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Wild monodromy action on the character variety of
the fifth Painleve equation
Martin Klimes
January 12, 2018
Abstract
The (wild) character variety of the Painleve V equation is constructed from thecharacter varieties of Painleve VI through confluence. The known description ofthe action of the monodromy of Painleve VI on the corresponding character varietyis used to construct a 1-parameter group action on the character variety of PainleveV. It is expected that this action corresponds to the wild monodromy of PainleveV, that is the group generated by the monodromy, the exponential torus and thenonlinear Stokes operators of the corresponding Hamiltonian foliation.
Key words: Painleve equations, character varieties, Stokes phenomenon, wild mon-odromy, confluence
1 Introduction
One approach to the Painleve equations is through the study of isomonodromic defor-mations of meromorphic connections on CP1. It is well known that the sixth Painleveequation PV I governs the isomonodromic deformations of Garnier systems (2×2 trace-less linear differential systems) with 4 Fuchsian singular points 0, 1, t,∞ (t being theindependent variable). It also governs isomonodromic deformations of 3×3 systems inOkubo normal form, and their dual through the Laplace transform, 3×3 systems inBirkhoff normal form with an irregular singularity of Poincare rank 1 at the origin (ofeigenvalues 0,1, t) and a Fuchsian singularity at the infinity. The fifth Painleve equationis obtained from PV I through the change of the independent variable t 7→ 1+ εt, accom-panied by a suitable change of parameters, and the confluence ε → 0. It is equivalentto the isomonodromy problems for the corresponding limit confluent linear systems. Inthis setting, the Riemann-Hilbert correspondence can be interpreted as a map betweenspace of local solutions of the given Painleve equation with fixed values of parameters,and the space of generalized monodromy representations (monodromy & Stokes data)with fixed local multipliers. This latter space is called the character variety. Hence,the Riemann-Hilbert correspondence conjugates the transcendental flow of the Painleveequations to a locally constant flow on the moduli spaces of monodromy representations(character varieties) [IIS06]. It is well-known that one can use this correspondence inorder to describe the non-linear monodromy action of the Painleve equation, i.e. theaction of analytic continuation of solutions along loops in the independent variable (timevariable). In the case of PV I this description of the non-linear monodromy action onthe equation is classical: it corresponds to a braid group action on the character variety[Iwa03, Iwa02]. The study of this action has many important applications, such as aconstruction and classification of algebraic solutions of PV I [Boa10, LT08], or a proof ofits irreducibility in the sense of Malgrange [CL09]. For the equation PV the monodromyalone does not carry enough information about the equation, one must also take in ac-count the nonlinear Stokes phenomenon of the underlying Hamiltonian foliation at itsirregular singularity at the infinity and introduce the notion of a wild monodromy.
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The goal of this article is to connect the character variety of PV with that of PV Ithrough a confluence, and to describe the wild monodromy action on it. Both the studyof the confluence of character varieties and the description of the wild monodromy ofPV seems to be completely new.
This article fits into the general program of study of “wild monodromy” actions onthe character varieties of isomonodromic deformations of linear differential systems thatwas sketched in [PR15].
1.1 The Painleve equations.
The Painleve equations originated from the effort of classifying all second order ordinarydifferential equations of type q′′ = R(q′, q, t), with R rational, possessing the so calledPainleve property which controls the ramification points of solutions:
Painleve property: Each germ of a solution can be meromorphically continued alongany path avoiding the singular points of the equation. In other words, solutions cannothave any other movable singularities other than poles.
Painleve and Gambier [Gam10] produced a list of 50 canonical forms of equations towhich any such equation can be reduced. Aside of equations corresponding to variousclassical special functions, the list contained six new families of equations, PI , . . . , PV I ,whose general solutions provided a new kind of special functions. In many aspects theymay be regarded as non-linear analogues of the hypergeometric equations [IKSY91].The equation PV I is a mother equation for the other Painleve equations, which can beobtained through degenerations and confluences following the diagram [OO06]
PD6
III → PD7
III → PD8
III↗ ↗↘ ↘
PV I → PV → P degV P JMII → PI↘ ↘↗ ↗
PIV → PFNII
This article studies some aspects of the confluence PV I → PV through the Riemann-Hilbert correspondence.
Each of the Painleve equations is equivalent to a time dependent Hamiltonian system
dq
dt=
∂H•(q, p, t)
∂p,
dp
dt= −∂H•(q, p, t)
∂q, • = I, . . . , V I, (1)
with a polynomial Hamiltonian function, from which it is obtained by reduction to thevariable q.
The general form of the sixth Painleve equation is
PV I : q′′ =1
2
(1
q+
1
q − 1+
1
q − t
)(q′)2 −
(1
t+
1
t− 1+
1
q − t
)q′
+q(q − 1)(q − t)
2 t2(t− 1)2
[(ϑ∞−1)2 − ϑ2
0
t
q2+ ϑ2
1
(t− 1)
(q − 1)2+ (1−ϑ2
t )t(t− 1)
(q − t)2
],
where ϑ = (ϑ0, ϑt, ϑ1, ϑ∞) ∈ C4 are complex constants. Its Hamiltonian function isgiven by
HV I =1
t(t− 1)
[q(q − 1)(q − t)p2 −
(ϑ0(q − 1)(q − t) + ϑ1q(q − t) + (ϑt − 1)q(q − 1)
)p
+1
4
((ϑ0 + ϑ1 + ϑt − 1)2 − ϑ2
∞
)(q − t)
].
The Hamiltonian system of PV I has three simple (regular) singular points on the Rie-mann sphere CP1 at t = 0, 1,∞.
2
The fifth Painleve equation PV is obtained from PV I by the change of the indepen-dent variable
t 7→ 1 + εt, ϑt 7→1
ε, ϑ1 7→ −
1
ε+ ϑ1 + 1,
which sends the three singularities to − 1ε , 0,∞, and then by taking the limit ε→ 0, so
that the two simple singular points − 1ε and∞merge into a double (irregular) singularity
at the infinity:1
PV : q′′ =( 1
2q+
1
q−1
)(q′)2− 1
tq′+
(q−1)2
2t2
((ϑ∞− 1)2q− ϑ
20
q
)+ (1 + ϑ1)
q
t− q(q+1)
2(q−1).
The same confluence procedure applied to HV I produces the corresponding polynomialHamiltonian:
HV =1
t
[q(q − 1)2p2 −
(ϑ0(q − 1)2 + ϑ1q(q − 1) + tq
)p+
1
4
((ϑ0 + ϑ1)2 − ϑ2
∞
)(q − 1)
].
1.2 Monodromy of PV I.
Let MV I(ϑ) be the (q, p, t)-space of the foliation given the Hamiltonian system of PV Iwith a parameter ϑ = (ϑ0, ϑt, ϑ1, ϑ∞); and letMV I,t(ϑ) be its fibre with respect to theprojection (q, p, t) 7→ t, which is transversal to the foliation. Naively,MV I,t(ϑ) would bethe (q, p)-space C2, but one must take in account that the solutions have poles, thereforeto define the foliation, it is necessary to add a bunch of points “at infinity”. Hence, thegood space
MV I,t(ϑ) = the Okamoto space of initial conditions,
is rather some complex surface (an 8-point blow-up of the Hirzebruch surface Σ2 minusan anti-canonical divisor, see [Oka79] and [IIS06], section 3.6).
The Painleve property of PV I means that there is a well-defined non-linear mon-odromy action on the foliation
π1(CP1 r {0, 1,∞}, t0)→ Aut(MV I,t0(ϑ)),
where the flow of the vector field (1) along each loop in the t-space CP1r{0, 1,∞} witha base-point t0 gives rise to an automorphism of MV I,t(ϑ). (See Figure 1.)
Figure 1: The Painleve flow and its monodromy.
In case of PV I this monodromy carries a lot of important information about the equa-tion. For example, algebraic solutions of PV I , which are very important in physical ap-plications, correspond to finite orbits of the monodromy action (see e.g. [Boa10, LT08])).The monodromy is also very important from the point of view of the differential Galois
1The equation (1.1) is the fifth equation of Painleve with a parameter η1 = −1. A general form of
this equation is obtained by a further change of variable t 7→ −η1t. The degenerate case P degV withη1 = 0 which has only a regular singular point at ∞ is not considered here.
3
theory: the Galois D-groupoid of Malgrange of the foliation is “generated” by the mon-odromy. Cantat and Loray [CL09] have showed the irreducibility of PV I in the sense ofMalgrange, i.e. maximality of the Malgrange groupoid (which implies transcendentnessof general solutions [Cas09]), by studying the action of this monodromy.
Similarly, there is a well-defined monodromy action also on PV and other Painleveequations. However the confluence of the two singularities − 1
ε ,∞ in the confluent familyof PV I into a single singularity∞ of PV means that an important part of the monodromygroup is lost and therefore also the information carried by it. This lost informationreappears in the (non-linear) Stokes phenomenon at the irregular singularity.
1.3 Nonlinear Stokes phenomenon of PV .
In the local coordinate s = t−1 at t =∞, the Hamiltonian system of PV has the form
s2 dds
( qp
)= s
(ϑ0
(ϑ0+ϑ1)2−ϑ2∞
4
)+
(1+ωs
−1−ωs
)( qp
)− s
(−(ϑ0+ϑ1)q2+2pq−4pq2+2pq3
2(ϑ0+ϑ1)pq−p2+4p2q−3p2q2
), (2)
ω = −2ϑ0 − ϑ1 (cf. [Tak83]), with a resonant irregular singularity at s = 0.
Proposition 1 (Takano [Tak83], Yoshida [Yos85]). The above system can be brought toa formal normal form
s2 dds
(v1
v2
)= (1 + ωs+ 4v1v2s)
( v1
−v2
), (3)
through an essentially unique sectoral change of coordinates
(q, sp) = T±(s, v1, sv2, v1v2),
bounded and analytic on a cone in the v-variable above two sectors in the s-domain
s ∈ Σ± = {| arg s∓ π2 | < π − ε0, |s| < ρ0}, |v1| < ρ1, |v2s| < ρ2, |v1v2| < ρ3,
for any 0 < ε < π2 and some ρl > 0, l = 0, . . . , 3, which has an analytic expan-
sion T±(s, v1, sv2, v1v2) =∑i,j,k T
(i,j,k)± (s) vi1(sv2)j(v1v2)k with respect to (v1, v2s, v1v2)
whose coefficients T(i,j,k)± (s) have asymptotic expansions in s.
The system (3), which is Hamiltonian for the Hamiltonian function
H(v1, v2, s) =(1 + ωs)v1v2 + 2s(v1v2)2
s2
with respect to the standard symplectic form dv1 ∧ dw2, has a 2-parameter family ofsolutions
v1 = C1e− 1s sω+4C1C2 ,
v2 = C2e1s s−ω−4C1C2 ,
C1, C2 ∈ C, and a first integral C1C2 = v1v2. The constants
C1 = v1e1s s−ω−4v1v2 ,
C2 = v2e− 1s sω+4v1v2 ,
are local first integrals of the system, defining coordinates on the space of leaves of thefoliation (3) above each of the sectors Σ±.
4
The Hamiltonian system (3) is preserved by the one-parameter group of symplecto-morphisms
E(κ) : (v1, v2) 7→ (v1κ,v2
κ), κ ∈ C∗,
which are given by the flow of the Hamiltonian vector field
v1∂v1 − v2∂v2
commuting with (3). We call the action of the family of automorphisms E(κ) “anexponential torus action”.
Remark 2. A recent works of Bittmann on a formal and sectoral normalization [Bit15,Bit16] of this kind of systems provides an improved result compared to [Yos85]. It inves-tigates transformations by a similar change of variables, defined on a polydisc in (v1, v2)above the sectors Σ±, providing also a finer formal classification in which the term v1v2
in (3) is replaced by a function c(v1v2). It also shows that the corresponding coefficients
T(i,j,k)± (s) in the analytic expansion of the transformation are not only asymptotic to
those of the formal transformation but they are in fact their Borel sums.
The pullback of the coordinates (C1, C2) on the foliation (3) through the sectoraltransformation T± then provides local coordinates on the restriction of the originalfoliation (2) to Σ±. On each of the two intersections of the overlapping sectors Σ+,Σ−,the map T± ◦ T−1
∓ that associates to a leaf of (2) of coordinate (C1, C2) on Σ+ theleaf of the same coordinates on Σ− (or vice-versa) provides an automorphism of thefoliation. These are the non-linear Stokes operators. Changing on one of the sectorsΣ± the coordinate by the action of E(κ), which happens for example if one changes thedetermination of log s on the sector, will produce a different pair of Stokes operators.This means that there is a parametric family of (pairs of) Stokes operators
T± ◦ E(κ) ◦ T ◦−1∓
depending on the parameter κ ∈ C∗. Let us remark, that unlike the monodromy, theseStokes operators are defined only locally on the foliation.
The (pseudo)-group action on the foliation of PV generated by both the usual mon-odromy and the Stokes operators is the wild monodromy of the title of this paper.
The central question of our work is how can one relate this wild monodromy of PVwith the monodromy of PV I?
It is not very surprising that the only monodromy operators that converge are thosecorresponding to the loops in π1(CP1 r {− 1
ε , 0,∞}, t0) that persist when ε → 0 as
loops in π1(CP1 r {0,∞}, t0), while those corresponding to the vanishing loops diverge.Indeed, for each ε 6= 0 the monodromy group is discretely generated, while for ε = 0 thewild monodromy group is generated by a continuous family. However, we believe thatthe wild monodromy is obtained by the accumulation of the monodromies when ε→ 0.
Conjecture 3. The monodromy of PV I in the confluent family possesses limits alongthe sequences {εn}n∈±N,
1εn
= 1ε0
+ n. (4)
and the monodromy accumulates toward the family of such limits, depending on theparameter ε0. This family generates the wild monodromy group of PV .
The proof of this conjecture will be the subject of a subsequent work. This articleshows how one can use it to describe the wild monodromy of PV .
5
1.4 Contents of the article.
The content of this article is the following: In Section 2, we describe confluence ofsingularities of linear systems: two simple (Fuchsian) singularities merging into a double(irregular) singularity, which provides a motivation for and a guide to understanding ofthe above Conjecture.
In Section 3, we will study the confluence PV I → PV through the Riemann–Hilbertcorrespondence, using the usual approach of isomonodromic deformations of 2×2 linearsystems with four Fuchsian singularities, and the description of confluence of Section2. We will show that the character variety (i.e. the space of equivalence classes ofmonodromy representations) corresponding to the equation Painleve V is obtained fromthe character varieties of PV I by birational changes of coordinates. Under the premise ofthe Conjecture we provide an explicit description of the action of the wild monodromyon the constructed character variety of PV . These actions correspond to limits of someof the modular actions on the character variety of PV I along different sequences εn → 0(4).
In the Appendix, we show how one can obtain the same description of the charactervarieties but this time using isomonodromic deformations 3×3 linear systems in Birkhoffnormal form with an irregular singularity of Poincare rank 1 at the origin and a Fuchsiansingularity at infinity. These systems can be obtained from the 2×2 linear systems by amiddle convolution and a Laplace transform [Bo14, HF07]. The degeneration from PV Ito PV corresponds to a confluence of eigenvalues. This description follows up on theprevious study by Boalch [Boa05].
2 Confluence of singularities in linear systems
Consider a confluence of two regular singular points to a non-resonant irregular singularpoint in a family of linear systems
dy
dx=
A(x, ε)
x(x− ε)y, y ∈ C2, (5)
with A a 2×2 traceless complex matrix depending analytically on (x, ε) ∈ (C×C, 0),such that A(0, 0) 6= 0 has two distinct eigenvalues ±λ(0)(0). For each fixed parameter εthe local differential Galois group (the Picard-Vessiot group) of the system is the Zariskiclosure of the group generated by
ε 6= 0: the monodromy around the two singular points 0 and ε,
ε = 0: by the Stokes operators, the formal monodromy, and the exponential torus[MR91, SP03].
The question is how are these two different descriptions related?
The above kind of confluence (5) have been studied by many authors, including Gar-nier [Gar19], Ramis [Ram89], Schafke [Sch98], Duval [Du98], Glutsyuk [Glu99], Zhang[Zha96], etc., here we follow mainly the approach of Parise [Par01] and of Lambert andRousseau [LR12] (see also [HLR13, Kli16]).
Proposition 4 (Parise, Lambert, Rousseau). 2
The family (5) is formally equivalent, by means of a formal transformation y = T (x, ε)φ,T ∈ GL2(C[[x, ε]]), to a model family of diagonal systems
dφ
dx=
Λ(x, ε)
x(x− ε)φ, Λ(x, ε) = Λ(0)(ε) + xΛ(1)(ε) =
(λ(x,ε) 0
0 −λ(x,ε)
), (6)
2See [HLR13] for a more general version of the theorem.
6
where ±λ(x, ε) = ±(λ(0)(ε) + xλ(1)(ε)
)are the eigenvalues of A(x, ε) modulo x(x− ε).
There also exist bounded analytic normalizing transformations between the systems
y = T±(x, ε)φ
defined on a family of domains (Figure 2), where ε is from one of two sectors of openingalmost 2π bisected by ±λ(0)(0)R+ respectively, and x from the domain spanned by thereal trajectories of the vector fields
±eiω x(x− ε)2λ(x, ε)
∂x
within some fixed neighborhood of the origin, where ω ∈] − π2 ,
π2 [ is allowed to vary
continuously as long as the phase portrait of the vector field does not bifurcate. Thelimit domain for ε = 0 consists of two overlapping sectors of opening > π, and therestriction of T±(x, 0) to each of them is the Borel sum of the formal transformationT (x, 0).
The diagonal model family (6) has a canonical fundamental matrix solution
Φ(x, ε) =
(x−εx
)Λ(0)(ε)ε (x− ε)Λ(1)(ε), ε 6= 0
e−Λ(0)(0)
x xΛ(1)(0), ε = 0,
(7)
whose monodromy matrices around the points 0, ε, and both 0, ε are respectively N0,N−1
0 N and N :
N0(ε) = e−2πiΛ(0)(ε)
ε , N(ε) = e2πiΛ(1)(ε).
Correspondingly, the family (5) has a canonical fundamental matrix solutions
Y ±(x, ε) = T±(x, ε)Φ(x, ε),
In order for Y ±(x, ε) to have a limit when ε→ 0, one has to split the domain of T± intwo parts by a cut between the points 0, ε, and take Φα, Φα+π two branches of Φ (7),such that Φα+π is a counter-clockwise continuation of Φα along a path exterior withrespect to the two singularities (the α here is a direction of summability of T (x, 0)).
There are three intersections of between the two parts of each domain and corre-spondingly three connection matrices between Y ±α = T±Φα and Y ±α+π = T±Φα+π, seeFigure 2:
- S+R (ε) (resp. S−R (ε)) on the intersection sector attached to x+
R = ε (resp. x−R = 0):Φ is continued analytically there, therefore it corresponds to the ramification of T±.S+R (0) = S−R (0) is a Stokes matrix of the limit system.
- N(ε)S+L (ε) (resp. N(ε)S−L (ε)) on the intersection sector attached to x+
L = 0 (resp.x−L = ε): N(ε) is the monodromy matrix of Φ around both x±L , x
±R, S+
L (0) = S−L (0)is a Stokes matrix of the limit system.
- N0(ε)−1N(ε) (resp. N0) on the center part for ε 6= 0: the transformation T± hasno ramification there, therefore this connection matrix comes purely from Φ as itsmonodromy around the point x±R.
The matrices S±L (ε), S±R (ε), called the “unfolded Stokes matrices”, converge when ε→ 0,and so does the formal monodromy N(ε). The formal monodromy matrix N0(ε) is thesource of divergence.
7
Figure 2: The fundamental matrix solutions of (5) and the unfolded Stokes matricesbetween them for λ(0)(0) ∈ R+.
The Differential Galois group: Let K be the field of meromorphic functions ofthe variable x on a fixed small neighbourhood of 0, equipped with the differentiationddx . For a fixed small ε, the differential Galois group of the system (5) is the group ofautomorphisms of the differential field K〈Y ±• (·, ε)〉, generated by the components of thefundamental matrix solution Y ±• , over K. Since its action must preserve the system (5),it acts on it by automorphisms. Namely it acts by left multiplication on the solutionspace (i.e. on the foliation associated to the system). Fixing a fundamental matrixsolution, say Y ±• , then each such automorphism is represented by a right multiplicationof Y ±• by a constant invertible matrix, hence the differential Galois group is representedby an (algebraic) subgroup of GL2(C) acting on the right.
- For ε 6= 0 from the corresponding sector the monodromy matrices of Y +α+π around
the points x+L = 0 and x+
R = ε in positive direction are
M+L (ε) = N0(ε)S+
L (ε), M+R (ε) = S+
R (ε)N(ε)N0(ε)−1. (8)
The representation of the differential Galois group of (5) with respect to Y +α+π is a
Zariski closure of the group generated by these two monodromies. The matrix N0(ε)diverges when ε→ 0, therefore so do the monodromies M+
L (ε) and M+R (ε).
- For ε = 0, the representation of the differential Galois group of (5) with respect toY +α+π is a Zariski closure of the wild monodromy group (cf. [MR91]), i.e. of the group
generated by the Stokes matrices SL(0), SR(0), and the exponential torus (which inthis case contains also the formal monodromy):
E(κ) = ( κ κ−1 ) , κ ∈ C∗,
whose action φ 7→ φE(κ) is infinitesimally generated by the vector field φ1∂φ1−φ2∂φ2
.For any element E(κ) of the exponential torus the pair of the connection matrices
8
between Y +α+π and Y +
α E(κ) on the two intersection sectors are
E(κ)−1N(0)SL(0), SR(0)E(κ)
which belong to the representation of the Galois group. We call them wild mon-odromy matrices. The infinite family of such pairs of wild monodromy matricesdepending on κ ∈ C∗ generates the wild monodromy group.
While the monodromies M+L (ε), M+
R (ε) diverge, there are particular sequences of valuesof ε tending to 0, along which the formal monodromy N0(ε) stays constant and thereforesuch particular limits exist. Supposing, for example, that the eigenvalues ±λ = ±(λ(0)+xΛ(1)) are independent of ε, let ε0 ∈ C r {0}, and define εn by 1
εn= 1
ε0+ n
λ(0) , n ∈ N,then N0(εn) = N0(ε0), which belongs to the exponential torus, and therefore
limn→+∞
M+L (εn) = N0(ε0)SL(0), lim
n→+∞M+R (εn) = SR(0)N(0)N0(ε0)−1
is a pair of wild monodromy matrices. The family of such limits depending on ε0generates the representation of the Galois group of the limit system.
Proposition 5. The wild monodromy group of the limit system (5) with ε = 0, isgenerated by the family of operators to which the monodromies accumulate when ε→ 0.
Remark 6. What we have essentially done here is that we have replaced the divergent
term e−2πiλ(0)(ε)
(ε) in the formal monodromy matrix N0(ε) by an independent parameter
κ = e−2πi
λ(0)(ε0)
(ε0) coming from the exponential torus.
3 The character variety of PV I and the monodromyaction on it
In this section we recall the usual approach to PV I through isomonodromic deformationsof 2×2 traceless linear systems with four Fuchsian singularities on CP1, the Riemann-Hilbert correspondence between Fuchsian systems and their monodromy representa-tions, the Fricke formulas for the character variety of PV I and for the modular groupaction on it. The main references for this part are the articles of Iwasaki [Iwa03] and ofInaba, Iwasaki and Saito [IIS06].
Notation 7. A triple of indices (i, j, k) will always denote a permutation of (0, t, 1),and (i, j, k, l) will denote a permutation of (0, t, 1,∞).
3.1 Isomonodromic deformations of Garnier systems and theRiemann-Hilbert correspondence
The sixth Painleve equation PV I(ϑ) with a parameter ϑ = (ϑ0, ϑt, ϑ1, ϑ∞) ∈ C4 governsisomonodromic deformations of Garnier systems (traceless 2×2 linear systems) with fourFuchsian singularities on CP1
dφ
dx=[A0(t)
x+At(t)
x− t+A1(t)
x− 1
]φ (9)
with the residue matrices Al ∈ sl2(C) having ±ϑl2 as eigenvalues. In general (if each Alis semi-simple and the system is irreducible), one can write
Ai =
(zi + ϑi
2 −uizizi+ϑiui
−zi − ϑi2
), i = 0, t, 1, −A0 −At −A1 = A∞ =
(ϑ∞2
−ϑ∞2
).
9
The isomonodromicity of such system is expressed by the Schlesinger equations
dA0
dt=
[At, A0]
t,
dAtdt
=[A0, At]
t+
[A1, At]
t− 1,
dA1
dt=
[At, A1]
t− 1, (10)
corresponding to the integrability conditions on the logarithmic connection in variablesx, t
∇(x, t) = d−[A0(t) d log(x) +At(t) d log(x−t) +A1(t) d log(x−1)
]on the rank 2 trivial vector bundle. Denoting A(x, t) =
(aij(x, t)
)the matrix of the
system (9), then, if the systems is irreducible, the 1-form a12(x, t) dx is non-null, so itmust have a unique zero at some point x = q(t). Denoting p(t) = a11(q, t)+ ϑ0
2q + ϑt2(q−t) +
ϑ1
2(q−1) , the Schlesinger equations (10) are equivalent to the Hamiltonian system of PV I
(1) [Oka80, JM81].3
Choosing a germ of a fundamental matrix solution Φ(x, t) of the system, one has amonodromy representation (anti-homomorphism)
ρ : π1(CP1 r {0, t, 1,∞}, x0)→ SL2(C),
such that the analytic continuation Φ(x, t) along a path γ defines another fundamentalmatrix solution Φ(x, t)ρ(γ). The conjugation class of such monodromy representationby SL2(C) is independent of the choice of Φ. The word isomonodromic means that theconjugation class of monodromy is locally constant with respect to t, or equivalently thatthere exists a fundamental matrix solution Φ(x, t) whose actual monodromy is locallyindependent of t [Bol97].
The Riemann–Hilbert correspondence is given by the monodromy map between thespace of linear systems (9) with prescribed poles and local eigenvalue data ±ϑl2 , moduloconjugation in SL2(C) on one side, to the space of monodromy representations withprescribed local exponents el,
1el
el := eπϑl , l ∈ {0, t, 1,∞}, (11)
modulo conjugation in SL2(C), on the other side (see [IIS06] for much more precisesetting of the correspondence). Therefore it can be also translated as a correspondencebetween solutions of PV I and equivalence classes of monodromy representations, betweenthe Painleve flow (1) on the moduli space of linear systems (9) and the locally constant“isomonodromic” flow on the moduli space of monodromy representations.
3.2 The character variety of PV I
Given a representation
ρ : π1(CP1 r {0, t, 1,∞}, x0)→ SL2(C),
let γ0, γt, γ1, γ∞ be simple loops in the x-space around 0, t, 1,∞ respectively such thatγ0γtγ1γ∞ = id, and Ml = ρ(γl) the corresponding monodromy matrices
M∞M1MtM0 = I.
3The change of variable
(φ1φ2
)=
(1 0a11 a12
)(φ1dφ1dx
)transforms the system (9) to a second order
equation for φ1, which has an additional apparent singularity at the point x = q(t) at which the abovetransformation is singular. The equation PV I controls the position of this apparent singularity in theisomonodromic family, which was the original approach of Fuchs [Fu07]. There are many other ways toconvert the system (9) to a single second order equation (every choice of a cyclic vector for the matrixA will do, and almost all vectors are cyclic), correspondingly there are many ways to define q(t) interms the matrix A.
10
The conjugacy class of an irreducible monodromy representation is completely deter-mined by its trace coordinates by a theorem of Fricke, Klein and Vogt. These coordinatesare given by the four parameters
al = tr(Ml) = el + 1el
= 2 cos(πϑl), l = 0, t, 1,∞, (12)
and the three variables
X0 = tr(M1Mt), Xt = tr(M0M1), X1 = tr(MtM0), (13)
satisfying the Fricke relation
F (X, a) := X0XtX1 +X20 +X2
t +X21 − θ0X0 − θtXt − θ1X1 + θ∞ = 0, (14)
where
θi = aia∞ + ajak, for i = 0, t, 1, and θ∞ = a0ata1a∞ + a20 + a2
t + a21 + a2
∞ − 4.
Definition 8. We call “the character variety of PV I” the complex surface
SV I(a) = {X ∈ C3 : F (X, a) = 0}. (15)
In this setting, the Riemann–Hilbert correspondence can be seen as a map betweenthe Hamiltonian flow of the Painleve system on one side and a locally constant Hamil-tonian flow on the character variety on the other side. Under this correspondence theOkamoto space of initial conditionsMV I,t(ϑ) is a minimal resolution of singularities ofSV I(a) [IIS06].
The character variety SV I(a) is equipped with a natural algebraic symplectic formgiven by the Poincare residue
ω =dX0 ∧ dXt
FX1
=dXt ∧ dX1
FX0
=dX1 ∧ dX0
FXt, (16)
whereFXi = dF
dXi= XjXk + 2Xi − θi.
3.3 Lines and singularities of SV I(a)
Proposition 9 (Lines of SV I(a)). The Fricke polynomial F (X, a) (14) can be decom-posed as
F (X, a) = (Xk − eiej− ej
ei)(FXk −Xk + ei
ej+
ejei
)
− 1eiej
(eiXi + ejXj − a∞ − eiejak)(eiXj + ejXi − ak − eieja∞),
= (Xk − eiej − 1eiej
)(FXk −Xk + eiej + 1eiej
)
− 1eiej
(eiejXi +Xj − eja∞ − eiak)(eiejXj +Xi − ejak − eia∞),
= (Xk − eke∞− e∞
ek)(FXk −Xk + ek
e∞+ e∞
ek)
− 1eke∞
(e∞Xi + ekXj − ai − eke∞aj)(ekXi + e∞Xj − aj − eke∞ai),
= (Xk − eke∞ − 1eke∞
)(FXk −Xk + eke∞ + 1eke∞
)
− 1eke∞
(Xj + eke∞Xi − ekai − e∞aj)(Xi + eke∞Xj − ekaj − e∞ai).
11
In particular, the following 24 lines are contained in SV I(a):
{Xk = eiej
+ejei, eiXi + ejXj = a∞ + eiejak},
{Xk = eiej
+ejei, eiXj + ejXi = ak + eieja∞},
{Xk = eiej + 1eiej
, Xi + eiejXj = ejak + eia∞},
{Xk = eiej + 1eiej
, Xj + eiejXi = eja∞ + eiak},
{Xk = eke∞
+ e∞ek, e∞Xi + ekXj = ai + eke∞aj},
{Xk = eke∞
+ e∞ek, ekXi + e∞Xj = aj + eke∞ai},
{Xk = eke∞ + 1eke∞
, Xi + eke∞Xj = ekaj + e∞ai},{Xk = eke∞ + 1
eke∞, Xj + eke∞Xi = ekai + e∞aj}.
The projective completion of SV I(a) in CP3 contains 3 additional lines at infinity,giving the total of 27 lines provided by the classical Cayley-Salmon theorem. They areall distinct if and only if SV I(a) is nonsingular.
Singular points of SV I(a). The surface SV I(a) is simply connected (cf. [CL09]),and it may or may not have singular points depending on a, but it never has morethan 4 singular points [Obl04]. The singularities that appear correspond to unstablemonodromy representations, which are of two kinds:
• Either Ml = ±I for some l ∈ {0, t, 1,∞}, hence el = ±1.
If l = i ∈ {0, t, 1}, then ai = ±2, Xi = ±a∞, Xj = ±ak, Xk = ±aj .
If l =∞, then a∞ = ±2, Xi = ±ai, i = 0, t, 1.
• Or the representation is reducible, in which case Mi =
(eδii ∗0 e
−δii
)for some triple
of signs (δ0, δt, δ1) ∈ {±1}3, and
eδ00 eδtt e
δ11 e∞ = 1, Xi = e
δjj e
δkk + e
−δjj e−δkk .
The surface is therefore singular if and only if∏l∈{0,t,1,∞}
(a2l − 4) · w(a) = 0,
where
w(a) := (a0+ at+ a1+ a∞)(a0+ a∞− at− a1)(at+ a∞− a1− a0)(a1+ a∞− a0− at)−− (a0a∞− ata1)(ata∞− a1a0)(a1a∞− a0at)
=(e0ete1e∞ − 1)(e0et − e1e∞)(ete1 − e0e∞)(e1e0 − ete∞)
(e0ete1e∞)4
∏l∈{0,t,1,∞}
(el − eiejek),
see [Iwa02]. All the singularities of the projective completion of SV I(a) are containedin its finite part, where they are situated on the intersection of 6 or more lines. LetS◦V I(a) = SV I(a) r Sing(SV I(a)) be the smooth locus.
3.4 The modular action on SV I(a)
The non-linear monodromy action on the space of SL2(C)-monodromy representations,is given by the action of moving t along loops in CP1 r {0, 1,∞} while keeping the
12
representation constant. When t returns to the initial position t0, the loops generatingπ1(CP1 r {0, t, 1,∞}, x0) will not be the same as before. It induces an automorphismof the fundamental group through which it acts on the space of monodromy represen-tations. The movement of t can be also seen as an action of the pure-braid group P3 onthree strands (0, t, 1), generated by the pure braids β2
0t, β2t1 ∈ P3 (Figure 3).
β20t :
0 t 1
0 t 1
β2t1 :
0 t 1
0 t 1
Figure 3: Elementary pure braids.
One can also consider the action of the whole braid group B3, generated by thebraids β0t, βt1 (Figure 4), where the action of βij on the three points (0, t, 1) (which arefor the moment considered as movable), is a composition of two commuting actions
1) turning the two points i, j, around each other by a half turn,
2) swapping their names i↔ j.
β0t :
0 t 1
0 t 1
βt1 :
0 t 1
0 t 1
Figure 4: Elementary braids.
This action was described by Iwasaki [Iwa03], see also [PR15].
Proposition 10 (Iwasaki [Iwa03]).1) The braid group on three strands B3, with the generators β0t, βt1, acts on x-spaceCP1r{0, t, 1,∞} as the mapping class group by fixing∞, therefore inducing an action onthe fundamental group π1(CP1r{0, t, 1,∞}, x0) through which it acts on the monodromyrepresentations:
β0t : M0 7→M−10 MtM0,
Mt 7→M0,
M1 7→M1,
βt1 : M0 7→M0,
Mt 7→M−1t M1Mt,
M1 7→Mt.
The induced “half-monodromy” actions g0t, gt1 on the character variety SV I(a) aregiven by
gij : ei 7→ ej , Xi 7→ Xj , FXi 7→ FXj − FXiXk,
ei 7→ ei, Xj 7→ Xi − FXi , FXj 7→ −FXi ,ek 7→ ek, Xk 7→ Xk, FXk 7→ FXk − FXiXj ,
e∞ 7→ e∞.
They preserve the Fricke relation: F ◦ gij = F , and therefore they preserve also the2-form ω. They satisfy
gij = g◦−1ji , gij ◦ gjk ◦ gij = gjk ◦ gij ◦ gjk, gki = gji ◦ gjk ◦ gij = gjk ◦ gij ◦ gkj ,
for any permutation (i, j, k) of (0, t, 1). The group
Γ = 〈g0t, gt1〉,
13
(a) Start. (b) Action of β0t. (c) Action of βt1.
Figure 5: Action of the elementary braids β0t, βt1 ∈ B3 on the fundamental groupπ1(CP1 r {0, t, 1,∞}, x0).
generated by them is isomorphic to the modular group PSL2(Z), with the standardgenerators
S = gt1 ◦ g◦20t , T = gt0, satisfying S◦2 = id = (S ◦ T )◦3. (17)
2) The action of the monodromy group of PV I on the character variety is inducedby the action of the pure braids β2
0t, β2t1 ∈ P3 on the monodromy representations. It is
isomorphic to the principal congruence subgroup of the modular group
Γ(2) = 〈g◦20t , g◦2t1 〉 ⊆ Aut(SV I(a)),
generated by
g◦20t : X0 7→ X0 − FX0
, FX07→ −FX0
−X1FXt +X21FX0
,
Xt 7→ Xt − FXt +X1FX0, FXt 7→ −FXt +X1FX0
,
X1 7→ X1, FX17→ FX1
−XtFX0−X0FXt + FX0
FXt +X1X0FX0−X1F
2X0,
g◦2t1 : X0 7→ X0, FX0
7→ FX0−X1FXt −XtFX1
+ FXtFX1+X0XtFXt −X0F
2Xt,
Xt 7→ Xt − FXt , FXt 7→ −FXt −X0FX1+X
20FXt ,
X1 7→ X1 − FX1+X0FXt , FX1
7→ −FX1+X0FXt ,
while preserving the parameters ei. The fixed points of this action are exactly the sin-gularities of SV I .
The modular group action on SV I has been studied in detail in [CL09].
Remark 11. i) The monodromy action of Γ(2) on SV I(a) corresponds through theRiemann-Hilbert correspondence to the non-linear monodromy action on the Painlevefoliation PV I given by the Poincare map. More precisely, the singular locus of SV I(a)corresponds to so called Riccati solutions of PV I , and the Γ(2)-action on the smoothlocus S◦V I(a) represents faithfully the non-linear monodromy action on the non-Riccatilocus of space of initial conditions MV I,t0(ϑ) [IIS06].
ii) The half-monodromy actions map an equation PV I with parameter to anotherequation PV I with a permuted parameter. Therefore they make sense only if one con-siders the totality of all equations PV I .
4 The confluence PV I → PV and the character vari-eties
4.1 Confluence of isomonodromic systems
The substitutiont 7→ 1 + ε, ϑt = 1
ε , ϑ1 = − 1ε + ϑ1 + 1, (18)
14
in the system (9) with
z1 = − z1εt , zt = z1εt − z0 + κ2, where κ2 = −ϑ0+ϑ1+1+ϑ∞
2 ,
gives a parametric family (depending on the parameter ε) of isomonodromic deforma-tions:
dφ
dx=[A0(t)
x+A
(0)1 (t) + (x−1−εt)A(1)
1 (t)
(x−1)(x−1−εt)
]φ, (19)
where
A(0)1 = −εtA1 =
z1 + t2 − εt
1+ϑ12 −u1z1
z1+t−εt(1+ϑ1)u1
−z1 − t2 + εt
1+ϑ12
,A
(1)1 = At +A1 =
−z0 − ϑ0+ϑ∞2 u0z0
− z0+ϑ0u0
z0 +ϑ0+ϑ∞
2
,which have well defined limits when ε→ 0. The matrix A
(0)1 has eigenvalues ± t−εt(1+ϑ1)
2with limit ± t
2 when ε→ 0.
Figure 6: Monodromy of the system (19).
4.2 Confluence on character varieties
Now we can use the description of the linear confluence given in Section 2. Under thesubstitution (18), we have
et = eπiε , e1 = −eπiϑ1−πiε .
The monodromy matrices with respect to the canonical fundamental matrix solutiondescribed in Section 2 (Proposition 4) near the two confluent singularities, for | arg ε
t | <π, are of the following form (see Figure 4.1)
M0 =(α βγ δ
), Mt = CtNt =
(et 0etct
1et
), M1 = N1C1 =
(e1 e1c10 1
e1
),
M∞ = (MtM1M0)−1 =
(ete1(βct+δctc1)+ δ
ete1−ete1(β+δc1)
−ete1(αct+γctc1)− γete1
ete1(α+γc1)
),
(20)
where Ct =(
1 0ct 1
), C1 =
(1 c10 1
), are the unfolded Stokes matrices, and Ni =
(ei 00 1
ei
),
i ∈ {t, 1}, for ε 6= 0, are the formal monodromies around the points 1 + εt and 1. Thesemonodromy matrices are subject to the conditions
det(M0) = 1 = αδ − βγ,tr(M0) = a0 = α+ δ,
tr(M∞) = a∞ = δete1
+ ete1(α+ ctβ + c1γ + ctc1δ).
15
This description is determined uniquely up to conjugation by diagonal matrices. Thetrace coordinates Xi (13) are given by
X0 = tr(M1Mt) = ete1 + 1ete1
+ ete1ctc1,
Xt = tr(M1M0) = e1(α+ γc1) + δe1,
X1 = tr(M0Mt) = et(α+ βct) + δet.
(21)
Only the parameters
a0 = 2 cos(πϑ0), ete1 = −eπiϑ1 := −e1, a∞ = 2 cos(πϑ∞),
have well defined limits when ε → 0, while et = eπiε and e1 = −eπiϑ1−πiε diverge.
Therefore the coordinate X0 passes well to the limit, but not Xt, X1 which need bereplaced by new coordinates (invariant with respect to diagonal conjugation).
Following [PS09], we choose as the new coordinates the low diagonal elements ofM∞ and M0:
Wt = (M∞)22 = e1et(α+ γc1),
U1 = (M0)22 = δ.(22)
A substitution in the identity
ete1ctc1(αδ − βγ − 1) = 0,
gives the Fricke relation in the new coordinates
F (X0,Wt, U1, a) := X0WtU1 +W 2t + U2
1 − θ0X0 − θtWt − θ1U1 + θ∞ = 0 (23)
where θl are functions of the parameter
a = (a0,−e1, a∞),
independent of ε,
θ0 = −e1, θt = a∞ − e1a0, θ1 = a0 − e1a∞, θ∞ = 1− e1a0a∞ + e21. (24)
The relation (23) is already known from [PS09, section 3.2].
Definition 12. The wild character variety of PV is the complex surface
SV (a) = {(X0,Wt, U1) ∈ C3 : F (X0,Wt, U1, a) = 0}, (25)
endowed with the algebraic symplectic form (28).
Remark 13. A very simple way to obtain the coordinates (X0,Wt, U1) on the wildcharacter variety SV (a) is by taking the following limit:
• When ε→ 0 radially with arg ε ∈]− π, 0[, then et → 0, e1 →∞, and
a1
at→ −e1,
Xt
at→ Wt := −e1(α+ γc1),
X1
at→ U1 := δ.
and 1a2tF (X0, Xt, X1, a)→ F (X0,Wt, U1, a).
• When ε→ 0 radially with arg ε ∈]0, π[, then et →∞, e1 → 0, and
ata1→ −e1,
Xt
a1→ Ut := δ,
X1
a1→ W1 := −e1(α+ βct),
and 1a2
1F (X0, Xt, X1, a)→ F (X0, Ut,W1, a).
16
These two limit coordinates are related by the limit of the action of the half-monodromyoperator gt1
gt1 : (X0,Wt, U1) 7→ (X0, Ut,W1) = (X0, U1,Wt − FWt).
However, this limit description of the character variety of PV , which has been more-less known (see for example [CMR15]), is too simple for our purpose. While the mon-odromy operator g◦2t1 has a well defined limit in this description, the monodromy operatorg◦20t diverges as expected. It is precisely the confluence ε→ 0 in the two omitted resonantdirections ±R+ that is interesting for us. Indeed, we want to be able to consider thelimits along sequences {εn}n∈±N with 1
εn= 1
ε0+ n, along which et is constant, in order
to obtain the generators of the wild monodromy.
The original variables (X0, Xt, X1) on the character variety SV I(a) and the newvariables (X0,Wt, U1) on the wild character variety SV (a) are related by the followingrational transformations:
Theorem 14 (ε 6= 0). i) The affine varieties SV I(a) and SV (a) are birationally equiv-alent through the change of variables
Φ : SV (a)→ SV I(a)
(X0,Wt, U1, e) 7→ (X0, Xt, X1, e),
given by
X0 = X0, FX0◦ Φ = U1
e1
( FU1
et− FWt
e1
)− FX0
ete1(X0 − et
e1− e1
et)
Xt = U1
e1+ Wt
et, FXt ◦ Φ =
FU1
e1+
FWtet− X0FWt
e1
X1 = U1
et+ Wt
e1− FWt
e1, FX1
◦ Φ =FU1
et− FWt
e1,
where
FX0:= ∂F
∂X0= U1Wt − θ0,
FWt:= ∂F
∂Wt= X0U1 + 2Wt − θt,
FU1:= ∂F
∂U1= X0Wt + 2U1 − θ1.
And the inverse map Φ◦(−1) : SV I(a) 99K SV (a) is given by
U1 = −etXt + e1X1 − θtX0 − et
e1− e1
et
= −ete1
X0 − ete1− e1
et− FX0
e1Xt + etX1 − θ1
, (cf. Proposition 9),
Wt = −e1Xt + etX1 − θ1 − etFX1
X0 − ete1− e1
et
= etXt − ete1U1.
(26)
which is singular on the line:
L0(a) := {X0 = ete1
+ e1et, e1Xt + etX1 = θ1}. (27)
The two Fricke relations are related by
F ◦ Φ = −1ete1
(X0 − ete1− e1
et) · F .
The restrictionΦ : SV (a)→ SV I(a) r L0(a)
is an isomorphism.
17
ii) The pull-back of the symplectic form ω (16) by Φ is a symplectic form on thesmooth locus of SV (a) given by the Poincare residue
ω =dX0 ∧ dWt
FU1
=dWt ∧ dU1
FX0
=dU1 ∧ dX0
FWt
. (28)
Proof. i) Follows from (21) and (22).ii) It can be verified by a direct calculation using the formulas of Proposition 14.
Remark 15. The singular line L0(a) on SV I(a) corresponds to a “line at infinity” in thecoordinates (X0,Wt, U1). In fact, if one sets
(X0,Wt, U1) = (Y0 + ete1
+ e1et, YtY0, Y1Y0
),
then the image of point X ∈ L0(a) by Φ◦(−1) is given by:
(Y0, Yt, Y1) = (0, etFX1 − etXt − e1X1 + θ1,−etXt − e1X1 + θt),
which satisfiesY0 = 0, Yt
et+ Y1
e1= 0,
and we may want to add this line to the surface {F (X0,Wt, U1, a) = 0}. This can be done inthe following way:
Let (Y0, Yt, Y1) be the coordinates on C3 and let C3 be its blowup at the origin given bythe quotient
C3 = (C3r{0})× C/ ∼,where (Z0 : Zt : Z1;Z∞) ∼ (Z0
λ: Zt
λ: Z1
λ;λZ∞) for any λ ∈ C∗, (Z0 : Zt : Z1;Z∞) ∈
(C3r{0})× C. Setting Yi = ZiZ∞ leads to the usual definition
C3 ' {(Y0, Yt, Y1;Z0 : Zt : Z1) ∈ C3 × CP2 : YiZj = YjZi}.
The local coordinates on C3 are given by
(Z0 : Zt : Z1;Z∞) ∼ (Y0 : Yt : Y1; 1) if Z∞ 6= 0,
∼ (1 : ZtZ0
: Z1Z0
;Y0) if Z0 6= 0,
∼ (Z0Zt
: 1 : Z1Zt
;Yt) if Zt 6= 0,
∼ (Z0Z1
: ZtZ1
: 1;Y1) if Z1 6= 0,
and we identify
(X0 − ete1− e1
et,Wt, U1) = (Z0Z∞,
ZtZ0, Z1Z0
) = (Y0,YtY0, Y1Y0
).
Then the intersection of the closure of the surface {F (X0,Wt, U1, a) = 0} in C3 with the “planeat infinity” {Z0 = 0} consists of two lines (disjoint if e2t 6= e21):
Z20 F ( et
e1+ e1
et, ZtZ0, Z1Z0
) = (etZt + e1Z1)(Ztet
+ Z1e1
) = 0.
The singular line L0(a) in SV I(a) is mapped by Φ◦(−1) to the line L0(a, et) = {Z0 = 0, Ztet
+Z1e1
= 0},X ∈ L0(a) 7→ (0 : − et
e1: 1; θt − etXt − e1X1) ∈ C3.
We define
SV I(a, et) = {(X0,Wt, U1) ∈ C3 : F (X0,Wt, U1, a) = 0} ∪ L0(a, et) ⊆ C3. (29)
If e2t 6= e21, then the transformation Φ (26) is an isomorphisms between SV I(a) and SV I(a, et).Indeed, if e2t 6= e21, then etXt+e1X1− θt is a coordinate on the line L0, whose image is thereforeL0. Hence the map Φ◦−1 extends to a smooth bijection SV I(a)→ SV I(a, et).
Remark 16 (ε 6= 0). In the trace coordinates (12) on the space of monodromy representations,the eigenvalues ei and 1
eiare interchangeable. On the other hand in the above description of the
monodromy data (20) this is no longer true for i = t, 1. While most monodromy representationscan be conjugated so that Mt is lower triangular and M1 is upper triangular, one cannot alwaysassure that et, e1 are on the (1, 1)-position. Those monodromy representations for which thisis not possible correspond exactly to the points of the line at infinity L0(a, et).
18
4.3 Lines and singularities of SV (a)
Proposition 17 (Lines of SV (a)). The polynomial F (23) can be decomposed as
F (X0,Wt, U1, a) =
= (X0 + e1 + 1e1
)FX0− e1(Wt − U1
e1− a∞)(U1 − Wt
e1− a0),
= (X0 − e0e∞ − 1e0e∞
)FX0+ (e0Wt + U1
e∞+ e1 − e0
e∞)(Wt
e0+ e∞U1 + e1 − e∞
e0),
= (X0 − e0e∞− e∞
e0)FX0
+ (e0Wt + e∞U1 + e1 − e0e∞)(Wt
e0+ U1
e∞+ e1 − 1
e0e∞),
= (Wt − e∞)(FWt −Wt + e∞) + (U1 + e1e∞
)(e∞X0 + U1 + e1e∞ − a0),
= (Wt − 1e∞
)(FWt −Wt + 1e∞
) + (U1 + e1e∞)(X0
e∞+ U1 + e1
e∞− a0),
= (Wt + e1e0)(FWt−Wt − e1e0) + (U1 − 1
e0)(−e1e0X0 + U1 − e0 + e1a∞),
= (Wt + e1e0
)(FWt−Wt − e1
e0) + (U1 − e0)(− e1e0X0 + U1 − 1
e0+ e1a∞),
= (U1 − e0)(FU1− U1 + e0) + (Wt + e1
e0)(e0X0 +Wt + e1e0 − a∞),
= (U1 − 1e0
)(FU1 − U1 + 1e0
) + (Wt + e1e0)(X0
e0+Wt + e1
e0− a∞),
= (U1 + e1e∞)(FU1 − U1 − e1e∞) + (Wt − 1e∞
)(−e1e∞X0 +Wt − e∞ + e1a0),
= (U1 + e1e∞
)(FU1− U1 − e1
e∞) + (Wt − e∞)(− e1
e∞X0 +Wt − 1
e∞+ e1a0),
defining thus 22 lines on SV (a).
The projective completion of SV (a) in CP3 contains 3 additional lines at infinity.
Proposition 18 (Singular points of SV (a)). The affine cubic variety SV (a) has singularpoints if and only if
(a20 − 4)(a2
∞ − 4)w(a) = 0,
where
w(a) = (a20 + a2
1 + a2∞ + a0a1a∞ − 4), with a1 = e1 + 1
e1,
=(e0ete1e∞ − 1)(e0ete1 − e∞)(ete1e∞ − e0)(ete1 − e0e∞)
(e0ete1e∞)2,
(cf. [PS09, section 3.2.2]). The corresponding possible singularities are the following:
• if a0 = ±2: X0 = ±a∞, Wt = ∓e1, U1 = ±1,
• if a∞ = ±2: X0 = ±a0, Wt = ±1, U1 = ∓e1,
• if −e1e0e∞ = 1: X0 = −e1 − 1e1, Wt = 1
e∞, U1 = 1
e0,
• if −e1 = e0e∞: X0 = −e1 − 1e1, Wt = e∞, U1 = e0.
Setting (X0,Wt, U1) = ( x0
v∞, wtv∞ ,
u1
v∞), the projective completion of SV (a) in CP3 has
also a singularity at the point (x0 : wt : u1 : v∞) = (1 : 0 : 0 : 0) for any value of theparameters.
Proof. The surface SV (a) is isomorphic to SV I(a) r L0(a), we can therefore use thedescription of the singular points of SV I(a) given in Section 3.3.
19
4.4 The wild monodromy action on SV (a)
The only monodromy actions on SV (a) that survive the confluence are those generatedby g◦2t1 . As was motivated in the introduction, we may instead consider limits along
sequences (εn)n∈±N, 1εn
= 1ε0
+ n for which the divergent parameter et = eπiε stays
constant. This amounts to replacing e2t by a new independent parameter ν = e
2πiε0
ete1 = −e1, e2t = ν ∈ C∗.
For ε 6= 0, let Φ be the corresponding map (7). In the coordinates (X0,Wt, U1) thepure braid group P3 (cf. Proposition 10) acts on SV (a) by the monodromy actions
g◦2ij = Φ−1 ◦ g◦2ij ◦ Φ,
preserving the Poincare residue form ω.The monodromy action
g◦2t1 : X0 7→ X0, FX0
7→ FX0−WtFU1
− U1FWt + FWt FU1+X0WtFWt −X0F
2Wt,
Wt 7→ Wt − FWt , FWt 7→ −FWt −X0FU1+X
20 FWt ,
U1 7→ U1 − FU1+X0FWt , FU1
7→ −FU1+X0FWt ,
g◦21t : X0 7→ X0, FX0
7→ FX0−WtFU1
− U1FWt + FWt FU1+X0U1FU1
−X0F2U1,
Wt 7→ Wt − FWt +X0FU1, FWt 7→ −FWt +X0FU1
,
U1 7→ U1 − FU1, FU1
7→ −FU1−X0FWt +X
20 FU1
,
is independent of et and preserves the Fricke relation:
F ◦ g2t1 = F .
The formulas for the wild monodromy action g◦20t = g◦20t (ν) and its inverse g◦2t0 =g◦2t0 (ν) are too complex to be written here. They depend only on ν = e2
t : indeed, theaction of g0t on (X0,
Xtet, X1
e1) depends only on ete1 and et
e1, and so does the transformation
(X0,Wt, U1) 7→ (X0,Xtet, X1
e1) and its inverse. Their action change the Fricke relation by
a factor:
F ◦ g◦20t (ν) =(X0 + e1
ν + νe1
)
(X0 + e1ν + ν
e1− FX0
◦ Φ)F .
Theorem 19 (Wild monodromy action on SV (a)). The wild monodromy actions arethose in the group generated by
〈g◦2t1 , g◦20t (ν) | ν ∈ C∗〉,
where g◦2t1 is the monodromy operator (independent of ν) and g◦2t0 (ν) = g◦−20t (ν), g◦2t1 ◦
g◦20t (ν) are wild monodromy operators, acting on SV (a) as ω-preserving birational trans-formations. For each ν fixed 〈g◦2t1 , g◦20t (ν)〉 ' Γ(2).
Theorem 20 (Torus action). The wild monodromy operators g◦2t0 (ν), and g◦2t1 ◦ g◦20t (ν),can be written as
g◦2t0 (ν) = τU1(ν) ◦ σ, and g◦2t1 ◦ g◦20t (ν) = τWt
(ν)◦−1 ◦ σ′ = σ′ ◦ τU1(ν)◦−1,
where
• σ, σ′ are “ Stokes operators” (independent of ν),
• g◦2t1 = σ′ ◦ σ is the monodromy operator,
• τU1(ν), τWt
(ν) are “ torus actions”.
20
The vector field (X0, Wt, U1) = ν ddν τU1(ν), which generates infinitesimally the action
of τU1(ν), is a Hamiltonian vector field on SV (a) with respect to the symplectic form ω
(28)4 and to the Hamiltonian function HU1= log(U1), given by
FWt
U1∂X0− FX0
U1∂Wt
. (30)
And the vector field (X0, Wt, U1) = ν ddν τWt
(ν), which generates infinitesimally the ac-tion of τWt
(ν), is a Hamiltonian vector field on SV (a) with respect to the symplecticform ω (28) and the Hamiltonian function HWt
= log(Wt), given by
− FU1
Wt∂X0
+FX0
Wt∂U1
. (31)
Proof. By formal calculations in SageMath using the formulas of Proposition 10 andProposition 14. One verifies that the vector field
ν ddν τU1
(ν) =(ν ddν g◦2t0 (ν)
)◦ g◦20t (ν)
is given by (30) modulo F , and that the vector field
ν ddν τWt
(ν) = −(ν ddν g◦2t1 ◦ g◦20t (ν)
)◦ g◦2t0 (ν) ◦ g◦21t
is given by (31) modulo F .
Remark 21. The action of the wild half-monodromy gt1(ν) = Φ−1 ◦ gt1 ◦ Φ and its inverseg1t(ν) on SV (a) are given by
gt1(ν) : X0 7→ X0, FX07→ FX0
− U1FWt −(U1 + ν
e1Wt − ν
e1FWt −
νe1
�)�,
Wt 7→ Wt − FWt −�, FWt 7→ −FWt −νe1
(X0 + 2e1ν )�,
U1 7→ U1 − νe1
�, FU17→ FU1
−X0FWt − (X0 + 2 νe1
)�,
ν
e17→
e1
ν,
where � :=FU1
− (X0 +e1ν )FWt
X0 +e1ν + ν
e1
, � 7→FU1
+ νe1FWt
X0 + +e1ν + ν
e1
+ νe1X0�,
g1t(ν) : X0 7→ X0, FX07→ FX0
−WtFU1−(Wt +
e1ν U1 − e1
ν FU1− e1
ν �)�,
Wt 7→ Wt − e1ν �, FWt 7→ FWt −X0FU1
− (X0 + 2e1ν )�,
U1 7→ U1 − FU1−�, FU1
7→ −FU1− e1
ν (X0 + 2 νe1
)�,
ν
e17→
e1
ν,
where � :=FWt − (X0 + ν
e1)FU1
X0 +e1ν + ν
e1
, � 7→FWt +
e1ν FU1
X0 +e1ν + ν
e1
+e1ν X0�.
The second iterate gt1(ν) ◦ gt1(ν) = g◦2t1 is is the monodromy operator.A different action on SV (a) whose square iterate is the monodromy operator g◦2t1 is obtained
as the limit of the half-monodromy operator gt1 in the description of Remark 13
gt1 : e0 7→ e0, X0 7→ X0, FX07→ FX0
− FWtU1,
e1 7→ e1, Wt 7→ Ut = U1, FWt 7→ FU1− FWtX0,
e∞ 7→ e∞, U1 7→ W1 = Wt − FWt , FU17→ −FWt .
The meaning of these transformations is not clear to the author.
4A Hamiltonian vector field on SV (a) with respect to the symplectic form ω (28) and a Hamiltonianfunction H(X0,Wt, U1) is the vector field
(FWtHU1− FU1
HWt ) ∂X0+ (FU1
HX0− FX0
HU1) ∂Wt + (FX0
HWt − FWtHX0) ∂U1
,
where (HX0, HWt , HU1
) = DH is the vector of partial derivatives of H.
21
5 Appendix: Painleve equations as isomonodromicdeformations of 3×3 systems
This section exposes how to derive the Fricke formula for the character variety of PV Iand for the modular group action on it as the space of Stokes data corresponding toisomonodromic deformations of 3×3-systems in Okubo and Birkhoff canonical forms.Some of this can be also found in a bit different form in the article of Boalch [Boa05](sections 2 and 3). Furthermore, we describe the confluence in these systems and showhow the Stokes data of the limit system ε = 0 are connected with those for ε 6= 0 (Figure10), providing thus another derivation of the character variety of PV (23) and of theformulas of the birational change of variables Φ (26).
5.1 Systems in Okubo and Birkhoff forms
The sixth Painleve equation PV I governs also isomonodromic deformations 3×3 linearsystems in Okubo form (
xI −(
0t
1
))dψdx
=[B(t) + λI
]ψ, (32)
where the matrix B(t) can be written in the following form
B(t) =
ϑ0 w0utzt − zt w0u1z1 − z1
wtu0z0 − z0 ϑt wtu1z1 − z1
w1u0z0 − z0 w1utzt − zt ϑ1
, wi =zi + ϑiuizi
, zi 6= 0,
where ϑi are the parameters, and the matrix B(t) has eigenvalues
0, −κ1 = 12 (ϑ0 + ϑt + ϑ1 − ϑ∞), −κ2 = 1
2 (ϑ0 + ϑt + ϑ1 + ϑ∞).
The system (32) can be obtained from the Garnier system (9) by the addition 5 of12 (ϑ0
x + ϑtx−t + ϑ1
x−1 )I to A(x, t), followed by the Katz’s operation of middle convolutionmcλ with a generic parameter λ different from 0, κ1, κ2 [HF07] (see also [Boa05, Maz02]).One may also replace B(t) by the conjugated matrix
(z0ztz1
)B(t)
(z0ztz1
)−1
=
ϑ0z0+ϑ0
u0ut − z0
z0+ϑ0
u0u1 − z0
zt+ϑtut
u0 − zt ϑtzt+ϑtut
u0 − ztz1+ϑ1
u1u0 − z1
z1+ϑ1
u1ut − z1 ϑ1
.
Another isomonodromic problem that will be considered here is that of the systemin a Birkhoff canonical form
ξ2 dy
dξ=[ (
0t
1
)+ ξB(t)
]y, (33)
which is dual to the Okubo system (32) through the Laplace transform
y(ξ) = ξ−1−λ∫ ∞
0
ψ(x) e−xξ dx, | arg(ξ)− arg(x)| < π
2. (34)
All three kinds of systems (9), (32), (33), and their isomonodromy problems areessentially equivalent (at least on a Zariski open set of irreducible systems (9)). Underan additional assumption that no ϑi is integral, the condition on (generalized) isomon-odromicity of the each of the above linear systems is equivalent to the Painleve equationPV I(ϑ) [HF07].
5Corresponding to the gauge transformation φ 7→ x−ϑ02 (x− t)−
ϑt2 (x− 1)−
ϑ12 φ.
22
Notation 22. The elements of all 3×3 matrices will be indexed by (0, t, 1) rather than
(1, 2, 3), corresponding to the eigenvalues of the matrix(
0t
1
). As before, the triple of
indices (i, j, k) will always denote a permutation of (0, t, 1), and (i, j, k, l) will denote apermutation of (0, t, 1,∞).
5.2 Stokes matrices of the Birkhoff system
The Birkhoff system (33) posses a canonical formal solution
Y (ξ, t) = T (ξ, t)
(ξϑ0
e− tξ ξϑt
e− 1ξ ξϑ1
),
with T (ξ, t) an invertible formal series in ξ (with coefficients locally analytic in t ∈CP1 r {0, 1,∞}), which is unique up to right multiplication by an invertible diagonalmatrix, and unique if one demands that T (0, t) = I [Sch01]. It is well known that thisseries is Borel summable in each non-singular direction α (we remind that a directionα ∈ R is singular for the system (33) if (i− j) ∈ eiαR+ for some i, j ∈ {0, t, 1}, i 6= j).This means that for each non-singular direction α, there is an associated canonicalfundamental matrix solution
Yα(ξ, t) = Tα(ξ, t)
(ξϑ0
e− tξ ξϑt
e− 1ξ ξϑ1
), | arg(ξ)− α| < π, (35)
where Tα is the Borel sum in ξ of T in the direction α. 6 This solution does not dependon α as long as α does not cross any singular direction [Bal00, IY08, MR88, MR91].
Let us restrict to α ∈]−π, π[, and suppose for a moment that 0, t, 1 are not collinear,i.e. there is six distinct singular rays (i − j)R+. When α crosses such a singulardirection (in clockwise sense) the corresponding sectoral basis Yα changes in a way thatcorresponds to multiplication by a constant (with respect to ξ) invertible matrix, socalled Stokes matrix,
Sij = I + sijEij , (36)
where Eij denotes the matrix with 1 at the position (i, j) and zero elsewhere. For thesingular ray −R+, one needs to take in account also the jump in the argument of ξbetween −π and π, therefore the change of basis is provided by a matrix NS01, whereN is the formal monodromy of Y :
N =
(e20e2te21
), where ej := eπiϑj . (37)
See Figure 7 (a).By definition, the (generalized) isomonodromicity of a family (33) demands that the
Stokes matrices are independent of t.Since in general the formal transformation T , and therefore also the collection of
the sectoral bases Yα, are unique only up to right multiplication by invertible diagonalmatrices, the collection of the Stokes matrices Sij is defined only up a simultaneous con-jugation by diagonal matrices. The obvious invariants with respect to such conjugationare
s0tst0, st1s1t, s10s01, s0tst1s10, s1tst0s01, (38)
6The Borel sum Tα is given by the Laplace integral
Tα(ξ, t) =1
ξ
∫ +∞eiα
0U(x, t) e
− xξ dx,
where U(x, t) =∑+∞k=0
Tk(t)k!
xk is the formal Borel transform of ξ T (ξ, t) =∑+∞k=0 Tk(t)ξk+1.
23
(a) Birhkhoff system (b) Okubo system
Figure 7: Singular directions and Stokes matrices.
subject to the relation
s0tst0 · st1s1t · s10s01 − s0tst1s10 · s1tst0s01 = 0. (39)
5.3 Monodromy of the Okubo system
Now consider the Okubo system (32), where we chose for simplicity
λ = 0.7
Corresponding to the canonical sectoral solutions bases Yα =(Yα,ij
)i,j
of (33), there
are canonical sectoral solutions bases 8 Ψα =(Ψα,ij
)i,j
of (32), related to Yα by the
Laplace transform (34)
Yα,ij(ξ, t) = 1ξ
∫ +∞eiα
j
Ψα,ij(x, t) e− xξ dx, i, j ∈ {0, t, 1}.
The sectors on which they are defined (see Figure 7 (b)) are the different components ofthe complement in C of ⋃
i,j∈{0,t,1}
(i+ (i−j)R+
)∪ [0, 1] ∪ [0, t].
When crossing one of the rays i + (i − j)R+ in clockwise sense the basis Ψα changesby the same Stokes matrix Sij as before, except for the ray −R+, where it changes byNS01 (cf. [Kli15]). When crossing the segments [0, t] the basis changes by Nt, and on[0, 1] by N1, where
Ni = I + (e2i − 1)Eii, ei = eπiϑi ,
is the monodromy matrix of
(xϑ0
(x−t)ϑt(x−1)ϑ1
)around the point i ∈ {0, t, 1}, and
N = N1NtN0 (37). See Figure 7 (b).
7For this choice of λ = 0 the system (32) is reducible but it does not matter here.8 The fundamental matrix solution Ψα is also given by the convolution integral
Ψα,ij(x, t) =1
Γ(ϑj + λ)
∫ x
jUij(z − j, t)(x− z)ϑj+λ−1dz,
where U(x, t) =∑+∞k=0
Tk(t)k!
xk is the formal Borel transform of ξT (ξ, t) =∑+∞k=0 Tk(t)ξk [Sch85, Kli15].
24
Fixing a base-point x0 and three simple loops in positive direction around the points0, t, 1 respectively, such that their composition gives a simple loop around the infinity innegative direction as in see Figure 7 (b), let Mi be the respective monodromy matrices:
M0 = N0S01S0t, Mt = N−11 St0St1NtN1, M1 = S1tS10N1,
determined up to simultaneous conjugation in GL3(C). We have
tr(Mi) = e2i + 2, i ∈ {0, t, 1}.
Denoting
Xi =tr(MjMk)−1
ejek,
we have
X0 =e2t+e
21+e21st1s1tete1
, Xt =e20+e21+e20s10s01
e0e1, X1 =
e20+e2t+e20s0tst0
e0et. (40)
The monodromy around all the three points equals
M−1∞ = M1MtM0 = S1tS10St0St1NS01S0t
=
e20 e20s0t e20s01
e20st0 e2t + e20st0s0t e21st1 + e20st0s01
e20s10 + e20s1tst0 e2ts1t + e20s10s0t + e20s1tst0s0t e21 + e21s1tst1 + e20s10s01 + e20s1tst0s01
We know that its eigenvalues are 1, e−2πiκ1 = e0ete1e∞
and e−2πiκ2 = e0ete1e∞. Express-ing the coefficients of the linear term E and the quadratic term E′ of the characteristicpolynomial of M−1
∞ leads to
ete1X0 +e0e1Xt+e0etX1 +e20s1tst0s01−e2
0−e2t −e2
1 = 1+ e0ete1e∞
+e0ete1e∞ := E, (41)
e20ete1X0 + e2
t e0e1Xt + e21e0etX1 − e2
0e21s0tst1s10 − e2
0e2t − e2
t e21 − e2
1e20 =
= e0ete1e∞
+ e0ete1e∞ + e20e
2t e
21 := E′.
(42)
Inserting the expression for s1tst0s01 (41) and for s0tst1s10 (42) into the relation (39)gives the Fricke relation (14)
X0XtX1 +X20 +X2
t +X21 − θ0X0 − θtXt − θ1X1 + θ∞,
with
θi =e2j+e
2k+E
ejek+ E′
e2i ejek= aia∞ + ajak,
θ∞ = 1 + Ee0
+ Eet
+ Ee1
+ E′
e0et+ E′
ete1+ E′
e1e0= a0ata1a∞ + a2
0 + a2t + a2
1 + a2∞ − 4.
The line{Xk = ei
ej+
ejei, eiXi + ejXj = a∞ + eiejak}
of Proposition 9 corresponds to sij = 0 in (42), while
{Xk = eiej
+ejei, eiXj + ejXi = ak + eieja∞}
corresponds to sji = 0 in (41), where (i, j, k) is a cyclic permutation of (0, t, 1).We can also derive the induced action of the braids β0t and βt1 (Figure 3.4) on the
Stokes matrices Sij .
Proof of Proposition 10. The action of the braids β0t and βt1 on the Stokes matrices isobtained by:
25
(a) Action of β0t. (b) Action of βt1.
Figure 8: Braid actions on the Stokes matrices.
1) Tracing the connection matrices of the Okubo system (32) as the two correspond-ing points turn around each other according to the braid β0t, resp. βt1, and seehow they change when the three points 0, t, 1 align. See Figure 8. We use the factthat SijSkl = SklSij if j 6= k and l 6= i.
2) Swapping the names of the points 0 ↔ t, resp. t ↔ 1. This permutes also the
corresponding positions of all the matrices by P0t =(
0 1 01 0 00 0 1
), resp. Pt1 =
(1 0 00 0 10 1 0
).
We found that the action on the Stokes matrices is given (up to simultaneous conjugationby diagonal matrices) by
β0t : N 7→ P0tNP0t,
St0 7→ P0tS0tP0t,
S−1t0 S1tS10St0S
−11t 7→ P0tS1tP0t,
S1t 7→ P0tS10P0t,
S0t 7→ P0tSt0P0t,
S−10t St1S01S0tS
−1t1 7→ P0tSt1P0t,
N−1St1N 7→ P0tS01P0t,
βt1 : N 7→ Pt1NPt1,
St1 7→ Pt1S1tPt1,
S−1t1 S10St0St1S
−110 7→ Pt1S10Pt1,
S10 7→ Pt1St0Pt1,
NS1tN−1 7→ Pt1St1Pt1,
S−11t S01S0tS1tS
−101 7→ Pt1S01Pt1,
S01 7→ Pt1S0tPt1,
from which the corresponding action of g0t, resp. gt1, on the invariant elements (38) canbe easily expressed, and subsequently re-expressed in terms of the coordinatesX0, Xt, X1
(40).
26
5.4 Confluence of the Birkhoff systems and their character va-rieties
The substitution (18) in the Birkhoff system (33) and a conjugation by Q =(εt
1 −1εt
),
corresponding to the change of variable y = Qy, gives a parametric family of isomon-odromic systems
ξ2 dy
dξ=[ (
01+εt 1
1
)+ ξB(t, ε)
]y, (43)
with
B = QBQ−1 =
ϑ0 εt(w0utzt − zt) −w0u0z0 − z1 − zt1εt (wt − w1)u0z0 ϑt + zt − w1utzt − 1
εt (wt − w1)u0z0
w1u0z0 − z0 εt(w1utzt − zt) ϑ1 − zt + w1utzt
=
ϑ0 u1z1
(z0+ϑ0)u0z0
− z1 − εt(κ2 + ϑ0) −κ2 − ϑ0
−bt1 1 + ϑ1 + κ2 + b10 bt1
b10 t− εt[1 + ϑ1 + κ2 − z0 +z1+tu1z1
] −κ2 − b10
,
where κ2 = −ϑ0+ϑ1+1+ϑ∞2 , and
b10=u0z0u1z1
(z1+t−εt(1+ϑ1))−z0, bt1=−u0z0u1z1
u0z0(z1+t−εt(1+ϑ1))−u1z1(z0−1−ϑ1−κ2)u1z1−εtu0z0
.
When ε 6= 0 the irregular singular point at the origin is non-resonant and the localdescription of the Stokes phenomenon is just the same as in the precedent section with sixStokes matrices Sij . But for ε = 0 the singularity becomes resonant and the descriptionchanges.
For |εt| small, there is a formal transformation y = T (x, ε)(y′
y′′
), y′ ∈ C, y′′ ∈ C2,
written as a formal power series in x with coefficients analytic in ε, that splits the systemin two diagonal blocks, one corresponding to the eigenvalue 0, other corresponding tothe other eigenvalues {1 + εt, 1} (cf. [Bal00]):
ξ2 dy′
dξ= ξϑ0 y
′, (44)
ξ2 dy′′
dξ=[(
1+εt 1
0 1
)+ ξB′′(ε) +O(ξ2)
]y′′, (45)
where B′′ =(btt bt1b1t b11
)is the submatrix of B. This formal transformation T is Borel
summable in all directions except of ±R+ and ±(1+εt)R+. Therefore it possesses Borelsums on the four sectors, overlapping on the singular directions, out of which only thoseon the two large sectors persist to the limit ε→ 0.
Confluence of eigenvalues in the subsystem (45). The phenomenon of confluenceof eigenvalues in 2×2 parametric systems at an irregular singular point of Poincare rank1 was studied previously by the author [Kli14]. This paragraph applies some of theresults to the system (45).
The matrix of the right side of the system has its eigenvalues equal to
λ(0) + ξλ(1) ±√α(0) + ξα(1) (mod ξ2),
where
λ(0) = 1 + εt2 , λ(1) = btt+b11
2 = 1+ϑ2 ,
α(0) =(εt2
)2, α(1) = b1t + εt(btt−b11)
2 = εtϑt−ϑ1
2 = t− εt 1+ϑ1
2 ,
27
constitute the formal invariants of the system. In [Kli14], it has been shown that (45)possess a fundamental matrix solutions of the form
Y ′′• = R′′•(ξ, t, ε) · e−λ
(0)
ξ ξλ(1)
(α(0) + ξα(1))− 1
4(α(0) + ξα(1)
) 14
(1 11 −1
)(eΘ
e−Θ
),
where
Θ(ξ, ε, t) =
∫ ξ
∞
√α(0)+ζα(1)
ζ2 dζ =
−√α(0)+ξα(1)
ξ − α(1)
2√α(0)
log
√α(0)+ξα(1)+
√α(0)√
α(0)+ξα(1)−√α(0)
, ε 6= 0,
− 2α(0)
ξ , ε = 0,
and R′′• , • = I±, O, are invertible analytic transformations defined on certain domainsin the ξ-space. These domains are delimited by the so called Stokes curves (in the senseof exact WKB analysis [KT06]): the separatrices of the real phase portrait of the vectorfield 9
eiω ξ2
2√α(0)+ξα(1)
∂
∂ξ, with some ω ∈]− π
2 ,π2 [
emanating either from ∞ or from the “turning point” at ξ = −α(0)
α(1) , if ε 6= 0.
There are two kinds of such sectoral domains, whose shape in the coordinate ξα(1)
depends only on a parameter µ = α(0)
(α(1))2 =(
ε2−ε(1+ϑ1)
)2:
- A pair of inner domains I± for ε 6= 0: these are sectors at 0 of radius proportionate
to µ ∼ ε2
4 , separated one from another by the singular directions ±εtR+. They
disappear at the limit. The connection matrices between Y ′′I+ and Y ′′I− are given by
S′′1t =(
1 0
s1t 1
), S′′t1 =
(1 st1
0 1
), submatrices of the Stokes matrices S1t, St1 (36).
- An outer domain O covering a complement of I+ ∪ I− in a disc of a fixed radius with
a cut in the direction α(0)α(1) R+ ∼ ε2tR+. We are mainly interested in the limit when
ε→ 0 along the sequences 1ε ∈
1ε0± 2N, hence the cut will be always in the direction
tR+. The connection matrix on this cut is S′′t =(X0 −i−i 0
). See Figure 9.
Returning now to the full system (43), one must intersect the domains I±, O with thesectors of the Borel summability of the transformation T . The full picture is thereforethat of Figure 10. There are six inner Stokes matrices Sij (36) between the canonicalsolutions on the inner domains, and five outer Stokes matrices between the canonicalsolutions on the outer domains. Only three of the outer domains persist to the limitε→ 0 together with the associated Stokes matrices:
S0 =
(1 s0t s01
1
1
), St =
(1
X0 −i−i 0
), S1 =
(1
st0 1
s10 1
), N =
(e20ete1
ete1
).
(46)The connection matrices between the canonical bases on the inner and outer domainsare provided by:
C+ =
(1
1 1s1t
0 −i ete1s1t
), C− =
1
0et
e1s1t
i −i e2te21s1t
. (47)
9This vector field is defined only on the Riemann surface of√α(0) + ξα(1) but the foliation by its
real trajectories projects well onto the ξ-space.
28
Figure 9: The inner and outer domains, and the connection matrices of the system (45).
Figure 10: The Stokes matrices of the confluent system, N (37), Sij (15), Si (46), C±(47).
Lemma 23. The coefficients of the outer Stokes matrices are equal to
s0t = s0t + s01s1t, s01 = −i e1ets0t,
st0 = st0 + s10s1t, s10 = −i et
e1
s10s1t.
(48)
Proof. We have S0 = C−N−1NS01S0tN
−1N(C−)−1, S1 = C+S10St0(C+)−1, see Figure 10.
Remark 24. The inner Stokes matrices Sij are considered only up to conjugation by
diagonal matrices
(d0
dtd1
). This corresponds through (48) to conjugation of the outer
Stokes matrices Si by
(d0
dtdt
).
The new variables (X0,Wt, U1) (22) are defined by
X0 = X0, Wt = ie0s10s0t + ete1e0, U1 = e0s10s01 + e0. (49)
29
They are invariant with respect to the conjugation of Remark 24.The monodromy matrix of an outer solution around the origin is given by
M−1∞ = S1StN S0 =
e20 e20s0t e20s01
e20st0 ete1X0 + e20st0s0t −iete1 + e20st0s01
e20s10 −iete1 + e20s10s0t e20s10s01
,
and we know that its eigenvalues are again 1, e−2πiκ1 = e0ete1e∞
and e−2πiκ2 = e0ete1e∞.Expressing the coefficients of the linear and the quadratic term of the characteristicpolynomial of M−1
∞ gives therefore
ete1X0 + e0U1 + e20s0tst0 = 1 + e0ete1
e∞+ e0ete1e∞ := E,
e0ete1X0U1 + e0ete1Wt + ie20ete1st0s01 = e0ete1
e∞+ e0ete1e∞ + e2
0e2t e
21 := E′.
Inserting these two identities into the obvious relation
0 = e20 · s01st0 · s10s0t − e2
0 · s0tst0 · s10s01
= −is01st0(e0Wt + ete1)− s0tst0(e0U1 − e20) = 0
(49), gives the Fricke relation (23)
0 = X0WtU1 +W 2t + U2
1 − θ0X0 − θtWt − θ1U1 + θ∞,
with
θ0 = ete1, θt = ete1e0
+ E′
e0ete1= a∞ + ete1a0,
θ∞ = E + E′
e20= 1 + ete1a0a∞ + e2
t e21, θ1 = e0 + E
e0= a0 + ete1a∞.
The formulas of Proposition 14 can be obtained from (48) and (40), (49). Thesingular line L0(a) (27) corresponds to s1t = 0.
30
References[BV89] D.G. Babbitt, V.S. Varadarajan, Local moduli for meromorphic differential equations,
Asterisque 169-170 (1989).
[Bal00] W. Balser, Formal power series and linear systems of meromorphic ordinary differentialequations, Springer, 2000.
[BJL81] W. Balser, W.B. Jurkat, D.A. Lutz, On the reduction of connection problems with an ir-regular singularity to ones with only regular singularities, I, II, SIAM J. Math. Anal. 12(1981), 691–721, SIAM J. Math. Anal. 19 (1988), 398–443.
[Bit15] A. Bittmann, Doubly-resonant saddle-nodes in C3 and the fixed singularity at infinity inthe Painleve equations. Part I: Formal classification, preprint arXiv:1505.06300.
[Bit16] A. Bittmann, Doubly-resonant saddle-nodes in C3 and the fixed singularity at infinity inthe Painleve equations. Part II: Sectorial normalization, preprint arXiv:1605.05052.
[Boa05] P. Boalch, From Klein to Painleve via Fourier, Laplace, Jimbo, Proc. London Math. Soc.90 (2005), 167–208.
[Bo14] P. Boalch, Geometry and braiding of Stokes data; Fission and wild character varieties, Ann.of Math. 179 (2014), 301–365.
[Boa10] P. Boalch, Towards a nonlinear Schwarz’s list, in: The many facets of geometry: a tributeto Nigel Hitchin (J-P. Bourguignon, O. Garcia-Prada, S. Salamon eds.), Oxford UniversityPress (2010), 210–236.
[Bol97] A. Bolibruch, On isomonodromic deformations of Fuchsian systems, J. Dynam. ControlSyst. 3 (1997), 589–604.
[CL09] S. Cantat, F. Loray, Dynamics on Character Varieties and Malgrange irreducibility ofPainleve VI equation, Ann. Inst. Fourier 59 (2009), 2927–2978.
[Cas09] G. Casale, Une preuve galoisienne de l’irreductibilite au sens de Nishioka-Umemura de lapremiere equation de Painleve, Asterisque 324 (2009), 83–100.
[CMR15] L.Chekhov, M. Mazzocco, V. Rubtsov, Painleve monodromy manifolds, decorated charactervarieties and cluster algebras, arXiv:1511.03851 (2015).
[Du98] A. Duval, Confluence procedures in the generalized hypergeometric family, J. Math. Sci.Univ. Tokyo 5 (1998), 597–625.
[Fu07] R. Fuchs, Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei imEndlichen gelegenen wesentlich singularen Stellen, Math. Ann. 63 (1907), 301-321.
[Gam10] B. Gambier, Sur les equations differentielles du second ordre et du premier degre dontl’integrale generale est a points critique fixes, Acta Math. 33 (1910), 1–55.
[Gar19] R. Garnier, Sur les singularites irregulieres des equations differentielles lineaires, J. demath. pures et appl. 8e serie (1919), 99–200.
[Glu99] A. Glutsyuk, Stokes Operators via Limit Monodromy of Generic Perturbation, J. Dyn.Control Syst. 5 (1999), 101–135.
[HF07] Y. Haraoka, G. Filipuk, Middle convolution and deformation for fuchsian systems, J. Lon-don Math. Soc. 76 (2007), 438–450.
[HLR13] J. Hurtubise, C. Lambert, C. Rousseau, Complete system of analytic invariants for unfoldeddifferential linear systems with an irregular singularity of Poincare rank k, Moscow Math.J. 14 (2013), 309–338.
[IY08] Y. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations, Grad. StudiesMath. 86, Amer. Math. Soc., Providence, 2008.
[IIS06] M. Inaba, K. Iwasaki, M.-H. Saito, Dynamics of the sixth Painleve equation, in: Seminaireset Congres 14 (2006), 103–167.
[IKSY91] K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painleve: a moderntheory of special functions, Vieweg, 1991.
[Iwa03] K. Iwasaki, An area-preserving action of the modular group on cubic surfaces and themonodromy of the Painleve VI equation, Commun. Math. Phys. 242 (2003), 185–219.
[Iwa02] K. Iwasaki, A modular group action on cubic surfaces and the monodromy of the PainleveVI equation, Proc. Japan. Acad., Serie A 78 (2002), 131–135.
[JM81] M. Jimbo, T. Miwa, Monodromy preserving deformations of linear ordinary differentialequations with rational coefficients II, Physica D 2 (1981), 407–448.
[KT06] H. Kimura, K. Takano, On confluences of general hypergeometric systems, Tohoku Math.J. 58 (2006), 1–31.
31
[Kli14] M. Klimes, Analytic classification of families of linear differential systems unfolding a res-onant irregular singularity, arXiv:1301.5228 (2014).
[Kli15] M. Klimes, Confluence of singularities in hypergeometric systems, arXiv:1511.00834 (2015).
[Kli16] M. Klimes, Confluence of singularities of non-linear differential equations via Borel-Laplacetransformations, J. Dynam. Contr. Syst. 22 (2016), 285–324.
[LR08] C. Lambert, C. Rousseau, The Stokes phenomenon in the confluence of the hypergeometricequation using Riccati equation, J. Differential Equations 244 (2008) 2641-–2664.
[LR12] C. Lambert, C. Rousseau, Complete system of analytic invariants for unfolded differentiallinear systems with an irregular singularity of Poincare rank 1, Moscow Math. Journal 12(2012), 77–138.
[LT08] O. Lysovyy, Yu. Tykhyy, Algebraic solutions of the sixth Painleve equation, arXiv:0809.4873(2009).
[MR88] J. Martinet, J.-P. Ramis, Theorie de Galois differentielle et resommation, in: ComputerAlgebra and Differential Equations (E.Tournier ed.), Acad. Press, 1988.
[MR91] J. Martinet, J.-P. Ramis, Elementary acceleration and multisummability. I, Ann. Inst. HenriPoincare (A) Physique theorique 54 (1991), 331–401.
[Maz02] M. Mazzocco, Painleve sixth equation as isomonodromic deformation equation of an irreg-ular system, CRM Proceedings and Lecture Notes 32, Amer. Math. Soc., 2002.
[Obl04] A. Oblomkov, Double affine Hecke algebras of rank 1 and affine cubic surfaces, IMRN 18(2004), 877–912.
[OO06] Y. Ohyama, S. Okumura, A coalescent diagram of the Painleve equations from the viewpointof isomonodromic deformations, J. Physics A: Math. Gen. 39 (2006), 12129–12151.
[Oka80] K. Okamoto, Polynomial Hamiltonians associated with Painleve Equations. I, Proc. JapanAcad. Ser. A, Math. Sci. 56 (1980), 264–268.
[Oka79] K. Okamoto, Sur les feuilletages associes aux equations du second ordre a points critiquesfixes de P. Painleve, Espaces des conditions initiales, Japan. J. Math 5 (1979), 1–79.
[Par01] L. Parise, Confluence de singularites regulieres d’equations differentielles en une singulariteirreguliere. Modele de Garnier, these de doctorat, IRMA Strasbourg (2001). [http://www-irma.u-strasbg.fr/annexes/publications/pdf/01020.pdf]
[PR15] E. Paul, J.-P. Ramis, Dynamics on wild character varieties, Symmetry, Integrability andGeometry: Methods and Applications (SIGMA) 11 (2015), 331–401.
[PS09] M. van der Put, M-H. Saito, Moduli spaces for linear differential equations and the Painleveequations, Ann. Inst. Fourier 59 (2009), 2611–2667.
[Ram85] J.-P. Ramis, Phenomene de Stokes et resommation, C. R. Acad. Sc. Paris 301 (1985),99–102.
[Ram89] J.-P. Ramis, Confluence and resurgence, J. Fac. Sci. Univ. Tokyo, Sec. IA 36 (1989), 703–716.
[RT08] C. Rousseau, L. Teyssier, Analytical moduli for unfoldings of saddle-node vector filds,Moscow Math. J. 8 (2008), 547–614.
[Sch85] R. Schafke, Uber das globale analytische Verhalten der Normallosungen von (s−B)v′(s) =(B + t−1A)v(s) und zweier Arten von assoziierten Funktionen, Math. Nachr. 121 (1985),123–145.
[Sch98] R. Schafke, Confluence of several regular singular points into an irregular singular one, J.Dyn. Control Syst. 4 (1998), 401–424.
[Sch01] R. Schafke, Formal Fundamental Solutions of Irregular Singular Differential Equations De-pending Upon Parameters, J. Dyn. Control Syst. 7 (2001), 501–533.
[SP03] M. Singer, M. van der Put, Galois Theory of Linear Differential Equations, Grundlehrender mathematische Wissenschaften 328, Springer, 2003.
[Tak83] K. Takano, A 2-parameter family of solutions of Painleve equation (V) near the point atinfinity, Funkcialaj Ekvacioj 26 (1983), 79–113.
[Yos84] S. Yoshida, A general solution of a nonlinear 2-system without Poincare’s condition at anirregular singular point, Funkcialaj Ekvacioj 27 (1984), 367–391.
[Yos85] S. Yoshida, 2-parameter family of solutions of Painleve equations (I)-(V) at an irregularsingular point, Funkcialaj Ekvacioj 28 (1985), 233–248.
[Zha96] C. Zhang, Confluence et phenomene de Stokes, J. Math. Sci. Univ. Tokyo 3 (1996), 91–107.
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