A model for Hopfions on the space-time S(3)xR

E. De Carli and L. A. Ferreira

Citation: Journal of Mathematical Physics 46, 012703 (2005); doi: 10.1063/1.1829911 View online: http://dx.doi.org/10.1063/1.1829911

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/46/1?ver=pdfcov Published by the AIP Publishing

A model for Hopfions on the space–time

S

3

ÃR

E. De Carli

Instituto de Física Teórica, IFT/UNESP, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, São Paulo–SP, Brazil and Departamento de Engenharia Mecânica, LMPT, Universidade Federal de Santa Catarina, UFSC, Campus Universitário, Trindade, 88040-900, Florianópolis-SC, Brazil

L. A. Ferreira

Instituto de Física de São Carlos, IFSC/USP, Universidade de São Paulo, Caixa Postal 369, CEP 13560-970, São Carlos–SP, Brazil and Instituto de Física Teórica, IFT/UNESP, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, São Paulo–SP, Brazil

(Received 7 October 2004; accepted 19 October 2004; published online 5 January 2005) We construct static and time dependent exact soliton solutions for a theory of scalar fields taking values on a wide class of two dimensional target spaces, and defined on the four dimensional space–time S3⫻R. The construction is based on an ansatz built out of special coordinates on S3. The requirement for finite energy introduce boundary conditions that determine an infinite discrete spectrum of frequencies for the oscillating solutions. For the case where the target space is the sphere S2, we obtain static soliton solutions with nontrivial Hopf topological charges. In addition, such Hopfions can oscillate in time, preserving their topological Hopf charge, with any of the frequencies belonging to that infinite discrete spectrum. © 2005

American Institute of Physics. [DOI: 10.1063/1.1829911]

I. INTRODUCTION

In this paper we consider a nonlinear theory of a complex scalar field u, on the space–time

S3⫻R. Our considerations apply to a wide class of target spaces, but the case of most interest is

that of the sphere S2, in which case the field u parametrizes a plane corresponding to the stereo-graphic projection of S2. The action is the integral of the square of the pull-back of the area form on the target space. Therefore, it is quartic in derivatives, but only quadratic in time derivatives. The theory is integrable in the sense that it possesses a generalized zero curvature representation of its equations of motion and an infinite number of local conservation laws.1,2 The conserved currents are associated to the invariance of the theory under the area preserving diffeomorphisms on the target space.

We construct an infinite number of static and time dependent exact soliton solutions using an ansatz2,3 that reduces the four dimensional nonlinear equations of motion into linear ordinary differential equations for the profile function. In the case of the target space S2_{the solitons carry} nontrivial Hopf topological charges. Although the topology is the same as that of other models possessing Hopfion solutions,3–5 the Derrick’s scaling argument is circumvented in a different manner. The stability of the static three dimensional solutions comes from the fact that the physi-cal space is S3 _{and that introduces a length scale given by its radius. A model in Euclidean space} where a similar stability mechanism occurs is discussed in Ref. 6.

The requirement for finite energy leads to boundary conditions that determine an infinite discrete set of allowed frequencies for the oscillations of the solutions. Those solutions can be linearly superposed since the profile function satisfies a linear equation. It turns out that the energy of the superposed solution is the sum of the energies of the modes, and in this sense the modes are decoupled. However, the profile function can take values only on some intervals on the real line, which depend on the choice of the target space. Therefore, not all superpositions are allowed and that introduces some sort of coupling among the modes. The allowed superpositions for the case of the target space S2 are discussed in detail. One of the most interesting superpositions

corre-JOURNAL OF MATHEMATICAL PHYSICS 46, 012703(2005)

46, 012703-1

sponds to oscillating Hopfion solutions. We show that it is possible to superpose to the static Hopfion soliton any number of oscillating modes with any of the frequencies belonging to that infinite discrete spectrum. Although the solution oscillates in time, its topological Hopf charge is preserved. The only constraint on such superposition appears on the intensity of each mode.

The paper is organized as follows: in Sec. II we introduce the model and discuss its integra-bility properties, in Sec. III we propose the ansatz and construct the exact soliton solutions, in Sec. IV we discuss the energy, boundary conditions and allowed frequencies. The case of the target space S2_{is discussed in Sec. V.}

II. THE MODEL

The metric on the space–time S3_{⫻R is given by}

ds2= dt2− r₀2

冉

dz 2

4z共1 − z兲+共1 − z兲d␸1 2

+ z d␸₂2

冊

, 共1兲

where t is the time, z, and ␸_i, i = 1, 2, are coordinates on the sphere S3_{, and 0艋z艋1, 0艋␸} i 艋2␲, and r₀ is the radius of the sphere S3_{. Embedding S}3 _{on R}4 _{we get that the Cartesian} coordinates of the points of S3 _are

x1= r0

冑

z cos␸2, x3= r0

冑

1 − z cos␸1,

x2= r0

冑

z sin␸2, x4= r0

冑

1 − z sin␸1. 共2兲 The model is defined by the action

S =

冕

dt

冕

S3

d⌺h␮␯ 2

␥2 , 共3兲

where the volume element on S3_{is d⌺=共r0}3_{/ 2兲dz d␸}

1d␸2, and

h_␮␯⬅⳵_␮u⳵_␯u*−⳵_␯u⳵_␮u*, 共4兲 where⳵_␮ denotes partial derivatives with respect to the four coordinates on S3⫻R, namely ␨␮ ⬅共t,z,␸1,␸2兲, u is a complex scalar field, and ␥⬅␥共兩u兩2兲, is a real functional of the squared modulus of u, and it defines the geometry of the target space. The metric on target space is given by

d␴2=du du *

␥ . 共5兲

Some cases of interest are the following: (a)␥= 1 corresponding to the plane with coordinates being the real and imaginary parts of u, (b) ␥=共1−兩u兩2_兲2_{, with 兩u兩}2_{⬍1, corresponding to the} Poincaré hyperbolic disc, and(c)␥=共1+兩u兩2_兲2_{corresponding to the sphere S}2_{. In such case, u is} related to the three dimensional unit vector nជ 共nជ2_{= 1兲 defining the sphere, through the} stereo-graphic projection

nជ= 1

1 +兩u兩2共u + u

*_{,− i共u − u}*_兲,兩u兩2_{− 1兲. 共6兲}

Notice that in the case of the target space being S2, the action(3) corresponds to the quartic term of the Skyrme–Faddeev action4which has been studied on the space–time S3⫻R in Ref. 7.

⳵␮

冉

K␮

␥

冊

= 0 共7兲

together with its complex conjugate, and where

K_␮= h_␮␯⳵␯u =共⳵u⳵u*兲⳵_␮u −共⳵u兲2⳵_␮u*. 共8兲 The model(3) possesses an infinite number of local conserved currents given by

J_␮=K␮ ␥ ␦G ␦u − K_␮* ␥ ␦G ␦u*, 共9兲

where G is a function of u and u*_{but not of its derivatives. Using the equations of motion(7) and} the identities

K_␮⳵␮u = 0, K_␮⳵␮u*− K_␮*⳵␮u = 0 共10兲

one can check that (9) is indeed conserved, i.e., ⳵␮J_␮= 0. The symmetries associated to such conservation laws are the area preserving diffeomorphisms of the target manifold. Indeed, the tensor h_␮␯/␥ is the pull-back of the area form on the target space

dA = idu∧du *

␥ . 共11兲

Therefore, the action(3) is invariant under diffeomorphisms preserving the area (11) and (9) are the corresponding Noether currents.2,8

III. THE ANSATZ

Following Refs. 2 and 3 we introduce the ansatz

u = f共t,z兲ei共m1␸1+m2␸2兲_, 共12兲

where mi, i = 1, 2, are arbitrary integers, and f is a real profile function. Replacing it into the

equation of motion(7) one gets ⳵t

冉

⳵tf2 ␥

冊

− 4z共1 − z兲 r₀2⍀ ⳵z

冉

⍀ ⳵zf2 ␥

冊

= 0, 共13兲 where ⍀ ⬅ m12_{z + m} 2 2_{共1 − z兲. 共14兲}

We now make a change of variable in the profile function, introducing a function g by

dg⬃df 2

␥ . 共15兲

For instance, in the particular cases discussed below (5) we get that (a) for the plane where ␥ = 1 one has g = f2_{, and g艌0; (b) for the Poincaré hyperbolic disk where␥=共1−兩u兩}2_兲2_{, one has g} = 1 /共1− f2_{兲, and g艌1; (c) for the sphere S}2_{where␥=共1+兩u兩}2_兲2_{one has g = 1 /共1+ f}2_{兲, and 0艋g} 艋1.

So, with the change(15) one gets that (13) becomes a linear partial differential equation in g ⳵t

2_{g −}4z共1 − z兲

r₀2⍀ ⳵z共⍀⳵zg兲 = 0. 共16兲

The solutions can be obtained by separation of variables introducing

g共t,z兲 = J共t兲H共z兲 共17兲

and then J and H must satisfy

⳵t 2 J +␻2J = 0 共18兲 and z共1 − z兲⳵z关共q2z +共1 − z兲

兲

⳵_zH

兴

+r0 2 ␻2 4

共

q 2_{z +共1 − z兲}

兲

_{H = 0, 共19兲}

where␻2is the separation of variables constant, and where q is

q⬅兩m1兩

兩m2兩. 共20兲

Equation (19) is what is called a Heun equation.9 It is a generalization of Gauss hypergeo-metric equation in the sense that it has one extra regular singular point. Indeed,(19) reduces to the hypergeometric equation for q = 1. Notice that (19) is invariant under the joint transformations

q↔1/q and z↔1−z. Therefore, if Hq,␻共z兲 is a solution of (19) for some value of q, so is

H1/q,␻共z兲 = Hq,␻共1 − z兲 共21兲

for the inverse value 1 / q. One can obtain solutions of(19) in powers series around z=0 and for 0艋q艋1. The solutions for q艌1, are then obtained from those using the above symmetry. The power series solutions are given by

Hq,␻共z兲 = 1 vq,␻

冉

z +

兺

n=2 ⬁ cnzn

冊

, 共22兲

where the positive constantsvq,␻are chosen such that the maximum absolute value of Hq,␻共z兲 in

the interval 0艋z艋1, is unity. The coefficients, for n艌2, are determined by the recursion relation (with c0= 0, and c1= 1) cn= − 1 n共n − 1兲

冋

共q 2_{− 1兲}

冉

␻ 2_r 0 2 4 −共n − 2兲 2

冊

_c n−2+

冉

␻2_r 0 2 4 −共n − 1兲共n − 2兲 + 共q 2_{− 1兲共n − 1兲}2

冊

_c n−1

册

. 共23兲 We will be interested in solutions with␻real, and therefore the solutions of(18) are trigonometric functions. Of course, if g is a solution of(16), so is␣g +␤, with␣and␤constants. We will then use the following normalization for the solutions of(16):

gq,␻= 1

2关sin共␻t +␦兲Hq,␻共z兲 + 1兴, ␻⫽ 0 共24兲

with Hq,␻共z兲 given by (22). The advantage of such normalization is that the solutions (24) take

values on the real line from zero to unity only. Therefore, they are admissible solutions for the case where the target space is the sphere S2[see discussion below (15)].

We normalize the static solutions of(16) as

g_q,0=ln共q

2_{z + 1 − z兲}

ln q2 . 共25兲

So, gq,0is a monotonic function varying from zero at z = 0 to unity at z = 1. Notice that gq,0→z as

q→1. A decreasing function can be obtained by the interchange gq,0→1−gq,0. Therefore,(25) are

also admissible solutions for the case where the target space is the sphere S2.

The admissible solutions for the cases of the plane or the Poincaré hyperbolic disk can then be written as

g_q,plane_␻ =␣gq,␻, gq,␻ Poinc.disk

=␣gq,␻+ 1 共26兲

with␣ being a real and positive constant, and gq,␻being given either by(24) or (25) (see Tables

I–III).

IV. THE ENERGY AND BOUNDARY CONDITIONS

The energy for the solutions obtained through the ansatz (12) and (15) is given by

E =

冕

S3 d⌺ H = 16␲2r0

冕

0 1 dz⍀

冉

共⳵tg兲 2 4z共1 − z兲+ 1 r0 2共⳵tg兲2

冊

, 共27兲

whereH is the Hamiltonian density associated to (3). Using the equation of motion (16) one gets that the energy of static configurations is

Estatic=

冏

16␲2

r0

共⍀g⳵zg兲兩z=0

z=1_{. 共28兲}

Therefore, the energy of the static solutions(25) is

E共gq,0兲 =

16␲2

r0

兩m1m2兩共q − 1/q兲

ln q2 . 共29兲

Notice that the solutions of (19), for ␻⫽0, which do not vanish at z=0 or z=1, have a logarithmic divergence on its first derivative, at those points. We then observe from(27) that such solutions do not have finite energy. Consequently, we shall impose the following boundary con-ditions:

Hq,␻共0兲 = Hq,␻共1兲 = 0 for␻⫽ 0. 共30兲

In addition, using the equation of motion(16) one obtains that

TABLE I. The first three frequencies leading to solutions(22) satisfying the

boundary conditions(30) for some chosen values of q. Due to the symmetry

(21) the frequencies are invariant under the interchange q↔1/q.

q ␻1 2_r 0 2_{/ 4 ␻} 2 2_r 0 2_{/ 4 ␻} 3 2_r 0 2_{/ 4} 1 2 6 12 3 / 4 1.983 765 266 031 5.987 866 563 654 11.988 820 961 965 1 / 2 1.913 643 550 920 5.923 271 548 463 11.930 495 946 441 1 / 4 1.733 644 829 967 5.672 599 570 202 11.668 111 143 086 1 / 10 1.517 649 738 276 5.238 029 617 258 11.066 608 078 277

TABLE II. Numerical values of⌳q,␻i, as defined in(34), for some chosen

values of q and for the three frequencies given in Table I. Due to(21),⌳q,␻

is invariant under the interchange q↔1/q.

q ⌳q,␻1 ⌳q,␻2 ⌳q,␻3 1 4 / 3 27/ 5 64/ 7 3 / 4 1.362 643 4.693 750 9.501 984 1 / 2 1.505 658 4.275 566 9.428 440 1 / 4 2.065 180 4.819 658 8.855 009 1 / 10 3.739 188 7.930 185 13.004 162

dE dt = 32␲2 r0 兩共⍀⳵tg⳵zg兲兩z=0 z=1 . 共31兲

Therefore, the energy is conserved for the static solutions, and for those solutions(24) satisfying the boundary conditions(30).

Multiplying(19) by Hq,␻and integrating in z, one gets that

冕

0 1 dz ␻ 2_⍀ 4z共1 − z兲 Hq,␻ 2 − 1 r₀2

冕

₀ 1 dz⍀ 共H_q,

⬘

_␻兲2= − 1 r₀2兩共⍀Hq,␻Hq,

⬘

␻兲兩z=0 z=1 . 共32兲

Consequently, the energy(27) for the solutions (24) satisfying (30) is

E共g_q,␻兲 =16␲ 2 r0 兩m1m2兩⌳_q,␻ , 共33兲 where ⌳q,␻=⌳1/q,␻= ␻2_r 0 2 16

冕

₀ 1 dz

冉

q 1 − z+ 1 qz

冊

Hq,␻ 2 ₌ 1 4

冕

₀ 1 dz

冉

qz +1 − z q

冊

共Hq,

⬘

␻兲 2_{. 共34兲}

The fact that⌳q,␻is invariant under q↔1/q, follows from the symmetry (21). In Table II we give

the values of⌳q,␻for some chosen values of q and some allowed frequencies␻. So, the energies (29) and (33) do not depend upon the signs of the integers m1and m2and it is invariant under the interchange m1↔m2 [see (20)].

Multiplying(19) by Hq,␻, subtracting from the same relation with␻interchanged with␻, and

integrating in z, one gets that

共␻2_−␻2_兲

冕

0 1 dz ⍀ 4z共1 − z兲Hq,␻Hq,␻= 1 r₀2兩共⍀共Hq,␻H

⬘

q,␻− Hq,

⬘

␻Hq,␻兲兲兩z=0 z=1 . 共35兲

Similarly, differentiating(19) with respect to z once, multiplying by H_q,

⬘

_␻, subtracting from the same relation with␻interchanged with␻, and integrating in z, one gets that

共␻2_−␻2_兲

冕

0 1 dz⍀H_q,

⬘

_␻H_q,

⬘

_␻= 1 r0 2兩共4z共1 − z兲⍀共Hq,

⬘

␻Hq,

⬙

␻− Hq,

⬙

␻Hq,

⬘

␻兲兲兩z=0 z=1_{. 共36兲}

Consequently, for the solutions(24) satisfying (30), and having finite first and second z-derivatives at z = 0 and z = 1, one gets the orthogonality relations

TABLE III. Numerical values of the normalization constantvq,␻i, as defined

in(22), for some chosen values of q and for the three frequencies given in

Table I. Due to(21), vq,␻is invariant under the interchange q↔1/q.

q vq,␻1 vq,␻2 vq,␻3 1 1 / 4 1 / 6冑3 1 / 16 3 / 4 0.285 189 93 0.119 156 83 0.070 789 33 1 / 2 0.332 611 65 0.152 779 52 0.087 010 28 1 / 4 0.409 991 03 0.204 149 92 0.126 957 32 1 / 10 0.502 749 25 0.256 726 47 0.167 310 92

冕

0 1 dz ⍀ 4z共1 − z兲Hq,␻Hq,␻=

冕

₀ 1 dz⍀H_q,

⬘

_␻H_q,

⬘

_␻= 0 for ␻⫽␻. 共37兲 It then follows that the energy(27) of a linear combination of solutions (24) satisfying (30), is just the sum of the energies of each solution, i.e.,

E共Agq,␻+ Bgq,␻兲 = A2E共gq,␻兲 + B2E共gq,␻兲 for␻⫽␻. 共38兲

Therefore, in that sense, the modes for the same value of q are decoupled. However, the intensities in which they enter in the superposition are not independent. As discussed below (15) the real values that the profile function g can take depend on the target space under consideration. So, when we take linear combinations of the solutions we have to respect those constraints. In Sec V we discuss those constrained linear combinations in detail in the case where the target space is the sphere S2_.

The boundary conditions(30) lead to a discrete spectrum of allowed frequencies ␻. Indeed, for the case where q = 1 the series(23) truncates whenever

r₀2␻2

4 = n共n + 1兲, n = 1,2,3,…, 共39兲

and the corresponding polynomials satisfy the boundary conditions (30). The first four of those polynomials are given by[with the normalization as in (22)]

H1,2= 4共z − z2兲, H_1,6= 6

冑

3共z − 3z2+ 2z3兲, H1,12= 16共z − 6z2+ 10z3− 5z4兲, H1,20= 2450共z − 10z2+ 30z3− 35z4+ 14z5兲

冑

7共3

冑

5 + 5

冑

6兲

冑

15 − 2

冑

30 , 共40兲

where the first index refers to q = 1 and the second to n共n+1兲 as given by (39).

For q⫽1 the series (22) does not truncate, and the frequencies␻leading to solutions satis-fying(30) can be found numerically. We give in Table I the first three frequencies for some chosen values of q. In addition, the same frequencies hold true under the exchange q↔1/q due to the symmetry (21). Therefore, the frequencies become smaller as q departs from unity, either to smaller or greater values than unity.

In Fig. 1 we exemplify the shape of the functions Hq,␻, by plotting the first three modes Hq,␻_i,

for q = 3 / 4 and q = 1 / 10 and the frequencies␻i, i = 1 , 2 , 3, given in Table I. Due to the symmetry (21), H1/q,␻i can be obtained by reflecting the plots around z = 1 / 2. Notice that the polynomial solutions for q = 1, given in(40), are invariant under (21). One observes that as q decreases, the functions Hq,␻ get deformed in a way that their first derivatives at z = 1 increase. For q⬎1, it

follows, due to(21), that the first derivative of Hq,␻at z = 0 increases as q increases.

V. THE CASE OFS2_{AS TARGET SPACE}

According to the comments below (15), in the case where the target space is the two dimen-sional sphere S2, we have that␥=共1+兩u兩2兲2, and so g = 1 /共1+ f2兲 and 0艋g艋1. The ansatz (12) becomes

u =

冑

1 − g g e

i共m₁␸₁+m₂␸₂兲_{. 共41兲}

As we have discussed, the solutions(24) and (25) are admissible solutions for the case where the target space is S2_{. However, one can construct more admissible solutions, i.e., with 0艋g} 艋1, by taking linear combinations of the solutions (24) and (25). An interesting case corresponds to linear combinations of the static solution gq,0given by(25), with one or more time dependent

solutions of the type(24). It leads to oscillating (in time) Hopfion solutions. Let us consider the case q艋1 first. The static solution gq,0 vanishes at z = 0, and has z-derivative equals to 共q2

− 1兲/ln q2_{there. The function f}

q共z兲⬅关共q2− 1兲/ln q2兴vq,␻Hq,␻共z兲, with Hq,␻given by (23) and

sat-isfying(30), has the same behavior at z=0 as gq,0. In addition, c2given by(23), is negative for the case of Hq,␻ satifying (30), and so fq共z兲 grows slower than gq,0 for small z. We do not have

rigorous proof, but by careful direct inspection we found that, as z increases from zero to unity the absolute value of fq共z兲 never exceeds the value of gq,0, which is a positive monotonic function of

z. It then follows that the combination

gq,0+␣ 共q2_{− 1兲}

ln q2 vq,␻sin共␻t +␦兲Hq,␻共z兲 共42兲 with 0艋␣艋1, takes values between zero and unity only, and so it is an admissible time dependent solution for the target space S2. Therefore, by adding up such types of solutions and dividing by their number, one gets that the time dependent solutions

FIG. 1. Plots of Hq,␻i共z兲, i=1, 2, 3, as normalized in (22), for q=3/4 (top) and q=1/10 (bottom) and for the three

frequencies given in Table I. The number of zeroes increase with the increase of␻i. The plots of Hq,␻and H1/q,␻are related

g_q共N兲共t,z兲 =ln共q 2_{z + 1 − z兲} ln q2 + 1 N 共q2_{− 1兲} ln q2

兺

_␻ i ␣ivq,␻_isin共␻it +␦i兲Hq,␻_i共z兲, 共43兲

where N is the number of modes(frequencies) entering into the sum, and␣i’s are real coefficients

satisfying 0艋␣i艋1, are admissible solutions for the target space S2. The energy of such solutions,

according to(24), (25), (29), (33), and (38), is given by

E =16␲ 2 r₀ 兩m1m2兩 共q − 1/q兲 ln q2

冉

1 + 4 q2 N2 共q − 1/q兲 ln q2

兺

_␻ i ␣i 2 vq,␻_i 2 _⌳ q,␻_i

冊

共44兲 with⌳q,␻_i given by(34).

Solutions of the type(43) for q艌1 can be obtained using the symmetry (21), i.e., g_1/q共N兲共t,z兲 = g_q共N兲共t,1−z兲. Solutions that decrease from unity at z=0 to zero at z=1 can be obtained by the symmetry g_q共N兲共t,z兲→1−g_q共N兲共t,z兲. The energy (44) is invariant under those two types of transfor-mations. As we now show, all these solutions correspond to oscillating Hopfions, i.e., solutions that oscillate in time and have a constant(in time) nontrivial Hopf number.

For any fixed time t our solutions define a map from the physical space S3_{to the target space}

S2_{, and so it is a Hopf map.}10

We now show that the Hopf invariant (the linking number) is independent of time for the admissible solutions we have constructed. In order to calculate the Hopf index of the solution we introduce

⌽1=

冑

g cos共m2␸2兲, ⌽3=

冑

1 − g cos共m1␸1兲,

⌽2= −

冑

g sin共m2␸2兲, ⌽4=

冑

1 − g sin共m1␸1兲, 共45兲 which defines another 3-sphere S_⌽3, since⌽12+⌽22+⌽32+⌽42= 1. The field u in(41) can be written as

u =⌽3+ i⌽4

⌽1+ i⌽2. 共46兲

Since u parametrizes the sphere S2 through the stereographic projection (6), we have that (46) gives the map S_⌽3→S2. So, the Hopf index is in fact evaluated through the map S3→S_⌽3→S2, as we now explain. We introduce the potential

Aជ− i 2共Z †_{ⵜជZ − ⵜជZ}†_{Z兲, 共47兲} where Z =

冉

Z1 Z2

冊

, Z1=⌽3+ i⌽4, Z2=⌽1+ i⌽2 共48兲 and the differential operatorⵜជ is the gradient on the physical space S3_{. The Hopf index is defined} by the integral10

QH=

1

4␲2

冕

d⌺ Aជ·curl Aជ, 共49兲

where d⌺ is the volume element on the physical space S3_{. Evaluating we get}

Aជ= − m₁共1 − g兲

冑

1 − zeˆ␸1+ m2

g

冑

zeˆ␸2 共50兲

and

curl Aជ= 2⳵zg共− m2

冑

1 − zeˆ␸1+ m1

冑

zeˆ␸2兲. 共51兲 Consequently we get that

QH= m1m2关g共t,z = 1兲 − g共t,z = 0兲兴. 共52兲 Therefore, the solutions(25) have Hopf index

QH共gq,0兲 = m1m2 共53兲

and so they are static Hopfion soliton solution. The time dependent solutions(24) are constant at

z = 0 and z = 1 due to the boundary condition (30), i.e., gq␻共t,0兲=gq␻共t,1兲=1/2. Therefore, they

carry no Hopf number

QH共gq,␻兲 = 0, ␻⫽ 0. 共54兲

The solutions(43) although time dependent, also have contant values at z=0 and z=1, determined by their static component. It then follows that

QH

共

gq共N兲

兲

= m1m2 共55兲 and so they do correspond to oscillating Hopfion soliton solutions.

The Hopf index can also be calculated as the linking number of the preimages of two points of S2.10Notice, from(6), that the north pole of S2, nជ=共0,0,1兲, corresponds to u→⬁, and so from (41) to g=0. On the other hand, the south pole of S2_{, nជ=共0,0,−1兲, corresponds to u=0, and so to}

g = 1. Therefore, from (45) and (48), we see that the pre-image on S_⌽3 of the north pole of S2 correponds to Z1= eim1␸1 and Z2= 0, while the preimage of the south pole to Z1= 0 and Z2 = e−im2␸2_{. For the static solution}(25) and the time dependent solutions (43), the preimages in the

spacial S3of these two circles in S_⌽3 are constant in time. In addition, those two circles in S_⌽3 pass through each other m1m2times as␸1and␸2varies from 0 to 2␲in S3. So their linking number is

m1m2, and that is the Hopf index. For the time dependent solutions(24) it is not possible to have

g = 0 and g = 1 on different points on the spatial S3_{at the same time t. Therefore, the preimages of} the north and south poles of S2_{never link, and so the Hopf index of such solutions vanishes.}

ACKNOWLEDGMENTS

The authors are grateful to Olivier Babelon for many helpful discussions throughout the development of this work. L.A.F. thanks the hospitality at LPTHE-Paris where part of this work was developed. The project was funded by a Capes/Cofecub agreement. L.A.F. is partially sup-ported by a CNPq research grant and E.D.C. by a Fapesp scholarship.

1

O. Alvarez, L. A. Ferreira, and J. Sanchez Guillen, Nucl. Phys. B 529, 689(1998).

2

O. Babelon and L. A. Ferreira, J. High Energy Phys. 0211, 020(2002).

3

H. Aratyn, L. A. Ferreira, and A. H. Zimerman, Phys. Rev. Lett. 83, 1723(1999); Phys. Lett. B 456, 162 (1999). 4

L. Faddeev, 40 Years in Mathematical Physics(World Scientific, Singapore, 1995); L. D. Faddeev and A. J. Niemi,

Nature(London) 387, 58 (1997); R. A. Battye and P. M. Sutcliffe, Phys. Rev. Lett. 81, 4798 (1998); J. Hietarinta and P. Salo, Phys. Rev. D 62, 081701(2000).

5

D. A. Nicole, J. Phys. G 4, 1363(1978).

6

L. A. Ferreira, “Euclidean 4d exact solitons in a Skyrme type model,” hep-th/0406227.

7

R. S. Ward, Nonlinearity 12, 241(1999). 8

L. A. Ferreira and A. V. Razumov, Lett. Math. Phys. 55, 143(2001).

9

A. Ronveaux, Heun’s Differential Equations(Oxford University Press, New York, 1995); S. Yu. Slavyanov and W. Lay, “The Heun class of equations,” Special Functions: A Unified Theory Based on Singularities(Oxford University Press, New York, 2000), Chap. 3, pp. 97–162.

10

R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82(Springer-Verlag,