Holography for field theory solitons

Journal of High Energy Physics, Jul 2017

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 AdS4 × S 2 background and admits sixteen supersymmetries. We also consider a one-parameter deformation that gives rise to a family of Yang-Mills theories on asymptotically AdS4 × S 2 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 AdS4 × S 2, 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 AdS4.

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Holography for field theory solitons

HJE Holography for eld theory solitons Sophia K. Domokos 0 1 3 Andrew B. Royston 0 1 2 Texas A 0 1 M University 0 1 Supersymmetric Gauge Theory 0 College Station , TX 77843 , U.S.A 1 300 Jay Street, Brooklyn, NY 11201 , U.S.A 2 George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy 3 Department of Physics, New York City College of Technology , USA 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 de ned S2 background and admits sixteen supersymmetries. We also consider a one-parameter deformation that gives rise to a family of Yang-Mills theories on asympS2 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 AdS4 nite-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 AdS4. AdS-CFT Correspondence; D-branes; Solitons Monopoles and Instantons - on a rigid AdS4 totically AdS4 S2, as well as systems of BPS equations for 2.1 2.2 2.3 2.4 2.5 3.1 3.2 4.1 4.2 4.3 4.4 6.1 6.2 6.3 1 Introduction and summary 1.1 Summary of results 2 Branes and holography Brane con guration Low energy limit and AdS/dCFT The non-abelian D5-brane action Yang-Mills as the low energy e ective theory Fermionic D-brane action 3 Supersymmetry Killing spinors in the bulk Killing spinors on the brane 3.3 Supersymmetry of the worldvolume theory 4 Classical vacua, boundary conditions, and asymptotic analysis Classical ( ux) vacua Perturbative spectrum Boundary conditions and consistency of the variational principle Conservation of supersymmetry on the boundary 5 A consistent truncation to N = 4 SYM on AdS4 6 Bogomolny equations and monopoles BPS equations as generalized self-duality equations Bogomolny bound on the energy Domain walls and dyonic octonionic instantons 6.4 Dimensional reduction and the Haydys-Witten equations 7 Conclusions and future directions A Fluctuation expansion of the Myers action A.1 The DBI action A.2 The CS action B Background geometry and Killing spinors B.1 Frame rotations B.2 D5-brane Killing spinor equation C Mode analysis C.1 Bosons C.2 Fermions { i { 1 Introduction and summary The original AdS/CFT duality [1] is nearing its twentieth anniversary. Even AdS/dCFT, which introduces defects in conformal eld theories with gravity duals [2{4], is fteen 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 eld 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, o ers rich physics and rich mathematics, which p respectively, and the string coupling by gs, the regime of parameters is Nc gsNc 1 and Nf Nc= gsNc. The rst 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 nal condition is a slight re nement of the oft-quoted `probe limit' Nf Nc. We will see that it arises naturally when we demand that corrections from gravity to the su(Nf ) sector of the D5-brane theory be suppressed. In this regime, therefore, the e ects 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{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 con ned 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, [5]. { 1 { We begin by constructing a six-dimensional (6D) SYM theory with osp(4j4) symmetry from a D3/D5 intersection. We assume that the number of D3-branes (Nc) 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 D-brane actions that already appear in the literature. For the bosonic theory on the D5-branes, we use the non-abelian Myers action [6]. We determine the kinetic and mass-like terms for the fermions using the abelian action of [7{13], and infer the nonabelian gauge and Yukawa couplings via a simple ansatz consistent with gauge invariance and supersymmetry. We then apply the Maldacena low-energy near-horizon limit. The resulting action is summarized in equations (3.24){(3.26). While we obtained this action from a D-brane model, it makes sense as a classical eld theory for arbitrary simple Lie groups. We go on to analyze the vacuum structure, perturbative spectrum, and the BPS equations satis ed 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 AdS4. Here are a few highlights from the road ahead: The space of vacua of the 6D theory has multiple components. There are, in fact, in nitely many when Nc ! 1. One component is a standard Coulomb branch labeled by vevs of Higgs elds. The other components are labeled by magnetic charges and are quite complicated: they have roughly the form of moduli spaces of singular monopoles bered over spaces of Higgs vevs. A D-brane picture (see gure 4 below) provides some intuition for these vacua. We perform a perturbative mode analysis around a class of vacua that carry magnetic ux. The background elds of this class are Cartan-valued and simple enough to make the linearized equations tractable. Furthermore, the background elds of any vacuum will asymptote to the same near the boundary, so the results for the asymptotic behavior of uctuations 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 o ers 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 [16], is required for holographic duality.1 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 1The paper [17] appeared when this work was nearing completion. Its authors make a closely related observation in maximally supersymmetric YM on AdS4. The consistent truncation of the 6D theory on the 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 rst order equations arise when we demand that eld con gurations preserve various amounts of supersymmetry. Solutions to the BPS equations saturate bounds on the energy functional. These bounds depend on a combination of the elds' boundary values as well as the magnetic and electric uxes 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 [18], and the extended Bogomolny equations [19]. All of these equations are de ned 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 elds. 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 satis ed 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 e ective theory on the D5-branes. 2.1 Brane con guration We begin with a non-abelian version of the brane con guration in [4]. Nf D5-branes and Nc D3-branes in the ten-dimensional IIB theory span the directions indicated in gure 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. The ten coordinates, x~M = (x ; r~i; z~i; y) are divided as follows: x , ; = 0; 1; 2, parameterizes the R1;2 spanned by both stacks; the triplet r~i = (r~1; r~2; r~3) = (x3; x4; x5) parameterizes the remaining directions along the D5-branes; the triplet z~i = (z~1; z~2; z~3) = (x6; x7; x8) parameterizes directions orthogonal to both stacks; and nally y = x9 parameterizes the remaining direction along the D3-branes and orthogonal to the D5-branes. We reserve the notation xM = (x ; ri; zi; y) for a rescaled version of these coordinates to be introduced below. We will sometimes use spherical coordinates (r~; ; ) to parameterize the r~i directions, and we denote the radial coordinate in the z~i directions by z~. We also write x~ a = (x ; r~i), a = 0; 1; : : : ; 5, and x~m = (z~i; y), m = 1; : : : ; 4, for the full set of directions parallel and transverse to the D5-branes respectively. { 3 { D3 D5 directions common to both types of brane worldvolume are suppressed in the gure on the left. The D5-branes can be separated from the D3-branes by a distance z~0 in the directions transverse to both stacks. The D3-branes are taken to be coincident and sitting at r~i = z~j = 0. The center of mass position of the D5-branes in the transverse x~m space is denoted x~0m = (z~0;i; y0). 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 z~i = 0 an SO(1; 2) SO(3)r SO(3)z subgroup of the ten-dimensional Lorentz group is preserved. Nonzero D5-brane displacements in z~ break SO(3)z. 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 z~0;i 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 N = 4 supersymmetry. The R-symmetry group is SO(4)R = SU(2)r SU(2)z with the two factors being realized geometrically as the double covers of the rotation groups in the r~i and z~i directions. The light degrees of freedom on the D3-branes and the D5branes are a four-dimensional N = 4 u(Nc)-valued vector-multiplet, and a six-dimensional N = (1; 1) u(Nf )-valued vector-multiplet. Each of these decompose into a 3D N = 4 vector-multiplet and hypermultiplet. For those D5-branes intersecting the D3-branes, the 3-5 strings localized at the intersection are massless. They furnish a 3D N = 4 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 [1] that, in the absence of D5-branes, establishes a correspondence between 4D N = 4 SYM and type IIB string theory on AdS5 S5. To arrive at the AdS/dCFT correspondence one considers the { 4 { low-energy e ective description of the D3/D5 system at energy scale and takes the limit `s ! 0, where `s is the string length. The dynamics of the massless degrees of freedom have two equivalent descriptions in terms of two di erent sets of eld 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, z~0 = 0. The rst set of variables that describes the D3/D5 intersection is based on an expansion around the at background: Minkowski space for the closed strings and constant values of the brane embedding coordinates for the open strings. In this case standard eld theory scaling arguments apply. After canonically normalizing the kinetic terms for open and closed string uctuations, 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 N = 4 SYM coupled to a co-dimension one planar interface, breaking half the supersymmetry and hosting a 3D N = 4 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 Nf and Nc | the fourdimensional Yang-Mills coupling | given in terms of the string coupling via gy2m := 2 gs. 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 osp(4j4), with bosonic subalgebra SO(2; 3) SO(4)R and sixteen odd generators. SO(2; 3) 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 z~0 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 uctuations around the supergravity background produced by the Nc D3branes. This background involves a nontrivial metric and Ramond-Ramond (RR) ve-form ux given in our coordinates by ds120 = f 1=2( dx dx + dy2) + f 1=2( dr~i dr~i + dz~i dz~i) ; F (5) = (1 + ?) dx0 dx1 dx2 dy df 1 ; with f = 1 + L4 (r~2 + z~2)2 ; where L4 = 4 gsNc`s4 : (2.1) The metric is asymptotically at and approaches AdS5 S5 with equal radii of L when v~ 2 r~2 + z~ 2 L2. The energy of localized modes in the throat region, as measured by an observer at position v~, is redshifted in comparison to the asymptotic xed energy according to Ev = f 1=4 (L=v~) , for v~ L. Hence, while closed string and D5-brane modes with Compton wavelengths large compared to L 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 v~=`s ! 0 in such { 5 { to sending v~=L ! 0 while holding v~=(L2 ) xed. a way that Ev`s remains xed. From the redshift relation it follows that we are sending v~=`s ! 0 while holding v~=(`s2 ) xed. For xed 't Hooft coupling gsNc, this is equivalent To facilitate taking this limit we introduce new coordinates ri = r~ i ; and write (r; ; ) for the corresponding spherical coordinates and z sometimes employ a vector notation ~r = (r1; r2; r3), ~z = (z1; z2; z3). One nds that with pzizi. We will also these new coordinates, the metric becomes2 =: (L )2GMN dxM dxN ; =: (L )4 dC(4) ; ds120 ! (L ) 2 2(r2 + z2) dx dx + dy2 + F (5) ! 4(L ) 4 4(r2 + z2)(1 + ?) dx0 dx1 dx2 dy (r dr + z dz) dr2 + r2 d 2( ; ) + d~z d~z where we've introduced a rescaled metric and four-form potential, GMN ; C(4). GMN is the metric on AdS5 S5 with radii 1 . 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{4]. This means the following: closed string modes in the (ambient) spacetime of the bulk side are dual to operators constructed from the 4D N = 4 SYM elds 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 gure 2. The validity of the supergravity approximation in the closed string sector requires that 1. The rst condition suppresses gs corrections to the low energy e ective action, while the second condition is equivalent to L `s, 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 e ective Yang-Mills theory on the D5-branes. This extends previous analyses of the D3/D5 system to the case of multiple D5branes, showing how the non-abelian interaction terms among open strings are dominant to the open-closed couplings, at least in the su(Nf ) sector of the theory. In subsection 2.3 2The metric can be brought to the form found in [4] by rst introducing standard spherical coordinates (z; ; ) in the ~z directions and then setting r = v cos and z = v sin with Then v is the AdS5 radial coordinate in the Poincare patch, with v ! 1 the asymptotic boundary, while ( ; ; ; ; ) parameterize the S5, viewed as an S 2 S 2 bration over the interval parameterized by . 2 [0; =2], and = 1. { 6 { S5, coupled to a defect composed of probe D5-branes. The boundary theory consists of an ambient N = 4 SYM on R1;3 coupled to a co-dimension one defect hosting localized modes. we will describe explicitly what these interactions look like (using the Myers non-abelian D-brane action). In preparation for that, consider the following rede nition of the relevant supergravity elds. Let SIIB[G; B; ; C(n); ] denote the type IIB supergravity action in Einstein frame. Here B is the Kalb-Ramond two-form potential and := 0 is the uctuation of the dilaton eld around its vev, 0, with e 0 gs. The C(n), n even, are the RamondRamond potentials, and is the ten-dimensional Newton constant, 2 = 12 (2 )7gs2`s8. Upon rescaling the metric and potentials according to GMN = (L )2G~MN ; BMN = (L )2B~MN ; C(n) = (L )nC~(n) ; { 7 { one nds that where the new Newton constant is SIIB[G; B; ; C(n); ] = SIIB[G~; B~; ; C~(n); ] ; = (L )4 = (2 )3p Nc 4 : Thus an expansion in canonically normalized closed string uctuations, (hMN ; bMN; '; c(n)), around the near-horizon background, (2.3), takes the form GMN = (L ) 2 GMN + hMN ; BMN = (L )2 bMN ; C(4) = (L ) 4 C(4) + c(4) ; C(n) = (L )n c(n) ; = ' ; n 6= 4 ; where GMN and C(4) were given in (2.3), and n-point couplings among closed string uctuations go as n 2. 2.3 The non-abelian D5-brane action The massless bosonic degrees of freedom on the D5-branes are a U(Nf ) gauge eld Aa, a = 0; 1; : : : ; 5, with eldstrength Fab, and four adjoint-valued scalars Xm = (Z1;2;3; Y ). (2.4) (2.5) (2.6) (2.7) The gauge eld carries units of mass while the Xm carry units of length. The eigenvalues of ( i times) the latter are to be identi ed with the displacements of the Nf D5-branes away from (~z0; y0). Our conventions are that elements of the u(Nf ) Lie algebra are represented by anti-Hermitian matrices, so there are no factors of i coming with the Lie bracket in covariant derivatives. The ` Tr ' operation denotes minus the trace in the fundamental representation, Tr := tr Nf , with the minus inserted so that it is a positive-de nite bilinear form on the Lie algebra. Later on we will generalize the discussion to a generic simple Lie algebra g, and then we de ne the trace through the adjoint representation via Tr := 2h1_ tr adj, where h_ is the dual Coxeter number. This reduces to the previous de nition for g = u(Nf ). The non-abelian D-brane action of Myers, [6], captures a subset of couplings between the 5-5 open string and ambient closed string modes. It takes the form := 2 `s2, and D5 := 2 =(gs(2 `s)6) is the D5-brane tension. Besides the factor of e in SDBI, the closed string elds are encoded in the two quantities EMN := e =2(GMN + BMN ) ; C = X C(n) n ^ exp e =2B : The factors of the dilaton are present here because we work in Einstein frame for the closed string elds. This action generalizes the non-abelian D-brane action of [24] to the case of a generic closed string background. The quantity P [TMN:::Q] denotes the gauge-covariant pullback P of a bulk tensor TMN:::Q to the worldvolume of the D5-branes. For instance, the pullback of the generalized metric to the brane is P [Eab] = Eab i(DaXm)Emb iEam(DbXm) (DaXm)Emn(DbXn) ; (2.13) with Da = @a +[Aa; ]. The closed string elds are to be taken as functionals of the matrixvalued coordinates, EMN (xP ) ! EMN (xa; iXm), de ned by power series expansion: EMN (xa; iXm) := EMN (xa; x0m) + X1 ( i)n @mn EMN ) (xa; x0m) ; The determinants in the DBI action (2.9) refer to spacetime indices a; b and m; n. n=1 n! { 8 { (2.8) (2.9) (2.10) (2.11) (2.12) (2.14) In the Chern-Simons (CS) action, (2.10), the symbol iX denotes the interior product with respect to Xm. This is an anti-derivation on forms, reducing the degree by one. Since the Xm are non-commuting one has, for example, 1 2 (i2X C(k+2))M1 Mk = [Xm; Xn]Cn(km+M2)1 Mk : (2.15) See [6] for further details. The `STr' stands for a fully symmetrized trace, de ned as follows [6]. After expanding the closed string elds in power series and computing the determinants, the arguments of the STr in (2.9) and (2.10) will take the form of an in nite sum of terms, each of which will involve powers of four types of open string variable: Fab; DaXm; [Xm; Xn], and individual Xm's from the expansion of the closed string elds. The STr notation indicates that one is to apply Tr 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 gs. Second, if F denotes any components of the `ten-dimensional' eldstrength, Fab, DaXm, or [Xm; Xn], (2.8) is known to yield results incompatible with open string amplitudes at O(F6) [25, 26], even in the limit of trivial closed string background. Finally, the action (2.8) is given directly in \static gauge," and there have been questions about whether it can be obtained from gauge xing a generally covariant action. This could lead to ambiguities in open-closed string couplings at O(F4) according to [27]. However, the results of [28] suggest that the Myers action can in fact be obtained by gauge- xing 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 e ective theory We now expand the action (2.8) in both closed and open string uctuations, where the closed string expansion is an expansion around the near-horizon geometry of the D3-branes, in accord with (2.7). This was already done in some detail in the abelian case [4], 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 SDBI D5(L ) g6 Tr 6 Z p 4 1 2(L ) 4FabF ab + 1 2 (Gmnjx0m )DaXmDaXn ; (2.16) where we recall that = 2 `s2. The factors of (L ) arise from writing the background metric in terms of the barred metric. We have introduced the notation g6 := det(gab), with gab := Gab(xa; x0m) the induced background metric on the worldvolume. It takes the form gab dxa dxb = deformation of it. Worldvolume indices will always be raised with the inverse, gab. We use the notation jx0m to indicate when other closed string elds are being evaluated at xm = x0m. 1, while z0 6= 0 gives a { 9 { The coe cient of the F 2 term determines the e ective six-dimensional Yang-Mills coupling: Note that the dimensionless coupling (gym6 ) is small in the regime Nc gsNc order to bring the scalar kinetic terms to standard form we de ne mass dimension-one scalar elds through m := 1(L )2Xm = p p gsNc 2Xm ; (2.18) (2.19) so that (Aa; m) carry the same dimension. Once the closed string elds in the D-brane action are expressed in terms of the rescaled quantities, one nds that Fab is always accompanied by a factor of (L ) 2, while [Xm; Xn] is always accompanied by the inverse factor. After changing variables to m for the scalars, all four types of open string quantities appearing in D5-brane action carry the same prefactor: (L )2 Fab; DaXm; (L ) 2 [Xm; Xn]; Xm = p p gsNc 2 (Fab; Da m; [ m ; n]; and this provides a convenient organizing principle for the expansion. Of course it is (Aa; m)c, de ned by (Aa; m ) = gym6 (Aa; m )c ; that are the canonically normalized open string modes. The open string expansion variables on the right-hand side of (2.20) 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. (Fab; Da Now, let C 2 (hMN ; bMN ; '; c(n)) denote a generic closed string uctuation, let O 2 m; [ m; n]; m) denote any of the open string expansion variables, and set op := (L )2 = p p gsNc 2 : Then the expansion of (2.8) can be written in the form SDbo5s = 1 2opgy2m6 Z d x 6 p g6 1 X no;nc=0 no nc Vno;nc ; op where Vno;nc is a sum of monomials of the form Cnc STr (Ono ), with rational coe cients. V2;0 = Tr 1 1 2 abcdef (6) m(@mcabcdef )jx0m + (Da m)c(m6b)cdef ; + + xm 0 ; 1 4 1 5! 1 3!2 (Da m)C(m4b)cdbef + 41!2 c(a4b)cdFef GmnDa mDa n + where abcdef is the Levi-Civita tensor with respect to the background metric, 012345 = ( g6) 1=2, and we have used that STr reduces to the ordinary trace when there are no more than two powers of the open string variables O. All closed string elds are to be understood as being evaluated at xm = x0m except for those in V1;1 that involve taking a transverse derivative before setting xm = x0m. There is a great deal of physics in the Vno;nc 's: V0;0 corresponds to the energy density of the background D5-brane con guration. Nf p gsNc 2, which is large when gsNc V0;1 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 elds. The strength of these tadpoles is Nf gym26 op2 / 1. However this does not necessarily mean that the probe approximation is bad! The e ects of these tadpoles on open and closed string processes will still be suppressed if the interaction vertices are su ciently weak. Consider, for example, the leading correction to the open string propagators due to these tadpoles. This corresponds to the diagram in gure 3. The correction is proportional to the product of the tadpole vertex with the cubic vertex for two open and one closed string uctuation. After canonically normalizing the open string modes via (2.21), the three-point vertex goes as . Therefore the product is proportional to Nf p gsNc 2 / (Nf p gsNc=Nc) 2 / Nf gy2m6 . 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 Nf Nc= gsNc, both the standard one-loop correction and this closed string correction will be suppressed. Note this is a slightly p created from the vacuum by a vertex in V0;1. It propagates to a three-point vertex in V2;1. This gives a correction to the open string propagator that is of the same order as a standard one-loop correction from virtual open string modes. ~z0 ~ z stronger restriction than the usual Nf Nc limit when the 't Hooft coupling gsNc is large, but nevertheless can be comfortably satis ed for a range of Nf in the regime Nc gsNc The vanishing of V1;0 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 xm = x0m, extremizes the equations of motion for the open string modes in the xed closed string background. Only the center-of-mass degrees of freedom corresponding to the central u(1) participate in V1;1 due to the trace. The strength of these interactions is gym16 op1 / gym6 , where we have made use of the convenient relation p Hence they can be treated perturbatively. Furthermore the u(1) and su(Nf ) degrees of freedom decouple in V2;0, so the couplings in V1;1 can only transmit the e ects of the closed string tadpoles to the su(Nf ) elds through higher order open string interactions. The rst three terms of V2;0 come from the DBI action, and comprise the usual YangMills action on a curved background. The nal term in V2;0, meanwhile, comes from the CS action and is non-vanishing because there is a the nontrivial RR ux in the supergravity background. It is also interesting to consider the form of terms in V3;0, or higher order open string is an STr (~ zF 2) coupling of the form interactions. V3;0 is nontrivial when z0 6= 0; V4;0 is always nontrivial. For example, there u(Nf ) (2.25) (2.26) Three- and four-point couplings in V3;0 and V4;0 come with extra factors of op relative to the three- and four-point couplings in the Yang-Mills terms, V2;0. Hence they will be suppressed relative to the Yang-Mills terms for eld variations at or below the scale . More precisely, if the elds vary on a scale 0 we merely require ( 0= )2 that these terms be suppressed relative to their counterparts in V2;0. p gsNc, in order In summary, there is a regime of parameters | namely Nc gsNc Nc= gsNc | where the leading interactions of the (bosonic) su(Nf ) open string modes are governed by V2;0. This forms the bosonic part of a six-dimensional super-Yang-Mills theory on the curved background (2.17). We can present this action in two di erent forms, both of which will prove useful below. HJEP07(21)65 First there is the form we have used to give V2;0, in which the scalars carry curved space indices. In order to be more explicit with regards to the C(4) term, we have from (2.3) that the relevant components are and so the last term of V2;0 contributes as follows: d x 6 p g6 Tr Here we have introduced ~, which should be thought of as the Levi-Civita tensor on the Euclidean R 3 spanned by ~r: ~r1r2r3 = 1, or if we work in spherical coordinates ~ r = (r2 sin ) 1. Then the bosonic part of the Yang-Mills action is We can also derive a more standard eld theoretic form for the action by rescaling the scalar elds 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 Gmn: y := (r2 + z02)1=2 y ; zi := 1 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 (2.29) and make use of the Bianchi identity, ~rirjrk Dri Frjrk = 0. We also switch to spherical coordinates, as the only surviving bulk term comes from the derivative of the (r2 + z02)2 prefactor. This integration by parts becomes Mz2 := 1 gy2m6 Z 1 2 2 r 2 b Sbndry = where d5xp R1;2 1 Z r 2 In the last term the indices ; correspond to coordinates ; along the two-sphere and = (gS2 ) 1=2 = 2(r2 + z02)=(r2 sin ). The mass parameters are de ned as follows: HJEP07(21)65 2; 4; 1 in units of the inverse AdS radius. When z0 = 0 they take these values everywhere. Although the squared mass of the Z scalars is negative, it satis es the Breitenlohner-Freedman bound [29] for AdS4. The reason for the notation M 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 S2 at r = rb ! 1.3 They are given by M 2 d5xp ( y)2 ij zi zj + F ; (2.33) 1 y 2 also generates a boundary term. After carrying out these manipulations, the bosonic action d x 6 p g6 Tr 1 4 FabF ab + 2 (Da m)(Da m) + 4 [ m; n][ m p (2.34) with d3x := dx0 dx1 dx2 and d := sin d d . If one works with the action in the form (2.31) 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 rb ! 1 of quantities computed using (2.33) 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{33] and references therein), the subject has not matured su ciently 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{10], and continuing with [11{13]. Here we follow the conventions of [12, 13]. This will provide the fermionic couplings that are quadratic order in open string uctuations | kinetic and mass-like terms. With these and 3We assume the elds are su ciently regular such that there is no boundary contribution from r = 0. This is discussed in some further detail for static con gurations later. See section 6.2. where matrices. Sf = D5 Z 2 the full set of bosonic couplings in hand, we will be able to deduce the remaining Yukawatype 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 elds are described most conveniently by introducing a doublet of ten-dimensional MajoranaWeyl spinors ^ = ( 1 ; 2) T 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 f the doublet structure. One introduces M ; N g = 2GMN , are likewise extended by 0123456789 is the ten-dimensional chirality operator and 1;2;3 are the Pauli The abelian fermionic D5-brane action, to quadratic order in ^ , takes the form d6xe p det (P [E] i F ) ^ (1 D5) h(M 1)ab ^(P )D^ a b i ^ ; (2.36) where EMN = e =2(GMN + BMN ) as before and the matrix M is Mab = e =2P [Gab] + Fab : ^ expression4 in terms of F : matrix Here we have also introduced the shorthand Fab := e D5 appearing in the -symmetry projector, 12 (1 =2P [Bab] i Fab. The idempotent D5), has a somewhat nontrivial (2.35) (2.37) D5 := p 1 X q+r=3 det(P [E] i F ) "a1 a2qb1 b2r q!2q(2r)! ( i)qFa1a2 Fa2q 1a2q (P ) b1 b2r ( i 2) (^)r ; (2.38) gauge, (aP ) = a where "012345 = 1, and the a (P ) are the pullbacks of The remaining couplings to closed string elds are encoded in the generalized derivative D^ and the mass-like operator, . We write only the terms that contribute when evaluated on the near-horizon background geometry (2.3); the full set of couplings can be found in [12, 13]. In this case M to the worldvolume. In static ^ Da = P [Da] 12 + 1 16 5! e F M(51) M5 M1 M5 (aP ) (i 2) + ; (2.39) where the terms represented by vanish when closed string uctuations are switched o , while ! 0 when closed string uctuations are switched o . The notation P [Da] is meant 4Our Mab; D5 are denoted Mfab; eD5 in [12, 13]. to indicate that one takes the pullback of DM 1;2 to the brane worldvolume, and DM is the standard covariant derivative on ten-dimensional Dirac spinors. Now we would like to argue that in the near-horizon geometry (2.3), the action (2.36) has an expansion in closed and open string uctuations controlled by the same parameters, op; , that appeared in the expansion of the bosonic action (2.23). Considering rst the rescaling of the closed string elds, (2.4), there are a few key points: After applying this rescaling under the determinant of (2.36) we can pull out a factor of (L )6, and we will have the usual factor of (L ) 2 = op accompanying Fab. The (P ) b1 b2r factor in D5 rescales according to (P ) b1 b2r = (L )2r ~(P ) b1 b2r , due to the implicit vielbein factors present in it. Taking into account the (L ) 6 from the determinant factor out front, D5 retains its form under the rescaling except that each factor of Fab picks up a corresponding (L ) 2 prefactor. This combines with the 's already present so that all Fab in D5 are accompanied by op. One can check that (M 1)ab ^(P )D^ a b gets a net factor of (L ) 1 when expressed in terms of the rescaled closed string elds, while Fab in Mab acquires an op prefactor. string terms of m via (2.19). elds around x0m is accompanied by a factor of op when we express Xm in simply for the fermion. Hence we write Together, these observations show that all open string interaction vertices between and powers of Fab; @aXm, and Xm are controlled by the expected power of op. The 2 overall prefactor of the leading 2 term is D5(L )5 = ( op gym6 ) 2(L ) 1. We can make a rescaling5 of ^ analogous to (2.19) such that the coe cient of this leading order term is i=gy2m6 . We will assume this has been done and continue using the same notation (2.40) (2.41) (2.42) where and Sf = i 2gy2m6 Z d x 6 p g6 ^ (1 (D05)) aD^ a(0) ^ (1 + O( op; )) ; (D05) := 012r 1 ; D^ a(0) := !MN;a MN 1 4 12 + 1 (D05) we took the q = 0, r = 3 term in (2.38) and used that ( g6) 1=2"b1 b6 (bP1 ) b6 = to leading order in open and closed string uctuations. In (2.42), !MN;P are the components of the spin connection with respect to the background metric GMN , evaluated 5The ^ in (2.36) must have units of (length)1=2. It would be natural to include a factor of 1=2 out in front of (2.36) so that they are dimensionless. Then the rescaling would be ^~ = o3=p4 ^ . at xm = x0m, and all gamma matrices with covariant indices are de ned using the vielbeine of the background metric. Let us evaluate (2.42) in more detail. It follows from the background (2.3) that 1 where we recall that (r; ; ) are spherical coordinates for the directions spanned by ~r. But the second term drops out of (2.40) because Regarding the ten-dimensional spin connection, there are nonzero components of the type !bm;a when z0 6= 0. (See appendix B for details.) However, the contribution of these aD^ a(0) = a (i 2) : (2.45) The projector in the last term of (2.45) will either give the identity or zero when acting on 1;2, depending on the 10D chirality of the latter. The two possibilities distinguish between a D5-brane and an anti-D5, and only one choice will lead to a supersymmetric worldvolume theory on the brane. We will see that the supersymmetric theory corresponds to Thus the coupling to the background F (5) provides a necessary mass-like term for the It is now straightforward to diagonalize the operator 12 (1 (D05)) aD^ a(0) with respect to the auxiliary doublet structure. Introducing the unitary transformation 1;2 = one nds U 1 (1 2 while of M U := p 1 2 1 ( (D05)) aD^ a(0) U y = aDa + r 2pr2 + z02 y(1 + ) ) 2 1 (1 + 3) ; (2.48) where Da := @a + 14 !bc;a bc. Thus, setting ( ; 0)T := U ^ , one sees that 0 is projected out encodes the physical degrees of freedom. Using (2.46), and recalling the de nition in (2.32), the nal result for (2.40) takes the form Sf = i 2gy2m6 Z d x 6 p g6 n aDa + M yo (1 + O( op; )) : (2.49) Note that for a ten-dimensional Majorana-Weyl spinor, the bilinear M1 Mp vanishes unless p = 3 (or 7), so the gamma matrix structure of the mass term is as it had to be. contains the degrees of freedom of a single six-dimensional Dirac fermion and we could write (2.49) in six-dimensional language, but for now it is more convenient to work directly with the `10D' form. Finally, we will infer from (2.49) and the bosonic Yang-Mills terms (2.31), the nonabelian analogs of the leading terms in (2.49) that complete (2.31) into a supersymmetric invariant. Clearly the covariant derivative Da should be generalized to a gauge covariant derivative, Da := @a + 14 !bc;a bc + [Aa; ]. We will, for convenience, continue to use the same notation for this covariant derivative as we did above. A natural ansatz that will yield the Yukawa couplings is simply to extend this to a ten-dimensional covariant derivative: aDa m[ m; ]. Our ultimate justi cation for this ansatz (detailed below) will be HJEP07(21)65 that supersymmetry requires it. Hence we take the fermionic terms of the Yang-Mills action to be Sym;f := i 2gy2m6 Z d x 6 p g6 Tr is now valued in the adjoint representation of su(Nf ). We've included a boundary action for the fermion, f Sbndry := i Z d5xp n y o : (2.50) (2.51) The analysis of [34] for fermions on anti-de Sitter space demonstrates that such boundary terms are necessary in order to have a well-de ned variational principle. We will see that the boundary action (2.51) is also required for supersymmetry. Without it, the supersymmetry variation of the action would produce boundary terms that do not vanish on their own. These points are analyzed in sections 4.3 and 4.4 below. In principle such boundary terms should have already been present in (2.36), but we are not aware of any previous work on this issue. 3 Supersymmetry As noted previously, the intersecting D-brane system of gure 1 preserves eight superp symmetries. In the near-horizon limit of the D3-brane geometry, the symmetry algebra is enhanced to osp(4j4) with sixteen odd generators, provided the D3 and D5-branes have zero transverse separation. The leading low-energy e ective description in the regime Nc gsNc Nc= gsNc consists of a six-dimensional Yang-Mills theory on the rigid background (2.17) in which the transverse separation appears as a parameter, (along with decoupled supergravity and u(1) sectors). Thus one expects the Yang-Mills theory to possess eight supersymmetries when z0 6= 0 and sixteen when z0 = 0. In this section we rst review the Killing spinors of the background geometry [35, 36] and the induced Killing spinors on the D5-brane worldvolume [37]. Then, using the latter as generators, we exhibit the full set of supersymmetry transformations on the Yang-Mills elds and establish the invariance of the action, (2.31) plus (2.50), modulo boundary terms. To analyze the spectrum of modes on the asymptotically AdS4 space we choose an adapted basis for the six-dimensional gamma matrices: The next step is then to diagonalize the operator set of eigenspinors on the two-sphere. This is an S2 Dirac operator coupled to a Dirac 2D~ over a complete monopole background. The eigenvalue equation ;r = ;r 3 ; is equivalent to the dim g equations ~ D = iM ; D 2 sin ips ( 1 cos ) 2 ( )s = iM ( )s : (C.31) (C.32) (C.33) (C.34) (C.36) (C.37) (C.38) +0 ; (C.35) = 0 spinors HJEP07(21)65 Here = speci es the northern or southern patch of the S2 respectively. The two solutions will be related by a transition function, (+)s = eips ( )s, on the overlap. This is a classic problem with a completely explicit solution. (See appendix C of [49] for a recent treatment.) The eigenspinors are labeled by three indices, ; j; m, where 2 f+; ; 0g and (j; m) are angular momentum quantum numbers. Let j := 1 2 (jpsj 1) : Then the eigenspinors with = have j-values starting at j + 1 and increasing integer steps, while m runs from j to j in integer steps as usual. They are given by 1 2 ( ;)js;m( ; ) = p N j m; 1 2ps ei(m+ ps=2) djm; 1 2ps ( )12 + i djm; 12 ps ( ) 1 where djm;m0 ( ) is a Wigner little d function38 and +0 = (1; 0)T . The correspond to the special value j = j only, and their form depends on the sign of ps: 0(;j)s;m( ; ) = ei(m+ ps=2) ( N mj; j djm; j ( ) +0 ; ps > 0 ; N mj;j djm;j ( ) 1 +0 ; ps < 0 : Note these solutions only exist when ps is nonzero; j takes an unphysical value when j ps = 0. If ps = 0 then the = Nm;m0 are normalization coe cients: solutions are a complete set with j 2 f 21 ; 23 ; : : :g. The where the choice of phase will be convenient below. The corresponding eigenvalues are is m0 Yjm( ; ) = 2j+1 1=2 eim djm;m0 ( ). 4 38These solutions can also be expressed in terms of spin-weighted spherical harmonics. The relationship j Nm;m0 = ei jm m0j=2 r 2j + 1 4 ; M s;j = 2 p(2j + 1)2 ps2 : The = 0 modes are zero modes of D~ , but this does not mean that they correspond to massless spinors on AdS4 as there are other terms in the equation (C.30) that must be taken into account. To nd the four-dimensional spectrum we insert the mode expansion into the linearized equation (C.30), using (C.31). Note that ( )s = X s ;j;m(x ; r) ( ;j)s;m( ; ) ; = 3 ; B6 = B4 Here B4 is the product over the imaginary ;r and satis es ( ;r) = B4 ;rB4 1. (We also used that it is necessarily the product of an odd number of 's, as charge conjugation reverses chirality for Spin(1; 3).) Hence we'll need the action of 3 and charge conjugation, on the eigenspinors. These are found to be 3 ( ;j);;sm = ( );s ;j;m ; = ; and 3 ( );s 0;j ;m = sgn(ps) 0(;j);s;m ; ps 2 1( ( ;j);s;m) = sgn m + ( ); s ;j; m ; = ; and 1( 0(;j);s;m) = 0(;j); ; sm : In order to obtain the latter one requires the property djm;m0 ( ) = ( 1)m m0 dj m; m0 ( ). The phase of (C.37) was chosen to make the action of charge conjugation as simple as possible. Remember also that p s = ps. See the discussion under (4.25). Using all these facts, we nd that (C.30) splits into two families of coupled systems for the modes s ;j;m. The coupled system for the = 0 modes (which exist when ps 6= 0) is (C.41) (C.42) (C.43) (C.44) (C.45) (C.46) 7! 1 and where B 0 0 D+ B D 0 0 ! s 0;j ;m s B4( 0;j ; m ) ! = 0 ; 0 = = D4 h ims;z3 D + rDr is the standard Dirac operator on the asymptotically AdS4 space. D= 4 = (r2 + z02)1=2 r 1 Inserting (C.45) into (C.43) and dividing through by (r2 + z02)1=2, we have where B0 := sgn(ps) 3r + imz3;s ; which is a more useful form for studying the large r asymptotics of solutions. At this point we will content ourselves with understanding the r ! 1 behavior of solutions. Then it is su cient to expand the matrix operator in (C.46) through O(1=r). To this order it diagonalizes and reduces to 2r + sgn(ps)my;s 1 + jpsj 1 2 r + O(1=r2) s 0;j ;m = 0 ; along with an equivalent equation for the conjugate spinor. The equation diagonalizes with respect to r. If we decompose into eigenspinors, 0;j ;m = 0;j ;m + 0s;;j ;m ; s s;+ with r s; 0;j ;m = s; 0;j ;m ; then the leading behavior of solutions is 0;j ;m / e sgn(ps)my;srr 23 m0 (1 + O(1=r)) ; s; m0 := 1 + jpsj : 2 When my;s 6= 0 we have exponential decay or blowup behavior. When my;s = 0 we have power-law behavior dictated by the mass m0, which we have de ned in such a way that it can be identi ed with a standard AdS4 mass for the fermion. In other words, the asymptotic behavior of solutions to (D= 4 + m) = 0 on AdS4 is / r 23 m . Since the ms0 s; are all negative, we see that the normalizable modes in the case my;s = 0 are necessarily associated with 0;j ;m. However the normalizable (exponentially decaying) modes when my;s 6= 0 could be associated with either psmy;s. It will be associated with s; 0;j ;m if this sign is negative. We will comment further s; 0;j ;m, depending on the sign of the product Taking similar steps, one nds that the coupled system for the = modes can be on this below. put in the following form: where (C.47) (C.48) (C.49) (C.50) (C.51) (C.52) 0 B BB C B 0 D B C D 3r ps 2r ; D C = my;s B = sgn m + ps (mz1;s s C C B B4( +;j; m s s +;j;m ;j;m B4( s ;j; m ) r r2 + z02 + 1 C A ) CCCC = 0 ; imz3;s ; This transformation diagonalizes (C.53) to the order we are working. The new variables ; satisfy the asymptotic equations 2r jC(r)j + O(1=r2) ( (sj;m); (sj;m)) = 0 ; 1 ps 2 2r where the +( ) is for ( ) respectively, and jC(r)j = my;s + jM s;j j2 = my2;s r psmy;s + r (2j + 1)2 4r2 = 21r (2j + 1) ; ( jmy;sj ps sgn(my;s) + O(1=r2) ; my;s 6= 0 ; 2r my;s = 0 : to r, as in (C.49). Then the asymptotic behavior of solutions to (C.55) is Let (j;m) and (sj;;m) denote the positive and negative chirality components with respect ;s Henceforth restrict our analysis to the r ! 1 behavior of solutions. Working through O(1=r) the B entries can be dropped and the system reduces to 20 ps 2 1 1 ijM s;j j r my;s p2s + ijM s;j j r 1 1 A + O(1=r2)5 s +;j;m ;j;m ! e i = C=jCj, and consider the unitary transformation along with an equivalent equation for the conjugates. Let (r) denote the phase of C, s (j;m) (j;m) := U s +;j;m ;j;m e i =2 ei =2 ! e i =2 ei =2 s +;j;m ;j;m ! : HJEP07(21)65 where the AdS4 masses are s; (j;m) / s; (j;m) / 8 < e jmy;sjrr 23 (1+ p2s sgn(my;s))(1 + O(1=r)) ; my;s 6= 0 ; : r 23 m( ) j (1 + O(1=r)) ; 8 < e jmy;sjrr 23 (1 p2s sgn(my;s))(1 + O(1=r)) ; my;s 6= 0 ; : r 23 m( ) j (1 + O(1=r)) ; my;s = 0 ; my;s = 0 ; m j ( ) = j 1 2 ; j j + : 3 2 The normalizable modes for while the normalizable modes of are those that have positive r chirality asymptotically, are those that have negative r chiarlity asymptotically. In both cases the normalizable modes along Lie algebra directions with my;s 6= 0 are exponentially decaying while those along directions with my;s = 0 are power-law decaying. (C.54) (C.55) (C.56) (C.57) (C.58) (C.59) s 0;j ;m modes as lling in a lower j = j rung for the tower in the sense that Recall that j starts at j + 1 = 12 (jpsj + 1) for these modes. However we can view the jpsj 2 1 + 3 2 = m0 : Also the asymptotic r-chiralities match provided sgn(ps)my;s < 0 whenever my;s 6= 0. Assuming this is the case, for the same reasons as discussed under (C.18), we can identify s (j ;m) s 0;j ;m ; 12 = i r ; as the lowest rung of the tower for those s such that ps 6= 0. Finally we note that the r-chirality condition can be translated back to a condition on the six-dimensional or on the ten-dimensional . First, since the action of r commutes with the rotation U relating ; to the s;j;m, we see that will be an asymptotic eigenspinor of when restricted to normalizable modes of or only. We will have = totically for the normalizable -type modes and = +i r asymptotically for the i r normalizable -type modes. One can then show from (C.20) and (C.27) that 1 i r = 0 () 1 r y = 0 : Hence positive (negative) r chirality corresponds to negative (positive) r y chirality. D Boundary supersymmetry In this appendix we provide some of the details of the asymptotic analysis that we quoted in subsection 4.4. We begin with Br and B bndry, appearing in (4.62). From (3.29), (C.60) (C.61) HJEP07(21)65 (C.62) asymp(C.63) + (D.1) B r = " Tr 1 2 + Tr Meanwhile B we infer B r y + 1 2 Fab abr + (Da m) m ar 1 2 r " : [ m; n] mn r M m m y r bndry is de ned in terms of the supersymmetry variation of the boundary action, (3.26), according to (4.61). Taking the variation of (3.26) with respect to (3.27), + 1 r 1 sin (D y ) (D y ) y r + Tr n y " o : (D.2) in the large r expansion of Br + B bndry. Since the boundary measure in (4.62) is O(r3) as r ! 1, we must work through O(r 3) For the moment we set aside the last terms of (D.1) and (D.2) involving the variation of the fermion, and we focus on the remaining terms. Since " = O(r1=2) and = O(r 3=2), we must compute the terms in square-brackets through O(r 2), utilizing the eld asymptotics (4.58). All terms can contribute at this order. We expand out, plug in vielbein factors, and collect terms together as follows: The rst four sets of terms are proportional to the projector 12 (1 zi term, the relevant spinor bilinear is " r y) acting to the = O(1=r2). However, (nn) is nonzero, then we must set the superconformal generators 0 to zero, which implies " = 0. Hence, we get an extra order of suppression from the zi does not contribute, and therefore this term can be neglected. The same reasoning applies to the remaining terms in this set involve the spinor bilinear " 1 part of y in the rst term. The +, which is O(1=r2). Thus we need to evaluate them through O(1=r), which corresponds precisely to the contribution from X~ in y; A ; A . Speci cally, 2 r y + r 2(r2 + z0 ) D 2 2(r2 + z02) D r sin y y 1 ! r ! r 1 r r^ X~(n) + ^ X~(n) + 1 ^ X~(n) + ; : ; + + 1 2 + Tr 2 r y + 1 1 2(r2 + z02) r 1 + zi zi (nn) part of + (D zi ) zi r + 2(r2 +z02) F y( y r r sin 1 2 F r y " F 1 r sin + D o : ( y r ) r2 +z02 r sin ) + (D zi ) zi r + 1 2 [ zi ; y] ziyr + (D y) y r+ zi zi + [ zi ; zj ] zizj r zi zi (1 r y)+ y( y r + HJEP07(21)65 + (D.3) (D.4) (D.5) Hence the relevant combination is 1 r y ^ + X~(n)(1 + r y ) rr^ + ^ + + = = 2 ry r r 2 ry(~ (r) X~(n)) + ; where we used r y + = + and r^ r + ^ + ^ = ( r1 ; r2 ; r3 ) ~ (r). (See (B.12).) Next consider the set of six terms inside the large round brackets of (D.3). It follows from the eld asymptotics (4.58) that all of these terms start at O(1=r2). Furthermore all of the gamma matrix structures associated with these terms commute with they involve "+ + = O(1=r) and " = O(1=r2), and we only need to worry about the former. The order O(1=r2) terms in the round brackets are all of the form D(nn) or ad( (znin)) acting on X~ , where D(nn) = @ + ad(a(nn)). Speci cally, the relevant combination . Hence of terms is 1 2r2 ( = r ^ 1 2r2 r ^ + y rr^) D(nn)X~(n) + ( zi r ^ zi r ^ ziyrr^) [ (znin); X~(n)] The remaining terms in the square brackets of (D.3) start at O(1=r2) and anti-commute with r y . Hence they involve the couplings "+ and " +, and we only need to keep the former. One simply needs to evaluate (A ; zi ) on their leading behavior, (a(nn); (znin)). Collecting results, we have " Tr 1 2 1 1 2r2 2r2 Tr "+ Tr "+ Tr n n y D(nn) + zi ad( (znin)) (~ (r) X~(n)) +o + 2 r y +D(nn) zi (nn) " o + O(r 7=2) : zi + 21 [ (znin); (znjn)] zizj + (D.7) (D.8) (D.9) Plugging in (3.21) and (4.56) leads to the result in the text, (4.63). The next step is to analyze the asymptotics of " , as given in (3.27). Our goal will + through O(r 3=2) since this is the only order that can contribute be to compute ( " ) to (D.7), given the asymptotics of give the supersymmetry variation of the non-normalizable mode, to set this eld to zero, its variation need not be zero. The reason is that we are allowing 0 (nn). Even if we choose certain non-normalizable modes of the bosonic be turned on, and they can source the supersymmetry variation of the non-normalizable elds | namely (a(nn); (znin); X~(nn)) | to , (4.58). We note that the O(r 3=2) terms of ( " ) + fermion modes. where We expand out (3.27) and collect terms as follows: " = fMr + M + M + Mrestg " ; Mr = (r2 + z02)1=2Dr y ry + M M (r2 + z02)1=2 r r sin D D y y + y y + r r y r sin 2(r2 + z02) Fr 2(r2 + z02) Fr + r ; r ; 2(r2 + z02) F r2 sin ; and Mrest = (r2 + z02)1=2Dr zi rzi + 1 r + F r r + F + + 1 D 1 r sin F sin zi + D 1 2 yzi 1 + (r2 + z02)1=2 D + [ y; zi ] yzi + + D zi zi + [ zi ; zj ] zizj : (D.10) Let's start with Mrest. It follows from the eld asymptotics that all seven terms in the big square-brackets are O(1=r2). Furthermore the gamma matrix structure of each of these terms is such that it maps the ( )-chirality eigenspace of r y to the ( )-chirality eigenspace. Hence, these terms acting on "+ give an O(r 3=2) contribution to ( " ) , while these terms acting on " give an O(r 5=2) contribution to ( " )+. Therefore these terms can be neglected to the order we are working. In contrast the terms in the last line preserve the chirality and so acting on "+ they give a contribution to ( " )+ that is O(r 3=2) that must be kept. Finally, consider the rst two terms of Mrest. Using (2.30) one nds (r2 + z02)1=2Dr zi rzi + (r2 + z02)1=2 zi yzi = The Dr zi term is O(1=r2) and exchanges r y chiralities. If (znin) is nonzero then the projector annihilates ", so the last term is also e ectively O(1=r2) and exchanges chiralities. Hence these terms are on the same footing as the square-bracketed terms and can be neglected. In summary, (Mrest")+ = (Mrest") = O(r 3=2) : 1 2r2 2 = Dr zi rzi + zi zir 1 r y : (D.11) + D(nn) zi (nn) zi + 21 [ (znin); (znjn)] zizj "+ + O(r 5=2) ; Now consider M y. Plugging in (2.30) we have Mr = 2(r2 + z02)Dr y ry + 2 r y( ry + ) + 2(r2 + z02)Dr y + 2 r y 2 r y1 + O(1=r) ry 1 ry + r y 2(r2 + z02) F 2(r2 + z02) Dr y 1 4r2(r2 + z02) P y ; where in the last step we recalled the de nition, (4.5). The rst term will drop out of (D.7) since it involves the opposite projector. The large r expansion of P in (4.47). Using that result here gives y was determined 1 Mr = O(1=r) ry 1 r y r^ X~(nn) + O(1=r) : (D.14) 1 2r2 = O(1=r) y 1 r y + ^ X~(nn) + O(1=r) r ; (D.15) = O(1=r) y 1 r y ^ X~(nn) + O(1=r) r : (D.16) y 1 2r2 2r2 r y 1 2r2 r y 1 2r2 + + sin 4r(r2 + z02) P 1 4r(r2 + z02) P r ^ r + ^ r "+ + O(r 5=2) y r^ r + ^ + ^ X~(nn) "+ + O(r 5=2) y~ (r) X~(nn) "+ + O(r 5=2) ; Similar manipulations lead to M and M r sin Thus we have ((Mr + M + M )")+ = ((Mr + M + M )") = O(r 3=2) ; (D.17) Combining (D.12) and (D.17) leads to the result quoted in the text, (4.65). Our nal goal is to derive the asymptotics of due to the massless AdS4 fermions, as given in (4.55) with (4.67). The leading behavior of these modes as r ! 1 is O(r 3=2) and the rst subleading behavior is O(r 5=2). They are solutions to the fermion equation of motion 0 = D + rDr + D + M y + zi ad( zi ) + y ad( y (D.18) Our analysis in appendix C.2 shows that the massless modes are in the simultaneous kernel of ad( y1) and ad(P ). Taking this into account with respect to the eld asymptotics (4.58), the large r form of the equation of motion is 0 = 3 2r 1 r r ad(a(rn)) + ~ D~ 1 h ^ + y + ^ + 1 h r y i r^ ; where ~ D~ is (the 10D embedding of) the standard Dirac operator on the two-sphere. The rst three terms give the leading order equation of motion while the remaining terms give O(1=r) corrections. Note that this equation only involves the asymptotics of the bosonic modes that we keep in the truncation, (5.1), and therefore the asymptotics of the solution to the order we need will be the same as in the truncated theory. Hence we will derive the equations of motion for the fermion in the truncated theory, which we quoted in (5.7), and then consider the asymptotics of it. We rst use results from appendix C.2 to determine the form of the 10D fermion, , restricted to the massless AdS4 modes. These are the j = 1=2 doublet ( 12 ;m)(x ; r). They satisfy (C.55) with the plus sign, and since my;s = ps = 0 for these modes, we have 2 jC(r)j = jM s; 1 j=r = 1=r. Hence ( 12 ;m)(x ; r) = ( r) 3=2 ~( 12 ;m)(x ) (1 + O(1=r)) : and therefore the corresponding ; 12 ;m modes are ; 12 ;m = p12 e i =4 spinor, (C.39), restricted to these modes, which we will denote by normalizable modes. The boundary data ~( 12 ;m) can be decomposed into eigenspinors of r, ~( 12 ;m) = and we will see that ~( 12 ;m) corresponds to the normalizable modes and ~( 12 ;m) to the non+ The phase of C(r) that appears in the unitary transformation of (C.54) is = =2, ( 12 ;m). Hence the 6D ( ) j=1=2, takes the form ( ) j=1=2 = X m= 1=2 ( 12 ;m)(x ; r) 1 p 2 e i =4 +; 12 ;m + ei =4 ; 12 ;m where the are given by Hence ; 12 ;m = N m1=;21=2eim 0 A : ( ) j=1=2 = X m ( 12 ;m)(x ; r) N m1=;21=2eim 0 d1m=;21=2( ) 1 A : Now, using d11==22;1=2 = d1=12=2; 1=2 = cos 2 , d1=12=2;1=2 = d11==22; 1=2 = sin 2 , and N11==22; 1=2 = iN11==22;1=2 iN1=2, one nds that this spinor can be expressed in the form ( ) j=1=2 = N1=2 cos 2 sin 2 sin 2 cos 2 ei 3 =2 0 i ( 12 ; 12 )(x ; r) A 1 = exp i exp 2 2 6D(x ; r) ; where in the last step we introduced the 6D spinor i ( 12 ; 12 )(x ; r) A ; 1 and wrote the expression in 6D notation with the de nitions (C.31). (D.20) r ~( 12 ;m) , (D.21) (D.22) (D.23) (D.24) (D.25) 6D has a large r expansion starting at O(r 3=2) with the leading behavior given in terms of the boundary spinors ~, (D.20). If one restricts the 4D spinors to r eigenspaces, , this corresponds to restricting 6D to 6D de ned by i r 6D = ( 12) 6D = 6D : We use this to express j(=)1=2 in the form ( ); 2 exp 2 6+D(x ; r) + exp exp 2 6D(x ; r) : This result is straightforwardly expressed in 10D notation via (C.20). We nd ( ) j=1=2 = hS2( ; ) +(x ; r) + hS2( ; ) (x ; r) ; where we made use of (3.8), + + is de ned in terms of 6D via (C.27), and r y : y This is (4.54), which is given in a natural basis with respect to the S2 frame in which are constant. Indeed, this was assumed throughout the analysis in appendix C.2. This is to be plugged into the full fermion equation of motion, E := expand the Dirac operator, with the bosonic elds restricted to (5.1) as well. The basic idea it to pull the factors of ; ) through to the left and collect the terms that are proportional to each. We satisfy ; ; identities: and (D.26) (D.27) (D.28) (D.29) (D.30) (D.32) (D.33) (D.34) aDa = D + rDr + D~= S2 + 1 sin ad(A ) + ad(A ) ; (D.31) with D~= S2 the standard Dirac operator on the unit S2. Then we make use of the following D~= S2h( ; ) = yh( ; ) h( ; ) r ; rh( ; ) = h( ; ) r ; = r r yh( ; ) = h( ; ) r ; hS2( ; ) hS2( ; ) hS2( ; ) hS2( ; ) ^ ( ; = hS2( ; ) ^ ( ; = hS2( ; ) r^ ( ; ; y) ; ; y) ; y) ; : + (r2 + z02)1=2( ; + hS2 ( ; ) D + rDr The mass-like term 1 r(r2 + z2) 0 h1h2h3 r r y r 1 ^A + 1 r sin ^A + r^ y = ( ^ ^ X~ + ^^ X~ + r^r^ X~ ) = 1 2r2 X~ : where we used (5.8) and (5.10), vanishes for the AdS4 background where z0 = 0, and in general the r-dependent mass vanishes asymptotically like O(1=r2). Plugging in the truncation ansatz (5.1) for the bosonic modes, observe that Hence the quantities in curly brackets in (D.35) are independent of ; on this ansatz. After introducing the triplet notation (5.10) and the metric (5.4), we obtain the result quoted in the text, (5.6) and (5.7): ; y 1 2r2 1 1 ^A + 1 r 1 ^A + r sin r sin 1 1 = + zi [ ~ zi ; + ] ^A + r^ y; r + + zi [ ~ zi ; ^A + r^ y; + : (D.35) z 2 0 r(r2 + z02)1=2 Note for these last three we are employing (D.29) as well. Then we nd E = hS2 ( ; ) D + + rDr (D.36) (D.37) (D.39) (D.40) O(1=r) in the operator acting on , one nds The asymptotics of 0 = 1 h D(nn) + hi ad(X(in))i + O(1=r2) : e = D + rDr h1h2h3 + zi [ zi ; ] + hi [X i; ] = 0 : (D.38) Now we analyze the large r asymptotics of this equation. Keeping terms through r(r2 + z02)1=2 z 2 are 2r 1 1 +(x ; r) = (x ; r) = ( r)3=2 0 ( r)3=2 (nn)(x ) + r (0n)(x ) + ( r)5=2 1+(x ) + O(r 7=2) ; ( r)5=2 1 (x ) + O(r 7=2) : 1 1 This is consistent with (4.56), remembering that ( r3 )cart = ( r)S2 . The 1 are found by plugging this expansion back into (D.39) and solving it at the rst subleading order. 1+ = 1 = h r h zi ad( (znin)) i (n) + h ad(a(rnn)) + 0 r hi ad(X(in))i (nn) ; 0 D(nn) + zi ad( (znin)) i (nn) + r h ad(a(rnn))+ hi r ad(X(in))i 0(n) : (D.41) 0 This can be expressed in terms of Cartesian frame quantities using ( r)S2 = ( r3 )cart and ( r~ (h))S2 + = ( r ; r ; ry)S2 + = ( y ; y; ry)S2 + = y(~ (r))cart + ; (D.42) which leads to the results for (D.40) quoted in (4.67). E Some details on the truncation F^ zi = D ones we have Here we collect expressions for the components of the non-abelian eldstrength and covariant derivatives evaluated on the truncation ansatz (5.1). We use a 10D notation A^M for the gauge eld and Higgs elds in which we identify (A^zi ; A^y) ( zi ; y) and, for example, zi . There is nothing to say about F ; F r; F^ zi ; F^rzi ; F^zizj . For the remaining HJEP07(21)65 and F F F^ y ^ Fzi ^ Fzi F^ziy trnc sin trnc trnc trnc trnc trnc sin 2r 1 2r2 2r 1 2r2 2r 1 ^ D X~ ; ^ D X~ ; r^ D X~ ; 2r 1 ^ [ zi ; X~ ] ; ^ [ zi ; X~ ] ; r^ [ zi ; X~ ] ; Fr Fr ^ Fry F ^ F y ^ F y trnc trnc trnc trnc trnc trnc sin 1 2r2 ^ ~ X sin 2r2 P 2r2 P 2 sin 1 2r2 2r2 X X 2r 1 ^ DrX~ ; ^ X~ + 2 2r3 r^ X~ + sin 2r sin 2r r^ ^ DrX~ ; 1 2r2 2X~ r^ DrX~ ; 1 2 2r [X~ ; 1 1 2 2r [X~ ; 2 2r [X~ ; X~ ] ; (E.1) (E.2) (E.3) We also list some formulae that are used in subsection 6.4 for the reduction of the BPS equations. From (E.3) one nds that and converting to the Cartesian coordinate system results in HJEP07(21)65 Dr y trnc 1 sin D Fr + D y trnc y trnc sin 2r2 DrX~ 1 ^ 2r 1 2 2r2 [X~ ; X~ ] ; DrX~ 1 2 2r2 [X~ ; X~ ] ; 1 ^ 2r DrX~ 1 2 2r2 [X~ ; X~ ] ; Dr3 Dr2 Dr2 y trnc y trnc y trnc 1 2r2 2r2 1 2r2 DrX DrX DrX 1 1 1 1 sin 1 i 2 p~q~ dxp~ dxq~ : Likewise, converting from Fpr; Fp ; Fp , to the Cartesian frame Fpri results in Fpr1 Fpr2 Fpr3 trnc trnc trnc 1 1 2r2 2r2 sin cos ( 2r2Fpr) + sin sin DpX cos DpX 2 ; sin cos DpX 3 + sin sin ( 2r2Fpr) + cos DpX 1 ; 2r2 sin cos DpX 2 sin sin DpX 1 + cos ( 2r2Fpr) ; while Fpy trnc 1 2r2 sin cos DpX 1 + sin sin DpX 2 + cos DpX 3 : Identical expressions hold for the F^zpri and F^zpy upon replacing Dp ! ad( zp ). F The BPS energy from (6.30). In this appendix we show how one obtains (6.65) from (6.62), and as a special case, (6.37) parameterize R4 with the standard orientation. Introduce a basis of self-dual two-forms, First we introduce some notation that exposes the structure of 04. Let xp~ = (x1; x2; z^1; z^2) !1 = dx2 dz^2 dx1 dz^1 ; !2 = dx2 dz^1 + dx1 dz^2 ; !3 = dx1 dx2 + dz^1 dz^2 : (F.1) These can be expressed in terms of 't Hooft matrices, (E.4) (E.5) (E.6) (E.7) (F.2) where our conventions are 0 0 0 1 = BBB 01 00 1 0 1 0 1 C C ; A 0 1 0 0 0 0 2 = BBB 00 0 1 0 C 1 0 0 C C ; A 1 0 0 0 1 0 0 0 C 3 = BBB 0 0 0 1 C C : A 0 0 1 0 Note this is a slightly di erent convention than the standard one given in [100] in that ( 1; 2; 3)here = ( 2; 1; 3)standard : With our convention matrix multiplication gives the quaternion algebra, i j = k, with a plus sign in front of the rather than a minus. Then, in terms of the two-forms (F.1), one has (F.3) (F.4) ij + (F.5) (F.6) (F.8) 04 = 1 4(r2 + z02)2 dy^ dr1 dr2 dr3 + ( dy^ dr1 + dr2 dr3) ^ !1+ + ( dy^ dr2 + dr3 dr1) ^ !2 + ( dy^ dr3 + dr1 dr2) ^ !3 : Dropping the !1 and !2 terms gives !0 . 4 Converting to spherical coordinates, (r; ; ), results in 04 = r2 sin d d r^i + r dy d ^i + r sin dy d ^ i ^ !i + dr terms : Here we have suppressed terms that have a leg along the radial direction since they will not contribute to the boundary integral. It follows that ( 04 ^ !CS)12 z^1z^2y^ = 2 1 ( i)p~q~ !CS A^y r2 sin r^i +A^ r ^i A^ r sin ^i ; A^p~ ; A^q~ ; (F.7) where we are using the notation !CS(A^A; A^B; A^C ) (!CS)ABC for the components of the Chern-Simons form. If we want !40 ^ !CS instead, then we drop the i = 1; 2 terms. These expressions integrated against dx1 dx2 d d dz^1 dz^2 dy^ at the boundary r ! 1. Hence we need the large r limit of (F.7). The leading behavior of the A^p~ is O(1) and given by the non-normalizable S2 singlet modes. Therefore the furthest we need to go in the subleading asymptotics of ( y; A ; ) is the X~(n) terms, which will yield a nite contribution to (F.7) as r ! 1. In fact, if one restricts to the X~(n) terms, the rst factor in !CS collapses A^y r2 sin r^ + A^ r ^ A^ r sin ^ ! 1 2 sin X~(n) : One might worry that the y1 and 't Hooft charge terms in the asymptotics of ( y; A ; ) will lead to a divergence, but this is not the case. The 't Hooft charge drops out of (F.7). The y1 term can contribute, but integration over the two-sphere will pick out subleading behavior in the A^p~ factors such that the result is nite. (The integration over S2 should be carried out before the r ! 1 limit is taken.) We thus have Z d x S2 d d sin ~ p~q~ 2 !CS(X~(n); A^p~; A^q~) + r2r^ !CS( y ; A^p~; A^q~) : 1 (F.9) such that is equivalent to 1 X m= 1 Here the normalization convention is consistent with the one taken in (4.44). Then (F.9) ap~;(1;m)(x ; r)Y1m( ; ) = r^ ~ap~(n)(x ) + O(r 3) : (F.13) Both Chern-Simons terms are of a similar structure in that they involve an adjointvalued scalar in one of the factors. When this is the case, one can obtain the following equivalent expression, starting from the de nition (6.28): !CS(X~(n); A^p~; A^q~) = 2 Tr nX~(n)F^p~q~o + @q~ h Tr fX~(n)Ap~g (F.10) is constant and that any power-law modes of A^p~ commute with Here it should be understood that the total derivative term is only present when p~; q~ = 1; 2. An analogous expression holds with X~(n) ! 1 y . However in this case we can use that y 1 y1 to observe that the total derivative terms just subtract o half of the rst term, resulting in: Z lim r!1 S2 d r2r^i !CS( y1; A^p~; A^q~) = lim Z r!1 S2 d r2r^i Tr n y1F^p~q~o : Now let us recall the mode expansion of A^p~ = (Ap; zp). The terms we need are (F.11) ap~;(1;m)(x ; r)Y1m( ; ) + (F.12) where ap~(x ; r) = a(p~nn)(x ) + O(r 1) as usual, and we introduce the triplet notation, ~ap~, 1 X m= 1 p 2r2 1 Z 04 ^ !CS = Z (nn) 2 Tr fX~(n)fp~q~ g p Tr f y1f~p~(q~n)g+ 1 3 (nn) i (nn) + [a(p~nn); a(q~nn)] and f~q~(q~n) = 2@[p~~aq~(]n), and we used the integral Z S2 d r^ir^j = 4 3 ij : (F.14) (F.15) This reproduces the magnetic contribution to the energy bound given in (6.65). Dropping the terms proportional to the rst two 't Hooft symbols will give the result for 04 ! !40. Furthermore there are some simpli cations if we plug in the explicit form of p3~q~: !40 ^ !CS = 2 Z d2x 2 Tr nX(3n) f1(2nn) + [ (zn1n); (zn2n)] o 1 3 p Tr f y1f132(n)g+ (nn) i @1 h Tr fX(3n)a2 g (nn) i : (F.16) For the second term we can pull y1 out of the integral. Then we are simply computing the total magnetic ux of the third component of the normalizable mode of the gauge eld triplet. (See (6.35).) Meanwhile by Stokes' theorem the last two terms give a line integral around the circle at in nity: Z R2 h 2 n (nn) i g h (nn) io = g Tr nX(3n)a(nn)o : (F.17) I S1 1 Taking these facts into account one nds that (F.16) reproduces the magnetic energy contribution to (6.37). For the electric energy contribution, we rst note that (?E)12 z^1z^2y^ = r2 sin grrFr0 : (F.18) (grr) 1 and the de nition (6.36), one quickly nds the remaining terms in (6.37) and (6.65). Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] J.M. Maldacena, The large-N limit of superconformal eld theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE]. [2] A. Karch and L. Randall, Localized gravity in string theory, Phys. Rev. Lett. 87 (2001) 061601 [hep-th/0105108] [INSPIRE]. [3] A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT's on branes with boundaries, JHEP 06 (2001) 063 [hep-th/0105132] [INSPIRE]. [4] O. DeWolfe, D.Z. Freedman and H. Ooguri, Holography and defect conformal eld theories, Phys. Rev. D 66 (2002) 025009 [hep-th/0111135] [INSPIRE]. [5] S.K. Domokos and A.B. Royston, to appear. [6] R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE]. [7] M. Aganagic, C. Popescu and J.H. Schwarz, D-brane actions with local kappa symmetry, Phys. Lett. B 393 (1997) 311 [hep-th/9610249] [INSPIRE]. [8] M. Cederwall, A. von Gussich, B.E.W. Nilsson, P. Sundell and A. Westerberg, The Dirichlet super p-branes in ten-dimensional type IIA and IIB supergravity, Nucl. Phys. B 490 (1997) 179 [hep-th/9611159] [INSPIRE]. [9] E. Bergshoe and P.K. Townsend, Super D-branes, Nucl. Phys. B 490 (1997) 145 [hep-th/9611173] [INSPIRE]. [10] M. Aganagic, C. Popescu and J.H. Schwarz, Gauge invariant and gauge xed D-brane actions, Nucl. Phys. B 495 (1997) 99 [hep-th/9612080] [INSPIRE]. [11] D. Marolf, L. Martucci and P.J. Silva, Fermions, T duality and e ective actions for D-branes in bosonic backgrounds, JHEP 04 (2003) 051 [hep-th/0303209] [INSPIRE]. [12] D. Marolf, L. Martucci and P.J. Silva, Actions and Fermionic symmetries for D-branes in bosonic backgrounds, JHEP 07 (2003) 019 [hep-th/0306066] [INSPIRE]. Press (2007). [INSPIRE]. [INSPIRE]. [13] L. Martucci, J. Rosseel, D. Van den Bleeken and A. Van Proeyen, Dirac actions for 037 [hep-th/0602174] [INSPIRE]. (2006) 066 [hep-th/0605017] [INSPIRE]. [14] D. Arean and A.V. Ramallo, Open string modes at brane intersections, JHEP 04 (2006) [15] R.C. Myers and R.M. Thomson, Holographic mesons in various dimensions, JHEP 09 [16] I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE]. [17] M. Mezei, S.S. Pufu and Y. Wang, A 2d=1d Holographic Duality, arXiv:1703.08749 [18] E. Corrigan, C. Devchand, D.B. Fairlie and J. Nuyts, First Order Equations for Gauge Fields in Spaces of Dimension Greater Than Four, Nucl. Phys. B 214 (1983) 452 [INSPIRE]. [19] A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE]. [20] J. Polchinski, String theory. Vol. 2: Superstring theory and beyond, Cambridge University [21] J. Erdmenger, Z. Guralnik and I. Kirsch, Four-dimensional superconformal theories with interacting boundaries or defects, Phys. Rev. D 66 (2002) 025020 [hep-th/0203020] [22] A. Karch and E. Katz, Adding avor to AdS/CFT, JHEP 06 (2002) 043 [hep-th/0205236] [hep-th/0207171] [INSPIRE]. Phys. B 501 (1997) 41 [hep-th/9701125] [INSPIRE]. [23] S. Yamaguchi, Holographic RG ow on the defect and g theorem, JHEP 10 (2002) 002 [24] A.A. Tseytlin, On nonAbelian generalization of Born-Infeld action in string theory, Nucl. [25] A. Hashimoto and W. Taylor, Fluctuation spectra of tilted and intersecting D-branes from the Born-Infeld action, Nucl. Phys. B 503 (1997) 193 [hep-th/9703217] [INSPIRE]. [26] P. Bain, On the nonAbelian Born-Infeld action, hep-th/9909154 [INSPIRE]. [27] J. de Boer, K. Schalm and J. Wijnhout, General covariance of the nonAbelian DBI action: Checks and balances, Annals Phys. 313 (2004) 425 [hep-th/0310150] [INSPIRE]. [28] P.S. Howe, U. Lindstrom and L. Wul , On the covariance of the Dirac-Born-Infeld-Myers action, JHEP 02 (2007) 070 [hep-th/0607156] [INSPIRE]. [29] P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE]. Phys. B 573 (2000) 703 [hep-th/9910052] [INSPIRE]. [30] W. Taylor and M. Van Raamsdonk, Multiple Dp-branes in weak background elds, Nucl. [31] J.M. Drummond, P.S. Howe and U. Lindstrom, Kappa symmetric nonAbelian Born-Infeld actions in three-dimensions, Class. Quant. Grav. 19 (2002) 6477 [hep-th/0206148] [32] P.S. Howe, U. Lindstrom and L. Wul , Superstrings with boundary fermions, JHEP 08 hep-th/9902137 [INSPIRE]. 40 (1999) 4518 [hep-th/9805151] [INSPIRE]. [33] I.A. Bandos, Superembedding approach to Dp-branes, M-branes and multiple D(0)-brane [34] M. Henneaux, Boundary terms in the AdS/CFT correspondence for spinor elds, [35] H. Lu, C.N. Pope and J. Rahmfeld, A construction of Killing spinors on Sn, J. Math. Phys. [36] P. Claus and R. Kallosh, Superisometries of the AdS x S superspace, JHEP 03 (1999) 014 [37] K. Skenderis and M. Taylor, Branes in AdS and p p wave space-times, JHEP 06 (2002) 025 [hep-th/9812087] [INSPIRE]. [hep-th/0204054] [INSPIRE]. [hep-th/0005098] [INSPIRE]. [hep-th/9806087] [INSPIRE]. [38] D. Arean, A.V. Ramallo and D. Rodriguez-Gomez, Mesons and Higgs branch in defect theories, Phys. Lett. B 641 (2006) 393 [hep-th/0609010] [INSPIRE]. [39] M. Blau, Killing spinors and SYM on curved spaces, JHEP 11 (2000) 023 [40] M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/0002230] [INSPIRE]. 5849 [hep-th/0209067] [INSPIRE]. [41] V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE]. [42] R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE]. [43] S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [44] K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) [45] I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE]. [46] I. Papadimitriou, Holographic renormalization as a canonical transformation, JHEP 11 (2010) 014 [arXiv:1007.4592] [INSPIRE]. B 138 (1978) 1 [INSPIRE]. [47] G. 't Hooft, On the Phase Transition Towards Permanent Quark Con nement, Nucl. Phys. [48] A. Kapustin, Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE]. [49] G.W. Moore, A.B. Royston and D. Van den Bleeken, Parameter counting for singular monopoles on R3, JHEP 10 (2014) 142 [arXiv:1404.5616] [INSPIRE]. [50] P.B. Kronheimer, Monopoles and Taub-NUT Metrics, MSc Thesis, Oxford (1985), http://www.math.harvard.edu/ kronheim/papers.html. [51] M. Pauly, Monopole moduli spaces for compact 3-manifolds, Math. Ann. 311 (1998) 125. [52] S.A. Cherkis and A. Kapustin, Singular monopoles and supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 525 (1998) 215 [hep-th/9711145] [INSPIRE]. [53] G.W. Moore, A.B. Royston and D. Van den Bleeken, Brane bending and monopole moduli, JHEP 10 (2014) 157 [arXiv:1404.7158] [INSPIRE]. [55] A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE]. [56] D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE]. [hep-th/9709027] [INSPIRE]. [58] C.G. Callan and J.M. Maldacena, Brane death and dynamics from the Born-Infeld action, Nucl. Phys. B 513 (1998) 198 [hep-th/9708147] [INSPIRE]. [59] M. de Leeuw, A.C. Ipsen, C. Kristjansen, K.E. Vardinghus and M. Wilhelm, Two-point functions in AdS/dCFT and the boundary conformal bootstrap equations, arXiv:1705.03898 [INSPIRE]. [60] V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in anti-de Sitter space-time, Phys. Rev. D 59 (1999) 046003 [hep-th/9805171] [INSPIRE]. [61] D.Z. Freedman, K. Pilch, S.S. Pufu and N.P. Warner, Boundary Terms and Three-Point Functions: An AdS/CFT Puzzle Resolved, JHEP 06 (2017) 053 [arXiv:1611.01888] [62] N. Banerjee, B. de Wit and S. Katmadas, The o -shell c-map, JHEP 01 (2016) 156 Phys. B 259 (1985) 460 [INSPIRE]. Nucl. Phys. B 281 (1987) 211 [INSPIRE]. [68] M. Gunaydin, L.J. Romans and N.P. Warner, Gauged N = 8 Supergravity in Five-Dimensions, Phys. Lett. B 154 (1985) 268 [INSPIRE]. [69] M. Pernici, K. Pilch and P. van Nieuwenhuizen, Gauged N = 8 D = 5 Supergravity, Nucl. [70] B. de Wit and H. Nicolai, The Consistency of the S7 Truncation in D = 11 Supergravity, [arXiv:1512.06686] [INSPIRE]. Rev. Lett. 38 (1977) 121 [INSPIRE]. Phys. 72 (1980) 15 [INSPIRE]. [63] G. 't Hooft, Magnetic Monopoles in Uni ed Gauge Theories, Nucl. Phys. B 79 (1974) 276 [64] A.M. Polyakov, Particle Spectrum in the Quantum Field Theory, JETP Lett. 20 (1974) 194 [65] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. B 59 (1975) 85 [INSPIRE]. [66] E. Witten, Some Exact Multi-Instanton Solutions of Classical Yang-Mills Theory, Phys. [67] P. Forgacs and N.S. Manton, Space-Time Symmetries in Gauge Theories, Commun. Math. [71] A. Baguet, O. Hohm and H. Samtleben, Consistent Type IIB Reductions to Maximal 5D Supergravity, Phys. Rev. D 92 (2015) 065004 [arXiv:1506.01385] [INSPIRE]. [72] K.-M. Lee and P. Yi, Dyons in N = 4 supersymmetric theories and three pronged strings, Phys. Rev. D 58 (1998) 066005 [hep-th/9804174] [INSPIRE]. [73] J.P. Gauntlett, N. Kim, J. Park and P. Yi, Monopole dynamics and BPS dyons N = 2 super Yang-Mills theories, Phys. Rev. D 61 (2000) 125012 [hep-th/9912082] [INSPIRE]. [74] S.K. Donaldson, Nahm's equations and the classi cation of monopoles, Commun. Math. [75] R.S. Ward, Completely Solvable Gauge Field Equations in Dimension Greater Than Four, [76] D.-s. Bak, K.-M. Lee and J.-H. Park, BPS equations in six-dimensions and eight-dimensions, Phys. Rev. D 66 (2002) 025021 [hep-th/0204221] [INSPIRE]. [77] O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N eld theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE]. [78] B. Julia and A. Zee, Poles with Both Magnetic and Electric Charges in Nonabelian Gauge [79] G.W. Gibbons and N.S. Manton, Classical and Quantum Dynamics of BPS Monopoles, [80] E. Witten and D.I. Olive, Supersymmetry Algebras That Include Topological Charges, Phys. [81] R. Reyes Carrion, A generalization of the notion of instanton, Di er. Geom. Appl. 8 (1998) Phys. 96 (1984) 387 [INSPIRE]. Nucl. Phys. B 236 (1984) 381 [INSPIRE]. Theory, Phys. Rev. D 11 (1975) 2227. Nucl. Phys. B 274 (1986) 183 [INSPIRE]. Lett. B 78 (1978) 97 [INSPIRE]. 1 [INSPIRE]. [INSPIRE]. 549 [INSPIRE]. [arXiv:1403.6836] [INSPIRE]. [82] C. Lewis, Spin(7) instantons, Ph.D. Thesis, Oxford University Press, Oxford (1998). [83] S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, in The geometric universe, Oxford (1996), Oxford University Press, Oxford (1998), pg. 31{47. [84] S. Donaldson and E. Segal, Gauge Theory in higher dimensions, II, arXiv:0902.3239 [85] S.A. Cherkis, Octonions, Monopoles and Knots, Lett. Math. Phys. 105 (2015) 641 [86] J.A. Harvey and A. Strominger, Octonionic superstring solitons, Phys. Rev. Lett. 66 (1991) [87] D.D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, vol. 12, Oxford University Press, Oxford (2007). [88] T. Max eld, D. Robbins and S. Sethi, A Landscape of Field Theories, JHEP 11 (2016) 162 [arXiv:1512.03999] [INSPIRE]. [arXiv:1010.2353] [INSPIRE]. [89] A. Haydys, Fukaya-Seidel category and gauge theory, J. Sympl. Geom. 13 (2015) 151 [90] E. Witten, Fivebranes and Knots, Quantum Topol. 3 (2012) 1 [arXiv:1101.3216] [INSPIRE]. [91] D. Gaiotto and E. Witten, Knot Invariants from Four-Dimensional Gauge Theory, Adv. Theor. Math. Phys. 16 (2012) 935 [arXiv:1106.4789] [INSPIRE]. ve-brane world volume action, Nucl. Phys. B 547 (1999) 127 [hep-th/9810092] [INSPIRE]. Compactness Theorem, arXiv:1608.07272 [INSPIRE]. qq-characters, arXiv:1701.00189 [INSPIRE]. holography, Phys. Rev. D 68 (2003) 106007 [hep-th/0211222] [INSPIRE]. Pseudoparticle, Phys. Rev. D14 (1976) 3432 [Erratum ibid. D 18 (1978) 2199] [INSPIRE]. D-branes on backgrounds with uxes , Class. Quant. Grav . 22 ( 2005 ) 2745 [hep-th/0504041] systems , Phys. Part. Nucl. Lett. 8 ( 2011 ) 149 [arXiv: 0912 .2530] [INSPIRE]. [54] D.-E. Diaconescu , D-branes, monopoles and Nahm equations , Nucl. Phys. B 503 ( 1997 ) 220 [57] G.W. Gibbons , Born-Infeld particles and Dirichlet p-branes, Nucl . Phys. B 514 ( 1998 ) 603 [92] E. Witten , Analytic Continuation Of Chern-Simons Theory , AMS/IP Stud. Adv. Math. 50 [93] C.G. Callan Jr. , A. Guijosa and K.G. Savvidy , Baryons and string creation from the [ 94] J.P. Gauntlett , N.D. Lambert and P.C. West , Branes and calibrated geometries, Commun. Math. Phys. 202 ( 1999 ) 571 [ hep -th/9803216] [INSPIRE]. [95] M. Aganagic , A. Karch , D. Lust and A. Miemiec, Mirror symmetries for brane con gurations and branes at singularities , Nucl. Phys. B 569 ( 2000 ) 277 [ hep -th/9903093]


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Sophia K. Domokos, Andrew B. Royston. Holography for field theory solitons, Journal of High Energy Physics, 2017, 65, DOI: 10.1007/JHEP07(2017)065