Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 077, 55 pages      arXiv:2309.15364      https://doi.org/10.3842/SIGMA.2024.077

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

Hidetoshi Awata a, Koji Hasegawa b, Hiroaki Kanno ac, Ryo Ohkawa de, Shamil Shakirov fg, Jun'ichi Shiraishi h and Yasuhiko Yamada i
a) Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
b) Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
c) Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan
d) Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, Osaka 558-8585, Japan
e) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
f) University of Geneva, Switzerland
g) Institute for Information Transmission Problems, Moscow, Russia
h) Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan
i) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received November 06, 2023, in final form August 07, 2024; Published online August 22, 2024

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.

Key words: affine Laumon space; quantum affine algebra; non-stationary difference equation; quantum Knizhnik-Zamolodchikov equation.

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