Higher Order Deformed Elliptic Ruijsenaars Operators

Abstract

We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a rigorous proof of the quantum integrability of the deformed Ruijsenaars model.

Conformal field theory, solitons, and elliptic Calogero--Sutherland models

“…Since the deformed eCS model is a limiting case of the deformed elliptic Ruijsenaars model, the results in Ref. [33] strongly suggest that the deformed eCS model is quantum integrable. However, the limit from the Ruijsenaars models to CS models is delicate and, for this reason, it requires non-trivial work to deduce from the quantum integrability of the former the quantum integrability of the latter (see [34,Section 4.3]); to our knowledge, this work has not been done for the deformed Ruijsenaars model.…”

“…The eCS model has a relativistic quantum integrable generalization known as the elliptic Ruijsenaars model [29]. There exists a deformed elliptic Ruijsenaars model which is quantum integrable as well [33]. Since the deformed eCS model is a limiting case of the deformed elliptic Ruijsenaars model, the results in Ref.…”

Wall-crossing for vortex partition function and handsaw quiver varierty

“…Furthermore, we obtain geometric interpretations of formula for multiple hypergeometric functions. They are considered as rational limits of the Kajihara transformation [12] and formula obtained in Langer-Schlosser-Warnaar [14] and Hallnäs-Langmann-Noumi-Rosengren [4], [5] from various contexts ( see §2.4).…”