Characteristic Classes and Hitchin Systems. General Construction

Levin, A.; Olshanetsky, M.; Smirnov, Andrey; Zotov, A.

doi:10.1007/s00220-012-1585-x

Cited by 30 publications

(74 citation statements)

References 59 publications

(87 reference statements)

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“…Ниже мы явно находим g 0 для всех простых алгебр Ли из нашего списка. Система кокорней Π в g 0 строится усреднением по ор-битам λ-действия на Π ∨ ext [45]. Суммируем информацию об инвариантных подалгебрах в табл.…”

Section: разложение алгебр лиunclassified

“…Изомонодромные уравнения на эллиптических кривых В этом разделе мы обобщаем результаты работ [45], [46] в двух направле-ниях. Во-первых, на случай многих отмеченных точек.…”

Section: классификация изомонодромных задачunclassified

“…В общем случае решения (1.5) классифицируются характеристическими клас-сами расслоений, т. е. элементами группы вторых когомологий H 2 (Σ τ , Z (G)), где Z (G) -центр G. Классификация таких решений, соответствующих рас-слоений Хиггса и интегрируемых систем (в случае единственной отмеченной точки) описана в наших недавних работах [45], [46] (см. также [47]- [49]).…”

Section: Introductionunclassified

“…В настоящей работе мы предлагаем обобщение классификации расслоений Хиггса и интегрируемых систем по характеристическим классам [45], [46] 2 на случай плоских связностей и произвольного числа отмеченных точек на эллип-тической кривой.…”

Section: Introductionunclassified

“…Интегрируемые семейства, отвечающие рассмат-риваемым задачам, были описаны в [45], [46]. Соответствие между интегрируе-мыми системами и уравнениями изомонодромных деформаций, названное в [44] соответствием Калоджеро-Пенлеве, использовалось еще П. Бутру при иссле-довании уравнений Пенлеве [66].…”

Section: Introductionunclassified

See 4 more Smart Citations

Classification of isomonodromy problems on elliptic curves

Levin¹,

Ольшанецкий²,

Olshanetsky³

et al. 2014

Uspekhi Mat. Nauk

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Section: разложение алгебр лиunclassified

Section: классификация изомонодромных задачunclassified

Section: Introductionunclassified

See 3 more Smart Citations

Classification of isomonodromy problems on elliptic curves

Levin¹,

Ольшанецкий²,

Olshanetsky³

et al. 2014

Uspekhi Mat. Nauk

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Spectrum of quantum transfer matrices via classical many-body systems

Gorsky

Zabrodin

Zotov

2014

J. High Energ. Phys.

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In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous gl n -invariant XXX spin chain on N sites with twisted boundary conditions can be found in terms of velocities of particles in the rational N -body Ruijsenaars-Schneider model. The possible values of the velocities are to be found from intersection points of two Lagrangian submanifolds in the phase space of the classical model. One of them is the Lagrangian hyperplane corresponding to fixed coordinates of all N particles and the other one is an N -dimensional Lagrangian submanifold obtained by fixing levels of N classical Hamiltonians in involution. The latter are determined by eigenvalues of the twist matrix. To support this picture, we give a direct proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars-Schneider model, where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians, calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with certain multiplicities. We also prove a similar statement for the gl n Gaudin model with N marked points (on the quantum side) and the Calogero-Moser system with N particles (on the classical side). The realization of the results obtained in terms of branes and supersymmetric gauge theories is also discussed.

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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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Characteristic Classes and Hitchin Systems. General Construction

Cited by 30 publications

References 59 publications

Classification of isomonodromy problems on elliptic curves

Classification of isomonodromy problems on elliptic curves

Spectrum of quantum transfer matrices via classical many-body systems

Relativistic classical integrable tops and quantum R-matrices

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