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AdS/CFT and BPS Strings in Four Dimensions

We consider N=2 superconformal theories defined on a 3+1 dimensional hyperplane intersection of two sets of M5 branes. These theories have (tensionless) BPS string solitons. We use a dual supergravity formulation to deduce some of their properties via the AdS/CFT correspondence.


Introduction
The N = 2 supersymmetry algebra in four dimensions contains a central extension term that corresponds to string charges in the adjoint representation of the SU(2) R part of the R-symmetry group. This implies that N = 2 supersymmetric gauge theories in four dimensions can have BPS string configurations at certain regions in their moduli space of vacua. In particular, at certain points in the moduli space of vacua these strings can become tensionless. A brane configuration that exhibits this phenomena consists of two sets of M5 branes intersecting on 3 + 1 dimensional hyperplane. The theory on the intersection is N = 2 supersymmetric. One can stretch M2 branes between the two sets of M5 branes in a configuration that preserves half of the supersymmetry. This can be viewed as a BPS string of the four dimensional theory. The purpose of this note is to study such brane configurations using the AdS/CFT correspondence [1,2]. Such a non-localized brane system has been discussed in [3], and it does not correpsond to an N = 2 SCFT. The fully localized brane system was given a DLCQ description as a supersymmetric quantum mechanics in [4]. This system does lead to an N = 2 SCFT on the 3 + 1 dimensional intersection and will be discussed in the paper. Most of our detailed analysis will be done for a semi-localized brane system which corresponds to an N = 2 SCFT too.
The paper is organized as follows. In the next section we will present the dual supergravity description of the four dimensional theory on the intersection and discuss the field theory on the intersection. In section 3 we will use the dual supergravity description to deduce some properties of these strings. We will argue that from the four dimensional field theory viewpoint they are simply BPS string configurations on the Higgs branch.
Consider two sets of fivebranes in M theory: N 1 coinciding M5 branes and N 2 coinciding M5' branes. Their worldvolume coordinates are M5 : (x || , y) and M5 ′ : (x || , x). Such a configuraion preserves eight supercharges. The eleven-dimensional supergravity background takes the form [5,6,7] with the 4-form field strength F where * defines the dual form in the three dimensional space (z 1 , z 2 , z 3 ). When the M5 and M5' branes are only localized along the overall transverse directions z the harmonic functions are H 1 = 1 + l p N 1 /2| z|, H 2 = 1 + l p N 2 /2| z|, where l p is the eleven dimensional Planck length. The near horizon geometry in this case does not have the AdS isometry group and thus cannot describe a dual SCFT. Moreover, there does not seem to be in this case a theory on the intersection which decouples from the bulk physics.
Consider the semi-localized case when the M5 branes are completely localized while the M5' branes are only localized along the overall transverse directions 1 . When the branes are at the origin of ( x, z) space the harmonic functions in the near core limit of the M5' branes take the form [8,9] It is useful to make a change of coordinates l p z = (r 2 sin α 2 )/2N 2 , x = r cos α, 0 ≤ α ≤ π/2. In the near-horizon limit we want to keep the energy U = r l 2 p fixed. This implies, in particular, that membranes stretched between the M5 snd M5' branes in the z direction have finite tension | z|/l 3 p . Since we are smearing over the y directions we should keep y/l p fixed. It is useful to make a change of variables y/l p → y. The near horizon metric is of the form of a warped product of AdS 5 and a six dimensional manifold M 6 The metric (2.4) has the AdS 5 isometry group [11]. Therefore, in the spirit of [1], M theory on the background (2.4), (2.5) should be dual to a four dimensional N = 2 SCFT. Similar types of metrics appear in the near-horizon limit of the localized D2-D6 [10,12] and D4-D8 brane systems [13].
Note that the curvature of the metric diverges for small α as R ∼ 1 Away from α = 0, eleven-dimensional supergravity can be trusted for large N 1 . The singularity at α = 0 is interpreted as a signal that some degrees of freedom have been effectively integrated and are needed in order to resolve the singularity. These presumably correspond to membranes that end on the M5' branes.
The above M5-M5' brane system can be understood as up lifting to eleven dimensions of an elliptic brane system of Type IIA. It consists of N 2 NS5-branes with worldvolume coordinates (0, 1, 2, 3, 4, 5) periodically arranged in the 6-direction and N 1 D4-branes with worldvolume coordinates (0, 1, 2, 3, 6) stretched between them as in figure 1. When we lift this brane configuration to eleven dimensions, we delocalize in the eleven coordinate (which in our notation is 7) and since we have delocalization in coordinate 6 as well, we end up with the semi-localized M5-M5' brane system.  The four dimensional theory at low energies on the D4-branes worldvolume is an SU(N 1 ) N 2 gauge theory with matter in the bi-fundamentals. The metric (2.4) can be viewed as providing the eleven-dimensional supergravity description of an M5 brane with worldvolume R 4 × Σ [14], where Σ is the Seiberg-Witten holomorphic curve (Riemann surface) associated with the four dimensional SCFT at the origin of the moduli space of vacua [15]. In this brane set-up the R-symmetry group SU(2) R × U(1) R is realized as the rotation group SU(2) 8910 × U(1) 45 . The dimensionless gauge coupling of each SU(N 1 ) part of the gauge group is g 2 Y M ∼ g s N 2 . The ten dimensional metric describing the elliptic Type IIA brane configuration is where (2.8) Its near-horizon limit is and R 2 = (4πg s N 1 N 2 ) 1/2 . The curvature of the metric is R ∼ 1 l 2 s R 2 and it has a dilaton e Φ = gsN 2 R sin α −1 . This supergravity solution can be trusted when both the curvature and the dilaton are small. Away from α = 0 we need large N 2 and N 2 ≪ N 1/3 1 . When the latter condition is not satisfied the dilaton is large and we should consider the elevendimensional description.
When both sets of M5 branes are fully localized the supergravity background is not known. The matrix description of intersecting M5 branes [4] describes this system. A discussion on this case can be found in [16].

BPS String Solitons
The centrally extended N = 2 supersymmetry algebra in four dimensions takes the form [17]  The BPS particles and BPS strings are realized by stretching M2 brane between the M5 and M5' branes. When the membrane worldvolume coordinates are (0, x, y) where x is one of the x components and y is one of the y components we get a BPS particle of the four dimensional theory. It is charged under U(1) R that acts on x and is a singlet under SU(2) R that acts on z. The BPS particles exist on the Coulomb branch of the gauge theory on which U(1) R acts, as in figure 2, and their mass is given by the Seiberg-Witten solution.
When the membrane worldvolume coordinates are (0, 1, z) where z is one of the z components we get a BPS string of the four dimensional theory. It is not charged under U(1) R and it transforms in the adjoint SU(2) R since z transforms in this representation. The BPS strings exist on the Higgs branch of the gauge theory on which SU(2) R acts, as in figure 2. The tension of the BPS string is given by | z| l 3 p where | z| is the distance between the M5 and M5' branes and it is finite in the field theory limit. It is easy to see that we do not have in this set-up BPS domain walls since stretching a membrane with worldvolume coordinates (0, 1, 2) breaks the supersymmetry completely. One expects BPS domain wall configurations when the moduli space of vacua has disconnected components.
In the Type IIA picture the BPS string is constructed by stretching a D2 brane between the D4 branes and NS5-brane. The seperation between these two types of branes in the z direction has two interpretations depending on whether the four dimensional gauge group has a U(1) part or not [18]. If is does then the separation is interpreted as the SU(2) R triplet FI parameters ζ. The BPS string tension is proportional to | ζ|. If the gauge group does not have a U(1) part the seperation between these two types of branes in the z direction is interpreted as giving a vev to an SU(2) R triplet component of the mesonQQ which transforms under SU(2) R as 2 × 2 = 3 ⊕ 1. The BPS string tension is proportional to this vacuum expectation value. At the origin of the moduli space, where we have SCFT, the string becomes tensionless.
We can use the AdS/CFT correspondence in order to compute the self energy of a string and a potential between two such strings of opposite orientation [19]. This is obtained, in the supergravity approximation, by a minimization of the M2 brane action in the background ( The SCFT is at the intersection of these two branches. x i = x i (σ). Then, the energy per unit length of the string is given by Thus, the string self-energy (3.4) The function E parametrizes the dependence of the self-energy on the coordinate α. From the field theory point of view, this parametrizes a dependence on the moduli that parametrize the space of vacua. δx || is a cut-off. In (3.4) we used the holographic relation between a distance δx || in field theory and the coordinate U which in our case reads 3 Similarly, the potential energy per units length between two strings of opposite orientation seperated by distance L is here L is distance between two strings. The results (3.4),(3.6) do not depend on N 2 which means that they do not depend on the gauge coupling of the four dimensional field theory g 2 Y M ∼ N 2 . This is not unexpected. The gauge coupling can be viewed as a 3 This correponds to the familiar relation δx || ∼ vacuum expectation value of a scalar in the vector multiplet and does not appear in the hypermultiplet metric. Since the BPS strings exist on the Higgs branch it is natural that their potential and self-energy do not depend on the gauge coupling. However, (3.4) and (3.6) are only the large N 1 results and will presumably have 1/N 1 corrections.
As noted above, the theory on the intersection of two sets of M5 branes containes BPS particles, which arise from M2 branes stretched in the directions (0, x, y). Minimization of the M2 brane action in this case yields the potential between these two such objects of opposite charge separated by distance L Again, this is expected since for the four dimensional field theory it reads as V ∼ suggesting that the number of degrees of freedom goes like N 2 1 N 2 . This is also deduced from the two-point function of the stress energy tensor. This is, of course, expected for SU(N 1 ) N 2 gauge theory. It is curious to note that there is similar growth (N 3 ) of the entropy for a system of N parallel M5 branes [21] 4 . The one loop correction has the following form l −9 p √ −gl 6 p R 4 ∼ N 2 which is suppressed by N 2 1 compared to the tree level action suggesting that the field theory has a 1/N 1 expansion, which again is in agreement with the field theory expectation.
The Type IIA background (2.10) is T-dual to Type IIB on AdS 5 × S 5 /Z N 2 [15]. The latter is the dual description of the Z N 2 orbifold of N = 4 theory [22]. For instance the number of degrees of freedom can be understood as c(N = 4)/N 2 ∼ N 2 1 N 2 . We can identify the spectrum of chiral primary operators with the supergravity Kaluza-Klein excitations as analysed in [23,24].