VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

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Bound States of String Networks and D-branes

Alok Kumar,* Rashmi Rekha Nayak, † and Kamal Lochan Panigrahi ‡

Institute of Physics, Bhubaneswar 751 005, India(Received 29 August 2001; published 6 March 2002)

We show the existence of nonthreshold bound states of �p, q� string networks and D3-branes, preserv-ing 1�4 of the full type-IIB supersymmetry, interpreted as string networks “dissolved” in D3-branes. Wealso explicitly write down the expression for the mass density of the system and discuss the extensionof the construction to other Dp-branes. Differences in our construction of string networks with the onesinterpreted as dyons in N � 4 gauge theories are also pointed out.

DOI: 10.1103/PhysRevLett.88.121601 PACS numbers: 11.25.Sq

Nonthreshold bound states of various D-branes [1,2]have been objects of much interest due to their applicationsto the nonperturbative dynamics of string theory and gaugetheory, including from the point of view of anti–de Sitterand conformal field theory correspondence [3,4]. Theyhave an interpretation as branes that are “dissolved” insideother branes, and preserve 1�2 supersymmetry. They arealso of importance in understanding the physics of blackholes from a microscopic point of view [5]. Bound statesof F strings with D-branes have been analyzed as well[6,7]. Such bound states are generally obtained by apply-ing T dualities [8] to delocalized brane solutions and haveexplicit realizations as supergravity solutions. In view oftheir wide applications, it is of importance to analyze theseresults further.

In this Letter, we generalize the above constructions andobtain the bound states, now interpreted as �p,q� stringnetworks [9,10] dissolved in D3-branes. They preserve1�4 of the full type-IIB supersymmetry and therefore de-scribe new nonperturbative objects in these theories. It willalso be pointed out later that our construction of the boundstates of string networks and D3-branes is different fromthe ones appearing in the context of N � 4 gauge theories,interpreted as dyons [11–13].

The existence of stable networks as well as weblike con-figurations for strings and branes is now known for sev-eral years [14] on the basis of charge conservation, tensionbalance and supersymmetry analysis. Although for largenumber of these configurations no explicit supergravity orworld volume realizations are known, several examples inthe context of string networks have been worked out froma world volume point of view [12,15]. The results in ourpaper give evidence for the existence of similar configura-tions when they are dissolved inside other D-branes.

We now start by writing the classical supergravity so-lution [3] corresponding to the D1-D3 bound state [2,7],preserving 1�2 supersymmetry:

ds2str � f21�2�2dx2

0 1 dx21 1 h�dx2

2 1 dx23 ��

1 f1�2�dr2 1 r2dV25 � ,

f � 1 1a02R4

r4, h21 � sin2ff21 1 cos2f ,

0031-9007�02�88(12)�121601(4)$20.00

B23 �sinf

cosff21h, e2F � g2h , (1)

F01r �1g

sinf≠rf21, F0123r �

1g

cosfh≠rf21,

where Bmn is the NS-NS antisymmetric tensor field. Fmnr

and Fmnrab are, respectively, R-R 3-form and 5-form fieldstrengths. The asymptotic value of the B field in Eq. (1)is B`

23 � tan�f� and gives the expression for the ratio ofcharge densities of (smeared) D1- and D3-branes. The pa-rameter R is defined by cosfR4 � 4pgn, with n beingthe number of D3-branes. Finally g � g` is the asymp-totic value of the string coupling.

To describe explicitly the 1�2 supersymmetry propertyof the D1-D3 bound state, we note from their explicit so-lution in Eq. (1) that they also have an alternative interpre-tation in terms of D3-branes in a constant NS-NS antisym-metric tensor background of magnetic type: B23 � tan�f�.The 1�2 supersymmetry condition is then written in thefollowing form [13]:

�eL 2 eR� � sinfG01�eL 2 eR� 1 cosfG0123�eL 1 eR� ,(2)

where eL and eR are two positive chirality space-timespinors arising from the left and right moving sectors ofthe type-IIB string theory. The above condition can alsobe written in an alternative form:

eL � 2 sinfG01eR 1 cosfG0123eR , (3)

or, equivalently,

eR � 2 sinfG01eL 2 cosfG0123eL . (4)

Supersymmetry conditions of Eqs. (3) and (4) reduceto that of a standard D3-brane for f � 0. From Eq. (1),we also notice that a D string in the above bound state liesalong the x1 axis and is smeared in the remaining spatial di-rections x2 and x3, giving it the interpretation of a D stringbeing dissolved in a D3-brane. A generalization of the su-pergravity solution in Eq. (1), representing ����F, D1�, D3���bound state, is known and corresponds to the case whenboth electric and magnetic type Bmn fields are turned on[3,16]. These solutions also preserve 1�2 supersymmetry,thereby ensuring their stability.

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VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

The mass density of these ����F, D1�, D3��� bound statescan be expressed (in string frame) as [2,17,18]

m2 � T20

∑n2

g2 1 jp 1 qtj2∏

, (5)

where we have one �p, q� string along, say, the x1 direc-tion per �2p�2a 0 area [19] over the x2-x3 plane, and n isthe D3-brane charge. Also, T0 � 1��2p�3a02 and axion-dilaton moduli are given as t � x 1

ig . We also notice

that mass density (5) is a sum of distinct energy densities,associated with a �p, q� string and that of a D3-brane.Moreover the contributions of a �p, q� string for differ-ent �p, q�’s remain identical to the one, when D3-brane isabsent.

We now discuss the construction of the bound state ofstring networks and the D3-brane from a supersymmetrypoint of view. Following the above reasoning, these objectscan also be viewed as �p, q� string networks dissolved ina D3-brane. To discuss the network construction we nowcomplexify Eqs. (3) and (4):

�eL 2 ieR� � i sinfG01�eL 1 ieR �1 i cosfG0123�eL 2 ieR� , (6)

giving the 1�2 supersymmetry projection of a (D1-D3)bound state. Then, to write down the supersymmetry pro-jection of a bound state ����F1, D1�, D3��� of a �p, q� stringand D3-brane, we use the fact that they can be generated byapplying SL�2, Z� duality [20] on the D1-D3 bound statediscussed above. This procedure also gives the 1�2 super-symmetry condition for the ����F1, D1�, D3��� bound state, byusing the fact that the spinors �eL 6 ieR� transform co-variantly under the maximal compact subgroup, SO�2� [SL�2, R�, with SL�2, R��SO�2� parametrizing the modulispace represented by axion-dilaton fields. The transforma-tion properties of spinors are given as

�eL 6 ieR � ! ei�a�2��eL 6 ieR� . (7)

To obtain the phase a for a given SL�2, Z� transformation,one notes that by using the vielbein E, corresponding tothe axion-dilaton moduli M � EET , any SL�2, R� vectorcan be turned into an SO�2� vector. As a result, the phasetransformation parameter a can be read off from the cor-responding SL�2, Z� parameters. The supersymmetry con-dition for a �p, q� string dissolved in a D3 brane can thenbe generated from the one in (6) and has a form

�eL 2 ieR� � eiQ�p,q,t� sinfG01�eL 1 ieR �1 i cosfG0123�eL 2 ieR� . (8)

We notice that a phase factor Q, dependent on axion-dilaton moduli �t� as well as SL�2, Z� quantum num-bers �p, q�, eiQ�p,q,t� � p 1 qt�jp 1 qtj, appears inthe first term in the right-hand side (rhs) representingthe supersymmetry condition of a �p,q� string. SinceD3-branes are SL�2, Z� invariant objects, the second termin the rhs of Eq. (8) remains unchanged with respect to theone for the D1-D3 case.

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Now, to show the possibility of string network construc-tion, we consider a �p, q� string lying in the x1-x2 planeat an angle u with the x1 axis. Then Eq. (8) is replaced by

�eL 2 ieR� � eiQ�p,q,t� sinfG0�G1 cosu 1 G2 sinu�3 �eL 1 ieR� 1 i cosfG0123�eL 2 ieR � .

(9)

It can be seen that, as in the case of string networksin the absence of a D3-brane, if one identifies the orienta-tion of the �p, q� string inside a D3-brane, with its phasein the internal space, u � Q�p, q, t�, then the abovesupersymmetry condition is solved by the followingprojections:

�eL 2 ieR� � sinfG01�eL 1 ieR�1 i cosfG0123�eL 2 ieR� , (10)

and

�eL 2 ieR� � i sin fG02�eL 1 ieR�1 i cosfG0123�eL 2 ieR� . (11)

We notice that the projection condition (10) correspondsto that of an F string along the x1 axis dissolved in aD3-brane. Similarly, the projection condition (11) corre-sponds to that of a D-string along the x2 axis, dissolvedin the same D3-brane. These together imply that super-symmetry is broken to 1�4 of the original one. Interest-ingly, the supersymmetry condition, Eq. (9), is satisfiedfor arbitrary �p, q� with only a finite number of projec-tions, provided the above identification of the phases, u �Q�p, q, t�, holds. The projection conditions [Eqs. (10)and (11)], also reduce to the ones in [9], for f �

p

2 ,which corresponds to the case when there is no D3-brane.We have therefore shown the existence of a [�p, q� stringnetwork, D3] bound state preserving 1�4 supersymmetry.

To confirm the 1�4 supersymmetry property of ourconfiguration further, we now show that simultaneoussolutions for eL and eR , of appropriate type, do existfor Eqs. (10) and (11). In this connection, we note thatEq. (10), representing the supersymmetry of an F stringdissolved in a D3-brane, can be written as

eL � sinfG01eL 1 cosfG0123eR , (12)

or, equivalently, as

eR � 2sinfG01eR 2 cosfG0123eL . (13)

The supersymmetry conditions of dissolved D strings,along x1, were already written in Eq. (3) or, equivalently,in (4). From Eq. (11), we get conditions that are identicalto the ones in Eqs. (3) and (4), when we replace G01 byG02. In particular, for our argument we use

eR � 2sinfG02eL 2 cosfG0123eL (14)

as well as Eq. (12) as independent conditions followingfrom (10) and (11). Now, by substituting eR from Eq. (14)

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VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

into (12), one obtains

eL � G0�G1 sinf 2 G3 cosf�eL . (15)

The 1�4 supersymmetry now directly follows fromEqs. (14) and (15).

We now obtain the mass density of the (string network,D3) bound state that we have constructed. For this pur-pose, one can start with the expression of the mass den-sity for a bound state of a �p, q� string with D3-branesas in Eq. (5). As already emphasized, �p, q� strings in-side D3-branes make a distinct contribution to the totalmass formula. Now, for the case of the (string network,D3) bound state the contribution to the total mass, comingfrom the network, can be written as the sum of contribu-tions from different strings in that network [9]:

m2network � �SiliTi�2, (16)

where li’s are the the lenghts of various links and Ti’s arethe corresponding tensions. The final expression for massdensity is then obtained by adding the contribution fromthe D3-branes as well. In other words, the modification tothe mass formula in the string network case is essentiallydue to the replacement of the �p, q� string tension, by thecorresponding network mass formula in Eq. (5).

To write the expression for the mass density in a concreteform, we consider the case when the string network aswell as the D3-brane are wrapped on a T2. In this context,for the wrapping of the string network, one defines latticevectors �a, �b, constructed out of the link vectors �li �i �1, 2, 3� of a 3-prong string junction in a periodic stringnetwork of strings with quantum numbers �pi, qi� �i �1, 2, 3� [9,21], obeying charge conservation on the junction.The T2 is parametrized by moduli, l1 � �a ? �b� �a2, l2 �j �a 3 �bj� �a2. Total mass density (per unit length) then turnsout to be of the form (now in Einstein frame)

m2 � T20 n2A2 1 m2

network , (17)

where A � j �a 3 �bj is the area of T2. Explicit expressionfor network contribution to this mass density, m2

network, isidentical to the one in [9], with appropriate replacementsof charges by charge densities, coming from the smearingof the resulting (delocalized) particlelike state in the un-wrapped direction of the D3-brane. By placing the factorsof a0, etc., appropriately,

m2network �

1�2p�4a03

A�p1 q1 p2 q2�

3 �M 6 L�

0BB@

p1q1p2q2

1CCA , (18)

where we have one wrapped string network per 2pp

a0

length along the unwrapped direction of a D3-brane. Also,

M �1l2

µM l1M

l1M jlj2M

∂, L �

µ0 L

2L 0

∂,

(19)

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M �1t2

µ1 t1t1 jtj2

∂, L �

µ0 1

21 0

∂. (20)

We have therefore given the expression for the mass den-sity of the bound states discussed above. We also noticethat the above mass formula reduces to the one for the con-ventional string networks [9] in the absence of D3-branes�n � 0�. Moreover, by setting charges �p2, q2� � �0, 0�,which implies the reduction to the case of a straight stringwith �p1, q1� charges, one can obtain the energy spec-trum of the ����F, D1�, D3��� bound state with 1�2 supersym-metry. In the compactified theory, the mass formula (17)also corresponds to that of a bound state of a particlewith U-duality charges, dissolved in a string with �0, 0, 1�charge. Our result therefore turns out to be consistent witha general supersymmetry analysis of objects satisfyingBogomolnyi-Prasad-Sommerfield (BPS) bound in eight di-mensions [22], implying 1�4 supersymmetry for particle-like objects and 1�2 supersymmetry for stringlike objectsin D � 8. Apart from the supersymmetry analysis thatwe have presented in this paper, we also note that the non-threshold BPS mass formula, written in Eq. (17), impliesthat such bound states of string networks and D3-branesare also energetically favorable, and therefore likely to beformed, when several �p,q� strings are dissolved insidethese branes. However, a more detailed analysis is neededin this context.

We emphasize that the BPS configuration obtained aboveis different from the ones obtained in [13] in the context ofnoncommutative gauge theory. The string networks of [13]preserve 1�4 [11] of the D3-brane supersymmetry, with aninterpretation as a dyon in these theories, whereas in ourcase we have 1�4 of the full type-II supersymmetry. Also,our string network lies completely inside the D3-brane,compared to the ones representing dyons which connectdifferent branes. It will certainly be interesting to examineour string network configurations from the point of viewof noncommutative world volume theory, appearing in thiscontext.

These results can also be generalized to other boundstates of fundamental and D objects discussed in the litera-ture [19,23] and will give rise to lower supersymmetriesthan the known ones. For example, one can generalize theconstruction of the D1-D5 system [3] to string networksdissolved in �p,q� webs of 5-branes. It will then be ofinterest to analyze the implications of these results to blackhole physics.

We are grateful to A. Misra and S. Mukherji for severaluseful discussions.

*Email address: kumar@iopb.res.in†Email address: rashmi@iopb.res.in‡Email address: kamal@iopb.res.in

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