Non-stationary difference equation and affine Laumon space III : Generalization to $\widehat{\mathfrak{gl}}_N$

TL;DR

The paper generalizes the non-stationary difference equation to the affine algebra ${\widehat{\mathfrak{gl}}}_N$, formulating a Hamiltonian in terms of $q$-commuting variables and proving the equivalence of multiple operator realizations via the pentagon identity and $q$-binomial theorem. It conjectures that the affine Laumon partition function of type $A_{N-1}^{(1)}$ provides a solution to this equation, with the ${\widehat{\mathfrak{gl}}}_2$ case established previously; in the mass-truncation limit the equation reduces to the finite-dimensional $R$-matrix of $U_q(A_{N-1}^{(1)})$, connected to the tetrahedron $R$-matrix through a key building block $\Phi_q$. The work also casts the AL partition function as a Jackson integral, derives the exchange symmetries of Nekrasov factors, and constructs a cocycle basis that captures the truncated theory’s solution space. In the four-dimensional limit, the framework recovers the Fuji-Suzuki-Tsuda system, aligning with known isomonodromic differential systems and supporting the conjectured dualities between higher-rank quantum groups and $q$-KZ structures. Overall, the paper links non-stationary $q$-difference equations, affine Laumon spaces, and finite $R$-matrices, offering a unified view of higher-rank isomonodromic-like dynamics in the quantum setting.

Abstract

In a series of papers we have considered a non-stationary difference equation which was originally discovered for the deformed Virasoro conformal block. The equation involves mass parameters and, when they are tuned appropriately, the equation is regarded as a quantum KZ equation for $U_q(A_{1}^{(1)})$. We introduce a $\widehat{\mathfrak{gl}}_N$ generalization of the non-stationary difference equation. The Hamiltonian is expressed in terms of $q$-commuting variables and allows both factorized forms and a normal ordered form. By specializing the mass parameters appropriately, the Hamiltonian can be identified with the $R$-matrix of the symmetric tensor representation of $U_q(A_{N-1}^{(1)})$, which in turn comes from the 3D (tetrahedron) $R$-matrix. We conjecture that the affine Laumon partition function of type $A_{N-1}^{(1)}$ gives a solution to our $\widehat{\mathfrak{gl}}_N$ non-stationary difference equation. As a check of our conjecture, we work out the four dimensional limit and find that the non-stationary difference equation reduces to the Fuji-Suzuki-Tsuda system.

Figures (3)

• Figure 1: The shifted residue $(\ell_I)$ for the column with label $I=(i,m)$ agrees with the number in the end box of the column, namely $(2,0,2,1,1;2,1,2,2;2,1)$ for the above case.
• Figure 2: After the mass truncation $d_i=q^{-m_i}$, the affine Laumon partition function becomes a Laurent polynomial in $(x_1, x_2)$, while it is still a formal power series in $\Lambda$. The circles represent the positions of allowed terms in the $(x_1,x_2)$-lattice. The total number of the circles agrees with the rank of the truncated $q$-difference equation.
• Figure 3: The triangle on the $(x_1, x_2)$ lattice which indicates possible terms in the instanton expansion after the mass truncation by $\mathbf{m}=(m_1,m_2,m_3)$.

Theorems & Definitions (50)

• Definition 1.1: Non-stationary ${\widehat{\mathfrak{gl}}_N}$ Hamiltonian
• Definition 1.2: Factorized form of simple root type
• Definition 1.3: Factorized form of higher root type
• Definition 1.4: Normal ordered form
• Definition 1.5: Affine Laumon partition function
• Conjecture 1.6
• Proposition 2.1: Kirillov
• Lemma 2.2
• proof
• Proposition 2.3