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Published by Institute of Physics Publishing for SISSA

Received: May 23, 2007

Accepted: August 8, 2007 Published: August 20, 2007 Geometry of non-supersymmetric three-charge bound states

Berkeley Center for Theoretical Physics, 366 Le Conte Hall #7300, Berkeley, CA 94720-7300, U.S.A.

E-mail: eggimon@lbl.gov

Center for Cosmology and Particle Physics, New York University, 4 Washington Place, New York, NY 10003, U.S.A.

E-mail: levi@physics.nyu.edu

Centre for Particle Theory, Department of Mathematical Sciences University of Durham, South Road, Durham DH1 3LE, U.K.

E-mail: S.F.Ross@durham.ac.uk
Abstract: We study the smooth non-supersymmetric three-charge microstates of Jejjala,
Madden, Ross and Titchener [

3.1 Adapted coordinates 3.2 Alternative coordinates 3.3 Relation to the supersymmetric case 4. Structure of the solution

4.1 Orbifold singularities 5. Reduction to four dimensions

The study of black holes and their thermodynamics has led to a number of important
advances in string theory. Recently, the construction of smooth geometries corresponding to
individual microstates of the black holes (more precisely, in a dual field theory description,
to individual pure states which contribute to the thermal ensemble) has received extensive
attention. Supersymmetric two-charge microstates are now reasonably well-understood [

In the three-charge case, an important step was the rewriting [

? 1 ? J H E P 0 8 ( 2 0 0 7 ) 0 5 5

We would like to apply the lessons learnt from this analysis to the non-supersymmetric
solutions of [

Implementing this program leads to some surprises: first, in the five-dimensional solution, the two centers are locally described by non-supersymmetric orbifolds. We had expected to find an orbifold singularity at these points in the five dimensional description, but it is a surprise that the supersymmetry is broken even locally: in the six-dimensional description, the full asymptotically flat solution is non-supersymmetric, but it has (for suitable choices of parameters) a near-core AdS3 × S3 geometry, in which supersymmetry is restored. This supersymmetry is broken by the choice of Kaluza-Klein reduction: even if we consider the Kaluza-Klein reduction from the exact AdS3 × S3 geometry, the fivedimensional solution contains non-supersymmetric orbifold singularities. This leads to the first general lesson we wish to draw: the picture of microstates as made up of half-BPS ?atoms? does not extend to the non-supersymmetric case. We must consider more general kinds of basic building blocks.

If we reduce to four dimensions, we find our second surprise: the four-dimensional solution is not smooth away from the centers. There is a conical singularity which connects the two centers. The ambiguity in the definition of the Kaluza-Klein gauge field on this conical line singularity makes it impossible for us to unambiguously associate brane charges with the two centers. This suggests that unlike in the supersymmetric case, we cannot treat the two centers independently, and the physical interpretation of these centers in the fourdimensional description is much less clear here.

One of the ultimate goals of this work was to make progress towards constructing
more general non-supersymmetric solutions. In general, constructing non-supersymmetric
solutions is much more difficult than supersymmetric ones. In the non-supersymmetric case
we have to deal directly with the non-linear, second order equations of motion, which makes
it impossible to construct solutions in terms of harmonic functions, as can be done in the
supersymmetric case. The results we have obtained show that the non-supersymmetric case
is quite different from the supersymmetric one, and suggest that it will be difficult to extend
the work on supersymmetric cases to construct multi-center non-supersymmetric solutions.
However, it may still be possible at least to construct non-supersymmetric solutions which
are asymptotically flat in four dimensions; work on this problem is ongoing [

The plan of the rest of the paper is as follows: in the next section, we review the
non-supersymmetric solitons of [

We start from the metric for the D1-D5-P black string, written in the fibered form in [

n?(f ? M ) £dt?? (f ? M )?1M cosh ?1 cosh ?5(a1 cos2 ?¯d? + a2 sin2 ?¯d?)¤2 M cos2 ?¯ ? H1

M sin2 ?¯ + ?

H1 + f £dy? + f ?1M sinh ?1 sinh ?5(a2 cos2 ?¯d? + a1 sin2 ?¯d?)¤2o + qH?1H?5 ½ (r¯2 + a21)(rr¯¯22d+r¯2a22) ? M r¯2 + d?¯2 + (f (f ? M ))?1 £¡f (f ? M ) + f a22 sin2 ?¯ ? (f ? M )a12 sin2 ?¯¢ sin2 ?¯d?2 + 2M a1a2 sin2 ?¯cos2 ?¯d?d? + ¡f (f ? M ) + f a12 cos2 ?¯ ? (f ? M )a22 cos2 ?¯¢ cos2 ?¯d?2¤ ¾ + s H??1 X4 dzi2,

H5 i=1 where t? = t cosh ?p ? y sinh ?p, y? = y cosh ?p ? t sinh ?p,

H?i = f + M sinh2 ?i, f = r¯2 + a12 sin2 ?¯ + a22 cos2 ?¯.

?

H1 e2? = ? .

H5 The 2-form gauge potential which supports this configuration is This metric is more usually written in terms of functions Hi = H?i/f . Writing it in this way instead makes it clear that there is no singularity at f = 0. The above metric is in the string frame, and the dilaton is

C2 = [(a2c1s5cp ? a1s1c5sp)dt + (a1s1c5cp ? a2c1s5sp)dy] ? d? (2.1) (2.2) (2.3) (2.4) [(a1c1s5cp ? a2s1c5sp)dt + (a2s1c5cp ? a1c1s5sp)dy] ? d? ? MH?s11c1 dt ? dy ? MH?s51c5 (r¯2 + a22 + M s12) cos2 ?¯d? ? d?, where ci = cosh ?i, si = sinh ?i. We take the T 4 in the zi directions to have volume V , and the y circle to have radius R, that is y ? y + 2?R.

This metric and fields describes a family of solutions with D1, D5 and P charges,
labelled by the seven parameters M, a1, a2, ?1, ?2, ?3, R. It was shown in [

? 1, j + j?1 s + s?1 = m ? n, j ? j?1 s ? s?1 = m + n, Which gives

s2 = m2 + n2 ? 1 ? p(m22?mnn2)2 ? 2(m2 + n2) + 1 . (2.9) Note that this constraint is invariant under the permutation of the three charges. We can rewrite the mass (2.5) as

M = a1a2(s2 ? j2)(j?2s?2 ? 1) = a1a2nm(s?2 ? s2)2,
(2.10)
so having solved for s in terms of m, n, we can use this to write a1a2 in terms of M . The
only remaining step is to solve for M in terms of m, n and the charges. This was left
entirely implicit in [

An unsatisfactory feature of this story is the highly implicit nature of the above conditions. We will see below that we can write the metric explicitly in terms of the integer quantised parameters m, n and a single length scale, which is related to M and hence determined in terms of the charges qi. This will bring the analysis much closer to the supersymmetric cases. It is worth pointing out at this stage that it would equally have been possible to rewrite it in this way in the type IIB form above; this rewriting is independent of the transformation to the M-theory form.

In what follows, we will usually assume m, n are coprime. If we do not, there will be additional orbifold singularities in the M-theory form of the solution.

The general study of solutions in the supersymmetric case is conducted in an M-theory picture, where the symmetry between the three charges is manifest. Our first task is ? 4 ? (2.5) (2.6) (2.7) (2.8)

J H E P 0 8 ( 2 0 0 7 ) 0 5 5 therefore to translate the type IIB form in which the non-supersymmetric solutions were first obtained to an M-theory form.

To pass to the M-theory form, we T-dualise on z3, z4 to get to a D3-D3-P solution, and then T-dualise on y to get to D2-D2-F1 in IIA. The IIA solution is then uplifted to M-theory. There is a symmetry between the charges in the M-theory picture, so we rename the charges to Q1 = q1, Q2 = q5, Q3 = qp. For the general family of rotating black string solutions (2.1), the resulting solution is

ds121 = ?(H?1H?2H?3)?2/3f (f ? M )(dt + k)2 + (H?1H?2H?3)1/3ds¯42 + ds2T 6 , where and the four-dimensional base metric is g(r¯) = (r¯2 + a12)(r¯2 + a22) ? M r¯2 ? (r¯2 ? r+2)(r¯2 ? r?2), f = r¯2 + a12 sin2 ?¯+ a22 cos2 ?¯. (3.4) Note that the coordinate r¯ lives on the interval [r+, ?); this fact will become clearer in some of the later coordinate systems. The metric is asymptotically flat in five dimensions. The coordinates ?¯, ?, ? are conventional coordinates on the S3 at large r¯. The metric on the T 6, which plays little part in our discussions, is ds2T 6 = (H?1H?2H?3)1/3(H?1?1(dz12 + dz22) + H?2?1(dz32 + dz42) + H?3?1(dz52 + dz62)). (3.1) (3.2) (3.5) (3.6)

J H E P 0 8 ( 2 0 0 7 ) 0 5 5 The gauge field in eleven dimensions is

C(3) = A1 ? dz1 ? dz2 + A2 ? dz3 ? dz4 + A3 ? dz5 ? dz6, where

Ai = ? H?1i [Qidt+M cj cksi(a1 cos2 ?¯d?+a2 sin2 ?¯d?)?M sj skci(a2 cos2 ?¯d?+a1 sin2 ?¯d?)],

The foregoing discussion applies to the general family of solutions (2.1); we would like to specialise to the smooth soliton solutions, and eliminate the dependence on the parameters to rewrite this solution in a way which makes the dependence on the integer parameters m, n manifest. This will be helpful in understanding the relation to the supersymmetric cases, and will also make the structure of our non-supersymmetric solution clearer. In this subsection, we introduce a new coordinate system adapted to this solution. In the next, we will consider coordinates which are as close as possible to the coordinates used in the supersymmetric cases.

We set

?2 = rr¯+22 ?? rr+2?2 , x = cos 2?¯.

The range of the coordinates is ? ? 0, ?1 ? x ? 1. In these coordinates, and where Thus, r¯2dr¯2 d?2 g(r¯) = ?2 + 1 . c ?

This now involves only the integer parameters m, n and the single length scale c. We can continue to calculate the full metric, and we find that we can in a similar fashion eliminate all the dependence on the parameters in favour of m, n and the single length scale c by fully exploiting (2.5)?(2.6). Some useful relations which follow from (2.5)?(2.6) are

M a1a2 = 16c2nm, a12 ? a22 = 4c(m2 ? n2), 1 M (a12 + a22) = 8c2[(m2 ? n2)2 ? (m2 + n2)], 2 1 M 2 = 4c2[(m2 ? n2)2 ? 2(m2 + n2) + 1].

4 Omitting the details of the calculation, we find k = ?4cQV1?Qm2Qn3 ©[2?2 ? (m2 ? n2)(1 + x)]n(1 ? x)d? ? [2?2 + (m2 ? n2)(1 ? x)]m(1 + x)d?ª , (3.15) ? 6 ?

(3.9) (3.10) (3.11) (3.12) (3.13) (3.14)

J H E P 0 8 ( 2 0 0 7 ) 0 5 5 and the base metric is

This fibered form is useful for comparison to the base metric in the supersymmetric case, which is conventionally written in a similar fibered form, and also for analysing the dimensional reduction to four dimensions. The coordinates ? and ? are 2? periodic. This implies that ? and ? have periodicities + m(m2 ? n2 ? 1 ? Ei)(1 + x)d?¤ ¾.

? 7 ? (?, ?) ? (? ? 2?, ? + 2?), ? ? ? + 4?.

The fermions will be antiperiodic under (?, ?) ? (? ? 2?, ? + 2?) and periodic under ? ? ? + 4?. The functions H?i appearing in the metric become

H?i = 2c[2?2 + 1 ? (m2 ? n2)x + Ei], where we introduce the convenient constants

Ei ? r Q2

4ci2 + [(m2 ? n2)2 ? 2(m2 + n2) + 1].

It was shown in [

Ai =H?i?1½Qidt ? s Qj Qk

Qi

? mcn £n(m2 ? n2 + 1 + Ei)(1 ? x)d?

We can also write the ADM mass and angular momenta of the five-dimensional asymptotically flat solution in the Einstein frame (in units where 4G(5)/? = 1) in terms of the integer parameters as

MADM = 2c(E1 + E2 + E3),

J? = ?r mn pQ1Q2Q3,

J? = r mn pQ1Q2Q3. (3.25)

This coordinate transformation provides us with a nice form of the metric, which
makes the special nature of the soliton solutions evident. We see that the base metric is
completely independent of the charges or any length scale. The full metric is written in
terms of m, n, the length scale c, and the charges Qi, which enter explicitly only through
the functions H?i. However, c is not an independent length scale: it should be determined
in terms of the charges and m, n. In the supersymmetric case m = n + 1, we had [

c = 4Mmn (s?2 ? s2)?1. (3.27) Since s2 is determined in terms of m, n by (2.9), the problem is simply to determine M in terms of the charges Qi and m, n. Using

Qi = M sinh ?i cosh ?i = M (3.28) tanh ?i 1 ? tanh2 ?i , we can express s2 = tanh ?1 tanh ?2 tanh ?3 in terms of the Qi and M . We want to solve this for M as a function of s2 and the charges. The equation can be rearranged to find that the combination M¯ = M (s?2 ? s2)?1 = 4mnc satisfies (s?2 ? s2)Q1Q2Q3M¯ 3(Q1M¯ + Q2Q3)(Q2M¯ + Q1Q3)(Q3M¯ + Q1Q2) +[2Q12Q22Q23(Q12 + Q22 + Q3) ? (Q14Q34 + Q14Q42 + Q24Q43)]M¯ 4 + 8Q13Q32Q33M¯ 3 2 +2Q12Q22Q32(Q12Q22 + Q12Q23 + Q22Q23)M¯ 2 ? Q14Q42Q34 = 0. (3.29) In the supersymmetric case, s = 1, the first line is absent, so one can see that (3.26) satisfies this equation.1 In the non-supersymmetric case, there is no explicit solution for c in general, but it is determined implicitly in terms of the charges by (3.29). Explicit solutions are possible in special cases: If all three charges are equal, we can write J H E P 0 8 ( 2 0 0 7 ) 0 5 5 c = 4mn

Q s?2/3 ? s2/3

s?2 ? s2 . ? 8 ? (3.30) If one of the charges is much smaller than the other two, say Q3 ? Q1, Q2, then This limit is interesting because it is the regime where the type IIB solution has a near-core AdS3 × S3 geometry.

The coordinates above provide the simplest description of the solution, but they are not
directly related to the coordinates used for the supersymmetric solutions in [

We therefore introduce new coordinates r, ? through (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) (3.37)

J H E P 0 8 ( 2 0 0 7 ) 0 5 5 r = (r¯2 ? r+2) + c sin2 ?¯

4 rc2 = r2 + 2rc cos ? + c2; g(r¯) = 4(r + rc)2 ? 4c2,

? 9 ? The range of the coordinates is r ? 0, 0 ? ? ? ?. One could similarly define coordinates centered on the singularity at ? = 0, x = ?1 by but we will focus henceforth on the r, ? coordinates. It is useful to retain rc, however, thought of as a function of r, ? given by then the inverse coordinate transformation is

2c sin2 ?¯ = (r ? rc) + c, r¯2 ? r+2 = 2(r + rc ? c), or

?2 = (r + 2rcc ? c) , x = (rc c? r) .

These coordinates have some nice properties even for the general metric (3.3): which can be used to show

g(r¯) sin2 ?¯cos2 ?¯ = 4r2 sin2 ? r¯2dr¯2 + d?¯2 = g(r¯) so this coordinate transformation casts this part of the metric in a conformally flat form (note that we get a conformally flat form even before restricting to the smooth soliton solutions). The base metric used in the supersymmetric solutions differs from the one used up to now by a conformal factor; define

The eleven-dimensional metric is then

ds121 = ?(Z1Z2Z3)?2/3(dt + k)2 + (Z1Z2Z3)1/3ds42 + ds2T 6 , Zi ? 2cH??iV = c?1V [r + rc + (m2 ? n2)(r ? rc) + cEi],

V = c12 [(m2 ? n2)(r ? rc) + (r + rc)]2 ? [(m2 ? n2)2 ? 2(m2 + n2) + 1(],3.42) k = ?8cQ3?1Qm2nQV3 ©?m[(m2 ? n2)(r ? rc + c) + (r + rc ? c)](rc ? r + c)(d? + d?) + n[(m2 ? n2)(r ? rc ? c) + (r + rc ? c)](r ? rc + c)(?d? + d?)ª .

The base metric here is (using the angular coordinates (3.17) appropriate for writing the base as a fibration) and with and with and ds42 = H?1?2 + H £?(dr2 + r2d?2) + r2 sin2 ?d?2¤ ,

2c?V H = [((r + rc)2 ? c2) ? (m ? n)2((r ? rc)2 ? c2)] , ? = [((r + rc)2 ? c2) ? (m ? n)2((r ? rc)2 ? c2)] ,

4rrc ? = d? ? [((r + rc)2 ? c2)(r ? rc) + ((r ? rc)2 ? c2)(m2 ? n2)(r + rc)] d?.

c[((r + rc)2 ? c2) ? (m ? n)2((r ? rc)2 ? c2)]

The metric (3.41) is in the form used in writing the supersymmetric solutions, so we can
make a detailed comparison to the supersymmetric case. If we set m = n + 1, this metric
should reduce to the familiar two-center supersymmetric solution of [

H = n + 1 rc

? nr , ? = d? + [(n + 1) cos ?c ? n cos ?] d?, (3.48) (3.49) and so the base space (3.44) reduces to the usual Gibbons-Hawking base of the two-center supersymmetric solution. This base space plays an important role in the study of the supersymmetric solutions.

Compared to the supersymmetric cases, the obvious novelty in these solutions is that the base metric (3.44) is no longer of the Gibbons-Hawking form. Indeed, although it is still a U(1) fibration over a three-dimensional base space, that space is no longer flat, or even conformally flat, and the function H does not appear to satisfy a harmonic equation on this base. There is no clear sign of any linear structure in the equations satisfied by this solution which could be exploited to generate multi-center solitons as in the supersymmetric case.

In the supersymmetric case, an important insight gained from the analysis of the
twocenter solutions in [

This suggests that if we wanted to look for further non-supersymmetric solutions using this kind of decomposition, we would need to allow possibilities which included a region where the base metric becomes imaginary. However, the absence of any nice harmonic behaviour in this form of the solution suggests that, unlike in the supersymmetric case, this decomposition may not be particularly useful.

We now analyse the structure of the solitons in this M-theory form, using the adapted
coordinates. In [

It is clear from (3.16) that at x = 1, the circle along ?? shrinks smoothly to zero size, while at x = ?1, the circle along ?? shrinks smoothly to zero size. The component of the gauge field along the circle which shrinks goes to zero, so the gauge fields are also smooth there.

At ? = 0, a circle which is a combination of these two shrinks; to analyse this region, it is convenient to introduce a different set of angular coordinates. We define ?? = ?l? ? k?, ?? = n? + m?, (4.1) where k, l are integers such that kn ? ml = 1. Since we assume m, n are coprime, we can always find such integers.2 We introduce them so that the identifications ? ? ? + 2?, ? ? ? + 2? are equivalent to ?? ? ?? + 2?, ?? ? ?? + 2?. In these coordinates, ?Q1Q2Q3 · ?mncV

mn?2d?? + 14 [?(m2 ? n2)(1 ? x2) ? 2?2x + 2(kn + ml)?2]d??¸ , (4.2) (?2d?+2 1) + 14 (1 d?x2x2) + ?V2 [?(m2 ? n2)(km ? nl)(1 ? x2) ? 2(km ? nl)x(?2 + 1)

+ ?2d??2 + ?V4 [?(m2 ? n2)x ? (m2 + n2) + 2(?2 + 1)]d??2 + ?V2 [?(k2 ? l2)(m2 ? n2)(1 ? x2) ? 2(k2 ? l2)x(?2 + 1) + 2(k2 + l2)(?2 + 1)]d??2. (4.3) (4.4)

J H E P 0 8 ( 2 0 0 7 ) 0 5 5 k = and ds¯42 = Thus at ? = 0, the circle along ??? shrinks smoothly to zero size.

The component of the gauge field along ??? is (Ai)?? = ?s QQjQi k ?mn µ1 ? 4Hc??i2 ¶ .

As ? ? 0, it goes to a constant, which gives a non-zero holonomy around the shrinking circle, H?? Ai = ?2?q QjQk ?mn. Can this be removed by a large gauge transformation?

Qi In the type IIB picture, the gauge field associated with qp comes from a Kaluza-Klein reduction, so we know that Ap ? Ap + Rd?? is an allowed large gauge transformation. Using (2.5)?(2.6), we can show R = q q1q5 ?mn. Regarding each of the gauge fields in qp turn as arising as the Kaluza-Klein gauge field in a distinct reduction from type IIB, we can argue that

Ai ? Ai + s QQjQi k ?mnd?? 2If m, n are not coprime, we could write m = a m¯, n = an¯, with m¯ , n¯ coprime, and proceed as above, with kn¯ ? lm¯ = 1. In this case, there will be a Za orbifold singularity at ? = 0. are indeed allowed large gauge transformations. That is, the gauge group is a circle of size 2?q QjQk ?mn, and the holonomy around any cycle takes values in this circle, so any

Qi holonomy that is an integer multiple of the size of the circle is actually equivalent to zero holonomy. Thus, the gauge fields have zero holonomy up to gauge transformations, and the gauge fields are also smooth at ? = 0.

We should finally consider what happens at ? = 0, x = ±1, where two circles shrink simultaneously. There will be orbifold singularities at these points.

The most interesting feature of the non-supersymmetric solitons in this five-dimensional
picture is the way in which they differ from the supersymmetric case in their local
structure near the points ? = 0, x = ±1. As in the supersymmetric case, there are orbifold
singularities at these points. However, in the present case these are non-supersymmetric
orbifolds. Hence, the supersymmetry is broken not just in the full asymptotically flat
solution, but even by the local solution describing the geometry near the singular points. This
is somewhat surprising, as supersymmetry is not broken locally in the type IIB solution. In
this section, we will explore the local structure, showing that the supersymmetry is broken
locally, and explain the relation to the type IIB solution. We conclude from this local
breaking of the supersymmetry that the picture of the smooth supersymmetric solutions
as made up of 1/2 BPS ?atoms? [

Let us consider the point ? = 0, x = 1, and study it in the r, ? coordinates introduced in section 3.2, where it corresponds to r = 0. Near r = 0, V ? 4n2, H?i ? 2c[?(m2?n2)+1+Ei] is a constant, and k ? 0, so the full geometry decomposes as a flat R1,6 cross the fourdimensional base space (3.44). In the base space, rc ? c + r cos ?,

1 ? ? 2 [(1 + cos ?) + (m ? n)2(1 ? cos ?)],

H? = 2c?V 4rrc ? nr . and Finally, the fiber is

? ? d? + 21? [(1 + cos ?) + (m2 ? n2)(1 ? cos ?)]d?.

We introduce a new radial coordinate r? = ?r, so the base space can be written, up to an irrelevant constant scale factor, as ds42 = dr?2 + r?2µ d?2 + 1 sin2 ?d?2 + 4 4? 4?n2 ?2¶.

The geometry looks locally like a cone over d?2 = d?2 + 1 sin2 ?d?2 + 4 4?

4?n2 ?2. (4.5) (4.6) (4.7) (4.8) (4.9)

J
H
E
P
0
8
(
2
0
0
7
)
0
5
5
This is an orbifold of S3. If we start from S3 in the standard form
d?2 =
we can obtain (4.9) by the transformation
The periodicities (3.21) of ? and ? in our solution then imply that ? ? and ?? are identified
under ?? ? ? ? 2? and (? ?, ??) ? (? ? + 4?/n, ?? ? (m ? n ? 1)2?/n). This defines a
freely?
acting Zn quotient of S3, which is referred to in the mathematical literature as a lens space,
denoted L(n, m) [

There is thus a Zn orbifold singularity at r = 0. We can make this manifest by defining complex coordinates (4.10) (4.11) (4.12) (4.13) (4.14) (4.15)

J H E P 0 8 ( 2 0 0 7 ) 0 5 5 u1 = r? sin 2? ei??/2 = r? sin ? ei(?+?)/2n,

2 u2 = r? cos ? ei(???2??)/2

2 = r? cos ? ei[(m?n)?+(m+n)?]/2n,

2 Ai = HQ?ii dt + s QQjQi k ?m2n

d? ?. in terms of which the local geometry is simply flat C2, and the identification ? ? ? + 4? acts as u1 ? u1e2?i/n,

u2 ? u2e2?i(m?n)/n.

The geometry is thus a non-supersymmetric C2/Zn orbifold of the kind discussed for
example in [

However, this is not the end of the story; we also need to consider the gauge fields. Near r = 0,

3Note that the lens spaces L(n, m) and L(n, m?) are homeomorphic for m = ±m? mod n or mm? = ±1 mod n.

Thus, the holonomy of the gauge field around the orbifold circle is

I ? ? Ai = 2?s QQjQi k ?mn n1 . (4.16) This is a fractional holonomy; it cannot be removed by a large gauge transformation, which can only shift the holonomy by an integer multiple of 2?q QjQk ?mn, as argued in Qi the previous section.4

The presence of this holonomy makes this orbifold qualitatively different from the one
considered in [

The fact that the total space of the gauge bundle is smooth also explains how this non-supersymmetric orbifold can arise from a smooth geometry in six dimensions. In the Kaluza-Klein reduction to five dimensions, the fiber always has finite size, so we might be surprised that there is a singularity in the base, but the fractional holonomy explains how this arises: even though the total space is smooth and the fiber is not degenerating, the connection is not well-behaved at this point.

Since the type IIB solution is smooth, it is also supersymmetric locally, in a neighbourhood of any point,5 but the orbifold we obtained locally at r = 0 is non-supersymmetric. It is useful to consider carefully what happens to the supersymmetry under the duality from the type IIB to the M-theory picture. The key step is the reduction to a five-dimensional solution from the type IIB solution (2.1) on S1 y × T 4. Recall that in the type IIB solution, there is a circle which shrinks smoothly to zero size at r¯ = r+. The solution thus has a unique spin structure, which is antiperiodic around this circle, and also around each of the two contractible S1s in S3 (parametrized by the coordinates ? and ? above). Imposing the latter two conditions, the Killing spinor must be of the general form ? = ei?1? 2yR ei?2 ?2 +i?3 ?2 f (r¯, ?¯)?0. (4.17) for some real ? and constant spinor ?0, with three independent sign choices ?1, ?2, ?3 = ±1. To impose the antiperiodicity on the degenerating circle, note that going once around the 4Note that the gauge field is however well-behaved in the covering space S3, as we would in general expect. That is, this gauge field can be thought of locally as arising from the orbifolding of a well-behaved gauge connection on the covering space.

5And if we take one charge small, the type IIB solution will have a near-core region which is approximately AdS3 × S3, so supersymmetry is actually restored in the whole near-core region. J H E P 0 8 ( 2 0 0 7 ) 0 5 5 circle which shrinks to zero at r¯ = r+ corresponds to going once around the y circle, ?m times around the ? circle, and n times around the ? circle. Thus, we must have ? = 1 + ?1?2m ? ?1?3n to produce the correct antiperiodic behaviour around this circle. That is, the Killing spinor will be

? = ei(?1+?2m??3n) 2yR ei?2 ?2 +i?3 ?2 f (r¯, ?¯)?0 = ei?1 2yR ei?2(m 2yR + ?2 )e?i?3(n 2yR ? ?2 )f (r¯, ?¯)?0. (4.18) We get a five-dimensional solution by reducing on the y circle. In the case m = n + 1, where the whole solution is supersymmetric, the Killing spinor (4.18) with ?2 = ?3 = ??1 is constant around the y circle. For the non-supersymmetric solutions, the Killing spinor (4.18) will not be constant around the y circle for any choice of signs. As a result, it does not give rise to a Killing spinor in the lower-dimensional theory. To see this, note that we obtain the five-dimensional gravitino from the constant mode of the six-dimensional gravitino.6 Thus, we cannot obtain a supersymmetry in the five-dimensional solution from the supersymmetry in the six-dimensional solution: the Killing spinor which provides the supersymmetry parameter for the six-dimensional solution descends to a spinor field on the five-dimensional solution which is charged under the Kaluza-Klein gauge field, so it cannot be the parameter for a variation of the gravitino, which is not charged under the Kaluza-Klein gauge field.

That is, although the local six-dimensional geometry has a supersymmetry even for
non-supersymmetric solitons, this supersymmetry is not visible in the five-dimensional
solution. This is an example of the phenomenon of ?supersymmetry without
supersymmetry? [

This relation to the type IIB solutions also implies that there is no instability of the
local geometry. That is, there will not be any winding tachyon modes of the type studied
in [

To summarize our analysis of the M-theory form of the solitons, we have found that the five-dimensional solution is smooth except at two points, where we have nonsupersymmetric orbifold singularities. Thus, the solution has a ?two-center? structure, which is similar to the simplest supersymmetric solitons. However, these centers do not 6This is true if we work in the sector where m + n is odd. If m + n is even, the spin structure on the six-dimensional solution does not give rise to a spin structure on the five-dimensional solution, as we are Kaluza-Klein reducing on a circle with antiperiodic boundary conditions for the fermions.

7In the full type IIB solution, this arises from the fact that in the asymptotically flat region, the only
possible Killing spinors are constant around the y circle.
J
H
E
P
0
8
(
2
0
0
7
)
0
5
5
preserve any supersymmetry, even locally. This is one of the main lessons from our
analysis: the picture of the supersymmetric solutions in [

In this section we examine Kaluza-Klein reductions of our solution to four dimensions. Our ambition in considering these reductions was to find a description of the non-BPS atoms in terms of brane systems in IIA. Unfortunately, we have failed to achieve this, but it is perhaps instructive to explain how it goes wrong.

We would like to have a reduction which is smooth away from the centers at ? = 0, x = ±1. Considering the local geometry near r = 0, this amounts to asking for a representation of the lens space (4.9) as an S1 bundle over some smooth two-dimensional manifold. In the supersymmetric case m = n + 1, ? = 1, and (4.9) is already of the desired form: the S?1 is fibered over S2. However, in the general case, there is a problem. Since ?(? = 0) = 1, the two-dimensional space is still smooth at ? = 0, but ?(? = ?) = (m ? n)2, so the two-dimensional space has a Z|m?n| orbifold singularity along ? = ?. In the full four-dimensional geometry, this will form a line conical singularity connecting the two centers.

Is there some other, inequivalent way to represent the lens space as an S1 bundle that avoids this problem? Start from the metric in the manifestly locally S3 form (4.10), and consider the general S1, which is for some coprime integers e, f .8 We introduce coordinates ?¯ and ?¯ through ? ? = ne ?¯ + 2 ng ?¯, ?? = 21 µ? (m ?n 1)e + f ¶ ?¯ + µ? (m ?n 1)g + h¶ ?¯, where g, h are integers such that eh ? f g = 1. The S1 cycle (5.1) is along ??¯ in these coordinates, and the identifications are ?¯ ? ?¯ + 4?, ?¯ ? ?¯ + 2?. Rewriting (4.10) in these coordinates, it becomes

d?2 = d4?2 + 41G sin2 ?d?¯2 + 4Gn2 ?¯2, (5.3) where ?¯ = d?¯ + F d?¯ (the precise form of F will not be necessary for our arguments), and

G = sin2 ?((m ? 1)e ? nf )2 + ·e + 12 (1 ? cos ?)((m ? 1)e ? nf )¸2.

8If e, f are not coprime, the cycle is not primitive ? it is an integer multiple of some other cycle, and we should consider instead the corresponding primitive cycle. (5.1) (5.2) (5.4)

J H E P 0 8 ( 2 0 0 7 ) 0 5 5 Thus, we can make the two-dimensional metric smooth at ? = 0 by choosing e = ±1, and we can make it smooth at ? = ? by choosing me ? nf = ±1.9 But we cannot satisfy both conditions simultaneously unless m = ±1 mod n, that is, unless we are considering the lens space L(n, 1) which arises in the supersymmetric case.

The conclusion is thus that in the general case, the best we can do is to make the solution smooth on one of the two axes near r = 0. It seems sensible to keep the solution smooth at large distances; we thus turn to considering in a little more detail the reduction along ?? , which has a Z|m?n| conical singularity connecting the two centers. To do so, we first need to rewrite the metric in the usual Kaluza-Klein form. The metric is of the general form ds52 = ?A(dt + ?(d? + ?d?) + ?d?)2 + B(d? + ?d?)2 + Cds32, (5.5) where ? was given in (3.19) and ds23 was given in (3.20). It can be written as ds52 = (B ? A?2) ·d? + ?d? ? B ? A?2 (dt + ?d?)

A?

AB ? B ? A?2 (dt + ?d?)2 + Cds32. (5.6) If we then reduce along ? , the last two terms will give the four-dimensional geometry.

Thus, the three-dimensional metric in (3.20) is the base space for the four-dimensional geometry: we repeat it here, ¸2 In the supersymmetric case m = n + 1, the base metric (3.20) is flat. In the nonsupersymmetric case, it is not even conformally flat. We can see directly that it is smooth along the axes which extend to infinity, at x = ±1, and has a conical singularity between the two centers, along the line at ? = 0. Also, there are curvature singularities in this base metric at ? = 0, x = ±1. These are in addition to the curvature singularities in the full four-dimensional metric which will arise from the fact that the dilaton is diverging at these points.

To simplify the Kaluza-Klein reduction, it is better to change to the coordinate ? ? = ? + ?, so that the identifications are simply ? ? ? + 2? and ? ? ? ? ? + 4?. From the Kaluza-Klein point of view, this is a gauge transformation which shifts ? ? ? ? = ? ? 1.

The Kaluza-Klein gauge field coming from this reduction is

A?

AKK = ? ?d? ? B ? A?2 (dt + ?d?), (5.8) where

?Q1Q2Q3 (m ? n)?2(1 ? x2) ? = ? 4c?mn [4?2(?2 + 1) + (m ? n)2(1 ? x2)] , (5.9) 9This corresponds to a reduction of the full solution along the ?? coordinate. We can easily see that the base metric (3.16) will be smooth at ? = 0 for such a reduction, but will have conical singularities along x = ±1. Locally near ? = 0, these two reductions are analogous to the reductions of a flat metric ds2 = dr2 + r2d?2 + R2(d? + d?)2 with the identifications ? ? ? + ?, ? ? ? + 2?. If we reduce this along ? , we get a flat space with a Z2 orbifold. If we reduce instead along ?, we get a smooth fluxbrane solution in two dimensions. J H E P 0 8 ( 2 0 0 7 ) 0 5 5 After a lot of algebra, and using the constraint (3.29) that determines c in terms of the Qi, the factor of V cancels, and this can be written as

B ? A?2 = This clearly vanishes at ? = 0, x = ±1. We will thus have singularities of the fourdimensional metric and dilaton at these points. It will not vanish at any other point, as Ei > (m2 ? n2) ? 1, so both terms are positive away from ? = 0, x = ±1.

In the supersymmetric case, we would view the singularities at ? = 0, x = ±1 as D6-branes. Here, however, an attempt to define the charge carried by these singularities is obstructed by non-trivial holonomies around the conical singularity. The Kaluza-Klein gauge field which arises in the reduction from five to four dimensions has a non-trivial holonomy around the conical line at ? = 0,

I ?

n AKK|?=0 = 4? (m ? n) . and after the above gauge transformation, ? ? = [4?2(?2 + 1)(x ?[14)?2+(?(2m+2 ?1)n+2)((m1 ?? xn2))22(?12?+x22n)](m ? n)(1 ? x2)] . (5.10) Since ? = 0 at ? = 0 and at x = ±1, the second factor in AKK will not contribute to holonomies at these points. We therefore do not write it more explicitly. The factor of B ? A?2, which will give the dilaton of the four-dimensional solution, is more complicated: B ? A?2 = [4?2(?2 + 1) + (m ? n)2(1 ? x2)]

{2?2[(m+n)+x(m?n)]+(m?n)2(m+n)(1?x2)}2.

Note that for m = n + 1, this is an integer multiple of 4?, and hence gauge-equivalent to zero. This fractional holonomy in the general case implies that there is a delta-function singularity in the field strength FKK along ? = 0, and we cannot associate separate KaluzaKlein charges with the two singularities at ? = 0, x = ±1. Thus, we cannot interpret the singularities as due to the presence of D-branes at these points, since we cannot unambiguously define brane charges associated with the singularities.10

Similarly, for the gauge fields Ai, we can decompose

· A? ¸

Ai = A?i + ? d? + ?d? ? B ? A?2 (dt + ?d?) , 10Note that the total charge carried by the two singularities is still well-defined. The integral of the flux over a sphere enclosing the whole line at ? = 0 is R FKK = H? AKK|x=1 ? H? AKK|x=?1 = 4?, so the structure carries one unit of KK monopole charge through a surface at large distance. That is, the solution is asymptotically flat in five dimensions. (5.11) (5.12) (5.13) (5.14)

J E P 0 8 ( 2 0 0 7 ) 0 5 5 and interpret A?i as the four-dimensional gauge field.11 We find that A?i? = 0 at x = ±1. Once again, ? vanishes along ?2 = 0, so near the conical line singularity,

A?i|?=0 = ?2?s QQjQi k ?mn 1

, m ? n (5.15) (5.16) showing the same fractional holonomy for these gauge fields as well. As for the KaluzaKlein gauge field, this implies that the charges associated with the two singularities at ? = 0, x = ±1 are ambiguous.

We saw in the previous section that the M-theory solution has non-supersymmetric orbifold singularities at ? = 0, x = ±1. We had hoped that the reduction to four dimensions would clarify the interpretation of these singularities. However, the four-dimensional solution has some new features (compared to the supersymmetric case) which make it difficult to extract a brane interpretation of these singularities. The four-dimensional solution has curvature singularities only at ? = 0, x = ±1; however, it also has a line conical singularity connecting these two curvature singularities, along ? = 0. This conical singularity makes it impossible to unambiguously assign charges to the two curvature singularities. Also, the curvature singularities at ? = 0, x = ±1 arise from a divergence of the dilaton (as in the supersymmetric case) but also from a singularity in the three-dimensional base metric at these points. It is not clear what interpretation we could give to this additional singularity from a brane construction point of view. The geometry describing a D6-brane has a flat base space, and just gets a curvature singularity from the divergence of the dilaton at the brane?s position. Thus, in the non-supersymmetric case, the centers do not have a clear D-brane interpretation. It would be interesting to see whether and how this local structure is changed for non-supersymmetric solutions which are asymptotically flat in four dimensions.

We thank Henriette Elvang, Ori Ganor, and Ashish Saxena for discussions. We would also like to thank the Aspen Center for Physics where part of this work was conducted. The work of SFR was supported in part by the EPSRC and PPARC. The work of EGG was supported by the US Department of Energy under contracts DE-AC03-76SF00098 and DE-FG03-91ER-40676 and by the National Science Foundation under grant PHY-00-98840. 11Note that A?i is invariant under the gauge transformation made on AKK, so it does not matter if we use ? and the ? of (3.19), or ? ? and the ?? from (5.10) in carrying out this calculation. We will use the former. J H E P 0 8 ( 2 0 0 7 ) 0 5 5 [2] J.M. Maldacena and L. Maoz, De-singularization by rotation, JHEP 12 (2002) 055 [hep-th/0012025]. J H E 0 8 ( 2 0 0 7 ) 0 5 5

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