ITEP-TH-2/99

Painlevé type equations and Hitchin systems^{1}

M.A.Olshanetsky Max-Plank-Institut für Matematik, Bonn Institut of Theoretical and Experimental Physics, Moscow, Russia, e-mail olshanet@mpim-bonn.mpg.de e-mail olshanet@heron.itep.ru

Abstract

In this survey we present the interpretation of isomondromy preserving equations on Riemann surfaces with marked points as reduced Hamiltonian systems. The upstairs space is the space of smooth connections of GL(N) bundles with simple poles in the marked points. We discuss relations of these equations with the Whitham quantization of the Hitchin systems and with the classical limit of the Knizhnik-Zamolodchikov-Bernard equations. The main example is the one-parameter family of Painlevé VI equation and its multicomponent generalization.

1 Introduction

The famous Painlevé VI equation depends on four free parameters (P V I_{α,β,γ,δ}) and has the form

\frac{d^{2}X}{dt^{2}} = \frac{1}{2} (\frac{1}{X} + \frac{1}{X − 1} + \frac{1}{X − t}) (\frac{dX}{dt} )^{2} − (\frac{1}{t} + \frac{1}{t − 1} + \frac{1}{X − t}) \frac{dX}{dt} +

+\frac{X(X − 1)(X − t)}{t^{2}(t − 1)^{2}} (α + β \frac{t}{X^{2}} + γ \frac{t − 1}{(X − 1)^{2}} + δ \frac{t(t − 1)}{(X − t)^{2}} ) .  (1.1)

It was discovered by B.Gambier [1] in 1906. He accomplished the Painlevé classification program of the second order differential equations whose solutions have not movable critical points. This equation and its degenerations P V − P I have a lot of applications in classical and quantum integrable systems (see, for example [2]), topological field theories [3], general relativity [4, 5], and in the Seiberg-Witten theory [6]. In this paper we discuss two important and interrelated aspects of PVI:

• PVI and isomonodromic deformations of linear differential equations;

• The Hamiltonian structure of PVI. The derivation of PVI equation as the preserving monodromy condition was given by R.Fuchs [7], while the Hamiltonian structure of PVI was introduced by J.Malmquist [8].

1Contribution in the Proceedings ”Integrability: the Seiberg-Witten and Whitham Equations”, Edinburgh, September 1998

We incorporate the one-parameter family PVI_{\frac{ν^{2}}{4} ,− \frac{ν^{2}}{4} , \frac{ν^{2}}{4} , \frac{1}{2} − \frac{ν^{2}}{4}} =PVI_{ν} in a wide class of nonlinear equations. They preserve monodromies of systems of linear equations on Riemann curves with marked points when the complex structures of curves are changed. These systems come from the flatness condition of vector bundles on the curves. We restrict ourself by considerations of smooth connections with simple poles only, and therefore don’t include the Stokes phenomena. In such general form the isomondromy preserving equations were considered in [9]. Our investigation of these systems is inspired by methods developed in classical and quantum integrable systems. In general all the systems can be derived using three different constructions: A)The symplectic reduction procedure from free infinite dimensional theory. This approach is very similar to the derivation of the Hitchin integrable systems [10]; B)The Whitham quantization of the Hitchin systems [11]; C)Classical limit of the Knizhnik-Zamolodchikov-Bernard (KZB) equations [12, 13]. We discuss these constructions separately and then demonstrate their application on the multicomponent generalization of PVI_{ν}. The presentation is based for the most part on a previously published paper [14]. First, in Sect.2 we consider the elliptic form of PVI. In this form the relations of PVI with the Hitchin systems and KZB equations become transparent. Then we discuss these three approaches to the isomonodromic deformations. In Sect.6 the multicomponent generalization of PVI is described. Finally, we discussed some open problems related to PVI and its generalizations.

2 Elliptic form of PVI

1. Elliptization procedure Soon after discovering of PVI (1.1) by Gambier, Painlevè presented it in terms of the Weierstrass elliptic functions [15]. This paper was almost forgotten for ninety years and the elliptic form was rediscovered recently in [16, 17]. We follow the derivation presented in [16], where the Hamiltonian form and symmetrices of PVI in terms of elliptic functions are treated.

Consider the family of elliptic curves

E_{τ} = C/(Z + Zτ)  (2.1)

where τ ∈ H = {Imτ > 0} Let ℘(u|τ) be the Weierstrass function

℘(u| = \frac{1}{u^{2}} + ∑^{′} (\frac{1}{(u + mω_{1} + nω_{2})^{2}} − \frac{1}{(mω_{1} + nω_{2})^{2}} ) , (τ = \frac{ω_{2}}{ω_{1}} ).  (2.2)

In the most part of the paper we put ω_{1} = 1 and ℘(u|τ) = ℘(u|1, ω_{2}). ℘(u|τ) uniformize the elliptic curve

℘_{u}(u|τ) = 4(℘(u|τ) − e_{1}(τ))(℘(u|τ) − e_{2}(τ))(℘(u|τ) − e_{3}(τ)),  (2.3)

e_{i} = ℘ (\frac{T_{i}}{2} |τ) , (T_{0}, …, T_{3}) = (0, 1, τ, 1 + τ).

We consider two kind of transformations. First one is the lattice action

u → u + m + nτ, τ → τ.  (2.4)

It leaves ℘(u|τ) and ℘_{u}(u|τ) invariant. The second is the modular transformation by PSL_{2}(Z)

℘ (\frac{u}{cτ + d}|\frac{aτ + b}{cτ + d}) = (cτ + d)^{2}℘(u|τ), ℘_{u} (\frac{u}{cτ + d}|\frac{aτ + b}{cτ + d}) = (cτ + d)^{3}℘_{u}(u|τ).  (2.5)

Now consider another family of elliptic curves E_{t} → B, Y^{2} = X(X−1)(X−t) parameterized by B = {t ∈ P^{1} \ (0, 1, ∞)}. There exists the morphism {E_{τ} } → {E_{t}} defined as

(u, τ) → (X = \frac{℘(u|τ) − e_{1}}{e_{2} − e_{1}} , Y = \frac{℘_{u}(u|τ)}{e_{2} − e_{1}}, t = \frac{e_{3} − e_{1}}{e_{2} − e_{1}} ) .  (2.6)

Theorem 2.1 In terms of (u, τ) P V I_{α,β,γ,δ} takes the form

\frac{d^{2}u}{dτ^{2}} = ∂_{u}U(u|τ), U(u|τ) = \frac{1}{(2πi)^{2}} ∑_{j=0}^{3} α_{j}℘(u + \frac{T_{j}}{2} |τ),  (2.7)

(α_{0}, …, α_{3}) = (α, −β, γ, \frac{1}{2} − δ).

The proof of the equivalence of (1.1) and (2.7) is based on the Picard-Fuchs equation on elliptic curves. The Picard-Fuchs operator

L_{t} = t(t − 1)\frac{∂^{2}}{∂t^{2}} + (1 − 2t)\frac{∂}{∂t} − \frac{1}{4}

acting on the holomorphic differential ω = (d_{E/B}x)/y yields the exact differential \frac{1}{2}d_{E/B} \frac{y}{(x−t)^{2}} . The Picard-Fuchs equation just means that periods of d_{E/B}x/y are annihilated by L_{t}. Using the Picard-Fuchs operator Fuchs proved that PVI (1.1) is equivalent to the following equation

t(1 − t)L_{t} ∫_{∞}^{X} d_{E/B}x/y = (α + β \frac{t}{X^{2}} + γ \frac{t − 1}{(X − 1)^{2}} + (δ − \frac{1}{2})\frac{t(t − 1)}{(X − t)^{2}} ) Y,  (2.8)

The equivalence of (1.1) and (2.8) follows from the following equality

t(1 − t)L_{t} ∫_{∞}^{X} d_{E/B}x/y = \frac{1}{2} \frac{t(t − 1)Y}{(X − t)^{2}} + \frac{d^{2}X}{dt^{2}} − \frac{1}{2} (\frac{1}{X} + \frac{1}{X − 1} + \frac{1}{X − t}) (\frac{dX}{dt} )^{2} +

(\frac{1}{t} + \frac{1}{t − 1} + \frac{1}{X − t}) \frac{dX}{dt} , Y^{2} = X(X − 1)(X − t).

The proof is straightforward.

Thus, PVI can be written in the form of the so-called μ-equation [16]

L_{t} ∫_{∞}^{X} ω = s_{(α,β,γ,δ)}(X),  (2.9)

where the right hand side is a special section of the bundle E_{t}. It can be fixed by the symmetries of the equation.

Under the morphism (2.6) the holomorphic differential d_{E/H}z on E_{τ} is transformed in d_{E/B}x/y, and \frac{d^{2}}{dτ^{2}} in L_{t}. More exactly, the left hand side of (2.9) takes the form

\frac{2π^{2}}{(e_{1} − e_{2})(e_{1} − e_{3})(e_{2} − e_{1})^{\frac{1}{2}}} \frac{d^{2}}{dτ^{2}} ∫_{0}^{u(τ)} dz.

Taking into account that

Y = \frac{1}{2}(e_{2} − e_{1})^{−3/2}℘_{u}(u, τ)

we come finally to (2.7).

2. Hamiltonian structure

The hamiltonian form of (2.7) is defined by the standard symplectic form

ω^{(0)} = δvδu,  (2.10)

and the Hamiltonian

H = \frac{v^{2}}{2} − U(u|τ).  (2.11)

Consider the bundle P over the moduli space M = H/PSL_{2}(Z) with the symplectic fibers parameterized by the local coordinates (v, u). It plays role of the extended phase space for the non-autonomous hamiltonian system (2.10),(2.11). The equation of motion (2.7) can be derived from the action F on P

δF = vδu − Hδτ.  (2.12)

The symmetries of the non-autonomous hamiltonian systems are determined by the invariance of the two-form ω on P

ω = ω^{(0)} − δHδτ = δvδu − δHδτ.  (2.13)

It follows from (2.2) and (2.5) that the symmetry group is the semi-direct product of Z+Zτ and the group Γ(2) ⊂ PSL_{2}(Z). We consider a simplified version of this action in Sect.7 in detail.

3. Calogero-Inozemtsev equation and PVI

Let us introduce the new parameter κ and instead of (2.13) consider

ω = ω_{0} − \frac{1}{κ}δHδτ.  (2.14)

It can be achieved by the rescaling the dynamical variables (v, u) and periods ω_{1}, ω_{2} v → κ^{− \frac{1}{2}} , u → κ^{\frac{1}{2}} , ω_{1} → κ^{\frac{1}{2}} , ω_{1} → κ^{\frac{1}{2}} . Then, (2.7) takes the form

κ^{2} \frac{d^{2}u}{dτ^{2}} = −∂_{u}U(u|τ).  (2.15)

Put τ = τ_{0} + κt^{H} and consider the system in the limit κ → 0. We come to the equation

\frac{d^{2}u}{(dt^{H})^{2}} = −∂_{u}U(u|τ_{0})  (2.16)

corresponding to the autonomous Hamiltonian system with the time-independent potential U(u|τ_{0}). It is just the rank one elliptic Calogero-Inozemtsev equation (CI_{α,β,γ,δ}) [18, 19]. The potential U(u|τ_{0}) was considered first by Darboux [20]. It arises also in the soliton theory [21]. Thus, we have in this limit

P V I_{α,β,γ,δ} ⟶^{κ→0} CI_{α,β,γ,δ}.  (2.17)

There is the inverse procedure (the Whitham quantization) that allows to construct approximations of non-autonomous systems starting from integrable autonomous systems. It will be discussed in Sect.4.

Inozemtsev considered degenerations of U(u|τ_{0}) playing with the coupling constants, the periods, and u. In this way he obtained the trigonometric, rational and exponential interactions.

Presumably, they describe the degenerations of PVI to PV-PI in terms of degenerations of elliptic functions. Here is one of his potentials:

α_{0} \frac{1}{sinh^{2} u} + α_{1} \frac{1}{sinh^{2} 2u} + α_{2} exp u + α_{3} exp 2u.

In what follows we consider only the subfamily PVI_{ν} corresponding to α_{j} = ν^{2}.

3 Isomonodromic deformations

Here we describe the monodromy preserving equations as reduced Hamiltonian systems. The original phase space is infinite-dimensional and almost all degrees of freedom are killed by the symplectic reduction. Our approach differs from [22], where PI-PVI equations are treated as a result of symplectic reduction from finite-dimensional space.

1. Hamiltonian approach Let Σ_{g} be a Riemann curve of genus g. Consider the space F Bun_{Σ,G} of flat vector bundle V_{G}, where G = GL(N, C) with smooth connection A. The flatness means that its curvature vanishes

F_{A} = dA + \frac{1}{2}[A, A] = 0.  (3.1)

Let us fix the complex structure on Σ_{g}. Then for A = (A, Ā) we have locally a consistent system of matrix differential equations

(∂ + A)Ψ = 0,
(¯∂ + Ā)Ψ = 0.

We modify this system in the following way. First, introduce formally a parameter κ ∈ R (the level) and consider the operator κ∂ instead of ∂ in the first equation. Let μ be a Beltrami differential on Σ_{g} (μ ∈ Ω^{(−1,1)}(Σ_{g})). It means that in local coordinates μ = μ(z, z̄)\frac{∂}{∂z} ⊗ dz̄. It allows to deform the complex structure on Σ_{g} such that the new complex coordinates are

w = z − ϵ(z, z̄), w̄ = z̄, μ(z, z̄) = \frac{¯∂ϵ(z, z̄)}{1 − ∂ϵ(z, z̄)}.

The holomorphic operator ∂w̄ = ¯∂+ μ∂ is annihilates the one-form dw likewise ¯∂ annihilates dz. We do not touch the anti-holomorphic operator κ∂. In the new coordinates (3.1) takes the form

F_{A} = (¯∂ + ∂μ)A − κ∂Ā + [Ā, A] = 0.  (3.2)

Thus, we come to the system

(κ∂ + A)Ψ = 0,  (3.3)
(¯∂ + μ∂ + Ā)Ψ = 0.  (3.4)

Represent the Beltrami differential as μ = ∑_{a=1}^{l} t_{a}μ_{a}^{0}, where μ_{1}^{0}, …, μ_{l}^{0} is the basis in the tangent space to the moduli space M_{g} of complex structures on Σ_{g}, (l = dim M_{g} = 3g − 3, for g > 1, l = 1, for g = 1). In other words, t = (t_{1}, …, t_{l}) are coordinates of the tangent vector to M_{g}.

To fix a fundamental solution of (3.3),(3.4), impose the following normalization for some reference point (z_{0}, z̄_{0}) ∈ Σ_{g}

Ψ(z_{0}, z̄_{0}) = I.

Let γ be a homotopically nontrivial cycle in Σ_{g} such that (z_{0}, z̄_{0}) ∈ γ and Y is the corresponding monodromy transformation

Y(γ) = Ψ(z_{0}, z̄_{0})|_{γ} = P exp ∮_{γ} A.

The set of matrices {Y(γ)} generates a representation of the fundamental group π_{1}(Σ_{g}, z_{0}) in GL(N, C). Independence the monodromy Y on the deformations of the complex structure means that the linear equations

∂_{a}Y = 0, (a = 1, …, l) (∂_{a} = ∂_{t_{a}}).  (3.5)

are consistent with (3.3),(3.4).

Proposition 3.1 Equations (3.5) are consistent with (3.3),(3.4) iff

∂_{a}A = 0, (a = 1, …, l),  (3.6)
∂_{a}Ā = \frac{1}{κ}Aμ_{a}^{0}, (a = 1, …, l).  (3.7)

The proof is straightforward.

Proposition 3.2 Equations of motion (3.6),(3.7) are Hamiltonian.

Endow the space F Bun_{Σ,G} with the symplectic form

ω^{(0)} = ∫_{Σ_{g}} < δA, δĀ >, (<, >= tr),  (3.8)

and the set of Hamiltonians

H_{a} = \frac{1}{2} ∫_{Σ_{g}} < A, A > μ_{a}^{(0)}, (a = 1, …, l).  (3.9)

Then (3.6),(3.7) are Hamiltonian equations with respect to ω^{(0)} and H_{s}.

Consider the bundle P over the moduli space M_{g} with F Bun_{Σ,G} as the fibers. The triple (A, Ā, t) can be considered as the local coordinates of the total space of the bundle. It is useful to consider P as the extended phase space [23]. There is a closed two-form on P

ω = ω^{(0)} − \frac{1}{κ} ∑_{a} δH_{a}δt_{a}.  (3.10)

Though ω is degenerated on P it produces the equations of motion (3.6),(3.7), since the form ω^{(0)} is non-degenerated along the fibers.

The gauge transformations in the deformed complex structure take the form

A → f^{−1}κ∂f + f^{−1}Af, Ā → f^{−1}(¯∂ + μ∂)f + f^{−1}Āf.  (3.11)

The form ω is invariant under these transformations, though its constituents ω^{(0)} and H_{s} separately are not invariant.

Introduce a new couple of the connection components A = (A, Ā′), where Ā′ = Ā − \frac{1}{κ}μA. In terms of (A, Ā′) the form ω (3.10) takes the canonical form

ω = ∫_{Σ_{g}} < δA, δĀ′ > .  (3.12)

2.Symplectic reduction The form ω is degenerated on F Bun_{Σ,G}, because it is invariant under the action of the group G of gauge transformations (3.11), generating by the flatness condition (3.2). The gauge fixing along with the flatness condition (3.2) is nothing else as the symplectic reduction from the space of smooth connections Sm_{Σ,G} in the bundle V_{G} to the reduced space

F̃ Bun_{Σ,G} = F Bun_{Σ,G}/G = Sm_{Σ,G}//G

The double slashes means that we impose the moment constraints (3.2) and fix the gauge. F̃ Bun_{Σ,G} is the moduli space of flat connections of the bundle V_{G}. In terms of the symplectic reduction procedure the flatness condition is called the moment constraint equation.

Let us fix the gauge in a such way that the Ā component of A becomes anti-holomorphic

∂L̄ = 0, (L̄ = f^{−1}(¯∂ + μ∂)f + f^{−1}Āf).  (3.13)

We can do it because the antiholomorphity of f^{−1}(¯∂ + μ∂)f + f^{−1}Āf amounts to the classical equations of motion for the Wess-Zumino-Witten functional S_{W ZW} (f, Ā) for the gauge field f in the external field Ā. Denote the gauge transformed field A as L

L = f^{−1}κ∂f + f^{−1}Af.

Then (3.2) takes the form

(¯∂ + ∂μ)L + [L̄, L] = 0.  (3.14)

Thus, the moduli space of flat connections F̃ Bun_{Σ,G} are characterized by the set of solutions of the linear differential equation (3.14) along with the condition (3.13). The moduli space F̃ Bun_{Σ,G} is finite-dimensional space

dim F̃ Bun_{Σ,G} = 2(N^{2} − 1)(g − 1), g > 1.

After the gauge fixing we come to the bundle P̃ over M_{g} with F̃ Bun_{Σ,G} as the fibers. The system of linear differential equations (3.3),(3.4) and (3.5) after the gauge fixing takes the form

(κ∂ + L)Ψ = 0,  (3.15)
(¯∂ + μ∂ + L̄)Ψ = 0,  (3.16)
(κ∂_{s} + M_{s})Ψ = 0,  (3.17)

where we replaced Ψ on f^{−1}Ψ and M_{s} = κ∂_{s}ff^{−1}.

The gauge transformations do not spoil the consistency of the system. The consistency (3.15) and (3.16) is provided by (3.14) and (3.13). In fact, the consistency (3.17) with (3.15) and (3.16) leads to the Lax form of the equations of isomonodromic deformations

∂_{s}L − κ∂M + [M, L] = 0,  (3.18)
κ∂_{s}L̄ − μ_{s}^{0}L = (¯∂ + μ∂)M_{s} − [M_{s}, L̄].  (3.19)

They play the role (3.6),(3.7) correspondingly. The last equation allows to find M_{s} in terms of dynamical variables L, L̄.

The symplectic form ω on the reduced phase space P̃ is

ω = ∫_{Σ_{g}} < δL, δL̄ > −\frac{1}{κ} ∑_{s} δH_{s}δt_{s}.  (3.20)

H_{s} = \frac{1}{2} ∫_{Σ_{g}} < δL, δL > μ_{s}^{(0)}  (3.21)

Introduce the local coordinates (v, u) in F̃ Bun_{Σ,G}:

L = L(v, u, t), L̄ = L̄(v, u, t),
v = (v_{1}, …, v_{(N^{2}−1)(g−1)}), u = (u_{1}, …, u_{(N^{2}−1)(g−1)}).

Assume for simplicity that this parameterization leads to the canonical form on F̃ Bun_{Σ,G}

ω^{(0)} = ∫_{Σ_{g}} < δL(v, u, t), δL̄(v, u, t) >= (δv, δu),  (3.22)

where the pairing in the right hand side is induced by the trace. The form on the extended phase space is

ω = (δv, δu) − \frac{1}{κ} ∑_{s} δK_{s}(v, u, t)δt_{s},  (3.23)

and the variations of the Hamiltonians K_{s} in the new variables take the form

δK_{s} = ∫_{Σ_{g}}[< L, δL > μ_{s}^{(0)} + κ(< δL, ∂_{s}L̄ > − < ∂_{s}L, δL̄ >)].  (3.24)

Now, due to (3.14), the hamiltonians depends explicitly on times. Consider the one-form (the integral invariant of Poincaré-Cartan)

θ = δ^{−1}ω = (v, δu) − \frac{1}{κ} ∑_{s} K_{s}(v, u, t)δt_{s}.

There exist 3g − 3 = dim M_{g}-dimensional space of vector fields V_{s} that annihilates θ

V_{s} = κ∂_{s} + {H_{s}, ·}, (s = 1, …, l).  (3.25)

It can be checked that V_{s} satisfy the following conditions

κ∂_{s}H_{r} − κ∂_{r}H_{s} + {H_{s}, H_{r}}_{ω^{(0)}} = 0.  (3.26)

Thereby, they define the flat connection in the bundle P̃. These conditions are called the Whitham hierarchy (WH). The equations for any function f(v, u, t) on P̃ take the form

\frac{df(v, u, t)}{dt_{s}} = κ\frac{∂f(v, u, t)}{∂t_{s}} + \{H_{s}, f\}  (3.27)

They are called the hierarchy of isomonodromic deformations (HID).

The both hierarchies can be derived from variations of the prepotential F. It is defined as the integral over the classical trajectories in the extended phase space P̃

F(u, t) = F(u_{0}, t_{0}) + ∫_{u_{0},t_{0}}^{u,t} L_{s}dt_{s},  (3.28)

where L_{s}(∂_{s}u, u, t) = (v, ∂_{s}u) − K_{s}(v, u, t), (∂_{s}u = \frac{δK_{s}}{δv} ) is the Lagrangian. F satisfies the set of the Hamilton-Jacobi equations

κ∂_{s}F + H_{s}(\frac{δF}{δu} , u, t) = 0.  (3.29)

The logarithm of F is called the tau-function of HID.

3. Singular curves

The singular curves are important for applications, since they produce nontrivial systems for the low genus curves (g = 0, 1). In these cases the explicit calculations of hamiltonians are available.

Consider a curve Σ_{g,n} of genus g with n marked points (x_{1}, …, x_{n}). The number of times is equal to dimension of the moduli space M_{g,n}. We extend the space of connections F Bun_{Σ,G} = {A, Ā} by adding the coadoint orbits of G = GL(N, C) in the marked points

(O_{1}, …, O_{n}), O_{b} = {p_{b} = gp_{b}^{0}g^{−1}},

where p_{b}^{0} fixes the conjugacy class of O_{b}. We allow the A component of connection to have simple poles at the marked points, while the Beltrami differentials vanish there. Then instead of (3.2) we obtain

(¯∂ + ∂μ)A − κ∂Ā + [Ā, A] = ∑_{b=1}^{n} δ^{2}(x_{b})p_{b}  (3.30)

The Hamiltonian formalism is provided by the modified symplectic form

ω^{(0)} = ∫_{Σ_{g,n}} < δA, δĀ > +2πi ∑_{b=1}^{n} < δ(p_{b}g_{b}^{−1}), δg_{b} > .  (3.31)

To derive HID one should start from the space of connections with simple poles in the marked points.

Finally, we come to the same linear system (3.15), (3.16),(3.17), but due to (3.30) the following relation between L and L̄ holds

(¯∂ + ∂μ)L + [L̄, L] = ∑_{b=1}^{n} δ^{2}(x_{b})p_{b}.  (3.32)

As before, the linear equations are equivalent to the equations of motion of HID coming from the symplectic form

ω = ω^{(0)}(v, u, p) − \frac{1}{κ} ∑_{s=1}^{l} δK_{s}(v, u, p, t)δt_{s}, (l = dim(M_{g,n}),  (3.33)

where p = (p_{1}, …, p_{n}),ω_{0} is determined by the reduction from (3.31)

ω^{(0)} = ∫_{Σ_{g,n}} < δL(v, u, p), δL̄(v, u, p) > +2πi ∑_{b=1}^{n} < δ(p_{b}g_{b}^{−1}), δg_{b} >,  (3.34)

and K_{s} (3.24).

4 Hitchin systems and their Whitham deformations

1. Hitchin systems Consider the moduli space R_{g,N} of stable holomorphic GL(N, C) vector bundles V over Σ_{g}. It is a smooth variety of dimension

dim R_{g,N} = g̃ = N^{2}(g − 1) + 1.  (4.1)

Let T^{∗}R_{g,N} be the cotangent bundle to R_{g,N} with the standard symplectic form on it. Hitchin [10] defined a completely integrable system on T^{∗}R_{g,N}.

The space T^{∗}R_{g,N} can be obtained by the symplectic reduction from the space T^{∗}R_{g,N}^{s} = (Φ, Ā), where Ā is a smooth connection of the stable bundle corresponding to ¯∂ + Ā and Φ is the Higgs field Φ ∈ Ω^{0}(Σ_{g}, EndV ⊗ K) (K is the canonical bundle of Σ_{g}). There is the well defined symplectic form on this space

ω^{(0)} = ∫_{Σ_{g}} < δΦ, δĀ > .  (4.2)

This form is invariant with respect to the gauge group G = C^{∞}Map(Σ_{g}, GL(N, C)) action

Φ → f^{−1}Φf, Ā → f^{−1} ¯∂f + f^{−1}Āf.  (4.3)

In particular, R_{g,N} = R_{g,N}^{s}/G. Let ρ_{s,k} = ρ_{s,k}∂_{z}^{k−1} ⊗ dz̄ be the (−k + 1, 1)-differentials (ρ_{s,k} ∈ H^{1}(Σ_{g}, Γ^{k−1} ⊗ K)), and s enumerates the basis in H^{1}(Σ_{g}, Γ^{k−1} ⊗ K). (ρ_{s,2} = μ_{s}). Due to the Riemann-Roch theorem

dim H^{1}(Σ_{g}, Γ^{k−1} ⊗ K) = (2k − 1)(g − 1).

These differentials allows to define the gauge invariant Hamiltonians

H_{s,k} = \frac{1}{k} ∫_{Σ_{g}} < Φ^{k} > ρ_{s,k}, (k = 1, …, N, s = 1, …, (2k − 1)(g − 1)).  (4.4)

The Hamiltonian equations take the form

∂_{a}Φ = 0, (∂_{a} = \frac{∂}{∂t_{a}} , a = (s, k)),  (4.5)
∂_{a}Ā = Φ^{k−1}ρ_{s,k}  (4.6)

The gauge action produces the moment map μ : T^{∗}R^{s} → Lie^{∗}(GL(N, C)). It follows from (4.2),(4.3) that μ = ¯∂Φ + [Ā, Φ]. The reduced phase space is the cotangent bundle we started with

T^{∗}R_{g,N} ∼ T^{∗}R^{s}//G := μ^{−1}(0)/G.

The Hitchin hierarchy (HH) is the set of Hamiltonian equations with H_{s,k} (4.4) on the reduced phase space T^{∗}R_{g,N}. Hitchin observed that the number of integrals H_{s,k}

∑_{k=1}^{N}(2k − 1)(g − 1) = N^{2}(g − 1) + 1

coincides with dimension g̃ of the coordinate space R_{g,N} (4.1). Since they are independent and Poisson-commute, HH is the set of completely integrable Hamiltonian systems.

Let fix the gauge of the field Ā

Ā = f ¯∂f^{−1} + fL̄f^{−1}.

Then

L = f^{−1}Φf.

is a solution of the moment constraint equation

¯∂L + [L̄, L] = 0.  (4.7)

The space of solutions of this equation is isomorphic to H^{0}(Σ_{g}, End V ⊗ K) - the cotangent space to the moduli space R_{g,N}.

The gauge transformation f defines the element M_{a} in Lie(GL(N, C)) M_{a} = ∂_{a}ff^{−1}.

Proposition 4.1 The system of linear equations

(λ + L)Y = 0,  (4.8)
(¯∂ + ∑_{s,k} λ^{k−1}t_{s,k}ρ_{s,k} + L̄)Y = 0,  (4.9)
(∂_{a} + M_{a})Y = 0  (4.10)

is consistent and defines the equations of motion for HH.

Proof. The consistency of (4.8) and (4.9) follows from (4.7). In terms of L the equations of motion (4.5),(4.6) take the form

∂_{a}L + [M_{a}, L] = 0 (the Lax equation),  (4.11)
∂_{a}L̄ − ¯∂M_{a} + [M_{a}, L̄] = L^{k−1}ρ_{s,k}, (a = (s, k)).  (4.12)

The Lax equation provides the consistency of (4.8) and (4.10), while the second equation of motion (4.12) plays the same role for the couple (4.9) and (4.10). This equation allows to determine M_{a} from L and L̄.

The bundles over singular curves can be incorporated in this approach as well [24]. The Higgs field has simple poles at the marked points and (4.7) is replaced by

¯∂L + [L̄, L] = 2πi ∑_{b=1}^{n} δ^{2}(x_{b})p_{b}.  (4.13)

The form ω^{(0)} on T^{∗}R_{g,N}

ω^{(0)} = ∫_{Σ_{g}} < δL, δL̄ > +2πi ∑_{b=1}^{n} < δ(p_{b}g_{b}^{−1}), δg_{b} >  (4.14)

Note that the space T^{∗}R_{g,N} and the space of flat bundles F̃ Bun_{Σ,G} have the same dimensions. Moreover, it follows from the comparison the moment constraints (3.32),(4.13) and the symplectic forms (3.34),(4.14) that they are isomorphic as symplectic manifolds.

2.Spectral description

Due to the Liouville theorem the phase flows of HH are restricted to the Abelian varieties, corresponding to a level set of the Hamiltonians H_{s,k} = c_{s,k}. The phase flow takes a simple form in terms of action-angle coordinates. They are defined in a such way that the angle type coordinates are angular coordinates on the Abelian variety, and the hamiltonians depend on the action coordinates only. To describe them consider the characteristic polynomial of the matrix L

P (λ, z) = det(λ + L) = λ^{N} + b_{1}λ^{N−1} + … + b_{j}λ^{N−j} + … + b_{N},  (4.15)
b_{j} = ∑ Min_{j}, (Min_{j} − principle minors of order j, b_{1} = trL, b_{N} = det L).

The spectral curve C ⊂ T^{∗}Σ_{g} is defined as the zero set of P

C = {P (λ, z) = 0}.

It is a well defined object, because the coefficients b_{j} are gauge invariant.

Since L ∈ H^{0}(Σ_{g}, End V ⊗ K), the coefficients b_{j} ∈ H^{0}(Σ_{g}, K^{j}) and we obtain the map

p : T^{∗}R_{g,N} → B = ⊕_{j=1}^{N}H^{0}(Σ_{g}, K^{j}).  (4.16)

The space B can be considered as the moduli space of the family of spectral curves parameterized by the Hamiltonians H_{s,k}. The fibers of p are Lagrangian subvarieties of T^{∗}R_{g,N}. The spectral curve C is the N-fold covering of the basis curve Σ_{g,N}

π : C → Σ_{g,N}.

Its genus g(C) is equal to dimension g̃ of R_{g,N}. There is a line bundle L with an eigenspace of L(z) corresponding to the eigenvalue λ as a fiber over a generic point (λ, z)

L ⊂ ker(λ + L) ⊂ π^{∗}(V).

It defines a point of the Jacobian Jac(C), the Liouvillean variety of dimension g̃ = g(C).

Conversely, if z ∈ Σ_{g} is not a branch point one can reconstruct V for a given line bundle on C as

V_{z} = ⊕_{v∈π^{−1}(z)}L_{v}.

Let ω_{j}, j = 1 …, g̃ be the canonical holomorphic one-differentials on C such that for the cycles α_{1}, …, αg̃; β_{1}, …, βg̃, α_{i} · α_{j} = β_{i} · β_{j} = 0, α_{i} · β_{j} = δ_{ij}, ∮_{α_{i}} ω_{j} = δ_{ij}. Then the symplectic form ω^{(0)} (4.2) can be written in the form

ω^{(0)} = ∫_{Σ_{g}} < δL, δL̄ >= ∑_{j=1}^{N} ∫_{Σ_{g}} δλ_{j}δξ_{j}.

Here ξ_{j} are diagonal elements of sL̄s^{−1}, where s diagonalizes L, sLs^{−1} = diag(λ_{1}, …, λ_{N}). Then we obtain

ω^{(0)} = ∫_{C} δλδξ.

Because λ is a holomorphic one-form on C, it can be decomposed as λ = ∑g̃ ; j=1 a_{j}ω_{j}. Thereby

ω^{(0)} = ∑_{j=1}^{g̃} δa_{j} ∫_{C} ω_{j}δξ.

The action variables can be identify with

a_{j} = ∮_{α_{i}} λω_{j}, (j = 1 …, g̃).  (4.17)

To define the angle variables, put locally ξ = ¯∂log ψ. If (p_{m}) is a divisor of ψ then

∫_{C} ω_{j}δξ = ∑_{m} ∫_{p_{0}}^{p_{m}} ω_{j} log ψ = δφ_{j}.

Thus φ_{j} are linear coordinates on Jac(C) and

ω^{(0)} = ∑_{j=1}^{g̃} δa_{j}δφ_{j}.

3. Scaling limit.

Consider HID in the limit κ → 0. The value κ = 0 is called critical. We prove that on the critical level HID coincide with part of HH relating to the quadratic Hamiltonians (4.4).

Note first, that in this limit the A-connection is transformed in the Higgs field Φ: A ⟶^{κ→0} Φ, and therefore

F̃ Bun_{Σ,G} ⟶^{κ→0} T^{∗}R_{g,N}.

But the form ω on the extended phase P appears to be singular (see (3.10),(3.20)). To get around we rescale the times

t = T + κt^{H},  (4.18)

where t^{H} are the fast (Hitchin) times and T are the slow times. Assume that only fast times are dynamical. It means that

δμ(t) = κ ∑_{s} μ_{s}^{(0)}δt_{s}^{H}, (μ_{s}^{(0)} = ¯∂n_{s}).

After this rescaling the forms (3.10),(3.20) become regular. The rescaling procedure means that we blow up a vicinity of the fixed point μ_{s}^{(0)} in M_{g,n} and the whole dynamic is developed in this vicinity. This fixed point is defined by the complex coordinates

w_{0} = z − ∑_{s} T_{s}ϵ_{s}(z, z̄), w̄_{0} = z̄.  (4.19)

Now compare the Baker-Akhiezer function of HID Ψ (3.15), (3.16),(3.17) with the Baker-Akhiezer function of HH Y (4.8),(4.9),(4.10). Using the WKB approximation, assume that

Ψ = Φ exp(\frac{S^{(0)}}{κ} + S^{(1)}),  (4.20)

where Φ is a group valued function and S^{(0)}, S^{(1)} are diagonal matrices. Let substitute (4.20) in the linear system (3.15),(3.16),(3.17). If

\frac{∂}{∂w̄_{0}} S^{(0)} = 0, \frac{∂}{∂t_{s}^{H}} S^{(0)} = 0.

there are no terms of order κ^{−1}. It follows from the definition of the fixed point in the moduli of complex structures (4.19) that

S^{(0)} = S^{(0)}(T_{1}, …, T_{l}|z − ∑_{s} T_{s}ϵ_{s}(z, z̄)).  (4.21)

We take also S^{(1)} = ∂S^{(0)} ∑_{s} t_{s}^{H}ϵ_{s}(z, z̄). In the quasi-classical limit we put

∂S^{(0)} = λ.  (4.22)

In the zero order approximation we come to the linear system of HH (3.15),(3.16),(3.17), defining by the Hamiltonians H_{k,s}, k = 1, 2. The Baker-Akhiezer function Y takes the form

Y = Φe^{∑_{s} t_{s}^{H} \frac{∂}{∂Ts} S^{(0)}}.  (4.23)

Our goal is the inverse problem. We need to reconstruct the dependence on the slow times T starting from solutions of HH. Since T is a vector in the tangent space to the moduli of curves M_{g,n}, it defines a deformation of the spectral curve in the space B (see (4.4),(4.16)). Solutions Y of the linear systems (4.8)(4.9),(4.10) take the form Y = Φe^{∑_{s} t_{s}^{H}Ω_{s}}, where Ω_{s} are diagonal matrices. Their entries are primitive functions of meromorphic differentials with singularities matching the corresponding poles of L. Then according with (4.23) we can assume that

\frac{∂}{∂T_{s}} dS = dΩ_{s}.

These equation define the approximation to the phase of Ψ in the linear problems (3.15),(3.16), (3.17) of HID along with

\frac{∂}{∂a_{j}} dS = ω_{j}.

The differential dS plays role of the Seiberg-Witten differential. Important point is that only part of the spectral moduli, connected with H_{k,s}, k = 1, 2, is deformed. As a result there is no matching between the action parameters of the spectral curve a_{j}, j = 1, …, g̃ (4.17) and deformed hamiltonians. The detailed analyses of this situation in the rational case is undertaken in [25].

Another object of the Whitham quantization is the prepotential F (3.28). It depends on the action variables a_{j} . This dependence is compatible with the Hamilton-Jacobi equation (3.29) with slow times T_{s} as the independent variables. These equations are discussed in [25, 26].

5 Classical limit of the Knizhnik-Zamolodchikov-Bernard equations

The Knizhnik-Zamolodchikov-Bernard equations (KZB) are the system of differential equations having the form of the non-stationer Schrödinger equations with the times coming from M_{g,n} (see, for example, [27]).

They arise in the geometric quantization of the moduli of flat bundles F̃ Bun_{Σ,G} [28, 29]. Let V = V_{1} × ⋯ ⊗ V_{n} be the tensor product of finite-dimensional irreducible representations associated with the marked points. The Hilbert space of the quantum system is a space of sections of the bundle E_{V,κ^{quant}}(Σ_{g,n}) over F̃ Bun_{Σ,G} depending on an negative number κ^{quant} with the V -fibers. It is the space of conformal blocks of the WZW theory on Σ_{g,n}.

The Hitchin systems are the classical limit of the KZB equations on the critical level [24, 27]. The classical limit means that one replaces operators by their symbols and generators of finitedimensional representations in the vertex operator acting in the spaces V_{j} by the corresponding elements of coadjoint orbits. To pass to the classical limit in the KZB equations

(κ^{quant}∂_{s} + Ĥ_{s})F = 0.  (5.1)

we replace the conformal block by its quasi-classical expression

F = exp \frac{F}{h̄} ,  (5.2)

where h̄ = (κ^{quant})^{−1}. Consider the classical limit κ^{quant} → ∞ and assume that values of the Casimirs C_{a}^{i}, (i = 1, …, rankG, a = 1, …, n) corresponding to the irreducible representations defining the vertex operators also go to infinity. Let all values lim \frac{C_{a}^{i}}{κ^{quant}} are finite. It allows to fix the coadjoint orbits in the marked points. In the classical limit (5.1) is transformed to the Hamilton-Jacobi equation for the action F = log τ (3.29) of HID.

The integral representations of conformal blocks are known for WZW theories over rational and elliptic curves [31, 32, 33, 34]. Then (5.2) allows to extract the prepotential F of HID.

The KZB operators (5.1) play role of flat connections in the bundle P^{quant} over the moduli of curves M_{g,n} with the fibers E_{V,κ^{quant}}(Σ_{g,n}) [4, 30]

[κ^{quant}∂_{s} + Ĥ_{s}, κ^{quant}∂_{r} + Ĥ_{r}] = 0.

These equations is the quantum counterpart of the Whitham hierarchy (3.26).

6 Multicomponent generalization of PVI_{ν}

Consider F̃ Bun_{Σ,G} over the family of elliptic curves with a one marked point M_{1,1}. The space M_{1,1} is one-dimensional, because the position of one point on a torus is irrelevant. Thus, we have only one time τ and M_{1,1} ∼ E_{τ} (2.1). In this case the Beltrami differential takes the form

μ = \frac{τ − τ_{0}}{τ − τ̄_{0}} .

Consider the most degenerated orbit O = (gp^{0}g^{−1}) of GL(N, C) sitting in the marked point z = 0 with

p^{0} = ν[(\underbrace{1, …, 1}_{N})^{T} ⊗ (\underbrace{1, …, 1}_{N}) − Id].  (6.1)

For stable bundles the gauge transforms allow to put Ā component in the diagonal form

L̄ = \frac{2πi}{τ − τ̄_{0}} u, u = diag(u_{1}, …, u_{N}) ∈ H − Cartan algebra.  (6.2)

It means that

∫_{E_{τ}} L̄dwdw̄ = u.  (6.3)

Let L̄ = ¯∂log ϕ. Then the integral

∫_{E_{τ}} L̄dwdw̄ = ∫_{P_{0}}^{P} log ϕdw.

defines the Abel map E_{τ} in the product of N Jacobians.

The remaining gauge transforms do not change the gauge fixing. These transformations are generated by the Weyl subgroup W of G and elements f(w, w̄) ∈ Map(T_{τ}^{2}, Cartan(G)). The orbit variables can be gauged away by these transforms and we are left with p^{(0)} (6.1). The solution L of the moment constraint

∂w̄L + [L̄, L] = 2πiδ^{2}(0)p^{(0)}

takes the form

L = P + X, P = 2πi(\frac{v}{1 − μ} − κ\frac{u}{ρ}),  (6.4)
u = diag(u_{1}, …, u_{N}), v = diag(v_{1}, …, v_{N})

X_{jk} = x(u_{j} − u_{k}) = (τ − τ̄_{0})ν exp 2πi\{\frac{w − w̄}{τ − τ̄_{0}} (u_{j} − u_{k})\}ϕ(α(u_{j} − u_{k}), w),

ϕ(u, z) = \frac{θ(u + z)θ′(0)}{θ(u)θ(z)}, θ(z|τ) = q^{\frac{1}{8}} ∑_{n∈Z}(−1)^{n}e^{πi(n(n+1)τ+2nz)}.

The operator M defining the phase flow according with the Lax equation (3.18) can be extracted from (3.19)

M = −D + Y, D = diag(d_{1}, …, d_{N}), d_{j} = ∑_{i≠j}^{N} s(u_{j} − u_{i}), s(u) = \frac{1}{κ}℘(u) + const. (6.5)

Y_{jk} = y(u_{j} − u_{k}), y(u, w, w̄) = \frac{ρ}{2πiκ(τ − τ̄_{0})}∂_{u}x(u, w, w̄).

The functions x, y, z satisfy the functional equations

x(u, z, z̄)y(v, z, z̄) − x(v, z, z̄)y(u, z, z̄) = (s(v) − s(u))x(u + v, z, z̄).  (6.6)

This equation is derived from the Lax equation (3.18).

The symplectic form ω is boiled down to

ω = (δv, δu) − \frac{1}{κ}δHδτ,

where

H = \frac{(δv, δv)}{2} − \frac{ν^{2}}{(2πi)^{2}} ∑_{j<k}^{N} ℘(u_{j} − u_{k}|τ).

They define the hamiltonian flow

\frac{d^{2}u_{j}}{dτ^{2}} = \frac{ν^{2}}{(2πi)^{2}} ∑_{k<j}^{N} ℘_{u}(u_{j} − u_{k}|τ).  (6.7)

ForN = 2 one can put u_{1} = −u_{2} = u. Then the potential

\frac{ν^{2}}{(2πi)^{2}} ℘(2u|τ) = \frac{ν^{2}}{(2πi)^{2}} ∑_{j=0}^{3} ℘(u + \frac{T_{j}}{2} |τ)

produces PVI_{ν} (see (2.7).

The remaining gauge symmetries implies that ω is invariant under the Weyl transformations W of (v, u) and the lattice actions (compare with (2.4)

v → sv, v + κn, u → su, u − m + τn, (s ∈ W, n ∈ Z^{N}).

It is also invariant under the PSL_{2}(Z) action on τ

τ → \frac{aτ + b}{cτ + d}, v → v(cτ + d) − κcu, u → u(cτ + d)^{−1}.

This invariance is follows from the invariance of the upstairs system under the diffeomorphisms of Σ_{g,n} (see [14] for detailes).

On the critical level κ → 0, τ − τ_{0} = κt we obtain the elliptic Calogero N-body system. This system is a particular example of the Hitchin systems [24]. Note, that the functions x, y and s defining the Lax matricies satisfy the same functional equation (6.6) as in the Calogero-Hitchin limit κ = 0 [18].

7 Conclusion

Here we propose a few open problems in the context of topics discussed above.

• The evident problem is a description of PVI with four arbitrary constants as a reduced Hamiltonian system. The first step in this direction is the Lax form of P V I_{α,β,γ,δ}. The Lax form is even unknown on the critical level, i.e. for the Calogero-Inozemtsev system. It will be interesting to generalize this approach to the N-body Calogero-Inozemtsev system and the N-component PVI with four coupling constants.

• The degenerations of PVI to PV-PI in terms of elliptic functions.

• There exists a generalization of the Calogero systems related to any simple group. In addition to degrees of freedom coming from the moduli of bundles (the coordinates of particles), these systems certainly contain degrees of freedom related to the coadjoint orbits. Recently a new Lax equations based on arbitrary root systems without the orbit coordinates were proposed [36, 37]. This construction is purely algebraic and does not use the symplectic reduction. How these systems can be incorporated in the Hitchin approach, or, more generally, in the isomonodromic deformation construction?

• Consider the N = 2 elliptic Calogero system. The solution u(t), corresponding to the fixed value h_{2} of the Hamiltonian

H = \frac{v^{2}}{2} + \frac{ν^{2}}{4π^{2}} ℘(2u|τ_{0}) = h_{2},

is implicitly described by the elliptic integral of the first kind

t − t_{0} = \frac{1}{2} ∫_{2u_{0}}^{2u} \frac{dx}{y′}, y′ = y\sqrt{2h_{2} − \frac{ν^{2}}{2π^{2}}} ,

where y = 4(x − e_{1}(τ_{0}))(x − e_{2}(τ_{0}))(x − e_{3}(τ_{0})). As it was mentioned at the end of Sect.6 it can serve for the calculations of solutions to PVI_{ν}. This procedure can be accomplished by the Krichever averaging method [35]. It will be interesting to compare this approximation with explicit solutions presented recently in [38, 39] for some particular value of the coupling constant ν.

As suggested in Sect.5 another way of approximation comes from the classical limit of conformal blocks for SL_{2}(C) theory on elliptic curves with one marked point [33, 40]. Which method gives the better approximation?

• We considered deformations with respect to the moduli of complex structures of curves. They describe only part of the moduli of the spectral curves C. The remaining moduli of C come from ρ_{s,k}, k > 2. They correspond to the so-called W-geometry of the basic curve Σ_{g,n}. This geometry is poorly understood. On the other side, there are no examples of isomonodromic deformation equations with respect to these moduli spaces, as well as the corresponding higher order KZB equations. Any progress in understanding of one of these subjects will shed light on another.

Acknowledgments The work is supported in part by grants RFFI-96-02-18046 INTAS 96-518 and 96-15-96455 for support of scientific schools. I am grateful to the Max-Planck-Institut für Mathemamatik in Bonn for the hospitality, where this paper was prepared.

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