Classical<i>r</i>-matrices and the Feigin–Odesskii algebra via Hamiltonian and Poisson reductions

Braden, H. W.; Dolgushev, Vasily A.; Olshanetsky, M.; Zotov, A.

doi:10.1088/0305-4470/36/25/306

Cited by 31 publications

(43 citation statements)

References 38 publications

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“…At the level of integrable systems, it was established by K.Hasegawa [133][134][135] (see also a trigonometric version of the correspondence in [143]) and claims an equivalence of the elliptic multi-point spin Ruijsenaars system given by (N, k, l) and the elliptic spin chain on k sites, given by the l-orbit of the SklyaninOdesskii-Feigin gl N [144][145][146][147][148][149]. In the particular case of l = 1 (the orbit of minimal dimension), one obtains the duality between SU(N ) k theory with fundamental matter in 6d and SU(k + 1) N necklace quiver theory in 5d.…”

Section: Spectral Dualitymentioning

confidence: 99%

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Миронов¹,

Морозов²,

Zenkevich³

2016

J. High Energ. Phys.

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We consider Dotsenko-Fateev matrix models associated with compactified Calabi-Yau threefolds. They can be constructed with the help of explicit expressions for refined topological vertex, i.e. are directly related to the corresponding topological string amplitudes. We describe a correspondence between these amplitudes, elliptic and affine type Selberg integrals and gauge theories in five and six dimensions with various matter content. We show that the theories of this type are connected by spectral dualities, which can be also seen at the level of elliptic Seiberg-Witten integrable systems. The most interesting are the spectral duality between the XYZ spin chain and the Ruijsenaars system, which is further lifted to self-duality of the double elliptic system.

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Section: Spectral Dualitymentioning

confidence: 99%

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Миронов¹,

Морозов²,

Zenkevich³

2016

J. High Energ. Phys.

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“…More detailed description can be found in [75]. Higher rank Sklyanin algebras in the context of integrable systems were also discussed in [18,23,37].…”

Section: Jhep07(2014)012mentioning

confidence: 99%

“…14 For the top-like (elliptic) Lax operators (1.19) a similar procedure was suggested in [18] and [23]. In all the description the Lax matrix appears through reduction from the group element.…”

Section: Poincaré Invariancementioning

confidence: 99%

Relativistic classical integrable tops and quantum R-matrices

Levin

Olshanetsky

Zotov

2014

J. High Energ. Phys.

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We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical R-matrices. A particular case of gl N system is gauge equivalent to the N -particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum R-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin's quantum R-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large N limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new gl N quantum rational non-dynamical R-matrix via the relativistic top, which we get in a different way -using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of gl 2 the quantum rational R-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained R-matrix.

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“…Здесь S a не являются, вообще говоря, независимыми из-за нали-чия условий фиксации калибровки. Таким образом, после редукции приходим к пуассонову многообразию P red с размерностью коприсоединенной орбиты O. Скобки Пуассона на P red вычисляются как скобки Дирака [84], [85]. На P red уравнения движения записываются в форме Лакса (6.13).…”

Section: редуцированная пуассонова алгебра есть факторалгебраunclassified