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Brane embeddings in sphere submanifolds

Nikos Karaiskos, Konstadinos Sfetsos and Efstratios Tsatis

Department of Engineering Sciences, University of Patras

26110 Patras, Greece

{nkaraiskos, sfetsos, etsatis

}@upatras.gr

Synopsis

Wrapping a D(8-p)-brane on AdS2 times a submanifold of S8p introduces point-like

defects in the context of AdS/CFT correspondence for a Dp-brane background. We

classify and work out the details in all possible cases with a single embedding angular

coordinate. Brane embeddings of the temperature and beta-deformed near horizon D3-

brane backgrounds are also examined. We comment on the relevance of our results to

holographic lattices and dimers.

arXiv:1106.1

200v1

[hep-th]6

Jun2011

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1 Prolegomena

When branes wrap internal manifolds have the tendency to shrink but they can be

stabilized by turning on the worldvolume gauge field with quantized flux. A pioneering

example of such flux stabilization of D-branes was presented in [1] for the wrapping

of a probe D2-brane inside an SU(2) group manifold and a connection with results

from an exact CFT approach was also made. There have been numerous works in the

literature considering brane embeddings in various backgrounds and dimensions [2]-[7]

(and references therein). In particular, the authors of [4] considered configurations where

a D(8-p)-background wraps an S7p inside an S8p in the background of a Dp-brane. The

stabilization occurs at quantized values of the equatorial angle of the bigger sphere. This

result, besides being aesthetically beautiful, is also relevant in a holographic approach to

condensed matter lattices and dimer systems [8].

Motivated by these works and the potential physical applications that the brane embed-

dings have, we realized that, for generic values of p there are many more possibilities

even when one considers the simplest case of one embedding coordinate. Instead ofS7p,

one could also select other submanifolds ofS8p that the D-brane could wrap. These are

essentially given by subgroups of SO(9p), which is the isometry group ofS8p. Thesesubmanifolds are presented in Table 1 for p = 0, 1, . . . , 5. The coloring is introduced for

later convenience.

Table 1: Submanifolds ofS8p

p = 0 S7, S3 S4, S2 S5, S1 S6p = 1 S6, S3 S3, S2 S4, S1 S5, CP3p = 2 S5, S2 S3, S1 S4p = 3 S4, S2 S2, S1 S3, CP2p = 4 S3, S1 S2p = 5 S2, S1 S1

In section 2 we minimize the action of the brane probe and calculate the semi-classical

energy for each one of the aforementioned configurations. In general the energy depends

on the ratio of the flux units n of the worldvolume gauge field to the number of the

Dp-branes N, that we stack together to form the background. For a given value of p

these energies depend on the specific submanifold that is wrapped. We have checked by

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using kappa symmetry that the embeddings in consideration are supersymmetric, hence

ensuring stability.

In section 3 we present brane embeddings in the deformed backgrounds of [9]. In this case

it turns out that the dependence of the deformation drops out completely in the probe

computation. Pertaining to the -deformation, which involves an S-duality, we formu-

lated the problem mathematically, but we were not able to find minimal configurations

explicitly due to its complexity.

In section 4, we turn on the temperature and examine its effect on the stability of our

constant embeddings. We conclude, by considering a small fluctuation analysis, that these

are perturbatively stable, even though there is no underlying supersymmetry. Finally, in

section 5 we present concluding remarks and comment on the possible use of our results

in the context of holographic lattices and dimers.

2 Brane embeddings in Ramond-Ramond backgrounds

The geometry created by a stack of N coincident Dp-branes in the near-horizon region

is described by the ten-dimensional metric [10]

ds2 = r

R

7p2

(dt2 + dx2||) +

R

r

7p2

(dr2 + r2d28p) , (1)

where d

2

p is generally the line element of a unit p-sphere and the parameter R is givenby

R7p = N gs 25p 5p2 ()

7p2 (7p

2) . (2)

The background is also supported by a dilaton, (r) and a non-zero RamondRamond

(RR) field strength F(8p) given by

e(r) =

R

r

(7p)(p3)4

,

F(8p) = (7p)R7p Vol(S(8p)) = dC(7p) , (3)where Vol(S(8p)) denotes the volume form of the unit p-sphere and C(7p) is the RR

potential. We split the (8 p) spherical coordinates as (, 1, . . . , 7p) and let be anembedding coordinate of the probe brane, along with x.

We concentrate first to the cases corresponding to the entries of the Table 1 that involve

solely spheres. For these cases the metric of the compact space will have the form

d28p = d2 + cos2 d2q + sin

2 d27pq , q = 0, 1, . . . ,

7p2

. (4)

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This parametrization of the metric corresponds to the split of the 9 p representationof the symmetry group SO(9 p) under the subgroup SO(q + 1) SO(8 p q) as(9 p) (q + 1, 1) (1, 8 p q). The variable [0,

2], unless q = 0 in which

case [0, ]. The restriction on the range that q takes independent values is due to thesymmetry under the interchange of cos and sin in the above metric. Consequently,

the RR potential will be written as

C(7p) = R7p f() (7p) , (5)

where

(7p) = dVol(Sq) dVol(S7pq) =

hd1 d2 . . . d 7p . (6)

The corresponding volume is

VM

= M d7p

h =4

9p2

1+q2

8pq2

. (7)For q = 0 we should divide this formula by two since the general expression for Vol(Sq)

gives 2 for q = 0. The function f() is given for q = 0 by

f() =7p

8p q (sin )8pq

2F1

1 q

2, 4 p + q

2, 5 p + q

2, sin2

(8)

and for q = 0 by

f() = 28p7p8

p sin

2

8p2F1

p

2 3, 4 p

2, 5 p

2, sin2

2 . (9)The difference has its origins in the different ranges of the angular variable for the two

cases that we mentioned above. Note also that this is not the most general form for the

RR potential, but it is the only consistent one for the particular embedding that we will

consider, as will become transparent below.

The D(8-p)-brane probe is described by the sum of a Dirac-Born-Infeld (DBI) and a

Wess-Zumino (WZ) term

S =

T8p d9pedet(g + F) + T8p C(7p) F , (10)

where g is the induced metric on the brane, F is an abelian gauge field strength living

on the world-volume of the brane and

T8p = (2)p8 ()p92 (gs)

1 , (11)

is the tension of the probe brane. The integration is performed over the world-volume

coordinates of the brane which are taken to be = (t,r,1, . . . , 7p). In general,

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the embedding coordinate may depend on any world-volume coordinate. Here, we shall

restrict to the case where depends only on the radial coordinate, which is also consistent

with the form of the RR potential (5). Since the WZ term acts as a source term for the

abelian gauge field strength F, the latter one is constrained to be F = Ftr dt dr. Wealso set the spacelike worldvolume coordinates x

||to constants which is consistent with

their equations of motion.

With the above conditions and assumptions the probe brane action assumes in general

the form

S =

M

d7p

dtdrL(, F) , (12)

where the Lagrangian density is computed to be

L(, F) = T8pR7p

h

f()7

p

1 F2tr + r22 f()Ftr

. (13)

By varying the Lagrangian density with respect to the worldvolume gauge potentials, At

and Ar, one observes that the following quantity is constant

LFtr

= const . (14)

Then in turns out that the gauge field assumes the form

Ftr =f()

(7p)2(f())2 + (f())2

. (15)

In order to attribute physical meaning to this constant, we consider the coupling of our

system to fundamental F-strings [4]. This is achieved by replacing F with FB, whereB is the KalbRamond field. Expanding the action, we pick out a term of the form

Md7p

dtdr

L

FtrBtr . (16)

We can interpret the coefficient in front of Btr as a charge (n units of Tf) that multi-

plies the Kalb-Ramond potential of the fundamental string. Therefore, the fundamental

strings feel a potential in this background, whose strength is proportional to theirnumber, n, and their tension Tf = 1/(2

). Consequently, one writesM

d7pL

Ftr= nTf , n Z . (17)

In order to find semiclassical minima of the embeddings, solving the equations of motion

arising from the Lagrangian density would suffice. However, since we are also interested

in computing the energies of our configurations, we will obtain the minima through

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the Hamiltonian procedure. By performing a Legendre transformation, which actually

removes the WZ part, the Hamiltonian of the system is given by

H =

M

d7p

dtdr

L

FtrFtr L

. (18)

Using the explicit form of the Lagrangian (13) and the quantization condition (17) the

Hamiltonian becomes

H = NTf

dtdr

1 + r22

f()7p

2+

1 f()2

, (19)

where we have defined

=n

N, =

7p2

8pq

2

1+q2

. (20)

Since the origin of the constant is VM, it turns out that, for reasons explained below (7),for q = 0 we should divide the above formula by two. It is obvious from the expression

for H that it is consistent to look for constant configurations, since the r-dependence

drops out in this case. Setting = 0 and requiring H/ = 0 gives the condition

f() = (7p)2(1 f()) . (21)

As one can see in Table 2 in some cases, depending on the specific values for p and q, this

equation admits an exact solution (), but in others it can only be solved numerically.

In the rest of the paper, in order to avoid a proliferation of symbols, we will denote by

the solution of (21). The energy density is defined by

H =

dr E. (22)

For general values of p and q it is given by

Ep,q = NTf

(cos )2q(sin )2(7pq) + (1 f())2 . (23)

For the case where a one-cycle is manifest, that is q = 1 the above formula as well as the

one for the minima have a much simpler form given by

sin =7p6p

15p

, Ep,1 =

2 + (sin )122p 2(sin )7p. (24)

Noting that f(0) = 0 and f(2 ) = 1, we find the limiting behaviours

Ep,q = nTf +O(2), Ep,q = (N n)Tf +O(1 )2 . (25)

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The results of our computations regarding the minima and the corresponding energies

are summarized in the Table 2 below. In all cases, the angle ranges from 0 to /2. We

have also included two more cases, apart from the products of spheres, which arise for

odd values ofp, by writing the S8p as a U(1) bundle over CP7p2 . We use the conventions

of [11] and [12] for the CP2 and CP3, respectively.1 The normalizations for the metrics

are such that R =16

p+1g (for the values p = 1 and p = 3 that are of interest to us).

Table 2: Minima and energies

Cycles Algebraic equations for minima Ep,q in units of NTfS3 S4 90sin + 25 sin 3 + 3 sin 5 = 112 E0,3

p = 0 S2 S5 450cos + 25 cos 3 27cos5 = 448(1 ) E0,2S1 S6 sin =

76

1/51

3536

76

2/5

S

3

S3

sin = 1/2

(1 )p = 1 S2 S4 18 + 8sin 2 + 5 sin 4 = 36 E1,2

S1 S5 sin = 651/4

1 2425651/2

CP3 as above as above

p = 2 S2 S3 cos = 21/3+(55+2750+252)2/3

22/3(55+2750+252)1/3 E2,2S1 S4 sin = 541/3

1 1532

252

21/3

S2 S2 = 2 1 sin p = 3 S1 S3 sin =

43

1/2

13227

CP2 as above as above

p = 4 S1 S2 sin = 32

1 27162p = 5 S1 S1 = 0 or = 2 singular solutions

We should clarify two subcases of the above table. First, the results for the CP2 and

the S1 S3 submanifolds coincide. This happens because the wrapping in the first caseinvolves the U(1) fiber with group structure S1 and a submanifold inside CP2, which

has a similar structure withS3

. The same happens with the results for theCP

3

andthe S1 S5 submanifolds. Second, for p = 5 we have the solutions = 0 and =

2

corresponding to a collapsing of the D-brane at the poles of the 3-sphere and are thus

singular. We also note that the respective equations that give the minima and energies

for the submanifolds S7p S8p can be found in [4], so that they are not reproduced1Our embedding coordinate (r) in these cases is identified with the coordinates and in equations

(5) and (4.1) in the references [11] and [12], respectively.

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here.

Having obtained the energies for the various values for p, we plot them together with

the submanifolds of [4] in the following five figures. The colors (black, blue, purple and

red) correspond to the entries with the same colors in Table 1. The energies are plotted

as functions of the ratio , in units of NTf. Curves with the same value for p, but a

different one for q, might intersect. We will use the obvious notation (q q, )

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

Figure 1: Submanifolds for p = 0. We have:

(12, 0.46), (13, 0.47) and (23, 0.48).

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.25

Figure 2: Submanifolds for p = 1. We have:

(12, 0.49), (13, 0.47) and (23, 0.51).

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.25

Figure 3: Submanifolds for p = 2. We have:

(12, 0.48).

0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.25

0.30

Figure 4: Submanifolds for p = 3. We have:

(12, 0.50).

0.2 0.4 0.6 0.8 1.0

0.1

0.2

0.3

Figure 5: Submanifolds for p = 4.

As is seen from the figures above, for a given value of p the maximally symmetric sub-

manifolds corresponding to q = 0 have the lowest energy. When the submanifold in

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consideration includes an S1 or a CP space the ratio cannot exceed6p7p found from

(24). At this maximum value, the corresponding value of the energy density is1

7p .

2.1 Kappa symmetry

We next check weather our embeddings preserve some portion of supersymmetry, by

examining the kappa symmetry. We will briefly present results for the p = 3 case, the

others follow in a similar manner. In order to have supersymmetric configurations of a

D-brane probe in a given background, the following condition [13]-[15]

= , (26)

should be satisfied. Here, is the Killing spinor of the background which for the maxi-

mally supersymmetric case of the AdS5S5 background, is unconstrained. We shall alsoput the two MajoranaWeyl spinors of type-IIB theory into a doublet of chiral spinors,

which transforms as a vector under SO(2)

=

1

2

. (27)

The kappa symmetry operator for the case of type-IIB theory is defined by

=1

det(g + F)

n=0

(1)n

2n

n!

m1n1mnnn

Fm1n1

F mnnn

(3)p32 +n(i2) (0) . (28)

Here g is the induced metric on the D-brane and

F= F P[B] , (29)

where F is the worldvolume gauge field and P[B] is the pullback of the KalbRamond

B field. The ms are the induced worldvolume gamma matrices defined as m = eAmA,

with m, A being curved and flat indices, respectively, while A are the ten-dimensionalflat Dirac matrices. Finally, (0) is defined by

(0) =1

(p + 1)!a11p+1a1ap+1 (30)

and i are the Pauli matrices which act in the usual way on the doublet of chiral spinors.

We next check the kappa symmetry for each one of the embeddings we considered in the

previous section. We consider first the kappa symmetry for the S2 S2 wrapping case.

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Note that Fhere is just the worldvolume field strength, since there is no KalbRamondfield in our background. Hence, the infinite sum in (28) has only two terms, since only

the Ftr components of the field strength have been turned on. We compute that

k =1

1 F2tr + r

22(1 P1P2 + iFtr2 P2) , (31)

where we have defined the commuting operators

P1 = 01 + r05 , P2 = 6789 . (32)

Consistency requires that 2 = I, which is easily proven. Breaking into components one

arrives at the following consistent algebraic system

(P1P2 + Ftr P2)2 =

1 F2tr + r221 ,

(P1P2 Ftr

P2)1 =

1 F

2

tr + r2

2

2 . (33)

We conclude that half of the components of the Killing spinor are related to the other half

and hence the configuration we considered here retains 1/2 of the original supersymmetry.

The cases of the S1 S3 and CP2 wrappings are similar to the one above and hence theresults will not be presented here. They are also found to be 1/2 BPS configurations and

similarly for other cases with p = 0, 1, 2, 4.

3 Brane embeddings in deformed backgrounds

In this section we consider -deformed solutions of type-IIB supergravity [9] and brane

embeddings inside these backgrounds. We begin with the -deformation of the AdS5S5background for which the AdS5 part of the metric remains the same, while the metric of

the -deformed 5-sphere is written as

d25, =3

i=1

(d2i +

G2id

2i ) +

GR4221

22

23(

i

di)2 , (34)

where

G1 = 1 + R42(2122 + 2223 + 2321) (35)and (1, 2, 3) (cos , sin cos , sin sin ). The NS sector of the background includesa dilaton and a KalbRamond two-form, given by

e2 = Ge20

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BNS = R4G(2122d1 d2 + cyclic) (36)

and the RR potential and field strengths

C2 = 4R4w1 (d1 + d2 + d3) , with dw1 = 12 cos sin3 sin2d d ,C4 = 4R

4(w4 +G

w1

d1

d2

d3) , Vol(AdS5) = dw4 ,

F5 = 4R4(Vol(AdS5) + G Vol(S5)) , Vol(S5) = dw1 d1 d2 d3 . (37)

We consider D5-brane embeddings in this deformed background. The brane will wrap

the four angles of the deformed sphere so that the world-volume coordinates will be

(t,r,,i) and the embedding coordinates are taken as x|| = const. and = (r). As

before, we also turn on an Abelian world-volume gauge field strength Ftr. The action

of the brane probe is given by a sum of a DBI and a WZ term. Some extra care is

needed since there are new terms arising from the induced KalbRamond field and the

RR potentials. The action assumes the generic form

S = T5D5

e

P[g] + F+ T5D5

p

Cp eF , (38)

where F is given in (29). After perfoming the computations, one concludes that theaction for the D5-brane becomes

S = T5R4

2 dd3 sin2 dtdr(cos sin

3 1 F2tr + r

22 Ftr sin4 ) . (39)

The entire -dependence has dropped out completely due to non-trivial cancelations in

the DBI and WZ terms, separately. In fact, this action is exactly the same as that

computed for the p = 3 and q = 1 case in which the D5-brane wraps the S1 S3submanifold of S5. Indeed, one may check that the above Lagrangian above falls in the

generic family (13) with f() = sin4 , which is the correct function appearing in the RR

potential for the aforementioned case.

The -deformed background hasN= 1 superconformal symmetry. Since for our embed-ding the action is actually -independent, we expect that the probe brane breaks one half

of it as it was shown before for the maximally supersymmetric cases. To demonstrate

this explicitly one has to work out (26) for our background using the fact that the corre-

sponding Killing spinor is no longer unconstrained, as in the maximally supersymmetric

case, but instead it is subject to two projections that reduce supersymmetry.

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3.1 Embeddings in the -deformed background

One may also consider a more general deformation of the background, by performing

an S-duality in the theory [9]. Apart from , the resulting background depends also on

which is an additional scaleless parameter. Searching for D5-brane embeddings, we

choose the embedding coordinates x|| = const. and = (). As opposed to the previouscases, here should depend on , since the latter enters in the computations in a non-

trivial way. Actually, as we shall explain later, this is related to the embedding that we

have chosen. The Hamiltonian of the system turns out to be

H = T5R4H

P2Q + (Psin2 f())2 , (40)

with

P =

1

2H sin2

sin2 , Q = H sin2

sin4

+H2

cos2

,f() =

2sin2 , H = 1 + 2R4(2122 + 2223 + 2321) (41)

and the is were defined in the previous section. As before, the parameter does

not appear at all, but does. It should be obvious that an attempt to find constant

minima, that is -independent, is inconsistent. Varying the Hamiltonian with respect

to gives a complicated nonlinear differential equation that one has to solve in order to

find configurations that minimize the energy. We were unable to find solutions of this

differential equation.

This increased level of complexity appears due to the particular embedding that we

considered. Had we chosen a similar embedding = () for the undeformed background,

that is the p = 3 case with manifest S1 S3,

Figure 6: Constant embedding Figure 7: Embeddings = ()

it would have also resulted to a similarly complicated differential equation. This happens

because the embedding that one chooses is supported by a corresponding patch on the

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sphere, to describe the forms locally. The two different embeddings in that case are

depicted in Figures 6 and 7 above.

In the case of non-constant embeddings = () it is obviously possible to rotate

the north pole in a way to obtain the first configuration. In practice, this is done by

performing an SO(6) transformation. In the -deformed case it is not obvious what the

corresponding transformation would be, given that the isometry group has been reduced.

The deformed sphere has the same Euler characteristic with the undeformed one, so that

their topology is the same. It makes sense, then, to assume that such a transformation

exists, although we were not able to find it.

4 Turning on temperature

It is natural to extend the discussion to asymptotically AdS spacetimes, which is relevant

to a holographic approach to dimers in condensed matter systems as pursued in [8].

We will briefly discuss the case of the near horizon black D3-brane in a way that the

submanifold S1 S3 ofS5 appears manifestly. The metric of the background is

ds2 = f(r)dt2 + dr2

f(r)+

r2

R2dx2|| + R

2(d2 + cos2 d21 + sin2 d23) , f(r) =

r4 4R2r2

(42)

and the RR potential changes also accordingly. We consider a D5-brane probe with

the same embedding coordinates as before, i.e. x|| = const. and = (r). It is a

straightforward task to show that the minima of the particular configuration remain the

same with the zero temperature case, a statement actually true for every p. This wouldnt

be the case for more general r-dependent solutions.

Supersymmetry is broken in all of these cases, due to the non-vanishing temperature.

Then, one cannot use kappa symmetry arguments to ensure the stability of the configu-

rations. Nonetheless, we can consider small fluctuations around the minima. Let

= + , Ftr = Ftr + , (43)

where the bars denote the minima and = tr rt. It should be stressed outthat this is consistent as long as one considers only the zero mode in an expansion in

harmonics of the S5. In order to find the complete spectrum, one should also turn

on fluctuations of the field strength in every possible direction (see also [16] for a prime

example). However, here we are only interested in demonstrating perturbative stability in

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for non-zero temperature and for that restricting to the zero-mode suffices. The effective

Lagrangian for quadratic fluctuations is computed to be

L = 12

hR4T5 cos sin3

11 F2tr

R2

f(r)1(t)2 f(r)(r)2

+2

1 F2tr+ A2 + B . (44)

The minima and the gauge field are given by

sin =

4

3 , Ftr =

9 1681 96 , (45)

where we have defined the constants

A = 4 +36

27 32 , B =

81 963 42 . (46)

We obtain the equations of motion by varying and . After combining them and

concentrating on a Fourier mode of the form = eit(r), we get

d

dr

f(r)

d

dr

+

2

f(r) C

2R2

(r) = 0 , C 24 + 72

32 27 , (47)

defined for r . Transforming this into a Schrodinger equation for , after changing

appropriately to a new variable z =r dr

f1(r), with z [0,) as r (, ]. Theassociated potential, that can be written explicitly only in terms or r, is

V = C2

f(r)R2

. (48)

Substituting the value for in C from (45) one sees that C is non-negative. Hence the

zero mode of the configuration is always positive. In fact, C vanishes for the critical

value = 3/4. In conclusion, the configuration that we considered is stable. Similar

arguments also hold for the other submanifolds and for the cases p = 0, 1, 2, 4 as well.

5 Concluding remarks

We classified and energetically compared all possible cases, with a single embedding

angular coordinate, in which a D(8-p) brane can wrap AdS2 times a submanifold ofS8p

in a Dp-brane background, thus producing a pointlike defect. We worked out the details

in all different cases that arise, performing also comparisons between them. We examined

similar constructions in the presence of temperature and in beta-deformed backgrounds.

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We demonstrated stability either by supersymmetry arguments or by a small fluctuation

analysis around the minima.

It will be interested to investigate and search for running solutions of the embedding

coordinate, i.e. = (r). This involves the classical equation of motion for the Hamil-

tonian (19). This is a highly non-linear equation but it should be possible to analyze it

numerically. Of particular interest will be solutions connecting minima corresponding to

different values of n, especially when they correspond to the same energy.

Our results should be relevant to models of holographic lattices and dimers. In par-

ticular, the position of minima for the case of D5-branes wrapped on AdS2 S4 in aD3-background plays an important role in determining the free energy of a square lattice

[8] made out of D5- and anti-D5-branes. In a condensed matter language, the dimeriza-

tion takes place at a lower than a critical value temperature where the configurations of

connected pairs of D5- and anti-D5-branes are favorable. We expect a richer structure

when we consider, besides S4, all submanifolds of S5 in Table 1. We hope to address

these and related issues in a forthcoming publication.

Acknowledgements

We would like to thank K. Siampos and D. Zoakos for useful discussions on the sub-

ject. N.K. acknowledges financial support provided by the Research Committee of the

University of Patras via a K. Karatheodori fellowship under contract number C. 915.

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