CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

Р, F O R G Á C S Z, H O R V Á T H L, P A L L A

KFKI-1981-21

SOLITON THEORETIC FRAMEWORK FOR GENERATING MULTIMONOPOLES

H ungarian ^Academy o f Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

P. Forgács

Central Research Institute for Physics H-1525 Budapest 114, P.O.B.49, Hungary

and

Z. Horváth and L. Palla

Institute for Theoretical Physics Roland Eötvös University H-1088 Budapest, Hungary

HU ISSN 0368 5330 ISBN 963 371 799 X

is based on the equivalence of the axially and mirror symmetric Bogomolny equations and the Ernst equation. The properties of the Ernst equation that are relevant formonopoles are also discussed.

The application of the method is illustrated on the example of the one and two monopolé solutions.

А Н Н О Т А Ц И Я

Предлагается систематический метод для генерации SU(2) монополей произ­

вольного заряда типа Янга-Милльса-Хиггса, основанный на преобразовании Бэклунда. Получена чисто алгебраическая итерационная формула для нашего пре­

образования Бэклунда. Наш метод основан на эквивалентности аксиально и зер­

кально симметричного уравнения Богомольного и уравенения Эрнста. Обсуждаются те свойства уравнения Эрнста, которые существенны для монополей.

Применение метода иллюстрируется на примере решений с одним и двумя мо- нополями.

KIVONAT

Az S U (2) Yang-Mills-Higgs elmélet véges energiájú, tetszőleges topológi- kus töltéssel rendelkező monopólus megoldásainak generálására szolgáló mód­

szert fejlesztünk ki Bäcklund transzformációk fölhasználásával. Levezetünk a Bäcklund transzformációk iterálására vonatkozó, tisztán algebrai formulát.

Módszerünk az axiállsan szimmetrikus Bogomolny és az Ernst egyenlet ekviva­

lenciáján alapul. Az Ernst egyenlet a monopólus elmélet szempontjából fontos tulajdonságait is diszkutáljuk.

A módszer alkalmazásaként megmutatjuk, hogyan lehet generálni az egysze­

res töltésű és egy uj, kétszeres töltésű monopólus megoldást.

Аз it was shown for the first time by ’tHooft and P o l y a ­ kov [l] spontaneously broken gauge theories w i t h a simple gauge group possess classical solutions w h ich may be identified as magnetic monopoles. These solutions can be interpreted as "soli- tons" in 3+1 dimensions in the sense that their e n e r g y is we l l

localised at any instant of time, so they can r e p r esent "particles".

These particles are stabilized by a quantum number, topological in origin w h i c h corresponds to the magnetic charge. Magn e t i c m o n o ­ poles arise naturally in the curre n t l y popular g r a n d unified t h e ­ ories u n i f y i n g the weak, electromagnetic and strong interactions.

In Ref, |2] it was suggested that particles w i t h b o t h magnetic and electric charges / dyons / m a y play an important role at the subconstituent level.

In the limit of v a n i s h i n g Higgs potential, wh e n the

Higgs field becomes massless, a considerable simp l i f i c a t i o n arises and the analytic fo r m of the static singly charged and spherically symmetric

*

Sommerfield jjj] and by B o g o m o l n y [

4

It has been shown by perturbative techniques |5,6j that the force b e t w e e n monopcles d e c r e a s e s faster than any inverse p o wer of the separation.

The existence of static, n o n i n t e r a c t i n g finite energy m o n o p o l e s has been r e c e n t l y established by Taubes [7] . The first

exact solution c o r r e s p o n d i n g to a d o u b l y charged solution / two m o n o p o l e s superimposed on each other / was found by Ward [в]

u s i n g twistor met h o d s and by the present authors [

9

[lOj and the d o u b l y charged m o n o p o l é / 2MP / from a simple

"vacuum" solution. These results are based on the equivalence of the axially and m i r r o r symmetric B o g o m o l n y equations w i t h a

r a t h e r well k n o w n e q u a t i o n of g e n e r a l relativity, the Ernst e q u a ­ tion« [ll] . F o r the E r n s t equation there are various solution g e n e r a t i n g techniques: Bäcklund t r a n s f o r m a t i o n s /ВТ/ found by H a r r i s o n [

12

13

14

The purpose of this paper is to d e v e l o p a systematic m e t h o d outlined in Ref. |9,10] to gene r a t e axially symmetric mult i m o n o p o l e s o f a r b i t r a r y charge. Our procedure is closely r e l a ­ ted w i t h the inverse scattering m e t h o d s applied in certain two d i m e n s i o n a l systems w h i c h are com p l e t e l y integrable.

We introduce a Bäcklund t r a n s f o r m a t i o n /slig h t l y d i f f e ­ rent fro m H a r r i s o n ’s В Т / to o btain the m o n o p o l é solutions. The m a j o r advantage o f our m e t h o d that it consists of entirely algebraic steps, and acts directly on the c o m p o n e n t s of vector potential. We have to solve a R i c c a t i type equation only once

which is easily done for o u r simple "vacuum" solution. We give in this article a very neat f o r m for the result of iterating an

arbitrary number of B T ’s. We show that by each s t e p the topological charge is increased by one unit.

Our paper is organized as follows! In Section 2. we

summarize our notation?and discuss the axially and mirror symmetric B o g o m o l n y equations. In Sect. 3. we discuss the relevant properties of the Ernst equation. In Sect. 4. we describe the relationship between the Bogomolny and the self-duality equations. In S e c t . 6.

we d i s p l a y the general formu l a e for an arbitrary number of B T ’s.

In Sect. 7. we give some special applications.

2. THE A X I A L L Y SYMMETRIC E Q U A T I O N S

We consider an S U (.2) gauge theory w i t h an isotriplet

Higgs field in the limit of v a n i shing Higgs potential. The L a g rangian d ensity is

»here F * = 'Э, ft' - 3 V ft“ - fty

/ We chose the coupling constant e = 1 /.

The Hamiltonian d ensity f o r static c o n f i g u r a t i o n s with no electric fields / A 0 = 0 / i s

* - Ац ^ F a i ^ + ^ D ^ f í D ^ V

( 2 )

The field equations of th i s theory are solved b y configurations s a t isfying the Bogomolny equations [

4

F % - 1 ■. Ш Т ^(V

The energy, E , can be w r i t t e n u s i n g the B o g o m o l n y equations as

E- ^(«)

N o w

(5)

u s i n g the equations of m o t i o n for the Higgs field,

so the energy c a n be cal c u l a t e d f r o m the H i ggs field alone

E- I

The topological charge, n , is g i v e n by

> h L - > 5 ‘ V K

У) я

*-*<0

Since the asymptotic b o u n d a r y c o n d i t i o n ^•v%r> V 4*— 9oo

imposed, for в solution w i t h topological c h a r g e n

(7)

is

a s

Л

С8)

However, condition (в) alone does not guarantee that the topologi­

cal charge is indeed n due to the presence o f possible singulari­

ties. Fro m n o w on, we take the v acuum expectation value of the Higgs field N/ -

A

When one is l o o k i n g for exact solutions of complicated equations, there is a need for a simplifying ansatz consistent w i t h the known properties of the c o nfiguration in quest. In the p resent case the simplest ansatz w o uld be a sphe r i c a l l y symmetric one. However, the as s u m p t i o n of spherical symm e t r y is too strong, it excludes all but the s i n g l y charged ’t H o o f t-Polyakov monopolé

[

15

/ e.g. a rotation around a n y axis / can be compensated b y a gauge transformation [lő] •

M a n t o n [

17

)

w here X ^ = <§

}

tions (

3

, and Ф

4

are

into the B o g o m o l n y equa-

(,11а) Э* Фа t W a<^» - g* (эИ г + W, (lib)

Ч Ч ~ Ч Ч = §_/1 ( Д Ч а - Ф л О (lie)

(na) + \ а /4Ф д =* § Л + Ч л О (lie)

We hav e five equations for the six u n k n o w n variables, however, equations (lla-e) still pos s e s s a residual U(l) gauge invariance:

V L = V/t + A

Ф*.

X'li.

CoS

A + tf.

V я!:

S l H A j

( ¹² )

2/|

This residual g a u g e freedom enables us to reduce the number of u n ­ known functions f r o m six to five. In fact, we c a n do even more than that, namely, we can sat i s f y one equa t i o n in (ll) by the f o l l o w i n g trick: it is possible to find such а Л that

W ’ A >

(l?a) (l?b)

are s i m u l t aneously true. This is mor e than a g a uge fixing, because (l^a) and (l3b) imply equations (lla-c) f o r <ф ^ ^ •

VI i

To sho w this explicitly, one should d erive the integrability con­

ditions for equations (l^a-b), w h i c h are easily found to be Ч Ч " Ч Ч = ^ г Ф л ) + ^ v > A ( Ф м +

+ в Л1>г,г + /у\4Н 1 ) + Ч г Ч ] )

(l4)

Eq.

(14)

Now, u s i n g (l3a-b) eqs. (lla-e) r educe to f o u r equations for four u n k n o w n functions. Next, we observe that (lib) implies the existence o f a f u n c t i o n ^ , such that

ь t

\

i V

Then it is easy to see that (lie) c a n be also satisfied by putting

VJ .= -

% s

Л i

VJ = - j k

>

1

(

16

and the r e m a ining two equations (lld-e) reduce to

(17.)

4 Л Ч - - 2 V í . V t - 0

where Л = e

I n t roducing 6 = £ + i. v|/

(17b)

, v = ( 3 „ o s )

eqs. (l7a-b) m a y be w r i t t e n as

R e e

Л е

which is the celebrated fo r m of the E r nst equation of general relativity. A geometrical d e r i v a t i o n of eq. (l8) fr o m (lla-e) was g i v e n in Ref. jio] •

The equivalence of eqs. (lla-e) and (l8) is of interest because it reveals a surprising / alth o u g h completely

f ormal / connection between general re l a t i v i t y and Sü(2) gauge theories, and w h a t is more important f o r us, it gave the clue to find n e w exact solutions of the B o g o m o l n y equations describing m u l t i p l y charged monopoles, using the various solution generating techniques w o r k e d out for the Ernst equation.

As we w o u l d like to make this paper somewhat self-c o n t a i n ­ ed, in the next section we summarize b riefly those properties of this equation that are r e l e v a n t in the study o f monopoles. For a moi'e detailed d i s c u s s i o n o f this important e q u a t i o n in general r e l a t i v i t y we r e f e r to the literature jl8,19] •

P R O P E R T I E S O F THE E R N S T E Q U A T I O N

The m o s t general line element of an axially symmetric, s t a t i o n a r y space-time can be w ritten as

(-19)

where

$

3

The v a c u u m E i n s t e i n equations c o rresponding to (19) are

Л in ^ ( x jиз) = 0 О 0

& Ü U ,

-

Г ?г s

/Í\T 0 o'

(21a-)

Once eqs. (20a,b) are solved, it is easy to integrate eqs.

(21a-c) f o r

Í)

sponds to the linear coordinate transformations:

t = ol 1 " b

^

^ =>

^ -

4

This SL(2,R) invariance g r oup wil l be denoted b y L. R e w r iting eqs. (20a,b) as

( ²³ )

■ Ц З - ^ - ^ v í - 0

w h e r e 4 + = — + u j . 'h + t r a n sforms u n d e r the Í “

a c t i o n o f L as

) c\ + b

4 + = ---- --- d £ с. ^ +

N o w we observe that eq. (20b) can be interpreted as the integrability c o n d i t i o n for the existence o f a n e w function

which is related to u3 in the following way:

о, »

f 0

j (25)

Intr o d u c i n g the complex E r n s t potential, € • t v v Hf , eqs.

(20a, b} take the form: ^e. € Д fc - ($7fe ) s О w h ich is just e q f (l8) . The Ernst eq. (l8) is invariant under a n e w SL(2,R) transformation group, P , / Ehlers transformations ^20] / defined by

€ >b ! L 1 1 1e ^ д о

5"- i í t

It is important to realize that there exists a d i rect map p i n g from eqs* (l8) to eqs. (20a,b) / this is the so called Neugebauer- K r amer mapping [2l] /. T h i s map p i n g is a g r oup element, I = l”\

of a cyclic g r o u p of order two, defined by

V - S > . %

+ - ■£ J ю я _L ^ ; 4/^=» i io

We s h all write (27) in a m o r e compact form

In fact, the g r o u p s L and P are r elated by I : X L I " 1 = P .

(27)

(28) The a c t i o n of the group o f coordinate transformations, L , does not c ommute w i t h the Ehlers transformations, P , and by the r e ­

peated applications of these transformations one can generate n e w solutions o f the Ernst equation f r o m the old ones. However,

properties of solutions g e n e r a t e d this w a y change in an u n c o n t r o l l a ­ ble way, e.g. starting w i t h an asymptotically f l a t solution the result will be an a s y m p t otically non-flat space-time w h ich is of little use in g eneral relativity. In fact, we h a v e an infinite d i ­ mensi o n a l symmetry group — the Geroch g r oup {

22

24

14

value problem. Ind e p e n d e n t l y of each other H a r r i s o n [l2] and N e u g ebauer [l3] found B ä c k l u n d transformations / В Т / f o r the Ernst equation. Neugebauer deduced a comp o s i t i o n theorem f o r two successive B T ’s , so once a В Т has b e e n d e t e rmined one c a n apply arbitrary many B T ’s by algebraic me t h o d s alone. Cosgrove [l9]

has found a composition t h e o r e m for H a r r i s o n ’s B T ’s using N e u g e ­ b a u e r ’ s work. A l l these techniques could be successfully applied to generate solutions of phys i c a l relevance / a s y m p t o t i c a l l y flat / for the Ernst equation c o n t a i n i n g arbitrary m a n y free parameters.

In g eneral relativity for m o s t solutions of interest, the Ernst potential, £ , can be w r i t t e n as

i I a u k ”'

£ = ,

П

(29)

/ are the u s u a l spherical coordinates, m is the / r e a l / Schwarzshild m a s s /. However, if one is interested in g e n e r a t i n g solutions of the Ernst e q u a t i o n c o r r e s p o n d i n g to m a g ­ netic monopoles, one needs a c o m p letely d i f f e r e n t c o n d ition of

(29) . The gauge invariant length o f the H i g g s field is given by

Ф " = ф \

Л

пП

í,* + ч->г

V

It is important to note h e r e that the Ehlers transformations (

26

Ф . Our boundary c o n d i t i o n requires that for a monopolé of c h a r g e n,

ф * _ (fö

w h i c h is just incompatible w i t h eq. (29

)

est in gauge theories and v i c e versa.

We think, it is c l ear f r o m the above that one should be somew h a t careful in a p p l y i n g the techniques w o r k i n g so beatifully in g e n e r a l relativity for g e n e r a t i n g mu l t i m o n o p o l e solutions due to the fact that the b o u n d a r y c o n ditions are so different. In fact, it is a v e r y intrigueing f eature of the E r nst equation that both the Kerr and the m o n o p o l é s o l u t i o n can be interpreted as

"solitons" in the sense that both c a n be g e n e r a t e d by BT» s starting wi t h the "vacuum" of g e n e r a l r e l a tivity / M i n k o w s k y space / or

with the Higgs v acuum / ф —

\

B e f o r e entering into the details of explaining h o w Böcklund transformations can be used to generate multimonopole

solutions, we make a short dig r e s s i o n to sh o w the c o n nection between the Ernst equa t i o n and its symmetry transformations on the one

han d and the selfduality equations and the non-linear transfor­

mat i o n s of C o r r i g a n et al (

26

4. THE C O N N E C T I O N BETWEEN N O N O P O L E S AND SELFVDUAL G A U G E FIELDS

Let u s consider a pure SU (

2

C ^ И

The Euclidean space field equations are solved by such configura­

tions w h ich satisfy the self-duality conditions

Г ” = *

/»v > v

^

w h ere b is the dual f i eld tensor /»V

* Г ' . ! t Г Я 4 5 '

V y

2

Ф )

(34)

In the case w h e n all fields, 0 ” , are independent of the Euclidean time the self-duality equations (ЗЗ) reduce to

F.“ = (D f t Y*

where we recognize the B o g o m o l n y eqations (

3

Yang h a s shown the existence of a particularly convenient gauge -n the R -gauge — in w h i c h the self-duality equations (33) take the f o l l o w i n g fo r m [

25

+ + T,»X V

(?

а)

Х

К з + О

2 Ь,Х>5 - 2 х)4Х,г-0 <?6Ь>

* ( Ъ - Л + х > » ) - 2 x >i

(,б с )

Гг^=хл*;хг П * - x 5 *íx 4

It is u seful to introduce a n e w funct i o n ТГ realizing that (?6b,c) are identically satisfied d e f i n i n g T as

- 2. — - 2 —

X 5 X ^;ij=

/ and similarly f o r the barred quantities /. In terms of X j T TV the self-duality equations take the form

X U )Vi-t 1,tl)-í^ + %s

= 0 (j6d)

+

(?6e>

1%Ч»),ч + ич*),»'0 C ,6f)

The striking s i m i larity of eqs. (j6a-c) to the Ernst equations (l7a,b) and of ^36d-f) to (|Oa,b) m a kes it obvious h o w to relate the former systems o f equations to the latter ones. Indeed, taking X to be real and assuming that both % and X depend only on we immediately obtain eqs. (l7a,b) [27^ • However, taking X =» S ^ X =

Vz

«

тг= ( 2 ы in eqs. фб^) / with vy and v>o being real f u n c tions / we again o b t a i n (l7a,b) and (20a,b)

respectively. This is important as it is also possible to show that u s ing these latter identifications f o r X 0 X we obtain the expressions (l3,15,l6) for the f u n c tions of the M a n t o n ansatz.

As it was pointed out by C o r r i g a n et al. |26j the self­

d uality equations (зба-с) have an S L (2, C ) invariance group w h i c h acts n o n l i n e a r l y on X and X :

X

____________X

(.♦ + ЬЬ XL ^(?Ta)

( c + d t ) ( 5 t b t ) + b d X 5"

-|T

5

(a + bl) + Ь x) + bb X 1

(?7b)

This transformation was denoted b y in Ref. ^26 • It is easy to see that (j57a,b) are the a n a l ogues of the Ehlers transformations for the Ernst equation discussed in S e c t . % It is important to

realize that in the SL(2,C) t r a n s f ormation g r o u p (?7) a,b,c,d can be taken not just simple par a m e t e r s but arbitrary analytical functions of ( y > z ) • ■^n w hat f ollows we shall use a simple subgroup of this Sb(2,C) group that maps ( т )

into (|й| Й í j R ' ^ ) where is an

arbitrary analytic function.

In Ref. [

26

if (

X

к j

36)

(

: -г

\ X. ) l T j

equations. Note that this is a discrete transformation y i e l d i n g the identity w h e n applied twice. This fac t and the structure of the t ransformation indicate that ^ c a n be related to the

N e u g ebauer-Kramer m a p p i n g (

27

This is yet not equivalent to eq. (

27

Q ( $ ) = 2 Га

£

— w i t h i n the M a n t o n ansatz — between the transformation o f Ref.

[26] and the N e u g e b a u e r - K r a m e r m a p p i n g eq. ( 2 7 ) : the latter is the product o f a ^ and a special Sb(2,C) t ransformation w i t h a g i ven analytic function.

In our opinion it would be m o r e appropriate to mak e a d i s t i n c t i o n between Bficklund t r a n s formations a n d the and

transformations of Ref. [26] , the l a t t e r being rather an invar­

iance property of the Yang equations. It seems reasonable to e x ­ pect that B T ’s for eqs. (зба-с) should reduce to those a lready k n o w n for the Ernst equation in a special case, just as the

Sb(2,C) transformations

(37)

)

28

that of Belinski and Zakharov for the Ernst equation was discovered. In Ref. j28j we constructed a four dimensional

"soliton" g e n e r a t i n g method to solve this linear eigenvalue p r o b ­ lem which is the g e n e r a l i z a t i o n of the method of Belinski and Zakharov. On the other hand, it has bee n proved by Cosgrove [l9]

that the soliton g e n e rating method of Belinski and Zakharov is in close connection w i t h H a r r i s o n ’s ВТ.

In Ref. ^28] it was shown that eqs. (_36a-c) possess an infinite p a r a meter invariance g r o u p w h ich is the analogue of the Geroch group. In applying this invariance g r o u p for generating n e w solutions for the Yang equations one encounters problems simi­

lar to the simpler case of the Ernst equation. A straightforward application does n o t lead ve r y far b ecause of the u ncontrollable singularities. Lohe [

29

X

/ which are of c ourse the B o g o m o l n y equations / carrying out this procedure on the B P S monopolé. He a pplied the special product

(*> £ to ensure the rea l i t y of the final r esult w h i c h turned out to be a singular configuration. This is not surprising r e m e m ­ bering that this m e t h o d does not lead to acceptable solutions in the case of gen e r a l relativity if we start from an asymptotically flat solution. To get physically interesting solutions / e.g.

those w i t h finite action / one should find suitable subgroups in a spirit similar to the w o r k of K i n n e r s l e y et al.

As it w a s shown by M a n t o n [

17

Corrigan-Fairli e - * t H o o f t - W i l c z e k / C F t H W / ansatz in a

com p l e x gauge, but he f o und that the C F t H W ansatz does not contain mul t i m o n o p o l e solutions.

A t i y a h and W a r d [

31

32

9

5. B Ä C K L U N D T R A N S F O R M A T I O N S F O R THE ERNST EQUATION

Since for g e n e r a t i n g mul t i m o n o p o l e solutions we found m o s t convenient to a p p l y a slightly modified f o r m of H a r r i s o h ’s В Т / HBT /, and the e x i s t i n g publications on this topic are some­

w h a t compressed , it m i g h t be of use to enter into some details.

We present the mater i a l f r o m a d i f f erent point o f v i e w f r o m that o f general relativity. We show h o w one can derive the composition the o r e m for H B T ’s fr o m f i r s t principles / wi t h o u t r e f e r r i n g to N e u g e b a u e r ’s c o m p o s i t i o n theorem / and derive the g eneral formulae f o r an arbitrary n u mber of H B T ’s.

A s a first step, we d iscuss H a r r i s o n ’s Bficklund trans­

f o r m a t i o n and derive the c o m p o s i t i o n theorem f o r two consecutive

B T ’s. This c o m position theorem is of fundamental importance since it enables us to construct m u l t i s o l i t o n solutions u s i n g pure

algebra once a Riccati equation for the first step has been solved»

We define the f o l l o w i n g variables:

*

4

) M - e M

x

c*

J

2 L *

* (38)

c 3 e "Э t -

u

)

These variables are different f r o m that of Harrison, the connection between (3S) and that of Ref. |l2^ w a s pointed out in Ref. [lO].

The Ernst equation (l8) can be recast into a s y stem of first order differential equations f o r the v a r i a b l e s (

38

«i,2 = - « А + « А - г ( Г + '0 ~ (Mi + * <-, 9 a )

«2,2 • - “A ♦ m 2 n i - г(с l Г)(N1 * мг) ■ C39b)

«1,1 = - « А

*

I

«2,1 ■ - ■ * * + «2 « 1 - Г ( С ~ Л 2Т M 1 + H 2 * (, 9 d )

Harrison derived h i s В Т following the m e t h o d of W a h l q u i s t and Estabrook [зз] • He started w i t h r e w r i t i n g eqs. (39a-d) as four 2-forms w h i c h are to v a n i s h for a solution. T h e n he looked for a p s e u d opotential q such that the 1 -form

ег = - d q + F ( q , N i>M i , S ^ d ^ + o ( q , N i ,Mi , ^ ) d C 2 satisfies dtr = 0 if eqs. (39) are satisfied and <a = 0.

Furthermore, he supposed that F and G depend only linearly on M ^ ’s and N ^ ’s. This a s s u m p t i o n guarantees that the r e s u l t i n g equa

tio n for the pseudopotential, q , w i l l be of Riccati type. Indeed this w a y one obtains a t o tal Riccati equation f o r q ( C ^ > ^

2

dq = [(М 2 - q + p(w) (М2 - Mlq2)] d X ^{, x +}

+ [ К - М 2)Ч + Р " Ч » ) ("l - V 2)] 2

w h e r e p(w) = у (w - i £, 2 )

(w

In order to f i n d the concrete fo r m o f the В Т he looked f o r a n e w solution of eqs. (,39) w h i c h d e p e n d s only on M^, N., q. This procedure l e a d s to the f o l l owing f o r m of the ВТ:

H(q,p)

H(q,p) M 2 = M 2Cl)

1 + pq

= - q --- - % p + q p + q q

(1

H (q, p) Nx = =

H(q,p) N2 = N 2Ü) =

1 + pq --- N, q ( p + q)

N,

q p2 -

1

1

1

1

2

q

(1

. ( « » )

. (41b)

, (41c)

4 s p C 1 + pq)

1 + pq

It will be convenient f o r us to s h o w at this stage h o w the Neugebauer-Kramer mapping, I , acts on the M^*s, N ^ ’s and o n q. A straightforward c a l culation u s i n g the d e f i n i t i o n s (

26

(

27

1 I M, = - + ---

1 Ni = * Ni + ₄

77 _%

1

I 1L = • M, + --- , 4 $

I N. N. _t

(42)

We define the action o f I on q in the f o i l w i n g way: Iq a q satisfies

dq = J- (q + p) + M2q (l + pq ) + — (l - q2)] d £ x +

. 4 5 (47)

+ [- Hi(5 + ;) + " г511 + ;) + “ “ С1 - 52)] a ( 2 .

Eq. (

43

40

It is not d i f f icult to v erify that q is g i v e n in terms o f p and q / satisfying eq. (40) / as

P + Q

4 ~ “ 1 + pq (44)

A t this stage it seems to be v e r y d i f f icult to apply a second ВТ, because we have to solve eq. (40) a gain replacing the M^*s, N ^ » s by * s and n P ^

»

q*

Usually this is a very hard task even for the simplest seed s o l u ­ tions, since after a single В Т the new c o n f i g urations can be

rather complicated. The great advantage o f the В Т is the existence of a c o m p o s i t i o n theorem stating that there is no n e e d to solve a n a l y t i c a l l y this complicated equation as its solution can be c o n ­ structed in an appropriate / algebraic / w a y from solutions of

(

40

We n o w proceed to d e r i v e this comp o s i t i o n theorem for two c o n secutive H B T ’s which w i l l enable us to find the general formulae f o r an arbitrary n u m b e r of steps b y algebraic methods alone. Let u s suppose that eq. (

40

Let us suppose that for the second ВТ q ’ depends only on (p-^, p 2 , q x , q 2) w h e r e (p^, q^) and (p2 , are b o t h solutions of (

40

w i t h the same M ^ ’s, N ^ ’s but d i f f erent constants w-^, w^, ^ 2 * This a s s u m p t i o n is motivated by other wor k in sol i t o n theory

/ Sine-Gordon, Korteweg-de Vries, nonlinear Schrödinger equations /.

This way we obtain two equations for d q ’, namely, eq. (

40

A f t e r e q u a t i n g them as it is s h o w n in d etail in A p p e n d i x A we obtain an o verdetermined system of equations for q' w hich can be consis t e n t l y solved yiel d i n g

, = 5 l pl “ 4 2 p 2 q l ( qlpl ” q 2 p 2)

Writing out (45) in terms of p^, p£, q-^, q 2 we get the formula derived by Cosg r o v e [l9j :

Рг(Х - pl K - Pli1 - p j K * (pf - pl)qlq2 _ л 6)

Ii[(pi - 4 ) * PiC1 - рг К - РгС1 - р1)Чг]

Since we w i l l use the transformations I H ( p ^ fqj) to

generate the m o n o poles we n o w turn our a t t e n t i o n to the e ffect of this transformation on the M ^ ' s and N^»s. Henceforth the I H ( p ^ fq ^ transformation w i l l be denoted by B ( p ^ , q ^ . It is easily veri f i e d that

В M^0’

В M

f

в

n[0 )

в K2<0)

(47a)

T ^ ° 5 + í t ) ’

(47b)

: ( ■ - r - ■ ) .

q \ q 4p § /

(47c)

q .

” 4P S

)

As we s h all iterate В we need the follo w i n g comm u t a t i o n property of the H and I transformations w h i c h can be derived w i t h ­ out difficulty f r o m (

41

B(p,q) = IH(p,q) = H(p,q)I = I"1B ( p , q ) I (

48

The pseudopotential in the argument of the second H t ransformation is of course not q* but q», since it is preceded by an I mapping.

So

I

4 2,5’) i HÍ pj . íj ) = н(р2,Ч>)н(р11Ч1) (49)

2

where we used that I = 1 .

6. S U P E R P O S I T I O N FORM U L A E F O R AN ARBITRARY N UMBER O F BÄCKLUND T R A N S F O R M A T I O N S

In this section we give the general formulae f o r n consecutive В transformations. Since the proof of this result is rather technical we present it in Appendix B.

It is important to deal w i t h the reality conditions for solutions g e n e rated by this method. The conditions

M * = N x , M * = N 2 (50)

wil l obviously ensure the rea l i t y of the n e w solution. F o r exam­

ple, after a single В step it i m m ediately follows fro m (47) that starting w i t h a real solution, € 0 , for w h i c h 4 ^ * 0 eq. (

50

q = q* > p*= p " 1 (51)

In a different case one should d e t e r m i n e what are the constraints following f r o m (

50

It is illuminating to s h o w h o w to derive the formulae

for two В steps which was applied in Ref. ^9} . W e proceed to carry out a second В transformation for (47)-.

N o w applying the du a l i t y principle we get for q* / see Appendix A /

9]^P2 “ ^2^1 Ql (q ].Pi “ q 2p 2^

Substituting expressions (45) (47b) and (

53

52

M.

(2)=

p lq l ~ p 2q2 M (o) + P 1 ~ p 2

p2q l " plq 2 4

s

(54 a)

Similarly for the other components

M (2) P lq l “ p 2q 2

q l P 2 “ p lq 2 LP lq l ” P 2q 2

p 2ql ~ plq 2 M (o) + q lq 2 (P1

~ Vl)

4 § (P2q2

, (54b)

N f2) _ p lql ~ p 2q 2 1 5 хр г -

plq l ~ p2q 2 N (o) + qlq 2 (p? ~ p

1

q lp 2 “ ?lq 2 4 § P

1

2

, (54c)

»,(2) _ q l p2 “ plq 2 w 2 I I "

P lql “ p 2q 2

p 2q l ~ plq 2 и (о) +

2 2

p 2 " P1

p lq l “ p 2q 2 4 p lp 2 (Plq l p 2q 2)

, (54d)

Fro m condition (

50

*

-1

-1

P 1 ~ p 2 » q l q 2 (55)

or

Pi = P i " 1 * Q* = Q i " 1 = 1 * 2 ) (56) Eqs. (55) and (

56

We n o w rewrite eqs. (

54

gestive. It is sufficient to consider (54a,d) only

M

(2) ;

=

q l q 2 pl p 2

1 p lql

1 P 2q 2

1 1

ql q 2

pl ql p2q 2 pl p 2

q l q 2 q l q 2

pi p : pl p2

1 1

plql p ?q 2 1

p lq l P 2q 2

M.. » ♦ —

N.^(o) 1 4 ?

j £

qi pi

pi p i

-1

p2

-1

1 1

plq l p 2q 2

(57a;

(57b)

The above for m is mos t useful if we w a n t to generalize it for an arbitrary number of steps. Having calculated the analogous f o r m u ­ lae for 5 В transformations it was quite evident wha t the c orres­

ponding result would be after iterating n-times the В trans f o r ­ mation. We realized that these В transformations hav e the

following compact form,for an even n u m b e r of steps

where the D * з / a =1,...4 / are 2 k X 2 k determinants and they are

8

completely characterized by their i-th r o w

4 % ) ¹ 1 Qi» Pi» PiQi» P p Pi^i» • •« ^{2 3 4} „ 2 k -1 1

•••> Pi 1

4 % ) - | l » PiQi» P?» P ^ i » Pi »«- «- П 2 k - l 1

4 % ) - |l» PiQi» P?, P iQ i* Pi»«««< _2k-2 2k

4 % ) - I

I 1 . pi> P i4 i> pi> P i V "

_ 2 k -l 1 ... pi 1

and i = 1 , ...2k;

for an odd number of steps.

K fek+i)

4 к Л ч )

4 к+*(ч) 4 k+\i)

1 4 ^ M

4 ?

4^\) I

N l?k+1l = . w2

here D g ’s are (2k+í)y;|j2k+l) determinants w r i t t e n in an analogous w a y as above

w here i = 1 , ...,2k+l.

We give an inductive p r o o f of these results in Appendix B.

Formulae (

58

as an output: just the components of the v e c t o r potential, and the physically most rele v a n t quantity, the length of the Higgs field \4>l , can be read off immediately. It should be noted here, that Neugebauer [

34

Needless to say, it w o u l d be very awkward to use this result in

our сазе.

We guarantee the re a l i t y of our solutions b y imposing the following conditions for and

* -1 * -1

q 2r-l q 2r * p 2r-l “ p 2r (60a)

for an even n u mber of steps, whe r e a s for an odd n umber of iterations

* -1 ql ~ q l * P1 = P1 q 2^ = 4 2r+l ,

p?*

^ r + l 1

(60b)

r = l,....,k. There exist other possibilities as well, however, as it is clear from Ref. [35] the physically interesting case for us is either (60a) or (60b) . We made the tacit assumption in (бОЬ)- that for the seed solution \^0 = О •

As 1<$>1 is the mo s t interesting quantity for us we find it useful to present here .

1Ф| = 2

D (2k+e)

D > k + 6 )

г

i 2 i l

<5i

)

(o) 1+fe

(2k+6) (q i) 4 2k+64'.i )

n2(o^ +

(6i)

4 §

* r % i )

D

felei О

D

W) I

Where (бОа)

G = 0 or 1, к is any integer and we used the reality or (60b) respectively.

To actually get multimonopole configurations one should start with a suitable seed solution for w h i c h one has to solve eq. (40) to obtain q. With these q ’s and the corresponding p's where the parameters ^ ’s and w ’s are arbitrary / up to the

constraints (бо) / one usually ends u p w i t h singularities in 1 ф | ,therefore, the energy for this configuration will not be in general finite. As it was found in [35^ the parameters

^ i ’ w i are u n iQu e ^y determined by imposing the condition of finite energy. In fact this constraint means that there be no singularities in eq. (6l) for \ Ф 1 •

7. GENERATING M U L T I M O N O P O L E S

We n o w show h o w can one make the first step in applying the method developped in this paper w h i c h amounts to generate the singly charged monopolé. We indicate here w h y the modified ВТ, В / = IH / , was singled out as the candidate for generating the nonlinear superposition of monopoles.

The moet naive expectation is that one could start w i t h the well known B P S one monopolé / IMP / as a seed solution and applying a ВТ one would end u p with a d o u b l y charged monopolé, etc. In the Sine-Gordon theory this is indeed the case, that is from a soliton a В Т generates a two soliton, etc. In practice it is not quite straightforward to proceed this way since the IMP is somewhat complicated in the R gauge; its Ernst potential, £

4nP

is given by [lO,29j & = F”*(.S ♦ ip) * where r

F = --- + r cosh z coth r - z sinh z , sinh r

P = z cosh z - г sinh z cot h z

C62)

It is not an easy task to solve eq. (

40

In our case the "vacuum" is not the trivial solution o f the Ernst equation, f = 1, 4' 0 w h i c h corresponds to M i n k o w s k i space, but rather a more "natural" v a c u u m would be the Higgs vacuum,

ф Х = 1. The simplest s o l u t i o n of the Ernstt equat i o n with this property is f

=

, Ч'о = О» ог апУ

sinh z

tained f r o m e by a complex E h l e r s transformation. E v e n for this complex v a c u u m state the solut i o n o f eq. (.40) is quite compli­

cated, so it was important to note that by the В transformation one obtains the IMP just fr o m f Q . The m

S°^

,

S°^

4^ =0 are given by

U^0) = u£0)

j

4

q = - tanh (^R (w)

2

ф = (c o th

(

) ---“ ) (64)

4 R (w) /

where we had to choose ^ =0 to avoid s in g u la r it ie s and we used (64)

p + q

(65)

= 1 1 + pq

One would now think the way is open to apply a second В transfor­

m a t i o n for the IMP to get a 2MP, etc. However, in contrast to the k n own cases it is impossible to generate the 2MP f r o m the IMP by a В Т / either by В or HBT /. The reason of this surprising fact wil l become clearer if we see how the 2MP can be generated. The

"vacuum" we start w i t h is again f Q = e z and the constants we had to choose to ensure the absence o f singularities in \ф1

a re : wi = w* = i

» ® = - i • 1Ф1 can now be c a lc u la ted a f t e r some algebra

where

+/>\ ^ CoS^o(^ _ (^Av'^')ooSoí ^ o< = í

and we introduced oblate spheroidal coordinates

\J(A-^ ) ( A ^ ) +

i

4

(<tl drastically simplifies on the z =

0

I 4 > U » 0 , ^ ) \ 1 +

2 * (^Co«,V>S4 -

°? *ьилС>

where

9

,г

l Ф С * ^v § “ ° 3 | а l tctvAfc ~ ^ ~

1 ы.г ¥ гг

This s o lu tio n describes a doubly charged monopolé s itu a te d a t the o r ig in .

Thus we see that the 2MP is generated by a double В transformation, as we claimed, but the second of these transfor­

mations is carried out on a state entirely different from the

IMP.

Finally, we show that applying (öl) to the simplest Higgs v acuum the asymptotic behaviour o f satisfies eq. (в) . First we observe that q of eq. (

63

^

2

while using an identity [зб] for the ratio o f two V a n dermonde- type determinants

N o w plugging these ratios as well as j into (öl) we obtain

n

This shows that every В t r a n s f ormation increases the topological charge by one. However, the r e s u lting configurations are in general singular, corresp o n d i n g to infinite energy.

Therefore, the only r e m a i n i n g task is to find those v a l u e s of w., ( b i for w h ich I ф \ given by (öl) is nonsingular. This was achieved in Ref. j?5] , where the structure of these new

solutions is thoroughly discussed.

8. CONCLUSIONS

We d e r i v e d the ge n e r a l form of the n-fold ВТ for the axially and m i r r o r symmetric Bogomolny equations. We have pointed out that applying these B T ’s on a simple Higgs v acuum one can generate monop o l e s of a r b i trary charge. We illustrated our method by showing how the one m o n o p o l é and a n e w doubly charged monopolé emerges from this process.

After our work w a s completed, we received the papers by Prasad, Sinha, W a n g [з7j in which they described another f r a m e ­ w o r k to generate multimonopoles, genera l i z i n g W a r d ’s construction.

F r o m the comparison of these two superficially ver y different me t h o d s we can immediately establish that our В Т connects the A £ and the A^ ans'átze w h i l e p r e s erving reality and regularity properties of the seed solu t i o n / at least for a suitable choice of the parameters /.

We give a detailed derivation of the composition theorem for two subsequent ВТ* $.

The Riccati equation ( 34 ) for the pseudopotential, q*

in the second В Т is

dq* = Г- K - - ~ q»(l ♦ P2q») + + P2) ♦

L Qi q i

Fro m the

dq* =

N o w substituting dq^ arid d q

2

Ъ ^{q* Э q* Э q* 9 q* , ,}

dqi + — -- dq2 + -- -- dpx + — -- dp2 .

[A.2 /

3 q x

q2 9

2

P2

ve

9 I* / . 3 q*

i

— -- q,(l + p^q,) + — -- q2 (l ♦ P2q2 ) = q»— (l + P2q'), (A.4a)

о

о

9 q*

3 q 1

3 q* q-i

(qx + Pi) + - г — (q2 + P

) = ~ ( q * +

2 )

о

* (a .4b)

a Ql - M V 1 5 4 * = * ^ l 4 ' ^ 4o)

3 q*

I

^ Ч1Г ^ Г

3 q * / 1 ] 9 q* / 1 \ 1 / 1

q2

1

, ÍA.4d)

3 q*

,2 \

2 s

" э Г P! (P! " l} + ~ Э Г ^ “ 1]

Pi P 2

( pi - 1j [ qi q 4 1+P2q,) ‘ fq ,+ P2)],

( A.4e)

3 q* Pi - 1 3 q ’ p2 2 - 1

P l Pi P2 P2

2

1 - Pi q l q ’ ( q*

1

—

i

Pi 1 + p-jqj \ p

2 )

'J

We see that we got an overdetermined system of differential equa­

tions for q*, and it is possible to reduce eqe» (ft#4a-dj just

to simple algebraic equations

- S g l = , ^^a .5^s)

ql^l “ q 2q 2 1 “ q lq lq ’q>

Pi q lq 2 ~ q 2q l P2 q lq l ” q 2q2

2

-2

q l q 2q * "* q lq 2q>

q 2q 2 ” qlq lq,q>

U . 5 b )

These two quadratic equations for q ’ / or q ’ / are easily solved and indeed they have one c o m m o n root, or alternatively one can reduce (A.5a-b) to a linear equation, the solution of which

is seen to satisfy both ( A . 5 a - b ) •

1 Q

1

2

( A . 6 )

I q ’ = q* is the "dual** of q* in the sense that one should simply replace q^, q 2 and q-^ in (a .6 ) by q^, q 2 and q^.

One can verify that ( A.6 ) r eally solves all of the eqs. in (A.

4