A Stable Non-BPS Configuration From Intersecting Branes and Antibranes
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hep-th/0003219TIFR/TH/00-12A Stable Non-BPS Con?guration From Intersecting Branes and Antibranes Sunil Mukhi and Nemani V.Suryanarayana Tata Institute of Fundamental Research,Homi Bhabha Rd,Mumbai 400005,India ABSTRACT We describe a tachyon-free stable non-BPS brane con?guration in type IIA string theory.The con?guration is an elliptic model involving rotated NS5branes,D4branes and anti-D4branes,and is dual to a fractional brane-antibrane pair placed at a conifold singularity.This con?guration exhibits an interesting behaviour as we vary the radius of
the compact direction.Below a critical radius the D4and anti-D4branes are aligned,but as the radius increases above the critical value the potential between them develops a minimum away from zero.This signals a phase transition to a con?guration with ?nitely separated branes.
March 2000
Introduction and Review
Much has been learned in recent times about the physics of brane-antibrane pairs and non-BPS branes in superstring theory[1-32].Parallel,in?nitely extended pairs attract each other,and can annihilate into the vacuum by a process of tachyon condensation into a constant minimum.An analogous decay process takes place for single or multiple non-BPS branes.Condensation of the tachyon as a kink,vortex or more general soliton is associated to brane-antibrane annihilation into branes of lower dimension.
The above considerations have been extended to backgrounds with lower supersym-metry(orientifolds,orbifolds and smooth Calabi-Yau manifolds),where one?nds new phenomena including the existence of stable,non-BPS branes.As parameters of the back-ground are varied,there can also be phase transitions between qualitatively di?erent con-?gurations.The reader may consult the reviews in Refs.[33-36].
A di?erent direction,explored in Ref.[37],is to consider non-BPS con?gurations of intersecting branes and antibranes in the fully supersymmetric type II spacetime back-ground.Here one encounters novel phenomena including both attractive and repulsive interactions among branes and antibranes.Such con?gurations could be useful to study non-supersymmetric?eld theories,and also to understand better the basic underlying structure of superstring theory.
In the present note we examine a variant of a con?guration of“adjacent brane-antibrane pairs”that was discussed in Ref.[37].Let us describe the original con?gura-tion.In type IIA theory we start with a pair of parallel NS5-branes extended along x1,x2,x3,x4,x5and separated along x6.The x6direction is compact,with circumference 2L.Now stretch a D4-brane along x6from the?rst NS5-brane to the second,and a
D4between parallel branes.
In Ref.[37]it was argued that the D4-brane and
D4end on the NS5brane from opposite sides,and their ends are charged 3-branes in the NS5world volume.If we dimensionally reduce everything over these three directions,then the ends become vortices living on the reduced NS5world volume.These vortices carry the same charge under the gauge?eld,hence they repel,and since the con?guration is non-supersymmetric there is no reason to expect that this repulsion is cancelled by exchange of other massless?elds.Because the NS5-branes are parallel,the repelling D4-and
D4-branes are actually two types of fractional branes(denoted1f and
D4in the other,so that the4-branes extend along x1,x2,x3and x6(Fig.2).This con?guration breaks all the supersymmetries of Type IIA,though each of the D4and
NS5NS5
NS5’D4D4
__Fig.2:Adjacent D4and
D4-branes repel.In the case of parallel NS5-branes,
the repulsion was identi?ed as coming from like charges carried by the ends of the 4-branes.In this picture the repulsive e?ect is localised on each NS5-brane separately,hence introducing a relative rotation should not matter.Moreover,from the string theory point of view,the repulsion between the D4and
D4across any one NS5brane.This again
suggests that the force is localized near one NS5brane at a time https://www.sodocs.net/doc/074790177.html,ing these arguments,we conclude that the D4-and
D4-branes when the NS5-branes are not
rotated with respect to each other.
With rotated NS5-branes,the important di?erence is that the the D4-branes no longer have moduli to move away from each other.As they move with their ends on the NS5-branes,the D4-branes get stretched.In the process their e?ective 3-brane tension increases,providing a restoring force for the repelling ends of the adjacent D4?
D4pair exactly cancel,giving rise to a con?guration
that is stable at least under small perturbations.
In fact,as we now show explicitly,such a stable con?guration exists for some range of values of the circumference R 6.For simplicity,let us assume that the NS5and NS5’-branes are located at diametrically opposite points on the compact x 6direction,with the
separation between them being L =1
D4-brane will also be displaced by an equal amount r ,and the
displacement of the other ends of the 4-branes along x 8(or x 9)will also be r
(Fig.3).
NS5NS5
D4D4
__r
NS5’Fig.3:Equilibrium con?guration after displacement of 4-branes.
With the above data,the net tension of the stretched D4(
L 2+2r 2where
V is the (in?nite)3-volume of the (x 1,x 2,x 3)directions and T 4=1D4-branes on an NS5-brane to the energy of the system is given by[37]:
V t 3
e ?2X 2t
D4along the NS5brane,and q =exp(?πt ).The
function F (q )is given by
F (q )=
f 4(q )8f 2(q )4f 4(q )4 (2)where the f i (q )are de?ned as:
f 1(q )=q 12q
1
24∞ n =1(1+q 2n ?1)f 4(q )=q ?1
Dropping the common factor V ,the total potential energy of the system of branes in Fig.3is
V (r )=1L 2+2r 2?1
t 3e ?8r 2t
16(2π)4
∞0
dt π(?16t 2),r ?1(6)(we are measuring distances in units of √(2π)4 ?γ?log(8r 2?π)n
(2π)4.From this the contribution of the second
term in
Eq.(4)to the force between the two D 4-branes,given by ?dV
(2)r ,r ?1(9)
Thus in the large-separation limit this contribution to the force between two 4-brane segments is repulsive,as expected.
Now let us look at the behaviour of this contribution for small values of r.In this limit we can expand the the exponential in Eqn.(4)in powers of r to get
V(2)(r)=C?Dr2,r?1(10)
where
C=1
t3F(q)
D=?1
t2F(q)
(11)
Notice that C is a divergent integral whereas D is convergent.D is also positive because F(q)is negative all through the range of integration.From Eqn.(10)the small-r behaviour
of the force turns out to be
F(2)(r)=2Dr,r?1(12) which is also repulsive,as expected.From Eqn.(4),the restoring force is:
F(1)(r)=?dV(1)
g s(2π)4
2r
2L2+(2r)2
(13)
which is attractive as explained above.The strength of attraction depends on the value of L,related to the size of the compact x6direction.
We want to know whether there is a stable minimum of the total potential,and under what conditions this minimum is attained away from r=0.In order to argue for the presence of a stable minimum at nonzero separation of the brane-antibrane pair,it is su?cient to show that the potential has an unstable turning point at the https://www.sodocs.net/doc/074790177.html,bined with the attractive behaviour for large r,this su?ces to show that the potential develops a stable minimum somewhere in between.
From Eqns.(4)and(10),we have for r?1,
V(r)?1
L?Dr2(14)
upto additive constants.Here,D is the positive constant given in Eqn.(11).It follows that V has a turning point at the origin that is unstable(tachyonic)when L is greater than a critical value L c,namely:
L>L c=
1
The function F(q)de?ned in Eqn.(2)tends to the constant value?8as t→∞.Hence an estimate for D can be made by approximating F in the integrand in Eqn.(11)by?16t2
for0 2and?8for1 2 (2π)4D～ 16 √ 2π～3.60(16) so the phase transition takes place at L c～0.28g?1s. We expect that there will be no loop corrections to the restoring potential V(1),as this depends only on the D-brane tension which is unrenormalized.The repulsive potential V(2)will,on the other hand,receive loop corrections,but they are independent of L,and can be expected to be small for su?ciently small g s.Hence we do not expect stringy corrections to invalidate the conclusions of this section. Some further analysis of this potential can be found in the Appendix. The T-dual Con?guration It has been argued that the elliptic con?guration involving rotated NS5-branes is T-dual to the conifold geometry[38,39].Above we have studied an adjacent D4? D3pair at a conifold,obtained by T-duality along the compact x6direction.Such a con?guration should describe a stable non-BPS system exhibiting a phase transition. As discussed above in the introduction,there is a simpler situation where the analogous T-duality relation holds:the elliptic model of two parallel NS5-branes[41],which is T-dual[42]to a Z2ALE geometry.An adjacent D4? locus inside an NS5-brane.On the other hand,in the T-dual conifold geometry we know[43] that D3-branes completely smoothen out the conifold singularity:the near-horizon geome-try becomes AdS5×T1,1.Thus in the latter picture the back reaction of the branes on the geometry is qualitatively very important.This can be traced to the fact that the branes completely?ll the space transverse to the singularity. The stable non-BPS con?guration discussed in the previous section is an example of this type.While it can be visualised explicitly in the brane-construction picture,it is not so easy to describe in terms of branes at a conifold.For very weak string coupling(and ?xed L)the problem is somewhat easier,since in this case the D4? D4separates,as discussed above.In this case the T-dual con?guration is harder to visualise.The fractional pair cannot separate in any direction transverse to the conifold,so it must be thought of as separating within the conifold directions. Asymptotically this should look like a D3? D4branes in type IIA string theory,suspended between relatively rotated NS5-branes,which corresponds to a stable non-BPS state.A crucial assumption was that the force between adjacent brane-antibrane pairs can be estimated using a“locality”property,according to which it originates from the repulsion between the ends of these4-branes in the NS5-brane worldvolumes.This repulsion can in turn be computed using standard orbifold techniques,valid for the model with parallel NS5-branes which is dual to a Z2ALE singularity. While we do not know at present how to estimate the validity of this assumption, it is encouraging that it gives a de?nite and physically reasonable answer.As we have indicated in the previous section,a supergravity calculation might be one route to provide an independent check of our conclusions. The3+1-dimensional?eld theory on the common worldvolume in our brane con-struction will be a non-supersymmetric,tachyon-free theory.Because the model is elliptic, it should?ow to a CFT.One can generalise the model to include N1D4-branes in one segment and N2 D4-branes. Above the critical radius,our model provides a situation where a brane and an an-tibrane are at a?nite separation that is calculable in terms of various parameters including the string coupling and the radius of a compact direction.Such con?gurations might per-haps be useful to construct novel“brane-world”type models. Acknowledgements: We would like to thank Atish Dabholkar and Sandip Trivedi for helpful discussions. We are particularly grateful to Sandip Trivedi for a careful reading of the manuscript. Appendix The numerical value of L c below Eqn.(16)is only approximate,since we have made a crude estimate for the integral in Eqn.(11).An improvement on this estimate can be made by taking F(q)～?16t2+16t4,t small F(q)～?8+45q,t large (17) and then?nding an intermediate value of t at which these two functions match.We?nd that t is shifted from1 2 to～0.76,and the value of(2π)4D decreases from3.60to3.02. As a result,L c moves up to about0.33g?1s,an increase of18%.This suggests that at least the order of magnitude of L c has been correctly estimated. One may wonder if the potential has a unique minimum away from0for L>L c or whether there are several minima,some of them metastable.For this,it is convenient to make the same approximation above,but not just for the r?1behaviour.We take the term V(2)(r)in Eqn.(4)and write it as follows: V(2)(r)=? 1 t3 e?8r2t 16(2π)4 120dtπ(?16t2)?1√t3e?8r2t 4 π r,then we?nd: V(2)～12(1?y2)e?y2+(1?y42?γ (19) Here,Ei is the exponential-integral function andγis the Euler constant.We have dropped an in?nite constant associated to the logarithmic term in the potential,and subtracted a ?nite constant?1 g s(2π)4 π4g s(2π)4 4 π L,and again a constant has been subtracted to make the function vanish at the origin.It is now straightforward to plot V(1)(y)+V(2)(y)for di?erent values of L (Figs.4,5).In these plots we have set g s =0.1.For this value of g s ,the phase transition is at L c ～2.65.We see that,at least for the L values in the plots,there seems to be a unique an d nonzero minimum when L > L c .0.050.10.150.2y 0.001 0.002 0.003 0.004 16(pi)4V 0.050.10.150.2 y -0.0002-0.00010.00010.000216(pi)4V Fig.4:Potential for L =2.6,2.7.0.51 1.52 2.53 y -0.50.5 116(pi)4V 50100150200250300 y 102030 16(pi)4V Fig.5:Potential for L =10,10000. 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