We compare the spectrum of the elliptic Ruijsenaars–Schneider model with the finite-size spectrum of the sine-Gordon model, highlighting both their similarities and differences. Our analysis focuses on the two-particle sector in the center-of-mass frame. At the free point, we carry out an analytic comparison, while at generic couplings we employ non-perturbative numerical calculations based on the truncated Hilbert space method adapted to difference operators. To benchmark this numerical approach, we first study the trigonometric limit, where analytic results are available. We then compare the Bethe–Yang momentum quantization conditions, derived from infinite-volume scattering phases, with the exact finite-volume solutions. We find that these conditions hold exactly in the rational and trigonometric cases, but acquire finite-size corrections in the hyperbolic and elliptic cases, however, these corrections are different in quantum field theories and in many body systems.

Hungary

Poland

Zoltan Bajnok

Apor Roth

Comparing the elliptic Ruijsenaars–Schneider model to the finite volume sine-Gordon theory

We discuss two physics-inspired approaches to derivation of the eigenfunctions and eigenvalues of $\textit{A}$ Ruijsenaars-Schneider model. First approach which was recently proposed by the authors relies on the computations of superconformal indices of class $$ \mathcal{S} $$ 4$\textit{d}$ $$ \mathcal{N} $$ = 2 theories with the insertion of surface defects. Second approach uses computations of Nekrasov-Shatashvili limit of 5$\textit{d}$ $$ \mathcal{N} $$ = 1 instanton partition functions in the presence of co-dimension two defect. We compare results of these two approaches for the low-lying levels of Ruijsenaars-Schneider model. We also discuss different previously proposed exact quantization conditions for the Coulomb branch parameters of the instanton partition functions and their interpretations in terms of index calculations.

United States

South Korea

Hee-Cheol Kim

United Kingdom

Anton Nedelin

Israel

Shlomo Razamat

On Ruijsenaars-Schneider spectrum from superconformal indices and ramified instantons

The Ruijsenaars-Schneider models are integrable dynamical realizations of the Poincaré group in 1 + 1 dimensions, which reduce to the Calogero and Sutherland systems in the nonrelativistic limit. In this work, a possibility to construct a one-parameter deformation of the Ruijsenaars-Schneider models by uplifting the Poincaré algebra in 1 + 1 dimensions to the anti de Sitter algebra is studied. It is shown that amendments including a cosmological constant are feasible for the rational variant, while the hyperbolic and trigonometric systems are ruled out by our analysis. The issue of integrability of the deformed rational model is discussed in some detail. A complete proof of integrability remains a challenge.

Russia

Anton Galajinsky

Rational Ruijsenaars-Schneider model with cosmological constant

Integrability of  $$\mathcal{N}$$  = 1 supersymmetric Ruijsenaars-Schneider three-body models based upon the potentials  $$W\left(x\right)=\frac{2}{x}$$ ,  $$W\left(x\right)=\frac{2}{{\text{sin}}x}$$ , and  $$W\left(x\right)=\frac{2}{{\text{sinh}}x}$$  is proven. The problem of constructing an algebraically resolvable set of Grassmann-odd constants of motion is reduced to finding a triplet of vectors such that all their scalar products can be expressed in terms of the original bosonic first integrals. The supersymmetric generalizations are used to build novel integrable (iso)spin extensions of the respective Ruijsenaars-Schneider three-body systems.

Remarks on integrability of  $$\mathcal{N}$$  = 1 supersymmetric Ruijsenaars-Schneider three-body models

An   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric extension of the Ruijsenaars-Schneider three-body model is constructed and its integrability is established. In particular, three functionally independent Grassmann-odd constants of the motion are given and their algebraic resolvability is proven. The supersymmetric generalization is used to build a novel integrable isospin extension of the Ruijsenaars-Schneider three-body system.

Integrability of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric Ruijsenaars-Schneider three-body system

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of L-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical elliptic spin chain. In this way, one obtains a lattice field analogue of the Ruijsenaars-Schneider model with continuous time. The second method is based on investigation of general elliptic families of solutions to the 2D Toda equation. We derive equations of motion for their poles, which turn out to be difference equations in space with a lattice spacing η, together with a zero curvature representation for them. We also show that the equations of motion are Hamiltonian. The obtained system of equations can be naturally regarded as a field generalization of the Ruijsenaars-Schneider system. Its lattice version coincides with the model introduced via the first method. The limit η → 0 is shown to give the field extension of the Calogero-Moser model known in the literature. The fully discrete version of this construction is also discussed.

A. Zabrodin

A. Zotov

Field analogue of the Ruijsenaars-Schneider model

We construct an <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supersymmetric extension of n-particle Ruijsenaars–Schneider models. The guiding feature is a deformation of the phase space. The supercharges have a “free” form linear in the fermions but produce an interacting four-fermion Hamiltonian. A field-dependent unitary transformation maps to standard fermions obeying conventional Poisson brackets. In this frame, the supercharges and Hamiltonian have long “fermionic tails”. We also comment on previous attempts in this direction.

JINR

Sergey Krivonos

Germany

Olaf Lechtenfeld

On supersymmetric Ruijsenaars–Schneider models

We conjecture the quantum analogues of the classical trace formulae for the integrals of motion of the quantum hyperbolic Ruijsenaars-Schneider model. This is done by departing from the classical construction where the corresponding model is obtained from the Heisenberg double by the Poisson reduction procedure. We also discuss some algebraic structures associated to the Lax matrix in the classical and quantum theory which arise upon introduction of the spectral parameter.

Gleb Arutyunov

Rob Klabbers

Enrico Olivucci

Quantum trace formulae for the integrals of the hyperbolic Ruijsenaars-Schneider model

We propose and test exact quantization conditions for the N-particle quantum elliptic Ruijsenaars-Schneider integrable system, as well as its Calogero-Moser limit, based on the conjectural correspondence to the five-dimensional   <math> <mi>N</mi> <mo>=</mo> <msup> <mn>1</mn> <mo>∗</mo> </msup> </math>  $$ \mathcal{N}={1}^{\ast } $$  SU(N) gauge theory in the Nekrasov-Shatashvili limit. We discuss two natural sets of quantization conditions, related by the electro-magnetic duality, and the importance of non-perturbative corrections in the Planck constant. We also comment on the eigenfunction problem, by reinterpreting the Separation of Variables approach in gauge theory terms.

Japan

Yasuyuki Hatsuda

Antonio Sciarappa

Switzerland

Szabolcs Zakany

Exact quantization conditions for the elliptic Ruijsenaars-Schneider model

The Ruijsenaars-Schneider models are conventionally regarded as relativistic generalizations of the Calogero integrable systems. Surprisingly enough, their supersymmetric generalizations escaped attention. In this work, N = 2 supersymmetric extensions of the rational and hyperbolic Ruijsenaars-Schneider three-body models are constructed within the framework of the Hamiltonian formalism. It is also known that the rational model can be described by the geodesic equations associated with a metric connection. We demonstrate that the hyperbolic systems are linked to non-metric connections.