Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation

TL;DR

This paper connects Shakirov's non-stationary difference equation with the quantum KZ equation for affine quantum groups, showing that affine Laumon space partition functions solve these equations and exploring related dualities and limits.

Contribution

It demonstrates that the affine Laumon space K-theoretic partition function solves the quantum KZ equation and confirms a conjecture linking Shakirov's equation to affine Laumon space.

Findings

The Hamiltonian matches the R-matrix and quantum 6j symbols.

Affine Laumon partition function solves the q-KZ equation.

Established base-fiber duality and four-dimensional limit relations.

Abstract

We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik-Zamolodchikov ($q$-KZ) equation for $U_{\mathsf v}\bigl(A_1^{(1)}\bigr)$ with generic spins. Namely, we 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 $6j$ symbols. On the other hand, we 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, we establish that the affine Laumon partition function gives a solution to Shakirov's equation, which was a conjecture in our previous paper. We also work out the base-fiber duality and four-dimensional limit in relation with the $q$-KZ equation.