup \cdots \sqcup R_{N-1}\}}(z) \nonumber \\ &= \prod_{k=0}^{N-1} \frac{1}{[r_k]_{q^{-1}}!} \cdot \Delta(z)^{-1} \nonumber \\ & \qquad \times \mathcal{A} \left( \prod_{i_0=1}^{r_0} f_0(z_{i_0}) \prod_{i_1=1}^{r_1} f_1(z_{r_0 + i_1}) \cdots \prod_{i_{N-1}=1}^{r_{N-1}} f_{N-1}(z_{r_0 + \cdots + r_{N-2}+ i_{N-1}})\Delta(q,z) \right). \end{align}\tag{51}\]

In summary we have²⁰

The functions \(\phi_{(r_0,r_1, \ldots, r_{N-1})}(z)\) are supposed to give a basis of the cocycle factors. In fact in [17] the set \(Z_{N,M}= \{ \mu = (\mu_1, \mu_2, \ldots, \mu_N) \in \mathbb{Z}^N_{\geq 0} \vert \mu_1 + \mu_2 + \cdots + \mu_N =M \}\) is introduced²¹ and the functions \(E_\lambda(z)\) labeled by \(\lambda \in Z_{N,M}\) are considered. We expect the functions \(\phi_{(r_0,r_1, \ldots, r_{N-1})}(z)\) gives a basis of solutions to \(t\)-difference equation of rank \(r:= \vert Z_{N,M} \vert = \binom{N+M-1}{M}= \binom{N+M-1}{N-1}\). Note that the rank agrees with the size of \(R_M\) block of the \(R\) matrix computed in Section 3. We can check the functions \(\phi_{(r_0,r_1, \ldots, r_{N-1})}(z)\) coincide with the basis of the cocycle functions for \(U_q(\widehat{\mathfrak{sl}}_2)\) \(q\)-KZ equation by Matsuo and Varchenko [15], [16].

In [2] we rely on the Jackson integral representation of the affine Laumon partition function to prove that it is a solution to the non-stationary difference equation 1 . As we have shown in this section the Jackson integral representation of the affine Laumon partition function is also valid for \({\widehat{\mathfrak{gl}}_N}\). Unfortunately it only tells us the relation to the \(q\)-KZ equation of \(U_q(\widehat{\mathfrak{sl}}_2)\). Namely, we can see the \({\widehat{\mathfrak{gl}}_N}\) affine Laumon partition function corresponds to \(N+2\) point correlation functions for the \(U_q(\widehat{\mathfrak{sl}}_2)\) \(q\)-KZ equation, where the shift operator acts on the mass parameters \(d_i\), not on the expansion parameters \(x_i\). In section 3, we have seen that after imposing the mass truncation condition the non-stationary difference equation for \({\widehat{\mathfrak{gl}}_N}\) is related to the \(R\)-matrix of \(U_q(A_{N-1}^{(1)})\). It is natural to expect a duality of \(q\)-KZ equations between \(U_q(\widehat{\mathfrak{sl}}_2)\) and \(U_q(\widehat{\mathfrak{sl}}_N)\). In fact when \(N=2\) we have a dual pair of the \(U_q(\widehat{\mathfrak{sl}}_2)\) KZ equations [18], [2]. Such a duality may be crucial for a proof of our conjecture.

5 Four dimensional limit and Fuji-Suzuki-Tsuda system↩︎

The following computation uses almost the same method as that in Appendix C to [1].

We start with the equation satisfied by the five dimensional \(\widehat{\mathfrak{sl}}_N\) affine Laumon function \(\psi_{5d}\) written in the normal ordered form \[\begin{align} \label{eq:6461} &(A_1 -A_2) \cdot \psi_{5d}=0, \\ &A_1=~: \prod_{a=1}^N \frac{(q^{\vartheta'_a+{\tilde{m}}_a} x_a;q)_\infty}{(x_a;q)_\infty} q^{\gamma_a \vartheta_a}: ,\quad A_2=~: \prod_{a=1}^N \frac{(q^{{\tilde{m}}_a-m_a} x_a;q)_\infty}{(q^{-\vartheta'_a-m_a} x_a;q)_\infty} q^{-\vartheta_a \vartheta'_a}: . \end{align}\tag{52}\] To take the four dimensional limit, we have set the parameters as \[d_a=q^{m_a}, \quad \overline{d}_{a} =q^{\tilde{m}_a}, \quad \kappa \frac{b_{a+1}}{b_a} =q^{\gamma_a}, \quad \vartheta_a=x_a \frac{\partial}{\partial x_a}, \quad \vartheta'_a=\vartheta_a-\vartheta_{a-1},\] where \(a \in \mathbb{Z}/ N \mathbb{Z}\).

For \(q=e^h\), \(h \to 0\), we have \[\label{4dexpansion} (q^a x;q)_b=(1-x)^{b} \left\{1+\frac{h x}{1-x}\left(a b +{ \binom{b}{2} }\right)+O(h^2)\right\}.\tag{53}\] Namely, by taking the limit of the \(q\)-binomial theorem (see [2] Eq.(5.2)) \[\begin{align} \frac{(q^\alpha x;q)_\infty}{(x;q)_\infty} &= \sum_{k=1}^\infty \frac{(q^\alpha;q)_k}{(q;q)_k} x^k =(1-x)^{-\alpha}\Big\{1+\frac{h}{2}\alpha(\alpha-1)\frac{x}{1-x}+O(h^2)\Big\}, \\ \frac{(-q^\alpha x;q)_\infty}{(-q^\beta x;q)_\infty} &=(1+x)^{\beta-\alpha}\Big\{1+\frac{h}{2}(\alpha-\beta)(\alpha+\beta-1)\frac{x}{1-x}+O(h^2)\Big\}. \end{align}\]

By using 53 , the operators \(A_1, A_2\) are expanded as \[\begin{align} A_1&=~: \prod_{a=1}^N (1-x_a)^{-\vartheta'_a-{\tilde{m}}_a} \left\{1+h \sum _{a=1}^N K_{1,a}+O(h^2)\right\}: ,\\ A_2&=~: \prod_{a=1}^N (1-x_a)^{-\vartheta'_a-{\tilde{m}}_a} \left\{1+h \sum_{a=1}^N K_{2,a} +O(h^2)\right\}: , \end{align}\] where \[\begin{align} &K_{1,a}=\gamma_a \vartheta_a+\frac{x_a}{1-x_a} \binom{\vartheta' _a+{\tilde{m}}_a}{2}, \\ & K_{2,a}= -\vartheta_a\vartheta'_a-\frac{x_a}{1-x_a} \left(({\tilde{m}}_a-m_a) (-\vartheta' _a-{\tilde{m}}_a)+\binom{-\vartheta'_a-{\tilde{m}}_a}{2} \right). \end{align}\] Then, since \(\binom{a}{2}+\binom{-a}{2}=a^2\), we have \[\begin{align} \label{eq:K12-lim} &K_1-K_2=\vartheta_a(\vartheta'_a+\gamma_a)+\frac{x_a}{1-x_a}(\vartheta'_a+m_a)(\vartheta'_a+\tilde{m}_a). \end{align}\tag{54}\]

Define an operator \(R\) acting only on \(x\)-variables (not on \(\vartheta\)’s) as \[R: x_a \mapsto x_a \frac{U_{a+1}}{U_{a}}, \quad a \in \mathbb{Z}/(N \mathbb{Z})\] \[\begin{align} &U_a=\sum_{i=0}^{N-1}\prod_{j=0}^{i-1}x_{a+j}=\underbrace{x_a+x_ax_{a+1}+\cdots+\Lambda}_{N}, \\ &\Lambda=x_a x_{a+1}\cdots x_n x_1 \cdots x_{a-1}=x_1x_2\cdots x_N. \end{align}\] The following relation is useful. \[\begin{align} U_a-x_a U_{a+1}&=(x_a+x_ax_{a+1}+\cdots+\Lambda) -x_a(x_{a+1}+x_{a+1}x_{a+2}+\cdots+\Lambda) \nonumber \\ &=x_a(1-\Lambda). \end{align}\]

Recently some constructions of the \(q\)-FST system through the cluster algebra were obtained (see e.g. [26] and [27]). Since such formulation will also give a natural way of the quantization (see e.g. [28]), the relation to our construction from the affine Laumon space is an interesting problem. The relation to the tetrahedron \(R\) matrix uncovered in section 3 also gives a link to the cluster algebra. In [29] it was argued that the conjugation by the tetrahedron \(R\) matrix can be described by a sequence of mutations (see also [30] for the quantum case).

In [31] it was shown that the \(W_N\) conformal block with a fully degenerate field gives a solution to the FST system. This opens a problem of a higher rank generalization of the relation of the Virasolo conformal block with a degenerate field and the conformal block of the \(SL(2)\) current algebra which is described by the KZ equation. We find it intriguing in what sense the affine Laumon partition function gives a conformal block of the deformed \(W\) algebra.

6 Symmetric form of \(\widehat{\mathfrak{gl}}_2\) Hamiltonian↩︎

The \(\widehat{\mathfrak{gl}}_2\)-non-stationary difference equation considered in [3] and [1] is \[\label{Shakirov59Appendix} {\mathcal{H}_{\mathrm S}}T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1} \cdot \Psi(\Lambda,x)=\Psi(\Lambda,x).\tag{55}\] The Hamiltonian is \[\begin{align} {\mathcal{H}_{\mathrm S}}=& \frac{1}{\varphi(qx)\varphi(\Lambda/x)} \cdot \mathcal{B}\cdot \frac{\varphi(\Lambda)\varphi(q^{-1} d_1d_2d_3d_4\Lambda)}{\varphi(-d_1x)\varphi(-d_2x)\varphi(-d_3 \Lambda/x)\varphi(-d_4 \Lambda/x)} \nonumber \\ &~~\cdot \mathcal{B}\cdot \frac{1}{\varphi(q^{-1}d_1d_2x)\varphi(d_3d_4 \Lambda/x)}, \end{align}\] where \(\varphi(z) := (z;q)_\infty\) and \(\mathcal{B}\) is the \(q\)-Borel transformation on a formal Laurent series in \(x\); \[\label{q-Borel} \mathcal{B}\left( \sum_{n} c_n x^n \right) = \sum_{n} q^{\frac{1}{2}n(n+1)} c_n x^n.\tag{56}\] In order to generalize the non-stationary difference equation 55 to higher rank, it is instructive to recast \({\mathcal{H}_{\mathrm S}}\) into more ‘’symmetric’’form. In terms of homogeneous coordinates \(x_1:=x\) and \(x_2:= q^{-1} \Lambda /x\), the total Hamiltonian becomes \[\begin{align} {\mathcal{H}_{\mathrm S}}T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1} =& \frac{1}{\varphi(qx_1)\varphi(qx_2)} \cdot \mathcal{B}\cdot \frac{\varphi(q x_1 x_2)\varphi(d_1d_2d_3d_4 x_1 x_2)}{\varphi(-d_1x_1)\varphi(-d_2x_1)\varphi(-q d_3 x_2)\varphi(-q d_4 x_2)} \nonumber \\ &~~\cdot \mathcal{B}\cdot \frac{1}{\varphi(q^{-1}d_1d_2x_1)\varphi(qd_3d_4 x_2)} T_{qtQ,x_1}^{-1}T_{qQ,x_2}. \end{align}\] Since \(x_1^{n} x_2^{m} \sim x^{n-m}\) the \(q\)-Borel transformation in coordinates \((x_1,x_2)\) is \[\label{relative} \mathcal{B}= q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)(\vartheta_1 - \vartheta_2 +1)} = q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2} p_1^{\frac{1}{2}} p_2^{-\frac{1}{2}},\tag{57}\] where \(\vartheta_i := x_i \frac{\partial}{\partial x_i}\) and \(p_i = q^{\vartheta_i}= T_{q, x_i}\). Hence, we have \[\begin{align} {\mathcal{H}_{\mathrm S}}T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1} =& \frac{1}{\varphi(qx_1)\varphi(qx_2)} \cdot q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2} \cdot \frac{\varphi(q x_1 x_2)\varphi(d_1d_2d_3d_4 x_1 x_2)}{\varphi(-q^{\frac{1}{2}} d_1x_1)\varphi(-q^{\frac{1}{2}} d_2x_1) \varphi(-q^{\frac{1}{2}} d_3 x_2)\varphi(-q^{\frac{1}{2}} d_4 x_2)} \nonumber \\ &~~\cdot q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2} \cdot \frac{1}{\varphi(d_1d_2x_1)\varphi(d_3d_4 x_2)} T_{tQ,x_1}^{-1}T_{Q,x_2}. \end{align}\] The shift operator \(p_1^{\frac{1}{2}} p_2^{-\frac{1}{2}}\) in 57 is combined with the original shift operator \(T_{qtQ,x_1}^{-1}T_{qQ,x_2}\). Note that the factors in the numerator commute with \(q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2}\). Parametrizing \(Q= \kappa \frac{b_2}{b_1}\) with \(\kappa:= t^{-\frac{1}{2}}\), we finally obtain \[\begin{align} \label{gl2} {\mathcal{H}_{\mathrm S}}T_{qtQ,x}^{-1}T_{t,\Lambda}^{-1} =& \frac{1}{\varphi(qx_1)\varphi(qx_2)} \cdot q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2} \cdot \frac{\varphi(q x_1 x_2)\varphi(d_1d_2d_3d_4 x_1 x_2)}{\varphi(-q^{\frac{1}{2}} d_1x_1)\varphi(-q^{\frac{1}{2}} d_2x_1) \varphi(-q^{\frac{1}{2}} d_3 x_2)\varphi(-q^{\frac{1}{2}} d_4 x_2)} \nonumber \\ &~~\cdot q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2} \cdot \frac{1}{\varphi(d_1d_2x_1)\varphi(d_3d_4 x_2)} T_{\frac{\kappa b_1}{b_2},x_1}T_{\frac{\kappa b_2}{b_1},x_2}. \end{align}\tag{58}\]

The rescaling \(x_i \to - q^{\frac{1}{2}} x_i\) and the exchange of mass parameters \(d_2 \leftrightarrow d_3\) implies a complete matching of the Hamiltonian ?? with \(N=2\) and 58 . When \(N=2\), \(\Delta = (\vartheta_1 - \vartheta_2)^2\) and the twist operation on \(\varphi(-\check{x}_0)^{-1}\) and \(\varphi(-\hat{x}_0)^{-1}\) is trivial. Hence the Hamiltonian ?? reduces to \[\label{gl2-Hamiltonian} \mathcal{H}^{\widehat{\mathfrak{gl}}_2}(x_i; b_i, d_i, q,t) = q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2} \cdot \mathcal{A}_L \cdot \mathcal{A}_C \cdot \mathcal{A}_R \cdot q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2} \cdot T_{\frac{\kappa b_1}{b_{2}}, x_1} T_{\frac{\kappa b_2}{b_{1}}, x_2},\tag{59}\] where \[\begin{align} \mathcal{A}_L &= \frac{1}{\varphi(-\check{x}_{2})} \frac{1}{\varphi(-\check{x}_{1})} \varphi(\Lambda), \\ \mathcal{A}_R &= \varphi(q^{-1} d_1d_2d_3d_4 \Lambda ) \frac{1}{\varphi(-d_1d_3\hat{x}_1)} \frac{1}{\varphi(-d_2d_4\hat{x}_2)}, \\ \mathcal{A}_C &= \frac{1}{\varphi(d_1 x_1)\varphi(d_3 x_1 )\varphi(d_2 x_2)\varphi(d_4 x_2)}, \end{align}\] and we have identified \(x_0\) with \(x_2\). In order to compare the Hamiltonian 59 with the symmetric form of the \({\widehat{\mathfrak{gl}}_2}\) Hamiltonian 58 , we have to commute \(q^{\frac{1}{2} (\vartheta_1 - \vartheta_2)^2}\) with \(\mathcal{A}_L\) or \(\mathcal{A}_R\) by using the formula 62 in the next subsection. This commutation removes the hat and the the check on \(x_i\); \(\check{x}_i \to x_i\) and \(\hat{x}_i \to x_i\) and also it scales \(x_i\) by \(q^{\pm \frac{1}{2}}\). Namely \[\begin{align} \mathcal{A}_L \longrightarrow \widetilde{\mathcal{A}}_L&= \frac{1}{\varphi(-q^{\frac{1}{2}} x_{2})} \frac{1}{\varphi(- q^{\frac{1}{2}} x_{1})} \varphi(\Lambda), \\ \mathcal{A}_R \longrightarrow \widetilde{\mathcal{A}}_R&= \varphi(q^{-1} d_1d_2d_3d_4 \Lambda ) \frac{1}{\varphi(-d_1d_3 q^{-\frac{1}{2}} x_1)} \frac{1}{\varphi(-d_2d_4 q^{-\frac{1}{2}} x_2)}, \end{align}\] Then as claimed above by the rescaling \(x_i \to - q^{\frac{1}{2}} x_i\) and the exchange of mass parameters, we see the agreement of the Hamiltonians 59 and 58 .

6.1 \(q\)-Borel transformation for multi-variables↩︎

In the higher rank generalization of \({\mathcal{H}_{\mathrm S}}\), there appear the operators of the form \(q^{\mathcal{L}}\), where \(\mathcal{L}\) is a second order polynomial in the Euler derivative \(\vartheta_i\). For later convenience let us work out the commutation relations with coordinate variables \(x_i\). Since \(q^{\frac{1}{2}\vartheta_i^2} x_i \cdot x_i^{n} = q^{\frac{1}{2}(n+1)^2} x_i^{n+1} = q^{n + \frac{1}{2}} x_i q^{\frac{1}{2}\vartheta_i^2} \cdot x_i^n = q^{\frac{1}{2}} x_i p_i q^{\frac{1}{2}\vartheta_i^2} \cdot x_i^n\), we see \[\label{xx-com} {\mathrm{Ad}}(q^{\pm \frac{1}{2}\vartheta_i^2}) \cdot x_i = q^{\pm \frac{1}{2}} x_i p_i^{\pm 1}.\tag{60}\] We also have \[\label{xy-com} {\mathrm{Ad}}(q^{\pm \vartheta_i \vartheta_j}) \cdot x_i = x_i p_j^{\pm 1}, \qquad (i \neq j).\tag{61}\] Let us introduce the relative \(q\)-Borel transformation \(\widetilde{\mathcal{B}}_{xy}\) by (see eq.@eq:relative ) \[\widetilde{\mathcal{B}}_{xy} := q^{\frac{1}{2}(\vartheta_x - \vartheta_y)(\vartheta_x - \vartheta_y +1)} = q^{\frac{1}{2}(\vartheta_x^2 + \vartheta_y^2)} q^{-\vartheta_x \vartheta_y} p_x^{\frac{1}{2}} p_y^{-\frac{1}{2}}.\] Namely \[\widetilde{\mathcal{B}}_{xy} \cdot x^n y^m = q^{\frac{1}{2}(n-m)(n-m+1)} x^n y^m.\] From 60 and 61 we obtain the commutation relation \[{\mathrm{Ad}}(\widetilde{\mathcal{B}}_{xy}) \cdot x = p_x p_y^{-1} x, \qquad {\mathrm{Ad}}(\widetilde{\mathcal{B}}_{xy}) \cdot y = y p_x^{-1} p_y.\] We note that for \(x'= p_x p_y^{-1} x\) and \(y' = y p_x^{-1} p_y\), we have \(x'y' = y' x' = xy\). On the power of \(x_i\), we have \[\label{adjoint-onpower} {\mathrm{Ad}}(q^{\pm \frac{1}{2}(\vartheta_i - \vartheta_j)^2}) \cdot x_i^n = q^{\pm \frac{1}{2}n^2} x_i^n p_{ij}^{\pm n} = q^{\pm\frac{n}{2}}(x_i p_{ij}^{\pm 1})^n, \quad p_{ij} = p_i/p_j, \qquad (i \neq j).\tag{62}\] This formula is useful in the computation of the normal ordered Hamiltonian.

7 Two types of the affine Laumon partition function↩︎

There are two types of the Nekrasov factor, which we call Pochhammer type and hyperbolic-sine (sinh) type. From the view point of the index theorem for the instanton moduli space, they come from the Dolbeault operator and the Dirac operator, respectively. Consequently we have the affine Laumon partition function of Pochhammer type and of sinh type. When the moduli space is hyperKähler, the index of the Dolbeault operator and the Dirac operator coincide, since the discrepancy is measured by the first Chern class. However, the affine Laumon space is not hyperKähler, because of the asymmetry of the chain-saw quiver [32], [33] and two types of the partition function are different in general. In this appendix we will show that the \(q\)-Borel transformation transforms the affine Laumon partition function of sinh type into of Pochhammer type.

7.1 \(q\)-Borel transformation from sinh type into Pochhammer type↩︎

Let \(\lambda\) be a Young diagram, i.e. a partition \(\lambda = (\lambda_1,\lambda_2,\cdots)\), which is a sequence of nonnegative integers such that \(\lambda_{i} \geq \lambda_{i+1}\) and \(|\lambda| = \sum_i \lambda_i < \infty\). \(\lambda^\vee\) denotes its conjugate (dual) diagram. We define \[{\vert \lambda \vert_{k}}:= \sum_{n \in \mathbb{Z}}\lambda_{k+nN}, \qquad k \in \mathbb{Z}/N\mathbb{Z},\] where we set \(\lambda_i = 0\) for \(i \leq 0\). Throughout Appendix 7, we use the notation \[\mathsf{v} := (q{ {\kappa} })^{{\frac{1}{2}}}.\] In this appendix the notation \(\equiv\) always means the congruence of integers modulo \(N\).

For a pair of Young diagrams \(\lambda\) and \(\mu\), we define the orbifolded Nekrasov factor of Pochhammer type as \[\begin{align} { N^{{\rm Poch }(k\vert N)}_{\lambda\mu}(\mathsf{v}Q\vert q, {\kappa} ) } :=\hskip-20pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv -k-{\frac{1}{2}}}} \hskip-20pt \left(1-Qq^{\lambda_i-j+{\frac{1}{2}}} { {\kappa^{-\mu^\vee_j+i-{\frac{1}{2}}}} }\right) \cdot \hskip-20pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} \hskip-20pt \left(1-Qq^{-\mu_i+j-{\frac{1}{2}}} { {\kappa^{\lambda^\vee_j-i+{\frac{1}{2}}}} }\right). \label{eq:ONekIpDef} \end{align}\tag{63}\] The orbifolded Nekrasov factor of sinh type (31 ) is written as \[{ \mathsf{N}^{(k\vert N)}_{\lambda\mu}(\mathsf{v}Q\vert q, {\kappa} ) } = \hskip-20pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv -k-{\frac{1}{2}}}} \hskip-20pt \left[Qq^{\lambda_i-j+{\frac{1}{2}}} { {\kappa^{-\mu^\vee_j+i-{\frac{1}{2}}}} }\right] \cdot \hskip-20pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} \hskip-20pt \left[Qq^{-\mu_i+j-{\frac{1}{2}}} { {\kappa^{\lambda^\vee_j-i+{\frac{1}{2}}}} }\right],\] with \([x]:=x^{-{\frac{1}{2}}}-x^{{\frac{1}{2}}}\) (see Appendix F to [2]). It satisfies²² \[{ \mathsf{N}^{(k\vert N)}_{\lambda\mu}(\mathsf{v}Q\vert q, {\kappa} ) } = (-1)^n{ \mathsf{N}^{(k\vert N)}_{\lambda\mu}(\mathsf{v}^{-1}Q^{-1}\vert q^{-1}, {\kappa} ^{-1} ) },\] where \(n= {\vert \mu \vert_{-k}}+ {\vert \lambda \vert_{1+k}}\), as we will show in (78 ).

For \(N\)-tuple of Young diagrams \(\lambda^{(i)}\) with \(i \in \mathbb{Z}/N\mathbb{Z}\), let \({\vec{\lambda}}:=\left(\lambda^{(1)}, \lambda^{(2)},\cdots, \lambda^{(N)}\right)\). For \(N\) variables \(x_i\in{\mathbb{C}}\) with \(i \in \mathbb{Z}/N\mathbb{Z}\), let \(\vec{x}:=\left(x_1,x_2,\cdots, x_N\right)\). For \(3N^2\) variables \({Q^{a}_{i,j}}, {Q^{b}_{i,j}}, {Q^{c}_{i,j}}\in{\mathbb{C}}\) with \(i,j \in \mathbb{Z}/N\mathbb{Z}\), let \[\begin{align} { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) } &:= \prod_{i,j=1}^N \frac{ {\mathsf{N}^{(j-i\vert N)}_{\emptyset\lambda^{(j)}}( {\mathsf{v}Q^{a}_{i,j}} \vert q, {\kappa} ) } {\mathsf{N}^{(j-i\vert N)}_{\lambda^{(i)}\emptyset}( {\mathsf{v}Q^{c}_{i,j}} \vert q, {\kappa} ) } }{ {\mathsf{N}^{(j-i\vert N)}_{\lambda^{(i)}\lambda^{(j)}}( {\mathsf{v}Q^{b}_{i,j}} \vert q, {\kappa} ) } } x_i^{ {\vert \lambda^{(j)}\vert_{1+i-j}} }, \tag{64} \\ { {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) } &:= \prod_{i,j=1}^N \frac{1}{ {\mathsf{N}^{(j-i\vert N)}_{\lambda^{(i)}\lambda^{(j)}}( {\mathsf{v}Q^{b}_{i,j}} \vert q, {\kappa} ) } } x_i^{ {\vert \lambda^{(j)}\vert_{1+i-j}} }. \tag{65} \end{align}\] We define \(Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}})\) and \(Z^{{\rm Poch, pure}}_{{\vec{\lambda}}}({\vec{x}})\) by replacing the orbifolded Nekrasov factor of sinh type with that of Pochhammer type.

As we will show in (?? ), the denominators and the numerators of (64 ) and (65 ) have even numbers of factors or brackets \([\quad]\). Therefore, even if we change the sign of \([x]\), \({ {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }\) and \({ {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) }\) are unchanged.

Let \(\vartheta_{x_i}:=x_i \frac{\partial{\partial x_i}}{None}\) and \[\Delta := {\frac{1}{2}}\sum_{i=1}^N \left({\vartheta_{x_{i-1}} }-{\vartheta_{x_i} }\right)^2 =\sum_{i=1}^N \left( \vartheta_{x_i}^2 -\vartheta_{x_{i-1}}\vartheta_{x_i} \right).\] Since \(\vartheta_{x} x = x (1+\vartheta_{x} )\), \(q^{\vartheta_{x}^2} x^n \cdot 1 = x^n q^{(n+\vartheta_{x})^2} \cdot 1 = x^n q^{n^2} \cdot 1\). Thus, for any \(c\in{\mathbb{C}}\), \({ {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }\) satisfies \[\begin{align} q^{ {\frac{c}{2}}\vartheta_{x_i} } { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) } &= { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }\, q^{ {\frac{c}{2}}\sum_{j=1}^{N} {\vert \lambda^{(j)}\vert_{1+i-j} } }, \tag{66} \\ q^{ {\frac{c}{2}}\Delta} { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) } &= { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }\, \prod_{i=1}^N q^{ {\frac{c}{4}} \left(\sum_{j=1}^{N}\left( {\vert \lambda^{(j)}\vert_{i-j} }-{\vert \lambda^{(j)}\vert_{1+i-j} } \right)\right)^2}. \tag{67} \end{align}\] The same relations are also valid for \({ {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) }\), \({ Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) }\) and \({ Z^{{\rm Poch, pure}}_{{\vec{\lambda}}}({\vec{x}}) }\).

Remark that, in the case of \(N=2\), \(\Delta=\vartheta_{x_1}^2 -2\vartheta_{x_{1}}\vartheta_{x_2} +\vartheta_{x_2}^2\). Remark also that, since \(\sum_{i=1}^{N}{\vert \lambda^{(j)}\vert_{i-j}} = {\vert \lambda^{(j)}\vert_{}}\),²³\[\begin{align} \label{eq:ZZpureS} &\hskip-23.5pt { Z^{{\rm Poch, pure}}_{{\vec{\lambda}}}({\vec{x}}) } \cr &\hskip-23.5pt= { {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) } \prod_{i=1}^N q^{\frac{1}{4} \left(\sum_{j=1}^{N}\left( {\vert \lambda^{(j)}\vert_{i-j} }-{\vert \lambda^{(j)}\vert_{1+i-j} } \right)\right)^2} b_{i}^{-{\frac{1}{2}}\sum_{j=1}^{N}\left( {\vert \lambda^{(j)}\vert_{i-j} }-{\vert \lambda^{(j)}\vert_{1+i-j} } \right) } \left(q{ {\kappa} }\right)^{{\frac{1}{2}}{\vert \lambda^{(i)}\vert_{}} }. \end{align}\tag{68}\]

For any function \(f\) of \(a_i,b_j,c_k,q,t\), let \[\begin{align} &\hskip-37pt \overline{f(a_1,a_2,\cdots, b_1,b_2,\cdots, c_1,c_2,\cdots,q,{ {\kappa} }) } \cr \hskip37pt &:= f(a_1^{-1},a_2^{-1},\cdots, b_1^{-1},b_2^{-1},\cdots, c_1^{-1},c_2^{-1},\cdots,q^{-1},{ {\kappa} ^{-1} }). \end{align}\] Since, \(\overline{ {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }={ {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }\) and \(\overline{ {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) }={ {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) }\), we have \[\begin{align} \overline{ Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) } &= q^{-{\frac{1}{2}}\Delta} \prod_{i=1}^N \left( \frac{a_i}{b_i} \frac{b_{i-1}}{c_{i-1}}\right)^{-{\frac{1}{2}}\vartheta_{x_{i-1}}} \cdot { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }, \\ \overline{ Z^{{\rm Poch, pure}}_{{\vec{\lambda}}}({\vec{x}}) } &= q^{-{\frac{1}{2}}\Delta} \prod_{i=1}^N \left(\frac{b_{i-1}}{b_i} q{ {\kappa} } \right)^{-{\frac{1}{2}}\vartheta_{x_{i-1}}} \cdot { {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) }. \end{align}\]

Let \([x]_c:=x^{\frac{c}{2}}[x]\) with \([x]=x^{-{\frac{1}{2}}}-x^{{\frac{1}{2}}}\). From \({ {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }\) and \({ {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) }\), we define \({ {\mathcal{Z}}_{c, {\vec{\lambda}}}({\vec{x}}) }\) and \({ {\mathcal{Z}}^{{\rm pure}}_{c, {\vec{\lambda}}}({\vec{x}}) }\) by replacing \([x]\) in the orbifolded Nekrasov factor of sinh type with \([x]_c\). For example, \({ {\mathcal{Z}}_{0, {\vec{\lambda}}}({\vec{x}}) }= {{\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }\), \({ {\mathcal{Z}}_{1, {\vec{\lambda}}}({\vec{x}}) }= {Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) }\) and \({ {\mathcal{Z}}_{-1, {\vec{\lambda}}}({\vec{x}}) }= \overline{Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) }\).

In view of 66 and 67 , it should be clear that even if we replace \({ Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) }\) and \({ Z^{{\rm Poch, pure}}_{{\vec{\lambda}}}({\vec{x}}) }\) with \({ {\mathcal{Z}}_{c, {\vec{\lambda}}}({\vec{x}}) }\) and \({ {\mathcal{Z}}^{{\rm pure}}_{c, {\vec{\lambda}}}({\vec{x}}) }\), respectly, Proposition 36 is true, if \({\frac{1}{2}}\Delta\) and \({\frac{1}{2}}\vartheta_{x_{i-1}}\) are replaced with \({\frac{c}{2}}\Delta\) and \({\frac{c}{2}}\vartheta_{x_{i-1}}\), respectively.

7.2 Proof of the Proposition 36↩︎

Before embarking on a proof of Proposition 36, it is convenient to introduce a few notations. Since \((1-x) = x^{\frac{1}{2}}[x]\), we have \[\label{Nekrasovfactor-rel} { N^{{\rm Poch }(k\vert N)}_{\lambda\mu}(\mathsf{v}Q\vert q, {\kappa} ) } = { \mathsf{N}^{(k\vert N)}_{\lambda\mu}(\mathsf{v}Q\vert q, {\kappa} ) } \left( {f^{(k\vert N)}_{\lambda,\mu} } {g^{(k\vert N)}_{\lambda,\mu}\! \left(Q\right) } \right)^{\frac{1}{2}},\tag{69}\] where \[\begin{align} {f^{(k\vert N)}_{\lambda,\mu}} &:= \hskip-15pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k-{\frac{1}{2}}}} \hskip-20pt q^{\lambda_i-j+{\frac{1}{2}}} { {\kappa^{-\mu^\vee_j+i-{\frac{1}{2}}}} } \cdot \hskip-15pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} \hskip-20pt q^{-\mu_i+j-{\frac{1}{2}}} { {\kappa^{\lambda^\vee_j-i+{\frac{1}{2}}}} }, \\ {g^{(k\vert N)}_{\lambda,\mu }\! \left(Q\right)} &:= \hskip-15pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k-{\frac{1}{2}}}} Q \hskip10pt\cdot \hskip-6pt\prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} Q. \label{eq:gDef} \end{align}\tag{70}\] Let us denote \[\begin{align} f_{\vec{\lambda}} &:= \prod_{i,j=1}^N \frac{ {f^{(j-i\vert N)}_{\emptyset,\lambda^{(j)}} } {f^{(j-i\vert N)}_{\lambda^{(i)},\emptyset }} }{ {f^{(j-i\vert N)}_{\lambda^{(i)},\lambda^{(j)}} } }, \qquadg_{\vec{\lambda}} :=\prod_{i,j=1}^N \frac{ {g^{(j-i\vert N)}_{\emptyset,\lambda^{(j)}}\! \left(Q^{a}_{i,j}\right) } {g^{(j-i\vert N)}_{\lambda^{(i)},\emptyset }\! \left(Q^{c}_{i,j}\right) } }{ {g^{(j-i\vert N)}_{\lambda^{(i)},\lambda^{(j)}}\! \left(Q^{b}_{i,j}\right) } }, \\f^{{\rm num}}_{\vec{\lambda}} &:= \prod_{i,j=1}^N {f^{(j-i\vert N)}_{\emptyset,\lambda^{(j)}} } {f^{(j-i\vert N)}_{\lambda^{(i)},\emptyset }} =\prod_{i,j=1}^N {f^{(j-i\vert N)}_{\emptyset,\lambda^{(j)}} } {f^{(i-j-1\vert N)}_{\lambda^{(j)},\emptyset }}, \\ g^{{\rm num}}_{\vec{\lambda}} &:= \prod_{i,j=1}^N {g^{(j-i\vert N)}_{\emptyset,\lambda^{(j)}}\! \left( {Q^{a}_{i,j}} \right) } {g^{(j-i\vert N)}_{\lambda^{(i)},\emptyset }\! \left( {Q^{c}_{i,j}} \right) } =\prod_{i,j=1}^N {g^{(j-i\vert N)}_{\emptyset,\lambda^{(j)}}\! \left( {Q^{a}_{i,j}} \right) } {g^{(i-j-1\vert N)}_{\lambda^{(j)},\emptyset }\! \left( {Q^{c}_{j,i-1}} \right) }, \\f^{{\rm pure}}_{\vec{\lambda}} &:= \frac{f_{\vec{\lambda}}}{f^{{\rm num}}_{\vec{\lambda}}} = \prod_{i,j=1}^N \frac{ 1}{ {f^{(j-i\vert N)}_{\lambda^{(i)},\lambda^{(j)}} } }, \qquad g^{{\rm pure}}_{\vec{\lambda}} :=\frac{g_{\vec{\lambda}}}{g^{{\rm num}}_{\vec{\lambda}}} = \prod_{i,j=1}^N \frac{ 1}{ {g^{(j-i\vert N)}_{\lambda^{(i)},\lambda^{(j)}}\! \left(Q^{b}_{i,j}\right) } }. \label{eq:gpvlambdaDef} \end{align}\tag{71}\] From 69 we have \[\begin{align} { Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) } &= \left(f_{\vec{\lambda}}g_{\vec{\lambda}}\right)^{\frac{1}{2}} { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }, \tag{72} \\ { Z^{{\rm Poch, pure}}_{{\vec{\lambda}}}({\vec{x}}) } &= \left(f^{{\rm pure}}_{\vec{\lambda}}g^{{\rm pure}}_{\vec{\lambda}}\right)^{\frac{1}{2}} { {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) }. \tag{73} \end{align}\] Hence to prove Proposition 36, it is enough to evaluate \(f_{\vec{\lambda}}\), \(g_{\vec{\lambda}}\), \(f^{{\rm num}}_{\vec{\lambda}}\) and \(g^{{\rm num}}_{\vec{\lambda}}\). This is achieved by the following four steps.

7.2.1 Step 1 : Good combination↩︎

We have\[{f^{(-k\vert N)}_{\mu,\lambda}} = \hskip-15pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k-{\frac{1}{2}}}} \hskip-15pt q^{\mu_i-j+{\frac{1}{2}}} { {\kappa^{-\lambda^\vee_j+i-{\frac{1}{2}}}} } \cdot \hskip-15pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k+{\frac{1}{2}}}} \hskip-20pt q^{-\lambda_i+j-{\frac{1}{2}}} { {\kappa^{\mu^\vee_j-i+{\frac{1}{2}}}} },\] and we can eliminate \(\kappa\) from \(f\)-factors by taking the following combinations: \[\begin{align} \frac{ {f^{(k\vert N)}_{\emptyset,\mu}}{f^{(k\vert N)}_{\lambda,\emptyset}} }{ {f^{(k\vert N)}_{\lambda,\mu}} } &= \hskip-15pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k-{\frac{1}{2}}}} \hskip-15pt q^{-\lambda_i} \hskip3pt \cdot \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} \hskip-10pt q^{\mu_i} , \label{eq:fffkqq} \\ \frac{ {f^{(-k\vert N)}_{\emptyset,\lambda}}{f^{(-k\vert N)}_{\mu,\emptyset}} }{ {f^{(-k\vert N)}_{\mu,\lambda}} } &= \hskip-10pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k-{\frac{1}{2}}}} \hskip-25pt ~q^{-\mu_i} \hskip3pt \cdot \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k+{\frac{1}{2}}}} \hskip-20pt q^{\lambda_i}. \end{align}\tag{74}\] We also have \[\begin{align} {f^{(k\vert N)}_{\lambda,\mu}} {f^{(-1-k\vert N)}_{\mu,\lambda}} &= 1, \tag{75} \\ {g^{(k\vert N)}_{\lambda,\mu }\! \left(Q\right)} {g^{(-1-k\vert N)}_{\mu,\lambda }\! \left(Q'\right)} &= \hskip-10pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k-{\frac{1}{2}}}} \hskip-25pt QQ' \hskip2pt \cdot \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} \hskip-20pt QQ' \hskip7pt = {g^{(k\vert N)}_{\lambda,\mu }\! \left(QQ'\right) }. \tag{76} \end{align}\] Note that, by (74 ), \[f_{\vec{\lambda}} = \prod_{i,j=1}^N \hskip-10pt \prod_{\genfrac{}{}{0pt}{}{(a,b)\in\lambda^{(j)} }{ {\lambda^{(j)} }^\vee_b-a+{\frac{1}{2}}\equiv i-j-{\frac{1}{2}}}} \hskip-26pt q^{-\lambda^{(i)}_a} \hskip3pt \cdot \prod_{\genfrac{}{}{0pt}{}{(a,b)\in\lambda^{(j)} }{ {\lambda^{(j)} }^\vee_b-a+{\frac{1}{2}}\equiv i-j+{\frac{1}{2}}} } \hskip-26pt q^{\lambda^{(i)}_a} .~~ \label{eq:fffkqqLemma}\tag{77}\]

7.2.2 Step 2 : Box product↩︎

Since \(\lambda^\vee_j-i\) is the leg length of the square \((i,j)\) in the Young diagram of a partition \(\lambda\), we have \[\sum_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+1 \equiv k}}1 = \sum_{n=0}^\infty\lambda_{k+nN} = {\vert \lambda \vert_{k}}.\] Hence, the power of \(Q\)’s in (70 ) is \({\vert \mu \vert_{-k}}+ {\vert \lambda \vert_{1+k}}\), and \[{g^{(k\vert N)}_{\lambda,\mu }\! \left(Q\right)} = Q^{{\vert \mu \vert_{-k}}+ {\vert \lambda \vert_{1+k}}}. \label{eq:gLemma}\tag{78}\]

By (78 ), \[\frac{ {g^{(k\vert N)}_{\emptyset,\mu }\! \left(Q^{a}\right) }{g^{(k\vert N)}_{\lambda,\emptyset }\! \left(Q^{c}\right) } }{ {g^{(k\vert N)}_{\lambda,\mu }\! \left(Q^{b}\right) } } = \frac{ \left(Q^{a}\right)^{\vert \mu\vert_{-k}} \left(Q^{c}\right)^{\vert \lambda\vert_{1+k}} }{ \left(Q^{b}\right)^{{\vert \mu \vert_{-k}}+ {\vert \lambda \vert_{1+k}}} },\] which gives (?? ). From (75 ), (76 ) and (78 ), we get (?? ).

For any \(j,s\in{\mathbb{Z}}_{>0}\), \(\lambda^\vee_j=s\) if and only if \(1+\lambda_{s+1}\leq j \leq \lambda_s\). Thus, for \(0\leq k \leq N-1\), \[\begin{align} \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} \hskip-10pt q^{\mu_i} &= \prod_{i\geq 1}\prod_{n\geq 0} \prod_{\genfrac{}{}{0pt}{}{j\geq 1}{\lambda^\vee_j=i+k+nN}} q^{\mu_i} \cr &= \prod_{i\geq 1}\prod_{n\geq 0} \prod_{\genfrac{}{}{0pt}{}{s\geq 1}{s=i+k+nN}} q^{\mu_i(\lambda_s-\lambda_{s+1})} \cr &= \prod_{n\geq 0} \prod_{\genfrac{}{}{0pt}{}{i,j\in{\mathbb{Z}}}{j-i=k+nN}} q^{\mu_{i-1}(\lambda_{j-1}-\lambda_{j})}, \label{eq:mu} \end{align}\tag{79}\] where we have used \(\lambda^\vee_j=s\Leftrightarrow 1+\lambda_{s+1}\leq j \leq \lambda_s\) for the second equality. By replacing \(q\), \(k\), \(\lambda\) and \(\mu\) in (79 ) with \(1/q\), \(N-1-k\), \(\mu\) and \(\lambda\), respectively, we have \[\prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k-{\frac{1}{2}}}} \hskip-20pt q^{-\lambda_i} = \prod_{n\geq 0} \prod_{\genfrac{}{}{0pt}{}{i,j\in{\mathbb{Z}}}{j-i+1=N-k+nN}} q^{-\lambda_{i-1}(\mu_{j-1}-\mu_{j})} = \prod_{n\geq 1} \prod_{\genfrac{}{}{0pt}{}{i,j\in{\mathbb{Z}}}{j-i=-k+nN}} q^{-\lambda_{i}(\mu_{j-1}-\mu_{j})}. \label{eq:mlambda}\tag{80}\] Similarly, by replacing \(q\), \(k\) in (79 ) and (80 ) with \(1/q\), \(k-1\), respectively, we obtain the formulas for \(1\leq k \leq N\). Combining them, we have \[\begin{align} \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} \hskip-15pt q^{\mu_i} \hskip3pt \cdot \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k-{\frac{1}{2}}}} \hskip-15pt q^{-\mu_i} &= \prod_{n\geq 0} \prod_{\genfrac{}{}{0pt}{}{i,j\in{\mathbb{Z}}}{j-i=k+nN}} q^{(\mu_{i-1}-\mu_{i})(\lambda_{j-1}-\lambda_{j})}, \cr \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k-{\frac{1}{2}}}} \hskip-15pt q^{-\lambda_i} \cdot \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k+{\frac{1}{2}}}} \hskip-15pt q^{\lambda_i} &= \prod_{n\geq 1} \prod_{\genfrac{}{}{0pt}{}{i,j\in{\mathbb{Z}}}{j-i=-k+nN}} q^{(\lambda_{i-1}-\lambda_{i})(\mu_{j-1}-\mu_{j})}. \nonumber \end{align}\] Then, for \(1\leq k \leq N-1\), \[\begin{align} \frac{ {f^{(k\vert N)}_{\emptyset,\mu}}{f^{(k\vert N)}_{\lambda,\emptyset}} {f^{(-k\vert N)}_{\emptyset,\lambda}}{f^{(-k\vert N)}_{\mu,\emptyset}} }{ {f^{(k\vert N)}_{\lambda,\mu}} {f^{(-k\vert N)}_{\mu,\lambda}} } &= \hskip-20pt \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k-{\frac{1}{2}}}} \hskip-20pt q^{-\lambda_i} \cdot \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k+{\frac{1}{2}}}} \hskip-15pt q^{\mu_i} \cdot \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\lambda}{\lambda^\vee_j-i+{\frac{1}{2}}\equiv k-{\frac{1}{2}}}} \hskip-15pt q^{-\mu_i} \cdot \prod_{\genfrac{}{}{0pt}{}{(i,j)\in\mu}{\mu^\vee_j-i+{\frac{1}{2}}\equiv N-k+{\frac{1}{2}}}} \hskip-20pt q^{\lambda_i} \cr &= \prod_{n\in{\mathbb{Z}}} \prod_{\genfrac{}{}{0pt}{}{i,j\in{\mathbb{Z}}}{j-i=k+nN}} q^{(\mu_{i-1}-\mu_{i})(\lambda_{j-1}-\lambda_{j})} \cr &= \prod_{\genfrac{}{}{0pt}{}{i,j=1}{j-i\equiv k}}^N \prod_{n,m\geq 0} q^{(\mu_{i-1+nN}-\mu_{i+nN})(\lambda_{j-1+mN}-\lambda_{j+mN})}, \nonumber \end{align}\] which reduces to (?? ). In \(\frac{ {f^{(k\vert N)}_{\emptyset,\mu}}{f^{(k\vert N)}_{\lambda,\emptyset}} }{ {f^{(k\vert N)}_{\lambda,\mu}} }\) and \(\frac{ {f^{(-k\vert N)}_{\emptyset,\lambda}}{f^{(-k\vert N)}_{\mu,\emptyset}} }{ {f^{(-k\vert N)}_{\mu,\lambda}} }\), \(k\) should be \(0\leq k \leq N-1\) and \(1\leq k \leq N\), respectively. But \[\frac{ {f^{(0\vert N)}_{\emptyset,\lambda}}{f^{(0\vert N)}_{\lambda,\emptyset}} }{ {f^{(0\vert N)}_{\lambda,\lambda}} } = \frac{ {f^{(-N\vert N)}_{\emptyset,\lambda}}{f^{(-N\vert N)}_{\lambda,\emptyset}} }{ {f^{(-N\vert N)}_{\lambda,\lambda}} } = \left( \frac{ {f^{(0\vert N)}_{\emptyset,\lambda}}{f^{(0\vert N)}_{\lambda,\emptyset}} }{ {f^{(0\vert N)}_{\lambda,\lambda}} } \frac{ {f^{(-N\vert N)}_{\emptyset,\lambda}}{f^{(-N\vert N)}_{\lambda,\emptyset}} }{ {f^{(-N\vert N)}_{\lambda,\lambda}} }\right)^{\frac{1}{2}}, \nonumber\] which gives (?? ). 0◻

7.2.3 Step 3 : \(N^2\) product↩︎

We have

Here is a remark on (81 ) in the case of \(N=2\). It should read, if \(y_{a+1}^i = -y_a^i\), \[\sum_{\genfrac{}{}{0pt}{}{ a,b,i,j=0 }{ j-i \equiv \pm(b-a) }}^{1} y_a^i y_b^j = \sum_{a=0}^{1} \left(\sum_{i=0}^{1} y_{a\pm i}^i \right)^2 = 2 \left(\sum_{i=0}^{1} y_{i}^i \right)^2. \nonumber\]

7.2.4 Final step↩︎

By using relations (72 ), (73 ), (71 ) and Lemma 38, we obtain \[\begin{align} { Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) } &= { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) } q^{\frac{1}{4} \sum_{i=1}^{N} \left(\sum_{j=1}^{N}\left( {\vert \lambda^{(j)}\vert_{i-j} }-{\vert \lambda^{(j)}\vert_{1+i-j} } \right)\right)^2} \prod_{i,j=1}^N \left( \frac{ {Q^{a}_{i,j}} {Q^{c}_{j,i-1}} }{ {Q^{b}_{i,j}} {Q^{b}_{j,i-1}} } \right)^{{\frac{1}{2}}{\vert \lambda^{(j)}\vert_{i-j}}} \cr &= q^{{\frac{1}{2}}\Delta} { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }\prod_{i,j=1}^N \left( \frac{ {Q^{a}_{i,j}} {Q^{c}_{j,i-1}} }{ {Q^{b}_{i,j}} {Q^{b}_{j,i-1}} } \right)^{{\frac{1}{2}}{\vert \lambda^{(j)}\vert_{i-j}}} \end{align}\] and \[\begin{align} { Z^{{\rm Poch, pure}}_{{\vec{\lambda}}}({\vec{x}}) } &= { {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) } q^{\frac{1}{4} \sum_{i=1}^{N} \left(\sum_{j=1}^{N}\left( {\vert \lambda^{(j)}\vert_{i-j} }-{\vert \lambda^{(j)}\vert_{1+i-j} } \right)\right)^2} \prod_{i,j=1}^N \left(\frac{ 1 }{ {Q^{b}_{i,j}} {Q^{b}_{j,i-1}} } \right)^{{\frac{1}{2}}{\vert \lambda^{(j)}\vert_{i-j}}} \cr &= q^{{\frac{1}{2}}\Delta} { {\mathcal{Z}}^{{\rm pure}}_{{\vec{\lambda}}}({\vec{x}}) } \prod_{i,j=1}^N \left(\frac{ 1 }{ {Q^{b}_{i,j}} {Q^{b}_{j,i-1}} } \right)^{{\frac{1}{2}}{\vert \lambda^{(j)}\vert_{i-j}}} , \end{align}\] where we also used (66 ) and (67 ) for recasting the first line to the second.

Since (?? ) implies \[\frac{ {Q^{a}_{i,j}} {Q^{c}_{j,i-1}} }{ {Q^{b}_{i,j}} {Q^{b}_{j,i-1}} } = \frac{a_i}{b_i} \frac{b_{i-1}}{c_{i-1}}, \qquad \frac{ 1 }{ {Q^{b}_{i,j}} {Q^{b}_{j,i-1}} } = \frac{b_{i-1}}{b_{i}} q{ {\kappa} }, \nonumber\] we finally obtain Proposition 36.

7.3 Inversion symmetry↩︎

By using the partition function of Pochhammer type \({ Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) }\), we can rewrite the non-stationary \({\widehat{\mathfrak{gl}}_N}\) equation (?? ) without \(q\)-Borel transformation \(q^{ {\frac{1}{2}} \Delta}\). Suppose \(q^n\neq 1\) for any integer \(n\). Let us define the \(q\)-exponential without using the \(q\)-Pochhammer symbol by \[e_q\left(xq^{\frac{1}{2}}\right) := \exp\left(-\sum_{n=1}^\infty \frac{x^n}{n} \frac{1}{q^{\frac{n}{2}} -q^{-\frac{n}{2}} } \right), \qquad q\in{\mathbb{C}}^{\times},\] which is a formal power series in \(x\). Then it satisfies \(e_q(xq^{\frac{1}{2}}) e_{q^{-1}}(xq^{-\frac{1}{2}})=1\).

Let \(d_i := q\kappa {b_i}/{a_{i+1}}\), \(\overline{d}_i :={b_i}/{c_i}\), \[T := \prod_{i=1}^N \left(\frac{\kappa b_i}{b_{i+1}} \right)^{\vartheta_i}, \qquad \mathcal{S}:=\prod_{i=1}^N \left( \frac{q}{ d_i { { \mathchoice {\setbox 0=\displaystyle \overline{d}_{i} \rule[.3ex]{0pt}{\ht 0}\lower\dp 0\rule[-0ex]{0pt}{\dp 0}\box 0} {\setbox 0=\textstyle \overline{d}_{i} \rule[.3ex]{0pt}{\ht 0}\lower\dp 0\rule[-0ex]{0pt}{\dp 0}\box 0} {\setbox 0=\scriptstyle \overline{d}_{i} \rule[.3ex]{0pt}{\ht 0}\lower\dp 0\rule[-0ex]{0pt}{\dp 0}\box 0} {\setbox 0=\scriptscriptstyle \overline{d}_{i} \rule[.3ex]{0pt}{\ht 0}\lower\dp 0\rule[-0ex]{0pt}{\dp 0}\box 0} } } } \right)^{\vartheta_i}, \qquad \mathcal{T} := \prod_{i=1}^N \left(\frac{d_i}{q { { \mathchoice {\setbox 0=\displaystyle \overline{d}_{i} \rule[.3ex]{0pt}{\ht 0}\lower\dp 0\rule[-0ex]{0pt}{\dp 0}\box 0} {\setbox 0=\textstyle \overline{d}_{i} \rule[.3ex]{0pt}{\ht 0}\lower\dp 0\rule[-0ex]{0pt}{\dp 0}\box 0} {\setbox 0=\scriptstyle \overline{d}_{i} \rule[.3ex]{0pt}{\ht 0}\lower\dp 0\rule[-0ex]{0pt}{\dp 0}\box 0} {\setbox 0=\scriptscriptstyle \overline{d}_{i} \rule[.3ex]{0pt}{\ht 0}\lower\dp 0\rule[-0ex]{0pt}{\dp 0}\box 0} } } } \right)^{\vartheta_i},\] then \[{ Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) } = \left( q^{\Delta} \mathsf{T}/\mathcal{T} \right)^{\frac{1}{2}} { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }, \qquad \sum_{{\vec{\lambda}}} { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) } =(T\mathcal{S})^{\frac{1}{2}}\psi\] with \(\psi\) in Conjecture 6. Also let \[\mathcal{A}'_{C\pm} := \prod_{i=1}^{N} e_q \left(x_i \sqrt{ q \left({ d_i} /{\overline{d}_i }\right)^{\pm1} } \right), \qquad \mathcal{A}'_{*} := \mathcal{S}^{\frac{1}{2}} \mathcal{A}_{*} \mathcal{S}^{-\frac{1}{2}}, \qquad *=L,C,R,\] then \[\mathcal{A}'_C = \mathcal{A}'_{C+} \mathcal{A}'_{C-} , \qquad \mathcal{A}'_{R} = \;:\prod_{i=1}^N e_q\left(x_i \sqrt{q d_i \overline{d}_i} q^{\vartheta_i - \vartheta_{i-1}} \right)^{-1} :.\]

The non-stationary \({\widehat{\mathfrak{gl}}_N}\) equation \((\ref{eq:EigenEq})\) can be rewritten by \[\left(q^{\Delta} \mathsf{T}/\mathcal{T} \right)^{\frac{1}{2}} \mathcal{T}^{\frac{1}{2}} \mathcal{A}'_L \mathcal{A}'_{C-} \mathcal{A}'_{C+} \mathcal{A}'_R \mathcal{T}^{\frac{1}{2}} \left( q^{\Delta}\mathsf{T}/\mathcal{T} \right)^{\frac{1}{2}} \sum_{{\vec{\lambda}}} { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) } = \sum_{{\vec{\lambda}}} { {\mathcal{Z}}_{{\vec{\lambda}}}({\vec{x}}) }.\] Let \(\psi^{\rm Poch} := \sum_{{\vec{\lambda}}}{ Z^{\rm Poch }_{{\vec{\lambda}}}({\vec{x}}) = \left( q^{\Delta} \mathsf{T}/\mathcal{T} \right)^{\frac{1}{2}} (T\mathcal{S})^{\frac{1}{2}}\psi }\). Since \[\mathcal{A'}_{C-}^{-1} = \overline{ \mathcal{A}'_{C+} },\qquad \mathcal{A'}_L^{-1} = \overline{ \mathcal{A}'_R },\] we have

Note that ?? is also equivalent to the equation where \(C+\) is replaced by \(C-\).

8 Instanton expansion with mass truncation↩︎

Let us examine the instanton expansion of the partition function with mass parameter truncation. For the \(\widehat{\mathfrak{gl}}_3\) case, the truncation condition is \[d_1 = q^{-m_1}, \qquad d_2 = q^{-m_2}, \qquad d_3 = q^{-m_3}.\] Recall that the partition function is a summation over triplets of the Young diagrams \(\vec{\lambda} = (\lambda^{(1)}, \lambda^{(2)}, \lambda^{(3)})\). After the mass parameter truncation, the summation is restricted to the triplets such that the length of the first row of \(\lambda^{(i)}\) is at most \(m_i\). Set \(M:= m_1 + m_2 + m_3\). In the main text we argued that the rank of the \(q\)-difference equation for the partition function is \(\frac{1}{2}(M+1)(M+2)\) which depends only on the sum of \((m_1, m_2, m_3)\). For each column of the Young diagram \(\lambda^{(i)}\) we define its shifted residue by \(((\lambda^{(i)})^\vee_k +i -1), k=1, \ldots m_i\), where \((\lambda^{(i)})^\vee_k\) is the length of the \(k\)-th column and \((\bullet)\) means the residue of the integer module \(3\). When the coloring of \(\lambda^{(i)}\) is such that the color of the first row is \(i\), the shifted residue agrees with the color (the number) of the end box of each column.

8.1 The case \(M=2\)↩︎

The rank of the \(q\)-difference system is \(6\).

1. \((m_1, m_2, m_3) = (2,0,0)\); There are 9 possibilities of the shifted resides of the first two columns of the first Young diagram.

   In the table below, \(r_k\) is the number of columns with the shifted residue \(k\) (see Subsection 4.3). Note that \(r_0+r_1+r_2=2=M\) and the set of possible \((r_0, r_1,r_2)\) has \(\frac{1}{2}(M+1)(M+2)\) elements, which agrees with the rank of the truncated \(q\)-difference system. We have introduced \(z_1=x_1, z_2= x_1x_2\) and \(z_3=x_1x_2x_3 \equiv 1\). Then the Young diagram with \((r_0, r_1, r_2)\) contributes to \(z_1^{r_1} z_2^{r_2} z_3^{r_0}\), which is a monomial in \(z_1\) with homogeneous degree \(M=2\).

   Table 1: No caption

   Shifted residues	Contribues to	\((r_0, r_1, r_2)\)	homogeneous monomial
   \((1,1)\)	\(x_1^2 \Lambda^k\)	\((0,2,0\))	\(z_1^2\)
   \((2,1)\)	\(x_1^2 x_2 \Lambda^k\)	\((0,1,1\))	\(z_1 z_2\)
   \((0,1)\)	\(x_1 \Lambda^k\)	\((1,1,0\))	\(z_1 z_3\)
   \((1,2)\)	\(x_1^2 x_2 \Lambda^k\)	\((0,1,1\))	\(z_1 z_2\)
   \((2,2)\)	\(x_1^2 x_2^2 \Lambda^k\)	\((0,0,2\))	\(z_2^2\)
   \((0,2)\)	\(x_1 x_2 \Lambda^k\)	\((1,0,1\))	\(z_2 z_3\)
   \((1,0)\)	\(x_1 \Lambda^k\)	\((1,1,0\))	\(z_1 z_3\)
   \((2,0)\)	\(x_1 x_2 \Lambda^k\)	\((1,0,1\))	\(z_2 z_3\)
   \((0,0)\)	\(\Lambda^k\)	\((2,0,0\))	\(z_3^2\)

2. \((m_1, m_2, m_3) = (1,1,0)\); There are 9 possibilities of the shifted resides of the first columns of two Young diagrams. The Young diagram with \((r_0, r_1, r_2)\) contributes to \(x_1^{-1} z_1^{r_1} z_2^{r_2} z_3^{r_0}\). Compared with the first case, the monomials are uniformly shifted by \(x_1^{-1}\).

   Table 2: No caption

   Shifted residues	Contribues to	\((r_0, r_1, r_2)\)	homogeneous monomial
   \((1;2)\)	\(x_1 x_2 \Lambda^k\)	\((0,1,1\))	\(x_1^{-1} z_1 z_2\)
   \((2;2)\)	\(x_1 x_2^2 \Lambda^k\)	\((0,0,2\))	\(x_1^{-1} z_2^2\)
   \((0;2)\)	\(x_2 \Lambda^k\)	\((1,0,1\))	\(x_1^{-1} z_2 z_3\)
   \((1;0)\)	\(\Lambda^k\)	\((1,1,0\))	\(x_1^{-1} z_1 z_3\)
   \((2;0)\)	\(x_2 \Lambda^k\)	\((1,0,1\))	\(x_1^{-1} z_2 z_3\)
   \((0;0)\)	\(x_1^{-1} \Lambda^k\)	\((2,0,0\))	\(x_1^{-1} z_3^2\)
   \((1;1)\)	\(x_1 \Lambda^k\)	\((0,2,0\))	\(x_1^{-1} z_1^2\)
   \((2;1)\)	\(x_1 x_2 \Lambda^k\)	\((0,1,1\))	\(x_1^{-1} z_1 z_2\)
   \((0;1)\)	\(\Lambda^k\)	\((1,1,0\))	\(x_1^{-1} z_1 z_3\)

Other four cases \((m_1, m_2, m_3)=(0,2,0),(0,0,2), (1,0,1), (0,1,1)\) are obtained by the cyclic permutation of \((x_1, x_2, x_3)\).

8.2 The case \(M=3\)↩︎

The rank of the \(q\)-difference system is \(10\). There are \(10\) possibilities of \((m_1, m_2,m_3)\), which coincides with the rank. They are \((3,0,0)\) and its cyclic permutations (3 cases), \((2,1,0)\) and its permutations (6 cases) and \((1,1,1)\). In each case, there are \(3^3 = 27\) possibilities of the three shifted residues, which are classified according to \((r_0, r_1,r_2)\) with \(r_0+r_1+r_2=3\) as follows;

1. \((m_1,m_2,m_3)=(3,0,0)\); The Young diagrams with \((r_0, r_1, r_2)\) contribute to \(z_1^{r_1} z_2^{r_2} z_3^{r_0} \equiv z_1^{r_1} z_2^{r_2}\).

2. \((m_1,m_2,m_3)=(2,1,0)\); The Young diagrams with \((r_0, r_1, r_2)\) contribute to \(x_1^{-1} z_1^{r_1} z_2^{r_2} z_3^{r_0} \equiv z_1^{r_1-1} z_2^{r_2}\).

3. \((m_1,m_2,m_3)=(1,1,1)\); The Young diagrams with \((r_0, r_1, r_2)\) contribute to \(x_1^{-2} x_2^{-1}z_1^{r_1} z_2^{r_2} z_3^{r_0} \equiv z_1^{r_1-1} z_2^{r_2-1}\).

The allowed terms in the Laurent polynomial in \((x_1, x_2)\) are plotted in Figure 2. The fundamental triangle for the case \(\mathbf{m} = (3,0,0)\) has the vertices \((0,0), (3,0), (3,3)\). The triangle for the general case \(\mathbf{m} = (m_1,m_2,m_3)\) is obtained from the fundamental triangle by \(-(m_2+m_3)\)-shift in \(x_1\) direction and \(-m_3\)-shift in \(x_2\) direction. We also note that these vertices come from the Young diagrams whose shifted resides are the same, namely \((r_0, r_1, r_2) = (3,0,0), (0,3,0), (0,0,3)\).

8.3 General \(M\)↩︎

From the above examples we now see the general rule for the possible terms of the partition function after the mass truncation. In the case \(\mathbf{m}= (M,0,0)\) the vertices of the triangle is \((0,0), (M,0)\) and \((M,M)\). In general case the three vertices are determined by considering the Young diagrams with \(\mathbf{r} = (M,0,0), (0,M,0)\) and \((0,0,M)\). Each case gives the following contribution;

1. \(\mathbf{r} = (M,0,0)\); the shifted residues are \((~\overbrace{0, \ldots, 0}^{m_1}~;~\overbrace{0, \ldots, 0}^{m_2}~;~\overbrace{0, \ldots, 0}^{m_3}~)\), which gives the terms with \((x_2x_3)^{m_2} x_3^{m_3} \Lambda^{k}=x_1^{-m_2-m_3} x_2^{-m_3} \Lambda^{k+m_2+m_3}\).

2. \(\mathbf{r} = (0,M,0)\); the shifted residues are \((~\overbrace{1, \ldots, 1}^{m_1}~;~\overbrace{1, \ldots, 1}^{m_2}~;~\overbrace{1, \ldots, 1}^{m_3}~)\), which gives the terms with \((x_1)^{m_1} (x_3x_1)^{m_3} \Lambda^{k}=x_1^{m_1} x_2^{-m_3} \Lambda^{k+m_3}\).

3. \(\mathbf{r} = (0,0,M)\); the shifted residues are \((~\overbrace{2, \ldots, 2}^{m_1}~;~\overbrace{2, \ldots, 2}^{m_2}~;~\overbrace{2, \ldots, 2}^{m_3}~)\), which gives the terms with \((x_1x_2)^{m_1} (x_2)^{m_2} \Lambda^{k}=x_1^{m_1} x_2^{m_1+m_2} \Lambda^{k}\).

Hence the vertices are \((-m_2-m_3, -m_3), (m_1, -m_3)\) and \((m_1, m_1+m_2)\), We see that they are \((-m_2-m_3, -m_3)\)-shift of \((0,0), (M,0)\) and \((M,M)\). The boundary of the shifted triangle is \(x_1=m_1, x_2 = - m_3\) and \(x_2=x_1+m_2\). (See Figure 3).

8.4 Generalization to \({\widehat{\mathfrak{gl}}_N}\)↩︎

Now it is easy to figure out the combinatorics for \({\widehat{\mathfrak{gl}}_N}\) case. It is convenient to introduce the following coordinates; \[\label{z-coordinates} z_1 = x_1, \quad z_2 = x_1 x_2, \quad \ldots \quad z_{N-1}=x_1x_2 \cdots x_{N-1}, \quad z_N = x_1 \cdots x_N = \Lambda \equiv 1.\tag{82}\] We also introduce the fundamental \((N-1)\)-dimensional polyhedron \(\Delta^{(N-1)}\) in \((x_1, \ldots, x_{N-1})\) space. The vertices of \(\Delta^{(N-1)}\) are \(\mathbf{v}_k:= (\overbrace{M, \ldots, M}^{k}, \overbrace{0, \ldots, 0}^{N-1-k}), k=0,\cdots, N-1\). In terms of the coordinates 82 , these vertices correspond to \(z_N^M, z_1^M, \ldots, z_{N-1}^M\), respectively. By the Pascal’s relation \[\binom{N+M-1}{M} - \binom{N+M-2}{M-1} = \binom{N+M-2}{M},\] one can check by induction that the number of the lattice points in \(\Delta^{(N-1)}\) or on the boundary of \(\Delta^{(N-1)}\) is \(\binom{N+M-1}{N-1}\) as it should be. Another way to see it is to note that under the identification \(z_N \equiv 1\), the lattice points in \(\Delta^{(N-1)}\) are in one to one correspondence with the monomials in \(z_i\) with homogeneous degree \(M\).

• When the mass truncation is given by \(\mathbf{m} = (m_1, m_2, \ldots, m_N)\), we make the shift by \((-m_{i+1}- \cdots -m_N)\) in \(x_i\) coordinate to obtain the shifted polyhedron \(\Delta^{(N-1)}(\mathbf{m})\). The possible terms in the instanton expansion of the partition function correspond to the lattice points in \(\Delta^{(N-1)}(\mathbf{m})\) or on its boundary.

• If the \(N\)-tuple of the Young diagrams has the shifted residue \(\mathbf{r}=(r_0, r_1, \ldots, r_{N-1})\), it contribute to the coefficient of \(z_1^{r_1-m_2}z_2^{r_2-m_3} \cdots z_{N-1}^{r_{N-1}-m_N} z_N^{r_0}\). This follows from the fact that a column of \(\lambda^{(i)}\) has the shifted residue \(k\) gives the factor \(z_{i-1}^{-1} z_k\) up to a power of \(\Lambda = z_N\).

References↩︎

---

1. There is a variety of the \(K\)-theoretic Nekrasov partition functions on the affine Laumon space (see for example [5]). Among them we consider the partition function with fundamental matter multiplets.↩︎

2. When \(N=2\) this is empty.↩︎

3. The arguments of the left block \(\mathcal{A}_L\) are \(\check{x}_i\) and we have set \(\beta_i=1\) in the definition of \(\mathcal{A}_L\).↩︎

4. The factor \(1/N\) in the definition of \(q_2\) comes from the \(\mathbb{Z}_N\) orbifold action on \(z_2\), which is an effective way of introducing a surface defect at the divisor \(z_2=0\).↩︎

5. Note that \([\, \hat{x}_{k}\, ,\;\hat{x}_{\ell} \, ]=0\) for \(k \neq \ell \pm 1\) (mod \(N\)) and \([\, \hat{x}_{i+N-1} \cdots \hat{x}_{i+2}\hat{x}_{i+1} \, , \, \hat{x}_{i+j} \, ]=0\) for \(j\neq \pm 1\) (mod \(N\)).↩︎

6. We have fixed the scaling parameters as \(\alpha_i = d_i \overline{d}_i\) and \(\beta_i =1\) to write down the building block of the Hamiltonian.↩︎

7. \(\mu_i := d_i\) are the remaining mass parameters after the mass truncation. When \(N=2\) this should be compared with eq.(2.12) in [2].↩︎

8. These are the remaining mass parameters after the mass truncation.↩︎

9. We do not assume for example \(\Lambda = x_1 x_2 \cdots x_N\).↩︎

10. See Eqs.(13.49) and (13.50) in [7] (or Eqs.(19) and (20) in [12]).↩︎

11. A similar formula \(\sum_{{\boldsymbol{j}}} \Phi_q({\boldsymbol{i}}|{\boldsymbol{j}};b,c)\Phi_q({\boldsymbol{j}}|{\boldsymbol{k}};a,b)=\Phi_q({\boldsymbol{i}}|{\boldsymbol{k}};a,c)\) also seems to be true.↩︎

12. See eq.(13.51) in [7], where \((q^2, z,l,m)_{\rm there} =(q,q^{-|{\boldsymbol{c}}|/2}\Lambda, |{\boldsymbol{c}}|-M,M)_{\rm here}\).↩︎

13. In the present case we take \(\ell=m=M\). In [7] \(I_\ell, I_m\) are denoted by \(B_\ell, B_m\).↩︎

14. We have changed the notation \((n,m) \to (M, N)\) from [18].↩︎

15. Originally the orbifolded Nekrasov factor is defined in terms of the finite \(q\)-shifted factorial \([u;q]_n\). But to see the relation to the Jackson integral it is convenient to employ the \(t=\kappa^{-\frac{1}{N}}\)-shifted factorial \([u;t]_\infty\).↩︎

16. \([\ell]_N\) and \([m]_N\) denote the residue class modulo \(N\). Not to be confused with the \(q\)-integer.↩︎

17. There is also a single product part coming from the boundaries of semi-infinite regions.↩︎

18. Compare this with Theorem 4.1 of [2] . The original \(q\)-KZ equation is reproduced by considering the simultaneous \(t\)-shift \(\mathsf{a}_k \to t \mathsf{a}_k, \mathsf{b}_k \to t^{-1}\mathsf{b}_k\), which is equivalent to the \(t\)-shift of the Coulomb moduli \(b_k \to t^{-1} b_k\).↩︎

19. Recall that we have exchanged \(q\) and \(t\), compared with the original literature.↩︎

20. Precisely speaking the Pochhammer type factors are used in the Jackson integral, while we employ the sinh type factors in the definition of the affine Laumon function. They agree up to some monomial factor (See Appendix 7).↩︎

21. We changed \(s \to N\) and \(n \to M\) from [17].↩︎

22. Such a simple symmetry is specific to the Nekrasov factor of sinh type. But both \({ \mathsf{N}^{(k\vert N)}_{\lambda\mu}(\mathsf{v}Q\vert q, {\kappa} ) }\) and \({ N^{{\rm Poch }(k\vert N)}_{\lambda\mu}(\mathsf{v}Q\vert q, {\kappa} ) }\) satisfy the inversion formula of type \[{ N^{{\rm Poch }(k\vert N)}_{\lambda\mu}(\mathsf{v}Q\vert q, {\kappa} ) } = { N^{{\rm Poch }(-k-1\vert N)}_{\mu\lambda}(\mathsf{v}^{-1}Q\vert q^{-1}, {\kappa} ^{-1} ) } , \nonumber\] which follows directly from the definition.↩︎

23. The inverse square of the last factor of (68 ) \[q^{-{\frac{1}{2}} \left(\sum_{j=1}^{N}\left( {\vert \lambda^{(j)}\vert_{i-j} }-{\vert \lambda^{(j)}\vert_{1+i-j} } \right)\right)^2} b_{i}^{\sum_{j=1}^{N}\left( {\vert \lambda^{(j)}\vert_{i-j} }-{\vert \lambda^{(j)}\vert_{1+i-j} } \right) } { {\kappa^{ - { \vert \lambda^{(i)}\vert_{ }} }} }\] up to \(q^{ {\vert \lambda^{(i)}\vert_{ }} }\) is the same as the prefactor \(s_i^{-m_i} q^{-m_i^2/2} \kappa^{-|\lambda^{(i)}|}\) of (12) in [34]. Here \(s_i := 1/b_i\) and \(m_i := \sum_{j=1}^{N}\left( {\vert \lambda^{(j)}\vert_{i-j} }-{\vert \lambda^{(j)}\vert_{1+i-j} } \right)\).↩︎

24. For the Nekrasov partition function without surface defect, the \(f\)-factors corresponding to (?? ) and (?? ) are equal to \(1\) and the \(g\)-factors corresponding to (?? ) and (?? ) are given by replacing \({\vert \lambda\vert_{k} }\) with \({\vert \lambda\vert_{}}\). Thus the Nekrasov partition function without surface defect satisfies (?? ) without \(q\)-Borel transformation \(q^{{\frac{1}{2}}\Delta}\) and by replacing \({\vert \lambda\vert_{k} }\) with \({\vert \lambda\vert_{}}\).↩︎