Ruijsenaars-Schneider Model Research Articles

Ground state wavefunctions of elliptic relativistic integrable Hamiltonians

We derive ground state eigenfunctions and eigenvalues of various relativistic elliptic integrable models. The models we discuss appear in computations of superconformal indices of four-dimensional theories obtained by compactifying six-dimensional models on Riemann surfaces. These include, among others, the Ruijsenaars-Schneider model and the van Diejen model. The derivation of the eigenfunctions builds on physical inputs, such as conjectured Lagrangian across dimensions IR dualities and assumptions about the behavior of the indices in the limit of compactifications on surfaces with large genus/number of punctures/flux.

Integrable Multi-Hamiltonian Systems from Reduction of an Extended Quasi-Poisson Double of {text {U}}(n)

We construct a master dynamical system on a {text {U}}(n) quasi-Poisson manifold, {mathcal {M}}_d, built from the double {text {U}}(n) times {text {U}}(n) and dge 2 open balls in mathbb {C}^n, whose quasi-Poisson structures are obtained from T^* mathbb {R}^n by exponentiation. A pencil of quasi-Poisson bivectors P_{underline{z}} is defined on {mathcal {M}}_d that depends on d(d-1)/2 arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the {text {U}}(n)-invariant functions. The master system on {mathcal {M}}_d is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle T^*!{text {U}}(n) times mathbb {C}^{ntimes d}. Its commuting Hamiltonians are pullbacks of the class functions on one of the {text {U}}(n) factors. We prove that the master system descends to a degenerate integrable system on a dense open subset of the smooth component of the quotient space {mathcal {M}}_d/{text {U}}(n) associated with the principal orbit type. Any reduced Hamiltonian arising from a class function generates the same flow via any of the compatible Poisson structures stemming from the bivectors P_{underline{z}}. The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of {text {U}}(n) provide a new real form of the complex, trigonometric spin Ruijsenaars–Schneider model of Krichever and Zabrodin. This generalizes the derivation of the compactified trigonometric RS model found previously in the d=1 case.

Poisson Reductions of Master Integrable Systems on Doubles of Compact Lie Groups

We consider three ‘classical doubles’ of any semisimple, connected and simply connected compact Lie group G: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of ‘master integrable systems’ and investigate their Poisson reductions. In the simplest cotangent bundle case, the reduction is defined by taking quotient by the cotangent lift of the conjugation action of G on itself, and this naturally generalizes to the other two doubles. In each case, we derive explicit formulas for the reduced Poisson structure and equations of motion and find that they are associated with well known classical dynamical r-matrices. Our principal result is that we provide a unified treatment of a large family of reduced systems, which contains new models as well as examples of spin Sutherland and Ruijsenaars–Schneider models that were studied previously. We argue that on generic symplectic leaves of the Poisson quotients the reduced systems are integrable in the degenerate sense, although further work is required to prove this rigorously.

Elliptic generalisation of integrable q-deformed anisotropic Haldane–Shastry long-range spin chain

We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Ruijsenaars–Macdonald operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range spin chain constructed by means of the elliptic Baxter-Belavin -matrix. Namely, we show that the freezing trick is reduced to a set of elliptic function identities, which are then proved. These identities can be treated as conditions for equilibrium position in the underlying classical spinless Ruijsenaars–Schneider model. Trigonometric degenerations are studied as well. For example, in M = 2 case our construction provides q-deformation for anisotropic XXZ Haldane–Shastry model. The standard Haldane–Shastry model and its Uglov’s q-deformation based on XXZ R-matrix are included into consideration by separate verification.

On the $$R$$-matrix identities related to elliptic anisotropic spin Ruijsenaars–Macdonald operators

We propose and prove a set of identities for ${\rm GL}_M$ elliptic $R$-matrix (in the fundamental representation). In the scalar case ($M=1$) these are elliptic function identities derived by S.N.M. Ruijsenaars as necessary and sufficient conditions for his kernel identity underlying construction of integral solutions to quantum spinless Ruijsenaars-Schneider model. In this respect the result of the present paper can be considered as the first step towards constructing solutions of quantum eigenvalue problem for the anisotropic spin Ruijsenaars model.

Lax equations for relativistic Gaudin models on elliptic curve

We describe the most general GL NM classical elliptic finite-dimensional integrable system, which Lax matrix has n simple poles on elliptic curve. For M = 1 it reproduces the classical inhomogeneous spin chain, for N = 1 it is the Gaudin type (multispin) extension of the spin Ruijsenaars–Schneider model, and for n = 1 the model of M interacting relativistic GL N tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on GL-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars–Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general GL NM model is also described through R-matrices satisfying associative Yang–Baxter equation. This description provides the trigonometric and rational analogues of GL NM models.

Field analogue of the Ruijsenaars-Schneider model

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.

Constrained Toda hierarchy and turning points of the Ruijsenaars–Schneider model

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^{\dag}$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities.

Quantum K-theory of quiver varieties and many-body systems

We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.

Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model

We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians which govern the dynamics of poles with respect to the m-th hierarchical times t_m and \bar t_m of the 2D Toda lattice hierarchy are obtained from expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points.

Superintegrable Systems on Moduli Spaces of Flat Connections

The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principal G-bundle on a surface. The moduli space is a Poisson variety with Atiyah–Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars–Schneider models.

A-type Quiver Varieties and ADHM Moduli Spaces

We study quantum geometry of Nakajima quiver varieties of two different types - framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of $A_n$ quiver varieties in a certain $n\to\infty$ limit reproduces equivariant K-theory of the Hilbert scheme of points on $\mathbb{C}^2$. We analyze the correspondence from the point of view of enumerative geometry, representation theory and integrable systems. We also propose a conjecture which relates spectra of quantum multiplication operators in K-theory of the ADHM moduli spaces with the solution of the elliptic Ruijsenaars-Schneider model.

$$({{\,\mathrm{\mathrm {SL}}\,}}(N),q)$$-Opers, the q-Langlands Correspondence, and Quantum/Classical Duality

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for $SL(N)$. We introduce a difference equation version of opers called $q$-opers and prove a $q$-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the $q$-Langlands correspondence. We also describe an application of $q$-opers to the equivariant quantum $K$-theory of the cotangent bundles to partial flag varieties.

Spin Ruijsenaars–Schneider Models from Reduction

Spin Ruijsenaars–Schneider model introduced by Krichever and Zabrodin is a generalisation of a finite-dimensional integrable system due to Ruijsenaars and Schneider, where particles carry additional internal degrees of freedom. We review how a Hamiltonian structure of the spin model with the hyperbolic potential can be obtained by means of Poisson reduction applied to a specially chosen Poisson manifold.

On [formula omitted] supersymmetric Ruijsenaars–Schneider models

We construct an N=2 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.

Hyperbolic Spin Ruijsenaars-Schneider Model from Poisson Reduction

We derive a Hamiltonian structure for the N-particle hyperbolic spin Ruijsenaars-Schneider model by means of Poisson reduction of a suitable initial phase space. This phase space is realised as the direct product of the Heisenberg double of a factorisable Lie group with another symplectic manifold that is a certain deformation of the standard canonical relations for Nl conjugate pairs of dynamical variables. We show that the model enjoys the Poisson-Lie symmetry of the spin group GLl(ℂ), which explains its superintegrability. Our results are obtained in the formalism of the classical r-matrix, and they are compatible with the recent findings on the different Hamiltonian structure of the model established in the framework of the quasi-Hamiltonian reduction applied to a quasi-Poisson manifold.

Induced Dynamics

Construction of new integrable systems and methods of their investigation is one of the main directions of development of the modern mathematical physics. Here we present an approach based on the study of behavior of roots of functions of canonical variables with respect to a parameter of simultaneous shift of space variables. Dynamics of singularities of the KdV and Sinh—Gordon equations, as well as rational cases of the Calogero– Moser and Ruijsenaars–Schneider models are shown to provide examples of such induced dynamics. Some other examples are given to demonstrates highly nontrivial collisions of particles and Liouville integrability of induced dynamical systems.

Toda lattice hierarchy and trigonometric Ruijsenaars–Schneider hierarchy

We consider solutions of the 2D Toda lattice hierarchy, which are trigonometric functions of the ‘zeroth’ time t0 = x. It is known that their poles move as particles of the trigonometric Ruijsenaars–Schneider model. We extend this correspondence to the level of hierarchies: the dynamics of poles with respect to the mth hierarchical time tm (respectively, ) of the 2D Toda lattice hierarchy is shown to be governed by the Hamiltonian which is proportional to the mth Hamiltonian (respectively, ) of the Ruijsenaars–Schneider model, where L is the Lax matrix.

Relativistic Interacting Integrable Elliptic Tops

We propose relativistic generalization of integrable systems describing $M$ interacting elliptic ${\rm gl}(N)$ tops of the Euler-Arnold type. The obtained models are elliptic integrable systems, which reproduce the spin elliptic ${\rm GL}(M)$ Ruijsenaars-Schneider model for $N=1$ case, while in the $M=1$ case they turn into relativistic integrable ${\rm GL}(N)$ elliptic tops. The Lax pairs with spectral parameter on elliptic curve are constructed.

Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone

We investigate the finite dimensional dynamical system derived by Braden and Hone in 1996 from the solitons of An−1 affine Toda field theory. This system of evolution equations for an Hermitian matrix L and a real diagonal matrix q with distinct eigenvalues was interpreted as a special case of the spin Ruijsenaars–Schneider models due to Krichever and Zabrodin. A decade later, Li re-derived the model from a general framework built on coboundary dynamical Poisson groupoids. This led to a Hamiltonian description of the gauge invariant content of the model, where the gauge transformations act as conjugations of L by diagonal unitary matrices. Here, we point out that the same dynamics can be interpreted also as a special case of the spin Sutherland systems obtained by reducing the free geodesic motion on symmetric spaces, studied by Pusztai and the author in 2006; the relevant symmetric space being . This construction provides an alternative Hamiltonian interpretation of the Braden–Hone dynamics. We prove that the two Poisson brackets are compatible and yield a bi-Hamiltonian description of the standard commuting flows of the model.