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Supergravity solutions for branes localized within branes
30 July 1998
Physics Letters B 432 Ž1998. 298–304
Supergravity solutions for branes localized within branes N. Itzhaki a
a,1
, A.A. Tseytlin
b,2
, S. Yankielowicz
a,3
School of Physics and Astronomy, Tel AÕiÕ UniÕersity, Ramat AÕiÕ, 69978, Israel b Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 31 March 1998; revised 13 May 1998 Editor: H. Georgi
Abstract We construct supergravity solutions describing branes ŽD2-branes or NS 5-branes or waves. localized within D6-branes in the region close to the core of the D6-branes. Other similar string-theory and M-theory ‘near-core’ localized solutions can be found by applying U-duality andror lifting D s 10 solutions to D s 11. In particular, the D2-branes localized on D6-branes is T-dual to a special case of the background describing ŽD.strings localized on ŽD.5-branes and thus is also related to a localized intersection of M2-branes and M5-branes. D6 q wave configuration is U-dual to a D0-brane localized on a Kaluza-Klein 5-brane or to the fundamental string intersecting a D5-brane with the point of intersection localized on D5-brane. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In view of the fundamental role of soliton solutions of supergravity there seems to be little need to justify construction of new such solutions. In particular, recent progress in relating the large N behavior of certain gauge theories to semi-classical supergravity Žsee, e.g., w1x and references there. serves as a motivation for looking for special solutions describing regions close to the cores of the branes. Also, finding supergravity solutions representing branes ending on branes like Hanany-Witten w2x configuration Žand other configurations in different dimensions and with less supersymmetries w3x. should be important. 1
E-mail: [email protected]. E-mail: [email protected]. 3 E-mail: [email protected]. 2
General solutions representing localized Žas opposed to smeared w4,5x. intersections of branes or branes ending on branes have not yet been constructed but some progress have already being made. The solution describing a string localized on a 5-brane was found Žin an implicit form. in w6,7x Žsee also w8x.. This is a special case of the solution corresponding to a string localized on an intersection w9x of two 5-branes w7,10x. This latter solution and similar composite solutions involving several branes related by U-duality and lifting to D s 11 Žsee also w11x. represent only partially localized intersections, while in the case of, e.g., the Hanany-Witten configuration the localization should be in all relevant dimensions. In this note we present several explicit solutions for branes which are completely localized within other branes. The 1r4 supersymmetric solutions we find describe, however, only the region close to the core of the ‘bigger’ brane. Our starting point is the
0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 7 0 - 4
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
D s 11 solutions ŽM2-brane, M5-brane and gravitational wave with ZN identifications in the transverse space. which we shall reduce down to D s 10 along a circular direction of S 3 part of the transverse space. 4 The M2-branes solution leads ŽSection 2. to D s 10 background representing D2-branes localized on D6-branes in the region close to the D6-branes core Žor, equivalently, it corresponds to the limit of very large charge of D6-brane.. The M5-branes and waves backgrounds lead ŽSection 3. to similar solutions describing the near-core region of D6-branes with, respectively, NS 5-branes and waves localized on it. Other similar solutions can be found by applying U-duality. In particular, T-dual to our D6 q D2 background gives explicit analytic expression for the ‘string localized on 5-brane’ solution Žwhich was previously found only in the general form parameterized by a function subject to a differential equation w7,6x. in the special case when the 5-brane is smeared in one of the transverse directions and in the region close to the core of the 5-brane ŽSection 4.. The system of D1-branes localized within D5-branes is relevant for the Ž4,4. two-dimensional field theory description of certain five dimensional black holes Žsee, e.g., w13x.. Applying T-duality twice one obtains a system of D0-branes localized within D4branes related to the DLCQ of the superconformal Ž0,2. theory w14x. Finally, D6 q D2 configuration is also related by T-duality to a D-instanton within D3-branes system which corresponds to a localized instanton in the N s 4 four-dimensional theory. Therefore, in the spirit of w1x, the N s 4 4d theory on a background of instantons should be dual to type IIB string on a D3 q D-instanton background T-dual to the D6 q D2 one we find here. There are several possible generalizations. In particular, one may start with configurations of several intersecting M-branes and reduce along cycle of transverse S 3 obtaining near-core form of localized intersections involving n ) 2 different D-branes. One example is provided by the explicit near-core form
299
of solution describing string localized on the intersection of the two 5-branes ŽSection 4. which is related w7x to localized 5 l 5 H 2 M-theory and 5NS l 5R H 3 type IIB theory configurations.
2. D2-brane within D6-brane We would like to find a supergravity solution representing n 2 D2-branes localized within a collection of N D6-brane in the decoupling Žor ‘near-core’. limit. From the point of view of the field theory on the D6-brane world-volume, it should correspond to localized 2-branes ‘instantons’. The simplification of considering D6-branes in this limit is that their Mtheory counterpart ŽKK monopole background w15x. becomes an ALE space with an A Ny 1 singularity w16x times a 7-dimensional Minkowski space M Ž6,1. w17,18x. This is just a Minkowski space M Ž10,1. with ZN identifications. Suppose we introduce n 2 M2branes along 2 q 1 directions in M Ž6,1.. Such M2branes are invariant under the ZN identifications. Therefore, the corresponding eleven dimensional background will be given by the M2-brane solution w19x with the ZN identifications in transverse space. It can then be reduced to ten dimensions to obtain a type IIA solution which, remarkably, can be interpreted as representing the near-core region of a configuration of D2-branes localized within a collection of D6-branes. Before we turn to the D6 q D2 case let us first recall how one can obtain the near-core region of type IIA solution for a collection of N D6-branes upon dimensional reduction of the A Ny 1 space w20x. The metric which corresponds to M Ž6,1. times an A Ny 1 singularity is 2 ds11 s dx <2< q d r 2
q r 2 du˜ 2 q sin2u˜ d w˜ 2 q cos 2u˜ d f˜ 2 ,
ž
/
Ž 1.
where dx <2< s ydt 2 q dx 12 q . . . qdx 62 , r 2 s x 72 2 q . . . qx 10 , 0 F u˜F pr2 and 0 F w˜ , f˜ F 2p with the Z N identification Ž w˜ , f˜ . ; Ž w˜ , f˜ . q Ž2prN,2 prN .. Defining the new variables
4
This reduction is a special case of the Hopf reduction discussed in detail in w12x which appeared while this paper was in preparation Žthe motivation and interpretation of the solutions found in w12x is different from that of the present work..
r2 Us
2 Nl p3
,
u s 2 u˜ ,
w s w˜ y f˜ ,
f s Nf˜ , Ž 2 .
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
300
dial coordinate r via r 2 q x i x i and not on u˜, f˜ and w˜ .. The metric is, therefore,
we obtain the metric 2 ds11 s dx <2< q
q
l p3 N 2U
2Ul p3
dU 2 q
df q
N
N 2
l p3 NU 2
Ž du 2 q sin2u d w 2 . 2
Ž cos u y 1 . d w ,
m 2
Ž dx 11 q Am dx .
y2 f r3
qe
2 ds10 ,
Ž 4.
we obtain 2 ds10 saX
q
Ž 2p . gYM gYM
Ž 2p . q
f
e s
g Y2 M 2p
Am dx m s
2
2
( (
ž
2U g Y2 M N
f2 s 1 q
2 5p 2 n 2 l p6
, rˆ 6 2 r 2 s x 72 q . . . qx 10 ,
rˆ 2 s x 32 q . . . qx 62 q r 2 ,
Ž 7.
and the 3-rank tensor has the standard form C012 s Ž . fy1 2 . Changing the coordinates as in 2 we get
q f 21r3
,
ž
q
l p3 NU
q
2 2Ul p3
l p3 N 2 2 dx 3 q . . . qdx 6 q 2U
Ž du 2 q sin2u d w 2 . df q
N
, f2 s 1 q
N
Ž cos u y 1 . d w , Ž 5. 2 where d V 22 s du 2 q sin2u d w 2 . This is the IIA supergravity solution representing a collection of D6branes w21x in the decoupling limit w22x < x< U s X s fixed, a
N 2
2
Ž cos u y 1 . d w
2 5p 2 n 2 l p6
Ž x 32 q . . . qx 62 q 2 Nl p3U .
3
/
.
,
Ž 8.
Ž 9.
As above, we can reduce the solution to ten dimensions along the isometric f direction. The result is the type IIA solution describing D2-branes localized within D6-branes Žin the region near the core of D6-branes. 2 ds10 s a X fy1r2 hy1r2 Ž ydt 2 q dx 12 q dx 22 . 2 6
g Y2 M s Ž 2p . l p3 s Ž 2p . g s a X 3r2 s fixed,
r2 qf 21r2 hy1 Ž dx 32 q . . . qdx 62 . 6
a X ™ 0. Ž 6. Let us now consider a configuration of M2-branes stretched along x 1 , x 2 directions. For simplicity we shall put all of them at the origin in the transverse space Žit is straightforward to generalize the solution to the case where the two-branes are separated along x 3 , . . . , x 6 .. The required eleven dimensional solution is obvious: it is essentially the same as in w19x but with the above ZN identification of angles of S 3 Žthe relevant harmonic function depends only on the ra-
2 2 2 qf 21r2 h1r2 6 Ž dU q U d V 2 . ,
4
dU 2
where now
3r4
/
,
/
where
dU 2
'N U 3r2 d V 22
2 Ž 2p . '2
ž
dx <2<
N
2U
gYM
qr 2 du˜ 2 q sin2u˜ d w˜ 2 q cos 2u˜ d f˜ 2
2 r3 ds11 s fy2 Ž ydt 2 q dx 12 q dx 22 . 2
2U
N
q f 21r3 dx 32 q . . . qdx 62 q d r 2
Ž 3.
where f has standard period f ; f q 2p . This metric has a Killing vector along the f direction and hence it can be reduce to ten dimensions along x 11 ' R 11 f . Using the relation between the eleven dimensional metric and the ten dimensional type IIA string metric, dilaton and the gauge field, 2 ds11 s e 4f r3
2 r3 ds11 s fy2 Ž ydt 2 q dx 12 q dx 22 . 2
4
g Y2 M
f
e s
Ž 2p .
Am dx m s
4
N 2
f 21r4 hy3r4 , 6
Ž cos u y 1 . d w ,
Ž 10 .
where h6 s
g Y2 M N 4
2 Ž 2p . U
,
Ž 11 .
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
and C012 s fy1 2 . Note that the presence of the D2branes does not modify the expression for gauge field Am and hence the Dirac string singularity can be removed for any integer N This background has the same ‘harmonic function rule’ form as the standard 1r4 supersymmetric asymptotically flat D6 q D2 bound state solution w5x for which the positions of D2-branes are smeared within the D6-brane directions so that f 2 and h 6 are given by f2 s 1 q
Q2 r
,
X2
h 6 s a f6 ,
f6 s 1 q
Q6 r
,
Ž x 32 q . . . qx 62 q 4Q6 r .
r s Ura X ,
,
q f52r3 dx 62 q d r 2
h6 s a X 2
,
/
where
Here instead we find 3
To find a solution describing n 5 NS5-branes within a collection of N D6-branes we follow the logic of the previous section starting now with the M5-brane solution w23x with the ZN identification in the transverse 3-sphere. Its metric is 2 r3 ds11 s fy1 Ž ydt 2 q dx 12 q . . . qdx 52 . 5
ž
Ž 12 . Q2
3. NS5-brane and wave within D6-brane
qr 2 du˜ 2 q sin2u˜ d w˜ 2 q cos 2u˜ d f˜ 2
Q6 s 12 Ng s a X1r2 .
f2 s 1 q
301
Q6 r
,
q f52r3
Ž 13 .
i.e. we are restricted to the region close to the D6-brane core Žor, equivalently, consider the decoupling limit Ž6.. but the D2-branes are completely localized within D6-branes Ž‘averaging’ over x 3 , . . . , x 6 leads back to the near-core region of the smeared solution.. Surprisingly, the D2-brane function f 2 in Ž13. ‘remembers’ its D s 11 membrane origin: its power of decay with distance is 1rx 6 compared to 1rx 5 for the usual D2-brane in ten dimensions. This behavior is characteristic of a string solution, and indeed the above D6 q D2 configuration is T-dual to a D-string localized on D5-brane Žsee Section 4.. Note that the 1rx 6 is the behavior at the near core region Žwhere our solution is valid.. Thus in the full solution the power of decay with distance should interpolates between 1rx 6 at the near core region and 1rx 5 far from the horizon. To summarize, the solution we have found applies only in the decoupling limit. It should be a ‘near-core’ approximation to a more general solution, yet to be found, describing D2-branes localized within D6branes and having also the asymptotically flat region Žor, equivalently, to the solution with finite a X , g s and r .. Trying to add a constant to h 6 in Ž13. does not, however, lead to a simple analytic expression for the background Žcf. Section 4..
p n 5 l p3
, rˆ 2 s x 62 q r 2 , Ž 14 . rˆ 3 and r was defined in Ž7.. The change of variables Ž2. gives 2 r3 ds11 s fy1 Ž ydt 2 q dx 12 q . . . qdx 52 . 5 f5 s 1 q
l p3 N 2 dx 6 q
ž
2U
q
l p3 NU
q
2 2Ul p3 N
dU 2
Ž du 2 q sin2u d w 2 . df q
N 2
2
Ž cos u y 1 . d w
/
,
Ž 15 .
where f5 s 1 q
p n 5 l p3
Ž x 62 q 2 Nl p3U .
3r2
,
Ž 16 .
Reducing to ten dimensions along f we obtain the type IIa solution of n 5 NS-fivebranes localized within N D6-branes, 5 2 ds10 s a X hy1r2 Ž ydt 2 q dx 12 q . . . qdx 52 . 6 r2 2 2 2 qf5 hy1 dx 62 q f5 h1r2 6 6 Ž dU q U d V 2 . ,
ef s
g Y2 M
Ž 2p .
Am dx m s
f 1r2 hy3r4 , 6 4 5
N
Ž cos u y 1 . d w , Ž 17 . 2 where h 6 was defined in Eq. Ž11.. Reducing instead 5
This is to be compared with the solution describing delocalised superposition of D6-brane and NS 5-brane which is a reduction of the ‘smeared’ superposition of M5-brane and KK monopole given in w5x.
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
302
along x 5 we find a superposition of D4-brane and the near-core region of KK monopole. T-duality in x 6 and f then leads back to D6 q NS5 solution. Next we turn to waves localized within D6-branes. Starting with the metric of the gravitational wave in D s 11 2 ds11 s ydt 2 q dx 12 q f 0 Ž dt y dx 1 .
2
2 2 2 ds10 s fy1 1 Ž x , y . Ž ydt q dz .
q dx 22 q . . . qdx 62 q d r 2
q dyn dyn q f5 Ž x . dx m dx m ,
ž
f 0s
rˆ7
rˆ
,
/
2
s x 22 q . . . qx 62 q r 2
,
and reducing it along f as above we find the background representing a wave localized on D6brane with the metric 2 ds10 s a X hy1r2 ydt 2 q dx 12 q f 0 Ž dt y dx 1 . 6
ž
2
Ž dU
2
qU
2
d V 22
./.
Ž 18 .
T-duality along the wave Ž x 1 . direction gives a configuration of D5-brane Žsmeared along x 1 . intersected by a fundamental string with the intersection point localized on D5. S-dual to this is NS5-brane intersected by a D-string. T-duality along D-string direction x 1 gives then a near-core configuration of Kaluza-Klein 5-brane Žwhich is T-dual to NS 5-brane smeared in one transverse direction with T-duality applied in that direction. with D0-brane completely localized on it in all internal coordinates. But this is just the background which is obtained directly by reducing Ž18. along the x 1 direction. Again, as usual, the reductions of D s 11 solutions along different isometric directions give U-dual D s 10 solutions.
Ž 20 .
Remarkably, this equation admits a simple analytic solution in the special case when Ži. the 5-brane is smeared in one of the transverse dimensions Že.g., x 2 . so that f5 s 1 q Q5rr, r 2 s x 12 q x 32 q x 42 , and Žii. one considers only the near-throat region of the 5-brane where f5 ™ Q5rr. Assuming that f 1 depends only on the radial coordinates r and Õ, Õ 2 ' ym ym , we can put Ž20. in the form ry1Er Ž r 2Er . q Q5 Õy3EÕ Ž Õ 3EÕ . f 1 Ž r ,Õ . s 0 .
Ž 21 .
(
In terms of the variable u s 2 Q5 r it becomes uy3Eu Ž u 3Eu . q Õy3EÕ Ž Õ 3EÕ . f 1 Ž u,Õ . s 0 ,
Ž 22 .
i.e. is formally the same as the radial part of the Laplace equation in flat 8 dimensions Žwith u 2 q Õ 2 as the total distance.. This equation is solved, in particular, by f1 s 1 q
4. String localized on 5-brane Applying T-duality to the localized D6 q D2 solution Ž10. along x 2 direction one expects to find the D5 q D1 solution or S-dual NS5 q NS1 solution describing a fundamental string w24x localized on a NS5-brane w25x. Let us show that, indeed, the back-
Ž 19 .
Here Ž z, yn . are the internal dimensions of the 5-brane Ž x 1 , x 3 , . . . , x 6 in Ž10.., x n are the dimensions transverse to the 5-brane Ž x 2 , U, u , w in Ž10.. and f5 Ž x . is the harmonic function Ž E mEm f5 ' Ex2 f5 s 0. which defines the position of the 5-braneŽs.. The string function f 1Ž x, y . must satisfy the condition ŽLaplace equation in the curved transverse space.
Ex2 q f5 Ž x . E y2 f 1 Ž x , y . s 0 .
qdx 22 q . . . qdx 62 qh1r2 6
e 2 f s fy1 1 f5 .
dB s dfy1 1 n dt n dz q ) df 5 ,
q r 2 du˜ 2 q sin2u˜ d w˜ 2 q cos 2u˜ d f˜ 2 , Q0
ground T-dual to Ž10. is a special case of the general 5 q 1 solution constructed in w6,7x. The exact string background describing NS 5 q 1 configuration is represented by the conformal sigma-model with the following metric, 2-form and dilaton couplings Ž m,n s 1,2,3,4.
s1q
Q1
Ž Õ 2 q u2 .
3
Q1
Ž y12 q . . . qy42 q 4Q5 r .
3
,
Ž 23 .
which has, indeed, the same form as f 2 in Ž13.. More general solutions are found by separating the string cores in the 5-brane yn directions. This is in
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
agreement with the possibility to choose a more general M2-brane harmonic function f 2 in Ž9. Že.g. a sum of several terms with different centers in x 3 , . . . , x 6 . and allows, in particular, to construct the solution smeared in these internal directions, thus returning back to f 2 in Ž12.. One straightforward generalization is obtained by replacing the product of the flat 4-space ym and curved 5-brane 4-space x n factors in Ž19. by the direct product of the two 5-brane factors w9x. The resulting exact conformal background describing a string localized on the intersection of the two 5-branes is w6,7x 2 2 2 ds10 s fy1 1 Ž x , y . Ž ydt q dz .
X dB s dfy1 1 n dt n dz q ) df 5 q ) df 5 , X e 2 f s fy1 1 f5 f5 ,
Ž 24 . f5X Ž
where the harmonic functions f5 Ž x . and y . Ž Ex2 f5 2 X s 0, E y f5 s 0. define the positions of the two 5branes Ž z, yn . and Ž z, x m . and the string function f 1Ž x, y . satisfies f5X Ž y . Ex2 q f5 Ž x . E y2 f 1 Ž x , y . s 0 .
f5 s
Q5 r
,
f5X s
r
r 2 s x 12 q x 32 q x 42 ,
, r X 2 s y 12 q y 32 q y42 ,
we find that Ž25. is solved by f1 s 1 q
Q1
Ž Q5 r q QX5 r X .
3
I4q 0 s yT0 dt n 4 q n 0
H ž
y1 q n 4 fy1 0 Ž x , y . q n 0 f4 Ž x .
(
= 1 y f 0 Ž x , y . y˙m y˙m y f 0 Ž x , y . f 4 Ž x . x˙ n x˙ n
/.
Here f 0 , f 4 are the same as f 1 , f5 in Ž20. or as f 2 , f6 in Ž13. Žoriginating from D6 q D2, the D4 q D0 background is smeared in the two transverse directions. and n 0 ,n 4 are the charges of the D4 q D0 probe. It is obvious that the motion of the probe in both the transverse Ž x n . and parallel Ž ym . directions to the D4 q D0 source depends on the detailed structure of f 0 Ž x, y ..
Ž 25 .
Assuming that the two 5-branes are smeared in one of the relative transverse directions Že.g., x 2 and y 2 . and considering the near-core region where QX5 X
source background. Representing the D4 q D0 probe by the standard Born-Indeld action with a self-dual gauge field background and ignoring the dependence on internal dimensions of D4-brane probe one finds the following expression for the probe action Žsee w26x.
y1
q f5X Ž y . dyn dyn q f5 Ž x . dx m dx m ,
303
Acknowledgements The work of N.I. and S.Y. was supported in part by the US-Israel Binational Science Foundation, by GIF – the German-Israeli Foundation for Scientific Research, and by the Israel Science Foundation. The work of A.A.T. was supported by PPARC and the European Commission TMR programme grant ERBFMRX-CT96-0045.
.
This gives a simple explicit solution describing a string localized on an intersection of the two 5-branes. Closely related D s 10 and D s 11 solutions may be found by using U-duality and lifts to D s 11 as in w7,10x. To study some consequences of localization of one brane on another one may consider the action of a D5 q D1 probe moving in the background produced by localized D5 q D1 source. Equivalent Tdual system is D4 q D0 probe moving in D4 q D0
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Physics Letters B 432 Ž1998. 298–304
Supergravity solutions for branes localized within branes N. Itzhaki a
a,1
, A.A. Tseytlin
b,2
, S. Yankielowicz
a,3
School of Physics and Astronomy, Tel AÕiÕ UniÕersity, Ramat AÕiÕ, 69978, Israel b Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 31 March 1998; revised 13 May 1998 Editor: H. Georgi
Abstract We construct supergravity solutions describing branes ŽD2-branes or NS 5-branes or waves. localized within D6-branes in the region close to the core of the D6-branes. Other similar string-theory and M-theory ‘near-core’ localized solutions can be found by applying U-duality andror lifting D s 10 solutions to D s 11. In particular, the D2-branes localized on D6-branes is T-dual to a special case of the background describing ŽD.strings localized on ŽD.5-branes and thus is also related to a localized intersection of M2-branes and M5-branes. D6 q wave configuration is U-dual to a D0-brane localized on a Kaluza-Klein 5-brane or to the fundamental string intersecting a D5-brane with the point of intersection localized on D5-brane. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In view of the fundamental role of soliton solutions of supergravity there seems to be little need to justify construction of new such solutions. In particular, recent progress in relating the large N behavior of certain gauge theories to semi-classical supergravity Žsee, e.g., w1x and references there. serves as a motivation for looking for special solutions describing regions close to the cores of the branes. Also, finding supergravity solutions representing branes ending on branes like Hanany-Witten w2x configuration Žand other configurations in different dimensions and with less supersymmetries w3x. should be important. 1
E-mail: [email protected]. E-mail: [email protected]. 3 E-mail: [email protected]. 2
General solutions representing localized Žas opposed to smeared w4,5x. intersections of branes or branes ending on branes have not yet been constructed but some progress have already being made. The solution describing a string localized on a 5-brane was found Žin an implicit form. in w6,7x Žsee also w8x.. This is a special case of the solution corresponding to a string localized on an intersection w9x of two 5-branes w7,10x. This latter solution and similar composite solutions involving several branes related by U-duality and lifting to D s 11 Žsee also w11x. represent only partially localized intersections, while in the case of, e.g., the Hanany-Witten configuration the localization should be in all relevant dimensions. In this note we present several explicit solutions for branes which are completely localized within other branes. The 1r4 supersymmetric solutions we find describe, however, only the region close to the core of the ‘bigger’ brane. Our starting point is the
0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 7 0 - 4
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
D s 11 solutions ŽM2-brane, M5-brane and gravitational wave with ZN identifications in the transverse space. which we shall reduce down to D s 10 along a circular direction of S 3 part of the transverse space. 4 The M2-branes solution leads ŽSection 2. to D s 10 background representing D2-branes localized on D6-branes in the region close to the D6-branes core Žor, equivalently, it corresponds to the limit of very large charge of D6-brane.. The M5-branes and waves backgrounds lead ŽSection 3. to similar solutions describing the near-core region of D6-branes with, respectively, NS 5-branes and waves localized on it. Other similar solutions can be found by applying U-duality. In particular, T-dual to our D6 q D2 background gives explicit analytic expression for the ‘string localized on 5-brane’ solution Žwhich was previously found only in the general form parameterized by a function subject to a differential equation w7,6x. in the special case when the 5-brane is smeared in one of the transverse directions and in the region close to the core of the 5-brane ŽSection 4.. The system of D1-branes localized within D5-branes is relevant for the Ž4,4. two-dimensional field theory description of certain five dimensional black holes Žsee, e.g., w13x.. Applying T-duality twice one obtains a system of D0-branes localized within D4branes related to the DLCQ of the superconformal Ž0,2. theory w14x. Finally, D6 q D2 configuration is also related by T-duality to a D-instanton within D3-branes system which corresponds to a localized instanton in the N s 4 four-dimensional theory. Therefore, in the spirit of w1x, the N s 4 4d theory on a background of instantons should be dual to type IIB string on a D3 q D-instanton background T-dual to the D6 q D2 one we find here. There are several possible generalizations. In particular, one may start with configurations of several intersecting M-branes and reduce along cycle of transverse S 3 obtaining near-core form of localized intersections involving n ) 2 different D-branes. One example is provided by the explicit near-core form
299
of solution describing string localized on the intersection of the two 5-branes ŽSection 4. which is related w7x to localized 5 l 5 H 2 M-theory and 5NS l 5R H 3 type IIB theory configurations.
2. D2-brane within D6-brane We would like to find a supergravity solution representing n 2 D2-branes localized within a collection of N D6-brane in the decoupling Žor ‘near-core’. limit. From the point of view of the field theory on the D6-brane world-volume, it should correspond to localized 2-branes ‘instantons’. The simplification of considering D6-branes in this limit is that their Mtheory counterpart ŽKK monopole background w15x. becomes an ALE space with an A Ny 1 singularity w16x times a 7-dimensional Minkowski space M Ž6,1. w17,18x. This is just a Minkowski space M Ž10,1. with ZN identifications. Suppose we introduce n 2 M2branes along 2 q 1 directions in M Ž6,1.. Such M2branes are invariant under the ZN identifications. Therefore, the corresponding eleven dimensional background will be given by the M2-brane solution w19x with the ZN identifications in transverse space. It can then be reduced to ten dimensions to obtain a type IIA solution which, remarkably, can be interpreted as representing the near-core region of a configuration of D2-branes localized within a collection of D6-branes. Before we turn to the D6 q D2 case let us first recall how one can obtain the near-core region of type IIA solution for a collection of N D6-branes upon dimensional reduction of the A Ny 1 space w20x. The metric which corresponds to M Ž6,1. times an A Ny 1 singularity is 2 ds11 s dx <2< q d r 2
q r 2 du˜ 2 q sin2u˜ d w˜ 2 q cos 2u˜ d f˜ 2 ,
ž
/
Ž 1.
where dx <2< s ydt 2 q dx 12 q . . . qdx 62 , r 2 s x 72 2 q . . . qx 10 , 0 F u˜F pr2 and 0 F w˜ , f˜ F 2p with the Z N identification Ž w˜ , f˜ . ; Ž w˜ , f˜ . q Ž2prN,2 prN .. Defining the new variables
4
This reduction is a special case of the Hopf reduction discussed in detail in w12x which appeared while this paper was in preparation Žthe motivation and interpretation of the solutions found in w12x is different from that of the present work..
r2 Us
2 Nl p3
,
u s 2 u˜ ,
w s w˜ y f˜ ,
f s Nf˜ , Ž 2 .
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
300
dial coordinate r via r 2 q x i x i and not on u˜, f˜ and w˜ .. The metric is, therefore,
we obtain the metric 2 ds11 s dx <2< q
q
l p3 N 2U
2Ul p3
dU 2 q
df q
N
N 2
l p3 NU 2
Ž du 2 q sin2u d w 2 . 2
Ž cos u y 1 . d w ,
m 2
Ž dx 11 q Am dx .
y2 f r3
qe
2 ds10 ,
Ž 4.
we obtain 2 ds10 saX
q
Ž 2p . gYM gYM
Ž 2p . q
f
e s
g Y2 M 2p
Am dx m s
2
2
( (
ž
2U g Y2 M N
f2 s 1 q
2 5p 2 n 2 l p6
, rˆ 6 2 r 2 s x 72 q . . . qx 10 ,
rˆ 2 s x 32 q . . . qx 62 q r 2 ,
Ž 7.
and the 3-rank tensor has the standard form C012 s Ž . fy1 2 . Changing the coordinates as in 2 we get
q f 21r3
,
ž
q
l p3 NU
q
2 2Ul p3
l p3 N 2 2 dx 3 q . . . qdx 6 q 2U
Ž du 2 q sin2u d w 2 . df q
N
, f2 s 1 q
N
Ž cos u y 1 . d w , Ž 5. 2 where d V 22 s du 2 q sin2u d w 2 . This is the IIA supergravity solution representing a collection of D6branes w21x in the decoupling limit w22x < x< U s X s fixed, a
N 2
2
Ž cos u y 1 . d w
2 5p 2 n 2 l p6
Ž x 32 q . . . qx 62 q 2 Nl p3U .
3
/
.
,
Ž 8.
Ž 9.
As above, we can reduce the solution to ten dimensions along the isometric f direction. The result is the type IIA solution describing D2-branes localized within D6-branes Žin the region near the core of D6-branes. 2 ds10 s a X fy1r2 hy1r2 Ž ydt 2 q dx 12 q dx 22 . 2 6
g Y2 M s Ž 2p . l p3 s Ž 2p . g s a X 3r2 s fixed,
r2 qf 21r2 hy1 Ž dx 32 q . . . qdx 62 . 6
a X ™ 0. Ž 6. Let us now consider a configuration of M2-branes stretched along x 1 , x 2 directions. For simplicity we shall put all of them at the origin in the transverse space Žit is straightforward to generalize the solution to the case where the two-branes are separated along x 3 , . . . , x 6 .. The required eleven dimensional solution is obvious: it is essentially the same as in w19x but with the above ZN identification of angles of S 3 Žthe relevant harmonic function depends only on the ra-
2 2 2 qf 21r2 h1r2 6 Ž dU q U d V 2 . ,
4
dU 2
where now
3r4
/
,
/
where
dU 2
'N U 3r2 d V 22
2 Ž 2p . '2
ž
dx <2<
N
2U
gYM
qr 2 du˜ 2 q sin2u˜ d w˜ 2 q cos 2u˜ d f˜ 2
2 r3 ds11 s fy2 Ž ydt 2 q dx 12 q dx 22 . 2
2U
N
q f 21r3 dx 32 q . . . qdx 62 q d r 2
Ž 3.
where f has standard period f ; f q 2p . This metric has a Killing vector along the f direction and hence it can be reduce to ten dimensions along x 11 ' R 11 f . Using the relation between the eleven dimensional metric and the ten dimensional type IIA string metric, dilaton and the gauge field, 2 ds11 s e 4f r3
2 r3 ds11 s fy2 Ž ydt 2 q dx 12 q dx 22 . 2
4
g Y2 M
f
e s
Ž 2p .
Am dx m s
4
N 2
f 21r4 hy3r4 , 6
Ž cos u y 1 . d w ,
Ž 10 .
where h6 s
g Y2 M N 4
2 Ž 2p . U
,
Ž 11 .
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
and C012 s fy1 2 . Note that the presence of the D2branes does not modify the expression for gauge field Am and hence the Dirac string singularity can be removed for any integer N This background has the same ‘harmonic function rule’ form as the standard 1r4 supersymmetric asymptotically flat D6 q D2 bound state solution w5x for which the positions of D2-branes are smeared within the D6-brane directions so that f 2 and h 6 are given by f2 s 1 q
Q2 r
,
X2
h 6 s a f6 ,
f6 s 1 q
Q6 r
,
Ž x 32 q . . . qx 62 q 4Q6 r .
r s Ura X ,
,
q f52r3 dx 62 q d r 2
h6 s a X 2
,
/
where
Here instead we find 3
To find a solution describing n 5 NS5-branes within a collection of N D6-branes we follow the logic of the previous section starting now with the M5-brane solution w23x with the ZN identification in the transverse 3-sphere. Its metric is 2 r3 ds11 s fy1 Ž ydt 2 q dx 12 q . . . qdx 52 . 5
ž
Ž 12 . Q2
3. NS5-brane and wave within D6-brane
qr 2 du˜ 2 q sin2u˜ d w˜ 2 q cos 2u˜ d f˜ 2
Q6 s 12 Ng s a X1r2 .
f2 s 1 q
301
Q6 r
,
q f52r3
Ž 13 .
i.e. we are restricted to the region close to the D6-brane core Žor, equivalently, consider the decoupling limit Ž6.. but the D2-branes are completely localized within D6-branes Ž‘averaging’ over x 3 , . . . , x 6 leads back to the near-core region of the smeared solution.. Surprisingly, the D2-brane function f 2 in Ž13. ‘remembers’ its D s 11 membrane origin: its power of decay with distance is 1rx 6 compared to 1rx 5 for the usual D2-brane in ten dimensions. This behavior is characteristic of a string solution, and indeed the above D6 q D2 configuration is T-dual to a D-string localized on D5-brane Žsee Section 4.. Note that the 1rx 6 is the behavior at the near core region Žwhere our solution is valid.. Thus in the full solution the power of decay with distance should interpolates between 1rx 6 at the near core region and 1rx 5 far from the horizon. To summarize, the solution we have found applies only in the decoupling limit. It should be a ‘near-core’ approximation to a more general solution, yet to be found, describing D2-branes localized within D6branes and having also the asymptotically flat region Žor, equivalently, to the solution with finite a X , g s and r .. Trying to add a constant to h 6 in Ž13. does not, however, lead to a simple analytic expression for the background Žcf. Section 4..
p n 5 l p3
, rˆ 2 s x 62 q r 2 , Ž 14 . rˆ 3 and r was defined in Ž7.. The change of variables Ž2. gives 2 r3 ds11 s fy1 Ž ydt 2 q dx 12 q . . . qdx 52 . 5 f5 s 1 q
l p3 N 2 dx 6 q
ž
2U
q
l p3 NU
q
2 2Ul p3 N
dU 2
Ž du 2 q sin2u d w 2 . df q
N 2
2
Ž cos u y 1 . d w
/
,
Ž 15 .
where f5 s 1 q
p n 5 l p3
Ž x 62 q 2 Nl p3U .
3r2
,
Ž 16 .
Reducing to ten dimensions along f we obtain the type IIa solution of n 5 NS-fivebranes localized within N D6-branes, 5 2 ds10 s a X hy1r2 Ž ydt 2 q dx 12 q . . . qdx 52 . 6 r2 2 2 2 qf5 hy1 dx 62 q f5 h1r2 6 6 Ž dU q U d V 2 . ,
ef s
g Y2 M
Ž 2p .
Am dx m s
f 1r2 hy3r4 , 6 4 5
N
Ž cos u y 1 . d w , Ž 17 . 2 where h 6 was defined in Eq. Ž11.. Reducing instead 5
This is to be compared with the solution describing delocalised superposition of D6-brane and NS 5-brane which is a reduction of the ‘smeared’ superposition of M5-brane and KK monopole given in w5x.
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
302
along x 5 we find a superposition of D4-brane and the near-core region of KK monopole. T-duality in x 6 and f then leads back to D6 q NS5 solution. Next we turn to waves localized within D6-branes. Starting with the metric of the gravitational wave in D s 11 2 ds11 s ydt 2 q dx 12 q f 0 Ž dt y dx 1 .
2
2 2 2 ds10 s fy1 1 Ž x , y . Ž ydt q dz .
q dx 22 q . . . qdx 62 q d r 2
q dyn dyn q f5 Ž x . dx m dx m ,
ž
f 0s
rˆ7
rˆ
,
/
2
s x 22 q . . . qx 62 q r 2
,
and reducing it along f as above we find the background representing a wave localized on D6brane with the metric 2 ds10 s a X hy1r2 ydt 2 q dx 12 q f 0 Ž dt y dx 1 . 6
ž
2
Ž dU
2
qU
2
d V 22
./.
Ž 18 .
T-duality along the wave Ž x 1 . direction gives a configuration of D5-brane Žsmeared along x 1 . intersected by a fundamental string with the intersection point localized on D5. S-dual to this is NS5-brane intersected by a D-string. T-duality along D-string direction x 1 gives then a near-core configuration of Kaluza-Klein 5-brane Žwhich is T-dual to NS 5-brane smeared in one transverse direction with T-duality applied in that direction. with D0-brane completely localized on it in all internal coordinates. But this is just the background which is obtained directly by reducing Ž18. along the x 1 direction. Again, as usual, the reductions of D s 11 solutions along different isometric directions give U-dual D s 10 solutions.
Ž 20 .
Remarkably, this equation admits a simple analytic solution in the special case when Ži. the 5-brane is smeared in one of the transverse dimensions Že.g., x 2 . so that f5 s 1 q Q5rr, r 2 s x 12 q x 32 q x 42 , and Žii. one considers only the near-throat region of the 5-brane where f5 ™ Q5rr. Assuming that f 1 depends only on the radial coordinates r and Õ, Õ 2 ' ym ym , we can put Ž20. in the form ry1Er Ž r 2Er . q Q5 Õy3EÕ Ž Õ 3EÕ . f 1 Ž r ,Õ . s 0 .
Ž 21 .
(
In terms of the variable u s 2 Q5 r it becomes uy3Eu Ž u 3Eu . q Õy3EÕ Ž Õ 3EÕ . f 1 Ž u,Õ . s 0 ,
Ž 22 .
i.e. is formally the same as the radial part of the Laplace equation in flat 8 dimensions Žwith u 2 q Õ 2 as the total distance.. This equation is solved, in particular, by f1 s 1 q
4. String localized on 5-brane Applying T-duality to the localized D6 q D2 solution Ž10. along x 2 direction one expects to find the D5 q D1 solution or S-dual NS5 q NS1 solution describing a fundamental string w24x localized on a NS5-brane w25x. Let us show that, indeed, the back-
Ž 19 .
Here Ž z, yn . are the internal dimensions of the 5-brane Ž x 1 , x 3 , . . . , x 6 in Ž10.., x n are the dimensions transverse to the 5-brane Ž x 2 , U, u , w in Ž10.. and f5 Ž x . is the harmonic function Ž E mEm f5 ' Ex2 f5 s 0. which defines the position of the 5-braneŽs.. The string function f 1Ž x, y . must satisfy the condition ŽLaplace equation in the curved transverse space.
Ex2 q f5 Ž x . E y2 f 1 Ž x , y . s 0 .
qdx 22 q . . . qdx 62 qh1r2 6
e 2 f s fy1 1 f5 .
dB s dfy1 1 n dt n dz q ) df 5 ,
q r 2 du˜ 2 q sin2u˜ d w˜ 2 q cos 2u˜ d f˜ 2 , Q0
ground T-dual to Ž10. is a special case of the general 5 q 1 solution constructed in w6,7x. The exact string background describing NS 5 q 1 configuration is represented by the conformal sigma-model with the following metric, 2-form and dilaton couplings Ž m,n s 1,2,3,4.
s1q
Q1
Ž Õ 2 q u2 .
3
Q1
Ž y12 q . . . qy42 q 4Q5 r .
3
,
Ž 23 .
which has, indeed, the same form as f 2 in Ž13.. More general solutions are found by separating the string cores in the 5-brane yn directions. This is in
N. Itzhaki et al.r Physics Letters B 432 (1998) 298–304
agreement with the possibility to choose a more general M2-brane harmonic function f 2 in Ž9. Že.g. a sum of several terms with different centers in x 3 , . . . , x 6 . and allows, in particular, to construct the solution smeared in these internal directions, thus returning back to f 2 in Ž12.. One straightforward generalization is obtained by replacing the product of the flat 4-space ym and curved 5-brane 4-space x n factors in Ž19. by the direct product of the two 5-brane factors w9x. The resulting exact conformal background describing a string localized on the intersection of the two 5-branes is w6,7x 2 2 2 ds10 s fy1 1 Ž x , y . Ž ydt q dz .
X dB s dfy1 1 n dt n dz q ) df 5 q ) df 5 , X e 2 f s fy1 1 f5 f5 ,
Ž 24 . f5X Ž
where the harmonic functions f5 Ž x . and y . Ž Ex2 f5 2 X s 0, E y f5 s 0. define the positions of the two 5branes Ž z, yn . and Ž z, x m . and the string function f 1Ž x, y . satisfies f5X Ž y . Ex2 q f5 Ž x . E y2 f 1 Ž x , y . s 0 .
f5 s
Q5 r
,
f5X s
r
r 2 s x 12 q x 32 q x 42 ,
, r X 2 s y 12 q y 32 q y42 ,
we find that Ž25. is solved by f1 s 1 q
Q1
Ž Q5 r q QX5 r X .
3
I4q 0 s yT0 dt n 4 q n 0
H ž
y1 q n 4 fy1 0 Ž x , y . q n 0 f4 Ž x .
(
= 1 y f 0 Ž x , y . y˙m y˙m y f 0 Ž x , y . f 4 Ž x . x˙ n x˙ n
/.
Here f 0 , f 4 are the same as f 1 , f5 in Ž20. or as f 2 , f6 in Ž13. Žoriginating from D6 q D2, the D4 q D0 background is smeared in the two transverse directions. and n 0 ,n 4 are the charges of the D4 q D0 probe. It is obvious that the motion of the probe in both the transverse Ž x n . and parallel Ž ym . directions to the D4 q D0 source depends on the detailed structure of f 0 Ž x, y ..
Ž 25 .
Assuming that the two 5-branes are smeared in one of the relative transverse directions Že.g., x 2 and y 2 . and considering the near-core region where QX5 X
source background. Representing the D4 q D0 probe by the standard Born-Indeld action with a self-dual gauge field background and ignoring the dependence on internal dimensions of D4-brane probe one finds the following expression for the probe action Žsee w26x.
y1
q f5X Ž y . dyn dyn q f5 Ž x . dx m dx m ,
303
Acknowledgements The work of N.I. and S.Y. was supported in part by the US-Israel Binational Science Foundation, by GIF – the German-Israeli Foundation for Scientific Research, and by the Israel Science Foundation. The work of A.A.T. was supported by PPARC and the European Commission TMR programme grant ERBFMRX-CT96-0045.
.
This gives a simple explicit solution describing a string localized on an intersection of the two 5-branes. Closely related D s 10 and D s 11 solutions may be found by using U-duality and lifts to D s 11 as in w7,10x. To study some consequences of localization of one brane on another one may consider the action of a D5 q D1 probe moving in the background produced by localized D5 q D1 source. Equivalent Tdual system is D4 q D0 probe moving in D4 q D0
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