Advances in Mathematical Physics
Volume 2009, Issue 1 461860
Research Article
Open Access

Generalization of Okamoto′s Equation to Arbitrary 2 × 2 Schlesinger System

Dmitry Korotkin, 

Corresponding Author

Dmitry Korotkin

Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke West, Montreal QC, Canada H4B 1R6, concordia.ca

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Henning Samtleben, 

Henning Samtleben

Laboratoire de Physique, Ecole Normale Supérieure de Lyon, Université de Lyon, 46, Allée d′Italie, 69364 Lyon Cedex 07, France universite-lyon.fr

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First published: 15 October 2009
https://doi.org/10.1155/2009/461860
Citations: 1
Academic Editor: Alexander P. Veselov

Abstract

The 2 × 2 Schlesinger system for the case of four regular singularities is equivalent to the Painlevé VI equation. The Painlevé VI equation can in turn be rewritten in the symmetric form of Okamoto′s equation; the dependent variable in Okamoto′s form of the PVI equation is the (slightly transformed) logarithmic derivative of the Jimbo-Miwa tau-function of the Schlesinger system. The goal of this note is twofold. First, we find a universal formulation of an arbitrary Schlesinger system with regular singularities in terms of appropriately defined Virasoro generators. Second, we find analogues of Okamoto′s equation for the case of the 2 × 2 Schlesinger system with an arbitrary number of poles. A new set of scalar equations for the logarithmic derivatives of the Jimbo-Miwa tau-function is derived in terms of generators of the Virasoro algebra; these generators are expressed in terms of derivatives with respect to singularities of the Schlesinger system.

1. Introduction

The Schlesinger system is the following nonautonomous system of differential equations for N unknown matrices Aj ∈ 𝔰𝔩(M) depending on N variables {λj}:
(1.1)
System (1.1) determines isomonodromic deformations of a solution of matrix ODE with meromorphic coefficients
(1.2)
if one assumes that , that is, λ = ∞ is a regular point of the function Ψ, and the function is normalized by the condition Ψ(∞) = I. The function Ψ solves a matrix Riemann-Hilbert problem with some monodromy matrices around the singularities λj.
The Schlesinger equations were discovered almost 100 years ago [1]; however, they continue to play a key role in many areas of mathematical physics such as the theory of random matrices, integrable systems and theory of Frobenius manifolds. System (1.1) is a nonautonomous Hamiltonian system with respect to the Poisson bracket
(1.3)
where are structure constants of 𝔰𝔩(M); with the 𝔰𝔩(M) generators ta; δjk is the Kronecker symbol (one assumes summation over repeating indices). Obviously, the traces are integrals of the Schlesinger system for any value of n. The commuting Hamiltonians defining evolution with respect to the times λj are given by
(1.4)
The generating function τJM({λj}) of the Hamiltonians Hj, defined by
(1.5)
was introduced by Jimbo et al. [2, 3]; it is called the τ-function of the Schlesinger system. The τ-function plays a key role in the theory of the Schlesinger equations; in particular, the divisor of zeros of the τ-function coincides with the divisor of singularities of the solution of the Schlesinger system.
In the simplest nontrivial case when the matrix dimension equals M = 2 and the number of singularities equals N = 4, the Schlesinger system can equivalently be rewritten as a single scalar differential equation of order two—the Painlevé VI equation:
(1.6)
where t is the cross-ratio of the four singularities λ1, …, λ4. Solution y is defined as follows. One maps the set of poles (λ1, λ2, λ3, λ3) to (0, t, 1, ∞) by a Möbius transformation and then multiplies Ψ from the left by a constant matrix such that A4 becomes diagonal. Then y coincides with the position of the (unique) zero of the upper right corner element of the transformed matrix A(λ).
Let us denote the eigenvalues of the matrices Aj by αj/2 and −αj/2. Then the constants α, β, γ, and δ from the Painlevé VI equation (1.6) are related to the constants αj as follows:
(1.7)
It was further observed by Okamoto [4, 5] that the Painlevé VI equation (and, therefore, the original 2 × 2 Schlesinger system with four singularities) can be rewritten alternatively in a simple form in terms of the so-called auxiliary Hamiltonian function h(t). To define this function we need to introduce first four constants bj, which are expressed in terms of the eigenvalues of the matrices Aj as follows:
(1.8)
The auxiliary Hamiltonian function h(t) is defined in terms of solution y of (1.6) and the constants bj as follows:
(1.9)
where
(1.10)
In terms of the function h, the Painlevé equation (1.6) can be represented in a remarkably symmetric form as follows:
(1.11)

Okamoto′s form (1.11) of the Painlevé VI equation turned out to be extremely fruitful for establishing the hidden symmetries of the equation (the so-called Okamoto symmetries). These symmetries look very simple in terms of the auxiliary Hamiltonian function h but are highly nontrivial on the level of the solution y of the Painlevé VI equation, the corresponding monodromy group, and the solution of the associated Fuchsian system [6, 7].

The goal of this paper is twofold. First, we show how to rewrite the Schlesinger system in an arbitrary matrix dimension in a symmetric universal form. Second, we use this symmetric form to find natural analogues of the Okamoto equation (1.11) for 2 × 2 Schlesinger systems with an arbitrary number of simple poles. Our approach is similar to the approach used by Harnad to derive analogues of the Okamoto equation for Schlesinger systems corresponding to higher-order poles (nonfuchsian systems) [8].

Namely, introducing the following differential operators (which satisfy the commutation relations of the Virasoro algebra):
(1.12)
and the following dependent variables:
(1.13)
one can show that the Schlesinger system (1.1) implies
(1.14)
for all n ≥ 0 and m ≥ −1. The infinite set of (1.14) is of course dependent for any given N. To derive the original Schlesinger system (1.1) from (1.14) it is sufficient to take the set of (1.14) for n ≤ N and m ≤ N. The advantage of system (1.14) is in its universality. Its form is independent of the number of the poles; the positions of the poles enter only the definition of the differential operators Lm.
Consider now the case of 2 × 2 matrices. To formulate the analog of the Okamoto equation for the case of an arbitrary number of poles we introduce the following “Hamiltonians”:
(1.15)
which can be viewed as symmetrised analogues of the Hamiltonians (1.4); they coincide with Lmlog  τJM up to an elementary transformation. The simplest equation satisfied by in the case of 2 × 2 system is given by
(1.16)
as we will show in the following.

Since the themselves are combinations of the first-order derivatives of the tau-function, this equation is of the third order; it also has cubic nonlinearity. In the case N = 4, (1.16) boils down to the standard Okamoto equation (1.11).

The paper is organized as follows. In Section 2 we derive the symmetrised form of the Schlesinger system. In Section 3 we derive the generalized Okamoto equations. In Section 4 we show how the usual Okamoto equation is obtained from the generalized equation (1.16) in the case N = 4. In Section 5 we discuss some open problems.

2. Universal Form of Schlesinger System in Terms of Virasoro Generators

Consider the differential operators Lm (1.12); these operators satisfy the commutation relations of the Virasoro algebra:
(2.1)
To represent the Schlesinger equations in a universal form we also introduce the symmetric dependent variables given by (1.13). The new variable
(2.2)
plays a distinguished role; it vanishes on-shell (i.e., on solutions of the Schlesinger system); however, off-shell plays the role of a generator (with respect to the Poisson bracket (1.3)) of constant gauge transformations (i.e., constant simultaneous similarity transformations of all matrices Aj).
To describe the dynamics under the action of the differential operators Lm we introduce the symmetrised Hamiltonians ℋm:
(2.3)
These Hamiltonians can be expressed in terms of the variables ℬk as follows:
(2.4)
where the modified Hamiltonians are given by
(2.5)
and . In particular, (taking into account that ℬ0 = 0), such that the first three symmetrised Hamiltonians take the form
(2.6)
In terms of the Virasoro generators Lm, (1.5) for the Jimbo-Miwa τ-function τJM looks as follows:
(2.7)
It is convenient to introduce also a modified τ-function, invariant under Möbius transformations.

Lemma 2.1. The modified τ-function defined by

(2.8)
is annihilated by the first three Virasoro generators:
(2.9)

Proof. By a straightforward computation.

In terms of the new variables (1.13), the Schlesinger system (1.1) takes a very compact form.

Theorem 2.2. The differential operators Lm act on the symmetrised variables ℬn as follows:

(2.10)
for m = −1,0, 1,2, …, n = 1,2, ….

Proof. Using the Schlesinger equations (1.1), we have

(2.11)
Expanding
(2.12)
we further rewrite this expression for Lmℬn as
(2.13)
which coincides with the right-hand side of (2.10).

Remark 2.3. System (2.10) can be equivalently rewritten as follows:

(2.14)
That is, the right-hand side of (2.10) does not change if the upper limit n − 1 is substituted by m.

The system of (2.10), or (2.14), can be viewed as universal form of the Schlesinger system; formally the number of poles does not enter the system any more. Each Schlesinger system written in the standard form (1.1) can be obtained from (2.10) if one makes the ansatz (1.13) for variables ℬk. Using (2.14) we can express the commutators [ℬm, ℬn] as follows:
(2.15)
Acting on the modified Hamiltonians by the operators Lm, we get the following equation:
(2.16)
In particular, we have
(2.17)
The same equation holds for the Hamiltonians ℋn as a corollary of the integrability of (2.7).
The Poisson bracket (1.3) induces the following Poisson bracket between variables ℬn, n = 0,1, 2, …:
(2.18)
Then (2.10) can then be written in the following form:
(2.19)
We note that formally the second term can be absorbed into the symplectic action {ℋm, ℬn} upon extending the affine Poisson structure (2.18) by the standard central extension.

Remark 2.4. Let us briefly discuss the geometric origin of the Virasoro generators Lm. The vectors ∂/∂λj span the tangent space to the space of N-punctured spheres with punctures at {λj}. On the other hand, there exist several universal ways to vary the moduli of a given Riemann surface. For example, one can vary the moduli by vector fields on a chosen closed contour l (see [9]). In the case of our punctured sphere the contour l can be chosen to be a circle containing all singularities λj; then variation of moduli by the vector field λm+1∂/∂λ on the circle coincides with variation by the generator Lm. The commutation relations between Lm are then inherited from the commutation relations of vector fields on the circle.

3. Generalized Okamoto Equations

Here we will use the symmetric form (2.10) of the Schlesinger equations to derive an analog of Okamoto′s equation (1.11) for an arbitrary 2 × 2 Schlesinger system. In fact, one can write down a whole family of scalar differential equations for the tau-function in terms of the Virasoro generators Lm. In the next theorem we prove two equations of this kind.

Theorem 3.1. The τ-function τJM (1.5) of an arbitrary 2 × 2 Schlesinger system satisfies the following two differential equations.

  • (i)

    The third-order equation with cubic nonlinearity:

    (3.1)

  • (ii)

    The fourth-order equation with quadratic nonlinearity:

    (3.2)
    where according to (2.4), (2.7),
    (3.3)

Proof. Inverting the system of (2.16), we can express tr ℬmℬn in terms of the Hamiltonians as follows:

(3.4)
From the Schlesinger system (2.10), we furthermore get
(3.5)
Inverting this relation, we obtain for k < m < n
(3.6)
Combining this equation with (3.4) we can thus express also tr(ℬk[ℬm, ℬn]) entirely in terms of the action of the operators Lm on the Hamiltonians , which can further be simplified upon using the commutation relations (2.1) and (2.17). This leads to the closed expression
(3.7)
In particular, for the lowest values of k, m, n we obtain
(3.8)
To derive from these relations the desired result, we make use of the following algebraic identity:
(3.9)
that is valid for an arbitrary set of six matrices Mj ∈ 𝔰𝔩(2), where the dots on the right-hand side denote complete antisymmetrisation of the expression with respect to the indices 1,2, 3. In terms of the structure constants of 𝔰𝔩(2), this identity reads
(3.10)
where adjoint indices a, b, … are raised and lowered with the Cartan-Killing form. Setting in (3.9) M1 = M4 = ℬ1, M2 = M5 = ℬ2, and M3 = M6 = ℬ3, and using (3.4), (3.8), we arrive (after some calculation) at (3.1).

Equation (3.2) descends from another algebraic identity

(3.11)
that is valid for any four 𝔰𝔩(2)-valued matrices Mj. In terms of the structure constants of 𝔰𝔩(2), this identity reads
(3.12)
and is obtained by contraction from (3.10). We consider the action of L2 on (3.8) which yields
(3.13)
The l.h.s. of this equation can be reduced by (3.6) while the first terms on the r.h.s. are reduced by means of the algebraic relations (3.11) together with (3.4). As a result we obtain (3.2).

4. Four Simple Poles: Reproducing the Okamoto Equation

As remarked before, the explicit form of the differential equations (3.1), (3.2) for the τ-function τJM is obtained upon expressing the modified Hamiltonians in terms of τJM by virtue of (2.4), (2.7). As an illustration, we will work out these equations for the Schlesinger system with four singularities and show that they reproduce precisely Okamoto′s equation (1.11). For N = 4, the modified τ-function from (2.8) depends only on the cross-ratio
(4.1)
We furthermore define the auxiliary function
(4.2)
Then (3.1) in terms of S after lengthy but straightforward calculation gives rise to the second-order differential equation
(4.3)
where σi[C] are the elementary symmetric polynomials of the Ci’s
(4.4)

Finally, it is straightforward to verify that with h(t) ≡ S(t) − (1/12)(1 − 2t)σ1[C], (4.3) is equivalent to Okamoto′s equation (1.11).

In turn, (3.2) leads to the following quadratic third-order differential equation in the function S:
(4.5)
Indeed, this equation can also be obtained by straightforward differentiation of (4.3) with respect to t. In terms of the function h, (4.5) takes the following form:
(4.6)
equivalently obtained by derivative of Okamoto′s equation (1.11).

5. Discussion and Outlook

We have shown in this paper that the symmetric form (2.10), (2.14) of the Schlesinger system gives rise to a straightforward algorithm that allows to translate the algebraic 𝔰𝔩(2) identities (3.10), (3.12) into differential equations for the τ-function of the Schlesinger system. In the simplest case of four singularities, the resulting equations reproduce the known Okamoto equation (1.11). In the case of more singularities, the same equations (3.1), (3.2) give rise to a number of nontrivial differential equations to be satisfied by the τ-function.

Apart from this direct extension of Okamoto′s equation, the link between the algebraic structure of 𝔰𝔩(2) and the Schlesinger system′s τ-function gives rise to further generalizations. Note that in the proof of Theorem 3.1, with (3.7), we have already given the analogue of (3.8) to arbitrary values of k, m, n. Combining this equation with identity (3.9) thus gives rise to an entire hierarchy of third-order equations that generalize (3.1). Likewise, the construction leading to the fourth-order equation (3.2) can be generalized straightforwardly upon applying (3.11) to other Virasoro descendants of the cubic equation.

As an illustration, we give the first three equations of the hierarchy generalizing (3.2):
(5.1)
Obviously, these equations are not all independent, but related by the action of the lowest Virasoro generators L±1, using that
(5.2)
and so forth, since . The number and structure of the independent equations in this hierarchy is thus organized by the structure of representations of the Virasoro algebra. For the case of N = 4 singularities, the explicit form of all the equations of the hierarchy reduces to equivalent forms of (1.11) and (4.6). With growing number of simple poles, the number of independent differential equations induced by the hierarchy increases.

Therefore, we arrive to a natural question: which set of derived equations for the tau-function is equivalent to the original Schlesinger system? We stress that all differential equations for the tau-function are PDE with respect to the variables λ1, …, λN. However, if one gets a sufficiently high number of independent equations, one can actually come to a set of ODEs for the tau-function. This situation resembles the situation with the original form of the Schlesinger system (1.1): if one ignores the second set of equations in (1.1); one gets a system of PDEs for the residues Aj; only upon adding the equations for ∂Aj/∂λj one gets a system of ODEs with respect to each λj (the flows with respect to different λj commute).

Let us finally note that the construction we have presented in order to derive the differential equations (3.1), (3.2) suggests a number of interesting further generalizations that deserve further study.

  • (i)

    At the origin of our derivation have figured the algebraic 𝔰𝔩(2) identities (3.10), (3.12) that we have translated into differential equations. Similar identities exist also for higher rank groups (e.g., M > 2, or the Schlesinger system for orthogonal, symplectic, and exceptional groups), where the number of independent tensors may be larger. It would be highly interesting to understand if equations analogous to (3.1), (3.2) can be derived from such higher rank algebraic identities. As those identities will be built from a larger number of invariant tensors (structure constants, etc.), the corresponding differential equations would be of higher order in derivatives.

  • (ii)

    Is it possible to combine our present construction applicable to Schlesinger systems with simple poles only with construction of [8] which requires the presence of higher-order poles? What would be the full set of equations for the tau-function with respect to the full set of deformation parameters in presence of higher-order poles?

  • (iii)

    The Schlesinger system (1.1) has also been constructed for various higher genus Riemann surfaces [10–13]. It would be interesting to first of all find the proper generalization of the symmetric form (2.10), (2.14) of the Schlesinger system to higher genus surfaces which in turn should allow to derive by an analogous construction the nontrivial differential equations satisfied by the associated τ-function. We conjecture that in some sense the form (2.10) should be universal; it should remain the same, although the definition of the Virasoro generators Lm and the variables ℬm may change.

  • (iv)

    As we have mentioned above, the extra term nℬm+n in the Hamiltonian dynamics of the symmetrised Schlesinger system (2.19) can be absorbed into the symplectic action upon replacing the standard affine Lie-Poisson bracket (2.18) by its centrally extended version. However, this central extension is not seen in any of the finite-N Schlesinger systems. This seems to suggest that system (2.10) should be considered not just as a symmetric form of the usual Schlesinger system with finite number of poles, but as a “universal” Schlesinger system which involves an infinite set of independent variables ℬn. Presumably, this full system involves the generators Ln and coefficients ℬn not only for positive, but also for negative n. In this setting, the centrally extended version of the bracket (2.18) should appear naturally. The most interesting problem would be to find the geometric origin of such a generalized system; a possible candidate could be the isomonodromic deformations on higher genus curves.

Acknowledgments

The work of H. Samtleben was supported in part by the Agence Nationale de la Recherche (ANR). The work of D. Korotkin was supported by NSERC, NATEQ, and Concordia University Research Chair grant.

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