# Generalization of Okamoto's Equation to Arbitrary2×2

Authors: Dmitry Korotkin; Henning Samtleben
Journal: Advances in Mathematical Physics
 (2009)

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## Abstract

The2×2
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.


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## Body

## 1. Introduction

The Schlesinger system is the following nonautonomous system of differential equations forN unknown matrices Aj∈𝔰𝔩(M) depending on N variables {λj}:
(1.1)∂Aj∂λi=[Aj,Ai]λj-λi,i≠j,∂Aj∂λj=-∑i≠j[Aj,Ai]λj-λi.
System (1.1) determines isomonodromic deformations of a solution of matrix ODE with meromorphic coefficients
(1.2)∂Ψ∂λ=A(λ)Ψ≡∑j=1NAjλ-λjΨ,
if one assumes that ∑j=1NAj=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 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){Aja,Akb}=δjkfcabAjc,
where fcab are structure constants of 𝔰𝔩(M); Aj=Ajata with the 𝔰𝔩(M) generators ta; δjk is the Kronecker symbol (one assumes summation over repeating indices). Obviously, the traces trAjn 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)Hj=14πi∮λitrA2(λ)dλ≡12∑k≠jtrAjAkλk-λj.
The generating function τJM({λj}) of the Hamiltonians Hj, defined by
(1.5)∂∂λjlogτJM=Hj,
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 equalsM=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)d2ydt2=12(1y+1y-1+1y-t)(dydt)2-(1t+1t-1+1y-t)dydt+y(y-1)(y-t)t2(t-1)2(α+βty2+γt-1(y-1)2+δt(t-1)(y-t)2),
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 matricesAj 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)α=(α1-1)22,β=-α222,γ=α322,δ=12-α422.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)b1=12(α2+α3),b2=12(α2-α3),b3=12(α4+α1),b4=12(α4-α1).
The auxiliary Hamiltonian function h(t) is defined in terms of solution y of (1.6) and the constants bj as follows:
(1.9)h=y(y-1)(y-t)(dydt)2-{(b1+b2)(y-1)(y-t)+(b1-b2)y(y-t)+(b3+b4)y(y-1)}dydt+{14(2b1+b3+b4)2-14(b3-b4)2}(y-t)+σ2′[b]t-12σ2[b],
where
(1.10)σ2′[b]:=b1b3+b1b4+b3b4,σ2[b]:=∑j,k=1j<k4bjbk.
In terms of the function h, the Painlevé equation (1.6) can be represented in a remarkably symmetric form as follows:
(1.11)dhdt[t(1-t)d2hdt2]2+[dhdt{2h-(2t-1)dhdt}+b1b2b3b4]2-∏k=14(dhdt+bk2)=0.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)Lm:=∑j=1Nλjm+1∂∂λj,m=-1,0,1,…,
and the following dependent variables:
(1.13)ℬn:=∑jλjnAj≡res∣λ=∞{λnA(λ)},n=0,1,…,
one can show that the Schlesinger system (1.1) implies
(1.14)Lmℬn=∑k=1n-1[ℬk,ℬm+n-k]+nℬm+n,
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 of2×2 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)ℋ̂m:=-14∑k=0mtrℬkℬm-k,which can be viewed as symmetrised analogues of the Hamiltonians (1.4); they coincide with LmlogτJM up to an elementary transformation. The simplest equation satisfied by ℋ̂m in the case of 2×2 system is given by
(1.16)18(L2L2ℋ̂2+2L3ℋ̂3-5L4ℋ̂2-2ℋ̂6)2=(L3ℋ̂3-L4ℋ̂2-ℋ̂6)((ℋ̂3)2+4ℋ̂2(L2ℋ̂2-ℋ̂4))-(L3ℋ̂2-ℋ̂5)(ℋ̂2L3ℋ̂2+ℋ̂3L2ℋ̂2-ℋ̂2ℋ̂5)+(L2ℋ̂2-ℋ̂4)(L2ℋ̂2)2,
as we will show in the following.Since theℋ̂m 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 Section2 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 operatorsLm (1.12); these operators satisfy the commutation relations of the Virasoro algebra:
(2.1)[Lm,Ln]=(n-m)Lm+n.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)ℬ0=∑jAj
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 operatorsLm we introduce the symmetrised Hamiltonians ℋm:
(2.3)ℋm:=-12res∣λ=∞λm+1trA2(λ)≡∑jλjm+1Hj.
These Hamiltonians can be expressed in terms of the variables ℬk as follows:
(2.4)ℋm=ℋ̂m+14(m+1)∑jλjmCj,
where the modified Hamiltonians ℋ̂ are given by
(2.5)ℋ̂m:=-14∑k=0mtrℬkℬm-k,
and Cj:=trAj2=(1/2)αj2. In particular, ℋ̂-1=ℋ̂0=ℋ̂1=0 (taking into account that ℬ0=0), such that the first three symmetrised Hamiltonians take the form
(2.6)ℋ-1=0,ℋ0=14∑jCj,ℋ1=12∑jλjCj.In terms of the Virasoro generatorsLm, (1.5) for the Jimbo-Miwa τ-function τJM looks as follows:
(2.7)Lm(logτJM)=ℋm.
It is convenient to introduce also a modified τ-function, invariant under Möbius transformations.Lemma 2.1.
The modifiedτ-function τ̃ defined by
(2.8)τ̃≡τJM∏i<j(λj-λi)-(1/2(N-2))(Ci+Cj)+(2/(N-1)(N-2))ℋ0
is annihilated by the first three Virasoro generators:
(2.9)L-1τ̃=L0τ̃=L1τ̃=0.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 operatorsLm act on the symmetrised variables ℬn as follows:
(2.10)Lmℬn=∑k=1n-1[ℬk,ℬm+n-k]+nℬm+n,
for m=-1,0,1,2,…, n=1,2,….Proof.
Using the Schlesinger equations (1.1), we have
(2.11)Lmℬn≡∑i=1Nλim+1∂∂λi{∑j=1NAjλjn}=∑i≠jλim+1λin-λjnλi-λj[Aj,Ai]+n∑i=1Nλim+nAi.
Expanding
(2.12)λin-λjnλi-λj=λjn-1+λjn-2λi+⋯+λjλin-2+λin-1,
we further rewrite this expression for Lmℬn as
(2.13)[ℬn-1,ℬm+1]+[ℬn-2,ℬm+2]+⋯+[ℬ1,ℬm+n-1]+nℬm+n,
which coincides with the right-hand side of (2.10).Remark 2.3.
System (2.10) can be equivalently rewritten as follows:
(2.14)Lmℬn=∑k=1m[ℬk,ℬm+n-k]+nℬm+n.
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)[ℬm,ℬn]=Lmℬn-Lm-1ℬn+1+ℬm+n.
Acting on the modified Hamiltonians ℋ̂n by the operators Lm, we get the following equation:
(2.16)Lmℋ̂n=-12∑k=1n-1ktrℬm+kℬn-k.
In particular, we have
(2.17)Lmℋ̂n-Lnℋ̂m=(n-m)ℋ̂m+n.
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){ℬna,ℬmb}=fcabℬm+nc.Then (2.10) can then be written in the following form:
(2.19)Lmℬn={ℋm,ℬn}+nℬm+n.
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 generatorsLm. 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.
Theτ-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)18(L2L2ℋ̂2+2L3ℋ̂3-5L4ℋ̂2-2ℋ̂6)2=(L3ℋ̂3-L4ℋ̂2-ℋ̂6)((ℋ̂3)2+4ℋ̂2(L2ℋ̂2-ℋ̂4))-(L3ℋ̂2-ℋ̂5)(ℋ̂2L3ℋ̂2+ℋ̂3L2ℋ̂2-ℋ̂2ℋ̂5)+(L2ℋ̂2-ℋ̂4)(L2ℋ̂2)2.(ii)
The fourth-order equation with quadratic nonlinearity:(3.2)L2L2L2ℋ̂2=8ℋ̂8-9L4ℋ̂4+10L5ℋ̂3-4L3L3ℋ̂2+10L4L2ℋ̂2-4(L2ℋ̂2(2ℋ̂4-3L2ℋ̂2)-ℋ̂3(ℋ̂5-2L3ℋ̂2))-8ℋ̂2(2ℋ̂6-3L3ℋ̂3+4L4ℋ̂2),
where according to (2.4), (2.7),
(3.3)ℋ̂m=LmlogτJM-m+14∑j=1NλjmCj.Proof.
Inverting the system of (2.16), we can express trℬmℬn in terms of the Hamiltonians ℋ̂n as follows:
(3.4)trℬmℬn=4Lmℋ̂n-2(Lm-1ℋ̂n+1+Lm+1ℋ̂n-1).
From the Schlesinger system (2.10), we furthermore get
(3.5)Lktr(ℬmℬn)=∑j=1k(tr(ℬm[ℬj,ℬn+k-j])+tr(ℬn[ℬj,ℬm+k-j])),+ntrℬmℬk+n+mtrℬnℬk+m.
Inverting this relation, we obtain for k<m<n(3.6)tr(ℬk[ℬm,ℬn])=∑j=mn-1(Lktrℬjℬm+n-j-12Lk-1trℬj+1ℬm+n-j-12Lk+1trℬjℬm+n-1-j)+trℬnℬk+m-trℬmℬk+n.
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 ℋ̂n, which can further be simplified upon using the commutation relations (2.1) and (2.17). This leads to the closed expression
(3.7)tr(ℬk[ℬm,ℬn])=2(Ln-1Lm+1-Ln+1Lm-1)ℋ̂k+2(Ln+1Lm-LnLm+1)ℋ̂k-1+2(LnLm-1-Ln-1Lm)ℋ̂k+1-4Lm+nℋ̂k+2Lm+n+1ℋ̂k-1-2Lm+n-1ℋ̂k+1-4Lk+mℋ̂n+2Lk+m-1ℋ̂n+1+2Lk+m+1ℋ̂n-1+4Lk+nℋ̂m-2Lk+n-1ℋ̂m+1-2Lk+n+1ℋ̂m-1.
In particular, for the lowest values of k,m,n we obtain
(3.8)tr(ℬ1[ℬ2,ℬ3])=-2L2L2ℋ̂2-4L3ℋ̂3+10L4ℋ̂2+4ℋ̂6.
To derive from these relations the desired result, we make use of the following algebraic identity:
(3.9)tr(M1[M2,M3])tr(M4[M5,M6])=-2(tr(M1M4)tr(M2M5)tr(M3M6)+⋯),
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)fabcfdef=6δ[a[dδbeδc]f],
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)tr([M1,M2][M3,M4])=-2(tr(M1M3)tr(M2M4)-tr(M2M3)tr(M1M4)),
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)fabffcdf=2δ[a[cδb]d]
and is obtained by contraction from (3.10). We consider the action of L2 on (3.8) which yields
(3.13)3tr(ℬ1[ℬ2,ℬ5])-2tr(ℬ1[ℬ3,ℬ4])=tr[ℬ1,ℬ3][ℬ1,ℬ3]-tr[ℬ1,ℬ2][ℬ1,ℬ4]-tr[ℬ1,ℬ2][ℬ2,ℬ3]-2L2L2L2ℋ̂2-4L2L3ℋ̂3+10L2L4ℋ̂2+4L2ℋ̂6.
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 ℋ̂m 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 τ̃ from (2.8) depends only on the cross-ratio
(4.1)t=(λ1-λ3)(λ2-λ4)(λ1-λ4)(λ2-λ3).
We furthermore define the auxiliary function
(4.2)S(t)=2t(1+t)ddtlogτ̃(t).
Then (3.1) in terms of S after lengthy but straightforward calculation gives rise to the second-order differential equation
(4.3)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)-118σ1[C]3+12σ1[C]σ2[C]-3σ3[C],
where σi[C] are the elementary symmetric polynomials of the Ci’s
(4.4)σ1[C]=C1+C2+C3+C4,σ2[C]=∑j<kCjCk,σ3[C]=∑j<k<lCjCkCl.Finally, it is straightforward to verify that withh(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)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.
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)(1-t)t(6h′2-(1-2t)h′′-(1-t)th(3))=4h2+8(1-2t)hh′-2(b12+b22+b32+b42)h′-∏i<jbi2bj2+2(1-2t)b1b2b3b4,
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 of𝔰𝔩(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)L3L2L2ℋ̂2=6ℋ̂9-6L5ℋ̂4+10L6ℋ̂3-5L7ℋ̂2-L3L3ℋ̂3+L4L3ℋ̂2+6L5L2ℋ̂2+8ℋ̂2(2L4ℋ̂3-3L5ℋ̂2-ℋ̂7)+4ℋ̂3(L3ℋ̂3-L4ℋ̂2)-8ℋ̂4L3ℋ̂2+12L2ℋ̂2L3ℋ̂2,L4L2L2ℋ̂2=2ℋ̂10+3L5ℋ̂5-8L6ℋ̂4+9L7ℋ̂3-9L8ℋ̂2-L4L3ℋ̂3+4L4L4ℋ̂2-3L5L3ℋ̂2+6L6L2ℋ̂2+8ℋ̂2(L4ℋ̂4+3L5ℋ̂3-3L6ℋ̂2)-4ℋ̂3(L4ℋ̂3-3L5ℋ̂2)+8ℋ̂4L4ℋ̂2-12L2ℋ̂2L4ℋ̂2,L3L3L2ℋ̂2=8ℋ̂10-6L5ℋ̂5+4L6ℋ̂4+6L7ℋ̂3-6L8ℋ̂2-L4L3ℋ̂3-2L4L4ℋ̂2+6L5L3ℋ̂2+4L6L2ℋ̂2-4(ℋ̂5-2L3ℋ̂2)L3ℋ̂2+4L2ℋ̂2L3ℋ̂3-8ℋ̂4L4ℋ̂2+4ℋ̂3(L4ℋ̂3-3L5ℋ̂2)-8ℋ̂2(2ℋ̂8-2L4ℋ̂4+L5ℋ̂3+2L6ℋ̂2).Obviously, these equations are not all independent, but related by the action of the lowest Virasoro generatorsL±1, using that
(5.2)L1(L2L2L2ℋ̂2)=4(L3L2L2ℋ̂2)+…,
and so forth, since ℋ̂1=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 variablesλ1,…,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 algebraic𝔰𝔩(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 termnℬm+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.



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*Source: 461860-2009-10-15.xml*