Q-operators for the Ruijsenaars model
Eric Rains, Hjalmar Rosengren
TL;DR
This work proves that the Ruijsenaars model possesses a one-parameter family of commuting Q-operators $Q_c$, with commutativity equivalent to a elliptic hypergeometric transformation conjectured by GRRY and linked to S-duality in quiver gauge theories. It provides two rigorous proofs: one showing that elliptic Macdonald polynomials are joint eigenfunctions of the Ruijsenaars operators and the $Q_c$, thereby deducing commutativity, and another deriving the same result from a transformation of elliptic hypergeometric integrals. The paper also clarifies how Noumi–Sano operators arise as degenerations of the $Q$-operators under gauge and parameter limits, unifying four operator families through elliptic hypergeometric identities. Overall, the results forge a deep link between integrable quantum systems, elliptic special functions, and dualities in supersymmetric gauge theories, offering robust tools for studying joint eigenfunctions and operator algebras in the elliptic regime.
Abstract
We prove that the Ruijsenaars model admits a one-parameter commuting family of Q-operators. The commutativity is equivalent to an elliptic hypergeometric integral transformation that was conjectured by Gadde et al., and has an alternative interpretation in terms of S-duality for quiver gauge theories. We present two proofs of this conjecture, one using the elliptic Macdonald polynomials of Langmann et al., and one using known results on elliptic hypergeometric integrals. We also explain how the Noumi-Sano operators appear as degenerations of Q-operators.