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

SIGMA 21 (2025), 107, 48 pages      arXiv:2507.09568      https://doi.org/10.3842/SIGMA.2025.107

A-Type Open ${\rm SL}(2,\mathbb{C})$ Spin Chain

Pavel V. Antonenko ab, Sergey É. Derkachov a and Pavel A. Valinevich a
a) Saint-Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
b) Leonhard Euler International Mathematical Institute, Pesochnaya nab. 10, 197022 St. Petersburg, Russia

Received August 13, 2025, in final form December 08, 2025; Published online December 21, 2025

Abstract
For the noncompact open ${\rm SL}(2, \mathbb{C})$ spin chain, the eigenfunctions of the special matrix element of monodromy matrix are constructed. The key ingredients of the whole construction are local Yang-Baxter $\mathcal{R}$-operators, $Q$-operator and raising operators obtained by reduction from the $Q$-operator. The calculation of various scalar products and the proof of orthogonality are based on the properties of $Q$-operator and demonstrate its hidden role. The symmetry of eigenfunctions with respect to reflection of the spin variable $s \to 1-s$ is established. The Mellin-Barnes representation for eigenfunctions is derived and equivalence with initial coordinate representation is proved. The transformation from one representation to another is grounded on the application of $A$-type Gustafson integral generalized to the complex field.

Key words: open spin chain; principal series representations; Mellin-Barnes integrals.

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