Contents

arXiv:0712.3625 [hep-th]

[22mm]

On the internal structure of dyons

[2mm] in  super Yang-Mills theories

K. Narayan

Chennai Mathematical Institute,

Plot H1, SIPCOT IT Park,

Padur PO, Siruseri - 603103, India.

Email:  narayan@cmi.ac.in

We use the low energy effective action on the Coulomb branch of super Yang-Mills theory to construct approximate field configurations for solitonic dyons in these theories, building on the brane prong description developed in hep-th/0101114. This dovetails closely with the corresponding description of these dyons as string webs stretched between D-branes in the transverse space. The resulting picture within these approximations shows the internal structure of these dyons (for fixed asymptotic charges) to be molecule-like, with multiple charge cores held together at equilibrium separations, which grow large near lines of marginal stability. Although these techniques do not yield a complete solution for the spatial structure (i.e. all core sizes and separations) of large charge multicenter dyons in high rank gauge theories, approximate configurations can be found in specific regions of moduli space, which become increasingly accurate near lines of marginal stability. We also discuss string webs with internal faces from this point of view.

## 1 Introduction

Understanding the internal structure of solitonic states in string theories and their low energy limits is an interesting and important question. In other words, for a localized soliton, given asymptotic quantum numbers (as seen by a distant observer), we would like to understand what internal structure one could associate with the soliton. This is often notoriously difficult in the regimes of most interest, even in supersymmetric theories. A key system exemplifying these difficulties is a black hole: given the large Bekenstein-Hawking entropy, it is tempting to imagine a rich internal structure that is “fuzzy” on approximately horizon size. A somewhat simpler but still fairly rich system in this context constitutes charged solitonic states in supersymmetric non-abelian gauge theories, which also helpfully admit realizations in terms of D-brane constructions.

It is well-known that there exist -BPS dyonic states in   SYM theories [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14] (see e.g. [15] for a recent review), represented as string webs [16, 17, 18] in brane constructions of these theories. These states can be labelled by their charges w.r.t. the long range fields, i.e. the abelian subsector (for convenience, we regard these as states in theories Higgsed to uncharged under an overall ). Unlike -BPS states (whose charges are “parallel” or mutually local), the electromagnetic and scalar forces between the constituent charge centers of a -BPS state do not precisely cancel except at specific equilibrium separations (i.e. there is a nontrivial potential for the dynamics of the charge cores). There is a rich structure of the decay of these -BPS states across lines of marginal stability (LMS) in the moduli space (Coulomb branch) of these theories. For instance, for a simple 2-center -BPS bound state, the abelian approximation shows that the spatial size of the state (i.e. the separation between the two charge centers) is inversely proportional to the distance from the LMS [11, 12, 13, 14]. Thus as one approaches the LMS, the separation between the centers grows and the two charges become more and more loosely bound until they finally unbind at the LMS. In terms of the density of states, we have a stable 1-particle state on one side of the LMS, which decays into the 2-particle state continuum as we cross the LMS. On the other side of the LMS, the 1-particle state does not exist, and we only have the 2-particle continuum. The field theory construction of these  string web states dovetails beautifully with “brane-prong” generalizations (see e.g. [9, 12, 13]) of the “brane-spike” [8]: these prongs in turn can be interpreted as string webs stretched between D-branes.

We expect that this picture holds for more general multicenter bound states too, i.e. near lines of marginal stability, some of these states become loosely bound. Then the internal structure in spacetime of these dyonic states can be described from the point of view of the low energy abelian effective gauge theory (i.e. the nonabelian microscopic physics becomes unimportant) as “molecule-like”, consisting of several solitonic charge centers bound together by electromagnetic and scalar forces, qualitatively somewhat similar to that of the split attractor black hole bound states of Denef et al [19, 20] in  string theories. In this paper, we make some modest attempts to make precise these general expectations, building on the construction in [12].

It is clear that as a function of the moduli values, a typical solitonic dyon (for fixed asymptotic charges) can have a complicated internal structure, potentially hard to pin down especially for large charge in a high rank  gauge theory Higgsed to at the generic point in moduli space. It is therefore interesting to ask if we can use D-brane constructions to glean insights into the internal structure of dyons, perhaps focussing on specific regions of moduli space where abelian approximations are reliable. Towards this end, we will analyze, in what follows, the spatial molecule-like structure of these states as it varies on the moduli space, in part using its limiting description as loosely bound configurations near decay across lines of marginal stability. While the decay of such a state into a 2-body endpoint admits an increasingly accurate description near the LMS, for a general dyon (where the subscript refers to some that these charges refer to), there are several 2-body decay endpoints (corresponding to multiple lines of marginal stability) depending on how we fix moduli values at charge cores and how we break up the dyon charges into constituents. Typically each constituent is itself a composite with further internal structure. Obtaining a description of such a state turns out to be easier if we take recourse to the brane construction: starting with a dyon of large charge which corresponds to a complicated string web, we decompose the web into smaller sub-webs representing constituent dyons and so on (sort of reminiscent of wee-partons in a hadron). This typically corresponds to a (nested) configuration in spacetime with multiple charge cores. The (approximate) internal structure of the parent dyon resulting from this tree-like process becomes increasingly reliable in regions of moduli space exhibiting a hierarchy of scales which enables the constituent dyons (at a given level in the tree) to be pointlike and widely separated within their parent dyons (immediately above in the tree). The structure of these states becomes increasingly more complicated as their charges (or alternatively the number of branes) increase.

We then use these techniques to study the internal structure of dyons corresponding to string webs with internal faces.

We first review dyons/webs in Sec. 2, and then describe transitions in the dyon internal structure as we move on the moduli space in Sec. 3. In Sec. 4.1, we describe dyon/web configurations, while Sec. 4.2, 4.3 describe more general dyon/web states of higher charge. Sec. 5 discusses webs with internal faces. Finally we discuss a few issues in Sec. 6, in particular pertaining to how reliable these configurations are. An Appendix reviews the basic framework we use here.

## 2 String webs from field theory

We give here a brief description of the construction of string web dyon states in SYM theories from their low energy (Higgsed from , with an overall decoupled ) effective theories, in part making transparent their corresponding D3-brane constructions (this mainly follows [12, 13]). The general field theory strategy is to extremize the energy functional and construct first-order Bogomolny equations which relate the electric and magnetic fields linearly to gradients of the scalar fields, which can then be solved subject to certain charged source boundary conditions, such that the resulting solutions extremize the mass of the charged states in question. The scalar field configurations obtained thus are maps , from spacetime to the moduli space of the gauge theory. These approximate solutions to the theory become more and more exact in the vicinity of lines of marginal stability and give a constructive answer to the question of the existence and stability of these states. They describe string webs on the moduli space of the field theory, which can then be shown to “fold” into string webs stretching between D-branes in transverse space. The minimax problem involved in the field theory construction is straightforward but not simple, especially for higher rank theories. However in known examples, it effectively reproduces brane-prong configurations (generalizations of -BPS brane-spikes [8]) that can be written down relatively simply and intuitively: it is these effective brane-prong field theory configurations that we will find useful here.

### 2.1 Su(3) dyons and webs: mostly a review

For simplicity and ease of illustration, we describe -BPS string web dyon states in SYM theory (along the lines discussed at length in [12, 13], and reviewed briefly in Appendix A), arising as the low energy theory on three non-coincident D3-branes, Higgsing (with an overall decoupled ). This is mainly a review (but presented slightly differently from [12]), meant to set our notation for what follows.

Let us first recall that -BPS states are only charged with respect to a single (relative) . Thus the electric/magnetic charge vectors of a -BPS charge state are

 Qe=pα=p(e1−e2) ,Qm=qα=q(e1−e2) , (1)

where we have defined simple roots , with , and are the (orthonormal) roots of , with . There of course exist -BPS states charged under different s as well. Labelling these states by their charges w.r.t. (the total charge is zero, decoupling the overall ), and writing the charge vectors in terms of the basis of makes transparent the connection to the brane constructions of these states. For example, the above -BPS state can be interpreted as a (oriented) string stretched between two D3-branes, with the two D3-branes carrying point-like dyons of charge and respectively. As an example, the scalar field configuration (for the two scalars of the two D-branes) representing say an electric charge in a theory is

 X1=e|→s−→s0|−X0 ,X2=−e|→s−→s0|+L , (2)

where is the unit of electric charge, and the two D3-branes are located at and . For magnetic charges, we have say , with the unit of magnetic charge. The two separate brane spikes join at , where : regulating the divergence in this abelian approximation111Note that these are approximate solutions in this abelian framework: e.g. the two sides and do not join smoothly. Note also that the Bogomolny bound equations for the BPS sector from the full nonlinear Born-Infeld action are the same as the ones from this leading order approximation (although their masses might differ by numerical factors). That these effective actions are insufficient is not surprising, since near a charge core, the field strengths are not slowly varying. In this region, new physics (higher derivative contributions, nonabelian physics etc) enters., this gives

 eα→s0,→s0−X0=X′0=−eα→s0,→s0+L ⇒ eα→s0,→s0=L+X02 , (3)

where is the (approximate) inverse core size. This gives the location of gluing on the moduli space as , which is the midpoint of the line joining the two branes, corresponding to the singularity of enhanced gauge symmetry. For the magnetic charge or monopole, this agrees with our expectation that the nonabelian fields are nontrivial inside the monopole core (roughly given by the Higgs vev), which via the scalar configurations is located at on the moduli space. In the abelian description here, these are approximated as Dirac monopoles.

Analysing the energy222A field theory vev and its corresponding coordinate length in transverse space are related as , as can be seen from e.g. the D-brane DBI action. of this state to compare with the mass  of a fundamental string of tension stretched between the D-branes (and a similar analysis for a D-string) shows that we must set

 e=gs,g=1 . (4)

(the numerical factors are not important for what follows.) This makes intuitive sense: at weak coupling (small ), we see as expected that the fundamental string ending on the brane at causes only a small deformation to the brane worldvolume (as shown by say ) away from , where we see a sharp spike. However the D-string is not a small perturbation as reflected by the -independence of the magnetic charge unit . In spacetime, this implies that the monopole core size is , i.e. set by the Higgs expectation value, while the electric charge core size is : thus for small , electric charge cores can be regarded as pointlike.

In contrast to -BPS states, -BPS states in SYM theory are charged under both s. The electric and magnetic charge vectors of the generic web labelled as are

 Qe=p1.α+p2.(α+β)=(p1+p2)e1+(−p1)e2+(−p2)e3 , Qm=q1.α+q2.(α+β)=(q1+q2)e1+(−q1)e2+(−q2)e3 . (5)

From the expressions in terms of the s, it is straightforward to interpret this as a string web (see Figure 1), with and strings constituting the three legs ending on three D3-branes. Regarding outgoing charge from a charge center as positive, the dyon charges on the three D3-brane worldvolumes are and .

The field theory description shows that one can think of this state as a charge dyon, spatially consisting of two constituent charge centers , separated by a distance inversely proportional to the distance from the LMS. Then as one approaches the LMS, the -BPS state (represented as ) decays as

 (p1.α+p2.(α+β), q1.α+q2.(α+β))  →  (p1α, q1α) + (p2(α+β), q2(α+β)) , (6)

into the two constituent -BPS dyons of charge and representing the two separate -BPS and strings. Here we describe the simplest such configuration . The configuration of scalars describing this string web (Figure 1) is

 X0=e|→s−→s1|−X00 , Y0=g|→s−→s2|−Y00 , X1=−e|→s−→s1|+X01 ,Y1=0 , X2=0 ,Y2=−g|→s−→s2|+Y02 . (7)

From the D-brane point of view, the interpretation is clear (see Figure 1): represent scalars of the three D-branes , with worldvolumes parametrized by , so that this configuration describes brane-prong deformations of the three brane worldvolumes. We have the boundary conditions on the scalars:

 →s→∞: (X0,Y0)→(−X00,−Y00) ,(X1,Y1)→(X01,0) ,(X2,Y2)→(0,Y02) , →s→→s1: (X0,Y0)→(X01′,0)←(X1,Y1) , →s→→s2: (X0,Y0)→(0,Y02′)←(X2,Y2) , (8)

Defining the inverse core separations and core sizes

 αij≡1|→si−→sj| ,  i≠j,αii=1ϵi , (9)

for some cutoffs on the charge core sizes, we obtain the following constraints on the existence of a solution to the above system within the approximation:

 →s→→s1:X01′=eα11−X00=−eα11+L1 , 0=gα12−Y00 , →s→→s2:0=eα12−X00 , Y02′=gα22−Y00=−gα22+Y02 , (10)

giving

 X01′=X01−X002 , Y02′=Y02−Y002 , eα11=X01−X01′=X01+X002 , gα22= Y02−Y02′=Y02+Y002 ,   eα12=X00 . (11)

This thus solves for the charge core sizes and the separation between the charge core centers

 r12=1α12∼gsX00 , (12)

and further gives also the constraint (since appears in both lines of (2.1))

 egY00=X00 . (13)

Clearly a solution with these charges exists only if the physical core separation , i.e. only for , i.e. on one side of the line of marginal stability, which passes through the junction at . Note that is inversely proportional to the length of the shortest leg (i.e. the -string) of the string web. As , we see that , i.e. we have two widely separated pointlike charge cores. In more detail, the abelian approximation is good when , i.e. in the region of moduli space where  and . In this region, the approximation neglecting the microscopic nonabelian physics of the charge cores becomes increasingly good, and this molecule-like description in terms of pointlike dyon constituents is reliable. Furthermore, the single from brane-0 with only charge boundary conditions from branes-1,2 yields an increasingly good description of the spatial structure of the dyon-web (see Figure 1).

In terms of the brane-prong interpretation, we see that the leg of the prong from brane-1 joins the corresponding leg from brane-2 at the location of the charge core , while the leg of the prong from brane-1 joins the corresponding leg from brane-3 at the location of the charge core . Note that the existence and structure of the dyon configuration is closely tied to the geometry of the D-branes in transverse space, e.g.  and so on.

The cutoff sizes () also shows that the charge cores are located at the enhanced symmetry points and on the moduli space. At these locations, there are light nonabelian fields which must be included into the low energy effective action for a nonsingular description. Clearly, there is also a in moduli space where the core sizes are comparable to their relative separations so that the charge centers effectively coalesce.

## 3 Su(3) dyons: moduli space transitions in internal structure

Now we discuss what seems to be a generic feature of these field theory dyons: this is the fact that the apparent internal structure of these dyons undergoes transitions as we move around on the moduli space. It is easiest to describe this in a specific example: consider the string web. Keeping explicitly the charges w.r.t. all of the s in the theory, the asymptotic charges of this dyon at spatial infinity can be read off from the charge vectors in (2.1) as

 { (1,1)0 (−1,0)1 (0,−1)2 } , (14)

where the subscript labels the that the charge values refer to.

Consider the region in moduli space where : here the leg of the web is short and the corresponding field configuration is described in (2.1), with the core sizes and separations given in (2.1). As we have seen, the internal structure of the dyon in this region can be essentially thought of as consisting of two charge cores of charges

 X00,Y00≪X01,Y02: {(1,0)0(−1,0)1} ,{(0,1)0(0,−1)2} . (15)

Now consider the region in moduli space where , i.e. the leg of the web is short. It is straightforward to study the spatial structure from constructing a similar field configuration as before. We then find that the dyon can now be thought of as consisting of two cores of charges

 Y02≪X00,Y00,X01: {(−1,−1)2(1,1)0} ,{(1,0)2(−1,0)1} . (16)

Similarly, the region  in moduli space where the leg of the web is short shows the dyon constituents to have charges

 X01≪X00,Y00,Y02: {(−1,−1)1(1,1)0} ,{(0,1)1(0,−1)2} . (17)

In going from one of these regions to another in moduli space, one does not cross any line of marginal stability: clearly the dyon exists and is stable in these transitions. Furthermore it is clear that there is no violation of charge conservation since the charges at spatial infinity are unchanged throughout. What is happening in the process of transiting between any two of these regions in moduli space is simply charge rearrangement. In the region where say the web leg ending on D-brane is shortest, i.e. where D-brane is closest to the LMS, the charges w.r.t. this break up into constituents, and similarly in the other regions of moduli space333It is worth recalling something similar in e.g. Seiberg-Witten theory [22]: the W-boson of charge , elementary at weak coupling, is best described as a string-web-like configuration (in the D-brane construction via F-theory [23]), a bound state of the monopole (charge ) and dyon (charge ), that decays across the line of marginal stability in the strong coupling region. A description of this decay in spacetime appears in e.g. [12] (using equilibrium scalar configurations as here) and [14] (using the long-range forces between the constituents)..

Let us look more closely at the transition between say the region , and . From the web, it is clear that we extend the leg and shrink the leg, while correspondingly in spacetime, the size of one of the cores increases with the core separation decreasing. At some intermediate point , we have : from (2.1), we see that this point, where , corresponds to the singularity of enhanced symmetry on the moduli space. This is not surprising since we expect that the nonabelian degrees of freedom become light and cannot be neglected when the dyon constituents approach each other. The Higgs vev hierarchies are different on either side of the transition.

Analyzing the other transitions shows similar structure: transitions in the internal structure pass through singularities of enhanced symmetry in the moduli space.

## 4 Su(n): general (n,m) dyon web states

Now let us try to understand the internal structure of more general dyons: for concreteness, we consider a dyon whose charge w.r.t. some is (where the subscript refers to the w.r.t. which this charge is defined), and study the corresponding field configuration, which should then give insights into the internal structure.

This state (unless -BPS), will typically exhibit a rich structure of decay across one or more lines of marginal stability, where it becomes loosely bound. The field configuration describing this can be written down reliably in the vicinity of the simultaneous location (coincidence) of the various lines of marginal stability at which this state can decay. A simple and intuitive way to obtain the field configuration without “deriving” it rigorously is to note that ultimately this state comprises and therefore decays into strings and strings. Far from any LMS, we expect that the dyon is pointlike so that the part of the scalar field configuration stemming from is

 X0=ne|→s−→s0|−X00 , Y0=mg|→s−→s0|−Y00 , (18)

i.e. a single spike emanating from , representing the string beginning on a D-brane located at , and ending on a stack of D-branes located at approximately . However as we move on the moduli space to split up the D-brane stack at (on the Coulomb branch), the dyon constituents break up and begin to separate, revealing some internal structure. In other words, the charge core at gets resolved into multiple distinct charge cores within as the single D-brane stack at splits to a multicenter solution on the Coulomb branch. Each core is charged w.r.t. as well as one or more other s: this corresponds to the fact that each constituent string (or string web) is stretched between two or more D-branes, thus carrying charge under the corresponding s. Clearly we can split up the D-brane stack in many distinct ways: this leads to correspondingly distinct internal structures for the dyon, depending on how we split up the charge.

As an example, consider say a dyon of charge . With no splitting, i.e. with a single stack at , we have a -BPS state in effectively an theory, corresponding to the string stretched between the D-branes. Now split the stack into two: depending on how we break up the charge into two, we get different string webs with three legs in an theory. Similarly starting with the dyon/string emanating from the D-brane at , and splitting up the stack into say centers on the Coulomb branch gives distinct string webs with legs in an theory, depending on how the charge is broken up into constituents. In other words, for each leg of the string web in transverse space, we have a charge core in spacetime. Assuming the final elementary constituents to be only and charges, we can split a dyon/string into irreducible string webs at most in an   gauge theory.

Such a maximally split dyon state corresponds to the charge vectors

 Qe=ne0+n∑i=1(−1)eEi ,Qm=me0+m∑i=1(−1)eMi , (19)

in an   gauge theory444These can also be regarded as dyonic states studied by Stern and Yi [7] (in the context of identifying their degeneracies), given by charge vectors (for appropriate ) or equivalently, using the roots , of , . The endpoint charges at each show that this state represents a string web with legs carrying charges

 {(n,m)0,(−1,0)E1,(−1,0)E2,…,(−1,0)En,(0,−1)M1,…,(0,−1)Mm},

and ending on the D-branes ( represents ). Diagrammatically this can be represented as the web in Figure 4 (the Figure shows the web described in more detail later). This state is classically BPS since its constituents are essentially and strings which together preserve a -th of the supersymmetry (see however the Discussion, Sec. 6, for more on this). This is vindicated by the field configuration we exhibit below as a solution to first order Bogomolny bound equations in the SYM theory. For , this reduces to a dyon, i.e. a monopole with electric charges attached: this is clearly the simplest such dyonic state and there are simplifications in its internal structure. It is thus efficient to glean insights into dyons of higher magnetic charge by splitting their charges to ultimately reduce to the form of an state. A systematic way to implement this is obtained by noticing the sequential decomposition

 (n,m)→ (n,m−1)+(0,1)→ (n,m−2)+(0,1)+(0,1)→ … (n,1)+m∑(0,1) (20)

of the state. This is of course reliable in specific regions of moduli space, which we will describe below. Also note that this decomposition gives non-degenerate constituents only if are prime (not just coprime).

The D-brane worldvolume scalars describing the string web representing this dyonic state in this theory can then be written, generalizing the simple 2-center web (2.1), as

 X0=En∑i=1e|→s−→si|−X00 , Y0=Mm∑i=1g|→s−→si|−Y00 , Xk=−e|→s−→sk|+X0k , Yk=Y0k ,  k=E1,…,En, Xk=X0k , Yk=−g|→s−→sk|+Y0k ,k=M1,…,Mm . (21)

This is valid in the region of moduli space where brane- is near one or more lines of marginal stability. The intuition for writing these field configurations is as we have mentioned above: for brane-, the scalars have prongs extending from each of the charge cores, which then glue onto single charge spikes from other branes-. The boundary conditions on these scalars are

 →s→∞: (X0,Y0)→(−X00,−Y00) ,(Xk,Yk)→(X0k,Y0k) ,k∈{Ei,Mj} , →s→→sEk: (X0,Y0)→ (X0Ek′,Y0Ek′)← (XEk,YEk) , →s→→sMk: (X0,Y0)→ (X0Mk′,Y0Mk′)← (XMk,YMk) , (22)

the being the locations in transverse space where the prongs glue onto each other (to be distinguished from the vacuum moduli values ). These then give constraint equations on the field configuration to exist:

 →s→→sEk: X0Ek′=eEn∑i=1αEk,Ei−X00=−eαEk,Ek+X0k , Y0Ek′=gMm∑i=1αEk,Mi−Y00=Y0k , →s→→sMk: X0Mk′=eEn∑i=1αMk,Ei−X00=X0k , (23) Y0Mk′=gMm∑i=1αMk,Mi−Y00=−gαMk,Mk+Y0k .

In what follows, we use the above equations to write out scalar field configurations and the corresponding constraints from boundary conditions for dyons in various  theories, and glean insights into their internal structure.

### 4.1 Su(4) dyons: 3-center configurations

In this section, we study string web states describing 3-center dyon bound states in the Higgsed theory stemming from (or more precisely ). Consider the dyon, represented by the string web . This can be described by the following 3-center scalar configuration (left side of Figure 3):

 X0=e|→s−→s1|+e|→s−→s2|−X00 , Y0=g|→s−→s3|−Y00 , X1=−e|→s−→s1|+X01 ,Y1=Y01 , X2=−e|→s−→s2|+X02 ,Y2=Y02 , X3=X03 , Y3=−g|→s−→s3|+Y03 . (24)

In the limit where all centers coalesce, i.e. , the deformation from brane-0 resembles a spike representing a -string. Thus the centers are electric charges, while is a monopole. We have the boundary conditions near each charge core:

 →s→∞: (X0,Y0)→(−X00,−Y00) ,(Xk,Yk)→(X0k,Y0k) , →s→→s1,2: (X0,Y0)→ (X01,2′,Y01,2′)← (X1,2,Y1,2) , →s→→s3: (X0,Y0)→ (X03′,Y03′)← (X3,Y3) . (25)

This gives the constraints:

 →s→→s1: X01′=eα11+eα12−X00=−eα11+X01 , Y01′=gα13−Y00=Y01 , →s→→s2: X02′=eα12+eα22−X00=−eα22+X02 , Y02′=gα23−Y00=Y02 , →s→→s3: X03′=eα13+eα23−X00=X03 , Y03′=gα33−Y00=−gα33+Y03 . (26)

It is straightforward to simplify these equations towards solving for the core sizes and separations, and we find in particular that the electric-magnetic core separations are fixed as

 gα13=Y00+Y01 ,gα23=Y00+Y02 , (27)

showing the two distinct lines of marginal stability to be at and , i.e. at , and . Note however that the electric-electric core separations are not determined completely. In this case, with a single monopole, the magnetic core size is also fixed, while the electric core sizes are fixed only upto the separations. This incompleteness in the solution turns out to be a generic feature as we go to higher charge, as we will see in what follows. Finally, consistency of the solution (compatibility of the equations above) gives the constraint

 eg(2Y00+Y01+Y02)=X00+X03 , (28)

on the moduli values of this solution. To elucidate the meaning of this constraint, notice that for large , this simplifies to , which describes the line of slope traced out by the -string in the transverse space, while the limit