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 , Dbranes
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 lowenergy dynamics of internally wrapped Dbranes in fourdimensional antide 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
Rsymmetry and Rframes
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 lowenergy 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 fourdimensional spacetime degrees
of freedom and label wrapped branes; the arrows correspond to internalspace degrees of
freedom and label open string modes. If X6 is a CalabiYau 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 onedimensional N = 4 quiver matrix
mechanics Lagrangian may be obtained by dimensional reduction of a fourdimensional
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 Dparticles 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 superYangMills 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
atspace quiver mechanics of [4] (particle masses mv, FayetIlopoulos 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 \Rframe"
eld rede nitions
the vector multiplets and N = 4 supersymmetry, assuming a at targetspace 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 atspace 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 fourdimensional gauged supergravity model was considered, with the
appropriate ingredients needed to lift previously known, asymptotically at bound states of black
holes carrying wrapped Dbrane 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 fourdimensional 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 Dbrane 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 niteenergy 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
FayetIliopoulos (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 atspace 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 twonode 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 codimensionone 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 Rcharge generator, namely the global AdS
Rframe 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 fourdimensional 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 fourdimensional 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 targetspace 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" Dbranes 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 FayetIliopoulos (FI) parameter v
setting the Dterm potential for the scalars a connected to the node v. The quiver model
may also have a superpotential, given by an arbitrary gaugeinvariant 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, atspace 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
Rsymmetry. If its Rsymmetry 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
Rcharge 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 Rsymmetry,
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 D0branes 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 smokinggun phenomena
usually associated with the presence of background uxes, including the Myers e ect
(section 3.1.4), magnetic trapping in the internal Dbrane moduli spaces (section 3.2.2) and
uxinduced 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 Rsymmetry and subtleties associated with the existence of di erent Rframes
(sections 3.1.3 and 3.2.3).
3.1
One node, no arrows
The onenode 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
Dparticles in a 3+1dimensional 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 Rframes
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 Rframe 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 Rcharge
1, the fermionic sector consists of a spin zero
state with Rcharge 0, one spin 12 doublet with Rcharge
1 and another spin 0 state with
Rcharge
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 Rcharge 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 normalmode 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 Rframe. 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 Rrotations 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 commutatorsquared term already appeared in the
atspace quiver matrix models.
It also universally appears in the worldvolume theories of N coincident Dbranes. Since
we are interpreting X as a position coordinate of a Dparticle 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 commutatorsquared term actually has the wrong dimension compared to, say, the
kinetic term. This can be traced to the way the original atspace 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 Dbrane 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 Dbrane 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 higherdimensional dielectric branes. Speci cally, for N coincident D0branes in a
background RR 4form
ux F , in the units of equation (3.14) and the conventions of [24],
this term in the D0brane 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 D0brane. 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 4form
1
= `AdS
in equation (3.5), this precisely agrees,
incations of 11dimensional 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 D0branes polarizing into a stable fuzzy sphere con guration, which,
at large N , approximates a spherical D2brane 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 RR 4form
ux is sourced by D2branes, or its uplift to Mtheory by M2branes.
Thus, the fuzzy sphere can be interpreted as a membrane which has separated itself from
the stack of membranes generating the AdS4 FreundRubin 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 singlenode 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 Dparticle under consideration is a D0brane in tendimensional IIA string theory,
the
a are complex coordinates on the physical compacti cation space, and F is then to
be identi ed with an RR 2form
ux. To more accurately describe such a situation, for
example a D0brane on the AdS4
CP 3 geometry dual to ABJM theory [7], we should
consider a generalization of our quiver models to general curvedspace 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 targetspace 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 2form
uxcarrying CP 3 also equals .
Before jumping to conclusions, we should recall, however, that this is the motion in
the original,
= 0, Rframe, whereas the natural, \inertial" Rframe for Dparticles 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 Rcharge 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 Rcharge is
identi ed with the Rcharge of the N = 2 AdS4 superalgebra OSp(2j4), which in turn
typically arises as the KaluzaKleincharge 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 Rsymmetry. 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 commutatorsquared 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 Rsymmetry, 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 twodimensional family of solutions, all of which provide Rsymmetries of the
system. This is a manifestation of the presence of an extended Rsymmetry. 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 D0brane 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 blockdiagonal antisymmetric matrix with "12 =
"56 = +1. If we call the space parametrized by the 3vector Xi external and the space
parametrized by the 6vector Y n internal, then this is precisely the 6d internalspace part
of the Lagrangian of a stack of D0branes with a at internal space threaded by the RR
2form 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 D0branes in string
ux
compacti cations. This could be improved further by generalizing the models to arbitrary
target space Kahler potentials.
This D0model is just one of many possible onenode 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 Dbranes in
ux compacti cations with
fewer isometries than AdS4
CP 3. More generally, instead of D0branes, 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 geometricdeformation 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 Dbrane moduli space; they no longer have a direct
physicalspace interpretation as in the case of D0branes. Nevertheless, their e ect will be similar,
causing oscillatory motion even in the absence of a potential on Dbrane moduli space.
Such dynamical features can presumably also be thought of as the result of the presence
of background magnetic RR uxes interacting with the Dbranes.
3.2.4
BMN matrix mechanics
The BMN matrix model describes Dbranes or Mbranes in a supersymmetric planewave
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 Rcharge 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 Rcharges allows for an enhancement of the
Rsymmetry 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 centerofmass (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 atspace 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 Dbranes, and the
a correspond to the lightest openstring modes
that exist between the two branes. Roughly speaking, this means that if we hold x
xed
and quantize the
a, the nth 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 nth 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 zeromode 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)rinvariant
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 fourdimensional 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 Rsymmetry is present, one is restricted to the
= 0
= 0
frame and simply drops the u(1)R factor and the Rgenerator. 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 timeindependent 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 Rframes.
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 Rframe 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 Rsymmetry
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 moleculelike 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 targetspace 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 S3reductions 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
atspace 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 atspace 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 DESC0011941. 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 ORISEORAU under contract no.
DEAC0506OR23100.
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
Rsymmetry, with Rcharge 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 LeviCivita symbol, i.e.,
. Fermion
bilinears are formed by contracting with the LeviCivita 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, atspace 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 targetspace 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 Rsymmetry 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 timeindependent 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
Myerslike term cubic in the Xi. The physical implications of this term were reviewed in
section 3.1.4.
C.2
Chiral multiplet and vectorchiral 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 nonquadratic 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 atspace,
= 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 bifundamental 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
(nonquadratic) +
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) Rsymmetry 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 Rframes, characterized by
in the presence of an Rsymmetry, 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 Rsymmetry 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, Rsymmetry
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 fourdimensional 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 Rcharges 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) Rsymmetry gauge eld which we expect to couple to the superparticle.
The subalgebra we want should then also have an Rgenerator.
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 Rcharge and anticommutation 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 Rcharge. Conjugation negates the value of the Rcharge. The
Dirac conjugate is then given by S = ( +; y _ ). The anticommutation 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 nonvanishing anticommutation 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 Rframe
= 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 LeviCivita 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 NWSE 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 LeviCivita 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
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