Symmetries of the 4-matrices Schlesinger system

M.V.Babich^{1}

January 11, 2000

Abstract

All types of symmetries, such as a permutation of parameters, Schlesinger and Okamoto transformations are considered. The new form of Schlesinger system and the action of modular group on the corresponding linear system are studied, the geometrical meaning of the Okamoto transformation is discussed.

0 Schlesinger system

Consider the following linear differential equation on SL(2, ℂ)-valued function Ψ of the complex variable λ:

\frac{d}{dλ}Ψ = ∑_{k} A_{k} \frac{1}{λ − λ_{k}} Ψ, Ψ ∈ SL(2, ℂ), A_{k} ∈ sl(2, ℂ), λ, λ_{k} ∈ ℂ.  (1)

The solution Ψ(λ) is a multivalued function with branch points λ_{k}. When Ψ is prolonged along a closed path going around λ_{k}, Ψ is changed on some (λindependent) matrix M_{k}. Matrices M_{k} are called the monodromy matrices. In general, monodromy matrices depend on the positions of branch points λ_{k}, if we consider them as parameters. How should we change matrices A_{k} for keeping constant all monodromy matrices M_{k} and eigenvalues of A_{k}?

L. Schlesinger [1] found that these conditions are equivalent to the following system of equations on matrices A_{k}, so called Schlesinger system (Ss):

\frac{∂A_{k}}{∂λ_{j}} = \frac{1}{λ_{j} − λ_{k}} [A_{j}, A_{k}] (j ≠ k), \frac{∂A_{k}}{∂λ_{k}} = ∑_{j≠k} \frac{1}{λ_{k} − λ_{j}} [A_{j}, A_{k}],  (2)

function Ψ depends on parameters λ_{k} in according to the equations

\frac{∂}{∂λ_{k}} Ψ = \frac{1}{λ_{k} − λ}A_{k}Ψ ∀k.  (3)

1Permanent address: St.Petersburg department of the Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 191011 Russia; E-mail: mbabich@pdmi.ras.ru

In the present paper I am considering the simplest non-trivial case of four matrices A_{1}, A_{2}, A_{3}, A_{0} = −(A_{1} + A_{2} + A_{3}). By the fractional-linear transformation we move points {λ_{1}, λ_{2}, λ_{3}, λ_{0}} into {0, x, 1, ∞}. Function Ψ(λ) depends on the complex parameter x and, as a function of λ, has four branch points on \overline{ℂ}: 0, 1, x and infinity.

Equations (1), (3) take the form

\{ \begin{matrix}\frac{∂}{∂λ}|_{x=const}Ψ & = (\frac{1}{λ} A_{1} + \frac{1}{λ − x}A_{2} + \frac{1}{λ − 1}A_{3}) Ψ \\ \frac{∂}{∂x}|_{λ=const}Ψ & = \frac{1}{x − λ}A_{2}Ψ, \end{matrix}  (4)

and Ss takes the form

\frac{d}{dx}A_{1} = \frac{1}{x} [A_{2}, A_{1}] , \frac{d}{dx}A_{2} = \frac{1}{x} [A_{1}, A_{2}] + \frac{1}{x − 1} [A_{3}, A_{2}] , \frac{d}{dx}A_{3} = \frac{1}{x − 1} [A_{2}, A_{3}] . (5)

The set (a_{1}, a_{2}, a_{3}, a_{0}), where a_{k}/2 is an eigenvalue of A_{k} is called the set of parameters of Ss.

The symmetries of Ss are under consideration. The concept of symmetry is treated as a correspondence between two Ss’s with different parameters. In this connection I want to mention here the works of D. Arinkin and S. Lysenko [2, 3], where using methods of theory of stacks, they found all the isomorphisms between moduli spaces corresponding to different Schlesinger systems.

1 Permutation of parameters

The source of this symmetry is evident on the level of a setting up the problem of isomonodromy deformation. The normalization dΨΨ^{−1} → 0 when λ → ∞ destroys this symmetry, and we make some gauge transformation for the clarifying the action of S_{4}-group on the Ψ-function, the solution of the linear system.

Consider the universal covering of \overline{ℂ}_{λ} \ \{0, 1, x, ∞\}, it is the complex plane ℂ_{u}. We define the correspondence between λ and u in the following way:

u = \frac{1}{4} ∫_{0}^{λ} \frac{dt}{\sqrt{t(1 − t)(x − t)}}/ ∫_{0}^{1} \frac{dt}{\sqrt{(1 − t^{2})(1 − xt^{2})}}, λ = \frac{θ_{2}^{2}(0) θ_{1}^{2}(u)}{θ_{3}^{2}(0) θ_{0}^{2}(u)}

τ = i ∫_{0}^{1} \frac{dt}{\sqrt{(1 − t^{2})(1 − (1 − x)t)}}/ ∫_{0}^{1} \frac{dt}{\sqrt{(1 − t^{2})(1 − xt^{2})}}, x = θ_{2}^{4}(0|τ)/θ_{3}^{4}(0|τ),

and the theta functions are calculated by the period τ:

θ[p, q](u|τ) = ∑_{m∈ℤ}exp{πi(m + p)^{2}τ + 2πi(u + q)(m + p)}

θ_{1}(u):=−θ[\frac{1}{2}, \frac{1}{2}](u|τ), θ_{2}(u):=θ[\frac{1}{2}, 0](u|τ), θ_{3}(u):=θ[0, 0](u|τ), θ_{0}(u):=θ[0, \frac{1}{2}](u|τ).

Theorem 1. Let G = G(x) be a solution of the equation

−2G^{−1} \frac{d}{dx}G = \frac{1}{x}A_{1} + \frac{2x − 1}{x(x − 1)}A_{2} + \frac{1}{x − 1}A_{3},

and Ψ be a solution of the system (4).

Then the function Ψ̂ = GΨ satisfies the system

\{ \begin{matrix}Ψ̂′ & = & 2 ∑_{k=0}^{3} log′ θ_{k}(u|τ)Â_{k}Ψ̂ \\ Ψ̂^{•} & = & 2 ∑_{k=0}^{3} log^{•} θ_{k}(u|τ)Â_{k}Ψ̂, \end{matrix}  (6)

where ...′ = ∂_{u}|_{τ=const} , ...^{•}= ∂_{τ}|_{u=const}, Â_{k} = GA_{k}G^{−1}.

Proof:

Consider (6); the change of variables (u, τ) ⟶ (λ, x) gives^{2}

\{ \begin{matrix}\frac{∂}{∂λ} |_{x=const}Ψ̂Ψ̂^{−1} & = \frac{1}{λ} Â_{1} + \frac{1}{λ − x}Â_{2} + \frac{1}{λ − 1}Â_{3} \\ \frac{∂}{∂x} |_{λ=const}Ψ̂Ψ̂^{−1} & = \frac{1}{x − λ}Â_{2} − \frac{1}{2x}Â_{1} − \frac{2x − 1}{2x(x − 1)}Â_{2} − \frac{1}{2(x − 1)}Â_{3}. \end{matrix}

If G = G(x) is a solution of the equation

−2\frac{d}{dx}GG^{−1} = \frac{1}{x}Â_{1} + \frac{2x − 1}{x(x − 1)}Â_{2} + \frac{1}{x − 1}Â_{3},

one can see that Ψ = G^{−1}Ψ̂ satisfy the system (4).

QED

It is worth to notice the puzzling resemblance of the linear representation (6) to the linear system arisen in description of isomonodromic deformations on the torus [7].

Remark 1. Using the analytical properties of Ψ̂_{u}Ψ̂^{−1} and Ψ̂_{τ} Ψ̂^{−1}, we can write the following representation for the second equation of (6):

4πi Ψ̂^{•}= (Â′ + Â log′ θ_{1}(2u|τ))Ψ̂ , where Â = ∑_{k=0}^{3} log′ θ_{k}(u|τ)Â_{k}.

2For the details see the Appendix

For the representation (6) the compatibility conditions, i.e. the version of Ss (5) take the following symmetrical form

\{ \begin{matrix}Â^{•}_{1} & = 2 [Â_{2} log^{•} θ_{2} + Â_{3} log^{•} θ_{3} + Â_{0} log^{•} θ_{0}, Â_{1}] \\ Â^{•}_{2} & = 2 [Â_{1} log^{•} θ_{2} + Â_{0} log^{•} θ_{3} + Â_{3} log^{•} θ_{0}, Â_{2}] \\ Â^{•}_{3} & = 2 [Â_{0} log^{•} θ_{2} + Â_{1} log^{•} θ_{3} + Â_{2} log^{•} θ_{0}, Â_{3}] \\ Â^{•}_{0} & = 2 [Â_{3} log^{•} θ_{2} + Â_{2} log^{•} θ_{3} + Â_{1} log^{•} θ_{0}, Â_{0}] \end{matrix}  (7)

with the standard notation θ_{k} = θ_{k}(0|τ).

Let us define the action of the modular group Σ on functions Ψ(u|τ) as follows:

if σ = (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ SL(2, ℤ) ≅ Σ, then σΨ(u|τ) = Ψ (\frac{u}{γτ + δ} |\frac{ατ + β}{γτ + δ} ) .

The system (6) is invariant with respect to the action of the modular group in the following sense

Theorem 2. Let σ = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ Σ be an element of the modular group, and Ψ̂(u|τ) be a solution of (6) for some set Â_{k}(τ), k = 1, 2, 3, 0 . Then the matrix Ψ̃ = σΨ̂ is a solution of

\{ \begin{matrix}Ψ̃′ & = 2 ∑_{k=0}^{3} log′ θ_{k}(u|τ)Ã_{k}Ψ̃ \\ Ψ̃^{•} & = 2 ∑_{k=0}^{3} log^{•} θ_{k}(u|τ)Ã_{k}Ψ̃, \end{matrix}  (8)

where Ã_{k}(τ) = Â_{k_{σ}}(στ); στ = (ατ + β)/(γτ + δ); k_{σ} is the permutation of the indices 2,3 and 0 that corresponds to the permutation of half-periods \{\frac{γτ+δ}{2}; \frac{(α+γ)τ+β+δ}{2}; \frac{ατ+β}{2} \} ⟶ \{\frac{1}{2}; \frac{1+τ}{2} ; \frac{τ}{2}\} (mod 1, τ).

Proof:

It is well known that Σ is generated by two elements S = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and T = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). It is sufficient to verify the statement for S and T, and to check that the matrix ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) produces the identical transformation Ã_{k}(τ) = Â_{k}(τ).

Consider S:

SΨ̂(u|τ) = Ψ̂(u|τ + 1)

Equalities

θ_{1}(u|τ) = e^{−iπ/4}θ_{1}(u|τ + 1), θ_{2}(u|τ) = e^{−iπ/4}θ_{2}(u|τ + 1),

and

θ_{3}(u|τ) = θ_{0}(u|τ + 1), θ_{0}(u|τ) = θ_{3}(u|τ + 1),

imply that Ψ̃ = SΨ̂ satisfies (8) with Ã_{1}(τ) = Â_{1}(τ + 1), Ã_{2}(τ) = Â_{2}(τ + 1), Ã_{3}(τ) = Â_{0}(τ + 1), Ã_{0}(τ) = Â_{3}(τ + 1).

Consider T:

T Ψ̂(u|τ) = Ψ̃(u|τ) = Ψ̂(−u/τ| − 1/τ) = Ψ̂(v|τ_{T}),

where τ_{T} = −1/τ, v = τ_{T}u.

\frac{1}{2}dΨ̃(u|τ) = ∑_{k=0}^{3} log′ θ_{k}(v|τ_{T})Â_{k}(τ_{T})Ψ̃dv + ∑_{k=0}^{3} log^{•} θ_{k}(v|τ_{T})Â_{k}(τ_{T})Ψ̃dτ_{T} =

∑_{k=0}^{3} log′ θ_{k}(v|τ_{T})Â_{k}(τ_{T})τ_{T} Ψ̃du + ∑_{k=0}^{3} (log^{•} θ_{k}(v|τ_{T}) + log′ θ_{k}(v|τ_{T})u) Â_{k}(τ_{T})(τ_{T})^{2}Ψ̃dτ

(9)

From the equalities

θ_{1}(u|τ) = −iμθ_{1}(v|τ_{T}), θ_{2}(u|τ) = μθ_{0}(v|τ_{T}), θ_{3}(u|τ) = μθ_{3}(v|τ_{T}),

and

θ_{0}(u|τ) = μθ_{2}(v|τ_{T}), with μ = \sqrt{−iτ} exp\{τ_{T}πiu^{2}\},

we get

d log θ_{k}(u|τ) = d log μ + d log θ_{k_{T}} (v|τ_{T}) = d log μ + log′ θ_{k_{T}} (v|τ_{T})dv+ log^{•} θ_{k_{T}} (v|τ_{T})dτ_{T} =

d log μ + log′θ_{k_{T}} (v|τ_{T})du + (log^{•} θ_{k_{T}} (v|τ_{T}) + log′θ_{k_{T}} (v|τ_{T})u) dτ_{T},

in which the index k_{T} takes values 1, 0, 3, 2 when k takes values 1, 2, 3, 0.

log′θ_{k_{T}} (v|τ_{T})τ_{T} = log′ θ_{k}(u|τ) − \frac{∂}{∂u} log μ
log^{•} θ_{k_{T}} (v|τ_{T})(τ_{T})^{2} = log^{•} θ_{k}(u|τ) − uτ_{T} (log′θ_{k}(u|τ) − \frac{∂}{∂u} log μ) − \frac{∂}{∂τ} log μ.

Substituting the last two equalities into (9) produces

\frac{1}{2}dΨ̃(u|τ) = ∑_{k=0}^{3} log′ θ_{k}(u|τ)Â_{k_{T}} (τ_{T})du + ∑_{k=0}^{3} log^{•} θ_{k}(u|τ)Â_{k_{T}} (τ_{T})dτ. (10)

Here we have used the identity ∑_{k} Â_{k} = 0 and the fact that μ does not depend on the index “k” of theta-functions. Equality (10) means that Ψ̃ satisfies (8) with

Ã_{1}(τ) = Â_{1}(−1/τ), Ã_{2}(τ) = Â_{0}(−1/τ), Ã_{3}(τ) = Â_{3}(−1/τ), Ã_{0}(τ) = Â_{2}(−1/τ).

To conclude the proof we must show that actions of “σ” and “−σ” coincide, but that is clear since log^{•} θ_{k}(u) =log^{•} θ_{k}(−u) and log′ θ_{k}(u) = log′ θ_{k}(−u).

We have defined the action of the modular group on functions of two variables and denoted this group of transformations by Σ. Let us add one generator more: σ′Ψ(u|τ) = Ψ(u + 1|τ) and denote the corresponding group of transformations by Σ′.

From Theorem 2 and the trivial observation that the function Ψ̃ = σ′Ψ(u|τ) = Ψ(u + 1|τ) satisfies (8) with Ã_{1,2}(τ) = Â_{2,1}(τ), Ã_{3,0}(τ) = Â_{0,3}(τ), there follows

Proposition 1.1. Let Ψ(u|τ) be a solution of (6) with parameters (a_{1}, a_{2}, a_{3}, a_{0}),

and let (1, 2, 3, 0) ⟶ (j_{1}, j_{2}, j_{3}, j_{0}) be any permutation of indices. Then there

is transformation σ ∈ Σ′ such that Ψ̃ = σΨ satisfies (8) with parameters

(a_{j_{1}}, a_{j_{2}}, a_{j_{3}}, a_{j_{0}}).

We say that system (6) is invariant under an arbitrary permutation of parameters.

2 Schlesinger transformations

The idea of the Schlesinger transformation belong to L. Schlesinger himself [1]. Different authors wrote down different formulae for such transformations designed for their own purposes (see [5, 9, 10, 11]). My intention is to explain the conception of the Schlesinger transformation and give such formulae that might be easily modified, as desired.

Consider function Ψ, which is a solution of (4) or (6). From the local considerations near the singular points (see [4, 5, 8])

Ψ(u + ω_{J}) = (G_{J} + O(u^{2})) (\begin{matrix}u^{a_{J}} & 0 \\ 0 & u^{−a_{J}} \end{matrix} ) (I + j_{J}^{±}σ_{±} log u)C_{J},  (11)

where G_{J} + O(u^{2}) is some power series in u^{2}, G_{J}, C_{J} ∈ SL(2, ℂ), σ_{+} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), σ_{−} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ), and j_{J}^{±} vanish for any ±a_{J} ∉ ℤ^{+} = \{0, 1, 2, 3 … \}. Matrices C_{J} and values j_{J} are constants. By ω_{J} we denoted a half-period, zero of the corresponding thetafunction: θ_{J}(ω_{J}) = 0.

It follows from (11), that Â_{J} = G_{J} æ_{J} (G_{J})^{−1}, where 2æ_{J} = a_{J} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) if a_{J} ≠ 0 and 2æ_{J} = j_{J}^{±}σ_{±} if a_{J} = 0.

The (constant) monodromy matrices are

M_{J} = C_{J}^{−1} (\begin{matrix}e^{πia_{J}} & 0 \\ 0 & e^{−πia_{J}} \end{matrix} ) (\begin{matrix}1 & πij_{J}^{±} \\ 0 & 1 \end{matrix}) C_{J}

Let F be the analytical and single-valued outside the singularities of Ψ matrixfunction. Suppose, we found such F, that in the singular points ω_{J} it has such a representation that the product F Ψ = Ψ̃ has the representation (11) again, maybe with some new G̃_{J}, ã_{J}. If so, the function Ψ̃ has the same (consequently constant!) monodromy matrices as Ψ. After the normalization “det Ψ̃ = 1” and proper gauge transformation, Ψ̃ satisfies systems (6) or (4).

Note that from the uniqueness theorem for the corresponding Riemann-Hilbert problem follows that essentially (up to normalizations and gauge transformations) function Ψ is determined by parameters a_{k} and the monodromy matrices M_{k}. Schlesinger transformation does not change the monodromy matrices, consequently different Schlesinger transformations commute (up to a gauge), and any Schlesinger transformation can be constructed as a sequence of so-called “elementary” Schlesinger transformations, which are constructed below.

We present such a matrix F which adds one to two parameters (for example a_{I} and a_{J}):

F_{I^{+},J^{+}} = (\begin{matrix}θ_{I}^{2}(u) & 0 \\ 0 & θ_{J}^{2}(u) \end{matrix}) (\begin{matrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{matrix}) ,

where vector-rows (f_{11}, f_{12}) and (f_{21}, f_{22}) are arbitrary vectors orthogonal to the vector-columns of the leading term of (11) ((G_{J})_{11}, (G_{J})_{21})^{T} and ((G_{I})_{11}, (G_{I})_{21})^{T} respectively; for example:

(\begin{matrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{matrix}) = (\begin{matrix}−(G_{J})_{21} & (G_{J})_{11} \\ −(G_{I})_{21} & (G_{I})_{11} \end{matrix})

One can easily verify that such a construction of the matrix (f_{ij}) preserves form (11) for the product F_{I^{+},J^{+}}Ψ = Ψ̃.

In the same manner we can write down matrix-functions F_{I^{−},J^{+}}, F_{I^{+},J^{−}} and F_{I^{−},J^{−}} that give all possible changes of the values a_{I}, a_{J} by ±1:

F_{I^{+},J^{−}} = (\begin{matrix}θ_{I}^{2}(u) & 0 \\ 0 & θ_{J}^{2}(u) \end{matrix}) (\begin{matrix}−(G_{J})_{22} & (G_{J})_{12} \\ −(G_{I})_{21} & (G_{I})_{11} \end{matrix})

F_{I^{−},J^{+}} = (\begin{matrix}θ_{I}^{2}(u) & 0 \\ 0 & θ_{J}^{2}(u) \end{matrix}) (\begin{matrix}−(G_{J})_{21} & (G_{J})_{11} \\ −(G_{I})_{22} & (G_{I})_{12} \end{matrix})

F_{I^{−},J^{−}} = (\begin{matrix}θ_{I}^{2}(u) & 0 \\ 0 & θ_{J}^{2}(u) \end{matrix}) (\begin{matrix}−(G_{J})_{22} & (G_{J})_{12} \\ −(G_{I})_{22} & (G_{I})_{12} \end{matrix})

Note that it is impossible to make the trivial generalization of this construction and get a half-integer changes of a_{J}, using some function F with simple zeros.

3 Okamoto transformation

In this section we consider the transformation that was found by K. Okamoto [6] for the Painlevé VI equation, which is equivalent to Ss. This transformation connects Schlesinger systems with parameters (a_{1}, a_{2}, a_{3}, a_{0}) and (a_{1}, a_{2}, a_{3}, a_{0})J, where

J = \frac{1}{2} ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & −1 & 1 & −1 \\ 1 & −1 & −1 & 1 \end{matrix})

To make this subject more visual we change the “sl(2, ℂ)-language” to the equivalent “ℂ^{3}-language”.

Let us introduce the scalar and vector products on the sl(2, ℂ):

⟨A, B⟩ = 2tr AB, [A, B] = AB − BA.

We need the following versions^{3} of the well known formulae from linear algebra

− < [A, B]C >^{2} = det (\begin{matrix}< A, A > & < A, B > & < A, C > \\ < B, A > & < B, B > & < B, C > \\ < C, A > & < C, B > & < C, C > \end{matrix})  (12)
−[A, [B, C]] = B < A, C > − C < A, B > .

In this interpretation equations (5) govern the evaluation of tetrahedron A_{1}, A_{2}, A_{3}, A_{0}, where A_{k} is the radius-vector of the same name vertex; the origin is in the masses center. A gauge transformation of the linear system (4) leads to the turn A_{k} ⟶ GA_{k}G^{−1} (where G = G(x) ∈ SL(2, ℂ)) of the tetrahedron as a solid body.

3sl(2, ℝ) ≅ ℝ^{2,1}, not to ℝ^{3}, that is the reason for the minus sign in the left-hand sides

Consider “the Gram-matrix”, i.e. the pair-wise scalar products of radius-vectors of vertices. Note, that the scalar product does not depend on the gauge transformations, and < A_{i}, A_{k} >=< Â_{i}, Â_{k} >= a_{ik}, so we consider (5) and (7) simultaneously:

a_{ik} :=< A_{i}, A_{k} >= 2tr (A_{i}A_{k}), a_{kk} = 2tr A_{k}^{2} = −4 det A_{k} = (a_{k})^{2}.

Matrix entries a_{ik} satisfy the relations

a_{ik} = a_{ki}
∑_{i=0}^{3} a_{ik} = 0 ∀k
a_{ik} + (a_{ii} + a_{kk})/2 = a_{jl} + (a_{jj} + a_{ll})/2 for mutually different i, j, k, l.

(13)

The second and the third rows follow from the equalities < ∑ A_{i}, A_{k} >= 0 and < A_{i} + A_{k}, A_{i} + A_{k} >=< −A_{j} − A_{l}, −A_{j} − A_{l} > which are hold since ∑ A_{i} = 0.

The equations which determine the evaluation of the matrix (a_{ik}) are

\{ \begin{matrix}\frac{d}{dτ} a_{12} & = \frac{d}{dτ} a_{03} = 4 (\frac{d}{dτ} log \frac{θ_{0}}{θ_{3}} ) a_{230} \\ \frac{d}{dτ} a_{13} & = \frac{d}{dτ} a_{20} = 4 (\frac{d}{dτ} log \frac{θ_{2}}{θ_{0}} ) a_{230} \\ \frac{d}{dτ} a_{10} & = \frac{d}{dτ} a_{32} = 4 (\frac{d}{dτ} log \frac{θ_{3}}{θ_{2}} ) a_{230} \\ a_{kk} & = const ∀k, \end{matrix}  (14)

or

\frac{d}{dτ} a_{1k} = \frac{d}{dτ} a_{ij} = −4 (\frac{d}{dτ} log \frac{θ_{i}}{θ_{j}} ) a_{ijk}, i, j, k ∈ \{2, 3, 0\},

where θ_{k} = θ_{k}(0|τ) are the theta-constants, a_{ijk} =< [A_{i}, A_{j}], A_{k} > is a one and a half of the volume of the tetrahedron A_{1}, A_{2}, A_{3}, A_{0} (with some sign).

Note that the sum of equations (14) is zero, so there are only two independent equations of the first order on two unknown functions (for example a_{12} and a_{13}); all other a_{ik} are linear combinations of them and constant parameters a_{kk} (equalities (13)). We can exclude one unknown and get an ordinary differential equation of the second order. After some (non-trivial!) change of variables it can be brought to the Painlevé VI equation.

If we know a_{ik}, the tetrahedron can be restored up to a position, and linear system (4) up to a gauge transformation. Let us set as a goal the following natural question:

What different Schlesinger systems are controlled by the same system of equations (14)?

The answer is:

They are Schlesinger systems, connected by Okamoto transformation.

Let us comment this statement. We say that Ss’s A_{1}, …, A_{0} and Ã_{1}, …, Ã_{0} have the same systems (14) if a_{ik} = ã_{ik} + const_{ik} and a_{230} = ã_{230}, where const_{ik} are independent on the variable τ.

From now on we will assign our tetrahedron A_{1}, A_{2}, A_{3}, A_{0} by three vectors X⃗, Y⃗ , Z⃗:

X⃗ := A_{1} + A_{2}, Y⃗ := A_{1} + A_{3}, Z⃗ := A_{1} + A_{0}

2A_{1} = X⃗ + Y⃗ + Z⃗, 2A_{2} = X⃗ − Y⃗ − Z⃗, 2A_{3} = −X⃗ + Y⃗ − Z⃗, 2A_{0} = −X⃗ − Y⃗ + Z⃗.

Vectors X⃗, Y⃗ , Z⃗ connect the antipodal (having no common vertexes) edges of the tetrahedron and form a basis of the ℂ^{3} space. The calculation of its Gram-matrix gives

(\begin{matrix}X^{2} & a_{10/23} & a_{13/20} \\ a_{10/23} & Y^{2} & a_{12/30} \\ a_{13/20} & a_{12/30} & Z^{2} \end{matrix}) := (\begin{matrix}< X⃗, X⃗ > & < X⃗, Y⃗ > & < X⃗, Z⃗ > \\ < Y⃗ , X⃗ > & < Y⃗ , Y⃗ > & < Y⃗ , Z⃗ > \\ < Z⃗, X⃗ > & < Z⃗, Y⃗ > & < Z⃗, Z⃗ > \end{matrix}) =

(\begin{matrix}< X⃗, X⃗ > & \frac{1}{2}(a_{11} + a_{00} − a_{22} − a_{33}) & \frac{1}{2}(a_{11} + a_{33} − a_{22} − a_{00}) \\ \frac{1}{2}(a_{11} + a_{00} − a_{22} − a_{33}) & < Y⃗ , Y⃗ > & \frac{1}{2}(a_{11} + a_{22} − a_{33} − a_{00}) \\ \frac{1}{2}(a_{11} + a_{33} − a_{22} − a_{00}) & \frac{1}{2}(a_{11} + a_{22} − a_{33} − a_{00}) & < Z⃗, Z⃗ > \end{matrix})

We use the notation a_{1i/jk} = (a_{11}+a_{ii}−a_{jj}−a_{kk})/2 = const, a_{Σ} = X^{2}+Y^{2}+Z^{2} = a_{11} +a_{22} +a_{33} +a_{00} = const. You can see that for any Ss all the pair-wise products of the vectors of this basis are constants, except the “squares of lengths” of the vectors X⃗, Y⃗ , Z⃗ which are connected by one relation: X^{2} + Y^{2} + Z^{2} = a_{Σ} = const. We see that the state of Ss can be naturally parameterized by a point of a sphere with radius \sqrt{a_{Σ}}. The squares of the Cartesian coordinates of this point satisfy equations (14) (a_{12} = X^{2}/2 + const, a_{13} = Y^{2}/2 + const, a_{10} = Z^{2}/2 + const). For the value a_{230} the following expression, using (12), can be written down:

−4a_{230}^{2} = det (\begin{matrix}X^{2} & a_{10/23} & a_{13/20} \\ a_{10/23} & Y^{2} & a_{12/30} \\ a_{13/20} & a_{12/30} & Z^{2} \end{matrix}) = X^{2}Y^{2}Z^{2}−\frac{1}{3} (X^{2}(2a_{12/30}^{2} − a_{10/23}^{2} − a_{13/20}^{2})+

Y^{2}(2a_{13/20}^{2} − a_{12/30}^{2} − a_{10/23}^{2}) + Z^{2}(2a_{10/23}^{2} − a_{12/30}^{2} − a_{13/20}^{2})) + 2a_{12/30}a_{13/20}a_{10/23}−

\frac{1}{3}a_{Σ}(a_{12/30}^{2} + a_{13/20}^{2} + a_{10/23}^{2}) = X^{2}Y^{2}Z^{2} + \frac{2}{3}(4σ_{2}^{2} − σ_{1}^{2})X^{2} − \frac{1}{3}(4σ_{2}^{2} − σ_{1}^{2} + 24\sqrt{σ_{4}})Y^{2}−

\frac{1}{3}(4σ_{2}^{2} − σ_{1}^{2} − 24\sqrt{σ_{4}})Z^{2} − 16σ_{3} + \frac{8}{3}σ_{1}σ_{2} − σ_{1}^{3},

where, following Okamoto [6], we denoted

a_{1} = b_{1} + b_{2}, a_{2} = b_{1} − b_{2}, a_{3} = b_{3} + b_{4}, a_{0} = b_{3} − b_{4},

σ_{1} = ∑_{i} b_{i}^{2}, σ_{2} = ∑_{i<j} b_{i}^{2}b_{j}^{2}, σ_{3} = ∑_{i<j<k} b_{i}^{2}b_{j}^{2}b_{k}^{2}, \sqrt{σ_{4}} = b_{1}b_{2}b_{3}b_{4}.

We can see that a_{230} is invariant under a permutation of values b_{i}. From the other hand we can see that the equality

det (\begin{matrix}X^{2} + const_{X} & ã_{10/23} & ã_{13/20} \\ ã_{10/23} & Y^{2} + const_{Y} & ã_{12/30} \\ ã_{13/20} & ã_{12/30} & Z^{2} + const_{Z} \end{matrix} ) = det (\begin{matrix}X^{2} & a_{10/23} & a_{13/20} \\ a_{10/23} & Y^{2} & a_{12/30} \\ a_{13/20} & a_{12/30} & Z^{2} \end{matrix})

is equivalent to the following set of equalities:

const_{X} = const_{Y} = const_{Z} = 0, σ_{1} = σ̃_{1}, σ_{2} = σ̃_{2}, σ_{3} = σ̃_{3}, b_{1}b_{2}b_{3}b_{4} = b̃_{1}b̃_{2}b̃_{3}b̃_{4},

where values with tilde are computed for ã_{kk}.

These equalities imply that the only possible transformations are permutations of b_{k} and simultaneous changes of signs of some pair b_{i}, b_{k}. Generators of this group act on the set of parameters of Ss (a_{1}, a_{2}, a_{3}, a_{0}) as a change of the sign of a_{k} (which is a trivial transformation since our equations contain only squares of a_{k}’s) and one non-trivial generator

(ã_{1}, ã_{2}, ã_{3}, ã_{0}) = (a_{1}, a_{2}, a_{3}, a_{0})\frac{1}{2} ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & −1 & 1 & −1 \\ 1 & −1 & −1 & 1 \end{matrix}) ,

that was the goal of our consideration. Note that it is difficult to imagine how this transformation of tetrahedron looks “geometrically”, because of its non-reality. In the case of orthogonal vectors X⃗, Y⃗ , Z⃗ (corresponding to equal parameters a_{k} = a_{i} ∀k, i; let for example a_{k} = 1):

det (\begin{matrix}< X⃗, X⃗ > & < X⃗, Y⃗ > & < X⃗, Z⃗ > \\ < Y⃗ , X⃗ > & < Y⃗ , Y⃗ > & < Y⃗ , Z⃗ > \\ < Z⃗, X⃗ > & < Z⃗, Y⃗ > & < Z⃗, Z⃗ > \end{matrix}) = det (\begin{matrix}X^{2} & 0 & 0 \\ 0 & Y^{2} & 0 \\ 0 & 0 & 1 − X^{2} − Y^{2} \end{matrix}) =

det (\begin{matrix}X^{2} & 1/2 & 1/2 \\ 1/2 & Y^{2} & 1/2 \\ 1/2 & 1/2 & 1 − X^{2} − Y^{2} \end{matrix}) = det (\begin{matrix}< X̃, X̃ > & < X̃, Ỹ > & < X̃, Z̃ > \\ < Ỹ , X̃ > & < Ỹ , Ỹ > & < Ỹ , Z̃ > \\ < Z̃, X̃ > & < Z̃, Ỹ > & < Z̃, Z̃ > \end{matrix}) ,

we can see that < X̃ + Ỹ − Z̃, X̃ + Ỹ − Z̃ >= 0. For real vectors it implies that Z̃ = X̃ + Ỹ, the determinant is identically zero, and all the values a_{ij} are constant.

4 Appendix

I want to change variables (τ, u) ⟶ (x, λ) in the system

Ψ̂′ = 2 ∑_{k=0}^{3} log′ θ_{k}(u|τ)Â_{k}Ψ̂
Ψ̂^{•} = 2 ∑_{k=0}^{3} log^{•} θ_{k}(u|τ)Â_{k}Ψ̂,

where

u = \frac{1}{4K} ∫_{0}^{λ} \frac{dt}{\sqrt{t(1 − t)(x − t)}} = \frac{1}{4K} ∫_{0}^{λ/x} \frac{xdt}{\sqrt{xt_{1}(1 − xt_{1})x(1 − t_{1})}} =

\frac{1}{2K} ∫_{0}^{√\overline{λ/x}} \frac{dt_{2}}{\sqrt{(1 − t_{2}^{2})(1 − xt_{2}^{2})}};

λ = \sqrt{x} θ_{1}^{2}(u)/θ_{0}^{2}(u) = x sn^{2}v, 1 − λ = \sqrt{1 − x} θ_{3}^{2}(u)/θ_{0}^{2}(u) = dn^{2}v,
x − λ = \sqrt{x(1 − x)} θ_{2}^{2}(u)/θ_{0}^{2}(u) = x cn^{2}v

K = \frac{1}{2} ∫_{0}^{x} \frac{dt}{\sqrt{t(1 − t)(x − t)}} = ∫_{0}^{1} \frac{dt_{2}}{\sqrt{(1 − t_{2}^{2})(1 − xt_{2}^{2})}}, v = 2Ku,

the theta functions are calculated by the period τ:

τ = ∫_{1}^{1/x} \frac{dt_{1}}{\sqrt{t_{1}(1 − t_{1})(1 − xt_{1})}}/ ∫_{0}^{1} \frac{dt_{1}}{\sqrt{t_{1}(1 − t_{1})(1 − xt_{1})}}, x = θ_{2}^{4}(0|τ)/θ_{3}^{4}(0|τ).

Consider the first equation.

2 ∑_{k} Â_{k} \frac{∂}{∂λ}|_{x=const} log θ_{k}(u) = 2 ∑_{k≠0} Â_{k} \frac{∂}{∂λ}|_{x=const} log \frac{θ_{k}(u)}{θ_{0}(u)} = \frac{1}{λ}Â_{1}+\frac{1}{λ − x}Â_{2}+\frac{1}{λ − 1}Â_{3}

For the transformation of the second equation we derive some formulae:

\frac{θk^{′′}(u)}{θ_{k}(u)} − \frac{θ0^{′′}(u)}{θ_{0}(u)} = log′′ \frac{θ_{k}(u)}{θ_{0}(u)} + (\frac{θk^{′}(u)}{θ_{k}(u)})^{2} − (\frac{θ0^{′}(u)}{θ_{0}(u)})^{2} = log′′ \frac{θ_{k}(u)}{θ_{0}(u)} + (log′ \frac{θ_{k}(u)}{θ_{0}(u)})^{2} +

2 log′ \frac{θ_{k}(u)}{θ_{0}(u)} log′ θ_{0}(u) = \frac{(θ_{k}(u)/θ_{0}(u))′′}{θ_{k}(u)/θ_{0}(u)} + 2 (log′ \frac{θ_{k}(u)}{θ_{0}(u)}) log′ θ_{0}(u).

Using the heat equation 4πi θ^{•}_{k} (u) = θk^{′′}(u) we get

log^{•} \frac{θ_{1}(u)}{θ_{0}(u)} = \frac{1}{4πi} (\frac{θ1^{′′}(u)}{θ_{1}(u)} − \frac{θ0^{′′}(u)}{θ_{0}(u)}) = \frac{K^{2}}{πi} (\frac{1}{sn v} \frac{∂^{2}}{∂v^{2}} sn v + 2\frac{∂}{∂v} log sn v Z) =

\frac{K^{2}}{πi} (−dn^{2} v − xcn^{2} v + 2\frac{cn vdn v}{sn v} Z) ,

log^{•} \frac{θ_{2}(u)}{θ_{0}(u)} = \frac{K^{2}}{πi} (−dn^{2} v + xsn^{2} v − 2\frac{sn vdn v}{cn v} Z) ,
log^{•} \frac{θ_{3}(u)}{θ_{0}(u)} = \frac{K^{2}}{πi} x (−cn^{2} v + sn^{2} v − 2\frac{sn vcn v}{sn v} Z) .

We denote Z := ∂/∂v log θ_{0}(v/(2K)). Using

\frac{dτ}{dx} = \frac{iπ}{4K^{2}x(x − 1)}

we rewrite these equalities:

\frac{∂}{∂x}|_{u=const} log \frac{θ_{1}(u)}{θ_{0}(u)} = \frac{1}{4x(x − 1)} (2λ − 1 − x + 2\frac{cn vdn v}{sn v} Z)
\frac{∂}{∂x}|_{u=const} log \frac{θ_{2}(u)}{θ_{0}(u)} = \frac{1}{4x(x − 1)} (2λ − 1 − 2\frac{sn vdn v}{cn v} Z)
\frac{∂}{∂x}|_{u=const} log \frac{θ_{3}(u)}{θ_{0}(u)} = \frac{1}{4x(x − 1)} (2λ − x − 2x\frac{sn vcn v}{dn v} Z) .

We pass on to the second equation:

\frac{∂}{∂x}|_{λ=const}Ψ̂Ψ̂^{−1} = \frac{∂}{∂x}|_{u=const}Ψ̂Ψ̂^{−1} − (\frac{∂λ}{∂x}|_{u=const}) \frac{∂}{∂λ}|_{x=const}Ψ̂Ψ̂^{−1}.

Let us calculate ∂λ/∂x:

\frac{∂λ}{∂x}|_{u=const} = λ\frac{∂}{∂x}|_{u=const} log \sqrt{x} \frac{θ_{1}^{2}(u)}{θ_{0}^{2}(u)} = λ (\frac{1}{2x} + 2\frac{∂}{∂x}|_{u=const} log \frac{θ_{1}(u)}{θ_{0}(u)}) =

λ (\frac{1}{2x} + \frac{1}{2x(x − 1)}(2λ − 1 − x + 2\frac{cn v dn v}{sn v} Z)) = \frac{λ}{x(x − 1)} (λ − 1 + \frac{cn v dn v}{sn v} Z) .

We get lastly

\frac{∂}{∂x}|_{λ=const}Ψ̂Ψ̂^{−1} = \frac{Â_{1}}{2x(x − 1)} (2λ − 1 − x + 2\frac{cn v dn v}{sn v} Z) + \frac{Â_{2}}{2x(x − 1)}( 2λ − 1 −

2\frac{sn v dn v}{cn v} Z) + \frac{Â_{3}}{2x(x − 1)} (2λ − x − 2x\frac{cn v sn v}{dn v} Z) − \frac{λ}{x(x − 1)} (λ − 1 + \frac{cn v dn v}{sn v} ·

Z )(\frac{1}{λ}Â_{1} + \frac{1}{λ − x}Â_{2} + \frac{1}{λ − 1}Â_{3}) = \frac{1}{x − λ}Â_{2} − \frac{1}{2x}Â_{1} − \frac{2x − 1}{2x(x − 1)}Â_{2} − \frac{1}{2(x − 1)}Â_{3}

QED

Acknowledgments

I am pleased to thank D. Korotkin for the interest in this work and valuable discussions. I am grateful to MPIM-Bonn for support and hospitality during my stay there.

This work is partially supported by the grant RFFI 99-01-00687 of Russian Academy of Sciences.

References

[1] L. Schlesinger, Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten , J. Reine Math., 141, 96–145 (1912).

[2] D. Arinkin, S. Lysenko, On the Moduli of SL(2)-bundles with Connections on P^{1} \ {x_{1}, x_{2}, x_{3}, x_{4}}, International Mathematics Research Notices, 19, 983–999 (1997).

[3] D. Arinkin, S. Lysenko, Isomorphisms between Moduli Spaces of SL(2)-bundles with Connections on P^{1} \ {x_{1}, x_{2}, x_{3}, x_{4}}, Mathematical Research Letters, 4, 181–190 (1997).

[4] M. Jimbo, T. Miwa, K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients I, Phisica 2D, 306–352 (1981).

[5] M. Jimbo, T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients II, III, Phisica 2D, 407–448 (1981), Phisica 4D, 26-46, (1981).

[6] K. Okamoto, Studies in the Painlevé equations I. Sixth Painlevé equation PVI, Annali Mat. Pura Appl., 146, 337–381 (1987).

[7] D. A. Korotkin and H. Samtleben, On the quantization of isomonodromic deformations on the torus, International Journal of Modern Physics A, v. 12, No. 11, 2013–2029 (1997).

[8] A.S.Fokas, A.R.Its, A.A.Kapaev and V.Yu.Novokshenov,Painlevé transcendents as Nonlinear Special Functions, Kluwer Ac.Pub.

[9] Mugan, U. and Sakka, A. Schlesinger transformations for Painlevé VI equation, J. Math. Phys. 36 (3), 1284–1298, (1995).

[10] A.Kitaev Special functions of the isomonodromy type, Preprint SFB-288-272 (Berlin), Jun 1997. 27pp.

[11] D. A. Korotkin, N. Manojlović and H. Samtleben, Schlesinger transformations for elliptic isomonodromic deformations, In preparation, (1999).