Holography for field theory solitons
MMI-TH-1756
Holography for field theory solitons
Sophia K. Domokos a and Andrew B. Royston b a Department of Physics, New York City College of Technology, 300 Jay Street, Brooklyn, NY 11201,USA b George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy, Texas A&MUniversity, College Station, TX 77843, USA
E-mail: [email protected] , [email protected] Abstract:
We extend a well-known D-brane construction of the AdS/dCFT correspondenceto non-abelian defects. We focus on the bulk side of the correspondence and show that thereexists a regime of parameters in which the low-energy description consists of two approx-imately decoupled sectors. The two sectors are gravity in the ambient spacetime, and asix-dimensional supersymmetric Yang–Mills theory. The Yang–Mills theory is defined on arigid
AdS × S background and admits sixteen supersymmetries. We also consider a one-parameter deformation that gives rise to a family of Yang–Mills theories on asymptotically AdS × S spacetimes, which are invariant under eight supersymmetries. With future holo-graphic applications in mind, we analyze the vacuum structure and perturbative spectrum ofthe Yang–Mills theory on AdS × S , as well as systems of BPS equations for finite-energy soli-tons. Finally, we demonstrate that the classical Yang–Mills theory has a consistent truncationon the two-sphere, resulting in maximally supersymmetric Yang–Mills on AdS . a r X i v : . [ h e p - t h ] J u l ontents N = 4 SYM on
AdS
496 Bogomolny equations and monopoles 55
A.1 The DBI action 78A.2 The CS action 81
B Background geometry and Killing spinors 83
B.1 Frame rotations 85B.2 D5-brane Killing spinor equation 87– i –
Mode analysis 88
C.1 Bosons 88C.2 Fermions 93
D Boundary supersymmetry 99
D.1 The massless fermion modes 104
E Some details on the truncation 107F The BPS energy 109
The original AdS/CFT duality [1] is nearing its twentieth anniversary. Even AdS/dCFT,which introduces defects in conformal field theories with gravity duals [2–4], is fifteen yearsold. 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 thatforms 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 ofD3-branes, we consider the seemingly simple non-abelian generalization, with several parallelD5-branes.The resulting model, when subjected to Maldacena’s low-energy limit and restricted to anappropriate regime of parameters, offers rich physics and rich mathematics, which we beginto uncover here. Denoting the numbers of D3- and D5-branes by N c and N f respectively, andthe string coupling by g s , the regime of parameters is N c (cid:29) g s N c (cid:29) N f (cid:28) N c / √ g s N c .The first two conditions are the ones that arise in the usual AdS/CFT correspondence. Theyensure 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’ N f (cid:28) N c . We will seethat it arises naturally when we demand that corrections from gravity to the su ( N f ) sectorof the D5-brane theory be suppressed. In this regime, therefore, the effects of closed stringscan 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 sensethat it relates bulk closed strings to operators in the ambient part of the boundary theory, andbulk 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 operatorsconfined to a defect in the boundary CFT. As the bulk SYM is dual to a (2+1)-dimensionalsystem, it is potentially relevant to holographic condensed matter applications. Indeed, the– 1 –ulk SYM admits a zoo of solitonic objects, whose masses and properties are constrained bysupersymmetry. 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 detailedconstruction of the dual boundary theory will appear elsewhere, [5]. We begin by constructing a six-dimensional (6D) SYM theory with osp (4 |
4) symmetry froma D3/D5 intersection. We assume that the number of D3-branes ( N c ) is large, so we canrepresent them with a Type IIB supergravity solution. We then consider the D5-branes asprobes 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 forthe fermions using the abelian action of [7–13], and infer the non-abelian gauge and Yukawacouplings via a simple ansatz consistent with gauge invariance and supersymmetry. We thenapply the Maldacena low-energy near-horizon limit.The resulting action is summarized in equations (3.24)-(3.26). While we obtained thisaction from a D-brane model, it makes sense as a classical field theory for arbitrary simpleLie groups.We go on to analyze the vacuum structure, perturbative spectrum, and the BPS equationssatisfied by solitons in the 6D SYM theory. We also show that the 6D theory has a nonlinearconsistent truncation to maximally supersymmetric YM theory on AdS .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 N c → ∞ . One component is a standard Coulomb branch labeledby vevs of Higgs fields. The other components are labeled by magnetic charges and arequite complicated: they have roughly the form of moduli spaces of singular monopolesfibered over spaces of Higgs vevs. A D-brane picture (see Figure 4 below) provides someintuition for these vacua. • We perform a perturbative mode analysis around a class of vacua that carry magneticflux. The background fields of this class are Cartan-valued and simple enough to makethe linearized equations tractable. Furthermore, the background fields of any vacuumwill asymptote to the same near the boundary, so the results for the asymptotic behaviorof fluctuations are robust. This is important for the holographic dictionary, where onemaps modes to local operators in the dual, based in part on their decay properties nearthe boundary. – 2 –ur analysis of the perturbative spectrum generalizes previous results for the abelianD5-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 trans-form of the on-shell action with respect to one of the low-lying modes, along the linesof [16], is required for holographic duality. We identify a set of low-lying non-normalizable modes that can be turned on withoutviolating the variational principle or supersymmetry. These modes form a natural classof boundary values for soliton solutions in the non-abelian D5-brane theory. In theholographic dual, meanwhile, they source a set of relevant operators—and in one casea distinguished irrelevant operator. • Having explored the vacua and perturbative structure of the bulk SYM theory, we thensurvey various systems of BPS equations. These first order equations arise when wedemand that field configurations preserve various amounts of supersymmetry.Solutions to the BPS equations saturate bounds on the energy functional. These boundsdepend on a combination of the fields’ boundary values as well as the magnetic andelectric fluxes through the asymptotic boundary.The BPS systems we obtain house a number of generalized self-duality equations thatare well known in mathematical physics, like (translationally invariant) octonionic in-stantons [18], and the extended Bogomolny equations [19]. All of these equations aredefined 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 andtake the low-energy limit of the action to arrive at a curved space SYM theory. In section 3 weverify the invariance of the action under supersymmetry. In section 4 we describe the vacuumstructure of the model, and formulate asymptotic boundary conditions on the fields. In section5 we derive the consistent truncation of our six-dimensional theory to four dimensions, whilein section 6 we derive the BPS equations satisfied by solitons in the system. We conclude anddiscuss future directions in section 7. Necessary but onerous details are relegated to a seriesof Appendices.
In this section we describe the brane set-up, the AdS/dCFT picture, and the low-energy limitand parameter regime that isolates six-dimensional SYM as the low-energy effective theoryon the D5-branes. The paper [17] appeared when this work was nearing completion. Its authors make a closely relatedobservation in maximally supersymmetric YM on
AdS . The consistent truncation of the 6D theory on thetwo-sphere, explained in section 5, shows that these results are in fact describing the same phenomenon. – 3 – 1 2 3 4 5 6 7 8 9D3 X X X XD5 X X X X X X x µ ˜ r i ˜ z i y Figure 1 . The intersecting brane system. The x µ directions common to both types of brane world-volume are suppressed in the figure on the left. The D5-branes can be separated from the D3-branesby a distance ˜ z in the directions transverse to both stacks. We begin with a non-abelian version of the brane configuration in [4]. N f D5-branes and N c D3-branes in the ten-dimensional IIB theory span the directions indicated in Figure 1.Standard arguments [20] show that the intersecting D3/D5 system preserves one quarter ofthe supersymmetry of 10D type IIB string theory, or eight supercharges.The ten coordinates, ˜ x M = ( x µ ; ˜ r i ; ˜ z i ; y ) are divided as follows: x µ , µ, ν = 0 , ,
2, parame-terizes the R , spanned by both stacks; the triplet ˜ r i = (˜ r , ˜ r , ˜ r ) = ( x , x , x ) parameterizesthe remaining directions along the D5-branes; the triplet ˜ z i = (˜ z , ˜ z , ˜ z ) = ( x , x , x ) param-eterizes directions orthogonal to both stacks; and finally y = x parameterizes the remainingdirection along the D3-branes and orthogonal to the D5-branes. We reserve the notation x M = ( x µ , r i , z i , y ) for a rescaled version of these coordinates to be introduced below. We willsometimes use spherical coordinates (˜ r, θ, φ ) to parameterize the ˜ r i directions, and we denotethe radial coordinate in the ˜ z i directions by ˜ z . We also write ˜ x a = ( x µ , ˜ r i ), a = 0 , , . . . , x m = (˜ z i , y ), m = 1 , . . . ,
4, for the full set of directions parallel and transverse to theD5-branes respectively.The D3-branes are taken to be coincident and sitting at ˜ r i = ˜ z j = 0. The center of massposition of the D5-branes in the transverse ˜ x m space is denoted ˜ x m = (˜ z ,i , y ). We will allowfor relative displacements of the D5-branes from each other, but assume that these distancesare small compared to the string scale. In other words, the separation is well-described byvev’s of non-abelian scalars in the D5-brane worldvolume theory. This will be explained inmore detail below. When all D5-branes are positioned at ˜ z i = 0 an SO (1 , × SO (3) r × SO (3) z subgroup of the ten-dimensional Lorentz group is preserved. Nonzero D5-brane displacements– 4 –n ˜ z break SO (3) z . This can be explicit or spontaneous from the point of view of the D5-braneworldvolume theory, depending on whether the center of mass position ˜ z ,i is, respectively,nonzero or zero.As noted above, eight of the original thirty-two Type IIB supercharges are preservedby the brane setup. From the point of view of the three-dimensional intersection, this isequivalent 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 groupsin the ˜ r i and ˜ z i directions. The light degrees of freedom on the D3-branes and the D5-branesare a four-dimensional N = 4 u ( N c )-valued vector-multiplet, and a six-dimensional N = (1 , u ( N f )-valued vector-multiplet. Each of these decompose into a 3D N = 4 vector-multipletand hypermultiplet. For those D5-branes intersecting the D3-branes, the 3-5 strings localizedat the intersection are massless. They furnish a 3D N = 4 hypermultiplet transforming inthe bi-fundamental representation of the appropriate gauge groups. Meanwhile the masslessclosed strings comprise the usual type IIB supergravity multiplet. Let us now consider the low-energy limit of the brane setup, that ultimately yields the defectAdS/CFT correspondence. This is the famous Maldacena limit [1] that, in the absence ofD5-branes, establishes a correspondence between 4D N = 4 SYM and type IIB string theoryon AdS × S . To arrive at the AdS/dCFT correspondence one considers the low-energyeffective description of the D3/D5 system at energy scale µ and takes the limit µ(cid:96) s → (cid:96) s is the string length. The dynamics of the massless degrees of freedom have twoequivalent descriptions in terms of two different sets of field variables. This fact is the essenceof the original AdS/dCFT correspondence.To simplify the present discussion we temporarily assume no separation between thebrane stacks – in other words, ˜ z = 0. The first set of variables that describes the D3/D5intersection is based on an expansion around the flat background: Minkowski space for theclosed 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 thekinetic terms for open and closed string fluctuations, interactions of the closed strings and 5-5open strings amongst themselves, as well as the interactions of the closed and 5-5 open stringswith the other open strings, vanish in the low-energy limit. These degrees of freedom decouplefrom the system. Meanwhile the 3-3 and 3-5 strings form an interacting system described byfour-dimensional N = 4 SYM coupled to a co-dimension one planar interface, breaking halfthe supersymmetry and hosting a 3D N = 4 hypermultiplet. The interface action, which canin principle be derived from the low energy limit of string scattering amplitudes, was obtainedin [4] by exploiting symmetry principles. The entire theory contains a single dimensionlessparameter in addition to N f and N c —the four-dimensional Yang–Mills coupling—given interms of the string coupling via g := 2 πg s .– 5 –he interface plus boundary ambient Yang–Mills theory is classically scale invariant, andit was argued in [4, 21] to be a superconformal quantum theory. The symmetry algebra is osp (4 | SO (2 , × SO (4) R and sixteen odd generators. SO (2 ,
3) isthe three-dimensional conformal group of the interface while the odd generators correspond tothe eight supercharges along with eight superconformal generators. This is the “defect CFT”side of the correspondence. Considering a nonzero separation ˜ z corresponds to turning on arelevant mass deformation in the dCFT [22, 23].Our focus here will be mostly on the other side of the correspondence, which is based onan expansion in fluctuations around the supergravity background produced by the N c D3-branes. This background involves a nontrivial metric and Ramond-Ramond (RR) five-formflux given in our coordinates byd s = f − / ( η µν d x µ d x ν + d y ) + f / ( d˜ r i d˜ r i + d˜ z i d˜ z i ) ,F (5) = (1 + (cid:63) ) d x d x d x d y d f − , with f = 1 + L (˜ r + ˜ z ) , where L = 4 πg s N c (cid:96) s . (2.1)The metric is asymptotically flat and approaches AdS × S with equal radii of L when˜ v ≡ ˜ r + ˜ z (cid:28) L . The energy of localized modes in the throat region, as measured by anobserver at position ˜ v , is redshifted in comparison to the asymptotic fixed energy µ accordingto E v = f / µ ∼ ( L/ ˜ v ) µ , for ˜ v (cid:28) L . Hence, while closed string and D5-brane modes withCompton wavelengths large compared to L decouple as before, excitations of arbitrarily highenergy can be achieved in the throat region. The near-horizon limit isolates the entire setof stringy degrees of freedom in the throat region by sending ˜ v/(cid:96) s → E v (cid:96) s remains fixed. From the redshift relation it follows that we are sending ˜ v/(cid:96) s → v/ ( (cid:96) s µ ) fixed. For fixed ’t Hooft coupling g s N c , this is equivalent to sending ˜ v/L → v/ ( L µ ) fixed.To facilitate taking this limit we introduce new coordinates r i = ˜ r i L µ , z i = ˜ z i L µ , (2.2)and write ( r, θ, φ ) for the corresponding spherical coordinates and z ≡ √ z i z i . We will alsosometimes employ a vector notation (cid:126)r = ( r , r , r ), (cid:126)z = ( z , z , z ). One finds that with these– 6 – igure 2 . The defect AdS/CFT correspondence. The bulk theory consists of an ambient IIB stringtheory on AdS × S , coupled to a defect composed of probe D5-branes. The boundary theory consistsof an ambient N = 4 SYM on R , coupled to a co-dimension one defect hosting localized modes. new coordinates, the metric becomes d s → ( Lµ ) (cid:26) µ ( r + z ) (cid:2) η µν d x µ d x ν + d y (cid:3) ++ 1 µ ( r + z ) (cid:2) d r + r dΩ ( θ, φ ) + d (cid:126)z · d (cid:126)z (cid:3) (cid:27) =: ( Lµ ) G MN d x M d x N ,F (5) → Lµ ) µ ( r + z )(1 + (cid:63) ) d x d x d x d y ( r d r + z d z )=: ( Lµ ) d C (4) , (2.3)where we’ve introduced a rescaled metric and four-form potential, G MN , C (4) . G MN is themetric on AdS × S with radii µ − .The degrees of freedom in the near-horizon geometry include both the closed strings andthe open strings on the D5-branes. String theory in this background is conjecturally dualto the dCFT system, with the duality ‘acting twice’ [2–4]. This means the following: closedstring modes in the (ambient) spacetime of the bulk side are dual to operators constructedfrom the 4D N = 4 SYM fields in the (ambient) spacetime on the boundary. Open stringmodes on the D5-branes, which form a defect in the bulk, are dual to operators localized onthe defect in the boundary theory. These operators are constructed from modes of the 3-5strings and modes of the 3-3 strings restricted to the boundary defect. See Figure 2. The metric can be brought to the form found in [4] by first introducing standard spherical coordinates( z, ζ, χ ) in the (cid:126)z directions and then setting r = v cos ψ and z = v sin ψ with ψ ∈ [0 , π/ µ = 1. Then v isthe AdS radial coordinate in the Poincare patch, with v → ∞ the asymptotic boundary, while ( ψ, θ, φ, ζ, χ )parameterize the S , viewed as an S × S fibration over the interval parameterized by ψ . – 7 –he validity of the supergravity approximation in the closed string sector requires that N c (cid:29) g s N c (cid:29)
1. The first condition suppresses g s corrections to the low energy effectiveaction, while the second condition is equivalent to L (cid:29) (cid:96) s , ensuring that higher derivativecorrections are suppressed as well.In subsection 2.4 we’ll see how this limit suppresses the interactions between closedstring and open string D5-brane modes, leading to an effective Yang–Mills theory on theD5-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 tothe open-closed couplings, at least in the su ( N f ) sector of the theory. In subsection 2.3 wewill describe explicitly what these interactions look like (using the Myers non-abelian D-braneaction).In preparation for that, consider the following redefinition of the relevant supergravityfields. Let S IIB [ G, B, ∆ φ , C ( n ) ; κ ] denote the type IIB supergravity action in Einstein frame.Here B is the Kalb–Ramond two-form potential and ∆ φ := φ − φ is the fluctuation of thedilaton field φ around its vev, φ , with e φ ≡ g s . The C ( n ) , n even, are the Ramond-Ramondpotentials, and κ is the ten-dimensional Newton constant, κ = (2 π ) g s (cid:96) s . Upon rescalingthe metric and potentials according to G MN = ( Lµ ) ˜ G MN , B MN = ( Lµ ) ˜ B MN , C ( n ) = ( Lµ ) n ˜ C ( n ) , (2.4)one finds that S IIB [ G, B, ∆ φ , C ( n ) ; κ ] = S IIB [ ˜ G, ˜ B, ∆ φ , ˜ C ( n ) ; κ ] , (2.5)where the new Newton constant is κ = κ ( Lµ ) = (2 π ) √ πN c µ − . (2.6)Thus an expansion in canonically normalized closed string fluctuations, ( h MN , b MN , ϕ, c ( n ) ),around the near-horizon background, (2.3), takes the form G MN = ( Lµ ) (cid:0) G MN + κh MN (cid:1) , B MN = ( Lµ ) κb MN , ∆ φ = κϕ ,C (4) = ( Lµ ) (cid:16) C (4) + κc (4) (cid:17) , C ( n ) = ( Lµ ) n κc ( n ) , n (cid:54) = 4 , (2.7)where G MN and C (4) were given in (2.3), and n -point couplings among closed string fluctu-ations go as κ n − . The massless bosonic degrees of freedom on the D5-branes are a U ( N f ) gauge field A a , a = 0 , , . . . ,
5, with fieldstrength F ab , and four adjoint-valued scalars X m = ( Z , , , Y ). Thegauge field carries units of mass while the X m carry units of length. The eigenvalues of– 8 – − i times) the latter are to be identified with the displacements of the N f D5-branes awayfrom ( (cid:126)z , y ). Our conventions are that elements of the u ( N f ) Lie algebra are represented byanti-Hermitian matrices, so there are no factors of i coming with the Lie bracket in covariantderivatives. The ‘ Tr ’ operation denotes minus the trace in the fundamental representation,Tr := − tr N f , with the minus inserted so that it is a positive-definite bilinear form on theLie algebra. Later on we will generalize the discussion to a generic simple Lie algebra g , andthen we define the trace through the adjoint representation via Tr := − h ∨ tr adj , where h ∨ is the dual Coxeter number. This reduces to the previous definition for g = u ( N f ).The non-abelian D-brane action of Myers, [6], captures a subset of couplings between the5-5 open string and ambient closed string modes. It takes the form S bosD5 = S DBI + S CS , with (2.8) S DBI = τ D5 (cid:90) d xe − ∆ φ ×× STr (cid:26)(cid:112) − det ( P [ E ab ] + P [ E am ( Q − − δ ) mn E nb ] − iλF ab ) det Q mn (cid:27) , (2.9) S CS = − τ D5 (cid:90) STr (cid:26) P (cid:104) e λ − i X i X C (cid:105) ∧ e − iλF (cid:27) , (2.10)where Q mn := δ mn + λ − [ X m , X k ] E kn , (2.11) λ := 2 π(cid:96) s , and τ D5 := 2 π/ ( g s (2 π(cid:96) s ) ) is the D5-brane tension. Besides the factor of e − ∆ φ in S DBI , the closed string fields are encoded in the two quantities E MN := e ∆ φ / ( G MN + B MN ) , C = (cid:88) n C ( n ) ∧ exp (cid:16) e ∆ φ / B (cid:17) . (2.12)The factors of the dilaton are present here because we work in Einstein frame for the closedstring fields. This action generalizes the non-abelian D-brane action of [24] to the case of ageneric closed string background.The quantity P [ T MN...Q ] denotes the gauge-covariant pullback P of a bulk tensor T MN...Q to the worldvolume of the D5-branes. For instance, the pullback of the generalized metric tothe brane is P [ E ab ] = E ab − i ( D a X m ) E mb − iE am ( D b X m ) − ( D a X m ) E mn ( D b X n ) , (2.13)with D a = ∂ a + [ A a , · ]. The closed string fields are to be taken as functionals of the matrix-valued coordinates, E MN ( x P ) → E MN ( x a ; − iX m ), defined by power series expansion: E MN ( x a , − iX m ) := E MN ( x a , x m ) + ∞ (cid:88) n =1 ( − i ) n n ! X m ··· m n ( ∂ m · · · ∂ m n E MN ) ( x a , x m ) , (2.14)The determinants in the DBI action (2.9) refer to spacetime indices a, b and m, n .– 9 –n the Chern-Simons (CS) action, (2.10), the symbol i X denotes the interior product withrespect to X m . This is an anti-derivation on forms, reducing the degree by one. Since the X m are non-commuting one has, for example,(i X C ( k +2) ) M ··· M k = 12 [ X m , X n ] C ( k +2) nmM ··· M k . (2.15)See [6] for further details.The ‘STr’ stands for a fully symmetrized trace, defined as follows [6]. After expandingthe closed string fields in power series and computing the determinants, the arguments ofthe STr in (2.9) and (2.10) will take the form of an infinite sum of terms, each of which willinvolve powers of four types of open string variable: F ab , D a X m , [ X m , X n ], and individual X m ’s from the expansion of the closed string fields. The STr notation indicates that one isto apply Tr to the complete symmetrization on these variables.The precise regime of validity of the Myers action is not a completely settled issue. Firstof all, like its abelian counterpart, it captures only tree-level interactions with respect to g s .Second, if F denotes any components of the ‘ten-dimensional’ fieldstrength, F ab , D a X m , or[ X m , X n ], (2.8) is known to yield results incompatible with open string amplitudes at O ( F )[25, 26], even in the limit of trivial closed string background. Finally, the action (2.8) is givendirectly in “static gauge,” and there have been questions about whether it can be obtainedfrom gauge fixing a generally covariant action. This could lead to ambiguities in open-closedstring couplings at O ( F ) according to [27]. However, the results of [28] suggest that theMyers action can in fact be obtained by gauge-fixing symmetries in a generally covariantformalism where the Chan–Paton degrees of freedom are represented by boundary fermionson the string worldsheet. As we will see below, none of these ambiguities pose a problem inthe scaling limit we are interested in. We now expand the action (2.8) in both closed and open string fluctuations, where the closedstring expansion is an expansion around the near-horizon geometry of the D3-branes, inaccord with (2.7). This was already done in some detail in the abelian case [4], but thereare some important new wrinkles that arise in the non-abelian case. We summarize the mainpoints here and provide further details in appendix A.First, the kinetic terms for the open string modes take the form S DBI ⊃ − τ D5 ( Lµ ) (cid:90) √− g Tr (cid:26) λ ( Lµ ) − F ab F ab + 12 ( G mn | x m ) D a X m D a X n (cid:27) , (2.16)where we recall that λ = 2 π(cid:96) s . The factors of ( Lµ ) arise from writing the backgroundmetric in terms of the barred metric. We have introduced the notation g := det( g ab ), with– 10 – ab := G ab ( x a , x m ) the induced background metric on the worldvolume. It takes the form g ab d x a d x b = µ ( r + z ) η µν d x µ d x ν + 1 µ ( r + z ) (cid:0) d r + r dΩ( θ, φ ) (cid:1) . (2.17)When z = 0 this is the metric on AdS × S with equal radii of µ − , while z (cid:54) = 0 gives adeformation of it. Worldvolume indices will always be raised with the inverse, g ab . We usethe notation | x m to indicate when other closed string fields are being evaluated at x m = x m .The coefficient of the F term determines the effective six-dimensional Yang–Mills cou-pling: g := 1 τ D5 λ ( Lµ ) = 4 π / √ g s N c N c µ − . (2.18)Note that the dimensionless coupling ( g ym µ ) is small in the regime N c (cid:29) g s N c (cid:29)
1. In orderto bring the scalar kinetic terms to standard form we define mass dimension-one scalar fieldsthrough Φ m := λ − ( Lµ ) X m = √ g s N c √ π µ X m , (2.19)so that ( A a , Φ m ) carry the same dimension.Once the closed string fields in the D-brane action are expressed in terms of the rescaledquantities, one finds that F ab is always accompanied by a factor of λ ( Lµ ) − , while [ X m , X n ]is always accompanied by the inverse factor. After changing variables to Φ m for the scalars,all four types of open string quantities appearing in D5-brane action carry the same prefactor: (cid:18) λ ( Lµ ) F ab , D a X m , ( Lµ ) λ [ X m , X n ] , µX m (cid:19) = √ π √ g s N c µ − ( F ab , D a Φ m , [Φ m , Φ n ] , µ Φ m ) , (2.20)and this provides a convenient organizing principle for the expansion. Of course it is ( A a , Φ m ) c ,defined by ( A a , Φ m ) = g ym ( A a , Φ m ) c , (2.21)that are the canonically normalized open string modes. The open string expansion variableson 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.Now, let C ∈ ( h MN , b MN , ϕ, c ( n ) ) denote a generic closed string fluctuation, let O ∈ ( F ab , D a Φ m , [Φ m , Φ n ] , µ Φ m ) denote any of the open string expansion variables, and set (cid:15) op := λ ( Lµ ) = √ π √ g s N c µ − . (2.22)Then the expansion of (2.8) can be written in the form S bosD5 = − (cid:15) g (cid:90) d x √− g ∞ (cid:88) n o ,n c =0 (cid:15) n o op κ n c V n o ,n c , (2.23)– 11 –here V n o ,n c is a sum of monomials of the form C n c STr ( O n o ), with rational coefficients. Thefirst few V n o ,n c ’s are V , = N f ,V , = N f (cid:26)
12 ( h aa + ϕ ) − (cid:15) abcdef c (6) abcdef (cid:27) ,V , = 0 ,V , = N f (cid:26)
18 ( h aa + ϕ ) + 14 ( b ab b ab − h ab h ab ) − (cid:15) abcdef c (4) abcd b ef (cid:27) ,V , = i Tr (cid:26) b ab F ab + h am D a Φ m + 12 Φ m (cid:16) ∂ m ( G ab h ab ) + ∂ m ϕ (cid:17)(cid:12)(cid:12)(cid:12) x m + − (cid:15) abcdef (cid:20) (cid:18)
16! Φ m ( ∂ m c (6) abcdef ) | x m + 15! ( D a Φ m ) c (6) mbcdef (cid:19) ++ 13!2 ( D a Φ m ) C (4) mbcd b ef + 14!2 c (4) abcd F ef (cid:21)(cid:27) ,V , = Tr (cid:26) F ab F ab + 12 G mn D a Φ m D a Φ n + 14 G mk G nl [Φ m , Φ n ][Φ k , Φ l ]+ − (cid:15) abcdef ( D a Φ m ) C (4) mbcd F ef (cid:27) , (2.24)where (cid:15) abcdef is the Levi–Civita tensor with respect to the background metric, (cid:15) =( − g ) − / , and we have used that STr reduces to the ordinary trace when there are no morethan two powers of the open string variables O . All closed string fields are to be understoodas being evaluated at x m = x m except for those in V , that involve taking a transversederivative before setting x m = x m .There is a great deal of physics in the V n o ,n c ’s: • V , corresponds to the energy density of the background D5-brane configuration. • V , 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 theD5-branes— i.e. we have not expanded around a solution to the equations of motion forthese closed string fields. The strength of these tadpoles is N f g − (cid:15) − κ ∝ N f √ g s N c µ ,which is large when g s N c (cid:29)
1. However this does not necessarily mean that the probeapproximation is bad! The effects of these tadpoles on open and closed string processeswill still be suppressed if the interaction vertices are sufficiently weak.Consider, for example, the leading correction to the open string propagators due to thesetadpoles. This corresponds to the diagram in Figure 3. The correction is proportionalto the product of the tadpole vertex with the cubic vertex for two open and one closedstring fluctuation. After canonically normalizing the open string modes via (2.21), thethree-point vertex goes as κ . Therefore the product is proportional to N f √ g s N c µ κ ∝ – 12 – N f √ g s N c /N c ) µ − ∝ N f g . Hence this process acts just like a standard one-loopcorrection to the Yang–Mills coupling that we would get from open string modes. Aslong as N f (cid:28) N c / √ g s N c , both the standard one-loop correction and this closed stringcorrection will be suppressed. Note this is a slightly stronger restriction than the usual N f (cid:28) N c limit when the ’t Hooft coupling g s N c is large, but nevertheless can becomfortably satisfied for a range of N f in the regime N c (cid:29) g s N c (cid:29) • The vanishing of V , indicates that open string tadpoles are absent. This simply vali-dates the fact (already implicitly assumed in the above discussion) that the D5-braneembedding, described by x m = x m , extremizes the equations of motion for the openstring modes in the fixed closed string background. • Only the center-of-mass degrees of freedom corresponding to the central u (1) ⊂ u ( N f )participate in V , due to the trace. The strength of these interactions is g − (cid:15) − κ ∝ g ym , where we have made use of the convenient relation (cid:15) op g = 2 √ πκ . (2.25)Hence they can be treated perturbatively. Furthermore the u (1) and su ( N f ) degrees offreedom decouple in V , , so the couplings in V , can only transmit the effects of theclosed string tadpoles to the su ( N f ) fields through higher order open string interactions. • The first three terms of V , come from the DBI action, and comprise the usual Yang–Mills action on a curved background. The final term in V , , meanwhile, comes fromthe CS action and is non-vanishing because there is a the nontrivial RR flux in thesupergravity background.It is also interesting to consider the form of terms in V , , or higher order open stringinteractions. V , is nontrivial when z (cid:54) = 0; V , is always nontrivial. For example, there isan STr ( (cid:126) Φ z F ) coupling of the form V , ⊃ STr (cid:40) (cid:126)z · (cid:126) Φ z r + z ) (cid:0) F r i r j F r i r j − F µν F µν (cid:1)(cid:41) . (2.26)Three- and four-point couplings in V , and V , come with extra factors of (cid:15) op relative to thethree- and four-point couplings in the Yang–Mills terms, V , . Hence they will be suppressedrelative to the Yang–Mills terms for field variations at or below the scale µ . More precisely,if the fields vary on a scale µ (cid:48) we merely require ( µ (cid:48) /µ ) (cid:28) √ g s N c , in order that these termsbe suppressed relative to their counterparts in V , .In summary, there is a regime of parameters—namely N c (cid:29) g s N c (cid:29) N f (cid:28) N c / √ g s N c —where the leading interactions of the (bosonic) su ( N f ) open string modes aregoverned by V , . This forms the bosonic part of a six-dimensional super-Yang–Mills theoryon the curved background (2.17). – 13 – V ℭ(cid:2)
Figure 3 . A virtual closed string correction to an open string propagator. The closed string iscreated from the vacuum by a vertex in V , . It propagates to a three-point vertex in V , . This givesa correction to the open string propagator that is of the same order as a standard one-loop correctionfrom virtual open string modes. 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 V , , in which the scalars carry curved spaceindices. In order to be more explicit with regards to the C (4) term, we have from (2.3) thatthe relevant components are C (4)012 y ( x a , x m ) = µ ( r + z ) , (2.27)and so the last term of V , contributes as follows: (cid:90) d x √− g Tr (cid:26) (cid:15) abcdef ( D a Φ m ) C (4) mbcd F ef (cid:27) == 12 (cid:90) d x √− g ˜ (cid:15) r i r j r k µ ( r + z ) Tr (cid:8) ( D r i Φ y ) F r j r k (cid:9) . (2.28)Here we have introduced ˜ (cid:15) , which should be thought of as the Levi–Civita tensor on the Eu-clidean R spanned by (cid:126)r : ˜ (cid:15) r r r = 1, or if we work in spherical coordinates ˜ (cid:15) rθφ = ( r sin θ ) − .Then the bosonic part of the Yang–Mills action is S ym ,b := − g (cid:90) d x √− g V , = − g (cid:90) d x √− g Tr (cid:26) F ab F ab + 12 G mn D a Φ m D a Φ m ++ 14 G mk G nl [Φ m , Φ n ][Φ k , Φ l ] −
12 ˜ (cid:15) r i r j r k µ ( r + z ) ( D r i Φ y ) F r j r k (cid:27) . (2.29)We can also derive a more standard field theoretic form for the action by rescaling thescalar fields in such a way that their kinetic terms are canonically normalized. To do this, we– 14 –ake use of a vielbein associated with the background metric G mn :Φ y := µ ( r + z ) / Φ y , Φ z i := 1 µ ( r + z ) / Φ z i . (2.30)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,˜ (cid:15) r i r j r k D r i F r j r k = 0. We also switch to spherical coordinates, as the only surviving bulk termcomes from the derivative of the ( r + z ) prefactor. This integration by parts also generatesa boundary term. After carrying out these manipulations, the bosonic action becomes S ym ,b := − g (cid:90) d x √− g Tr (cid:26) F ab F ab + 12 ( D a Φ m )( D a Φ m ) + 14 [Φ m , Φ n ][Φ m , Φ n ]++ 12 M z δ ij Φ z i Φ z j + 12 M y (Φ y ) + 2 M Ψ (cid:15) αβ F αβ Φ y (cid:27) + S bndry b . (2.31)In the last term the indices α, β correspond to coordinates θ, φ along the two-sphere and (cid:15) θφ = ( g S ) − / = µ ( r + z ) / ( r sin θ ). The mass parameters are defined as follows: M z := µ (cid:18) r r + z − (cid:19) , M y := µ (cid:18) r r + z + 3 (cid:19) , M Ψ := µr (cid:112) r + z . (2.32)As r → ∞ they approach the values − , , z = 0they take these values everywhere. Although the squared mass of the Z scalars is negative, itsatisfies the Breitenlohner–Freedman bound [29] for AdS . The reason for the notation M Ψ 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 ∂M ∼ = { r b } × R , × S at r = r b → ∞ . They are given by S bndry b = 1 g (cid:90) ∂M d x (cid:112) − g ( ∂ ) Tr (cid:26) M Ψ (cid:0) (Φ y ) − δ ij Φ z i Φ z j (cid:1) + 12 Φ y (cid:15) αβ F αβ (cid:27) , (2.33)where d x (cid:112) − g ( ∂ ) is the induced volume form on the boundary, (cid:112) − g ( ∂ ) d x = µr b ( r b + z ) / d x dΩ , (2.34)with d x := d x d x d x and dΩ := sin θ d θ d φ . 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 establishingthe consistency of the variational principle and in the supersymmetry invariance of the Yang–Mills action. The limit r b → ∞ of quantities computed using (2.33) is understood to be takenat the end of any calculation (when it exists). We assume the fields are sufficiently regular such that there is no boundary contribution from r = 0. Thisis discussed in some further detail for static configurations later. See section 6.2. – 15 – .5 Fermionic D-brane action Ideally, one would like to obtain non-abelian super-Yang–Mills theory on the D5-branes viathe limiting behavior of a κ -symmetric non-abelian super D-brane action for general closedstring backgrounds. While important progress toward constructing such actions has beenmade ( see e.g. [30–33] and references therein), the subject has not matured sufficiently to beof practical use for our purposes.Instead, we will fall back on abelian fermionic D-brane actions that have been discussedextensively, starting with the initial work of [7–10], and continuing with [11–13]. Here wefollow the conventions of [12, 13]. This will provide the fermionic couplings that are quadraticorder in open string fluctuations—kinetic and mass-like terms. With these and the full set ofbosonic couplings in hand, we will be able to deduce the remaining Yukawa-type couplingsand 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, Ψ. The couplings of Ψ to the IIB closed string supergravity fields are describedmost conveniently by introducing a doublet of ten-dimensional Majorana–Weyl spinors ˆΨ =(Ψ , Ψ ) T of the same 10D chirality. One linear combination will be projected out by the κ -symmetry projector while the other will be the physical Ψ. The ten-dimensional gammamatrices, satisfying { Γ M , Γ N } = 2 G MN , are likewise extended by the doublet structure. Oneintroduces ˆΓ M := Γ M ⊗ , ˆΓ := Γ ⊗ σ , (2.35)where Γ = Γ is the ten-dimensional chirality operator and σ , , are the Pauli ma-trices.The abelian fermionic D5-brane action, to quadratic order in ˆΨ, takes the form S f = τ D5 (cid:90) d xe − ∆ φ (cid:112) − det ( P [ E ] − iλF ) ˆΨ( − Γ D5 ) (cid:104) ( M − ) ab ˆΓ ( P ) b ˆ D a − ∆ (cid:105) ˆΨ , (2.36)where E MN = e ∆ φ / ( G MN + B MN ) as before and the matrix M is M ab = e ∆ φ / P [ G ab ] + F ab ˆΓ . (2.37)Here we have also introduced the shorthand F ab := e ∆ φ / P [ B ab ] − iλF ab . The idempotentmatrix Γ D5 appearing in the κ -symmetry projector, ( − Γ D5 ), has a somewhat nontrivialexpression in terms of F :Γ D5 := 1 (cid:112) − det( P [ E ] − iλF ) ×× (cid:88) q + r =3 ε a ··· a q b ··· b r q !2 q (2 r )! ( − i ) q F a a · · · F a q − a q (Γ ( P ) b ··· b r ⊗ ( − iσ ))(ˆΓ) r , (2.38) Our M ab , Γ D5 are denoted (cid:102) M ab , (cid:101) Γ D5 in [12, 13]. – 16 –here ε = 1, and the Γ ( P ) a are the pullbacks of Γ M to the worldvolume. In static gauge,Γ ( P ) a = Γ a − i ( ∂ a X m )Γ m .The remaining couplings to closed string fields are encoded in the generalized derivativeˆ D and the mass-like operator, ∆. We write only the terms that contribute when evaluated onthe near-horizon background geometry (2.3); the full set of couplings can be found in [12, 13].In this case ˆ D a = P [ D a ] ⊗ + 116 · e ∆ φ F (5) M ··· M (cid:16) Γ M ··· M Γ ( P ) a ⊗ ( iσ ) (cid:17) + · · · , (2.39)where the terms represented by · · · vanish when closed string fluctuations are switched off,while ∆ → P [ D a ] is meantto indicate that one takes the pullback of D M Ψ , to the brane worldvolume, and D M is thestandard 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) hasan expansion in closed and open string fluctuations controlled by the same parameters, (cid:15) op , κ ,that appeared in the expansion of the bosonic action (2.23). Considering first the rescalingof the closed string fields, (2.4), there are a few key points: • After applying this rescaling under the determinant of (2.36) we can pull out a factorof ( Lµ ) , and we will have the usual factor of ( Lµ ) − λ = (cid:15) op accompanying F ab . • The Γ ( P ) b ··· b r factor in Γ D5 rescales according to Γ ( P ) b ··· b r = ( Lµ ) r ˜Γ ( P ) b ··· b r , due to theimplicit vielbein factors present in it. Taking into account the ( Lµ ) − from the deter-minant factor out front, Γ D5 retains its form under the rescaling except that each factorof F ab picks up a corresponding ( Lµ ) − prefactor. This combines with the λ ’s alreadypresent so that all F ab in Γ D5 are accompanied by (cid:15) op . • One can check that ( M − ) ab ˆΓ ( P ) b ˆ D a − ∆ gets a net factor of ( Lµ ) − when expressed interms of the rescaled closed string fields, while F ab in M ab acquires an (cid:15) op prefactor. • Finally, each appearance of ∂ a X m from pullbacks and X m from expanding the closedstring fields around x m is accompanied by a factor of (cid:15) op when we express X m in termsof Φ m via (2.19).Together, these observations show that all open string interaction vertices between Ψ andpowers of F ab , ∂ a X m , and X m are controlled by the expected power of (cid:15) op . The overallprefactor of the leading Ψ term is τ D5 ( Lµ ) = ( (cid:15) op g ym ) − ( Lµ ) − . We can make a rescaling of ˆΨ analogous to (2.19) such that the coefficient of this leading order term is simply − i/g .We will assume this has been done and continue using the same notation for the fermion. The ˆΨ in (2.36) must have units of (length) / . It would be natural to include a factor of λ / out in frontof (2.36) so that they are dimensionless. Then the rescaling would be ˜ˆΨ = (cid:15) / ˆΨ. – 17 –ence we write S f = − i g (cid:90) d x √− g ˆΨ( − Γ (0)D5 )Γ a ˆ D (0) a ˆΨ × (1 + O ( (cid:15) op , κ )) , (2.40)where Γ (0)D5 := Γ rθφ Γ ⊗ σ , (2.41)and ˆ D (0) a := (cid:18) ∂ a + 14 ω MN,a Γ MN (cid:19) ⊗ + 116 · F (5) M ··· M Γ M ··· M Γ a ⊗ ( iσ ) . (2.42)For Γ (0)D5 we took the q = 0, r = 3 term in (2.38) and used that ( − g ) − / ε b ··· b Γ ( P ) b ··· b =6! Γ rθφ to leading order in open and closed string fluctuations. In (2.42), ω MN,P are thecomponents of the spin connection with respect to the background metric G MN , evaluatedat x m = x m , and all gamma matrices with covariant indices are defined using the vielbeineof the background metric.Let us evaluate (2.42) in more detail. It follows from the background (2.3) that116 · F (5) M ··· M Γ M ··· M = − µ (cid:40) r (cid:112) r + z Γ ry + z ,i (cid:112) r + z Γ z i y (cid:41) ( − Γ) , (2.43)where we recall that ( r, θ, φ ) are spherical coordinates for the directions spanned by (cid:126)r . Butthe second term drops out of (2.40) becauseΓ a Γ ry Γ a = 2Γ ry , Γ a Γ z i y Γ a = 0 . (2.44)Regarding the ten-dimensional spin connection, there are nonzero components of the type ω bm,a when z (cid:54) = 0. (See appendix B for details.) However, the contribution of these compo-nents to Γ a ω MN,a Γ MN cancels out. HenceΓ a ˆ D (0) a = Γ a (cid:18) ∂ a + 14 ω bc,a Γ bc (cid:19) ⊗ − µr (cid:112) r + z Γ ry ( + Γ) ⊗ ( iσ ) . (2.45)The projector in the last term of (2.45) will either give the identity or zero when acting onΨ , , depending on the 10D chirality of the latter. The two possibilities distinguish betweena D5-brane and an anti-D5, and only one choice will lead to a supersymmetric worldvolumetheory on the brane. We will see that the supersymmetric theory corresponds toΓΨ , = Ψ , . (2.46)Thus the coupling to the background F (5) provides a necessary mass-like term for the fermion.It is now straightforward to diagonalize the operator ( − Γ (0)D5 )Γ a ˆ D (0) a with respect tothe auxiliary doublet structure. Introducing the unitary transformation U := 1 √ (cid:16) − Γ rθφ Γ ⊗ ( iσ ) (cid:17) , (2.47)– 18 –ne finds U (cid:20)
12 ( − Γ (0)D5 )Γ a ˆ D (0) a (cid:21) U † = (cid:40) Γ a D a + µr (cid:112) r + z Γ θφy ( + Γ) (cid:41) ⊗
12 ( + σ ) , (2.48)where D a := ∂ a + ω bc,a Γ bc . Thus, setting (Ψ , Ψ (cid:48) ) T := U ˆΨ, one sees that Ψ (cid:48) is projected outwhile Ψ encodes the physical degrees of freedom. Using (2.46), and recalling the definition of M Ψ in (2.32), the final result for (2.40) takes the form S f = − i g (cid:90) d x √− g Ψ (cid:110) Γ a D a + M Ψ Γ θφy (cid:111) Ψ × (1 + O ( (cid:15) op , κ )) . (2.49)Note that for a ten-dimensional Majorana–Weyl spinor, the bilinear ΨΓ M ··· M p Ψ vanishesunless p = 3 (or 7), so the gamma matrix structure of the mass term is as it had to be. Ψcontains 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 withthe ‘10D’ form.Finally, we will infer from (2.49) and the bosonic Yang–Mills terms (2.31), the non-abelian analogs of the leading terms in (2.49) that complete (2.31) into a supersymmetricinvariant. Clearly the covariant derivative D a should be generalized to a gauge covariantderivative, D a := ∂ a + ω bc,a Γ bc + [ A a , · ]. We will, for convenience, continue to use thesame notation for this covariant derivative as we did above. A natural ansatz that will yieldthe Yukawa couplings is simply to extend this to a ten-dimensional covariant derivative:Γ a D a Ψ + Γ m [Φ m , Ψ]. Our ultimate justification for this ansatz (detailed below) will be thatsupersymmetry requires it.Hence we take the fermionic terms of the Yang–Mills action to be S ym ,f := − i g (cid:90) d x √− g Tr (cid:26) Ψ (cid:16) Γ a D a + M Ψ Γ θφy (cid:17) Ψ + i ΨΓ m [Φ m , Ψ] (cid:27) + S bndry f , (2.50)where Ψ is now valued in the adjoint representation of su ( N f ).We’ve included a boundary action for the fermion , S bndry f := − i g (cid:90) ∂M d x (cid:112) − g ( ∂ ) Tr (cid:110) ΨΓ θφy Ψ (cid:111) . (2.51)The analysis of [34] for fermions on anti-de Sitter space demonstrates that such boundaryterms are necessary in order to have a well-defined variational principle. We will see that theboundary action (2.51) is also required for supersymmetry. Without it, the supersymmetryvariation of the action would produce boundary terms that do not vanish on their own. Thesepoints are analyzed in sections 4.3 and 4.4 below. In principle such boundary terms shouldhave already been present in (2.36), but we are not aware of any previous work on this issue.– 19 – Supersymmetry
As noted previously, the intersecting D-brane system of Figure 1 preserves eight supersymme-tries. In the near-horizon limit of the D3-brane geometry, the symmetry algebra is enhancedto osp (4 |
4) with sixteen odd generators, provided the D3 and D5-branes have zero transverseseparation. The leading low-energy effective description in the regime N c (cid:29) g s N c (cid:29) N f (cid:28) N c / √ g s N c 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 decoupledsupergravity and u (1) sectors). Thus one expects the Yang–Mills theory to possess eightsupersymmetries when z (cid:54) = 0 and sixteen when z = 0.In this section we first 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 asgenerators, we exhibit the full set of supersymmetry transformations on the Yang–Mills fieldsand establish the invariance of the action, (2.31) plus (2.50), modulo boundary terms. AdS × S is a maximally supersymmetric background admitting thirty-two linearly indepen-dent Killing spinors—that is, solutions (cid:15) to the vanishing of the gravitino variation, (cid:20) D (0) M − i ·
5! (Γ M ··· M F (5) M ··· M )Γ M (cid:21) (cid:15) = 0 . (3.1)Here (cid:15) is complex Weyl and our conventions are that it has positive chirality, Γ (cid:15) = (cid:15) . Explicitsolutions can be found in various references, going back to [35, 36]. The form of thesesolutions depends of course on the choice of coordinate system and frame. Most referencesemploy a frame adapted to some type of spherical coordinate parameterization of the S .This is inconvenient for the applications we have in mind here. We provide two alternativedescriptions that are better-suited to the analysis in subsequent sections; both of them willbe useful below.The first is based on coordinates ( x µ , (cid:126)r, (cid:126)z, y ) in which the metric takes the form G MN d x M d x N = ( µv ) ( η µν d x µ d x ν + d y ) + d (cid:126)r + d (cid:126)z ( µv ) , (3.2)and a maximally Cartesian-like choice of orthonormal frame: e µ = ( µv ) d x µ , e y = ( µv ) d y , e r i = 1 µv d r i , e z j = 1 µv d z j , (3.3) For example, a naive application of these formulae will lead to expressions for worldvolume Killing spinorson the D5-brane that appear to depend on angular coordinates parameterizing the transverse space that arenot well-defined on the brane locus. – 20 –ere v should be understood as shorthand for v := √ r + z and we recall that µ is theinverse AdS radius. The equations (3.1) are straightforwardly integrated to yield the solutions (cid:15) → ( (cid:15) ) cart , with( (cid:15) ) cart = 1 √ µv (cid:18) r i Γ r i + z i Γ z i v (cid:19) (cid:15) − + √ µv (cid:104) (cid:15) − µ ( x µ Γ µ + y Γ y ) (cid:15) − (cid:105) , (3.4)where (cid:15) ± are constant complex Weyl spinors satisfying an additional projection condition:Γ (cid:15) ± = ± (cid:15) ± & i Γ y (cid:15) ± = ± (cid:15) ± . (3.5)We provide some details of the analysis in appendix B. Each of the (cid:15) ± contain eight complex(sixteen real) free parameters, for a total of thirty-two.The notation ( (cid:15) ) cart is meant to emphasize that these are the components of the Killingspinor with respect to a specific basis of sections (or class of bases) on the Dirac spinor bundle.The basis is such that the gamma matrices { Γ µ , Γ y , Γ r i , Γ z j } associated with the frame (3.3)have constant matrix elements.The second presentation makes use of spherical coordinates ( r, θ, φ ) and the frame e r = 1 µv d r , e θ = 1 µv r d θ , e φ = 1 µv r sin θ d φ , (3.6)in place of the e r i . The two frames are related by a local rotation, and the components ofthe Killing spinor with respect to the new frame, which we denote by ( (cid:15) ) S , are related tothe components of the Killing spinor with respect to the old frame, (3.4), by the lift of thisrotation to the Dirac spinor bundle. The result is( (cid:15) ) S = 1 √ µv (cid:18) r Γ r + z i Γ z i v (cid:19) h S ( θ, φ ) (cid:15) − + √ µv h S ( θ, φ ) (cid:104) (cid:15) − µ ( x µ Γ µ + y Γ y ) (cid:15) − (cid:105) , (3.7)where h S ( θ, φ ) := exp (cid:18) θ rθ (cid:19) exp (cid:18) φ θφ (cid:19) . (3.8)In this presentation we are implicitly working with respect to a basis where the matrix ele-ments of Γ r , Γ θ , Γ φ are constant. (cid:15) ± denote the same column vectors of constant entries inboth expressions. Additional details on the relationship between ( (cid:15) ) cart , ( (cid:15) ) S and between thegamma matrices { Γ r i } and { Γ r , Γ θ , Γ φ } can be found in appendix B, where we also describesome further transformations that bring the Killing spinors to the form typically found in theliterature. The subset of supersymmetries preserved by the D5-brane embedding is generated by thoseKilling spinors (cid:15) that additionally satisfy a κ -symmetry projection condition [7–10]. Let (cid:15) = ε + iε (cid:48) , (3.9)– 21 –here ε, ε (cid:48) are Majorana–Weyl and introduce the doublet ˆ ε = ( ε, ε (cid:48) ) T . Then the conditioncan be expressed as [12, 13] Γ D5 ˆ ε = ˆ ε , (3.10)where Γ D5 is given by (2.38). If we restrict to the leading-order effective description of theD5-brane, i.e. the Yang–Mills theory, then it will be sufficient to work with the leading orderexpression Γ (0)D5 , (2.41). Since Γ ε ( (cid:48) ) = ε ( (cid:48) ) , this condition is equivalent to ε (cid:48) ( x a ; x m ) = Γ rθφ ε ( x a ; x m ) . (3.11)Here we also emphasize that the Killing spinors are to be evaluated on the backgroundembedding defined by x m = x m .Let us analyze this condition on the explicit solutions (3.7). First we extract the Majorana–Weyl components of (3.7). Let ε , ε (cid:48) be constant Majorana–Weyl spinors of positive chiralityand η , η (cid:48) be constant Majorana–Weyl spinors of negative chirality such that (cid:15) − = η + iη (cid:48) , (cid:15) = ε + iε (cid:48) + ( µy )Γ y ( η + iη (cid:48) ) . (3.12)Here y is the asymptotic y -value of the D5-brane stack; this shift has been included forconvenience below. With these definitions, the second of the conditions (3.5) is equivalent to ε (cid:48) = Γ y ε , η (cid:48) = − Γ y η , (3.13)and the Majorana–Weyl components of (3.7) are( ε ) S = 1 √ µv (cid:18) r Γ r + z i Γ z i v (cid:19) h S ( θ, φ ) η ++ √ µv h S ( θ, φ ) (cid:104) ε − µ ( x µ Γ µ + ( y − y )Γ y ) η (cid:105) , (3.14)and ( ε (cid:48) ) S of the same form with ( ε , η ) S → ( ε (cid:48) , η (cid:48) ) S .Now we impose (3.11). This must hold for all values of x a ; in particular, the terms thatgo as v − / and the ones that go as v / must match independently. Matching the v − / terms leads to the condition( r Γ r + z ,i Γ z i ) h S ( θ, φ ) η (cid:48) = Γ rθφ ( r Γ r + z ,i Γ z i ) h S ( θ, φ ) η . (3.15)Since Γ rθφ commutes with Γ z i but anticommutes with Γ r , this leads to two (additional)conditions on η , η (cid:48) that are incompatible with each other when (cid:126)z (cid:54) = 0, unless we set η =0 = η (cid:48) . However if (cid:126)z = 0 then we only obtain one additional relation between η , η (cid:48) , (sinceΓ rθφ commutes with h S ): η , η (cid:48) = 0 if (cid:126)z (cid:54) = 0 , η (cid:48) = − Γ rθφ η if (cid:126)z = 0 . (3.16)Matching the v / terms, meanwhile, leads to the requirement ε (cid:48) − ( µx µ )Γ µ η (cid:48) = Γ rθφ (cid:16) ε − ( µx µ )Γ µ η (cid:17) . (3.17)– 22 –he x µ Γ µ terms cancel because of (3.16). Thus equality can be achieved by taking ε (cid:48) = Γ rθφ ε . (3.18)The conditions (3.16) and (3.18) are compatible with (3.13). Together they impose thefollowing projection conditions on the Majorana–Weyl spinors η , ε : ε = Γ rθφy ε , η = Γ rθφy η . (3.19)In summary, when (cid:126)z = 0 the D5-brane embedding preserves sixteen supersymmetries,parameterized by the Majorana–Weyl spinors ε , η of positive and negative chirality, respec-tively, and additionally satisfying the projections (3.19). When (cid:126)z (cid:54) = 0, the η must be setto zero and the embedding preserves eight supersymmetries. In the following we will use theMajorana–Weyl spinor ε ( x a ; x m ) as our generator of supersymmetry transformations in theYang–Mills theory. ( ε (cid:48) is determined in terms of it.) We will simply write ε henceforth, withthe understanding that we are alway evaluating at x m = x m . With respect to a frame inwhich { Γ µ , Γ r , Γ θ , Γ φ } are constant, it has components( ε ) S = ( µr ) − / Γ r h S ( θ, φ ) η + ( µr ) / h S ( θ, φ ) (cid:104) ε − ( µx µ )Γ µ η (cid:105) , (cid:126)z = 0 , √ µ ( r + z ) / h S ( θ, φ ) ε , (cid:126)z (cid:54) = 0 . (3.20)(Note that v → r when (cid:126)z → { Γ µ , Γ r i } are constant and the bulk Killing spinor has the form (3.4). Theanalogous result is( ε ) cart = ( µr ) − / ˆ r i Γ r i η + ( µr ) / (cid:104) ε − ( µx µ )Γ µ η (cid:105) , (cid:126)z = 0 , √ µ ( r + z ) / ε , (cid:126)z (cid:54) = 0 , (3.21)where we used the shorthand ˆ r i = r i /r . The conditions on ε , η are the same; in particularthe projection conditions (3.19) can equivalently be written as ε = Γ r r r y ε , η = Γ r r r y η . (3.22)Finally, we would like to derive a ‘Killing spinor equation’ for ε alone, the solutions ofwhich can be equivalently represented by (3.20) or (3.21). This can be done by looking at thereal and imaginary parts of (3.1), restricting to x m = x m , and imposing (3.11). We relegatethe details to appendix B.2 and state the final result here: (cid:20) D a + M Ψ θφy Γ a (cid:21) ε = 0 , (3.23)where D a = ∂ a + ω bc,a Γ bc is the spinor covariant derivative with respect to the 6D metric(but still utilizing the 10D gamma matrices).– 23 – .3 Supersymmetry of the worldvolume theory We now turn the supersymmetry of the worldvolume theory. Recall that the six-dimensionalYang–Mills action is the sum of (2.31) and (2.50) and can be written S ym := 1 g (cid:90) d x √− g L + S bndry , (3.24)where the ‘bulk’ Lagrangian density is L := − Tr (cid:26) F ab F ab + 12 D a Φ m D a Φ m + i (cid:16) Γ a D a + M Ψ Γ θφy (cid:17) Ψ + 2 M Ψ (cid:15) αβ F αβ Φ y ++ 12 M y (Φ y ) + 12 M z ( (cid:126) Φ z ) + i m [Φ m , Ψ] + 14 [Φ m , Φ n ][Φ m , Φ n ] (cid:27) , (3.25)where the background metric is given by (2.17), and the ( r -dependent) masses by (2.32). Theboundary action, which we discuss in the next subsection, is the sum of (2.33) and (2.51): S bndry := 1 g (cid:90) ∂M d x (cid:112) − g ( ∂ ) Tr (cid:26) M Ψ (cid:16) (Φ y ) − ( (cid:126) Φ z ) (cid:17) + 12 Φ y (cid:15) αβ F αβ − i θφy Ψ (cid:27) . (3.26)Here we focus on supersymmetry invariance of the bulk action modulo boundary terms. Noteagain that while we derived this action from a D-brane system where the relevant Lie algebrais su ( N f ), it describes super Yang-Mills theory on AdS × S for any simple Lie group. Motivated by the form of the fermion mass term, the Killing spinor equation (3.23),and the philosophy espoused in [38], we make the following ansatz for the supersymmetryvariations of the fields: δ ε A a = − i ε Γ a Ψ , δ ε Φ m = − i ε Γ m Ψ ,δ ε Ψ = (cid:20) F ab Γ ab + D a Φ m Γ am + 12 [Φ m , Φ n ]Γ mn + α Γ θφy Γ m Φ m (cid:21) ε , (3.27)where α is a parameter—possibly a function of r —to be determined. with the appropriately defined Tr . See the comments in the first paragraph of section 2.3. – 24 –tandard manipulations, without making use of the Killing spinor equation, lead to iδ ε L = ∇ a B a ++ D a ε Tr (cid:26) (cid:18) F bc Γ bc + ( D b Φ m )Γ bm + 12 [Φ m , Φ n ]Γ mn + α Φ m Γ m Γ θφy (cid:19) Γ a Ψ (cid:27) ++ ε Tr (cid:26) − M Ψ F ab Γ ab Γ θφy Ψ − M Ψ (cid:15) αβ F αβ Γ y Ψ++ ( D a Φ m ) (cid:16) α Γ m Γ θφy Γ a − M Ψ Γ am Γ θφy (cid:17) Ψ + 4 M Ψ ( D α Φ y ) (cid:15) αβ Γ β Ψ+ − [Φ m , Φ n ] (cid:18) M Ψ Γ mn Γ θφy + α Γ m Γ θφy Γ n (cid:19) Ψ++ Φ m Γ m (cid:20) − (cid:18) ∂ r α + 2 z r ( r + z ) α (cid:19) Γ r Γ θφy + αM Ψ − M m (cid:21) Ψ (cid:27) . (3.28)The terms appearing here are naturally divided into three sets. First there are the totalderivative terms of the first line; the boundary current B a is B a := ε Tr (cid:26) − F bc Γ bca Ψ + ( D b Φ m )Γ m Γ ba Ψ+ −
12 [Φ m , Φ n ]Γ mn Γ a Ψ − α Φ m Γ m Γ θφy Γ a Ψ (cid:27) + 12 Tr (cid:8) ΨΓ a δ ε Ψ (cid:9) , (3.29)The contribution of these terms to δ ε S ym will have to be canceled by the variation of theboundary action; we analyze this in subsection 4.4 below. Then there are the terms propor-tional to D a ε in the second line, and those proportional to ε in the remaining lines. Some cancellations have already occurred to arrive a (3.28)—namely those that wouldhave occurred in the flat space limit, µr →
0. The remaining terms are present preciselybecause we are working on a nontrivial background. They involve either the derivative ofthe supersymmetry parameter, or the mass-type couplings. From the D-brane point of viewthe latter are induced from the non-flat normal bundle to the brane worldvolume and thebackground RR five-form flux.The next step is to make use of the Killing spinor equation, or rather its conjugate: D a ε = M Ψ ε Γ a Γ θφy . (3.30)Supersymmetry invariance then requires that all of the resulting terms from the second lineof (3.28) cancel with the terms in the remaining lines. This must be checked by explicitlyworking out the coefficients for each of the possible index structures of the worldvolume fields, The appearance of ∂ r α term and especially the term that accompanies it inside the round brackets in thelast line of (3.28) is somewhat subtle. It arises in the process of moving the D a in − αε Tr (cid:0) Φ m Γ m Γ θφy Γ a D a Ψ (cid:1) off of Ψ. When z (cid:54) = 0, the covariant derivative does not commute with Γ θφy due to nonzero mixed componentsof the spin connection of the type ω αr,β . The commutator of D a with Γ θφy is what gives rise to the term withthe 2 z / ( r ( r + z )) factor. – 25 – .g. F µν F µr , F µα , F αβ , D µ Φ y , etc . One indeed finds that supersymmetry is preserved, moduloboundary terms, provided the following three conditions on α , the masses, and the Killingspinor hold: iδ ε L = ∇ a B a ⇐⇒ M Ψ − α , ε (cid:104)(cid:16) ∂ r α + z r ( r + z ) α (cid:17) Γ r Γ θφy + 4 αM Ψ − M y (cid:105) , ε (cid:104) − (cid:16) ∂ r α + z r ( r + z ) α (cid:17) Γ r Γ θφy − αM Ψ − M z (cid:105) . (3.31)All three conditions are met by taking α = M Ψ = µr (cid:112) r + z . (3.32)When z = 0 this follows directly from the expressions for the masses (2.32). When z (cid:54) = 0one can show that with α = M Ψ the latter two equations are proportional to the projector ( − Γ rθφy ). But when z (cid:54) = 0 we must set the η to zero in ε , and then ε is indeed annihilatedby this projector acting to the left. (See (B.33).) In this section we describe the (classical) vacuum structure of the Yang–Mills theory andformulate appropriate boundary conditions on the fields. The latter is based on consistencyof the variational principle and a large- r asymptotic analysis of field modes with an eyetowards holographic applications.It will be convenient to start with the action (2.29) given in terms of the Φ m (rather thanthe Φ m we have been using so far). We have S ym = 1 g (cid:90) d x √− g L (cid:48) + S bndry f , (4.1)with L (cid:48) = − Tr (cid:26) F ab F ab + 12 G mn D a Φ m D a Φ m + 14 G mk G nl [Φ m , Φ n ][Φ k , Φ l ]+ −
12 ˜ (cid:15) r i r j r k µ ( r + z ) ( D r i Φ y ) F r j r k ++ i (cid:16) Γ a D a + M Ψ Γ θφy (cid:17) Ψ + i m [Φ m , Ψ] (cid:27) . (4.2)Only the fermion has an explicit boundary action, (2.51).– 26 – first variation of (4.1) leads to the equations of motion,0 = D a F ab − G mn [Φ m , D b Φ n ] + i , Γ b Ψ] + ˜ (cid:15) rαβ µ (cid:2) ∂ r ( r + z ) (cid:3) ( D α Φ y ) δ bβ , D a ( G yy D a Φ y ) + G mn G yy [Φ m , [Φ n , Φ y ]] + i , Γ y Ψ] −
12 ˜ (cid:15) rαβ µ (cid:2) ∂ r ( r + z ) (cid:3) F αβ , D a ( G z i z j D a Φ z j ) + G mn G z i z j [Φ m , [Φ n , Φ z j ]] + i , Γ z i Ψ] , (cid:16) /D + M Ψ Γ θφy (cid:17) Ψ + Γ m [Φ m , Ψ] . (4.3)On a solution to these equations the variation reduces to a set of boundary terms:( δS ym ) o-s = − lim r →∞ µ g (cid:90) d x dΩ Tr (cid:26) P µ δA µ + P β δA β ++ δ ij P z i δ Φ z i + P y δ Φ y + P Ψ δ Ψ (cid:27) , (4.4)where P ν := µ r F rµ η µν , P β := µ ( r + z ) (cid:104) F rα ˜ g αβ + ( D α Φ y )˜ (cid:15) αβ (cid:105) , P z i := µ r ( D r Φ z i ) , P y := µ ( r + z ) (cid:20) r D r Φ y −
12 ˜ (cid:15) αβ F αβ (cid:21) , P Ψ := − iµ r ( r + z ) / Ψ (cid:16) Γ r + Γ θφy (cid:17) . (4.5)Here we have made all powers of r explicit. In particular, ˜ g αβ and ˜ (cid:15) αβ are the inverse metricand Levi–Civita tensor on the round S of unit radius. We also note that there is no δA r term and that the Ψ-term receives contributions from the variation of both bulk and boundaryactions.The on-shell variation, (4.4), must vanish in order to ensure a consistent variationalprinciple. This, in turn, restricts the asymptotic behavior of bulk field configurations. Intaking this approach we are excluding the additional boundary action that arises in thecontext of holographic renormalization. Holographic renormalization is a procedure thatintroduces a cutoff surface at large r and determines a set of boundary counterterms that areto be added to the action [39–43]. Originally these terms were determined from the conditionof having a finite on-shell action, but later it was understood that the same terms are requiredto render the variational principle well-defined when one allows for field configurations withdivergent behavior, (i.e. non-normalizable modes), as r → ∞ [44, 45]. This procedurecorresponds to the standard renormalization of UV divergences in the holographic dual, and itis appropriate for constructing the generator of correlation functions. In this paper, however,we are interested in classical finite-energy BPS field configurations of the 6D Yang–Millstheory which, roughly speaking, should be the appropriate leading-order description of BPSstates in the holographic dual in the limit N c (cid:29) g s N c (cid:29)
1. In this case, notions of finite– 27 –nergy, consistency of the variational principle, and the like should descend from the classicalYang–Mills action, without the cut-off boundary and associated boundary terms.Before examining this in detail it is useful to first consider the vacuum structure andperturbative spectrum of the theory.
The classical vacua are the absolute minima of the energy functional. We construct theYang–Mills energy functional, or Hamiltonian, by performing a Legendre transform of theLagrangian in (4.1) with respect to the natural time coordinate t = x of the Minkowskifoliation of AdS. Let x p , p = 1 ,
2, denote the spatial coordinates in x µ so that x a = ( t, x p , r i ),and let ( E p , E r i ) = ( F p , F r i ) be the components of the non-abelian electric field. Then thebosonic part of the Yang–Mills Hamiltonian takes the form H bosym = 1 g (cid:90) d x √− g (cid:0) K + V − g Tr (cid:8) E p D p A + E r i D r i A + G mn D Φ m [ A , Φ n ] (cid:9)(cid:1) , (4.6)where the kinetic and potential energy densities are K := − g Tr (cid:8) E p E p + E r i E r i + G mn D Φ m D Φ n (cid:9) , (4.7) V := Tr (cid:26) F r i r j F r i r j + 12 ( F F + F pr i F pr i ) + 12 G mn ( D p Φ m D p Φ n + D r i Φ m D r i Φ n )++ 14 G mn G m (cid:48) n (cid:48) [Φ m , Φ m (cid:48) ][Φ n , Φ n (cid:48) ] − µ ( r + z ) ˜ (cid:15) r i r j r k ( D r i Φ y ) F r j r k (cid:27) . (4.8)Together, the last terms of (4.6), proportional to g , are a total derivative when werestrict the space of field configurations to the constraint surface defined by the Gauss Law, D r i ( g E r i ) + g (cid:0) D p E p − G mn [Φ m , D Φ n ] (cid:1) = 0 . (4.9)We refer to this as the local Gauss Law constraint. It is equivalent to the A equation ofmotion in (4.3) when the fermi field is set to zero. There is also a boundary Gauss Lawconstraint that comes from demanding that the total derivative term vanishes: (cid:90) d x (cid:2) ∂ r (cid:0) √− g Tr { E r A } (cid:1)(cid:3) = 0 . (4.10)We will return to this condition after we have analyzed the field asymptotics, but for now wesimply assume it is satisfied. It will merely amount to imposing appropriate fall-off conditionson A .Restricting to field configurations that satisfy both of these constraints, our energy func-tional for the bosonic fields is H bosym = 1 g (cid:90) d x √− g ( K + V ) . (4.11)– 28 –n general grounds, supersymmetry implies that this functional is positive semi-definite.The presence of the last term in the potential (4.8), however, makes this property slightlynon-obvious. Recall that this term originates from the Chern–Simons part of the D5-braneaction and is present because of the RR-flux of the string background. Positivity is establishedby noting that it can be combined with two other terms to make a complete square:14 F r i r j F r i r j + 12 G mn D r i Φ y D r i Φ y − µ ( r + z ) ˜ (cid:15) r i r j r k ( D r i Φ y ) F r j r k = 14 µ ( r + z ) (cid:0) F r i r j − ˜ (cid:15) r i r j r k D r k Φ y (cid:1) . (4.12)Here, repeated downstairs indices are contracted with a flat Euclidean metric δ r i r j , and wehave taken advantage of the fact that g r i r j = µ ( r + z ) δ r i r j = G yy δ r i r j . With this observationit is then manifest that K + V is a positive sum of squares.Hence the space of classical vacua of the Yang–Mills theory is the space of gauge-inequivalent zero energy configurations: M vac := { [( A a , Φ m )] | K = V = 0 } . (4.13)This is an extremely interesting space, mostly because of the observation (4.12). One of itscomponents is the usual sort of field theory Coulomb branch that we expect to have in asupersymmetric Yang–Mills theory—mutually commuting, constant vevs for the Higgs fields,Φ m ∞ . Since the vevs are mutually commuting they can be simultaneously diagonalized to aCartan subalgebra t ⊂ g . Residual gauge transformations act by Weyl conjugation. Hencethis component of the vacuum has the form M = t ⊗ /W , (4.14)where W is the Weyl group.However, the vacuum space has an infinite number of additional components associatedwith nontrivial zeros of (4.12). These consist of field configurations ( A r i , Φ y ) that solve(B) : F r i r j − ˜ (cid:15) r i r j r k D r k Φ y = 0 , (4.15)together with ( A , A p , Φ z i ) that ensure the remaining terms in H bosym vanish. The condition(4.15) is none other than the Bogomolny equation for Euclidean monopoles on the R param-eterized by r i !It might sound strange that monopole moduli spaces are part of the vacuum manifoldof the theory. Ordinarily, they parameterize local, but not global, minima of the energyfunctional, which are associated with soliton masses. The reason the story is different here isagain due to the ‘extra’ term in the action arising from the background RR flux. Normally,one has to complete the square by hand, adding and subtracting such a term. Also, this crossterm is usually a total derivative, so the term that is added in this process is topological and– 29 –rovides the mass of the soliton. Here, this is not the case. The cross term is dynamical, andalready present in the action from the beginning.Monopole moduli spaces are defined as spaces of gauge-inequivalent solutions to (4.15),and therefore they play an essential role in defining the vacuum manifold of (4.11). Thedata that goes into specifying an ordinary monopole moduli space are the asymptotic Higgsvev, Φ y ∞ , and a magnetic charge P . They determine asymptotic boundary conditions on thesolutions such that(bc ∞ ) : Φ y = Φ y ∞ − P r + · · · , F = P ω S + · · · , r → ∞ , (4.16)where ω S = sin θ d θ d φ is the standard volume form on the two-sphere and the ellipsesrepresent subleading terms.For reasons to be explained shortly, we should actually consider a more general notionof monopole moduli space that allows for magnetic singularities at specified points (cid:126)r = (cid:126)v σ ,corresponding to the insertion of ’t Hooft defects [46]. These singularities are defined byimposing boundary conditions on the fields as (cid:126)r → (cid:126)v σ of the form [47, 48](bc σ ) : Φ y = − P σ | (cid:126)r − (cid:126)v σ | + · · · , F = P σ ω ( σ ) S + · · · , | (cid:126)r − (cid:126)v σ | → , (4.17)where ω ( σ ) S is the standard volume form on a two-sphere centered on (cid:126)v σ and P σ is the ’t Hooftcharge of the defect. The associated moduli space of singular monopoles is defined as thespace of gauge-inequivalent solutions to (4.15) obeying the asymptotic boundary conditions(4.16) as r → ∞ and the ’t Hooft defect boundary conditions (4.17) as (cid:126)r → (cid:126)v σ : M ( { P σ , (cid:126)v σ } ; P, Φ y ∞ ) := { ( A r i , Φ y ∞ ) | (B) & (bc ∞ ) & (bc σ ) } (cid:30) G { P σ } . (4.18)Here G { P σ } is the group of gauge transformations that approach the identity at infinity andleave the ’t Hooft charges invariant. These spaces have been studied intensely since the initialwork of Kronheimer [49], with important contributions in [19, 50, 51], to name a few. See[48] and references therein for a more complete discussion.Note that the ’t Hooft and asymptotic charges are quantized. Factoring out the cen-ter of mass U (1) results in the su ( N f ) charges taking values in the co-character lattice of U ( N f ) /U (1) ∼ = P SU ( N f ). This lattice consists of integer linear combinations of the funda-mental magnetic weights. For su (2) the fundamental magnetic weight is half of the simpleco-root.A detailed description of how the space of vacua, M vac , is defined in terms of the modulispaces M ( { P σ , (cid:126)v σ } , P, Φ y ∞ ) is beyond the scope of the present paper. We will limit ourselves,instead, to using the intersecting D3/D5 system to indicate what the various data defining M vac correspond to in terms of branes. This will also lead to a natural description of the– 30 – y z y D3D3 D5
Figure 4 . The brane configuration corresponding to a generic point of the vacuum M vac . The thickred line along the y -axis represents the original color D3-branes. The blue dots are the D5-branes.Their relative separations in the y and z i directions are dictated by the vevs Φ y ∞ , Φ z i ∞ . Additionalfinite and/or semi-infinite D3-branes can begin and end on the D5-branes. The precise numbers ofthese will be controlled by the asymptotic and ’t Hooft charges. corresponding vacua in the holographic dual, though we leave a detailed matching to futurework.A generic vacuum configuration is depicted in Figure 4. The vevs (Φ z i ∞ , Φ y ∞ ) characterizethe relative separation of the D5-branes from each other in the z i and y directions respec-tively. The center-of-mass position of the D5-branes is instead parameterized by ( y , (cid:126)z ).We are free to set y = 0 since translations in y are an isometry of the background.It is well known that finite-length D3-branes stretched between D5-branes appear assmooth monopoles in the worldvolume theory of the D5-branes [53], while semi-infinite D3-branes ending on D5-branes appear as ’t Hooft defects [51, 52, 54]. Thus, nontrivial configu-rations ( A r i , Φ y ) carrying asymptotic magnetic and ’t Hooft charges, correspond to additionalfinite-length and semi-infinite D3-branes, stretching between and ending on the D5-branes.The additional D3-branes should run parallel to the color D3-branes so as to preserve thesame supersymmetries as the vacua without flux, (4.14).To gain intuition for the properties of these vacua, let us consider some special cases.Suppose first that all of the vevs Φ z i ∞ are vanishing, so the D5-branes are separated from each Giving a precise transcription from the vevs to the separations requires a little more Lie algebra notationthan we need in the rest of this paper, so we refer the reader to [52] for details. – 31 –ther in the y direction only. A detailed description of this moduli space M ( { P σ , (cid:126)v σ } ; P, Φ y ∞ )in terms of brane configurations was given in [52]. This includes an accounting for the dimen-sion of the moduli space, which was derived in [48], in terms of mobile D3-brane segmentsthat can slide along parallel D5-branes. It also includes formulae for the ’t Hooft chargesin terms of the numbers of semi-infinite D3-branes ending on each D5-brane, and for thedecomposition of these charges into u (1) and su ( N f ) components.Second, consider the case that Φ y ∞ is vanishing, meaning that the D5-branes are separatedfrom each other in the z i directions only. Then these brane configurations are those studiedin [55]. The authors of [55] focused on the description of the vacua from the point of view ofthe D3-branes, with the goal of classifying all half-BPS boundary conditions for N = 4 SYMon the half-space. Note that N = 4 SYM on the half space can be used to describe N = 4SYM with a defect at y = 0 using the folding trick, as they discussed.From the perspective of the D3-brane theory, semi-infinite D3-branes ending on a stackof D5-branes at y = 0 are described by solutions to Nahm’s equation with a pole at y = 0.The six adjoint-valued scalars of the N = 4 SYM are divided into two triplets, (cid:126)R D3 and (cid:126)Z D3 , which encode transverse fluctuations in the r i and z i directions respectively. The (cid:126)R D3 solve Nahm’s equation on the semi-infinite interval y ∈ ( −∞ , y ] with the pole at y = 0specified in terms of an su (2) representation. The components of the ’t Hooft charge, whichdictate the number of D3-branes ending on each D5-brane, are encoded in the dimensions ofthe irreducible components of the su (2) representation. The scalars (cid:126)Z D3 are required to takeconstant values that commute with the (cid:126)R D3 , and will be related to the vevs Φ z i ∞ . Finally theposition of the ’t Hooft charges, (cid:126)v σ , will be related to the asymptotic values of the (cid:126)R D3 as y → −∞ .Then, based on [52, 55], and the known relation between singular monopoles and solutionsto Nahm’s equations on a semi-infinite interval [51], we expect that the generic configurationdepicted in Figure 4 corresponds to a solution to the ( (cid:126)R D3 , (cid:126)Z D3 )-system analogous to thosedescribed in [55], but with multiple parallel defects at different values of y , as dictated by thevev Φ y ∞ . The solution will involve a solution to Nahm’s equation on a union of semi-infiniteand finite intervals, with appropriate boundary conditions at each of the defects. This willprovide the holographically dual description of the space of vacua, (4.13). It would behighly desirable to have a complete description of this space from both points of view, andwe will return to this issue in future work.Let us note that in the abelian case, a ’t Hooft defect is also known as a BIon spike[56, 57]. The case of a single BIon spike of charge p at r = 0 has been studied extensively The description used in [52] employed a D1/D3 system that is T-dual to the D3/D5 system here. Therelationship between monopoles and brane configurations is identical for the two. These scalars were denoted (cid:126)X and (cid:126)Y respectively in [55]. All of these vacua preserve eight supersymmetries, so we do not expect the space of classical vacua to belifted by quantum effects. See (4.20) below. – 32 –n the AdS/dCFT literature. See e.g. [58] and references therein. On the gravity side of thecorrespondence, a U (1) magnetic flux through the S is accompanied by a modification ofthe D5-brane embedding [3, 37, 59]. This is described by the nontrivial profile for the abelianΦ y in (4.16) with P → p , where now the displayed terms are the full solution. As r → AdS × S , but the AdS slice is different. Itsradius depends on the flux and is different than the AdS radius of the ambient AdS . In theholographic dual, nontrivial U (1) flux of charge p corresponds to a defect that implementsa jump in the rank of the 4D N = 4 SYM gauge group from SU ( N c ) to SU ( N c − p ). The triplet (cid:126)R D3 obeys Nahm’s equations with a pole at the defect given in terms of the p -dimensional irreducible su (2) representation, as pointed out in [59]. A generic configurationof branes like Figure 4 will involve an abelian defect of the type just described together witha singular su ( N f ) monopole configuration.A simple non-abelian generalization of the BIon spike discussed above is a Cartan-valuedflux vacuum, in which the fields take the form˜ F = P θ d θ d φ , ˜Φ y = Φ y ∞ − P r , ˜Φ z i = Φ z i ∞ , (4.19)where all of the Higgs vevs and the charge P are constant and mutually commuting. Thiscan be viewed as a Dirac monopole embedded in the non-abelian gauge group via the ho-momorphism U (1) → P SU ( N f ) specified by P . See [48] for a detailed discussion of thesesolutions in the context of singular monopole moduli spaces. It is a convenient backgroundto use for the analysis of the perturbative spectrum below since the linearized equations ofmotion around this vacuum are tractable and the fields of any vacuum configuration will takethis form asymptotically. Note that for these solutions P plays the role of both an asymptoticmagnetic charge and a ’t Hooft defect charge at r = 0. For this reason, and because therewill be other magnetic charges that make an appearance below, we will often refer to P inthis context as a ’t Hooft charge.Finally, let us consider the supersymmetry of the vacua (4.13), starting with the exampleof the flux vacua (4.19). Making use of the relations (2.30), one finds that the variation ofthe Fermi field evaluated on (4.19) can be expressed in the form δ ε Ψ (cid:12)(cid:12)(cid:12)(cid:12) ( ˜ A, ˜Φ) = (cid:26) µ z r P Γ ry + µ r Φ y ∞ Γ ry − rr + z Φ z i ∞ Γ rz i (cid:27) (cid:16) − Γ rθφy (cid:17) ε = (cid:26) µ r Φ y ∞ Γ ry − rr + z Φ z i ∞ Γ rz i (cid:27) (cid:16) − Γ rθφy (cid:17) ε . (4.20)The second step followed because if z (cid:54) = 0 then ε satisifes Γ rθφy ε = ε , while if z = 0 the P term vanishes trivially. Hence we learn the following. On the one hand, if z (cid:54) = 0, then Since we work at leading order in the large N c limit, we can’t distinguish the difference between a jumpfrom SU ( N c ) to SU ( N c − p ) versus a jump from SU ( N c + p ) to SU ( N c ). All that matters for this discussionis that the rank changes by p . – 33 – ε Ψ = 0. In other words no further supersymmetry is broken when we turn on Φ m ∞ beyondthat broken by z already. On the other, if z = 0 then turning on any nonzero Φ m ∞ breaksthe supersymmetries generated by η . This is expected since they generate supersymmetriesassociated with the superconformal symmetries in the holographic dual, and separating theD5-branes breaks scale invariance. Notice that P completely drops out. Hence the flux vacua(4.19) with vanishing Higgs vevs preserve all sixteen supersymmetries for any P . The sameconclusion was previously shown to hold for the U (1) magnetic flux vacua in [37].More general monopole vacua preserve all eight supersymmetries generated by ε . Toshow this we proceed as follows. Assuming η vanishes, the supersymmetry variation of thefermion can be simplified. In this case ε is an eigenspinor of Γ rθφy = Γ r r r y , and this allowsus to absorb the M Ψ term into the D a Φ terms by switching to scalars with coordinate indicesrather than tangent space indices. We work in the Cartesian frame where( ε ) cart = √ µ ( r + z ) / ε , (4.21)and it is the Γ r i that are constant. Then the relations we need, following from (2.30) andΓ r r r y ε = ε , are (cid:104) D r i Φ y Γ r i y + M Ψ Γ θφy Γ y Φ y (cid:105) ε = µ ( r + z ) D r i Φ y Γ r i y ε (cid:104) D r i Φ z j Γ r i z j + M Ψ Γ θφy Γ z j Φ z j (cid:105) ε = D r i Φ z j Γ r i z j ε , (4.22)where we have also used that Γ θφy = Γ r Γ rθφy = (ˆ r i Γ r i )Γ r r r y .Since Φ z i must be covariantly constant on a global minimum of the energy functional,the D r i Φ z j terms drop out. The D µ Φ y terms as well as the terms involving the componentsof F ab that do not participate in the Bogomolny equation (4.12) drop out as well. Then thesupersymmetry variation reduces to δ Ψ (cid:12)(cid:12)(cid:12)(cid:12) vac = µ ( r + z ) (cid:20) F r i r j Γ r i r j + D r k Φ y Γ r k y (cid:21) ε = 12 µ / ( r + z ) / (cid:0) F r i r j − ˜ (cid:15) r i r j r k D r k Φ y (cid:1) Γ r i r j ε = 0 , (4.23)where in the second step we used Γ r r r y ε = ε , and in the last step we used the Bogomolnyequation. Hence all of the vacua in M vac preserve (at least) eight supersymmetries. In factthey preserve the full 3D N = 4 super-Poincar´e algebra.Next we turn to the analysis of the perturbative spectrum around these vacua. The spectrum of linearized fluctuations on the D5-brane has been computed in the abeliantheory. Each field gives rise to a Kaluza–Klein tower of modes on the (asymptotically)
AdS – 34 –pace after expanding in an orthogonal basis on the S . The modes organize into shortmultiplets of the superconformal algebra and are holographically dual to a tower of operatorsin the dCFT localized on the defect, identified in [4]. This analysis was originally carriedout for the AdS background with z = 0, and extended to the z (cid:54) = 0 case in [14, 15]. Inthe latter case z provides a scale, breaking conformal invariance in the IR, and leading to adiscrete set of normalizable radial modes for each KK mode.These analyses can be further extended to the flux vacua, (4.19), of the non-abeliantheory, and the results pertaining to the large r behavior of the modes will be valid aroundany of the monopole vacua of the previous subsection. In order to describe the results we firstintroduce some notation. We use ( a, φ ) to denote field fluctuations around (4.19), so that A a = ˜ A a + a a , Φ m = ˜Φ m + φ m . (4.24)Then we expand the fluctuations in components along a basis of the Lie algebra { T s } , writing e.g. a a = a sa T s .Basis elements of the real Lie algebra g are represented by anti-Hermitian matrices in ourconventions, but it is more convenient to employ a basis of the complexified Lie algebra g C that utilizes raising and lowering operators associated with a root decomposition of g C . Sincethe background data Φ m ∞ , P are mutually commuting, they can be taken to lie in a Cartansubalgebra such that their adjoint action is diagonal. Thus we introduce masses ( (cid:126)m z,s , m y,s )and charges p s such that[Φ z i ∞ , T s ] = − im z i ,s T s , [Φ y ∞ , T s ] = − im y,s T s , [ P, T s ] = − ip s T s . (4.25)The quantization of P implies that the p s are integers. The index s runs over values of theform {± α , i } , where the { α } are a set of positive roots and i in an index labeling a basis ofgenerators for the Cartan subalgebra. We can choose the basis such that Tr ( T − α T β ) = δ αβ ,Tr ( T i T j ) = δ ij , with the rest vanishing. If Φ y ∞ is a regular element of g we fix the Cartansubalgebra uniquely by requiring that it be in the fundamental Weyl chamber of t . Thismeans that all m y,s > s corresponding to a positive root, and all D5-branes havedistinct y -positions in Figure 4. In any case we partially fix the choice of Cartan by requiring m y,s ≥ s . The masses and charges vanish for s = i , and satisfy p − α = − p α etc . for thenonzero roots. The components of a real adjoint-valued field Φ = Φ s T s satisfy Φ − α = (Φ α ) ∗ and (Φ i ) ∗ = Φ i . In an effort to keep the notation manageable, we will avoid making thedecomposition into root and Cartan directions explicit, and instead just write p − s = − p s and(Φ s ) ∗ = Φ − s .We plug these expansions back into the equations of motion (4.3) and linearize in thefluctuations. Some details of this procedure are given in appendix C. Here we summarize the This was further generalized in [59] to the case of the abelian D5-brane embedding that includes bothnonzero z and a U (1) magnetic flux on the worldvolume. There, unlike in our case, the presence of U (1) fluxtogether with nonzero z leads to a continuous 3D spectrum. – 35 –ey points. First, the equations for a µ and φ z i can be decoupled from the rest and take anidentical form after choosing a convenient gauge-fixing condition (described in the appendix),so it is useful to start with them: (cid:26) ∂ r + 2 r ∂ r + 1 r ˜ D S − (cid:16) m y,s − p s r (cid:17) + ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) µ ( r + z ) (cid:27) ( a sµ , φ z i ,s ) = 0 . (4.26)Here ˜ D S ≡ ˜ g αβ ˜ D α ˜ D β is the covariant background Laplacian on the two-sphere constructedfrom the background gauge field, ˜ D α = ∂ α + [ ˜ A α , · ]. When (cid:126)m z,s , m y,s , and p s = 0 thisequation coincides with the analogous one given in [14, 15], as do the equations for theremaining fluctuations. The effects of the nontrivial background (4.19) are qualitatively thesame for all the fluctuations, so we describe them in the context of (4.26).Regarding the large r behavior of the solution, the most important question is whether m y,s is zero or positive— i.e. whether the fluctuation commutes with Φ y ∞ or not. When m y,s is positive then r = ∞ is an essential singularity of the ODE. Solutions either blow up ordecay exponentially, ( a µ , φ z i ) ∼ e ± m y,s r , and the boundary conditions we impose below willallow for the decaying behavior only. Due to their exponential rather than power-law fall-off,these modes are not dual to local operators in the holographic dual. Henceforth we restrictattention to those Lie algebra components such that m y,s = 0. (If Φ y ∞ is generic— i.e. aregular element of g —then these will be the components along the Cartan subalgebra that isuniquely determined by it.)When m y,s is zero, the effects of nonzero (cid:126)m z,s , p s are easily accommodated by makingslight modifications to the analysis of [4, 14, 15]. First, (cid:126)m z,s always appears as a shift of the R , wave operator: η µν ∂ µ ∂ ν → η µν ∂ µ ∂ ν − (cid:126)m z,s . This just leads to a constant shift for the3D spectrum of each mode. It does not affect the leading order large r behavior of solutions.The background flux on the two-sphere is dealt with by making a mode expansion thatdiagonalizes ˜ D S . The background gauge field ˜ A α is that of a Dirac monopole, and this isa well-known problem with a complete and explicit solution. When acting on a scalar, theeigenfunctions are spin-weighted spherical harmonics, with eigenvalues˜ D S (cid:12)(cid:12)(cid:12)(cid:12) T s m (cid:48) Y jm ( θ, φ ) = (cid:18) − j ( j + 1) + p s (cid:19) m (cid:48) Y jm ( θ, φ ) , where m (cid:48) = − p s . (4.27)Here ( j, m ) are the usual angular momentum quantum numbers, but j is restricted to start atthe minimum value | p s | / j ’s are all integers orall half-integers depending on whether p s is even or odd respectively. m runs from − j to j ininteger steps as usual. If p s = 0 then these reduce to the ordinary spherical harmonics. Thecombination ˜ D S − p s in (4.26) almost always appears together: the second term comes fromthe background ˜Φ y . Hence, after expanding, say, a sµ in spin-weighted spherical harmonics theresulting equation for each KK mode is (cid:26) ∂ r + 2 r ∂ r − j ( j + 1) r + ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) µ ( r + z ) (cid:27) a sµ, ( j,m ) = 0 , (4.28)– 36 –nd similarly for φ z i ,s ( j,m ) . (Here we are assuming m y,s = 0.) One can further make a Fourierexpansion along R , so that η µν ∂ µ ∂ ν → − k , and then this equation becomes identical inform to the corresponding ones in [14, 15].Exact solutions to (4.28) for the radial dependence are available in terms on hyperge-ometric functions when z (cid:54) = 0. Demanding that the series solution truncate (in order tohave normalizability) leads to the introduction of a radial quantum number n and a discretespectrum of masses, k = M n,j for ‘meson’ states in the holographic dual. See [14, 15] forfurther details. (These masses will now all be shifted from the values given in [14, 15] by (cid:126)m z,s .) If z = 0 we instead get the usual continuous spectrum indicative of a conformal dual,and the radial wavefunctions can be given in terms of Bessel functions.Our focus is on the large r behavior of these modes, where z can be neglected and theanalysis reduces to the AdS case studied in [4]. At large r one has from (4.28) that ( a sµ, ( j,m ) , φ z i ,s ( j,m ) ) ∼ (cid:40) r − ( j +1) = r − ∆ j ,r j = r − j , (4.29)where ∆ j , the conformal dimension of the dual operator, is given by ∆ j = 2 + j . Thefirst behavior corresponds to the ‘normalizable mode’ solution for the given KK mode, andthe second behavior corresponds to the ‘non-normalizable mode’. In the dual theory, the( x µ -dependent) coefficient of the non-normalizable mode is the source for the dual operator,while the coefficient of the normalizable mode is its vev. This holographic interpretationof the field asymptotics is an important guide to the type of boundary conditions that oneshould consider. We will have more to say about this in the next subsection. First, however,let us summarize the rest of the perturbative spectrum.Like a µ and φ z i , the Lie algebra components of the other field fluctuations along directionswith nonzero m y,s have exponential decay at large r . Thus we focus on the case m y,s = 0.Then the equation for the radial component of the gauge field fluctuations, a sr, ( j,m ) , decouplesfrom the rest. The asymptotic behavior of solutions is a sr, ( j,m ) ∼ (cid:40) r − ( j +2) = r − ∆ j ,r j − = r − j , (4.30)for the normalizable and non-normalizable mode respectively.The remaining bosonic degrees of freedom are contained in a α and φ y . These are usefullyrepackaged into three adjoint-valued scalars λ, f, y defined by a α = ˜ D α λ + ˜ (cid:15) αβ ˜ g βγ ˜ D γ f , φ y = − r [ P, λ ] + y , (4.31)which can then be expanded in Lie algebra components and in spin-weighted spherical har-monics on the S . The λ modes are nondynamical and determined by a gauge-fixing condition. This agrees with [4] after taking into account that a µ ∼ ra µ and (cid:126)φ z ∼ r(cid:126)φ z . – 37 –he j = 0 modes (which are only possible when p s = 0) can be set to zero while the higher j modes are given by λ s ( j,m ) = 1 j ( j + 1) (cid:20) ∂ r (cid:0) r a sr (cid:1) + ip s (cid:16) r y s ( j,m ) − f s ( j,m ) (cid:17)(cid:21) . (4.32)Notice that, if a r = 0, then λ s is only present when there is a nontrivial flux, p s . In the absenceof flux, λ is pure gauge and can be set to zero. This is consistent with the assumptions madein [4], which did not consider turning on flux. The y and f modes form a coupled system that must be diagonalized, as in [4, 14, 15].The diagonal combinations are φ ± defined by φ +( j,m ) = 1 √ j + 1 (cid:0) jf ( j,m ) + r y ( j,m ) (cid:1) , φ − ( j,m ) = 1 √ j + 1 (cid:0) ( j + 1) f ( j,m ) − r y ( j,m ) (cid:1) . (4.33)The respective asymptotics at large r are φ ± ,s ( j,m ) ∼ r − ∆ ± j ,r − ± j , (4.34)where ∆ + j = j + 4 , ∆ − j = j . (4.35)Here it is important to note that only the + mode is physical for the lowest, j = | p s | /
2, rungof the KK tower: one can show that the j = | p s | / φ − drops out of the expressions(4.31) for a α and φ y . (This includes the j = 0 modes along directions with p s = 0.)Finally there are two KK towers of four-dimensional fermions η s ( j,m ) , χ s ( j,m ) , coming fromΨ. The spectrum of these modes has not been previously discussed in the literature. Weprovide the details in appendix C.2. Let us denote by Ψ ( χ ) s ( j,m ) the restriction of Ψ to the casein which only the χ s ( j,m ) mode is turned on, and analogously for Ψ ( η ) s ( j,m ) . For each mode, thetwo possible behaviors—normalizable and non-normalizable—are correlated with a specificchirality of Ψ with respect to Γ rθφy . (This is true for the leading order behavior of the mode;the other chirality will be turned on at subleading order.) DefiningΨ ± := 12 ( ± Γ rθφy )Ψ , (4.36)the leading asymptotics areΨ ( χ ) s, ± ( j,m ) ∼ r − ± m ( χ ) j , Ψ ( η ) s, ± ( j,m ) ∼ r − ± m ( η ) j , (4.37) Reference [4] also works in a different gauge, related to ours by a shift involving a r that removes the a r dependence from λ at the price of introducing it back into e.g. the a µ equation. The gauge choice we use isconsistent with the analysis of [15]. See appendix C for further details. – 38 –here the AdS masses are given by m ( χ ) j = j − , m ( η ) j = − (cid:18) j + 32 (cid:19) . (4.38)For the χ -type modes, j starts at ( | p s | + 1) and increases in integers steps. For the η -type modes, j starts at ( | p s | −
1) if p s (cid:54) = 0, or if p s = 0, and increases in integer steps.The quantum number m runs from - j to j in integer steps as usual. These asymptotics arevalid when m y,s = 0, otherwise the normalizable (non-normalizable) modes are exponentiallydecaying (blowing up). The conformal dimensions of the dual operators are ∆ ( χ,η ) j = + | m ( χ,η ) j | .Observe that the masses of the η -type modes are always ≤ − , while the masses of the χ -type modes are always ≥
0. For each s such that p s = 0 (and m y,s = 0), the j = 1 / χ -typemodes provide a doublet of massless fermions on the asymptotically AdS space. Also, sincethe masses have opposite sign for the χ - and η -type modes, the leading behavior of Ψ sits inopposite Γ rθφy eigenspaces. For example, the normalizable χ -type modes correspond to Ψ − while the normalizable η -type modes correspond to Ψ + .In fact, for the massless χ modes it is not obvious from this analysis whether Ψ ( χ ) , − or Ψ ( χ ) , + should be identified with the normalizable mode, as both have the same O ( r − / )behavior. We will see in subsection 4.4 below that supersymmetry dictates that Ψ ( χ ) , − is thenormalizable mode while Ψ ( χ ) , + is the non-normalizable mode.The S singlet modes of the fields will appear often in the following. We will use asimplified notation for them,( a, φ ) (0 , ( x µ , r ) Y ≡ ( a, φ )( x µ , r ) , (4.39)absorbing the constant factor Y = (4 π ) − / into the definition of the modes. It will be clearfrom the context whether we are using lowercase ( a, φ ) to refer to the S singlet mode only,or to the sum over all modes as in (4.24). With the asymptotic behavior of the linearized perturbations in hand, we can return to thequestions of boundary conditions and consistency of the variational principle. As we sawabove, each KK mode has an associated normalizable and non-normalizable mode. Corre-spondingly, one can consider two types of boundary condition. For the first type one turnson the non-normalizable mode and holds it fixed. Then, according to the AdS/CFT corre-spondence, the on-shell action as a function of these sources gives (a leading saddle point ap-proximation to) the generator of correlation functions for the dual operators. This boundarycondition generally requires the addition of boundary terms in order to maintain consistency– 39 –ith the variational principle, and the systematic procedure for doing this is known as holo-graphic renormalization. (See the discussion following (4.5).) The second type of boundarycondition sets the coefficient of the non-normalizable mode to zero and lets the coefficientof the normalizable mode fluctuate. This is appropriate for the construction of the Hilbertspace of states in the bulk theory, which is equivalent to the Hilbert space of states in the dualCFT. Here we follow the Lorentzian-signature version of the correspondence as described in[60].In this paper we generally want to consider the latter type of boundary condition. How-ever for some special modes at the bottom of the KK towers it is possible (and useful) toturn on the non-normalizable mode without spoiling consistency of the variational principle.In fact we have already done so for two types of mode. The vevs Φ z i ∞ and Φ y ∞ can be viewedas finite, constant coefficients for the j = 0 non-normalizable modes of φ z i and y = r − φ + respectively. In contrast, the flux P is not associated with any non-normalizable (or normal-izable) mode; this is consistent with the fact that it is quantized and cannot be adiabaticallytuned. Rather, P determines a superselection sector of the theory under consideration.We will not turn on any further non-normalizable modes for the φ z i . However, we willallow for the S singlet modes corresponding to the vevs to be spacetime varying. We writeΦ z i = φ z i ( x µ , r ) + (cid:88) j ≥ ,m,s φ z i ,s ( j,m ) ( x µ , r ) (cid:16) − ps Y jm ( θ, φ ) (cid:17) T s , with φ z i ( x µ , r ) = φ z i (nn) ( x µ ) + 1 µr φ z i (n) ( x µ ) + O ( r − ) , φ z i ,s ( j,m ) = O ( r − ( j +1) ) , (4.40)The S singlets φ z i are Lie algebra-valued and commute with P, Φ y ∞ . The more generalasymptotics are encoded by the φ z i (nn) ( x µ ), which can be mutually non-commuting. This willstill give rise to finite energy field configurations provided they have sufficient decay propertiesat large x µ . Specifically, we require them to approach the (mutually commuting) constantvevs Φ z i ∞ at spatial infinity and have vanishing time derivatives as t → ±∞ . Since the non-normalizable mode is held fixed in the variational principle, δ Φ z i = O (1 /r ), D r Φ z i = O (1 /r ),and it follows that the δ ij P z i δ Φ z i term drops out of (4.4), as required for consistency.There are additional types of non-normalizable modes we wish to consider. The first typeis the j = 0 non-normalizable mode of the gauge field components A µ = a µ . According to(4.29), these modes have a finite limit as r → ∞ . We set all higher j non-normalizable modesof A µ to zero. We also set all non-normalizable modes of A r to zero, so that the leadingbehavior of this field at large r is O (1 /r ), corresponding to the normalizable S singlet.– 40 –hus our boundary conditions for the AdS part of the gauge field are A µ = a µ ( x ν , r ) + (cid:88) j ≥ ,m,s a µ, ( j,m ) ( x ν , r ) (cid:16) − ps Y jm ( θ, φ ) (cid:17) T s , with a µ ( x ν , r ) = a (nn) µ ( x ν ) + 1 µr a (n) µ ( x ν ) + O (1 /r ) , a µ, ( j,m ) = O ( r − ( j +1) ) , and A r = a r ( x ν , r ) + (cid:88) j ≥ ,m,s a r, ( j,m ) ( x ν , r ) (cid:16) − ps Y jm ( θ, φ ) (cid:17) T s , with a r ( x ν , r ) = 1 µ r a (n) r ( x ν ) + O ( r − ) , a r, ( j,m ) = O ( r − ( j +2) ) . (4.41)The singlets a µ,r commute with P, Φ y ∞ but need not commute with one another. The data a (nn) µ is held fixed, and we assume it approaches a pure gauge configuration as x µ → ∞ . Theseboundary conditions imply that F rµ = O (1 /r ) and the variation δA µ = O (1 /r ), ensuringthat the P µ δA µ term drops out of (4.4).Next, consider the j = 1 angular momentum triplet of modes at the bottom of the KKtower for φ − . In general the low j modes of φ − are subtle, due to the small value of theconformal dimension, ∆ − j = j . We can immediately assume that all non-normalizable modesof φ + are set to zero (besides the singlet giving the vev Φ y ∞ ) and all non-normalizable modesof φ − for j > are set to zero—these conditions ensure that the fluctuations are subleadingto background ( ˜Φ y , ˜ A α ). The j ≤ modes of φ − are all decaying, and a closer look atconsistency of the variational principle is required.Restricting consideration to the φ − type modes and utilizing (4.24) and (4.31) through(4.33), one obtains the following results for the KK modes of the conjugate variables P y and P α , (4.5): P y,s ( j,m ) = µ r (cid:26) − (cid:18) j − p s j (cid:19) ( r∂ r + j ) φ s, − ( j,m ) (2 j + 1) / + r [ a r , φ y ] s ( j,m ) − θ [ a θ , a φ ] s ( j,m ) + O ( r − ) (cid:27) , (4.42)and P θ,s ( j,m ) = µ r (cid:26) r (cid:20) θ ˜ D φ − ip s j ∂ θ (cid:21) ( r∂ r + j ) φ s, − ( j,m ) (2 j + 1) / + [ a r , a θ ] s ( j,m ) + − θ [ a θ , φ y ] s ( j,m ) + O ( r − ) (cid:27) P φ,s ( j,m ) = µ r sin θ (cid:26) − r (cid:20) ∂ θ + ip s j sin θ ˜ D φ (cid:21) ( r∂ r + j ) φ s, − ( j,m ) (2 j + 1) / + [ a r , a φ ] s ( j,m ) ++ [ a θ , φ y ] s ( j,m ) + O ( r − ) (cid:27) . (4.43)– 41 –e have computed these to the order necessary for taking the r → ∞ limit in (4.4), takinginto account that δ Φ y = O ( r − ) and δA α = O (1 /r ). In obtaining these results we droppedterms proportional to ∂ r ( r a r, ( j,m ) ) − j ( j + 1) a r, ( j,m ) . One can use the linearized equation ofmotion for a r, ( j,m ) and the allowed asymptotics (4.41) to conclude that this combination ofterms is O ( r − ).We can now argue that the non-normalizable modes for j = 5 / , , / φ − ( j,m ) , which leads to( r∂ r + j ) φ − ( j,m ) = O ( r j − ), implying contributions to the P of P y ∼ O ( r j +1 ), P α ∼ O ( r j ).However δφ − , which is the order of the normalizable mode, since the non-normalizable modeis held fixed, contributes to the variations according to δ Φ y = O ( r − − j ) and δA α = O ( r − j ).Hence, P y δ Φ y + P α δA α in (4.4) would have a finite limit as r → ∞ . Thus for these valuesof j , as with all higher values, the non-normalizable mode of φ − ( j,m ) should be set to zero, orelse one must resort to holographic renormalization.In the case j = 1 the dominant asymptotics correspond to the normalizable mode, φ − ∼ r − j = r − . The operator ( r∂ r + 1) annihilates this, however. Hence it is still the non-normalizable mode that gives the leading contribution to ( r∂ r +1) φ − . It is useful to introducea real basis for the j = 1 triplet of scalars, (cid:126) X = (cid:126) X ( x µ , r ), such that (cid:88) m = − φ s, − (1 ,m ) ( x µ , r ) Y m ( θ, φ ) =: − √ µ r ˆ r · (cid:126) X s ( x µ , r ) , (4.44)where ˆ r = (sin θ cos φ, sin θ sin φ, cos θ ). Note that these modes only exist for those s suchthat p s = 0. In other words the adjoint-valued (cid:126) X = (cid:126) X s T s satisfies [ (cid:126) X , P ] = 0 = [ (cid:126) X , Φ y ∞ ]. Wedenote the coefficients of the two modes by (cid:126) X (n) and (cid:126) X (cid:48) nn such that (cid:126) X ( x µ , r ) = (cid:126) X (n) ( x ν ) + 1 µr (cid:126) X (cid:48) (nn) ( x ν ) + O (1 /r ) . (4.45)As we will see, (cid:126) X (cid:48) (nn) is not quite the conjugate of (cid:126) X (n) . One then has the following expansions:Φ y = Φ y ∞ − P r + 1 µ r ˆ r · (cid:126) X ( x ν , r ) + O ( r − / ) ,A θ = − µ r ˆ φ · (cid:126) X ( x ν , r ) + O ( r − / ) ,A φ = P ± − cos θ ) + sin θµ r ˆ θ · (cid:126) X ( x ν , r ) + O ( r − / ) , (4.46)where ˆ θ = ∂ θ ˆ r and sin θ ˆ φ = ∂ φ ˆ r . We do not specify the mode expansion of the subleadingterms in detail since they receive contributions from both φ ± ( j,m ) and λ ( j,m ) in a rather non-trivial way. In particular the φ − (2 ,m ) modes will contribute at the same order in 1 /r as the (cid:126) X (cid:48) nn piece of (cid:126) X . This does not lead to any ambiguities below, however, as they correspond toorthogonal harmonics. – 42 –sing these one finds that the conjugate momenta (4.42), (4.43) take the form P y = µ r (cid:26) − µ r ˆ r · (cid:126) X (cid:48) (nn) + 1 µ r [ a (n) r , ˆ r · (cid:126) X (n) ] + 1 µ r [ ˆ φ · (cid:126) X (n) , ˆ θ · (cid:126) X (n) ] + O ( r − / ) (cid:27) ≡ µ r ˆ r · (cid:16) (cid:126) X (nn) + O ( r − / ) (cid:17) , (4.47)and similarly P θ = − µ r ˆ φ · (cid:16) (cid:126) X (nn) + O ( r − / ) (cid:17) , P φ = µ r sin θ ˆ θ · (cid:16) (cid:126) X (nn) + O ( r − / ) (cid:17) , (4.48)where (cid:126) X (nn) := lim r →∞ (cid:18) µ r D r (cid:126) X −
12 [ (cid:126) X , × (cid:126) X ] (cid:19) = − µ (cid:126) X (cid:48) (nn) + [ a (n) r , (cid:126) X (n) ] −
12 [ (cid:126)X (n) , × (cid:126) X (n) ] . (4.49)The × notation refers to the Cartesian cross product on Euclidean R such that ([ (cid:126) X , × (cid:126) X ]) i =˜ (cid:15) ijk [ X j , X k ]. We have that X (nn) = − µ (cid:126) X (cid:48) (nn) plus non-abelian terms. X (nn) is indeed themomentum conjugate to (cid:126) X (n) in the sense that (cid:110) P y δ Φ y + P θ δA θ + P φ δA φ (cid:111) = (cid:126) X (nn) · δ (cid:126) X (n) + O ( r − / ) . (4.50)We see from (4.50) that in order for the variational principle to be well-defined, wemust either hold (cid:126) X (n) fixed, or set (cid:126) X (nn) = 0. This however is not consistent with theholographic interpretation. The modes (cid:126) X were identified with a triplet of conformal dimensionone operators in the dCFT in [4], and (cid:126) X (nn) is the source dual to these operators. The actionshould be extremized when (cid:126) X (nn) is held fixed. Hence, following [16], it is not the originalon-shell Yang–Mills action that provides a holographic description of the dCFT, but ratherits Legendre transform with respect to the pair ( (cid:126) X (n) , (cid:126) X (nn) ): S hol [ (cid:126) X (nn) , . . . ] := (cid:34) ( S ym ) o-s [ (cid:126) X (n) , . . . ] + 4 πg µ (cid:90) d x Tr (cid:110) (cid:126) X (n) · (cid:126) X (nn) (cid:111)(cid:35) (cid:126) X (n) = (cid:126) X (n) [ (cid:126) X (nn) ] , (4.51)where (cid:126) X (n) [ (cid:126) X (nn) ] is the (cid:126) X (n) that extremizes the quantity in square brackets. The additionof this term ensures that the variation of S hol is proportional to δ (cid:126) X (nn) and vanishes whenwe hold (cid:126) X (nn) fixed. In the next subsection we will show that the additional term is alsonecessary for the cancelation of boundary contributions to the supersymmetry variation. Avery similar situation is nicely analyzed in the recent works [17, 61].Finally, the normalizable modes of the fermion can be isolated by requiring Ψ − = O ( r − / ) and Ψ + = O ( r − / ). These conditions ensure that all normalizable modes of χ and η type are admissible while none of the non-normalizable ones are. In particular, we areidentifying the normalizable mode of the j = 1 / χ -type modes with the O ( r − / ) behavior of– 43 – − asymptotically. These modes have a vanishing 4D mass, and it would also be consistentwith the variational principle to identify the normalizable mode with an O ( r − / ) Ψ + compo-nent instead. We will see below that supersymmetry requires the identifications we have made.Nonetheless it is useful to turn on the non-normalizable modes for the massless fermions since,as we will see, they sit in a supermultiplet with some of the other non-normalizable modes.Let Ψ ( χ ) j =1 / := Ψ ( χ )( , ) + Ψ ( χ )( , − ) , (4.52)be the restriction of Ψ to the j = 1 / χ -type modes. ThenΨ + = Ψ ( χ ) , + j =1 / + O ( r − ) , Ψ − = Ψ ( χ ) , − j =1 / + O ( r − / ) , (4.53)and we show appendix D.1 that Ψ ( χ ) j =1 / takes the form(Ψ ( χ ) j =1 / ) S = h S ( θ, φ ) ψ + ( x µ , r ) + h S ( − θ, φ ) ψ − ( x µ , r ) , (4.54)with respect to a basis in which Γ r , Γ θ , Γ φ are constant, and h S is given by (3.8). Here ψ ± = ( ± Γ rθφy ) ψ , where ψ ( x µ , r ) is Majorana–Weyl. If we instead work in a natural basiswith respect to the Cartesian frame in which the Γ r i are constant, we have(Ψ ( χ ) j =1 / ) cart = ψ + ( x µ , r ) + (ˆ r · (cid:126) Γ ( r ) )Γ r ψ − ( x µ , r ) , (4.55)where we’ve introduced the notation (cid:126) Γ ( r ) := (Γ r , Γ r , Γ r ). The asymptotics of ψ ± are ψ + = 1( µr ) / ψ (nn)0 ( x µ ) + O ( r − / ) , ψ − = 1( µr ) / Γ r ψ (n)0 ( x µ ) + O ( r − / ) , (4.56)where ( ψ (nn)0 , ψ (n)0 ) encode the non-normalizable and normalizable modes of the j = 1 / χ -type doublet. They are 10D Majorana–Weyl spinors satisfying the same chirality andprojection conditions as ( ε , η ).The non-normalizable mode is to be held fixed so ( δ Ψ) + = O ( r − / ). Since Γ r anticom-mutes with Γ rθφy we then have12 ΨΓ r (cid:16) rθφy (cid:17) δ Ψ = Ψ − ( δ Ψ) + = O ( r − / ) , (4.57)which implies that the P Ψ δ Ψ term drops out of (4.4). The addition of the fermion boundaryaction (2.51) was crucial for this to work. – 44 –ummarizing, the field asymptotics are given by (4.40), (4.41), (4.46), and (4.56), whichwe collect here: A µ = a µ ( x ν , r ) + O ( r − / ) , A r = a r ( x ν , r ) + O ( r − / ) , Φ z i = φ z i ( x ν , r ) + O ( r − / ) ,A θ = − µ r ˆ φ · (cid:126) X ( x ν , r ) + O ( r − / ) ,A φ = P ± − cos θ ) + sin θµ r ˆ θ · (cid:126) X ( x ν , r ) + O ( r − / ) , Φ y = Φ y ∞ − P r + 1 µ r ˆ r · (cid:126) X ( x ν , r ) + O ( r − / ) , Ψ + = Ψ ( χ ) , + j =1 / + O ( r − ) , Ψ − = Ψ ( χ ) , − j =1 / + O ( r − / ) , (4.58)with a µ = a (nn) µ + 1 µr a (n) µ + O ( r − ) , a r = 1 µ r a (n) r + O ( r − ) ,φ z i = φ z i (nn) + 1 µr φ z i (n) + O (1 /r ) ,(cid:126) X ( x ν , r ) = (cid:126) X (n) + 1 µr (cid:126) X (cid:48) (nn) + O ( r − ) , Ψ ( χ ) j =1 / = 1( µr ) / (cid:16) ψ (nn)0 + (ˆ r · (cid:126) Γ ( r ) ) ψ (n)0 (cid:17) + O ( r − / ) . (4.59)The non-normalizable data ( a (nn) µ , φ z i (nn) , (cid:126) X (nn) , ψ (nn)0 ; Φ y ∞ ) and the ’t Hooft flux P are to beheld fixed while all remaining modes vary. This is consistent with the variational principlefor the Legendre transformed action, (4.51). Here (cid:126) X (nn) is related to (cid:126) X through (4.49). With the aid of the field asymptotics (4.58), (4.59) we can now complete the supersymme-try analysis. In section 3.3 we established that the variation of S ym with respect to thesupersymmetry transformations, (3.27), reduces to a set of boundary terms: δ ε S ym = − ig (cid:90) d x √− g ∇ a B a + δ ε S bndry , (4.60)where the boundary current B a is given in (3.29) and the boundary action in (3.26). We willassume the fields are sufficiently regular at r → r → ∞ boundary. Defining B bndry such that δ ε S bndry = − ig (cid:90) ∂M d x (cid:112) − g ( ∂ ) B bndry , (4.61)– 45 –e have δ ε S ym = − ig (cid:90) ∂M d x (cid:112) − g ( ∂ ) (cid:16) B r + B bndry (cid:17) . (4.62)Recall that the boundary measure is given in (2.34) and goes as r as r → ∞ .The large r behavior of B r + B bndry following from the field asymptotics (4.58) is analyzedin appendix D. It is useful to separate the contribution from the variation of the fermion fromthe rest. We eventually obtain the following expression: B r + B bndry == − µr ) ε ( x ν ) Tr (cid:26)(cid:18) f (nn) µν Γ µν + D (nn) µ φ z i (nn) Γ µz i + 12 [ φ z i (nn) , φ z j (nn) ]Γ z i z j (cid:19) ψ (n)0 (cid:27) ++ 1( µr ) Tr (cid:110)(cid:104)(cid:16) µη Γ y + ε ( x ν )Γ y (Γ µ D (nn) µ + Γ z i ad( φ z i (nn) )) (cid:17) ( (cid:126) Γ ( r ) · (cid:126) X (n) ) (cid:105) ψ (nn)0 (cid:111) + 12 Tr (cid:110) ΨΓ r (cid:16) + Γ rθφy (cid:17) δ ε Ψ (cid:111) + O ( r − / ) , (4.63)where f (nn) µν = 2 ∂ [ µ , a (nn) ν ] +[ a (nn) µ , a (nn) ν ] is the fieldstrength of the boundary gauge field, D (nn) µ = ∂ µ + ad( a (nn) µ ) is the corresponding covariant derivative, and the terms we neglected decaysufficiently fast so as not to contribute to (4.62). We also introduced the shorthand ε ( x µ ) := ε + µx ν η Γ ν . (4.64)The leading O ( r − / ) asymptotics of ( δ ε Ψ) + , which encode the supersymmetry variationof ψ (nn)0 , turn out to be( δ ε Ψ) + = 1( µr ) / δ ε ψ (nn)0 + O ( r − )= 1( µr ) / (cid:20) f (nn) µν Γ µν + D (nn) µ φ z i (nn) Γ µz i ++ 12 [ φ z i (nn) , φ z j (nn) ]Γ z i z j − Γ y (cid:126) Γ ( r ) · (cid:126) X (nn) (cid:21) ε ( x ν ) + O ( r − ) . (4.65)Three terms cancel when ( δ ε Ψ) + is plugged back into (4.63), but the (cid:126) X (nn) term remainsand gives an additional finite contribution to δ ε S ym , (4.62). All terms that contribute in the r → ∞ limit are independent of θ, φ so we can trivially integrate over the S , leading to δ ε S ym = 4 πig µ (cid:90) R , d x Tr (cid:26) ε ( x ν )Γ y ( (cid:126) Γ ( r ) · (cid:126) X (nn) ) ψ (n)0 + − (cid:104)(cid:16) µη Γ y + ε ( x ν )Γ y (Γ µ D (nn) µ + Γ z i ad( φ z i (nn) )) (cid:17) ( (cid:126) Γ ( r ) · (cid:126) X (n) ) (cid:105) ψ (nn)0 (cid:27) . (4.66)This is our final result for the supersymmetry variation of S ym . We see that supersymmetryinvariance of S ym can be achieved by taking, for example, (cid:126) X (nn) = ψ (nn)0 = 0.– 46 –owever it is the Legendre transformed action, (4.51), that is relevant for the holographicdual. We wish to show, therefore, that the supersymmetry variation of the (cid:126) X (n) · (cid:126) X (nn) termcancels (4.66). The variation of the boundary data is determined from the large r asymptoticsof the variations of the bosons in (3.27). In order to extract the necessary information, onemust describe the asymptotic behavior of the fermi field Ψ in some detail. Specifically, inorder to extract the supersymmetry variation of (cid:126) X (cid:48) (nn) in (cid:126) X (nn) , we will need to determinethe first subleading corrections in (4.56). This is done by solving the fermion equation ofmotion asymptotically, in terms of the boundary data. Note it is to be expected that theequations of motion must be used, as the Legendre transform in (4.51) takes place at the levelof the on-shell action, viewed as a functional of boundary data. The solution is obtained inappendix D.1 and takes the form ψ + = 1( µr ) / (cid:26) (cid:20) + 1 µ r (cid:16) ad( a (n) r ) − Γ y (cid:126) Γ ( r ) · ad( (cid:126) X (n) ) (cid:17)(cid:21) ψ (nn)0 ( x µ )+ − µ r (cid:104) Γ µ D (nn) µ + Γ z i ad( φ z i nn ) (cid:105) ψ (n)0 ( x µ ) (cid:27) + O ( r − / ) , ψ − = 1( µr ) / Γ r (cid:26) (cid:20) + 1 µ r (cid:16) ad( a (n) r ) + Γ y (cid:126) Γ ( r ) · ad( (cid:126) X (n) ) (cid:17)(cid:21) ψ (n)0 ( x µ )++ 1 µ r (cid:104) Γ µ D (nn) µ + Γ z i ad( φ z i nn ) (cid:105) ψ (nn)0 ( x µ ) (cid:27) + O ( r − / ) . (4.67)Equations (4.67) and (4.55) in conjunction with (3.21) can be straightforwardly used toobtain the supersymmetry variations of the bosonic boundary data. One simply comparesthe asymptotic expansions of the left- and right-hand sides of (3.27) order by order. Theresults are δ ε a (nn) µ = − iε ( x ν )Γ µ ψ (nn)0 ,δ ε φ z i (nn) = − iε ( x ν )Γ z i ψ (nn)0 , (4.68)and δ ε a (n) r = − iε ( x ν ) ψ (n)0 ,δ ε (cid:126) X (n) = − iε ( x ν )Γ y (cid:126) Γ ( r ) ψ (n)0 ,δ ε (cid:126) X (cid:48) (nn) = i (cid:104) η Γ y (cid:126) Γ ( r ) − µ − ε ( x ν )Γ y (cid:126) Γ ( r ) (cid:16) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:17)(cid:105) ψ (nn)0 + − iµ − ε ( x ν ) (cid:104) Γ y (cid:126) Γ ( r ) ad( a (n) r ) + Γ y (cid:126) Γ ( r ) × ad( (cid:126) X (n) ) − ad( (cid:126) X (n) ) (cid:105) ψ (n)0 . (4.69) One simply finds δ ε Φ y ∞ = 0 under our assumptions. It would be sourced by the non-normalizable modeof a massive fermi field on AdS . This is consistent with findings in [4], which identified the dual operator asthe lowest component in a different supermultiplet associated with a higher KK mode of the S expansion. – 47 –ith the aid of the last three and (4.49), one can show that the variation of (cid:126) X (nn) is sourcedby ψ (nn)0 only: δ ε (cid:126) X (nn) = − i (cid:104) µη Γ y (cid:126) Γ ( r ) − ε ( x ν )Γ y (cid:126) Γ ( r ) (cid:16) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:17)(cid:105) ψ (nn)0 . (4.70)Notice that { a (nn) µ , φ z i (nn) , Ψ , (cid:126) X (nn) } is a closed system under the supersymmetry trans-formations, (4.65), (4.68), and (4.70). This cements our identification of the non-normalizablemodes of the massless fermions. In fact these are the transformations of an off-shell 3D N = 4vector-multiplet, with (cid:126) X (nn) playing the role of the triplet of auxiliary fields. (See e.g. [62].)This is consistent with the supersymmetry discussion in [4] which identifies SU (2) z as the SU (2) V under which the triplet of scalars in an N = 4 vector-multiplet is charged. Theauxiliary fields of the vector-multiplet transform as a triplet of the other SU (2) r ≡ SU (2) H ,under which the scalars in an N = 4 hypermultiplet are charged. The non-normalizabledata { a (nn) µ , φ z i (nn) , Ψ , (cid:126) X (nn) } is a vector-multiplet of sources for the bottom KK multipletof relevant operators in the dCFT. Using these results, the variation of the (cid:126) X (n) · (cid:126) X (nn) term in (4.51) takes the form δ ε (cid:32) πg µ (cid:90) R , d x Tr (cid:110) (cid:126) X (nn) · (cid:126) X (n) (cid:111)(cid:33) == − πig µ (cid:90) R , d x Tr (cid:26) ε ( x ν )Γ y ( (cid:126) Γ ( r ) · (cid:126) X (nn) ) ψ (n)0 ++ (cid:104) µη Γ y ( (cid:126) Γ ( r ) · (cid:126) X (n) ) − ε ( x ν )Γ y ( (cid:126) Γ ( r ) · (cid:126) X (n) ) (cid:16) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:17)(cid:105) ψ (nn)0 (cid:27) . (4.71)Adding this to (4.66), one sees that the ψ (n)0 term cancels. Remarkably, the ψ (nn)0 termscombine into a total derivative, so that δ ε S hol = 4 πig µ (cid:90) R , d x ∂ µ Tr (cid:110) ε ( x ν )Γ µ Γ y ( (cid:126) Γ ( r ) · (cid:126) X (n) ) ψ (nn)0 (cid:111) . (4.72)In particular we used ∂ µ ε ( x ν )Γ µ = 3 µη . Hence δ ε S hol = 0 , (4.73)provided we assume sufficient fall-off conditions on ψ (nn)0 as we go to infinity in the Minkowskispace on the boundary. ABR thanks Dan Butter for an enlightening discussion on this point. See the last column in the table on page 32 of [4]. The map of notation for the modes is a µ → b µ , φ z i → ψ ,and (cid:126) X → ( b + z ) ( − ) . – 48 – A consistent truncation to N = 4 SYM on
AdS In this section we show that the six-dimensional Yang–Mills theory can be consistently trun-cated to a four-dimensional theory by keeping the modes ( a µ , a r , φ z i , (cid:126) X , ψ ). When z = 0 thisgives a consistent truncation of the six-dimensional theory on AdS × S to maximally super-symmetric N = 4 SYM on AdS . Turning on z yields a one-parameter family of consistenttruncations to Yang–Mills on asymptotically AdS spaces preserving half of the supersymme-try. The gauge group of the reduced theory is generated by the centralizer C ( P, Φ y ∞ ) ⊆ g . In the extreme cases this will be the full group if Φ y ∞ , P are vanishing, or a Cartan torus ifeither is generic.Having a consistent truncation means that every solution to the equations of motion ofthe lower-dimensional theory can be uplifted to a solution of the parent theory. In particular,the fields we want to keep must not source the modes we want to discard in the full nonlinearequations of motion.Consistent truncations of gauge theories on coset spaces are implicit in ansatze for instan-ton and monopole configurations that are based on spherical symmetry. See [63–66]. Theseideas were formalized and generalized in [67]. In this approach one identifies the action of theisometry group of the internal space with the action of gauge transformations on the fields,and as a result the gauge group of the truncated theory is reduced.The consistent truncation described here is different in that the gauge group need not bereduced. We can start with any simple gauge group in the parent theory and it need not bereduced at all in the truncated theory. This is despite the fact that some of the modes wekeep have nontrivial dependence on the two-sphere, namely the fermions and the triplet ofscalars parameterized by (cid:126) X . We do not have a clear conceptual understanding of why thistruncation works, but we observe that the Chern–Simons-like term in the 6D theory playsa crucial role. The 6D Lagrangian evaluated on the reduction ansatz would not be an S singlet without it, and the S dependence of the 6D equations of motion would not factorout. Recall this term originates from (2.10) and is present thanks to the Ramond-Ramondflux of the string background.The ansatz for the 6D degrees of freedom in terms of the 4D degrees of freedom is A µ,r = a µ,r ( x ν , r ) , Φ z i = φ z i ( x ν , r ) ,A θ = − µ r ˆ φ · (cid:126) X ( x ν , r ) , A φ = P ± − cos θ ) + sin θµ r ˆ θ · (cid:126) X ( x ν , r ) , Φ y = Φ y ∞ − P r + 1 µ r ˆ r · (cid:126) X ( x ν , r ) , Ψ = h S ( θ, φ ) ψ + + h S ( − θ, φ ) ψ − , (5.1) or more generally, the commutant of the vacuum monopole configuration we are expanding around. Seesection 4.1. – 49 –here ( a µ,r , φ z i , (cid:126) X , ψ ) are taken to commute with Φ y ∞ and P . Note we are using (4.54) forthe fermion ansatz. It will be much more convenient in this section to work in a natural basiswith respect to the S frame in which Γ r , Γ θ , Γ φ are constant.In the following we combine the AdS directions into a single notation, x ˆ µ = ( x µ , r ) , (5.2)with indices ˆ µ running over 0 , , , r , and lowered with the metric g ˆ µ ˆ ν d x ˆ µ d x ˆ ν = µ ( r + z ) η µν d x µ d x ν + d r µ ( r + z ) . (5.3)We also introduce a new transverse metric that will be used to contract the j = 1 tripletindices of X i : G h i h j := ( r + z ) µ r δ ij , (5.4)and collect the transverse metric and scalars as follows: φ I = ( φ z , φ z , φ z , X , X , X ) , G IJ = diag( G z i z j , G h i h j ) , (5.5)where I, J = 1 , . . .
6. The letter h is for ‘hypermultiplet’ in the new triplet of indices.Let us denote the right-hand sides of the 6D equations of motion (4.3) by EOM a , EOM y ,EOM z i , and EOM Ψ respectively. We insert A = a + δA , etc . into these equations, where δA collectively represents all remaining degrees of freedom that we wish to discard. Asan intermediate step, in appendix E one can find expressions for the components of thefieldstrength and covariant derivatives on the ansatz 5.1. Some tedious but straightforwardcomputations lead toEOM ˆ µ = eom ˆ µ + O ( δA ) , EOM z i = eom z i + O ( δA ) , EOM θ = − µ r ˆ φ · −−→ eom ( h ) + O ( δA ) , EOM φ = µ r sin θ ˆ θ · −−→ eom ( h ) + O ( δA ) , EOM y = µ r ˆ r · −−→ eom ( h ) + O ( δA ) , EOM Ψ = h S ( θ, φ )eom + ψ + h S ( − θ, φ )eom − ψ + O ( δA ) , (5.6)where −−→ eom ( h ) = (eom h , eom h , eom h ), and all of these eom’s are S -independent quantities The Φ y ∞ and P terms in ( A r i , Φ y ) could be generalized to any of the monopole vacua described in section4.1 provided we restrict ( a µ,r , φ z i , (cid:126) X , ψ ) to the commutant of the vacuum configuration. – 50 –iven byeom ˆ ν := 1 r D ˆ µ ( r F ˆ µ ˆ ν ) − G IJ [ φ I , D ˆ ν φ J ] + i ψ , Γ ˆ ν ψ ] , eom z i := 1 r D ˆ µ ( r G z i z j D ˆ µ φ z j ) + G z i z j G IJ [ φ I , [ φ J , φ z j ]] + i ψ , Γ z i ψ ] , eom h i := 1 r D ˆ µ ( r G h i h j D ˆ µ X j ) + G h i h j G IJ [ φ I , [ φ J , X j ]] − m ψ (cid:15) h i h j h k [ X j , X k ] + i ψ , Γ h i ψ ] , eom ψ := (cid:16) Γ ˆ µ D ˆ µ + m ψ Γ h h h (cid:17) ψ + Γ I [ φ I , ψ ] . (5.7)Some details of the derivation of the fermion equation can be found in appendix D.1. F and D are to be understood as the fieldstrength and covariant derivative associated with a . The( r -dependent) fermion mass is m ψ := − µz r ( r + z ) / , (5.8)and an orthonormal frame is employed along the new h i directions such that (cid:15) h i h j h k = ( r + z ) / µ r ˜ (cid:15) ijk , Γ h i = ( r + z ) / µr Γ h i , (5.9)with ˜ (cid:15) = 1 as usual. The new triplet of gamma matrices is related to the old one by (cid:126) Γ ( h ) := (Γ h , Γ h , Γ h ) := (Γ φ , − Γ θ , Γ y ) . (5.10)When acting on a spinor of definite Γ rθφy -chirality, this triplet is related to the (cid:126) Γ ( r ) thathave appeared before: (cid:126) Γ ( h ) ψ ± = ( ± Γ rθy , ± Γ rφy , Γ r Γ ry ) ψ ± = Γ y (cid:40) Γ r (Γ θ , Γ φ , Γ r ) ψ + (Γ θ , Γ φ , Γ r )Γ r ψ − = Γ y (cid:40) ( U Γ r (cid:126) Γ ( r ) U − ) ψ + ( U (cid:126) Γ ( r ) Γ r U − ) ψ − , (5.11)where U = h S ( θ, φ ) − is the unitary transformation sending (Γ θ , Γ φ , Γ r ) to (cid:126) Γ ( r ) . (See (B.12).)For the consistent truncation, it is more natural, however, to use (cid:126) Γ ( h ) and the S -based framebecause this makes it clear that the directions associated with X i can be viewed as internaland independent of the radial direction r of the four-dimensional spacetime. Note also thatΓ h h h = Γ θφy is the same combination that appears in the 6D fermion mass term, (4.3).Since the eom in (5.7) are S -independent, the 6D equations of motion restricted to thetruncation ansatz, (5.1), we haveEOM (cid:12)(cid:12)(cid:12)(cid:12) δA =0 = 0 ⇐⇒ eom = 0 . (5.12)– 51 –he truncation will be consistent iff the equations of motion of the reduced action are equiv-alent to eom = 0.We insert the truncation ansatz into the 6D Yang–Mills action in the form (4.1), with(4.2) and (2.51). After some effort one finds that the density (4.2) can be put in the form L (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) δA =0 = − Tr (cid:26) F ˆ µ ˆ ν F ˆ µ ˆ ν + 12 G IJ D ˆ µ φ I D ˆ µ φ J + 14 G IJ G KL [ φ I , φ K ][ φ J , φ L ]+ − ( r + z ) µ r ˜ (cid:15) ijk ( D r X i )[ X j , X k ]++ i ψ (cid:16) Γ ˆ µ D ˆ µ + m ψ Γ h h h (cid:17) ψ + i ψ Γ I [ φ I , ψ ] (cid:27) . (5.13)The 6D measure can be expressed in terms of the 4D measure associated with (5.3), at theprice of introducing an r -dependent 4D Yang–Mills coupling. Since (5.13) is an S -invariantwe can carry out the integral over S as well:1 g (cid:90) d x √− g = 1 g (cid:90) d x d r dΩ r → (cid:90) d x √− g g ym ( r ) , (5.14)where g = det( g ˆ µ ˆ ν ) is the determinant of (5.3) and g ym ( r ) = g µ ( r + z )4 πr = π / √ g s N c N c · ( r + z ) r . (5.15)Meanwhile the boundary action (2.51) at r = r b reduces to S bndry f (cid:12)(cid:12)(cid:12)(cid:12) δA =0 = − i (cid:90) d x (cid:112) − g ( ∂ g ym ( r b ) Tr (cid:110) ψ Γ h h h ψ (cid:111) , (5.16)where (cid:112) − g ( ∂ = µ ( r b + z ) / is the induced measure. However the asymptotics of ψ implythat the leading r b → ∞ behavior is finite and hence the z ’s in the measure and couplingcan be dropped in the limit.The ˜ (cid:15) ijk ( D r X i )[ X j , X j ] term in (5.13) descends from the Chern–Simons-like term in the6D action. The ad( a r ) part of D r actually drops out of this term by the Jacobi identity. Wethen integrate by parts. Keeping in mind the factor of r out in front, (5.14), the overallprefactor is asymptotically constant, or exactly constant if z = 0. Hence we get a boundaryterm and a bulk term that vanishes when z → r → ∞ . The bulk term can be expressedin terms of m ψ . The boundary term is finite in the r b → ∞ limit and adds to the boundaryterm we already have, (5.16). We write the result for the truncated action as follows: S trnc := S ym (cid:12)(cid:12)(cid:12)(cid:12) δA =0 = (cid:90) d x √− g g ym ( r ) L (cid:48) trnc + S (cid:48) bndrytrnc , (5.17)– 52 –ith L (cid:48) trnc := − Tr (cid:26) F ˆ µ ˆ ν F ˆ µ ˆ ν + 12 G IJ D ˆ µ φ I D ˆ µ φ J + 14 G IJ G KL [ φ I , φ K ][ φ J , φ L ]++ 2 m ψ (cid:15) h i h j h k X i [ X j , X k ] + i ψ (cid:16) Γ ˆ µ D ˆ µ + m ψ Γ h h h (cid:17) ψ + i ψ Γ I [ φ I , ψ ] (cid:27) , (5.18)and S (cid:48) bndrytrnc = 1 g (cid:90) d x (cid:112) − g ( ∂ Tr (cid:26) (cid:15) h i h j h k X i [ X j , X k ] − i ψ Γ h h h ψ (cid:27) , (5.19)where g denotes the r → ∞ limit of (5.15). The equations of motion one derives from(5.17) are precisely the vanishing of the eom’s, (5.7): δS trnc = 0 ⇐⇒ eom = 0 . (5.20)Hence the truncation (5.1) is consistent.We can also express the truncated action in terms of scalars with canonical kinetic termsby setting φ z i = µ ( r + z ) / φ z i , X i = µr ( r + z ) / X i . (5.21)This introduces both mass terms and additional boundary terms. The latter are linearlydivergent and necessary to cancel divergences in the variational principle, energy, etc , if weuse this form of the action. The result is S trnc = (cid:90) d x √− g g ym ( r ) L trnc + S bndrytrnc , (5.22)with L trnc := − Tr (cid:26) F ˆ µ ˆ ν F ˆ µ ˆ ν + 12 D ˆ µ φ I D ˆ µ φ I + 12 (cid:0) m z ( φ z i ) + m X ( X i ) (cid:1) + 14 [ φ I , φ J ] ++ 4 m ψ X [ X , X ] + i ψ (cid:16) Γ ˆ µ D ˆ µ + m ψ Γ h h h (cid:17) ψ + i ψ Γ I [ φ I , ψ ] (cid:27) , (5.23)and S bndrytrnc = 1 g (cid:90) d x (cid:112) − g ( ∂ Tr (cid:26) X [ X , X ] − µ φ I ) − i ψ Γ h h h ψ (cid:27) , (5.24)where the new r -dependent masses are m z ( r ) := − µ (cid:18) r + 3 z r + z (cid:19) , m X ( r ) := − µ (cid:18) r + r z − z r ( r + z ) (cid:19) . (5.25)– 53 –inally we must take into account the Legendre transform, (4.51), which involves onlythe degrees of freedom we are keeping in the truncation. Thus we define S holtrnc := S trnc + 1 g (cid:90) d x Tr (cid:110) (cid:126) X (nn) · (cid:126) X (n) (cid:111) . (5.26)Now recall that X i (n) is the boundary value of X i , while X i (nn) can be expressed in terms ofthe boundary values of X i and D r X i via (4.49). Taking into account the rescaling (5.21) wecan write (cid:126) X (nn) · (cid:126) X (n) = lim r →∞ (cid:0)(cid:112) − g ( ∂ Tr (cid:8) µr X i D r X i + µ ( X i ) − X [ X , X ] (cid:9)(cid:1) , (5.27)and hence S holtrnc = (cid:90) d x √− g g ym ( r ) L trnc + S hol-bndrytrnc , (5.28)where the boundary action is S hol-bndrytrnc = 1 g (cid:90) d x (cid:112) − g ( ∂ Tr (cid:26) µr X i D r X i + µ X i ) − µ φ z i ) + − X [ X , X ] − i ψ Γ h h h ψ (cid:27) . (5.29)We have expressed the result in this fashion because (5.29) is precisely the set of boundaryterms obtained recently in [17]. These authors carried out a supersymmetric localizationcomputation in maximally supersymmetric Yang–Mills on AdS , and found that this set ofterms is necessary and sufficient for the preservation of supersymmetry in the presence ofnon-normalizable mode boundary data. We have arrived at the same set of terms via anindependent analysis.One can check that the supersymmetry variations close on the truncation (5.1), so thereduced theory enjoys the same amount of supersymmetry as the parent theory. Indeed,when z = 0 we see that the background is AdS , the Yang–Mills coupling is constant, m ψ vanishes, the bosonic masses go to the conformally coupled value, m = − µ , and the bulkLagrangian takes the N = 4 maximally supersymmetric formlim z → L trnc = − Tr (cid:26) F ˆ µ ˆ ν F ˆ µ ˆ ν + 12 D ˆ µ φ I D ˆ µ φ I − µ ( φ I ) + 14 [ φ I , φ J ] ++ i ψ (cid:16) Γ µ D µ + Γ I ad( φ I ) (cid:17) ψ (cid:27) . (5.30)Turning on z gives a deformation that preserves half of the supersymmetries—namely thosesatisfying the 3D N = 4 super-Poincar´e algebra.We make two further comments on this result before concluding the section. First, thediscussion in section 2.4 shows that the on-shell six-dimensional action, S hol , viewed as a– 54 –unctional of the non-normalizable modes ( a (nn) µ , φ z i (nn) , ψ (nn)0 , (cid:126) X (nn) ), is the leading approxi-mation to the generating functional of correlators amongst the lowest KK multiplet defectoperators in the dual, in the regime g s N c (cid:29) N c (cid:29) N f (cid:28) N c / √ g s N c . In principle itis the generating functional for the full tower of KK multiplets of defect operators, but onewould have to include the additional boundary terms required by holographic renormalizationfor the higher multiplets. And more precisely, it is the generating functional for those oper-ators transforming in the su ( N f ) subalgebra of the u ( N f ) flavor symmetry of the dual. Theresults of this section imply that we can just as well use S holtrnc as the leading-order generatingfunctional in this regime. The fact that the truncation is consistent implies that the effectsof the higher KK multiplets on the lowest multiplet can only enter through loops.Second, recall that, on the one hand, that ten-dimensional type IIB supergravity has aconsistent truncation on S to the five-dimensional maximally supersymmetric gauged super-gravity of [68, 69]. On the other hand, we have just shown that the low-energy effectivetheory on the D5-branes has a consistent truncation in the regime where the coupling togravity is suppressed. Although we will not provide any evidence here, it seems natural toconjecture that the coupled IIB supergravity plus D5-brane system has a consistent trunca-tion to maximal 5D gauged supergravity with a half-BPS codimension one defect hosting a U ( N f ) super-Yang–Mills theory. Specifically, we suggest that the bosonic couplings of thedefect theory to gravity should be obtained by applying the combined reduction ansatze forthe supergravity and D5-brane modes to the non-abelian Myers action. One should expandthe Myers action to second order in the open string field variables, (2.20), but keep all ordersin closed string fluctuations. We now turn to a preliminary study of BPS field configurations in both the 6D Yang–Millstheory and its 4D truncation. These are solutions to the equations of motion that preservesome supersymmetry. We are especially interested in finite energy configurations that can beinterpreted as solitons in
AdS , and that can serve as the starting point for a description ofBPS particle states in the corresponding quantum (string) theory. We set the fermion to zero and derive a system of first order equations for the bosonic fieldsby demanding that the fermion’s supersymmetry variation (parameterized by ε ) vanish as This had long been suspected since the work of [70] on eleven-dimensional supergravity on S , and aftera series of partial results in this direction it was recently demonstrated for type IIB on S in full generality.See [71] for the complete nonlinear reduction ansatze, and for further discussion and references. In the language of appendix A, one makes the open string expansion as in (A.12) through (A.14) and(A.26) through (A.28), and evaluates the result on the reduction ansatze. – 55 –ell: δ ε Ψ = (cid:20) F ab Γ ab + D a Φ m Γ am + 12 [Φ m , Φ n ]Γ mn + M Ψ Γ θφy Γ m Φ m (cid:21) ε ! = 0 . (6.1)What conditions should we impose on ε in order to satisfy this requirement? One certainlyexpects that superconformal symmetries are broken in the presence of BPS states, and we canargue for this as follows. If we expand δ ε Ψ near r → ∞ , then we already know the leadingorder equation—namely that δ ε Ψ , (4.65), should vanish: (cid:20) f (nn) µν Γ µν + D (nn) µ φ z i (nn) Γ µz i + 12 [ φ z i (nn) , φ z j (nn) ]Γ z i z j − Γ y (cid:126) Γ ( r ) · (cid:126) X (nn) (cid:21) (cid:0) ε − ( µx ν )Γ ν η (cid:1) ! = 0 . (6.2)We know from (4.20) that having a covariantly constant φ z i (nn) = Φ z i ∞ breaks superconformalsymmetry. Then we see from (6.2) that having any other nontrivial boundary data will requirea condition on ε ( x ν ) = ε − ( µx ν )Γ ν η of the form Γ M M ε + = ± Γ M M ε + where the M ’s aredifferent and at least one of them is 0 ,
1, or 2. This condition should hold for all values of x µ and leads to incompatible projection conditions on η . Therefore we set η = 0.With η set to zero, ε becomes an eigenspinor of Γ rθφy = Γ r r r y . Then one can use theprevious observation, (4.22), to simplify (6.1).Now, ε has eight real independent parameters corresponding to the 3D N = 4 Poincar´esupersymmetry of the boundary theory. The most general BPS (particle) states in thistheory are 1 / ε beyond Γ r r r y ε = ε , such that all three projectionsare mutually compatible and such that the SO (2) little group of the 3D Lorentz group ispreserved. There is an S × S family of choices parameterized by two fixed unit vectorsˆ n ( r ) ∈ R r ) and ˆ n ( z ) ∈ R z ) , given byΓ ε = 12 (cid:15) ijk ˆ n i ( r ) Γ r j r k ε = 12 (cid:15) ijk ˆ n i ( z ) Γ z j z k ε . (6.3)We need to keep track of the parameters ˆ n ( r ) , ˆ n ( z ) . The Bogomolny bound we derive belowwill depend on them, and they must be allowed to vary to achieve the strongest possiblebound. However there is no need to carry them around explicitly: we can always choose our { r i } and { z i } axes so that the ˆ n point in the respective ‘3’ directions, and then restore thedependence on them at the end using covariance. Working in such a basis for now, the fullset of projections satisfied by ε areΓ r r r y ε = ε , Γ r y ε = ε , Γ r r ε = − ε , Γ z z ε = − ε , Γ r r z z = − ε , Γ r z z y ε = ε , Γ z ε = ε . (6.4)With the aid of (6.4) and (4.22) we collect terms in δ ε Ψ that are proportional to equivalentΓ M M ε structures. Setting the coefficient of each linearly independent structure to zero– 56 –ields the following system of 22 BPS equations: F + [Φ z , Φ z ] + µ ( r + z ) ( F r r − D r Φ y ) = 0 ,F r + D Φ y = 0 , F r − D Φ y = 0 ,F r r + D r Φ y = 0 , F r r − D r Φ y = 0 ,F r − F r = 0 , F r + F r = 0 ,D Φ z − D Φ z = 0 , D Φ z + D Φ z = 0 ,D r Φ z − D r Φ z = 0 , D r Φ z + D r Φ z = 0 ,D r Φ z + [Φ y , Φ z ] = 0 , D r Φ z − [Φ y , Φ z ] = 0 , (6.5)and F p − D p Φ z = 0 , F r i − D r i Φ z = 0 , D Φ z p − [Φ z , Φ z p ] = 0 ,D Φ y − [Φ z , Φ y ] = 0 , D Φ z = 0 , (6.6)where we recall that the indices p, q = 1 ,
2. We refer to the first set of equations, (6.5), as the primary or magnetic system of BPS equations and the second set, (6.6), as the secondary or electric system of BPS equations. This is due to their close analogy with BPS equations for4D N = 4 , e.g. [72, 73].The primary equations do not involve A , Φ z and can be solved independently of thesefields. By working in generalized temporal gauge , defined by A = Φ z , (6.7)one sees that the secondary equations reduce to time-independence for the fields participatingin the primary equations: ∂ ( A p , A r i , Φ z i ) = 0 . (6.8)This in particular applies to the boundary data, ( a (nn) p , (cid:126) X (nn) , φ z p (nn) , Φ y ∞ ; P ).Given a solution to the primary system, the secondary BPS equations are satisfied with(6.7). We clearly need one further equation to specify an independent solution for A or Φ z .This can be taken to be the Gauss Law constraint, (4.9). Using (6.6), this equation takes theform of a gauge-covariant Laplacian, constructed from the solution ( A p , A r i , Φ z p , Φ y ) of (6.5),annihilating Φ z .One nice way to repackage the primary BPS system introduces complex covariant deriva-tives, D := ∂ + i∂ + ad( A + iA ) , D := ∂ r + ad( A r − i Φ y ) , D := ∂ r + i∂ r + ad( A r + iA r ) , D := ad(Φ z + i Φ z ) . (6.9)Then the last twelve of (6.5) are equivalent to the six complex equations[ D p , D q ] = 0 , (6.10)– 57 –or p , q = 1 , . . . ,
4. The first equation can be written in the form[ D , D † ] + [ D , D † ] + µ ( r + z ) (cid:16) [ D , D † ] + [ D , D † ] (cid:17) = 0 . (6.11)The advantage of this approach, which applies to many of the standard self-duality typeequations, is that a subset of the equations—(6.10) in this case—have an extended gaugeinvariance. They are invariant under gauge transformations of the complexified gauge group, G C . This can be a powerful tool, both for studying the space of solutions and for constructingmodel solutions— e.g. Donaldson’s approach to monopoles via rational maps [74]. The resultsof such analyses, of course, depend heavily on the boundary conditions, and we have describeda class of boundary conditions that are natural from the holographic perspective in (4.58).However we prefer to leave a detailed analysis of these issues to future work.The primary system (6.5) also has the structure of a generalized self-duality equation ofthe type introduced in [18, 75]. Indeed these equations were studied from a supersymmetrypoint of view in [76], and we recognize our system as a curved space version of the eight-dimensional ‘ -BPS’ case given in their equation (53). Let us view( ˆ A z , ˆ A z , ˆ A y ) := (Φ z , Φ z , Φ y ) , (6.12)as the remaining three components of a gauge field ˆ A A = ( A p , A r i , ˆ A z p , ˆ A y ) on the Riemannianeight-manifold with metricdˆ s := ˆ g AB dˆ x A dˆ x B := µ ( r + z ) δ pq ( d x p d x p + dˆ z p dˆ z q ) + 1 µ ( r + z ) ( δ ij d r i d r j + dˆ y ) . (6.13)Here A, B = 1 , . . . ,
8, and we use hatted coordinates (ˆ z p , ˆ y ) to emphasize that these are notcoordinates of the original 10D spacetime. Note that ˆ g ˆ z p ˆ z q = G z p z q and ˆ g ˆ y ˆ y = G yy . Now introduce the four-form ω := µ ( r + z ) d x d x dˆ z dˆ z + 1 µ ( r + z ) dˆ y d r d r d r ++ (cid:0) d x d x + dˆ z dˆ z (cid:1) ∧ ( dˆ y d r + d r d r ) , (6.14)and observe that it is anti-self-dual with respect to (6.13):ˆ (cid:63) ω = − ω . (6.15)Then one can check that the primary BPS system (6.5) is equivalent toˆ (cid:63) ˆ F = ω ∧ ˆ F , (6.16) These relations suggest an interpretation in terms of T-duality, but we will not explore that possibilityhere. – 58 –estricted to configurations that are translation-invariant with respect to ˆ z , ˆ z and ˆ y . Takingthe dual of (6.16) we find F = i F (ˆ (cid:63)ω ) = − i F ω . The first step is valid generally, and in thesecond step we used that ω and anti-self-dual. In components, this result takes the form F AB = − ( ω ) ABCD F CD , which is the form given in [18, 76]. This equation is consistentthanks to the identity( ω ) A A AB ( ω ) A A AB = 4( ω ) A A A A + 6( δ A A δ A A − δ A A δ A A ) . (6.17)The secondary BPS equations can also be written in terms of (6.12) and (6.13). Letˆ E A := ˆ F A be the electric field vector associated with the gauge field ( A , ˆ A ) on the 9Dspacetime with metric g d t + dˆ s . Then the equations (6.6) are equivalent toˆ E A − D A Φ z = 0 , D Φ z = 0 . (6.18)We will discuss some of the mathematical background of these generalized self-dualityequations further in subsection 6.3 below, after introducing one additional generalization. Inthe next subsection we will see how the repackaging (6.16) is useful for obtaining a Bogomolnybound on the energy functional.A generic solution to (6.5) and (6.6) will preserve two supersymmetries, but specialtypes of solutions can preserve additional supersymmetry. For example, one can repeat thesupersymmetry analysis imposing only the first projection in (6.3). This leads to the systemof equations F + µ ( r + z ) ( F r r − D r Φ y ) = 0 ,F r + D Φ y = 0 , F r − D Φ y = 0 ,F r r + D r Φ y = 0 , F r r − D r Φ y = 0 ,F r − F r = 0 , F r + F r = 0 ,D p Φ z q = D r i Φ z q = [Φ y , Φ z q ] = [Φ z , Φ z ] = 0 , (6.19)together with the conditions that F p = F r i = D Φ m = 0 , D p Φ z = D r i Φ z = [Φ m , Φ z ] = 0 . (6.20)In other words the electric field vanishes and the Φ z i are covariantly constant, mutuallycommuting, and commuting with Φ y . Field configurations satisfying (6.19) and (6.20) alsosolve (6.5) and (6.6), but preserve twice as many supersymmetries— i.e. four supercharges.We can also write (6.19) in the form (6.16) with ω → ˜ ω , given by˜ ω := µ ( r + z ) d x d x dˆ z dˆ z + dˆ z dˆ z ( dˆ y d r + d r d r ) . (6.21)This four-form is neither self-dual nor anti-self-dual. This, however, is not required. All thatis required for consistency of (6.16) is that − ˆ (cid:63) ˜ ω satisfies (6.17), and one can check that itdoes. We note that ω and ˜ ω are related by ω = (1 − ˆ (cid:63) )˜ ω .– 59 – .2 Bogomolny bound on the energy Recall that the Yang–Mills energy functional takes the form (4.11), provided the Gauss Lawconstraints (4.9) and (4.10) hold. Now that we have a handle on the field asympototics, letus take a closer look at the boundary constraint, (4.10).First we explain why the r = 0 term can always be dropped in such boundary integrals.Any constant time slice of the asymptotically AdS × S spacetime is asymptotically H × S ,where H is hyperbolic three-space. Thus the boundary we are integrating over is ∂H × S .The boundary of (the conformal compactification of) H is a two-sphere, viewed as the one-point compactification R ∪ {∞} . The R × S part of ∂H × S is reached by sending r → ∞ for any fixed x p . When z = 0, the {∞} × S part of ∂H × S can be reached by eithersending x p → ∞ for any fixed r (including r = ∞ ), or by sending r → x p .(See, for example, Figure 2.8 in the review [77].) In contrast, when z (cid:54) = 0, the points at r = 0 with any finite x p are regular points in the interior, and {∞} × S is only reached bysending x p → ∞ (for any fixed r ). The reason is that, when z (cid:54) = 0, the two-sphere shrinksto zero size and the space smoothly caps off as r → r = 0 with the boundary ∂H × S is merelyproviding the set of points {∞} × S that compactifies R × S . Therefore r = 0 makes nocontribution to boundary integrals provided the integral over R × S at r → ∞ is finite. Wewill always impose asymptotic conditions on boundary data, as we go to infinity in the R parameterized by x p , such that this integral is finite.Therefore in analyzing (4.10) we can set (cid:90) d x (cid:2) ∂ r (cid:0) √− g Tr { E r A } (cid:1)(cid:3) = lim r →∞ (cid:90) R × S d x dΩ r Tr { F r A } . (6.22)The field asymptotics in (4.58), (4.59) allow for F r = O ( r − ), and thus the boundary Gaussconstraint requires us to set the non-normalizable S singlet mode of A to zero. Thiscondition is not compatible with generalized temporal gauge, (6.7), when φ z (nn) (cid:54) = 0, but it canbe easily accommodated by making a time-dependent gauge transformation that eliminates a (nn)0 . This is completely analogous to going from the Julia–Zee form of the dyon solution in4D Yang–Mills–Higgs theory [78] to the Gibbons–Manton form [79], in which the dyon fieldconfiguration takes the form of the monopole field configuration dressed with a simple timedependence that generates the requisite electric field.Hence we can assume that the Gauss constraints hold, and therefore the energy functionalis given by (4.11). We make use of the notation (6.12), (6.13), to write the latter as H bosym = 1 g (cid:90) d ˆ x (cid:112) ˆ g Tr (cid:26) − g ( ˆ E A ˆ E A + D A Φ z D A Φ z ) + 12 ( g ) ( D Φ z ) ++ 14 ˆ F AB ˆ F AB − µ ( r + z ) ( ˆ F ∧ ˆ F ) r r r y (cid:27) . (6.23)– 60 –ince nothing depends on (ˆ z p , ˆ y ) the integral over these directions is trivial. We take themto be periodic with periodicity one so that (6.23) reproduces (4.11). The first line gives thecontribution from K and the second line gives the contribution from V . We’ve also used that G z z = − g .Now consider the quantity (cid:12)(cid:12)(cid:12) ˆ (cid:63) ˆ F − ω ∧ ˆ F (cid:12)(cid:12)(cid:12) := (ˆ (cid:63) ˆ F − ω ∧ ˆ F ) ∧ ˆ (cid:63) (ˆ (cid:63) ˆ F − ω ∧ ˆ F )= (cid:12)(cid:12)(cid:12) ˆ F (cid:12)(cid:12)(cid:12) − ω ∧ ˆ F ∧ ˆ F + (cid:12)(cid:12) i ˆ F ω (cid:12)(cid:12) = 2 ˆ F AB ˆ F AB d ˆ x − ω ∧ ˆ F ∧ ˆ F , (6.24)where we used (6.17). Similarly, with ˆ E = ˆ E A dˆ x A and D Φ z = ( D A Φ z ) dˆ x A , we have (cid:12)(cid:12)(cid:12) ˆ E − D Φ z (cid:12)(cid:12)(cid:12) = (cid:16) ˆ E A ˆ E A + D A Φ z D A Φ z (cid:17) d ˆ x − D Φ z ∧ ˆ (cid:63) ˆ E . (6.25)Therefore H bosym = 1 g (cid:90) Tr (cid:26) (cid:12)(cid:12)(cid:12) ˆ (cid:63) ˆ F − ω ∧ ˆ F (cid:12)(cid:12)(cid:12) − g (cid:12)(cid:12)(cid:12) ˆ E − D Φ z (cid:12)(cid:12)(cid:12) + 12 ( g ) | D Φ z | ++ 12 (cid:0) ω − µ ( r + z ) d x d x dˆ z dˆ z (cid:1) ∧ ˆ F ∧ ˆ F − g D Φ z ∧ ˆ (cid:63) ˆ E (cid:27) . (6.26)The last line of (6.26) is in fact a boundary term. While ω is not closed, the shiftedfour-form ω (cid:48) := ω − µ ( r + z ) d x d x dˆ z dˆ z = 1 µ ( r + z ) dˆ y d r d r d r + (cid:0) d x d x + dˆ z dˆ z (cid:1) ∧ ( d r d r + d y d r ) , (6.27)clearly is. Since Tr ( ˆ F ∧ ˆ F ) = d ω CS ( ˆ A ), we have ω (cid:48) ∧ Tr ( ˆ F ∧ ˆ F ) = d( ω (cid:48) ∧ ω CS ( ˆ A )), wherethe Chern–Simons three-form is ω CS ( ˆ A ) := Tr (cid:16) ˆ F ∧ ˆ A − ˆ A ∧ ˆ A ∧ ˆ A (cid:17) . (6.28)Furthermore the last term is a total derivative by the local Gauss constraint (4.9).Hence we have brought the Hamiltonian to Bogomolny form: H bosym = 1 g (cid:90) Tr (cid:26) (cid:12)(cid:12)(cid:12) ˆ (cid:63) ˆ F − ω ∧ ˆ F (cid:12)(cid:12)(cid:12) + 12 ( − g ) (cid:12)(cid:12)(cid:12) ˆ E − D Φ z (cid:12)(cid:12)(cid:12) + 12 ( g ) | D Φ z | (cid:27) ++ 1 g (cid:90) d (cid:18) ω (cid:48) ∧ ω CS ( ˆ A ) − g Tr { Φ z ∧ ˆ (cid:63) ˆ E } (cid:19) . (6.29)– 61 –he first line is a sum of squares with positive coefficients, so we can immediately infer thebound H bosym ≥ g (cid:90) ∂ ˆ M (cid:18) ω (cid:48) ∧ ω CS ( ˆ A ) − g Tr { Φ z ∧ ˆ (cid:63) ˆ E } (cid:19) , (6.30)which is saturated on field configurations satisfying the first order equations (6.16), (6.18):ˆ (cid:63) ˆ F = ω ∧ ˆ F , ˆ E A − D A Φ z = 0 , D Φ z = 0 . (6.31)Finally, on a solution to these equations the local Gauss constraint takes the form D A (cid:0) G z z D A Φ z (cid:1) = 0 . (6.32)In appendix F we evaluate the BPS energy (6.30) in terms of the field asymptotics (4.58),(4.59). The results are summarized here. The magnetic energy (the ω (cid:48) ∧ ω CS term) receivesto types of contribution in general. First, there is a contribution proportional to the vev Φ y ∞ .It has the form of a standard monopole mass term, where the relevant magnetic charge isexpressed in terms the magnetic flux through the R boundary associated to one of the j = 1triplet modes of gauge fields. Let us introduce a vector notation for this triplet, analogous to(4.44), such that ˆ r · (cid:126)a µ ( x ν , r ) := − √ m (cid:88) m = − a µ, (1 ,m ) ( x ν , r ) Y m ( θ, φ ) . (6.33)The mode analysis determined that the leading behavior of this triplet is (cid:126)a µ ( x ν , r ) = 1 µ r (cid:126)a (n) µ ( x ν ) + O ( r − ) , (6.34)where (cid:126)a (n) µ commutes with the vev Φ y ∞ and the ’t Hooft charge P . Then we define a tripletof magnetic fluxes in terms of these: (cid:126)γ m := 12 π (cid:90) R (cid:126)f (n) , (cid:126)f (n) µν := ∂ µ (cid:126)a (n) ν − ∂ ν (cid:126)a (n) µ . (6.35)The first contribution to the magnetic energy is proportional to Tr { Φ y ∞ γ } . The fact thatthe third component of the flux vector is picked out can be traced back to our choice for theunit vector ˆ n ( r ) in the supersymmetry projection (6.3) to be along the three-direction in (cid:126)r space.The second type of magnetic contribution is proportional to X , the third componentof the triplet (cid:126) X (n) . In fact there are two types of terms—one that depends on the local valueof X and is integrated over R , and a line integral around the circle at infinity, i.e. theboundary of the boundary, that hence only depends on the asymptotic value of X . In bothof these terms, X is traced against quantities constructed from the non-normalizable S singlet modes { a (nn) µ , φ z i (nn) } . – 62 –hen there is the contribution from the electric energy term. This can be expressed interms of the non-normalizable S singlet φ z (nn) , and the radial component of the normalizable S singlet electric field. Note that the z component is picked out in (6.30) because of ourchoice of unit vector ˆ n ( z ) . Only the coefficient of the leading O (1 /r ) component of the electricfield contributes. We define this coefficient by f (n) r ( x µ ) := lim r →∞ ( µ r F r ) . (6.36)On a solution to the BPS equations one can use F r = D r Φ z , and so this quantity dependson the first subleading, O ( r − ) behavior of Φ z , where Φ z is required to solve (6.32) subjectto the boundary condition Φ z → φ z (nn) as r → ∞ .In terms of these quantities one then finds the following expression for the Bogomolnybound: H bosym ≥ πµ g (cid:90) R d x Tr (cid:110) X (cid:16) f (nn)12 + [ φ z (nn) , φ z (nn) ] (cid:17) + φ z (nn) f (n) r (cid:111) + − π √ µ g Tr (cid:8) Φ y ∞ γ (cid:9) − πµ g (cid:73) S ∞ Tr (cid:110) X a (nn) (cid:111) . (6.37)This gives the bound on the Hamiltonian associated with S ym . Recall, however, that itis S hol , (4.51), rather than the on-shell value of S ym , that is the relevant functional for theholographic correspondence. This means that the holographic energy for static solutions tothe equations of motion is given by the Legendre transform H hol = (cid:34) ( H ym ) o-s − πg µ (cid:90) ∂H d x Tr (cid:110) (cid:126) X (nn) · (cid:126) X (n) (cid:111)(cid:35) (cid:126) X (n) = (cid:126) X (n) [ (cid:126) X (nn) ] . (6.38)Extremization with respect to (cid:126) X (n) leads to X , , f (nn)12 + [ φ z (nn) , φ z (nn) ] − X . (6.39)In particular, these relations are consistent with the asymptotics of the BPS equations. Thereare some cancellations in H hol upon using them.Note that in the extremization with respect to (cid:126) X (n) , we hold the asymptotics of (cid:126) X (n) fixed, as we go to infinity on the two-plane. Therefore the last term of (6.37) does not vary.Finiteness of the energy suggests that the appropriate boundary conditions at S ∞ should be Write ( H ym ) o-s = H pos + H BPSym where H pos is the on-shell value of the positive-definite sum-of-squaresterm in H ym and H BPSym is the right-hand side of (6.37). Any static solution to the equations of motion forfixed boundary data (cid:126) X (n) will be a local minimum of H pos , and hence the first variation of the on-shell valuewith respect to (cid:126) X (n) will vanish: δH pos δ X (n) = δH pos δφ · δφδ X (n) = 0. – 63 –f vortex type. Letting ( (cid:37), ϕ ) be plane-polar coordinates, we impose the following asymptoticbehavior: (cid:126) X (n) = (cid:126) v + O ( (cid:37) − ) , a (nn) = γ m d ϕ + O ( (cid:37) − ) , as (cid:37) → ∞ , (6.40)where the triplet of vevs (cid:126) v is constant and mutually commuting with the magnetic charge, γ m . Note that the asymptotics of a (nn) are consistent with the definition γ m := 12 π (cid:90) R f (nn) = 12 π (cid:73) S ∞ a (nn) . (6.41)We then see that the last term of (6.37) is proportional to Tr { v γ m } . The bound on theholographic energy takes the form H hol ≥ πµ g (cid:26) (cid:90) R Tr (cid:110) φ z (nn) f (n) r (cid:111) − π √ (cid:8) Φ y ∞ γ (cid:9) − π Tr (cid:8) v γ m (cid:9) (cid:27) , (6.42)which is saturated on solutions to (6.31).Finally we must restore the dependence on the unit vectors ˆ n ( r,z ) , as discussed under(6.3), and vary to achieve the strongest bound. The ‘3’ component of the triplets (cid:126)γ m and (cid:126)v refers to their component along ˆ n ( r ) , while φ z (nn) is the component of the triplet (cid:126)φ z (nn) alongˆ n ( z ) . Thus we have the bound H hol ≥ ˆ n ( r ) · (cid:126)M m + ˆ n ( z ) · (cid:126)M e , (6.43)where we’ve introduced the triplets of magnetic and electric masses (cid:126)M m := 4 π µ g Tr (cid:26) √ y ∞ (cid:126)γ m + (cid:126) v γ m (cid:27) ,(cid:126)M e := − πµ g (cid:90) R d x Tr (cid:110) (cid:126)φ z (nn) f (n) r (cid:111) . (6.44)The strongest bound is achieved by takingˆ n ( r ) = (cid:126)M m | (cid:126)M m | , ˆ n ( z ) = (cid:126)M e | (cid:126)M e | , (6.45)which gives H hol ≥ H BPS := | (cid:126)M m | + | (cid:126)M e | , (cid:0) -BPS (cid:1) . (6.46)Here we’ve emphasized that this bound is saturated on solutions preserving two supersym-metries, i.e. / N = 4 Poincare superalgebra.The masses (6.44) transform in the ( , ) and ( , ) of SU (2) r × SU (2) z = SU (2) V × SU (2) H respectively. This is consistent with the central charges of the 3D N = 4 superal-gebra. It would be nice to derive these charges independently, a la [80], by computing thecommutator of Noether charges associated with the supersymmetry transformations.– 64 –he story can be repeated for the 1 / ω defined in(6.21), in place of ω in the energy bound. The result is the same, except that the electriccontribution vanishes: H hol ≥ H BPS := | (cid:126)M m | , (cid:0) -BPS (cid:1) . (6.47)This is analogous to 4D N = 4 supersymmetry: monopoles are 1/2-BPS while dyons are1 / The BPS equations discussed above are the most general ones giving rise to configurationsthat have a soliton-particle interpretation. If we consider extended objects, however, wecan impose one further projection condition on ε , bringing us all the way down to a singlepreserved supersymmetry. There is a U (1) family of choices, corresponding to choosingdirections in the x - x , r - r , and z - z planes. The latter two are the planes orthogonal toˆ n ( r ) ∈ R r ) and ˆ n ( z ) ∈ R z ) respectively. For now we will take these directions to be along therespective 1-axes and then restore the dependence on this choice at the end. Hence our finalprojection condition is Γ r z y ε = ε , (6.48)which is mutually compatible with all previous ones, (6.4). We will see that the correspondingfield configurations give a holographic description of codimension-one domain walls within thedefect CFT—that is, (1 + 1)-dimensional strings inside the (1 + 2)-dimensional defect CFT—and more generally soliton-domain wall junctions.By combining the new projection with the previous ones we find that the 28 Γ AB breakinto seven sets of four, where each member of a given set is equivalent when acting on ε : (cid:8) Γ , Γ r r , Γ z z , − Γ r y (cid:9) , (cid:8) Γ z y , Γ r , − Γ r , − Γ z r (cid:9) , (cid:8) Γ r y , − Γ z , Γ z , − Γ r r (cid:9) , (cid:8) Γ r , Γ z y , Γ z r , Γ r (cid:9) , (cid:8) Γ r z , Γ r , − Γ y , − Γ r z (cid:9) , (cid:8) Γ z , − Γ r y , − Γ r r , Γ z (cid:9) , (cid:8) Γ r , Γ r z , Γ r z , Γ y (cid:9) . (6.49)We also still have the electric-type projection, Γ z ε = ε , which is unaffected by the above.Setting the supersymmetry variation of the fermion to zero, we get the same set of electricBPS equations as before, but the magnetic equations are modified. The four-term equation we– 65 –ad in (6.5) remains, as it corresponds to the first quadruplet in (6.49). The twelve two-termequations combine into six four-term equations, so that the new magnetic system is F + [Φ z , Φ z ] + µ ( r + z ) ( F r r − D r Φ y ) = 0 ,D Φ z − D Φ z + µ ( r + z ) ( F r r − D r Φ y ) = 0 ,D Φ z + D Φ z + µ ( r + z ) ( F r r − D r Φ y ) = 0 ,F r + D Φ y + D r Φ z + D r Φ z = 0 ,F r − D Φ y + D r Φ z − D r Φ z = 0 ,F r − F r − ( D r Φ z − [Φ y , Φ z ]) = 0 ,F r + F r − ( D r Φ z + [Φ y , Φ z ]) = 0 . (6.50)This is another example of a generalized self-duality equation in eight dimensions, [18,75, 76]—the “ -BPS” case in the latter reference. Utilizing (6.12) and (6.13), one can showthat (6.50) is equivalent to ˆ (cid:63) ˆ F = Ω ∧ ˆ F , (6.51)where the new anti-self-dual four-form, Ω = − (cid:63) Ω , has some additional terms relative to(6.14): Ω = ω + ( dˆ y d r + d r d r ) ∧ ( d x dˆ z − d x dˆ z )++ ( dˆ y d r + d r d r ) ∧ ( d x dˆ z + d x dˆ z ) . (6.52)More precisely, (6.50) is equivalent to (6.51) when the latter is restricted to configurationswith R invariance corresponding to translations of ˆ z , ˆ z , ˆ y .Equation (6.51) with (6.52) is also known as the octonionic instanton equation, or the Spin (7) instanton equation, [18, 75, 81–85]. Let us briefly review the connection to octonions,following [18]. An arbitrary element q ∈ O can be written q = (cid:88) A (cid:48) =1 q A (cid:48) e A (cid:48) , (6.53)where e = 1 and e a (cid:48) a (cid:48) = 1 , . . . , R with the tangent space of our eight-manifold, (6.13),where the coordinates x A (cid:48) are a simple reshuffling of the x A . The unit octonions satisfy e a (cid:48) e b (cid:48) = − δ a (cid:48) b (cid:48) + C a (cid:48) b (cid:48) c (cid:48) e c (cid:48) , (6.54)where the structure constants are totally antisymmetric and satisfy { C c (cid:48) , C d (cid:48) } a (cid:48) b (cid:48) = δ c (cid:48) a (cid:48) δ d (cid:48) b (cid:48) + δ c (cid:48) b (cid:48) δ d (cid:48) a (cid:48) − δ c (cid:48) d (cid:48) δ a (cid:48) b (cid:48) , (6.55)with ( C c (cid:48) ) a (cid:48) b (cid:48) = C c (cid:48) a (cid:48) b (cid:48) . Then one way to write the octonionic instanton equation isˆ F a (cid:48) = 12 C a (cid:48) b (cid:48) c (cid:48) ˆ F b (cid:48) c (cid:48) , (6.56)– 66 –here ( C c (cid:48) ) A (cid:48) B (cid:48) = 0 when A (cid:48) or B (cid:48) = 8. Taking the basis for the structure constants to be1 = C = C = C = C = C = C = C , (6.57)one finds the equations ˆ F + ˆ F + ˆ F + ˆ F = 0 , ˆ F + ˆ F + ˆ F + ˆ F = 0 , ˆ F + ˆ F + ˆ F + ˆ F = 0 , ˆ F + ˆ F + ˆ F + ˆ F = 0 , ˆ F + ˆ F + ˆ F + ˆ F = 0 , ˆ F + ˆ F + ˆ F + ˆ F = 0 , ˆ F + ˆ F + ˆ F + ˆ F = 0 . (6.58)The system (6.50) is equivalent to this upon using (6.12), going to an orthonormal frameassociated with (6.13), and relabeling indices according to { , , r , r , ˆ y, r , ˆ z , ˆ z } ↔ { , , , , , , , } . (6.59)Eight-manifolds with Spin (7) structure provide the natural geometric setting for theseequations. The structure constants for the octonions yield a canonical self-dual four-formon R = R ⊕ R , known as the Cayley form:Ω = (cid:63) ϕ + ϕ ∧ d x , with ϕ := 13! C a (cid:48) b (cid:48) c (cid:48) d x a (cid:48) d x b (cid:48) d x c (cid:48) . (6.60)Ω is a Spin (7) structure on R : It is invariant under an irreducible Spin (7) subgroup ofthe SO (8) rotation group. This subgroup is the little group of a constant unit spinor inthe positive chirality Weyl spinor representation of Spin (8). A general self-dual four-formtransforms in a of SO (8), which decomposes into ⊕ ⊕ under Spin (7). The Cayleyform sits in the singlet. Note that ϕ in (6.60) is a G -structure on R and that G is theautomorphism group of the octonion algebra.We used the same notation for the Cayley form in (6.60) as for the anti-self-dual four-form,(6.52). They are indeed the same upon identifying the canonical frame { d x A (cid:48) } on R withthe natural orthonormal frame { e A } associated to (6.13) via (6.59). Hence Ω is a Spin (7)structure on ( ˆ M , ˆ g AB ). More precisely, it is a non-integrable Spin (7), or almost - Spin (7)structure, [84], because it is not closed. If it would have been closed then ( ˆ M , ˆ g AB , Ω ) wouldhave been a Spin (7)-holonomy manifold. See e.g. [87]. A non-integrable
Spin (7) structure,however, is already enough to define the octonionic, or
Spin (7), instanton equation, and is Another interesting occurrence of these equations in everyday physics is observed in [86], where theydescribe heterotic string solitons. The map (6.59) sends the canonical orientation on R to the negative of the orientation we chose on ˆ M ,hence the anti-self-duality of Ω on ˆ M . – 67 –ufficient to guarantee certain nice properties of this equation. For example, on a closedeight-manifold, the linearized equations determining gauge-inequivalent deformations of thisequation form an elliptic complex whose index has been computed in [81, 82].A new Bogomolny bound on the energy functional, which is saturated on solutionsto (6.51), can be derived by repeating an identical sequence of steps as before, but with { ω , ω (cid:48) } → { Ω , Ω (cid:48) } , whereΩ (cid:48) := Ω − µ ( r + z ) d x d x dˆ z dˆ z . (6.61)The shift is exactly what is needed to guarantee that Ω (cid:48) is closed. The bound on the Yang–Mills functional is H bosym ≥ g (cid:90) ∂ ˆ M (cid:18)
12 Ω (cid:48) ∧ ω CS ( ˆ A ) − g Tr { Φ z ∧ ˆ (cid:63) ˆ E } (cid:19) . (6.62)The shift of Ω to Ω (cid:48) in the boundary term is again ultimately due to the presence of thebackground RR flux.Let us comment on this further. If the Spin (7) structure Ω had been closed, then the Spin (7) instanton equations would have implied the standard Yang–Mills equations:
D (cid:63) F = Ω ∧ DF = 0 , if dΩ = 0 . (6.63)In the first step we used that Ω is closed and in the second step we used the Bianchi identity.Our Ω is not closed, but Ω (cid:48) is. Correspondingly, our second-order equations of motion arenot the standard Yang–Mills equations. They have an extra piece,
D (cid:63) F = dΩ ∧ F = 4 µ ( r + z ) r d r d x d x dˆ z dˆ z ∧ F . (6.64)This ‘extra’ term in the equations of motion comes precisely from the coupling to the back-ground RR flux. One can view this as another instance of background fluxes in string theorynaturally leading to modified or generalized geometric structures. Here, this occurs in thecontext of field theories on curved backgrounds, and dovetails nicely with the philosophyrecently presented in [88].The bound (6.62) is evaluated on the field asymptotics (4.58) in appendix F. In orderto describe the result in a relatively compact fashion we introduce a little notation. Let x ˜ p = { x , x , ˆ z , ˆ z } parameterize a Euclidean R with standard orientation, and let ( η i ) ˜ p ˜ q ,or equivalently (cid:126)η ˜ p ˜ q , denote the triplet of self-dual ’t Hooft matrices. (See appendix F forour conventions.) We collect the gauge field A p and the scalars Φ z p , p = 1 , , into a 4Dgauge field A ˜ p = { A , A , Φ z , Φ z } . Since the A p and Φ z p have the same asymptotics, we This situation, in which one has a pair (Ω , Ω (cid:48) ) consisting of a non-integrable Spin (7) structure and aclosed four-form such that the Yang–Mills energy functional is given by a boundary integral in terms of it, wasactually considered in [84], where it was referred to as having a tamed (non-integrable)
Spin (7) structure. – 68 –an consistently define all of the corresponding normalizable and non-normalizable modes forthis 4D gauge field. Then, with these definitions, the bound (6.62) takes the form H bosym ≥ πµ g (cid:90) R d x (cid:126)η ˜ p ˜ q · (cid:26) Tr { (cid:126) X (n) f (nn)˜ p ˜ q } − √ { Φ y ∞ (cid:126)f (n)˜ p ˜ q } − ∂ ˜ p (cid:104) Tr { (cid:126) X (n) a (nn)˜ q } (cid:105) (cid:27) ++ 4 πµ g (cid:90) R d x Tr { φ z (nn) f (n) r } . (6.65)In this expression it should be understood that nothing depends on the coordinates ˆ z p , so forexample the total derivative term vanishes those values of the ˜ p index.The previous bound, (6.37), can be recovered by dropping the terms proportional to thefirst two ’t Hooft matrices. This corresponds to restricting Ω (cid:48) to ω (cid:48) , as discussed in appendixF. The bound (6.65) can be Legendre transformed to a bound on the energy functional H hol ,(6.38). Variation with respect to (cid:126) X (n) now results in the triplet of equations12 (cid:126)η ˜ p ˜ q f (nn)˜ p ˜ q − (cid:126) X (nn) = 0 , (6.66)if we assume that the total derivative term in the first line of (6.65) does not vary. It can beshown that (6.66) is consistent with the asymptotics of (6.50). (This will be demonstrated inthe next subsection. See the paragraph below (6.68).) Plugging (6.66) back in, one finds anexpression for H hol that looks identical to (6.65) except that the first term is absent.However it is likely that the total derivative term of (6.65) does in fact contribute to thevariation with respect to (cid:126) X (n) . This quantity evaluates to (cid:126)η ˜ p ˜ q · ∂ ˜ p (cid:104) Tr { (cid:126) X (n) a (nn)˜ q } (cid:105) = ∂ (cid:104) Tr (cid:110) X φ z (nn) − X φ z (nn) (cid:111)(cid:105) ++ ∂ (cid:104) Tr (cid:110) X φ z (nn) + X φ z (nn) (cid:111)(cid:105) ++ ∂ (cid:104) Tr {X a (nn)2 } (cid:105) − ∂ (cid:104) Tr {X a (nn)1 } (cid:105) . (6.67)The last line comes from the η terms, and produces the line integral we found previously in(6.37). The first two lines are new, relative to (6.37), and are clearly related to the possiblepresence of a domain wall in the plane. Integrating the first term over R reduces it to aline integral of a tension over the x direction, for example, where the tension is given by adiscontinuity in x . Which discontinuity conditions are consistent with the BPS equationsrequires further study.One must also restore the dependence on the parameters determining the supersymmetryprojections—the unit vectors ˆ n ( r ) , ˆ n ( z ) and the three U (1) rotations in the x - x plane andthe planes orthogonal to these vectors. This can be done by defining appropriately rotatedversions of the ’t Hooft symbols. One should then vary with respect to all of these parametersto achieve the strongest bound. We postpone both of these analyses to a future publication.– 69 – .4 Dimensional reduction and the Haydys–Witten equations Given the consistent truncation of section 5, it is natural to ask how the BPS equations reducewhen restricted to the ansatz (5.1). We apply the ansatz directly to the system (6.50). Withthe help of the formulae collected in appendix E, the first three equations reduce to F + [ φ z , φ z ] − µ ( r + z ) r (cid:18) D r X − µ r [ X , X ] (cid:19) = 0 ,D φ z − D φ z − µ ( r + z ) r (cid:18) D r X − µ r [ X , X ] (cid:19) = 0 ,D φ z + D φ z − µ ( r + z ) r (cid:18) D r X − µ r [ X , X ] (cid:19) = 0 . (6.68)Let x ˜ p = ( x , x , ˆ z , ˆ z ) and a ˜ p = ( a , a , φ z , φ z ), and let (cid:126)η ˜ p ˜ q denote the ’t Hooft matricesas in the previous subsection. Then the first two terms in each of these equations can bewritten as ( η i ) ˜ p ˜ q F ˜ p ˜ q for i = 3 , , (cid:126) X (nn) as well, one sees that the leading terms in the r → ∞ limit of these equations reproducethe constraint (6.66) on the non-normalizable boundary data. Since the leading asymptoticbehavior in the full theory is controlled by precisely the degrees of freedom we are keepingin the dimensional reduction, (6.66) gives the leading asymptotics of the first three BPSequations in the full theory too.Each of the last four equations in (6.50) implies a triplet of equations on the truncationansatz, due to dependence on the three j = 1 modes of the two-sphere. However thereare redundancies in these twelve equations such that only four are independent. Hence theremaining BPS equations in the reduced theory are D X + D X + µ r D r φ z − [ X , φ z ] = 0 ,D X − D X + µ r D r φ z + [ X , φ z ] = 0 ,µ r F r + D X + [ X , φ z ] − [ X , φ z ] = 0 ,µ r F r − D X − [ X , φ z ] − [ X , φ z ] = 0 . (6.69)We can work in the generalized temporal gauge a = φ z to solve the electric equationsas before. The Gauss law constraint for φ z is then found to be0 = 1 µ ( r + z ) ( D p D p + ad( φ z p ) ad( φ z p )) φ z ++ µ ( r + z ) (cid:18) D r + 2 r D r + 1 µ r ad( (cid:126) X ) · ad( (cid:126) X ) (cid:19) φ z . (6.70)The form of these equations simplifies a bit if we work with the inverse radial coordinate s = 1 µ r , (6.71)– 70 –uch that s ∈ [0 , ∞ ) with the holographic boundary at s = 0. One finds that the first threeequations are 12 (cid:126)η ˜ p ˜ q F ˜ p ˜ q + (1 + µ z s ) (cid:18) D s (cid:126) X + 12 [ (cid:126) X , × (cid:126) X ] (cid:19) = 0 , (6.72)while the remaining four equations can be put in the form F s + D X − D z X + D z X = 0 , F s − D X + D z X + D z X = 0 ,F sz + D X − D X + D z X = 0 , F sz − D X − D X − D z X = 0 . (6.73)Here the indices ˜ p, ˜ q = 1 , , z , z , are raised and lowered with the flat Euclidean metric on R . These equations are closely related to ones written down recently by Haydys in [89], andby Witten in [90]. In particular, the latter reference used them to develop a gauge theoryformulation of Khovanov homology for knot invariants. In order to put the equations in theform given in [90], we first set B ˜ p ˜ q := − (cid:126) X · (cid:126)η ˜ p ˜ q . (6.74) B is then an adjoint-valued self-dual two-form on the R parameterized by x ˜ p . We define thecross product ( B × B ) ˜ p ˜ q := [ B ˜ p ˜ r , B ˜ q ˜ r ] . (6.75)This gives another adjoint-valued self-dual two-form, a fact that can be seen from the productformula for the ’t Hooft matrices, η i η j = − δ ij + (cid:15) ijk η k . Explicitly,( B × B ) ˜ p ˜ q = − [ (cid:126) X , × (cid:126) X ] · (cid:126)η ˜ p ˜ q . (6.76)Then, using also that14 (cid:126)η ˜ p ˜ q · (cid:126)η ˜ r ˜ s = (Π + ) ˜ p ˜ q ˜ r ˜ s := 14 (cid:0) δ ˜ p ˜ r δ ˜ q ˜ s − δ ˜ p ˜ s δ ˜ q ˜ r + (cid:15) ˜ p ˜ q ˜ r ˜ s (cid:1) , (6.77)the projector onto the self-dual forms, one finds that (6.72) and (6.73) can be written in theform F + −
12 (1 + µ z s ) (cid:18) D s B + 12 B × B (cid:19) = 0 ,F s ˜ q + D ˜ p B ˜ p ˜ q = 0 , (6.78)where F + := Π + ( F ). When z = 0 these are the Haydys–Witten (HW) equations. Moreprecisely, when z = 0 they are the HW equations on M × R + where the four-manifoldis M = R . Furthermore we only obtain these equations when they are restricted totranslationally invariant configurations along two of the directions in R , such that D z p → ad( φ z p ) for p = 1 , Reference [90] uses y for the coordinate parameterizing R + , so s here = y there . – 71 –onzero z (which, we recall, is the separation between the D3-branes and D5-branes`a la Figure 1), apparently leads to a rather interesting deformation of the HW equations.This deformation modifies the form of the equations in the interior, but the equations quicklyapproach their standard form as we approach the boundary at s = 0. The case of nonzero z deserves further study, but we leave it to the future and henceforth set z = 0 , (6.79)for the rest of this subsection.Translationally invariant forms of the HW equations of the type appearing here are infact closely related to another set of equations introduced earlier by Kapustin and Witten[19]. These equations also play an important role in the study of Khovanov homology andknot invariants [90, 91]. We follow the discussion in [90].Suppose we start with the HW equations on M × R + . Now suppose that M = R × M and we look for solutions that are translationally invariant along the first factor. Then (6.78)reduces to the KW equations on M × R + . These are equations for a gauge field a and anadjoint-valued one-form b given by F − b ∧ b + (cid:63)Db = 0 = D (cid:63) b . (6.80)The one-form is constructed from the components of B and the component of the gauge fieldalong the first R factor. The components of B provide the legs along M while the componentof the gauge field along the direction associated with translation invariance is reinterpretedas the component of b along R + . (The precise details can be found in [90]; they will not beimportant here.)Now suppose we have a second translation invariance. We look for solutions to (6.80) on M × R + where M is of the form M = R × M and we assume translation invariance alongthe first factor. This corresponds to solutions of the HW equations on R × M × R + that aretranslationally invariant in the two-plane associated with the first factor. This is precisely thesituation we have here, where the two-plane is parameterized by (ˆ z , ˆ z ) and the M factor is M = R parameterized by ( x , x ).The extended Bogomolny equations, also introduced in [19], arise from this system, i.e. the system we have in (6.72), (6.73), upon specializing to Φ z = Φ z = 0. Explicitly, they are F + D s X + [ X , X ] = 0 ,D s X + [ X , X ] = 0 , D s X + [ X , X ] = 0 ,F s + D X = 0 , F s + D X = 0 ,D X − D X = 0 , D X − D X = 0 . (6.81) These are the KW equations with the parameter t = 1. This parameter can be restored by restoringdependence of the supersymmetry projection (6.48) on the choice of U (1) phase in the z - z plane. – 72 –e have them on the half-space R × R + , parameterized by ( x , x , s ), with flat Euclidean met-ric. They are an interesting mishmash of the (ordinary) Bogomolny equations for monopoles,the Hitchin equations, and the Nahm equations, and can be reduced to all of these uponfurther specializations. The Bogomolny equations arise by setting X = X = 0, the Hitchinequations arise by setting X = a s = 0 and restricting to s -independent field configurations,and the Nahm equations arise by setting a = a = 0 and restricting to x - and x -independentconfigurations.Although (6.81) arises from (6.72) and (6.73) upon setting φ z p = 0, this is not as signifi-cant of a restriction as it sounds. In fact it is sufficient to set the boundary values φ z p (nn) = 0.A vanishing theorem then implies that φ z p = 0 identically [90]. The vanishing theorem can be seen from the Bogomolny bounds we have derived in thispaper. When we set φ z p (nn) = 0, the terms that depend on φ z p (nn) do not contribute to theBPS energy on the right-hand side of (6.65). We can also set the electric contribution to theenergy to zero, since φ z does not participate in the extended Bogomolny equations. In thissituation, the bound (6.65) is the same as the bound we derived earlier for configurationspreserving four supercharges, (6.47). (The Φ y ∞ term drops out of both because the tripletof fieldstrengths, (cid:126)f , has been set to zero by the truncation ansatz, (5.1).) But if the fieldconfiguration saturates (6.47), then it must satisfy the stronger system of BPS equations thatwere used in deriving that bound—namely (6.19).Indeed, if we evaluate (6.19) on the truncation ansatz and set z = 0, we recover theextended Bogomolny equations, (6.81), together with the conditions D p φ z q = D s φ z q = [ (cid:126) X , φ z q ] = [ φ z , φ z ] = 0 . (6.82)However if φ z p is covariantly constant in s and vanishing as s = 0, then it vanishes everywhere.In summary, we have shown that the extended Bogomolny equations on R × R + ariseas equations for finite-energy BPS field configurations, preserving four supercharges in max-imally supersymmetric Yang–Mills on AdS . Furthermore we have shown how maximallysupersymmetric Yang–Mills on AdS is obtained from a consistent truncation of SYM on AdS × S . The latter is the low energy effective description of D5-branes on the bulk sideof the defect AdS/CFT correspondence, as depicted in Figure 2. This opens the door to thepossibility of using holography to study knot invariants. However in order to do so we willneed to generalize the class of boundary conditions we have considered so far in this paper.We sketch this idea a little further in the conclusions.We make one final comment in closing. After setting z = 0, the three-dimensionalequations we obtained in this section are defined on the half-space with Euclidean metric.Since these are BPS equations for solitons in supersymmetric Yang–Mills on
AdS , one mighthave expected to find equations on the half-space with hyperbolic metric. This was the initial See also the paragraph containing equation (4.11) in [92]. – 73 –xpection of the authors, at least. Indeed, one of the initial motivations for this project was toembed hyberbolic monopoles into a string theory brane system. With hindsight, the reason weget Euclidean self-duality equations seems clear. In order to have a supersymmetric theoryon
AdS , the Higgs fields have to be conformally coupled. (See (5.30).) This means thattheir second-order equations of motion can be mapped to flat space equations by a conformaltransformation. In light of this, it is not surprising that the BPS equations also appear asflat space equations.Nevertheless the equations are defined on a manifold with boundary, and the boundaryis the holographic boundary of the AdS/dCFT set-up. There are many exciting directions topursue, a few of which are sketched next. In this paper we constructed a six-dimensional supersymmetric Yang–Mills theory on
AdS × S , with osp (4 |
4) symmetry. We showed that, for g = su ( N f ) and in the regime N c (cid:29) g s N c (cid:29) N f (cid:28) N c / √ g s N c , this is a good low-energy effective description of N f D5-branes probing the near-horizon geometry of N c D3-branes. The probe D5-branes are defectson the bulk side of an AdS/dCFT correspondence that generalizes the original, single probeset-up of [2–4]. The primary motivation driving this work is the application of holography tothe study of curved space Yang–Mills solitons (described in greater detail below), within thecontext of a controlled, top-down string theory framework.With that goal in mind, we analyzed the vacuum structure and perturbative spectrumof the 6D SYM theory. We also derived systems of first order equations for finite-energyBPS solitons with various fractions of supersymmetry. Solutions to these equations saturatebounds on the energy functional, and we evaluated these bounds in terms of asymptoticdata at the holographic boundary. We left questions about the existence of solutions, andthe structure of the space of solutions, to future work. We believe that the holographicperspective will be useful here. We describe some ongoing work along these lines below.We also showed that the 6D theory has a nonlinear consistent truncation on the two-sphere to maximally supersymmetric Yang–Mills on
AdS . This means in particular thatevery solution of the lower-dimensional theory can be uplifted to a solution of the higher-dimensional one — though of course the higher-dimensional theory contains many more so-lutions.We now sketch three avenues for future work: The holographic dual.
In this paper we alluded to general features of the holographicdual on several occasions, but mostly focused on the bulk side of the correspondence. Inforthcoming work [5], we will construct the holographic dual in detail. We study its vacuum– 74 –tructure and compare with the picture described in section 4.1. For those vacua that preservesuperconformal symmetry, the dCFT is a simple extension of the one in [4], in which theglobal U (1) “baryon” symmetry is enhanced to a global U ( N f ). The basic structure of thedefect theory consists of a 3D N = 4 hypermultiplet, which contains a doublet of complexscalars ( q , q ). These transform in the bi-fundamental of SU ( N c ) × U ( N f ), where the firstfactor is gauged by including couplings to (the restriction to the defect of) the 4D N = 4vector-multiplet on the D3-branes.We will also use the dual theory to elaborate on the structure of BPS states. Thenon-normalizable modes we identified in the field asymptotics (4.58), (4.59), play a doublerole. On the one hand they provide boundary values for the D5-brane fields participatingin the various generalized self-duality equations of section 6. On the other, they appear assources for a class of dual operators constructed out of the scalars ( q , ) in the boundarydefect theory. The correspondence suggests that supersymmetric solutions for the q ’s in thepresence of these sources will exist if and only if the same sources serve as boundary valuesfor a supersymmetric bulk solution. In this way we obtain a characterization of boundaryvalues that lead to SYM solitons in the bulk. This characterization will be given in terms ofthe integrability of a different system of equations for the q ’s.In carrying out this analysis, we will be guided by two key points. The first is supersym-metry: the action of supersymmetry on the defect theory will determine the relevant systemof first-order BPS equations. We are also using supersymmetry to motivate the comparisonbetween supersymmetric solutions in the two systems. Holography is of course a strong/weakduality, so one does not expect a priori that semiclassical techniques will be useful on bothsides of the correspondence. Our working assumption is that supersymmetry supplies thenecessary rigidity to justify the comparison.The second point that guides the analysis is the decoupling of ambient modes. Thedecoupling of closed string fluctuations in the bulk should be mirrored by the decoupling ofD3-brane fluctuations in the boundary theory. We can thus look for solutions to the boundaryequations in which the D3-brane fields are restricted to their vacuum configuration. Note thatthe latter can still involve a nontrivial solution to Nahm’s equations, as described in section4.1. D-brane interpretation of solitons.
One advantage of embedding Yang–Mills–Higgs the-ory into a D-brane system is that the D-brane system provides a geometric interpretation forsolitons, in terms of branes extending in extra dimensions. The basic example is the onewe already mentioned in section 4.1: the D3/D5 system makes manifest the equivalent de-scriptions of monopoles as solutions to the Bogomolny equations or as solutions to the Nahmequations [53]. In our case, the picture of [53] is merely describing the vacua of the 6D theory.But what about the soliton configurations of section 6?The supersymmetry projections (6.3) and (6.48), as well as the charges that appear in– 75 – able 1 . Solitonic D-branes r r r z z z y D3 c,v X X X XD5 ym X X X X X XD3 m X X X XD3 e X X X XF1 e X XD3 d X X X Xthe Bogomolny bounds, suggest the identifications in Table 1. As we work our way down thelist we decrease the supersymmetry by half at each stage. Starting at the top we have thevacuum configurations of D5-branes, the original color D3-branes, and additional semi-infiniteor finite-length ‘vacuum’ D3-branes, as depicted in Figure 4. These configurations preserveeight supercharges, or sixteen in special cases.The ‘magnetic’ D3-branes denoted by D3 m correspond to solutions to the system (6.19)from the D5-brane worldvolume point of view, and preserve four supercharges. They providethe purely magnetic contribution to the BPS energy (6.46). This contribution has two pieces,one proportional to the Higgs vev Φ y ∞ and one proportional to the vev | (cid:126) v | that gives theasymptotic value of X (n) as we send | x | to infinity on the two-plane boundary. Recall that (cid:126) X (n) is in the commutant of Φ y ∞ . (See (6.40).) These observations suggest that the Φ y ∞ -contributionto the energy is associated with finite length D3 m -branes stretched between D5-branes atdifferent y -positions, while naively the | (cid:126) v | contribution is associated with infinitesimal D3 m -branes stretched between coincident D5-branes.In the case of the finite-length D3 m -branes, solutions will necessarily depend on both x p and ( r, θ, φ ) directions. The general solution will describe a combination of D3 m ’s and D3 v ’s.In contrast, solutions describing the infinitesimal D3 m ’s can be obtained in the truncatedtheory, where the relevant BPS equations are the extended Bogomolny equations, (6.81). Configurations with both D3 m ’s and ‘electric’ D3-branes, D3 e , preserve two supersym-metries. We expect that they correspond to solutions to the system (6.5) in the D5-braneworldvolume theory. We’ve included macroscopic fundamental strings as part of this sys-tem. They are responsible for depositing electric charge on the D5-brane worldvolume. Theyshould stretch along the z direction with one end on the D3 e -branes and the other end onthe D5-branes. An abelian version of this D5/D3 /F1 e system is described in detail in [93].Further evidence for the identification of this set of branes is provided by the following ob- The corresponding uplifted solutions in the D5-brane theory will also depend on θ, φ , but the dependenceis specified by the truncation ansatz (5.1) and is relatively simple. – 76 –ervation. Any pair of distinct members from the set { D3 v , D3 m , D3 e } have a 1+1-dimensionalcommon worldvolume. For a given pair, let the two sets of orthogonal directions be parame-terized by two complex coordinates. Abelian intersecting D3-brane systems of this type canbe deformed into a ‘diamond,’ described by a holomorphic profile in the corresponding C [94, 95]. This should be an abelian version of the observation we made in (6.10) regardingthe complex form of the BPS equations.Finally we come to the ‘defect’ D3-branes. When these are present with all of the othertypes of branes the supersymmetry is reduced to one preserved supercharge. We expect thatgeneral solutions to the system (6.50) are described by such configurations. In particular,we showed how the BPS bound for this system, (6.62) with (6.67), receives contributionsassociated with domain walls in the boundary defect. The D3 d ’s are mutually BPS with allother branes listed in the table, and we expect that they can be interpreted as these domainwalls.One interesting direction going forward would be to use the various D-brane identifica-tions suggested here to provide dual descriptions of BPS solitons – analogous to the Nahmdescription of monopoles. In other words, we can analyze the supersymmetry conditionson the worldvolume theories of these various probe D3-branes, as they stretch between D5-branes and sit in the background geometry of the color D3’s. In particular, an analysis ofthe D3 d ’s should provide insight into the appropriate implementation of singularity/jumpingconditions discussed under (6.67). The bouquet of branes appearing in Table 1 is reminiscentof Nekrasov’s brane origami for the construction of spiked and crossed instantons [96–98]. Itwould be interesting to investigate possible connections between them. A holographic construction for knot invariants.
As we discussed in section 6.4, theextended Bogomolny equations (6.81) on R × R + play a prominent role in Witten’s gaugetheory approach to Khovanov homology [90]. A critical part of the construction of [90]is the Nahm pole boundary condition on the triplet of scalar fields, and its generalizationrepresenting the insertion of a knot.We are pursuing the implementation of this boundary condition in the holographic set-updeveloped in this paper. Note that a Nahm pole in (cid:126) X as s →
0, corresponds to an O (1 /r )term in the asymptotic behavior of Φ y . This has the same fall-off as the term involving themagnetic charge, P , but it would be in the commutant of P and described by a triplet of su (2) matrices. Given this, as well as the D-brane identifications of Table 1, we suspect thatthe Nahm pole and knot boundary conditions are related to placing D3 m -branes at r = 0.If so, these would be probe D3-branes of the type considered in another case of the defectAdS/CFT correspondence [99] — and results from that analysis could be used to understandthe holographic dual of the Nahm pole boundary condition! This would be a first step towardsconstructing a holographically dual description of knot invariants. Ultimately, one would like to promote the Higgs fields, φ z p , in the translationally-invariant form of the HW – 77 – cknowledgements We thank Ofer Aharony, Nima Arkani-Hamed, Ibrahima Bah, Katrin Becker, Melanie Becker,Dan Butter, Luca Delacr´etaz, Hadi Godazgar, Mahdi Godazgar, Aki Hashimoto, WilliamLinch, Nick Manton, Greg Moore, Chris Pope, Michael Singer, and Paul Sutcliffe for helpfuldiscussions. SKD thanks the Mitchell Institute at Texas A&M University for hospitalityduring the preparation of this work, as well as the Center for Cosmology and Particle Physicsat NYU, where she was a employed when this work began. ABR thanks the Center forCosmology and Particle Physics at NYU for hospitality. He is supported by the MitchellFamily Foundation.
A Fluctuation expansion of the Myers action
In this appendix we fill in some of the details in going from (2.8) to (2.23) and (2.24).
A.1 The DBI action
We want to make a double expansion of the DBI action (2.9) in the open string variables O ∈ ( F ab , D a Φ m , [Φ m , Φ n ] , µ Φ m ) and the closed string fluctuations C ∈ ( h MN , b MN , ϕ, c ( n ) ).This is carried out in three steps. The first step is to write the NS sector closed stringfields G MN , B MN in terms of their “near-horizon” analogs: G MN = ( Lµ ) ˜ G MN and B MN =( Lµ ) ˜ B MN . Equivalently, E MN = ( Lµ ) ˜ E MN . The dilaton, ∆ φ , does not require rescaling.This will enable us to write the open string fields in terms of the O , and we will see thatfactors of O are accompanied by factors of (cid:15) op .The matrix Q mn , (2.11), becomes Q mn = δ mn + ( Lµ ) λ − [ X m , X k ] ˜ E kn ( X )= δ mn + (cid:15) op [Φ m , Φ k ] ˜ E kn ( X ) . (A.1)In the second step we introduced the scalars Φ m according to (2.19), and then the open stringexpansion parameter (cid:15) op according to (2.22). We also emphasize that the closed string fieldsstill depend on the transverse fluctuation scalars; E ( X ) is shorthand for E ( x a ; − iX m ). Wewill also need the inverse,( Q − ) mn = δ mn − (cid:15) op [Φ m , Φ k ] ˜ E kn ( X ) + (cid:15) [Φ m , Φ k ] ˜ E kl ( X )[Φ l , Φ m (cid:48) ] ˜ E m (cid:48) n ( X ) + O ( (cid:15) ) . (A.2) equations, (6.72), (6.73), to honest covariant derivatives— i.e. incorporate dependence on the corresponding ˆ z p directions. As we also remarked in footnote 24, T-duality might be of relevance. – 78 –he second index of ( Q − − δ ) mn is raised using E mn which is, by definition, the inverse of E km [6]. Hence( Q − − δ ) mn = ( Lµ ) − (cid:110) − (cid:15) op [Φ m , Φ n ] + (cid:15) [Φ m , Φ k ] ˜ E kl ( X )[Φ l , Φ n ] + O ( (cid:15) ) (cid:111) . (A.3)Applying the pullback operation (2.13), the quantity appearing in the first determinantis G ab := P [ E ab ( X )] + P (cid:2) E am ( X )( Q − − δ ) mn E nb ( X ) (cid:3) − iλF ab = ( Lµ ) (cid:26) ˜ E ab ( X )+ − i(cid:15) op (cid:104) ( D a Φ m ) ˜ E mb ( X ) + ˜ E am ( X )( D b Φ m ) − i ˜ E am ( X )[Φ m , Φ n ] ˜ E nb ( X ) + F ab (cid:105) + − (cid:15) (cid:20) ( D a Φ m ) ˜ E mn ( X )( D b Φ n ) − i ( D a Φ k ) ˜ E km ( X )[Φ m , Φ n ] ˜ E nb ( X )+ − i ˜ E am ( X )[Φ m , Φ n ] ˜ E nk ( X )( D b Φ k ) − ˜ E am ( X )[Φ m , Φ k ] ˜ E kl ( X )[Φ l , Φ n ] ˜ E nb ( X ) (cid:21) ++ O ( (cid:15) ) (cid:27) . (A.4)Before we can take the determinant, however, we must extract the transverse fluctuationscalars from the closed string functionals ˜ E MN ( X ) according to (2.14). This finally brings usto G ab = ( Lµ ) (cid:110) ˜ E ab − i(cid:15) op G (1) ab − (cid:15) G (2) ab + O ( (cid:15) ) (cid:111) , (A.5)where G (1) ab := Φ m ( ∂ m ˜ E ab ) | x m + ( D a Φ m ) ˜ E mb + ˜ E am ( D b Φ m ) − i ˜ E am [Φ m , Φ n ] ˜ E nb + F ab , (A.6)and G (2) ab := 12 Φ m Φ n ( ∂ m ∂ n ˜ E ab ) | x m + ( D a Φ m )Φ k ( ∂ k ˜ E mb ) | x m + Φ k ( ∂ k ˜ E am ) | x m ( D b Φ m )++ ( D a Φ m ) ˜ E mn ( D b Φ n ) − i (cid:16) Φ k ( ∂ k ˜ E am ) | x m + ( D a Φ k ) ˜ E km (cid:17) [Φ m , Φ n ] ˜ E nb + − i ˜ E am [Φ m , Φ n ] (cid:16) Φ k ( ∂ k ˜ E nb ) | x m + ˜ E nk ( D b Φ k ) (cid:17) − ˜ E am [Φ m , Φ k ] ˜ E kl [Φ l , Φ n ] ˜ E nb . (A.7)In these expressions, factors of ˜ E MN without explicit | x m are to be understood as evaluatedat x m = x m . In particular the first term, ˜ E ab , in (A.5) is a scalar with respect to U ( N f ).Similarly, for the second determinant in the DBI action we need the expansion Q mn = δ mn − i(cid:15) op ( Q (1) ) mn − (cid:15) ( Q (2) ) mn + O ( (cid:15) ) , (A.8) The reason is as follows. The standard manipulation, det ( E ab + M ab ) = det ( E ab ) det ( + E − M ), is notvalid under the STr if E ab is a functional of the matrix-valued X m , due to the fact that STr ( A, A − , B, C ) (cid:54) =STr ( B, C ). In fact this is not an issue for us since we only work to second order in the open string O ’s, butwe prefer to present a systematic approach that could be carried out to higher orders. – 79 –ith ( Q (1) ) mn := i [Φ m , Φ k ] ˜ E kn , ( Q (2) ) mn = i [Φ m , Φ k ]Φ l ( ∂ l ˜ E kn ) | x m . (A.9)The next step is to evaluate the determinants perturbatively in (cid:15) op : (cid:112) − det( G ab ) = ( Lµ ) (cid:113) − det( ˜ E ab ) (cid:26) − i(cid:15) op E ab ( G (1) ) ba + − (cid:15) (cid:20)
12 ˜ E ab ( G (2) ) ba −
14 ˜ E ab ( G (1) ) bc ˜ E cd ( G (1) ) da + 18 (cid:16) ˜ E ab ( G (1) ) ba (cid:17) (cid:21) ++ O ( (cid:15) ) (cid:27) , (A.10)where ˜ E ab is defined to be the inverse of ˜ E bc . A similar formula applies for (cid:112) det( Q mn ). Theintegrand of the DBI action is a product of these two determinants and the dilaton factor,which must also be expanded in open string fluctuations: e − ∆ φ ( X ) = e − ∆ φ (cid:26) i(cid:15) op Φ m ( ∂ m ∆ φ ) | x m + − (cid:15) m Φ n ( ∂ m ∆ φ ∂ n ∆ φ − ∂ m ∂ n ∆ φ ) | x m + O ( (cid:15) ) (cid:27) . (A.11)This results in the open string expansion of the DBI action, S DBI = − τ D5 ( Lµ ) (cid:90) d x (cid:110) V DBI(0) + (cid:15) op V DBI(1) + (cid:15) V DBI(2) + O ( (cid:15) ) (cid:111) , (A.12)with V DBI(0) = − (cid:113) − det( ˜ E ab ) e − ∆ φ STr (1) ,V DBI(1) = i (cid:113) − det( ˜ E ab ) e − ∆ φ STr (cid:26) (cid:16) ˜ E ab ( G (1) ) ba + ( Q (1) ) mm (cid:17) − Φ m ( ∂ m ∆ φ ) | x m (cid:27) , (A.13)and V DBI(2) = (cid:113) − det( ˜ E ab ) e − ∆ φ STr (cid:26) (cid:16) ˜ E ab ( G (2) ) ba + ( Q (2) ) mm − Φ m Φ n ( ∂ m ∂ n ∆ φ ) | x m (cid:17) + − (cid:16) ˜ E ab ( G (1) ) bc ˜ E cd ( G (1) ) da + ( Q (1) ) mn ( Q (1) ) nm (cid:17) ++ 18 (cid:16) ˜ E ab ( G (1) ) ba + ( Q (1) ) mm − m ( ∂ m ∆ φ ) | x m (cid:17) (cid:27) . (A.14)The final step is to expand in closed string fluctuations. This is straightforward using˜ E MN = e κϕ/ (cid:0) G MN + κ ( h MN + b MN ) (cid:1) , (A.15)– 80 –nd gives expansions V DBI( n o ) = √− g (cid:88) n c κ n c V DBI n o ,n c , (A.16)for each of the V DBI( n o ) above, where g ≡ det( G ab ( x a , x m )). This results in the contributionsfrom the DBI action to the V n o ,n c ’s appearing in (2.24). A.2 The CS action
Now we carry out the analogous steps for the CS action, (2.10). First consider the near-horizon rescaling. Given the form of C ( n ) and B MN in (2.7), it is natural to define ˜ C ( n ) suchthat C = (cid:88) n ( Lµ ) n ˜ C ( n ) . (A.17)Then, for example, ˜ C (4) = ˜ C (4) + ˜ C (2) ∧ e ∆ φ / ˜ B + 12 ˜ C (0) ∧ e ∆ φ ˜ B , (A.18)where ˜ C (4) = C (4) + κc (4) , ˜ C ( n ) = κc ( n ) otherwise, and ˜ B = κb .Now observe that exp (cid:0) λ − i X i X (cid:1) = exp (cid:0) ( Lµ ) − (cid:15) op i Φ i Φ (cid:1) , (A.19)when acting on any n -form, and thatexp( − iλF ) = exp (cid:0) − i ( Lµ ) (cid:15) op F (cid:1) . (A.20)Hence each power of i comes with a factor of ( Lµ ) − (cid:15) op and reduces the degree of the formit acts on by two. Meanwhile each power of F comes with a factor of ( Lµ ) (cid:15) op and increasesthe degree of the form by two. Since the integral picks out the six-form part, it follows thatevery term scales as ( Lµ ) and we have the equality τ D5 (cid:90) STr (cid:26) P (cid:104) e λ − i X i X C (cid:105) ∧ e − iλF (cid:27) = τ D5 ( Lµ ) (cid:90) STr (cid:26) P (cid:104) e (cid:15) op i Φ i Φ ˜ C (cid:105) ∧ e − i(cid:15) op F (cid:27) , (A.21)where ˜ C := (cid:80) n ˜ C ( n ) . Since each power of i will produce a factor of [Φ m , Φ n ], and each powerof ( D a Φ m ) from the pullback operation comes with an (cid:15) op , it is clear that the expansion inopen string variables O is organized by (cid:15) op . The sum over n starts at n = 0, runs over even values, and truncates at n = 10, the top degree for a formon spacetime. – 81 –he next step is to carry out the open string expansion, which is simply a matter ofpealing away the various operations on ˜ C . Working to O ( (cid:15) ) we first have that (cid:26) P (cid:20) e (cid:15) o i ˜ C ( X ) (cid:21) e − i(cid:15) o F (cid:27) (6) = (cid:26) P (cid:20) e i(cid:15) op i ˜ C ( X ) (cid:21)(cid:27) (6) − i(cid:15) op (cid:26) P (cid:20) e (cid:15) op i ˜ C ( X ) (cid:21)(cid:27) (4) ∧ F + − (cid:15) (cid:26) P (cid:20) e (cid:15) op i ˜ C ( X ) (cid:21)(cid:27) (2) ∧ F + O ( (cid:15) ) . (A.22)Next we expand e (cid:15) op i Φ i Φ as far as necessary in each term: (cid:20) e (cid:15) op i ˜ C ( X ) (cid:21) (6) = ˜ C (6) ( X ) + (cid:15) op i ˜ C (8) ( X ) + 12 (cid:15) i i ˜ C (10) ( X ) , (cid:20) e (cid:15) op i ˜ C ( X ) (cid:21) (4) = ˜ C (4) ( X ) + (cid:15) op i ˜ C (6) ( X ) + O ( (cid:15) ) , (cid:20) e (cid:15) op i ˜ C ( X ) (cid:21) (2) = ˜ C (2) ( X ) + O ( (cid:15) op ) . (A.23)Then the pullbacks that need to be computed are P [ ˜ C (6) ] abcdef = ˜ C (6) abcdef − i(cid:15) op ( D [ a Φ m ) ˜ C (6) | m | bcdef ] + − · (cid:15) ( D [ a Φ m )( D b Φ n ) ˜ C (6) | mn | cdef ] + O ( (cid:15) ) , (cid:104) P (cid:16) i ˜ C (8) (cid:17)(cid:105) abcdef = (cid:16) i ˜ C (8) (cid:17) abcdef − i(cid:15) op ( D [ a Φ m ) (cid:16) i ˜ C (8) (cid:17) | m | bcdef ] + O ( (cid:15) ) ,P [ ˜ C (4) ] abcd = ˜ C (4) abcd − i(cid:15) op ( D [ a Φ m ) ˜ C (4) | m | bcd ] + O ( (cid:15) ) , (A.24)while the remaining ones can be evaluated at leading order.Assembling the pieces brings us to the following expression for the CS integrand: (cid:26) P (cid:20) e (cid:15) op i ˜ C (cid:21) e (cid:15) op F (cid:27) (6) == (cid:15) abcdef (cid:26)
16! ˜ C (6) abcdef + − i(cid:15) op (cid:20)
15! ( D a Φ m ) ˜ C (6) mbcdef + i (cid:16) i ˜ C (8) (cid:17) abcdef + 14!2! ˜ C (4) abcd F ef (cid:21) + − (cid:15) (cid:20)
14! ( D a Φ m )( D b Φ n ) ˜ C (6) mncdef + 15! ( D a Φ m ) (cid:16) i ˜ C (8) (cid:17) mbcdef + − (cid:16) i i ˜ C (10) (cid:17) abcdef + (cid:20) D a Φ m ) ˜ C (4) mbcd + i (cid:16) i ˜ C (6) (cid:17) abcd (cid:21) F ef ++ 116 ˜ C (2) ab F cd F ef (cid:21)(cid:27) √− g d x + O ( (cid:15) ) . (A.25)– 82 –e have suppressed the arguments of the ˜ C ( n ) in (A.23), (A.24), and (A.25), but it shouldbe understood that they are still functionals of the transverse scalars X m at this point. Thefinal step in the open string expansion is to expand them according to (2.14). This brings usto the result S CS = τ D5 ( Lµ ) (cid:90) d x √− g (cid:110) V CS(0) + (cid:15) op V CS(1) + (cid:15) V CS(2) + O ( (cid:15) ) (cid:111) , (A.26)where V CS(0) = − (cid:15) abcdef ˜ C (6) abcdef STr (1) , and V CS(1) = i(cid:15) abcdef STr (cid:26)
16! Φ m ( ∂ m ˜ C abcdef ) | x m + 15! ( D a Φ m ) ˜ C (6) mbcdef + i (cid:16) i ˜ C (8) (cid:17) abcdef ++ 14!2! ˜ C (4) abcd F ef (cid:27) . (A.27)Rather than writing out the full V CS(2) we just give the single term that will contribute atleading order in the closed string expansion: V CS(2) = (cid:15) abcdef STr (cid:26) D a Φ m ) ˜ C (4) mbcd F ef + · · · (cid:27) . (A.28)The final step is the expansion of the above in closed string fluctuations, V CS( n o ) = (cid:88) n c κ n c V CS n o ,n c . (A.29)Some relevant observations are˜ C (6) abcdef = κ c (6) abcdef + κ c (4)[ abcd b ef ] + O ( κ ) , ˜ C (6) mbcdef = κ (cid:18) c (6) mbcdef + 5!3!2 C (4) m [ bcd b ef ] (cid:19) + O ( κ ) , ˜ C (4) mbcd = C (4) mbcd + κ c (4) mbcd + O ( κ ) , ˜ C (4) abcd = κ c (4) abcd + O ( κ ) . (A.30)The rest is straightforward and these results together with the V DBI n o ,n c give the V n ,n c quotedin (2.24) through V n o ,n c = V DBI n o ,n c − V CS n o ,n c . (A.31) B Background geometry and Killing spinors
Let x ˜ µ , ˜ µ = 0 , . . . , x ˜ µ = ( x µ , y ), let v I , I = 1 , . . . , v I = ( r i , z j ), and set v = √ v I v I . The 10D backgroundmetric is d s = ( µv ) η ˜ µ ˜ ν d x ˜ µ d x ˜ ν + δ IJ d v I d v J ( µv ) , (B.1)– 83 –nd we take the frame to be e ˜ µ = ( µv ) d x ˜ µ , e I = d v I ( µv ) . (B.2)This is the ‘Cartesian-like’ frame introduced in (3.3). The components of the spin connectionare ω ˜ µI, ˜ ν = ( µ ˆ v I ) η ˜ µ ˜ ν , ω IJ,K = µ (ˆ v I δ JK − ˆ v J δ IK ) , (B.3)and so the covariant derivatives, ˆ D (0) P := ∂ M + ω MN,P Γ MN , areˆ D (0)˜ µ = ∂ ˜ µ + µ ˜ µ (ˆ v I Γ I ) , ˆ D (0) I = ∂ I + µ (cid:0) (ˆ v J Γ J )Γ I − Γ I (ˆ v J Γ J ) (cid:1) , (B.4)where we use the shorthand ˆ v I := v I /v .Now F (5) = 4 µ (1 + (cid:63) ) vol AdS , and in these coordinatesvol AdS = ( µv ) d x ∧ d v = e ∧ e ∧ e ∧ e y ∧ ˆ v I e I . (B.5)It follows that 15! Γ M ··· M F (5) M ··· M = 4 µ ˆ v I (cid:18) Γ yI + 15! (cid:15) I yI ··· I Γ I ··· I (cid:19) = 4 µ ˆ v I Γ I y (cid:0) − Γ (cid:1) , (B.6)where we are using that (cid:15) r r r z z z y = 1 and Γ := Γ r r r z z z y is the 10D chiralityoperator.Hence the M = ˜ µ components of the Killing spinor equation (3.1) take the form (cid:104) ∂ ˜ µ + µ ˜ µ (ˆ v I Γ I ) (cid:0) − i Γ y (cid:1)(cid:105) (cid:15) = 0 . (B.7)We write (cid:15) = (cid:15) + + (cid:15) − with (cid:15) ± = ± i Γ y (cid:15) ± and project the equation onto i Γ y eigenspaces: ∂ ˜ µ (cid:15) − = 0 , ∂ ˜ µ (cid:15) + = − µ Γ ˜ µ (ˆ v I Γ I ) (cid:15) − . (B.8)The solutions can be parameterized as (cid:15) − = (ˆ v J Γ J )˜ (cid:15) − ( v ) , (cid:15) + = ˜ (cid:15) + ( v ) − ( µv )( µx µ Γ µ )˜ (cid:15) − ( v ) , (B.9)where ˜ (cid:15) ± are functions of the v I only and satisfy i Γ y ˜ (cid:15) ± = ± ˜ (cid:15) ± . Note that ˜ (cid:15) + has the same10D chirality as (cid:15) itself, but that ˜ (cid:15) − has the opposite. In our conventions, Γ˜ (cid:15) ± = ± ˜ (cid:15) ± .Turning to the M = I components of (3.1) and projecting the equation onto i Γ y eigenspaces gives (cid:104) ∂ I + µ (ˆ v J Γ J )Γ I − v I v (cid:105) (cid:15) − = 0 , (cid:104) ∂ I − v I v (cid:105) (cid:15) + = 0 , (B.10)and one finds that these equations are solved by taking˜ (cid:15) + ( v ) = 1 √ µv (cid:15) , ˜ (cid:15) − ( v ) = √ µv (cid:15) − , (B.11)where (cid:15) ± are constant spinors satisfying the same projection conditions as the ˜ (cid:15) ± . Pluggingthese back into (B.9), one finds that (cid:15) = (cid:15) + + (cid:15) − takes the form given in (3.4).– 84 – .1 Frame rotations The relationship between the triplets { Γ r i } and { Γ r , Γ θ , Γ φ } is expressed in terms of the usualrotation sending the { ˆ r , ˆ r , ˆ r } frame to the { ˆ r, ˆ θ, ˆ φ } frame in R :Γ r = Γ r cos θ + (Γ r cos φ + Γ r sin φ ) sin θ = U ( θ, φ )Γ r U ( θ, φ ) − , Γ θ = − Γ r sin θ + (Γ r cos φ + Γ r sin φ ) cos θ = U ( θ, φ )Γ r U ( θ, φ ) − , Γ φ = − Γ r sin φ + Γ r cos φ = U ( θ, φ )Γ r U ( θ, φ ) − , (B.12)where U ( θ, φ ) − = exp (cid:18) θ r r (cid:19) exp (cid:18) φ r r (cid:19) . (B.13)These relationships are basis independent and hold for any matrix representations of the Γ.If we work in a basis (of sections of the Dirac spinor bundle) where the matrix elements of theΓ r i are constant then, as is clear from (B.12), the matrix elements of Γ r , Γ θ , Γ φ will not be.Conversely, if we work in a basis where the matrix elements of Γ r , Γ θ , Γ φ are constant, thenthose of the Γ r i will not be. Typically, one assumes a basis with respect to which gammamatrices with tangent frame indices are constant. Such an assumption was implicit in e.g. writing the solutions (B.9) above—when we said the spinors (cid:15) ± are constant, this meantconstant with respect to such a basis. A basis in which gamma matrices carrying the sametangent space indices as the frame have constant matrix elements will be referred to as a natural basis associated with the given frame.In expressions like (B.9) containing ‘constant’ spinors a natural basis, with respect towhich the Γ M that appear in the expression have constant matrix elements, will always beassumed unless explicitly stated otherwise. When we want to emphasize the choice of basis,we will write brackets, ( · ) with a subscript label, around the quantity in question. We write( · ) cart , for ‘Cartesian,’ for a natural basis with respect to the frame { e r i } , and we write ( · ) S for a natural basis with respect to the frame { e r , e θ , e φ } .The transformation (B.12) takes an active point of view: we are rotating the Γ them-selves, rather than any basis we may choose to express their matrix elements with respectto. However, when we wish to understand how the presentation of a solution such as (B.9)changes when we change our choice of frame, then we must take a passive point of view.The change of basis transformation, h S ( θ, φ ), that maps components with respect to theCartesian basis to components with respect to the S frame is precisely the (lift to the Diracbundle of the) inverse of the frame rotation:(Γ r,θ,φ ) S = h S ( θ, φ )(Γ r,θ,φ ) cart h S ( θ, φ ) − , with h S ( θ, φ ) = U − ( θ, φ ) . (B.14)Note that the relationship h S = U − allows us to express h S in terms of Γ r , Γ θ , Γ φ insteadof Γ r i : On the one hand, U h S U − = U U − U − = U − = h S , while on the other hand from(B.12) we have U h S U − = exp (cid:18) θ rθ (cid:19) exp (cid:18) φ θφ (cid:19) . (B.15)– 85 –his gives h S as we defined it in (3.8). Note also that h S = U − implies the relations(Γ θ , Γ φ , Γ r ) S = (Γ r , Γ r , Γ r ) cart , (B.16)among the matrix elements of different Γ’s referred to different bases.Thus, if ( (cid:15) ) cart and ( (cid:15) ) S denote the components of the Killing spinor with respect to anatural basis associated with the { e r i } -frame and { e r , e θ , e φ } -frame respectively, then theyare related by h S . To see how this gives (3.7) from (3.4), we first observe from (B.12) that(Γ r ) cart = ˆ r i (Γ r i ) cart , (B.17)so we can write ( (cid:15) ) cart as( (cid:15) ) cart = 1 √ µv (cid:18) r (Γ r ) cart + z i Γ z i v (cid:19) (cid:15) − + √ µv (cid:104) (cid:15) − µ ( x p Γ p + y Γ y ) (cid:15) − (cid:105) , (B.18)Then we set ( (cid:15) ) S = h S ( θ, φ )( (cid:15) ) cart , (B.19)and use (B.14). This results in the expression (3.7). Note that the remaining gamma matrices, { Γ p , Γ z i , Γ y } , are the same with respect to both bases.Additional frame rotations can be made to bring the bulk Killing spinors into a form foundmore commonly in the literature. First consider introducing spherical coordinates ( z, ζ, χ )for the (cid:126)z directions such that ( z , z , z ) = z (sin ζ cos χ, sin ζ sin χ, cos ζ ). Then ˆ z i (Γ z i ) cart =(Γ z ) cart and the Cartesian basis is transformed to an S basis by a completely analogous setof formulae. Referring to this 10D frame and its associated natural bases by the subscript S × S one finds that the components of the Killing spinor are( (cid:15) ) S × S = h S ( ζ, χ ) h S ( θ, φ )( (cid:15) ) cart = 1 √ µv (cid:18) r Γ r + z Γ z v (cid:19) h S ( ζ, χ ) h S ( θ, φ ) (cid:15) − ++ √ µvh S ( ζ, χ ) h S ( θ, φ ) (cid:104) (cid:15) − µ ( x p Γ p + y Γ y ) (cid:15) − (cid:105) , (B.20)where h S ( ζ, χ ) = exp (cid:18) ζ zζ (cid:19) exp (cid:16) χ ζχ (cid:17) . (B.21)Then one can exchange coordinates ( r, z ) for ( v, ψ ) via r = v cos ψ , z = v sin ψ , (B.22)for ψ ∈ [0 , π/ s = ( µv ) η µν d x µ d x ν + d v ( µv ) + µ − dΩ , withdΩ = d ψ + cos ψ (cid:0) d θ + sin θ d φ (cid:1) + sin ψ (cid:0) d ζ + sin ζ d χ (cid:1) . (B.23)– 86 –aking e v = d v/ ( µv ) and e ψ = µ − d ψ , the frames (and hence gamma matrices) are relatedby Γ v = cos ψ Γ r + sin ψ Γ z = exp (cid:18) − ψ rz (cid:19) Γ r exp (cid:18) ψ rz (cid:19) Γ ψ = − sin ψ Γ r + cos ψ Γ z = exp (cid:18) − ψ rz (cid:19) Γ z exp (cid:18) ψ rz (cid:19) . (B.24)We will use the subscript S to refer to a natural basis associated with the ten-dimensionalframe { e µ , e v , e ψ , e θ , e φ , e ζ , e χ } . The change of basis transformation is the inverse of the activeone appearing in (B.24): { (Γ v ) S , (Γ ψ ) S } = h ψ { (Γ v ) S × S , (Γ ψ ) S × S } h − ψ , with h ψ := exp (cid:18) ψ vψ (cid:19) = exp (cid:18) ψ rz (cid:19) . (B.25)Then noting first that (B.20) can be written( (cid:15) ) S × S = 1 √ µv (Γ v ) S × S h S ( ζ, χ ) h S ( θ, φ ) (cid:15) − ++ √ µvh S ( ζ, χ ) h S ( θ, φ ) (cid:104) (cid:15) − µ ( x p Γ p + y Γ y ) (cid:15) − (cid:105) , (B.26)we find that ( (cid:15) ) S = h ψ ( (cid:15) ) S × S = 1 √ µv Γ v h S ( θ A ) (cid:15) − + √ µvh S ( θ A ) (cid:104) (cid:15) − µ ( x p Γ p + y Γ y ) (cid:15) − (cid:105) , (B.27)where we use the shorthand θ A = ( ψ, θ, φ, ζ, χ ) and with h S ( θ A ) := exp (cid:18) ψ vψ (cid:19) exp (cid:18) θ vθ (cid:19) exp (cid:18) φ θφ (cid:19) exp (cid:18) ζ ψζ (cid:19) exp (cid:16) χ ζχ (cid:17) . (B.28)This has the form of the Killing spinors found in e.g. [35, 37]. B.2 D5-brane Killing spinor equation
Taking the real part of (3.1) and utilizing (3.11) for the F (5) term gives (cid:20) ∂ a + 14 ω MN,a Γ MN (cid:21) ε + σ ·
5! (Γ M ··· M F (5) M ··· M )Γ a Γ r r r ε = 0 , (B.29)where we are working in the Cartesian-like frame. One computes that116 ·
5! Γ M ··· M F (5) M ··· M = µ (cid:32) r i Γ r i + z ,i Γ z i (cid:112) r + z (cid:33) Γ y ( − Γ) , (B.30)– 87 –n the brane worldvolume. There are also contributions to the spin connection from directionstransverse to the brane. The nonzero components are ω νz i ,µ = µ e µµ η νµ z ,i (cid:112) r + z , ω r j z k ,r i = − µ e r i r i δ r i r j z ,k (cid:112) r + z , (B.31)in addition to the ω bc,a . The z ,i terms of (B.30) and (B.31) can be combined in (B.29), suchthat that equation becomes (cid:34) D a ∓ µ z ,i Γ z i (cid:112) r + z Γ a ( − Γ r r r y ) + µ r i Γ r i (cid:112) r + z Γ r r r y Γ a (cid:35) ε = 0 , (B.32)where we recall that D a := ∂ a + ω bc,a Γ bc , and the top (bottom) sign is for the case a = p ( a = r i ).In fact the middle term of (B.32) annihilates ε , as we can see from e.g. (3.21). On theone hand, if (cid:126)z = 0 then this term is simply not present. On the other hand, if (cid:126)z (cid:54) = 0 then ε itself is actually an eigenspinor of Γ r r r y :Γ r r r y ε = ε when (cid:126)z = 0 . (B.33)Hence the Killing spinor equation satisfied by ε is (cid:34) D a + µ r i Γ r i (cid:112) r + z Γ r r r y Γ a (cid:35) ε = 0 . (B.34)Working in terms of { Γ r , Γ θ , Γ φ } instead, using that ˆ r i Γ r i = Γ r , and recalling the definitionof M Ψ in (2.32), gives the desired result, (3.23). One can directly check that (3.21) and (3.20)are the solutions of this equation. C Mode analysis
C.1 Bosons
Our starting point is the equations of motion (4.3). For the gauge field equations we takeadvantage of the identity ∇ a T ab = √ | g | ∂ a (cid:16)(cid:112) | g | T ab (cid:17) , which holds for any antisymmetrictensor T ab = T [ ab ] on a (pseudo-)Riemannian manifold with metric g ab with determinant g .Then plugging in (4.24) with the background (4.19), using (4.25), and working to linear order Equivalently, Γ rθφy ε = ε when (cid:126)z (cid:54) = 0, which is evident from (3.20) upon noting that Γ rθφy commuteswith h S . – 88 –n ( a, φ ), we eventually find the following results for the a µ , φ z i , and a r equations:0 = (cid:26) ∂ r + 2 r ∂ r + 1 r ˜ D S − (cid:16) m y,s − p s r (cid:17) + ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) µ ( r + z ) (cid:27) a sµ + − µ ( r + z ) ∂ µ (cid:16) ˜ D M a M,s (cid:17) , (cid:26) ∂ r + 2 r ∂ r + 1 r ˜ D S − (cid:16) m y,s − p s r (cid:17) + ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) µ ( r + z ) (cid:27) φ z i ,s + − µ ( r + z ) (cid:104) ˜Φ z i , ˜ D M a M,s (cid:105) , r ∂ r (cid:0) r a sr (cid:1) + 1 r ˜ D S a sr − (cid:16) m y,s − p s r (cid:17) a sr + ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) µ ( r + z ) a sr − im y,s r φ y,s ++ 1( r + z ) (cid:20) ∂ r ( r + z ) r (cid:21) (cid:18) ∂ r ( r a sr ) + 1 √ ˜ g ˜ D α ( (cid:112) ˜ g ˜ g αβ a sβ ) + r [ ˜Φ y , φ y ] s (cid:19) + − µ ( r + z ) ∂ r (cid:16) g rr ˜ D M a M,s (cid:17) . (C.1)The new object in these expressions is the ten-dimensional covariant divergence, based on themetric G MN = diag( g ab , G mn ) evaluated at x m = x m , with determinant satisfying ( − G ) / = r / ( µ ( r + z )):˜ D M a M = ( − G ) − / ∂ M (cid:16) ( − G ) / G MN a N (cid:17) + G MN [ ˜ A M , a N ]= µ ( r + z ) (cid:18) r ∂ r (cid:0) r a r (cid:1) + 1 r √ ˜ g ˜ D α (cid:16)(cid:112) ˜ g ˜ g αβ a β (cid:17) + [ ˜Φ y , φ y ] (cid:19) ++ 1 µ ( r + z ) (cid:16) η µν ∂ µ a ν + δ ij [ ˜Φ z i , φ z j ] (cid:17) . (C.2)Here we have collected the bosonic fluctuations into a ten-dimensional gauge field, a M =( a a , φ m ), which is translation invariant along the x m directions.It is useful to fix a gauge before proceeding further. A natural gauge-fixing condition is˜ D M a M = 0 , (C.3)as it decouples the equations for a µ and φ z i from the rest of the fluctuations. This stillleaves us the freedom to make gauge transformations a M → a M − D M (cid:15) that preserve thiscondition. The gauge-fixing condition will be preserved if the parameter (cid:15) is annihilated bythe ten-dimensional background covariant Laplacian:0 = ( − G ) − / D M (cid:16) ( − G ) / G MN D N (cid:15) (cid:17) = (cid:26) ∂ r + 2 r ∂ r + 1 r ˜ D − (cid:16) m y,s − p s r (cid:17) + ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) µ ( r + z ) (cid:27) (cid:15) s . (C.4)– 89 –otice that this the same operator appearing in the a µ and φ z i equations. Hence we coulduse residual gauge freedom to set any one component of these fields to zero. However a betterchoice is the following. Observe that the combination η µν ∂ µ a ν + δ ij [ ˜Φ z i , φ z j ] will also beannihilated by ˜∆ , using the condition (C.3) and the equations of motion, since both ∂ ν andad(Φ z i ∞ ) commute with it. Thus we can use the residual gauge freedom to additionally set η µν ∂ µ a ν + δ ij [ ˜Φ z i , φ z j ] = 0 . (C.5)This condition together with (C.3) also imply ∂ r (cid:0) r a r (cid:1) + 1 √ ˜ g ˜ D α (cid:16)(cid:112) ˜ g ˜ g αβ a β (cid:17) + r [ ˜Φ y , φ y ] = 0 . (C.6)Note that this quantity is exactly what appears in the second line of the a r equation of motion.These conditions define the gauge that we work in. In this gauge the equations (C.1) simplifyto0 = (cid:26) ∂ r + 2 r ∂ r + 1 r ˜ D S − (cid:16) m y,s − p s r (cid:17) + ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) µ ( r + z ) (cid:27) ( a sµ , φ z i ,s ) , r ∂ r (cid:0) r a sr (cid:1) + 1 r ˜ D S a sr − (cid:16) m y,s − p s r (cid:17) a sr + ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) µ ( r + z ) a sr − im y,s r φ y,s . (C.7)Next there are the equations for φ y and a α . We immediately plug in the parameterizationof a α in terms of adjoint valued scalars λ, f given in (4.31). Using (C.3), we eventually obtainthe φ y equation,0 = (cid:26) r ( r + z ) ∂ r (cid:2) r ( r + z ) ∂ r (cid:3) + 1 r ˜ D S − (cid:16) y σ − p σ r (cid:17) + η µν ∂ µ ∂ ν − (cid:126)m z,s µ ( r + z ) (cid:27) φ y,s + − (cid:104) ∂ r ˜Φ y , a r (cid:105) s − rr + z (cid:18) [ ˜Φ y , a r ] s + [ ∂ r ˜Φ y , λ ] s − r ˜ D f s (cid:19) , (C.8)and the a α equation,0 = ˜ g αβ ˜ D β (cid:26) ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) r λ s + µ r ∂ r (cid:2) ( r + z ) ∂ r λ s (cid:3) − µ ( r + z ) r (cid:16) y σ − p σ r (cid:17) λ s ++ µ ( r + z ) r (cid:16) ˜ D S λ s + λ s + 2 r [ ∂ r ˜Φ y , f ] s (cid:17) − µ (cid:20) ∂ r ( r + z ) r (cid:21) a sr + − µ ( r + z ) r λ s + (cid:20) f, µ ( r + z ) r ∂ r ˜Φ y + µ r (cid:2) ∂ r ( r + z ) (cid:3) ˜Φ y (cid:21) s (cid:27) ++ ˜ (cid:15) αβ ˜ D β (cid:26) ( η µν ∂ µ ∂ ν − (cid:126)m z,s ) r f s + µ r ∂ r (cid:2) ( r + z ) ∂ r f s (cid:3) − µ ( r + z ) r (cid:16) y σ − p σ r (cid:17) f s ++ µ ( r + z ) r (cid:16) ˜ D S f s + f s − r [ ∂ r ˜Φ y , λ ] s (cid:17) − µ r (cid:2) ∂ r ( r + z ) (cid:3) φ y,s + − µ ( r + z ) r f s − (cid:20) λ, µ ( r + z ) r ∂ r ˜Φ y + µ r (cid:2) ∂ r ( r + z ) (cid:3) ˜Φ y (cid:21) s (cid:27) . (C.9)– 90 –n order to obtain these results we used, for example,˜ (cid:15) αβ ˜ D α a β = ˜ (cid:15) αβ ˜ D α ˜ D β λ − √ ˜ g ˜ D α ( (cid:112) ˜ g ˜ g αβ ˜ D β f )= 12 ˜ (cid:15) αβ [ ˜ F αβ , λ ] − ˜ D S f = r [ ∂ r ˜Φ y , λ ] − ˜ D S f , (C.10)and˜ D S (cid:16) ˜ g αβ a β (cid:17) = ˜ g αβ ˜ D β (cid:16) ˜ D S λ + λ + 2 r [ ∂ r ˜Φ y , f ] (cid:17) + ˜ (cid:15) αβ ˜ D β (cid:16) ˜ D f + f − r [ ∂ r ˜Φ y , λ ] (cid:17) . (C.11)Terms in the latter arise from the commutator of covariant derivatives, which involves botha Riemann curvature term for the two-sphere and a fieldstrength term for the backgroundgauge field.The a α equation has the form 0 = ˜ g αβ ˜ D β Λ + ˜ (cid:15) αβ ˜ D β F . This leads to two separateequations, Λ = 0 and F = 0, which can be viewed as the equations associated with λ and f respectively. One can show, however, that the λ equation follows from the gauge-fixingcondition (C.6) together with the equations of motion for a r , φ y , and f . Hence we candrop the λ equation and instead use the constraint (C.6) to solve for λ . Hence the remainingsecond order equations in addition to (C.7) are0 = (cid:26) r ( r + z ) ∂ r (cid:2) r ( r + z ) ∂ r (cid:3) + 1 r ˜ D S − (cid:16) y σ − p σ r (cid:17) + η µν ∂ µ ∂ ν − (cid:126)m z,s µ ( r + z ) (cid:27) φ y,s + − ∂ r ˜Φ y , a r ] s − rr + z (cid:18) [ ˜Φ y , a r ] s + [ ∂ r ˜Φ y , λ ] s − r ˜ D S f s (cid:19) , (cid:26) r + z ) ∂ r (cid:2) ( r + z ) ∂ r (cid:3) + 1 r ˜ D S − (cid:16) y σ − p σ r (cid:17) + η µν ∂ µ ∂ ν − (cid:126)m z,s µ ( r + z ) (cid:27) f s + − rr + z (cid:16) φ y,s − [ ˜Φ y , λ ] s (cid:17) , (C.12)and the gauge constraint expressed in terms of these variables is0 = ∂ r ( r δa r ) + ˜ D S λ + r [ ∂ r ˜Φ y , f ] + r [ ˜Φ y , φ y ] . (C.13)It appears that after using (C.13) to eliminate λ , the equations for a r , φ y , and f form acoupled system. However it is actually possible to remove the a r dependence from the φ y - f system by one further shift of variables. We introduce new fluctuations y and f by setting φ y,s = y s + ip s r λ s , f s ( j,m ) = f s ( j,m ) + im y,s r j ( j + 1) − p s m y,s r a sr . (C.14) Apply the operator ∂ r (cid:2) r ( r + z ) · (cid:3) to the a r equation and go from there. – 91 –lugging these back into the pair (C.12), all dependence on λ and a r remarkably drops out,and the two equations can be cast into the form0 = (cid:26) (cid:34) r + z ) ∂ r (cid:2) ( r + z ) ∂ r (cid:3) − m y,s + p s m y,s r + η µν ∂ µ ∂ ν − (cid:126)m z,s µ ( r + z ) (cid:35) + − J ( r ) − Y ( r ) (cid:27) (cid:32) r y s ( j,m ) f s ( j,m ) (cid:33) , (C.15)where the two-by-two matrices J , Y are J = j ( j +1) r + r + z j ( j +1) r + z r + z j ( j +1) r , (C.16)and Y = − p σ m y,s rr + z m y,s γ + m y,s r (cid:0) m y,s − p σ r (cid:1) (cid:16) γ (cid:48) + rγr + z (cid:17) p σ m y,s (cid:16) γ (cid:48) + rγr + z (cid:17) , (C.17)with γ ( r ) ≡ (cid:0) j ( j + 1) − p s m y,s r (cid:1) − . Hence we can in principle solve the system (C.15). Thenthe solutions will appear as inhomogeneous sources in the equation for a r , (C.7). Finally from(C.13) (and remembering (4.27)) we find that λ is given by λ s ( j,m ) = ∂ r (cid:34) r a sr, ( j,m ) j ( j + 1) − p s m y,s r (cid:35) + ip s (cid:16) r y s ( j,m ) − f s ( j,m ) (cid:17) − ir m y,s y s ( j,m ) j ( j + 1) − p s m y,s r . (C.18)The appearance of the quantity j ( j + 1) − p s m y,s r in denominators might seem disturbing:If the sign of p s m y,s is positive then such expressions become singular at a physical value of theradial coordinate. We believe these singularities are indicative of an instability in the systemthat occurs when p s m y,s > s . Recall that positive m y,s indicates a separationbetween consecutive D5-branes in the y -direction, while p s indicates the presence of vacuumD3-branes, parallel to the color D3-branes, stretched between these D5-branes or ending onthem. The sign of p s is what determines whether these are D3-branes or D3-branes, andsupersymmetry requires a specific choice. Hence we expect that supersymmetry leads to thecondition p s m y,s < Y , which is not obvious from (C.17), is that it vanishes as r → ∞ . More precisely, the off-diagonal entries are O (1 /r ) and the lower right entry is O (1 /r ). Hence the dominant term in (C.15) at large r is the − m y,s term, when m y,s (cid:54) = 0.The same is true of the other fluctuation equations, (C.7). It follows that the asymptoticbehavior of all fluctuations is e ± m y,s r when m y,s is nonzero.– 92 –he matrix J can be diagonalized by an r -independent similarity transformation: S = (cid:32) j − j + 1 (cid:33) , SJS − = j ( j +1) r + j +1) r + z j ( j +1) r − jr + z . (C.19)In general this is not useful since S does not diagonalize Y . However in the case m y,s = 0, Y vanishes and then S can be used to diagonalize the system (C.15). S acting on ( r y , f ) T defines the scalar fluctuations φ ± introduced in (4.33). Also in the m y,s = 0 case we have f = f and (C.18) reduces to (4.32). C.2 Fermions
We now turn to the last of (4.3). The fermion is already first order in fluctuations, so weevaluate the gauge field and the scalars on their background values (4.19). The first step isto write the equation in terms of a six-dimensional Dirac spinor ψ . In general we follow theconventions in appendix B of Polchinski, [20]. We decompose the ten-dimensional gammamatrices according to Γ a = Σ a ⊗ , Γ m = Σ ⊗ ρ m , (C.20)where Σ a are Spin (1 ,
5) gamma matrices,Σ := − Σ = − Σ rθφ (C.21)is the corresponding chirality operator, and the ρ m are Spin (4) gamma matrices. Specifically,for the ρ m we take ρ m := (cid:32) τ m τ m (cid:33) , τ m := ( (cid:126)σ, − i ) , τ m := ( (cid:126)σ, i ) , (C.22)where the (cid:126)σ are the Pauli matrices. Then with ρ := ρ = diag( , − ) we haveΓ = − Σ ⊗ ρ . (C.23)The Majorana condition Ψ ∗ = B Ψ (C.24)is implemented with the intertwiner B defined such that (Γ M ) ∗ = B Γ M B − . It can betaken as B := (cid:89) imag Γ M , (C.25)and we can always choose our basis such that B = B ∗ = B T = B − . (This choice meansin particular that we take Γ to be real, and hence antisymmetric.) The six-dimensionalcounterpart, B := (cid:89) imag Σ a , (C.26)– 93 –atisfying (Σ a ) ∗ = B Σ a B − , will then have B = B ∗ = − B T = − B − .With these conventions a MW spinor Ψ, satisfying (C.24) and Ψ = ΓΨ, takes the formΨ = ψ − B ψ ∗− ψ + − B ψ ∗ + , (C.27)where ψ ± = ± Σ ψ ± are the positive and negative chirality components of a six-dimensional(complex Dirac) spinor ψ = ψ + + ψ − . One finds that the fermionic action (2.50) expressedin terms of ψ takes the form S ym ,f = − i g (cid:90) d x √− g Tr (cid:26) ¯ ψ /D ψ − ¯ ψ ←− /D ψ + 2 iM Ψ ¯ ψ Σ θφ ψ + 2 iψ [Φ y , ψ ]++ 2 ψ [Φ z , Σ ψ ] − ψ T Σ [Φ z + i Φ z , B ψ ] − ψ [Φ z − i Φ z , B ψ ∗ ] (cid:27) + S bndry f , (C.28)where /D := Σ a D a , ψ := ψ † Σ , and iψ ←− /D ψ is the conjugate of − iψ /D ψ . They are equal upto a total derivative, (but the total derivative can be nonvanishing). Varying with respect to ψ gives the equation of motion,0 = (cid:16) /D + iM Ψ Σ θφ (cid:17) ψ + i [Φ y , ψ ] + [Φ z , Σ ψ ] − [Φ z − i Φ z , B ψ ∗ ] , (C.29)which is equivalent to the last of (4.3). Evaluating the bosonic fields on their backgroundvalues gives the linearized equation0 = (cid:40) Σ µ D µ + Σ r D r + µ ( r + z ) / r ˜Σ α ˜ D α + iM Ψ Σ θφ + µ ( r + z ) / (cid:16) y σ − p σ r (cid:17)(cid:41) ψ + − iµ ( r + z ) / (cid:8) z ,σ Σ ψ − ( z ,σ − iz ,σ ) B ψ ∗ (cid:9) , (C.30)where ˜Σ α is constructed using zweibein on the unit-radius S .To analyze the spectrum of modes on the asymptotically AdS space we choose an adaptedbasis for the six-dimensional gamma matrices:Σ µ,r = γ µ,r ⊗ σ , Σ θ,φ = ⊗ σ , . (C.31)The next step is then to diagonalize the operator σ α ˜ D α := σ ˜ D θ + θ σ ˜ D φ over a completeset of eigenspinors on the two-sphere. This is an S Dirac operator coupled to a Diracmonopole background. The eigenvalue equation σ α ˜ D α ξ = iM ξ , (C.32)– 94 –s equivalent to the dim g equations (cid:20) σ α D α − ip s θ ( (cid:15) − cos θ ) σ (cid:21) ξ ( (cid:15) ) s = iM ξ ( (cid:15) ) s . (C.33)Here (cid:15) = ± specifies the northern or southern patch of the S respectively. The two solutionswill be related by a transition function, ξ (+) s = e ip s φ ξ ( − ) s , on the overlap.This is a classic problem with a completely explicit solution. (See appendix C of [48] for arecent treatment.) The eigenspinors are labeled by three indices, σ , j, m , where σ ∈ { + , − , } and ( j, m ) are angular momentum quantum numbers. Let j ∗ := 12 ( | p s | − . (C.34)Then the eigenspinors with σ = ± have j -values starting at j ∗ + 1 and increasing integersteps, while m runs from − j to j in integer steps as usual. They are given by ξ ( (cid:15) ) s ± ,j,m ( θ, φ ) = 1 √ N jm, − ps e i ( m + (cid:15)p s / φ (cid:18) d jm, − ps ( θ ) + i σ d jm, − − ps ( θ ) σ (cid:19) ξ , (C.35)where d jm,m (cid:48) ( θ ) is a Wigner little d function and ξ = (1 , T . The σ = 0 spinors correspondto the special value j = j ∗ only, and their form depends on the sign of p s : ξ ( (cid:15) ) s ,j ∗ ,m ( θ, φ ) = e i ( m + (cid:15)p s / φ (cid:40) N j ∗ m, − j ∗ d j ∗ m, − j ∗ ( θ ) ξ , p s > ,N j ∗ m,j ∗ d j ∗ m,j ∗ ( θ ) σ ξ , p s < . (C.36)Note these solutions only exist when p s is nonzero; j ∗ takes an unphysical value when p s = 0.If p s = 0 then the σ = ± solutions are a complete set with j ∈ { , , . . . } . The N jm,m (cid:48) arenormalization coefficients: N jm,m (cid:48) = e iπ | m − m (cid:48) | / (cid:114) j + 14 π , (C.37)where the choice of phase will be convenient below. The corresponding eigenvalues are M s σ ,j = σ (cid:112) (2 j + 1) − p s . (C.38)The σ = 0 modes are zero modes of σ α ˜ D α , but this does not mean that they correspond tomassless spinors on AdS as there are other terms in the equation (C.30) that must be takeninto account.To find the four-dimensional spectrum we insert the mode expansion ψ ( (cid:15) ) s = (cid:88) σ ,j,m ψ s σ ,j,m ( x µ , r ) ⊗ ξ ( (cid:15) ) s σ ,j,m ( θ, φ ) , (C.39) These solutions can also be expressed in terms of spin-weighted spherical harmonics. The relationship is m (cid:48) Y jm ( θ, φ ) = (cid:0) j +14 π (cid:1) / e imφ d jm,m (cid:48) ( θ ). – 95 –nto the linearized equation (C.30), using (C.31). Note thatΣ = γ ⊗ σ , B = B ⊗ σ σ = − iB ⊗ σ . (C.40)Here B is the product over the imaginary γ µ,r and satisfies ( γ µ,r ) ∗ = − B γ µ,r B − . (We alsoused that it is necessarily the product of an odd number of γ ’s, as charge conjugation reverseschirality for Spin (1 , σ and charge conjugation, ξ (cid:55)→ σ ξ ∗ on the eigenspinors. These are found to be σ ξ ( (cid:15) ) ,s σ ,j,m = ξ ( (cid:15) ) ,s − σ ,j,m , σ = ± , and σ ξ ( (cid:15) ) ,s ,j ∗ ,m = sgn( p s ) ξ ( (cid:15) ) ,s ,j ∗ ,m , (C.41)and σ ( ξ ( (cid:15) ) ,s σ ,j ∗ ,m ) ∗ = σ sgn (cid:16) m + p s (cid:17) ξ ( (cid:15) ) , − s − σ ,j, − m , σ = ± , and σ ( ξ ( (cid:15) ) ,s ,j ∗ ,m ) ∗ = ξ ( (cid:15) ) , − s ,j ∗ , − m . (C.42)In order to obtain the latter one requires the property d jm,m (cid:48) ( θ ) = ( − m − m (cid:48) d j − m, − m (cid:48) ( θ ). Thephase of (C.37) was chosen to make the action of charge conjugation as simple as possible.Remember also that p − s = − p s . See the discussion under (4.25).Using all these facts, we find that (C.30) splits into two families of coupled systems forthe modes ψ s σ ,j,m . The coupled system for the σ = 0 modes (which exist when p s (cid:54) = 0) is0 = (cid:26)(cid:20) /D − im s,z µ ( r + z ) / γ (cid:21) + M (cid:27) (cid:32) ψ s ,j ∗ ,m B ( ψ − s ,j ∗ , − m ) ∗ (cid:33) , (C.43)where M = (cid:104) − M Ψ + sgn( p s ) µ ( r + z ) / (cid:16) m y,s − p s r (cid:17)(cid:105) σ + − sgn( p s ) µ ( r + z ) / (cid:0) m z ,s σ + m z ,s σ (cid:1) , (C.44)and /D = γ µ D µ + γ r D r is the standard Dirac operator on the asymptotically AdS space.Explicitly, one finds /D = µ ( r + z ) / γ r ∂ r + 1 µ ( r + z ) / γ µ ∂ µ + 3 µr r + z ) / γ r . (C.45)Inserting (C.45) into (C.43) and dividing through by µ ( r + z ) / , we have (cid:32) D B B ∗ D − (cid:33) (cid:32) ψ s ,j ∗ ,m B ( ψ − s ,j ∗ , − m ) ∗ (cid:33) = 0 , (C.46)where D ± := γ r (cid:18) ∂ r + 3 r r + z ) (cid:19) ± (cid:18) sgn( p s ) m y,s − | p s | r − rr + z (cid:19) + γ µ ∂ µ − im z ,s γµ ( r + z ) , B := − sgn( p s ) ( m z ,s − im z ,s ) µ ( r + z ) , (C.47)– 96 –hich is a more useful form for studying the large r asymptotics of solutions.At this point we will content ourselves with understanding the r → ∞ behavior ofsolutions. Then it is sufficient to expand the matrix operator in (C.46) through O (1 /r ). Tothis order it diagonalizes and reduces to (cid:20) γ r (cid:18) ∂ r + 32 r (cid:19) + sgn( p s ) m y,s − (cid:18) | p s | (cid:19) r + O (1 /r ) (cid:21) ψ s ,j ∗ ,m = 0 , (C.48)along with an equivalent equation for the conjugate spinor. The equation diagonalizes withrespect to γ r . If we decompose ψ into eigenspinors, ψ s ,j ∗ ,m = ψ s, +0 ,j ∗ ,m + ψ s, − ,j ∗ ,m , with γ r ψ s, ± ,j ∗ ,m = ± ψ s, ± ,j ∗ ,m , (C.49)then the leading behavior of solutions is ψ s, ± ,j ∗ ,m ∝ e ∓ sgn( p s ) m y,s r r − ∓ m (1 + O (1 /r )) , m := − (cid:18) | p s | (cid:19) . (C.50)When m y,s (cid:54) = 0 we have exponential decay or blowup behavior. When m y,s = 0 we havepower-law behavior dictated by the mass m , which we have defined in such a way that itcan be identified with a standard AdS mass for the fermion. In other words, the asymptoticbehavior of solutions to ( /D + m ) ψ = 0 on AdS is ψ ± ∝ r − ∓ m . Since the m s are allnegative, we see that the normalizable modes in the case m y,s = 0 are necessarily associatedwith ψ s, − ,j ∗ ,m . However the normalizable (exponentially decaying) modes when m y,s (cid:54) = 0 couldbe associated with either ψ s, ± ,j ∗ ,m , depending on the sign of the product p s m y,s . It will beassociated with ψ s, − ,j ∗ ,m if this sign is negative. We will comment further on this below.Taking similar steps, one finds that the coupled system for the σ = ± modes can be putin the following form: D + C B C ∗ D + −BB ∗ D − C −B ∗ C ∗ D − ψ s + ,j,m ψ s − ,j,m B ( ψ − s + ,j, − m ) ∗ B ( ψ − s − ,j, − m ) ∗ = 0 , (C.51)where D ± = γ r (cid:18) ∂ r + 3 r r + z ) (cid:19) ∓ rr + z + γ µ ∂ µ − im z ,s γµ ( r + z ) , C = m y,s − i | M s σ ,j | r − p s r , B = sgn (cid:16) m + p s (cid:17) ( m z ,s − im z ,s ) µ ( r + z ) . (C.52)– 97 –enceforth restrict our analysis to the r → ∞ behavior of solutions. Working through O (1 /r ) the B entries can be dropped and the system reduces to γ r (cid:0) ∂ r + r (cid:1) − r m y,s − (cid:16) p s + i | M s σ ,j | (cid:17) r m y,s − (cid:16) p s − i | M s σ ,j | (cid:17) r γ r (cid:0) ∂ r + r (cid:1) − r + O (1 /r ) (cid:32) ψ s + ,j,m ψ s − ,j,m (cid:33) = 0 , (C.53)along with an equivalent equation for the conjugates. Let α ( r ) denote the phase of C , e iα = C / |C| , and consider the unitary transformation (cid:32) χ s ( j,m ) η s ( j,m ) (cid:33) := U (cid:32) ψ s + ,j,m ψ s − ,j,m (cid:33) := 1 √ (cid:32) e − iα/ e iα/ e − iα/ − e iα/ (cid:33) (cid:32) ψ s + ,j,m ψ s − ,j,m (cid:33) . (C.54)This transformation diagonalizes (C.53) to the order we are working. The new variables χ, η satisfy the asymptotic equations (cid:20) γ r (cid:18) ∂ r + 32 r (cid:19) − r ± |C ( r ) | + O (1 /r ) (cid:21) ( χ s ( j,m ) , η s ( j,m ) ) = 0 , (C.55)where the +( − ) is for χ ( η ) respectively, and |C ( r ) | = (cid:114)(cid:16) m y,s − p s r (cid:17) + | M s σ ,j | = (cid:114) m y,s − p s m y,s r + (2 j + 1) r = (cid:40) | m y,s | − p s sgn( m y,s )2 r + O (1 /r ) , m y,s (cid:54) = 0 , r (2 j + 1) , m y,s = 0 . (C.56)Let χ ± ,s ( j,m ) and η s, ± ( j,m ) denote the positive and negative chirality components with respectto γ r , as in (C.49). Then the asymptotic behavior of solutions to (C.55) is χ s, ± ( j,m ) ∝ e ∓| m y,s | r r − ± ( ps sgn( m y,s ) )(1 + O (1 /r )) , m y,s (cid:54) = 0 ,r − ∓ m ( χ ) j (1 + O (1 /r )) , m y,s = 0 , (C.57) η s, ± ( j,m ) ∝ e ±| m y,s | r r − ± ( − ps sgn( m y,s ) )(1 + O (1 /r )) , m y,s (cid:54) = 0 ,r − ∓ m ( η ) j (1 + O (1 /r )) , m y,s = 0 , (C.58)where the AdS masses are m ( χ ) j = j − , m ( η ) j = − (cid:18) j + 32 (cid:19) . (C.59)The normalizable modes for χ are those that have positive γ r chirality asymptotically, whilethe normalizable modes of η are those that have negative γ r chiarlity asymptotically. In bothcases the normalizable modes along Lie algebra directions with m y,s (cid:54) = 0 are exponentiallydecaying while those along directions with m y,s = 0 are power-law decaying.– 98 –ecall that j starts at j ∗ + 1 = ( | p s | + 1) for these modes. However we can view the ψ s ,j ∗ ,m modes as filling in a lower j = j ∗ rung for the η tower in the sense that m ( η ) j ∗ = − (cid:18) | p s | −
12 + 32 (cid:19) = m . (C.60)Also the asymptotic γ r -chiralities match provided sgn( p s ) m y,s < m y,s (cid:54) = 0. As-suming this is the case, for the same reasons as discussed under (C.18), we can identify η s ( j ∗ ,m ) ≡ ψ s ,j ∗ ,m , (C.61)as the lowest rung of the η tower for those s such that p s (cid:54) = 0.Finally we note that the γ r -chirality condition can be translated back to a condition onthe six-dimensional ψ or on the ten-dimensional Ψ. First, since the action of γ r commuteswith the rotation U relating χ, η to the ψ s,j,m , we see that ψ will be an asymptotic eigenspinorof γ r ⊗ = − i Σ rθφ , (C.62)when restricted to normalizable modes of χ or η only. We will have ψ = − i Σ rθφ ψ asymptoti-cally for the normalizable χ -type modes and ψ = + i Σ rθφ ψ asymptotically for the normalizable η -type modes. One can then show from (C.20) and (C.27) that (cid:16) ± i Σ rθφ (cid:17) ψ = 0 ⇐⇒ (cid:16) ± Γ rθφy (cid:17) Ψ = 0 . (C.63)Hence positive (negative) γ r chirality corresponds to negative (positive) Γ rθφy chirality. D Boundary supersymmetry
In this appendix we provide some of the details of the asymptotic analysis that we quoted insubsection 4.4. We begin with B r and B bndry , appearing in (4.62). From (3.29), B r = ε Tr (cid:26)(cid:20) − F ab Γ abr + ( D a Φ m )Γ m Γ ar −
12 [Φ m , Φ n ]Γ mn Γ r − M Ψ Φ m Γ m Γ θφy Γ r (cid:21) Ψ (cid:27) ++ 12 Tr (cid:8) ΨΓ r δ ε Ψ (cid:9) . (D.1)Meanwhile B bndry is defined 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), we infer B bndry = ε Tr (cid:26)(cid:20) (cid:18) µ r Φ y + µ ( r + z ) r sin θ F θφ (cid:19) Γ y − rr + z Φ z i Γ z i ++ 1 r (cid:18) θ ( D φ Φ y )Γ θ − ( D θ Φ y )Γ φ (cid:19) (cid:21) Ψ (cid:27) + 12 Tr (cid:110) ΨΓ θφy δ ε Ψ (cid:111) . (D.2)– 99 –ince the boundary measure in (4.62) is O ( r ) as r → ∞ , we must work through O ( r − ) inthe large r expansion of B r + B bndry .For the moment we set aside the last terms of (D.1) and (D.2) involving the variation ofthe fermion, and we focus on the remaining terms. Since ε = O ( r / ) and Ψ = O ( r − / ), wemust compute the terms in square-brackets through O ( r − ), utilizing the field asymptotics(4.58). All terms can contribute at this order. We expand out, plug in vielbein factors, andcollect terms together as follows: B r + B bndry = ε Tr (cid:26)(cid:20) (cid:18) µ r Φ y + µ ( r + z ) r sin θ F θφ (cid:19) (Γ y − Γ rθφ ) − rr + z Φ z i Γ z i ( − Γ rθφy )++ µ ( r + z ) r D θ Φ y (Γ yθr − Γ φ ) + µ ( r + z ) r sin θ D φ Φ y (Γ yφr + Γ θ )++ (cid:18) − r F µθ Γ µθr − r sin θ F µφ Γ µφr − [Φ z i , Φ y ]Γ z i yr + ( D µ Φ y )Γ yµr ++ 1 r ( D θ Φ z i )Γ z i θr + 1 r sin θ ( D φ Φ z i )Γ z i φr (cid:19) + − µ ( r + z ) (cid:18) F µν Γ µν + D µ Φ z i Γ µz i + 12 [Φ z i , Φ z j ]Γ z i z j (cid:19) Γ r (cid:21) Ψ (cid:27) ++ 12 Tr (cid:110) ΨΓ r (cid:16) + Γ rθφy (cid:17) δ ε Ψ (cid:111) . (D.3)The first four sets of terms are proportional to the projector ( − Γ rθφy ) acting to theleft. In the case of the Φ z i term, the relevant spinor bilinear is ε − Ψ − = O (1 /r ). However,recall that if the vev φ z i (nn) is nonzero, then we must set the superconformal generators η tozero, which implies ε − = 0. Hence, we get an extra order of suppression from the fact thatthe leading φ z i (nn) part of Φ z i does not contribute, and therefore this term can be neglected.The same reasoning applies to the Φ y ∞ part of Φ y in the first term. The remaining termsin this set involve the spinor bilinear ε − Ψ + , which is O (1 /r ). Thus we need to evaluatethem through O (1 /r ), which corresponds precisely to the contribution from (cid:126) X in Φ y , A θ , A φ .Specifically, µ r Φ y + µ ( r + z ) r sin θ F θφ → − r ˆ r · (cid:126) X (n) + · · · ,µ ( r + z ) r D θ Φ y → r ˆ θ · (cid:126) X (n) + · · · ,µ ( r + z ) r sin θ D φ Φ y → r ˆ φ · (cid:126) X (n) + · · · . (D.4)Hence the relevant combination is1 r (cid:16) − Γ y ˆ r − Γ φ ˆ θ + Γ θ ˆ φ (cid:17) · (cid:126) X (n) (1 + Γ rθφy )Ψ = 2 r Γ ry (cid:16) Γ r ˆ r + Γ θ ˆ θ + Γ φ ˆ φ (cid:17) · (cid:126) X (n) Ψ + = 2 r Γ ry ( (cid:126) Γ ( r ) · (cid:126) X (n) )Ψ + , (D.5)– 100 –here we used Γ rθφy Ψ + = Ψ + and ˆ r Γ r + ˆ θ Γ θ + ˆ φ Γ φ = (Γ r , Γ r , Γ r ) ≡ (cid:126) Γ ( r ) . (See (B.12).)Next consider the set of six terms inside the large round brackets of (D.3). It followsfrom the field asymptotics (4.58) that all of these terms start at O (1 /r ). Furthermore all ofthe gamma matrix structures associated with these terms commute with Γ rθφy . Hence theyinvolve ε + Ψ + = O (1 /r ) and ε − Ψ − = O (1 /r ), and we only need to worry about the former.The order O (1 /r ) terms in the round brackets are all of the form D (nn) µ or ad( φ z i (nn) ) actingon (cid:126) X , where D (nn) µ = ∂ µ + ad( a (nn) µ ). Specifically, the relevant combination of terms is1 µ r (cid:16) (Γ µθr ˆ φ − Γ µφr ˆ θ + Γ yµr ˆ r ) · D (nn) µ (cid:126) X (n) + (Γ z i θr ˆ φ − Γ z i φr ˆ θ − Γ z i yr ˆ r ) · [ φ z i (nn) , (cid:126) X (n) ] (cid:17) Ψ + = 1 µ r Γ y (cid:16) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:17) ( (cid:126) Γ ( r ) · (cid:126) X (n) )Ψ + . (D.6)The remaining terms in the square brackets of (D.3) start at O (1 /r ) and anti-commutewith Γ rθφy . Hence they involve the couplings ε + Ψ − and ε − Ψ + , and we only need to keepthe former. One simply needs to evaluate ( A µ , Φ z i ) on their leading behavior, ( a (nn) µ , φ z i (nn) ).Collecting results, we have B r + B bndry = 2 r ε − Tr (cid:110) Γ ry ( (cid:126) Γ ( r ) · (cid:126) X (n) )Ψ + (cid:111) ++ 1 µ r ε + Tr (cid:110) Γ y (cid:16) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:17) ( (cid:126) Γ ( r ) · (cid:126) X (n) )Ψ + (cid:111) + − µ r ε + Tr (cid:26) Γ r (cid:18) f (nn) µν Γ µν + D (nn) µ φ z i (nn) Γ µz i + 12 [ φ z i (nn) , φ z j (nn) ]Γ z i z j (cid:19) Ψ − (cid:27) ++ 12 Tr (cid:110) ΨΓ r (cid:16) + Γ rθφy (cid:17) δ ε Ψ (cid:111) + O ( r − / ) . (D.7)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 δ ε Ψ, as given in (3.27). Our goal willbe to compute ( δ ε Ψ) + through O ( r − / ) since this is the only order that can contribute to(D.7), given the asymptotics of Ψ − , (4.58). We note that the O ( r − / ) terms of ( δ ε Ψ) + givethe supersymmetry variation of the non-normalizable mode, ψ (nn)0 . Even if we choose to setthis field to zero, its variation need not be zero. The reason is that we are allowing certainnon-normalizable modes of the bosonic fields—namely ( a (nn) µ , φ z i (nn) , (cid:126) X (nn) )—to be turned on,and they can source the supersymmetry variation of the non-normalizable fermion modes.We expand out (3.27) and collect terms as follows: δ ε Ψ = { M r + M φ + M φ + M rest } ε , (D.8)– 101 –here M r = µ ( r + z ) / D r Φ y Γ ry + µr ( r + z ) / Φ y Γ θφ + µ ( r + z ) r sin θ F θφ Γ θφ ,M θ = µ ( r + z ) / r D θ Φ y Γ θy + µ ( r + z ) r sin θ F rφ Γ rφ ,M φ = µ ( r + z ) / r sin θ D φ Φ y Γ φy + µ ( r + z ) r F rθ Γ rθ , (D.9)and M rest = (cid:18) µ ( r + z ) / D r Φ z i Γ rz i + µr ( r + z ) / Φ z i Γ θφyz i (cid:19) ++ (cid:20) F µr Γ µr + 1 r F µθ Γ µθ + 1 r sin θ F µφ Γ µφ + 1 µ ( r + z ) / D µ Φ y Γ µy ++ µ ( r + z ) / r (cid:18) D θ Φ z i Γ θz i + 1sin θ D φ Φ z i Γ φz i (cid:19) + [Φ y , Φ z i ]Γ yz i (cid:21) ++ 1 µ ( r + z ) (cid:18) F µν Γ µν + D µ Φ z i Γ µz i + 12 [Φ z i , Φ z j ]Γ z i z j (cid:19) . (D.10)Let’s start with M rest . It follows from the field asymptotics that all seven terms in thebig square-brackets are O (1 /r ). Furthermore the gamma matrix structure of each of theseterms is such that it maps the ( ± )-chirality eigenspace of Γ rθφy to the ( ∓ )-chirality eigenspace.Hence, these terms acting on ε + give an O ( r − / ) contribution to ( δ ε Ψ) − , while these termsacting on ε − give an O ( r − / ) contribution to ( δ ε Ψ) + . Therefore these terms can be neglectedto the order we are working. In contrast the terms in the last line preserve the chirality and soacting on ε + they give a contribution to ( δ ε Ψ) + that is O ( r − / ) that must be kept. Finally,consider the first two terms of M rest . Using (2.30) one finds µ ( r + z ) / D r Φ z i Γ rz i + µr ( r + z ) / Φ z i Γ θφyz i == D r Φ z i Γ rz i + rr + z Φ z i Γ z i r (cid:16) − Γ rθφy (cid:17) . (D.11)The D r Φ z i term is O (1 /r ) and exchanges Γ rθφy chiralities. If φ z i (nn) is nonzero then theprojector annihilates ε , so the last term is also effectively O (1 /r ) and exchanges chiralities.Hence these terms are on the same footing as the square-bracketed terms and can be neglected.In summary,( M rest ε ) + = 1 µ r (cid:18) f (nn) µν Γ µν + D (nn) µ φ z i (nn) Γ µz i + 12 [ φ z i (nn) , φ z j (nn) ]Γ z i z j (cid:19) ε + + O ( r − / ) , ( M rest ε ) − = O ( r − / ) . (D.12)– 102 –ow consider M y . Plugging in (2.30) we have M r = µ ( r + z ) D r Φ y Γ ry + µ r Φ y (Γ ry + Γ θφ ) + µ ( r + z ) r sin θ F θφ Γ θφ = (cid:2) µ ( r + z ) D r Φ y + µ r Φ y (cid:3) (cid:16) Γ ry + Γ θφ (cid:17) − µ ( r + z ) (cid:20) D r Φ y − r sin θ F θφ (cid:21) Γ θφ = (cid:2) µ r Φ y ∞ + O (1 /r ) (cid:3) Γ ry (cid:16) − Γ rθφy (cid:17) − µ r ( r + z ) P y Γ θφ , (D.13)where in the last step we recalled the definition, (4.5). The first term will drop out of (D.7)since it involves the opposite projector. The large r expansion of P y was determined in (4.47).Using that result here gives M r = O (1 /r ) · Γ ry (cid:16) − Γ rθφy (cid:17) − µ r (cid:16) ˆ r · (cid:126) X (nn) + O (1 /r ) (cid:17) Γ θφ . (D.14)Similar manipulations lead to M θ = µ ( r + z ) r D θ Φ y Γ θy (cid:16) − Γ rθφy (cid:17) + sin θµ r ( r + z ) P φ Γ rφ = O (1 /r ) · Γ θy (cid:16) − Γ rθφy (cid:17) + 1 µ r (cid:16) ˆ θ · (cid:126) X (nn) + O (1 /r ) (cid:17) Γ rφ , (D.15)and M φ = µ ( r + z ) r sin θ D φ Φ y Γ φy (cid:16) − Γ rθφy (cid:17) + 1 µ r ( r + z ) P θ Γ rθ = O (1 /r ) · Γ φy (cid:16) − Γ rθφy (cid:17) − µ r (cid:16) ˆ φ · (cid:126) X (nn) + O (1 /r ) (cid:17) Γ rθ . (D.16)Thus we have(( M r + M θ + M φ ) ε ) + = − µ r (cid:126) X (nn) · (cid:16) ˆ r Γ θφ − ˆ θ Γ rφ + ˆ φ Γ rθ (cid:17) ε + + O ( r − / )= − µ r Γ y (cid:16) ˆ r Γ r + ˆ θ Γ θ + ˆ φ Γ φ (cid:17) · (cid:126) X (nn) ε + + O ( r − / )= − µ r Γ y (cid:126) Γ ( r ) · (cid:126) X (nn) ε + + O ( r − / ) , (( M r + M θ + M φ ) ε ) − = O ( r − / ) , (D.17)Combining (D.12) and (D.17) leads to the result quoted in the text, (4.65).Our final goal is to derive the asymptotics of Ψ due to the massless AdS fermions, asgiven in (4.55) with (4.67). The leading behavior of these modes as r → ∞ is O ( r − / )and the first subleading behavior is O ( r − / ). They are solutions to the fermion equation ofmotion 0 = (cid:26) Γ µ D µ + Γ r D r + Γ α D α + M Ψ Γ θφy + Γ z i ad(Φ z i ) + Γ y ad(Φ y ) (cid:27) Ψ . (D.18)– 103 –ur analysis in appendix C.2 shows that the massless modes are in the simultaneous kernelof ad(Φ y ∞ ) and ad( P ). Taking this into account with respect to the field asymptotics (4.58),the large r form of the equation of motion is0 = (cid:26) µr Γ r (cid:18) ∂ r + 32 r (cid:19) + µ ˜Γ α ˜ D α + µ Γ θφy + 1 µr (cid:104) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:105) ++ 1 µr Γ r ad( a (n) r ) + 1 µr (cid:104) − Γ θ ˆ φ + Γ φ ˆ θ + Γ y ˆ r (cid:105) · ad( (cid:126) X (n) ) + O (1 /r ) (cid:27) Ψ , (D.19)where ˜Γ α ˜ D α is (the 10D embedding of) the standard Dirac operator on the two-sphere. Thefirst 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 keepin the truncation, (5.1), and therefore the asymptotics of the solution to the order we needwill be the same as in the truncated theory. Hence we will derive the equations of motionfor the fermion in the truncated theory, which we quoted in (5.7), and then consider theasymptotics of it. D.1 The massless fermion modes
We first use results from appendix C.2 to determine the form of the 10D fermion, Ψ, restrictedto the massless
AdS modes. These are the j = 1 / χ ( ,m ) ( x µ , r ). They satisfy (C.55)with the plus sign, and since m y,s = p s = 0 for these modes, we have |C ( r ) | = | M s σ , | /r = 1 /r .Hence χ ( ,m ) ( x µ , r ) = ( µr ) − / ˜ χ ( ,m ) ( x µ ) (1 + O (1 /r )) . (D.20)The boundary data ˜ χ ( ,m ) can be decomposed into eigenspinors of γ r , ˜ χ ± ( ,m ) = ± γ r ˜ χ ± ( ,m ) ,and we will see that ˜ χ − ( ,m ) corresponds to the normalizable modes and ˜ χ +( ,m ) to the non-normalizable modes.The phase of C ( r ) that appears in the unitary transformation of (C.54) is α = − π/ ψ σ , ,m modes are ψ ± , ,m = √ e ∓ iπ/ χ ( ,m ) . Hence the 6Dspinor, (C.39), restricted to these modes, which we will denote by ψ ( χ ) j =1 / , takes the form ψ ( χ ) j =1 / = (cid:88) m = ± / χ ( ,m ) ( x µ , r ) ⊗ √ (cid:16) e − iπ/ ξ + , ,m + e iπ/ ξ − , ,m (cid:17) , (D.21)where the ξ are given by ξ ± , ,m = N / m, / e imφ d / m, / ( θ ) ± id / m, − / ( θ ) . (D.22)– 104 –ence ψ ( χ ) j =1 / = (cid:88) m χ ( ,m ) ( x µ , r ) ⊗ N / m, / e imφ d / m, / ( θ ) d / m, − / ( θ ) . (D.23)Now, using d / / , / = d / − / , − / = cos θ , d / − / , / = − d / / , − / = sin θ , and N / / , − / = iN / / , / ≡ iN / , one finds that this spinor can be expressed in the form ψ ( χ ) j =1 / = N / (cid:32) cos θ sin θ − sin θ cos θ (cid:33) e iσ φ/ χ ( , ) ( x µ , r ) iχ ( , − ) ( x µ , r ) = exp (cid:18) i Σ φ θ (cid:19) exp (cid:18) Σ θφ φ (cid:19) ψ ( x µ , r ) , (D.24)where in the last step we introduced the 6D spinor ψ ( x µ , r ) := N / χ ( , ) ( x µ , r ) iχ ( , − ) ( x µ , r ) , (D.25)and wrote the expression in 6D notation with the definitions (C.31). ψ has a large r expansion starting at O ( r − / ) with the leading behavior given in termsof the boundary spinors ˜ χ , (D.20). If one restricts the 4D spinors χ to γ r eigenspaces, χ ± ,this corresponds to restricting ψ to ψ ± defined by ∓ i Σ rθφ ψ ± = ( ± γ r ⊗ ) ψ ± = ψ ± . (D.26)We use this to express ψ ( χ ) j =1 / in the form ψ ( χ ) , ± j =1 / = exp (cid:18) Σ rθ θ (cid:19) exp (cid:18) Σ θφ φ (cid:19) ψ +6D ( x µ , r ) + exp (cid:18) − Σ rθ θ (cid:19) exp (cid:18) Σ θφ φ (cid:19) ψ − ( x µ , r ) . (D.27)This result is straightforwardly expressed in 10D notation via (C.20). We findΨ ( χ ) j =1 / = h S ( θ, φ ) ψ + ( x µ , r ) + h S ( − θ, φ ) ψ − ( x µ , r ) , (D.28)where we made use of (3.8), ψ = ψ + + ψ − is defined in terms of ψ via (C.27), and ψ ± satisfy Γ rθφy ψ ± = ± ψ ± . (D.29)This is (4.54), which is given in a natural basis with respect to the S frame in which Γ r , Γ θ , Γ φ 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 Ψ := (cid:16) Γ a D a + M Ψ Γ θφy (cid:17) Ψ + Γ m [Φ m , Ψ] = 0 , (D.30)– 105 –ith the bosonic fields restricted to (5.1) as well. The basic idea it to pull the factors of h S ( ± θ, φ ) through to the left and collect the terms that are proportional to each. Weexpand the Dirac operator,Γ a D a = Γ µ D µ + Γ r D r + µ ( r + z ) / r (cid:18) ˜ /D S + Γ θ ad( A θ ) + 1sin θ Γ φ ad( A φ ) (cid:19) , (D.31)with ˜ /D S the standard Dirac operator on the unit S . Then we make use of the followingidentities: ˜ /D S h ( ± θ, φ ) = ∓ h ( ∓ θ, φ )Γ r , Γ r h ( ± θ, φ ) = h ( ∓ θ, φ )Γ r , (D.32)Γ θφy h ( ± θ, φ ) ψ ± = Γ r Γ rθφy h ( ± θ, φ ) ψ ± = ± h ( ∓ θ, φ )Γ r , (D.33)and Γ θ h S ( ± θ, φ ) ψ ± = − h S ( ∓ θ, φ ) ˆ φ · (Γ φ , − Γ θ , Γ y ) ψ ± , Γ φ h S ( ± θ, φ ) ψ ± = h S ( ∓ θ, φ ) ˆ θ · (Γ φ , − Γ θ , Γ y ) ψ ± , Γ y h S ( ± θ, φ )Ψ ± = h S ( ∓ θ, φ ) ˆ r · (Γ φ , − Γ θ , Γ y ) ψ ± . (D.34)Note for these last three we are employing (D.29) as well. Then we find E Ψ = h S ( θ, φ ) (cid:26) Γ µ D µ ψ + + Γ r D r ψ − + (cid:18) r − rr + z (cid:19) Γ r ψ − + Γ z i [ ˜Φ z i , ψ + ]+ µ ( r + z ) / (Γ φ , − Γ θ , Γ y ) · (cid:20) − r ˆ φA θ + 1 r sin θ ˆ θA φ + ˆ r Φ y , ψ − (cid:21) (cid:27) ++ h S ( − θ, φ ) (cid:26) Γ µ D µ ψ − + Γ r D r ψ + − (cid:18) r − rr + z (cid:19) Γ r ψ + + Γ z i [ ˜Φ z i , ψ − ]+ µ ( r + z ) / (Γ φ , − Γ θ , Γ y ) · (cid:20) − r ˆ φA θ + 1 r sin θ ˆ θA φ + ˆ r Φ y , ψ + (cid:21) (cid:27) . (D.35)The mass-like term ± (cid:18) r − rr + z (cid:19) Γ r ψ ∓ = − z r ( r + z ) Γ r Γ rθφy ψ ∓ = − µz r ( r + z ) / Γ θφy ψ ∓ = − m ψ Γ h h h ψ ∓ , (D.36)where we used (5.8) and (5.10), vanishes for the AdS background where z = 0, and in generalthe r -dependent mass vanishes asymptotically like O (1 /r ). Plugging in the truncation ansatz(5.1) for the bosonic modes, observe that − r ˆ φA θ + 1 r sin θ ˆ θA φ + ˆ r Φ y = 1 µ r ( ˆ φ ˆ φ · (cid:126) X + ˆ θ ˆ θ · (cid:126) X + ˆ r ˆ r · (cid:126) X ) = 1 µ r (cid:126) X . (D.37)– 106 –ence the quantities in curly brackets in (D.35) are independent of θ, φ on this ansatz. Afterintroducing the triplet notation (5.10) and the metric (5.4), we obtain the result quoted inthe text, (5.6) and (5.7): e ψ = (cid:18) Γ µ D µ + Γ r D r − µz r ( r + z ) / Γ h h h (cid:19) ψ + Γ z i [ φ z i , ψ ] + Γ h i [ X i , ψ ] = 0 . (D.38)Now we analyze the large r asymptotics of this equation. Keeping terms through O (1 /r )in the operator acting on ψ , one finds0 = (cid:26) µr Γ r (cid:18) ∂ r + 32 r (cid:19) + 1 µr (cid:104) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:105) ++ 1 µr (cid:104) Γ r ad( a (n) r ) + Γ h i ad( X i (n) ) (cid:105) + O (1 /r ) (cid:27) ψ . (D.39)The asymptotics of ψ ± are ψ + ( x µ , r ) = 1( µr ) / ψ (nn)0 ( x µ ) + 1( µr ) / ψ +1 ( x µ ) + O ( r − / ) , ψ − ( x µ , r ) = 1( µr ) / Γ r ψ (n)0 ( x µ ) + 1( µr ) / ψ − ( x µ ) + O ( r − / ) . (D.40)This is consistent with (4.56), remembering that (Γ r ) cart = (Γ r ) S . The ψ ± are found byplugging this expansion back into (D.39) and solving it at the first subleading order. We find µ ψ +1 = − (cid:104) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:105) ψ (n)0 + (cid:104) ad( a (nn) r ) + Γ r Γ h i ad( X i (n) ) (cid:105) ψ (nn)0 ,µ ψ − = Γ r (cid:104) Γ µ D (nn) µ + Γ z i ad( φ z i (nn) ) (cid:105) ψ (nn)0 + Γ r (cid:104) ad( a (nn) r ) + Γ h i Γ r ad( X i (n) ) (cid:105) ψ (n)0 . (D.41)This can be expressed in terms of Cartesian frame quantities using (Γ r ) S = (Γ r ) cart and(Γ r (cid:126) Γ ( h ) ) S ψ + = (Γ rφ , − Γ rθ , Γ ry ) S ψ + = (Γ θy , Γ φy , Γ ry ) S ψ + = − Γ y ( (cid:126) Γ ( r ) ) cart ψ + , (D.42)which leads to the results for (D.40) quoted in (4.67). E Some details on the truncation
Here we collect expressions for the components of the non-abelian fieldstrength and covariantderivatives evaluated on the truncation ansatz (5.1). We use a 10D notation ˆ A M for thegauge field and Higgs fields in which we identify ( ˆ A z i , ˆ A y ) ≡ (Φ z i , Φ y ) and, for example,ˆ F µz i = D µ Φ z i . There is nothing to say about F µν , F µr , ˆ F µz i , ˆ F rz i , ˆ F z i z j . For the remainingones we have F µθ trnc −−→ − µ r ˆ φ · D µ (cid:126) X ,F µφ trnc −−→ sin θµ r ˆ θ · D µ (cid:126) X , ˆ F µy trnc −−→ µ r ˆ r · D µ (cid:126) X , (E.1)– 107 – F z i θ trnc −−→ − µ r ˆ φ · [Φ z i , (cid:126) X ] , ˆ F z i φ trnc −−→ sin θµ r ˆ θ · [Φ z i , (cid:126) X ] , ˆ F z i y trnc −−→ µ r ˆ r · [Φ z i , (cid:126) X ] , (E.2)and F rθ trnc −−→ µ r ˆ φ · (cid:126) X − µ r ˆ φ · D r (cid:126) X ,F rφ trnc −−→ − sin θµ r ˆ θ · (cid:126) X + sin θµ r ˆ θ · D r (cid:126) X , ˆ F ry trnc −−→ P r − µ r ˆ r · (cid:126) X + 1 µ r ˆ r · D r (cid:126) X ,F θφ trnc −−→ P θ − sin θµ r ˆ r · (cid:18) (cid:126) X − µ r [ (cid:126) X , × (cid:126) X ] (cid:19) , ˆ F θy trnc −−→ µ r ˆ θ · (cid:18) (cid:126) X − µ r [ (cid:126) X , × (cid:126) X ] (cid:19) , ˆ F φy trnc −−→ sin θµ r ˆ φ · (cid:18) (cid:126) X − µ r [ (cid:126) X , × (cid:126) X ] (cid:19) . (E.3)We also list some formulae that are used in subsection 6.4 for the reduction of the BPSequations. From (E.3) one finds that1 r sin θ F θφ − D r Φ y trnc −−→ − sin θµ r ˆ r · (cid:18) D r (cid:126) X − µ r [ (cid:126) X , × (cid:126) X ] (cid:19) ,F rθ − θ D φ Φ y trnc −−→ − µ r ˆ φ · (cid:18) D r (cid:126) X − µ r [ (cid:126) X , × (cid:126) X ] (cid:19) , θ F rφ + D θ Φ y trnc −−→ µ r ˆ θ · (cid:18) D r (cid:126) X − µ r [ (cid:126) X , × (cid:126) X ] (cid:19) , (E.4)and converting to the Cartesian coordinate system results in F r r − D r Φ y trnc −−→ − µ r (cid:18) D r X − µ r [ X , X ] (cid:19) ,F r r − D r Φ y trnc −−→ − µ r (cid:18) D r X − µ r [ X , X ] (cid:19) ,F r r − D r Φ y trnc −−→ − µ r (cid:18) D r X − µ r [ X , X ] (cid:19) . (E.5)– 108 –ikewise, converting from F pr , F pθ , F pφ , to the Cartesian frame F pr i results in F pr trnc −−→ µ r (cid:0) sin θ cos φ ( µ r F pr ) + sin θ sin φD p X − cos θD p X (cid:1) ,F pr trnc −−→ µ r (cid:0) − sin θ cos φD p X + sin θ sin φ ( µ r F pr ) + cos θD p X (cid:1) ,F pr trnc −−→ µ r (cid:0) sin θ cos φD p X − sin θ sin φD p X + cos θ ( µ r F pr ) (cid:1) , (E.6)while ˆ F py trnc −−→ µ r (cid:0) sin θ cos φD p X + sin θ sin φD p X + cos θD p X (cid:1) . (E.7)Identical expressions hold for the ˆ F z p r i and ˆ F z p y upon replacing D p → ad(Φ z p ). F The BPS energy
In this appendix we show how one obtains (6.65) from (6.62), and as a special case, (6.37)from (6.30).First we introduce some notation that exposes the structure of Ω (cid:48) . Let x ˜ p = ( x , x , ˆ z , ˆ z )parameterize R with the standard orientation. Introduce a basis of self-dual two-forms, ω = d x dˆ z − d x dˆ z , ω = d x dˆ z + d x dˆ z , ω = d x d x + dˆ z dˆ z . (F.1)These can be expressed in terms of ’t Hooft matrices, ω i := 12 η i ˜ p ˜ q d x ˜ p d x ˜ q . (F.2)where our conventions are η = − − , η = − − , η = − − . (F.3)Note this is a slightly different convention than the standard one given in [100] in that( η , η , η ) here = ( η , η , η ) standard . (F.4)With our convention matrix multiplication gives the quaternion algebra, η i η j = − δ ij + (cid:15) ijk η k ,with a plus sign in front of the (cid:15) rather than a minus.Then, in terms of the two-forms (F.1), one hasΩ (cid:48) = 1 µ ( r + z ) dˆ y d r d r d r + ( dˆ y d r + d r d r ) ∧ ω ++ ( dˆ y d r + d r d r ) ∧ ω + ( dˆ y d r + d r d r ) ∧ ω . (F.5)– 109 –ropping the ω and ω terms gives ω (cid:48) .Converting to spherical coordinates, ( r, θ, φ ), results inΩ (cid:48) = (cid:16) r sin θ d θ d φ ˆ r i + r d y d θ ˆ θ i + r sin θ d y d φ ˆ φ i (cid:17) ∧ ω i + d r terms . (F.6)Here we have suppressed terms that have a leg along the radial direction since they will notcontribute to the boundary integral. It follows that(Ω (cid:48) ∧ ω CS ) θφ ˆ z ˆ z ˆ y = 12 ( η i ) ˜ p ˜ q ω CS (cid:16) ˆ A y r sin θ ˆ r i + ˆ A φ r ˆ θ i − ˆ A θ r sin θ ˆ φ i , ˆ A ˜ p , ˆ A ˜ q (cid:17) , (F.7)where we are using the notation ω CS ( ˆ A A , ˆ A B , ˆ A C ) ≡ ( ω CS ) ABC for the components of theChern–Simons form. If we want ω (cid:48) ∧ ω CS instead, then we drop the i = 1 , x d x d θ d φ dˆ z dˆ z dˆ y at the boundary r → ∞ .Hence we need the large r limit of (F.7). The leading behavior of the ˆ A ˜ p is O (1) and given bythe non-normalizable S singlet modes. Therefore the furthest we need to go in the subleadingasymptotics of (Φ y , A θ,φ ) is the (cid:126) X (n) terms, which will yield a finite contribution to (F.7) as r → ∞ . In fact, if one restricts to the (cid:126) X (n) terms, the first factor in ω CS collapses nicely:ˆ A y r sin θ ˆ r + ˆ A φ r ˆ θ − ˆ A θ r sin θ ˆ φ → µ sin θ (cid:126) X (n) . (F.8)One might worry that the Φ y ∞ and ’t Hooft charge terms in the asymptotics of (Φ y , A θ,φ ) willlead to a divergence, but this is not the case. The ’t Hooft charge drops out of (F.7). The Φ y ∞ term can contribute, but integration over the two-sphere will pick out subleading behavior inthe ˆ A ˜ p factors such that the result is finite. (The integration over S should be carried outbefore the r → ∞ limit is taken.) We thus have (cid:90) ∂ ˆ M Ω (cid:48) ∧ ω CS = lim r →∞ (cid:90) R d x (cid:90) S d θ d φ sin θ ×× (cid:126)η ˜ p ˜ q · (cid:26) µ ω CS ( (cid:126) X (n) , ˆ A ˜ p , ˆ A ˜ q ) + r ˆ r ω CS (Φ y ∞ , ˆ A ˜ p , ˆ A ˜ q ) (cid:27) . (F.9)Both Chern–Simons terms are of a similar structure in that they involve an adjoint-valuedscalar in one of the factors. When this is the case, one can obtain the following equivalentexpression, starting from the definition (6.28): ω CS ( (cid:126) X (n) , ˆ A ˜ p , ˆ A ˜ q ) = 2 Tr (cid:110) (cid:126) X (n) ˆ F ˜ p ˜ q (cid:111) + ∂ ˜ q (cid:104) Tr { (cid:126) X (n) ˆ A ˜ p } (cid:105) − ∂ ˜ p (cid:104) Tr { (cid:126) X (n) ˆ A ˜ q } (cid:105) . (F.10)Here it should be understood that the total derivative term is only present when ˜ p, ˜ q = 1 , (cid:126) X (n) → Φ y ∞ . However in this case we can use that Φ y ∞ is constant and that any power-law modes of ˆ A ˜ p commute with Φ y ∞ to observe that the totalderivative terms just subtract off half of the first term, resulting in:lim r →∞ (cid:90) S dΩ r ˆ r i ω CS (Φ y ∞ , ˆ A ˜ p , ˆ A ˜ q ) = lim r →∞ (cid:90) S dΩ r ˆ r i Tr (cid:110) Φ y ∞ ˆ F ˜ p ˜ q (cid:111) . (F.11)– 110 –ow let us recall the mode expansion of ˆ A ˜ p = ( A p , Φ z p ). The terms we need areˆ A ˜ p ( x µ , r, θ, φ ) = a ˜ p ( x µ , r ) + · · · + (cid:88) m = − a ˜ p, (1 ,m ) ( x µ , r ) Y m ( θ, φ ) + · · · , (F.12)where a ˜ p ( x µ , r ) = a (nn)˜ p ( x µ ) + O ( r − ) as usual, and we introduce the triplet notation, (cid:126)a ˜ p , suchthat (cid:88) m = − a ˜ p, (1 ,m ) ( x µ , r ) Y m ( θ, φ ) = − √ µ r ˆ r · (cid:126)a (n)˜ p ( x µ ) + O ( r − ) . (F.13)Here the normalization convention is consistent with the one taken in (4.44). Then (F.9) isequivalent to12 (cid:90) ∂ ˆ M Ω (cid:48) ∧ ω CS = πµ (cid:90) R d x (cid:126)η ˜ p ˜ q · (cid:26) { (cid:126) X (n) f (nn)˜ p ˜ q } − √ { Φ y ∞ (cid:126)f (n)˜ p ˜ q } ++ ∂ ˜ q (cid:104) Tr { (cid:126) X (n) a (nn)˜ p } (cid:105) − ∂ ˜ p (cid:104) Tr { (cid:126) X (n) a (nn)˜ q } (cid:105) (cid:27) , (F.14)where f (nn)˜ p ˜ q = 2 ∂ [˜ p a (nn)˜ q ] + [ a (nn)˜ p , a (nn)˜ q ] and (cid:126)f (n)˜ q ˜ q = 2 ∂ [˜ p (cid:126)a (n)˜ q ] , and we used the integral (cid:90) S dΩˆ r i ˆ r j = 4 π δ ij . (F.15)This reproduces the magnetic contribution to the energy bound given in (6.65).Dropping the terms proportional to the first two ’t Hooft symbols will give the result forΩ (cid:48) → ω (cid:48) . Furthermore there are some simplifications if we plug in the explicit form of η p ˜ q :12 (cid:90) ∂ ˆ M ω (cid:48) ∧ ω CS = 2 πµ (cid:90) R d x (cid:26) (cid:110) X (cid:16) f (nn)12 + [ φ z (nn) , φ z (nn) ] (cid:17)(cid:111) − √ { Φ y ∞ f } ++ ∂ (cid:104) Tr {X a (nn)1 } (cid:105) − ∂ (cid:104) Tr {X a (nn)2 } (cid:105) (cid:27) . (F.16)For the second term we can pull Φ y ∞ out of the integral. Then we are simply computingthe total magnetic flux of the third component of the normalizable mode of the gauge fieldtriplet. (See (6.35).) 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