Finite gap theory of the Clifford torus

Iskander A. TAIMANOV ^{∗}

Abstract

We construct the spectral curve and the Baker–Akhiezer function for the Dirac operator which corresponds to the Clifford torus via the Weierstrass representation. This torus appears in many conjectures in differential geometry. By constructing this Baker–Akhiezer function we demonstrate a general procedure for constructing Dirac operators and their Baker–Akhiezer functions corresponding to singular spectral curves

1 Introduction and main results

In this paper we construct the spectral curve and the Baker–Akhiezer function for the Dirac operator which form the data of the Weierstrass representation of the Clifford torus. This torus appears in many conjectures in differential geometry (see Section 2).

By constructing this Baker–Akhiezer function we demonstrate a general procedure for constructing Dirac operators and their Baker– Akhiezer functions corresponding to singular spectral curves. This procedure is exposed in Section 3.

The Clifford torus is a torus embedded into ℝ^{3}. The Dirac operator which corresponds to it via the Weierstrass representation (see [11, 12]) is

D = ( \begin{matrix}0 & ∂ \\ −¯∂ & 0 \end{matrix} ) + ( \begin{matrix}U & 0 \\ 0 & U \end{matrix} )

with the potential

U(z, z̄) = \frac{sin y}{2\sqrt{2}(sin y − \sqrt{2})}, z = x + iy.  (1)

We have Theorem 1 The Baker–Akhiezer function of the Dirac operator D with the potential (1) is a vector function ψ(z, z̄, P), where z ∈ ℂ and P belongs to the spectral curve Γ of this operator, such that

∗Institute of Mathematics, 630090 Novosibirsk, Russia; e-mail: taimanov@math.nsc.ru

1) the spectral curve Γ is a sphere ℂP^{1} = ℂ̄ with two marked points ∞_{+} = (λ = ∞), ∞_{−} = (λ = 0) where λ is an affine parameter on ℂ ⊂ ℂP^{1} and with two double points obtained by stacking together the points from the following pairs:

(\frac{1 + i}{4}, \frac{−1 + i}{4}) and (−\frac{1 + i}{4}, \frac{1 − i}{4});

2) the function ψ is meromorphic on Γ \ {∞_{±}} and has at the marked points (“infinities”) the following asymptotics:

ψ ≈ ( \begin{matrix}e^{k_{+}z} \\ 0 \end{matrix} ) as k_{+} = λ → ∞; ψ ≈ ( \begin{matrix}0 \\ e^{k_{−}z̄} \end{matrix} ) as k_{−} = −\frac{|u|^{2}}{λ} → ∞

where k_{±}^{−1} are local parameters near ∞_{±} and u = \frac{1+i}{4};

3) ψ has three poles on Γ \ {∞_{±}} which are independent on z and have the form

p_{1} = \frac{−1 + i + \sqrt{−2i − 4}}{4\sqrt{2}}, p_{2} = \frac{−1 + i − \sqrt{−2i − 4}}{4\sqrt{2}}, p_{3} = \frac{1}{\sqrt{8}}.

Therewith the geometric genus p_{g}(Γ) and the arithmetic genus p_{a}(Γ) of Γ are as follows:

p_{g}(Γ) = 0, p_{a}(Γ) = 2.

The Baker–Akhiezer function satisfies the Dirac equation

Dψ = 0

at any point of Γ \ {∞_{+}, ∞_{−}, p_{1}, p_{2}, p_{3}} where the potential U of the Dirac operator takes the form (1).

The Clifford torus is constructed via the Weierstrass representation (2) from the function

ψ = ψ (z, z̄, \frac{1 − i}{4}) .

Remark that in the proof of this theorem which will be given in Section 4 it is actually will be showed that the Baker–Akhiezer function ψ takes the form

ψ_{1}(z, z̄, λ) = e^{λz− \frac{|u|^{2}}{λ} z̄} (q_{1} \frac{λ}{λ − p_{1}} + q_{2} \frac{λ}{λ − p_{2}} + (1 − q_{1} − q_{2})\frac{λ}{λ − p_{3}} ) ,

ψ_{2}(z, z̄, λ) = e^{λz− \frac{|u|^{2}}{λ} z̄} (t_{1} \frac{p_{1}}{p_{1} − λ} + t_{2} \frac{p_{2}}{p_{2} − λ} + (1 − t_{1} − t_{2})\frac{p_{3}}{p_{3} − λ})

where u = \frac{1+i}{4} and q_{1}, q_{2}, t_{1}, t_{2} are functions of z, z̄ which are uniquely defined by the conditions

ψ (z, z̄, \frac{1 + i}{4}) = ψ (z, z̄, \frac{−1 + i}{4}) ,

ψ (z, z̄, −\frac{1 + i}{4}) = ψ (z, z̄, \frac{1 − i}{4}) .

It appears that q_{1}, q_{2}, t_{1}, t_{2} are 2π-periodic functions of y and are independent on x.

We would like to mention one interesting feature:

• the spectral curve Γ admits a holomorphic involution σ(λ) = −λ which preserves the infinities ∞_{±}. Although both the Jacoby variety J(Γ/σ) of the quotient space Γ/σ and the Prym variety of this involution are noncomplete Abelian varieties, the potential is a smooth function.

This is explained by some effect unfamiliar for other operators. It is as follows.

The potential U is written in the terms of Prym functions (as in the case of the two-dimensional Schrödinger operator and some other operators, see [3]). Although the Prym variety is isomorphic to ℂ^{∗}, the potential depends on one real-valued variable y in a way that it is a restriction of some meromorphic function on the Prym variety onto a compact circle S^{1} ⊂ ℂ^{∗}. This circle is a compact subgroup of ℂ^{∗}.

The correspondence between tori in ℝ^{3} and Dirac operators D with real-valued potentials is established by the Weierstrass representation.

It is based on a local representation of any surface immersed into ℝ^{3} by the formulas

\begin{matrix}x^{k} = x^{k}(0) + ∫ (x_{z}^{k}dz + \overline{x_{z}^{k}}dz̄) , k = 1, 2, 3, \\ x_{z}^{1} = \frac{i}{2}(ψ̄_{2}^{2} + ψ_{1}^{2}), x_{z}^{2} = \frac{1}{2}(ψ̄_{2}^{2} − ψ_{1}^{2}), x_{z}^{3} = ψ_{1}ψ̄_{2}, \end{matrix}  (2)

where ψ meets the Dirac equation

Dψ = 0.

In this event z defines a conformal parameter z = x+iy on the surface, the first fundamental form equals e^{2α}dzdz̄ and

U = \frac{He^{α}}{2}

where H is the mean curvature (see [7, 11]).

After a globalization we obtain for a closed surface in ℝ^{3} a representation by these formulas where ψ is a solution of the Dirac equation and the Dirac operator acts on smooth sections of some spinor bundles over a conformally equivalent surface of constant curvature [11, 12].

It appears that the spectral curve of D on the zero energy level defined initially for the two-dimensional Schrödinger operator in [2] has to have some geometric meaning.

Let us briefly recall the origin of the spectral curves in theory of differential operators with double-periodic coefficients.

The spectral curve Γ is a complex curve which parameterizes the Floquet functions on the zero energy level, i.e. joint eigenfunctions of D and translations T_{γ} by periods

T_{γ}f(z, z̄) = f(z + γ, z̄ + γ̄), U(z + γ, z̄ + γ̄) = U(z, z̄).

Here Dψ = 0 (the “eigenvalue” equals zero) and ψ is considered as a formal analytic solution to this equation not necessary belonging to some fixed functional space. Floquet functions are glued together into a meromorphic function ψ(z, z̄, P) on the spectral curve. If Γ is of finite genus, it is also completed by two “infinities” at which ψ has exponential asymptotics. This would be the Baker–Akhiezer function. Given a pair of generators γ_{1}, γ_{2} of the period lattice, to every Floquet function ψ(z, z̄, P) where corresponds two functions on the spectral curve holomorphic outside the “infinities”, the multipliers μ_{1}(P) and μ_{2}(P), P ∈ Γ, such that

T_{γ_{j}}ψ(P) = μ_{j}ψ(P).

We correspond to a torus in ℝ^{3} an operator D via the Weierstrass representation of the torus and the spectral curve of this operator and define the spectral genus of a torus as the geometric genus of the normalization of the spectral curve (see [12]).

We conjectured that • the spectral curve Γ corresponding to a torus in ℝ^{3} and the

multipliers μ_{1}, μ_{2} : Γ → ℂ are invariant under conformal transformations of the ambient space ℝ^{3} (that was confirmed in [5]);

• given a conformal class of a torus, the Willmore functional attains its minima on tori with the minimal value of the spectral genus.

In geometric problems the spectral curves related to integrable surfaces are not always smooth. This was discussed for minimal tori in S^{3} in [6]. For general tori in ℝ^{3} we have to define the spectral curves via Baker–Akhiezer functions ψ. We show how this is done for the Clifford torus in Theorem 1.

The Willmore conjecture reads that the global minimum is attained on the Clifford torus for which the spectral genus vanishes (as it is showed by Theorem 1). We think that

• given a conformal class of a torus, the Willmore functional attains its minima on tori with the minimal value of the spectral genus and the minimal value of the arithmetic genus of the spectral curve Γ_{ψ} defined via the Baker–Akhiezer function.

We shall discuss this conjecture elsewhere together with the expected relation between the spectral curves of tori in S^{3} (as they are defined in [14]) and the spectral curves of their stereographic images in ℝ^{3}. For the Clifford torus the mapping

ℂ^{∗} → ℂ^{∗}/ \{\frac{1 + i}{4} ∼ \frac{−1 + i}{4}, −\frac{1 + i}{4} ∼ \frac{1 − i}{4}\}

establishes an isomorphism of the corresponding spectral curves and the equivalence of the multipliers μ_{1}, μ_{2}. This was the basic idea of the computations in Section 4, i.e. in the proof of Theorem 1.

We summarize the contents of the paper:

in Section 2 we explain the geometry of Clifford torus in ℝ^{3}; in Section 3 we expose the procedure for constructiing Dirac op-

erators and their Baker–Akhiezer functions corresponding to singular spectral curves;

in Section 4 we prove Theorem 1.

This work was supported by RFBR (grant 03-01-00403), the Programme “Leading Scientific Schools” (NS-2185.2003.1), and Max-Planck-Institute on Mathematics in Bonn.

2 The Clifford torus

In fact, in differential geometry two objects are called the Clifford torus:

1) (a torus in S^{3}) the product of circles of the same radii which lies in the unit 3-sphere S^{3} ⊂ ℝ^{4} and therewith is defined by the equations:

x_{1}^{2} + x_{2}^{2} = x_{3}^{2} + x_{4}^{2} = \frac{1}{2}

where (x_{1}, …, x_{4}) are Euclidean coordinates in ℝ^{4};

2) (a torus in ℝ^{3}) the following torus of revolution: take in the x_{1}x_{3} plane a circle γ of radius r = 1 such that the distance between its center and the x_{1} axis equals R = \sqrt{2} and obtain a torus of revolution in ℝ^{3} by rotating this circle γ around the x_{1} axis.

The torus in S^{3} is considered up to isometries of S^{3} and in symplectic geometry it is considered as a torus in a symplectic 4-space ℝ^{4} and is also called by this name.

The torus in ℝ^{3} is considered up to conformal transformations of ℝ^{3}. In this event it is distinguished among tori of revolution by the ratio R/r = \sqrt{2}.

These tori are related to many conjectures in geometry:

• the Willmore functional defined on immersed surfaces in ℝ^{3} by

the formula

W(M) = ∫_{M}(H^{2} − K)dμ,

where dμ is the area form in the induced metric, H and K are the mean curvature and the Gaussian curvature, respectively, for tori is not less than 2π^{2} and attains its minimum on the Clifford torus (and its images under conformal transformations of ℝ^{3}) (the Willmore conjecture);

• the Clifford torus in S^{3} is the unique (up to isometries) embedded minimal torus in S^{3} (the Lawson conjecture);

• the area of any minimal torus in S^{3} is not less than 2π^{2} which is the area of the Clifford torus;

• the Clifford torus in S^{3} ⊂ ℝ^{4} minimizes the Willmore functional

W(M) = \frac{1}{4} ∫_{M} |H|^{2}dμ,

where H is the mean curvature vector, for tori in ℝ^{4}, or at least for Lagrangian tori in ℝ^{4} with the standard symplectic structure; • the Clifford torus in S^{3} ⊂ ℝ^{4} minimizes the area in its Hamil-

tonian isotopy class of Lagrangian tori in ℝ^{4}, i.e. among tori obtained from the Clifford torus by Hamiltonian deformations (the Oh conjecture).

The relation between two different notions of the Clifford torus and the corresponding conjectures is supplied by the stereographic projection F of S^{3} onto ℝ^{3} ∪ {∞} = ℝ̄^{3}. For that project S^{3} \ {(0, 0, 0, 1)} onto ℝ^{3} = {x_{4} = 0} from the north pole N = (0, 0, 0, 1) ∈ S^{3} mapping N into ∞ ∈ ℝ̄^{3}. This mapping establishes the conformal equivalence of S^{3} and ℝ̄^{3} and in coordinates takes the form

F (x_{1}, x_{2}, x_{3}, x_{4}) = (\frac{x_{1}}{1 − x_{4}} , \frac{x_{2}}{1 − x_{4}} , \frac{x_{3}}{1 − x_{4}} ) , x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = 1.

It is known that for a minimal torus T^{2} ⊂ S^{3} its area coincides with the value of the Willmore functional on its stereographic image:

Area(T^{2}) = W(F (T^{2})).

Therefore the third conjecture follows from the Willmore conjecture although The opposite is not true (these conjectures are not equivalent). Moreover the Lawson conjecture also implies the third conjecture (for expositions of the conjectures, some related results and their relations for tori in ℝ^{3} see, for instance, [1, 11] and for tori in ℝ^{4} see [9]).

Let us parameterize the Clifford torus in S^{3} as follows:

x_{1} = \frac{cos x}{\sqrt{2}}, x_{2} = \frac{sin x}{\sqrt{2}}, x_{3} = \frac{cos y}{\sqrt{2}} , x_{4} = \frac{sin y}{\sqrt{2}}

where 0 ≤, x, y ≤ 2π. Then the Clifford torus in ℝ^{3} is given by the formulas

r(x, y) = (\frac{cos x}{\sqrt{2} − sin y} , \frac{sin x}{\sqrt{2} − sin y} , \frac{cos y}{\sqrt{2} − sin y} ) .  (3)

It is easy to compute that x, y define a conformal parameter z = x+iy on the torus in which the induced metric takes the form

ds^{2} = \frac{dzdz̄}{(\sqrt{2} − sin y)^{2}} ,

the normal vector (assuming that the orientation on the surface is defined by the positively oriented frame (r_{x}, r_{y})) equals

N = (\frac{cos x(1 − \sqrt{2} sin y)}{\sqrt{2} − sin y}, \frac{sin x(1 − \sqrt{2} sin y)}{\sqrt{2} − sin y}, \frac{− cos y}{\sqrt{2} − sin y} ) ,

the second fundamental form is

\frac{(\sqrt{2} sin y − 1)}{(\sqrt{2} − sin y)^{2}}dx^{2} + \frac{1}{(\sqrt{2} − sin y)^{2}} dy^{2},

the principal curvatures take very simple forms:

κ_{1} = 1, κ_{2} = \sqrt{2} sin y − 1,

and therefore the mean curvature equals

H = \frac{sin y}{\sqrt{2}} .

The potential of the Weierstrass representation is

U = \frac{sin y}{2\sqrt{2}(\sqrt{2} − sin y)}.  (4)

Other important functions related to this representation are

\frac{∂}{∂z} (x_{1} + ix_{2}) = iψ̄_{2}^{2}, \frac{∂x_{3}}{∂z} = ψ_{1}ψ̄_{2}

and for the Clifford torus they are equal to

\frac{∂}{∂z} (x_{1} + ix_{2}) = \frac{∂}{∂z} \frac{cos x + i sin x}{\sqrt{2} − sin y} = \frac{i}{2}e^{ix} \frac{\sqrt{2} − cos y − sin y}{(\sqrt{2} − sin y)^{2}} ,

\frac{∂x_{3}}{∂z} = \frac{∂}{∂z} \frac{cos y}{\sqrt{2} − sin y} = \frac{i}{2} \frac{\sqrt{2} sin y − 1}{(\sqrt{2} − sin y)^{2}} .

(5)

3 Baker–Akhiezer functions and Dirac operators

Let us recall the definition of the Baker–Akhiezer (vector) function ψ corresponding to the Dirac operator

D = ( \begin{matrix}0 & ∂ \\ −¯∂ & 0 \end{matrix} ) + ( \begin{matrix}U & 0 \\ 0 & V \end{matrix} ) .

By definition it depends on a complex variable z ∈ ℂ and on a parameter P on a complex curve Γ of finite arithmetic genus g = p_{a}(Γ) and meets the following conditions:

1) ψ is meromorphic in P outside a couple of marked points ∞_{±} ∈ Γ and has poles at g+1 points P_{1}+⋯+P_{g+1} (the points ∞_{+}, ∞_{−}, P_{1}, …, P_{g+1} are nonsingular);

2) ψ has the following asymptotics near ∞_{±}:

ψ ≈ e^{k_{+}z} [( \begin{matrix}1 \\ 0 \end{matrix} ) + ( \begin{matrix}ξ_{1}^{+} \\ ξ_{2}^{+} \end{matrix} ) k_{+}^{−1} + O(k_{+}^{−2})] as P → ∞_{+},

ψ ≈ e^{k_{−}z̄} [( \begin{matrix}0 \\ 1 \end{matrix} ) + ( \begin{matrix}ξ_{1}^{−} \\ ξ_{2}^{−} \end{matrix} ) k_{−}^{−1} + O(k_{−}^{−2})] as P → ∞_{−},

where k_{±}^{−1} are local parameters near infinities such that

k_{±}^{−1}(∞_{±}) = 0.

It follows from the general theory of Baker–Akhiezer functions ([8, 3]) that the such function is unique (for a generic divisor D = P_{1} + ⋯ + P_{g+1}) and therefore we can find an operator D with potentials U and V such that Dψ is meromorphic with analogous singularities but with the asymptotics

Dψ = e^{k_{+}z}O(k_{+}^{−1}) as P → ∞_{+}, Dψ = e^{k_{−}z̄}O(k_{−}^{−1}) as P → ∞_{−}.

The uniqueness of ψ implies that Dψ = 0. We have

Theorem 2 ([12, 13]) The Baker–Akhiezer function ψ satisfies the Dirac equation

Dψ = 0 with U = −ξ_{2}^{+}, V = ξ_{1}^{−}.

If the curve Γ is singular we assume that it has some special form. Let us now define what it is this form and recall some algebrogeometric properties of such singular curves following [10].

Let Γ_{nm} be a nonsingular complex curve. Take on Γ_{nm} effective divisors D_{1}, …, D_{n}, i.e. for any k = 1, …, k such a divisor D_{k} is a formal sum of finitely many points on Γ_{nm} with positive coefficients:

D_{k} = a_{k1}Q_{k1} + ⋯ + a_{km_{k}}Q_{km_{k}},
Q_{kj} ∈ Γ_{nm}, a_{kj} > 0, a_{kj} ∈ ℤ, j = 1, …, m_{k}, k = 1, …, n.

We assume that the supports suppD_{k} = Q_{k1}+⋯+Q_{km_{k}}, k = 1, …, n, of these divisors are pairwise nonintersecting. Then we denote by

Γ_{D_{1},...,D_{n}}

the singular curve obtained by contracting all points from each divisor D_{k} into one singular point S_{k} ∈ Γ. This is done with respect to the multiplicities. This means that a function f which is meromorphic on Γ is, by definition, a meromorphic function f : Γ_{nm} → ℂ such that it has no poles at ∪_{k}suppD_{k}, and for each divisor

D_{k} = a_{k1}Q_{k1} + ⋯ + a_{km_{k}}Q_{km_{k}}

we have

f(Q_{k1}) = ⋯ = f(Q_{km_{k}}), ∂^{m}f(Q_{kl}) = 0 for l = 1, …, (a_{kl} − 1),

where k = 1, ….n. Here ∂ is the Cauchy derivation with respect to a complex parameter on Γ.

The natural projection

π : Γ_{nm} → Γ

is the normalization mapping.

If Γ is singular we assume that it takes the form

Γ = Γ_{D_{1},...,D_{n}}.

Let T ⊂ Γ_{D_{1},...,D_{n}} and T is empty or contains only nonsingular points. Recall that a one-form ω is called a regular form on Γ_{D_{1},...,D_{n}}\T if ω is a meromorphic form on Γ_{nm}\π^{−1}(T) which may have poles only at the supports of D_{1}, …, D_{k} and at each point Q_{kl} the degree of the pole is not greater than a_{kl} and moreover for every k the inequality

∑_{l} Resfω(Q_{kl}) = 0

which holds for all meromorphic functions on Γ. For a nonsingular curve Γ the notions of regular and holomorphic forms coincide.

For nonsingular curves the values of the arithmetic genus p_{a} and the geometric genus p_{g} coincide. For a singular curve Γ = Γ_{D_{1},...,D_{k}} we have

p_{g}(Γ) = p_{g}(Γ_{nm}), p_{a}(Γ) = p_{g}(Γ) + ∑_{k}(deg D_{k} − 1)

where the degree of the divisor D_{k} equals to

deg D_{k} = deg(a_{k1}Q_{k1} + ⋯ + a_{km_{k}}Q_{km_{k}}) = a_{k1} + ⋯ + a_{km_{k}}.

We say that σ : Γ → Γ is an involution of Γ if σ^{1} = 1 and its pullback defines an involution on Γ_{nm} such that the singularity divisors D_{1}, …, D_{n} falls into two groups:

1) divisors which are preserved: D_{j} = σ(D_{j});

2) divisors which are interchanges with others: D_{j} = σ(D_{k}), j ≠ k. The first group corresponds to fixed points S_{j} = π(D_{j}) of an invo-

lution on Γ although the points from D_{j} could be permuted.

Theorem 3 ([12, 13]) 1) Let σ be a holomorphic involution σ : Γ → Γ such that

σ(∞_{±}) = ∞_{±}, σ(k_{±}) = −k_{±}

under which the singular points S_{1} = π(D_{1}), …, S_{k} = π(D_{k}) are fixed and other singular points are no fixed.

If there exists a meromorphic differential ω on Γ_{2D_{1},...,2D_{k},D_{k+1},...,D_{n}} such that ω has two poles at ∞_{±} with the principal parts ±k_{±}^{2}(1 + O(k_{±}^{−1}))dk_{±}^{−1}, it is regular outside ∞_{±} and it has zeros at D + σ(D) then the potentials U and V coincide: U = V.

2) Let τ be an antiholomorphic involution τ : Γ → Γ such that

τ(∞_{±}) = ∞_{∓}, τ(k_{±}) = −k̄_{∓}

under which the singular points S′_{1} = π(D′_{1}), …, S′_{l} = π(D′_{l}) are fixed and other singular points are no fixed (here (D′_{1}, …, D′_{k}) is obtained by a permutation of the set (D_{1}, …, D_{n})).

If there exists a meromorphic differential ω′ on Γ_{2D′_{1},...,2D′_{l},D′_{l+1},...,D′_{n}} such that it has two poles at ∞_{±} with the principal parts k_{±}^{2}(1 + O(k_{±}^{−1}))dk_{±}^{−1}, it is regular outside ∞_{±} and it has zeros at D + τ(D), then the potentials U and V are real-valued: U = Ū, V = V̄.

Proof. We proved this theorem for smooth curves in [12] and for singular curves for which all singular points are fixed by involution in [13] (here we correct some inaccuracy and typos in the conditions on the antiholomorphic involution). In general, the proof is the same for all these cases: we have to compute the residues of the following differentials with poles only in ∞_{+} and ∞_{−} and remember that their sum taken over all poles vanishes:

1) the sum of the residues of the differential ψ_{1}(P)ψ_{2}(σ(P))ω equals −2πi(ξ_{2}^{+} + ξ_{1}^{−}) = 0 which implies that U = V;

2) the sums of the residues of the differentials ψ_{1}(P)\overline{ψ(τ(P))}ω′ and ψ_{2}\overline{ψ_{2}(τ(P))}ω′ are equal to 2πi(ξ_{1}^{−} − ξ̄_{1}^{−}) and 2πi(ξ_{2}^{+} − ξ̄_{2}^{+} respectively which implies that U = Ū and V = V̄.

This proves the theorem.

There is a general procedure for constructing Baker–Akhiezer functions corresponding to different operators and smooth curves Γ [8]. In [12] we applied it to the Dirac operator and derived the explicit formulas for ψ and the potentials U and V in terms of the theta function of Γ.

For Γ = Γ_{D_{1},...,D_{n}} we have to do the following:

• take a divisor D = P_{1} + ⋯ + P_{s} of degree s = p_{a}(Γ) + 1 = p_{g}(Γ_{nm}) + ∑_{k}(deg D_{k} − 1) + 1;

• for each divisor D′_{j} = P_{1} + ⋯ + P_{g} + P_{g+j} where g = p_{g}(Γ) = p_{a}(Γ_{nm}) and j = 1, …, r = ∑_{k}(deg D_{k} −1) construct the Baker-Akhiezer φ_{j} on Γ_{nm} with D′_{j} as the divisor of poles (as it was done in [12]);

• construct the function ψ in the form

( \begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix} ) = ( \begin{matrix}q_{1}φ_{1,1} + ⋯ + q_{r}φ_{r,1} \\ t_{1}φ_{1,2} + ⋯ + t_{r}φ_{r,2} \end{matrix} )

by solving the following equations on q_{i}, t_{j}:

ψ_{α}(Q_{k1}) = ⋯ = ψ_{α}(Q_{km_{k}}), ∂^{m}ψ_{j}(Q_{kl}) = 0,

∑_{i} q_{i} = ∑_{j} t_{j} = 1

where k = 1, …, n, α = 1, 2, l = 1, …, m_{k}, and m = 1, …, (a_{kl}− 1).

Notice that we have exactly 2r equations on the same number parameters. The first two systems means that ψ is defined on Γ and the latter two equations implies the correct asymptotics of ψ at the infinities. The solutions q_{i}, t_{j} depend on the spatial parameter z as well as on the parameter on Γ.

Remark that complex curves with more general singularities were considered in [4] as the spectral curves of the operator L = i∂_{y}−∂_{x}^{2}+u. However we do not know do such singularities appear on the spectral curve of a Dirac operator and if they do what have to be analogs of the conditions on the involutions σ and τ. In our case such conditions involve not only the spectral curve Γ_{D_{1},...,D_{n}} but also another curve Γ_{2D_{1},...,2D_{k},D_{k+1},...,D_{n}}.

4 The spectral curve of the Clifford torus

In this section we prove Theorem 1.

Let ψ(z, z̄, λ) be a function on ℂP^{1} which is meromorphic in λ ∈ ℂ ⊂ ℂP^{1} on ℂP \ {0, ∞}, has a pole at the point λ = p and has the following asymptotics at two “infinities”:

ψ ≈ ( \begin{matrix}e^{k_{+}z} \\ 0 \end{matrix} ) as k_{+} = λ → ∞; ψ ≈ ( \begin{matrix}0 \\ e^{k_{−}z̄} \end{matrix} ) as k_{−} = −\frac{|u|^{2}}{λ} → ∞.

This means that it is a Baker–Akhiezer (vector) function corresponding to the data

Γ_{u} = ℂP^{1}, D = p, ∞_{±}, k_{±}.

It is known from the general theory that such a function is unique and we easily compute it obtaining the following formula:

ψ = \frac{λ}{λ − p} e^{λz− \frac{|u|^{2}}{λ} z̄} ( \begin{matrix}1 \\ −\frac{p}{λ} \end{matrix} ) .

The function ψ satisfies the Dirac equation

Dψ = 0

where the potentials of the Dirac operator D are

U = p, V = \frac{|u|^{2}}{p} .

If p = u we have the operator

D = ( \begin{matrix}0 & ∂ \\ −¯∂ & 0 \end{matrix} ) + ( \begin{matrix}u & 0 \\ 0 & ū \end{matrix} ) .

We have two involutions on Γ_{u}: a holomorphic involution σ and an antiholomorphic involution τ which act as follows

σ(λ) = λ, τ(λ) = \frac{|u|^{2}}{λ̄} .

We shall look for the spectral data of the Clifford torus in the following form:

• the normalized spectral curve Γ_{nm} is ℂP^{1} with two infinities ∞_{±}; • the actions of σ and τ descend to involutions on Γ;

• u = \frac{1+i}{4}.

Consider the following singular curve:

Γ = ℂP^{1}/\{±u ∼ ∓ū\}, u = \frac{1 + i}{4},

i.e. it is obtained from ℂP^{1} by gluing u with −ū and −u with ū. We see that Γ is a sphere with a pair of double points. We have

p_{g}(Γ) = 0, p_{a}(Γ) = 2.

Therefore we look for the Baker–Akhiezer function on Γ in the form

ψ_{1}(z, z̄, λ) = e^{λz− \frac{|u|^{2}}{λ} z̄} (q_{1} \frac{λ}{λ − p_{1}} + q_{2} \frac{λ}{λ − p_{2}} + (1 − q_{1} − q_{2})\frac{λ}{λ − p_{3}} ) ,

ψ_{2}(z, z̄, λ) = e^{λz− \frac{|u|^{2}}{λ} z̄} (t_{1} \frac{p_{1}}{p_{1} − λ} + t_{2} \frac{p_{2}}{p_{2} − λ} + (1 − t_{1} − t_{2})\frac{p_{3}}{p_{3} − λ})

where p_{1}, p_{2}, p_{3} are the poles of ψ and q_{1}, q_{2}, t_{1}, and t_{2} are functions of z, z̄ obtained from the conditions:

ψ(z, z̄, u) = ψ(z, z̄, −ū), ψ(z, z̄, −u) = ψ(z, z̄, ū).

This leads to the following equations for q_{1} and q_{2}:

q_{1} [e^{uz−ūz̄}u (\frac{1}{u − p_{1}} − \frac{1}{u − p_{3}} ) − e^{uz̄−ūz}ū (\frac{1}{ū + p_{1}} − \frac{1}{ū + p_{3}} )] +

+q_{2} [e^{uz−ūz̄}u (\frac{1}{u − p_{2}} − \frac{1}{u − p_{3}} ) − e^{uz̄−ūz}ū (\frac{1}{ū + p_{2}} − \frac{1}{ū + p_{3}} )] =

= −e^{uz−ūz̄} \frac{u}{u − p_{3}} + e^{uz̄−ūz} \frac{ū}{ū + p_{3}} ,

q_{1} [e^{ūz−uz̄}ū (\frac{1}{ū − p_{1}} − \frac{1}{ū − p_{3}} ) − e^{ūz̄−uz}u (\frac{1}{u + p_{1}} − \frac{1}{u + p_{3}} )] +

+q_{2} [e^{ūz−uz̄}ū (\frac{1}{ū − p_{2}} − \frac{1}{ū − p_{3}} ) − e^{ūz̄−uz}u (\frac{1}{u + p_{2}} − \frac{1}{u + p_{3}} )] =

= e^{ūz̄−uz} \frac{u}{u + p_{3}} − e^{ūz−uz̄} \frac{ū}{ū − p_{3}} .

The formula for the potential V = ξ_{1}^{−} takes the form

V = |u|^{2} (\frac{q_{1}}{p_{1}} + \frac{q_{2}}{p_{2}} + \frac{1 − q_{1} − q_{2}}{p_{3}} ) .

We can also compute t_{1} and t_{2} solving analogous equations. The formula for the potential U is

U = t_{1}p_{1} + t_{2}p_{2} + (1 − t_{1} − t_{2})p_{3}.  (6)

In both cases we have to know the points p_{1}, p_{2}, p_{3} to make computations. It follows from Theorem 3 that if these points are chosen in a rather convenient way then the potentials U and V are real-valued and moreover coincide U = V. This is the case when there are meromorphic differentials ω and ω′ with properties exposed in Theorem 3.

Let us look for such differentials.

The general form of a differential ω on Γ which has poles at ∞_{±} with the principal parts ±k_{±}^{2}(1 + O(k_{±}^{−1}))dk_{±}^{−1} and is regular outside these points is

ω = − [\frac{λ^{2} − |u|^{2}}{λ^{2}} + a (\frac{1}{λ − u} − \frac{1}{λ + ū}) + b (\frac{1}{λ + u} − \frac{1}{λ − ū})] dλ

and the general form of a differential ω′ on Γ which has poles at ∞_{±} with the principal parts k_{±}^{2}(1 + O(k_{±}^{−1}))dk_{±}^{−1} and is regular outside these points is

ω′ = − [\frac{λ^{2} + |u|^{2}}{λ^{2}} + c (\frac{1}{λ − u} − \frac{1}{λ + ū}) + d (\frac{1}{λ + u} − \frac{1}{λ − ū})] dλ.

We have to find differentials ω and ω′ such that their zero sets would be of the form D + σ(D) and D + τ(D) respectively.

Notice that the zero sets of ω and ω′ are the zero sets of the polynomials Q(λ) and Q′(λ) respectively where

Q(λ) = (λ^{2} − |u|^{2})(λ^{2} − u^{2})(λ^{2} − ū^{2})+
+λ^{2}(u + ū)[(a − b)(λ^{2} − |u|^{2}) + (a + b)(u − ū)λ],
Q′(λ) = (λ^{2} + |u|^{2})(λ^{2} − u^{2})(λ^{2} − ū^{2})+
+λ^{2}(u + ū)[(c − d)(λ^{2} − |u|^{2}) + (c + d)(u − ū)λ].

Since we look for a differential ω with the zero set of the form D+σ(D) the polynomial Q(λ) has to have only terms with even powers. This implies that b = −a and we have

Q(λ) = (λ^{2} − |u|^{2})[(λ^{2} − u^{2})(λ^{2} − ū^{2}) + 2a(u + ū)λ^{2}].

We see that Q(|u|) = 0 and put

p_{3} = |u|.

Other poles p_{1} and p_{2} satisfy the equation

(λ^{2} − u^{2})(λ^{2} − ū^{2}) + 2a(u + ū)λ^{2} = 0

together with −p_{1}, −p_{2}. Since τ(p_{3}) = p_{3}, the point p_{3} = |u| has to be at least a double root of the polynomial Q′(λ).

We skip some simple calculations which show that if we assume that c = d and that (λ − |u|)^{2} divides Q′(λ) then we have

Q′(λ) = (λ−|u|)^{2} [λ^{4} + 2|u|λ^{3} + (4|u|^{2} − (u^{2} + ū^{2}))λ^{2} + 2|u|^{3}λ + |u|^{4}]

and

c = d = \frac{(u^{2} + ū^{2})|u| − 2|u|^{3}}{u^{2} − ū^{2}} .

We have

Q′(λ) = (λ − |u|)^{2}|u|^{4}P (\frac{λ}{|u|}) ,

where P (μ) = (μ^{2} − Aμ + 1)(μ^{2} − Āμ + 1),

A + Ā = −2, |A|^{2} = 2 − \frac{u^{2} + ū^{2}}{|u|^{2}} .

and

A = t + t^{−1}, Ā = t̄+ t̄^{−1},

where t, t̄, t^{−1} and t̄^{−1} are the roots of the polynomial P (μ).

Until now we did all computations for a general value of u. let us consider the special case when

u = \frac{1 + i}{4}, u^{2} + ū^{2} = 0.

The polynomial P (μ) takes a very simple form and we can easily find all its roots:

P (μ) = μ^{4} + 2μ^{3} + 4μ^{2} + 2μ + 1,

μ_{1,2} = \frac{−(1 − i) ± \sqrt{−2i − 4}}{2}, μ_{3,4} = μ̄_{1,2} = \frac{−(1 + i) ± \sqrt{−2i − 4}}{2}

and we obtain four other zeros of the differential ω′:

s_{1} = \frac{−1 + i + \sqrt{−2i − 4}}{4\sqrt{2}}, s_{2} = \frac{−1 + i − \sqrt{−2i − 4}}{4\sqrt{2}},

τ(s_{1}) = \frac{−1 − i − \sqrt{2i − 4}}{4\sqrt{2}}, τ(s_{2}) = \frac{−1 − i − \sqrt{2i − 4}}{4\sqrt{2}}.

Therefore for c = d = \frac{i}{\sqrt{8}} the differential ω′ has zeroes exactly at D + τ(D) where D = s_{1} + s_{2} + p_{3} where p_{3} = |u| and τ(p_{3}) = p_{3}.

Notice that the zeros of ω have to be invariant with respect to the involution σ and their product equals |u|^{6}. This implies that we can put either p_{1} = s_{1}, p_{2} = s_{2} either p_{1} = τ(s_{1}), p_{2} = τ(s_{2}).

Let us make our choice as follows:

p_{1} = \frac{−1 + i + \sqrt{−2i − 4}}{4\sqrt{2}}, p_{2} = \frac{−1 + i − \sqrt{−2i − 4}}{4\sqrt{2}}, p_{3} = |u| = \frac{1}{\sqrt{8}}.

In this event a = −b = \frac{1+i}{\sqrt{8}}.

Substituting these values of p_{k}, k = 1, 2, 3, into the equations for q_{1} and q_{2} we compute them as functions of x, y and by that obtain the explicit formula for ψ_{1} and V. Now the functions ψ_{2} and U can be found by solving analogous equations for t_{1}, t_{2} or from the following formulas which follows from Theorem 3:

U = V, ψ_{2} = \frac{¯∂ψ_{1}}{V} .

We omit the explicit formulas for q_{1}, q_{2}, t_{1}, t_{2} and just mention that these functions appear to be functions of y which are 2π-periodic. The ψ-function at the point λ = u equals:

ψ_{1}(z, z̄, ū) = \frac{1 + i}{4}e^{− \frac{ix}{2}} \frac{2e^{\frac{iy}{2}} + \sqrt{2}(1 − i)e^{− \frac{iy}{2}}}{sin y − \sqrt{2}} ,

ψ_{2}(z, z̄, ū) = −\frac{\sqrt{2}}{4}e^{− \frac{ix}{2}} \frac{2e^{\frac{iy}{2}} − \sqrt{2}(1 + i)e^{− \frac{iy}{2}}}{sin y − \sqrt{2}} .

We see that

iψ̄_{2}^{2} = −\frac{(1 + i)e^{ix}}{2} \frac{\sqrt{2} − cos y − sin y}{(sin y − \sqrt{2})^{2}} ,

ψ_{1}ψ̄_{2} = \frac{\sqrt{2}i}{2} \frac{1 − \sqrt{2} sin y}{(sin y − \sqrt{2})^{2}}

and comparing these formulas with the formulas (5) we conclude that a surface constructed from the function \frac{1}{\sqrt[4]{2}}ψ(z, z̄, ū) by the Weierstrass representation is a torus which is mapped to the Clifford torus defined by the formulas (3) by the orthogonal transformation

T = ( \begin{matrix}−\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ −\frac{1}{\sqrt{2}} & −\frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & −1 \end{matrix} ) .

This transformation changes the orientation of ℝ^{3} which results in the change of the sign of the potential U: the formula (6) gives us

U = \frac{sin y}{2\sqrt{2}(sin y − \sqrt{2})}

and this expression differs by the sign from (4).

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