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

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 𝐴_𝑗∈𝔰𝔩(𝑀) depending on 𝑁 variables {𝜆_𝑗}: 𝜕𝐴_𝑗𝜕𝜆_𝑖=[𝐴_𝑗,𝐴_𝑖]𝜆_𝑗−𝜆_𝑖,𝑖≠𝑗,𝜕𝐴_𝑗𝜕𝜆_𝑗=−_𝑖≠𝑗[𝐴_𝑗,𝐴_𝑖]𝜆_𝑗−𝜆_𝑖. (1.1) System (1.1) determines isomonodromic deformations of a solution of matrix ODE with meromorphic coefficients 𝜕Ψ𝜕𝜆=𝐴(𝜆)Ψ≡^𝑁_𝑗=1𝐴_𝑗𝜆−𝜆_𝑗Ψ, (1.2) if one assumes that ∑^𝑁_𝑗=1𝐴_𝑗=0, 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 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 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 {𝐴^𝑎_𝑗,𝐴^𝑏_𝑘}=𝛿_𝑗𝑘𝑓_𝑐^𝑎𝑏𝐴^𝑐_𝑗, (1.3) where 𝑓_𝑐^𝑎𝑏 are structure constants of 𝔰𝔩(𝑀); 𝐴_𝑗=𝐴^𝑎_𝑗𝑡^𝑎 with the 𝔰𝔩(𝑀) generators 𝑡^𝑎; 𝛿_𝑗𝑘 is the Kronecker symbol (one assumes summation over repeating indices). 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 𝐻_𝑗=14𝜋i_𝜆𝑖tr𝐴²1(𝜆)𝑑𝜆≡2_𝑘≠𝑗tr𝐴_𝑗𝐴_𝑘𝜆_𝑘−𝜆_𝑗. (1.4) The generating function 𝜏_JM({𝜆_𝑗}) of the Hamiltonians 𝐻_𝑗, defined by 𝜕𝜕𝜆_𝑗log𝜏_JM=𝐻_𝑗, (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: 𝑑²𝑦𝑑𝑡²=12(1𝑦+1+1𝑦−1𝑦−𝑡)(𝑑𝑦)𝑑𝑡²1−(𝑡+1+1𝑡−1)𝑦−𝑡𝑑𝑦+𝑑𝑡𝑦(𝑦−1)(𝑦−𝑡)𝑡²(𝑡−1)²𝑡(𝛼+𝛽𝑦²+𝛾𝑡−1(𝑦−1)²+𝛿𝑡(𝑡−1)(𝑦−𝑡)²), (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 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: 𝛼=(𝛼₁−1)²2𝛼,𝛽=−²²2𝛼,𝛾=²³21,𝛿=2−𝛼²₄2. (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 𝑏_𝑗, which are expressed in terms of the eigenvalues of the matrices 𝐴_𝑗 as follows: 𝑏₁=12(𝛼₂+𝛼₃),𝑏₂=12(𝛼₂−𝛼₃𝑏),₃=12(𝛼₄+𝛼₁),𝑏₄=12(𝛼₄−𝛼₁). (1.8) The auxiliary Hamiltonian function ℎ(𝑡) is defined in terms of solution 𝑦 of (1.6) and the constants 𝑏_𝑗 as follows: ℎ=𝑦(𝑦−1)(𝑦−𝑡)(𝑑𝑦)𝑑𝑡²−{(𝑏₁+𝑏₂)(𝑦−1)(𝑦−𝑡)+(𝑏₁−𝑏₂)𝑦(𝑦−𝑡)+(𝑏₃+𝑏₄)𝑦(𝑦−1)}𝑑𝑦1𝑑𝑡+{4(2𝑏₁+𝑏₃+𝑏₄)²−14(𝑏₃−𝑏₄)²}(𝑦−𝑡)+𝜎^′₂1[𝑏]𝑡−2𝜎₂[𝑏], (1.9) where 𝜎^′₂[𝑏]∶=𝑏₁𝑏₃+𝑏₁𝑏₄+𝑏₃𝑏₄,𝜎₂[𝑏]∶=⁴_{𝑗,𝑘=1𝑗<𝑘}𝑏_𝑗𝑏_𝑘. (1.10) In terms of the function ℎ, the Painlevé equation (1.6) can be represented in a remarkably symmetric form as follows: 𝑑ℎ𝑑𝑑𝑡[𝑡(1−𝑡)²ℎ𝑑𝑡²]²+[𝑑ℎ𝑑𝑡{2ℎ−(2𝑡−1)𝑑ℎ𝑑𝑡}+𝑏₁𝑏₂𝑏₃𝑏₄]²−⁴_𝑘=1(𝑑ℎ𝑑𝑡+𝑏²_𝑘)=0. (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): 𝐿_𝑚∶=^𝑁_𝑗=1𝜆_𝑗^𝑚+1𝜕𝜕𝜆_𝑗,𝑚=−1,0,1,…, (1.12) and the following dependent variables: ℬ_𝑛∶=_𝑗𝜆^𝑛_𝑗𝐴_𝑗≡res∣_𝜆=∞{𝜆^𝑛𝐴(𝜆)},𝑛=0,1,…, (1.13) one can show that the Schlesinger system (1.1) implies 𝐿_𝑚ℬ_𝑛=^𝑛−1_𝑘=1[ℬ_𝑘,ℬ_{𝑚+𝑛−𝑘}]+𝑛ℬ_𝑚+𝑛, (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 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”: ℋ_𝑚1∶=−4^𝑚_𝑘=0trℬ_𝑘ℬ_𝑚−𝑘, (1.15)

which can be viewed as symmetrised analogues of the Hamiltonians (1.4); they coincide with 𝐿_𝑚log𝜏_JM up to an elementary transformation. The simplest equation satisfied by ℋ_𝑚 in the case of 2×2 system is given by 18(𝐿₂𝐿₂ℋ₂+2𝐿₃ℋ₃−5𝐿₄ℋ₂−2ℋ₆)²=(𝐿₃ℋ₃−𝐿₄ℋ₂−ℋ₆)((ℋ₃)²+4ℋ₂(𝐿₂ℋ₂−ℋ₄))−(𝐿₃ℋ₂−ℋ₅)(ℋ₂𝐿₃ℋ₂+ℋ₃𝐿₂ℋ₂−ℋ₂ℋ₅)+(𝐿₂ℋ₂−ℋ₄)(𝐿₂ℋ₂)², (1.16) as we will 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 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 𝐿_𝑚 (1.12); these operators satisfy the commutation relations of the Virasoro algebra: [𝐿_𝑚,𝐿_𝑛]=(𝑛−𝑚)𝐿_𝑚+𝑛. (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 ℬ₀=_𝑗𝐴_𝑗 (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 𝐴_𝑗).

To describe the dynamics under the action of the differential operators 𝐿_𝑚 we introduce the symmetrised Hamiltonians ℋ_𝑚: ℋ_𝑚1∶=−2res∣_𝜆=∞𝜆^𝑚+1tr𝐴²(𝜆)≡_𝑗𝜆_𝑗^𝑚+1𝐻_𝑗. (2.3) These Hamiltonians can be expressed in terms of the variables ℬ_𝑘 as follows: ℋ_𝑚=ℋ_𝑚+14(𝑚+1)_𝑗𝜆^𝑚_𝑗𝐶_𝑗, (2.4) where the modified Hamiltonians ℋ are given by ℋ_𝑚1∶=−4^𝑚_𝑘=0trℬ_𝑘ℬ_𝑚−𝑘, (2.5) and 𝐶_𝑗∶=tr𝐴²_𝑗=(1/2)𝛼²_𝑗. In particular, ℋ₋₁=ℋ₀=ℋ₁=0 (taking into account that ℬ₀=0), such that the first three symmetrised Hamiltonians take the form ℋ₋₁=0,ℋ₀=14_𝑗𝐶_𝑗,ℋ₁=12_𝑗𝜆_𝑗𝐶_𝑗. (2.6)

In terms of the Virasoro generators 𝐿_𝑚, (1.5) for the Jimbo-Miwa 𝜏-function 𝜏_JM looks as follows: 𝐿_𝑚(log𝜏_JM)=ℋ_𝑚. (2.7) It is convenient to introduce also a modified 𝜏-function, invariant under Möbius transformations.

Lemma 2.1. The modified 𝜏-function 𝜏 defined by 𝜏≡𝜏_{J𝑀𝑖<𝑗}(𝜆_𝑗−𝜆_𝑖)^{−(1/2(𝑁−2))(𝐶𝑖+𝐶𝑗)+(2/(𝑁−1)(𝑁−2))ℋ0} (2.8) is annihilated by the first three Virasoro generators: 𝐿₋₁𝜏=𝐿₀𝜏=𝐿₁𝜏=0. (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 𝐿_𝑚 act on the symmetrised variables ℬ_𝑛 as follows: 𝐿_𝑚ℬ_𝑛=^𝑛−1_𝑘=1[ℬ_𝑘,ℬ_{𝑚+𝑛−𝑘}]+𝑛ℬ_𝑚+𝑛, (2.10) for 𝑚=−1,0,1,2,…, 𝑛=1,2,….

Proof. Using the Schlesinger equations (1.1), we have 𝐿_𝑚ℬ_𝑛≡^𝑁_𝑖=1𝜆_𝑖^𝑚+1𝜕𝜕𝜆_𝑖{^𝑁_𝑗=1𝐴_𝑗𝜆^𝑛_𝑗}=_𝑖≠𝑗𝜆_𝑖^𝑚+1𝜆^𝑛_𝑖−𝜆^𝑛_𝑗𝜆_𝑖−𝜆_𝑗[𝐴_𝑗,𝐴_𝑖]+𝑛^𝑁_𝑖=1𝜆_𝑖^𝑚+𝑛𝐴_𝑖. (2.11) Expanding 𝜆^𝑛_𝑖−𝜆^𝑛_𝑗𝜆_𝑖−𝜆_𝑗=𝜆_𝑗^𝑛−1+𝜆_𝑗^𝑛−2𝜆_𝑖+⋯+𝜆_𝑗𝜆_𝑖^𝑛−2+𝜆_𝑖^𝑛−1, (2.12) we further rewrite this expression for 𝐿_𝑚ℬ_𝑛 as [ℬ_𝑛−1,ℬ_𝑚+1]+[ℬ_𝑛−2,ℬ_𝑚+2]+⋯+[ℬ₁,ℬ_{𝑚+𝑛−1}]+𝑛ℬ_𝑚+𝑛, (2.13) which coincides with the right-hand side of (2.10).

Remark 2.3. System (2.10) can be equivalently rewritten as follows: 𝐿_𝑚ℬ_𝑛=^𝑚_𝑘=1[ℬ_𝑘,ℬ_{𝑚+𝑛−𝑘}]+𝑛ℬ_𝑚+𝑛. (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 ℬ_𝑘. Using (2.14) we can express the commutators [ℬ_𝑚,ℬ_𝑛] as follows: [ℬ_𝑚,ℬ_𝑛]=𝐿_𝑚ℬ_𝑛−𝐿_𝑚−1ℬ_𝑛+1+ℬ_𝑚+𝑛. (2.15) Acting on the modified Hamiltonians ℋ_𝑛 by the operators 𝐿_𝑚, we get the following equation: 𝐿_𝑚ℋ_𝑛1=−2^𝑛−1_𝑘=1𝑘trℬ_𝑚+𝑘ℬ_𝑛−𝑘. (2.16) In particular, we have 𝐿_𝑚ℋ_𝑛−𝐿_𝑛ℋ_𝑚=(𝑛−𝑚)ℋ_𝑚+𝑛. (2.17) The same equation holds for the Hamiltonians ℋ_𝑛 as a corollary of the integrability of (2.7).

The Poisson bracket (1.3) induces the following Poisson bracket between variables ℬ_𝑛, 𝑛=0,1,2,…: {ℬ^𝑎_𝑛,ℬ^𝑏_𝑚}=𝑓_𝑐^𝑎𝑏ℬ^𝑐_𝑚+𝑛. (2.18)

Then (2.10) can then be written in the following form: 𝐿_𝑚ℬ_𝑛={ℋ_𝑚,ℬ_𝑛}+𝑛ℬ_𝑚+𝑛. (2.19) We note that formally the second term can be absorbed into the symplectic action {ℋ_𝑚,ℬ_𝑛} upon extending the affine Poisson structure (2.18) by the standard central extension.

Remark 2.4. Let us briefly discuss the geometric origin of the Virasoro generators 𝐿_𝑚. The vectors 𝜕/𝜕𝜆_𝑗 span the tangent space to the space of 𝑁-punctured spheres with punctures at {𝜆_𝑗}. 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 𝜆_𝑗; then variation of moduli by the vector field 𝜆^𝑚+1𝜕/𝜕𝜆 on the circle coincides with variation by the generator 𝐿_𝑚. The commutation relations between 𝐿_𝑚 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 𝐿_𝑚. In the next theorem we prove two equations of this kind.

Theorem 3.1. The 𝜏-function 𝜏_JM (1.5) of an arbitrary 2×2 Schlesinger system satisfies the following two differential equations.(i)The third-order equation with cubic nonlinearity: 18(𝐿₂𝐿₂ℋ₂+2𝐿₃ℋ₃−5𝐿₄ℋ₂−2ℋ₆)²=(𝐿₃ℋ₃−𝐿₄ℋ₂−ℋ₆)((ℋ₃)²+4ℋ₂(𝐿₂ℋ₂−ℋ₄))−(𝐿₃ℋ₂−ℋ₅)(ℋ₂𝐿₃ℋ₂+ℋ₃𝐿₂ℋ₂−ℋ₂ℋ₅)+(𝐿₂ℋ₂−ℋ₄)(𝐿₂ℋ₂)². (3.1)(ii)The fourth-order equation with quadratic nonlinearity: 𝐿₂𝐿₂𝐿₂ℋ₂=8ℋ₈−9𝐿₄ℋ₄+10𝐿₅ℋ₃−4𝐿₃𝐿₃ℋ₂+10𝐿₄𝐿₂ℋ₂−4(𝐿₂ℋ₂(2ℋ₄−3𝐿₂ℋ₂)−ℋ₃(ℋ₅−2𝐿₃ℋ₂ℋ))−8₂(2ℋ₆−3𝐿₃ℋ₃+4𝐿₄ℋ₂), (3.2) where according to (2.4), (2.7), ℋ_𝑚=𝐿_𝑚log𝜏_JM−𝑚+14^𝑁_𝑗=1𝜆^𝑚_𝑗𝐶_𝑗. (3.3)

Proof. Inverting the system of (2.16), we can express trℬ_𝑚ℬ_𝑛 in terms of the Hamiltonians ℋ_𝑛 as follows: trℬ_𝑚ℬ_𝑛=4𝐿_𝑚ℋ_𝑛−2(𝐿_𝑚−1ℋ_𝑛+1+𝐿_𝑚+1ℋ_𝑛−1). (3.4) From the Schlesinger system (2.10), we furthermore get 𝐿_𝑘tr(ℬ_𝑚ℬ_𝑛)=^𝑘_𝑗=1(tr(ℬ_𝑚[ℬ_𝑗,ℬ_{𝑛+𝑘−𝑗}])+tr(ℬ_𝑛[ℬ_𝑗,ℬ_{𝑚+𝑘−𝑗}])),+𝑛trℬ_𝑚ℬ_𝑘+𝑛+𝑚trℬ_𝑛ℬ_𝑘+𝑚. (3.5) Inverting this relation, we obtain for 𝑘<𝑚<𝑛tr(ℬ_𝑘[ℬ_𝑚,ℬ_𝑛])=^𝑛−1_𝑗=𝑚(𝐿_𝑘trℬ_𝑗ℬ_{𝑚+𝑛−𝑗}−12𝐿_𝑘−1trℬ_𝑗+1ℬ_{𝑚+𝑛−𝑗}−12𝐿_𝑘+1trℬ_𝑗ℬ_{𝑚+𝑛−1−𝑗})+trℬ_𝑛ℬ_𝑘+𝑚−trℬ_𝑚ℬ_𝑘+𝑛. (3.6) Combining this equation with (3.4) we can thus express also tr(ℬ_𝑘[ℬ_𝑚,ℬ_𝑛]) entirely in terms of the action of the operators 𝐿_𝑚 on the Hamiltonians ℋ_𝑛, which can further be simplified upon using the commutation relations (2.1) and (2.17). This leads to the closed expression tr(ℬ_𝑘[ℬ_𝑚,ℬ_𝑛])=2(𝐿_𝑛−1𝐿_𝑚+1−𝐿_𝑛+1𝐿_𝑚−1)ℋ_𝑘+2(𝐿_𝑛+1𝐿_𝑚−𝐿_𝑛𝐿_𝑚+1)ℋ_𝑘−1+2(𝐿_𝑛𝐿_𝑚−1−𝐿_𝑛−1𝐿_𝑚)ℋ_𝑘+1−4𝐿_𝑚+𝑛ℋ_𝑘+2𝐿_𝑚+𝑛+1ℋ_𝑘−1−2𝐿_{𝑚+𝑛−1}ℋ_𝑘+1−4𝐿_𝑘+𝑚ℋ_𝑛+2𝐿_{𝑘+𝑚−1}ℋ_𝑛+1+2𝐿_𝑘+𝑚+1ℋ_𝑛−1+4𝐿_𝑘+𝑛ℋ_𝑚−2𝐿_{𝑘+𝑛−1}ℋ_𝑚+1−2𝐿_𝑘+𝑛+1ℋ_𝑚−1. (3.7) In particular, for the lowest values of 𝑘,𝑚,𝑛 we obtain tr(ℬ₁[ℬ₂,ℬ₃])=−2𝐿₂𝐿₂ℋ₂−4𝐿₃ℋ₃+10𝐿₄ℋ₂+4ℋ₆. (3.8) To derive from these relations the desired result, we make use of the following algebraic identity: tr(𝑀₁[𝑀₂,𝑀₃])tr(𝑀₄[𝑀₅,𝑀₆])=−2(tr(𝑀₁𝑀₄)tr(𝑀₂𝑀₅)tr(𝑀₃𝑀₆)+⋯), (3.9) that is valid for an arbitrary set of six matrices 𝑀_𝑗∈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 𝑓_𝑎𝑏𝑐𝑓^𝑑𝑒𝑓=6𝛿^[𝑑_[𝑎𝛿^𝑒_𝑏𝛿^𝑓]_𝑐], (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 tr([𝑀₁,𝑀₂][𝑀₃,𝑀₄])=−2(tr(𝑀₁𝑀₃)tr(𝑀₂𝑀₄)−tr(𝑀₂𝑀₃)tr(𝑀₁𝑀₄)), (3.11) that is valid for any four 𝔰𝔩(2)-valued matrices 𝑀_𝑗. In terms of the structure constants of 𝔰𝔩(2), this identity reads 𝑓_𝑎𝑏𝑓𝑓^𝑐𝑑𝑓=2𝛿^[𝑐_[𝑎𝛿^𝑑]_𝑏] (3.12) and is obtained by contraction from (3.10). We consider the action of 𝐿₂ on (3.8) which yields 3tr(ℬ₁[ℬ₂,ℬ₅])−2tr(ℬ₁[ℬ₃,ℬ₄])=tr[ℬ₁,ℬ₃][ℬ₁,ℬ₃]−tr[ℬ₁,ℬ₂][ℬ₁,ℬ₄]−tr[ℬ₁,ℬ₂][ℬ₂,ℬ₃]−2𝐿₂𝐿₂𝐿₂ℋ₂−4𝐿₂𝐿₃ℋ₃+10𝐿₂𝐿₄ℋ₂+4𝐿₂ℋ₆. (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 𝜏_JM is obtained upon expressing the modified Hamiltonians ℋ_𝑚 in terms of 𝜏_JM 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 𝜏 from (2.8) depends only on the cross-ratio 𝑡=(𝜆₁−𝜆₃)(𝜆₂−𝜆₄)(𝜆₁−𝜆₄)(𝜆₂−𝜆₃). (4.1) We furthermore define the auxiliary function 𝑑𝑆(𝑡)=2𝑡(1+𝑡)𝑑𝑡log𝜏(𝑡). (4.2) Then (3.1) in terms of 𝑆 after lengthy but straightforward calculation gives rise to the second-order differential equation 3(𝑡(1−𝑡)𝑆^′′)²=3(𝐶₂−𝐶₃)(𝐶₁−𝐶₄)(𝑆−𝑡𝑆^′)−3(𝐶₁−𝐶₃)(𝐶₂−𝐶₄)𝑆^′+12(𝑆−𝑡𝑆^′)𝑆^′2+12(𝑆−𝑡𝑆^′)²𝑆^′+2𝜎₁[𝐶]((𝑆−𝑡𝑆^′)²+(𝑆−𝑡𝑆^′)𝑆^′+𝑆^′2)−1𝜎18¹[𝐶]³+12𝜎₁[𝐶]𝜎₂[𝐶]−3𝜎₃[𝐶], (4.3) where 𝜎_𝑖[𝐶] are the elementary symmetric polynomials of the 𝐶_𝑖’s 𝜎₁[𝐶]=𝐶₁+𝐶₂+𝐶₃+𝐶₄,𝜎₂[𝐶]=_𝑗<𝑘𝐶_𝑗𝐶_𝑘,𝜎₃[𝐶]=_{𝑗<𝑘<𝑙}𝐶_𝑗𝐶_𝑘𝐶_𝑙. (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: 6𝑡(1−𝑡)((1−2𝑡)𝑆^′′+𝑡(1−𝑡)𝑆⁽³⁾)=3((𝐶₁−𝐶₃)(𝐶₂−𝐶₄)−(𝐶₂−𝐶₃)(𝐶₁−𝐶₄)𝑡)−12𝑆²+2(1−2𝑡)𝑆(𝜎₁[𝐶]−12𝑆^′)+4𝜎₁[𝐶](1−𝑡+𝑡²)𝑆^′+36𝑡(1−𝑡)𝑆^′2. (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: (1−𝑡)𝑡(6ℎ^′2−(1−2𝑡)ℎ^′′−(1−𝑡)𝑡ℎ⁽³⁾)=4ℎ²+8(1−2𝑡)ℎℎ^′−2(𝑏²₁+𝑏²₂+𝑏²₃+𝑏²₄)ℎ^′−_𝑖<𝑗𝑏²_𝑖𝑏²_𝑗+2(1−2𝑡)𝑏₁𝑏₂𝑏₃𝑏₄, (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): 𝐿₃𝐿₂𝐿₂ℋ₂=6ℋ₉−6𝐿₅ℋ₄+10𝐿₆ℋ₃−5𝐿₇ℋ₂−𝐿₃𝐿₃ℋ₃+𝐿₄𝐿₃ℋ₂+6𝐿₅𝐿₂ℋ₂+8ℋ₂(2𝐿₄ℋ₃−3𝐿₅ℋ₂−ℋ₇)+4ℋ₃(𝐿₃ℋ₃−𝐿₄ℋ₂)−8ℋ₄𝐿₃ℋ₂+12𝐿₂ℋ₂𝐿₃ℋ₂,𝐿₄𝐿₂𝐿₂ℋ₂=2ℋ₁₀+3𝐿₅ℋ₅−8𝐿₆ℋ₄+9𝐿₇ℋ₃−9𝐿₈ℋ₂−𝐿₄𝐿₃ℋ₃+4𝐿₄𝐿₄ℋ₂−3𝐿₅𝐿₃ℋ₂+6𝐿₆𝐿₂ℋ₂+8ℋ₂(𝐿₄ℋ₄+3𝐿₅ℋ₃−3𝐿₆ℋ₂)−4ℋ₃(𝐿₄ℋ₃−3𝐿₅ℋ₂)+8ℋ₄𝐿₄ℋ₂−12𝐿₂ℋ₂𝐿₄ℋ₂,𝐿₃𝐿₃𝐿₂ℋ₂=8ℋ₁₀−6𝐿₅ℋ₅+4𝐿₆ℋ₄+6𝐿₇ℋ₃−6𝐿₈ℋ₂−𝐿₄𝐿₃ℋ₃−2𝐿₄𝐿₄ℋ₂+6𝐿₅𝐿₃ℋ₂+4𝐿₆𝐿₂ℋ₂−4(ℋ₅−2𝐿₃ℋ₂)𝐿₃ℋ₂+4𝐿₂ℋ₂𝐿₃ℋ₃−8ℋ₄𝐿₄ℋ₂+4ℋ₃(𝐿₄ℋ₃−3𝐿₅ℋ₂)−8ℋ₂(2ℋ₈−2𝐿₄ℋ₄+𝐿₅ℋ₃+2𝐿₆ℋ₂). (5.1)

Obviously, these equations are not all independent, but related by the action of the lowest Virasoro generators 𝐿_±1, using that 𝐿₁(𝐿₂𝐿₂𝐿₂ℋ₂)=4(𝐿₃𝐿₂𝐿₂ℋ₂)+…, (5.2) and so forth, since ℋ₁=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 (4.6). 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,…,𝜆_𝑁. 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 𝐴_𝑗; only upon adding the equations for 𝜕𝐴_𝑗/𝜕𝜆_𝑗 one gets a system of ODEs 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.

(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 𝐿_𝑚 and the variables ℬ_𝑚 may change.(iv)As we have mentioned above, the extra term 𝑛ℬ_𝑚+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 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 (2.18) 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.