Massive quiver matrix models for massive charged particles in AdS

Journal of High Energy Physics, Jan 2016

We present a new class of \( \mathcal{N}=4 \) supersymmetric quiver matrix models and argue that it describes the stringy low-energy dynamics of internally wrapped D-branes in four-dimensional anti-de Sitter (AdS) flux compactifications. The Lagrangians of these models differ from previously studied quiver matrix models by the presence of mass terms, associated with the AdS gravitational potential, as well as additional terms dictated by supersymmetry. These give rise to dynamical phenomena typically associated with the presence of fluxes, such as fuzzy membranes, internal cyclotron motion and the appearance of confining strings. We also show how these models can be obtained by dimensional reduction of four-dimensional supersymmetric quiver gauge theories on a three-sphere.

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Massive quiver matrix models for massive charged particles in AdS

JHE Massive quiver matrix models for massive charged Curtis T. Asplund 0 1 2 4 Frederik Denef 0 1 2 4 Eric Dzienkowski 0 1 2 3 0 sions , D-branes 1 Santa Barbara , California 93106 , U.S.A 2 538 West 120 th Street, New York, New York 10027 , U.S.A 3 Department of Physics, Broida Hall, University of California Santa Barbara 4 Department of Physics, Columbia University We present a new class of N = 4 supersymmetric quiver matrix models and argue that it describes the stringy low-energy dynamics of internally wrapped D-branes in four-dimensional anti-de Sitter (AdS) ux compacti cations. The Lagrangians of these models di er from previously studied quiver matrix models by the presence of mass terms, associated with the AdS gravitational potential, as well as additional terms dictated by supersymmetry. These give rise to dynamical phenomena typically associated with the presence of uxes, such as fuzzy membranes, internal cyclotron motion and the appearance of con ning strings. Matrix Models; Extended Supersymmetry; Field Theories in Lower Dimen- - HJEP01(26)5 3.1.1 3.1.2 3.1.3 3.1.4 3.2.1 3.2.2 3.2.3 3.2.4 3.3.1 3.3.2 3.3.3 3.3.4 4.1 4.2 4.3 4.4 4.5 5.1 5.3 5.4 5.5 5.6 6 Outlook 1 Introduction and summary 2 General Lagrangian and supersymmetry 2.1 2.2 2.3 Lagrangian Supersymmetry transformations R-symmetry and R-frames 3 Examples and physical interpretations One node, no arrows { i { 3.2 Single particle: internal space magnetic and Coriolis forces Nonabelian case with superpotential BMN matrix mechanics Lagrangian The Gauss's law constraint and con nement Ground state structure Concluding comments 4 Comparison to wrapped branes on AdS4 CP 3 Notation and conventions AdS4 CP 3 Probe brane masses Con ning strings Comparison ux compacti cation 5 Derivation from dimensional reduction on R S3 Reduction on R S3: general idea 5.2 Interpretation of the dimensionally reduced model N = 1 supersymmetric Lagrangians on R Dimensional reduction of the Lagrangian S3 Dimensional reduction of the supersymmetry transformations Comparison to other models and generalizations 1 4 Conventions and identities for the dimensional reduction F.1 Spherical harmonics on S3 F.2 F.3 Curved and at spinor indices Killing equations and explicit representations F.4 Spinor and vector identities 39 40 The low-energy dynamics of these modes is captured by quiver matrix mechanics [1{4]. A quiver is an oriented graph of which the vertices are called nodes and the edges arrows. In quiver matrix mechanics, the nodes correspond to four-dimensional spacetime degrees of freedom and label wrapped branes; the arrows correspond to internal-space degrees of freedom and label open string modes. If X6 is a Calabi-Yau manifold without uxes then M4 is at Minkowski space and the bulk superalgebra has eight supercharges, of which the branes preserve four. In this case, the corresponding one-dimensional N = 4 quiver matrix mechanics Lagrangian may be obtained by dimensional reduction of a four-dimensional N = 1 quiver gauge theory. On the other hand if X6 is an Einstein manifold carrying magnetic uxes, compacti cations with eight or more supersymmetries to M4 = AdS4 are possible. A standard example is the type IIA AdS4 CP 3 compacti cation [5, 6] holographically dual to ABJM theory [7]. Thus, a natural question is what the analogous quiver matrix mechanics description is for D-particles in AdS4. In this paper we answer this question. We will argue that the low energy, short distance dynamics of particles in AdS4, obtained as internally wrapped branes preserving at least four supercharges, is captured by a tightly constrained N = 4 supersymmetric massive quiver matrix mechanics. By \massive" we mean the brane position degrees of freedom are trapped near the origin by a { 1 { harmonic potential, interpreted here as the AdS gravitational potential well. These N = 4 massive quiver matrix models generalize the N = 16 BMN matrix model [8], which is a mass deformation of the BFSS matrix model [9]. Although the standard interpretation of the BMN model is quite di erent from the interpretation we consider here, its Lagrangian can nevertheless be viewed a special case of our general class of models, after a suitable eld rede nition. As was pointed out in [10], the BMN model can be obtained by dimensionally reducing N = 4 super-Yang-Mills theory on R N = 4 massive quiver matrix models we present can be obtained by dimensional reduction S3. Similarly, we will see that the S3. The details of this reduction are given in of N = 1 quiver gauge theories on R section 5. The core result of the paper is the general Lagrangian of these N = 4 massive quiver matrix models, presented in section 2. Besides the parameters already present in the at-space quiver mechanics of [4] (particle masses mv, Fayet-Ilopoulos parameters v and superpotential data), they depend on just one additional mass deformation parameter , appearing in harmonic potentials for the particle positions ~xv, such as fuzzy membranes, internal space cyclotron motion, and branes con ned by strings. will elaborate on these in the introduction and especially in sections 3.1.2, 3.1.4, 3.2.2, and 3.3.2, respectively. V (x) = X 1 v 2 mv 2 ~xv2 ; = c `AdS ; (1.1) (1.2) HJEP01(26)5 as well as in a number of other terms related by supersymmetry. The parameter has the dimension of frequency. In the context of our AdS interpretation, it equals the global time oscillation frequency of a particle in the AdS gravitational well: where c is the speed of light and `AdS is the AdS radius. Under some simplifying assumptions stated in section 2, we conjecture that this captures, in fact, the most general case consistent with the symmetries imposed.1 In the context of the AdS interpretation, the isotropic harmonic potential is due to the AdS gravitational potential well. The fact that the deformation introduces just one new parameter, uniform across all connected nodes, can be physically understood as the equality of gravitational and inertial mass, i.e., the 1More precisely, we conjecture that for connected quivers, and modulo \R-frame" eld rede nitions the vector multiplets and N = 4 supersymmetry, assuming a at target-space metric for both vector and chiral multiplets, is given by the Lagrangian (2.1). { 2 { equivalence principle. In view, however, of the very di erent (short distance) regime of validity of the quiver picture and the (long distance) bulk supergravity picture, it is by no means a priori obvious that the quiver should retain this feature of gravity. It does so as a consequence of the structure of the interactions and the constraints of supersymmetry. Further remarkable consequences are highlighted in gure 1. Turning on the mass deformation for the position degrees of freedom and requiring N = 4 supersymmetry automatically implies all of the peculiar dynamical phenomena typically featured by branes in ux backgrounds, including noncommutative fuzzy membranes, magnetic cyclotron motion in the internal space, and con nement of particles by fundamental strings. In section 3 we discuss examples explicitly exhibiting these phenomena in simple quiver models. The supergravity counterpart of this is, essentially, that supersymmetric compacti cations to AdS require ux [11{13]. We devote particular attention to the emergence of con ning strings, as this is perhaps the most dramatic di erence with the at-space quiver models of [4], and one of the main motivations for this work, prompted by problems raised in [14]. The goal of [14] was to demonstrate the existence of multicentered black hole bound states in AdS4 ux compacti cations and to investigate their potential use as holographic models of structural glasses. A simple four-dimensional gauged supergravity model was considered, with the appropriate ingredients needed to lift previously known, asymptotically at bound states of black holes carrying wrapped D-brane charges [4, 15{17] to AdS4 with minimal modi cations. However, as was pointed out already in [14], this model actually misses an important universal feature of ux compacti cations of string theory; the fact that particles obtained by wrapping branes on certain cycles are con ned by fundamental strings. In the example of AdS4 CP 3, dual to ABJM theory, it was explained in [7] how this can be understood from a four-dimensional e ective eld theory point of view; it is because these particles have a nonzero magnetic charge with respect to a Higgsed U(1). The Higgs condensate forces the magnetic ux lines into ux tubes, which act as con ning strings. Alternatively, their inevitability can be inferred directly from the D-brane action. In the presence of background ux, the Gauss's law constraint for the brane worldvolume gauge eld gets a contribution equal to the quantized ux threading the brane, which must be canceled by an equal amount of endpoint charge of fundamental strings attached to the brane. This shows that the con ning strings are fundamental strings, and a rather universal feature of ux compacti cations. If the brane is considered in isolation, the attached strings extend out from it all the way to the boundary of AdS. For this reason, such branes are often called baryonic vertices [18]. Note, however, that suitable pairs of charges may allow the strings emanating from one brane to terminate on the other, thus producing a nite-energy con guration. In section 3.3 we show that all of this is elegantly reproduced by massive quiver matrix mechanics. Gauss's law for the quiver gauge elds forces charged elds to have a nonzero minimal excitation energy that grows linearly with particle separation. The tension of this string is a multiple of the fundamental string tension. More precisely, the number of fundamental strings NF;v terminating on a brane corresponding to a quiver node v, with { 3 { HJEP01(26)5 Fayet-Iliopoulos (FI) parameter v, is given by the universal formula NF;v = v : (1.3) Quantum consistency requires NF;v to be an integer and hence v to be quantized, in contrast to the case of at-space quivers, where the FI parameters are related to continuously tunable bulk moduli. This is consistent with the fact that bulk moduli are typically stabilized in ux compacti cations. As in the at space case, the FI parameters also control supersymmetric bound state formation. In particular, for a two-node quiver with all arrows oriented in one direction, a supersymmetric bound state exists for one sign of the FI parameter but not the other. An interesting immediate consequence is that the boundary of the region in constituent charge space where supersymmetric bound states cease to exist is the same as the codimension-one slice through charge space where con ning strings between the constituents are absent. In section 4 we interpret these ndings in some detail for internally wrapped branes in AdS4 CP 3. Most of the analysis in this paper is classical, but we provide the complete quantum Hamiltonian and supersymmetry algebra in appendix D. The supersymmetry algebra is su(2j1). If the Lagrangian has a U(1)R symmetry, the algebra is extended to the semidirect product su(2j1) o u(1)R. This algebra arises naturally on the worldline of a superparticle in an N = 2 AdS4 background, as shown in appendix E. This con rms our AdS interpretation and provides the appropriate identi cations of the global AdS energy with a particular linear combination of the Hamiltonian and the R-charge generator, namely the global AdS R-frame identi ed in section 3.1.3. We note that the chiral multiplet part of the massive quiver matrix mechanics Lagrangian in equation (2.1) has been given before, as part of a systematic construction of supersymmetric quantum mechanics models with su(2j1) supersymmetry [19, 20]. This part can also be obtained by dimension reduction of the general four-dimensional N = 1 chiral multiplet Lagrangian of [21] on R S3, and it has been obtained this way in [22], for the purpose of computing Casimir energies in conformal eld theories on curved spaces. The dimensional reduction of the four-dimensional vector multiplet is also known from [10], but we explain how to create a general gauged su(2j1) quantum mechanics with coupled vector and chiral multiplets and an arbitrary superpotential. The massive quiver Lagrangian in equation (2.1) is a special case of these models as is the BMN matrix model (see 3.2.4). We explain how to perform the dimensional reduction in section 5 and give additional details in appendix F. In section 5.6, we give a more detailed comparison of our models and those given in the works [19, 20]. 2 General Lagrangian and supersymmetry In this section we give the core results of the paper, the general massive quiver matrix mechanics Lagrangian and its supersymmetries. It represents a general deformation of the quiver models of [4] (see especially appendix C) preserving SO(3) rotation symmetry and N = 4 supersymmetry. For simplicity, we also restrict to a at target-space metric for both { 4 { the vector and chiral multiplets. We conjecture that this is the most general Lagrangian having these properties. The eld content remains the same as in [4]. It is encoded in a quiver, with nodes v 2 V , directed edges (arrows) a 2 A, and dimension vector N = (Nv)v2V . To each node is assigned a vector, or linear, multiplet (Av; Xvi ; v ; Dv) with i = 1; 2; 3. The eld Av is the gauge eld for the group U(Nv). The elds Xvi , v , Dv transform in the adjoint of U(Nv). To each edge a : v ! w is assigned a chiral multiplet ( a; a ; F a) transforming in the bifundmental (Nw; Nv) of U(Nw) U(Nv). In a string theory context the nodes v can be thought of as labeling di erent \parton" D-branes wrapped on internal cycles with multiplicity Nv, and arrows a : v ! w as labeling light open string modes polarized in the internal dimensions, connecting the parton branes. The Lagrangian depends on a number of parameters that are already present in the atspace quiver models. For each node v, there is an inertial mass parameter mv determining the kinetic terms for the vector multiplet elds, and a Fayet-Iliopoulos (FI) parameter v setting the D-term potential for the scalars a connected to the node v. The quiver model may also have a superpotential, given by an arbitrary gauge-invariant holomorphic function W ( a) of the a . Before imposing any supersymmetry, a general SO(3)-symmetric and gauge invariant mass deformation of the vector multiplets consists of adding harmonic potential terms of the form 2v Tr(Xvi )2 to the Lagrangian, and similarly for the fermions. Requiring N = 4 supersymmetry to be preserved dictates the inclusion of additional terms for the vector and chiral multiplets, and reduces the a priori arbitrary deformation parameters v to functions of a single deformation parameter , namely v = mv 2 . In the AdS interpretation discussed in section 3, we identify = 1=`AdS. In appendix C, we provide more details on how supersymmetry xes the form of the mass deformation. In section 5, we explain how it can be obtained from dimensional reduction of N = 1 quiver gauge theories on R 2.1 Lagrangian given by The Lagrangian of massive quiver matrix mechanics with deformation parameter is L = L0 + L0 ; where L0 is the original, undeformed, at-space quiver Lagrangian, identical to the Lagrangian in appendix C of [4], and L and de nitions for this Lagrangian in detail in appendix A. 0 is the mass deformation. We give our conventions S3. (2.1) (2.2) b a + h.c. ) a a v a L0W = X Tr LI0 = X a:v!w The covariant derivatives are given by i[Av; Xvi ]; iAv ay iAv ay; with the arrow a : v ! w. 2.2 Supersymmetry transformations The action is supersymmetric with respect to the transformations 2 1 ijk[Xvi ; Xvj ] k + iDv i Xvi i A notable di erence with [4] is that the supersymmetry parameter in equation (2.4) is time dependent. A given massive quiver matrix model Lagrangian may or may not possess R-symmetry. If its R-symmetry group contains a U(1) subgroup then can be made timeindependent by a eld rede nition, which we give below. Without loss of generality, we can take the subgroup U(1)R to act on the elds Y as (2.5) (2.6) (2.7) (2.8) (2.9) for some real parameter , the only change to the Lagrangian and supersymmetry transformations is that all covariant derivatives are e ectively shifted by constant U(1)R background connections, because DtYold = e i QY 2 t Dt i QY 2 Ynew. Thus, all of the above expressions for the Lagrangian and the supersymmetry transformations remain unchanged provided we replace DtY ! D~tY Dt i QY 2 Y : In particular, picking = 1 renders the supersymmetry parameter time independent. As we will see below, in an example in section 3.1.3, and establish in general in appendix E, the appropriate value to identify the quiver Hamiltonian with the AdS global energy in N = 2 compacti cations is Y ! eiQY # Y ; Y y ! e iQY # Y y ; with the charges QY given in table 1. For the Lagrangian to be invariant under equation (2.5), the superpotential must satisfy a homogeneity condition so that it has overall R-charge QW = 2: W ( qa a) = 2 W ( a) ; Yold = e i QY 2 t Ynew ; or equivalently Pa qa a@aW = 2W . If we rede ne the elds, including , by a timedependent R-symmetry, 3 Examples and physical interpretations The massive quiver matrix mechanics Lagrangian presented in section 2.1 arises naturally in a number of string theoretic contexts. One is already well known; as we show in section 3.2.4, the BMN matrix model [8], describing the dynamics of D0-branes in a plane wave background, arises as a special case. In this paper we will, however, focus on a different interpretation; the nonrelativistic limit of massive particles living in global AdS4, AdS = 2 : { 7 { In this section we will substantiate and explore this interpretation by studying various simple examples. The interpretation of the harmonic potentials as the gravitational potential of AdS, and the corresponding identi cation = 1=`AdS, is explained in section 3.1.2. More interestingly, the model exhibits a number of smoking-gun phenomena usually associated with the presence of background uxes, including the Myers e ect (section 3.1.4), magnetic trapping in the internal D-brane moduli spaces (section 3.2.2) and ux-induced background charges on wrapped branes, forcing a nonzero number of con ning strings to end on the branes (section 3.3). Finally, we will also use the examples to clarify the role of R-symmetry and subtleties associated with the existence of di erent R-frames (sections 3.1.3 and 3.2.3). 3.1 One node, no arrows The one-node quiver without arrows and dimension vector (N ), see gure 2, has just one vector multiplet with gauge group U(N ) and no chiral multiplets. It describes N identical D-particles in a 3+1-dimensional spacetime. 3.1.1 Lagrangian The Lagrangian is The covariant derivative acts in the adjoint of U(N ), DtX = @tX i[A; X], so the diagonal U(1) part of A does not couple to anything in the Lagrangian except to the constant Varying L with respect to A = a 1 thus leads to the constraint = 0, hence for we get the consistency condition = 0. We will give an interpretation of this later in section 3.3.4, after we have studied examples of quivers in which can be nonzero. { 8 { (3.1) 3.1.2 For N = 1 the Lagrangian (3.1) describes a nonrelativistic superparticle in an isotropic 3d harmonic oscillator potential with frequency : L = m 2 1 (X_ i)2 1 2 2(Xi)2 + ( _ _ ) i 2 We can interpret this as a massive nonrelativistic superparticle near the bottom (origin) of global AdS4. Using isotropic coordinates t; x1; x2; x3, with x space of radius ` has a metric [ai; ajy] = ij ; fb ; by g = ; where i = 1; 2; 3 and = 1; 2 is a spinor index. Then the Hamiltonian derived from (3.2) and the generator of the U(1)R symmetry b ! ei#b can be expressed as H^ = aya + R^ = byb : 2 3 byb ; pm b the algebra The normal ordering constants of the bosonic and fermionic parts cancel each other in H^ , so the ground state energy is zero. The Hilbert space is spanned by the Fock eigenbasis { 9 { The action of a massive particle in this background is If x ` and x_ 1 at any given time, the classical motion of such a particle remains nonrelativistic at all times. In this regime, the action becomes S m dt We have explicitly reinstated the speed of light c, to emphasize that is most naturally viewed in this nonrelativistic setting as the universal oscillation frequency of a particle in the AdS gravitational potential well. The action obtained here reproduces the bosonic part of equation (3.2), con rming the interpretation of announced earlier. 3.1.3 Fermionic excitations, de nitions of energy and R-frames To extract the physics of the fermionic degrees of freedom, we have to quantize the system. This is straightforward in the case at hand, and done in general in appendix D. We introduce canonical bosonic and fermionic annihilation operators, ai p m p 2 Xi + i p21m Pi and respectively, together with their conjugate creation operators, which satisfy (3.3) (3.4) (3.5) (3.6) (3.7) The equation of motion for Ynew is then Ynew = ei 2 R t Yolde i 2 R t : d i dt Ynew = H^ + 2 R^; Ynew : Thus, the operator H^ = H + 2 the energy operator in that frame. The Hamiltonian H^ becomes R^ de nes time translation in a di erent R-frame and is H ^ = H^ + and the fermionic energy gap becomes built by acting with the aiy and by on the ground state. Since by transforms in the spin 12 representation of SO(3) and has R-charge 1, the fermionic sector consists of a spin zero state with R-charge 0, one spin 12 doublet with R-charge 1 and another spin 0 state with R-charge 2. Browsing through the tables of OSp(2j4) multiplets in, e.g., the appendix of [23], one sees that our superparticle, together with its antiparticle, produces exactly the spin and R-charge content of a massive N = 2 hypermultiplet in AdS4 (table 7). However, at rst sight, there seems to be a mismatch with the energy spectrum. For our superparticle, the energy gaps for bosonic and fermionic excitations are, respectively, Under the identi cation (3.5), EB is exactly the scalar normal-mode energy gap in AdS, with respect to standard AdS global time. However, according to table 7 in the appendix of [23], we should then nd EF = 12 instead of 32 . The solution to this puzzle is that our identi cation of H^ as \the" energy is ambiguous, since we can always shift to a di erent R-frame. In the Heisenberg picture, the transformation in equation (2.7) becomes EB = ; EF = From the particle mechanics point of view, these di erent notions of energy H^ are all equally valid and just amount to a relabeling of conserved charges without a ecting the bosonic part of the symmetry algebra (see appendix D). However, only one choice of corresponds to the bulk gravitational AdS energy de ned with respect to global AdS time, used in [23]. Matching energy gaps EF , we see this is the case for = 2, so we are led to identify H^AdS = H^ j =2 = H^0 + R^ = aya + 2 1 byb : The relation HAdS = H0 + R turns out to hold in general, as we show in appendix E. We reiterate that does not label truly di erent models the way, for example, di erent values of do. Rather, it labels di erent reference frames, related to each other by the constant R-rotations given in equation (2.7). This is similar to ordinary particle mechanics in a cylindrically symmetric potential described in di erent rotating frames. The rotation shifts the Hamiltonian of the system by the corresponding angular momentum. Physically, observers in di erent frames will perceive di erent centrifugal and Coriolis forces, but water remains water, and wine remains wine. One particular frame may be singled out if the system is part of a larger context, like a lab or distant stars. This is the role played by AdS4, which singles out the = 2 frame. When N > 1, the Lagrangian in equation (3.1) contains interesting nonabelian interactions. more precisely l = p 2 The commutator-squared term already appeared in the at-space quiver matrix models. It also universally appears in the worldvolume theories of N coincident D-branes. Since we are interpreting X as a position coordinate of a D-particle in AdS, we want to assign it the dimension of length. Similarly, we want t to be time and m to be a mass. However, then the commutator-squared term actually has the wrong dimension compared to, say, the kinetic term. This can be traced to the way the original at-space quiver matrix model was obtained in [4]. It was essentially by dimensional reduction from a four dimensional gauge theory, in which the dimension of Xi = Ai is naturally inverse length and m = g12 Vol3 is length cubed. To get the dimensions we want, we can rescale the elds and parameters by powers of a reference length l. This will produce explicit powers of l at various places, including a factor l 2 in front of the commutator term. Equivalently and more conveniently, we can pick units such that besides c 1 and ~ 1, we also have l 1. Matching the kinetic and commutator terms with the standard expressions for D-brane worldvolume theories in the literature [24], we see the appropriate length scale is the string length, or 0. Thus, throughout we will be working in units with Ftijk = 3 ijk : Perhaps the most interesting new element in the massive quiver Lagrangian in equation (3.1) is the presence of a Myers term in L0V , L0V = i m ijk Tr(XiXj Xk) + : Such terms arise in D-brane physics from the presence of background ux with legs in the directions transversal to the brane [24]. They allow multiple coincident branes to polarize into higher-dimensional dielectric branes. Speci cally, for N coincident D0-branes in a background R-R 4-form ux F , in the units of equation (3.14) and the conventions of [24], this term in the D0-brane Lagrangian is LMyers = i m 1 3 Ftijk Tr(XiXj Xk) ; the deformed quiver has a Myers coupling to an e ective ux background where m = T0 = 1 gsp 0 is the mass of the D0-brane. Comparing to equation (3.15), we see (3.14) (3.15) (3.16) (3.17) 1 2 U(N) . . . κ cluding the coe cient,2 with the 4-form 1 = `AdS in equation (3.5), this precisely agrees, incations of 11-dimensional supergravity [11], such as, for example, the 11d AdS4 or 10d AdS4 CP 3 compacti cations [5, 6] dual to ABJM theory [7]. The most striking consequence of the presence of such cubic terms is the Myers effect [24]; N coincident D0-branes polarizing into a stable fuzzy sphere con guration, which, at large N , approximates a spherical D2-brane with N units of worldvolume ux. In the case at hand, unlike the at space case studied in [24], the fuzzy sphere is supersymmetric. This can be seen from the supersymmetry variations of the gaugino given in section 2.2: S7=Zk = (DtXi) i + 2 1 ijk[Xi; Xj ] k + iD i Xi i : The con guration is supersymmeric provided = 0, that is to say, provided X is time independent, D = 0 (trivially the case here, by the equations of motion for D), and (3.18) (3.19) (3.20) A maximally nonabelian solution to this equation is given by the N -dimensional representation of SU(2), yielding, at large N , a fuzzy sphere of radius [24] [Xi; Xj ] = i ijk Xk : Rfuzzy = N = 1 2 0 `AdS N : Notice that R-R 4-form ux is sourced by D2-branes, or its uplift to M-theory by M2-branes. Thus, the fuzzy sphere can be interpreted as a membrane which has separated itself from the stack of membranes generating the AdS4 Freund-Rubin compacti cation, supported by its worldvolume ux. Such membranes in AdS4, known as dual giant gravitons, were studied in, e.g., [ 25 ]. 3.2 3.2.1 sort out. One node, arrows Lagrangian and consistent truncation To focus speci cally on the new features brought by adding chiral multiplets and to simplify things as much as possible, let us consider a single-node quiver with arrows from the node to itself, see gure 3, and let us put X = 0 ; = 0 ; = 0 : (3.21) 2The sign depends on a number of conventions for orientations and charges, which we did not try to Classically, this is a consistent truncation since there are no terms in the Lagrangian involving one of these elds coupled linearly to the other elds or to constants. Note that it is not possible to set A or D to zero in this way, since they do couple linearly to, for example, . Thus, the truncated theory consists of a U(N ) gauge eld A and auxiliary D together with adjoint scalars a, a = 1; : : : ; . The covariant derivative acts as Dt i[A; ], so the diagonal U(1) part does not couple to anything in the Lagrangian except to the constant . Thus, as in the pure vector case discussed earlier, for get the consistency condition = 0. Integrating out the auxiliary elds then puts 6= 0 we D = m a 1 P [ ay; a] ; (3.22) and the Lagrangian (2.1) becomes, i 2 L = L0 + L0 L 0 = Tr jDt aj2 L0 = Tr Dt ay a ayDt a : 1 2m The Gauss's law constraint obtained by varying A is Having derived the Gauss's law constraint, we can pick a gauge A Single particle: internal space magnetic and Coriolis forces Let us rst have a look at the N = 1 case with W = 0. Then the Lagrangian collapses to This describes a charged particle with complex position coordinates a moving on a at Kahler manifold C , in a background magnetic eld F proportional to the Kahler form: L = j _aj2 + _a a a _a : F = d a ^ d a : If the D-particle under consideration is a D0-brane in ten-dimensional IIA string theory, the a are complex coordinates on the physical compacti cation space, and F is then to be identi ed with an R-R 2-form ux. To more accurately describe such a situation, for example a D0-brane on the AdS4 CP 3 geometry dual to ABJM theory [7], we should consider a generalization of our quiver models to general curved-space Kahler potentials. This can be done, but we won't do it here. Sticking with our simple model, the classical solution to the equations of motion is a cyclotron motion of frequency with arbitrary center C and amplitude A: a(t) = Ca + Aa e i t : In particular, in the limit jAj ! 0 the e ect of target-space curvature should become negligible, so we expect that in this limit the frequency of motion should be independent i 2 i 2 (3.24) (3.25) (3.26) (3.27) of the curvature. Comparing to the case AdS4 CP 3 mentioned above and reviewed in detail below in section 4, it can be checked that the cyclotron frequency (with respect to global AdS time) of a D0 moving in the 2-form ux-carrying CP 3 also equals . Before jumping to conclusions, we should recall, however, that this is the motion in the original, = 0, R-frame, whereas the natural, \inertial" R-frame for D-particles in N = 2 global AdS4 corresponds to = 2. This was illustrated in section 3.1.3 and shown in general in appendix D. Thus, for proper comparisons we should transform the motion in equation (3.27) by a eld rede nition, as in equation (2.7), with = 2, yielding a(t) anew(t) = ei qat oald(t) = Ca ei qa t + Aa ei(qa 1) t ; where qa is the R-charge of a. The Lagrangian in equation (3.25) becomes L = j _ aj2 + i (1 2qa) 2 _ a a a _ a + qa(qa 1) 2 j aj2 : The new terms in the Lagrangian can be thought of as centrifugal (or electric) and Coriolis (or magnetic) forces due to the rotating frame transformation. An interesting symmetry can be observed between qa = 0 and qa = 1. In both cases the Lagrangian describes free motion in a magnetic eld of equal magnitude, but of opposite sign. In line with this, the transformed motion in equation (3.28) is so ! a symmetry qa $ 1 magnetic force. a(t)jqa=0 = Ca + Aa e i t ; a(t)jqa=1 = Ca ei t + Aa ; (3.31) and the roles of amplitude and center get switched. More generally, there is qa. The xed point qa = 12 is distinguished by the absence of any In the context of an actual N = 2 ux compacti cation to AdS4, the R-charge is identi ed with the R-charge of the N = 2 AdS4 superalgebra OSp(2j4), which in turn typically arises as the Kaluza-Klein-charge of an isometry of the compacti cation manifold. Hence, in such a setup we could, in principle, determine the actual qa. Here our discussion is more general and we do not know, a priori, the values of the qa. In the present case we have assumed W = 0, so any choice would give an R-symmetry. On the other hand, if W 6= 0, the qa are not arbitrary, but constrained by the homogeneity condition in equation (2.6). We consider this case next. 3.2.3 Nonabelian case with superpotential A natural example of a system with a nonzero superpotential is the case of three arrows, = 3, and N > 1, with superpotential with c some constant. This choice leads to a commutator-squared type potential: W = c 3 abc Tr a b c ; c 2 2 Tr [ a; b] [ a; b]y : (3.28) (3.29) (3.30) This superpotential actually has an extended, nonabelian R-symmetry, but in line with the above discussion let us consider just the U(1)R subgroup. The constraint in equation (2.6) becomes This has a two-dimensional family of solutions, all of which provide R-symmetries of the system. This is a manifestation of the presence of an extended R-symmetry. One possible choice is After transforming to the = 2 AdS frame by rede ning, as before, a(t) e i qat a(t), with (s1; s2; s3) = ( 1; 1; +1). To compare this to the nonabelian D0-brane Lagrangian in real coordinates, we decompose the complex matrices a into real and imaginary parts as a = ua + iva. Then we have Tr [ a; b] [ a; b]y = Tr [ua; ub]2 + [va; vb]2 + [ua; vb]2 + [va; ub]2 + 2 Tr [ua; va][ub; vb] ; Tr [ ay; a] [ by; b] = 4 Tr [ua; va][ub; vb] ; and thus, denoting (y1; y2; y3; y4; y5; y6) (u1; v1; u2; v2; u3; v3), 1 2 1 2m Tr [ a; b] [ a; b]y + [ ay; a] [ by; b] = Tr [ym; yn]2 : (3.37) Hence we see that for this particular combination of complex eld commutators, which corresponds to setting c p2=m in (3.36), we obtain an SO(6)-symmetric commutator squared term. Finally, to bring the kinetic term into the same form as the kinetic term of the vector X as in (3.1), we rescale yn = p m2 Y n, so (3.36) becomes where "mn is a block-diagonal antisymmetric matrix with "12 = "56 = +1. If we call the space parametrized by the 3-vector Xi external and the space parametrized by the 6-vector Y n internal, then this is precisely the 6d internal-space part of the Lagrangian of a stack of D0-branes with a at internal space threaded by the R-R 2-form eld F = dY 1 ^ dY 2 dY 3 ^ dY 4 + dY 5 ^ dY 6 : Apart from the orientation reversal of the (12) and (34) planes, this is the same magnetic eld as in equation (3.26), leading to cyclotron motions with frequency similar to those in equation (3.27). As noted there, this is also the D0 cyclotron frequency in ux compacti cation dual to ABJM theory, so we see that this simple quiver model already captures quite accurately the dynamics of D0-branes in string ux compacti cations. This could be improved further by generalizing the models to arbitrary target space Kahler potentials. This D0-model is just one of many possible one-node quiver models. Di erent values of c, for example, lead to models with the SO(6) symmetry of the Lagrangian in equation (3.38) broken so some subgroup, modeling D-branes in ux compacti cations with fewer isometries than AdS4 CP 3. More generally, instead of D0-branes, we can model internally wrapped branes of di erent dimensions, possibly carrying worldvolume uxes, and so on. The adjoint scalars a will then correspond to geometric-deformation moduli of these brane con gurations. From the general form of the deformed quiver Lagrangian it is clear that magnetic elds like (3.26) will be a generic feature. However, in general these magnetic elds live on D-brane moduli space; they no longer have a direct physicalspace interpretation as in the case of D0-branes. Nevertheless, their e ect will be similar, causing oscillatory motion even in the absence of a potential on D-brane moduli space. Such dynamical features can presumably also be thought of as the result of the presence of background magnetic R-R- uxes interacting with the D-branes. 3.2.4 BMN matrix mechanics The BMN matrix model describes D-branes or M-branes in a supersymmetric plane-wave background [8]. Although its interpretation is quite di erent from the nonrelativistic Dparticles in AdS we have been considering so far, its Lagrangian is nevertheless a special case of our general massive quiver Lagrangian. It corresponds to a quiver with one node and three arrows and a cubic superpotential of the form in equation (3.32), but now with q1 = q2 = q3 = 2 3 ; = 3 2 ; (3.40) instead of = 2 and the R-charge assignments in equation (3.35). Thus, the eld rede nition of equation (2.7) becomes a = e i 12 t a, which, for the chiral scalar eld, is e ectively the same as the case qa = 12 in equation (3.28). From equation (3.30) we can then immediately read o that the magnetic interaction vanishes for the transformed Lagrangian for a in this case, and that instead a harmonic oscillator potential V ( ) = 14 2 j aj2 appears. Performing the same changes of variables as those leading up to equation (3.38), and combining this with the Langrangian in equation (3.1) for the vector multiplet scalars Xi, we obtain the Lagrangian i ijkXiXj Xk + ; (3.41) where the ellipsis denotes the fermionic and the commutator squared terms. (Some of these arise from the interaction part LI0 in equation (2.1), which we have ignored so far in this section.) This correctly matches the BMN model, and it can be checked that the same holds for the fermions. The equality of the R-charges allows for an enhancement of the R-symmetry group from U(1) to SO(6). Consequently, the superalgebra is also enhanced from su(2j1) to su(2j4). HJEP01(26)5 Our next example is a quiver with two nodes and arrows, and dimension vector (1; 1), hence two U(1) vector multiplets containing the scalars x multiplets containing the scalars a, see gure 4. Since we will not need the fermions in our discussion here, we just give the bosonic part of the Lagrangian, Lb = LbV + LbC , with iv and charge ( 1; 1) chiral aDt a (Dt a) a ; 1D1 2D2 1A1 + 2A2 ; A2) a. Since a superpotential is forbidden by gauge invariance, the equations of motion for the auxiliary elds F a are trivial; that is F a = 0. Varying the gauge elds together, i.e., A1 = A2, gives the consistency constraint ( 1 + 2) = 0 : This is analogous to the = 0 consistency constraint we found for the single node quiver. In what follows we will therefore assume 1 + 2 = 0. As in the single node case, the xiv may be thought of as the positions of two massive particles near the bottom of an AdS potential well, with = 1=`AdS. Despite the absence of translation invariance when 6= 0, the exact proportionality of potential and kinetic terms for the vector multiplet means it is still possible to separate the Lagrangian into a decoupled center-of-mass (c.o.m.) part and a relative part, L = Lc:o:m: + Lrel. Given the gravitational interpretation of the potential, this can be interpreted as a consequence of the equivalence principle. De ne c.o.m. variables Y0 and relative variables Y for the vector multiplet as Y0 (m1Y1 + m2Y2)=(m1 + m2) and Y Y2 the bosonic part of the c.o.m. Lagrangian is given by Y1, for Y = xi; A; D; . Then b Lc.o.m. = m1 + m2 x_ 2 0 2 2x20 + D02 ; (3.42) (3.43) (3.44) (3.45) where We may also integrate out the auxiliary D eld, leading to the following potential: HJEP01(26)5 iA) a j j 2 : > 0, the potential attains its (zero) minimum at x = 0, j < 0, the potential reaches its (nonzero) minimum at x = 0, a = 0. If x is held xed aj2 = , whereas at some su ciently large xed value, V ( ) is minimized at = 0. In the at-space case, these observations essentially determine the bound state formation properties of the system [4]. In the massive case however, the Gauss's law constraint will signi cantly alter this analysis. We turn to this next. 3.3.2 The Gauss's law constraint and con nement The Gauss's law constraint will turn out to have rather dramatic consequences; when 6 = 0, it causes the particles to be con ned by fundamental strings. The Gauss's law constraint is obtained by varying the Lagrangian with respect to A. Working in the gauge A 0, we obtain: j j 2 + i a _a _a a = 0 ; and the bosonic part of the relative Lagrangians is Lrbel = 2 x_ 2 2x2 + D2 + ( A + jDt aj2 (x2 D)j aj2 aDt a aj2. One immediate consequence is that and that more generally, the Gauss's law constraint will force the a to be in some excited state. Clearly then, the naive energy minimization analysis under (3.49) must be modi ed. = 0 is inconsistent if In what follows we will interpret the constraint in string theory as a lower bound nmin on the number of physical open strings stretched between the two particles. More speci cally, we will nd nmin = . Recall that in a 10d string context, the two particles in AdS correspond to two distinct internally wrapped D-branes, and the a correspond to the lightest open-string modes that exist between the two branes. Roughly speaking, this means that if we hold x xed and quantize the a, the n-th energy level can be thought of as a state containing n stretched strings, all in their string oscillator ground state. To check this, let us assume a large xed separation jxj, so we can view the frequency jxj. Their n-th level excitation energy is then equal to njxj. Keeping in mind our choice of units, equation (3.14), this is indeed the energy of n fundamental strings stretched over a distance jxj. In the same vein, classical eld excitations of a may be thought of as quantum coherent states, which are superpositions of in nitely many di erent string number eigenstates. For large amplitudes however, the string number probability a as simple harmonic oscillators with The mode expansion implies we are in Coulomb gauge since raBa = B~a^raV a^+ = 0 a (using (F.48)) and is relevant to other places in the reduction. An immediate simpli cation arises from this expansion. Any terms in the Lagrangian which contain the complex scalars, auxiliary elds, or time component of the gauge eld are constant over the S3 and can immediately be pulled out of the spatial integral. If these are the only elds present in these terms, then these terms retain their form in the reduced Lagrangian. In particular, we have the zero-mode action i Rs S0 = 2 2R3 s dt D^t ~iyD^t ~i ~iyD^t ~i (D^t ~iy) ~i + F~iyF~i + F~iWi( ~) + F~iyWi( ~y) + Tr(D~ 2) + ~iyD~ ~i + Tr D ~ elds there is a nite number of SU(2)r invariant modes. It is possible to go beyond this invariant sector and consider general mode expansions of the elds in S3 spherical harmonics, classi ed by SU(2)l SU(2)r representations. See appendix F for details and conventions; in particular, table 2 lists all scalar, spinor and vector harmonics. The SU(2)r-invariant sector corresponds to ` = 0 in this table. The truncated mode expansions we substitute are thus The only terms left to integrate over are those containing spatial derivatives, spatial components of the gauge eld, or both. We do not show the remaining calculations here, although we note that the reduction of the bosonic part of the vector Lagrangian was already given in section 5.1. 2 3 ^=1 a^=1 1 2 1 2 2 ~ B0 Rs : (5.42) (5.43) (5.45) i(t; ~x) = ~i(t) ; B0(t; ~x) =B~0(t); 2 ^=1 (t; ~x) = X ~ ^(t) S ^+(~x); D =D~ (t) : i (t; ~x) = X ~i^(t) S ^+(~x) ; F i =F~i(t) Ba(t; ~x) = X B~a^(t) Vaa^+(~x) (5.39) (5.40) (5.41) Next let us consider terms with fermion bilinears. Due to (F.31) and (F.33), fermion bilinears also reduce to the zero mode on the sphere. In particular, 1 Thus, we can add the following fermion terms to our reduced action S1=2 = 2 2R3 s dt i ~iyD^t ~i Wij ( ~) ~i ~j Wij ( ~y) ~iy ~jy + 1 2 gYM iTr(~yD^t ~) + p2i( ~iy ~ ~i 1 2 ~iy ~y ~i) : The nal steps are to get the Lagrangian of the massive quivers in equation (2.1) are few in number. All chiral multiplets are coupled to two vector multiplets, one with (cw)i = +1 and the other with (cv)i = 1, so that it is in a bifundamental represenation of the gauge group. This amounts to replacing the single spatial gauge eld by a di erence of gauge elds Ba ! Ba1 Ba2 and contracting the gauge indices appropriately. The parameters from the four-dimensional theory are related to the quiver by = 2 Rs mv = 1 2 gYM ; v = and for the elds B0 ! Av; ~iy ! Ba^ ! Xvi ; a ~iy ! D~ ! Dv F ~i ! F a; Note the minus sign on the spatial components of the gauge eld. The reasons for this minus sign are to identify with the at space quivers when we take ! 0. This convention is due to how one chooses to implement gauge invariance along with various other conventions for the Pauli matrices and fermion kinetic term. Dimensional reduction of the supersymmetry transformations Here we give one example of how to obtain the supersymmetry tranformation for the eld B~a^. It is performed by projecting onto the proper spherical harmonic. One has F~iy ! F ay : (5.48) HJEP01(26)5 B~a^ = = = = 1 1 1 2 2Rs3 a^^b S3 2 2Rs3 a^^b S3 2 2Rs3 a^^b S3 V ^b+a( Ba) (i~y^ ~^ (i~y^ ~^ (i~y a^ i ~y^ ~ ^)(S+^)y_ a_ S ^+V ^b+a i ~y^ ~ ^)( 1)( ^b) ^ ^ (5.46) (5.47) (5.49) (5.50) (5.51) (5.52) (5.53) (5.54) After making the replacements (5.47) we have which matches (B.5). For scalar elds, one integrates against the mode Y(000) = 1. For the fermions, one integrates against the mode (S+^)y_ 0_ . For example Comparison to other models and generalizations The superalgebra for the massive quiver matrix models is su(2j1) o u(1)R in the frame, as noted in appendix D. If no R-symmetry is present, one is restricted to the = 0 = 0 frame and simply drops the u(1)R factor and the R-generator. Quantum mechanics with su(2j1) symmetry has been studied before [19, 20]. The algebra listed in equation (2.1) of [19] is our algebra in the = 1 frame.12 The supersymmetries are time-independent and consequently the superalgebra is a centrally extended su(2j1) where the Hamiltonian plays the role of the central charge. They also comment on the ability to shift the Hamiltonian by \mF ," which in our language is accounted for by di erent R-frames. They also study di erent multiplets. In particular, their (2; 4; 2) multiplet corresponds to our chiral multiplets, but they have Lagrangian expressions for an arbitrary Kahler potential. After changing to the = 1 R-frame and removing all instances of the vector multiplets, our chiral Lagrangian agrees with equation (5.11) of [19] with the choice of a at Kahler potential. On the other hand, our superpotential is arbitrary up to R-symmetry constraints. To generalize our results to an arbitrary Kahler potential, we can perform the dimensional reduction on the chiral Lagrangian (5.11) with abitrary Kahler potential, that is, equation (6.5) of [21], with the background supergravity elds set to the appropriate values. The calculation is simpli ed by the fact that the Kahler potential, a function of the scalars, will only contain the zero mode scalar harmonic of the S3. Furthermore, the identities of appendix F show that fermion bilinears (e.g., ) reduce to the scalar mode on the sphere even before integration. The result matches equation (5.11) of [19]. Since the reduction preserves the SU(2j1) symmetry, the dimensional reduction remains classically consistent for arbitrary Kahler potential as well. In contrast to other su(2j1) quantum mechanical systems, we have gauge invariance. The vector multiplet acts as the (3; 4; 1) multiplet coupled to the `gauge' multiplet, as described in [38], except that our models take into account the mass deformation parameter . Without chiral multiplets, the abelian vector multiplet Lagrangian is just that of a supersymmetric harmonic oscillator, as described in section 3.1.2. Nonabelian gauge invariance allows for greater complexity in the form of cubic and quartic terms. We should be able to obtain even more general Lagrangians using an abitrary Kahler potential and kinetic gauge function in the dimensional reduction. Gauge invariance for an arbitrary Kahler potential is possible with constraints relating the Kahler potential, kinetic gauge theory on R function, the structure constants, and moment maps [29]. The reduction of an N = 1 S3 with arbitrary Kahler potential, superpotential, and kinetic gauge function, obtained directly from the \new minimal" supergravity [36], would yield some very interesting su(2j1) quantum mechanical models. 6 We have discussed the signi cance of our results in the introduction and throughout, so we conclude with a number of interesting questions that we leave for future work. One key question, raised in [14], is whether the molecule-like quantum Coulomb branch bound states of [4] persist even in the presence of con ning strings. Another set of questions is related to generalizations. We imposed SO(3) symmetry and N = 4 supersymmetry, and as in [4] we restricted to a at target-space for simplicity. Relaxing any of these will lead to new models. As mentioned before, generalization to arbitrary Kahler metrics for the chiral 12One must also make the identi cations m = , F = 12 R, and I = Jk k . multiplets can be obtained by dimensional reduction on R the vector multipet scalars more work will be needed; the (3; 4; 1) multiplet of [20] and the deformed S3-reductions of [22] suggest some possible directions. In further explorations of the model itself, it would be useful to check, for example, the bound state predictions discussed in section 4.5 against independent results. In the at-space case it is known that quiver predictions can be unreliable if there is no point in the physical moduli space where the FI parameters all vanish [16]; analogous subtleties may arise in the present case. Finally, it would be very interesting to apply this model to understand aspects of the microscopic dynamics of black holes, generalizing some of the successes of quiver matrix mechanics in at space. For example, some of the fuzzy membrane ideas explored in, e.g., [39{42], might nd a more natural home in the current setup, in view of the existence of the supersymmetric, nonabelian, fuzzy membrane ground states of massive quiver matrix mechanics, in contrast to the at-space case. Additionally, all of the above can be reinterpreted as questions in a holographically dual CFT. Acknowledgments We thank D. Anninos, T. Anous, D. Berenstein, R. Monten and C. Toldo for helpful discussions. CA and FD were supported in part by a grant from the John Templeton Foundation, and are supported in part by the U.S. Department of Energy (DOE) under DOE grant DE-SC0011941. ED is supported by the DOE O ce of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC0506OR23100. A De nitions and conventions We use the following index conventions for the quivers: quiver nodes, v; w; quiver arrows, a; b; c; : : : ; SO(3) vector indices, i; j; k; : : : ; SO(3) spinor indices, ; ; ; : : : ; gauge indices, A; B; C; : : : The eld content of an N = 4, d = 1 quiver matrix model Q is described by a graph with nodes v 2 V and directed edges (arrows) a 2 A, and dimension vector N = (Nv)v2V . To each node is assigned a linear, or vector, multiplet (Aiv; Xvi ; v ; Dv) with i = 1; 2; 3. The eld Av is the gauge eld for the group U(Nv). The adjoint of U(Nv). To each edge a : v ! w is assigned a chiral multiplet ( a; a ; F a). All elds of a chiral multiplet live in the bifundmental (Nw; Nv) of U(Nw) U(Nv). The quiver model is described by the additional real scalar parameters: the mass mv of the particle represented by each node; an FI parameter for each node v; the mass deformation parameter or coupling constant . The quiver may also have a superpotential W ( a), a holomorphic function of the chiral multiplet scalars a . If the superpotential satis es the homogeneity condition W ( qa a) = 2W ( a), then the quiver model has an R-symmetry, with R-charge qa for a and for the other elds as in table 1. elds Xvi , v , Dv live in the Gauge transformations exist for each node individually and act on gauge elds, adjoint elds gv 2 (Xvi ; v ; Dv), and bifundamental elds ha 2 ( a; a ; F a) as gv ! UvgvUvy; h ! UwhaU y v with a : v ! w. The covariant derivatives are given by iAw a + i aAv; The fermions are Grassmann valued SO(3) Weyl spinors. The complex conjugate of a spinor is denoted with a dagger and has upper indices y = ( ) We always explicitly write out the Levi-Civita symbol, i.e., . Fermion bilinears are formed by contracting with the Levi-Civita symbol or the Pauli matrices ( ) = y i = y i = 1 = 12 = 21 = 1 ( y ) = y 2 = 0 i 0 i 3 = 1 0 ! 0 1 Complex conjugation on a vector multiplet only a ects the fermion indices as those elds are represented as Hermitian matrices. On chiral multiplets, complex conjugation changes the representation from (Nw; Nv) of U(Nw) U(Nv) to the (Nw; Nv). It additionally acts on fermion indices. Thus, when we consider an arbitrary eld Y y, it is unambiguous to have the dagger act on the gauge indices of elds as matrix Hermitian conjugation and spinor indices as complex conjugation, should any be present. B Matrix model Lagrangian For convenience, we restate the Lagrangian and supersymmetry transformations given in section 2.1. The Lagrangian of massive quiver mechanics with deformation parameter is L = L0 + L0 where L0 is the original, undeformed, at-space quiver Lagrangian, given in appendix C of [4], and L0 is the mass deformation: L 0 = L0V + LF I + L0C + LI0 + LW 0 0 1 i + ( yvDt v (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (B.1) (B.2) L0C = X Tr jDt aj2 + jF aj2 + ( ayDt a aXvi ) v ay) L0W = LI0 = L0V = a X a:v!w L 0 = L0V + L0F I + L0C X mvTr 1 2 TrAv 2 v X v X Tr i y v i y i v a p 2 i 2 y (Dt a) Dt ay a The action is supersymmetric with respect to the transformations i Xvi i Dv = y v 3i 2 p p We obtained the Lagrangian and its supersymmetries given in section 2 by dimensional reduction as described in section 5, but also by more direct methods; that is to say, by considering the most general SO(3)-symmetric and gauge invariant mass deformations and inferring, through direct computation, the parameter constraints and additional terms needed to preserve N = 4 supersymmetry. Our analysis along those lines leads us to conjecture that the Lagrangian in equation (2.1) is the most general N = 4 mass deformation preserving SO(3)-symmetry, assuming a at target-space metric for both vector and chiral scalars. In this appendix we will demonstrate in some detail how supersymmetry constrains the Lagrangian, using direct methods. Since the computation is lengthy and not particularly illuminating, we will not reproduce it in its entirety here, focusing instead on clarifying the supersymmetric origin of the terms responsible for some of the most interesting physical features of the model, as explored in section 3. C.1 Vector multiplet The supersymmetry of the FI Lagrangian LF I = Tr ( A D) is easy to check: 2 LF I = y : Here the commutator term result is a total derivative, and similarly for i[Xvi ; yv] i in Dv vanishes once one takes the trace. The y LF I . Hence, the FI action is separately invariant. Notice that the mass deformation requires the FI parameter to couple to the gauge connection. As discussed in section 3.3, it is this coupling that leads to particle con nement. To check the supersymmetry of the remainder of the vector multiplet Lagrangian, it is useful to exploit its R-symmetry and rst rede ne the elds as in (2.7) with = 1, so that the supersymmetry parameter becomes time independent. We can then save ourselves some e ort by making use of the fact that we already know the undeformed ( = 0) model is supersymmetric under time-independent supersymmetry transformations. Denote L = L 0 + L0, where L0 is L evaluated at = 0, and similarly supersymmetry variation at = 0 and 0 the additional variation due to the extra terms proportional to in the supersymmetry variations. Then we already know so what remains to be shown is 0 0 = 0 + total derivative; L L 0 + 0 0 = 0 + total derivative : (C.1) (C.2) If did depend on time, then we could not separate the variations order by order in because time derivatives would yield addition factors of . This is why we consider here a frame in which is time independent. Performing the substitutions in equation (2.8) with = 1 we obtain 0Av = 0 0Xvi = 0 0 v = i Xvi i 0Dv = i y v i y v 0 a = 0 0 a = 0F a = p 2 p 2 2 2 qa (qa a y 2) y a Without loss of generality, we may consider the case of a single vector multiplet, say of mass mv = 1. Then after the substitutions in equation (2.8) with = 1 we have L0V = Tr L0V = Tr 1 2 1 4 1 D2 + 1 [Xi; Xj ]2 + 2 ijkXiXj Xk : ( yDt (Dt y) ) Notice that the coe cient of y is di erent from the one in (2.3), due to the shift, in equation (2.8), of the Dt terms in the full Lagrangian. The relevant variations are now relatively easy to compute, using the cyclic property of the trace and other algebraic and 0 L0V turns out to be minus this, up to a total derivative. As the sum of these is therefore a total derivative, this establishes the supersymmetry of the vector multiplet sector. Notice, in particular, that the mass deformation necessitated the addition of a Myers-like term cubic in the Xi. The physical implications of this term were reviewed in section 3.1.4. C.2 Chiral multiplet and vector-chiral interaction terms Checking the supersymmetry of the full Lagrangian is long and elaborate, but it can be organized by, for example, collecting all terms in the variation with the same elds in the same degree, which must cancel among each other. As an example, in the full variation of LI , which we give below in equation (C.3), one can see that there are terms with the same elds in the same degree among the non-quadratic terms in L0C and L0C . We give all these terms in equation (C.3), then collect all terms proportional to . We then show how these collected terms cancel with other terms in the variation, up to a total derivative. The remaining terms (those that are not in boxes in equation (C.3)) are exactly those one would obtain in the variation of the at-space, = 0 quiver Lagrangian, which are known to cancel [4]. For simplicity, we rst integrate by parts, so the chiral fermion kinetic term is i ayDt a and we start with the variation with respect to . We discuss the variation with respect to y afterwards. The computation is implicitly under a trace of color indices; the adjoint of U(Nv) for the vector multiplets and the bi-fundamental for the chiral multiplets. Blue text color in these portions indicate the variation of the elds. Terms are groups by colored boxes. Except for the groups immediately after the variation, the terms in the following lines have the same text color as the boxes they came from. LC (non-quadratic) + LI = Dt ay[ i(i yw ) a + i a(i yv )] + [i ay(i yw ) i(i yv ) ay]Dt a (C.3) + i [i ay(i yw ) i(i yv ) ay] a p + i 2(( ) i)( ayXwi Xvi ay)Dt a +i ay( i(i yw ) a +i a(i yv )) 1 [ p2 i( )( ayXwi Xvi ay)] 2 3i p = Dt i 2( ayXwi Xvi ay)( )( i a) 1 2 ( ) ay i(Xwi a aXvi) p2( ayXwi Xvi ay)(( ) i) j(Xwj a aXvj) ay i (i yw i ) a a(i yv i ) ay i Xwi(p2 Fa) ( 2 Fa)Xvi p ip2 ay (DtXwi i Xwi) i + 1 ijk[Xwi;Xwj] k +iDw 2 (DtXvi i Xvi) i + 1 ijk[Xvi;Xvj] k iDv 2 ay a p i 2( ay w p v ay) ( 2 Fa) p p +i 2 i 2(Dt ay)( ) 2( ayXwi Xvi ay)(( ) i) ( yw a a y) v = ip2( ayXwi Xvi ay)( ) i(Dt a) p 1 + 2 ( ) ay i(Xwi a aXvi) 2 2 i(Dt ay) ( ) j(Xwj a aXvj) ip2 ay (DtXwi) + i Xwi (DtXvi) + i Xvi ay ( )( i a) (C.4) (C.5) (C.6) ay + 2i = Dt = Dt + i 1 2 variation. i ayDt = Dt ay (( yw ) a a( yv )) ay(( yw ) a a( yv )) 3 2 (Dt yw) + i y w y 2 (Dt ay)(( yw ) a a( yv )) ( ay( yw ) ( yv ) ay)(Dt a) HJEP01(26)5 i(Dt ay) (( yw ) a a( yv )) ay((Dt yw) a (Dt yv) ) + i + i 3 2 ay(( yw ) a a( yv )) + ay(( yw ) a a( yv )) ay(( yw ) a a( yv )) : ay(( yw ) a a( yv )) For the variation with respect to y, because the total Lagrangian is hermitian, we only need to consider those terms which are asymmetric in time derivatives. The variation of all the other terms will be conjugate to those above. Thus, we only need to redo the 2 i( y )(Xwi a aXvi ) ay[ 2 i( y )(Xwi a aXvi )] 1 2 ay i Xwi ip2( y )(Dt a) ip2( y )(Dt a) Xvi + i 2 ay ( y i)(DtXwi) + i ( y i)Xwi a ( y i)(DtXvi ) + i ( y i)Xvi = 0 : 1 2 ( y ) (Xwi a aXvi ) D Quantum mechanics and operator algebra The elds satisfy the (anti-)commutation relations The adjoint bosonic elds obey the reality condition ((Xvi )AB)y = (Xvi )BA. [(Xvi )AB; (Pwj )CD] = i ij vw A D C B f( w )AB; ( yv )CDg = [( a)AB; ( b)CD] = i ab f ( a )AB; ( by ) D C g = ab 1 mv vw A D A B C D A C D B C B (C.7) (C.8) (C.9) (C.10) (C.11) (C.12) (C.13) (D.1) (D.2) (D.3) (D.4) 3 4 1 2 1 2 1 2mv ay a + + jXwi a +ip2 ( ay w a a 2 1 2 2 ay Tr X mv yv v + X h iqa( ay ay a a) + (qa iPvk + mv Xvk + mv ijk[Xvi ; Xvj ] ( yv k ) v( yv ) 4 2 X a: !v 2 a ay + X b:v! by b 2 aXvi j2 + 2 1) ay ai iWa( ay ) i( ayXwi Xvi ay)( i a + X Tr a + X Tr(jWaj2) 2 ab 1 X Tr(Wab a b + Wab M ij = X Tr Xvi Pvj PviXvj + mv ijn y n v v v v J i = R = Q = 2 1 ijkM jk X Tr + p Qy = X Tr ( a v 1 + p 2 k v) 1 + p 2 The supersymmetry algebra for the massive quivers is spanned by the Hamiltonian H, the generators of the internal SO(3) symmetry J i, the U(1) R-symmetry generator R, and the supercharges Q and Qy . They are given in terms of the elds by H X Tr 1 2mv Pv2i + mv 2(Xvi )2 mv [Xvi ; Xvj ]2 + imv v yv) + mv yv i[Xvi ; v] They satisfy the following algebra [J i; H ] = [J i; R] = [H ; R] = 0 [J i; J j ] = i ijkJ k ay ( yw ) a a( yv ) 2 iPvk + mv Xvk + mv ijk[Xvi ; Xvj ] v( v) + p ay + i 2 + iWa( a) + i( ay i)(Xwi a aXvi ) ay( w) ( v) ay (D.6) (D.7) (D.8) (D.9) (D.10) (D.11) (D.12) (D.13) (D.14) (D.15) (D.16) (D.17) H 2 ) Q ; R + 2 J i i [J i; Qy ] = Qy i 2 [R; Qy ] = + Qy (1 ) Qy All other (anti-)commutators vanish. For convenience, the algebra listed here accounts for all R-frames, characterized by in the presence of an R-symmetry, as discussed in section 2.3. In the = 0 frame, the algebra is easily recognizable as su(2j1) o u(1)R. There are also the generators of U(Nv) gauge transformations (G^v)AB, one for each node of the quiver. We write them in normal ordered form HJEP01(26)5 (G^v)AB = (Xvi )AC (Pvi)CB (Xvi )CB(Pvi)AC mv ( y ) C A ( )CB ( y ) B C ( )AC i X i X a: !v X i ( ay)CB( ay)AC ( a) C A ( a)CB + ( ay ) B C ( a )AC i ( b) B C ( b)AC ( by)AC ( by)CB ( by ) C A ( b )CB : (D.22) A state j i is physical if it satis es all U(Nv) gauge constraints (G^v)ABj i = ABj i : The supersymmetry algebra closes up to gauge terms and thus only closes on physical states. E AdS superalgebras The superalgebra of AdS4, with its fermionic counterpart, and N supersymmetries is osp(N j4). The spacetime algebra is sp(4) ' so(3; 2) and the R-symmetry algebra is so(N ). Let A; B; C; = 0; 1; 2; 3; 1 denote so(3; 2) indices; ; ; ; = 1; : : : ; 4 denote spinor indices of so(3; 1); ; ; ; = 0; : : : ; 3 denote so(3; 1) vector indices; and I; J; K; L = 1; : : : ; N denote so(N ) vector indices. De ne AB = diag( 1; 1; 1; 1; 1), gamma matrices satisfying the Cli ord algebra f I4, and symbols AB by 1 2 = [ ; ]; g = 2 ; 1 tors TIJ = The superalgebra is spanned by Lorentz generators M AB = M BA, R-symmetry genera TJI , and Majorana supercharges QI . The supercommutation relations are [M AB; M CD] = i( ADM BC BDM AC (D.20) (D.21) (D.23) (E.1) (E.2) (E.3) ; M ]= 2 M P M P ); Q ( QI ; ) ; [TIJ ; TKL]= M [P ; P ] = [P ; QI ] = [P ; QI ] = 2i `2 M 2` ( ) 2` I TIJ QI Q ( ) (E.5) (E.6) (E.7) (E.8) (E.9) (E.10) (E.11) (E.12) (E.13) (E.14) ! (E.15) + , [TIJ ; QK ] = i( IK QJ JK QI ); Q ( AB) ; with the Dirac bar de ned as = i y 0. All other supercommutators vanish. We can write this in a more recognizable four-dimensional notation by de ning momentum generators P M ; 1. To connect the algebra with the physical spacetime we introduce the AdS length ` and rescale the momenta and supercharges as Q ! supercommutation relations become p`Q and P ! `P . The S = m d S = m d q _2 t q 1 ~x_ 2 : ~x_ 2 : [TIJ ; QK ] = i( IK QJ JK QI ); [TIJ ; QK ] = i( IK QJ JK QI ) fQI ; QJ g= IJ 2i( ) P + ( M As we take ` ! 1 we obtain the at space super Poincare algebra. If we additionally scale the R-charges T ! `T , then the algebra contracts to the super Poincare algebra with central charges. From our interpretation of the massive quiver matrix models as descriptions of wrapped branes in AdS4, we expect the above algebra to be related to the operator algebra we gave in appendix D. We describe that relationship here. Even though the worldline action of a superparticle in AdS4 is symmetric under the full osp(2j4), gauge xing will leave only some of the symmetry manifest, e.g., reparametrization invariance and -symmetry [8, 31]. For example, the action of a bosonic particle in at space with coorindates (t( ); ~x( )) is given by This action is invariant under the full Poincare group. One can remove the reparameterization invariance by setting t( ) = . The action becomes The remaining symmetry of the action is some reparametrization invariance translation invariance, and rotation invariance. The symmetry + is related to time translations since t( ) = and thus should be identi ed with the Hamiltonian. Indeed, ! the quantum Hamiltonian of a free particle H^ = 21m p^ip^i commutes with the momenta and angular momentum generators. In quantizing the worldline of the free bosonic particle, we have lost boost invariance. The same story is true for the superparticle, except we have an additional fermionic -symmetry. The general action for superparticles in an arbitrary N = 2 (asymptotically at) supergravity background was written down in [31]. We do not possess the superparticle action for an AdS supergravity background, but we expect it to have some of the same features. In particular, gauge xing the -symmetry should remove half of the fermionic degrees of freedom. The resulting algebra of the worldline should then only have four supersymmetries. For the bosonic part of the action, we choose to write the AdS metric in isotropic coordinates as in equation (3.3) and x the reparametrization invariance t( ) = . Once we gauge x, the only remaining symmetries are time translations and rotations; we lose spatial translations and boosts. Thus, we seek a subalgebra of osp(2j4) that contains time translations, rotations, and four of the supersymmetries. Supergravity in AdS has a background U(1) R-symmetry gauge eld which we expect to couple to the superparticle. The subalgebra we want should then also have an R-generator. S = (Q1 We begin with the superalgebra osp(2j4) listed above and form a Dirac supercharge iQ2 )=p2. We additionally de ne boosts and angular momenta generators Ki = M 0i, J i = 12 ijkM jk. The R-charge and anti-commutation relations for the supercharges are [T12; S] = S; [T12; S] = S fS ; S g = fS ; S g = 0 fS ; S g = 2i( ) P + M + T12 : We now switch to a Weyl representation of the gamma matrices ) ; _ = ( I2; i); 1 2 = i 0 0 i 2 =( I2; _ ) _ = ( = 1 2 0 0 ! _ ) _ : We parametrize the Dirac spinor in terms of two Weyl spinors as ST = ( the subscript denotes the R-charge. Conjugation negates the value of the R-charge. The Dirac conjugate is then given by S = ( +; y _ ). The anti-commutation relations become ; +y_ ), where _ ; y f y g = f + ; +g = f _ ; +y g = 0 ( M 2 _ P + 2` T12 2 M P + 2` _ _ T12 (E.16) (E.17) (E.18) (E.19) (E.20) (E.21) (E.22) 1 A : (E.23) f 1 ; y1 _ g = ( _ + 0 f 1 ; y2 _ g = ( _ f 2 ; y1 _ g = 2 Pi f 2 ; y2 _ g = 2 0 _ P0 = 2 Pi + i Ki 1 1 2 2 0 i Ki _ 0 _ )P _ 0 _ )P Lastly, we de ne the spinors 1 = p ( + 0 _ +y ); 2 = p ( y1 _ = p ( y _ + + 0 _ ); y2 _ = p ( y _ 0 _ +y ) _ + _ 0 ) and the non-vanishing anti-commutation relations are = 2 0 _ P0 + 2 i _ J i + 2 0 _ T12 1 2 1 2 0 _ )M 0 _ )M + ` 2 0 _ T12 (E.26) 2 i _ J i + 2 0 _ T12 : The Hamiltonian and angular momenta act diagonally on 1 and 2, but the momenta and boosts mix the two spinors (E.24) (E.25) (E.27) (E.28) (E.29) (E.30) (E.31) (E.32) (E.33) 1 ; y1g [P 0; 1] = [J i; 1] = [P i; 1] = [Ki; 1] = 1 2` 1 2 1 i 1; 2` 1 i 2; 2 i i 2; residual symmetry of the worldline. There are two possible supersymmetry subalgebras to choose from, fP 0; J i; T12; 2 y2g. Both sets contain all the generators needed to describe the The set we identify with that of the massive quivers is the one containing the 1 supersymmetries. To complete the identi cation with eqs. (D.16){(D.21), we should choose ` = 1= , = 2, and P0 = P 0 ! Hj =2 = H0 + R; T12 ! R; J i ! J i; 1 ! Q; y1 ! Qy (E.34) The superalgebras of the gauge- xed worldline of the superparticle traveling in an AdS4 supergravity background and the massive quiver thus match identically in the R-frame = 2. B and C. F Conventions and identities for the dimensional reduction This appendix outlines the conventions used for the spherical harmonics on S3 along with the coordinate systems used. Many of these results are taken from [10], mostly appendices Spin Harmonic Rep of SU(2)` 0 1 2 1 Y(`0m) Y(`1m=2+) Y(`1m=2) Y(`1m)a+ Y(`1m)a (` + 1; ` + 1) (` + 2; ` + 1) (` + 1; ` + 2) (` + 3; ` + 1) (` + 1; ` + 3) (` + 1)(` + 3) Spherical harmonics on S3 The indices a; b; : : : will be `curved' indices while the hatted indices ^a; ^b; : : : will be ` at'. The curved indices can be raised with the inverse metric gab. Flat indices can be raised or lowered with the Kronecker delta a^^b, a^^b . The metric on the sphere S3 of radius Rs is given by ds2 = Rs2(d 2 + sin2( )d 2 + sin2( ) sin2( )d 2) : The Ricci curvature tensor and scalar are given by The curved di erentials are given by and the triads ea^ = eaa^dxa by Rab = 2 R2 gab; s R = gab Rab = 6 R2 : s dx1 = d ; dx2 = d ; dx3 = d e^1 = d ; e^2 = sin( )d ; e^3 = sin( ) sin( )d : The matrices a are the Pauli matrices pulled back to the sphere a^ a = a^ea with a^ the numerical Pauli matrices. Scalar, spinor, and vector functions on S3 can be expanded in a complete basis Y(`0m), Y(`1m=2) , Y(`1m)a . The representation of the harmonics under the full SU(2)` SU(2)r are given in table 2. They are normalized such that 1 2 2Rs3 S3 1 1 2 2Rs3 S3 Y(`0m)Y(`00)m0 = ``0 mm0 Y(`1m=2) Y(`10=m2)0 = ``0 mm0 2 2Rs3 S3 Y(`1m)a Y(`10)m0 a = ``0 mm0 : (F.1) (F.2) (F.3) (F.4) (F.5) (F.6) (F.7) (F.8) The truncation process of section 5 relies on the ` = 0 spherical harmonics and so the remainder of this section will mostly outline their properties and identities. The lowest spherical scalar harmonic is just a constant, For convenience we introduce new variables for the lowest spherical spinor and vector harmonics. Y(000) = 1 : S V a^ a 0 ^ Y(1=2) ; Y(01a^)a ; ^ = 1; 2 a^ = 1; 2; 3 : 2 = r ra. For the vector harmonics, a we have the additional identities Rs2 Y(`1m)b : F.2 Curved and at spinor indices The curved spinor indices for the spinor harmonics as currently de ned are of SO(3) type. To make connection to a theory on R S3 and we promote them to SO(4) SU(2)l SU(2)r type. The conjugates now come with dotted indices and we can use the four dimensional Pauli matrices with greek indices ( , , . . . ) ranging from 0 to 3. Our conventions for the Pauli matrices are those used by Wess and Bagger [32] = ( I2; i); = ( I2; i) : r2Y(`0m) = raY(`1m=2) = (` + )Y(`1m=2) ; 2 r2Y(`1m)a = R2 [`(` + 4) + 2]Y(`1m)a ; s 1 Rs 1 raY(`1m)a = 0 They satisfy the eigenvalue equations 1 m m m (` + 1)(` + 3); A conjugate spinor can be expanded, for example, as After the reduction, we will be left with the SU(2)` symbols ^ and ( a^)^ . ^ ^ The hatted spinor indices become at indices for a representation of SU(2)l SU(2)r. We will not choose to raise or lower at spinor indices with the Levi-Civita tensor. Instead, we will lower them automatically upon conjugation. That is (S^ )y_ (S ^ ) : y_ = (S+^)y_ ~y^ : (F.9) (F.10) (F.11) (F.12) (F.13) (F.14) (F.15) (F.16) (F.17) (F.18) (F.19) Flat spinor indices are contracted according the NW-SE convention. If and are at spinors, we have ^ = ^ ^ y y = y^ y ^ = ^^ y^ y The quiver matrix models are written in terms of the at spinor indices, but the hats have been removed. Further, the Levi-Civita symbol is explicitly written between spinor variables and accounts for the appearance of various minus signs. F.3 Killing equations and explicit representations Using equation (F.10), the spinors S satisfy the Killing spinor equation (F.20) (F.21) (F.22) (F.23) (F.24) (F.25) (F.26) (F.27) (F.28) (F.29) (F.30) (S^ )y aS = ( a^)^ ^V a^ : a 1 2 V a^ a Tr( a^ S y aS ) : + rbVaa^ = 0 : Va^1 dxa = cos( )e1 ^ ^ cos( ) sin( )e2 ^ sin( ) sin( )e3 Va^2 dxa = sin( ) cos( )e1 + (cos( ) cos( ) cos( ) ^ sin( ) sin( ))e2^ (cos( ) sin( ) sin( ) cos( ) cos( ))e3^ Va^3 dxa = sin( ) sin( )e1 + (cos( ) cos( ) sin( ) ^ sin( ) cos( ))e2^ + (cos( ) cos( ) sin( ) cos( ) sin( ))e3^ : Explicitly the Killing spinors are given by raS ^ a 0S ^ : 2Rs 2 S ^ S = exp ^3 exp S The Killing vectors on S3 are given by the lowest mode vector spherical harmonic and are related to the spinors S ^ via the Pauli matrices Inverting this relation we have They are given explicitly by It is can be shown using (F.22) that the Vaa^ satisfy the Killing vector equations (S^ )y_ 0_ S = (S^ )y 0S = ^ X(S ^ )(S^ )y_ = 0 _ : S ^ S = ^ ^ : _ _ (S^ )y_ (S ^ )y_ = ^ ^ = ^^ : They also can be chosen to transform the epsilon symbol in curved space to that of at space Conjugating this equation we have This property is very important because it guarantees that spinor bilinears do not produce higher mode spherical harmonics. Contracting each side with a Killing spinor relates the two kinds of spinors. 0 ^^S ^ = 0_ _ _ (S^ )y_ (S ^ )y_ S ^ = 0_ 0 _ _ _ _ (S ^ )y_ = _ _ (S ^ )y_ : Contracting with another spinor we have ^ ^S ^ S 0 _ _ _ (S ^ )y_ S ^ 0 _ 0 _ _ _ = : Conjugating we have also orthonormal Using (F.26) we can prove ^ ^(S^ )y_ (S ^ )y_ = _ _ : Vaa^ V ^b a = a^^b : Spinor and vector identities The solutions of (F.22) are orthonormalized and form a complete basis (F.31) (F.32) (F.33) (F.34) (F.37) (F.38) (F.39) (F.40) (F.41) (F.42) (F.43) (F.44) (F.45) As a consequence of the orthonormality of the Killing spinors, the Killing vectors are Vaa^ V a^ = 1 ( a^)^ ^(S ^ y( 1) aS ^ )( a^)^^(S^ y( 1) bS^ ) b ^ ^ = 4 ( ^ ^ + 2 ^ ^^)((S ^ )y_ a _ S ^ )((S^ )y_ b _ S^ ) 1 1 1 2 a 2 a 1 2 1 2 = gab : b _ _ _ _ _ _ b _ _ _ Tr( a b) ( 2gab) As a consequence of (F.25) and (F.38) we have ^ V a^ a = ( ^b)^ ^Va^b V a^ a = ( a^) ^ : Contracting each side with a Killing spinor we have the relation As a result of equation (F.12) the Killing vectors are divergenceless As a result of equation (F.11) we have aS ^ V a^ a = ( 0S ^+)( a^) ^ : raV a^ a = 0 : r2V a^ a R2 a s V a^ : 1 abcV a^ c : Using (F.25) and the Killing spinor equation, one can show As a result of this we have V a^ aV ^b b abcV a^ aV ^b bV c^ c = det(V ) a^^bc^ = 1 Rs Rs 1 a^^bc^ : Contracting each side with a vector harmonic and using the Killing vector equation we have V ^b b Rs 1 a^^bc^V c^ : a Open Access. 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Curtis T. Asplund, Frederik Denef, Eric Dzienkowski. Massive quiver matrix models for massive charged particles in AdS, Journal of High Energy Physics, 2016, 55, DOI: 10.1007/JHEP01(2016)055