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Closed Loop Solitons and Sigma Functions: Classical and Quantized Elasticas with Genera One and Two

Closed loop solitons in a plane, whose curvatures obey the modified Korteweg-de Vries equation, were investigated. It was shown that their tangential vectors are expressed by ratio of Weierstrass sigma functions for genus one case and ratio of Baker's sigma functions for the genus two case. This study is closely related to classical and quantized elastica problems.


§1. Introduction
This article is on loop solitons of genera one and two. We give them as ratios of sigma functions; for the case of genus one, the sigma functions are Weierstrass sigma functions while for the case of genus two, they are Baker's hyperelliptic sigma functions.
First we mention background of the study, which is related to elastica problem. Elastica problem was proposed by James Bernoulli in ending of seventeenth century and Euler and his nephew Daniel Bernoulli investigated the problem [T, L, W]. The elastica problem is to determine all allowed shapes of a thin elastic beam in a plane. The allowed shape is realized as a curve with the local minimum of the energy (1-1) where s is its arclength and k is its curvature. For its tangential angle φ and g := exp( √ −1φ), the integrand is given by k 2 ds = −(g −1 dg)( * g −1 dg) = (∂ s φ) 2 ds, (1)(2) where * is the Hodge star, a map from one-form to zero-form, and ∂ s := d/ds. Thus the elastica problem is the oldest harmonic map (sigma-model) problem from a circle S 1 or a real line R to U(1). This problem was essentially solved by Euler using numerical computations [T,L]. For a given elastica in two dimensional euclidean space, it is characterized by the affine coordinate (X 1 (s), X 2 (s)) or Z(s) = X 1 (s)+ √ −1X 2 (s). Then its curvature k is given by k := 1 √ −1 ∂ s log ∂ s Z.
The shape locally minimizing the energy (1-1) is given by the differential equation [Ma1,Ma2], This is known as the static modified Korteweg-de Vries (SMKdV) equation. As I showed in ref. [Ma2], when we deal with a non-stretching closed elastica in thermal bath or investigate the partition function of temperature 1/β, (1-4) we also need information of curves which have the excited energy. We will call such curves "quantized elastica" following [Ma4]. This problem is associated with large polymer problem like DNA in biology and polymer physics. As shown in [Ma2,Ma3,Ma4], the quantized elastica is classified by the modified Korteweg-de Vries (MKdV) equation, where ∂ s := ∂/∂s, ∂ t := ∂/∂t and t is a deformation parameter. The orbits obeying the MKdV equation preserve local length and their first integral (1-1), i.e., [∂ s , ∂ t ] = 0 and ∂ t E = 0. Since the MKdV equation is an initial value problem, for a given smooth curve, there exists, at least, an MKdV orbit including the curve; we can classify quantized elastica using the orbits. Hence the partition function (1-4) is reduced to an integration, where dΩ(E) is the density of states. The evaluation of dΩ(E) means the determination of the orbits of the MKdV equation whose first integral is E. In other words, in order to evaluate (1-5), we must explicitly construct the periodic solutions of MKdV equation, which are expressed by hyperelliptic functions. The hyperelliptic function is characterized by genus. As mentioned in [Ma2], orbits with small genus is expected to largely contributes to the partition functions (1-4). Thus as an attempt, we will investigate the classical and quantized elastica of genera one and two respectively. Since the curve whose curvature obeys the MKdV equation was called loop soliton [WIK, I], our investigation is, just, a study of closed loop solitons. Here we will also note that the time development of classical elastica does not obey the MKdV equation except genus one solution case [Ma1]. (From physical point of view, investigation of a loop soliton with higher genus is somewhat nonsense in pure kinematic study.) In this article, we will show that the tangential vectors of closed loop solitons with genera one and two are expressed by ratio of Weierstrass sigma functions in §3 and ratio of Baker's sigma functions §4 respectively. This result is very interesting as we will discuss in §5. The sigma function, which is well-tuned theta function, plays very important roles in the studies of elliptic and hyperelliptic functions [B1,B2,B3,BEL,G,Ma5,Ô1,Ô2,Ô3].
Further our result has an effect on historical interpretation of history of science. According to Truesdell and Love [T,L], Euler did illustrate the shape of loop soliton with genus one, our result means that Euler essentially found the elliptic functions and sigma functions even though he dealt only with elliptic integrations. In fact, the tangential vector ∂ s Z(s) is periodic function of s. I believe that Euler noticed its properties and thus after he read Fagnano's article, his mind felt the addition formula [W].
The construction of closed loop soliton with genus two is based upon Baker's hyperelliptic function theory [B1,B2,B3]. Recently Baker's approach is reevaluated from various viewpoints [BEL,G,Ma5,Ô1,Ô2,Ô3]. Buchstaber, Enolskii and Leykin has been actively studying the Baker's theory from soliton theory and developing it. Grant andÔnishi found interesting relations in hyperelliptic function in terms of Baker's theory from point of view of number theory. As on Baker's theory of the hyperelliptic function, we start from a hyperelliptic curve and find an explicit function form without ambiguous parameter, it is very effective. §2. Basic Properties of Loop Soliton Let us consider a smooth immersion of a circle S 1 into the two dimensional euclidean space E 2 ≈ C. The immersed curve C is characterized by the affine coordinate (X 1 (s), X 2 (s)). Here s is a parameter of S 1 and is, now, chosen as the arclength so that ds 2 = (dX 1 ) 2 + (dX 2 ) 2 . We will also use the complex expression, Then |∂ s Z(s)| = 1 and the curvature of C is given by Definition 2-1.
A one parameter family of curves {C t } for real parameter t ∈ R is called a loop soliton, if its curvature obeys the MKdV equation; for q := k/2, Here we will describe the Miura map for later convenience.
For the solutions p ± of the KdV equation, if we find a quantity q satisfying p ± ≡ q 2 ± ∂ s q over C t and for all t, q obeys the MKdV equation (2-3). §3. Loop Soliton with Genus One and Weierstrass Sigma Function In this section, we will deal with a loop soliton with genus one. In other words, we are consider the case of t = s/C, i.e., the SMKdV equation, First we will set up the Weierstrass ℘-functions and σ-functions.
The shape of loop soliton with genus one is given by where δ is a constant parameter.
Later part in this section devotes the proof of this theorem but first we will summary the properties of Weierstrass (elliptic) ℘ and σ functions. Let " ′ " be derivative in u.
Further we will note the addition relations.
From the definition of loop soliton, the theorem 3-2 is reduced to the next lemma.
Hence the theorem 3-2 is also satisfied.
Remark 3-6. We note that (3-8) is a necessary condition. If one tunes δ in (3-8) and the g 2 and g 3 in (3-2) to find a real line s in the complex plane {u} = C so that it satisfies the reality condition |∂ s Z| = 1 and the closed condition Z(s) = Z(s + 2ω 1 ), (3-8) becomes the closed loop soliton. §4. Loop Soliton with Genus Two and Baker's Sigma Function In this section, we will deal with a hyperelliptic solution of the loop soliton whose genus is two. We will redefine the quantities σ, ℘, ρ, ω, η, τ in the previous section. These quantities are altered to genus two versions of corresponding quantities in the previous section.
For a hyperelliptic curve of genus two, where λ 5 ≡ 1 and λ j 's are complex numbers, we will give definitions as follows.
(1) We choose a r , c r and c (r = 1, 2) for so that curves (a r , c r ) or (c, ∞) does not intersect each other and Re(a r ) ≤ Re(c r ), Re(a 1 ) ≤ Re(a 2 ), Re(c 2 ) ≤ Re(c).
Before proving this theorem, we will also review the properties of Baker's ℘ and σ-functions.
From the definition 2-1, the theorem 4-4 is reduced to next lemma.

Discussion
We showed that the loop soliton is directly connected with σ-functions. In ref. [Ma1], I partially showed the fact but I could not reach the concrete function form and the difference between numerator and denominator in the ratio of σ or (τ ) function. This study determines that the difference comes from the half of period or so-called theta characteristics [B2, M]. Even though we could not find a solution with higher genus, it is not difficult to conjecture the function form of loop soliton in terms of the theta characteristics.
As we used the definition of loop soliton as a relation of MKdV equation, our study is just that of MKdV equation. In other words, we have concretely constructed the function forms of the genus one and the genus two periodic solutions of MKdV equation in the section 3 and 4. In ordinary study of elliptic function solution of MKdV equation [Ma1], one uses Jacobi elliptic functions, sn, cn, and dn functions. Our solution (3-15) becomes the Jacobi elliptic function by means of Landen transformation, Jacobi transformation, Gauss transformation and so on. The relation between Weierstrass and Jacobi elliptic functions is derived from (3-14) [WW]. ℘(u) − e a plays very important roles; sn(z) = e 1 − e 3 ℘(u) − e 3 , cn(z) = ℘(u) − e 1 ℘(u) − e 3 , dn(z) = ℘(u) − e 2 ℘(u) − e 3 , (5-1) where z = (e 1 −e 3 ) 1/2 u. Since sn function is connected with 1/sn by phase shift and associated with dn and cn functions by algebraic equations, the Jaobi elliptic function could roughly be considered as ℘(u) − e 3 . Similarly we might regard √ ρ in (4-25) as a Jacobi-type hyperelliptic function due to (4-20); ρ/a 2 is essentially ℘(u) − e 3 when we set ∂ u 1 σ = 0 and u := u 2 . We should also notice the square in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and (4-20). As described in [Ma1 and reference therein], it is closely related to fermion. In fact, ψ := σ 2 /σ ∼ √ ∂ s Z in (3-14) ( ψ := σ 2 /σ 0 in (4-20) ) obeys the Dirac equation, The square root of ∂ s Z means double covering of the tangent bundle. (5-2) is related to Schwarz derivative and moduli space as mentioned in [Ma1]. We recognize that there is still open problem what is the orign of the square root. Finally we will comment upon the closed condition mentioned in the remark 3-6. We could not touch the condition. In general, an integration of periodic function is not periodic. However if the function satisfies some conditions, its integral is also periodic. The closed condition is a such condition. It suppresses two degree of freedom in the parameter spaces. For the study of quantized elastica problem, it is very important to determine the relations between the conditions and the coefficients of the algebraic curve λ j in (4-1). In future, we wish to determine them.