Non-stationary difference equation and affine Laumon space. II: Quantum Knizhnik-Zamolodchikov equation (Q6614601)

The quantum (or \(q\)-deformed) Knizhnik-Zamolodchikov (\(q\)-KZ) equation typically arises as a linear difference equation satisfied by matrix elements of the products of intertwining operators of highest weight representations of quantum affine algebra [\textit{I. B. Frenkel} and \textit{N. Yu. Reshetikhin}, Commun. Math. Phys. 146, No. 1, 1--60 (1992; Zbl 0760.17006)]. In this paper, the authors show that Shakirov's non-stationary difference equation, when it is truncated, implies the \(q\)-KZ equation for \(U_v(A^{(1)}_1\)) with generic spins. Namely, they can tune mass parameters so that the hamiltonian acts on the space of finite Laurent polynomials. Then the representation matrix of the hamiltonian agrees with the \(R\)-matrix, or the quantum 6\(j\) symbols. On the other hand, they prove that the \(K\) theoretic Nekrasov partition function from the affine Laumon space is identified with the well-studied Jackson integral solution to the \(q\)-KZ equation. Combining these results, the authors establish that the affine Laumon partition function gives a solution to Shakirov's equation, which was a conjecture in their previous paper [\textit{H. Awata} et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 19, Paper 089, 47 p. (2023; Zbl 1528.14039)] (they show that the non-stationary difference equation proposed in [\textit{Sh. Shakirov}, Lett. Math. Phys. 114, No. 5, Paper No. 115, 25 p. (2024; Zbl 07930035)] is transformed into a quantization of the discrete Painlevé VI equation). They also work out the base-fiber duality and four-dimensional limit in relation with the \(q\)-KZ equation. \N\NThe paper is organized as follows. Section 1, is an introduction to the subject. Section 2 deals with Shakirov's equation as a \(q\)-KZ equation with generic spins. Here, the authors introduce a truncation of a non-stationary difference equation by tuning two of mass parameters among four (called mass truncation). After the mass truncation the hamiltonian of equation acts on the space of Laurent polynomials. They find that the matrix elements satisfy the same relation as the quantum 6\(j\)-symbol derived by \textit{H. Rosengren} [Ramanujan J. 13, No. 1--3, 131--166 (2007; Zbl 1125.33015)]. Section 3 is devoted to Jackson integral representation of solutions to the \(q\)-KZ equation. The authors review the \(q\)-KZ equations satisfied by the Jackson integral which is a paring of an integration cycle and a cocycle function. They also study a special integration cycle which reduces the bilateral sum over the lattice in the definition of the Jackson integral to a sum over a positive cone of the lattice. This truncation (called lattice truncation) has been considered in literature. The relevant cycles are called the characteristic cycles or the \(\alpha\)-stable or \(\alpha\)-unstable cycles by Aomoto, and play crucial roles in the study of asymptotic behavior. The lattice truncation is crucial, when they make an identification of the Jackson integral with the Nekrasov partition function which is expressed as a sum over a pair of Young diagrams. \N\NSection 4 deals with Nekrasov partition function as Jackson integral. The authors consider the Nekrasov partition function with a surface defect, which can be obtained from the equivariant character at fixed points of the toric action on the affine Laumon space. They show that the expansion of the partition function in \(x\) is in accord with a basis, which is called Matsuo basis, of the cocycle functions in the Jackson integral representation of solutions to the \(q\)-KZ equation. Section 5 is devoted to four-dimensional limit and KZ equation. Here, the authors compute the four-dimensional limit of the \(R\)-matrix that appears in the truncation of the non-stationary difference equation and show that the limit correctly reproduces the KZ equation for the conformal block of the \(sl_2\) current algebra. \N\NThe paper is supported by some appendices where some helpful clarifications and technical details are presented. These appendices cover the following : \(q\)-Borel transformation and refined Chern-Simons theory, a lemma on the \(q\)-Borel transformation, a list of various \(R\) matrices and their characterization, a direct proof on a commutativity relation of Ito's matrices, truncation by tuning mass parameters, affine Laumon space and orbifolded Nekrasov factor, and finally (anti-)symmetrization in a factorized form, Shakirov's equation as a coupled system.