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Exact solutions of supergauge invariant Yang-Mills equations
Volume 84B, number 2
PHYSICS LETTERS
18 June 1979
EXACT SOLUTIONS OF SUPERGAUGE INVARIANT YANG-MILLS EQUATIONS T. DERELI Department o f Physws, Mtddle East Technical Umversity, Ankara, Turkey
and R. GUVEN Department o f Mathemattcs, Bo~aztqt Umverstty, Bebek, Istanbul, Turkey
Recewed 29 March 1979
We point our a new class of solutions of the supersymmetrlc Yang-Mdls equations. This class provides solutions which cannot be generated from the solutions of the ordinary Yang-Mdls equations by flmte supersymmetry transformations and contains the supersymmetric generahzatlon of the non-abehan plane waves.
Although s u p e r s y m m e m c gauge theories have been considerably investigated during the last few years, our knowledge o f their actual solutions still remains poor. One o f the simplest bxamples of a supersymmetrlc gauge theory is furnished by the lagrangian describing the interaction o f a Yang-Mills field with a Majorana spmor belonging to the adjolnt representation o f the internal symmetry group [ 1]. In this case it as clear that any solution o f the source-free Y a n g Mills equations may also be considered as a trivial solution o f the super-symmetric version with the spmor fields set to zero. Thus, it is possible to generate exact supersymmetric solutions simply by applying finite supersymmetry transformations on a gwen solution of the source-free Yang-Mllls equations. This class of solutions has been previously studied [2]. The main purpose of the present note is to point out a new famaly o f exact solutions o f the s u p e r s y m m e m c Y a n g Malls equations which is not restricted to lie m the above category. This set o f solutions includes as a special case the supersymmetrlc generahzatlon of the recently found non-abehan plane waves [3] and provides solutions that cannot be reduced to Coleman's solution by finite supersymmetry transformations. Hence the present problem resembles the case o f plane waves of supergravity which are also non-generated in the above sense [4].
The dynamics of the supersymmetric Yang-Mllls theory is described through the lagrangian density * i (1)
£ = _ ~1 F a u v F a u v + ~I l ~ a T u D u h a
where a
a
b c
(2)
Fauv = ~ u A a - ~ v A u + C b c A u A v ,
is the field strength tensor constructed from the gauge potentials A au. Both A~ and the Majorana spinor fields h a transform under the adjolnt representation of a semi-simple Lie algebra whose structure constants are Cabc. The gauge couphng constant is absorbed into the field variables by a redefimtion. The 7-matrices are in the Majorana representation, all 3,u's are real and 72 = - 1 . The gauge covariant derivative is defined in the usual way: ~a b e~a bu h c " D u h a = 0 u h a + L"
(3)
The variational principle yields the field equations a u F a u v + ~a
aUD u h a = 0 .
nbr'cuv 1 1Cabe~b3tvhc bc.~ ~t-=~
(4a) (4b)
+1 For the sake of completeness we should have added a term {DaD a to the lagranglan corresponding to auxiliary scalar fields D a However, the variational principle yields D a = 0 and for our present purposes the exclusion o f D a from the argument is of no consequence. 201
Volume 84B, number 2
PHYSICS LETTERS
It ls easy to check that the action I = f £ d4x remains invaraant under the infinitesimal supersymmetry transformations (see footnote 1): 8Xa = F a u v ' o U V e ,
8Aau = ie-yuX a ,
(5)
18 June 1979
and to three algebraic conditions: Cabc[Xl)t3 +
=
o,
Cabc [)tl~ b c1 + )tb )t~ -- )tb2 ~kc2 -- xb)t~] = 0 ,
where e is a constant Majorana spinor and Our = 1 [~',, % ] . There is a Noether current associated with this global invariance, and it as gwen by 1
(6)
S u = 7 iFaKX" °KX3' u h a .
Later we shall also be using the conserved (symmetrized) e n e r g y - m o m e n t u m tensor which may be written as ruv = Tgu(F) + f u r ( x ) ,
(7)
where
On the other hand, Introducing the new Majorana vanables defined by
.a=x
( x ) -- -
(8b)
o = 2-1/2(z + t),
(9)
so that the space-time metric takes the form ds 2 = 2dudo + dx 2 + dy 2 ,
(10)
and we assume that the potentials A au satisfy A~a dxU = A a ( u , x , y )
(1 1)
du ,
a = g ax Of~
Ij xa + u ~ : x / ~ V u o a
We start by adopting the light-cone coordinates u = 2-1/2(z - t),
- - g ya ~
(16)
and
1 l ~ a ( , Y u D v X a + ,YvDuXa)
1
=
(8a)
'
+ g iguv ~a ,yKD~ka "
(15)
Dirac's equations (4b) may be written as O xa
F a t':X "treaKh Tuu ( F ) = FauX FavX - 4I 1,75 U V -,#a - UP r~
(14)
where A a are certain functions that remain to be determined. This form of the potenUals was first employed by Coleman in the study of non-abelian plane waves [3] and also leads to their generalizataon to the curved space-time [5]. In the present case, after writing the Majorana splnors exphcltly as
'
, ya _ Vxa = V , ~ D u K a
,
(17)
where the subscript x ( y ) denotes differentiation with respect to x 0 ' ) a n d D u o a = Ouoa + C a b c A b a c. The couplings between the various field variables are now transparent and this gives rise to the following simple scheme" With the choice o a = ~a = 0 eq. (16) Is trwially satisfied and the algebraic restrictions (14) are all fullfilled. Then, if the new complex, antlcommutmg variables w a = ~(ya + ll)a) are defined, eqs. (17) are just ocoa/Oz = 0 ,
(18)
where z = x + ly. Furthermore, eq. (13) becomes ( ~ 2 / 3 ~ - b z ) A a = 2-3/2 iCabc ff)c.
(19)
If not written over a spinor, a bar over a quantity denotes ordinary complex conjugation. We are thus left only with the task of integrating the funcUons coa = coa (u, ~) which are analytic with respect to ~-, as the general solution of eq. (19) is formally A a _ ~1 ( a ~ + ~ a ) z
\xl/
+2-3/21 f
we find that the Yang-Mllls equatmns (4a) reduce to (02/0x 2 + 0 2 / O y 2 ) A a = 2-3/2 iCabc X [(Xbl 202
b
c
-- ~K2)(X 1 -- ~k~) +
(?tb+xb)(xS+X~)]
z
dz' f dz dbc'~ (u,~')~o (u,z ), -t
b
--c
(20)
where ~2a (u, z) are certain homogeneous solutions of eq. (19) which are analytic with respect to the variable (13)
Z,
Volume 84B, number 2
PHYSICS LETTERS
We have thus shown that, for an N-parameter gauge group, any set o f N anticommuting, analytic functions w a together with the functions A a, given by eq. (20), provide an exact solution of the field equations. We next factor out those cases which can be generated from the ordinary Yang-Mills equations by finite supersymmetry transformations. If AT~(1) denotes a homogeneous solution corresponding to the chmce Xa = 0, then a fimte supersymmetry transformation yields [2] At,a = A~(1) +1~ le~uKx ~-3`5 3`v~FaKX(1) ,
(21a)
V = Fauv(1)ouv~,
(21b)
where Fat*v (1) is constructed from AT~(1) and ~ is a constant Majorana spanor. Let us assume for the moment that the new solutions can be generated from an arbitrary ATL(1). It may readily be verified that our solutions obey "tvX a = 0 where % = 2 -1/2 (3`0 + 3'3) and therefore, eq. (21b) reqmres FaKa (1) % o KX~ = 0.
(22)
This in turn implies that Fat* v (1) = F a y (I) = 0. These conditions together with the assumptions (11) and (12) restrict the allowable AT~(1)'s (modulo a Yang-Mdls gauge transformation) to the form ATz(1) dxU = 2 -1 (xa + ~-a) du, where X a (u, z) are analytm with respect to z. Hence it is sufficient to check whether our supersymmetric solutions can be generated from the trivial ones coa = 0, A a = 2 -1 (X a + ~a). It may be noted that Coleman's solution is a member of this subclass with X a = a a ( u ) z + 3a(u), where c~a(u) and 3a(u) are bounded, complex functions. Returning back to eq. (21), wrltlng ~ as
co a = ~
18 June 1979 ,
(24)
where the subscript denotes differentiation with rea whose spect to ~. We also have the transformed At* components may be read off from the Lie algebra valued one-form At*a dxta = [½ (xa + ~a)
1
~ a(pqx a + pq~)]
+ 2 -5/21 [q, q-] [Xzadz
X-~ dz-].
¢2 e3 e4
(23) '
and Introducing, for convenience, the complex anticommuting parameters p = _2-1/2@1 _ e 2 + le 3 + ie4), q = _ 2 - 1 / 2 ( e l + e 2 + i e 3 - ie4) , we find that the solutions w a = 0, A a = 2-1(X a + ~-a) can produce a non-zero coa which is of the form
(25)
The terms in the above expression which involve the commutator [q, q] can be removed by the action of the Yang-Mdls transformation matrix S = exp{2 -5/2 l[q, ~] (~-a
_
xa) T a } ,
(26)
where T a are the group generators. This transformation leaves eq. (24) unaltered and the resultmg A a may be cast into the form given by eq. (20) with g2a = X a - i p q x a + 2 -3/2 l[q, q] xu-a.
(27)
It is clear from the above argument that eq. (24) together with eq. (27) completely specify the set of generated solutions. Hence, any coa which cannot be brought into the form (24) yields a non-generated family of solutions. Moreover, even if one starts with an coa given by eq. (24) one does not necessarily acqmre from eq. (20) a generated solution unless ~ a is selected in the prescribed form. For example, if we pick N complex functions 3`a (u) and let
.,a = p3`a (u),
(28)
and also adopt the following solution of eq. (19): A a = R e [ a a ( u ) z + 3a(u)] + 2-5/2 lZ~ [p, p] Cabc 3`b (u) q-c (u),
=
du
(29)
we get one of the many supersymmetric generahzations of non-abehan plane waves which cannot be reduced to Coleman's solution by a supergauge transformation. Therefore, either by choosing co a or ~2a or both in a more general form one obtains solutions of the field equations (4) which cannot be generated from the source-free Yang-Mllls equations. We finally list all the non-vanishing components of the conserved densities. For any solution of eqs. (18) and (19) the only surviving component o f the splnor current (6) is
203
Volume 84B, number 2
PHYSICS LETTERS
18 June 1979
T . ~ ( F ) = 4Aa _ ~ _A 7a ,
couragement. One of us (T.D.) thanks Dr. P.C. Aichelburg for the warm hospitality extended to him at the University of Vienna where part of this work was done. His stay there was made possible by a fellowship through the Einstein Memorial Foundation. The research reported in this paper has in part been supported by the Turkish Scientific and Technical Research Council.
Tuu (X) = 2X,'~(ooaDu ~a + goaDu coa ),
References
1
1
(30)
and using eqs. (8a) and (8b) we find
Tzu (X) = x/~ 6oa D a ,
Tz-u (X) = XQ- D a ,
(31)
as the non-zero components of the e n e r g y - m o m e n tum tensor. We are grateful to Prof. E. Inonu for constant en-
204
[1] A. Salam andJ. Strathdee, Phys. Lett. 51B (1974) 353; S. Ferrara and B. Zummo, Nucl. Phys. B79 (1974) 413. [2] N.S. Baakhnl, Nucl. Phys. B129 (1977) 354. [3] S. Coleman, Phys. Lett 70B (1977) 59 [4] P.C Aichelburg and T. Dereli, Phys. Rev. D18 (1978) 1754. [5] R. G/aven, Phys. Rev. D19 (1979) 471.
PHYSICS LETTERS
18 June 1979
EXACT SOLUTIONS OF SUPERGAUGE INVARIANT YANG-MILLS EQUATIONS T. DERELI Department o f Physws, Mtddle East Technical Umversity, Ankara, Turkey
and R. GUVEN Department o f Mathemattcs, Bo~aztqt Umverstty, Bebek, Istanbul, Turkey
Recewed 29 March 1979
We point our a new class of solutions of the supersymmetrlc Yang-Mdls equations. This class provides solutions which cannot be generated from the solutions of the ordinary Yang-Mdls equations by flmte supersymmetry transformations and contains the supersymmetric generahzatlon of the non-abehan plane waves.
Although s u p e r s y m m e m c gauge theories have been considerably investigated during the last few years, our knowledge o f their actual solutions still remains poor. One o f the simplest bxamples of a supersymmetrlc gauge theory is furnished by the lagrangian describing the interaction o f a Yang-Mills field with a Majorana spmor belonging to the adjolnt representation o f the internal symmetry group [ 1]. In this case it as clear that any solution o f the source-free Y a n g Mills equations may also be considered as a trivial solution o f the super-symmetric version with the spmor fields set to zero. Thus, it is possible to generate exact supersymmetric solutions simply by applying finite supersymmetry transformations on a gwen solution of the source-free Yang-Mllls equations. This class of solutions has been previously studied [2]. The main purpose of the present note is to point out a new famaly o f exact solutions o f the s u p e r s y m m e m c Y a n g Malls equations which is not restricted to lie m the above category. This set o f solutions includes as a special case the supersymmetrlc generahzatlon of the recently found non-abehan plane waves [3] and provides solutions that cannot be reduced to Coleman's solution by finite supersymmetry transformations. Hence the present problem resembles the case o f plane waves of supergravity which are also non-generated in the above sense [4].
The dynamics of the supersymmetric Yang-Mllls theory is described through the lagrangian density * i (1)
£ = _ ~1 F a u v F a u v + ~I l ~ a T u D u h a
where a
a
b c
(2)
Fauv = ~ u A a - ~ v A u + C b c A u A v ,
is the field strength tensor constructed from the gauge potentials A au. Both A~ and the Majorana spinor fields h a transform under the adjolnt representation of a semi-simple Lie algebra whose structure constants are Cabc. The gauge couphng constant is absorbed into the field variables by a redefimtion. The 7-matrices are in the Majorana representation, all 3,u's are real and 72 = - 1 . The gauge covariant derivative is defined in the usual way: ~a b e~a bu h c " D u h a = 0 u h a + L"
(3)
The variational principle yields the field equations a u F a u v + ~a
aUD u h a = 0 .
nbr'cuv 1 1Cabe~b3tvhc bc.~ ~t-=~
(4a) (4b)
+1 For the sake of completeness we should have added a term {DaD a to the lagranglan corresponding to auxiliary scalar fields D a However, the variational principle yields D a = 0 and for our present purposes the exclusion o f D a from the argument is of no consequence. 201
Volume 84B, number 2
PHYSICS LETTERS
It ls easy to check that the action I = f £ d4x remains invaraant under the infinitesimal supersymmetry transformations (see footnote 1): 8Xa = F a u v ' o U V e ,
8Aau = ie-yuX a ,
(5)
18 June 1979
and to three algebraic conditions: Cabc[Xl)t3 +
=
o,
Cabc [)tl~ b c1 + )tb )t~ -- )tb2 ~kc2 -- xb)t~] = 0 ,
where e is a constant Majorana spinor and Our = 1 [~',, % ] . There is a Noether current associated with this global invariance, and it as gwen by 1
(6)
S u = 7 iFaKX" °KX3' u h a .
Later we shall also be using the conserved (symmetrized) e n e r g y - m o m e n t u m tensor which may be written as ruv = Tgu(F) + f u r ( x ) ,
(7)
where
On the other hand, Introducing the new Majorana vanables defined by
.a=x
( x ) -- -
(8b)
o = 2-1/2(z + t),
(9)
so that the space-time metric takes the form ds 2 = 2dudo + dx 2 + dy 2 ,
(10)
and we assume that the potentials A au satisfy A~a dxU = A a ( u , x , y )
(1 1)
du ,
a = g ax Of~
Ij xa + u ~ : x / ~ V u o a
We start by adopting the light-cone coordinates u = 2-1/2(z - t),
- - g ya ~
(16)
and
1 l ~ a ( , Y u D v X a + ,YvDuXa)
1
=
(8a)
'
+ g iguv ~a ,yKD~ka "
(15)
Dirac's equations (4b) may be written as O xa
F a t':X "treaKh Tuu ( F ) = FauX FavX - 4I 1,75 U V -,#a - UP r~
(14)
where A a are certain functions that remain to be determined. This form of the potenUals was first employed by Coleman in the study of non-abelian plane waves [3] and also leads to their generalizataon to the curved space-time [5]. In the present case, after writing the Majorana splnors exphcltly as
'
, ya _ Vxa = V , ~ D u K a
,
(17)
where the subscript x ( y ) denotes differentiation with respect to x 0 ' ) a n d D u o a = Ouoa + C a b c A b a c. The couplings between the various field variables are now transparent and this gives rise to the following simple scheme" With the choice o a = ~a = 0 eq. (16) Is trwially satisfied and the algebraic restrictions (14) are all fullfilled. Then, if the new complex, antlcommutmg variables w a = ~(ya + ll)a) are defined, eqs. (17) are just ocoa/Oz = 0 ,
(18)
where z = x + ly. Furthermore, eq. (13) becomes ( ~ 2 / 3 ~ - b z ) A a = 2-3/2 iCabc ff)c.
(19)
If not written over a spinor, a bar over a quantity denotes ordinary complex conjugation. We are thus left only with the task of integrating the funcUons coa = coa (u, ~) which are analytic with respect to ~-, as the general solution of eq. (19) is formally A a _ ~1 ( a ~ + ~ a ) z
\xl/
+2-3/21 f
we find that the Yang-Mllls equatmns (4a) reduce to (02/0x 2 + 0 2 / O y 2 ) A a = 2-3/2 iCabc X [(Xbl 202
b
c
-- ~K2)(X 1 -- ~k~) +
(?tb+xb)(xS+X~)]
z
dz' f dz dbc'~ (u,~')~o (u,z ), -t
b
--c
(20)
where ~2a (u, z) are certain homogeneous solutions of eq. (19) which are analytic with respect to the variable (13)
Z,
Volume 84B, number 2
PHYSICS LETTERS
We have thus shown that, for an N-parameter gauge group, any set o f N anticommuting, analytic functions w a together with the functions A a, given by eq. (20), provide an exact solution of the field equations. We next factor out those cases which can be generated from the ordinary Yang-Mills equations by finite supersymmetry transformations. If AT~(1) denotes a homogeneous solution corresponding to the chmce Xa = 0, then a fimte supersymmetry transformation yields [2] At,a = A~(1) +1~ le~uKx ~-3`5 3`v~FaKX(1) ,
(21a)
V = Fauv(1)ouv~,
(21b)
where Fat*v (1) is constructed from AT~(1) and ~ is a constant Majorana spanor. Let us assume for the moment that the new solutions can be generated from an arbitrary ATL(1). It may readily be verified that our solutions obey "tvX a = 0 where % = 2 -1/2 (3`0 + 3'3) and therefore, eq. (21b) reqmres FaKa (1) % o KX~ = 0.
(22)
This in turn implies that Fat* v (1) = F a y (I) = 0. These conditions together with the assumptions (11) and (12) restrict the allowable AT~(1)'s (modulo a Yang-Mdls gauge transformation) to the form ATz(1) dxU = 2 -1 (xa + ~-a) du, where X a (u, z) are analytm with respect to z. Hence it is sufficient to check whether our supersymmetric solutions can be generated from the trivial ones coa = 0, A a = 2 -1 (X a + ~a). It may be noted that Coleman's solution is a member of this subclass with X a = a a ( u ) z + 3a(u), where c~a(u) and 3a(u) are bounded, complex functions. Returning back to eq. (21), wrltlng ~ as
co a = ~
18 June 1979 ,
(24)
where the subscript denotes differentiation with rea whose spect to ~. We also have the transformed At* components may be read off from the Lie algebra valued one-form At*a dxta = [½ (xa + ~a)
1
~ a(pqx a + pq~)]
+ 2 -5/21 [q, q-] [Xzadz
X-~ dz-].
¢2 e3 e4
(23) '
and Introducing, for convenience, the complex anticommuting parameters p = _2-1/2@1 _ e 2 + le 3 + ie4), q = _ 2 - 1 / 2 ( e l + e 2 + i e 3 - ie4) , we find that the solutions w a = 0, A a = 2-1(X a + ~-a) can produce a non-zero coa which is of the form
(25)
The terms in the above expression which involve the commutator [q, q] can be removed by the action of the Yang-Mdls transformation matrix S = exp{2 -5/2 l[q, ~] (~-a
_
xa) T a } ,
(26)
where T a are the group generators. This transformation leaves eq. (24) unaltered and the resultmg A a may be cast into the form given by eq. (20) with g2a = X a - i p q x a + 2 -3/2 l[q, q] xu-a.
(27)
It is clear from the above argument that eq. (24) together with eq. (27) completely specify the set of generated solutions. Hence, any coa which cannot be brought into the form (24) yields a non-generated family of solutions. Moreover, even if one starts with an coa given by eq. (24) one does not necessarily acqmre from eq. (20) a generated solution unless ~ a is selected in the prescribed form. For example, if we pick N complex functions 3`a (u) and let
.,a = p3`a (u),
(28)
and also adopt the following solution of eq. (19): A a = R e [ a a ( u ) z + 3a(u)] + 2-5/2 lZ~ [p, p] Cabc 3`b (u) q-c (u),
=
du
(29)
we get one of the many supersymmetric generahzations of non-abehan plane waves which cannot be reduced to Coleman's solution by a supergauge transformation. Therefore, either by choosing co a or ~2a or both in a more general form one obtains solutions of the field equations (4) which cannot be generated from the source-free Yang-Mllls equations. We finally list all the non-vanishing components of the conserved densities. For any solution of eqs. (18) and (19) the only surviving component o f the splnor current (6) is
203
Volume 84B, number 2
PHYSICS LETTERS
18 June 1979
T . ~ ( F ) = 4Aa _ ~ _A 7a ,
couragement. One of us (T.D.) thanks Dr. P.C. Aichelburg for the warm hospitality extended to him at the University of Vienna where part of this work was done. His stay there was made possible by a fellowship through the Einstein Memorial Foundation. The research reported in this paper has in part been supported by the Turkish Scientific and Technical Research Council.
Tuu (X) = 2X,'~(ooaDu ~a + goaDu coa ),
References
1
1
(30)
and using eqs. (8a) and (8b) we find
Tzu (X) = x/~ 6oa D a ,
Tz-u (X) = XQ- D a ,
(31)
as the non-zero components of the e n e r g y - m o m e n tum tensor. We are grateful to Prof. E. Inonu for constant en-
204
[1] A. Salam andJ. Strathdee, Phys. Lett. 51B (1974) 353; S. Ferrara and B. Zummo, Nucl. Phys. B79 (1974) 413. [2] N.S. Baakhnl, Nucl. Phys. B129 (1977) 354. [3] S. Coleman, Phys. Lett 70B (1977) 59 [4] P.C Aichelburg and T. Dereli, Phys. Rev. D18 (1978) 1754. [5] R. G/aven, Phys. Rev. D19 (1979) 471.