Quantum Trigonometric Spin Ruijsenaars-Schneider Models from $K$-theoretic Coulomb Branches
arXiv:2607.28043
The paper quantizes the trigonometric spin Ruijsenaars‑Schneider model using the K‑theoretic Coulomb branch of a 4d N=2 necklace quiver gauge theory, constructing an algebra of L‑operators that yields commuting Hamiltonians and a quantum spin chain description.
Abstract
We quantize the trigonometric spin Ruijsenaars-Schneider model of $N$ particles each with $\ell$ spin states using the recently developed description of the classical model in terms of the $K$-theoretic Coulomb branch of the 4d $\mathcal{N}=2$ quiver gauge theory for the necklace quiver with $\ell$ nodes of rank $N$. The main algebraic tool is an algebra of $L$-operators derived from abelianized monopole operators of minuscule charge, which turns the necklace quiver into an integrable spin chain by producing a family of commuting Hamiltonians. We show that the lowest Hamiltonian coincides with the first mode of the quantum determinant of the horizontal quantum loop algebra living inside the $K$-theoretic Coulomb branch algebra, whose Bethe subalgebra generates a maximal family of commuting Hamiltonians. Finally, we derive the commutation relations and quantum equations of motion of the quantized physical spin variables.
17 pages, 3 figures, 1 table