ITEP-TH-56/01

Hitchin Systems - Symplectic Hecke Correspondence and Two-dimensional Version

April 22, 2002

A.M.Levin^{1} Max Planck Institute of Mathematics, Bonn, Germany, e-mail alevin@wave.sio.rssi.ru M.A.Olshanetsky^{2} Max Planck Institute of Mathematics, Bonn, Germany, e-mail olshanet@gate.itep.ru A.Zotov Institute of Theoretical and Experimental Physics, Moscow, Russia, e-mail zotov@gate.itep.ru

Abstract The aim of this paper is two-fold. First, we define symplectic maps between Hitchin sys-

tems related to holomorphic bundles of different degrees. We call these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs bundles. They are constructed by means of the Hecke correspondence of the underlying holomorphic bundles. SHC allows to construct Bäcklund transformations in the Hitchin systems defined over Riemann curves with marked points. We apply the general scheme to the elliptic Calogero-Moser (CM) system and construct SHC to an integrable SL(N, ℂ) Euler-Arnold top (the elliptic SL(N, ℂ)-rotator). Next, we propose a generalization of the Hitchin approach to 2d integrable theories related to the Higgs bundles of infinite rank. The main example is an integrable two-dimensional version of the two-body elliptic CM system. The previous construction allows to define SHC between the two-dimensional elliptic CM system and the Landau-Lifshitz equation.

Contents

1 Introduction 2

2 Hitchin systems in the C̆ech description 4 2.1 The moduli space of holomorphic quasi-parabolic bundles in the C̆ech description. 4 2.2 Hitchin systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Standard description of the Hitchin system. . . . . . . . . . . . . . . . . . . . . . 9 2.4 Modified C̆ech description of the moduli space. . . . . . . . . . . . . . . . . . . . 9

3 Symplectic Hecke correspondence 10 3.1 Hecke correspondence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Symplectic Hecke correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 SHC and skew-conormal bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Bäcklund transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Elliptic CM system - elliptic SL(N, ℂ)-rotator correspondence 14

1On leave from Institute of Oceanology, Moscow, Russia 2On leave from Institute of Theoretical and Experimental Physics, Moscow, Russia

4.1 Elliptic CM system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 The elliptic SL(N, ℂ)-rotator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 A map R^{CM} → R^{rot}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.4 Bäcklund transformations in the CM systems. . . . . . . . . . . . . . . . . . . . 19

5 Hitchin systems of infinite rank 20 5.1 Holomorphic L̂(GL(N, ℂ))-bundles. . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Gauge symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3 Phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.4 Symplectic reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.5 Coadjoint orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.6 Conservation laws I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.7 Equations of motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.8 Conservation laws II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.9 Hamiltonians in SL(2, ℂ) case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 L̂(SL(N, ℂ))-bundles over elliptic curves with marked points. 26 6.1 General case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.2 L̂(SL(2, ℂ))-bundles over elliptic curves with marked points. . . . . . . . . . . . . 27 6.3 Hamiltonians for the 2d elliptic Gaudin model. . . . . . . . . . . . . . . . . . . . 30

7 Conclusion 32

8 Acknowledgments 32

9 Appendix 33 9.1 Appendix A. Sin-Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 9.2 Appendix B. Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 9.3 Appendix C: 2d sl(2, ℂ) Calogero L-M pair . . . . . . . . . . . . . . . . . . . . . 36

1 Introduction

Nowadays many examples of integrable one-dimensional and two-dimensional models are known. The problem of listing all of them, up to some equivalence, was solved for some particular forms of two-dimensional models [1]. The recently developed conception of duality for one-dimensional models [2] can shed light on the classification problem in analogy with string theory. In spite of this progress we are still far from understanding the structure of this universe. Therefore, the classification of integrable systems, apart from solving any individual equation, continues to be an actual task. We will consider integrable systems that have the Lax or Zakharov-Shabat representations. In these cases the gauge transformations of the accompanying linear equations lead essentially to the same systems, though their equations of motion differ in a significant way. For example, the non-linear Schrödinger model is gauge equivalent to the isotropic Heisenberg magnetic [3]. In such a manner the integrable system should be classified up to gauge equivalence, though it is not the only equivalence principle in their possible classifications. The crucial and delicate point of this approach is the exact definition of allowed gauge transformations, and it will be discussed here.

We restrict ourselves to Hitchin systems [4] and their two-dimensional generalizations that we will construct. The Hitchin construction establishes relations between finite dimensional integrable systems and the moduli space of holomorphic vector bundles over Riemann curves. The phase space of the integrable system is the cotangent bundle to the moduli space and the dual variables Φ are called the Higgs fields. The pair (E, Φ), where E is a holomorphic bundle, is called the Higgs bundle. The Lax representation arises immediately in this scheme as the equation of motion and the Lax operator is just the Higgs field defined on shell. The C^{∞} gauge transformations of the Lax pair define the equivalent holomorphic bundles. The different gauge fixing conditions give equivalent integrable systems.

We consider the generalization of the Hitchin systems based on the quasi-parabolic Higgs bundles [5], where the Higgs fields are allowed to have the first order poles at the marked points on the base curve. The gauge transformations preserve the flag structures that arise at the marked points. The corresponding integrable systems were considered in [6, 7, 8, 9]. We loosen the smoothness condition of the gauge transformations and allow them to have a simple zero or a pole at one of the marked points. This type of gauge transformations (the upper and lower Hecke correspondence (HC) or the modifications) is suggested by the geometric Langlands program. HC changes the degree of the underlying bundles on ±1. We assume, that HC is consistent with flag structures on the source and target bundles. It allows to choose a canonical form of the modifications. HC can be lifted as the symplectic correspondence (SHC) to the Higgs bundles. In this way SHC define a map of Hitchin systems related to bundles of different degrees. One can consider an arbitrary chain of consecutive SHC attributed to different marked points. If the resulting transformation preserves the degree of bundle, then it defines the Bäcklund transformations of the Hitchin system related to the initial bundle, or the integrable discrete time map [21].

As an example, we consider a trivial holomorphic SL(N, ℂ)-bundle E^{CM} (deg(E^{CM}) = 0) over an elliptic curve with a marked point. The corresponding quasi-parabolic Higgs bundle leads to the elliptic N-body Calogero-Moser system (CM system). The upper SHC defines a map of the Higgs bundle related to E^{CM} to the Higgs bundle (E^{rot}, Φ^{rot}) with deg(E^{rot}) = 1. SHC is generated by the N-th order matrix Ξ with theta-functions depending on coordinates of the particles as the matrix elements. The system (E^{rot}, Φ^{rot}) is the integrable SL(N, ℂ)-Euler-Arnold top (SL(N, ℂ)-elliptic rotator). The Lax pair for this top was proposed earlier [10]. The consecutive upper and lower SHC define the Bäcklund transformations of the both systems. Another way to construct a Bäcklund transformation is achieved by applying N consecutive upper modifications, since they lead to equivalent Higgs bundle.

In the second part of the paper we try to gain insight into interrelation between integrable theories in dimension one and two. It is known that some one-dimensional integrable systems can be extended to the two-dimensional case without sacrificing the integrability. For example, the Toda field theory comes from the corresponding Toda lattice. To understand this connection we apply the Hitchin construction to two-dimensional systems. For this purpose we consider infinite rank bundles over the Riemann curves with marked points. The transition group of the bundles is the central extended loop group L̂(GL(N, ℂ)). If the central charge vanishes the theory in essence becomes one-dimensional. In the two-dimensional situation the Higgs field is a gl(N, ℂ) connection on a circle S^{1}. In addition, we put coadjoint orbits of L̂(GL(N, ℂ)) at the marked points and in this way introduce the quasi-parabolic structure on the Higgs bundle of infinite rank. The monodromy of the Higgs field is a generating function for the infinite number of conservation laws. The equations of motion on the reduced phase space are the Zakharov-Shabat equations. The similar class of the Hitchin type systems from a different point of view was introduced recently by Krichever [11]. We consider in detail the case of L̂(SL(2, ℂ))-bundle over an elliptic curve with n marked points. The Higgs bundle corresponds to the two-dimensional version of the elliptic Gaudin system. For the 1 marked point case we come to the 2d two-body elliptic CM theory. The upper SHC is working in the two-dimensional situation as well. It leads to the map of the 2-body elliptic CM field theory to the Landau-Lifshitz equation.^{3} To summarize we consider here the following diagram:

2 − body elliptic CM

system

⟶ SL(2, ℂ) − elliptic rotator

↓  ↓
2 − body elliptic CM

field theory

⟶ Landau-Lifshitz equation

Figure 1: Interrelation in integrable theories

In fact, the upper SHC can be applied to the SL(N, ℂ) case. The quadratic Hamiltonian of the N-body elliptic CM field theory was constructed in [11], but the SL(N, ℂ) generalization of the Landau-Lifshitz equation is unknown.

It should be mentioned that the quantum version of SL(N, ℂ) SHC has appeared in a different context long ago [12]. It was defined as a twist transformation of the quantum R-matrices, and Hasegawa [13] has constructed such types of twists that transform the dynamical elliptic R-matrix of Felder [14] to the non-dynamical R-matrix of Belavin [15]. It was proved [16] that the dynamical R-matrix corresponds to the elliptic Ruijsenaars system [17]. The later is the relativistic deformation of the elliptic CM system. In this way the Hasegawa twist is the quantization of SHC we have constructed, since the elliptic CM system and the elliptic Ruijsenaars system are governed by the same R-matrix [18].

2 Hitchin systems in the C̆ech description

In this section we consider vector bundles with structure group G = GL(N, ℂ), or any simple complex Lie group.

2.1 The moduli space of holomorphic quasi-parabolic bundles in the C̆ech description.

Let E be a trivial rank r holomorphic vector bundle over a Riemann curve Σ_{n} with n marked points. Consider a covering of Σ_{n} by open disks U_{a}, a = 1, 2 . . .. Some of them may contain one marked point w_{α}. The holomorphic structure on E can be described by the differential d′′. On U_{a} it can be represented as

d′′ = ¯∂_{a} + Ā_{a}, Ā_{a} = h_{a}^{−1} ¯∂_{a}h_{a}, ¯∂_{a} = \frac{∂}{∂z̄_{a}}

where z_{a} is a local coordinate on U_{a} , and h_{a} is a C^{∞} G-valued function on U_{a}. It is a section of the local sheaf Ω_{C^{∞}}^{0}(Σ_{n}, Aut E).

3The equivalence of these models was pointed out by A.Shabat.

The transition functions g_{ab} = h_{a}h_{b}^{−1} are defined on the intersections U_{ab} = U_{a} ∩ U_{b}. They are holomorphic since Ā_{a} = Ā_{b} on U_{ab}

g_{ab} ∈ Ω_{hol}^{0}(U_{ab}, Aut E) .

The transformation h_{a} → f_{a}h_{a} by a function holomorphic on U_{a} (f_{a} ∈ Ω_{hol}^{0}(U_{a}, Aut E)) does not change Ā_{a}. Similarly, the transformation h_{b} → f_{b}h_{b} by f_{b} ∈ Ω_{hol}^{0}(U_{b}, Aut E) does not change Ā_{b}. Then the holomorphic structures described by the transition functions g_{ab} and f_{a}g_{ab}f_{b}^{−1} are equivalent. Globally we have the collection of transition maps

L_{Σ}^{C} = {g_{ab}(z_{a}) = h_{a}(z_{a})h_{b}^{−1}(z_{b}(z_{a})), z_{a} ∈ U_{ab}, a, b = 1, 2 …, } .  (2.1)

They define holomorphic structures on E or P = AutE depending on the choice of the representations.

The definition of the holomorphic structures by the transition functions works as well in the case if deg(E) ≠ 0 (G = GL(N, ℂ)). They should satisfy the cocycle condition

g_{ab}(z)g_{bc}(z)g_{ca}(z) = Id, z ∈ U_{a} ∩ U_{b} ∩ U_{c} ,

and

g_{ab} = g_{ba}^{−1}.  (2.2)

The degree of the bundle E is defined as the degree of the linear bundle L = det g.

We choose an open subset of stable holomorphic structures L_{Σ}^{C,st} in L_{Σ}^{C}. The gauge group G_{Σ}^{hol} acts as the automorphisms of L_{Σ}^{C,st}

g_{ab} → f_{a}g_{ab}f_{b}^{−1}, f_{a} = f(z_{a}), f_{b} = f_{b}(z_{b}(z_{a})), f ∈ G_{Σ}^{hol} .  (2.3)

We prescribe the local behavior of the gauge transformations G_{Σ}^{hol} at the marked points. Let

P_{1}, …, P_{α}, …, P_{n}

be parabolic subgroups of G attributed to the marked points. Then we assume that

f_{a} = \{ \begin{matrix}f̃_{α}^{(0)} + z_{α}f_{α}^{(1)} + …, & f̃_{α} ∈ P_{α}, if z_{α} = z − w_{α}, w_{α} is a marked point, \\ f_{a}^{(0)} + z_{a}f_{a}^{(1)} + …, & f_{a} ∈ G if a ≠ α, (U_{a} does not contain a marked point) . \end{matrix} (2.4)

It follows from (2.3) that the left action of the gauge group at the marked points preserves the flags

E_{α}(s) ∼ P_{α} \ G, E_{α} = F_{1}(α) ⊃ ⋯ ⊃ F_{s_{α}}(α) ⊃ F_{s_{α}+1}(α) = 0 .  (2.5)

The moduli space of the stable holomorphic bundles M_{n}(Σ, G) with the quasi-parabolic structure at the marked points is defined in Ref.[19] as the factor space under this action

M_{n} = G_{Σ}^{hol}\L_{Σ}^{C,st} .  (2.6)

For G = GL(N, ℂ) we have a disjoint union of components labeled by the corresponding degrees d = c_{1}(det E) : M_{n}(Σ, G) = ⨆ M_{n}^{(d)}.

The tangent space to M_{n}(Σ, G) is isomorphic to h^{1}(Σ, EndE). Its dimension can be extracted from the Riemann-Roch theorem and for curves without marked points (n = 0)

dim h^{0}(Σ, EndE) − dim h^{1}(Σ, EndE) = (1 − g) dim G .

For stable bundles h^{0}(Σ, EndE) = 1 and

dim M_{0}(Σ, G) = (g − 1)N^{2} + 1

for GL(N, ℂ), and

dim M_{0}(Σ, G) = (g − 1) dim G

for simple groups. For elliptic curves one has

dim h^{1}(Σ, EndE) = dim h^{0}(Σ, EndE),

and

dim M_{0}^{d} = g.c.d.(N, d) .  (2.7)

In this case the structure of the moduli space for the trivial bundles (i.e. with deg(E) = 0 and, for example, for bundles with deg(E) = 1 are different. We use this fact below.

For the quasi-parabolic bundles we have

dim M_{n}^{d} = dim M_{0}^{d} + ∑_{α=1}^{n} f_{α} ,  (2.8)

where f_{α} is the dimension of the flag variety E_{α}. In particular, for G = GL(N, ℂ), we get

f_{α} = \frac{1}{2} (N^{2} − ∑_{i=1}^{s_{α}} m_{i}^{2}(α)) , m_{i}(α) = dim F_{i}(α) − dim F_{i+1}(α) .  (2.9)

The space L_{Σ}^{C} is a sort of a lattice 2d gauge theory. Consider the skeleton of the covering {U_{a}, a = 1, …}. It is an oriented graph whose vertices V_{a} are some fixed inner points in U_{a} and edges L_{ab} connect those V_{a} and V_{b} for whose U_{ab} ≠ ∅. We choose an orientation of the graph, saying that a > b on the edge L_{ab} and put the holomorphic function z_{b}(z_{a}) which defines the holomorphic map from U_{a} to U_{b}. Then the space L_{Σ}^{C} can be defined by the following data. To each edge L_{ab}, a > b we attach a matrix valued function g_{ab} ∈ G along with z_{b}(z_{a}). The gauge fields f_{a} are living on the vertices V_{a} and the gauge transformation is given by (2.3).

2.2 Hitchin systems.

The Hitchin systems in the C̆ech description can be constructed in the following way [20]. We start from the cotangent bundle T^{∗}L_{Σ_{n}}^{C} to the holomorphic structures on P = AutE (2.1). Now

T^{∗}L_{Σ_{n}}^{C} = {η_{ab}, g_{ab}| η_{ab} ∈ Ω_{hol}^{(1,0)}(U_{ab}, (EndE)^{∗}), g_{ab} ∈ Ω_{hol}^{0}(U_{ab}, P)} .  (2.10)

This bundle can be endowed with a symplectic structure by means of the Cartan-Maurer oneforms on Ω_{hol}^{0}(U_{ab}, P).

Let Γ^{βγ} = Γ_{a}^{b}(βγ) be a path in U_{ab} with the end points in the triple intersections β ∈ U_{abc} = U_{a} ∩ U_{b} ∩ U_{c}, γ ∈ U_{abd}. We can put the data (2.10) on the fat graph corresponding to the covering {U_{a}}. Its edges are Γ_{a}^{b}(βγ) and Γ_{b}^{a}(γβ) with the opposite orientations. We assume that the covering is such that the orientation of edges defines the oriented contours around the faces U_{a}. The fields η_{ab}, g_{ab} are attributed to the edge Γ_{a}^{b}(βγ), while η_{ba}, g_{ba} to Γ_{b}^{a}(γβ). The last pair is not independent because g_{ab}^{−1} = g_{ba}. Its counterpart in the dual space is

η_{ab}(z_{a}) = g_{ab}(z_{a})η_{ba}(z_{b}(z_{a}))g_{ab}^{−1}(z_{a}) .  (2.11)

The symplectic structure is defined by the two-form

ω^{C} = ∑_{edges} ∫_{Γ_{a}^{b}(βγ)} Dtr (η_{ab}(z_{a})Dg_{ab}g_{ab}^{−1}(z_{a})) .  (2.12)

Since η_{ab} and g_{ab} are both holomorphic in U_{ab}, the integral is independent on the choice of the path Γ_{a}^{b} within U_{ab}.

The symplectic form is invariant under the gauge transformations (2.3) supplemented by

η_{ab} → f_{a}η_{ab}f_{a}^{−1}.  (2.13)

The set of invariant commuting Hamiltonians on T^{∗}L_{Σ}^{C} is

I_{j,k}^{C} = ∑_{edges} ∫_{Γ_{a}^{b}(βγ)} ν_{(j,k)}^{C}(z_{a})tr(η_{ab}^{d_{j}}(z_{a})) , (k = 1, …, n_{j})  (2.14)

where d_{j} are the orders of the basic invariant polynomials corresponding to G and ν_{j,k}^{C} are (1 − d_{j}, 0)-differentials. They are related locally to the (1 − j, 1)-differentials by ν_{j,k}^{D} = ¯∂ν_{j,k}^{C} and

n_{j} = h^{1}(Σ, T^{⊗(d_{j}−1)}) = \{ \begin{matrix}(2d_{j} − 1)(g − 1) + (d_{j} − 1)n, & (j = 1, …, r) \\ g, & j = 1 for G = GL(N, ℂ) . \end{matrix}

The total number of independent Hamiltonians is equal to

∑_{j=1}^{N} n_{j} = M_{0}^{d} + \frac{1}{2}r(r + 1)n .

This number is greater than the dimension of the moduli space M_{n}^{d} (2.8). There are rn highest weight integrals, (j = r), that become Casimir elements of coadjoint orbits after the symplectic reduction, that we will consider below.

Perform the symplectic reduction with respect to the gauge action (2.3), (2.13) of G_{Σ_{n}}^{hol} (2.4). The moment map is

μ_{G_{Σ}^{hol}}(η_{ab}, g_{ab}) : T^{∗}L_{Σ}^{C} → Lie^{∗}(G_{Σ}^{hol}).

Here the Lie coalgebra Lie^{∗}(G_{Σ}^{hol}) is defined with respect to the pairing

∑_{edges} ∫_{Γ_{a}^{b}(βγ)} tr(ξ_{a}ϵ_{a}), ϵ_{a} ∈ Lie(G_{Σ}^{hol}).

Then locally we have

ξ_{a} = \{ \begin{matrix}(z_{a}^{−1}ξ̃_{a} + z_{a}^{−2}ξ_{a}^{(−2)} + …) dz_{a}, & ξ̃_{a} ∈ Lie^{∗}(P_{α}), (U_{a} contains a marked point w_{α}) \\ (z_{a}^{−1}ξ_{a}^{(−1)} + z_{a}^{−2}ξ_{a}^{(−2)} + …) dz_{a}, & ξ_{a}^{(−1)} ∈ Lie^{∗}(G)(U_{a} does not contain a marked point) . \end{matrix}

(2.15)

The canonical gauge transformations (2.3),(2.13) of the symplectic form (2.12) are generated by the Hamiltonian

F_{ϵ^{hol}} = ∑_{edges} ∫_{Γ_{a}^{b}(βγ)} tr(η_{ab}(z_{a})ϵ_{a}^{hol}(z_{a})) − tr(η_{ab}(z_{a})g_{ab}(z_{a})ϵ_{b}^{hol}(z_{b}(z_{a}))g_{ab}(z_{a})^{−1}) =

= ∑_{a} ∫_{Γ_{a}} ∑_{b} tr(η_{ab}(z_{a})ϵ_{a}^{hol}(z_{a})) ,

where Γ_{a} is an oriented contour around U_{a}.

The non-zero moment is fixed in a special way at the neighborhoods of the marked points. Let G̃_{α} ⊂ P_{α} be the maximal semi-simple subgroup of the parabolic group P_{α} defined at the marked point w_{α}. We drop for a moment the index α for simplicity. We choose an ordering in the Cartan subalgebra 𝔥 ∈Lie(G), which is consistent with the embedding P ⊂ G. Let 𝔥̃ = 𝔥∩G̃ be the Cartan subalgebra in G̃. Consider the orthogonal decomposition of 𝔥^{∗}

𝔥^{∗} = 𝔥̃^{∗} + 𝔥^{′∗} .

We fix a vector p^{(0)} ∈ 𝔥^{∗} such that it is a generic element in 𝔥^{′∗} and

⟨p^{(0)}, 𝔥̃^{∗}⟩ = 0 ,  (2.16)

where ⟨ , ⟩ is the Killing scalar product in 𝔥^{∗}. Since 𝔥^{′∗} ⊂Lie^{∗}(P), we can take μ_{G_{Σ}^{hol}} in the form

μ_{G_{Σ}^{hol}} = μ_{0} = ∑_{α=1}^{n} p_{α}^{(0)}z_{α}^{−1}dz_{α}, p^{(0)} ∈ 𝔥^{′∗} ,  (2.17)

where z_{α} = z − w_{α} is a local coordinate in U_{α}. The moment equation μ_{G_{Σ}^{hol}} = μ_{0} can be read off

from F_{ϵ^{hol}}. It follows from the definition of Lie^{∗}(G_{Σ}^{0,hol}) that η_{ab} is the boundary value of some holomorphic or meromorphic one-form H_{a} defined on U_{a} via

η_{ab}(z_{a}) = H_{a}(z_{a}), for z_{a} ∈ U_{ab}, H_{a} ∈ Ω_{hol}^{(1,0)}(U_{a}, End^{∗}(E)) ,  (2.18)

where

H_{a} = \{ \begin{matrix}z_{a}^{−1}p_{α}^{(0)} + H_{a}^{(0)} + z_{a}H_{a}^{(1)} + …, & if U_{a} contains a marked point w_{α} \\ H_{a}^{(0)} + z_{a}H_{a}^{(1)} + …, & if U_{a} does not contain a marked point . \end{matrix}

The gauge fixing means that the transition functions g_{ab} are elements of the moduli space M_{n}^{d}(Σ, E). Thus, the reduced system is a symplectic quotient

G_{Σ}^{hol}\\T^{∗}L_{Σ}^{C} = G_{Σ}^{hol}\μ^{−1}(μ_{0}) .

We set off the zero modes g_{αb}^{(0)} ∈ G̃_{α} of the transition functions in the symplectic form on the reduced space (see (2.12))

ω^{C} = ∑_{edges} ∫_{Γ_{a}^{b}(βγ)} Dtr (η_{ab}(z_{a})Dg_{ab}g_{ab}^{−1}(z_{a})) + 2πi ∑_{α=1}^{n} ∑_{b} Dtr (p_{α}^{(0)}Dg_{αb}^{(0)}(g_{αb}^{(0)})^{−1}) .

The last sum defines the Kirillov-Kostant symplectic forms on the set of coadjoint orbits O(n) = (O_{1}, … O_{α}, …, O_{n}), where

O_{α} = {p_{α} ∈ Lie^{∗}(G) | p_{α} = (g_{α}^{(0)})^{−1}p_{α}^{(0)}g_{α}^{(0)}} .  (2.19)

Note that dim(O_{α}) = 2f_{α} (2.9).

Remark 2.1 It is possible to construct another type of orbit O′_{α} of the same dimension. There exist elements p_{α}^{′(0)} that belong to the complements of Lie^{∗}(G̃_{α}) in Lie^{∗}(P_{α}) such that the orbit

O′_{α} = {p_{α} = (g_{α}^{(0)})^{−1}p_{α}^{′(0)}g_{α}^{(0)}} .

is symplectomorphic to the cotangent bundles T^{∗}E_{α} to the corresponding flags E_{α} (2.5), while O_{α} (2.19) is a torsor over O′_{α}.

2.3 Standard description of the Hitchin system.

The standard approach of the Hitchin systems [4] is based on the description of the holomorphic bundles in terms of the operator d′′. The upstairs phase space has the form

T^{∗}L_{Σ_{n}}^{D} = {Φ, d′′ | Φ ∈ Ω_{C^{∞}}^{(1,0)}(Σ_{n}, End^{∗} E)} ,  (2.20)

where Φ is called the Higgs field. The symplectic form

ω^{D} = ∫_{Σ_{n}} tr(DΦ ∧ DĀ)  (2.21)

is invariant under the action of the gauge group

G_{Σ}^{C^{∞}} = {f ∈ Ω_{C^{∞}}^{0}(Σ_{n}, Aut V)},
Φ → f^{−1}Φf , Ā → f^{−1} ¯∂f + f^{−1}Āf .  (2.22)

The gauge invariant integrals take the evident form (compare with (2.14))

I_{j,k}^{D} = ∫_{Σ_{n}} ν_{(j,k)}^{D}tr(Φ^{d_{j}}) , (k = 1, …, n_{j})  (2.23)

where ν_{j,k}^{D} are (1 − j, 1)-differentials on Σ_{n}. The symplectic reduction with respect to this action leads to the moment map

μ : T^{∗}L_{Σ_{n}}^{D} → Lie^{∗}(G_{Σ}^{C^{∞}}) μ = ¯∂Φ + [Ā, Φ] .

The Higgs field Φ is related to η in a simple way

η_{ab} = h_{a}^{−1}Φh_{a}|_{U_{ab}} ,

and Ā_{a} = h_{a}^{−1} ¯∂_{a}h_{a}. The holomorpheity of η is equivalent to the equation μ(Φ, Ā) = 0, and Φ has the same simple poles as H_{a} (2.18). For simplicity, we call η the Higgs field. The bundle E equipped with the one-form η is called the Higgs bundle.

2.4 Modified C̆ech description of the moduli space.

We modify the C̆ech description of the moduli space of GL(N, ℂ)-vector bundles in the following way. Consider a formal (or rather small) disk D embedded into Σ in such way that its center maps to the point w.

Consider first the case of G = PGL(N, ℂ)-bundles. The moduli space M_{n}^{d} is the quotient of the space G_{D^{∗}} of G-valued functions g on the punctured disk D^{∗} by the right action of the group G_{out} of G-valued holomorphic functions on the complement to w and by the left action of the group G_{int} of G-valued holomorphic functions on the disk:

M_{n}^{d} = G_{int}\G_{D^{∗}}/G_{out}, g → h_{int}gh_{out} .

We assume that these transformations preserve the quasi-parabolic structure of the vector bundle E.

Now consider GL(N, ℂ)-bundles. The group GL(N, ℂ) is not semi-simple. One has an action of the Jacobian Jac(Σ) on the moduli space of vector bundles by the tensor multiplication, and the quotient is equal to the space of PGL(N, ℂ)-bundles. This follows from the exact sequence

0 → O^{∗} → GL(N, ℂ)(O) → PGL(N, ℂ)(O) → 0 .

Hence locally the moduli space of vector bundles is product of the Jacobian of the curve and moduli space of PGL(N, ℂ)-bundles. We associate to the pair (g, L) the bundle which is equal to ℂ^{N} ⊗ L on the complement of a point, and the transition function on the punctured disk is g.

Assume for simplicity that there is only one marked point and it coincides with the center of D^{∗}. Let z be the local coordinate on D^{∗}. Then the gauge group G_{D^{∗}} can be identify with the loop group L(GL(N, ℂ)). A parabolic subgroup of L(GL(N, ℂ)) has the form

G_{int} ∼ P · exp L^{+}(gl(N, ℂ)), L^{+}(gl(N, ℂ)) = ∑_{j>0} 𝔤_{j}z^{j}, 𝔤_{j} ∈ gl(N, ℂ) ,

where P is a parabolic subgroup in GL(N, ℂ) . The quotient G_{int}\G_{D^{∗}} is the infinite-dimensional flag variety, corresponding to the finite-dimensional flag E(s) (see (2.5))

⋯ ⊃ F_{s}z^{−1} + ∑_{j≥0} Ez^{j} ⊃ ∑_{j≥0} Ez^{j} ⊃ F_{1} + ∑_{j>0} Ez^{j} ⊃ ⋯  (2.24)

⊃ F_{s} + ∑_{j>0} Ez^{j} ⊃ ∑_{j>1} Ez^{j} ⊃ · · · .

The moduli space T^{∗}M_{n}^{d} of the GL(N, ℂ) Higgs bundles can be locally identified with the Hamiltonian quotient

G_{in}\\T^{∗}G_{D^{∗}} × T^{∗}Jac(Σ)//G_{out} .

The cotangent bundle of T^{∗}G_{D^{∗}} is identified with the space of pairs (g, η), where η is a Lie^{∗}(G)- valued one-form. The canonical one-form is equal to res_{w}(tr(ηDgg^{−1})). The second component T^{∗}Jac(Σ) is the pair (t, L), where L is a point of Jac(Σ) and t is the corresponding co-vector. The canonical one-form is ⟨t, DL⟩_{Jac} and the brackets denote the pairing between vectors and co-vectors on the Jacobian.

The group G_{out} acts as (g, η) → (gh_{out}, η). The corresponding momentum constraint can be reformulated as the following condition: gηg^{−1} is the restriction of some Lie^{∗}(G)-valued form on the complement to w. The group G_{int} acts as (g, η) → (h_{int}g, h_{int}ηh_{int}^{−1}). The momentum constraint means that η is holomorphic in U_{w} if w is a generic point, or it has the first order pole if w is a marked point.

3 Symplectic Hecke correspondence

In this section we consider only GL(N, ℂ)-bundles.

3.1 Hecke correspondence.

Let E and Ẽ be two quasi-parabolic bundles over Σ of the same rank. Assume that there is a map Ξ^{+} : E → Ẽ such that it is an isomorphism on the complement to w and it has one-dimensional cokernel at w :

0 → E →^{Ξ^{+}} Ẽ → ℂ|_{w} → 0 .

Definition 3.1 The upper Hecke correspondence (HC) at the point w ∈ Σ of order 1 is an auto-correspondence ℌ_{w}^{+} on M related to the map Ξ^{+}.

The correspondence ℌ_{w}^{+} has components placed only at M^{(d)} × M^{(d+1)}. On the complement to the point w consider the map

E ←^{Ξ^{−}} Ẽ ,

such that Ξ^{−}Ξ^{+} =Id. It defines the lower HC ℌ_{w}^{−} at the point w.

Let w = w_{α} be a marked point and the corresponding flag E_{α} (2.5) has a one-dimensional subspace F_{s_{α}}. In this case the HC ℌ^{±} can be chosen in the following way. Let

Ẽ_{α} = F̃_{1}(α) ⊃ ⋯ ⊃ F̃_{s_{α}}(α) ⊃ F̃_{s_{α}+1}(α) = 0

be the image of the flag variety E_{α} under the action of Ξ_{α}^{+}. In agreement with Definition 3.1 the map F_{s_{α}}(α) → F̃_{1}(α) ∼ Ẽ_{α} has a one-dimensional cokernel. Moreover, we assume that Ξ_{α}^{+} acts as

Ξ_{α}^{+} : F_{1}(α) → F̃_{2}(α), …, F_{s_{α}}(α) → F̃_{1}(α).

Then Ξ_{α}^{+} can be gauge transformed to the canonical form

Ξ_{α}^{+} = ( \begin{matrix}0 & Id_{N−1} \\ z_{α} & 0 \end{matrix} ) .  (3.1)

It is just the Coxeter transformation in the loop algebra L(gl(N, ℂ)), that has been defined on the punctured disk D_{α}^{∗} ⊂ U_{α} in Subsection 2.4. The Coxeter transformation provides the map E^{d} → Ẽ^{d+1}. On the infinite-dimensional flag variety chain (2.24) it acts as the left shift operator. In a similar way the lower HC can be transformed to the form

Ξ_{α}^{+} = ( \begin{matrix}0 & z_{α}^{−1} \\ Id_{N−1} & 0 \end{matrix} ) .  (3.2)

In this case E^{d+1} → Ẽ^{d} and the flag variety is shifted to the left.

3.2 Symplectic Hecke correspondence

. Consider two Higgs bundles (E, η) and (Ẽ, η̃), where E, Ẽ are in the HC ℌ_{a} at w_{a} ∈ Σ.

Definition 3.2 The upper symplectic Hecke correspondence (SHC) is the map of the Higgs bundles

𝔖_{a}^{+} : (E, η) → (Ẽ, η̃) ,

where E and Ẽ are in Hecke correspondence ℌ_{w_{a}}^{+}.

The lower SHC 𝔖_{a}^{−} is defined in a similar way.

Let w_{α} be a marked point. The Higgs field η has the first order poles at w_{α} and the residue p_{α}^{(0)} of η defines an orbit O_{α}. Then the correctness of the definition in this case follows from the following

Lemma 3.1 The gauge transforms Ξ_{α}^{±} corresponding to ℌ_{α}^{±}

• do not change singularity of the Higgs field at w_{α};

• are symplectic.

Proof. The choice of p_{α}^{(0)} (2.16) is consistent with the canonical forms (3.1),(3.2) of Ξ_{α}^{±} and their action does not change the order of the pole. The action is symplectic since Ξ_{α}^{±} depends only on p_{α}^{(0)}. □

3.3 SHC and skew-conormal bundles.

For any smooth correspondence Z between equi-dimensional varieties X an Y we define a skewconormal bundle SN^{∗}Z of Z as follows. Let

ν = (ν_{X}, ν_{Y}) ∈ T_{z}^{∗}(X × Y) = T_{x}^{∗}X ⊕ T_{y}^{∗}Y

be a co-vector attached to a point z = (x, y) ∈ Z ⊂ X × Y. It belongs to the fiber SN _{z}^{∗}Z of the skew-conormal bundle SN^{∗}Z at the point z iff for any vector v = (v_{X}, v_{Y}) tangent to Z

ν_{X}(v_{X}) = ν_{Y} (v_{Y}) .

Note that for the conormal bundle one has the opposite sign: ν_{X}(v_{X}) = −ν_{Y} (v_{Y}).

The total space of the skew-conormal bundle is a Lagrangian subvariety of the total space of the cotangent bundle T^{∗}(X × Y) with respect to the symplectic form ω_{X} − ω_{Y}, where ω denotes the canonical symplectic form on the cotangent bundle. So, the skew-conormal bundle of a correspondence is rather close to the graph of a symplectic map between cotangent bundles.

Proposition 3.1 The graph of the SHC 𝔖_{w} is isomorphic to the skew-conormal bundle SN^{∗}ℌ_{w} of the usual Hecke correspondence ℌ_{w}.

Proof. As it was explained in Subsection 2.4, a GL(N, ℂ)-bundle E is determined by the pair (g, L) in a neighborhood of a point w ∈ Σ. An upper HC of E corresponds to (g̃, L), where g̃ = Ξg and

¯∂Ξ = 0 in U_{w}, ord_{w}(det(Ξ)) = 1 .  (3.3)

Therefore, the skew-conormal bundle SN^{∗}ℌ_{w} of the HC SN^{∗}ℌ_{w} can be described by the data

(g, g̃; η, η̃; t, t̃, L), g̃ = Ξg ,

where Ξ satisfies (3.3), and

⟨t, DL⟩_{Jac} = ⟨t̃, DL⟩_{Jac}  (3.4)
res_{w}(Tr(η̃Dg̃g̃^{−1})) = res_{w}(Tr(ηDgg^{−1}))  (3.5)

for any variations of g and g̃, that preserve properties of Ξ = g^{−1}g̃. The first condition (3.4) means that t = t̃.

The condition (3.5) can be rewritten as

res_{w} (tr(η̃DΞΞ^{−1} + (Ξ^{−1}η̃Ξ − η)g^{−1}D g)) = 0 .

Since variations of g and Ξ are independent, both terms in the last expression must vanish separately:

res_{w}(tr(η̃DΞΞ^{−1})) = 0 ,  (3.6)
res_{w} (tr(Ξ^{−1}η̃Ξ − η)g^{−1}D g)) = 0 .  (3.7)

Consider first the case when w is not a marked point. Then we will demonstrate that (3.6) means that η′ = Ξ^{−1}η̃Ξ is holomorphic in w. Consider the value of Ξ at zero, this matrix has rank N − 1. Denote by K its kernel and by I its image. An essential variation of Ξ corresponds to the variations of its image, so it is a map I → ℂ^{N}/I. This variation corresponds to the right action: D Ξ = Ξϵ. The singular part Ξ_{sing}^{−1} of Ξ^{−1} at w is an operator of rank 1. Its kernel equals I and its image equals K, so the singular part η′_{sing} of η′ is a map from ℂ^{N}/Ker(Ξ_{sing}^{−1}) = ℂ^{N}/I to Im(Ξ_{sing}^{−1}) = I. The first condition can be rewritten as:

0 = res_{w}(tr(η̃DΞΞ^{−1})) = res_{w}(tr(η′Ξ^{−1}DΞ)) = tr(η′_{sing}ϵ)

for any ϵ ∈ Hom(I, ℂ^{r}/I). The space Hom(I, ℂ^{r}/I) is dual to Hom(ℂ^{r}/I, I), so η′_{sing} vanishes and η′ is holomorphic.

Note that η′ determines some Higgs field for g. Indeed, it is holomorphic in U_{w} and g^{−1}η′g = g̃^{−1}η̃g̃ is the restriction of some one-form on the complement to w. As the canonical one-form tr(ηDgg^{−1}) is non-degenerate on T^{∗}M_{n}^{d}, from the second condition (3.7) we conclude that η′ = η.

If w is a marked point then Ξ is fixed and it maps the Higgs field η into the Higgs field η̃ (Lemma 3.1). There is no variation of Ξ and we immediately have that again η′ = η. □

3.4 Bäcklund transformation.

The gauge transformations Ξ_{α}^{±} related to the SHC 𝔖_{α}^{±} depend on the marked point w_{α} and the choice of the eigenvectors e^{±} of p_{α}^{(0)}. Then the gauge transformations define the maps of the Hitchin systems

𝔖_{α}^{+} ∼ ξ^{e_{j}^{+},α} : T^{∗}M^{(d)}(Σ_{n}, G) → T^{∗}M^{(d+1)}(Σ_{n}, G) ,  (3.8)
𝔖_{α}^{−} ∼ ξ_{e_{j}^{−},β} : T^{∗}M^{(d)}(Σ_{n}, G) → T^{∗}M^{(d−1)}(Σ_{n}, G) .  (3.9)

Consider consecutive upper and lower modifications

ξ_{e_{j_{2}},α_{2}}^{e_{j_{1}},α_{1}} = ξ^{e_{j_{1}},α_{1}} · ξ_{e_{j_{2}},α_{2}} .  (3.10)

Since deg(E) does not change it is a symplectic transform T^{∗}M_{n}^{d}(Σ, E). In this way ξ_{e_{j_{2}},α_{2}}^{e_{j_{1}},α_{1}} maps solutions of the Hitchin hierarchy into solutions.

Corollary 3.1 The map (3.10) is the Bäcklund transformation, parameterized by a pair of marked points (w_{α_{1}}, w_{α_{2}}) and a pair of the eigenvectors (e_{j_{1}}, e_{j_{2}}).

We can generalize (3.10) as

ξ_{e_{i_{1}},α_{i_{1}};...;e_{is},α_{is}}^{e_{j_{1}},α_{j_{1}};...;e_{js},α_{js}} = ξ^{e_{j_{1}},α_{j_{1}}} · ξ_{e_{i_{1}},α_{i_{1}}} · · · .

For the generic orbits we have N parameters at a marked point. Because the Bäcklund transformation is a canonical one we can consider a discrete Hamiltonian system defined on the phase space T^{∗}M_{n}^{d}(Σ, E). They pairwise commute and in terms of the angle variables generate a lattice in the Liouville torus [21, 22]. In our case the dimension of the Liouville torus is equal to dim M_{n}^{d} (2.8), but the lattice we have constructed has in general a smaller dimension.

Note that when Σ_{n} is an elliptic curve, the Hitchin systems corresponding to d = kN and d = 0 (d =deg(V)) are equivalent. Hence, in this case one can construct some Bäcklund transformations by applying the upper SHC N times.

4 Elliptic CM system - elliptic SL(N, ℂ)-rotator correspondence

4.1 Elliptic CM system

. The elliptic CM system was first introduced in the quantum version [23]. It is defined on the phase space

R^{CM} = (v = (v_{1}, …, v_{N}), u = (u_{1}, …, u_{N}), ∑_{j} v_{j} = 0, ∑_{j} u_{j} = 0) ,  (4.1)

with the canonical symplectic form

ω^{CM} = (Dv ∧ Du) .

The second order with respect to the momenta v Hamiltonian is

H_{2}^{CM} = \frac{1}{2} ∑_{j=1}^{N} v_{j}^{2} + ν^{2} ∑_{j>k} ℘(u_{j} − u_{k}; τ) .

It was established in [7, 24] that the elliptic CM system can be derived in the Hitchin approach. The Lax operator L^{CM} is the reduced Higgs field η over the elliptic curve

E_{τ} = ℂ/ℤ + τℤ

with a marked point z = 0. In this way the phase space R^{CM} is the space of pairs

(SL_{N}-bundle V over E_{τ}, the Higgs field L^{CM} on this bundle (4.4)).

The bundle is determined by the transition functions (the multipliers)

Id_{N} : z → z + 1 ,  (4.2)
e(u) = diag(e(u_{1}), …, e(u_{N})) : z → z + τ ,

where e is defined in (A.1). The Lax operator L^{CM}(z) is the quasiperiodic one-form

L^{CM}(z + 1) = L^{CM}(z), L^{CM}(z + τ) = e(−u)L^{CM}(z)e(u) .  (4.3)

It is the N-th order matrix

L^{CM} = P + X, where P = diag(v_{1}, …, v_{N}), X_{jk} = νϕ(u_{j} − u_{k}, z) ,  (4.4)

and ϕ is defined as (B.5). It has the first order pole at the marked point with the residue in the minimal orbit

Res_{z=0}(L^{CM}(z)) = L_{−1}^{CM} = ν ( \begin{matrix}0 & 1 & ⋯ & 1 \\ 1 & 0 & ⋯ & 1 \\ ⋮ & ⋮ & ... & ⋮ \\ 1 & 1 & ⋯ & 0 \end{matrix} ) ,  (4.5)

so the column-vector e_{1} = (1, 1, ⋯, 1) is an eigen-vector e_{1}

L_{−1}^{CM}e_{1} = (N − 1)νe_{1}.  (4.6)

There is also an (N − 1)-dimensional eigen-subspace T_{N−1} corresponding to the degenerate eigen-value −ν

L_{−1}^{CM}e_{a} = −νe_{a}, e_{a} = (a_{1}, …, a_{N}), (∑_{n} a_{n} = 0) .  (4.7)

The M^{CM} operator corresponding to H_{2}^{CM} has the form

M^{CM} = −D + Y, where D = diag(Z_{1}, …, Z_{N}), Y_{jk} = y(u_{j} − u_{k}, z) ,  (4.8)

Z_{j} = ∑_{k≠j} ℘(u_{j} − u_{k}), y(u, z) = \frac{∂ϕ(u, z)}{∂u}, .

4.2 The elliptic SL(N, ℂ)-rotator.

The elliptic SL(N, ℂ)-rotator is an example of the Euler-Arnold top [16]. It is defined on a coadjoint orbit of SL(N, ℂ):

R^{rot} = {S ∈ sl(N, ℂ), S = g^{−1}S^{(0)}g} .  (4.9)

The phase space R^{rot} is equipped with the Kirillov-Kostant symplectic form

ω^{rot} = tr(S^{(0)}Dgg^{−1}Dgg^{−1}) .

The Hamiltonian is defined as

H^{rot} = −\frac{1}{2}tr(SJ(S)) ,  (4.10)

where J is a linear operator on Lie(SL(N, ℂ)). The inverse operator is called the inertia tensor. The equation of motion takes the form

∂_{t}S = [J(S), S].  (4.11)

We consider here a special form J, that provides the integrability of the system. Let

J(S) = J · S = ∑_{mn} J_{mn}S_{mn} ,

where J is a N-th order matrix

J = \{J_{mn}\} = \{℘ [ \begin{matrix}m \\ n \end{matrix} ]\} , (m, n = 1, …, N), ℘ [ \begin{matrix}m \\ n \end{matrix} ] = ℘(\frac{m + nτ}{N}; τ) .  (4.12)

We write down (4.11) in the basis of the sin-algebra S = S_{mn}E_{mn} (see (A.4))

∂_{t}S_{mn} = \frac{N}{π} ∑_{k,l} S_{k,l}S_{m−k,n−l}℘ [ \begin{matrix}k \\ l \end{matrix} ] sin \frac{π}{N} (kn − ml) .  (4.13)

The elliptic rotator is a Hitchin system [7]. We give a proof of this statement.

Lemma 4.1 The elliptic SL(N, ℂ)-rotator is a Hitchin system corresponding to the GL(N, ℂ)- bundle E (deg(E)=1) over the elliptic curve E_{τ} with the marked point z = 0.

Proof.

It can be proved that (4.13) is equivalent to the Lax equation. The Lax matrices in the basis of the sin-algebra take the form

L^{rot} = ∑_{m,n} S_{mn}φ [ \begin{matrix}m \\ n \end{matrix} ] (z)E_{mn}, φ [ \begin{matrix}m \\ n \end{matrix} ] (z) = e(−\frac{nz}{N})ϕ(−\frac{m + nτ}{N}; z) ,  (4.14)

M^{rot} = ∑_{m,n} S_{mn}f [ \begin{matrix}m \\ n \end{matrix} ] (z)E_{mn}, f [ \begin{matrix}m \\ n \end{matrix} ] (z) = e(−\frac{nz}{N})∂_{u}ϕ(u; z)|_{u=− \frac{m+nτ}{N}} .  (4.15)

They lead to the Lax equation for the matrix elements

∂_{t}S_{mn}φ [ \begin{matrix}m \\ n \end{matrix} ] (z) = \sqrt{−1} ∑_{k,l} S_{m−k,n−l}S_{kl}φ [ \begin{matrix}m − k \\ n − l \end{matrix} ] (z)f [ \begin{matrix}k \\ l \end{matrix} ] (z) sin \frac{π}{N} (nk − ml) .

Using the Calogero functional equation (B.27) we rewrite it in the form (4.13). Since

\frac{1}{N} tr(L^{rot})^{2} = −2H^{rot} + trS^{2}℘(z) ,

H^{rot} is the Hitchin quadratic integral.

The Lax operator satisfies the Hitchin equation

¯∂L^{rot} = 0, ResL^{rot}|_{z=0} = 2π\sqrt{−1}S

and is quasiperiodic

L^{rot}(z + 1) = Q(τ)L^{rot}(z)Q^{−1}(τ),  (4.16)
L^{rot}(z + τ) = Λ̃(z, τ)L^{rot}(z)(Λ̃(z, τ))^{−1},  (4.17)

where Λ̃(z, τ) = −e(\frac{−z− \frac{1}{2} τ}{N})Λ and the matrices Q and Λ are defined in (A.2),(A.3). The transition functions

Q(τ) : z → z + 1,  (4.18)
Λ̃(z, τ) : z → z + τ  (4.19)

define GL(N, ℂ)-bundle over E_{τ} with deg(V) = 1. Thus the phase space of the SL_{N}-rotator is the space of the Higgs fields L^{rot} on the bundle determined by multipliers Q, Λ̃ with the first order singularities at zero. □

4.3 A map R^{CM} → R^{rot}.

We construct a map from the phase space of the elliptic CM system R^{CM} into the phase space of the SL_{N}-rotator R^{rot}. The SL_{N}-rotator is living on the most degenerate orbit corresponding to L_{−1}^{CM} (4.5). The phase space of CM systems with spins is mapped into the general coadjoint orbits. This generalization is straightforward. In this way, for N = 2 we describe the upper horizontal arrow in Fig.(1.1).

The map is defined as the conjugation of L^{CM} by some matrix Ξ(z):

L^{rot} = Ξ × L^{CM} × Ξ^{−1}.  (4.20)

It follows from comparing (4.3) with (4.16) and (4.17) that Ξ must intertwine the multipliers of bundles:

Ξ(z + 1, τ) = Q × Ξ(z, τ) ,  (4.21)
Ξ(z + τ, τ) = Λ̃(z, τ) × Ξ(z, τ) × diag(e(u_{j})) .  (4.22)

The matrix Ξ(z) degenerates at z = 0, and the column-vector (1, ⋯, 1), in accordance with Lemma 3.1, should belong to the kernel of Ξ(0). In this case, Ξ × L^{CM} × Ξ^{−1} has a first order pole at z = 0.

Consider the following (N × N)- matrix Ξ̃(z, u_{1}, …, u_{N}; τ) :

Ξ̃_{ij}(z, u_{1}, …, u_{N}; τ) = θ[ \frac{i}{N} − \frac{1}{2}_{\frac{N}{2}} ](z − Nu_{j}, Nτ),  (4.23)

where θ[ \begin{matrix}a \\ b \end{matrix} ](z, τ) is the theta function with a characteristic (B.31). Sometimes we omit

nonessential arguments of Ξ for brevity.

Lemma 4.2 The matrix Ξ̃ is transformed under the translations z → z + 1, z → z + τ and u_{j} → u_{j} + 1, u_{j} → u_{j} + τ as :

Ξ̃(z + 1, τ) = −Q × Ξ̃(z, τ) ,  (4.24)
Ξ̃(z + τ, τ) = Λ̃(z, τ) × Ξ̃(z, τ) × diag(e(u_{j})),  (4.25)

Λ̃(z, τ) = −e (−\frac{τ}{2N} − \frac{z}{N} ) Λ ;

Ξ̃(u_{j} + 1,; τ) = Ξ̃(u_{j}; τ) × diag(1, ⋯, (−1)^{N}, ⋯, 1) ,  (4.26)
Ξ̃(u_{j} + τ; τ) = Ξ̃(u_{j}; τ) × diag(1, ⋯, (−1)^{N}e (−\frac{Nτ}{2} + z − Nu_{j}) ⋯, 1) .  (4.27)

Proof. The statement of the lemma follows from the properties of the theta functions with characteristics (B.33)-(B.35). □

Now we assume that ∑ u_{j} = 0, so u_{N} is no more an independent variable, but it is equal to − ∑_{j=1}^{N−1} u_{j}.

Lemma 4.3 The determinant of the matrix Ξ̃ has the form

detΞ̃ = C(τ)ϑ(z, τ) ∏_{j<k} ϑ(u_{j} − u_{k}, τ) ,  (4.28)

for some non-zero function C(τ), where ϑ is defined in (B.1).

Proof. It is easy to check that the lhs and the product of the theta function on the rhs are quasiperiodic with respect to translations z → z + 1, z → z + τ and u_{j} → u_{j} + 1, u_{j} → u_{j} + τ, j < N with multiplicators −1, −e(−\frac{τ}{2} − z), 1 and e(−Nτ − u_{j} + u_{N}) respectively. So, their quotient is an elliptic function in the variables z, u_{j}, (j = 1, …, N − 1. As a function in the variable z, this quotient may have a unique pole of first order at zero, so it does not depend on the variable z.

The determinant is an antisymmetric function of the columns of a matrix, so the lhs vanishes if u_{j} = u_{k}; hence the quotient has no poles as a function in the u_{j} and does not depend on this variable. Consequently, the quotient C(τ) depends only on τ.

One can calculate the leading term of C(τ) when τ tends to \sqrt{−1}∞: the leading term of Ξ̃_{ij} equals to

\{ \begin{matrix}e ((\frac{i}{N} − \frac{1}{2})^{2} \frac{Nτ}{2} + (\frac{i}{N} − \frac{1}{2}) (z + Nu_{j} + \frac{N}{2} )) , & i ≠ N , \\ e ((\frac{1}{2})^{2} \frac{Nτ}{2} + (\frac{1}{2}) (z + Nu_{j} + \frac{N}{2} )) × (1 + (−1)^{N}e(−z − Nu_{j})) , & i = N . \end{matrix}

So the leading term of the determinant equals

e (\frac{(N^{2} + 2)}{24}τ + \frac{1}{2} (z + \frac{N}{2} ) − \frac{N}{2} ∑ u_{j})

×det (e(iu_{j})) + (−1)^{N}e(−z)det(e((i − Nδ_{i,N})u_{j})) .

Both determinants are Vandermonde determinants, and we get as an answer (by taking into account the condition ∑ u_{j} = 0):

(−\sqrt{−1})^{N}e (\frac{(N^{2} + 2)}{24}τ − \frac{1}{2}z) (1 − e(z)) ∏_{j>k}(e(u_{j}) − e(u_{k})).

On the other hand, the leading term of the theta function ϑ(x, τ) is equal to

−\sqrt{−1}e (\frac{1}{8}τ − \frac{1}{2}x) (1 − e(x)).

Therefore, the leading term of the product of the theta functions on the rhs is equal to

(−\sqrt{−1})^{N(N−1)/2+1}e (\frac{N(N − 1)/2 + 1}{8}τ − \frac{1}{2}z) (1 − e(z)) ∏_{j>k}(e(u_{j}) − e(u_{k})).

Hence, its quotient has the leading term

(\sqrt{−1})^{\frac{(N−2)(N−1)}{2}} e (−\frac{(N − 2)(N − 1)}{48}τ) .

Thus, the leading term of C(τ) not being zero, and C(τ) does not vanish.

For general τ, z ≠ mτ + n and u_{j} − u_{k} ≠ mτ + n (m, n ∈ ℤ) the matrix Ξ̃ is invertible, but for z = 0 it degenerates. □

Lemma 4.4 The kernel of Ξ̃ at z = 0 is generated by the following column-vector :

{(−1)^{l} ∏_{j<k;j,k≠l} ϑ(u_{k} − u_{j}, τ)} , l = 1, 2, ⋯, N.

Proof. We must prove that for any i the following expression

∑_{l=1}^{N}(−1)^{l}θ[ \frac{i}{N} − \frac{1}{2}_{\frac{N}{2}} ](z − Nu_{l}, Nτ) ∏_{j<k;j,k≠l} ϑ(u_{k} − u_{j}, τ)  (4.29)

vanishes. First, the symmetric group S_{N} acts on u by permutation of u_{1}, …, u_{N} and (4.29) is antisymmetric with respect to the S_{N} action. Hence it vanishes on the hyperplanes u_{i} = u_{j}. As a function on u_{1}, (4.29) has 2N zeroes: N − 2 zeroes u_{1} = u_{k}, k ≠ 1, N, N − 2 zeroes u_{N} = u_{k}, k ≠ 1, N and four zeroes u_{1} = u_{N} (the last equation is 2u_{1} = − ∑_{j=2}^{N−1} u_{j}).

Second, (4.29) is quasiperiodic with respect to the shifts u_{1} → u_{1} + 1, u_{1} → u_{1} + τ with multiplicators 1 and e (−(N − 1)τ − (N − 1)(u_{1} − u_{N})). Any quasiperiodic function with such multiplicators is either zero or has 2N − 2 zeroes. Since our expression vanishes in 2N points it vanishes identically. □

It follows from the previous lemmas that the matrix

Ξ(z) = Ξ̃(z) × diag ((−1)^{l} ∏_{j<k;j,k≠l} ϑ(u_{k} − u_{j}, τ))  (4.30)

is the singular gauge transform from Lemma 2.1 that maps L^{CM} to L^{rot}. This transformation leads to the symplectic map

R^{CM} → R^{rot}, (v, u) ↦ S.  (4.31)

Consider in detail the case N = 2. Let

S = S_{a}σ_{a},

where σ_{a} denote the sigma matrices subject to the commutation relations

[σ_{a}, σ_{b}] = 2\sqrt{−1}ε_{abc}σ_{c} .

Then the transformation has the form

\{ \begin{matrix}S_{1} = −v \frac{θ_{10}(0)}{ϑ′(0)} \frac{θ_{10}(2u)}{ϑ(2u)} − ν \frac{θ_{10}^{2}(0)}{θ_{00}(0)θ_{01}(0)} \frac{θ_{00}(2u)θ_{01}(2u)}{ϑ^{2}(2u)} , \\ S_{2} = −v \frac{θ_{00}(0)}{\sqrt{−1}ϑ′(0)} \frac{θ_{00}(2u)}{ϑ(2u)} − ν \frac{θ_{00}^{2}(0)}{\sqrt{−1}θ_{10}(0)θ_{01}(0)} \frac{θ_{10}(2u)θ_{01}(2u)}{ϑ^{2}(2u)} , \\ S_{3} = −v \frac{θ_{01}(0)}{ϑ′(0)} \frac{θ_{01}(2u)}{ϑ(2u)} − ν \frac{θ_{01}^{2}(0)}{θ_{00}(0)θ_{10}(0)} \frac{θ_{00}(2u)θ_{10}(2u)}{ϑ^{2}(2u)} . \end{matrix}  (4.32)

Formulae of this kind were obtained in [12].

4.4 Bäcklund transformations in the CM systems.

We now use the map (4.31) to construct the Bäcklund transformation in the CM systems

ξ : (v, u) → (ṽ, ũ) .

Let the Lax matrix depends on the new coordinates and momenta L = L(ṽ, ũ). Consider the upper modification Ξ(z) (4.30). To construct the Bäcklund transformation ξ, we map (v, u) and (ṽ, ũ) to the same point S ∈ R^{rot}:

L^{rot}(S)

↗ Ξ(z, u)  ↖ Ξ(z, ũ)
L^{CM}(v, u)  ⟶^{ξ}  L^{CM}(ṽ, ũ)

In this way we reproduce implicitly the general formula (3.10) for the Bäcklund transformations. This transformation defines an integrable discrete time dynamics of a CM system. One example of this dicretization was proposed in [25]. It can be supposed to correspond to ξ.

Another way to construct new solutions from (v, u) is to act by N consecutive upper modifications

Ξ(N) = D_{N}Ξ_{N} ⋯ Ξ_{j} ⋯ Ξ_{2} · Ξ .  (4.33)

Here the matrices Ξ_{j}, j = 2, …, N, satisfy the quasi-periodicity conditions

Ξ_{j}(z + τ) = Λ̃^{j}Ξ_{j}(z)Λ̃^{1−j} ,

and D_{N} is an arbitrary diagonal matrix. We come back to the N-dimensional moduli space M^{(N)} (see (2.5)) and to the map

L^{CM}(v, u) ⟶^{Ξ(N)} L^{CM}(ṽ, ũ) .

If we break the chain (4.33) on a step k < L, then we obtain the map

L^{CM} → L^{rot,k} ,

where L^{rot,k} is the Lax operator for the elliptic rotator related to the holomorphic bundle of degree k. It satisfies the quasi-periodicity condition (4.16) and

L^{rot,k}(z + τ) = Λ̃^{j}L^{rot,k}(z)Λ̃^{−j}

instead of (4.17).

5 Hitchin systems of infinite rank

Here we generalize the derivation of finite-dimensional integrable systems in the form (2.20)- (2.23) on two-dimensional integrable field theories.

5.1 Holomorphic L̂(GL(N, ℂ))-bundles.

Let L(gl(N, ℂ)) be the loop algebra of C^{∞}-maps L(gl(N, ℂ)) : S^{1} → gl(N, ℂ), and L̂(gl(N, ℂ)) be its central extension with the multiplication

(g, c) × (g′, c′) = (gg′, cc′ exp C(g, g′)) ,  (5.1)

where exp C(g, g′) is a 2-cocycle of L̂(GL(N, ℂ)) providing the associativity of the multiplication.

Consider a holomorphic bundle V of an infinite rank over a Riemann curve Σ_{n} with n marked points. The bundle is defined by the transition functions from L̂(GL(N, ℂ)). Its fibers are isomorphic to the Lie algebra L̂(gl(N, ℂ)). The holomorphic structure on V is defined by the operator

d′′ : Ω_{C^{∞}}^{(0)}(Σ_{n}, End V) → Ω_{C^{∞}}^{(0,1)}(Σ_{n}, End V) .

It has two components d′′ = dĀ^{′′} + dλ^{′′}. The first component is

dĀ^{′′} : Ω_{C^{∞}}^{(0)}(Σ_{n}, L(gl(N, ℂ))) → Ω_{C^{∞}}^{(0,1)}(Σ_{n}, L(gl(N, ℂ))) .

Locally

dĀ^{′′} = ¯∂ + Ā, ¯∂ = ∂z̄, Ā = Ā(x, z, z̄), x ∈ S^{1}.

The second component is defined by the connection dλ^{′′} on a trivial linear bundle L on Σ_{n}, given by

dλ^{′′} = ¯∂ + λ .

The field λ is a map from Σ_{n} to the central element of the Lie algebra L̂(gl(N, ℂ)). A local section σ of V is holomorphic if d′′σ = 0. The sections allow to define the transition functions. We assume that Ā and λ are smooth at the marked points.

In addition we define n copies of the central extended loop groups located at the marked points

L̂G_{α} = (g_{α}(x), c_{α}), G_{α} = GL(N, ℂ), (α = 1, …, n), x ∈ S^{1},

with the multiplication (5.1).

Thus, we have the set R of fields playing the role of the ”coordinate space”:

R = {Ā, λ, (g_{1}, c_{1}), …, (g_{n}, c_{n})} .  (5.2)

5.2 Gauge symmetries.

Let G be the group of automorphisms of R (the gauge group).

G = C^{∞}Map(Σ_{n} → L̂(GL(N, ℂ))) = {f(z, z̄, x), s(z, z̄)} ,

where f(z, z̄, x) takes values in GL(N, ℂ), and s(z, z̄) is the map to the central element of L̂(GL(N, ℂ))). The multiplication is pointwise with respect to Σ_{n}

(f_{1}, s_{1}) × (f_{2}, s_{2}) = (f_{1}f_{2}, s_{1}s_{2} exp C(f_{1}, f_{2})) ,

where exp C(f_{1}, f_{2}) is a map from Σ_{n} to the 2-cocycle of L̂(GL(N, ℂ)).

Let (f_{α} = f_{α}(x), s_{α}) be the value of the gauge fields at the marked point w_{α}. The action of G on R takes the following form

Ā → f^{−1} ¯∂f + f^{−1}Āf,  (5.3)
λ → λ + s^{−1} ¯∂s + ∮ tr(Āf^{−1}∂_{x}f)dx,  (5.4)
c_{α} → c_{α}s_{α}, g_{α} → g_{α}f_{α}.  (5.5)

The quotient space N = R/G is the moduli space of infinite rank holomorphic bundles over Riemann curves with marked points.

5.3 Phase space.

The cotangent space to R has the following structure. Consider the analog of the Higgs field Φ ∈ Ω_{C^{∞}}^{(1,0)}(Σ_{n}, (End V)^{∗}). It is a one-form Φ on Σ_{n} taking values in the Lie coalgebra L^{∗}(gl(N, ℂ)). Let k be a scalar one-form on Σ_{n}, k ∈ Ω_{C^{∞}}^{(1,0)}(Σ_{n}). It is dual to the field λ. At the marked points we have the Lie coalgebras Lie^{∗}(G_{α}) ∼ L(gl(N, ℂ)) along with the central elements r_{α}, dual to c_{α}. Thus the cotangent bundle T^{∗}R contains the fields

T^{∗}R = {(Ā, Φ), (λ, k); (p_{1}, r_{1}), …, (p_{n}, r_{n})} .  (5.6)

There is a canonical symplectic structure on T^{∗}R. For F ∈ Ω_{C^{∞}}^{(1,0)}(Σ_{n}, (End V)^{∗}) and G ∈ Ω_{C^{∞}}^{(0,1)}(Σ_{n}, L(gl(N, ℂ))) define the pairing

< F |G >= ∫_{Σ_{n}} ∮ tr(F G)dx.

Then

ω =< DΦ ∧ DĀ > + ∫_{Σ_{n}} Dk ∧ Dλ + ∑_{α=1}^{n} ω_{α} ,  (5.7)

where ω_{α} is a canonical form on T^{∗}L̂(G_{α}). It is constructed in the canonical way by means of the Maurer-Cartan form on L̂(G_{α}) = {g_{α}, c_{α}}. The result is

ω_{α} = ∮_{S_{α}^{1}} tr(D(p_{α}g_{α}^{−1})Dg_{α}) + D(r_{α}c_{α}^{−1})Dc_{α} + \frac{r_{α}}{2} ∮_{S_{α}^{1}} tr (g_{α}^{−1}Dg_{α}∂_{x}(g_{α}^{−1}Dg_{α})) .  (5.8)

5.4 Symplectic reduction.

Now consider the lift of G to the global canonical transformations of T^{∗}R. In addition to (5.3),(5.4),(5.5) we have the following action of G

Φ → f^{−1}k∂_{x}f + f^{−1}Φf, k → k,  (5.9)
p_{α} → f_{α}^{−1}p_{α}f_{α} + r_{α}f_{α}^{−1}∂_{x}f_{α}, r_{α} → r_{α} .  (5.10)

This transformation leads to the moment map from the phase space to the Lie coalgebra of the gauge group μ : T^{∗}R → Lie^{∗}(G). It takes the form

μ = (¯∂Φ − kp_{x}Ā + [Ā, Φ] + ∑_{α=1}^{n} p_{α}δ(z_{α}), ¯∂k + ∑_{α=1}^{n} r_{α}δ(z_{α})) .  (5.11)

We assume that μ = (0, 0). Therefore, we have the two holomorpheity conditions

¯∂Φ − kp_{x}Ā + [Ā, Φ] + ∑_{α=1}^{n} p_{α}δ(z_{α}) = 0 ,  (5.12)

¯∂k + ∑_{α=1}^{n} r_{α}δ(z_{α}) = 0 .  (5.13)

The constraint equation (5.13) means that the k-component of the Higgs field is a holomorphic one-form on Σ with first order poles at the marked points.

Let us fix a gauge

L̄ = f^{−1} ¯∂f + f^{−1}Āf.  (5.14)

The same gauge action transform Φ as

L = kf^{−1}∂_{x}f + f^{−1}Φf.  (5.15)

We preserve the same notations g_{α}, p_{α} for the gauge transformed variables. The moment constraint equation (5.12) has the same form in terms of L̄ and L

¯∂L − k∂_{x}L̄ + [L̄, L] + ∑_{α=1}^{n} p_{α}δ(z_{α}) = 0 .  (5.16)

Solutions of this equation along with (5.13) define the reduced phase space

T^{∗}R//G ∼ T^{∗}N .

The symplectic form (5.7) on T^{∗}N becomes

ω =< δL|δL̄ > + ∫_{Σ_{n}} δkδλ + ∑_{α=1}^{n} ω_{α} .  (5.17)

5.5 Coadjoint orbits.

Consider in detail the symplectic form ω (5.8) on T^{∗}L̂(G) ∼ {(p, r); (g, c)}. We omit the subscript α below. The following canonical transformation of ω by (f, s) ∈ L̂(G), where s is a central element,

g → fg, p → p, r → r, c → sc, f ∈ L(G) ,  (5.18)

has not been used so far. The symplectic reduction with respect to this transformation leads to the coadjoint orbits of L̂(GL(N, ℂ)). In fact, the moment map

μ : T^{∗}L̂(G) → Lie^{∗}(L̂(GL(N, ℂ)))

takes the form

μ = (−gpg^{−1} + r∂_{x}gg^{−1}, r).

Let us fix the moment μ = (p^{(0)}, r^{(0)}). The result of the symplectic reduction of T^{∗}L̂(G) is the coadjoint orbit

O(p^{(0)}, r^{(0)}) = (p = −g^{−1}p^{(0)}g − r^{(0)}g^{−1}∂_{x}g, r^{(0)}) = μ^{−1} (T^{∗}L̂(SL(N, ℂ))) /G_{0},

where G_{0} is the subgroup of L̂(GL(N, ℂ)) that preserves μ

G^{0} = {g ∈ L(GL(N, ℂ), s is arbitrary) | p^{(0)} = −g^{−1}p^{(0)}g + r^{(0)}g^{−1}∂_{x}g}.

The symplectic form (5.8) being pushed forward on O takes the form

ω = ∮ tr(D(pg^{−1})Dg) + \frac{r^{(0)}}{2} ∮ tr (g^{−1}DgD(g^{−1}∂_{x}g)) .  (5.19)

In what follows we will consider the collection of the orbits O_{α}(p_{α}^{(0)}, r_{α}^{(0}) at the marked points instead of the cotangent bundles T^{∗}L̂(G_{α}). In this way we modify the notion of the cotangent bundle T^{∗}R (5.6)

T^{∗}R = {(Ā, Φ), (λ, k), O_{1}(p_{1}^{(0)}, r_{1}^{(0}), …, O_{n}(p_{n}^{(0)}, r_{n}^{(0})} .  (5.20)

5.6 Conservation laws I

.

The Higgs field Φ is transformed as a connection with respect to the circles S^{1} (5.9). If the central charge k ≠ 0, the standard Hitchin integrals (2.23) cease to be gauge invariant. Invariant integrals are generated by the traces of the monodromies of the Higgs field Φ. The generating function of Hamiltonians is given by

H(z) = tr (P exp \frac{1}{k} ∮_{S_{1}} Φ) ,  (5.21)

where z is a local coordinate of an arbitrary point. At a marked point, Φ has a first order pole and

H(z) = ∑_{j=−N}^{+∞} H_{j}z^{j} .  (5.22)

Since H(z) is gauge invariant one can replace Φ by L in (5.21)

H(z) = tr (P exp \frac{1}{k} ∮_{S_{1}} L) .  (5.23)

5.7 Equations of motion.

Consider the equations of motion on the ”upstairs” space T^{∗}R (5.20). They are derived by means of the symplectic form ω (5.7), where ω_{α} is replaced by (5.19), and the Hamiltonians (5.21), (5.22). Let t_{j} be a time variable corresponding to the Hamiltonian H_{j}. Taking into account that H_{j} is a functional depending on the Higgs field and the central charge k only, we arrive to the following free system

∂_{j}Φ = 0,  (5.24)
∂_{j}Ā = \frac{δH_{j}}{δΦ} ,  (5.25)
∂_{j}k = 0, ∂_{j}λ = \frac{δH_{j}}{δk} , ∂_{j}p_{α} = 0 .  (5.26)

After the symplectic reduction we are led to the fields L̄ (5.14) and L (5.15). For simplicity, we keep the same notation for the coadjoint orbits variables p_{α}, so they are transformed as in (5.10). Substituting (5.15) in (5.24) we obtain the Zakharov-Shabat equation

∂_{j}L − k∂_{x}M_{j} + [M_{j}, L] = 0 , (∂_{j} = ∂_{t_{j}}) ,  (5.27)

where M_{j} = ∂_{j}ff^{−1}. The operator M_{j} can be restored partly from the second equation (5.25)

¯∂M_{j} − ∂_{j}L̄ + [M_{j}, L̄] = \frac{δH_{j}}{δL} .  (5.28)

The last two equations along with the moment constraint equation (5.16) are the consistency conditions for the linear system

(k∂_{x} + L)Ψ = 0 ,  (5.29)
(¯∂ + L̄)Ψ = 0 ,  (5.30)
(∂_{j} + M_{j})Ψ = 0 .  (5.31)

5.8 Conservation laws II

.

The matrix equation (5.29) allows to write down the conservation laws. Its generic solutions can be represented in the form

Ψ(x) = (I + R) exp \frac{1}{k} ∫_{0}^{x} Sdx′ ,  (5.32)

where I is the identity matrix, R is an off-diagonal periodic matrix and S = diag(S^{1}, …, S^{N}) is a diagonal matrix. The equation (5.29) means that L can be gauge transformed to the diagonal form

(I + R)S = k∂_{x}(I + R) + L(I + R).  (5.33)

Consider this equation in neighborhood of a point on Σ_{n} with a local coordinate z. Assume, for simplicity, that it is a pole of the Lax operator and k is a constant. In particular, it follows from (5.13) that r_{α}^{0} = 0 and the coadjoint orbits have the form O_{α} = {p_{α} = −gp_{α}^{0}g^{−1}}. Then substitute into (5.33) the series expansions

L(z) = L_{−1}z^{−1} + L_{0} + L_{1}z + …, (L_{−1} = resL = p),
S(z) = S_{−1}z^{−1} + S_{0} + S_{1}z + …,
(I + R)(z) = h + R_{1}z + R_{2}z^{2} + …, (diag(R_{m}) ≡ 0).

It follows from (5.32) that the diagonal matrix elements S_{j}^{m} are the densities of the conservation laws

log H_{j,l} = ∮ S_{j}^{l}dx.

We present a recurrence procedure to define the diagonal matrices S_{j}. On the first step we find that

S_{−1} = h^{−1}L_{−1}h = h^{−1}ph, p = L_{−1} = Res L_{z=0} .  (5.34)

In other words the diagonal matrix S_{−1} determines the orbit located at the point z = 0. In the general case we get the following equation

S_{k} + [h^{−1}R_{k+1}, S_{−1}] = h^{−1}k∂_{x}R_{k} + h^{−1} ∑_{l=0}^{k} L_{l}R_{k−l} − R_{l}S_{k−l} .

Separating the diagonal and the off-diagonal parts allows us to express S_{k} and R_{k} in terms of the lower coefficients

S_{k} = (h^{−1}k∂_{x}R_{k} + h^{−1} ∑_{l=0}^{k−1} L_{l}R_{k−l} + h^{−1}L_{k}hR_{l}S_{k−l})_{diag} ,  (5.35)

[h^{−1}R_{k}, S_{−1}] = (h^{−1}k∂_{x}R_{k−1} + h^{−1}L_{k−1}hh^{−1} ∑_{l=0}^{k−2} L_{l}R_{k−l} − h^{−1} ∑_{l=1}^{k−1} R_{l}S_{k−l})_{nondiag} . (5.36)

In particular,

S_{0} = h^{−1}k∂_{x}h ,  (5.37)
S_{1} = (h^{−1}k∂_{x}R_{1} + h^{−1}L_{1}h − h^{−1}R_{1}S_{0})_{diag} ,  (5.38)

where R_{1} is defined from the equation

R_{1}S_{−1} − pR_{1} = k∂_{x}h .  (5.39)

5.9 Hamiltonians in SL(2, ℂ) case.

Let us perform the gauge transformation

f^{−1}Lf + kf^{−1}∂_{x}f = L′ ,  (5.40)

with f defined as follows:

f = ( \begin{matrix}\sqrt{L_{12}}  0 \\ −\frac{L_{11}}{\sqrt{L_{12}}} − k \frac{∂_{x}\sqrt{L_{12}}}{L_{12}}  \frac{1}{\sqrt{L_{12}}} \end{matrix} ) .  (5.41)

Then the Lax matrix L is transformed into

L′ = ( \begin{matrix}0 1 \\ T 0 \end{matrix} ) ,  (5.42)

where

T = L_{21}L_{12} + L_{11}^{2} + k \frac{L_{11}∂_{x}L_{12}}{L_{12}} − k∂_{x}L_{11} − \frac{1}{2}k^{2} \frac{∂_{x}^{2}L_{12}}{L_{12}} + \frac{3}{4}k^{2} \frac{(∂_{x}L_{12})^{2}}{L_{12}^{2}} .  (5.43)

The linear problem

\{ \begin{matrix}(xk∂_{x} + L′)ψ = 0 , \\ (∂_{j} + M′_{j})ψ = 0 , \end{matrix}  (5.44)

where ψ is the Bloch wave function ψ = exp{−i ∮ χ}, leads to the Riccati equation:

ik∂_{x}χ − χ^{2} + T = 0.  (5.45)

The decomposition of χ(z) provides densities of the conservation laws (see [26]):

χ = ∑_{k=−1}^{∞} z^{k}χ_{k} ,  (5.46)
H_{k} ∼ ∮ dxχ_{k−1} .  (5.47)

The values of χ_{k} can be found from (5.45) using the expression (5.43) for T (z) = ∑_{k=−2}^{∞} z^{k}T_{k} in a neighborhood of zero. For k = −2, −1 and 0 we have:

\{ \begin{matrix}χ_{−1} = \sqrt{T_{−2}} = \sqrt{h} , \\ 2\sqrt{h}χ_{0} = T_{−1} + ik∂_{x}χ_{−1} = T_{−1} , \\ 2\sqrt{h}χ_{1} = T_{0} + ik∂_{x}χ − χ_{0}^{2} . \end{matrix}  (5.48)

In Subsections 7.2, 7.3 below, explicit formulae for T_{k} are used for the computation of the Hamiltonians for the elliptic 2d Calogero-Moser and the elliptic Gaudin models.

6 L̂(SL(N, ℂ))-bundles over elliptic curves with marked points.

6.1 General case.

We apply the general construction to the L̂(SL(N, ℂ))-bundle over elliptic curve E_{τ} with marked points w_{α}, α = 1, …, n. It is a two-dimensional generalization of the elliptic Gaudin model [8]. In particular, for one marked point z = 0 we come to the N-body elliptic CM field theory.

Let us construct solutions of the moment equations (5.11), taking for simplicity at the marked points the orbits with vanishing central charges

O_{α} = {p_{ij}^{α}, r_{α} = 0} .

For elliptic curves one can fix the central charge as k = 1. For the stable bundles the gauge transformation (5.3) allows to diagonalize Ā:

Ā_{ij} = δ_{ij} \frac{2π\sqrt{−1}}{τ − τ̄}u_{i} .  (6.1)

Then the Lax operator L^{G} should satisfy (5.16). It takes the form:

\begin{matrix}L_{ij}^{G} = −\frac{δ_{ij}}{2π\sqrt{−1}} (\frac{v_{i}}{2} + ∑_{α} p_{ii}^{α} (2π\sqrt{−1}\frac{z−z̄}{τ−τ̄} + E_{1}(z − w_{α}))) − \\ −\frac{1−δ_{ij}}{2π\sqrt{−1}} ∑_{α} p_{ij}^{α}e (\frac{z−w_{α}−(z̄−w̄_{α})}{τ−τ̄}u_{ij}) ϕ(u_{ij}, z − w_{α}), (u_{ij} = u_{i} − u_{j}) . \end{matrix}  (6.2)

By the quasiperiodic gauge transform

f = diag\{e(\frac{z − z̄}{τ − τ̄}u_{i})\}  (6.3)

one comes to the holomorphic quasiperiodic Lax operator

l_{ij}^{G}(z) = −\frac{δ_{ij}}{2π\sqrt{−1}} (\frac{v_{i}}{2} + ∑_{α} p_{ii}^{α}E_{1}(z − w_{α})) − \frac{1 − δ_{ij}}{2π\sqrt{−1}} ∑_{α} p_{ij}^{α}ϕ(u_{ij}, z − w_{α}) .  (6.4)

reducing the moment map equation to the diagonal gives the additional constraint

\frac{1}{2π\sqrt{−1}} ∑_{α} p_{ii}^{α} = ∂_{x}u_{i} .  (6.5)

6.2 L̂(SL(2, ℂ))-bundles over elliptic curves with marked points.

In this subsection we study 2-body elliptic Calogero field theory in details.

The operator L. According to (6.4) the holomorphic Lax operator is

\{ \begin{matrix}l_{11}^{G} = −\frac{v}{4π\sqrt{−1}} − ∑_{α} \frac{p_{11}^{α}}{2π\sqrt{−1}}E_{1}(z − w_{α}) , \\ l_{12}^{G} = − ∑_{α} \frac{p_{12}^{α}}{2π\sqrt{−1}}ϕ(2u, z − w_{α}) , \\ l_{21}^{G} = − ∑_{α} \frac{p_{21}^{α}}{2π\sqrt{−1}}ϕ(−2u, z − w_{α}) , \end{matrix}  (6.6)

with the additional constraint (6.5)

\frac{1}{2π\sqrt{−1}} ∑_{α} p_{11}^{α} = u_{x} .  (6.7)

We still have the freedom to fix the gauge with respect to the action of the diagonal subgroup. The corresponding moment map is (6.7).

For the one marked point w_{1} = 0 the corresponding orbit is

p = 2π\sqrt{−1} ( \begin{matrix}u_{x} − ν \\ −ν − u_{x} \end{matrix} ) ,  (6.8)

where ν =const., is the result of the gauge fixing. In this case the Lax operator is a 2d generalization of the Lax operator for the two-body CM model:

L_{2D}^{CM} = ( \begin{matrix}−\frac{1}{4π\sqrt{−1}}v − u_{x}E_{1}(z)  νϕ(2u, z) \\ νϕ(−2u, z)  \frac{1}{4π\sqrt{−1}}v + u_{x}E_{1}(z) \end{matrix} ) ,  (6.9)

This operator is still periodic under the shift z → z + 1 and

L_{2D}^{CM}(z + τ) = e(u)L_{2D}^{CM}(z)e(−u) + e(u)∂_{x}e(−u) ,

where e(u) = diag(exp u, exp −u).

Hamiltonians for the 2d elliptic sl(2, ℂ) CM model. In this case the coefficients T_{k} are (see (5.43)-(5.48)):

\{ \begin{matrix}T_{−2}^{CM} = u_{x}^{2} + ν^{2} = h \\ T_{−1}^{CM} = 2\frac{v}{4π\sqrt{−1}}u_{x} − \frac{ν_{x}}{ν} u_{x} + u_{xx} \\ T_{0}^{CM} = −\frac{v^{2}}{16π^{2}} + (2u_{x}^{2} − ν^{2})℘(2u) − \frac{v}{4π\sqrt{−1}} \frac{ν_{x}}{ν} + \frac{1}{4}(\frac{ν_{x}}{ν})^{2} \end{matrix} ,  (6.10)

where h is the Casimir function, fixing the coadjoint orbit at the marked point. It can be chosen as a constant. Thus, we have

ν^{2} = h − u_{x}^{2}.

The next order Hamiltonian is quadratic

H_{−1}^{CM} = ∮ \frac{v}{2π\sqrt{−1}}u_{x} − \frac{ν_{x}}{ν} u_{x} .  (6.11)

It can be written in the following way:

H_{−1}^{CM} = ∮ \frac{v}{2π\sqrt{−1}}u_{x} + \frac{u_{xx}h}{ν^{2}} .  (6.12)

Since \{∮ dx\frac{u_{xx}}{ν^{2}} , v(y)\} = 0, the equations of motion are:

\{ \begin{matrix}u_{t} = \frac{1}{2π\sqrt{−1}}u_{x} , \\ v_{t} = \frac{1}{2π\sqrt{−1}}v_{x} . \end{matrix}  (6.13)

Note that the L-M pair is simple in this case: M = \frac{1}{2π\sqrt{−1}}L.

The first nontrivial Hamiltonian H_{0} is quadratic in the momenta field v. It is a twodimensional generalization of the quadratic CM Hamiltonian

H_{0}^{CM} = ∮ dx2\sqrt{h}χ_{1} = ∮ dx(T_{0} − \frac{1}{4h}T_{−1}^{2}) .  (6.14)

A direct evaluation yields:

T_{0}^{CM} − \frac{1}{4h}(T_{−1}^{CM})^{2} = −\frac{v^{2}}{16π^{2}} (1 − \frac{u_{x}^{2}}{h}) + (3u_{x}^{2} − h)℘(2u) − \frac{u_{xx}^{2}}{4ν^{2}} .  (6.15)

The equations of motion produced by H_{0}^{CM} are:

u_{t} = −\frac{v}{8π^{2}} (1 − \frac{u_{x}^{2}}{h}) ,  (6.16)

v_{t} = \frac{1}{8π^{2}h}∂_{x}(v^{2}u_{x}) − 2(3u_{x}^{2} − h)℘′(2u) + 6∂_{x}(u_{x}℘(2u)) + \frac{1}{2}∂_{x}(\frac{u_{xxx}ν − ν_{x}u_{xx}}{ν^{3}} ) .

It is reduced to the two-body elliptic CM system for the x-independent fields.

The L-M pair for the 2d elliptic sl(2, ℂ) CM model. The equations of motion (6.16) produced by the quadratic Hamiltonian H_{0}^{CM} can be represented in a form of the Zakharov-Shabat equation with the L matrix defined by (6.9) and the M matrix given as follows:

\{ \begin{matrix}M_{11} = −u_{t}E_{1}(z) − \frac{1}{4π\sqrt{−1}} (\frac{1}{8π^{2}h}v^{2}u_{x} + 6u_{x}℘(2u) + \frac{u_{xxx}ν−ν_{x}u_{xx}}{2ν^{3}} ) + \\ +\frac{u_{x}}{2π\sqrt{−1}}(E_{2}(2u) − E_{2}(z)) , \\ M_{12} = −\frac{ν}{2π\sqrt{−1}}ϕ′(2u, z) + (\frac{ν}{2π\sqrt{−1}}E_{1}(z) + \frac{vu_{x}ν}{8π^{2}h} − \frac{1}{4π\sqrt{−1}} \frac{u_{xx}}{ν} ) ϕ(2u, z) , \\ M_{21} = −\frac{ν}{2π\sqrt{−1}}ϕ′(−2u, z) + (\frac{ν}{2π\sqrt{−1}}E_{1}(z) + \frac{vu_{x}ν}{8π^{2}h} + \frac{1}{4π\sqrt{−1}} \frac{u_{xx}}{ν} ) ϕ(−2u, z) . \end{matrix}  (6.17)

See Appendix C for details of the proof. This construction completes the description of right vertical arrow in Fig.1

2d CM - LL correspondence . The upper modification that produces the map of the elliptic CM system into the elliptic rotator (4.9), (4.10) works in the two-dimensional case as well.

The two-dimensional extension of the SL(2, ℂ)-elliptic rotator is the Landau-Lifshitz (LL) equation

∂_{t}S = \frac{1}{2}[S, J(S)] + \frac{1}{2}[S, ∂_{xx}S].  (6.18)

This equation can be fitted in the Zakharov-Shabat form [28]. The Lax operator L^{LL} has the same form as for the SL(2, ℂ) elliptic rotator L^{rot} (4.14). For sl(2, ℂ) the basis of the sigma matrices coincides with the basis of the sin-algebra and L^{LL} takes the form

L = ∑_{a} u_{a}(z)S_{a}σ_{a},

u_{1} = φ [ \begin{matrix}0 \\ 1 \end{matrix} ] (z) , u_{2} = φ [ \begin{matrix}1 \\ 1 \end{matrix} ] (z) , u_{3} = φ [ \begin{matrix}1 \\ 0 \end{matrix} ] (z) .

The M^{LL} operator has a very simple extension

M^{LL} = M^{rot} + ∑_{a} u_{a}(z)tr(σ_{a}[S, ∂_{x}S])σ_{a} .

It is easy to check that the Zakharov-Shabat equation leads to (6.18) if

∑_{a} S_{a}^{2} = 1 .

Thereby be have defined the right vertical arrow in Fig.1.

Now the upper modification acts as

L^{LL} = Ξ∂_{x}Ξ^{−1} + ΞL_{2D}^{CM}Ξ^{−1} ,

where L_{2D}^{CM} is defined by (6.9). The gauge transformation Ξ has the same quasi-periodicity properties as in the one-dimensional situation. Therefore, it is defined by (4.30) and (4.23) for N = 2. It means that (4.32) allows to pass from the CM fields v(x, t), u(x, t) and the constant ν to the LL fields S = (S_{1}, S_{2}, S_{3}) with the orbit fixing condition

∑_{a} S_{a}^{2} = −\frac{1}{2π^{2}} (u_{x}^{2} + ν^{2}) = 1 .

It completes the description of the diagram on Fig.1.

Limit to the Sinh-G equation. It is known that the LL model is universal; it contains as a special limit the Sinh-Gordon and the Nonlinear Schrödinger models [3]. In this way they can be derived within the 2d CM system. The scaling limit in the CM model is a combination of the trigonometric limit Imτ → ∞ with shifts of coordinates: u = U + \frac{1}{2}Imτ and renormalization of the coupling constant ν = ν̄e^{\frac{1}{2} Imτ} [27]. This procedure applied to the 2d elliptic CM Hamiltonian yields the sinh-Gordon system:

H_{SG} = −\frac{v^{2}}{16π^{2}} − ν̄^{2}(e^{2U} + e^{−2U}) + \frac{U_{x}^{2}}{4} .  (6.19)

The equations of motion are:

\{ \begin{matrix}U_{t} = −\frac{v}{8π^{2}} , \\ v_{t} = 2ν̄^{2}(e^{2U} − e^{−2U}) + \frac{1}{2}U_{xx} . \end{matrix}  (6.20)

The L-M pair is:

L^{SG} = ( \begin{matrix}−\frac{v}{4π\sqrt{−1}} − \frac{1}{2}U_{x}  ν̄(1 − e^{2U}Z) \\ ν̄(\frac{1}{Z} − e^{−2U})  \frac{v}{4π\sqrt{−1}} + \frac{1}{2}U_{x} \end{matrix} ) .  (6.21)

M^{SG} = ( \begin{matrix}−\frac{U_{t}}{2} − \frac{1}{8π\sqrt{−1}}U_{x}  \frac{ν̄}{4π\sqrt{−1}}(1 + e^{2U}Z) \\ \frac{ν̄}{4π\sqrt{−1}}(e^{−2U} + \frac{1}{Z})  \frac{U_{t}}{2} + \frac{1}{8π\sqrt{−1}}U_{x} \end{matrix} ) .  (6.22)

6.3 Hamiltonians for the 2d elliptic Gaudin model.

Using (B.28) we obtain the Hamiltonian:

H_{−1,a}^{G} = 2\frac{v}{4π\sqrt{−1}} \frac{p_{11}^{a}}{2π\sqrt{−1}} + 2 ∑_{b} \frac{p_{11}^{a}}{2π\sqrt{−1}} \frac{p_{11}^{b}}{2π\sqrt{−1}}E_{1}(z_{a} − z_{b})−  (6.23)

− ∑_{a≠b} \frac{p_{12}^{a}p_{21}^{b}}{(2π\sqrt{−1})^{2}} ϕ(2u, z_{b} − z_{a}) + ∑_{a≠b} \frac{p_{12}^{b}p_{21}^{a}}{(2π\sqrt{−1})^{2}} ϕ(2u, z_{a} − z_{b}) − \frac{p_{11}^{a}}{2π\sqrt{−1}} \frac{∂_{x}p_{12}^{a}}{p_{12}} .

The last term makes the above Hamiltonian different from the one-dimensional version. Let us consider the sl(2, ℂ) case with two marked points on the elliptic curve. We will use the following notations:

\{ \begin{matrix}p_{11}^{1} = 2π\sqrt{−1}γ_{1}, p_{11}^{2} = 2π\sqrt{−1}γ_{2} , \\ p_{12}^{1} = −2π\sqrt{−1}ν_{+}, p_{21}^{1} = −2π\sqrt{−1}ν_{−} , \\ p_{12}^{2} = −2π\sqrt{−1}μ_{+}, p_{21}^{2} = −2π\sqrt{−1}μ_{−} . \end{matrix}  (6.24)

The L matrix is:

\{ \begin{matrix}l_{11}^{G} = −\frac{v}{4π\sqrt{−1}} − γ_{1}E_{1}(z − z_{1}) − γ_{2}E_{1}(z − z_{2}) , \\ l_{12}^{G} = νϕ(2u, z − z_{1}) + μ_{+}ϕ(2u, z − z_{2}) , \\ l_{21}^{G} = νϕ(−2u, z − z_{1}) + μ_{−}ϕ(−2u, z − z_{2}) . \end{matrix}  (6.25)

The solution exists if

γ_{1} + γ_{2} = u_{x} .  (6.26)

The gauge fixing condition is chosen to be

ν_{+} = ν_{−} = ν.  (6.27)

We fix the Casimir elements h_{1} = γ_{1}^{2} + ν^{2} and h_{2} = γ_{2}^{2} + μ_{+}μ_{−} to be constants: h_{1}, h_{2} ∈ ℂ.

On the reduced phase space there are two independent fields besides u and v. Let them be for example ν and μ_{+}, then

\{ \begin{matrix}γ_{1} = \sqrt{h_{1} − ν^{2}} , \\ γ_{2} = u_{x} − \sqrt{h_{1} − ν^{2}} , \\ μ_{−} = \frac{1}{μ_{+}} (h_{2} − (u_{x} − \sqrt{h_{1} − ν^{2}})^{2}) . \end{matrix}  (6.28)

However we are going to use all kinds of variables in order to make the formulae more transparent. The non-trivial brackets on the reduced phase space are:

\begin{matrix}\{v(x), u(y)\} = δ(x − y) , \{v(x), γ_{1}(y)\} = −δ′(x − y) , \{v(x), ν(y)\} = \frac{γ_{1}}{ν} δ′(x − y) , \\ \{μ_{+}(x), γ_{1}(y)\} = −\frac{1}{2π\sqrt{−1}}μ_{+}δ(x − y) , \{μ_{+}(x), μ_{−}(y)\} = −2\frac{1}{2π\sqrt{−1}}γ_{2}δ(x − y) , \\ \{μ_{+}(x), γ_{2}(y)\} = \frac{1}{2π\sqrt{−1}}μ_{+}δ(x − y) , \{μ_{+}(x), ν(y)\} = \frac{1}{2π\sqrt{−1}} \frac{γ_{1}}{ν} μ_{+}δ(x − y) , \\ \{ν(x), μ_{−}(y)\} = \frac{1}{2π\sqrt{−1}} \frac{γ_{1}}{ν} μ_{−}δ(x − y) . \end{matrix} (6.29)

The Hamiltonian is:

\begin{matrix}H_{−1}^{G} = ∮ dx (2γ_{1} \frac{v}{4π\sqrt{−1}} − γ_{1} \frac{ν_{x}}{ν} + ∂_{x}γ_{1} + νμ_{+}ϕ(2u, z_{1} − z_{2}) − νμ_{−}ϕ(2u, z_{2} − z_{1})+ \\ −2γ_{1}γ_{2}E_{1}(z_{1} − z_{2})) . \end{matrix}  (6.30)

The equations of motion are:

\{ \begin{matrix}∂_{t}u(x) = \frac{1}{2π\sqrt{−1}}γ_{1}(x) , \\ ∂_{t}v(x) = \frac{1}{2π\sqrt{−1}}v_{x} − ∂_{x}(\frac{γ_{1}μ_{+}}{ν} ϕ(2u, z_{1} − z_{2})) + ∂_{x}(\frac{γ_{1}μ_{−}}{ν} ϕ(2u, z_{2} − z_{1}))− \\ −2νμ_{+}ϕ′(2u, z_{1} − z_{2}) + 2νμ_{−}ϕ′(2u, z_{2} − z_{1}) − ∂_{x} (\frac{γ_{1}∂_{x}γ_{1}}{ν^{2}} ) , \\ ∂_{t}ν = −\frac{1}{2π\sqrt{−1}}∂_{x} (\frac{γ_{1}^{2}}{ν} ) + \frac{γ_{1}}{2π\sqrt{−1}ν} (μ_{+}ϕ(2u, z_{1} − z_{2}) − μ_{−}ϕ(2u, z_{2} − z_{1})) , \\ ∂_{t}μ_{+} = \frac{1}{2π\sqrt{−1}}(2\frac{v}{4π\sqrt{−1}}μ_{+}2μ_{+}(γ_{2} − γ_{1})E_{1}(z_{1} − z_{2})) − \frac{2νγ_{2}}{2π\sqrt{−1}}ϕ(2u, z_{2} − z_{1})− \\ −\frac{γ_{1}μ_{+}}{2π\sqrt{−1}ν} (μ_{+}ϕ(2u, z_{1} − z_{2}) − μ_{−}ϕ(2u, z_{2} − z_{1})) . \end{matrix}  (6.31)

The quadratic Hamiltonian is the direct generalization of (6.15)

H_{0}^{G} = ∮ dx2\sqrt{h_{1}}χ_{1} = ∮ dx(T_{0} − \frac{1}{4h_{1}} T_{−1}^{2}) ,  (6.32)

where

\begin{matrix}2\sqrt{h_{1}}χ_{1} = \\ −\frac{v^{2}}{16π^{2}} (1 − \frac{γ_{1}^{2}}{h_{1}} ) + (2u_{x}γ_{1} − ν^{2})℘(2u) − \frac{(∂_{x}γ_{1})^{2}}{4ν^{2}} + μ_{+}μ_{−}(E_{2}(z_{1} − z_{2}) − E_{2}(2u))+ \\ +4η_{1}γ_{1}γ_{2} + νμ_{−}ϕ(2u, z_{2} − z_{1})(E_{1}(z_{1} − z_{2}) − E_{1}(2u) + E_{1}(2u + z_{2} − z_{1}))− \\ −νμ_{+}ϕ(2u, z_{1} − z_{2})(E_{1}(z_{1} − z_{2}) + E_{1}(2u) − E_{1}(2u + z_{1} − z_{2})) + γ_{2}^{2}E_{1}^{2}(z_{1} − z_{2})+ \\ +2γ_{2} \frac{v}{4π\sqrt{−1}}E_{1}(z_{1} − z_{2}) − γ_{2} \frac{ν_{x}}{ν} E_{1}(z_{1} − z_{2}) + γ_{1} \frac{μ_{+}}{ν^{2}} ϕ(2u, z_{1} − z_{2})− \\ −γ_{1}ϕ(2u, z_{1} − z_{2})[\frac{∂_{x}μ_{+}}{ν} + 2u_{x} \frac{μ_{+}}{ν} (E_{1}(z_{1} − z_{2} + 2u) − E_{1}(2u))]− \\ −\frac{1}{4h_{1}} (νμ_{+}ϕ(2u, z_{1} − z_{2}) − νμ_{+}ϕ(2u, z_{2} − z_{1}) + 2γ_{1}γ_{2}E_{1}(z_{1} − z_{2}))^{2}− \\ −\frac{1}{2h_{1}} (νμ_{+}ϕ(2u, z_{1} − z_{2}) − νμ_{+}ϕ(2u, z_{2} − z_{1})+ \\ +2γ_{1}γ_{2}E_{1}(z_{1} − z_{2})) (2γ_{1} \frac{v}{4π\sqrt{−1}} − γ_{1} \frac{ν_{x}}{ν} + ∂_{x}γ_{1}) . \end{matrix}  (6.33)

7 Conclusion

Here we briefly summarize the results of our analysis and discuss some unsolved related problems. The following two subjects were investigating in the paper.

(i) We have constructed symplectic maps between Hitchin systems related to holomorphic bundles of different degrees. It allowed us to construct the Bäcklund transformations in the Hitchin systems defined over Riemann curves with marked points. We applied the general scheme to the elliptic CM systems and constructed the symplectic map to an integrable SL(N, ℂ) Euler-Arnold top (the elliptic SL(N, ℂ)-rotator). The open problem is to write down the explicit expressions for the spin variables in terms of the CM phase space for an arbitrary N as it was done for the case N = 2 (4.32). It should help to construct the Bäcklund transformations for the CM systems explicitly, and more generally, to construct the generating function for them. The later can be considered as the integrable discrete time mapping [21].

(ii) We have proposed a generalization of the Hitchin approach to 2d integrable theories related to holomorphic bundles of infinite rank. The main example is the integrable twodimensional version of the two-body elliptic CM system. The upper modification allows to define the symplectic map to the Landau-Lifshitz equation and to find, in principle, the Bäcklund transformations in the field theories. It will be extremely interesting to find the 2d generalization of the SL(N, ℂ)-rotator for N > 2 (the matrix LL equation). There is another point of view on the 2d generalizations of the Hitchin systems. One can try to define them starting from holomorphic bundles over complex surfaces, that are fibrations over Riemann curves. In this case the spectral parameter lives on the base of the fibration, while the space variable lives on the fibers. It will be interesting to analyze, for example, the known solutions of the LL equation from this point of view.

8 Acknowledgments

We would like to thank K.Hasegawa and I.Krichever for illuminating discussions and A.Shabat, T.Takebe, V.Sokolov and A.Zabrodin for useful remarks and H.Gangl for the careful reading the manuscript. We are grateful for the hospitality of the Max-Planck-Institut für Mathematik (Bonn), where the paper was partly prepared during the visits of A.L. and M.O. The work of all authors was supported by the grant 00-15-96557 of the scientific schools and RFBR-01-01-00539 (A.L.), RFBR-00-02-16530 (M.O.,A.Z.), INTAS-00-00561 (A.Z.), INTAS-99-01782 (M.O.).

9 Appendix

9.1 Appendix A. Sin-Algebra.

e(z) = exp(2π\sqrt{−1}z) ,  (A.1)
Q = diag(e(1/N), …, e(m/N), …, 1) ,  (A.2)

Λ = ( \begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 \\ 1 & 0 & 0 & ⋯ & 0 \end{matrix} ) ,  (A.3)

E_{mn} = e(\frac{mn}{2N})Q^{m}Λ^{n}, (m = 0, …, N − 1, n = 0, …, N − 1, (modN) m^{2} + n^{2} ≠ 0) (A.4)

is the basis in sl(N, ℂ). The commutation relations in this basis take the form

[E_{sk}, E_{nj}] = 2\sqrt{−1} sin \frac{π}{N} (kn − sj)E_{s+n,k+j} ,  (A.5)
tr(E_{sk}E_{nj}) = δ_{s,−n}δ_{k,−j}N .  (A.6)

9.2 Appendix B. Elliptic functions

We summarize the main formulae for elliptic functions, borrowed mainly from [29] and [30]. We assume that q = exp 2πiτ, where τ is the modular parameter of the elliptic curve E_{τ}.

The basic element is the theta function:

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

q^{\frac{1}{8}} e^{− \frac{iπ}{4}} (e^{iπz} − e^{−iπz}) ∏_{n=1}^{∞}(1 − q^{n})(1 − q^{n}e^{2iπz})(1 − q^{n}e^{−2iπz}) .

The Eisenstein functions

E_{1}(z|τ) = ∂_{z} log ϑ(z|τ), E_{1}(z|τ) ∼ \frac{1}{z} − 2η_{1}z,  (B.2)

where

η_{1}(τ) = ζ(\frac{1}{2}) =  (B.3)

\frac{3}{π^{2}} ∑_{m=−∞}^{∞} ∑_{n=−∞}^{∞′} \frac{1}{(mτ + n)^{2}} = \frac{24}{2πi} \frac{η′(τ)}{η(τ)} ,

where

η(τ) = q^{\frac{1}{24}} ∏_{n>0}(1 − q^{n}) .

is the Dedekind function.

E_{2}(z|τ) = −∂_{z}E_{1}(z|τ) = ∂_{z}^{2} log ϑ(z|τ), E_{2}(z|τ) ∼ \frac{1}{z^{2}} + 2η_{1} .  (B.4)

The next important function is

ϕ(u, z) = \frac{ϑ(u + z)ϑ′(0)}{ϑ(u)ϑ(z)} .  (B.5)

It has a pole at z = 0 and

ϕ(u, z) = \frac{1}{z} + E_{1}(u) + \frac{z}{2}(E_{1}^{2}(u) − ℘(u)) + …,  (B.6)

and

ϕ(u, z)^{−1}∂_{u}ϕ(u, z) = E_{1}(u + z) − E_{1}(u) .  (B.7)

The following formula plays an important role in checking of the zero curvature equation:

ϕ′′(u, z) = ϕ(u, z)(E_{2}(z) − E_{1}^{2}(z) + 2E_{1}(z)(E_{1}(u + z) − E_{1}(u)) + 2E_{2}(u) − 6η_{1}) .  (B.8)

It follows from:

(E_{1}(z) + E_{1}(u) − E_{1}(z + u))^{2} = E_{2}(u) + E_{2}(z) + E_{2}(u + z) − 6η_{1} .  (B.9)

Relations to the Weierstrass functions

ζ(z|τ) = E_{1}(z|τ) + 2η_{1}(τ)z ,  (B.10)
℘(z|τ) = E_{2}(z|τ) − 2η_{1}(τ) ,  (B.11)
ϕ(u, z) = exp(−2η_{1}uz)\frac{σ(u + z)}{σ(u)σ(z)} ,  (B.12)
ϕ(u, z)ϕ(−u, z) = ℘(z) − ℘(u) = E_{2}(z) − E_{2}(u) .  (B.13)

Particular values

E_{1}(\frac{1}{2}) = 0, E_{1}(\frac{τ}{2} ) = E_{1}(\frac{1 + τ}{2} ) = −π\sqrt{−1} .  (B.14)

Series representations

E_{1}(z|τ) = −2πi(\frac{1}{2} + ∑_{n≠0} \frac{e^{2πiz}}{1 − q^{n}}) =  (B.15)

−2πi(∑_{n<0} \frac{1}{1 − q^{n}e^{2πiz}} + ∑_{n≥0} \frac{q^{n}e^{2πiz}}{1 − q^{n}e^{2πiz}} + \frac{1}{2}), ,

E_{2}(z|τ) = −4π^{2} ∑_{n∈Z} \frac{q^{n}e^{2πiz}}{(1 − q^{n}e^{2πiz})^{2}} ,  (B.16)

ϕ(u, z) = 2πi ∑_{n∈Z} \frac{e^{−2πinz}}{1 − q^{n}e^{−2πiu}} .  (B.17)

Parity

ϑ(−z) = −ϑ(z) ,  (B.18)
E_{1}(−z) = −E_{1}(z) ,  (B.19)
E_{2}(−z) = E_{2}(z) ,  (B.20)
ϕ(u, z) = ϕ(z, u) = −ϕ(−u, −z) ,  (B.21)

Quasi-periodicity

ϑ(z + 1) = −ϑ(z) , ϑ(z + τ) = −q^{− \frac{1}{2}} e^{−2πiz}ϑ(z) ,  (B.22)
E_{1}(z + 1) = E_{1}(z) , E_{1}(z + τ) = E_{1}(z) − 2πi ,  (B.23)
E_{2}(z + 1) = E_{2}(z) , E_{2}(z + τ) = E_{2}(z) ,  (B.24)
ϕ(u + 1, z) = ϕ(u, z) , ϕ(u + τ, z) = e^{−2πiz}ϕ(u, z) .  (B.25)

Addition formula

ϕ(u, z)∂_{v}ϕ(v, z) − ϕ(v, z)∂_{u}ϕ(u, z) = (E_{2}(v) − E_{2}(u))ϕ(u + v, z) ,  (B.26)

or

ϕ(u, z)∂_{v}ϕ(v, z) − ϕ(v, z)∂_{u}ϕ(u, z) = (℘(v) − ℘(u))ϕ(u + v, z) .  (B.27)

The proof of (B.26) is based on (B.6),(B.21), and (B.25). In fact, ϕ(u, z) satisfies more general relation which follows from the Fay three-section formula

ϕ(u_{1}, z_{1})ϕ(u_{2}, z_{2}) − ϕ(u_{1} + u_{2}, z_{1})ϕ(u_{2}, z_{2} − z_{1}) − ϕ(u_{1} + u_{2}, z_{2})ϕ(u_{1}, z_{1} − z_{2}) = 0 . (B.28)

A particular case of this formula is

ϕ(u_{1}, z)ϕ(u_{2}, z) − ϕ(u_{1} + u_{2}, z)(E_{1}(u_{1}) + E_{1}(u_{2})) + ∂_{z}ϕ(u_{1} + u_{2}, z) = 0 .  (B.29)

Integrals

∫_{E_{τ}} E_{1}(z|τ)dzdz̄ = 0 .  (B.30)

Theta functions with characteristics. For a, b ∈ ℚ put :

θ[ \begin{matrix}a \\ b \end{matrix} ](z, τ) = ∑_{j∈ℤ}e ((j + a)^{2} \frac{τ}{2} + (j + a)(z + b)) .  (B.31)

In particular, the function ϑ (B.1) is the theta function with a characteristic

ϑ(x, τ) = θ [ \begin{matrix}1/2 \\ 1/2 \end{matrix} ] (x, τ) .  (B.32)

One has

θ[ \begin{matrix}a \\ b \end{matrix} ](z + 1, τ) = e(a)θ[ \begin{matrix}a \\ b \end{matrix} ](z, τ) ,  (B.33)
θ[ \begin{matrix}a \\ b \end{matrix} ](z + a′τ, τ) = e (−a′^{2} \frac{τ}{2} − a′(z + b)) θ[ \begin{matrix}a + a′ \\ b \end{matrix} ](z, τ) ,  (B.34)
θ[ \begin{matrix}a + j \\ b \end{matrix} ](z, τ) = θ[ \begin{matrix}a \\ b \end{matrix} ](z, τ) , j ∈ ℤ .  (B.35)

For the simplicity we denote θ [ \begin{matrix}a/2 \\ b/2 \end{matrix} ] = θ_{ab}.

The following identities are useful for the upper modification procedure in sl(2, ℂ) case:

\begin{matrix}θ_{01}(x, τ)θ_{00}(y, τ) + θ_{01}(y, τ)θ_{00}(x, τ) = 2θ_{01}(x + y, 2τ)θ_{01}(x − y, 2τ) , \\ θ_{01}(x, τ)θ_{00}(y, τ) − θ_{01}(y, τ)θ_{00}(x, τ) = 2ϑ(x + y, 2τ)ϑ(x − y, 2τ) , \\ θ_{00}(x, τ)θ_{00}(y, τ) + θ_{01}(y, τ)θ_{01}(x, τ) = 2θ_{00}(x + y, 2τ)θ_{00}(x − y, 2τ) , \\ θ_{00}(x, τ)θ_{00}(y, τ) − θ_{01}(y, τ)θ_{01}(x, τ) = 2θ_{10}(x + y, 2τ)θ_{10}(x − y, 2τ) . \end{matrix}  (B.36)

\begin{matrix}2ϑ(x, 2τ)θ_{01}(y, 2τ) = ϑ(\frac{x+y}{2} , τ)θ_{10}(\frac{x−y}{2} , τ) + θ_{10}(\frac{x+y}{2} , τ)ϑ(\frac{x−y}{2} , τ) , \\ 2θ_{00}(x, 2τ)θ_{10}(y, 2τ) = ϑ(\frac{x+y}{2} , τ)ϑ(\frac{x−y}{2} , τ) + θ_{10}(\frac{x+y}{2} , τ)θ_{10}(\frac{x−y}{2} , τ) , \\ 2θ_{00}(x, 2τ)θ_{00}(y, 2τ) = θ_{00}(\frac{x+y}{2} , τ)θ_{00}(\frac{x−y}{2} , τ) + θ_{01}(\frac{x+y}{2} , τ)θ_{01}(\frac{x−y}{2} , τ) , \\ 2θ_{10}(x, 2τ)θ_{10}(y, 2τ) = θ_{00}(\frac{x+y}{2} , τ)θ_{00}(\frac{x−y}{2} , τ) − θ_{01}(\frac{x+y}{2} , τ)θ_{01}(\frac{x−y}{2} , τ) . \end{matrix}  (B.37)

9.3 Appendix C: 2d sl(2, ℂ) Calogero L-M pair

The Zakharov-Shabat equations in sl(2, ℂ) case are:

\{ \begin{matrix}11 : ∂_{t}L_{11} − ∂_{x}M_{11} = M_{21}L_{12} − M_{12}L_{21} , \\ 12 : ∂_{t}L_{12} − ∂_{x}M_{12} = 2L_{11}M_{12} − 2L_{12}M_{11} , \\ 21 : ∂_{t}L_{21} − ∂_{x}M_{21} = 2M_{11}L_{21} − 2L_{11}M_{21} . \end{matrix}  (C.1)

Let the non diagonal terms in the M matrix be of the form:

\{ \begin{matrix}M_{12} = c(x)Φ′(2u, z) + (f_{12}^{1}(x)E_{1}(z) + f_{12}^{0}(x))Φ(2u, z) , \\ M_{21} = c(x)Φ′(−2u, z) + (f_{21}^{1}(x)E_{1}(z) + f_{21}^{0}(x))Φ(−2u, z) . \end{matrix}  (C.2)

Then from the diagonal part of (C.1) we conclude:

M_{11} = −u_{t}E_{1}(z) + α(x) + △M_{11} ,  (C.3)

where

α(x) = −\frac{1}{4π\sqrt{−1}} (\frac{1}{8π^{2}h}v^{2}u_{x} + 6u_{x}℘(2u) + \frac{u_{xxx}ν − ν_{x}u_{xx}}{2ν^{3}} ) ,  (C.4)

and △M_{11} will be defined in the following. It is supposed to be dependent on E_{2}(2u) in order to cancel terms proportional to E_{2}(2u) and E′_{2}(2u) in (C.1).

Using formula (B.8) in the non diagonal part of (C.1), we arrive to some conditions equivalent to cancellations of the terms proportional to functions ξ(2u, z) = E_{1}(2u + z) − E(2u), E_{1}(z), E_{1}^{2}(z), E_{1}(z)ξ(−2u, z):

(12) \{ \begin{matrix}E_{1}(z)ξ(2u, z) : f_{12}^{1} = −c , \\ E_{1}^{2}(z) : f_{12}^{1} = −c , \\ ξ(2u, z) : 2νu_{t} − c_{x} − 2u_{x}f_{12}^{0} = −2c\frac{v}{4π\sqrt{−1}} , \\ E_{1}(z) : −∂_{x}f_{12}^{1} = −2\frac{v}{4π\sqrt{−1}}f_{12}^{1} − 2u_{x}f_{12}^{0} + 2u_{t}ν . \end{matrix}  (C.5)

(21) \{ \begin{matrix}E_{1}(z)ξ(−2u, z) : f_{12}^{1} = −c , \\ E_{1}^{2}(z) : f_{12}^{1} = −c , \\ ξ(−2u, z) : −2νu_{t} − c_{x} + 2u_{x}f_{21}^{0} = 2c\frac{v}{4π\sqrt{−1}} , \\ E_{1}(z) : −∂_{x}f_{21}^{1} = 2\frac{v}{4π\sqrt{−1}}f_{12}^{1} + 2u_{x}f_{21}^{0} − 2u_{t}ν . \end{matrix}  (C.6)

Thus

\{ \begin{matrix}f_{12}^{1} = f_{21}^{1} = −c , \\ 2u_{x}f_{12}^{0} = 2νu_{t} − c_{x} + 2c\frac{v}{4π\sqrt{−1}} , \\ 2u_{x}f_{21}^{0} = 2νu_{t} + c_{x} + 2c\frac{v}{4π\sqrt{−1}} . \end{matrix}  (C.7)

\{ \begin{matrix}f_{+} = f_{21}^{0} + f_{12}^{0} = \frac{2}{u_{x}} (νu_{t} + c\frac{v}{4π\sqrt{−1}}) , \\ f_{−} = f_{21}^{0} − f_{12}^{0} = \frac{c_{x}}{u_{x}} . \end{matrix}  (C.8)

The remaining parts of the non diagonal equations are:

\{ \begin{matrix}ν_{t} + 12cu_{x}η_{1} − ∂_{x}f_{12}^{0} − 2cu_{x}E_{2}(z) − 4cu_{x}E_{2}(2u) = \\ = −2\frac{v}{4π\sqrt{−1}}f_{12}^{0} − 2να − 2ν△M_{11} , \\ −ν_{t} + 12cu_{x}η_{1} + ∂_{x}f_{21}^{0} − 2cu_{x}E_{2}(z) − 4cu_{x}E_{2}(2u) = \\ = −2\frac{v}{4π\sqrt{−1}}f_{21}^{0} − 2να − 2ν△M_{11} . \end{matrix}  (C.9)

Subtracting the above equations we have:

2\frac{u_{x}}{ν} ∂_{x}u_{t} + ∂_{x}f_{+} = −2\frac{v}{4π\sqrt{−1}}f_{−} .  (C.10)

Substituting f_{+} and f_{−} from (C.8) into (C.10) we arrive to the equation for c:

\frac{u_{x}}{ν} ∂_{x}u_{t} + ∂_{x}(\frac{1}{u_{x}} (νu_{t} + c\frac{v}{4π\sqrt{−1}})) = −\frac{v}{4π\sqrt{−1}} \frac{c_{x}}{u_{x}} .  (C.11)

Now some concrete equations of motion should be used. For H_{−1}^{CM} this equation yields c ∼ \sqrt{\frac{u_{x}}{v}}. However the coefficient of the proportionality appear to be equal zero. For H_{0}^{CM} (6.15) we have c = −\frac{ν}{2π\sqrt{−1}}.

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