On the explicit solutions of the elliptic Calogero system

L. Gavrilov

Laboratoire Emile Picard , CNRS UMR 5580, Université Paul Sabatier

118, route de Narbonne, 31062 Toulouse Cedex, France E-mail: gavrilov@picard.ups-tlse.fr

A.M. Perelomov ^{1}

Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Abstract

Let q_{1}, q_{2}, ..., q_{N} be the coordinates of N particles on the circle, interacting with the integrable potential ∑_{j<k}^{N} ℘(q_{j} − q_{k}), where ℘ is the Weierstrass elliptic function. We show that every symmetric elliptic function in q_{1}, q_{2}, ..., q_{N} is a meromorphic function in time. We give explicit formulae for these functions in terms of genus N − 1 theta functions.

1 Introduction

The elliptic Calogero system [Ca 1975]

\frac{d^{2}}{dt^{2}} q_{i} = − ∑_{j≠i} ℘′(q_{i} − q_{j}), i = 1, 2, ..., N  (1.1)

is a canonical Hamiltonian system, describing the motion of N particles on the circle S^{1} = ℝ/ωℤ, ω ∈ ℝ, with Hamiltonian (energy)

H = \frac{1}{2} ∑_{j=1}^{N} p_{j}^{2} + ∑_{j<k}^{N} ℘(q_{j} − q_{k}),  (1.2)

where ℘(q) = ℘(q|ω, ω′) is the Weierstrass elliptic function

℘(q|ω, ω′) = ∑_{m,n∈ℤ}(q + mω + nω′)^{−2}, ω′/ω ∉ ℝ.  (1.3)

1On leave of absence from Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia. Current e-mail address: perelomo@posta.unizar.es

Denote by Γ_{1} the elliptic curve ℂ/{2ωℤ+2ω′ℤ} with period lattice generated by 2ω and 2ω′. The Hamiltonian H is invariant under the obvious action of the permutation group S_{n}, so the phase space of the compexified system is the cotangent bundle T^{∗}(S^{N}Γ_{1}) of the Nth symmetric product S^{N}Γ_{1}.

It is known that this system has two Lax representations ( [Ca 1975], [Kr 1980], see also [Pe 1990] for details ). The Lax operator L defines N integrals of motion I_{k}(p, q) = k^{−1}tr(L^{k}), k = 1, ..., N. It was proved in [Pe 1977] that these integrals are in involution and hence this system is completely integrable in the Jacobi–Liouville sense [Ja 1842/43], [Li 1855].

The Krichever Lax pair has a spectral parameter. This means that the equations of motion of the system under consideration are equivalent to the matrix equation

i L̇(λ) = [L(λ), M(λ)],  (1.4)

where L(λ) = L(p, q; λ) and M(λ) = M(p, q; λ) are two matrices of order N:

{L(λ)}_{jk} = p_{j} δ_{jk} + i (1 − δ_{jk}) Φ(q_{j} − q_{k}, λ);  (1.5)
{M(λ)}_{jk} = δ_{jk} (∑_{l≠j} ℘(q_{j} − q_{l}) − ℘(λ))

+ (1 − δ_{jk}) Φ′(q_{j} − q_{k}, λ);  (1.6)
Φ(q, λ) = \frac{σ(q − λ)}{σ(q) σ(λ)} exp (ζ(λ) q) ;  (1.7)

σ(q) = q ∏_{m,n}′ (1 − \frac{q}{ω_{mn}} ) exp [\frac{q}{ω_{mn}} + \frac{1}{2} (\frac{q}{ω_{mn}} )^{2}] , (1.8)

ζ(q) = \frac{σ′(q)}{σ(q)}, ω_{mn} = mω + nω′.

As it was shown by Krichever [Kr 1980], the equations of motion may be “linearized” on the Jacobian of the spectral curve

Γ^{N} = {(λ, μ) : f(λ, μ) ≡ det (L(λ) − μI) = 0}.  (1.9)

Namely, let

θ(z|B) = ∑_{N∈ℤ^{N}} e^{πi⟨N,BN⟩+2πi⟨N,z⟩}, z ∈ ℂ^{N}  (1.10)

be the Riemann theta function with period matrix B, where

B = (B_{ij}), B = B^{t}, Im B > 0, ⟨x, y⟩ = ∑_{j} x_{j}y_{j} , i, j = 1, ..., N.

It has been shown by Krichever [Kr 1980] that, if B is the period matrix of the curve Γ^{N}, then for suitable constant vectors U, V, W ∈ ℂ^{N} and for a fixed parameter t ∈ ℂ, the equation

θ(Uq + V t + W) = 0, q ∈ ℂ  (1.11)

has exactly N solutions q = q_{j}(t) on the Jacobian Jac (Γ^{N}) of the curve Γ^{N}. The functions q_{j}(t) provide solutions of the elliptic Calogero system (1.1). The equation (1.11) for these solutions is, however, not explicit and seems to be not well understood.

The aim of the present paper is to give “the effectivization” of these formulae based on the projection method by Olshanetsky and Perelomov ([OP 1976], [OP 1977]) of explicit integration of the equations of motion in the rational and the trigonometric cases, as well on the algebro-geometric approach of Krichever [Kr 1978], [Kr 1980].

2 Explicit solutions

Let Γ_{N} be a genus N Riemann surface which is an N-sheeted covering of an elliptic curve Γ_{1}

Γ_{N} →^{π} Γ_{1} .  (2.12)

It follows from a theorem of Weierstrass (see for example [Ko 1884], [Po 1884], [Po 1886] and [BBEIM 1994], Theorem7.4) that the period matrix of the curve Γ_{N} in a suitable basis has the form (I, B) where I = diag(1, 1, ..., 1), and

B = ( \begin{matrix}\frac{τ}{N} & \frac{k}{N} & 0 & … & 0 \\ \frac{k}{N} & b_{22} & b_{23} & … & b_{2N} \\ 0 & b_{32} & b_{33} & … & b_{3N} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & b_{N2} & b_{N3} & … & b_{NN} \end{matrix} ) .  (2.13)

for a suitable positive integer k. Consider the Riemann theta function θ(x, t) = θ(x, t|B), where t = (t_{1}, t_{2}, ..., t_{N−1}), (x, t) ∈ ℂ^{N}. We have

θ(x + 1, t) = θ(x, t), θ(x + τ, t) = e^{−2πiNx−πiNτ}θ(x, t), i = \sqrt{−1} (2.14)

and therefore for any fixed t the function θ(x, t) is an elliptic theta function of order N [Du 1981]. In particular it has exactly N zeros on Γ_{1} = ℂ/{ℤ + τℤ} which we denote by x_{i}(t), i = 1, 2, ..., N.

Lemma 2.1 . The following identity holds

\frac{∂^{2}}{∂x^{2}} log θ(x, t|B) = ∑_{i=1}^{N} ℘(x − x_{i}(t)|τ) + N \frac{θ′′′_{1}(0)}{3θ′_{1}(0)} ,

where [BE 1955]

θ_{1}(x|τ) = θ [ \begin{matrix}1/2 \\ 1/2 \end{matrix} ] (x|τ)

Proof. The relations

θ_{1}(x + 1) = −θ_{1}(x), θ_{1}(x + τ) = −e^{−2πix−πiτ}θ_{1}(x)  (2.15)

compared to (2.14) imply that

(\frac{θ(x, t)}{∏_{i=1}^{N} θ_{1}(x − x_{i}(t))})^{2}  (2.16)

is a meromorphic function in x on Γ_{1} which has no poles, and hence it is a constant (in x). It follows that

\frac{∂^{2}}{∂x^{2}} log \frac{θ(x, t)}{∏_{i=1}^{N} θ_{1}(x − x_{i}(t))} ≡ 0 .

Finally we use that

℘(x) = −\frac{∂^{2}}{∂x^{2}} log σ(x), θ_{1}(x) = c exp(ηx^{2}) σ(x)  (2.17)

where

η = −\frac{θ′′′_{1}(0)}{6θ′_{1}(0)}

and c is a suitable constant [BE 1955]. □

Theorem 2.2 . The Krichever curve Γ^{N} is an N-sheeted covering of an elliptic curve Γ_{1} = ℂ/{ℤ+τℤ}. There exists a canonical homology basis and a normalized basis of holomorphic one-forms on Γ^{N}, such that the corresponding period matrix of Γ_{N} takes the form (I, B), where I = diag(1, 1, ..., 1), and

B = ( \begin{matrix}\frac{τ}{N} & \frac{1}{N} & 0 & … & 0 \\ \frac{1}{N} & b_{22} & b_{23} & … & b_{2N} \\ 0 & b_{32} & b_{33} & … & b_{3N} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & b_{N2} & b_{N3} & … & b_{NN} \end{matrix} ) .  (2.18)

In the same basis the vectors U and V in (1.11) read

U = (1, 0, ..., 0), V = (0, V_{2}, ..., V_{N}) .  (2.19)

A direct proof (without using the Weierstrass theorem) of the above Theorem will be given in the last section. From now on we make the convention that 2ω = 1 so the period lattice of Γ_{1} is

ℤ + τℤ, τ = 2ω′/2ω = 2ω′ .

Corollary 2.3 . The symmetric functions

f_{k}(t) = ∑_{i=1}^{N} ℘^{(k)}(q_{i}(t))

are meromorphic in t. Explicit formulae for them are obtained from Lemma 2.1:

f_{0}(t) = \frac{∂^{2}}{∂x^{2}} log θ(x, t)|_{x=0} − N \frac{θ′′′_{1}(0)}{3θ′_{1}(0)}

f_{k}(t) = (−1)^{k} \frac{∂^{k+2}}{∂x^{k+2}} log θ(x, t)|_{x=0}, k > 0,

where

t = (V_{2}t + W_{2}, V_{3}t + W_{3}, ..., V_{N}t + W_{N}) .

Our next construction is motivated by [OP 1976], [OP 1977] and [Kr 1980]. Let us define the function

F (x, t) = ∏_{j=1}^{N} \frac{σ(x − q_{j}(t))}{σ(x)σ(q_{j}(t))} = [θ′_{1}(0)]^{−N}] ∏_{j=1}^{N} \frac{θ_{1}(x − q_{j}(t))}{θ_{1}(x)θ_{1}(q_{j}(t))}, ∑_{j=1}^{N} q_{j}(t) = 0

(2.20)

where

q_{j}(t), t ∈ ℂ, j = 1, 2, ..., N,

is a solution of the elliptic Calogero system.

Lemma 2.4 . F (x, t) is a meromorphic function in x on Γ_{1} and meromorphic function in t on ℂ, explicitly given by

F (x, t) = [−θ′_{1}(0)]^{−N}]\frac{θ(Ux + V t + W)}{θ_{1}(x)^{N}θ(V t + W)} .  (2.21)

Proof. We already noted that the function (2.16) is a constant in x, and hence

\frac{θ(x, t)}{∏_{i=1}^{N} θ_{1}(x − x_{i}(t))} ≡ \frac{θ(0, t)}{∏_{i=1}^{N} θ_{1}(−x_{i}(t))} .

This combined with (2.19) gives

\frac{∏_{i=1}^{N} θ_{1}(x − q_{i}(t))}{∏_{i=1}^{N} θ_{1}(q_{i}(t))} = (−1)^{N} \frac{θ(Ux + V t + W)}{θ(V t + W)} .

□

The expansion of F (x, t) on the basis of first order theta functions in x defines (N − 1) meromorphic functions in the variables q_{1}, …, q_{N} which are also meromorphic functions in t with only simple poles. Hence we can take them as new “good” variables. The expansion of F (x, t) can be obtained by making use of the addition formulae for elliptic functions. In the case N = 2, we have the following “addition formula” [BE 1955]

F (x, t) = −\frac{σ(x − q) σ(x + q)}{σ^{2}(x)σ^{2}(q)} = ℘(x) − ℘(q) ,  (2.22)

which generalizes for arbitrary N in the following way Lemma 2.5 . For any q = (q_{1}, q_{2}, ..., q_{N}), x, such that ∑ q_{j} = 0 define

F (x, q) = ∏_{j=1}^{N} \frac{σ(x − q_{j})}{σ(x)σ(q_{j})}  (2.23)

Δ(q) = (N −1)! det | \begin{matrix}1 & ℘(q_{1}) & ℘′(q_{1}) & … & ℘^{(N−3)}(q_{1}) \\ 1 & ℘(q_{2}) & ℘′(q_{2}) & … & ℘^{(N−3)}(q_{2}) \\ … & … & … & … & … \\ 1 & ℘(q_{N−1}) & ℘′(q_{N−1}) & … & ℘^{(N−3)}(q_{N−1}) \end{matrix} | . (2.24)

The following identity holds

F (x, q)Δ(q) ≡ det | \begin{matrix}1 & ℘(x) & ℘′(x) & … & ℘^{(N−2)}(x) \\ 1 & ℘(q_{1}) & ℘′(q_{1}) & … & ℘^{(N−2)}(q_{1}) \\ … & … & … & … & … \\ 1 & ℘(q_{N−1}) & ℘′(q_{N−1}) & … & ℘^{(N−2)}(q_{N−1}) \end{matrix} | (2.25)

Remark. The substitution x = q_{N} in (2.25) gives the following addition formula for the Weierstrass ℘-function

det | \begin{matrix}1 & ℘(q_{1}) & ℘′(q_{1}) & … & ℘^{(N−2)}(q_{1}) \\ 1 & ℘(q_{2}) & ℘′(q_{2}) & … & ℘^{(N−2)}(q_{2}) \\ … & … & … & … & … \\ 1 & ℘(q_{N}) & ℘′(q_{N}) & … & ℘^{(N−2)}(q_{N}) \end{matrix} | ≡ 0 .  (2.26)

Proof. For fixed q = (q_{1}, q_{2}, ..., q_{N}) the functions in the left and right-hand side of the identity (2.25) are meromorphic in x on the elliptic curve Γ_{1}. Both of them have a pole of order N at x = 0 and simple zeros at x = q_{1}, ..., q_{N−1}. It follows that their ratio is a first order elliptic function, and hence a constant in x. To compute this constant we use that σ(x) = x + ..., ℘(x) = 1/x^{2} + ..., and then compare the Laurent series of the two functions in a neighborhood of x = 0. □

Note finally that if for fixed q and q̃ holds F (x, q) ≡ F (x, q̃), then up to a permutation q = q̃. Therefore there is a one-to-one correspondence between the coefficients of ℘^{k}(x) in the expansion of F (x, q), and the points of the (N − 1)th symmetric power of the elliptic curve Γ_{1}\{0}. In particular every meromorphic function on this symmetric power is a rational function in the above coefficients. This implies the following

Corollary 2.6 . Let f(x) be a meromorphic function on the elliptic curve Γ_{1}, and let S be a symmetric rational function in N − 1 variables. If q_{1}(t), q_{2}(t), ..., q_{N}(t), ∑ q_{i} ≡ 0 is a solution of the elliptic Calogero system, then S(f(q_{1}(t), f(q_{2}(t)), ..., f_{N−1}(q_{N−1}(t))) is a meromorphic function in t.

The further analysis of the explicit formulae for the solutions of the elliptic Calogero system can be based on Lemma 2.4, Lemma 2.5, and the identity

F (x, t) ≡ F (x, q(t)) .

Consider the seemingly trivial case of two particles (N = 2). Let us give first an explanation of the Krichever formula (1.11) for the solutions q_{1}(t) = −q_{2}(t). Put q_{1} − q_{2} = q and p_{1} = −p_{2} = p. The Hamiltonian H becomes H(p, q) = p^{2} + ℘(q), and the reduced Hamiltonian system is

\frac{d}{dt}q = 2p, \frac{d}{dt}p = − ℘′(q), (q, p) ∈ T^{∗}Γ_{1} .  (2.27)

The Lax matrix L is

L(λ) = ( \begin{matrix}p & iΦ(q, λ) \\ iΦ(−q, λ) & −p \end{matrix} )

and the corresponding spectral polynomial

det(L(λ) − μI) = μ^{2} − p^{2} + Φ(q, λ)Φ(−q, λ)
= μ^{2} − p^{2} + ℘(λ) − ℘(q) = μ^{2} + ℘(λ) − H(p, q)

defines a spectral curve

Γ_{2} = {(μ, λ) : μ^{2} + ℘(λ) = h} .

Suppose that h = H(p, q) is fixed in such a way, that the meromorphic function ℘(λ) − h has two distinct zeros on Γ_{1}. The spectral curve Γ_{2} is a double ramified covering over the elliptic curve Γ_{1} with projection map π : Γ_{2} → Γ_{1} : (μ, λ) → λ. It follows that Γ_{2} is a genus two curve with holomorphic differentials

ω_{1} = dλ, ω_{2} = \frac{dλ}{μ} .

On the other hand Γ_{2} is identified to the orbit

{(p, q) ∈ T^{∗}Γ_{1} =: H(p, q) = h}

under the map

(p, q) → (μ, λ) .

Consider further the embedding of the orbit Γ_{2} into its Jacobian variety Jac(Γ_{2})

Γ_{2} → Jac(Γ_{2}) : P → (∫_{P_{0}}^{P} dλ, ∫_{P_{0}}^{P} \frac{dλ}{μ}) .  (2.28)

By the Riemann theorem [GH, 1978], the curve Γ_{2} ⊂ Jac(Γ_{2}) defines a divisor which coincides, up to addition of a constant, with the Riemann theta divisor Θ ⊂ Jac(Γ_{2}) on the Jacobian variety Jac(Γ_{2}).

Let (p(t), q(t)) be a solution of the elliptic Calogero system, with initial condition (p(t_{0}), q(t_{0})) = P_{0}. Taking into consideration that

\frac{dλ}{μ} = 2 dt, d λ = dq , (λ, μ) ∈ Γ_{2}

the formula (2.28) takes the form

T^{∗}Γ_{1} = ℂ × Γ_{1} ∋ (p(t), q(t)) → (2t − 2t_{0}, q(t) − q(t_{0})) ∈ Jac(Γ_{2}) . (2.29)

It follows that there exist constant vectors a, b, c ∈ ℂ^{2} such that

θ(aq(t) + bt + c) ≡ 0 .  (2.30)

Of course these constants depend on the choise of symplectic homology basis and the choice of normalized basis of holomorphic one-forms. Namely, let a, b be two loops on Γ_{1}, such that π^{−1}(a) = {a_{1}, a_{2}}, π^{−1}(b) = {b_{1}, b_{2}}, where a_{i}, b_{j} represent an integer symplectic homology basis on Γ_{2}: a_{i} ∘ b_{j} = δ_{ij}, a_{i} ∘ a_{j} = 0, b_{i} ∘ b_{j} = 0. Then

∫_{a_{1}} dλ = ∫_{a_{2}} dλ, ∫_{b_{1}} dλ = ∫_{b_{2}} dλ

∫_{a_{1}} \frac{dλ}{μ} = − ∫_{a_{2}} \frac{dλ}{μ}, ∫_{b_{1}} \frac{dλ}{μ} = − ∫_{b_{2}} \frac{dλ}{μ} .

If we define a new symplectic basis

ã_{1} = a_{1} + a_{2}, ã_{2} = b_{1} − b_{2}, b̃_{1} = b_{1}, b̃_{2} = a_{2}

and normalize the two holomorphic one-forms as

dλ → \frac{dλ}{∫ã π^{∗}dλ} = \frac{dλ}{2 ∫_{a} dλ}, \frac{dλ}{μ} → \frac{dλ/μ}{∫ã_{2} dλ/μ},

then the period matrix of Γ_{2} takes the form

( \begin{matrix}1 & 0 & τ_{1}/2 & 1/2 \\ 0 & 1 & 1/2 & τ_{2}/2 \end{matrix} )

where

τ_{1} = \frac{∫_{b} dλ}{∫_{a} dλ}, τ_{2} = \frac{∫_{a_{2}} dλ/μ}{∫_{b_{1}} dλ/μ} .

This, together with (2.29) implies that

a = (\frac{1}{∫ã_{1} dλ}, 0) = (\frac{1}{2 ∫_{a} dλ}, 0), b = (0, \frac{1}{∫_{b_{1}} dλ/μ}) .

Finally the vector c is arbitrary and plays the role of initial condition. The function F (x, t) defined in (2.20) takes the form

F (x, t) = −\frac{σ(x − q(t)) σ(x + q(t))}{σ^{2}(x)σ^{2}(q(t))}  (2.31)

and hence [BE,1955], [WW 1927]

F (x, t) = ℘(x) − ℘(t) .  (2.32)

So the elliptic function ℘(q|ω, ω′), and also

sn^{2}(q, k) ∼ \frac{θ_{1}^{2}(q|k)}{θ_{4}^{2}(q|k)}, cn^{2}(q, k) ∼ \frac{θ_{2}^{2}(q|k)}{θ_{4}^{2}(q|k)}, dn^{2}(q, k) ∼ \frac{θ_{3}^{2}(q|k)}{θ_{4}^{2}(q|k)} (2.33)

are “good” variables (in the sense that they are meromorphic in t). The equation of motion for them takes a very simple form. We get

sn^{2}(q, k) = 1 − a^{2} + a^{2} sn^{2}(γt, k̃),  (2.34)

where

a^{2} = \frac{h − 1}{h}, γ = 2(h − k^{2}), k̃^{2} = \frac{h − 1}{h − k^{2}} k^{2} .  (2.35)

One can easily show that the even functions cn (q, k) and dn (q, k) (but not sn (q, k)) are “good” variables and we get as in [Pe 1990]

cn (q, k) = α cn (γt, k̃),  (2.36)
dn (q, k) = β dn (γt, k̃), b = (k/k̃)a.  (2.37)

3 Reduction of theta functions

The reduction theory was elaborated by Weierstrass (see for example [Ko 1884]) and Poincaré [Po 1884], [Po 1886]. Consider first the case N = 2. The Riemann theta function associated with the Riemann matrix (2.18) has the form:

θ(z_{1}, z_{2}) = ∑_{n_{i},n_{j}} exp{iπ [B_{ij}n_{i}n_{j} + 2n_{j}z_{j}]}, i, j = 1, 2.  (3.1)

where

B_{11} = τ_{1}/2, B_{22} = τ_{2}/2, B_{12} = B_{21} = 1/2.

A straightforward computation gives

θ(z_{1}, z_{2}) = ∑_{n_{1},n_{2}} exp\{iπ [τ_{1} \frac{n_{1}^{2}}{2} + n_{1}n_{2} + τ_{2} \frac{n_{2}^{2}}{2} + 2n_{1}z_{1} + 2n_{2}z_{2}]\}

= ∑_{k_{1},n_{2}∈ℤ}exp\{iπ [2τ_{1}k_{1}^{2} + 4k_{1}z_{1}]\} exp\{iπ [τ_{2} \frac{n_{2}^{2}}{2} + 2n_{2}z_{2}]\}

+ ∑_{k_{1},n_{2}∈ℤ}exp\{iπ [2τ_{1}(k_{1} + \frac{1}{2})^{2} + 4(k_{1} + \frac{1}{2})z_{1}]\} exp\{iπ [τ_{2} \frac{n_{2}^{2}}{2} + 2(n_{2} + \frac{1}{2})z_{2}]\}

= θ_{3}(2z_{1}|2τ_{1}) θ_{3}(z_{2}|\frac{τ_{2}}{2} ) + θ_{2}(2z_{1}|2τ_{1}) θ_{4}(z_{2}|\frac{τ_{2}}{2} )

where θ_{1}, θ_{2}, θ_{3} and θ_{4} are defined by formulae:

θ_{1}(z|τ) = θ [ \begin{matrix}1/2 \\ 1/2 \end{matrix} ] (z|τ) = 2q^{1/4} ∑_{n=1}^{∞}(−1)^{n}q^{n(n+1)} sin[(2n + 1)πz];(3.2)

θ_{2}(z|τ) = θ [ \begin{matrix}1/2 \\ 0 \end{matrix} ] (z|τ) = 2q^{1/4} ∑_{n=1}^{∞} q^{n(n+1)} cos [(2n + 1)πz];  (3.3)

θ_{3}(z|τ) = θ [ \begin{matrix}0 \\ 0 \end{matrix} ] (z|τ) = 1 + 2 ∑_{n=1}^{∞} q^{n^{2}} cos(2πnz); q = exp(iπτ); (3.4)

θ_{4}(z|τ) = θ [ \begin{matrix}0 \\ 1/2 \end{matrix} ] (z, τ) = 1 + 2 ∑_{n=1}^{∞}(−1)^{n} q^{n^{2}} cos (2πnz).  (3.5)

So in this case, the equation θ(z_{1}, z_{2}) = 0 is equivalent either to

A dn (2z_{1}|4τ_{1}) dn (z_{2}|τ_{2}) + cn (2z_{1}|4τ_{1}) = 0,  (3.6)

or to

A dn (2z_{2}|4τ_{2}) dn (z_{1}|τ_{1}) + cn (2z_{2}|4τ_{2}) = 0,  (3.7)

where

A = \frac{θ_{3}(0|4τ_{1}) θ_{3}(0|τ_{2})}{θ_{2}(0|4τ_{1}) θ_{4}(0|τ_{2})}  (3.8)

or

dn (z_{1}|τ_{1}) = B dn (2iz_{2} + K|τ̃_{2}).  (3.9)

Let us give also a more symmetric form of the theta divisor for this case:

dn(2z_{1}, k_{1}) dn(2z_{2}, k_{2}) + dn(2z_{1}, k_{1}) cn(2z_{2}, k_{2}) + cn(2z_{1}, k_{1}) dn(2z_{2}, k_{2})

− cn(2z_{1}, k_{1}) cn(2z_{2}, k_{2}) = 0.  (3.10)

Using the constraint θ(ax+bt+c) = 0 and taking z_{1} = q, z_{2} = (1/2)K + iγt, we get once again (2.36), (2.37).

Consider now the case of arbitrary N. Let θ(z_{1}, z_{2}, ..., z_{N}|B) be the Riemann theta function with period matrix as in Theorem 2.2. In a quite similar way we get

θ(z_{1}, z_{2}, …, z_{N}) = ∑_{j=0}^{N−1} θ_{j}(z_{1}) Θ_{j}(z_{2}, …, z_{N}),  (3.11)

where

θ_{j}(z_{1}) = θ [ \begin{matrix}j/N \\ 0 \end{matrix} ] (Nz_{1}|N^{2}τ_{1}),  (3.12)

Θ_{j}(z_{2}, …, z_{N}) = Θ [ \begin{matrix}0 & 0 & … & 0 \\ j/N & 0 & … & 0 \end{matrix} ] (z_{2}, …, z_{N}|B̂).  (3.13)

In the above formula B̂ is the right lower (N − 1) × (N − 1) minor of B (2.18), and the theta functions with fractional characteristics are defined for example in [Kr 1903], [Ko 1976], [Du 1981], [BBEIM 1994]. A reduction formula similar to (3.11), but containing N^{2} terms, can be found in [BBEIM 1994],Corollary 7.3.

4 Geometry of the spectral curve

In this section we prove Theorem 2.2.

Let Γ_{N} be a genus N Riemann surface which is an N-sheeted covering of an elliptic curve Γ_{1}

Γ_{N} →^{π} Γ_{1} .  (4.1)

Choose two loops a, b which generate the fundamental group π_{1}(Γ_{1}, P), P ∈ Γ_{1}, and denote Γ̌_{1} = Γ_{1}\{a ∪ b}. Let us suppose for simplicity that the ramification points of the projection map π are distinct. Connect further these ramification points by non-intersecting arcs γ_{i} ⊂ Γ̌_{1}. The set π^{−1}(Γ̌_{1}\ ∪_{i} γ_{i}) is a disjoint union of N “sheets”. To reconstruct the topological covering (4.1) we have to indicate how the opposite borders of the cuts γ_{i} are glued, as well how the opposite borders of the (pre-images of the) cuts a and b respectively are glued together. Thus there is only a finite number of topologically different coverings (4.1). It may be shown that the Krichever curve (1.9) is of genus at most N, and for generic (p_{i}, q_{i}) its genus is exactly N. The projection map π (4.1) is defined then by π(μ, λ) = λ. ¿From now on we shall always assume that (p_{i}, q_{i}) are generic. In the case when Γ_{N} is the genus N Krichever spectral curve (1.9), and Γ_{1} is the elliptic curve with half periods ω, ω′, the covering (4.1) has a number of special properties.

To prove (2.18) we shall need the following

Proposition 4.1 . Let Γ_{N} be the Krichever curve (1.9). There exist loops a, b ∈ π_{1}(Γ_{1}, P) such that, if Γ̌_{1} = Γ_{1}\{a ∪ b}, ∂Γ̌_{1} = a ∘ b ∘ a^{−1} ∘ b^{−1}, then i. π^{−1}(Γ̌_{1}) is connected ii. π^{−1}(∂Γ̌_{1}) has exactly N connected components.

On its hand the above proposition implies the following

Proposition 4.2 . There exist loops a, b ∈ π_{1}(Γ_{1}, P), P ∈ Γ_{1}, such that

π^{−1}(a) = {a_{1}, a_{2}, ..., a_{N}}, π^{−1}(b) = {b_{1}, b_{2}, ..., b_{N}}

where a_{i}, b_{i} represent a symplectic homology basis of H_{1}(Γ_{N}, ℤ), a_{i} ∘ b_{j} = δ_{ij}.

Proof of (2.18) assuming Proposition 4.2. Let dλ be the holomorphic one-form on Γ_{1}. Then the pullback π^{∗}dλ of dλ is a holomorphic one-form on Γ_{N} and we have

∫_{a_{i}} π^{∗}dλ = ∫_{a} dλ, ∫_{b_{i}} π^{∗}dλ = ∫_{b} dλ .

Choose the following new integer homology basis of Γ_{N}

ã_{1} = a_{1} + a_{2} + ...a_{N}, b̃_{1} = b_{1},
ã_{2} = Nb_{1} − b_{1} − b_{2} − ... − b_{N}, b̃_{2} = a_{2}

and

ã_{i} = b_{i} − b_{1}, b̃_{i} = a_{2} − a_{i}, i = 3, ..., N .

This is also a symplectic basis of H_{1}(Γ_{N}, ℤ), as

∑_{i=1}^{N} ã_{i} ∧ b̃_{i} = ∑_{i=1}^{N} a_{i} ∧ b_{i} .

Let ω_{1}, ω_{2}, ..., ω_{N} be a basis of holomorphic one-forms on Γ_{N}, such that

ω_{1} = \frac{dλ}{∫ã_{1} dλ}, ∫ã_{i} ω_{j} = δ_{ij} .

Then B = (∫b̃_{j} ω_{i})_{i,j}^{N,N} is a symmetric matrix with positive definite imaginary part, such that

∫b̃_{1} ω_{1} = \frac{τ}{N} , ∫b̃_{2} ω_{1} = \frac{1}{N} , ∫b̃_{i} ω_{1} = 0, i ≥ 3

which completes the proof of 2.18. Proof of Proposition 4.1. First of all let us note that if the claim holds for some Krichever curve, then it holds for any Krichever curve. Indeed, the space of all such curves is parameterized by ℂ^{N−1} (the first integrals of the integrable Hamiltonian system (1.4)) and hence it is connected. Let us fix a generic point (p_{i}, q_{i}), i = 1, 2, ..., N. It is enough to prove now our proposition for at least one pair of half-periods ω, ω′, for example for |ω|, |ω′| ∼ ∞.

Let us represent Γ̌_{1} ⊂ ℂ = IP^{1}\∞ as the interior of the period parallelogram formed by 2ω and 2ω′. When |ω| → ∞, |ω′| → ∞, the boundary. ∂Γ̌_{1} = a ∘ b ∘ a^{−1} ∘ b^{−1} tends to ∞ ∈ IP^{1}, and Γ̌_{1} tends to Γ̌_{1}^{∞} = ℂ. In a similar way we define the “limit” curve Γ̌_{N}^{∞} which is explicitly described in the following way. When |ω| → ∞, |ω′| → ∞, then on any compact set the Weierstrass functions σ(q), ζ(q), ℘(q) tend to q, 1/q, 1/q^{2} respectively, and hence the function Φ(q, λ) tends to

\frac{q − λ}{qλ} exp(q/λ).

Denote the corresponding “limit” Lax matrix (1.5) by L^{∞}(λ). The curve Γ̌_{N}^{∞} is the affine curve

{(λ, μ) : det(L^{∞}(λ) − μI_{N}) = 0}

completed with N distinct points corresponding to λ = 0. The last holds true if and only if the ramification points of the projection map π (4.1) tend to some values different from λ = 0 (it is easy to check that this is a generic condition on (p_{i}, q_{i})). We shall also suppose that these values are different from λ = ∞ (another generic condition). Under these restrictions one may prove (as in [Kr 1980]) that Γ̌_{N}^{∞} is a Riemann sphere, with N punctures (the pre-images of λ = ∞). We obtain thus a map π : IP^{1} → IP^{1} with 2N − 2 ramification points different from λ = 0, ∞. The fact that π^{−1}(ℂ) is connected implies the part i. of the proposition, and the fact that π^{−1}(∞) is a disjoint union of N points implies ii. Proof of Proposition 4.2. Let us represent Γ̌_{N} by a graph with N vertices. A vertex corresponds to a sheet (see the beginning of this section), an edge connects two vertices if and only if the corresponding sheets have a common ramification point. Proposition 4.1. i. implies that the graph is connected, and ii. that each sheet contains an even number of ramification points. As the total number of ramification points is 2N −2 and each point belongs to exactly two sheets, then in addition the graph of Γ̌_{N} is simply connected.

Consider now the punctured curve

Γ̃_{1} = Γ̌_{1}\ ∪_{i} R_{i} ,

where R_{i}, i = 1, ..., 2N − 2 are the ramification points of π. The fundamental group π_{1}(Γ̃_{1}, P) has a natural representation in the permutation group S_{n}. Namely, when a point Q ∈ Γ_{1} makes one turn along a loop a ∈ π_{1}(Γ̃_{1}, P), the set π^{−1}(P) = ∪_{i=1}^{N}P_{i} is transformed to itself. If the loops a and b induce the identity permutation, then π^{−1}(a),π^{−1}(b) are disjoint unions of N loops with obvious intersections, which implies Proposition 4.2. If not, we shall modify a and b.

Let c ∈ π_{1}(Γ̃_{1}, P) be a loop which makes one turn around some ramification point of π. Then c induces a permutation which exchanges the two sheets containing the ramification point. As the graph of Γ̌_{N} is connected then all such transpositions generate the permutation group S_{n}. Thus for suitable c the loop a ∘ c induces the identity permutation. It remains to substitute a → a ∘ c and to note that a = a ∘ c in π_{1}(Γ_{1}, P). Proof of (2.19) (compare to [Be 1994), Theorem 7.14). Let 0 ∈ Γ_{1} be the pole of ℘(z). We denote

π^{−1}(0) = {∞_{1}, ∞_{2}, ..., ∞_{N}}, ∞_{i} ∈ Γ_{N} .

In a neighborhood of each point ∞_{i} on the Krichever curve {(λ, μ) : f(λ, μ) = 0} the meromorphic function μ has the following Laurent expansion [Kr 1978]

μ = −\frac{1}{λ} + O(1), i = 1, 2, ..., N − 1,

μ = \frac{N − 1}{λ} + O(1) .

It follows that if

ω_{j} = f_{j}(P)dλ, P = (λ, μ) ∈ Γ_{N}

is a differential of first kind (i.e. holomorphic) on Γ_{N}, then μω_{j} is a differential of third kind with simple poles at ∞_{i}. The sum of the residues of μω_{j} is equal to

∑_{i=1}^{N−1} f_{j}(∞_{i}) − (N − 1)f_{j}(∞_{N}) = 0 .  (4.2)

Let Ω be a differential of second kind on Γ_{N} with a single pole at ∞_{N}. Such is for example the differential

\frac{μ^{2} − ℘(λ)}{\frac{∂f}{∂μ}(λ, μ)}dλ .

If moreover Ω s normalized as

∫ã_{i} Ω = 0

then it is well known that the vector V is co-linear to

(∫b̃_{1} Ω, ∫b̃_{2} Ω, ..., ∫b̃_{N} Ω)

(see for example [BBEIM 1994]). Equivalently, if we apply the reciprocity law to the differentials of second and first kind Ω, ω_{i}, we get that V is co-linear to

(f_{1}(∞_{N}), f_{2}(∞_{N}), ..., f_{N}(∞_{N})) .

On the other hand

ã_{1} = a_{1} + a_{2} + ... + a_{N} = π^{−1}(a)

and hence

∫ã_{1} ω_{i} = ∑_{k=1}^{N} ∫_{a} f_{i}(λ, μ_{k})dλ

where (λ, μ_{k}) ∈ Γ_{N} are the N pre-images of λ ∈ Γ_{1}. It is clear that ∑_{k=1}^{N} f_{i}(λ, μ_{k}) is a single-valued function on Γ_{1}. As ω_{i} is a holomorphic differential on Γ_{N} and dλ is the holomorphic differential on Γ_{1}, then ∑_{k=1}^{N} f_{i}(λ, μ_{k}) is a holomorphic function on Γ_{1} and hence a constant. As ω_{i} is a normalized basis of holomorphic forms, then ∫ã_{1} ω_{i} = 0 for i ≥ 2, and hence

∑_{k=1}^{N} ∫_{a} f_{i}(λ, μ_{k})dλ = ∑_{k=1}^{N} f_{i}(λ, μ_{k}) ∫_{a} dλ ≡ 0, (λ, μ) ∈ Γ_{N}, i ≥ 2 .

Therefore

∑_{k=1}^{N} f_{i}(λ, μ_{k}) ≡ 0, i ≥ 2

which combined with (4.2) implies that

f_{i}(∞_{N}) = 0, i ≥ 2

and hence the vector V is co-linear to (1, 0, ..., 0). In fact V is equal to this vector, because q_{i}(t) ∈ Γ_{1} = ℂ/{ℤ + τℤ}. Finally, we may always suppose that U = (0, U_{2}, ..., U_{N}). Indeed the Calogero system (1.1) is invariant under the translation

q_{i} → q_{i} − V_{1}t .

□

Acknowledgments. One of authors (A. P.) would like to thank the Laboratoire Emile Picard at the University of Toulouse III and Max-Planck-Institut für Mathematik, Bonn for the hospitality, and B.A. Dubrovin for the useful remarks.

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