The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

Generic Painlevé VI tau function τ (t) can be interpreted as four-point correlator of primary fields of arbitrary dimensions in 2D CFT with c = 1. Using AGT combinatorial representation of conformal blocks and determining the corresponding structure constants, we obtain full and completely explicit expansion of τ (t) near the singular points. After a check of this expansion, we discuss examples of conformal blocks arising from Riccati, Picard, Chazy and algebraic solutions of Painlevé VI.

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“…This results into the following statement (cf observations made in [32]): Proposition 1. Under assumption tr A(z) = 0, the matrices…”

Section: Jhep10(2012)038mentioning

confidence: 74%

Conformal field theory of Painlevé VI

Gamayun

Iorgov

Lisovyy

2012

J. High Energ. Phys.

152

267

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“…For some other purposes this change employed earlier by D.P. Novikov in [36], see also formula (2.3.36) in [25]. …”

Section: Resultsmentioning

confidence: 96%

“…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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“…In the above description of "quantizations" of two higher analogues of Painlevé equations, the key role is played by change (21), which was previously used in close situations for somewhat different purposes by Novikov in [7] (see also (2.3.36) in [39]). It is not for certain that a similar change alone is sufficient to as easily clarify the question of whether "quantizations" of the form (20) are valid for all higher Hamiltonian isomonodromic analogues of Painlevé equations with two degrees of freedom to which 2× 2 matrix IDM equations correspond (in particular, for all analogues considered in [37]).…”

Section: Discussionmentioning

confidence: 99%

“…The results of papers [2] and [3] received further development in [4] and [7]- [11]. However, similar "quantizations" for higher analogues of the Painlevé equations have not been studied so far.…”

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confidence: 99%

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

Suleǐmanov

2014

Funct Anal Its Appl

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We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian H 1 (z, t, q 1 , q 2 , p 1 , p 2 ) corresponding to the second equation P 2 1 in the Painlevé I hierarchy. This solution is obtained by an explicit change of variables from a solution of systems of linear equations whose compatibility condition is the ordinary differential equation P 2 1 with respect to z . This solution also satisfies an analogue of the Schrödinger equation corresponding to the Hamiltonian H 2 (z, t, q 1 , q 2 , p 1 , p 2 ) of a Hamiltonian system with respect to t compatible with P 2 1 . A similar situation occurs for the P 2 2 equation in the Painlevé II hierarchy.For all of the six Painlevé ordinary differential equations (ODEs) q zz = f (z, q, q z ), an explicit change in the solutions of the corresponding pairs of linear systems of the isomonodromic deformation method (IDM) given in Garnier's paper [1] yields solutions of the equations (see [2] and [3])(see also the beginning of the conclusion of this paper). These equations are determined by the Hamiltonians H = H(z, q, p) of Hamiltonian systems q z = H p (z, q, p) and p z = −H q (z, q, p); the Hamiltonians are quadratic in the momenta p, and eliminating p from the systems, we obtain the six Painlevé ODEs. From the Schrödinger equationswhich depend on the Planck constant h = 2π = −2πiε, the six evolution equations (1) are obtained by the formal change ε = 1. (In the cases of the Painlevé III and V equations, expressions given in [2] and [3] contain inaccuracies, which, however, are easy to correct.) Note that, for the ODEs obtained from the first and second Painlevé equations, the corresponding IDM equations can be rescaled into compatible systems of linear ODEs, which determine, for any fixed complex number ε, exact solutions of Eqs. (2) with Hamiltonians H depending on ε. Let us clarify this point. The pair of Garnier IDM equations [1] for the ODE(the a j are any constants), whose special cases are the first and second Painlevé equations, has the form W yy = P (τ, y)W, W τ = B(τ, y)W y − 1 2 B y (τ, y)W, (4) *

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The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

Cited by 20 publications

References 22 publications

Conformal field theory of Painlevé VI

Conformal field theory of Painlevé VI

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

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

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The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

Cited by 20 publications

References 22 publications

Conformal field theory of Painlevé VI

Conformal field theory of Painlevé VI

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

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

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