BUDAPEST PHYSICS INSTITUTE FOR RESEARCH CENTRAL

/SJ~- (h h

KFKI-1980-17

Hungarian ‘Academy of S cien ces CENTRAL

RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

P. FORGÁCS Z, HORVÁTH L . PALLA

GENERATING THE BPS ONE MONOPOLE

BY A BÄCKLUND TRANSFORMATION

GENERATING THE BPS ONE MONOPOLE BY A BACKLUND TRANSFORMATION

P. Forgács, Z. Horváth*, L. Palla*

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

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

HU ISSN 0368 5330 ISBN 963 371 644 6

ABSTRACT

It is shown that the Bogomolny equations for the simplest static, axially symmetric gauge fields are equivalent to the Ernst equation. The BPS one monopolé is obtained via Harrison's Bäcklund transformation.

АННОТАЦИЯ

Показывается, что в наиболее простом статическом аксиально симметриче­

ском случае уравнения Богомольного эквивалентны уравнению Эрнста. Монополь БПШ генерируется при помощи преобразования Бэклунда, данного Гаррисоном.

K I V O N A T

Megmutatjuk, hogy a Bogomolny egyenletek legegyszerűbb sztatikus, ten­

gelyszimmetrikus esetben az Ernst egyenlettel ekvivalensek. A BPS egy mono­

pólust Harrison Backlund transzformációja segitségével generáljuk.

We are looking for axially symmetric SU /2/ monopoles in the BPíD°\imit in a classical non-Abelian gauge theory with a Higgs field in the adjoint representation, such^hat we can treat it as the Rt component of the potential . Until now the only finite energy solution of this theory is the spherically symmet­

ric BPS one monopolé /1МР/. It is not known whether the theory has any other classical solution with finite energy, such as

multimonopoles. In the BPS limit the scalar field becomes massless and can mediate a long range force which can cancel the magnetic forces. It has been showrf2,3^ that the force between monopoles decreases faster than any inverse power of the separation. This encourages one to hope that static, noninteracting, finite energy monopoles may exist. On the other hand Montonen and Olive conjec­

tured that there may exist a dual theory which looks like the original one with monopoles аз basic constituents .C4) Furthermore

from an entirely different point of view Adler^ also investigated the axially symmetric case and conjectured the existence of solu­

tions with an extended zero set of the Higgs field within the IMP sector.

So far no systematic method has been given to generate solu­

tions for the axially symmetric case. Our aim here is to show that there are such methods: Bäcklund transformations and the method of inverse scattering To do this we first reduce our axi- ally symmetric equations to the Ernst equation , for which these (6)

transformations are worked out. Then as a first application of these techniques we generate the IMP from a suitably chosen solution.

Manton constructed an ansatz for the Euclidean static, axi­

ally, symmetric, selfdual gauge fields in order to find multi­

monopole solutions of the Bogomolny equations.^

2

His ansatz in polar coordinates

R j * _(о,,, <t>t) Ф • _{i 4t}

H i

where X«* fcosf , _{х г} * f a n d Ф«‘, IV« ere functions of f 1 only. The Bogomolny eqs. take the following form

* - ГЧ'Э*!« - К 1.) (2 a) + * - Г*( ?,*)* f _Wi *J4 ) (2 b) f y W i - Q x W t - Г ‘( Ф Л г c )

r ' { Q ! \ - М г Л х ) (2 d )

^хФг + И/,ф, = Г Ч + _{W t} ^ 4) _^

Eqs. /2а-е/ are invariant under Abelian gauge transformations

W,: * W; + A ,

/<* \ ГФЛ 'ФЛ cosA +

M Л

* _{1 *} г1«

Now our first observation is that eqs /2а-е/ are in fact

equivalent to the Ernst equatioii4 To show this let us consider the line element

<As4 R d § * + 2Bc\jcli + C d z 1 = elfolyt

Ы where R * V/$l » Ъ = -ф*',|/? . С = ф* using the notation

fn ' '

of ref> 'The "zweibeins" for this metric are , ef. “ (*„<*,)

I n t r o d u c in g th e on e form e с*>* a e \ cl<$*

t i o n one fo r m s t o 4 *. * -to1 4 = - \x/t- cly*

d oo* + coe v, A cow * 0 cl -Ь со1 Лсог e 0

,and the connec- , eqs./2a-c/ are

(4) (5)

3

which means the Gaussian curvature of the metric /3/ is just Re-1, / i -е- it describes a pseudosphere/. The remaining two equations /2d-e/ can be written as

dlööe + c o <,t A c 5 l:> e 0 where cD" * ol y * >

= ? ( Ф«,Ф,) , = C7Í 4 , ' iJt )

The geometrical meaning of these equations is not quite clear.

On can now proceed by solving eqs /2а-с/, choosing an appro­

priate parametrization of the pseudosphere, and the remaining two equations will determine the dependence of the parameter functions on the coordinates ( f , z ) . In what follows, we adopt a well-known form of the Poincaré metric on the pseu­

dosphere t г

a * 1 * s i i + á i , ч * Ч ( ? , г ) i 1

corresponding to Co' = T ( f _ifГ eft₎

* <f,.cU)

Now expressing V , W V in terms of ^{{ ,} , we get

♦ , ' Г Ч т ^•₎ + . ~ Г Ч . 1 ; 1 . '-f f ‘V,r ;

i t ' f f V . r ^• ; W x x - f ' 4 - í (6)

Substituting /6/ into /2d-e/ we obtain the celebrated form of the Ernst equation of General Relativity

■Re c Ae -Л РЧ )1 = 0 (7)

where £ * f + ; 9 /

One remark is in order in connection with /6/. The two condi­

tions VC/1 = _{~ 4} implied in /6/ cannot be Imposed simultaneously as a gauge fixing, however, for a solution of /2а-с/ this is possible.

4

This is so because adopting a certain parametrization of the pseudosphere is more than fixing a gauge; in fact - as was men»

tioned above - it simultaneously ensures the solution of some of the selfduality equations, Now, if for some $«,*]<• and

configuration these equations are satisfied, this fact guaran- teas that the configuration can be gauge transformed into a gauge where both relations are valid at onci7^.

It is straightforward to determine the functions -f and for the IMP: we have to transform the monopolé into the R

(-

gauge and integrate /6/ once; the result ii8)

F "F

F =

У _%'\v\\ T coth v- v=\ZP>?

(

)

P

The direct equivalence of eqs /2а-е/ and /7/ is important because it makes possible the use of solution generating

techniques that exist for the Ernst equation. Recently several authors proposed various group-theoretic or soliton-theoretic methods for generating new solutions of _{/ 7 / .} These techniques offer the possibility of iterations, thus are capable of ge­

nerating infinitely many solutions from an initial one. Next we want to indicate how the soliton-theoretic techniques

/Bäcklund transformations found by Harrison / H ^ ° ^ a n d Neugebauer /NB/^- ancj the "inverse scattering method" of Belinsky and Zakharov / B Z / A ^ m a y be applied to the monopolé problem. However, before entering into the details we must clarify the meaning of

Bäcklund transformation here.

Corrigan et al.^13^ have found nonlinear transformations-

5

that they also called Backlund transformation - which connect solutions of the selfduality equations in the R gauge. These

transformations were adopted to the static case in Ref.8. in an attempt to generate new solutions from IMP / the result was a singular ЗМР/. Having realized that the system /2а-е/ reduce to the Ernst equation /7/, one can identify the analogues of these transformations for the Ernst equation. In fact one finds that the transf ormationsi (5) and (y) in Ref.8. are the so called discrete Neugebauer-Kramer mapping _{/ г М} and Ehlers transfor­

mations respectively /Lohe applied the special product fiyßf.

Now, theee two transformations are elements of the infinite dimensional Geroch group^6^. Applying a finite number of these transformations usually changes the properties of the solution in an uncontrollable way. If we want to preserve certain pro­

perties of the seed solution, then we have to find suitable subgroups of the Geroch group. In practice, it means summing up in a suitable way an infinite number of similar infinite­

simal transformations. See Kinnersley et alP’7^. On the other hand - as it was shown by Cosgrov^9^- the /НВ/ and/NB/ trans­

formations are not contained in the Geroch group, and a finite number of these BTs preserve certain asymptotic properties of the solution^8^. Therefore, the transformations we are consi­

dering are different from those of Ref.8.

To apply /НВ/ or /NB/ transformations to a given solution requires to solve Riccati-type equations /see below/. These equations depend essentially on the form of the known solu­

tion, and given the fact that the IMP is somewhftt complicated it does not appear straightforward to apply these transfor­

mations directly.

6

Knowing that the Bts can be iterated algebraically it eeene to be sufficient to generate the IMP solution from a suitable ground state as a first step. By carrying this out we achie­

ve two things. On the one hand we see that the IMP /8/ can be interpreted as a "soliton" of the Ernst equation as well. On the other side, it may make it possible to find a finite energy nonlinear superposition of monopoles. We applied both the /НВ/ and the /BZ/ transformations to generate the IMP, here we give in some details the application of /НВ/ because we think it is more transparent and selfcontained.

To apply /НВ/ transformation one defines from the known solu­

tion e* f 1 i ijl the quatities

and solves the total Riccati equation for the pseudopotential

*V ( 5«, )

* f ' ( i ) ( N r N , ' f ‘ ) ] o l h (10)

where

Si

follows

и JL

= _

JJX-

f(H7f)г 1*УЧ'

q and yts,) as

i

S

f

7

%

9

It 19 and advantage of the /НВ/ transformation that it acts on the M^ — s as these are in direct connection - via /6/ - with the fields

Ф;, '»K and W * of our interest. In particular the gauge in­

variant length of the Higgs field / that determines the magnetic charge/ ф г * f is 9iven as

Ф г - * ъ*

i 1 - M H , Hi )

(

)

Now, the particular solution of /7/ from which the IMP can be obtained by а /НВ/ transformation is C o - У ^ f 2 • The naive expectation would be to choose a solution for which

ф г --1 , but it does not w o r k ^ 4 In fact applying the /НВ/ once to a family of these states we obtained singularities in the gauge invariant quantity ф г . It is important to realize that the Ehlers transformations^5^do not change ф г , however, the final reeult of the ВТ depends on which Ehlers equivalent state we start with. Note that for €„ ф г= _coilt^-z. ;however, it can be obtained by a Neugebauer-Kramer mapping /I/ from a complex solution f' = siи ~1 xy </« i t o t h i having ф г*1.

If we allow complex solutions we can say that we generated the IMP by the product /НВ/ I from this "natural" ground state. In the case of 6 0 the solution of /10/ is given by

f (*>5/1(2 + $ + £) i i s m1D * ( s) cojK( _{t f i )}

fcosk(2 i r /0 2(s)coik(^ ifi)

8

where _fi is a constant of integration and

while D * U ) « _{i ( l} -SZ. Combining /13/ and /11/ choosing

which is really the length of the Higgs field in the case of IMP.

This is the main result of our paper. As we mentioned earlier the fact that we generated the IMP from a suitable state may open the way to obtain a decisive answer to the question of the existence of multimonopole solutions by the repeated applications of these

transformations.

In conclusion we have shown the equivalence of Menton's

equations /2а-е/ to the Ernst equation /7/. This enabled us to generate the IMP from a complex state with using

Harrison's Bäcklund transformation and the Neugebauer-Kramer mapping. Details of applying the other techniques to the mono­

polé problem will be published elsewhere we obtain

/

Acknowledgemen ts

We would like to thank Prof. Z. Perjés for his considerable help in this work. One of us /P.F./ would like to thank Dr. N.

Manton for his contribution at an early stage of this work.

9

References and Footnotes 0. Bogomolny - Prasad - Sommerfield

1. N.S. Manton, Nucl. Phys. B135. 319 /1978/.

2. N.S. Menton, Nucl. Phys. B126, 525 /1977/.

3. W. Nahm, CERN Report No. TH 2642 /unpublished/.

4. C. Montonen and D. Olive, Phys. Letters 7 2 B ,117 /1977/.

5. S.L. Adler, Phys.Rev. D 2 0 . 1386 /1979/.

6. F.3. Ernst, Phys. Rev. 167, 1175 /1968/.

7. C.N.Yang, Phye. Rev.Lett. 38, 1377 /1977/. Note that _ a combining /1/ and /6/ one obtains expressions for r j ^

that are identical to the formulae of vectorpotentials in the R gauge with the special choice for the ф and f functions

= f = / 2 у у 1 Г , 2 ) ; «j = * 4 •

Indeed substituting this aneatz into the selfduality equa­

tions in the R gauge we obtain the Ernst equation /7/.

8. M .A . Lohe, Nucl. Phys. В 142, 236 /1978/. We take the vacuum expectation value of the Higgs field v«l.

9. C.M. Coegrove, Lecture given at the Second Marcel Grossmann Meeting on Recent developments of general relativity

/Trieste, Italy, Duly 1979/: Montana State University Report /1980, unpublished/.

10. B.K. Harrison, Phys.Rev.Lett. 4l, 1197 /1978/.

11. G. Neugebauer, D.Phys. A.12, L67 /1979/.

12. V. A. Belinski and V.E, Zakharov, Z h , Eksp. Teor. Fiz. 7 5 , 1953/1978/; ibid. 77, 3 /1979/.

Г

10 -

13. E. Corrigan, D .В . Fairlie, P. Goddard and R.G. Yates, Comm.

Math. Phys. 58. 223 /1978/.

14. G. Necgebauer and D. Kramer, Ann. Phya. Lpz. 24, 62 /1969/.

One defines a function w by

then (I) acte as : (X) { _{f }}w / } ж { X X _j , - . w j _t 15. 0. Ehlers in Las Theories Relativistáé de la Gravitation

/CNRS, Paris, 1959/. It acts on € as

. * </ e - . .

ei ^ f t 6

16. R. Geroch, D. Math. Phys. 12, 918 /1971/; ibid. 13, 394 /1972/

17. W. Kinnersley, 0. Math. Phys. 18, 1529 /1977/; W. Kinnersley and D.M. Chit re, ibid. 18, 1538 /1977/; 19, 1926 /1978/;

19, 2037 /1978/; C. Hoenselaers, W. Kinnersley and B.C.

Xanthopoulos, ibid. 20, 2530 /1979/.

18. In particular, an even number of /НВ/ transformations pre­

serves asymptotic flatness /Ref. 9./, however, since our asymptotic conditions are different it is far from being obvious what the implications are in our case

19. Even in general relativity one /НВ/ changes the asymptotic flatness, sec. Ref. 9.

1

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Szakmai lektor: Perjés Zoltán Nyelvi lektor: Margaritisz Tamás Példányszám: 360 Törzsszám: 80-202 Készült a KFKI sokszorosító üzemében Budapest, 1980. április hó