Generalization of Okamoto's equation to arbitrary $2\times 2$ Schlesinger systems

The $2\times 2$ Schlesinger system for the case of four regular singularities is equivalent to the Painlev\'e VI equation. The Painlev\'e 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 symmetric uniform 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\times2 $ 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.


Introduction
The Schlesinger system is the following non-autonomous system of differential equations for N unknown matrices A j ∈ sl(M ) depending on N variables {λ j }: The system (1.1) determines isomonodromic deformations of a solution of matrix ODE with meromorphic coefficients The solution of this system normalized at a fixed point λ 0 by Ψ(λ 0 ) = I solves a matrix Riemann-Hilbert problem with some monodromy matrices around the singularities λ j . The Schlesinger equations were discovered almost 100 years ago [1]; however, they continue to play a key role in many areas of mathematical physics: the theory of random matrices, integrable systems, theory of Frobenius manifolds, etc. . The system (1.1) is a non-autonomous hamiltonian system with respect to the Poisson bracket where f ab c are structure constants of sl(M ); δ jk is the Kronecker symbol. Obviously, the traces trA n j are integrals of the Schlesinger system for any value of n. The commuting Hamiltonians defining evolution with respect to the times λ j are given by The generating function τ JM ({λ j }) of the hamiltonians H j , defined by ∂ ∂λ j log τ JM = H j , (1.5) was introduced by Jimbo, Miwa and their co-authors [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; on the same divisor the underlying Riemann-Hilbert problem looses its solvability.
In the simplest non-trivial 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 where t is the cross-ratio of the four singularities λ 1 , . . . , λ 4 , and y is the position of a zero of the upper right corner element of the matrix 4 k=1 A k λ−λ k . Let us denote the eigenvalues of the matrices A j by α j /2 and −α j /2. Then the constants α, β, γ and δ from the Painlevé VI equation (1.6) are related to the constants α j as follows: 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 b j , which are expressed in terms of the eigenvalues of the matrices A j as follows: The auxiliary hamiltonian function h(t) is defined in terms of solution y of equation (1.6) and the constants b j as follows: In terms of the function h, the Painlevé equation (1.6) can be represented in a remarkably symmetric form as follows: 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 non-trivial 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 J. Harnad to derive analogues of the Okamoto equation for Schlesinger systems corresponding to higher order poles (non-fuchsian systems) [8].
Namely, introducing the following differential operators (which satisfy the commutation relations of the Virasoro algebra): and the following dependent variables: one can show that the Schlesinger system (1.1) implies for all n ≥ 0 and m ≥ −1. The infinite set of equations (1.12) is of course dependent for any given N . To derive the original Schlesinger system (1.1) from (1.12) it is sufficient to take the set of equations (1.12) for n ≤ N and m ≤ N . The advantage of the system (1.12) 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 L m . 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": which can be viewed as symmetrised analogues of the Hamiltonians (1.4); they coincide with L m log τ JM up to an elementary transformation. The simplest equation satisfied by H m in the case of 2 × 2 system is given by as we shall show in the following.
Since the H m themselves are combinations of the first order derivatives of the tau-function, this equation is of the third order; it also has cubic non-linearity. In the case N = 4 the equation (1.13) 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 (1.13) in the case N = 4. In section 5 we discuss some open problems.

Symmetrisation of Schlesinger system in terms of Virasoro generators 2.1 Variation of a Riemann surface by vector fields on a contour
The "times" λ j of the Schlesinger system can be viewed as coordinates on the space of genus zero Riemann surfaces with N punctures (we do not speak about the moduli space since we do not identify two configurations of punctures related by a Möbius transformation). The vectors ∂/∂λ j span the tangent space to the space of N -punctured spheres. However, there exist many different ways to parametrize the tangent space to the space of Riemann surfaces of given genus with a fixed number of punctures. One example is the variation of a Riemann surface by vector fields on a chosen closed contour l enclosing a disc D (see [9]). To vary an unpunctured Riemann surface L of genus g by a vector field v on the contour l one cuts the disc D and attaches it back with infinitesimal shift of the boundary along the vector field v. The complex structure of the Riemann surface also changes infinitesimally (or remains the same). The infinite-dimensional space V of vector fields on l can be represented as direct sum of three subspaces, V in (which consists of vector fields which can be analytically continued inside of the disc D), V out (which consists of vector fields which can be analytically continued outside of the disc D) and V 0 (which consists of vector fields whose analytical continuation is impossible neither inside nor outside of the disc D). The moduli of Riemann surface L are not changed by vector fields from V in and V out ; the dimension of V 0 turns out to be equal to 3g − 3 i.e. to dimension of the moduli space, and the vector fields from V 0 do change the complex structure of the Riemann surface.
If the Riemann surface L has punctures (which are assumed to lie outside of D), the vector fields from V out should not only admit analytical continuation to the exterior of D, but also vanish at all the punctures.
To apply this general scheme in our present framework we choose the contour l to be a circle around λ = ∞ (such that all points λ j lie "outside" of this circle). However, our goal will be to describe all Schlesinger equations (independently of the number of singularities) within one setting. Therefore, we shall vary λ j by all v m with m = −1, 0, 1, 2, . . . in spite of the fact that for any given N all these vectors can be expressed as linear combinations of N vectors ∂/∂λ j .
Namely, the action of the vector field v m on {λ j } is given by the following linear combination of the tangent vectors ∂/∂λ m : L m and v m coincide as tangent vector to the space of N -punctured spheres. The vectors L m also satisfy commutation relations of Virasoro algebra:

Schlesinger system in terms of Virasoro generators
To symmetrise the Schlesinger equations we also introduce the symmetric dependent variables: The new variable To describe the dynamics under the action of the differential operators L m we introduce the symmetrised Hamiltonians H m : These Hamiltonians can be expressed in terms of the variables B k as follows: where the modified Hamiltonians H are given by and C j := trA 2 j = 1 2 α 2 j . In particular, H −1 = H 0 = H 1 = 0 (taking into account that B 0 = 0), such that the first three symmetrised Hamiltonians take the form In terms of the Virasoro generators L m the equations (1.5) for the Jimbo-Miwa τ -function τ JM look as follows: It is convenient to introduce also a modified τ -function, invariant under Möbius transformations:

9)
is annihilated by the first three Virasoro generators: Proof: by straightforward computation.

2
In terms of the new variables (2.2), the Schlesinger system (1.1) takes a very compact form: Theorem 1 The differential operators L m act on the symmetrised variables B n as follows:
Proof. Using the Schlesinger equations (1.1), we have we further rewrite this expression for L m B n as which coincides with the right hand side of (2.11).
2 Remark 1 The system (2.11) can be equivalently rewritten as follows: i.e. the right-hand side of (2.11) does not change if the upper limit n − 1 is substituted by m.
The system of equations (2.11), or (2.12) is the symmetric form of the Schlesinger system. Using (2.12) we can express the commutators [B m , B n ] as follows: Acting on the modified hamiltonians H n by the operators L m , we get the following equation: (2.14) In particular, we have Then equations (2.11) can then be written in the following form: We note that formally the second term can be absorbed into the symplectic action {H m , B n } upon extending the affine Poisson structure (2.16) by the standard central extension.

Generalized Okamoto equations
Here we shall use the symmetric form (2.11) 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 L m . In the next theorem we prove two equations of this kind.

Theorem 2
The τ -function τ JM (1.5) of an arbitrary 2 × 2 Schlesinger system satisfies the following two differential equations: • The third order equation with cubic non-linearity: • The fourth order equation with quadratic non-linearity: where according to (2.5), (2.8) Proof. Inverting the system of equations (2.14), we can express tr B m B n in terms of the Hamiltonians H n as follows: (3.3) ¿From the Schlesinger system (2.11), we furthermore get Inverting this relation, we obtain for k < m < n In particular, for the lowest values of k, m, n we obtain To derive from these relations the desired result, we make use of the following algebraic identity  (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 valid for any four sl(2)-valued matrices M j . In terms of the structure constants of sl (2), this identity reads 11) and is obtained by contraction from (3.9). We consider the action of L 2 on (3.7) which yields The l.h.s. of this equation can be reduced by (3.5) while the first terms on the r.h.s. are reduced by means of the algebraic relations (3.10) together with (3.3). As a result we obtain equation (3.2).

2
4 Four simple poles: reproducing the Okamoto equation As remarked above, the explicit form of the differential equations (3.1), (3.2) for the τ -function τ JM is obtained upon expressing the modified Hamiltonians H m in terms of τ JM by virtue of (2.5), (2.8). 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 -functionτ from (2.9) depends only on the cross-ratio We furthermore define the auxiliary function Then equation (3.1) in terms of S after lengthy but straightforward calculation gives rise to the second order differential equation where σ i [C] are the elementary symmetric polynomials of the C i 's Finally, it is straightforward to verify that with h(t) ≡ S(t) − 1 12 (1 − 2t) σ 1 [C], equation (4.3) is equivalent to Okamoto's equation (1.11).
In turn, equation (3.2) leads to the following quadratic, third order differential equation in the function S: (4.4) Indeed, this equation can also be obtained by straightforward differentiation of (4.3) with respect to t. In terms of the function h equation (4.4) takes the following form

Discussion and Outlook
We have shown in this paper that the symmetric form (2.11), (2.12) of the Schlesinger system gives rise to a straightforward algorithm that allows to translate the algebraic sl(2) identities (3.9), (3.11) 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 non-trivial 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 2, with equation (3.6) we have already given the analogue of (3.7) to arbitrary values of k, m, n. Combining this equation with the identity (3.8) 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 (3.10) to other Virasoro descendants of the cubic equation.
As an illustration, we give the first three equations of the hierarchy generalizing (3.2): Obviously, these equations are not all independent, but related by the action of the lowest Virasoro generators L ±1 , using that etc., since H 1 = 0. 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 (4.5). 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 taufunction is equivalent to the original Schlesinger system? We stress that all differential equations for the tau-function are PDE with respect to the variables λ 1 , . . . , λ N . However, if one gets a sufficiently high number of independent equations, one can actually come to a set of ODE's 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 PDE's for the residues A j ; only upon adding the equations for ∂A j /∂λ j one gets a system of ODE's with respect to each λ j (the flows with respect to different λ j 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.
• At the origin of our derivation have figured the algebraic sl(2) identities (3.9), (3.11) 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.
• 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?
• The Schlesinger system (1.1) has also been constructed for various higher genus Riemann surfaces [10,11,12,13]. It would be interesting to first of all find the proper generalization of the symmetric form (2.11), (2.12) of the Schlesinger system to higher genus surfaces which in turn should allow to derive by an analogous construction the non-trivial differential equations satisfied by the associated τ -function. We conjecture that in some sense the form (2.11) should be universal: it should remain the same, although the definition of the Virasoro generators L m and the variables B m may change.
• As we have mentioned above, the extra term nB m+n in the Hamiltonian dynamics of the symmetrised Schlesinger system (2.17) can be absorbed into the symplectic action upon replacing the standard affine Lie-Poisson bracket (2.16) 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 the system (2.11) 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 B n . Presumably, this full system involves the generators L n and coefficients B n not only for positive, but also for negative n.
In this setting, the centrally extended version of the bracket (2.16) 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.