Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence

Michi-aki Inaba^{∗}, Katsunori Iwasaki^{†} and Masa-Hiko Saito^{‡}

July 7, 2003

Abstract

It is well known that the sixth Painlevé equation P_{VI} admits a group of Bäcklund transformations which is isomorphic to the affine Weyl group of type D_{4}^{(1)}. Although various aspects of this symmetry have been discussed by many authors, there still remains a basic problem yet to be considered, that is, the problem of characterizing the Bäcklund transformations in terms of Riemann-Hilbert correspondence. In this direction, we show that the Bäcklund transformations are just the pull-back of very simple transformations on the moduli of monodromy representations by the Riemann-Hilbert correspondence. This result gives a natural and clear picture of the Bäcklund transformations.

Key words: Bäcklund transformation, the sixth Painlevé equation, Riemann-Hilbert correspondence, isomonodromic deformation, affine Weyl group of type D_{4}^{(1)}.

2000 Mathematics Subject Classification: Primary 33E17; Secondary 37K20, 37J35

1 Introduction

It is well known that the sixth Painlevé equation P_{VI} admits a group of Bäcklund transformations which is isomorphic to the affine Weyl group of type D_{4}^{(1)}. Various aspects of this unexpectedly large symmetry have been discussed by many authors, e.g. Okamoto [21], Arinkin and Lysenko [2], Noumi and Yamada [18]. See also Conte and Musette [3], Fokas and Yorsos [5], Manin [16], Sakai [25], Watanabe [30] and others. However, there still seems to remain a basic problem yet to be considered with a special attention, namely, the problem of characterizing the Bäcklund transformations in terms of Riemann-Hilbert correspondence. This problem naturally arises from the work of Iwasaki [11, 12], which exploited the standpoint of studying P_{VI} based on a geometry of moduli spaces of monodromy representations via the Riemann-Hilbert correspondence; this standpoint had still previously been hinted at in Iwasaki [9, 10].

∗Faculty of Mathematics, Kyushu University, Hakozaki, Higashi-ku, Fukuoka 812-8581 Japan, E-mail address: inaba@math.kyushu-u.ac.jp

†Faculty of Mathematics, Kyushu University, Hakozaki, Higashi-ku, Fukuoka 812-8581 Japan, E-mail address: iwasaki@math.kyushu-u.ac.jp

‡Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657-8501 Japan, E-mail address: mhsaito@math.kobe-u.ac.jp

The Riemann-Hilbert correspondence is a map from a moduli of Fuchsian differential equations to a moduli of monodromies, associating to each Fuchsian equation its monodromy representation. On the other hand, P_{VI} is a differential equation on the moduli of Fuchsian equations that describes the isomonodromic deformation. Moreover, a Bäcklund transformation is a discrete transformation on the moduli of Fuchsian equations that commutes with the solution flow of P_{VI}. Therefore, it is very natural to ask what kind of discrete transformation on the moduli of monodromies is induced from a Bäcklund transformation by the Riemann-Hilbert correspondence. It is expected that the induced transformation looks simpler than the original, that is, a Bäcklund transformation looks simpler when viewed as a transformation on the moduli of monodromies. So the story should proceed in the other way round: Start with some simple transformations on the moduli of monodromies, pull them back via the Riemann-Hilbert correspondence to the moduli of Fuchsian equations and obtain the Bäcklund transformations.

The aim of this paper is to verify that the speculation above is true; the Bäcklund transformations are just the pull-back of almost identity transformations on the moduli of monodromies, namely, of those transformations which may alter local monodromy data but do not change “global monodromy data”. The affine Weyl group structure of the Bäcklund transformations will also be transparent from our point of view. We hope that our result and viewpoint give a natural and clear picture of the Bäcklund transformations for P_{VI}.

We remark that there is a geometric way to understand Bäcklund transformations of P_{VI} by means of families of spaces of initial conditions constructed by Okamoto [19]. As explained in [17, 27, 25, 22], there exists a family of open algebraic surfaces parametrized by the 4dimensional space K of local exponents, which correspond to the spaces of initial conditions of P_{VI}. Then the affine Weyl group W (D_{4}^{(1)}) acts on K in a natural way and the actions can be lifted to birational transformations of the total space of the family of Okamoto spaces. Saito and Umemura [23] pointed out that a Bäcklund transformation corresponding to a reflection of W (D_{4}^{(1)}) is nothing but a flop whose center is a family of (−2)-rational curves contained in Okamoto spaces lying over the reflection hyperplanes in K. Moreover, Saito and Terajima [24] clarified the relation between (−2)-curves in Okamoto spaces and Riccati solutions of P_{VI}. Since a flop of family of algebraic surfaces appears as a simultaneous resolution of rational double points, one can expect that transformations of reflection type are related to the simultaneous resolutions of versal deformations of rational double points.

On the other hand, Arinkin and Lysenko [1] introduced the moduli space of SL(2)-bundles with connections on ℙ^{1} parametrized by the local exponents and describe the Bäcklund transformations in [2]. From the viewpoint of the theory of isomonodromic deformations of flat connections (Fuchsian connections), the moduli space should corresponds to Okamoto spaces. Unfortunately, they treated the moduli space mostly as stack and restricted the parameter space to the complement of all reflection hyperplanes in order to avoid the reducible connections. In a forthcoming paper [8], we shall construct moduli spaces of stable parabolic connections over ℙ^{1} for all parameters. By using our moduli spaces, we can give a more geometric and conceptual picture of the Riemann-Hilbert correspondence and the Bäcklund transformations of P_{VI}.

2 Sketch of Main Result

We roughly state the main result of this paper. Here we are content with a sketch of outline, since a complete statement is possible only after all the necessary ingredients are prepared.

[figure: σ_{0}; σ_{1}; σ_{2}; σ_{3}; σ_{4}; C = ( \begin{matrix}2 & −1 & −1 & −1 & −1 \\ −1 & 2 & 0 & 0 & 0 \\ −1 & 0 & 2 & 0 & 0 \\ −1 & 0 & 0 & 2 & 0 \\ −1 & 0 & 0 & 0 & 2 \end{matrix} )]

Figure 1: Dynkin diagram and Cartan matrix of type D_{4}^{(1)}

Detailed explanations will be supplied in the subsequent sections.

The sixth Painlevé equation P_{VI} is a second order nonlinear ordinary differential equation

\begin{aligned}q_{xx} & = & \frac{1}{2} (\frac{1}{q} + \frac{1}{q − 1} + \frac{1}{q − x}) q_{x}^{2} − (\frac{1}{x} + \frac{1}{x − 1} + \frac{1}{q − x}) q \\ & + & \frac{q(q − 1)(q − x)}{2x^{2}(x − 1)^{2}} \{κ_{4}^{2} − κ_{1}^{2} \frac{x}{q^{2}} + κ_{2}^{2} \frac{x − 1}{(q − 1)^{2}} + (1 − κ_{3}^{2})\frac{x(x − 1)}{(q − x)^{2}}\} \end{aligned}  (2.1)

for an unknown function q = q(x), with complex parameters κ = (κ_{1}, κ_{2}, κ_{3}, κ_{4}). P_{VI} is written P_{VI}(κ) when the dependence on parameters κ should be indicated explicitly. In describing the Bäcklund transformations, it is convenient to think of the parameter space as an affine space

K = {κ = (κ_{0}, κ_{1}, κ_{2}, κ_{3}, κ_{4}) ∈ ℂ^{5} : 2κ_{0} + κ_{1} + κ_{2} + κ_{3} + κ_{4} = 1}.  (2.2)

We realize the affine Weyl group of type D_{4}^{(1)} as an affine reflection group acting on K,

W (D_{4}^{(1)}) = ⟨σ_{0}, σ_{1}, σ_{2}, σ_{3}, σ_{4}⟩,

where σ_{i} is the reflection in the hyperplane κ_{i} = 0. In terms of the Cartan matrix C = (c_{ij}) of type D_{4}^{(1)} (see Figure 1), the reflection σ_{i} is expressed as

σ_{i}(κ_{j}) = κ_{j} − κ_{i}c_{ij}.  (2.3)

A remarkable fact is that the affine Weyl group W (D_{4}^{(1)}) lifts up to a transformation group of P_{VI}. Namely, each reflection σ_{i} admits a lift s_{i} that is a transformation of P_{VI}(κ) to P_{VI}(σ_{i}(κ)), and the correspondence σ_{i} ↦ s_{i} induces an isomorphism between W (D_{4}^{(1)}) and

G = ⟨s_{0}, s_{1}, s_{2}, s_{3}, s_{4}⟩.  (2.4)

The group G is called the group of Bäcklund transformations for P_{VI}. The explicit form of s_{i} will be given in §4 after some Hamiltonian formalisms for P_{VI} are introduced in §3.

Let us turn to a monodromy problem. P_{VI}(κ) is the isomonodromic deformation equation for a class of second order linear Fuchsian differential equations on ℙ^{1} having four regular singular points with prescribed local exponents, where κ is used to assign local exponents. We denote by E_{t}(κ) the moduli of relevant Fuchsian equations with regular singular points at t = (t_{1}, t_{2}, t_{3}, t_{4}). The precise setting of E_{t}(κ) will be mentioned in §7. Let R_{t}(a) be the moduli of monodromy representations π_{1}(ℙ^{1}\{t_{1}, t_{2}, t_{3}, t_{4}}) → SL_{2}(ℂ), up to conjugacy, having prescribed local monodromy data a = (a_{1}, a_{2}, a_{3}, a_{4}), where a_{i} is defined to be the trace of the monodromy matrix M_{i} around the singular point t_{i}, namely,

a_{i} = Tr M_{i} (i = 1, 2, 3, 4).  (2.5)

Then the monodromy map or the Riemann-Hilbert correspondence

RH : E_{t}(κ) → R_{t}(a)  (2.6)

is defined by associating to each Fuchsian equation its monodromy representation class. In our setting, which will be detailed in §7, the correspondence of parameters κ ↦ a is given by

a_{i} = 2 cos πκ_{i} (i = 1, 2, 3),  a_{4} = −2 cos πκ_{4},  (2.7)

The minus sign for a_{4} is not a misprint; a_{4} is distinguished for a reason to be explained in §7.

In Iwasaki [11, 12], the representation space R_{t}(a) is realized as an affine cubic surface. Let us recall this construction. We introduce variables x = (x_{1}, x_{2}, x_{3}) by

x_{i} = Tr(M_{j}M_{k}) for {i, j, k} = {1, 2, 3}.  (2.8)

They are referred to as global monodromy data, since they carry a global information about monodromy; the product M_{j}M_{k} is the monodromy matrix along a “global” loop surrounding the two singular points t_{j} and t_{k} simultaneously. Let f(x, θ) be a polynomial defined by

f(x, θ) = x_{1}x_{2}x_{3} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} − θ_{1}x_{1} − θ_{2}x_{2} − θ_{3}x_{3} + θ_{4},  (2.9)

where the coefficients θ = (θ_{1}, θ_{2}, θ_{3}, θ_{4}) are given by

θ_{i} = \{ \begin{matrix}a_{i}a_{4} + a_{j}a_{k} & (i = 1, 2, 3), \\ a_{1}a_{2}a_{3}a_{4} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} − 4 & (i = 4). \end{matrix}  (2.10)

Then the representation space R_{t}(a) can be identified with an affine cubic surface

S(θ) = { x = (x_{1}, x_{2}, x_{3}) ∈ ℂ^{3} : f(x, θ) = 0}.  (2.11)

Therefore, the Riemann-Hilbert correspondence (2.6) is recast into

RH : E_{t}(κ) → S(θ),  (2.12)

where the correspondence of parameters κ ↦ θ is defined through (2.7) and (2.10). As a solution to the Riemann-Hilbert problem, the map (2.12) is an analytic isomorphism onto a Zariski open subset of S(θ). The following simple but fundamental observation is due to Terajima [29].

Lemma 2.1 Viewed as functions of κ, the coefficients θ are W (D_{4}^{(1)})-invariants.

The proof is just by calculations. Note that the local monodromy data a = (a_{1}, a_{2}, a_{3}, a_{4}) are invariants of the reflections σ_{1}, σ_{2}, σ_{3}, σ_{4}, but not of σ_{0}, so that the unexpected switch a ↦ θ has been necessary to obtain invariants of all five reflections σ_{i}. In this sense, the “true” local monodromy data might be attributed to θ rather than to a. Thanks to Lemma 2.1, each Bäcklund transformation s ∈ G induces an automorphism r of S(θ) such that the diagram

\begin{matrix}E_{t}(κ) & ⟶^{s} & E_{t}(σ(κ)) \\ ^{RH}↓ & & ↓^{RH} \\ S(θ) & ⟶_{r} & S(θ) \end{matrix}  (2.13)

is commutative, where σ ∈ W (D_{4}^{(1)}) is the transformation of parameters κ underlying the Bäcklund transformation s. Now a natural question occurs to us:

Problem 2.2 What is the transformation r ?

As one may expect naturally, this problem has a very simple solution:

Theorem 2.3 (Main Theorem) The transformation r in diagram (2.13) is just the identity; that is to say, the Bäcklund transformations are those transformations which cover the identity transformation on the moduli of monodromies through the Riemann-Hilbert correspondence.

The main result is far from trivial to the effect that the Bäcklund transformations, viewed as automorphisms of the cubic surface S(θ), are distinguished to be the identity from many other automorphisms. In fact, S(θ) admits a large number of automorphisms, even if the class is limited to algebraic automorphisms. In this respect, Iwasaki [11, 12] showed that the Riemann-Hilbert correspondence transforms the nonlinear monodromy of P_{VI} into a modular group action on S(θ) realized as a polynomial automorphism group on it. So S(θ) admits at least infinitely many algebraic automorphisms labeled by the elements of a modular group. It may be said that the Riemann-Hilbert correspondence is such a map that sends a transcendental object on the phase space of P_{VI} (the nonlinear monodromy of P_{VI}) to an algebraic object on the space of monodromies (a polynomial automorphism group on it), while collapsing an algebraic object on the former (the Bäcklund transformations) to a trivial object on the latter (the identity).

In this paper we shall present an analytic proof of Theorem 2.3, which is very simple in its essential idea (see §8) but requires some elaborate calculations in technicalities (see §9). It is desirable that there exists an alternative geometrical proof of the theorem.

This paper is organized as follows. Some Hamiltonian formalisms for P_{VI} are presented in §3, which enable us to describe the group of Bäcklund transformations in a symmetrical way in §4. The Hamiltonian systems are characterized as isomonodromic deformation equations of second order Fuchsian differential equations with four regular singular points in §5. The Bäckulnd transformations s_{1}, s_{2}, s_{3}, s_{4} are constructed as elementary gauge transformations in §6. The Riemann-Hilbert correspondence is formulated in §7. With these preliminaries, our main result, Theorem 2.3, is established in §8. The proof is based on the idea of coalescence of regular singular points along isomonodromic deformation. Here a key observation is that, by a coalescence procedure, a Fuchsian equation with four singular points degenerates into a Fuchsian equation with three singular points, and the difference of the two local exponents at the coalescent singular point is an invariant of the Bäcklund transformations; see Lemma 8.2.

In order for the idea to work, a certain technical lemma is needed concerning accumulation points of trajectories of the Hamiltonian system. This lemma is established in §9 (Lemma 9.1). The finial section, §10, is devoted to heuristics on finding Bäcklund transformations from our point of view. It provides a new way of discovering the hidden Bäcklund transformation s_{0}.

3 Hamiltonian Systems

In discussing Bäcklund transformations and monodromy problems related to P_{VI}, it is convenient to use a Hamiltonian system equivalent to the original single equation (2.1). For this purpose, we shall employ the following three systems:

(H1) a Hamiltonian system with single time variable; see (3.1); (H4) a completely integrable Hamiltonian system with four time variables; see (3.2); (H3) a completely integrable Hamiltonian system with three time variables; see (3.6).

We will flexibly use one or another of them depending upon contexts and purposes. The first system (H1) is the most conventional one which often appears in the literature:

\frac{∂q}{∂x} = \frac{∂h}{∂p},  \frac{∂p}{∂x} = −\frac{∂h}{∂q},  (3.1)

where the Hamiltonian h = h(q, p, x, κ) is given by

\begin{aligned}x(x − 1)h & = & q(q − 1)(q − x)p^{2} − \{(κ_{3} − 1)q(q − 1) + κ_{1}(q − 1)(q − x) + κ_{2}q(q − x)\}p \\ & & +κ_{0}(κ_{0} + κ_{4})(q − x). \end{aligned}

Indeed, equation (2.1) is recovered from system (3.1) by eliminating the variable p. A more symmetric description is feasible in terms of the second system (H4):

\frac{∂q}{∂t_{i}} = \frac{∂H_{i}}{∂p} ,  \frac{∂p}{∂t_{i}} = −\frac{∂H_{i}}{∂q} ,  (i = 1, 2, 3, 4),  (3.2)

with four time variables t = (t_{1}, t_{2}, t_{3}, t_{4}), the Hamiltonians H_{i} = H_{i}(q, p, t, κ) being given by

\begin{aligned}(t_{ij}t_{ik}t_{il})H_{i} & = & (q_{i}q_{j}q_{k}q_{l})p^{2} − \{(κ_{i} − 1)q_{j}q_{k}q_{l} + κ_{j}q_{i}q_{k}q_{l} + κ_{k}q_{i}q_{j}q_{l} + κ_{l}q_{i}q_{j}q_{k}\}p \\ & & +κ_{0}q_{i}\{(κ_{i} − 1)q_{i} + (κ_{j} + κ_{0})q_{j} + (κ_{k} + κ_{0})q_{k} + (κ_{l} + κ_{0})q_{l}\}, \end{aligned}  (3.3)

with q_{i} = q − t_{i}, t_{ij} = t_{i} − t_{j} and {i, j, k, l} = {1, 2, 3, 4}. We remark that (H1) is recovered from (H4) by a symplectic reduction. Indeed, we can observe that the diagonal action on t = (t_{1}, t_{2}, t_{3}, t_{4}) of the Möbius transformations lifts symplectically up to system (3.2) and the associated symplectic reduction takes (3.2) into (3.1) having the cross ratio

x = \frac{(t_{1} − t_{3})(t_{2} − t_{4})}{(t_{1} − t_{2})(t_{3} − t_{4})}  (3.4)

as the only time variable. (H3) is obtained from (H4) by letting t_{4} tend to infinity. Indeed,

H_{i}(q, p, t, κ) → h_{i}(q, p, t, κ) (i = 1, 2, 3),  H_{4}(q, p, t, κ) → 0 as t_{4} → ∞,  (3.5)

and system (3.2) reduces to the Hamiltonian system

\frac{∂q}{∂t_{i}} = \frac{∂h_{i}}{∂p} ,  \frac{∂p}{∂t_{i}} = −\frac{∂h_{i}}{∂q} ,  (i = 1, 2, 3).  (3.6)

with three time variables t = (t_{1}, t_{2}, t_{3}), where the Hamiltonians h_{i} = h_{i}(q, p, t, κ) are given by

(t_{ij}t_{ik})h_{i} = (q_{i}q_{j}q_{k})p^{2} − {(κ_{i} − 1)q_{j}q_{k} + κ_{j}q_{k}q_{i} + κ_{k}q_{i}q_{j}}p + κ_{0}(κ_{0} + κ_{4})q_{i},  (3.7)

with {i, j, k} = {1, 2, 3}. The symplectic reduction mentioned above amounts to taking the normalization t_{1} = 0, t_{2} = 1, t_{3} = x, t_{4} = ∞. Then system (3.6) with i = 3 yields (3.1).

4 Group of Bäcklund Transformations

Now we can state (recall) the explicit form of the Bäcklund transformations. For a symmetric description, we shall represent it in terms of the Hamiltonian system with four time variables (3.2). Necessary modifications for the systems (3.1) and (3.6) are a routine work. The Bäcklund transformation s_{i} corresponding to the reflection σ_{i} is expressed as

\begin{matrix}s_{0} & \{ \begin{matrix}s_{0}(t_{j}) & = & t_{j}, \\ s_{0}(q) & = & q + \frac{κ_{0}}{p} , \\ s_{0}(p) & = & p, \end{matrix} & (i = 0), \\ s_{i} & \{ \begin{matrix}s_{i}(t_{j}) & = & t_{j}, \\ s_{i}(q) & = & q, \\ s_{i}(p) & = & p − \frac{κ_{i}}{q − t_{i}} , \end{matrix} & (i = 1, 2, 3, 4). \end{matrix}  (4.1)

Noumi and Yamada [18] expressed (4.1) in a unified manner by introducing variables

q_{i} = \{ \begin{matrix}p & (i = 0), \\ q − t_{i} & (i = 1, 2, 3, 4). \end{matrix}  (4.2)

We remark that the fourth variable q_{4} was degenerating in [18], since they employed system (3.1) as their representation of P_{VI}, for which t_{4} = ∞ and hence q_{4} = ∞ was not observable. In any case, in terms of the variables q_{i}, formula (4.1) together with (2.3) is expressed as

s_{i}(κ_{j}) = κ_{j} − κ_{i}c_{ij},  s_{i}(t_{j}) = t_{j},  s_{i}(q_{j}) = q_{j} + \frac{κ_{i}}{q_{i}} u_{ij},  (4.3)

where C = (c_{ij}) is the Cartan matrix indicated in Figure 1 and u_{ij} is defined by

u_{ij} = {q_{i}, q_{j}} (i, j = 0, 1, 2, 3, 4).

Here {f, g} denotes the Poisson bracket,

\{f, g\} = \frac{∂f}{∂p} \frac{∂g}{∂q} − \frac{∂f}{∂q} \frac{∂g}{∂p}.

Explicitly, the matrix U = (u_{ij}) is given by

U = ( \begin{matrix}0 & 1 & 1 & 1 & 1 \\ −1 & 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \end{matrix} ) .

Remark 4.1 When t_{4} = ∞, formula (4.1) with i = 4 should be interpreted as

s_{4}(t_{j}) = t_{j},  s_{4}(q) = q,  s_{4}(p) = p.

Namely, the lift s_{4} of the reflection σ_{4} acts on (t, q, p) trivially.

The Bäcklund transformations s_{1}, s_{2}, s_{3}, s_{4} are not so difficult to understand; as will be seen in §6, they are none other than elementary gauge transformations of Fuchsian differential equations. Much more difficult and hence intriguing is the transformation s_{0}, whose existence is a sort of mystery. Hence the main body of this paper is devoted to understanding s_{0}.

The symmetry of P_{VI} with respect to the Bäcklund transformations is stated as follows.

Theorem 4.2 The Hamiltonian system (3.2) is invariant under the action of the Bäcklund transformation group G, that is, the action of G commutes with the Hamiltonian vector fields

X_{i} = \frac{∂}{∂t_{i}} + \frac{∂H_{i}}{∂p} \frac{∂}{∂q} − \frac{∂H_{i}}{∂q} \frac{∂}{∂p}.  (i = 1, 2, 3, 4)  (4.4)

Proof. The proof is by a direct check of condition (4.5) in the following lemma. □

Lemma 4.3 The transformation s_{i} commutes with the vector field X_{j} if and only if

s_{i}(H_{j}) − H_{j} + δ_{ij} \frac{κ_{i}}{q_{i}} is independent of (q, p),  (4.5)

where δ_{ij} is Kronecker’s delta symbol.

Proof. We prove the lemma for i = 1, 2, 3, 4. The case i = 0 can be treated in a similar manner. Write t̄_{j} = s_{i}(t_{j}), q̄ = s_{i}(q), p̄ = s_{i}(p) and H̄_{j} = s_{i}(H_{j}). The transformation rule (4.1) reads

t_{j} = t̄_{j},  q = q̄,  p = p̄ + \frac{κ_{i}}{q̄_{i}} ,

where q̄_{i} = q̄ − t̄_{i}. Applying the chain rule of partial differentiations, we have

\frac{∂}{∂t̄_{j}} = \frac{∂}{∂t_{j}} + δ_{ij} \frac{κ_{i}}{q_{i}^{2}} \frac{∂}{∂p},  \frac{∂}{∂q̄} = \frac{∂}{∂q} − \frac{κ_{i}}{q_{i}^{2}} \frac{∂}{∂p},  \frac{∂}{∂p̄} = \frac{∂}{∂p},

and hence

s_{i}(X_{j}) = \frac{∂}{∂t̄_{j}} + \frac{∂H̄_{j}}{∂p̄} \frac{∂}{∂q̄} − \frac{∂H̄_{j}}{∂q̄} \frac{∂}{∂p̄}
= (\frac{∂}{∂t_{j}} + δ_{ij} \frac{κ_{i}}{q_{i}^{2}} \frac{∂}{∂p}) + \frac{∂H̄_{j}}{∂p} (\frac{∂}{∂q} − \frac{κ_{i}}{q_{i}^{2}} \frac{∂}{∂p}) − (\frac{∂H̄_{j}}{∂q} − \frac{κ_{i}}{q_{i}^{2}} \frac{∂H̄_{j}}{∂p} ) \frac{∂}{∂p}
= \frac{∂}{∂t_{j}} + \frac{∂H̄_{j}}{∂p} \frac{∂}{∂q} − (\frac{∂H̄_{j}}{∂q} − δ_{ij} \frac{κ_{i}}{q_{i}^{2}} ) \frac{∂}{∂p}

Therefore, we have s_{i}(X_{j}) = X_{j} if and only if

\frac{∂H̄_{j}}{∂p} = \frac{∂H_{j}}{∂p} , \frac{∂H̄_{j}}{∂q} − δ_{ij} \frac{κ_{i}}{q_{i}^{2}} = \frac{∂H_{j}}{∂q} .

The last condition is equivalent to the assertion (4.5). The proof is complete. □

5 Isomonodromic Deformation

We discuss isomonodromy problems related to P_{VI}. Corresponding to the Hamiltonian systems with four time variables (3.2) and with three time variables (3.6), we set up two classes of Fuchsian differential equations, namely, (5.1) and (5.4) respectively. Then we consider their isomonodromic deformations. The Riemann-Hilbert correspondence for (5.4) will be formulated in §7; a similar formulation for (5.1) is possible, but omitted.

For the Hamiltonian system (3.2), we consider second order Fuchsian differential equations

\frac{d^{2}f}{dz^{2}} − u_{1}(z)\frac{df}{dz} + u_{2}(z)f = 0  (5.1)

on ℙ^{1} with six singular points at z = t_{1}, t_{2}, t_{3}, t_{4}, q, ∞ such that

(A1) z = t_{i} is a regular singular point with exponents 0 and κ_{i}; (A2) z = q is an apparent singular point with exponents 0 and 2; (A3) z = ∞ is a removable singular point, that is, (5.1) can be converted into a differential

equation without singular point at z = ∞ by some transformation of the form f = z^{−κ_{0}}g.

Here we recall the notion of an apparent singular point: A regular singular point q of a second order Fuchsian differential equation is said to be resonant if the difference of the two exponents at q is an integer. A resonant singular point q falls into two cases; one is the generic case where a solution basis at q involves the logarithmic function, and the other is the nongeneric case without logarithmic term in any solution basis. In the latter case, q is called an apparent singular point. We remark that a removable singular point mentioned in (A3) is nothing but an apparent singular point such that the difference of exponents is one. We also remark that the number κ_{0} in (A3) is uniquely determined, since equation (5.1) must satisfy Fuchs’ relation

2κ_{0} + κ_{1} + κ_{2} + κ_{3} + κ_{4} = 1.  (5.2)

Note that the affine linear relation (5.2) is a source of the parameter space K in (2.2). By conditions (A1), (A2), (A3), the coefficients of equation (5.1) must be of the form

u_{1}(z) = \frac{1}{z − q} + ∑_{i=1}^{4} \frac{κ_{i} − 1}{z − t_{i}},  u_{2}(z) = \frac{p}{z − q} − ∑_{i=1}^{4} \frac{H_{i}}{z − t_{i}} ,  (5.3)

where (q, p, t, κ) are free parameters, while H_{i} is a function of (q, p, t, κ), whose explicit form can be sought out by Frobenius’ method in the theory of Fuchsian differential equations; the result is that H_{i} = H_{i}(q, p, t, κ) must be the Hamiltonians (3.3) of the system (3.2).

It is well known that Painlevé-type equations arise from isomonodromic deformations of linear ordinary differential equations; see e.g. Fuchs [6], Schlesinger [26], Jimbo, Miwa and Ueno [13], Fokas and Its [4]. In the original case of P_{VI}, we have the following:

Theorem 5.1 The isomonodromic deformation of the Fuchsian differential equations (5.1) is described by the completely integrable Hamiltonian system (3.2).

Writing P_{VI} as a Hamiltonian system is due to Malmquist [15], Okamoto [20] and others. Iwasaki [9, 10] exploited an intrinsic (topological) reasoning for the Hamiltonian structure.

Similarly, for the Hamiltonian system (3.6), we consider Fuchsian differential equations

\frac{d^{2}f}{dz^{2}} − v_{1}(z)\frac{df}{dz} + v_{2}(z)f = 0  (5.4)

on ℙ^{1} with five singular points at z = t_{1}, t_{2}, t_{3}, q, ∞ such that

(B1) z = t_{i} is a regular singular point with exponents 0 and κ_{i}; (B2) z = q is an apparent singular point with exponents 0 and 2; (B3) z = ∞ is a regular singular point with exponents κ_{0} and κ_{4} + κ_{0}.

By conditions (B1), (B2), (B3), the coefficients of equation (5.4) must be of the form

v_{1}(z) = \frac{1}{z − q} + ∑_{i=1}^{3} \frac{κ_{i} − 1}{z − t_{i}},  v_{2}(z) = \frac{p}{z − q} − ∑_{i=1}^{3} \frac{h_{i}}{z − t_{i}} ,  (5.5)

where (q, p, t, κ) are free parameters, while h_{i} = h_{i}(q, p, t, κ) are the Hamiltonians (3.7) of the system (3.6). In view of (3.5), formula (5.5) is obtained by taking the limit t_{4} → 0 in (5.3). The counterpart of Theorem 5.1 for equations (5.4) is stated as follows.

Theorem 5.2 The isomonodromic deformation of the Fuchsian differential equation (5.4) is described by the completely integrable Hamiltonian system (3.6).

6 Gauge Transformations

For i = 1, 2, 3, 4, we can easily construct the Bäcklund transformation s_{i} as an elementary gauge transformation of the Fuchsian equation (5.1). Indeed, by the gauge transformation,

f = (z − t_{i})^{κ_{i}} f̄ (i = 1, 2, 3, 4),  (6.1)

equation (5.1) is transformed into another Fuchsian equation, say,

\frac{d^{2}f̄}{dz^{2}} − ū_{1}(z)\frac{df̄}{dz} + ū_{2}(z)f̄ = 0,

whose coefficients ū_{1} and ū_{2} must be of the same form as (5.3), say,

ū_{1}(z) = \frac{1}{z − q̄} + ∑_{j=1}^{4} \frac{κ̄_{j} − 1}{z − t̄_{j}},  ū_{2}(z) = \frac{p̄}{z − q̄} − ∑_{j=1}^{4} \frac{H̄_{j}}{z − t̄_{j}} ,  (6.2)

where H̄_{j} = H_{j}(q̄, p̄, t̄, κ̄). On the other hands, substituting (6.1) into (5.1) implies that

ū_{1} = u_{1} − \frac{2κ_{i}}{z − t_{i}} ,  ū_{2} = u_{2} − \frac{κ_{i}}{z − t_{i}} u_{1} + \frac{κ_{i}(κ_{i} − 1)}{(z − t_{i})^{2}}.  (6.3)

Then substituting (5.3) into (6.3) and comparing the result with (6.2), we have

κ̄_{j} = κ_{j} − κ_{i}c_{ij},  t̄_{j} = t_{j},  q̄ = q,  p̄ = p − \frac{κ_{i}}{q − t_{i}} ,

which is none other than the transformation s_{i} in (4.1) for i = 1, 2, 3, 4.

7 Riemann-Hilbert Correspondence

Let us formulate the Riemann-Hilbert correspondence for Fuchsian equations of the form (5.4). It is convenient to define it in such a manner that monodromy matrices have values in the special linear group SL_{2}(ℂ). To this end, by applying the gauge transformation

f = ϕF  with ϕ = (z − q) ∏_{i=1}^{3}(z − t_{i})^{κ_{i}/2},  (7.1)

equation (5.4) should be normalized into a Fuchsian equation

\frac{d^{2}F}{dz^{2}} − V_{1}(z)\frac{dF}{dz} + V_{2}(z)F = 0  (7.2)

with singular points at z = t_{1}, t_{2}, t_{3}, q, ∞ satisfying the following conditions:

(C1) z = t_{i} is a regular singular point with exponents ±κ_{i}/2; (C2) z = q is an apparent singular point with exponents ±1; (C3) z = ∞ is a regular singular point with exponents (3 ± κ_{4})/2.

Note that the coefficients of equation (7.2) and those of (5.4) are related by

V_{1} = v_{1} − 2\frac{ϕ′}{ϕ} , V_{2} = v_{2} − \frac{ϕ′}{ϕ} v_{1} + \frac{ϕ′′}{ϕ} .

Equation (7.2) is called the normal form of (5.4) and the two are often identified. Then the (normalized) monodromy of (5.4) will be defined to be the monodromy of its normal form (7.2).

Let E_{t}(κ) be the set of all Fuchsian equations of the form (5.4), or equivalently of their normal form (7.2), with a prescribed value of parameters κ and location of singular points t = (t_{1}, t_{2}, t_{3}, t_{4}), where t_{4} = ∞. Associating (q, p) to equation (5.4) yields an identification

E_{t}(κ) ≅ (ℂ \ {t_{1}, t_{2}, t_{3}}) × ℂ : (q, p)-space,  (7.3)

through which E_{t}(κ) is thought of as a complex manifold. In an algebro-geometrical framework developed by Inaba, Iwasaki and Saito [8], the space E_{t}(κ) is a Zariski open chart of a certain moduli space M_{t}(κ) of rank two stable parabolic bundles with Fuchsian connections on ℙ^{1}.

We proceed to moduli of monodromies. Let γ_{i} be a simple loop encircling the singular point t_{i} as in Figure 2, γ_{4} being a loop around t_{4} = ∞. Let M_{i} be the monodromy matrix along the loop γ_{i} of the Fuchsian equation (7.2). In view of (C1) and (C3), the matrix M_{i} has eigenvalues

exp(±π\sqrt{−1}κ_{i}) for i = 1, 2, 3;  − exp(±π\sqrt{−1}κ_{4}) for i = 4.  (7.4)

Hence det M_{i} = 1, viz., M_{i} ∈ SL_{2}(ℂ), and the trace a_{i} = Tr M_{i} is expressed as (2.7). Let R_{t}(a) be the space of monodromy representations of π_{1}(ℙ^{1} \ {t_{1}, t_{2}, t_{3}, t_{4}}) into SL_{2}(ℂ), up to conjugacy, whose monodromy matrices along the loop γ_{i} have trace a_{i}. Then the Riemann-Hilbert correspondence RH : E_{t}(κ) → R_{t}(a) in (2.6) is defined by associating to each Fuchsian equation (5.4) the SL_{2}(ℂ)-monodromy representation class of its normal form (7.2), where two parameters κ and a are related by (2.7). Then, composed with a natural map R_{t}(a) → S(θ),

[figure: t_{1}; t_{2}; t_{3}; γ_{1}; γ_{2}; γ_{3}; γ_{4}]

Figure 2: The loops γ_{1}, γ_{2}, γ_{3}, γ_{4}

where S(θ) is the cubic surface in (2.11), the Riemann-Hilbert correspondence is realized more concretely as a map into the cubic surface, namely, as the map RH : E_{t}(κ) → S(θ) in (2.12).

The construction above can be made relatively over the parameter spaces. Let E_{t} be the space of all Fuchsian equations of the form (5.4), or equivalently of their normal form (7.2), where the regular singular points are fixed at t = (t_{1}, t_{2}, t_{3}, t_{4}), t_{4} = ∞, but the parameters κ may vary. Let π_{1} : E_{t} → K be the natural projection associating to each equation in E_{t} its parameters κ. Note that E_{t}(κ) is a fiber of this projection. The space E_{t} is a Zariski open subset of the family M_{t} of moduli spaces M_{t}(κ) over K, which is constructed in Inaba, Iwasaki and Saito [8]. Let R_{t} be the space of all monodromy representations π_{1}(ℙ^{1} \ {t_{1}, t_{2}, t_{3}, t_{4}}) → SL_{2}(ℂ) up to conjugacy and let π_{2} : R_{t} → ℂ_{a}^{4} be the natural projection associating to each representation in R_{t} its local monodromy data a. Recall that R_{t} is the categorical quotient of the diagonal adjoint action of SL_{2}(ℂ) on SL_{2}(ℂ)^{3} and that π_{2} : R_{t} → ℂ_{a}^{4} is a family of affine cubic surfaces defined by the equation f(x, θ(a)) = 0 in (2.9) with the relation θ = θ(a) in (2.10). Moreover, let

S = { (x, θ) ∈ ℂ^{3} × ℂ^{4} : f(x, θ) = 0 },

and π_{3} : S → ℂ_{θ}^{4} be the natural projection down to parameters θ. Note that π : R_{t} → ℂ_{a}^{4} is obtained by the pullback of the family π_{3} : S → ℂ_{θ}^{4} by the finite morphism ℂ_{a}^{4} → ℂ_{θ}^{4}. Now the (relative) Riemann-Hilbert correspondence RH is formulated as a commutative diagram:

M_{t} ⟶^{RH} R_{t} ⟶^{φ} S
π_{1}↓  ↓^{π_{2}}  ↓^{π_{3}}
K ⟶_{u} ℂ_{a}^{4} ⟶_{v} ℂ_{θ}^{4},

where the maps u : κ ↦ a and v : a ↦ θ are defined by (2.7) and (2.10); or as its contraction:

\begin{matrix}M_{t} & ⟶^{RH} & S \\ ^{π_{1}}↓ & & ↓^{π_{3}} \\ K & ⟶_{vu} & ℂ_{θ}^{4}, \end{matrix}  (7.5)

8 Coalescence of Regular Singular Points

A principal idea for establishing our main result, Theorem 2.3, is to consider a coalescence of two regular singular points of Fuchsian equations, along an isomonodromic deformation. We first discuss the coalescence process only and then take the isomonodromic deformation into account. For this purpose we shall work with Fuchsian equations (5.4). Of course, working with Fuchsian equations (5.1) would lead us to the same conclusion; the latter choice would allow a more symmetrical discussion, but require somewhat heavier calculations. Here we employ (5.4), preferring simpler calculations at the cost of a minor symmetry breaking.

Lemma 8.1 By the coalescence t_{k} → t_{j} of the singular points t_{j} and t_{k}, equation (5.4) with (5.5) degenerates into a Fuchsian equation with singular points at z = t_{i}, t_{j}, q, ∞,

\frac{d^{2}f}{dz^{2}} − w_{1}(z)\frac{df}{dz} + w_{2}(z)f = 0,  (8.1)

whose coefficients w_{1}(z) and w_{2}(z) are expressed as

w_{1}(z) = \frac{1}{z − q} + \frac{κ_{i} − 1}{z − t_{i}} + \frac{κ_{j} + κ_{k} − 2}{z − t_{j}},  (8.2)
w_{2}(z) = \frac{p}{z − q} − \frac{L}{z − t_{i}} + \frac{M}{z − t_{j}} + \frac{N}{(z − t_{j})^{2}} ,  (8.3)

where L, M and N are given by

\begin{aligned}t_{ij}^{2}L & = & q_{i}q_{j}^{2}p^{2} − \{(κ_{i} − 1)q_{j} + (κ_{j} + κ_{k})q_{i}\}q_{j}p + κ_{0}(κ_{0} + κ_{4})q_{i}, \\ t_{ij}^{2}M & = & q_{i}q_{j}^{2}p^{2} − \{q_{i}^{2} + (κ_{j} + κ_{k} − 2)q_{i}q_{j} + κ_{i}q_{j}^{2}\}p + κ_{0}(κ_{0} + κ_{4})q_{i}, \\ t_{ij}N & = & q_{i}q_{j}^{2}p^{2} − \{(κ_{j} + κ_{k} − 1)q_{i} + κ_{i}q_{j}\}q_{j}p + κ_{0}(κ_{0} + κ_{4})q_{j}. \end{aligned}

Proof. Formula (8.2) readily follows from the first formula of (5.5) by letting t_{k} → t_{j}. Next we shall show formula (8.3). In view of (3.7), we notice that there is a relation h_{i} + h_{j} + h_{k} = p and that L, M and L are defined in such a manner that

L = lim_{t_{k}→t_{j}} h_{i},  M = L − p,  N = − lim_{t_{k}→t_{j}} t_{jk}h_{j}.

In particular, we have h_{j} + h_{k} = p − h_{i} → p − L = −M as t_{k} → t_{j}. Hence we have

\begin{aligned}\frac{h_{i}}{z − t_{i}} & → & \frac{L}{z − t_{i}} , \\ \frac{h_{j}}{z − t_{j}} + \frac{h_{k}}{z − t_{k}} = \frac{h_{j} + h_{k}}{z − t_{k}} + \frac{t_{jk}h_{j}}{(z − t_{j})(z − t_{k})} & → & −\frac{M}{z − t_{j}} − \frac{N}{(z − t_{j})^{2}} , \end{aligned}

as t_{k} → t_{j}. Therefore the second formula of (5.5) leads to (8.3). The proof is complete. □

Note that the local exponents at z = t_{i}, q, ∞ and the apparentness of q are preserved in the process of coalescence (5.4) → (8.1). Let λ_{1} and λ_{2} be the exponents of (8.1) at the coalescent singularity z = t_{j}, which are the roots of the quadratic equation λ^{2} − (κ_{j} + κ_{k} − 1)λ + N = 0. We are interested in its discriminant, that is, the squared difference between λ_{1} and λ_{2}:

Δ = (λ_{1} − λ_{2})^{2} = (κ_{j} + κ_{k} − 1)^{2} − 4N.  (8.4)

Then the following lemma will play a key role in establishing our main theorem.

[figure: t_{i}; t_{j}; t_{k}; γ_{i}; γ_{j}γ_{k}; γ_{4}; coalescence]

Figure 3: Coalescence of singular points t_{k} → t_{j}

Lemma 8.2 (Key Lemma) The discriminant Δ is G-invariant.

Proof. Since t_{ij} is G-invariant, it is sufficient to show the G-invariance of D = −t_{ij}Δ. Using t_{ij} = q_{j} −q_{i} we have D = (κ_{j} +κ_{k}−1)^{2}(q_{i}−q_{j})+4t_{ij}N. Since t_{ij}N is a polynomial of (q_{i}, q_{j}, p, κ), so is D; explicitly, D = D(q_{i}, q_{j}, p, κ) is given by

\begin{aligned}D(q_{i}, q_{j}, p, κ) & = & (κ_{j} + κ_{k} − 1)^{2}(q_{i} − q_{j}) \\ & + & 4q_{j}[q_{i}q_{j}p^{2} − \{(κ_{j} + κ_{k} − 1)q_{i} + κ_{i}q_{j}\}p + κ_{0}(κ_{0} + κ_{4})]. \end{aligned}  (8.5)

For the moment, we think of (q_{i}, q_{j}, p, κ) as independent variables, namely, we do not assume the relations (4.2) and (5.2), and put q_{k} = q_{j}. Then a direct check shows that

s_{0}(D) − D = −4κ_{0}(2q_{j}p + κ_{0})η,
s_{i}(D) − D = 4κ_{i}q_{j}η,
s_{j}(D) − D = 4κ_{j}q_{j}η,
s_{k}(D) − D = 4κ_{k}q_{j}η,
s_{4}(D) − D = 0,

where η = 2κ_{0} + κ_{1} + κ_{2} + κ_{3} + κ_{4} − 1. Here we must take q_{k} = q_{j} and Remark 4.1 into account in the evaluations of s_{k}(D)−D and s_{4}(D)−D, respectively. Hence, under Fuchs’ relation (5.2), D is an invariant of s_{0}, s_{1}, s_{2}, s_{3}, s_{4} and so is an invariant of G. The proof is complete. □

The normalization procedure (5.4) → (7.2) passes through the coalescence process, leading to a parallel normalization of equation (8.1). The corresponding gauge transformation is

f = ψF  with ψ = (z − q)(z − t_{i})^{κ_{i}/2}(z − t_{j})^{(κ_{j}+κ_{k})/2},  (8.6)

by which equation (8.1) is normalized into a Fuchsian equation

\frac{d^{2}F}{dz^{2}} − W_{1}(z)\frac{dF}{dz} + W_{2}(z)F = 0,  (8.7)

having singular points at z = t_{i}, t_{j}, q, ∞, such that (D1) z = t_{i} is a regular singular point with exponents ±κ_{i}/2; (D2) z = q is an apparent singular point with exponents ±1; (D3) z = ∞ is a regular singular point with exponents (3 ± κ_{4})/2; (D4) z = t_{j} is a regular singular point with exponents (−1 ± \sqrt{Δ})/2.

It follows from (D4) that the trace of the monodromy matrix around z = t_{j} is −2 cos π\sqrt{Δ}.

We proceed to taking the isomonodromic deformation into account and shall complete the proof of Theorem 2.3, leaving a certain technical issue to the next section.

Proof of Theorem 2.3. The notation of §2 will be retained in the subsequent discussions, except that the transformation r in diagram (2.13) will be written s upon identifying r and s. Establishing Theorem 2.3 amounts to showing that

s(x_{i}) = x_{i} (i = 1, 2, 3),  (8.8)

where x_{i} = x_{i}(q, p, t, κ), which was defined in (2.8), is a holomorphic function of (q, p, t, κ) and s(x_{i}) is understood to be x_{i}(s(q), s(p), s(t), s(κ)). It is sufficient to show (8.8) for the generators s = s_{j} of the Bäcklund transformation group G. But this claim for j = 1, 2, 3, 4 is trivial, since s_{j} in this case is a gauge transformation as seen in §6, which does not change monodromy representation and hence the value of x_{i}. So the substantial part of the proof is to verify the claim for j = 0, though the reasoning below will carry over for j = 1, 2, 3, 4 as well. Moreover, we have only to establish it for generic values of κ; the validity of (8.8) for each κ in a dense open subset leads to the validity for every κ due to the continuous dependence of x_{i} = x_{i}(t, q, p, κ) upon κ. As such a dense open condition on κ, we employ

1 − κ_{j} − κ_{k} ∈ ℂ \ ℝ (1 ≤ j < k ≤ 3),  (8.9)

for a certain technical reason to be explained in Lemma 8.3 below.

The idea of coalescence along isomonodromic deformation proceeds as follows. Given a point (q, p) ∈ (ℂ\{t_{1}, t_{2}, t_{3}})×ℂ, let (q(t), p(t)) be the solution trajectory to the Hamiltonian system (3.6) starting from the initial point (q, p). Since (3.6) describes the isomonodromic deformation of Fuchsian equations (7.2), their monodromy matrix along the loop γ_{j}γ_{k} (see Figure 3),

M_{j}M_{k} = (M_{j}M_{k})(q(t), p(t), t, κ),

is independent of t on the trajectory (q(t), p(t)). Assume that the trajectory admits an accumulation point (q̄, p̄) as t_{k} tends to t_{j} along a curve. As t_{k} tends to t_{j} in such a manner, the Fuchsian equation (7.2) degenerates into (8.7), where (q, p) in (8.7) should be replaced by (q̄, p̄). In this process, the role of the matrix M_{j}M_{k} changes from being the global monodromy matrix of (7.2) along the loop γ_{j}γ_{k} into being the local monodromy matrix of (8.7) at the coalescent singular point z = t_{j}. The former role gives x_{i} = Tr(M_{j}M_{k}), while the latter yields Tr(M_{j}M_{k}) = −2 cos π\sqrt{Δ} by virtue of the remark after (D4). Therefore we have

x_{i} = −2 cos π\sqrt{Δ}.  (8.10)

We can apply the same procedure with the initial point (q, p) replaced by (s(q), s(p)). Since (s(q(t)), s(p(t))) has an accumulation point (s(q̄), s(p̄)), the same reasoning as above yields

s(x_{i}) = −2 cos π\sqrt{s(Δ)}.  (8.11)

Thanks to Lemma 8.2, which asserts that Δ is an s-invariant, (8.10) and (8.11) lead to the desired equality (8.8). This completes the proof of Theorem 2.3.

For the true end of the proof, however, there is an extra issue yet to be argued, namely, the existence of the accumulation point (q̄, p̄). More precisely, the existence must be assured in such a manner that (q̄, p̄) is located in a general position. To understand what this means, we should notice that the arguments leading to Lemmas 8.1 and 8.2 are valid only under the condition that the apparent singular point q is different from the other singular points t_{i}, t_{j}, ∞ and that p is a finite complex number. In the current situation, this condition should be applied to (q, p) = (q̄, p̄) as well as to (q, p) = (s(q̄), s(p̄)). For the generators s = s_{0}, s_{1}, s_{2}, s_{3}, s_{4} of G, formula (4.1) implies that the condition applied to them is expressed as

q̄ ∈ ℂ \ \{t_{i}, t_{j}\},  p̄ ∈ ℂ \ \{0\},  q̄ + \frac{κ_{0}}{p̄} ∈ ℂ \ \{t_{i}, t_{j}\}.  (8.12)

Thus an accumulation point (q̄, p̄) is said to be in a general position if it satisfies condition (8.12). Intuitively the existence of such a point is quite likely, but logically nontrivial. It is at this stage that the generic condition (8.9) on κ is used to provide the following:

Lemma 8.3 Suppose that κ = (κ_{0}, κ_{1}, κ_{2}, κ_{3}, κ_{4}) ∈ K satisfies condition (8.9). Then there exists an open subset V of (ℂ \ {t_{1}, t_{2}, t_{3}}) × ℂ such that for every (q, p) ∈ V, the solution trajectory (q(t), p(t)) to the Hamiltonian system (3.6) starting from the initial point (q, p) admits an accumulation point (q̄, p̄) satisfying condition (8.12) as t_{k} tends to t_{j} along a curve.

Lemma 8.3 guarantees the existence of an accumulation point (q̄, p̄) in a general position only when the initial point (q, p) belongs to an open subset V. But this is enough for proving (8.8). Indeed, by Lemma 8.3, equation (8.8) is valid at least for (q, p) ∈ V. Then the unicity theorem for holomorphic functions implies that it remains valid for every (q, p), since x_{i} = x_{i}(q, p, t, κ) is holomorphic in (q, p) and the space E_{t}(κ) ≅ (ℂ \ {t_{1}, t_{2}, t_{3}}) × ℂ is connected. To establish Theorem 2.3, it only remains to prove Lemma 8.3. This final task will be done in §9. □

9 Accumulation Points in a General Position

We shall establish Lemma 8.3 for j = 1 and k = 3, namely, for the coalescence process t_{3} → t_{1}; due to the symmetry in t_{i}, t_{j}, t_{k}, the other cases can be treated in a similar manner. For this purpose, we may work with the Hamiltonian system (3.1) with single time variable x instead of the system (3.6) with three time variables t = (t_{1}, t_{2}, t_{3}). In view of (3.4), letting t_{3} → t_{1} means x → 0. With these remarks, Lemma 8.3 is reduced to the following:

Lemma 9.1 Suppose that κ = (κ_{0}, κ_{1}, κ_{2}, κ_{3}, κ_{4}) ∈ K satisfies a generic condition

1 − κ_{1} − κ_{3} ∈ ℂ \ ℝ.  (9.1)

Then there exists a 2-parameter family of solutions to the Hamiltonian system (3.1),

(q(x, c), p(x, c)),  c = (c_{1}, c_{2}) ∈ U,  (9.2)

where U is an open subset of ℂ^{2}, such that the following conditions are satisfied:

(1) The correspondence c = (c_{1}, c_{2}) ↦ (q(x_{0}, c), p(x_{0}, c)) defines a biholomorphic map of U onto an open subset V of (ℂ \ {0, 1}) × ℂ, where x_{0} is a point in ℂ \ {0, 1}.

(2) The solution (9.2) admits an accumulation point (q(c), p(c)) as x → 0 such that

q(c) ∈ ℂ \ \{0, 1\},  p(c) ∈ ℂ \ \{0\},  q(c) + \frac{κ_{0}}{p(c)} ∈ ℂ \ \{0, 1\}.  (9.3)

Note that condition (9.3) corresponds to (8.12) in the last section. For a proof of Lemma 9.1, we utilize a result by Takano [28] and Kimura [14], who established a reduction theorem and constructed a 2-parameter family of solutions to P_{VI} around its fixed singular points. We recall their construction in a manner suitable for our purpose. Put

\begin{aligned}E(r, ρ) & = & \{ (x, Q, P) ∈ ℂ^{3} : |x| < r, |Q| < ρ, |xP | < ρ, |QP | < ρ \}, \\ E(ρ) & = & \{ (Q, P) ∈ ℂ^{2} : |Q| < ρ, |QP | < ρ \}. \end{aligned}

Then the following lemma is an easy consequence of Theorems 1 and 2 in Takano [28].

Lemma 9.2 Suppose that κ = (κ_{0}, κ_{1}, κ_{2}, κ_{3}, κ_{4}) ∈ K satisfies condition (9.1). Then there exist positive constants r, ρ > 0 and a unique canonical transformation

q = b(x, Q, P), p = a(x, Q, P)

that reduces system (3.1) into a Hamiltonian system with Hamiltonian

h_{0}(x, Q, P) = {(QP)^{2} + (1 − κ_{1} − κ_{3})(QP)}/x,  (9.4)

where b(x, Q, P) and a(x, Q, P) are holomorphic functions in E(r, ρ) such that

|b(0, Q, P) − Q| ≤ M|Q|^{2},  |a(0, Q, P) − P | ≤ M  for (Q, P) ∈ E(ρ),  (9.5)

with some positive constant M; we may and shall assume that M > 2.

The Hamiltonian system with Hamiltonian (9.4) has a first integral QP and is settled as

Q(x, c) = c_{1}x^{λ}, P (x, c) = c_{2}x^{−λ} with λ = λ(c) = 1 − κ_{1} − κ_{3} + 2c_{1}c_{2},  (9.6)

where c = (c_{1}, c_{2}) ∈ ℂ^{2} is an arbitrary constant. Then Lemma 9.2 asserts that

q(x, c) = b(x, Q(x, c), P (x, c)),  p(x, c) = a(x, Q(x, c), P (x, c)),  (9.7)

yields a solution to system (3.1), provided that (x, Q(x, c), P (x, c)) ∈ E(r, ρ), namely,

0 < |x| < r,  |Q(x, c)| < ρ,  |xP (x, c)| < ρ,  |c_{1}c_{2}| < ρ.

In terms of arg x and log |x|, the second and third conditions are expressed as in Table 1, the exhibition of which is divided into five cases according to the values of Re λ.

Proof of Lemma 9.1. By assumption (9.1), we can choose a number ρ_{0} so that

0 < ρ_{0} < min{ρ, |Im(1 − κ_{1} − κ_{3})|/2, 1}.  (9.8)

case  λ  arg x  log |x|

1  Re λ > 1 (Im λ)(arg x) < L_{0}  L_{2}(arg x) < log |x| < L_{1}(arg x)

2  Re λ = 1 (Im λ)(arg x) < L_{0}  log |x| < L_{1}(arg x)

3 0 < Re λ < 1 no constraints log |x| < min{L_{1}(arg x), L_{2}(arg x)}

4  Re λ = 0 (Im λ)(arg x) > L_{0}  log |x| < L_{2}(arg x)

5  Re λ < 0 (Im λ)(arg x) > L_{0}  L_{1}(arg x) < log |x| < L_{2}(arg x)

L_{0} = (Re λ) log(ρ/|c_{2}|) + (Re λ − 1) log(ρ/|c_{1}|),
L_{1}(α) = \frac{(Im λ)α + log(ρ/|c_{1}|)}{Re λ},
L_{2}(α) = \frac{(Im λ)α − log(ρ/|c_{2}|)}{Re λ − 1}.

Table 1: The conditions |Q(x, c)| < ρ and |xP (x, c)| < ρ

Out of later necessity, if κ_{0} ≠ 0, we take ρ_{0} so as to satisfy an additional condition

ρ_{0} < \frac{|κ_{0}|}{2},  (9.9)

which should be neglected if κ_{0} = 0. As the open subset U mentioned in (9.2), we put

U = { c = (c_{1}, c_{2}) ∈ ℂ^{2} : 0 < |c_{1}c_{2}| < ρ_{0} }.  (9.10)

For each c ∈ U, let D(c) be the set of those points on the universal covering of 0 < |x| < r which satisfy the conditions in Table 1. By (9.6), (9.8) and (9.10), one has Im λ(c) ≠ 0, which implies that D(c) is a nonempty domain that contains a curve tending to the origin x = 0. Then the solution (q(x, c), p(x, c)) in (9.7) makes sense on the domain D(c).

For each c ∈ U, we shall show that (q(x, c), p(x, c)) admits an accumulation point (q(c), p(c)) satisfying condition (9.3) as x → 0 along a curve in D(c). Let μ be a number such that

0 < μ < \frac{|c_{1}c_{2}|}{M(|κ_{0}| + 8)}.  (9.11)

(Re λ)(Im λ) < 0  Re λ = 0  (Re λ)(Im λ) > 0

Figure 4: The curve γ: a spiral or a line segment

and consider a curve in the x-plane defined by

γ = \{ x : (Re λ) log |x| − (Im λ)(arg x) = log \frac{μ}{|c_{1}|}, |x| < \frac{μρ}{|c_{1}c_{2}|} \}

Note that (9.8), (9.10), (9.11) and M > 2 imply μ < ρ. Along the curve γ one has

|Q(x, c)| = μ,  |P (x, c)| = \frac{|c_{1}c_{2}|}{μ}.  (9.12)

In particular, |Q(x, c)| < ρ and |xP (x, c)| < ρ, and hence the curve γ lies in the domain D(c). The shape of γ is indicated in Figure 4; if Re λ ≠ 0, it is a spiral curve winding around and tending to the origin, while if Re λ = 0, it is a line segment terminating at the origin.

In view of (9.12), as x tends to the origin along the curve γ, there exists an accumulation point (Q(c), P (c)) ∈ E(ρ) of (Q(x, c), P (x, c)) such that Q(c)P (c) = c_{1}c_{2} and

|Q(c)| = μ,  |P (c)| = \frac{|c_{1}c_{2}|}{μ}.  (9.13)

Accordingly, as x → 0 along γ, (q(x, c), p(x, c)) has an accumulation point (q(c), p(c)) with

q(c) = b(0, Q(c), P (c)), p(c) = a(0, Q(c), P (c)).

We shall show that (q(c), p(c)) satisfies the desired property (9.3). Hereafter (Q(c), P (c)) and (q(c), p(c)) will be abbreviated to (Q, P) and (q, p) respectively. The arguments below are rather technical, but the idea itself is very simple: If μ > 0 is sufficiently small, then (9.13) implies that Q is sufficiently small but not zero and P = c_{1}c_{2}/Q is sufficiently large but finite. Now (9.5) means that (q, p) equals (Q, P) up to a first order error term. Then one can show that q is small but not zero, p is large but finite and also q + κ_{0}/p is small but not zero.

We verify the first and second conditions of (9.3). Note that (9.8), (9.10) and (9.11) yield

M < \frac{|c_{1}c_{2}|}{2μ} < \frac{1}{2μ}, μ < \frac{1}{2}.

Then (9.5) and (9.13), combined with these inequalities, lead to

\begin{aligned}|q| & ≤ & |Q| + |q − Q| & ≤ & |Q| + M|Q|^{2} & = & (1 + Mμ)μ & < & \frac{3μ}{2} & < & \frac{3}{4}, \\ |q| & ≥ & |Q| − |q − Q| & ≥ & |Q| − M|Q|^{2} & = & (1 − Mμ)μ & > & \frac{μ}{2} & > & 0, \\ |p| & ≥ & |P | − |p − P | & ≥ & |P | − M & = & \frac{|c_{1}c_{2}|}{μ} − M & > & \frac{|c_{1}c_{2}|}{2μ} & > & 0, \end{aligned}

which verifies the first and second conditions of (9.3) as desired.

We proceed to verify the third condition of (9.3). We may assume κ_{0} ≠ 0; for otherwise the third condition is the same as the first one. We observe that

|p| > \frac{|c_{1}c_{2}|}{2μ},  |c_{1}c_{2}| < \frac{|κ_{0}|}{2},  μ < \frac{|c_{1}c_{2}|}{2|κ_{0}|}.  (9.14)

Indeed, the first one is already seen in the last paragraph, the second one is obtained from (9.9) and (9.10), and the last one follows from (9.11) and M > 2, respectively. Let

R = (q + \frac{κ_{0}}{p} ) − (1 + \frac{κ_{0}}{c_{1}c_{2}} ) Q = (q − Q) + (\frac{κ_{0}}{p} − \frac{κ_{0}}{P} ) = (q − Q) + \frac{κ_{0}(P − p)}{pP}.

By applying (9.5), (9.13), (9.14) and (9.11) successively, R is estimated as

|R| ≤ |q − Q| + \frac{|κ_{0}||p − P |}{|p||P |} ≤ M|Q|^{2} + \frac{|κ_{0}|M}{|p||P |}

< Mμ^{2} + \frac{2|κ_{0}|Mμ^{2}}{|c_{1}c_{2}|^{2}} = \frac{Mμ^{2}(|c_{1}c_{2}|^{2} + 2|κ_{0}|)}{|c_{1}c_{2}|^{2}}

< \frac{Mμ^{2}|κ_{0}|(|κ_{0}| + 8)}{4|c_{1}c_{2}|^{2}} < \frac{|κ_{0}|μ}{4|c_{1}c_{2}|}.

On the other hand, the second inequality in (9.14) leads to

\begin{aligned}|1 + \frac{κ_{0}}{c_{1}c_{2}} | & ≤ & \frac{|κ_{0}|}{|c_{1}c_{2}|} + 1 & < & \frac{|κ_{0}|}{|c_{1}c_{2}|} + \frac{|κ_{0}|}{2|c_{1}c_{2}|} & = & \frac{3|κ_{0}|}{2|c_{1}c_{2}|}, \\ |1 + \frac{κ_{0}}{c_{1}c_{2}} | & ≥ & \frac{|κ_{0}|}{|c_{1}c_{2}|} − 1 & > & \frac{|κ_{0}|}{|c_{1}c_{2}|} − \frac{|κ_{0}|}{2|c_{1}c_{2}|} & = & \frac{|κ_{0}|}{2|c_{1}c_{2}|}. \end{aligned}

These preliminary estimates, (9.12) and the third inequality in (9.14) yield

\begin{aligned}|q + \frac{κ_{0}}{p} | & = & |(1 + \frac{κ_{0}}{c_{1}c_{2}} ) Q + R| ≤ |1 + \frac{κ_{0}}{c_{1}c_{2}} | |Q| + |R| < \frac{3|κ_{0}|μ}{2|c_{1}c_{2}|} + \frac{|κ_{0}|μ}{4|c_{1}c_{2}|} = \frac{7|κ_{0}|μ}{4|c_{1}c_{2}|} < \frac{7}{8}, \\ |q + \frac{κ_{0}}{p} | & = & |(1 + \frac{κ_{0}}{c_{1}c_{2}} ) Q + R| ≥ |1 + \frac{κ_{0}}{c_{1}c_{2}} | |Q| − |R| > \frac{|κ_{0}|μ}{2|c_{1}c_{2}|} − \frac{|κ_{0}|μ}{4|c_{1}c_{2}|} = \frac{|κ_{0}|μ}{4|c_{1}c_{2}|} > 0. \end{aligned}

Hence the third condition of (9.3), i.e., assertion (2) of Lemma 9.1 is verified. The proof of assertion (1) is omitted, since it is a standard inverse function argument, in which the open set U will be replaced by a smaller one, if necessary. The proof is complete. □

10 Finding Bäcklund Transformations

The Bäcklund transformations s_{i} have been found by Okamoto [21], Arinkin and Lysenko [2], Noumi and Yamada [18] and others by various methods. In any case, s_{1}, s_{2}, s_{3}, s_{4} are easy to find, since they are elementary gauge transformations as constructed in §6. But things are different with the transformation s_{0}; it cannot be a gauge transformation of rank two differential equations, since it does change local monodromy data. Now we wish to propose another way of finding s_{0} from our point of view.

Our idea is to revisit Lemma 8.2 together with formula (8.5), which is a key observation in this paper. It asserts that the discriminant Δ in (8.4), or equivalently the function D in (8.5), is G-invariant. Here we write D = D(q, p, t, κ), since D is a polynomial of (q, p, t, κ) with t_{j} = t_{k}. An essence of the reasoning in §8 is the implication that if there is an invariance relation

D(Q, P, t, σ_{0}(κ)) = D(q, p, t, κ),  (10.1)

then one has x_{i}(Q, P, t, σ_{0}(κ)) = x_{i}(q, p, t, κ) for the global monodromy data x_{i} = Tr(M_{j}M_{k}). Now our principle — a Bäcklund transformation should be a pull-back of the identity transformation on the moduli of global monodromy data — suggests that relation (10.1) would give us a Bäcklund transformation (q, p) ↦ (Q, P). So it must be promising to find out (Q, P) as a function of (q, p) satisfying (10.1). This thought brings our attention to the difference

E = E(Q, P; q, p; t, κ) = D(Q, P, t, σ_{0}(κ)) − D(q, p, t, κ).  (10.2)

Consider E as a polynomial of (t_{i}, t_{j}) and let E_{mn} denote the coefficient of the term t_{i}^{m}t_{j}^{n} in E. If E ≡ 0, each coefficient E_{mn} must vanish. We especially take a look at E_{12} and E_{11}:

\{ \begin{matrix}E_{12} & = & 4(p − P)(p + P), \\ E_{11} & = & 4p(κ_{j} + κ_{k} − 1 − 2qp) + 4P (κ_{i} + κ_{4} + 2QP). \end{matrix}

As is easily seen, the system of equations E_{12} = E_{11} = 0 has two solutions

Q = q + \frac{κ_{0}}{p} ,  P = p,  (10.3)
Q = q + \frac{κ_{0} + κ_{i} + κ_{4}}{p} ,  P = −p.  (10.4)

The first solution (10.3) is none other than the Bäcklund transformation s_{0} we seek. In this case we can check that substituting (10.3) into (10.2) yields E ≡ 0.

On the other hand, the second solution (10.4) does not imply E ≡ 0 in general. Indeed, (10.4) leads to E_{02} = 4(2κ_{i} − κ_{j} − κ_{k} + 1)p, and hence E does not vanish unless κ satisfies

2κ_{i} − κ_{j} − κ_{k} + 1 = 0.  (10.5)

Assume that (10.5) is the case. Then, substituting (10.4) and (10.5) into (10.2), we have E = 2κ_{i}(κ_{4} − κ_{i})(κ_{4} + κ_{i})/p. Therefore, (10.4) yields E ≡ 0 if and only if κ also satisfies

κ_{i}(κ_{4} − κ_{i})(κ_{4} + κ_{i}) = 0.  (10.6)

We wonder whether the transformation (10.4) has any meaning under the restriction of parameters, (10.5) and (10.6). But this point will not be touched in this paper.

It is worth considering whether there is any other Bäcklund transformation than those are already known. As to this question, the following proposition shows that the Bäcklund transformations are exhausted by the known ones, even if the class is enlarged to the analytic category.

Proposition 10.1 Let π : M_{t} → K be the family of moduli spaces of stable parabolic bundles with connections in §7. Let σ be an analytic automorphism of K such that θ(κ) = θ(σ(κ)) for κ ∈ K and let s be a bimeromorphic automorphism of M_{t} such that σπ = πs. Assume that for general values of κ ∈ K, the analytic isomorphisms s_{κ} : M_{t}(κ) → M_{t}(σ(κ)) induce the identity on S(θ(κ)) via the Riemann-Hilbert correspondence (7.5). Then σ ∈ W (D_{4}^{(1)}) and s is a known Bäcklund transformation, the unique lift of σ.

Proof. From the invariant-theoretical argument in Terajima [29], it is not difficult to see that the analytic quotient K/W (D_{4}^{(1)}) is biholomorphic to the complex 4-space ℂ_{θ}^{4} with coordinates θ = (θ_{1}, θ_{2}, θ_{3}, θ_{4}). In particular, distinct W (D_{4}^{(1)})-orbits in K have distinct values of θ. This observation shows that any analytic automorphism σ of K such that θ(κ) = θ(σ(κ)) for κ ∈ K is necessarily an element of W (D_{4}^{(1)}). Then clearly the transformation s is obtained as the unique lift of σ relative to the Riemann-Hilbert correspondence. □

We conclude this paper by putting some questions to ourselves. We were able to characterize the Bäcklund transformations in a natural manner in terms of the Riemann-Hilbert correspondence based on the following nice observation: The difference of the two local exponents at a coalescent regular singular point happens to be an invariant of the Bäcklund transformations (Lemma 8.2). Does this phenomenon occur just by chance or more universally? Do similar phenomena occur for other Painlevé equations than P_{VI} or for Garnier systems? If so, do they help us find Bäcklund transformations for those equations?

Acknowledgments. The authors would like to thank Kyoichi Takano for helpful discussions.

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