Holography for field theory solitons
Abstract
We extend a wellknown Dbrane construction of the AdS/dCFT correspondence to nonabelian defects. We focus on the bulk side of the correspondence and show that there exists a regime of parameters in which the lowenergy description consists of two approximately decoupled sectors. The two sectors are gravity in the ambient spacetime, and a sixdimensional supersymmetric Yang–Mills theory. The Yang–Mills theory is defined on a rigid background and admits sixteen supersymmetries. We also consider a oneparameter deformation that gives rise to a family of Yang–Mills theories on asymptotically spacetimes, which are invariant under eight supersymmetries. With future holographic applications in mind, we analyze the vacuum structure and perturbative spectrum of the Yang–Mills theory on , as well as systems of BPS equations for finiteenergy solitons. Finally, we demonstrate that the classical Yang–Mills theory has a consistent truncation on the twosphere, resulting in maximally supersymmetric Yang–Mills on .
a] Sophia K. Domokos b] and Andrew B. Royston \affiliation[a]Department of Physics, New York City College of Technology, 300 Jay Street, Brooklyn, NY 11201, USA \affiliation[b] George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA \emailAddsophia.domokos@gmail.com \emailAddaroyston@physics.tamu.edu
1 Introduction and summary
The original AdS/CFT duality [1] is nearing its twentieth anniversary. Even AdS/dCFT, which introduces defects in conformal field theories with gravity duals [2, 3, 4], is fifteen years old. A small corner of this Dbrane universe, however, remains relatively unexplored.
In this paper, we describe a simple generalization of the D3/D5brane intersection that forms the basis of the original antide Sitter/defect conformal field theory correspondence (AdS/dCFT). Instead of studying a single probe D5brane in the presence of a large number of D3branes, we consider the seemingly simple nonabelian generalization, with several parallel D5branes.
The resulting model, when subjected to Maldacena’s lowenergy limit and restricted to an appropriate regime of parameters, offers rich physics and rich mathematics, which we begin to uncover here. Denoting the numbers of D3 and D5branes by and respectively, and the string coupling by , the regime of parameters is and . The first two conditions are the ones that arise in the usual AdS/CFT correspondence. They ensure that gravity is weakly coupled and curvatures are small relative to the string scale. The final condition is a slight refinement of the oftquoted ‘probe limit’ . We will see that it arises naturally when we demand that corrections from gravity to the sector of the D5brane theory be suppressed. In this regime, therefore, the effects of closed strings can be neglected relative to the treelevel Yang–Mills interactions.
As in the original AdS/dCFT correspondence, the duality ‘acts twice’ [2, 3, 4] in the sense that it relates bulk closed strings to operators in the ambient part of the boundary theory, and bulk open strings on the D5branes to operators localized on a defect in the boundary theory. Hence the curvedspace superYang–Mills theory (SYM) describes the physics of operators confined to a defect in the boundary CFT. As the bulk SYM is dual to a (2+1)dimensional system, it is potentially relevant to holographic condensed matter applications. Indeed, the bulk SYM admits a zoo of solitonic objects, whose masses and properties are constrained by supersymmetry. We expect that these correspond to vortexlike states on the dual defect. Conversely, holography should provide a new tool for studying SYM solitons in the bulk.
In this paper, however, we focus on the bulk side of the correspondence. A detailed construction of the dual boundary theory will appear elsewhere, [5].
1.1 Summary of results
We begin by constructing a sixdimensional (6D) SYM theory with symmetry from a D3/D5 intersection. We assume that the number of D3branes () is large, so we can represent them with a Type IIB supergravity solution. We then consider the D5branes as probes in this background. We arrive at the SYM action by combining and extending Dbrane actions that already appear in the literature. For the bosonic theory on the D5branes, we use the nonabelian Myers action [6]. We determine the kinetic and masslike terms for the fermions using the abelian action of [7, 8, 9, 10, 11, 12, 13], and infer the nonabelian gauge and Yukawa couplings via a simple ansatz consistent with gauge invariance and supersymmetry. We then apply the Maldacena lowenergy nearhorizon limit.
The resulting action is summarized in equations (97)(100). While we obtained this action from a Dbrane model, it makes sense as a classical field theory for arbitrary simple Lie groups.
We go on to analyze the vacuum structure, perturbative spectrum, and the BPS equations satisfied by solitons in the 6D SYM theory. We also show that the 6D theory has a nonlinear consistent truncation to maximally supersymmetric YM theory on .
Here are a few highlights from the road ahead:

The space of vacua of the 6D theory has multiple components. There are, in fact, infinitely many when . One component is a standard Coulomb branch labeled by vevs of Higgs fields. The other components are labeled by magnetic charges and are quite complicated: they have roughly the form of moduli spaces of singular monopoles fibered over spaces of Higgs vevs. A Dbrane picture (see Figure 4 below) provides some intuition for these vacua.

We perform a perturbative mode analysis around a class of vacua that carry magnetic flux. The background fields of this class are Cartanvalued and simple enough to make the linearized equations tractable. Furthermore, the background fields of any vacuum will asymptote to the same near the boundary, so the results for the asymptotic behavior of fluctuations are robust. This is important for the holographic dictionary, where one maps modes to local operators in the dual, based in part on their decay properties near the boundary.
Our analysis of the perturbative spectrum generalizes previous results for the abelian D5brane defect [4, 14, 15], and offers a number of new results. We display, for instance, the complete KK spectrum of fermionic modes. We also observe that a Legendre transform of the onshell action with respect to one of the lowlying modes, along the lines of [16], is required for holographic duality.
^{1} We identify a set of lowlying nonnormalizable modes that can be turned on without violating the variational principle or supersymmetry. These modes form a natural class of boundary values for soliton solutions in the nonabelian D5brane theory. In the holographic dual, meanwhile, they source a set of relevant operators—and in one case a distinguished irrelevant operator.

Having explored the vacua and perturbative structure of the bulk SYM theory, we then survey various systems of BPS equations. These first order equations arise when we demand that field configurations preserve various amounts of supersymmetry.
Solutions to the BPS equations saturate bounds on the energy functional. These bounds depend on a combination of the fields’ boundary values as well as the magnetic and electric fluxes through the asymptotic boundary.
The BPS systems we obtain house a number of generalized selfduality equations that are well known in mathematical physics, like (translationally invariant) octonionic instantons [18], and the extended Bogomolny equations [19]. All of these equations are defined on a manifold with boundary, where the boundary is the holographic boundary.
The paper is structured as follows: In section 2 we describe the Dbrane intersection and take the lowenergy limit of the action to arrive at a curved space SYM theory. In section 3 we verify the invariance of the action under supersymmetry. In section 4 we describe the vacuum structure of the model, and formulate asymptotic boundary conditions on the fields. In section 5 we derive the consistent truncation of our sixdimensional theory to four dimensions, while in section 6 we derive the BPS equations satisfied by solitons in the system. We conclude and discuss future directions in section 7. Necessary but onerous details are relegated to a series of Appendices.
2 Branes and holography
In this section we describe the brane setup, the AdS/dCFT picture, and the lowenergy limit and parameter regime that isolates sixdimensional SYM as the lowenergy effective theory on the D5branes.
2.1 Brane configuration
We begin with a nonabelian version of the brane configuration in [4]. D5branes and D3branes in the tendimensional IIB theory span the directions indicated in Figure 1. Standard arguments [20] show that the intersecting D3/D5 system preserves one quarter of the supersymmetry of 10D type IIB string theory, or eight supercharges.
0  1  2  3  4  5  6  7  8  9  
D3  X  X  X  X  
D5  X  X  X  X  X  X  
The ten coordinates, are divided as follows: , , parameterizes the spanned by both stacks; the triplet parameterizes the remaining directions along the D5branes; the triplet parameterizes directions orthogonal to both stacks; and finally parameterizes the remaining direction along the D3branes and orthogonal to the D5branes. We reserve the notation for a rescaled version of these coordinates to be introduced below. We will sometimes use spherical coordinates to parameterize the directions, and we denote the radial coordinate in the directions by . We also write , , and , , for the full set of directions parallel and transverse to the D5branes respectively.
The D3branes are taken to be coincident and sitting at . The center of mass position of the D5branes in the transverse space is denoted . We will allow for relative displacements of the D5branes from each other, but assume that these distances are small compared to the string scale. In other words, the separation is welldescribed by vev’s of nonabelian scalars in the D5brane worldvolume theory. This will be explained in more detail below. When all D5branes are positioned at an subgroup of the tendimensional Lorentz group is preserved. Nonzero D5brane displacements in break . This can be explicit or spontaneous from the point of view of the D5brane worldvolume theory, depending on whether the center of mass position is, respectively, nonzero or zero.
As noted above, eight of the original thirtytwo Type IIB supercharges are preserved by the brane setup. From the point of view of the threedimensional intersection, this is equivalent to supersymmetry. The symmetry group is with the two factors being realized geometrically as the double covers of the rotation groups in the and directions. The light degrees of freedom on the D3branes and the D5branes are a fourdimensional valued vectormultiplet, and a sixdimensional valued vectormultiplet. Each of these decompose into a 3D vectormultiplet and hypermultiplet. For those D5branes intersecting the D3branes, the  strings localized at the intersection are massless. They furnish a 3D hypermultiplet transforming in the bifundamental representation of the appropriate gauge groups. Meanwhile the massless closed strings comprise the usual type IIB supergravity multiplet.
2.2 Low energy limit and AdS/dCFT
Let us now consider the lowenergy limit of the brane setup, that ultimately yields the defect AdS/CFT correspondence. This is the famous Maldacena limit [1] that, in the absence of D5branes, establishes a correspondence between 4D SYM and type IIB string theory on . To arrive at the AdS/dCFT correspondence one considers the lowenergy effective description of the D3/D5 system at energy scale and takes the limit , where is the string length. The dynamics of the massless degrees of freedom have two equivalent descriptions in terms of two different sets of field variables. This fact is the essence of the original AdS/dCFT correspondence.
To simplify the present discussion we temporarily assume no separation between the brane stacks – in other words, . The first set of variables that describes the D3/D5 intersection is based on an expansion around the flat background: Minkowski space for the closed strings and constant values of the brane embedding coordinates for the open strings. In this case standard field theory scaling arguments apply. After canonically normalizing the kinetic terms for open and closed string fluctuations, interactions of the closed strings and 55 open strings amongst themselves, as well as the interactions of the closed and 55 open strings with the other open strings, vanish in the lowenergy limit. These degrees of freedom decouple from the system. Meanwhile the 33 and 35 strings form an interacting system described by fourdimensional SYM coupled to a codimension one planar interface, breaking half the supersymmetry and hosting a 3D hypermultiplet. The interface action, which can in principle be derived from the low energy limit of string scattering amplitudes, was obtained in [4] by exploiting symmetry principles. The entire theory contains a single dimensionless parameter in addition to and —the fourdimensional Yang–Mills coupling—given in terms of the string coupling via .
The interface plus boundary ambient Yang–Mills theory is classically scale invariant, and it was argued in [4, 21] to be a superconformal quantum theory. The symmetry algebra is , with bosonic subalgebra and sixteen odd generators. is the threedimensional conformal group of the interface while the odd generators correspond to the eight supercharges along with eight superconformal generators. This is the “defect CFT” side of the correspondence. Considering a nonzero separation corresponds to turning on a relevant mass deformation in the dCFT [22, 23].
Our focus here will be mostly on the other side of the correspondence, which is based on an expansion in fluctuations around the supergravity background produced by the D3branes. This background involves a nontrivial metric and RamondRamond (RR) fiveform flux given in our coordinates by
(1)  
(2)  
(3) 
The metric is asymptotically flat and approaches with equal radii of when . The energy of localized modes in the throat region, as measured by an observer at position , is redshifted in comparison to the asymptotic fixed energy according to , for . Hence, while closed string and D5brane modes with Compton wavelengths large compared to decouple as before, excitations of arbitrarily high energy can be achieved in the throat region. The nearhorizon limit isolates the entire set of stringy degrees of freedom in the throat region by sending in such a way that remains fixed. From the redshift relation it follows that we are sending while holding fixed. For fixed ’t Hooft coupling , this is equivalent to sending while holding fixed.
To facilitate taking this limit we introduce new coordinates
(4) 
and write for the corresponding spherical coordinates and . We will also sometimes employ a vector notation , . One finds that with these new coordinates, the metric becomes
(5)  
(6)  
(7)  
(8)  
(9) 
where we’ve introduced a rescaled metric and fourform potential, . is the metric on with radii .
The degrees of freedom in the nearhorizon geometry include both the closed strings and the open strings on the D5branes. String theory in this background is conjecturally dual to the dCFT system, with the duality ‘acting twice’ [2, 3, 4]. This means the following: closed string modes in the (ambient) spacetime of the bulk side are dual to operators constructed from the 4D SYM fields in the (ambient) spacetime on the boundary. Open string modes on the D5branes, which form a defect in the bulk, are dual to operators localized on the defect in the boundary theory. These operators are constructed from modes of the 35 strings and modes of the 33 strings restricted to the boundary defect. See Figure 2.
The validity of the supergravity approximation in the closed string sector requires that . The first condition suppresses corrections to the low energy effective action, while the second condition is equivalent to , ensuring that higher derivative corrections are suppressed as well.
In subsection 2.4 we’ll see how this limit suppresses the interactions between closed string and open string D5brane modes, leading to an effective Yang–Mills theory on the D5branes. This extends previous analyses of the D3/D5 system to the case of multiple D5branes, showing how the nonabelian interaction terms among open strings are dominant to the openclosed couplings, at least in the sector of the theory. In subsection 2.3 we will describe explicitly what these interactions look like (using the Myers nonabelian Dbrane action).
In preparation for that, consider the following redefinition of the relevant supergravity fields. Let denote the type IIB supergravity action in Einstein frame. Here is the Kalb–Ramond twoform potential and is the fluctuation of the dilaton field around its vev, , with . The , even, are the RamondRamond potentials, and is the tendimensional Newton constant, . Upon rescaling the metric and potentials according to
(10) 
one finds that
(11) 
where the new Newton constant is
(12) 
Thus an expansion in canonically normalized closed string fluctuations, , around the nearhorizon background, (5), takes the form
(13)  
(14) 
where and were given in (5), and point couplings among closed string fluctuations go as .
2.3 The nonabelian D5brane action
The massless bosonic degrees of freedom on the D5branes are a gauge field , , with fieldstrength , and four adjointvalued scalars . The gauge field carries units of mass while the carry units of length. The eigenvalues of ( times) the latter are to be identified with the displacements of the D5branes away from . Our conventions are that elements of the Lie algebra are represented by antiHermitian matrices, so there are no factors of coming with the Lie bracket in covariant derivatives. The ’’ operation denotes minus the trace in the fundamental representation, , with the minus inserted so that it is a positivedefinite bilinear form on the Lie algebra. Later on we will generalize the discussion to a generic simple Lie algebra , and then we define the trace through the adjoint representation via , where is the dual Coxeter number. This reduces to the previous definition for .
The nonabelian Dbrane action of Myers, [6], captures a subset of couplings between the 55 open string and ambient closed string modes. It takes the form
(15)  
(16)  
(17)  
(18) 
where
(19) 
, and is the D5brane tension. Besides the factor of in , the closed string fields are encoded in the two quantities
(20) 
The factors of the dilaton are present here because we work in Einstein frame for the closed string fields. This action generalizes the nonabelian Dbrane action of [24] to the case of a generic closed string background.
The quantity denotes the gaugecovariant pullback of a bulk tensor to the worldvolume of the D5branes. For instance, the pullback of the generalized metric to the brane is
(21) 
with . The closed string fields are to be taken as functionals of the matrixvalued coordinates, , defined by power series expansion:
(22) 
The determinants in the DBI action (17) refer to spacetime indices and .
In the ChernSimons (CS) action, (18), the symbol denotes the interior product with respect to . This is an antiderivation on forms, reducing the degree by one. Since the are noncommuting one has, for example,
(23) 
See [6] for further details.
The ‘STr’ stands for a fully symmetrized trace, defined as follows [6]. After expanding the closed string fields in power series and computing the determinants, the arguments of the in (17) and (18) will take the form of an infinite sum of terms, each of which will involve powers of four types of open string variable: , and individual ’s from the expansion of the closed string fields. The notation indicates that one is to apply to the complete symmetrization on these variables.
The precise regime of validity of the Myers action is not a completely settled issue. First of all, like its abelian counterpart, it captures only treelevel interactions with respect to . Second, if denotes any components of the ‘tendimensional’ fieldstrength, , , or , (15) is known to yield results incompatible with open string amplitudes at [25, 26], even in the limit of trivial closed string background. Finally, the action (15) is given directly in “static gauge,” and there have been questions about whether it can be obtained from gauge fixing a generally covariant action. This could lead to ambiguities in openclosed string couplings at according to [27]. However, the results of [28] suggest that the Myers action can in fact be obtained by gaugefixing symmetries in a generally covariant formalism where the Chan–Paton degrees of freedom are represented by boundary fermions on the string worldsheet. As we will see below, none of these ambiguities pose a problem in the scaling limit we are interested in.
2.4 Yang–Mills as the low energy effective theory
We now expand the action (15) in both closed and open string fluctuations, where the closed string expansion is an expansion around the nearhorizon geometry of the D3branes, in accord with (13). This was already done in some detail in the abelian case [4], but there are some important new wrinkles that arise in the nonabelian case. We summarize the main points here and provide further details in appendix A.
First, the kinetic terms for the open string modes take the form
(24) 
where we recall that . The factors of arise from writing the background metric in terms of the barred metric. We have introduced the notation , with the induced background metric on the worldvolume. It takes the form
(25) 
When this is the metric on with equal radii of , while gives a deformation of it. Worldvolume indices will always be raised with the inverse, . We use the notation to indicate when other closed string fields are being evaluated at .
The coefficient of the term determines the effective sixdimensional Yang–Mills coupling:
(26) 
Note that the dimensionless coupling is small in the regime . In order to bring the scalar kinetic terms to standard form we define mass dimensionone scalar fields through
(27) 
so that carry the same dimension.
Once the closed string fields in the Dbrane action are expressed in terms of the rescaled quantities, one finds that is always accompanied by a factor of , while is always accompanied by the inverse factor. After changing variables to for the scalars, all four types of open string quantities appearing in D5brane action carry the same prefactor:
(28) 
and this provides a convenient organizing principle for the expansion. Of course it is , defined by
(29) 
that are the canonically normalized open string modes. The open string expansion variables on the righthand side of (28) do not scale homogeneously when expressed in terms of these, and this point must be kept in mind when comparing the strength of interaction vertices below.
Now, let denote a generic closed string fluctuation, let denote any of the open string expansion variables, and set
(30) 
Then the expansion of (15) can be written in the form
(31) 
where is a sum of monomials of the form , with rational coefficients. The first few ’s are
(32)  
(33)  
(34)  
(35)  
(36)  
(37)  
(38)  
(39)  
(40) 
where is the Levi–Civita tensor with respect to the background metric, , and we have used that reduces to the ordinary trace when there are no more than two powers of the open string variables . All closed string fields are to be understood as being evaluated at except for those in that involve taking a transverse derivative before setting .
There is a great deal of physics in the ’s:

corresponds to the energy density of the background D5brane configuration.

gives closed string tadpoles for the metric, dilaton, and RR sixform potential. These are present because we have not included the gravitational backreaction of the D5branes—i.e. we have not expanded around a solution to the equations of motion for these closed string fields. The strength of these tadpoles is , which is large when . However this does not necessarily mean that the probe approximation is bad! The effects of these tadpoles on open and closed string processes will still be suppressed if the interaction vertices are sufficiently weak.
Consider, for example, the leading correction to the open string propagators due to these tadpoles. This corresponds to the diagram in Figure 3. The correction is proportional to the product of the tadpole vertex with the cubic vertex for two open and one closed string fluctuation. After canonically normalizing the open string modes via (29), the threepoint vertex goes as . Therefore the product is proportional to . Hence this process acts just like a standard oneloop correction to the Yang–Mills coupling that we would get from open string modes. As long as , both the standard oneloop correction and this closed string correction will be suppressed. Note this is a slightly stronger restriction than the usual limit when the ’t Hooft coupling is large, but nevertheless can be comfortably satisfied for a range of in the regime .

The vanishing of indicates that open string tadpoles are absent. This simply validates the fact (already implicitly assumed in the above discussion) that the D5brane embedding, described by , extremizes the equations of motion for the open string modes in the fixed closed string background.

Only the centerofmass degrees of freedom corresponding to the central participate in due to the trace. The strength of these interactions is , where we have made use of the convenient relation
(41) Hence they can be treated perturbatively. Furthermore the and degrees of freedom decouple in , so the couplings in can only transmit the effects of the closed string tadpoles to the fields through higher order open string interactions.

The first three terms of come from the DBI action, and comprise the usual Yang–Mills action on a curved background. The final term in , meanwhile, comes from the CS action and is nonvanishing because there is a the nontrivial RR flux in the supergravity background.
It is also interesting to consider the form of terms in , or higher order open string interactions. is nontrivial when ; is always nontrivial. For example, there is an coupling of the form
(42) 
Three and fourpoint couplings in and come with extra factors of relative to the three and fourpoint couplings in the Yang–Mills terms, . Hence they will be suppressed relative to the Yang–Mills terms for field variations at or below the scale . More precisely, if the fields vary on a scale we merely require , in order that these terms be suppressed relative to their counterparts in .
In summary, there is a regime of parameters—namely and —where the leading interactions of the (bosonic) open string modes are governed by . This forms the bosonic part of a sixdimensional superYang–Mills theory on the curved background (25).
We can present this action in two different forms, both of which will prove useful below. First there is the form we have used to give , in which the scalars carry curved space indices. In order to be more explicit with regards to the term, we have from (5) that the relevant components are
(43) 
and so the last term of contributes as follows:
(44)  
(45) 
Here we have introduced , which should be thought of as the Levi–Civita tensor on the Euclidean spanned by : , or if we work in spherical coordinates . Then the bosonic part of the Yang–Mills action is
(46)  
(47)  
(48) 
We can also derive a more standard field theoretic form for the action by rescaling the scalar fields in such a way that their kinetic terms are canonically normalized. To do this, we make use of a vielbein associated with the background metric :
(49) 
Both mass terms and boundary terms arise when we integrate by parts in the kinetic terms. One can also integrate by parts on the last term of (46) and make use of the Bianchi identity, . We also switch to spherical coordinates, as the only surviving bulk term comes from the derivative of the prefactor. This integration by parts also generates a boundary term. After carrying out these manipulations, the bosonic action becomes
(50)  
(51) 
In the last term the indices correspond to coordinates along the twosphere and . The mass parameters are defined as follows:
(52) 
As they approach the values in units of the inverse AdS radius. When they take these values everywhere. Although the squared mass of the scalars is negative, it satisfies the Breitenlohner–Freedman bound [29] for . The reason for the notation will become clear below when we consider the fermionic part of the action.
The boundary terms arise due to the integration by parts and the boundary component at .
(53) 
where is the induced volume form on the boundary,
(54) 
with and . If one works with the action in the form (50) then it is important to keep these terms. They play a crucial role both in establishing the consistency of the variational principle and in the supersymmetry invariance of the Yang–Mills action. The limit of quantities computed using (53) is understood to be taken at the end of any calculation (when it exists).
2.5 Fermionic Dbrane action
Ideally, one would like to obtain nonabelian superYang–Mills theory on the D5branes via the limiting behavior of a symmetric nonabelian super Dbrane action for general closed string backgrounds. While important progress toward constructing such actions has been made ( see e.g. [30, 31, 32, 33] and references therein), the subject has not matured sufficiently to be of practical use for our purposes.
Instead, we will fall back on abelian fermionic Dbrane actions that have been discussed extensively, starting with the initial work of [7, 8, 9, 10], and continuing with [11, 12, 13]. Here we follow the conventions of [12, 13]. This will provide the fermionic couplings that are quadratic order in open string fluctuations—kinetic and masslike terms. With these and the full set of bosonic couplings in hand, we will be able to deduce the remaining Yukawatype couplings and the nonabelian supersymmetry transformations via a simple ansatz.
The massless fermionic degrees of freedom on a D5brane are the same as those in tendimensional superYang–Mills, and can be packaged into a single tendimensional Majorana–Weyl fermion, . The couplings of to the IIB closed string supergravity fields are described most conveniently by introducing a doublet of tendimensional Majorana–Weyl spinors of the same 10D chirality. One linear combination will be projected out by the symmetry projector while the other will be the physical . The tendimensional gamma matrices, satisfying , are likewise extended by the doublet structure. One introduces