Canonical Structure and Symmetries of the Schlesinger Equations

Dubrovin, Boris; Mazzocco, Marta

doi:10.1007/s00220-006-0165-3

Cited by 55 publications

(95 citation statements)

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“…Using this expression for log f , we can write the isomonodromy deformation equations for f , the Garnier system [24], [25], a scalar version of system (2). The equation…”

Section: Hyperelliptic Theta Functions and Solutions Of The Belavin-pmentioning

confidence: 99%

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

Novikov

2009

Theor Math Phys

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We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 matrices. This generalizes Suleimanov's result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function. Systems of linear equations accompanying the Painlevé andSchlesinger equations The Schlesinger system [1] for the matriceswhere i, j = 1, m (the second group of equations can be replaced with the condition that the matrix A 1 + · · · + A m = A ∞ is constant in t 1 , . . . , t m ), was discovered as the compatibility condition for the system(2)All algebraic integrals of motion of system (1) are known. They are the traces tr A k i , k = 1, 2, . . . (equivalently, the characteristic polynomials of the matrices A i ). A closed form ω = H i dt i was found in [2],and the τ -function (log τ ) ti = H i was also determined.Here, we consider only one aspect of the problem of the complete integrability of system (1) in terms of special functions. Namely, in the case of 2×2 matrices A i , we seek a second-order linear equation for

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“…Using this expression for log f , we can write the isomonodromy deformation equations for f , the Garnier system [24], [25], a scalar version of system (2). The equation…”

Section: Hyperelliptic Theta Functions and Solutions Of The Belavin-pmentioning

confidence: 99%

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

Novikov

2009

Theor Math Phys

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“…It is highly desirable to derive this result in a more systematic way. As regards the Garnier system, such a systematic explanation is implicit in the work of the Montreal group [28,29], and presented (in a more general form) by Dubrovin and Mazzocco [31]. Let us recall its essence.…”

Section: Resultsmentioning

confidence: 97%

Hamiltonian Structure of PI Hierarchy

Takasaki¹

2007

SIGMA

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Abstract. The string equation of type (2, 2g + 1) may be thought of as a higher order analogue of the first Painlevé equation that corresponds to the case of g = 1. For g > 1, this equation is accompanied with a finite set of commuting isomonodromic deformations, and they altogether form a hierarchy called the PI hierarchy. This hierarchy gives an isomonodromic analogue of the well known Mumford system. The Hamiltonian structure of the Lax equations can be formulated by the same Poisson structure as the Mumford system. A set of Darboux coordinates, which have been used for the Mumford system, can be introduced in this hierarchy as well. The equations of motion in these Darboux coordinates turn out to take a Hamiltonian form, but the Hamiltonians are different from the Hamiltonians of the Lax equations (except for the lowest one that corresponds to the string equation itself).

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“…The Euler method of "integrals with a parameter" can be generalized to the nth-order Fuchsian equations that have m singular points and enter the representation for the general Schlesinger system [19].…”

Section: Theoremmentioning

confidence: 99%