Quantum Painlevé-Calogero correspondence

Zabrodin, A.; Zotov, A.

doi:10.1063/1.4732532

Cited by 44 publications

(45 citation statements)

References 49 publications

(63 reference statements)

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“…Zabrodin and Zotov, in [34,35], provided a quantized version of the Calogero-Painlevé correspondence. Namely, they proved that for each of the Painlevé equation it exists a Lax pair such that the first component ψ(z; t) of its eigenfunction satisfies the equation 4…”

Section: Quantization β-Models and Open Questionsmentioning

confidence: 99%

Noncommutative Painlevé Equations and Systems of Calogero Type

Bertola

Cafasso

Rubtsov

2018

Commun. Math. Phys.

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All Painlevé equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of β-models. An almost two decade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlevé correspondence. In this paper we answer in the affirmative by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlevé equation.

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Section: Quantization β-Models and Open Questionsmentioning

confidence: 99%

Noncommutative Painlevé Equations and Systems of Calogero Type

Bertola

Cafasso

Rubtsov

2018

Commun. Math. Phys.

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“…3 The gauge theory interpretation of the four-point conformal block with an additional degenerate insertion has been proposed in [10]: it is conjectured to reproduce the instanton partition function in the presence of an elementary surface operator. We will demonstrate that this equation (before taking the semi-classical limit) essentially coincides with the quantum Painlevé VI equation [11][12] [13]. The Painlevé VI equation is the most general (rational) ordinary second order differential equation with no movable singularities except poles [14][15] [16].…”

Section: Introductionmentioning

confidence: 99%

Transformations of Spherical Blocks

Kashani-Poor

Troost

2013

J. High Energ. Phys.

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Abstract:We further explore the correspondence between N = 2 supersymmetric SU(2) gauge theory with four flavors on -deformed backgrounds and conformal field theory, with an emphasis on the -expansion of the partition function natural from a topological string theory point of view. Solving an appropriate null vector decoupling equation in the semi-classical limit allows us to express the instanton partition function as a series in quasi-modular forms of the group Γ(2), with the expected symmetry W (D 4 ) S 3 . In the presence of an elementary surface operator, this symmetry is enhanced to an action of W (D (1) 4 ) S 4 on the instanton partition function, as we demonstrate via the link between the null vector decoupling equation and the quantum Painlevé VI equation.

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“…During the last decade, there were written quite a lot of works on relations of the equations of IDM for Painlevé type ODEs with evolution linear equations of quantum mechanics and, starting from work [36], of quantum field theory [1]- [3], [7], [8], [14]- [17], [19]- [24], [26], [27], [32], [35]- [38], [41], [43]- [45].…”

Section: Introductionmentioning

confidence: 99%

“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

Pavlenko¹,

Suleǐmanov²

2017

Ufimskii Matematicheskii Zhurnal

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Abstract. We consider two compatible linear evolution equations with times 1 and 2 depending on two spatial variables. These evolution equations are the analogues of the non-stationary Schrödinger equations determined by the two Hamiltonians 7 2 +1 ( 1 , 2 , 1 , 2 , 1 , 2 ) ( = 1, 2) of the Hamilton system +1 . They arise by the formal replacement of the Planck constant by the imaginary unit. We construct explicit solutions of these analogues of Schrödinger equations in terms of the solutions of the corresponding linear systems of ordinary differential equations in the isomonodromic deformations method, whose compatibility condition is the Hamiltonian system 7 2 +1 . The key role in the construction of these explicit solutions is played by the change, which was used earlier in constructing the solutions of non-stationary Schrödinger equation determined by the Hamiltonians of isomonodromic Hamiltonian Garnier system with two degrees of freedom as well as of two isomonodromic degenerations of the latter. We discuss the applicability of this change for constructing the solutions to analogues of non-stationary Schrödinger equations determined by the Hamiltonians of the entire hierarchy of isomonodromic Hamiltonian systems with two degrees of freedom being the degenerations of this Garnier system. We mention also a relation of solutions to Hamilton systems +1 with some problems of modern nonlinear mathematical physics. In particular, we show that the solutions of these Hamiltonian systems are determined explicitly by the simultaneous solutions to the Korteweg-de Vries equation + + = 0 and a non-autonomous fifth order ordinary differential equations, which are used in universal description of the influence of a small dispersion on the transformation of weak hydrodynamical discontinuities into the strong ones.

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Quantum Painlevé-Calogero correspondence

Cited by 44 publications

References 49 publications

Noncommutative Painlevé Equations and Systems of Calogero Type

Noncommutative Painlevé Equations and Systems of Calogero Type

Transformations of Spherical Blocks

“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

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Quantum Painlevé-Calogero correspondence

Cited by 44 publications

References 49 publications

Noncommutative Painlevé Equations and Systems of Calogero Type

Noncommutative Painlevé Equations and Systems of Calogero Type

Transformations of Spherical Blocks

&#x201C;Quantizations&#x201D; of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

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“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$