Brane Configurations of BPS Domain Walls for the Gauge Theory
Abstract
We study supersymmetric domain walls in gauge theory with 3 massive adjoint representation chiral multiplets. This theory, known as , can be obtained as a massive deformation of YangMills theory. Following Polchinski and Strassler, we consider the string dual of this theory in terms of spherical 5branes and construct BPS domain walls interpolating between the many vacua. We compare our results to field theoretic domain walls and also find that this work is related to the physics of expanded “dielectric” branes near zero radius.
1 Introduction
Recently, Polchinski and Strassler [1] found the string theory dual to an gauge theory with adjoint matter, which can be obtained by giving masses to three chiral superfields in the gauge theory. They called this field theory and obtained the string dual by generalizing the AdS/CFT correspondence [2]. On the string side of the duality, the D3branes that source the AdS geometry arrange themselves into 5branes with units of D3 charge due to the RR 6form corresponding to the mass perturbation in the CFT (as first discussed in [3]). Following Myers [3], they suggested that the D3branes form a 5brane extended in the dimensions of the D3branes with the other two dimensions wrapped on an . The numbers and types of 5branes in the configuration correspond to specific vacua in the gauge theory, and their vacuum (coordinate) radii and orientations are determined by a superpotential on the brane. In particular, the totally Higgsed vacuum corresponds to a single D5brane, various Coulomb vacua correspond to multiple D5branes, and the confining and oblique confining vacua correspond respectively to a single NS5 or 5brane (for ). For specifics of the brane description, see [1]; other studies of the theory include [4, 5, 6, 7, 8, 9].
Because the theory has a large (but finite for finite ) number of vacua, there should generically be domain walls interpolating between pairs of those vacua. In supersymmetric theories with domain walls, there has been much interest in finding BPS domain wall solutions – domain walls that preserve some supersymmetry. In particular, for gauge theories, studies of BPS domain walls include [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]; see [12] for further references on BPS domain walls. Recently, [8, 20] studied BPS domain walls in the theory from the field theory perspective.
In this paper, we investigate the brane configurations corresponding to BPS domain walls that interpolate between different vacua in the gauge theory. On the string side, the 5branes bend from one vacuum state to the other, and when branes with nonzero net 5brane charge intersect, another 5brane fills the at the intersection [1]. Polchinski and Strassler [1] discussed two examples of domain walls in the small coupling limit and compared the vacuum state superpotentials and domain wall tensions to exact field theoretic calculations of [7, 8], finding agreement within their approximations.
We expand those results to finite string coupling and construct domain wall brane configurations. In section 2, we review the 5brane actions of [1] and establish some approximations. In section 3, we find conditions necessary for the mechanical equilibrium and supersymmetry of the 5brane junctions, and section 4 uses the general results of sections 2 and 3 to confirm that the BPS bound for the domain wall tension on the string side matches the field theoretic bound. In section 5, we construct a number of BPS domain walls and note interesting examples. Finally, in section 6, we summarize our results and discuss their relation to the body of research on BPS domain walls in field theory. In both sections 5 and 6, we will discuss the supersymmetric minimum in the brane potential at vanishing 5brane sphere size [1, 21] and its relation to the brane picture of BPS domain walls. In the end, though, we will have to confess ignorance as to the meaning or existence of a zero size state for the 5brane spheres.
While preparing this paper, we became aware of work by C. Bachas, J. Hoppe, and B. Pioline [20] that has some overlap with this work. Specifically, they find BPS domain wall configurations interpolating between Coulombic vacua (and the Higgs vacuum) within field theory. We discuss these domain walls in section 5.1 and compare our results to those of [20] in section 6.3.
2 Brane Actions
In this section, we will discuss the action that describes the 5brane bending for the domain walls. Through the rest of this paper, we will follow the conventions of Polchinski and Strassler [1], working to leading order in their small parameter, the ratio of 5brane to D3 charge. In doing so, we ignore the nearshell corrections to the metric and supergravity fields, so we take the dilaton to be constant and the Einstein frame metric to be equal to the string frame metric.
First, we rederive the action for brane bending given in equation (126) of [1]. The part of the 5brane twists and contracts (or expands) as we pass from one vacuum state to another (moving in, without loss of generality, the direction with translational invariance in the and directions). To start, we note that the induced metric in the directions parallel to the D3branes is
(1) 
The DiracBornInfeld action of a 5brane is therefore
(2)  
Here, with coupling for the brane, and is the metric on the wrapped of the 5brane [1].
Under the assumption of slow bending (small derivatives of brane position), the factor in the directions gives
(3) 
In the last step, we have substituted the scalar field introduced in [1] to describe the size and orientation of the . For a real unit vector on the , is defined by
(4) 
This may seem like a somewhat suspect expansion, given that diverges at the brane. However, it corresponds to the selfreaction of a charge, which should be ignored, as discussed in [1]. We can think of the 5brane as built up from infinitesimal 5branes, each of which acts as a probe brane to the rest of the geometry. Since each probe action is independent of , the expansion follows for the full 5brane.
The rest of the action follows as in [1]; we can expand both the determinant and the ChernSimons action in powers of the D3 charge, which is assumed to dominate. Then the action for D3 charges (integrated over the ) is
(5) 
where is the derivative of the superpotential
(6) 
A few comments are in order. First, the leading terms from the DiracBornInfeld and ChernSimons actions cancel, as they should for parallel D3branes. However, when we consider the tension of the 5brane, it is precisely the D3 tension that dominates. Also, the vacuum configuration of the 5branes occur at the roots of the potential , where the configuration and superpotential are
(7) 
For a configuration with multiple branes, the superpotential is summed over the branes, and the vacuum configuration of each brane is unaffected by the other branes [1].
Since we hope to find supersymmetric domain walls (with configurations varying only in the direction), it is useful to write the action (eqn. 5) as
(8) 
with a complex phase. From the above, we can see that brane bending that follows the “BPS equation”
(9) 
both satisfies the equations of motion and preserves supersymmetry. In this case, the domain wall tension due to the brane bending is given by the surface terms. Further, each 5brane in the domain wall must bend with the same phase for supersymmetry to be preserved. Also, since the BPS equations for different 5branes (in the same vacuum state, say) are decoupled except for having the same value of , each 5brane bends independently. We emphasize that each 5brane follows its own BPS bending equation independently of any other 5branes present and note that we will consider only domain walls with BPS brane bending.
At some point in the transition from one vacuum to another with a differently charged 5brane(s), charge conservation requires that an additional 5brane be present, as discussed in [1]. If the domain wall interpolates between, for example, a 5brane and a 5brane, then the third 5brane has charge . Further (taking the vacua to change along the direction), this extra 5brane extends in the directions and fills the at the intersection of the two “vacuum branes.”
The DBI part of the action of this 5brane ball is (at lowest order in the perturbations)
(10) 
where is the radius of the and is the 5brane charge as above. The metric factors have canceled because the 5brane is extended in 3 dimensions each of factor and . Then the domain wall tension due to this ballfilling brane is
(11) 
for the configuration for all the vacuum branes at the brane junction.
3 ForceBalancing Equations
We consider in this section the forcebalancing equations at the 5brane junction, which are needed to maintain mechanical equilibrium and preserve supersymmetry. These are analogous to conditions derived in [22, 23] but are quantitatively different due to the D3 tension here. We specialize to a static domain wall along (hereafter ) and translationally invariant in . We consider any number of 5branes labeled by the index with 5brane charges and units of D3 charge each approaching the junction from smaller and 5branes labeled by approaching from larger .
For a 5brane junction to occur, the 5branes must first intersect. On first glance, this means that the brane configuration variables must be the same for all the vacuum 5branes at the junction. However, we note that and describe the same sphere, so there could be a brane junction between, for example, two 5branes with opposite . We deal with this case by noting that the action (8) (and BPS equation (9)) is invariant under , so a 5brane is equivalent to the corresponding anti5brane with opposite configuration variable. Physically, taking leaves the invariant while reversing its orientation and therefore the 5brane charge. Then we can always choose 5brane charges at a junction so that all the 5branes have the same configuration variable, which takes the value at the junction (though we might need to consider different junctions separately in each domain wall).
The forcebalancing conditions should be derived in an inertial reference frame. In this flat metric, the D3 and 5brane tensions are simply the flat space values and give the force per proper area on the junction in the directions of extent of those branes. Thus, for D3branes on an of coordinate radius , the total D3 tension on a unit area of the is
(12) 
Technically, we are working slightly away from the 5branes (so is finite) – this substitutes for building up the domain wall out of infinitesimal charges – in an orthonormal basis aligned along the coordinate axes. We consider the north pole of the spherical brane junction, which is in the plane with . Henceforth, we denote .
Now consider a unit vector along each vacuum brane in the direction out of the junction. In the small bending approximation,
(13) 
where bold indices are the orthonormal frame indices and the signs are for the indices respectively. A ballfilling 5brane has a tangent vector
(14) 
leaving the north pole of the brane junction. Then mechanical equilibrium requires
(15) 
for . Equation (15) gives the conditions
(16) 
in the direction and
(17) 
in the directions.
The first of these equations simply gives conservation of D3 charge across the junction. For BPS brane bending according to equation (9), D3 charge conservation requires the linear terms from the potential to cancel. Then we find simply that
(18) 
This implies that the discontinuity of the superpotential at the junction is
(19) 
Note, however, if there is no 5brane ball at the junction, there is no condition on the phase of .
To this point, we have considered only positive D3brane charge, as negative D3 charges would break supersymmetry and add a high energy cost. We can also see this from mechanical equilibrium considerations; if we allow negative D3 charges, equation (16) becomes
(20) 
which conflicts with charge conservation at the junction. Thus, we conclude that negative D3 charges will not appear in any BPS domain wall.
4 Domain Walls: Generalities
In this section, we discuss the BPS bound for the domain wall tension and confirm that it matches the BPS bound from field theory for a domain wall interpolating between any two vacua of the theory. We also make some observations that will allow us to discuss some specific explicit domain wall solutions in the following section.
As discussed above, we consider domain walls which follow the BPS equation (9) for bending of the 5branes. Combining the BPS equation and its conjugate, we find
(21) 
or that the imaginary part of is conserved. This implies that, up to discontinuities in at brane junctions, the BPS trajectory follows a straight line in the complex plane with a tangent vector of complex phase (in the direction of increasing ). Due to the forcebalancing condition equation (19), the discontinuous jump in at a brane junction also has the phase , so the trajectory of the superpotential, including discontinuities, is then a straight line directly from to (ie, between the two vacuum values). Thus we have . In the case that there is no ballfilling 5brane at a junction, there is no constraint from mechanical equilibrium, but there is also no discontinuity in the superpotential, so we still have a straight line. We should note that, in field theory, the superpotential trajectory would also follow this straight line but without discontinuities.
Now consider the domain wall tension. From the action (8) and BPS equation (9), the tension from brane bending is
(22) 
along any single 5 brane. We see that gives a negative tension, which comes from a positive definite Hamiltonian, so that choice is unphysical. Taking , the brane bending contributes to the domain wall tension. Further, twice the discontinuity in the superpotential at a 5brane junction is equal in magnitude to the tension of the 5brane ball at that junction:
(23) 
(compare to eqn. (11)). Thus, assuming , we find the BPS bound for domain wall tension
(24) 
in agreement with field theoretic results since [1] found that the vacuum superpotentials (eqn. (7)) match the exact field theoretic results (within our approximations). Since the BPS bound on the domain wall tension follows from the supersymmetry algebra (see, for example, [12] for a review of the central charges of domain walls), it is not surprising to find the same bound; however, it is satisfying to see new physics give the same tension.
5 Domain Walls: Specifics
In this section, we will discuss explicit domain wall solutions and comment on them. We will begin by discussing domain walls between vacua with only one type of 5brane (for example, D5branes only on both the left and the right), which we can discuss analytically because the 5brane spheres keep the same orientation throughout the domain wall. We will proceed to consider domain walls between vacua with single 5branes of different types and will finally discuss some examples of domain walls between vacua with multiple 5branes of different types. Throughout, we take the mass to be positive.
5.1 Single Type of 5Brane Charge
Here, we consider only domain walls between vacua with only one type of 5brane charge given by . As discussed above, each 5brane has a vacuum configuration and superpotential given by equation (7) (inserting the appropriate value of for each 5brane). Therefore, we see that all the 5branes have the same orientation and same phase of the superpotential. For definiteness, we will always take domain walls with the magnitude of the superpotential increasing from negative to positive , which gives
(25) 
Additionally, we can rotate the phase of the brane configuration variables to with vacuum . Then the BPS equation becomes
(26) 
Without losing generality, we can take all the vacuum 5branes to be D5branes from this point forward; the only difference would be the size of .
To begin, we consider domain walls in which remains real for all the 5branes, so the BPS equation has the following solutions:
(27)  
(28)  
(29) 
In the above, is an integration constant, and the vacuum value is the appropriate value for a 5brane with D3 charges. The two positive solutions are plotted in figure 1. It is important to note that none of these solutions goes to as , so all the vacuum D5branes at negative must remain in their vacuum configurations until they reach a brane junction (for the real solutions that we consider here). We will come back to this point later. Also, the forcebalancing condition (eqn. (18)) for implies that any 5brane junction should have more D5branes on the left (lesser ) than on the right (or an equal number) in a BPS configuration.
Another useful solution to the BPS equation is that for a sphere of D3branes with no 5brane charge (henceforth a “zero5brane”), which bends according to the BPS equation with . With and defined as above, the BPS equation for a zero5brane and its real solution are
(30)  
(31) 
A few words are necessary about the zero5brane configuration. In a vacuum state, the D3 branes with no 5brane charge would collapse to a point. This situation seems physically similar to the minimum of the potential at for any 5brane charge, which is usually considered an unphysical minimum [1, 21]. However, inside a domain wall, we need not be concerned whether the zero5brane corresponds to a physical vacuum or not. We will discuss this point further below.
Now we can construct explicit domain wall solutions. Because we are taking all the vacuum 5branes to be D5s, we are discussing domain walls between various Coulomb vacua and the Higgsed vacuum – Coulomb vacua are vacua with multiple D5branes and or gauge symmetry unbroken, while the Higgs vacuum is a single D5brane with all gauge symmetry broken. In the following domain walls, we will define , the vacuum configuration for a single 5brane vacuum, for convenience.
The first domain wall we will consider is between a vacuum with two D5 branes with and D3 charges each () and a single D5 with all D3 charges. In this domain wall, the smaller D5 on the left remains in its vacuum state for , where there is a junction with a ballfilling D5brane and a zero5brane that follows the solution . The larger vacuum brane on the left stays in its vacuum state for , where it enters a junction with the other end of the zero5brane and the vacuum brane from the right. The single D5brane on the right follows for . This configuration is shown in figure 2. We should note that when , there is no zero5brane.
Another domain wall of interest is that between two Coulomb vacua, such as the vacuum discussed above, with D3 charges given by on the left and on the right. Because the magnitude of the superpotential of such a vacuum is proportional to , we see that we should take to have the larger superpotential at positive . The BPS domain wall corresponding to these vacua is shown in figure 3(a). It has no 5brane balls and has brane junctions between the two smaller D5s and between the two larger D5s, which are connected by a zero5brane. If we choose to have the junction of the two smaller branes at , the small D5 on the right bends according to for , the zero5brane bends according to for , and the large D5 on the right bends according to for . Here, .
Some interesting physics arises if we consider other domain walls interpolating between these two vacua. If we just consider BPS brane bending (as in eqns. (27,28)), it appears that we could have BPS domain walls such as those shown in figure 3(b), where the small D5 on the right does not connect to the small D5 on the left. In fact, it appears that there is a continuous family of such domain walls with a D5brane branching off of a zero5brane. However, these domain walls are not BPS and are not mechanically stable (despite the fact that they have BPS brane bending) because the brane junction involving the small vacuum brane on the right does not satisfy the forcebalancing condition (18). These domain walls actually have a higher tension than the BPS bound due to “backtracking” of the domain wall trajectory in the complex plane. Essentially, the smaller D5brane at positive has to bend too much. So, given one of these nonBPS domain walls, the D5brane balls, which are antibranes of each other, are free to attract (because there is a continuous family of domain walls) and annihilate, leaving the BPS solution, classically at least.
A final illustrative case to consider is a domain wall interpolating between Coulomb vacua with two and three D5branes respectively. If none of the three 5branes is larger than both of the two 5branes, then a BPS domain wall can be constructed similar to the first domain wall discussed above (see figure 4 for some BPS domain walls). However, in cases where one of the three D5branes is larger than both of the D5 branes in the other vacuum, there appears to be no BPS domain wall constructed out of real solutions to the BPS equation. If, for example, the three D5brane vacuum has a smaller magnitude superpotential, the domain wall would require a zero5brane with negative D3 charge, which we have seen are ruled out. Similarly, if the three D5 vacuum has a greater magnitude superpotential, there would be at least one 5brane junction with more D5branes at larger , violating the condition of equation (18).
Might there be BPS domain walls between these vacua with complex values of ? Considering the BPS equation (26) for (for real), we have
(32)  
(33) 
The boundary conditions on the vacuum branes at positive require that they do not twist (that remains real). To see this, consider a perturbation of equations (32,33) for a single 5brane around at and note that such a perturbation vanishes at only if it is real. Then equation (33) requires the solution to remain real. For the vacuum branes at negative , any imaginary perturbation is unstable (because as ), leading one to suspect that there is no BPS domain wall between these vacua. Figure 5 shows a vector field of in the plane, which seems to indicate that an imaginary perturbation would not flow back to the real axis. It is possible that some other type of 5brane plays a role in these “complex” domain walls; however, such a configuration, if it exists, would be difficult to find.
It is, however, not altogether surprising that some pairs of vacua do not have BPS domain walls; consider one vacuum with D5branes of D3 charge D3 and another vacuum with D5branes of D3 charges respectively . These two vacua have the same superpotential, so any BPS domain wall between them would be tensionless. On the other hand, the vacua are not identical, so there must be some positive tension due to brane bending in any domain wall. Thus, there is no BPS domain wall^{1}^{1}1Thanks to I. Bena and M. Patel for discussions of this point.. It is worth noting, though, that higher order effects in the 5brane charge or in the expansion might or might not lift the degeneracy of these vacua and permit a BPS domain wall. Also, it is known that supersymmetric theories with matter in the fundamental representation do not have BPS domain walls if the matter mass is too large [18, 17, 16]^{2}^{2}2We need not worry that the value of the mass will alter the spectrum of BPS domain walls in our case; since we work with a deformation of a conformal theory, is the only length scale..
We can now conjecture several conditions necessary for BPS domain walls to exist between a pair of Coulomb (or Higgs) vacua, based on the discussion above. We should stress that these conditions would be proved if we require to be real, but we cannot absolutely rule out domain walls with complex . First, the superpotential must take different values in the two vacua, . Next, because of the forcebalancing condition, the number of 5branes must not increase as (more physically, the ) increases. And further, the size of the largest 5brane (meaning its D3 charge) must not decrease as () increases; this follows because the size of a zero5brane, which carries D3 charge from one 5brane junction to another, increases with . If we considered vacua with a larger number of 5branes, we would find a larger number of conditions that must be satisfied; from the brane physics, it would seem that those conditions could be described as the conditions above applied to subsets of the 5branes in the two vacuum states.
5.2 Domain Walls Between Single 5Branes
In this section, we consider domain walls between vacua with one 5brane. As discussed in section 4, we know that the BPS trajectory along the domain wall follows a straight line in the complex superpotential plane, even including discontinuities at 5brane junctions, and . This allows us to solve the full complex BPS equation (9) numerically; we can solve the cubic superpotential for the configuration variable and plot the trajectory in configuration space as a function of on the line segment from to . If we do this for the two vacuum branes, we can find the configuration where they intersect to find the BPS domain wall. For single 5brane vacua, D3 charge is automatically conserved at the brane junction, and, for the specified value of , the phase condition (19) is satisfied up to a sign, which we check numerically.
One issue is that there are three solutions for as a function of because the superpotential is cubic. However, only two of these approach the vacuum configuration as the superpotential goes to the vacuum value (see eqn. (7)). Those two solutions correspond to the two solutions and discussed above, essentially. We can choose whichever solution will intersect with the other 5brane.
(An additional check on numerical solutions is provided by the argument of at the vacuum state. By considering linear perturbations of the BPS equation (9) around the vacuum, we find
(34)  
(35) 
for and real and imaginary parts respectively. Only one of the eigenvectors has for (or , as desired), so this gives the phase of . This checks numerically.)
The first case to consider are for a domain wall interpolating between a D5brane and a 5brane (5brane vacua are sometimes called oblique confining vacua, while the NS5brane vacuum is the confining vacuum). Figure 6 shows graphs of the brane configuration parameter for a domain wall interpolating between a D5brane and a 5brane with several values of the string coupling . For small , the 5brane looks like an NS5brane, while for large , it looks like a D5brane. Thus, for small , most of the domain wall tension is from brane bending, while, for large , it is mostly from the 3ball 5brane. The NS5 domain wall is Sdual to this case, and the other D5/NS5 to 5brane domain walls are similar. All these configurations satisfy the forcebalancing condition, and they are the only BPS domain walls found between each pair of vacua.
The domain wall interpolating between a D5brane vacuum and an NS5brane vacuum is an interesting case. For a value of the RR scalar , we can find the domain wall just as before (see figure 7(a)). For , on the other hand, the D5 and NS5 branes intersect with opposite values of , meaning that we can interpret the domain wall as interpolating between an antiD5brane and an NS5brane (or between a D5 and an antiNS5brane) (see figure 7(b)). For , though, the domain wall does not seem to exist. We can understand this from the BPS equation solutions found in section 5.1; for the appropriate value of in this case, both of the 5branes follow the solution (eqn. (27)). (See equations (34,35) to see that the appropriate perturbations are along the real and imaginary axes for the NS5 and D5branes respectively.) Thus, the NS5(D5)brane stays on the real (imaginary) axis, and they intersect only at the origin, which their solutions reach only in an infinite distance. Figure 8 shows a vector field of the BPS equation flow for both the D5 and NS5brane, and they seem not to intersect. Another argument that the D5 and NS5brane (with ) have no BPS domain wall connecting them is that such a domain wall would seem to violate the 5brane/anti5brane symmetry of the physics. This second argument should hold even when higher order corrections are taken into account. It is also notable that the tension of this domain wall would scale as in the ’t Hooft limit, indicating that it may not exist [8].
Physically, we seem to have a case of BPS spectrum restructuring in different regions of parameter space, as was discussed recently by [24]. In their language, is the curve of marginal stability for the domain wall between the D5 and NS5branes. For positive , the “stable” BPS object is a domain wall between D5 and NS5branes, while for negative the BPS domain wall is between an antiD5 and an NS5brane. At , if the analogy with the results of [24] holds, the domain wall supposedly becomes a composite of widely separated BPS solitons, presumably the domain walls between the D5/NS5branes and the “vacuum” at . As mentioned above, most sources consider the vacuum to be unphysical, and we certainly do not have a good description of its physics if it does exist, since corrections due to the 5brane charge and quantum string physics would play a large role there. Another possibility is that near shell corrections or quantum effects glue together the D5 and NS5branes at a small size, leaving a very thick but finite size domain wall between the Higgs and confining vacua. That would be a specific example of the transformation of 5brane charge at zero size conjectured in [1].
5.3 Multiple 5branes of Different Types
To find domain walls between the most general pairs of vacua, namely those with several 5branes of different types, we will typically need to use the numerical methods of section 5.2 above. These are applicable basically without modification, since the superpotential is additive over different 5branes and the BPS branebending equations are decoupled for the separate branes. It is typically difficult to find explicit solutions in the most general cases, because three or more curves in the complex plane will generically not all intersect at a point. Such general BPS domain walls, if they exist, require extra 5branes, similar to the zero5branes in section 5.1 but without exact solutions to the BPS equation that make it possible to find the domain walls. Here, we will discuss two cases with extra symmetry that allow us to find BPS domain walls.
First, we can discuss domain walls connecting D5branes to 5branes. Assuming that all of the 5branes in each of the vacua have the same number of D3 charges (that is, not all of the gauge symmetry is broken), these domain walls are more or less rescaled versions of those discussed above. These domain walls are uncomplicated because the vacua are essentially single 5branes with multiple 5brane charges.
The case of a D5 and NS5 each with D3brane charges going to a 5brane also has a special symmetry; it is selfSdual for and , and there is no ballfilling 5brane. This domain wall is shown in figure 9 for . For other values of the string coupling, the three vacuum branes never all have the same , so any domain wall would be more complicated, as in more general cases. However, one would expect that BPS domain walls would exist at least for a range of near because one does exist at that special value of the coupling.
6 Summary
6.1 Domain Walls
In summary, we have discussed BPS domain walls in the string theory dual of the theory. Using a smallbending approximation for the vacuum state 5branes, we found, as in [1], that the 5branes bend independently of each other (that is, independently of the configuration variables of the other 5branes, including the metric factor ). We were also able to establish, using conditions for mechanical equilibrium, that the BPS bound for the domain wall tension is the same in the string theory as in the field theory dual, given that the vacuum superpotentials are the same.
We then discussed domain wall solutions for a number of pairs of vacua. For domain walls between Coulomb (and Higgs) vacua, we gave analytic solutions for BPS brane bending and demonstrated a general construction of BPS domain walls. We gave an example of a nonBPS domain wall and a mechanism through which it can decay classically to a BPS domain wall. We were further able to find conditions in which BPS domain walls do not seem to exist. For a BPS domain wall between Coulomb vacua to exist:

the vacuum superpotentials must differ,

the number of 5branes must not increase as increases, and

the size of the largest radius 5brane must not decrease as increases.
We also found numerical BPS domain wall solutions interpolating between the Higgs and oblique confining vacua, as well as for some special cases with multiple 5branes.
6.2 Zero Radius 5Brane Spheres
Importantly, we also encountered physics involving the zero radius configuration of the 5brane spheres. In section 5.1, we used zero5branes to carry D3 charge without carrying 5brane charge; the vacuum state for such a zero5brane would have a collapsed sphere, if it exists. We reemphasize that a vacuum state for the zero5brane need not be physical for zero5branes to occur in nonvacuum configurations, such as domain walls. We also found that the domain wall between the Higgs (D5) and confining (NS5) vacua with zero RR scalar seems not to exist in this approximation. One interpretation of this is that the zerosize state (which is a minimum of the potential) does exist and that is a curve of marginal stability on which the D5/NS5 domain wall decomposes into two domain walls. However, recent literature is of the opinion that the zerosize state is unphysical (as supported by the string exclusion principle [21]^{3}^{3}3Thanks to J. McGreevy for a discussion of this point.). In that case, it is possible that effects due to the 5brane charge or quantum string physics connect the D5 and NS5branes at a finite size. At this time, the physics behind such a transition is not understood and remains a point of interest for future study.
It may also be interesting to compare the possible zerosize vacuum of brane physics to the controversial chirally symmetric KovnerShifman vacuum of supersymmetric theory [25]. For example, both appear as minima of effective potentials that describe the theory around a particular vacuum state (for a discussion of the VenezianoYankielowicz Lagrangian for YangMills theory [26] and an extension of it, see [27]) which may or may not be valid near the zerosize or chirally symmetric vacuum respectively. Assuming they exist, the zerosize and KovnerShifman vacua do have some at least superficial similarities; first, they both have a zero superpotential, as opposed to the (oblique) confining vacua. They also would both have BPS domain walls connecting them to the oblique confining vacua in which one real field varies (in the KovnerShifman case, that is the gluino condensate [10]). It would be interesting to calculate the gluino condensate in the zerosize vacuum through the AdS/CFT correspondence in order to see if it is also chirally symmetric, but – if that vacuum exists – it lies outside of the approximations of [1]. However, both of these vacua are generally considered to be unphysical (see [28] for a recent critique of the KovnerShifman vacuum). Perhaps a physical understanding of why one of these vacua fails to exist (or a determination that it does indeed exist) would shed light on the physics of the other.
6.3 Comparison to Field Theoretic Results
It is also possible to connect our results to previously known field theoretic results. First, we will compare our results to the recent work of [20], which found BPS domain wall configurations interpolating between Coulomb and Higgs vacua in the theory, as in section 5.1 here. In general, our results agree (including the solutions to the BPS equation, when given), but there are two comments to be made here. One comment is that the configuration (2.14) of [20] does not connect the Coulomb vacua to the confining vacuum, as we have discussed extensively above. Because [20] considered only the classical vacuum structure, they neglected quantum effects that give the confining vacuum a gluino condensate and nonzero superpotential. The other comment is a detailed comparison of our conditions for the existence of BPS domain walls to the conditions (4.21) (or equivalently (4.19)) of [20]^{4}^{4}4Thanks to B. Pioline for pointing out revisions to [20] regarding these conditions.. The condition that the number of 5branes (that is, the number of irreducible representations) not increase with increasing is precisely equivalent to the condition that not increase (following the notation of [20]). The condition on the size of the 5branes is more difficult to translate. Suppose that the largest 5brane in the vacuum with smaller has D3 charge , which is larger than any of the 5branes in the other vacuum. Then , while , where is the number of 5branes with D3 charge , violating the condition that not increase. Thus, we find agreement with [20], despite the different physics used to find our results. We should note that neither we nor [20] have shown that BPS domain walls with complex values of (corresponding to nonzero gluino condensate) connecting Coulomb vacua are ruled out, although they seem not to exist numerically.
Recent studies of BPS domain walls in supersymmetric gluodynamics in the large limit have stressed that the BPS tension between adjacent oblique confining vacua (such as an NS5 and 5brane) scales as , while the natural energy density scales as , leading to the conclusion that the domain wall thickness must vanish as [11, 12, 13, 14]. These studies have also noted that such a scaling is precisely that expected for a Dbrane, in line with a suggestion by Witten [15] that a domain wall would act as a Dbrane for the QCD string. As [1] stated, the domain walls considered here demonstrate that domain walls in the theory are indeed Dbranes. If we consider, for example, the BPS domain wall between an NS5brane and a 5brane in the ’t Hooft limit, then the vacuum states differ only at order , so the dominant contribution to the tension is from the ballfilling D5brane, which has a vanishing thickness and on which precisely the correct flux tube can end [1].
6.4 Future Directions
Finally, we should note a few future directions to take. In terms of string physics, an understanding of the domain wall between D5 and NS5branes or a definitive determination whether it exists would be important in understanding the physics of Dbrane spheres at small radius in RRform backgrounds. With the motivation of studying domain walls, this work could be extended to BPS domain wall junctions [29, 30], which have been of increasing interest in the literature [14, 31, 32, 33, 34]. Another direction might be to eliminate the smallbending approximation, which would correspond to finding a more general form of the Kähler potential for the configuration variable but would make finding explicit solutions much more difficult.
Acknowledgements
I am greatly endebted to Joseph Polchinski for many discussions and for reading this manuscript. I would also like to thank S. Hellerman, B. Pioline, I. Bena, M. Patel, and J. McGreevy for useful discussions. This material is based upon work supported by a National Science Foundation Graduate Research Fellowship.
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