- Email: [email protected]
A multimonopole solution in string theory
Physics Letters B 294 (1992) 325-330 North-Holland
PHYSICS LETTERS B
A multimonopole solution in string theory R a m z l R. K h u n 1 Center for Theorettcal Phystcs, Texas A&M Umverstty, College Statton, TX 77843, USA Received 11 May 1992
A multlmonopole solution in Yang-Mllls field theory is obtained by a modxfication of the 't Hooft ansatz for a four-dimensional instanton Ahhough this solution has divergent action near each source, it can be used to construct a finite action multlmonopole solution of heterotlc string theory, in which the divergences from the Yang-Mills sector are precisely cancelled by those from the gravity sector We argue that this solution is exact to all orders in a '
1. Introduction In recent work [ 1-7 ], several classical sohtonlc solutions of string theory with instanton structure have been presented Exact solutions in bosonlc [ 1,2 ] and heterotic [6,7 ] string theory have been obtained by observing that a generalized curvature [8,9 ] combining the metric and antisymmetrlc tensor possesses lnstanton structure [ 1 ] In this paper we present what we argue is an exact multlmonopole solution ofheterotlc string theory We first obtain an analogous multlmonopole solution in Yang-Mllls field theory, whose action diverges near each source We show that the string theory solution, however, is finite as a result of the cancellation o f divergent terms from the gauge and gravitational sectors In section 2 we derive a multlmonopole solution of four-dimensional Y M + s c a l a r field theory via a modification o f the 't Hooft ansatz [ 10-14 ] for the Yang-Mills lnstanton This solution is not in the Prasad-Sommerfield [15] limit, but has the topology of Q = l monopole sources, saturates the Bogomoln'yl bound [16] and exhibits the far field behavlour o f multimonopole sources The action for this solution, however, diverges near each source We use this solution in section 3 to construct a heterotic multlmonopole solution, and argue that this ~r Work supported in part by NSF grant PHY-9106593 Supported by a World Laboratory Fellowship Elsevier Scmnce Publishers B V
solution is exact to all orders in a ' This string solution has finite action, as the divergences coming from the Yang-Mills sector are precisely cancelled by those from the gravitational sector The resultant action reduces to the tree-level form and is easily calculated A more detailed version of this letter is presented in ref [ 17 ]
2. Multimonopole solution in field theory Consider the four-dimensional euclidean action
S= - ~g 2
d4x Tr G..G ~,
/~,v=1,2,3,4 (21)
For gauge group SU (2), the fields may be written as
A~= (g/2l)aaA~ and G~.= (g/21)aaG~. (where a a, a = 1, 2, 3, are the 2 × 2 Pauh matrices) The equation of motion derived from this action is solved by the 't Hooft ansatz [ 10-14]
Au=lZ.u. 0. l n f ,
(2 2)
where Z ; . =O '~" la' for z= 1, 2, 3, where 0 ' ~ ' = - O " u = e '~", =-J%
/t, v = 1, 2, 3, v=4,
(2 3)
and where f - i r--lf= 0 The ansatz for the anti-self-dual solution is similar, with the J-term in (2 3) changing sign 325
Volume 294, number 3,4
PHYSICS LETTERS B
To obtain a multl-lnstanton solution, one solves for f i n the four-dimensional space to obtain f=l+
~ p2 ,=1 I x - a , I 2 '
(24)
w h e r e p ,2 is the m s t a n t o n scale size and a, the location in four-space o f the tth lnstanton We obtain a m u l t l m o n o p o l e solution by m o d i f y i n g this procedure as follows We single out a direction in the transverse four-space (say x4) a n d assume all fields are i n d e p e n d e n t o f this coordinate Then the solutton f o r f c a n be written as f=l+
,~1 = I x m, -a,l'
(2 5)
where m, is the charge a n d a, the location in the threespace (123) o f the tth m o n o p o l e If we m a k e the identification t2~-A4 (we loosely refer to this field as a Hlggs field in this paper, even though there IS no a p p a r e n t s y m m e t r y breaking m e c h a n i s m ) , then the lagranglan density for the above ansatz can be rewntten as a a a a G a~.G a~ __ - GvG, s + 2Gk4 Gk4
= GgGg + 2 DktJ9a Dkqb a ,
fields are singular near the sources and vanish as r - - . ~ This solution can be thought o f as a m u l t l h n e source m s t a n t o n solution, each m o n o p o l e being interpreted as an " l n s t a n t o n string" [ 18 ] The topological charge o f each source is easily c o m p u t e d ( (J~ a~__ ~ al l ~ l ) to be Q = f d3x ko __ 1 f d 3 x f.uk~ab c O (2~a Odtj~b Ok t~ c
8n
The magnetic charge o f each source is then given by m, = Q/g= 1/g It is also straightforward to show that the Bogomoln'yl [ 16 ] b o u n d a
G,j --e.,jk D k fl) a
It"7.a t'7.1tva
!1-) ¢f)al'~#d)a
(2 7)
and show that the above m u l t l m o n o p o l e ansatz is a static solution with A~ = 0 and all time derivatives vanish The equations o f motion in this limit are given by
D, Gm~=g~ab~(D'S~b)~ c, D , D ' ~ = 0
(2 8)
It ~s then straightforward to verify that the above equations are solved by the m o d i f i e d 't Hooft ansatz I~a=-T- 1 6 a t O, o,), a~=~.akl oj09,
g
(29)
where o g = l n f This solution represents a multim o n o p o l e configuration with sources at a , = 1, 2, , N A simple observation o f far field and near field b e h a v l o u r shows that this solution does not arise in the P r a s a d - S o m m e r f i e l d [ 15 ] limit In particular, the 326
(2
11)
is saturated by this solution Finally, it is easy to show that the magnetic field B,=½%kF jk (where F j , . aGaa. - ( 1/g) e abcqbaD~ t~ b D~ ~ c IS the gauge-mvariant electromagnetic field tensor defined by 't Hooft [ 19 ] ) has the far field h m l t b e h a v l o u r o f a m u l t l m o n o p o l e configuration
B(x)--.,=1
~.
(2 10)
= 1
(2 6)
which has the same form as the lagranglan density for Y M + massless scalar field in three d i m e n s i o n s We now go to 3 + 1 d i m e n s i o n s with the lagranglan density [signature ( - + + + ) ]
19 November 1992
m,(x-a,) I x - a , 13 , as
r~oo
(212)
As usual, the existence o f this static m u l t l m o n o p o l e solution owes to the cancellation o f the gauge a n d Hlggs forces o f exchange - the "zero-force" condition We have presented all the m o n o p o l e properties o f this solution Unfortunately, this solution as it stands has divergent action near each source, and this singulanty cannot be simply removed by a unitary gauge t r a n s f o r m a t i o n This can be seen for a single source by notmg that as r-,O , Ak--, ½( U - I O k U ) ' where U is a unitary 2 × 2 matrix The expression in parentheses represents a pure gauge, a n d there is no way to get a r o u n d the ½ factor in attempting to "gauge away" the singularity [20] The field theory solution is therefore not very interesting physically As we shall see in the next section, however, we can obtain an analogous finite action solution in string theory
3. Heterotic multimonopole solution
We use the above solution to construct a multi-
Volume 294, number 3,4
PHYSICS LETTERS B
monopole solutxon o f heterotlc string theory We present arguments based on a conjecture of Bergshoeff and de Roo [ 21,22 ] that thxs solution is exact to all orders m a ' The derivation of this solution closely parallels that o f the multl-lnstanton solutmn presented in refs [6,7], but m this case, the solution possesses three-dimensional (rather than four-dimensional) spherical symmetry near each source Again the reductmn Is effected by slnghng out a d~rectlon m the transverse space The tree-level supersymmetric vacuum equatmns for the heterotlc string are given by 6~M = (VM - IMraasFAn)e=O,
(3 1 )
~2= (F A 0 a 0 - ~ H A s c F A n C ) E = O ,
(3 2)
5z=GaBFaB~=O,
(3 3)
where ~'M, 2 and g are the gravltmo, dllatlno and gauglno fields The Branch1 identity is given by d H = o t ' ( t r R ^ R - ~o T r F ^ F )
(3 4)
The ( 9 + 1 )-dimensional Majorana-Weyl fermlons decompose down to chlral splnors according to SO(9, 1 ) = S O ( 5 , 1 ) ® S O ( 4 ) for the Mg,1---~ M s'l × M4 decomposition Let#, v, 2, a = 1,2, 3, 4 and a, b = 0 , 5, 6, 7, 8, 9 Then the ansatz
guv=eZ° ~uv, gab=?]ab, Hu,,a = + Eu,,,l~ 0°0,
(3 5 )
19 November 1992
satz, the curvature of the generahzed connection can be written m the c o v a n a n t form [ 1 ] R(I2+ )u%"=J.~ VmV#O--(~n.u VmVuO"~(~rn.uVnVv~
--(~mv VnV,uO"["E.umnotVotVvO'T-E. . . . Vo~V#O, (38) from which it easily follows that R(12+)u. ~ m=n~ ~eu~ ~°R (12+)~n_
(39)
Thus we have a solutton with the ansatz (3 5) such that F um" = R ( O + )um" ,
(3 10)
where both F and R are (ant1) self-dual We claim that th~s solution is exact since Au=t2+ u amphes that all the higher order corrections vanish [23,21,22 ] The self-dual solutmn for the gauge connectmn is then gxven by the 't Hooft ansatz
Au=t~u~ 0 ~ l n f
(3 11)
A multlmonopole is obtained as in the previous section by f=e-2°°e2°=l+
N m, ,=1 ~ Ix--a,I '
(3 12)
where m, ~s the charge and a, the locatmn m the threespace (123) o f the lth monopole If we again identify the Hxggs field as ~ib=-A4, then the gauge and HlggS fields may be simply written an terms o f the dflaton
w~th constant chlral splnors e_+ solves the supersymmerry equations with zero background Fermi fields provided the YM gauge field satisfies the mstanton (ant1) self-duahty c o n & t m n
as
Fup= -+~ ,1. ~
for the self-dual solution For the antx-self-dual soluuon, the Hlggs field simply changes sign Here g is the YM coupling constant Note that 0o drops out m (3 13) The above solutmn [with the gravitational fields obtained &rectly from (3 5 ) and (3 12) ] represents what we have argued is an exact mult~monopole solution o f heterotic stnng theory and has the same structure in the four-&menslonal transverse space as the multlmonopole solution of the YM + scalar field action o f s e c t m n 2 If we ~dentlfy the (123) subspace of the transverse space as the space part o f the fourdimensional spacetlme (with some toroldal compactlficatmn, similar to that used m ref [ 24 ] ) and take
2o~~4o
(36)
Based on the conjecture o f Bergshoeff and de Roo [ 21,22] on the form of the effectwe actmn of heterOtlC string theory, we argue that an exact solution is obtained as follows Define a generalized connection by s'2A_+ s =o9~s -+HI~s
(3 7)
embedded m an SU (2) subgroup of the gauge group, and equate It to the gauge connection A u [23 ] so that d H = 0 and the corresponding curvature R (£2+) cancels against the Yang-Mdls field strength F The crucml point is that for e -2o I~e2¢= 0 with the above an-
I~)a=-- 2(~ta Oto, A~=-- 2~"k: O,O g
g
(313)
327
Volume 294, number 3,4
PHYSICS LETTERS B
the timehke direction as the usual X °, then the monopole properties described in the previous section carry over directly into the string solution The string action contains a term - or' F 2 which also &verges as in the field theory solution However, this divergence is precisely cancelled by the term or'R2(12_+ ) in the O ( a ' ) action This result follows from the exactness c o n d m o n A,=I2_+~, ,which leads to d H = 0 and the vanishing of all higher order corrections m or' Another way of seeing this is to consider the higher order corrections to the bosonlc action as argued in refs [21,22] All such terms were claimed to contain the t e n s o r a generahzed curvature incorporating both R (t2_+) and F The ansatz is constructed precisely so that this tensor vamshes identically [ 1,2 ] The action thus reduces to its lowest order form and can be calculated directly for a multisource solution from the expressions for the massless fields in the gravity sector The divergences in the gravitational sector in heterotic string theory thus serve to cancel the divergences stemming from the field theory solution This solution thus provides an interesting example o f how this type of cancellation can occur in string theory, and supports the promise of string theory as a finite theory of q u a n t u m gravity Another point of interest is that the string solution represents a supersymmetrlc multlmonopole solution coupled to gravity, whose zero-force con&tlon m the gravity sector (cancellation o f the attractive gravitational force and repulsive antlsymmetric field force) arises as a direct result o f the zero-force condition in the gauge sector (cancellation of gauge and Hlggs forces o f exchange) once the gauge connection and generahzed connection are identified We now calculate the mass of the heterotlc multimonopole configuration Naively, the mass can be calculated from the tree-level action
TMNPQ,
S=-
l I d 3 x v / g e -2# [ R + 4 ( V 0 ) 2 -
~x2
derivatives This problem was solved in general relativity by Gibbons and Hawking [ 26,27 ], who added a surface term which precisely cancelled the double derivative terms in the action in general relativity The addition of a surface term does not, o f course, affect the equations of motion It turns out that there is a relatively straightforward generahzatlon of the Gibbons-Hawking surface term ( G H S T ) to string theory [28,29] By antlsymmetry, the axlon field does not contribute to the GHST and the surface term in this case can be written in the simple form
if
(e-2*K-Ko),
Scusa-= - ~
(3 15)
0M where 0M is the surface boundary and K and Ko are the traces of the fundamental form o f the boundary surface embedded in the metric g and the mmkowsklan metric r/respectively The correct effective action is thus obtained by adding the surface term o f (3 15 ) to the volume term of (3 14)
S=-
1 (fd3xx/~e-2*[R+4(V~)V-~H2]
~-~x~
(e-2~'K-Ko))
+2f
(316)
0M
By using the equations of motion, the volume term Sv can be written as a surface term (see ref [25 ] ) Sv=-
~-i
ti'Ve-2~
(3 17)
0M
Note that x/g has been absorbed into the surface measure o f 0M Since we have separabdlty of sources in the hmlt o f surfaces o f infinite radius, we may therefore compute Sv for a single monopole configuration in three-space
1-~H2] (3 14)
There is one subtlety we must consider, however (see ref [25 ] ) From the term x / g e -2~R in the lntegrand o f the action, the action density in (3 14) contains double derivative terms of the metric component fields In general, one would like to work with an action which depends only on the fields and their first 328
19 November 1992
e2~= 1 + -m- ,
gzj=e2~ll
(3 18)
,
F
and simply add the contributions o f an arbitrary number ofsources The contribution of a single monopole to the static volume action is given by 1 ( 0 Sv=-~k,~
e
-
2¢~ ]A(M)=
4~rm ~
(3 19)
Volume 294, number 3,4
PHYSICS LETTERS B
an the r ~ oo hm]t, where A ( M ) = 4zrr 2 ( 1 -I- m / r ) IS the area o f the b o u n d a r y surface We now turn to the G H S T A simple calculatmn o f the extrinsic curvature K for a single m o n o p o l e configuratmn (3 18) gives
K = ~2~e-3O
(r+½m)
(3 20)
r-
W h e n the surface 0M as e m b e d d e d an flat space, the radius o f curvature R as given by R = r e ~ T h e extrinsic curvature Ko as then given by 2
K°= R
2
-
r e -~
(321)
T he G H S T is therefore given by
SGnsw=--K2-~r[e-SO(l+~r)--e-OlA(M) -
12rtm x2
(3 22)
an the r ~ ~ h m l t The total static actmn for a multlsollton configuration, equal to the total mass o f the sohtons, can then be obtained by adding the static contributions to the actmn o f the v o l u m e part and the G H S T T h e result lS
M T = ~5
m.
(3 23)
n=l
For our multxmonopole configuration, however, it should be noted that m n = 1 / g for n = 1, 2, , N
4. Conclusion In thas paper, we presented a m u l t l m o n o p o l e soluUon o f heterotac s t n n g theory, and argued that thxs solution was exact to all orders in a ' This solution represents a s u p e r s y m m e t n c extensaon o f the bosomc s t n n g m u l t l m o n o p o l e solutaon outlined an ref [ 30 ], and as obtaaned by a modification o f the 't H o o f t ansatz for a four-dimensional m s t a n t o n Exactness was argued by applying to the m o n o p o l e solution the generahzed curvature m e t h o d used m refs [ 1,2,6,7 ] to obtain exact m s t a n t o n solutions in bosonac and heterotic string theory U n l i k e the instanton solu-
19 November 1992
tlon, however, the m o n o p o l e solutions do not seem to be easdy describable in terms o f conformal field theories, an unfortunate state o f affairs from the point o f view o f stnng theory An analogous mult~monopole solution o f the fourdimensional field theory o f Y M + massless scalar field was first written down This solution possesses the properties o f a m u l t l m o n o p o l e solution (topology, far-field h m l t and Bo g o m o l n 'y l b o u n d ) but has divergent action near each source We used this solution to construct the string theory solution, for which the divergences in the Y M sector are cancelled by similar divergences m the grawty sector, resulting in a finite action solution Th~s finding is significant in that it represents an example o f how string theory incorporates gravity m such a way as to cancel anfinaties anherent m gauge theories, thus supporting its promise as a finate theory o f q u a n t u m gravity
References [ 1 ] RR Khurl, Phys Lett B259 (1991)261 [2] R R Khun, Some lnstanton solutxons in stnng theory, m Proc XXth Intern Conf on Differential geometric methods m theoreucal physics (World Scientific, Singapore, October 1992 ), to appear [ 3 ] M J Duff and J X Lu, Nucl Phys B 354 ( 1991 ) 141 [4 ] M J Duff and J X Lu, Nucl Phys B 354 ( 1991 ) 129 [5] A Stromlnger, Nucl Phys B 343 (1990) 167 [6 ] C G Callan, J A Harvey and A Stromynger,Nucl Phys B 359 (1991) 611 [ 7 ] C G Callan, J A Harvey and A Stromlnger, Nucl Phys B 367 (1991) 60 [8] R R MetsaevandAA Tseythn, Phys Lett B 191 (1987) 354 [9] R R Metsaev and A A Tseythn, Nucl Phys B 293 (1987) 385 [ 10] G 't Hooft, Phys Rev Lett 37 (1976) 8 [ 11 ] F W]lczek,m Quark confinement and field theory, eds D Stump and D Wemgarten (Wiley, New York, 1977) [ 12 ] E Corrlgan and D B Fayrlle,Phys Len B 67 ( 1977 ) 69 [ 13 ] R Jacklw, C Nohl and C Rebbl, Phys Rev D 15 (1977) 1642 [ 14] R Jacklw, C Nohl and C Rebbi, m Particles and fields, eds D Boa1and A N Kamal (Plenum, New York, 1978) p 199 [ 15] M K Prasad and C M Sommerfield, Phys Rev Lett 35 (1975) 760 [ 16] E B Bogomoln'yl, Sov J Nucl Phys 24 (1976) 449 [ 17] R R K_hun, A heterotlc mulUmonopole soluUon, Texas A&M prepnnt CPT/TAMU-35/92 [18] P Rossl, Phys Rep 86 (1982) 317 329
Volume 294, number 3,4
PHYSICS LETTERS B
[ 1 9 ] G 't Hooft, Nucl Phys B79 (1974)276 [20] R Rajaraman, Sohtons and mstantons (North-Holland, Amsterdam, 1982) [21 ] E A Bergshoeffand M de Roo, Nucl Phys B 328 (1989) 439 [22] E A Bergshoeff and M de Roo, Phys Lett B 218 (1989) 210 [ 2 3 ] M Dme, Lectures dehvered at TASI 1988 (Brown Umversxty, Providence, RI, 1988) p 653 [24] J A Harvey and J Llu, Phys Lett B 268 ( 1991 ) 40
330
19 November 1992
[ 2 5 ] R R Khun, Nucl Phys B 376 (1992)350 [ 26 ] G W Gibbons and S W Hawking, Phys Rev D 15 ( 1977 ) 2752 [27] G W Gibbons, S W Hawking and M J Perry, Nucl Phys B318 (1978) 141 [28 ] S B Glddmgs and A Strommger, Nucl Phys B 306 ( 1988 ) 890 [ 29 ] D Brlll and G T Horowltz, Phys Lett B 262 ( 1991 ) 437 [ 3 0 ] R R Khurl, Sohtons and lnstantons m stnng theory, Pnnceton University Doctoral Thesis (August 1991 )
PHYSICS LETTERS B
A multimonopole solution in string theory R a m z l R. K h u n 1 Center for Theorettcal Phystcs, Texas A&M Umverstty, College Statton, TX 77843, USA Received 11 May 1992
A multlmonopole solution in Yang-Mllls field theory is obtained by a modxfication of the 't Hooft ansatz for a four-dimensional instanton Ahhough this solution has divergent action near each source, it can be used to construct a finite action multlmonopole solution of heterotlc string theory, in which the divergences from the Yang-Mills sector are precisely cancelled by those from the gravity sector We argue that this solution is exact to all orders in a '
1. Introduction In recent work [ 1-7 ], several classical sohtonlc solutions of string theory with instanton structure have been presented Exact solutions in bosonlc [ 1,2 ] and heterotic [6,7 ] string theory have been obtained by observing that a generalized curvature [8,9 ] combining the metric and antisymmetrlc tensor possesses lnstanton structure [ 1 ] In this paper we present what we argue is an exact multlmonopole solution ofheterotlc string theory We first obtain an analogous multlmonopole solution in Yang-Mllls field theory, whose action diverges near each source We show that the string theory solution, however, is finite as a result of the cancellation o f divergent terms from the gauge and gravitational sectors In section 2 we derive a multlmonopole solution of four-dimensional Y M + s c a l a r field theory via a modification o f the 't Hooft ansatz [ 10-14 ] for the Yang-Mills lnstanton This solution is not in the Prasad-Sommerfield [15] limit, but has the topology of Q = l monopole sources, saturates the Bogomoln'yl bound [16] and exhibits the far field behavlour o f multimonopole sources The action for this solution, however, diverges near each source We use this solution in section 3 to construct a heterotic multlmonopole solution, and argue that this ~r Work supported in part by NSF grant PHY-9106593 Supported by a World Laboratory Fellowship Elsevier Scmnce Publishers B V
solution is exact to all orders in a ' This string solution has finite action, as the divergences coming from the Yang-Mills sector are precisely cancelled by those from the gravitational sector The resultant action reduces to the tree-level form and is easily calculated A more detailed version of this letter is presented in ref [ 17 ]
2. Multimonopole solution in field theory Consider the four-dimensional euclidean action
S= - ~g 2
d4x Tr G..G ~,
/~,v=1,2,3,4 (21)
For gauge group SU (2), the fields may be written as
A~= (g/2l)aaA~ and G~.= (g/21)aaG~. (where a a, a = 1, 2, 3, are the 2 × 2 Pauh matrices) The equation of motion derived from this action is solved by the 't Hooft ansatz [ 10-14]
Au=lZ.u. 0. l n f ,
(2 2)
where Z ; . =O '~" la' for z= 1, 2, 3, where 0 ' ~ ' = - O " u = e '~", =-J%
/t, v = 1, 2, 3, v=4,
(2 3)
and where f - i r--lf= 0 The ansatz for the anti-self-dual solution is similar, with the J-term in (2 3) changing sign 325
Volume 294, number 3,4
PHYSICS LETTERS B
To obtain a multl-lnstanton solution, one solves for f i n the four-dimensional space to obtain f=l+
~ p2 ,=1 I x - a , I 2 '
(24)
w h e r e p ,2 is the m s t a n t o n scale size and a, the location in four-space o f the tth lnstanton We obtain a m u l t l m o n o p o l e solution by m o d i f y i n g this procedure as follows We single out a direction in the transverse four-space (say x4) a n d assume all fields are i n d e p e n d e n t o f this coordinate Then the solutton f o r f c a n be written as f=l+
,~1 = I x m, -a,l'
(2 5)
where m, is the charge a n d a, the location in the threespace (123) o f the tth m o n o p o l e If we m a k e the identification t2~-A4 (we loosely refer to this field as a Hlggs field in this paper, even though there IS no a p p a r e n t s y m m e t r y breaking m e c h a n i s m ) , then the lagranglan density for the above ansatz can be rewntten as a a a a G a~.G a~ __ - GvG, s + 2Gk4 Gk4
= GgGg + 2 DktJ9a Dkqb a ,
fields are singular near the sources and vanish as r - - . ~ This solution can be thought o f as a m u l t l h n e source m s t a n t o n solution, each m o n o p o l e being interpreted as an " l n s t a n t o n string" [ 18 ] The topological charge o f each source is easily c o m p u t e d ( (J~ a~__ ~ al l ~ l ) to be Q = f d3x ko __ 1 f d 3 x f.uk~ab c O (2~a Odtj~b Ok t~ c
8n
The magnetic charge o f each source is then given by m, = Q/g= 1/g It is also straightforward to show that the Bogomoln'yl [ 16 ] b o u n d a
G,j --e.,jk D k fl) a
It"7.a t'7.1tva
!1-) ¢f)al'~#d)a
(2 7)
and show that the above m u l t l m o n o p o l e ansatz is a static solution with A~ = 0 and all time derivatives vanish The equations o f motion in this limit are given by
D, Gm~=g~ab~(D'S~b)~ c, D , D ' ~ = 0
(2 8)
It ~s then straightforward to verify that the above equations are solved by the m o d i f i e d 't Hooft ansatz I~a=-T- 1 6 a t O, o,), a~=~.akl oj09,
g
(29)
where o g = l n f This solution represents a multim o n o p o l e configuration with sources at a , = 1, 2, , N A simple observation o f far field and near field b e h a v l o u r shows that this solution does not arise in the P r a s a d - S o m m e r f i e l d [ 15 ] limit In particular, the 326
(2
11)
is saturated by this solution Finally, it is easy to show that the magnetic field B,=½%kF jk (where F j , . aGaa. - ( 1/g) e abcqbaD~ t~ b D~ ~ c IS the gauge-mvariant electromagnetic field tensor defined by 't Hooft [ 19 ] ) has the far field h m l t b e h a v l o u r o f a m u l t l m o n o p o l e configuration
B(x)--.,=1
~.
(2 10)
= 1
(2 6)
which has the same form as the lagranglan density for Y M + massless scalar field in three d i m e n s i o n s We now go to 3 + 1 d i m e n s i o n s with the lagranglan density [signature ( - + + + ) ]
19 November 1992
m,(x-a,) I x - a , 13 , as
r~oo
(212)
As usual, the existence o f this static m u l t l m o n o p o l e solution owes to the cancellation o f the gauge a n d Hlggs forces o f exchange - the "zero-force" condition We have presented all the m o n o p o l e properties o f this solution Unfortunately, this solution as it stands has divergent action near each source, and this singulanty cannot be simply removed by a unitary gauge t r a n s f o r m a t i o n This can be seen for a single source by notmg that as r-,O , Ak--, ½( U - I O k U ) ' where U is a unitary 2 × 2 matrix The expression in parentheses represents a pure gauge, a n d there is no way to get a r o u n d the ½ factor in attempting to "gauge away" the singularity [20] The field theory solution is therefore not very interesting physically As we shall see in the next section, however, we can obtain an analogous finite action solution in string theory
3. Heterotic multimonopole solution
We use the above solution to construct a multi-
Volume 294, number 3,4
PHYSICS LETTERS B
monopole solutxon o f heterotlc string theory We present arguments based on a conjecture of Bergshoeff and de Roo [ 21,22 ] that thxs solution is exact to all orders m a ' The derivation of this solution closely parallels that o f the multl-lnstanton solutmn presented in refs [6,7], but m this case, the solution possesses three-dimensional (rather than four-dimensional) spherical symmetry near each source Again the reductmn Is effected by slnghng out a d~rectlon m the transverse space The tree-level supersymmetric vacuum equatmns for the heterotlc string are given by 6~M = (VM - IMraasFAn)e=O,
(3 1 )
~2= (F A 0 a 0 - ~ H A s c F A n C ) E = O ,
(3 2)
5z=GaBFaB~=O,
(3 3)
where ~'M, 2 and g are the gravltmo, dllatlno and gauglno fields The Branch1 identity is given by d H = o t ' ( t r R ^ R - ~o T r F ^ F )
(3 4)
The ( 9 + 1 )-dimensional Majorana-Weyl fermlons decompose down to chlral splnors according to SO(9, 1 ) = S O ( 5 , 1 ) ® S O ( 4 ) for the Mg,1---~ M s'l × M4 decomposition Let#, v, 2, a = 1,2, 3, 4 and a, b = 0 , 5, 6, 7, 8, 9 Then the ansatz
guv=eZ° ~uv, gab=?]ab, Hu,,a = + Eu,,,l~ 0°0,
(3 5 )
19 November 1992
satz, the curvature of the generahzed connection can be written m the c o v a n a n t form [ 1 ] R(I2+ )u%"=J.~ VmV#O--(~n.u VmVuO"~(~rn.uVnVv~
--(~mv VnV,uO"["E.umnotVotVvO'T-E. . . . Vo~V#O, (38) from which it easily follows that R(12+)u. ~ m=n~ ~eu~ ~°R (12+)~n_
(39)
Thus we have a solutton with the ansatz (3 5) such that F um" = R ( O + )um" ,
(3 10)
where both F and R are (ant1) self-dual We claim that th~s solution is exact since Au=t2+ u amphes that all the higher order corrections vanish [23,21,22 ] The self-dual solutmn for the gauge connectmn is then gxven by the 't Hooft ansatz
Au=t~u~ 0 ~ l n f
(3 11)
A multlmonopole is obtained as in the previous section by f=e-2°°e2°=l+
N m, ,=1 ~ Ix--a,I '
(3 12)
where m, ~s the charge and a, the locatmn m the threespace (123) o f the lth monopole If we again identify the Hxggs field as ~ib=-A4, then the gauge and HlggS fields may be simply written an terms o f the dflaton
w~th constant chlral splnors e_+ solves the supersymmerry equations with zero background Fermi fields provided the YM gauge field satisfies the mstanton (ant1) self-duahty c o n & t m n
as
Fup= -+~ ,1. ~
for the self-dual solution For the antx-self-dual soluuon, the Hlggs field simply changes sign Here g is the YM coupling constant Note that 0o drops out m (3 13) The above solutmn [with the gravitational fields obtained &rectly from (3 5 ) and (3 12) ] represents what we have argued is an exact mult~monopole solution o f heterotic stnng theory and has the same structure in the four-&menslonal transverse space as the multlmonopole solution of the YM + scalar field action o f s e c t m n 2 If we ~dentlfy the (123) subspace of the transverse space as the space part o f the fourdimensional spacetlme (with some toroldal compactlficatmn, similar to that used m ref [ 24 ] ) and take
2o~~4o
(36)
Based on the conjecture o f Bergshoeff and de Roo [ 21,22] on the form of the effectwe actmn of heterOtlC string theory, we argue that an exact solution is obtained as follows Define a generalized connection by s'2A_+ s =o9~s -+HI~s
(3 7)
embedded m an SU (2) subgroup of the gauge group, and equate It to the gauge connection A u [23 ] so that d H = 0 and the corresponding curvature R (£2+) cancels against the Yang-Mdls field strength F The crucml point is that for e -2o I~e2¢= 0 with the above an-
I~)a=-- 2(~ta Oto, A~=-- 2~"k: O,O g
g
(313)
327
Volume 294, number 3,4
PHYSICS LETTERS B
the timehke direction as the usual X °, then the monopole properties described in the previous section carry over directly into the string solution The string action contains a term - or' F 2 which also &verges as in the field theory solution However, this divergence is precisely cancelled by the term or'R2(12_+ ) in the O ( a ' ) action This result follows from the exactness c o n d m o n A,=I2_+~, ,which leads to d H = 0 and the vanishing of all higher order corrections m or' Another way of seeing this is to consider the higher order corrections to the bosonlc action as argued in refs [21,22] All such terms were claimed to contain the t e n s o r a generahzed curvature incorporating both R (t2_+) and F The ansatz is constructed precisely so that this tensor vamshes identically [ 1,2 ] The action thus reduces to its lowest order form and can be calculated directly for a multisource solution from the expressions for the massless fields in the gravity sector The divergences in the gravitational sector in heterotic string theory thus serve to cancel the divergences stemming from the field theory solution This solution thus provides an interesting example o f how this type of cancellation can occur in string theory, and supports the promise of string theory as a finite theory of q u a n t u m gravity Another point of interest is that the string solution represents a supersymmetrlc multlmonopole solution coupled to gravity, whose zero-force con&tlon m the gravity sector (cancellation o f the attractive gravitational force and repulsive antlsymmetric field force) arises as a direct result o f the zero-force condition in the gauge sector (cancellation of gauge and Hlggs forces o f exchange) once the gauge connection and generahzed connection are identified We now calculate the mass of the heterotlc multimonopole configuration Naively, the mass can be calculated from the tree-level action
TMNPQ,
S=-
l I d 3 x v / g e -2# [ R + 4 ( V 0 ) 2 -
~x2
derivatives This problem was solved in general relativity by Gibbons and Hawking [ 26,27 ], who added a surface term which precisely cancelled the double derivative terms in the action in general relativity The addition of a surface term does not, o f course, affect the equations of motion It turns out that there is a relatively straightforward generahzatlon of the Gibbons-Hawking surface term ( G H S T ) to string theory [28,29] By antlsymmetry, the axlon field does not contribute to the GHST and the surface term in this case can be written in the simple form
if
(e-2*K-Ko),
Scusa-= - ~
(3 15)
0M where 0M is the surface boundary and K and Ko are the traces of the fundamental form o f the boundary surface embedded in the metric g and the mmkowsklan metric r/respectively The correct effective action is thus obtained by adding the surface term o f (3 15 ) to the volume term of (3 14)
S=-
1 (fd3xx/~e-2*[R+4(V~)V-~H2]
~-~x~
(e-2~'K-Ko))
+2f
(316)
0M
By using the equations of motion, the volume term Sv can be written as a surface term (see ref [25 ] ) Sv=-
~-i
ti'Ve-2~
(3 17)
0M
Note that x/g has been absorbed into the surface measure o f 0M Since we have separabdlty of sources in the hmlt o f surfaces o f infinite radius, we may therefore compute Sv for a single monopole configuration in three-space
1-~H2] (3 14)
There is one subtlety we must consider, however (see ref [25 ] ) From the term x / g e -2~R in the lntegrand o f the action, the action density in (3 14) contains double derivative terms of the metric component fields In general, one would like to work with an action which depends only on the fields and their first 328
19 November 1992
e2~= 1 + -m- ,
gzj=e2~ll
(3 18)
,
F
and simply add the contributions o f an arbitrary number ofsources The contribution of a single monopole to the static volume action is given by 1 ( 0 Sv=-~k,~
e
-
2¢~ ]A(M)=
4~rm ~
(3 19)
Volume 294, number 3,4
PHYSICS LETTERS B
an the r ~ oo hm]t, where A ( M ) = 4zrr 2 ( 1 -I- m / r ) IS the area o f the b o u n d a r y surface We now turn to the G H S T A simple calculatmn o f the extrinsic curvature K for a single m o n o p o l e configuratmn (3 18) gives
K = ~2~e-3O
(r+½m)
(3 20)
r-
W h e n the surface 0M as e m b e d d e d an flat space, the radius o f curvature R as given by R = r e ~ T h e extrinsic curvature Ko as then given by 2
K°= R
2
-
r e -~
(321)
T he G H S T is therefore given by
SGnsw=--K2-~r[e-SO(l+~r)--e-OlA(M) -
12rtm x2
(3 22)
an the r ~ ~ h m l t The total static actmn for a multlsollton configuration, equal to the total mass o f the sohtons, can then be obtained by adding the static contributions to the actmn o f the v o l u m e part and the G H S T T h e result lS
M T = ~5
m.
(3 23)
n=l
For our multxmonopole configuration, however, it should be noted that m n = 1 / g for n = 1, 2, , N
4. Conclusion In thas paper, we presented a m u l t l m o n o p o l e soluUon o f heterotac s t n n g theory, and argued that thxs solution was exact to all orders in a ' This solution represents a s u p e r s y m m e t n c extensaon o f the bosomc s t n n g m u l t l m o n o p o l e solutaon outlined an ref [ 30 ], and as obtaaned by a modification o f the 't H o o f t ansatz for a four-dimensional m s t a n t o n Exactness was argued by applying to the m o n o p o l e solution the generahzed curvature m e t h o d used m refs [ 1,2,6,7 ] to obtain exact m s t a n t o n solutions in bosonac and heterotic string theory U n l i k e the instanton solu-
19 November 1992
tlon, however, the m o n o p o l e solutions do not seem to be easdy describable in terms o f conformal field theories, an unfortunate state o f affairs from the point o f view o f stnng theory An analogous mult~monopole solution o f the fourdimensional field theory o f Y M + massless scalar field was first written down This solution possesses the properties o f a m u l t l m o n o p o l e solution (topology, far-field h m l t and Bo g o m o l n 'y l b o u n d ) but has divergent action near each source We used this solution to construct the string theory solution, for which the divergences in the Y M sector are cancelled by similar divergences m the grawty sector, resulting in a finite action solution Th~s finding is significant in that it represents an example o f how string theory incorporates gravity m such a way as to cancel anfinaties anherent m gauge theories, thus supporting its promise as a finate theory o f q u a n t u m gravity
References [ 1 ] RR Khurl, Phys Lett B259 (1991)261 [2] R R Khun, Some lnstanton solutxons in stnng theory, m Proc XXth Intern Conf on Differential geometric methods m theoreucal physics (World Scientific, Singapore, October 1992 ), to appear [ 3 ] M J Duff and J X Lu, Nucl Phys B 354 ( 1991 ) 141 [4 ] M J Duff and J X Lu, Nucl Phys B 354 ( 1991 ) 129 [5] A Stromlnger, Nucl Phys B 343 (1990) 167 [6 ] C G Callan, J A Harvey and A Stromynger,Nucl Phys B 359 (1991) 611 [ 7 ] C G Callan, J A Harvey and A Stromlnger, Nucl Phys B 367 (1991) 60 [8] R R MetsaevandAA Tseythn, Phys Lett B 191 (1987) 354 [9] R R Metsaev and A A Tseythn, Nucl Phys B 293 (1987) 385 [ 10] G 't Hooft, Phys Rev Lett 37 (1976) 8 [ 11 ] F W]lczek,m Quark confinement and field theory, eds D Stump and D Wemgarten (Wiley, New York, 1977) [ 12 ] E Corrlgan and D B Fayrlle,Phys Len B 67 ( 1977 ) 69 [ 13 ] R Jacklw, C Nohl and C Rebbl, Phys Rev D 15 (1977) 1642 [ 14] R Jacklw, C Nohl and C Rebbi, m Particles and fields, eds D Boa1and A N Kamal (Plenum, New York, 1978) p 199 [ 15] M K Prasad and C M Sommerfield, Phys Rev Lett 35 (1975) 760 [ 16] E B Bogomoln'yl, Sov J Nucl Phys 24 (1976) 449 [ 17] R R K_hun, A heterotlc mulUmonopole soluUon, Texas A&M prepnnt CPT/TAMU-35/92 [18] P Rossl, Phys Rep 86 (1982) 317 329
Volume 294, number 3,4
PHYSICS LETTERS B
[ 1 9 ] G 't Hooft, Nucl Phys B79 (1974)276 [20] R Rajaraman, Sohtons and mstantons (North-Holland, Amsterdam, 1982) [21 ] E A Bergshoeffand M de Roo, Nucl Phys B 328 (1989) 439 [22] E A Bergshoeff and M de Roo, Phys Lett B 218 (1989) 210 [ 2 3 ] M Dme, Lectures dehvered at TASI 1988 (Brown Umversxty, Providence, RI, 1988) p 653 [24] J A Harvey and J Llu, Phys Lett B 268 ( 1991 ) 40
330
19 November 1992
[ 2 5 ] R R Khun, Nucl Phys B 376 (1992)350 [ 26 ] G W Gibbons and S W Hawking, Phys Rev D 15 ( 1977 ) 2752 [27] G W Gibbons, S W Hawking and M J Perry, Nucl Phys B318 (1978) 141 [28 ] S B Glddmgs and A Strommger, Nucl Phys B 306 ( 1988 ) 890 [ 29 ] D Brlll and G T Horowltz, Phys Lett B 262 ( 1991 ) 437 [ 3 0 ] R R Khurl, Sohtons and lnstantons m stnng theory, Pnnceton University Doctoral Thesis (August 1991 )