In recent work a multimonopole solution of heterotic string theory was
obtained. The monopoles are noted to be stable, in contrast with analogous
solutions of Einstein-Maxwell or Yang-Mills-dilaton theory. The existence of
this and other classes of stable solitonic solutions in string theory thus
provides a possible test for low-energy string theory as distinct from other
gauge + gravity theories.Comment: 5 page

Bartnik

Bergshoeff

Bizon

Callan

Charap

Dabholkar

Dine

Duff

Felce

Gauntlett

Gross

Khuri

Lavrelashvili

Ramzi R. Khuri

Sorkin

Strominger

Physics Letters B

English

arXiv

Khuri, Ramzi R.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:hep-th/9301035v1  11 Jan 1993CTP/TAMU-81/92A Comment on the Stability of String Monopoles †Ramzi R. Khuri∗Center for Theoretical PhysicsTexas A&M UniversityCollege Station, TX 77843In recent work a multimonopole solution of heterotic string theory was obtained. Themonopoles are noted to be stable, in contrast with analogous solutions of Einstein-Maxwellor Yang-Mills-dilaton theory. The existence of this and other classes of stable solitonicsolutions in string theory thus provides a possible test for low-energy string theory asdistinct from other gauge + gravity theories.12/92†Work supported in part by NSF grant PHY-9106593.∗ Supported by a World Laboratory Fellowship.In recent work[1], a multimonopole solution of heterotic string theory was presented.An analogous solution in Yang-Mills field theory was found to have divergent action neareach source. In the string solution, however, the divergences from the Yang-Mills sectorare precisely cancelled by those from the gravity sector, so that the action is finite andeasily computed[1,2].In this letter we comment briefly on the stability of this solution. We note that thestring monopole solution, inasmuch as it saturates a Bogomol’nyi bound between ADMmass and charge, is stable. The stability of the resultant low-energy YM + dilaton gravityfield theory contrasts with the recently demonstrated instability of analogous non-stringEinstein-Maxwell or YM-dilaton solutions and thus represents a possible low-energy testfor string theory. We also briefly comment on the dynamics of these solitons.The bosonic fields for the self-dual multimonopole solution of heterotic string theorywith zero background fermi fields are given by[1]gµν = e2φδµν , gab = ηab,Hµνλ = ±ǫµνλσ∂σφ,e2φ = e2φ0f,Aµ = iΣµν∂ν ln f,(1)where µ, ν, λ, σ = 1, 2, 3, 4, a, b = 0, 5, 6, 7, 8, 9, Σµν = ηiµν(σi/2) for i = 1, 2, 3 (σi, i =1, 2, 3 are the 2× 2 Pauli matrices) whereηiµν = −ηiνµ = ǫiµν , µ, ν = 1, 2, 3,= −δiµ, ν = 4(2)and wheref = 1 +N∑n=1mn|~x− ~an|, (3)where mn is the charge and ~an the location in the three-space (123) of the nth monopole.The anti-self-dual solution is similar, with the δ-term in (2) changing sign. This solutionwas shown to have multimonopole structure[1] in the four-space (0123), each source havingtopological charge Q = 1 and magnetic charge m = 1/g, where g is the YM couplingconstant. The four-dimensional metric line-element strongly resembles that of the Kaluza-Klein monopole [3,4]. We argued in [1] that this solution is exact to all orders in α′.1If we make the identification Φ ≡ A4, then the gauge and Higgs fields may be simplywritten in terms of the dilaton asΦa = −2gδia∂iφ,Aak = −2gǫakj∂jφ(4)for the self-dual solution. For the anti-self-dual solution, the Higgs field simply changessign. A toroidal compactification can be adopted so that we consider the dynamics of oursolution in the spacetime (0123). As usual, the existence of a static multi-soliton solutiondepends on the “zero force” condition.A similar solution for Yang-Mills-dilaton theory was found in [5], with analogousdilaton behaviour and corresponding to a Dirac magnetic monopole with unit magneticcharge. It was noted that this solution has infinitely many unstable modes, but has finiteaction resulting from the cancellation between divergences stemming from the YM fieldand dilaton respectively. In fact, magnetically charged solutions to the coupled Einstein-Maxwell equations are typically found to be unstable [6,7,8], with or without the presenceof a dilaton.The ansatz of (4) represents an infinite-action solution of YM + scalar field theory in3+1 dimensions which satisfies the Bogomol’nyi bound Gaij = ǫijkDkΦa. Since the action isfinite away from the singularity, the Bogomol’nyi bound guarentees stability of the solutionoutside arbitrary finite enclosures around each source. Since the string solution has finiteaction owing to the cancellation between gauge and gravitational divergences, we can usea Bogomol’nyi bound to demonstrate stability everywhere. We see this by noting thatthe ADM mass of the multimonopole solution written in terms of the canonical metricsaturates a Bogomol’nyi bound in terms of the charges of the string monopoles[9]M =2πκ2eφ0/2N∑i=1mi. (5)Thus the string multimonopole solution sits at a minimum energy point and is automati-cally stable against perturbations independent of x4 (presumably if we allow perturbationsdepending on the compactified direction the string monopoles will decay into heterotic in-stanton fivebranes [10–14], which possess a higher symmetry1). As an exercise, one can1 This was pointed out to me by Jianxin Lu.2explicitly show that there are no unstable modes of the string monopole away from the sin-gularity, from either the gauge or gravitational sectors. The same divergence cancellationfeature which leads to a finite action solution then allows us to circumvent the singularityin each sector when considering the stability of the string solution.Since the divergence cancellation in the action is also possessed by the unstable so-lution in [5], however, this property is clearly insufficient in itself to explain the stabilityof the string solution, which is rooted in the (4,4) superconformal invariance of its un-derlying sigma-model [13,14]. It is also linked to the existence of a zero-dynamical forcecondition obeyed by this and other classes of heterotic multi-soliton solutions [15–19], inaddition to the usual zero-static force condition. Since the string solutions all saturateBogomol’nyi bounds and are hence stable, their existence in suitably compactified formprovides a possible test for string theory as distinct from other gauge + gravity models.A study of the low-energy dynamics of the string monopoles was done in [20], where itwas found from both a test-monopole approach and a computation of the Manton metricon moduli space that the monopoles scatter trivially in the low-energy limit. The lattermethod relied on the construction of anO(β) solution to the constraint equations of motion,which had the piecewise Lorentz boosted forme2φ(~x,t) = 1 +N∑n=1mn|~x− ~an(t)|,g00 = −1, g00 = −1, gij = e2φδij , gij = e−2φδij ,g0i = −N∑n=1mn~βn · xˆi|~x− ~an(t)|, g0i = e−2φg0i,Hijk = ǫijkm∂me2φ,H0ij = ǫijkm∂mg0k = ǫijkm∂kN∑n=1mn~βn · xˆm|~x− ~an(t)|,(6)where i, j, k,m = 1, 2, 3, 4 and we use a flat space ǫ-tensor. Note that g00, gij and Hijkare unaffected to order β. According to the conjecture of Bergshoeff and de Roo[21,22,1],from the exactness condition Aµ = Ω±µ[23,24] (where Ω±µ is the generalized connectiondefined in [1]), the higher order in α′ terms drop out from the action. The contributionsof these terms also cancel in the equations of motion, provided we assume a similar O(β)solution for the YM field Aµ to balance the piecewise-boosted generalized connection Ω±µ.We therefore argue that our low-energy scattering analysis in [20] is also exact, suggesting3that the dynamic YM force for these string monopoles is precisely cancelled by the dynamicgravity sector force to all orders in α′. This argument can equally well be applied to theheterotic instanton fivebranes.AcknowledgementsI would like to thank M. Duff, A. Felce, Z. Khviengia, J. Lu, V. P. Nair, M. Ortiz andE. Weinberg for helpful discussions.4References[1] R. R. Khuri, Phys. Lett. B294 (1992) 325.[2] R. R. Khuri, Nucl. Phys. B387 (1992) 315.[3] D. J. Gross and M. J. Perry, Nucl. Phys. B226 (1983) 29.[4] R. D. Sorkin, Phys. Rev. Lett 51 (1983) 87.[5] P. Bizon, “Saddle Point Solutions in Yang-Mills-Dilaton theory” UWThPh-1992-46,September 1992.[6] R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141.[7] K. Lee, V. P. Nair and E. J. Weinberg, Phys. Rev. Lett. 68 (1992) 1100.[8] G. Lavrelashvili and D. Maison, “Static Spherically Symmetric Solutions of a Yang-Mills Field Coupled to a Dilaton”, MPI-Ph/92-62, August 1992.[9] J. P. Gauntlett, J. A. Harvey and J. T. Liu, “Magnetic Monopoles in String Theory”IFP-434-UNC, November 1992.[10] M. J. Duff and J. X. Lu, Nucl. Phys. B354 (1991) 141.[11] M. J. Duff and J. X. Lu, Nucl. Phys. B354 (1991) 129.[12] A. Strominger, Nucl. Phys. B343 (1990) 167.[13] C. G. Callan, J. A. Harvey and A. Strominger, Nucl. Phys. B359 (1991) 611.[14] C. G. Callan, J. A. Harvey and A. Strominger, Nucl. Phys. B367 (1991) 60.[15] A. Dabholkar, G. Gibbons, J. A. Harvey and F. Ruiz Ruiz, Nucl. Phys. B340 (1990)33.[16] C. G. Callan and R. R. Khuri, Phys. Lett. B261 (1991) 363.[17] A. G. Felce and T. M. Samols, “Low-Energy Dynamics of String Solitons” NSF-ITP-92-155, November 1992.[18] R. R. Khuri, “Geodesic Scattering of Solitonic Strings”, Texas A&M preprintCTP/TAMU-79/92.[19] R. R. Khuri, “Classical Scattering of Macroscopic Strings”, Texas A&M preprintCTP/TAMU-80/92.[20] R. R. Khuri, Phys. Lett. B294 (1992) 331.[21] E. A. Bergshoeff and M. de Roo, Nucl. Phys. B328 (1989) 439.[22] E. A. Bergshoeff and M. de Roo, Phys. Lett. B218 (1989) 210.[23] J. M. Charap and M. J. Duff, Phys. Lett. B69 (1977) 445.[24] M. Dine, Lectures delivered at TASI 1988, Brown University (1988) 653.5

A comment on the stability of string monopoles

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CTP/TAMU-81/92
A Comment on the Stability of String Monopoles †
Ramzi R. Khuri∗
Center for Theoretical Physics
Texas A&M University
College Station, TX 77843
In recent work a multimonopole solution of heterotic string theory was obtained. The
monopoles are noted to be stable, in contrast with analogous solutions of Einstein-Maxwell
or Yang-Mills-dilaton theory. The existence of this and other classes of stable solitonic
solutions in string theory thus provides a possible test for low-energy string theory as
distinct from other gauge + gravity theories.
12/92
†
Work supported in part by NSF grant PHY-9106593.
∗ Supported by a World Laboratory Fellowship.
In recent work[1], a multimonopole solution of heterotic string theory was presented.
An analogous solution in Yang-Mills field theory was found to have divergent action near
each source. In the string solution, however, the divergences from the Yang-Mills sector
are precisely cancelled by those from the gravity sector, so that the action is finite and
easily computed[1,2].
In this letter we comment briefly on the stability of this solution. We note that the
string monopole solution, inasmuch as it saturates a Bogomol’nyi bound between ADM
mass and charge, is stable. The stability of the resultant low-energy YM + dilaton gravity
field theory contrasts with the recently demonstrated instability of analogous non-string
Einstein-Maxwell or YM-dilaton solutions and thus represents a possible low-energy test
for string theory. We also briefly comment on the dynamics of these solitons.
The bosonic fields for the self-dual multimonopole solution of heterotic string theory
with zero background fermi fields are given by[1]
gµν = e
2φδµν , gab = ηab,
Hµνλ = ±ǫµνλσ∂
σφ,
e2φ = e2φ0f,
Aµ = iΣµν∂ν ln f,
(1)
where µ, ν, λ, σ = 1, 2, 3, 4, a, b = 0, 5, 6, 7, 8, 9, Σµν = η
iµν(σi/2) for i = 1, 2, 3 (σi, i =
1, 2, 3 are the 2× 2 Pauli matrices) where
ηiµν = −ηiνµ = ǫiµν , µ, ν = 1, 2, 3,
= −δiµ, ν = 4
(2)
and where
f = 1 +
N∑
n=1
mn
|~x− ~an|
, (3)
where mn is the charge and ~an the location in the three-space (123) of the nth monopole.
The anti-self-dual solution is similar, with the δ-term in (2) changing sign. This solution
was shown to have multimonopole structure[1] in the four-space (0123), each source having
topological charge Q = 1 and magnetic charge m = 1/g, where g is the YM coupling
constant. The four-dimensional metric line-element strongly resembles that of the Kaluza-
Klein monopole [3,4]. We argued in [1] that this solution is exact to all orders in α′.
1
If we make the identification Φ ≡ A4, then the gauge and Higgs fields may be simply
written in terms of the dilaton as
Φa = −
2
g
δia∂iφ,
Aak = −
2
g
ǫakj∂jφ
(4)
for the self-dual solution. For the anti-self-dual solution, the Higgs field simply changes
sign. A toroidal compactification can be adopted so that we consider the dynamics of our
solution in the spacetime (0123). As usual, the existence of a static multi-soliton solution
depends on the “zero force” condition.
A similar solution for Yang-Mills-dilaton theory was found in [5], with analogous
dilaton behaviour and corresponding to a Dirac magnetic monopole with unit magnetic
charge. It was noted that this solution has infinitely many unstable modes, but has finite
action resulting from the cancellation between divergences stemming from the YM field
and dilaton respectively. In fact, magnetically charged solutions to the coupled Einstein-
Maxwell equations are typically found to be unstable [6,7,8], with or without the presence
of a dilaton.
The ansatz of (4) represents an infinite-action solution of YM + scalar field theory in
3+1 dimensions which satisfies the Bogomol’nyi bound Gaij = ǫijkDkΦ
a. Since the action is
finite away from the singularity, the Bogomol’nyi bound guarentees stability of the solution
outside arbitrary finite enclosures around each source. Since the string solution has finite
action owing to the cancellation between gauge and gravitational divergences, we can use
a Bogomol’nyi bound to demonstrate stability everywhere. We see this by noting that
the ADM mass of the multimonopole solution written in terms of the canonical metric
saturates a Bogomol’nyi bound in terms of the charges of the string monopoles[9]
M =
2π
κ2
eφ0/2
N∑
i=1
mi. (5)
Thus the string multimonopole solution sits at a minimum energy point and is automati-
cally stable against perturbations independent of x4 (presumably if we allow perturbations
depending on the compactified direction the string monopoles will decay into heterotic in-
stanton fivebranes [10–14], which possess a higher symmetry1). As an exercise, one can
1 This was pointed out to me by Jianxin Lu.
2
explicitly show that there are no unstable modes of the string monopole away from the sin-
gularity, from either the gauge or gravitational sectors. The same divergence cancellation
feature which leads to a finite action solution then allows us to circumvent the singularity
in each sector when considering the stability of the string solution.
Since the divergence cancellation in the action is also possessed by the unstable so-
lution in [5], however, this property is clearly insufficient in itself to explain the stability
of the string solution, which is rooted in the (4,4) superconformal invariance of its un-
derlying sigma-model [13,14]. It is also linked to the existence of a zero-dynamical force
condition obeyed by this and other classes of heterotic multi-soliton solutions [15–19], in
addition to the usual zero-static force condition. Since the string solutions all saturate
Bogomol’nyi bounds and are hence stable, their existence in suitably compactified form
provides a possible test for string theory as distinct from other gauge + gravity models.
A study of the low-energy dynamics of the string monopoles was done in [20], where it
was found from both a test-monopole approach and a computation of the Manton metric
on moduli space that the monopoles scatter trivially in the low-energy limit. The latter
method relied on the construction of anO(β) solution to the constraint equations of motion,
which had the piecewise Lorentz boosted form
e2φ(~x,t) = 1 +
N∑
n=1
mn
|~x− ~an(t)|
,
g00 = −1, g
00 = −1, gij = e
2φδij , g
ij = e−2φδij ,
g0i = −
N∑
n=1
mn~βn · xˆi
|~x− ~an(t)|
, g0i = e−2φg0i,
Hijk = ǫijkm∂me
2φ,
H0ij = ǫijkm∂mg0k = ǫijkm∂k
N∑
n=1
mn~βn · xˆm
|~x− ~an(t)|
,
(6)
where i, j, k,m = 1, 2, 3, 4 and we use a flat space ǫ-tensor. Note that g00, gij and Hijk
are unaffected to order β. According to the conjecture of Bergshoeff and de Roo[21,22,1],
from the exactness condition Aµ = Ω±µ[23,24] (where Ω±µ is the generalized connection
defined in [1]), the higher order in α′ terms drop out from the action. The contributions
of these terms also cancel in the equations of motion, provided we assume a similar O(β)
solution for the YM field Aµ to balance the piecewise-boosted generalized connection Ω±µ.
We therefore argue that our low-energy scattering analysis in [20] is also exact, suggesting
3
that the dynamic YM force for these string monopoles is precisely cancelled by the dynamic
gravity sector force to all orders in α′. This argument can equally well be applied to the
heterotic instanton fivebranes.
Acknowledgements
I would like to thank M. Duff, A. Felce, Z. Khviengia, J. Lu, V. P. Nair, M. Ortiz and
E. Weinberg for helpful discussions.
4
References
[1] R. R. Khuri, Phys. Lett. B294 (1992) 325.
[2] R. R. Khuri, Nucl. Phys. B387 (1992) 315.
[3] D. J. Gross and M. J. Perry, Nucl. Phys. B226 (1983) 29.
[4] R. D. Sorkin, Phys. Rev. Lett 51 (1983) 87.
[5] P. Bizon, “Saddle Point Solutions in Yang-Mills-Dilaton theory” UWThPh-1992-46,
September 1992.
[6] R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141.
[7] K. Lee, V. P. Nair and E. J. Weinberg, Phys. Rev. Lett. 68 (1992) 1100.
[8] G. Lavrelashvili and D. Maison, “Static Spherically Symmetric Solutions of a Yang-
Mills Field Coupled to a Dilaton”, MPI-Ph/92-62, August 1992.
[9] J. P. Gauntlett, J. A. Harvey and J. T. Liu, “Magnetic Monopoles in String Theory”
IFP-434-UNC, November 1992.
[10] M. J. Duff and J. X. Lu, Nucl. Phys. B354 (1991) 141.
[11] M. J. Duff and J. X. Lu, Nucl. Phys. B354 (1991) 129.
[12] A. Strominger, Nucl. Phys. B343 (1990) 167.
[13] C. G. Callan, J. A. Harvey and A. Strominger, Nucl. Phys. B359 (1991) 611.
[14] C. G. Callan, J. A. Harvey and A. Strominger, Nucl. Phys. B367 (1991) 60.
[15] A. Dabholkar, G. Gibbons, J. A. Harvey and F. Ruiz Ruiz, Nucl. Phys. B340 (1990)
33.
[16] C. G. Callan and R. R. Khuri, Phys. Lett. B261 (1991) 363.
[17] A. G. Felce and T. M. Samols, “Low-Energy Dynamics of String Solitons” NSF-ITP-
92-155, November 1992.
[18] R. R. Khuri, “Geodesic Scattering of Solitonic Strings”, Texas A&M preprint
CTP/TAMU-79/92.
[19] R. R. Khuri, “Classical Scattering of Macroscopic Strings”, Texas A&M preprint
CTP/TAMU-80/92.
[20] R. R. Khuri, Phys. Lett. B294 (1992) 331.
[21] E. A. Bergshoeff and M. de Roo, Nucl. Phys. B328 (1989) 439.
[22] E. A. Bergshoeff and M. de Roo, Phys. Lett. B218 (1989) 210.
[23] J. M. Charap and M. J. Duff, Phys. Lett. B69 (1977) 445.
[24] M. Dine, Lectures delivered at TASI 1988, Brown University (1988) 653.
5


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A Comment on the Stability of String Monopoles

A Comment on the Stability of String Monopoles

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