Note on the Ruijsenaars-Schneider model
英文摘要：
We study vector bundles with some additional structures on an elliptic curve and show how there are related to the elliptic Ruijsenaars-Schneider model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
Structures in BC_N Ruijsenaars-Schneider models
We construct the classical r-matrix structure for the Lax formulation of BC_N Ruijsenaars-Schneider systems proposed in hep-th 0006004. The r-matrix structure takes a quadratic form similar to the A_N Ruijsenaars-Schneider Poisson bracket behavior, although the dynamical dependence is more complicated. Commuting Hamiltonians stemming from the BC_N Ruijsenaars-Schneider Lax matrix are shown to be linear combinations of particular Koornwinder-van Diejen ``external fields'' Ruijsenaars-Schneider models, for specific values of the exponential one-body couplings. Uniqueness of such commuting Hamilt…
hep-thmath.QAnlin.SI
The dynamical twisting and nondynamical r-matrix structure of elliptic Ruijsenaars-Schneider model
From the dynamical twisting of the classical r-matrix, we obtain a new Lax operator for the elliptic Ruijsenaars-Schneider model (cf. Ruijsenaars'). The corresponding r-matrix is shown to be the classical $Z_n$-symmetric elliptic r-matrix, which is the same as that obtained in the study of the nonrelativistic version---the $A_{n-1}$ Calogero-Moser model.
hep-thmath.QAnlin.SI
Dynamical r-matrix for the elliptic Ruijsenaars-Schneider system
The classical r-matrix structure for the generic elliptic Ruijsenaars-Schneider model is presented. It makes the integrability of this model as well as of its discrete-time version that was constructed in a recent paper manifest.
hep-thmath.QAnlin.SI
Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models
We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case $m=1$ corresponds to the tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for $m>1$ we obtain new integrable systems that generalise the Ruijsenaars-Schneider …
math-phmath.MPmath.QA
Quantum vs Classical Integrability in Ruijsenaars-Schneider Systems
The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider systems, which are one parameter deformation of Calogero-Moser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) at equilibrium are reported in a previous paper (Corrigan-Sasaki). For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". In this paper we report that similar features and results hold for the Ruijsenaar…
hep-thmath-phmath.MP
Cherednik operators and Ruijsenaars-Schneider model at infinity
Heckman introduced $N$ operators on the space of polynomials in $N$ variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these $N$ operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. We compute the limits of our operators at $N\to\infty$. These limits yield the same Lax operator …
math.COmath.QAmath.RT
更多相关论文收起更多论文
R-matrix Quantization of the Elliptic Ruijsenaars--Schneider model
It is shown that the classical L-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical r and $\bar{r}$-matrices satisfying a closed system of equations. The corresponding quantum R and $\overline{R}$-matrices are found as solutions to quantum analogs of these equations. We present the quantum L-operator algebra and show that the system of equations on R and $\overline{R}$ arises as the compatibility condition for…
hep-thmath.QAnlin.SI
On 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.
hep-thmath-phmath.MP
Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials
The Ruijsenaars-Schneider systems are `discrete' version of the Calogero-Moser (C-M) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e^{\pmβ' p} (β' is a deformation parameter) instead of an ordinary polynomial in p in the hierarchies of C-M systems. We determine the polynomials describing the equilibrium positions of the rational and trigonometric Ruijsenaars-Schneider systems based on classical root systems. These are deformation of the classical orthogonal polynomials, the Hermite, Laguerre and Jacobi polynomials which describe the equilibriu…
hep-thmath.QA
Quantum Dynamical R-matrices and Quantum Frobenius Group
We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parameterization of the cotangent bundle over GL(N,C). In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical $\bar{r}$-matrix. Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parameterization of the cotangent bundle we also derive the algebra for the L-operato…
hep-thmath.QAnlin.SI
Universal R-matrix formalism for the spin Calogero-Moser system and its difference counterpart
The expression of the quantum Ruijsenaars-Schneider Hamiltonian is obtained in the framework of the dynamical $R$-matrix formalism. This generalizes to the case of $U_q(sl_n)$ the result obtained by O. Babelon, D. Bernard and E. Billey for $U_q(sl_2)$ which is the higher difference Lame operator. The general method involved is the universal $R$-matrix construction.
hep-thmath.QA
从冻结获得模形式的椭圆长程自旋链
Modular families of elliptic long-range spin chains from freezing
Humboldt-Universität zu Berlin（柏林洪堡大学）；School of Mathematics and Statistics, University of Glasgow（格拉斯哥大学数学与统计学院）
本文研究通过冻结可积量子多体系统构造具有q变形长程相互作用的量子可积自旋链，聚焦椭圆情况，展示如何通过模群作用和变形量子化框架实现模形式的自旋链构造。
hep-thmath-phmath.MP
On Elliptic Algebras and Large-n Supersymmetric Gauge Theories
In this note we further develop the duality between supersymmetric gauge theories in various dimensions and elliptic integrable systems such as Ruijsenaars-Schneider model and periodic intermediate long wave hydrodynamics. These models arise in instanton counting problems and are described by certain elliptic algebras. We discuss the correspondence between the two types of models by employing the large-n limit of the dual gauge theory. In particular we provide non-Abelian generalization of our previous result on the intermediate long wave model.
hep-thmath-phmath.MP
Elliptic generalization 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 ${\rm GL}_M$ $R$-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 fo…
cond-mat.str-elhep-thmath-ph
关于Nichols代数的讲义
Lecture notes on Nichols algebras
University of Hamburg（汉堡大学）
本文从范畴论视角介绍了Nichols代数的定义、性质及其在量子群表示范畴构建中的作用，同时探讨其在共形场论中的应用。
hep-thmath.QAmath.RA
