## Monodromy of certain Painlevé-VI transcendents and reflection groups

Venue: | Invent. Math |

Citations: | 40 - 6 self |

### BibTeX

@ARTICLE{Dubrovin_monodromyof,

author = {B. Dubrovin and M. Mazzocco},

title = {Monodromy of certain Painlevé-VI transcendents and reflection groups},

journal = {Invent. Math},

year = {},

pages = {55--147}

}

### OpenURL

### Abstract

Abstract. We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β = γ = 0, δ = 1 and α arbitrary. We 2 introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painlevé VI equation.

### Citations

488 |
Ordinary Differential Equations
- Ince
- 1956
(Show Context)
Citation Context ... − 1) (y − x) 2 ) 1 + y − x ] , yx PV Iµ in the complex plane, µ is an arbitrary complex parameter satisfying the condition 2µ ̸∈ Z. This is a particular case of the general Painlevé VI equation (see =-=[Ince]-=-) PVI(α, β, γ, δ), that depends on four parameters α, β, γ, δ, specified by the following choice of the parameters: (2µ − 1)2 α = , β = γ = 0 δ = 2 1 2 . The general solution y(x; c1, c2) of PVI(α, β,... |

262 |
Higher transcendental functions
- Bateman, Erdelyi
- 1953
(Show Context)
Citation Context ... contiguity transformations, are of fifteen types (the first type consists of an infinite sequence of solutions). The rows (2 − 15) of Schwartz’s list (see, for example, the table in Section 2.7.2 of =-=[Bat]-=-) correspond to the triples of generating reflections of the symmetry groups of regular polyhedra in the three-dimensional Euclidean space (we are grateful to E. Vinberg for bringing this point to our... |

231 |
Monodromy preserving deformation of linear ordinary differential equations with rational coefficients
- Jimbo, Miwa
- 1981
(Show Context)
Citation Context ...lows to classify the solutions, but also to obtain the explicit formulae, as we do in Section 2.4. The main tool to obtain these results is the isomonodromy deformation method (see [Fuchs], [Sch] and =-=[JMU]-=-, [ItN], [FlN]). The Painlevé VI is represented as the equation of isomonodromy deformation of the auxiliary Fuchsian system dY dz = ( A0 z For PVIµ, the 2 × 2 matrices A0, A1, Ax are nilpotent and ( ... |

224 |
Regular Polytopes
- Coxeter
- 1963
(Show Context)
Citation Context ...he symmetries of the equations described in Section 1.2, with the reciprocal pairs of the three-dimensional regular polyhedra and star-polyhedra (the description of the star-polyhedra can be found in =-=[Cox]-=-). The solutions corresponding to the regular tetrahedron, cube and icosahedron are the ones obtained in 4[Dub] using the theory of polynomial Frobenius manifolds. The solutions corresponding to the ... |

160 |
Geometry of 2D topological field theories
- Dubrovin
- 1996
(Show Context)
Citation Context ...inlevé equations which we do not discuss here. We mention only the paper [Tod] where our PVIµ appears in the problem of the construction of self-dual Bianchi-type IX Einstein metrics, and the 2paper =-=[Dub]-=- where the same equation was used to classify the solutions of WDVV equation in 2D-topological field theories. The name of transcendents could be misleading; indeed, for some particular values of (c1,... |

100 |
Studies on the Painlevé Equations
- Okamoto
- 1987
(Show Context)
Citation Context ...tions, that can be expressed via hypergeometric functions, of PVI were first constructed by Lukashevich [Luka]. A general approach to study the classical solutions of PVI was proposed by Okamoto (see =-=[Ok1]-=-[Ok2]). One of the main tools of this approach is the symmetry group of PVI: the particular solutions are those being invariant with respect to some symmetry of PVI. The symmetries act in a non trivia... |

90 |
les équations différentielles du second ordre à points critiques fixes, Oeuvres de Paul Painlevé. Tome III. (French) Équations différentielles du second ordre. Mécanique. Quelques documents. Éditions du Centre National de la Recherche Scientifique
- Painlevé, Sur
- 1975
(Show Context)
Citation Context ..., specified by the following choice of the parameters: (2µ − 1)2 α = , β = γ = 0 δ = 2 1 2 . The general solution y(x; c1, c2) of PVI(α, β, γ, δ) satisfies the following two important properties (see =-=[Pain]-=-): 1) The solution y(x; c1, c2) can be analytically continued to a meromorphic function on the universal covering of Cl \{0, 1, ∞}. 2) For generic values of the integration constants c1, c2 and of the... |

80 |
les équations différentielles du second ordre dont l’intégrale générale a ses points critiques fixes
- Malmquist
- 1922
(Show Context)
Citation Context ...y, yx), where R is rational in yx and meromorphic in x and y and satisfies the Painlevé property of absence of movable critical singularities, were classified by Painlevé and Gambier (see [Pain], and =-=[Gamb]-=-). Only six of these equations, which are given in the Painlevé-Gambier list, satisfy the property 2), i.e. they can not be reduced to known differential equations for elementary and classical special... |

67 | A.C., Monodromy- and spectrum-preserving deformations - Flaschka, Newell - 1980 |

67 | les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes - Chazy, Sur - 1911 |

61 |
Twistor spaces, Einstein metrics and isomonodromic
- Hitchin
(Show Context)
Citation Context ...e expressed via classical functions. For example Picard (see [Pic] and [Ok]) showed that the general solution of PVI(0, 0, 0, 1) can 2 be expressed via elliptic functions, and, more recently, Hitchin =-=[Hit]-=- obtained the general solution of PVI( 1 1 1 3 , , , ) in terms of the Jacobi theta-functions (see also [Man]). Partic8 8 8 8 ular examples of classical solutions, that can be expressed via hypergeome... |

56 |
Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegene wesentlich singulären Stellen
- Fuchs
- 1907
(Show Context)
Citation Context ... method not only allows to classify the solutions, but also to obtain the explicit formulae, as we do in Section 2.4. The main tool to obtain these results is the isomonodromy deformation method (see =-=[Fuchs]-=-, [Sch] and [JMU], [ItN], [FlN]). The Painlevé VI is represented as the equation of isomonodromy deformation of the auxiliary Fuchsian system dY dz = ( A0 z For PVIµ, the 2 × 2 matrices A0, A1, Ax are... |

49 |
From Gauss to Painlevé – A modern theory of special functions, Vieweg Verlag
- Iwasaki, Kimura, et al.
- 1991
(Show Context)
Citation Context ...oose the arbitrary constants b, T11, T12, σ in such a way that − T12 4bT11 = a0, σ = σ0, for any fixed a0 and σ0. Remark 2.3. Other existence results for σ ∈ Cl \{] − ∞, 0] ∪ [1, +∞[} can be found in =-=[IKSY]-=- and [S1], [S2], [S3]. For indices with Reσ ̸∈ [0, 1], the asymptotics obtained in these papers are valid in more complicated domains near 0. 2.1.3. Proof of the uniqueness. Now we prove that the solu... |

49 |
The isomonodromic deformation method in the theory of Painlevé equations
- Novokshenov
- 1986
(Show Context)
Citation Context ... classify the solutions, but also to obtain the explicit formulae, as we do in Section 2.4. The main tool to obtain these results is the isomonodromy deformation method (see [Fuchs], [Sch] and [JMU], =-=[ItN]-=-, [FlN]). The Painlevé VI is represented as the equation of isomonodromy deformation of the auxiliary Fuchsian system dY dz = ( A0 z For PVIµ, the 2 × 2 matrices A0, A1, Ax are nilpotent and ( ) −µ 0 ... |

42 |
Sixth Painlevé equation, universal elliptic curve, and mirror of P 2
- Manin
- 1998
(Show Context)
Citation Context ... of PVI(0, 0, 0, 1) can 2 be expressed via elliptic functions, and, more recently, Hitchin [Hit] obtained the general solution of PVI( 1 1 1 3 , , , ) in terms of the Jacobi theta-functions (see also =-=[Man]-=-). Partic8 8 8 8 ular examples of classical solutions, that can be expressed via hypergeometric functions, of PVI were first constructed by Lukashevich [Luka]. A general approach to study the classica... |

35 | Rings of Fricke characters and automorphism groups of free groups - Magnus - 1980 |

28 |
Painlevé property of monodromy preserving equations and the analyticity of τfunctions, Publ
- Miwa
- 1981
(Show Context)
Citation Context ...\{diags} := { (u1, u2, u3) ∈ Cl 3 | ui ̸= uj for i ̸= j } , and { (z, u1, u2, u3) ∈ Cl 4 | ui ̸= uj for i ̸= j andz ̸= ui, i = 1, 2, 3 } , respectively. The proof can be found, for example, in [Mal], =-=[Miwa]-=-, [Sib]. We recall the theorem of solvability of the inverse problem of the monodromy (see [Dek]): Theorem 1.2. Given three arbitrary matrices, satisfying (1.7) and (1.8), with M∞ of the form (1.4), a... |

23 |
Sur les déformations isomonodromique. I: Singularité régulières. Séminaire de l’ École Morm. Sup. Birkhäuser
- Malgrange
- 1982
(Show Context)
Citation Context ...f Cl 3 \{diags} := { (u1, u2, u3) ∈ Cl 3 | ui ̸= uj for i ̸= j } , and { (z, u1, u2, u3) ∈ Cl 4 | ui ̸= uj for i ̸= j andz ̸= ui, i = 1, 2, 3 } , respectively. The proof can be found, for example, in =-=[Mal]-=-, [Miwa], [Sib]. We recall the theorem of solvability of the inverse problem of the monodromy (see [Dek]): Theorem 1.2. Given three arbitrary matrices, satisfying (1.7) and (1.8), with M∞ of the form ... |

23 |
Birational canonical transformations and classical solutions of the sixth Painlevé equation
- Watanabe
- 1998
(Show Context)
Citation Context ...own at that moment, fit into the boundary of this fundamental region. The theory of the classical solutions of the Painlevé equations was developed by Umemura and Watanabe ([Um], [Um1], [Um2], [Um3], =-=[Wat]-=-); in particular, all the oneparameter families of classical solutions of PVI were classified in [Wat]. Watanabe also proved that, loosely speaking, all the other classical solutions of PVI (i.e. not ... |

21 |
Self-dual Einstein metrics from the Painlevé VIequation.Phys
- TOD
- 1994
(Show Context)
Citation Context ..., β, γ, δ) by a confluence procedure (see [Ince] §14.4). There are many physical applications of particular solutions of the Painlevé equations which we do not discuss here. We mention only the paper =-=[Tod]-=- where our PVIµ appears in the problem of the construction of self-dual Bianchi-type IX Einstein metrics, and the 2paper [Dub] where the same equation was used to classify the solutions of WDVV equat... |

17 |
On the irreducibility of the first differential equation of Painlevé, Algebraic geometry and commutative algebra
- Umemura
- 1988
(Show Context)
Citation Context ...lassical solutions known at that moment, fit into the boundary of this fundamental region. The theory of the classical solutions of the Painlevé equations was developed by Umemura and Watanabe ([Um], =-=[Um1]-=-, [Um2], [Um3], [Wat]); in particular, all the oneparameter families of classical solutions of PVI were classified in [Wat]. Watanabe also proved that, loosely speaking, all the other classical soluti... |

15 |
Rational solutions of the second and the fourth Painlevé equations, Funkcial
- Murata
- 1985
(Show Context)
Citation Context ... to elaborate a tool to classify all the algebraic solutions of the Painlevé VI equation (for the other five Painlevé equations, algebraic solutions have been classified, see [Kit], [Wat1], [Mur] and =-=[Mur1]-=-). Our idea is very close to the main idea of the classical paper of Schwartz (see [Schw]) devoted to the classification of the algebraic solutions of the Gauss hypergeometric equation. Let y(x; c1, c... |

13 |
The Matrix of a Connection having Regular Singularities on a Vector Bundle
- Dekkers
- 1979
(Show Context)
Citation Context ...uj for i ̸= j andz ̸= ui, i = 1, 2, 3 } , respectively. The proof can be found, for example, in [Mal], [Miwa], [Sib]. We recall the theorem of solvability of the inverse problem of the monodromy (see =-=[Dek]-=-): Theorem 1.2. Given three arbitrary matrices, satisfying (1.7) and (1.8), with M∞ of the form (1.4), and given a point u 0 = (u 0 1 , u0 2 , u0 3 ) ∈ Cl 3 \{diags}, for any neighborhood U of u 0 , t... |

12 |
Ueber eine Klasse von Differentsial System Beliebliger Ordnung mit Festen Kritischer Punkten
- Schlesinger
(Show Context)
Citation Context ...ot only allows to classify the solutions, but also to obtain the explicit formulae, as we do in Section 2.4. The main tool to obtain these results is the isomonodromy deformation method (see [Fuchs], =-=[Sch]-=- and [JMU], [ItN], [FlN]). The Painlevé VI is represented as the equation of isomonodromy deformation of the auxiliary Fuchsian system dY dz = ( A0 z For PVIµ, the 2 × 2 matrices A0, A1, Ax are nilpot... |

11 | V.:Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific - AGARWAL, LAKSHMIKANTHAM - 1993 |

11 |
Mémoire sur la theorie des functions algébriques de deux variables
- Picard
(Show Context)
Citation Context ... The name of transcendents could be misleading; indeed, for some particular values of (c1, c2, α, β, γ, δ), the solution y(x; c1, c2) can be expressed via classical functions. For example Picard (see =-=[Pic]-=- and [Ok]) showed that the general solution of PVI(0, 0, 0, 1) can 2 be expressed via elliptic functions, and, more recently, Hitchin [Hit] obtained the general solution of PVI( 1 1 1 3 , , , ) in ter... |

10 |
Poncelet polygons and the Painlevé transcendents, Geometry and Analysis
- Hitchin
- 1996
(Show Context)
Citation Context ..., loosely speaking, all the other classical solutions of PVI (i.e. not belonging to the one-parameter families) can only be given by algebraic functions. Examples of algebraic solutions were found in =-=[Hit1]-=-, for PVI( 1 1 1 1 8 , −1 8 , 2k2, 2 − 2k2), for an arbitrary integer k. Other examples for PVIµ were constructed in [Dub]. They turn out to be related to the group of symmetries of the regular polyhe... |

8 |
The Painlevé equations and the Dynkin diagrams, Painlevé transcendents (SainteAdèle
- Okamoto
- 1990
(Show Context)
Citation Context ..., that can be expressed via hypergeometric functions, of PVI were first constructed by Lukashevich [Luka]. A general approach to study the classical solutions of PVI was proposed by Okamoto (see [Ok1]=-=[Ok2]-=-). One of the main tools of this approach is the symmetry group of PVI: the particular solutions are those being invariant with respect to some symmetry of PVI. The symmetries act in a non trivial way... |

6 |
Classical solutions of the third Painlevé equation
- Murata
- 1995
(Show Context)
Citation Context ...ur work is to elaborate a tool to classify all the algebraic solutions of the Painlevé VI equation (for the other five Painlevé equations, algebraic solutions have been classified, see [Kit], [Wat1], =-=[Mur]-=- and [Mur1]). Our idea is very close to the main idea of the classical paper of Schwartz (see [Schw]) devoted to the classification of the algebraic solutions of the Gauss hypergeometric equation. Let... |

5 |
On the tau-function of the Painlevé equations, Physica 2D
- Okamoto
- 1981
(Show Context)
Citation Context ...of transcendents could be misleading; indeed, for some particular values of (c1, c2, α, β, γ, δ), the solution y(x; c1, c2) can be expressed via classical functions. For example Picard (see [Pic] and =-=[Ok]-=-) showed that the general solution of PVI(0, 0, 0, 1) can 2 be expressed via elliptic functions, and, more recently, Hitchin [Hit] obtained the general solution of PVI( 1 1 1 3 , , , ) in terms of the... |

4 |
Über Diejenigen Fälle in Welchen die Gaussische Hypergeometrische Reihe einer Algebraische Funktion iheres vierten Elementes Darstellit
- Schwartz
(Show Context)
Citation Context ...(for the other five Painlevé equations, algebraic solutions have been classified, see [Kit], [Wat1], [Mur] and [Mur1]). Our idea is very close to the main idea of the classical paper of Schwartz (see =-=[Schw]-=-) devoted to the classification of the algebraic solutions of the Gauss hypergeometric equation. Let y(x; c1, c2) be a branch of a solution of PVI; its analytic continuation along any closed path γ av... |

3 |
Frobenius manifolds from Yang-Mills instantons
- Segert
- 1998
(Show Context)
Citation Context ...ucted in [Dub]. They turn out to be related to the group of symmetries of the regular polyhedra in the three dimensional space. Other algebraic solutions of PVI can be extracted from the recent paper =-=[Seg]-=-. The main aim of our work is to elaborate a tool to classify all the algebraic solutions of the Painlevé VI equation (for the other five Painlevé equations, algebraic solutions have been classified, ... |

2 |
On the Theory of the Third Painlevé Equation
- Lukashevich
- 1967
(Show Context)
Citation Context ...ms of the Jacobi theta-functions (see also [Man]). Partic8 8 8 8 ular examples of classical solutions, that can be expressed via hypergeometric functions, of PVI were first constructed by Lukashevich =-=[Luka]-=-. A general approach to study the classical solutions of PVI was proposed by Okamoto (see [Ok1][Ok2]). One of the main tools of this approach is the symmetry group of PVI: the particular solutions are... |

2 |
The logarithmic solutions of the hypergeometric equation
- Nörlund
- 1963
(Show Context)
Citation Context ...ions of the Gauss equation around z = 1 for both the systems ( ˆ Σ) and ( ˜ Σ), and around z = 0 for ( ˜ Σ); moreover, we shall use the extension of the Kummer relations to this logarithmic case (see =-=[Nor]-=-). 66Im(z) Re(z) Fig.9. The branch cut |arg(z)| < π. In what follows we denote F(a, b, c, z) the hypergeometric function and with g(a, b, z) its logarithmic counterpart for c = 1, namely: ∞∑ (a)k(b)k... |

2 |
Painlevé Transcendents in the neighbourhood of fixed sjngular points
- Shimomura
- 1982
(Show Context)
Citation Context ...bitrary constants b, T11, T12, σ in such a way that − T12 4bT11 = a0, σ = σ0, for any fixed a0 and σ0. Remark 2.3. Other existence results for σ ∈ Cl \{] − ∞, 0] ∪ [1, +∞[} can be found in [IKSY] and =-=[S1]-=-, [S2], [S3]. For indices with Reσ ̸∈ [0, 1], the asymptotics obtained in these papers are valid in more complicated domains near 0. 2.1.3. Proof of the uniqueness. Now we prove that the solution y(x)... |

2 |
Series expansions of Painlevé transcendents in the neighbourhood of a fixed singular
- Shimomura
- 1982
(Show Context)
Citation Context ...y constants b, T11, T12, σ in such a way that − T12 4bT11 = a0, σ = σ0, for any fixed a0 and σ0. Remark 2.3. Other existence results for σ ∈ Cl \{] − ∞, 0] ∪ [1, +∞[} can be found in [IKSY] and [S1], =-=[S2]-=-, [S3]. For indices with Reσ ̸∈ [0, 1], the asymptotics obtained in these papers are valid in more complicated domains near 0. 2.1.3. Proof of the uniqueness. Now we prove that the solution y(x), x ∈ ... |

2 |
Solutions of the second and fourth differential equations of Painlev'e
- Umemura, Watanabe
- 1997
(Show Context)
Citation Context ...l solutions known at that moment, fit into the boundary of this fundamental region. The theory of the classical solutions of the Painlevé equations was developed by Umemura and Watanabe ([Um], [Um1], =-=[Um2]-=-, [Um3], [Wat]); in particular, all the oneparameter families of classical solutions of PVI were classified in [Wat]. Watanabe also proved that, loosely speaking, all the other classical solutions of ... |

2 |
Model Theory
- Weyl
- 2005
(Show Context)
Citation Context ... n be the least common multiple of n1, n2 and n3. Put: Lemma 1.18. The numbers ζ = 2 cos π n . xi = −2 cos π mi , i = 1, 2, 3, belong to the ring K0 of integers of the field K := Q[ζ]. ni Recall (see =-=[Wey]-=-) that K is the normal extension of Q generated by ζ and K0 is the ring of all the algebraic integer numbers of K, namely it consists of all the elements x ∈ K satisfying an algebraic equation of the ... |

1 |
endliche Gruppen linearer Transformationen einer Veranderlichen
- Gordan, Uber
(Show Context)
Citation Context ...e classify all the monodromy data of the algebraic solutions of PVIµ. To this end, we classify all the rational solutions of certain trigonometric equations using the method of a paper by Gordan (see =-=[Gor]-=-). In Section 1.4, we parameterize the monodromy data of PVIµ by ordered triples of planes in the three-dimensional space, considered modulo rotations. The structure of the analytic continuation of th... |

1 |
Talk at I.C.T.P
- Hitchin
- 1993
(Show Context)
Citation Context ...nonical form (1.20), are: M0 = ( ) 1 1 0 1 Mx = ( 1 ) 0 −1 1 M1 = ( ) 1 0 . 1 1 This solution was found in [Dub] in the implicit form (E.29). This was also obtained, independently, by N. Hitchin (see =-=[Hit2]-=-). To reduce (E.29) to the above form, we have to solve the cubic equation (E.29 b) with the substitution: t = Then the three roots of (E.29 b) are: 32(1 − 18s 2 + 81s 4 ) 27(1 + 9s 2 + 27s 4 + 27s 6 ... |

1 |
and Chazy Solutions to PVI Equation, preprint
- Mazzocco, Picard
- 1998
(Show Context)
Citation Context ...lts separately, postponing the investigation of the general case to another paper (in an effort to keep the paper within a reasonable size, we also postpone the study of the resonant case 2µ ∈ Z, see =-=[Ma]-=-). One of the motivations for the present publication is a nice geometrical interpretation of the structure of the analytic continuation (0.1), that seems to disappear in the general PVI equation. We ... |

1 |
Birational Automorphism Groups and Differential
- Umemura
- 1990
(Show Context)
Citation Context ... the classical solutions known at that moment, fit into the boundary of this fundamental region. The theory of the classical solutions of the Painlevé equations was developed by Umemura and Watanabe (=-=[Um]-=-, [Um1], [Um2], [Um3], [Wat]); in particular, all the oneparameter families of classical solutions of PVI were classified in [Wat]. Watanabe also proved that, loosely speaking, all the other classical... |

1 |
Solutions of the Third Differential Equation of Painlevé, preprint
- Umemura, Watanabe
- 1997
(Show Context)
Citation Context ...ions known at that moment, fit into the boundary of this fundamental region. The theory of the classical solutions of the Painlevé equations was developed by Umemura and Watanabe ([Um], [Um1], [Um2], =-=[Um3]-=-, [Wat]); in particular, all the oneparameter families of classical solutions of PVI were classified in [Wat]. Watanabe also proved that, loosely speaking, all the other classical solutions of PVI (i.... |