Monopoles and Dyons in the Pure Einstein-Yang-Mills Theory

9 Pages·0.21 MB·English

by  Yutaka Hosotani

UMN-TH-1839/00
NUC-MINN-00/2-T
BNL-NT-00/1
(31 January 2000)
Monopoles and Dyons
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0 in the Pure Einstein-Yang-Mills Theory
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2

n
Yutaka HOSOTANI and Jefferson BJORAKER∗
a
J School of Physics and Astronomy, University of Minnesota

1 Minneapolis, MN 55455, U.S.A.
3

1
v
To appear in theProceedings for
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The 6th International Wigner Symposium
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1 16 - 22 August 1999, Istanbul,Turkey
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0
0
0
/
c Abstract
q
- In thepureEinstein-Yang-Mills theory in four dimensions thereexist monopole
r
g and dyon solutions. The spectrum of the solutions is discrete in asymptotically
: flat or de Sitter space, whereas it is continuous in asymptotically anti-de Sitter
v
space. Thesolutionsareregulareverywhereandspecifiedwiththeirmass,andnon-
i
X Abelianelectricandmagneticcharges. Inasymptoticallyanti-deSitterspaceaclass
of monopole solutions have no node in non-Abelian magnetic fields, and are stable
r
a against spherically symmetric perturbations.
1. Introduction
In flat space there cannot be any static solution in the pure Yang-Mills theory in four
dimensions[1]. Only with scalar fields monopole solutions exist, the topology of the
scalarfieldplayingacrucialrolethere. The inclusionofthe gravityopensapossibilityof
having a soliton solution. The gravitational force, being always attractive, may balance
therepulsiveforceofthenon-Abeliangaugefields. Suchconfigurationswereindeedfound
inasymptoticallyflatanddeSitterspacesometimeago[2]-[9]. Unfortunatelyallofthem
turned out unstable against small perturbations [10]-[13].
∗Currentaddress: Brookhaven National Laboratory, Building 510A, Upton, NY 11973, U.S.A.
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HOSOTANIand BJORAKER
Thesituationdrasticallychangesinasymptoticallyanti-deSitterspace. Thenegative
cosmological constant (Λ < 0) provides negative pressure and energy density, making a
class of monopole and dyon configurations stable [14]-[17]. We review the current status
of such solutions.
2. Equations
The equations of motion in the Einstein-Yang-Mills theory are
1
Rµν gµν(R 2Λ)=8πG Tµν
− 2 −
Fµν +e[A ,Fµν]=0 . (1)
;µ µ
To find soliton solutions we make a spherically symmetric, static ansatz:
H(r) dr2
ds2 = dt2+ +r2(dθ2+sin2θdφ2)
−p(r)2 H(r)
τj x 1 w(r)
j
A= u(r) dt − ǫ x dx (2)
2e(cid:26) r − r2 jkl k l(cid:27)
with boundary conditions u(0) = 0 and H(0) = p(0) = w(0) = 1. It is convenient to
parameterize
2m(r) Λr2
H(r)=1 , (3)
− r − 3
wherem(r)isthemasscontainedinsidetheradiusr. p(r)=1andm(r)=0corresponds
to the Minkowski, de Sitter, or anti-de Sitter space for Λ = 0, > 0, or < 0, respectively.
u(r) and w(r) represent the electric and magnetic Yang-Mills fields, respectively.
The equations in (1) reduce to
r2pu′ ′ = 2puw2
H
(cid:0) (cid:1)
H ′ w(w2 1) p
w′ = − u2w
(cid:16) p (cid:17) r2p − H
4πG 1 (1 w2)2 p2
m′ = r2p2u′2+ − + u2w2+Hw′2
e2 (cid:26)2 2r2 H (cid:27)
8πG p p2
p′ = u2w2+Hw′2 . (4)
− e2 rH (cid:26)H (cid:27)
Neartheoriginu=ar,w =1 br2,m=v(a2+4b2)r3/2,andp=1 v(a2+4b2)r2,where
− −
(a,b) are two parameters to be fixed and v =4πG/e2. With given(a,b) the equations in
(4) are numerically integrated from r =0 to r = .
∞
Ingeneral,solutionsblowupatfiniter,unless(a,b)takespecialvalues. Wearelooking
for everywhere regular soliton configurations with a finite ADM mass M =m( ).
∞
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HOSOTANIand BJORAKER
3. Conserved charges
Non-Abelian solitons are characterized by the ADM mass and non-Abelian electric and
magnetic charges. The kinematical identities (Fµν ) = 0 and (F˜µν ) = 0 lead to
;µ ;ν ;µ ;ν
conservedchargesgivenby dS √ g(Fk0,F˜k0). They aregaugevariant,andtherefore
k
−
there are infinitely many cRonserved charges. Most of them vanish for solutions under
consideration. Non-vanishing charges are
Q e Fk0 xjτj
E = dS √ gTr τ , τ =
(cid:18)QM(cid:19) 4π Z k − r(cid:18)F˜k0(cid:19) r r
u p
= 1 0 . (5)
(cid:18)1 w2(cid:19)
− 0
In the second equality the coefficients u , p , and w are defined by the asymptotic
1 0 0
expansion u u +(u /r)+ etc.. These two charges are conserved as there exists a
0 1
unitary matr∼ix S satisfying τ··=· Sτ S−1.
r 3
4. Solutions in the Λ=0 or Λ>0 case
Ithasbeenshownin[18]and[17]thatsolutionsareelectricallyneutral(a=0,u(r)=0).
Solutions exist only for a discrete set of values of b. In the Λ=0 case, w =w( )= 1
0
∞ ±
so that Q = 0. In the Λ > 0 case, Q = 0. The Bartnik-McKinnon solution in the
M M
6
Λ=0 case is depicted in fig. 1.
H(r)
1
m(r)
0.5
p(r)
0
Bartnik-McKinnon solution
-0.5
Λ = 0. a=0, b=0.45372 w(r)
-1
0.01 0.1 1 10 100
r
Figure 1. Bartnik-McKinnon solution in theΛ=0 theory.
Solutions are characterizedby the number of nodes, n, in w(r); n=1,2,3, . All of
···
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HOSOTANIand BJORAKER
these solutions are shown to be unstable against small spherically symmetric perturba-
tions [10]-[12].
5. Solutions in the Λ<0 case
5.1. Configurations
The Λ < 0 case is qualitatively different in many respects from the Λ = 0 or Λ > 0
case. First there are dyonic solutions in which electric fields are non-vanishing; u(r)=0.
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Secondly solutions exist in a finite continuum region in the parameter space (a,b) as
opposed to discrete points. Thirdly there exist solutions with no node (n=0) in w(r).
A typical monopole solution with no node in w is depicted in fig. 2. Depending on
the value ofb,the asymptoticvalue w( ) canbe either greaterthan1,orbetween0and
∞
1, or negative.
1.4
H(r)
1.2
p(r)
1
0.8
0.6 Monopole in AdS
w(r)
0.4
Λ = −0.01 a=0, b=0.001
0.2
m(r)
0
-0.2
0.01 0.1 1 10 100 1000
r
Figure 2. Monopole solution in the Λ < 0 theory. There are a continuum of solutions. Dyon
solutions have similar behavior, with the additional u(r) monotonically increasing from zero to
theasymptotic value in the range (0∼0.2).
5.2. Monopole and dyon spectrum
When a = 0 and b is varied, a continuum of monopole solutions are generated. With Λ
given,solutionsappearinafinite numberofbranches. Thenumberofbranchesincreases
as Λ 0. For Λ= 0.01 there are only two branches, which are displayed in fig. 3.
→ −
The upper branch ends near the point (Q =1,M =1). The end point corresponds
M
to the critical spacetime geometry discussed in the next subsection.
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HOSOTANIand BJORAKER
Solutionswithnonodeinw(r)arespecial. Theyarestableagainstsmallperturbations
asdiscussedinSection6. Theyexistinalimitedregionintheparameterspaceasdepicted
in fig. 4.
1.4
Monopole spectrum
1.2
n=2 n=3
1
0.8 n=1
M Λ = −0.01
0.6
0.4
n=1
0.2
n=0
0
-4 -3 -2 -1 0 1 2
QM
Figure 3. Monopole spectrum at Λ=−0.01. n is the numberof thenodes in w(r).
5.3. Critical spacetime
When the parameter b is increased, the solution either blows up or reaches a critical
configuration. The end point in the upper branch in fig. 3 represents such a critical
spacetime with b = b = 0.70. H(r) vanishes at r = r . However, r = r is not a
c h h
standard event horizon appearing in black hole solutions.
Whenbisveryclosetob ,H(r)almostvanishesatr r . Oneofsuchconfigurations
c h
∼
is displayed in fig. 5.
At b = b , H(r) becomes tangent to the axis at r . Further p(r) vanishes. In other
c h
wordsthe spaceendsatr =r . Itisanopenquestionwhethersuchconfigurationsreally
h
represent possible spacetime.
Thesecriticalspacetimeshaveuniversalbehavior. Theirmagneticchargeisquantized,
Q = 1, whereas their electric charge Q is not. There are two additional parameters,
M E
Λ and v =4πG/e2. When v Λ 1, the critical spacetime is described near r by
h
| |≪
1
r = 1+4v Λ 1 √v
h
2|Λ|(cid:16)p | |− (cid:17)∼
w(r) 2y1/2
∼
H(r) 4y2
∼
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HOSOTANIand BJORAKER
0.003
0.0025 Spectrum of nodeless (n=0) dyons
Λ = −0.01
0.002
0.0015
b
r 0.001
e
met 0.0005
a
ar 0
p
-0.0005
-0.001
-0.0015
-0.002
0 0.001 0.002 0.003 0.004 0.005 0.006
parameter a
Figure 4. Spectrum of nodeless dyons.
1
m(r) √v(1 y)
∼ 2 −
p(r) p y2
0
∼
r
where y =1 0 . (6)
− r ≥
h
Exceptforp allcoefficientsandcriticalexponentsareindependentofQ and Λ( 1/v).
0 E
| | ≪
6. Stability
The stability of the solutions is examined by considering small perturbations. If they
exponentiallygrowintime,thesolutionsareunstable,whereasiftheyremainsmall,they
are stable. In the linearized theory, the problem is reduced to finding eigenvalues of a
Schr¨odinger equation.
The analysis is simplified in the tortoise coordinate ρ defined by dρ/dr = p/H. The
range of ρ is finite for Λ<0 ; 0 ρ ρ . For monopole solutions
max
≤ ≤
d2
+U(ρ) χ=ω2χ (7)
(cid:26)− dρ2 (cid:27)
(i) Odd parity perturbations
2
H 2 dw
U = (1+w2)+ , (8)
odd r2p2 w2 (cid:18)dρ(cid:19)
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HOSOTANIand BJORAKER
1.4
Near the critical spacetime
1.2 H(r)
a=0.01, b=0.69, v=1.0
m(r)
1
0.8
u(r)
0.6
0.4
0.2 w(r)
p(r)
0
-0.2
0.01 0.1 1 10 100 1000
r
Figure 5. Dyon solution very close to thecritical spacetime. At the critical value bc the space
endsat rh. p(r)=0 for r≥rh.
w 1 d
δν = χ , δw˜ = (r2pδν)
r2p odd −2w dρ
(ii) Even parity perturbations
H d Hw′2
U = (3w2 1)+4v (9)
even r2p2 − dρ(cid:18) pr (cid:19)
δH 4v δp ′ 4v
′ ′ ′
δw =χ , = w δw , = w δw
even
H − r (cid:16) p (cid:17) − r
′
Here denotes a r-derivative. The boundary condition for odd parity perturbations
is given by χ = 0 at ρ = 0 and d(wχ )/dρ = 0 at ρ = ρ . For even parity
odd odd max
perturbations χ = 0 at both ends. If Eq. (7) admits no boundstate (ω2 < 0), then
even
the solution is stable.
AlthoughU (ρ) is positive definite, the correspondingeigenvaluesmay not be posi-
odd
tiveduetothenontrivialboundaryconditionimposedonχ . Forsolutionswithnonode
odd
in w(r), both U and U behave as 2/ρ2 near the origin, but are regular elsewhere.
odd even
One can prove that all ω2 ,ω2 > 0. In other words nodeless monopole solutions are
odd even
stable against spherically symmetric perturbations.
When w has n nodes at ρ (k =1, ,n), U become singular there. There appear
k odd
···
n negative ω2 modes, which generally diverges at the singularities. U also becomes
odd even
negative in the vicinity of the nodes, and there appear negative ω2 . Solutions with
even
nodes in w(r) are unstable.
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HOSOTANIand BJORAKER
7. The Λ 0 limit
→
When the cosmological constant approaches zero, more and more branches of monopole
solutions emerge. In the Λ 0 limit the spectrum becomes discrete, there appearing
→
infinitely many (unstable) solutions.
How is it possible? The nodeless, stable solutions must disappear. One parameter
family ofsolutionsmustcollapseinto onepointinthe modulispaceofsolutions. In fig.6
we have plotted the monopole spectrum with various values of Λ. One can see that as Λ
approaches zero, new branches of solutions appear, and each branch collapses to a point
in the Λ 0 limit. The nodeless solutions disappear as their ADM mass vanishes.
→
The result is indicative of a fractal structure in the moduli space of the solutions.
2
1.8
Monopole spectrum
1.6
Λ = −.01
1.4
−.001
1.2
M 1
−.0001
0.8
0.6
−.001 Λ = −.01
0.4
−.0001
0.2
0
-14 -12 -10 -8 -6 -4 -2 0 2
QM
Figure 6. Monopole spectrum with varyingΛ.
8. Summary
WehaveshownthatthereexiststablemonopoleanddyonsolutionsintheEinstein-Yang-
Mills theory in asymptotically anti-de Sitter space. They have a continuous spectrum.
Their implication to physics is yet to be examined.
Acknowledgments
This work was supported in part by the U.S. Department of Energy under contracts
DE-FG02-94ER-40823and DE-FG02-87ER40328.
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