Isomonodromic deformation with an irregular singularity and the Theta function

0.23 MB·English

by  Kazuhide Matsuda

Isomonodromic deformation with
Θ
an irregular singularity and the function
7
Kazuhide Matsuda
0
0
2 Department of Mathmatics, Graduate School of Information Science and Technology,

n Osaka University,
a
J 1-1 Machikaneyama-machi, Toyonaka, 560-0043.

5
1
Abstract: We will study a monodromy preserving deformation with an irregular sin-

] gular point and determine the τ function of the monodromy preserving deformation by
A
the elliptic θ function moving the argument z and the period Ω.
C
Key words: τ function; θ function; monodromy preserving deformation; irregular singu-
.
h
lar point.
t
a
m
[ 1 Introduction

1
v In this paper, we construct a monodromy preserving deformation whose τ function is rep-
3
0 resentedbytheθ functionmovingboththeargumentz andtheperiodΩ. Themonodromy
4 preserving deformation has an irregular singularity of the Poincar´e rank one.
1
In [3], Miwa-Jimbo-Ueno extended the work of Schlesinger in [1] and established a
0
7 general theory of monodromy preserving deformation for a first order matrix system of
0
ordinary linear differential equations,
/
h
at dY n rν A r∞
m = A(x)Y, A(x) = ν,−k A , kxk−1, (1.1)
: dx ν=1 k=0 (x−aν)k+1 − k=1 ∞−
v XX X
i
X having regular or irregular singularities of arbitrary rank.
r The monodromy data to be preserved is
a
(i) Stokes multipliers S(ν) (j = 1,...,2r ),
j ν
(ii) connection matrices C(ν),
(iii) “exponents of formal monodromy” T(ν).
0
Miwa-Jimbo found a deformation equation as a necessary and sufficient condition
for the monodromy data to be invariant for deformation parameters and defined the τ
function for the deformation equation.
1
We will explain the relationship between the τ function and the θ function. In [4],
Miwa-Jimbo expressed the τ function with the θ function by moving the argument z of
the θ function. The monodromy preserving deformation has irregular singularities. The
Riemann surface is a ramified covering of CP1 with the covering degree m and its genus is
g. Especially the case of genus one is written in [2], which we will describe in Appendix.
In [6], Kitaev-Korotkin calculated the τ function with the θ function by moving the
period Ω of the θ function. They consider monodromy preserving deformation of a second
order Fuchsian equation with 2g+2 regular singular points. In this case the τ function is
represented by the theta null value of a hyperelliptic curve with genus g. In case g = 1,
Kitaev-Korotkin’s monodromy preserving deformation is equivalent to Picard’s solution
of the sixth Painlev´e equation by the B¨acklund transformation. The aim of this paper is
to unify Miwa-Jimbo’s result and Kitaev-Korotkin’s result in the elliptic case.
It is well known that the elliptic θ function satisfies the heat equation, whose solution
space is infinite dimensional. The equation (5.4) of isomonodromic deformation is solved
by the τ function. The elliptic θ function also satisfies the nonlinear equation (5.4), whose
solution space is finite dimensional.
We will explain this paper in detail. In section 2, we show some definitions and facts
of elliptic functions, which is necessary to construct the τ-function.
In section 3, we will study the following ordinary differential equation:
3
dY B B A
= −1 + 0 + ν Y(x), (1.2)
dx (x a)2 x a x e
ν!
− − ν=1 −
X
and define the τ-function.
In section 4, we construct a fundamental solution of (1.2) and calculate the following
monodromy data:
(i) Stokes multipliers around x = a Sa = Sa = 1,
1 2
(ii) connection matrices around x = e C(ν)(ν = 1,2,3, ),
ν
∞
(iii) the exponents of formal monodromy Ta = 0,T(ν) = diag( 1, 1)(ν = 1,2,3, ),
0 0 −4 4 ∞
where e is .
∞ ∞
The behavior of the fundamental solution near the singular points is as follows
Y(x) = (1+O(x a))expT(a)(x), (1.3)
−
Y(x) = G(ν)(1+O(x e ))expT(ν)(x) (ν = 1,2,3), (1.4)
ν
−
1
Y(x) = G( )(1+O( ))expT( )(x) (1.5)
∞ ∞
x
2
where
℘′(α)t (x a) 1
T(a)(x) = 2 ℘′(α)t −1 − +T0(a)log(x−a), (1.6)
!
− 2 −
T(ν)(x) = T(ν)log(x e ) (ν = 1,2,3), (1.7)
0 − ν
1
T(∞)(x) = T0(∞)log(x). (1.8)
In section 5, we concretely calculate the monodromy preserving deformation from the
fundamental solution and compute the coefficients, B ,B ,A (ν = 1,2,3).
1 0 ν
−
In section 6, we calculate the τ-function. Section 6 consists of three subsections.
Subsection 6.1 is devoted to H , the Hamiltonian on the deformation parameter t. Sub-
t
section 6.2, 6.3 is about H (ν = 1,2,3), the Hamiltonians on the deformation parameters
ν
e (ν = 1,2,3). In subsection 6.2, we show some facts about elliptic functions in order to
ν
calculate theHamiltoniansH (ν = 1,2,3). Insubsection 6.3, wecalculate H (ν = 1,2,3)
ν ν
and the τ-function. Our main theorem is as follows:
Theorem 1.1. For the monodromy preserving deformation (1.2), the τ function is
τ(e1,e2,e3,t) = θ[p,q] ωt ;Ω ω1−21 (eν −eµ)−18 exp(η21ωt2)exp(t42f(e1,e2,e3)),
(cid:18) 1 (cid:19) 1 ν<µ 3 1
≤Y≤
where
3 3 3
1 1 1
f(e ,e ,e ) = a+ e + (a e ) .
1 2 3 − 3 ν 2 − ν (a e )2
µ
ν=1 ν=1 µ=1 −
X Y X
Acknowledgments. The author thanks Professor Yousuke Ohyama for the careful guid-
ance. This work is partially done by the author in the Isaac Newton Institute for Mathe-
matical sciences, Cambridge University. The author is grateful to Professor D. Korotkin
for a fruitful discussion.
2 Elliptic Function Theory
We will review the elliptic function theory to fix our notation. See [5]. We will consider
the elliptic curve E defined by the equation
y2 = 4(x e )(x e )(x e )
1 2 3
− − −
3
δ γ
> <
q q q q
e e e
1 2 3
∞
Figure 1: Branch cuts and canonical basis of cycles on the elliptic curve, E. Continuous
and dashed paths lie on the first and second sheet of E, respectively.
with arbitrary constants e C (e = e ), where we do not assume that e +e +e = 0.
i i j 1 2 3
∈ 6
We set the new coordinate:
3 3
1 1
x˜ = x e , e˜ = e e for j = 1,2,3
i j j i
− 3 − 3
i=1 i=1
X X
and define the Abel map in the following way:
x dx x˜ dx˜
u := = . (2.1)
4(x e )(x e )(x e ) 4(x˜ e˜ )(x˜ e˜ )(x˜ e˜ )
Z∞ − 1 − 2 − 3 Z∞ − 1 − 2 − 3
p p
Let us define periods ω ,ω by
1 2
dx dx
ω = , ω = . (2.2)
1 y 2 y
Zγ Zδ
From the bilinear relation of Riemann, we get
ω
Ω := 2, (Ω) > 0. (2.3)
ω ℑ
1
x˜ = x˜(u), the inverse function of the Abel map, can be expressed with the Weierstrass
℘-function:
3
1
x e = x˜(u) = ℘(u;ω ,ω ). (2.4)
i 1 2
− 3
i=1
X
℘(u) satisfies the following differential equations:
℘(u)2 = 4(℘(u) e˜)(℘(u) e˜)(℘(u) e˜) (2.5)
′ 1 2 3
− − −
= 4℘(u)3 g ℘(u) g (2.6)
2 3
− −
4
We have
η ω u
σ(u;ω ,ω ) = exp 1 u2 1θ ;Ω . (2.7)
1 2 2ω θ 11 2ω
(cid:18) 1 (cid:19) 1′1 (cid:18) 1 (cid:19)
We set the σ function with characteristic p,q C in the following way:
∈
η ω u
σ[p,q](u) := exp 1 u2 1θ[p,q] ;Ω . (2.8)
2ω θ ω
(cid:18) 1 (cid:19) 1′1 (cid:18) 1 (cid:19)
σ[p,q](u) has the following periodicity properties:
ω
σ[p,q](u+ω ) = exp(2πip)expη u+ 1 σ[p,q](u), (2.9)
1 1
2
(cid:16) ω(cid:17)
σ[p,q](u+ω ) = exp( 2πiq)expη u+ 2 σ[p.q](u). (2.10)
2 2
− 2
(cid:16) (cid:17)
3 The Schlesinger System
We will study the following ordinary differential equation:
3
dY B B A
= −1 + 0 + ν Y(x), (3.1)
dx (x a)2 x a x e
ν!
− − ν=1 −
X
where A ,A ,A ,B ,B sl(2,C) are independent of x.
1 2 3 1 0
− ∈
The monodromy data of (3.1) is as follows:
(i) Stokes multipliers around x = a Sa = Sa;
1 2
(ii) connection matrices around ,e C ,C(ν)(ν = 1,2,3);
ν ∞
∞
(iii) the exponents of formal monodromy Ta,T(∞),T(ν)(ν = 1,2,3).
0 0 0
In the next section, we will get convergent series around x = a, ,e (ν = 1,2,3)
ν
∞
in the following way:
Y(x) = (1+O(x a))expT(a)(x) = Yˆ(a)(x)expT(a)(x), (3.2)
−
1
Y(x) = G( )(1+O( ))expT( )(x) = G( )Yˆ( )(x)expT( )(x), (3.3)
∞ ∞ ∞ ∞ ∞
x
Y(x) = G(ν)(1+O(x e ))expT(ν)(x) = G(ν)Yˆ(ν)(x)expT(ν)(x),
− ν 0
(ν = 1,2,3). (3.4)
5
where
℘′(α)t (x a) 1
T(a)(x) = 2 ℘′(α)t −1 − +T0(a)log(x−a), (3.5)
!
− 2 −
1
T( )(x) = T log( ) (3.6)
∞ 0∞ x
T(ν)(x) = T(ν)log(x e ) (ν = 1,2,3). (3.7)
0 − ν
For the deformation parameters t,e ,e ,e , we set the following closed 1-form
1 2 3
ω = ω +ω +ω +ω (3.8)
a e1 e2 e3
= H dt+H de +H de +H de , (3.9)
t 1 1 2 2 3 3
where
∂Yˆ(a)
ω = Res trYˆ(a)(x) 1 (x)dT(a)(x), (3.10)
a x=a −
− ∂x
∂Yˆ(ν)
ω = Res trYˆ(ν)(x) 1 (x)dT(ν)(x) (ν = 1,2,3), (3.11)
eν − x=eν − ∂x
andd is the exterior differentiation with respect to the deformationparameters t,e ,e ,e .
1 2 3
Especially, we can write
1 dY 2
ω = Res tr Y 1 de . (3.12)
eν x=eν2 dx − ν
" #
(cid:18) (cid:19)
From the closed 1-form ω, we define the τ-function in the following way:
ω := dlogτ(e ,e ,e ,t). (3.13)
1 2 3
4 The Riemann-Hilbert Problem for Special Param-
eters
In this section, we will concretely construct a 2 2 matrix valued function Y(x), whose
×
monodromy data, (i) Stokes multipliers, (ii) connection matrices, (iii) exponents of formal
monodromies, is independent of the deformation parameters t,e ,e ,e .
1 2 3
We denote the Abel map u = u(x) by u = u(P), because we regard it as a function of
the Riemann surface E defined by
y2 = 4(x e )(x e )(x e ). (4.1)
1 2 3
− − −
6
x
q 0
(cid:0)(cid:19)(cid:1)(cid:3)@SCA
(cid:0)(cid:19)(cid:1)(cid:3) CAS@
(cid:0)(cid:19)(cid:1)(cid:3) CAS@
(cid:0)(cid:19)(cid:1) (cid:3) C AS@
(cid:0)(cid:19) (cid:1) (cid:3) C A S@
(cid:0) (cid:19) (cid:1) (cid:3) C A S @
(cid:0) (cid:19) (cid:1) (cid:3) C A S @
(cid:0) (cid:19) (cid:1) (cid:3) C A S @
(cid:0) (cid:19) (cid:1) (cid:3) C A S @
(cid:0) (cid:19) (cid:1) (cid:3) C A S @
(cid:0) (cid:19) (cid:1) (cid:3) C A S @
(cid:0) (cid:19) (cid:1) (cid:3) C A S @
q q q q
e e e
(cid:26)∞> (cid:25)(cid:26)>1 (cid:25) (cid:26)>2 (cid:25)(cid:26)>3 (cid:25)
l l l l
1 2 3
∞
Figure 2: Generators of π (CP1 ,e ,e ,e ,x )
1 1 2 3 0
\{∞ }
We set the involution of E in the following way:
: (x,y) (x, y). (4.2)
∗ −→ −
We definethe 2 2matrixvalued functionΦ(P)with theσ functionandtheζ function
×
in the following way:
ϕ(P) ϕ(P )
Φ(P) = ∗ , (4.3)
ψ(P) ψ(P )
∗
(cid:18) (cid:19)
where
t
ϕ(P) = σ[p,q](u+u +t)σ(u u )exp ζ(u α)+ζ(u+α)
ϕ ϕ
− −2{ − }
t
ψ(P) = σ[p,q](u+u +t)σ(u u )exp ζ(u α)+ζ(u+α)
ψ ψ
− −2{ − }
u = u(P ), u = u(P ), α = u(a),
ϕ ϕ ψ ψ
with arbitrary P ,P E.
ϕ ψ
∈
Proposition 4.1. The function Φ(P) is holomorphic and invertible outside of the branch
points ,e ,e ,e and transforms as follows with respect to the tracing along the basic
1 2 3
∞
cycles γ,δ:
exp πi(2p+1)
T Φ(P) = Φ(P) { } exp η (2u+ω )
γ exp πi(2p+1) { 1 1 }
(cid:18) {− } (cid:19)
(cid:2) (cid:3) exp πi(2q +1)
T Φ(P) = Φ(P) {− } exp η (2u+ω )
δ exp πi(2q+1) { 2 2 }
(cid:18) { } (cid:19)
(cid:2) (cid:3)
where by T we denote the operator of analytic continuation along the contour l.
l
7
Proof. By using the periodicity (2.9) and (2.10), we get
exp πi(2p+1)
T [Φ(P)] = Φ(P) { } exp η (2u+ω )
γ exp πi(2p+1) { 1 1 }
(cid:18) {− } (cid:19)
exp πi(2q +1)
T [Φ(P)] = Φ(P) {− } exp η (2u+ω ) .
δ exp πi(2q +1) { 2 2 }
(cid:18) { } (cid:19)
Then we obtain
T [detΦ(P)] = exp2η (2u+ω )detΦ(P)
γ 1 1
T [detΦ(P)] = exp2η (2u+ω )detΦ(P).
δ 2 2
By integrating d(detΦ(P)) along the fundamental polygon of E and using the Legendre
detΦ(P)
relation, we get
1 d(detΦ(P)) 1
= 4(η ω η ω ) = 4. (4.4)
1 2 2 1
2πi detΦ(P) 2πi −
I∂Eˆ
Therefore, four zeros of detΦ(P) on E are branch points, ,e ,e ,e . We proved the
1 2 3
∞
proposition.
We will normalize Φ(P) at x = a. We can easily show the local behavior of the Abel
map.
Lemma 4.2.
1 ℘ (α) ℘ (α)2 ℘ (α)
u α = (x a) ′′ (x a)2 + ′′ ′′′ (x a)3 + . (4.5)
− ℘(α) − − 2℘ (α) − 2℘(α)5 − 6℘(α)4 − ···
′ ′′′ (cid:18) ′ ′ (cid:19)
From Lemma 4.2, Φ(x) can be developed near x = a in the following way:
℘′(α)t
Φ(x) = G(a) +O(x a) exp −2(x a) ,
− − ℘′(α)t
!
2(x a)
(cid:0) (cid:1) −
where G(a) is a 2 2 matrix with the matrix elements:
×
t ℘ (α)
(G(a)) = σ[p,q](α+u +t)σ(α u )exp ′′ +ζ(2α) ,
11 ϕ − ϕ − 2 2℘(α)
(cid:18) ′ (cid:19)
n o
t ℘ (α)
(G(a)) = σ[p,q](α+u +t)σ(α u )exp ′′ +ζ(2α) ,
21 ψ ψ
− − 2 2℘(α)
(cid:18) ′ (cid:19)
n o
t ℘ (α)
(G(a)) = σ[p,q]( α+u +t)σ( α u )exp ′′ +ζ(2α) ,
12 ϕ ϕ
− − − 2 2℘(α)
(cid:18) ′ (cid:19)
n o
t ℘ (α)
(G(a)) = σ[p,q]( α u +t)σ( α u )exp ′′ +ζ(2α) .
22 − − ψ − − ψ 2 2℘(α)
(cid:18) ′ (cid:19)
8 n o
From Proposition 4.1,
detG(a) = detΦ(a) = 0.
6
We define a matrix valued function Y(x) in the following way:
detΦ(a)
Y(P) = (G(a)) 1Φ(P), (4.6)
−
detΦ(P)
p
By the normalization near x = ap, we obtain the following lemma:
Lemma 4.3.
1 0
Y(x) = +O(x a) expT(a)(x)
0 1 −
(cid:18)(cid:18) (cid:19) (cid:19)
(x a) 1 ℘′(α)t
T(a)(x) = T(a1) −1 − , T(a1) = 2 ℘′(α)t .
− − − !
− 2
Especially, if u = α,u = α,
ϕ ψ
−
1 0
Y(x) = +Y(a)(x a)+ expT(a)(x)
0 1 1 − ···
(cid:18)(cid:18) (cid:19) (cid:19)
1 σ[p,q](t) t 1 ℘ (α) 2
Y(a) = ′ 4℘(α) ′′
1 11 ℘′(α) σ[p,q](t) − 2 − 2 (cid:18)℘′(α)(cid:19) !
(cid:16) (cid:17) n o
σ[p,q](2α+t) ℘ (α)
Y(a) = exp t ′′ +ζ(2α)
1 21 σ[p,q](t)σ[p,q](2α)℘′(α) − (cid:18)2℘′(α) (cid:19)
(cid:16) (cid:17) n o
σ[p,q]( 2α+t) ℘ (α)
Y(a) = − − exp t ′′ +ζ(2α)
1 σ[p,q](t)σ[p,q](2α)℘(α) 2℘(α)
12 ′ (cid:18) ′ (cid:19)
(cid:16) (cid:17) n o
1 σ[p,q](t) t 1 ℘ (α) 2
Y(a) = − ′ 4℘(α) ′′ .
1 22 ℘′(α) σ[p,q](t) − 2 − 2 (cid:18)℘′(α)(cid:19) !
(cid:16) (cid:17) n o
Y(x) has the following monodromy and Stokes matrices.
Theorem 4.4. For ν = ,1,2,3, we set
∞
0 m
M = ν . (4.7)
ν m 1 0
(cid:18) − −ν (cid:19)
The matrix elements corresponding to the contour l (ν = ,1,2,3) are given by the
ν
∞
expressions
m = i, m = iexp( 2πip)
1
∞ − −
m = iexp2πi( p+q)
2
− −
m = iexp(2πiq).
3
9
Stokes matrices are as follows
1 0
S = S = .
1 2 0 1
(cid:18) (cid:19)
Proof. From Proposition 4.1, we note that the branch points ,e ,e ,e on E are zero
1 2 3
∞
points of detΦ(P) of the first order. According to Proposition 4.1, we obtain
T [Y(x)] = Y(x)M M
γ 3 2
exp 2πip
= Y(x) { } ,
exp 2πip
(cid:18) {− } (cid:19)
Tδ−1 [Y(x)] = Y(x)M2M1
exp 2πiq
= Y(x) { } .
exp 2πiq
(cid:18) {− } (cid:19)
By the involution *, we get
i i
Y(x)M = Y(x) , or = Y(x) − .
i i
∞
(cid:18) (cid:19) (cid:18) − (cid:19)
We set
m i
M = m 1 ∞ = i − .
∞ −
(cid:18) − (cid:19) (cid:18) − (cid:19)
∞
From the property of the monodromy group, we obtain
M M M = M 1. (4.8)
3 2 1 −
∞
From
M M = diag(exp 2πip ,exp 2πip ),
3 2
{ } {− }
M M = diag(exp 2πiq ,exp 2πiq ),
2 1
{ } {− }
we get
m exp 2πip
M1 = m 1exp 2πip − ∞ {− } ,
−
(cid:18) { } (cid:19)
∞
m exp 2πiq
M3 = m 1exp 2πiq − ∞ { } .
−
(cid:18) {− } (cid:19)
∞ 10