Holography for field theory solitons
We extend a well-known D-brane construction of the AdS/dCFT correspondence to non-abelian defects. We focus on the bulk side of the correspondence and show that there exists a regime of parameters in which the low-energy description consists of two approximately decoupled sectors. The two sectors are gravity in the ambient spacetime, and a six-dimensional supersymmetric Yang–Mills theory. The Yang–Mills theory is defined on a rigid background and admits sixteen supersymmetries. We also consider a one-parameter 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 finite-energy solitons. Finally, we demonstrate that the classical Yang–Mills theory has a consistent truncation on the two-sphere, 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.firstname.lastname@example.org \emailAddaroyston@physics.tamu.edu
1 Introduction and summary
The original AdS/CFT duality  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 D-brane universe, however, remains relatively unexplored.
In this paper, we describe a simple generalization of the D3/D5-brane intersection that forms the basis of the original anti-de Sitter/defect conformal field theory correspondence (AdS/dCFT). Instead of studying a single probe D5-brane in the presence of a large number of D3-branes, we consider the seemingly simple non-abelian generalization, with several parallel D5-branes.
The resulting model, when subjected to Maldacena’s low-energy 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 D5-branes 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 oft-quoted ‘probe limit’ . We will see that it arises naturally when we demand that corrections from gravity to the sector of the D5-brane theory be suppressed. In this regime, therefore, the effects of closed strings can be neglected relative to the tree-level 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 D5-branes to operators localized on a defect in the boundary theory. Hence the curved-space super-Yang–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 vortex-like 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, .
1.1 Summary of results
We begin by constructing a six-dimensional (6D) SYM theory with symmetry from a D3/D5 intersection. We assume that the number of D3-branes () is large, so we can represent them with a Type IIB supergravity solution. We then consider the D5-branes as probes in this background. We arrive at the SYM action by combining and extending D-brane actions that already appear in the literature. For the bosonic theory on the D5-branes, we use the non-abelian Myers action . We determine the kinetic and mass-like terms for the fermions using the abelian action of [7, 8, 9, 10, 11, 12, 13], and infer the non-abelian gauge and Yukawa couplings via a simple ansatz consistent with gauge invariance and supersymmetry. We then apply the Maldacena low-energy near-horizon limit.
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 D-brane 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 Cartan-valued 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 D5-brane 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 on-shell action with respect to one of the low-lying modes, along the lines of , is required for holographic duality.111The paper  appeared when this work was nearing completion. Its authors make a closely related observation in maximally supersymmetric YM on . The consistent truncation of the 6D theory on the two-sphere, explained in section 5, shows that these results are in fact describing the same phenomenon.
We identify a set of low-lying non-normalizable 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 non-abelian D5-brane 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 self-duality equations that are well known in mathematical physics, like (translationally invariant) octonionic instantons , and the extended Bogomolny equations . 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 D-brane intersection and take the low-energy 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 six-dimensional 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 set-up, the AdS/dCFT picture, and the low-energy limit and parameter regime that isolates six-dimensional SYM as the low-energy effective theory on the D5-branes.
2.1 Brane configuration
We begin with a non-abelian version of the brane configuration in . D5-branes and D3-branes in the ten-dimensional IIB theory span the directions indicated in Figure 1. Standard arguments  show that the intersecting D3/D5 system preserves one quarter of the supersymmetry of 10D type IIB string theory, or eight supercharges.
The ten coordinates, are divided as follows: , , parameterizes the spanned by both stacks; the triplet parameterizes the remaining directions along the D5-branes; the triplet parameterizes directions orthogonal to both stacks; and finally parameterizes the remaining direction along the D3-branes and orthogonal to the D5-branes. 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 D5-branes respectively.
The D3-branes are taken to be coincident and sitting at . The center of mass position of the D5-branes in the transverse space is denoted . We will allow for relative displacements of the D5-branes from each other, but assume that these distances are small compared to the string scale. In other words, the separation is well-described by vev’s of non-abelian scalars in the D5-brane worldvolume theory. This will be explained in more detail below. When all D5-branes are positioned at an subgroup of the ten-dimensional Lorentz group is preserved. Nonzero D5-brane displacements in break . This can be explicit or spontaneous from the point of view of the D5-brane worldvolume theory, depending on whether the center of mass position is, respectively, nonzero or zero.
As noted above, eight of the original thirty-two Type IIB supercharges are preserved by the brane setup. From the point of view of the three-dimensional 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 D3-branes and the D5-branes are a four-dimensional -valued vector-multiplet, and a six-dimensional -valued vector-multiplet. Each of these decompose into a 3D vector-multiplet and hypermultiplet. For those D5-branes intersecting the D3-branes, the - strings localized at the intersection are massless. They furnish a 3D hypermultiplet transforming in the bi-fundamental 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 low-energy limit of the brane setup, that ultimately yields the defect AdS/CFT correspondence. This is the famous Maldacena limit  that, in the absence of D5-branes, establishes a correspondence between 4D SYM and type IIB string theory on . To arrive at the AdS/dCFT correspondence one considers the low-energy 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 5-5 open strings amongst themselves, as well as the interactions of the closed and 5-5 open strings with the other open strings, vanish in the low-energy limit. These degrees of freedom decouple from the system. Meanwhile the 3-3 and 3-5 strings form an interacting system described by four-dimensional SYM coupled to a co-dimension 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  by exploiting symmetry principles. The entire theory contains a single dimensionless parameter in addition to and —the four-dimensional 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 three-dimensional 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 D3-branes. This background involves a nontrivial metric and Ramond-Ramond (RR) five-form flux given in our coordinates by
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 D5-brane modes with Compton wavelengths large compared to decouple as before, excitations of arbitrarily high energy can be achieved in the throat region. The near-horizon 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
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 becomes222The metric can be brought to the form found in  by first introducing standard spherical coordinates in the directions and then setting and with , and . Then is the radial coordinate in the Poincare patch, with the asymptotic boundary, while parameterize the , viewed as an fibration over the interval parameterized by .
where we’ve introduced a rescaled metric and four-form potential, . is the metric on with radii .
The degrees of freedom in the near-horizon geometry include both the closed strings and the open strings on the D5-branes. 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 D5-branes, 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 3-5 strings and modes of the 3-3 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 D5-brane modes, leading to an effective Yang–Mills theory on the D5-branes. This extends previous analyses of the D3/D5 system to the case of multiple D5-branes, showing how the non-abelian interaction terms among open strings are dominant to the open-closed 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 non-abelian D-brane 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 two-form potential and is the fluctuation of the dilaton field around its vev, , with . The , even, are the Ramond-Ramond potentials, and is the ten-dimensional Newton constant, . Upon rescaling the metric and potentials according to
one finds that
where the new Newton constant is
Thus an expansion in canonically normalized closed string fluctuations, , around the near-horizon background, (5), takes the form
where and were given in (5), and -point couplings among closed string fluctuations go as .
2.3 The non-abelian D5-brane action
The massless bosonic degrees of freedom on the D5-branes are a gauge field , , with fieldstrength , and four adjoint-valued 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 D5-branes away from . Our conventions are that elements of the Lie algebra are represented by anti-Hermitian 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 positive-definite 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 non-abelian D-brane action of Myers, , captures a subset of couplings between the 5-5 open string and ambient closed string modes. It takes the form
, and is the D5-brane tension. Besides the factor of in , the closed string fields are encoded in the two quantities
The factors of the dilaton are present here because we work in Einstein frame for the closed string fields. This action generalizes the non-abelian D-brane action of  to the case of a generic closed string background.
The quantity denotes the gauge-covariant pullback of a bulk tensor to the worldvolume of the D5-branes. For instance, the pullback of the generalized metric to the brane is
with . The closed string fields are to be taken as functionals of the matrix-valued coordinates, , defined by power series expansion:
The determinants in the DBI action (17) refer to spacetime indices and .
In the Chern-Simons (CS) action, (18), the symbol denotes the interior product with respect to . This is an anti-derivation on forms, reducing the degree by one. Since the are non-commuting one has, for example,
See  for further details.
The ‘STr’ stands for a fully symmetrized trace, defined as follows . 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 tree-level interactions with respect to . Second, if denotes any components of the ‘ten-dimensional’ 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 open-closed string couplings at according to . However, the results of  suggest that the Myers action can in fact be obtained by gauge-fixing 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 near-horizon geometry of the D3-branes, in accord with (13). This was already done in some detail in the abelian case , but there are some important new wrinkles that arise in the non-abelian 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
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
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 six-dimensional Yang–Mills coupling:
Note that the dimensionless coupling is small in the regime . In order to bring the scalar kinetic terms to standard form we define mass dimension-one scalar fields through
so that carry the same dimension.
Once the closed string fields in the D-brane 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 D5-brane action carry the same prefactor:
and this provides a convenient organizing principle for the expansion. Of course it is , defined by
that are the canonically normalized open string modes. The open string expansion variables on the right-hand 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
Then the expansion of (15) can be written in the form
where is a sum of monomials of the form , with rational coefficients. The first few ’s are
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 D5-brane configuration.
gives closed string tadpoles for the metric, dilaton, and RR six-form potential. These are present because we have not included the gravitational backreaction of the D5-branes—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 three-point vertex goes as . Therefore the product is proportional to . Hence this process acts just like a standard one-loop correction to the Yang–Mills coupling that we would get from open string modes. As long as , both the standard one-loop 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 D5-brane embedding, described by , extremizes the equations of motion for the open string modes in the fixed closed string background.
Only the center-of-mass 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
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 non-vanishing 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
Three- and four-point couplings in and come with extra factors of relative to the three- and four-point 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 six-dimensional super-Yang–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
and so the last term of contributes as follows:
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
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 :
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
In the last term the indices correspond to coordinates along the two-sphere and . The mass parameters are defined as follows:
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  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 .333We assume the fields are sufficiently regular such that there is no boundary contribution from . This is discussed in some further detail for static configurations later. See section 6.2. They are given by
where is the induced volume form on the boundary,
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 D-brane action
Ideally, one would like to obtain non-abelian super-Yang–Mills theory on the D5-branes via the limiting behavior of a -symmetric non-abelian super D-brane 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 D-brane 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 mass-like terms. With these and the full set of bosonic couplings in hand, we will be able to deduce the remaining Yukawa-type couplings and the non-abelian supersymmetry transformations via a simple ansatz.
The massless fermionic degrees of freedom on a D5-brane are the same as those in ten-dimensional super-Yang–Mills, and can be packaged into a single ten-dimensional Majorana–Weyl fermion, . The couplings of to the IIB closed string supergravity fields are described most conveniently by introducing a doublet of ten-dimensional 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 ten-dimensional gamma matrices, satisfying , are likewise extended by the doublet structure. One introduces
where is the ten-dimensional chirality operator and are the Pauli matrices.
The abelian fermionic D5-brane action, to quadratic order in , takes the form