Holography for field theory solitons
HJE
Holography for eld theory solitons
Sophia K. Domokos 0 1 3
Andrew B. Royston 0 1 2
Texas A 0 1
M University 0 1
Supersymmetric Gauge Theory
0 College Station , TX 77843 , U.S.A
1 300 Jay Street, Brooklyn, NY 11201 , U.S.A
2 George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy
3 Department of Physics, New York City College of Technology , USA
We extend a wellknown Dbrane construction of the AdS/dCFT correspondence to nonabelian defects. We focus on the bulk side of the correspondence and show that there exists a regime of parameters in which the lowenergy description consists of two approximately decoupled sectors. The two sectors are gravity in the ambient spacetime, and a sixdimensional supersymmetric YangMills theory. The YangMills theory is de ned S2 background and admits sixteen supersymmetries. We also consider a oneparameter deformation that gives rise to a family of YangMills theories on asympS2 spacetimes, which are invariant under eight supersymmetries. With future holographic applications in mind, we analyze the vacuum structure and perturbative spectrum of the YangMills theory on AdS4 niteenergy solitons. Finally, we demonstrate that the classical YangMills theory has a consistent truncation on the twosphere, resulting in maximally supersymmetric YangMills on AdS4.
AdSCFT Correspondence; Dbranes; Solitons Monopoles and Instantons

on a rigid AdS4
totically AdS4
S2, as well as systems of BPS equations for
2.1
2.2
2.3
2.4
2.5
3.1
3.2
4.1
4.2
4.3
4.4
6.1
6.2
6.3
1 Introduction and summary
1.1 Summary of results
2 Branes and holography
Brane con guration
Low energy limit and AdS/dCFT
The nonabelian D5brane action
YangMills as the low energy e ective theory
Fermionic Dbrane action
3 Supersymmetry
Killing spinors in the bulk
Killing spinors on the brane
3.3 Supersymmetry of the worldvolume theory
4 Classical vacua, boundary conditions, and asymptotic analysis
Classical ( ux) vacua
Perturbative spectrum
Boundary conditions and consistency of the variational principle
Conservation of supersymmetry on the boundary
5 A consistent truncation to N = 4 SYM on AdS4
6 Bogomolny equations and monopoles
BPS equations as generalized selfduality equations
Bogomolny bound on the energy
Domain walls and dyonic octonionic instantons
6.4 Dimensional reduction and the HaydysWitten equations
7 Conclusions and future directions
A Fluctuation expansion of the Myers action
A.1 The DBI action
A.2 The CS action
B Background geometry and Killing spinors
B.1 Frame rotations
B.2 D5brane Killing spinor equation
C Mode analysis
C.1 Bosons
C.2 Fermions
{ i {
1
Introduction and summary
The original AdS/CFT duality [1] is nearing its twentieth anniversary. Even AdS/dCFT,
which introduces defects in conformal eld theories with gravity duals [2{4], is fteen years
old. A small corner of this Dbrane universe, however, remains relatively unexplored.
In this paper, we describe a simple generalization of the D3/D5brane intersection that
forms the basis of the original antide Sitter/defect conformal eld theory correspondence
(AdS/dCFT). Instead of studying a single probe D5brane in the presence of a large
number of D3branes, we consider the seemingly simple nonabelian generalization, with
several parallel D5branes.
The resulting model, when subjected to Maldacena's lowenergy limit and restricted
to an appropriate regime of parameters, o ers rich physics and rich mathematics, which
p
respectively, and the string coupling by gs, the regime of parameters is Nc
gsNc
1 and
Nf
Nc= gsNc. The rst two conditions are the ones that arise in the usual AdS/CFT
correspondence. They ensure that gravity is weakly coupled and curvatures are small
relative to the string scale. The
nal condition is a slight re nement of the oftquoted
`probe limit' Nf
Nc. We will see that it arises naturally when we demand that corrections
from gravity to the su(Nf ) sector of the D5brane theory be suppressed. In this regime,
therefore, the e ects of closed strings can be neglected relative to the treelevel YangMills
interactions.
As in the original AdS/dCFT correspondence, the duality `acts twice' [2{4] in the
sense that it relates bulk closed strings to operators in the ambient part of the boundary
theory, and bulk open strings on the D5branes to operators localized on a defect in the
boundary theory. Hence the curvedspace superYangMills theory (SYM) describes the
physics of operators con ned to a defect in the boundary CFT. As the bulk SYM is
dual to a (2+1)dimensional system, it is potentially relevant to holographic condensed
matter applications. Indeed, the bulk SYM admits a zoo of solitonic objects, whose masses
and properties are constrained by supersymmetry. We expect that these correspond to
vortexlike states on the dual defect. Conversely, holography should provide a new tool for
studying SYM solitons in the bulk.
In this paper, however, we focus on the bulk side of the correspondence. A detailed
construction of the dual boundary theory will appear elsewhere, [5].
{ 1 {
We begin by constructing a sixdimensional (6D) SYM theory with osp(4j4) symmetry
from a D3/D5 intersection. We assume that the number of D3branes (Nc) is large, so
we can represent them with a Type IIB supergravity solution. We then consider the
D5branes as probes in this background. We arrive at the SYM action by combining and
extending Dbrane actions that already appear in the literature. For the bosonic theory
on the D5branes, we use the nonabelian Myers action [6].
We determine the kinetic
and masslike terms for the fermions using the abelian action of [7{13], and infer the
nonabelian gauge and Yukawa couplings via a simple ansatz consistent with gauge invariance
and supersymmetry. We then apply the Maldacena lowenergy nearhorizon limit.
The resulting action is summarized in equations (3.24){(3.26). While we obtained this
action from a Dbrane model, it makes sense as a classical eld theory for arbitrary simple
Lie groups.
We go on to analyze the vacuum structure, perturbative spectrum, and the BPS
equations satis ed by solitons in the 6D SYM theory. We also show that the 6D theory has a
nonlinear consistent truncation to maximally supersymmetric YM theory on AdS4.
Here are a few highlights from the road ahead:
The space of vacua of the 6D theory has multiple components. There are, in fact,
in nitely many when Nc ! 1. One component is a standard Coulomb branch labeled
by vevs of Higgs elds. The other components are labeled by magnetic charges and are
quite complicated: they have roughly the form of moduli spaces of singular monopoles
bered over spaces of Higgs vevs. A Dbrane picture (see
gure 4 below) provides
some intuition for these vacua.
We perform a perturbative mode analysis around a class of vacua that carry magnetic
ux. The background elds of this class are Cartanvalued and simple enough to make
the linearized equations tractable. Furthermore, the background elds of any vacuum
will asymptote to the same near the boundary, so the results for the asymptotic
behavior of uctuations are robust. This is important for the holographic dictionary,
where one maps modes to local operators in the dual, based in part on their decay
properties near the boundary.
Our analysis of the perturbative spectrum generalizes previous results for the abelian
D5brane defect [4, 14, 15], and o ers a number of new results.
We display, for
instance, the complete KK spectrum of fermionic modes. We also observe that a
Legendre transform of the onshell action with respect to one of the lowlying modes,
along the lines of [16], is required for holographic duality.1
We identify a set of lowlying nonnormalizable modes that can be turned on without
violating the variational principle or supersymmetry. These modes form a natural
class of boundary values for soliton solutions in the nonabelian D5brane theory. In
1The paper [17] appeared when this work was nearing completion. Its authors make a closely related
observation in maximally supersymmetric YM on AdS4. The consistent truncation of the 6D theory on the
the holographic dual, meanwhile, they source a set of relevant operators  and in
one case a distinguished irrelevant operator.
Having explored the vacua and perturbative structure of the bulk SYM theory, we
then survey various systems of BPS equations. These rst order equations arise when
we demand that eld con gurations preserve various amounts of supersymmetry.
Solutions to the BPS equations saturate bounds on the energy functional. These
bounds depend on a combination of the elds' boundary values as well as the magnetic
and electric uxes through the asymptotic boundary.
The BPS systems we obtain house a number of generalized selfduality equations that
are well known in mathematical physics, like (translationally invariant) octonionic
instantons [18], and the extended Bogomolny equations [19]. All of these equations
are de ned on a manifold with boundary, where the boundary is the holographic
boundary.
The paper is structured as follows: in section 2 we describe the Dbrane intersection
and take the lowenergy limit of the action to arrive at a curved space SYM theory. In
section 3 we verify the invariance of the action under supersymmetry. In section 4 we
describe the vacuum structure of the model, and formulate asymptotic boundary conditions
on the elds. In section 5 we derive the consistent truncation of our sixdimensional theory
to four dimensions, while in section 6 we derive the BPS equations satis ed by solitons in
the system. We conclude and discuss future directions in section 7. Necessary but onerous
details are relegated to a series of appendices.
2
Branes and holography
In this section we describe the brane setup, the AdS/dCFT picture, and the lowenergy
limit and parameter regime that isolates sixdimensional SYM as the lowenergy e ective
theory on the D5branes.
2.1
Brane con guration
We begin with a nonabelian version of the brane con guration in [4]. Nf D5branes and
Nc D3branes in the tendimensional IIB theory span the directions indicated in
gure 1.
Standard arguments [20] show that the intersecting D3/D5 system preserves one quarter
of the supersymmetry of 10D type IIB string theory, or eight supercharges.
The ten coordinates, x~M = (x ; r~i; z~i; y) are divided as follows: x , ;
= 0; 1; 2,
parameterizes the R1;2 spanned by both stacks; the triplet r~i = (r~1; r~2; r~3) = (x3; x4; x5)
parameterizes the remaining directions along the D5branes; the triplet z~i = (z~1; z~2; z~3) =
(x6; x7; x8) parameterizes directions orthogonal to both stacks; and
nally y = x9
parameterizes the remaining direction along the D3branes and orthogonal to the D5branes. We
reserve the notation xM = (x ; ri; zi; y) for a rescaled version of these coordinates to be
introduced below. We will sometimes use spherical coordinates (r~; ; ) to parameterize the
r~i directions, and we denote the radial coordinate in the z~i directions by z~. We also write
x~
a = (x ; r~i), a = 0; 1; : : : ; 5, and x~m = (z~i; y), m = 1; : : : ; 4, for the full set of directions
parallel and transverse to the D5branes respectively.
{ 3 {
D3
D5
directions common to both types of brane
worldvolume are suppressed in the
gure on the left. The D5branes can be separated from the
D3branes by a distance z~0 in the directions transverse to both stacks.
The D3branes are taken to be coincident and sitting at r~i = z~j = 0. The center of
mass position of the D5branes in the transverse x~m space is denoted x~0m = (z~0;i; y0). We
will allow for relative displacements of the D5branes from each other, but assume that
these distances are small compared to the string scale. In other words, the separation is
welldescribed by vev's of nonabelian scalars in the D5brane worldvolume theory. This
will be explained in more detail below. When all D5branes are positioned at z~i = 0 an
SO(1; 2)
SO(3)r
SO(3)z subgroup of the tendimensional Lorentz group is preserved.
Nonzero D5brane displacements in z~ break SO(3)z. This can be explicit or spontaneous
from the point of view of the D5brane worldvolume theory, depending on whether the
center of mass position z~0;i is, respectively, nonzero or zero.
As noted above, eight of the original thirtytwo Type IIB supercharges are preserved
by the brane setup. From the point of view of the threedimensional intersection, this is
equivalent to N = 4 supersymmetry. The Rsymmetry group is SO(4)R = SU(2)r
SU(2)z
with the two factors being realized geometrically as the double covers of the rotation groups
in the r~i and z~i directions. The light degrees of freedom on the D3branes and the
D5branes are a fourdimensional N = 4 u(Nc)valued vectormultiplet, and a sixdimensional
N = (1; 1) u(Nf )valued vectormultiplet. Each of these decompose into a 3D N = 4
vectormultiplet and hypermultiplet. For those D5branes intersecting the D3branes, the
35 strings localized at the intersection are massless. They furnish a 3D N = 4
hypermultiplet transforming in the bifundamental representation of the appropriate gauge groups.
Meanwhile the massless closed strings comprise the usual type IIB supergravity multiplet.
2.2
Low energy limit and AdS/dCFT
Let us now consider the lowenergy limit of the brane setup, that ultimately yields the
defect AdS/CFT correspondence. This is the famous Maldacena limit [1] that, in the
absence of D5branes, establishes a correspondence between 4D N = 4 SYM and type IIB
string theory on AdS5
S5. To arrive at the AdS/dCFT correspondence one considers the
{ 4 {
lowenergy e ective description of the D3/D5 system at energy scale
and takes the limit
`s ! 0, where `s is the string length. The dynamics of the massless degrees of freedom
have two equivalent descriptions in terms of two di erent sets of eld variables. This fact
is the essence of the original AdS/dCFT correspondence.
To simplify the present discussion we temporarily assume no separation between the
brane stacks  in other words, z~0 = 0. The rst set of variables that describes the D3/D5
intersection is based on an expansion around the at background: Minkowski space for the
closed strings and constant values of the brane embedding coordinates for the open strings.
In this case standard
eld theory scaling arguments apply. After canonically normalizing
the kinetic terms for open and closed string
uctuations, interactions of the closed strings
and 55 open strings amongst themselves, as well as the interactions of the closed and 55
open strings with the other open strings, vanish in the lowenergy limit. These degrees of
freedom decouple from the system. Meanwhile the 33 and 35 strings form an interacting
system described by fourdimensional N = 4 SYM coupled to a codimension one planar
interface, breaking half the supersymmetry and hosting a 3D N = 4 hypermultiplet. The
interface action, which can in principle be derived from the low energy limit of string
scattering amplitudes, was obtained in [4] by exploiting symmetry principles. The entire
theory contains a single dimensionless parameter in addition to Nf and Nc  the
fourdimensional YangMills coupling  given in terms of the string coupling via gy2m := 2 gs.
The interface plus boundary ambient YangMills theory is classically scale invariant,
and it was argued in [4, 21] to be a superconformal quantum theory.
The symmetry
algebra is osp(4j4), with bosonic subalgebra SO(2; 3)
SO(4)R and sixteen odd generators.
SO(2; 3) is the threedimensional conformal group of the interface while the odd generators
correspond to the eight supercharges along with eight superconformal generators. This
is the \defect CFT" side of the correspondence. Considering a nonzero separation z~0
corresponds to turning on a relevant mass deformation in the dCFT [22, 23].
Our focus here will be mostly on the other side of the correspondence, which is based on
an expansion in uctuations around the supergravity background produced by the Nc
D3branes. This background involves a nontrivial metric and RamondRamond (RR) veform
ux given in our coordinates by
ds120 = f 1=2(
dx dx + dy2) + f 1=2( dr~i dr~i + dz~i dz~i) ;
F (5) = (1 + ?) dx0 dx1 dx2 dy df 1
;
with
f = 1 +
L4
(r~2 + z~2)2
;
where
L4 = 4 gsNc`s4 :
(2.1)
The metric is asymptotically at and approaches AdS5
S5 with equal radii of L when
v~
2
r~2 + z~
2
L2. The energy of localized modes in the throat region, as measured by
an observer at position v~, is redshifted in comparison to the asymptotic
xed energy
according to Ev = f 1=4
(L=v~) , for v~
L. Hence, while closed string and D5brane
modes with Compton wavelengths large compared to L decouple as before, excitations of
arbitrarily high energy can be achieved in the throat region. The nearhorizon limit isolates
the entire set of stringy degrees of freedom in the throat region by sending v~=`s ! 0 in such
{ 5 {
to sending v~=L ! 0 while holding v~=(L2 ) xed.
a way that Ev`s remains xed. From the redshift relation it follows that we are sending
v~=`s ! 0 while holding v~=(`s2 ) xed. For xed 't Hooft coupling gsNc, this is equivalent
To facilitate taking this limit we introduce new coordinates
ri =
r~
i ;
and write (r; ; ) for the corresponding spherical coordinates and z
sometimes employ a vector notation ~r = (r1; r2; r3), ~z = (z1; z2; z3). One nds that with
pzizi. We will also
these new coordinates, the metric becomes2
=: (L )2GMN dxM dxN ;
=: (L )4 dC(4) ;
ds120 ! (L )
2
2(r2 + z2)
dx dx + dy2 +
F (5)
! 4(L )
4 4(r2 + z2)(1 + ?) dx0 dx1 dx2 dy (r dr + z dz)
dr2 + r2 d 2( ; ) + d~z d~z
where we've introduced a rescaled metric and fourform potential, GMN ; C(4). GMN is the
metric on AdS5
S5 with radii
1
.
The degrees of freedom in the nearhorizon geometry include both the closed strings
and the open strings on the D5branes. String theory in this background is conjecturally
dual to the dCFT system, with the duality `acting twice' [2{4]. This means the following:
closed string modes in the (ambient) spacetime of the bulk side are dual to operators
constructed from the 4D N = 4 SYM
elds in the (ambient) spacetime on the boundary.
Open string modes on the D5branes, which form a defect in the bulk, are dual to operators
localized on the defect in the boundary theory. These operators are constructed from
modes of the 35 strings and modes of the 33 strings restricted to the boundary defect.
See gure 2.
The validity of the supergravity approximation in the closed string sector requires that
1. The rst condition suppresses gs corrections to the low energy e ective
action, while the second condition is equivalent to L
`s, ensuring that higher derivative
corrections are suppressed as well.
In subsection 2.4 we'll see how this limit suppresses the interactions between closed
string and open string D5brane modes, leading to an e ective YangMills theory on the
D5branes. This extends previous analyses of the D3/D5 system to the case of multiple
D5branes, showing how the nonabelian interaction terms among open strings are dominant
to the openclosed couplings, at least in the su(Nf ) sector of the theory. In subsection 2.3
2The metric can be brought to the form found in [4] by rst introducing standard spherical coordinates
(z; ; ) in the ~z directions and then setting r = v cos
and z = v sin
with
Then v is the AdS5 radial coordinate in the Poincare patch, with v ! 1 the asymptotic boundary, while
( ; ; ; ; ) parameterize the S5, viewed as an S
2
S
2 bration over the interval parameterized by .
2 [0; =2], and
= 1.
{ 6 {
S5, coupled to a defect composed of probe D5branes. The boundary
theory consists of an ambient N = 4 SYM on R1;3 coupled to a codimension one defect hosting
localized modes.
we will describe explicitly what these interactions look like (using the Myers nonabelian
Dbrane action).
In preparation for that, consider the following rede nition of the relevant supergravity
elds. Let SIIB[G; B;
; C(n); ] denote the type IIB supergravity action in Einstein frame.
Here B is the KalbRamond twoform potential and
:=
0 is the
uctuation of
the dilaton eld
around its vev,
0, with e 0
gs. The C(n), n even, are the
RamondRamond potentials, and
is the tendimensional Newton constant, 2 = 12 (2 )7gs2`s8. Upon
rescaling the metric and potentials according to
GMN = (L )2G~MN ;
BMN = (L )2B~MN ;
C(n) = (L )nC~(n) ;
{ 7 {
one nds that
where the new Newton constant is
SIIB[G; B;
; C(n); ] = SIIB[G~; B~;
; C~(n); ] ;
=
(L )4 =
(2 )3p
Nc
4
:
Thus an expansion in canonically normalized closed string uctuations, (hMN ; bMN; '; c(n)),
around the nearhorizon background, (2.3), takes the form
GMN = (L )
2 GMN + hMN ; BMN = (L )2 bMN ;
C(4) = (L )
4 C(4) + c(4) ;
C(n) = (L )n c(n) ;
=
' ;
n 6= 4 ;
where GMN and C(4) were given in (2.3), and npoint couplings among closed string
uctuations go as n 2.
2.3
The nonabelian D5brane action
The massless bosonic degrees of freedom on the D5branes are a U(Nf ) gauge eld Aa,
a = 0; 1; : : : ; 5, with eldstrength Fab, and four adjointvalued scalars Xm = (Z1;2;3; Y ).
(2.4)
(2.5)
(2.6)
(2.7)
The gauge eld carries units of mass while the Xm carry units of length. The eigenvalues of
( i times) the latter are to be identi ed with the displacements of the Nf D5branes away
from (~z0; y0). Our conventions are that elements of the u(Nf ) Lie algebra are represented
by antiHermitian matrices, so there are no factors of i coming with the Lie bracket in
covariant derivatives. The ` Tr ' operation denotes minus the trace in the fundamental
representation, Tr :=
tr Nf , with the minus inserted so that it is a positivede nite
bilinear form on the Lie algebra. Later on we will generalize the discussion to a generic
simple Lie algebra g, and then we de ne the trace through the adjoint representation via
Tr :=
2h1_ tr adj, where h_ is the dual Coxeter number. This reduces to the previous
de nition for g = u(Nf ).
The nonabelian Dbrane action of Myers, [6], captures a subset of couplings between
the 55 open string and ambient closed string modes. It takes the form
:= 2 `s2, and D5 := 2 =(gs(2 `s)6) is the D5brane tension. Besides the factor of e
in SDBI, the closed string elds are encoded in the two quantities
EMN := e
=2(GMN + BMN ) ;
C =
X C(n)
n
^ exp e
=2B :
The factors of the dilaton are present here because we work in Einstein frame for the closed
string elds. This action generalizes the nonabelian Dbrane action of [24] to the case of
a generic closed string background.
The quantity P [TMN:::Q] denotes the gaugecovariant pullback P of a bulk tensor
TMN:::Q to the worldvolume of the D5branes. For instance, the pullback of the generalized
metric to the brane is
P [Eab] = Eab
i(DaXm)Emb
iEam(DbXm)
(DaXm)Emn(DbXn) ;
(2.13)
with Da = @a +[Aa; ]. The closed string elds are to be taken as functionals of the
matrixvalued coordinates, EMN (xP ) ! EMN (xa; iXm), de ned by power series expansion:
EMN (xa; iXm) := EMN (xa; x0m) + X1 ( i)n
@mn EMN ) (xa; x0m) ;
The determinants in the DBI action (2.9) refer to spacetime indices a; b and m; n.
n=1
n!
{ 8 {
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.14)
In the ChernSimons (CS) action, (2.10), the symbol iX denotes the interior product
with respect to Xm. This is an antiderivation on forms, reducing the degree by one. Since
the Xm are noncommuting one has, for example,
1
2
(i2X C(k+2))M1 Mk =
[Xm; Xn]Cn(km+M2)1 Mk :
(2.15)
See [6] for further details.
The `STr' stands for a fully symmetrized trace, de ned as follows [6]. After expanding
the closed string elds in power series and computing the determinants, the arguments of
the STr in (2.9) and (2.10) will take the form of an in nite sum of terms, each of which will
involve powers of four types of open string variable: Fab; DaXm; [Xm; Xn], and individual
Xm's from the expansion of the closed string elds. The STr notation indicates that one
is to apply Tr to the complete symmetrization on these variables.
The precise regime of validity of the Myers action is not a completely settled issue. First
of all, like its abelian counterpart, it captures only treelevel interactions with respect to gs.
Second, if F denotes any components of the `tendimensional' eldstrength, Fab, DaXm,
or [Xm; Xn], (2.8) is known to yield results incompatible with open string amplitudes at
O(F6) [25, 26], even in the limit of trivial closed string background. Finally, the action (2.8)
is given directly in \static gauge," and there have been questions about whether it can be
obtained from gauge xing a generally covariant action. This could lead to ambiguities in
openclosed string couplings at O(F4) according to [27]. However, the results of [28] suggest
that the Myers action can in fact be obtained by gauge xing symmetries in a generally
covariant formalism where the ChanPaton degrees of freedom are represented by boundary
fermions on the string worldsheet. As we will see below, none of these ambiguities pose a
problem in the scaling limit we are interested in.
2.4
YangMills as the low energy e ective theory
We now expand the action (2.8) in both closed and open string
uctuations, where the
closed string expansion is an expansion around the nearhorizon geometry of the D3branes,
in accord with (2.7). This was already done in some detail in the abelian case [4], but there
are some important new wrinkles that arise in the nonabelian case. We summarize the
main points here and provide further details in appendix A.
First, the kinetic terms for the open string modes take the form
SDBI
D5(L )
g6 Tr
6
Z p
4
1 2(L ) 4FabF ab +
1
2
(Gmnjx0m )DaXmDaXn ;
(2.16)
where we recall that
= 2 `s2. The factors of (L ) arise from writing the background
metric in terms of the barred metric. We have introduced the notation g6 := det(gab), with
gab := Gab(xa; x0m) the induced background metric on the worldvolume. It takes the form
gab dxa dxb =
deformation of it. Worldvolume indices will always be raised with the inverse, gab. We use
the notation jx0m to indicate when other closed string elds are being evaluated at xm = x0m.
1, while z0 6= 0 gives a
{ 9 {
The coe cient of the F 2 term determines the e ective sixdimensional YangMills
coupling:
Note that the dimensionless coupling (gym6 ) is small in the regime Nc
gsNc
order to bring the scalar kinetic terms to standard form we de ne mass dimensionone
scalar elds through
m :=
1(L )2Xm =
p
p
gsNc 2Xm ;
(2.18)
(2.19)
so that (Aa; m) carry the same dimension.
Once the closed string elds in the Dbrane action are expressed in terms of the rescaled
quantities, one nds that Fab is always accompanied by a factor of (L ) 2, while [Xm; Xn]
is always accompanied by the inverse factor.
After changing variables to
m for the
scalars, all four types of open string quantities appearing in D5brane action carry the
same prefactor:
(L )2 Fab; DaXm; (L )
2
[Xm; Xn]; Xm
= p
p
gsNc
2 (Fab; Da
m; [ m
; n];
and this provides a convenient organizing principle for the expansion. Of course it is
(Aa; m)c, de ned by
(Aa; m
) = gym6 (Aa; m
)c ;
that are the canonically normalized open string modes. The open string expansion variables
on the righthand side of (2.20) do not scale homogeneously when expressed in terms of
these, and this point must be kept in mind when comparing the strength of interaction
vertices below.
(Fab; Da
Now, let C 2 (hMN ; bMN ; '; c(n)) denote a generic closed string uctuation, let O 2
m; [ m; n];
m) denote any of the open string expansion variables, and set
op :=
(L )2 = p
p
gsNc
2
:
Then the expansion of (2.8) can be written in the form
SDbo5s =
1
2opgy2m6
Z
d x
6 p
g6
1
X
no;nc=0
no nc Vno;nc ;
op
where Vno;nc is a sum of monomials of the form Cnc STr (Ono ), with rational coe cients.
V2;0 = Tr
1
1
2
abcdef
(6)
m(@mcabcdef )jx0m +
(Da
m)c(m6b)cdef
;
+
+
xm
0
;
1
4
1
5!
1
3!2
(Da
m)C(m4b)cdbef +
41!2 c(a4b)cdFef
GmnDa
mDa n +
where abcdef is the LeviCivita tensor with respect to the background metric, 012345 =
( g6) 1=2, and we have used that STr reduces to the ordinary trace when there are no
more than two powers of the open string variables O. All closed string
elds are to be
understood as being evaluated at xm = x0m except for those in V1;1 that involve taking a
transverse derivative before setting xm = x0m.
There is a great deal of physics in the Vno;nc 's:
V0;0 corresponds to the energy density of the background D5brane con guration.
Nf p
gsNc 2, which is large when gsNc
V0;1 gives closed string tadpoles for the metric, dilaton, and RR sixform potential.
These are present because we have not included the gravitational backreaction of
the D5branes  i.e. we have not expanded around a solution to the equations of
motion for these closed string elds. The strength of these tadpoles is Nf gym26 op2
/
1. However this does not necessarily mean
that the probe approximation is bad! The e ects of these tadpoles on open and closed
string processes will still be suppressed if the interaction vertices are su ciently weak.
Consider, for example, the leading correction to the open string propagators due to
these tadpoles. This corresponds to the diagram in
gure 3. The correction is
proportional to the product of the tadpole vertex with the cubic vertex for two open and
one closed string
uctuation. After canonically normalizing the open string modes
via (2.21), the threepoint vertex goes as . Therefore the product is proportional
to Nf
p
gsNc 2
/ (Nf p
gsNc=Nc)
2
/ Nf gy2m6 . Hence this process acts just like
a standard oneloop correction to the YangMills coupling that we would get from
open string modes. As long as Nf
Nc= gsNc, both the standard oneloop
correction and this closed string correction will be suppressed. Note this is a slightly
p
created from the vacuum by a vertex in V0;1. It propagates to a threepoint vertex in V2;1. This
gives a correction to the open string propagator that is of the same order as a standard oneloop
correction from virtual open string modes.
~z0 ~ z
stronger restriction than the usual Nf
Nc limit when the 't Hooft coupling gsNc
is large, but nevertheless can be comfortably satis ed for a range of Nf in the regime
Nc
gsNc
The vanishing of V1;0 indicates that open string tadpoles are absent. This simply
validates the fact (already implicitly assumed in the above discussion) that the
D5brane embedding, described by xm = x0m, extremizes the equations of motion for the
open string modes in the xed closed string background.
Only the centerofmass degrees of freedom corresponding to the central u(1)
participate in V1;1 due to the trace. The strength of these interactions is gym16 op1
/
gym6 , where we have made use of the convenient relation
p
Hence they can be treated perturbatively. Furthermore the u(1) and su(Nf ) degrees
of freedom decouple in V2;0, so the couplings in V1;1 can only transmit the e ects
of the closed string tadpoles to the su(Nf ) elds through higher order open string
interactions.
The rst three terms of V2;0 come from the DBI action, and comprise the usual
YangMills action on a curved background. The nal term in V2;0, meanwhile, comes from
the CS action and is nonvanishing because there is a the nontrivial RR
ux in the
supergravity background.
It is also interesting to consider the form of terms in V3;0, or higher order open string
is an STr (~ zF 2) coupling of the form
interactions. V3;0 is nontrivial when z0 6= 0; V4;0 is always nontrivial. For example, there
u(Nf )
(2.25)
(2.26)
Three and fourpoint couplings in V3;0 and V4;0 come with extra factors of op relative
to the three and fourpoint couplings in the YangMills terms, V2;0. Hence they will be
suppressed relative to the YangMills terms for eld variations at or below the scale .
More precisely, if the elds vary on a scale 0 we merely require ( 0= )2
that these terms be suppressed relative to their counterparts in V2;0.
p
gsNc, in order
In summary, there is a regime of parameters  namely Nc
gsNc
Nc= gsNc  where the leading interactions of the (bosonic) su(Nf ) open string modes
are governed by V2;0. This forms the bosonic part of a sixdimensional superYangMills
theory on the curved background (2.17).
We can present this action in two di erent forms, both of which will prove useful below.
HJEP07(21)65
First there is the form we have used to give V2;0, in which the scalars carry curved space
indices. In order to be more explicit with regards to the C(4) term, we have from (2.3) that
the relevant components are
and so the last term of V2;0 contributes as follows:
d x
6 p
g6 Tr
Here we have introduced ~, which should be thought of as the LeviCivita tensor on
the Euclidean R
3 spanned by ~r: ~r1r2r3 = 1, or if we work in spherical coordinates
~
r
= (r2 sin ) 1. Then the bosonic part of the YangMills action is
We can also derive a more standard eld theoretic form for the action by rescaling the
scalar elds in such a way that their kinetic terms are canonically normalized. To do this,
we make use of a vielbein associated with the background metric Gmn:
y := (r2 + z02)1=2 y
;
zi :=
1
Both mass terms and boundary terms arise when we integrate by parts in the kinetic terms.
One can also integrate by parts on the last term of (2.29) and make use of the Bianchi
identity, ~rirjrk Dri Frjrk = 0. We also switch to spherical coordinates, as the only surviving
bulk term comes from the derivative of the (r2 + z02)2 prefactor. This integration by parts
becomes
Mz2 :=
1
gy2m6
Z
1
2
2
r
2
b
Sbndry =
where d5xp
R1;2
1
Z
r
2
In the last term the indices ;
correspond to coordinates ; along the twosphere and
= (gS2 ) 1=2 =
2(r2 + z02)=(r2 sin ). The mass parameters are de ned as follows:
HJEP07(21)65
2; 4; 1 in units of the inverse AdS radius. When
z0 = 0 they take these values everywhere. Although the squared mass of the Z scalars is
negative, it satis es the BreitenlohnerFreedman bound [29] for AdS4. The reason for the
notation M
will become clear below when we consider the fermionic part of the action.
The boundary terms arise due to the integration by parts and the boundary component
S2 at r = rb ! 1.3 They are given by
M
2
d5xp
( y)2
ij zi zj +
F
;
(2.33)
1 y
2
also generates a boundary term. After carrying out these manipulations, the bosonic action
d x
6 p
g6 Tr
1
4 FabF ab + 2 (Da m)(Da m) + 4 [ m; n][ m
p
(2.34)
with d3x := dx0 dx1 dx2 and d
:= sin d d . If one works with the action in the
form (2.31) then it is important to keep these terms. They play a crucial role both in
establishing the consistency of the variational principle and in the supersymmetry
invariance of the YangMills action. The limit rb !
1 of quantities computed using (2.33) is
understood to be taken at the end of any calculation (when it exists).
2.5
Fermionic Dbrane action
Ideally, one would like to obtain nonabelian superYangMills theory on the D5branes via
the limiting behavior of a symmetric nonabelian super Dbrane action for general closed
string backgrounds. While important progress toward constructing such actions has been
made (see e.g. [30{33] and references therein), the subject has not matured su ciently to
be of practical use for our purposes.
Instead, we will fall back on abelian fermionic Dbrane actions that have been discussed
extensively, starting with the initial work of [7{10], and continuing with [11{13]. Here
we follow the conventions of [12, 13]. This will provide the fermionic couplings that are
quadratic order in open string uctuations  kinetic and masslike terms. With these and
3We assume the elds are su ciently regular such that there is no boundary contribution from r = 0.
This is discussed in some further detail for static con gurations later. See section 6.2.
where
matrices.
Sf =
D5 Z
2
the full set of bosonic couplings in hand, we will be able to deduce the remaining
Yukawatype couplings and the nonabelian supersymmetry transformations via a simple ansatz.
The massless fermionic degrees of freedom on a D5brane are the same as those
in tendimensional superYangMills, and can be packaged into a single tendimensional
MajoranaWeyl fermion, . The couplings of
to the IIB closed string supergravity elds
are described most conveniently by introducing a doublet of tendimensional
MajoranaWeyl spinors ^ = ( 1
; 2)
T of the same 10D chirality. One linear combination will be
projected out by the symmetry projector while the other will be the physical
. The
tendimensional gamma matrices, satisfying f
the doublet structure. One introduces
M ; N
g = 2GMN , are likewise extended by
0123456789 is the tendimensional chirality operator and
1;2;3 are the Pauli
The abelian fermionic D5brane action, to quadratic order in ^ , takes the form
d6xe
p
det (P [E] i F ) ^ (1
D5) h(M 1)ab ^(P )D^ a
b
i ^ ; (2.36)
where EMN = e
=2(GMN + BMN ) as before and the matrix M is
Mab = e
=2P [Gab] + Fab :
^
expression4 in terms of F :
matrix
Here we have also introduced the shorthand Fab := e
D5 appearing in the symmetry projector, 12 (1
=2P [Bab] i Fab. The idempotent
D5), has a somewhat nontrivial
(2.35)
(2.37)
D5 :=
p
1
X
q+r=3
det(P [E]
i F )
"a1 a2qb1 b2r
q!2q(2r)!
( i)qFa1a2
Fa2q 1a2q
(P )
b1 b2r
( i 2) (^)r ;
(2.38)
gauge, (aP ) =
a
where "012345 = 1, and the a
(P ) are the pullbacks of
The remaining couplings to closed string elds are encoded in the generalized derivative
D^ and the masslike operator, . We write only the terms that contribute when evaluated
on the nearhorizon background geometry (2.3); the full set of couplings can be found
in [12, 13]. In this case
M to the worldvolume. In static
^
Da = P [Da]
12 +
1
16 5!
e
F M(51) M5
M1 M5 (aP )
(i 2) +
;
(2.39)
where the terms represented by
vanish when closed string uctuations are switched o ,
while
! 0 when closed string uctuations are switched o . The notation P [Da] is meant
4Our Mab; D5 are denoted Mfab; eD5 in [12, 13].
to indicate that one takes the pullback of DM
1;2 to the brane worldvolume, and DM is
the standard covariant derivative on tendimensional Dirac spinors.
Now we would like to argue that in the nearhorizon geometry (2.3), the action (2.36)
has an expansion in closed and open string uctuations controlled by the same parameters,
op; , that appeared in the expansion of the bosonic action (2.23). Considering rst the
rescaling of the closed string elds, (2.4), there are a few key points:
After applying this rescaling under the determinant of (2.36) we can pull out a factor
of (L )6, and we will have the usual factor of (L ) 2
= op accompanying Fab.
The
(P )
b1 b2r factor in
D5 rescales according to
(P )
b1 b2r = (L )2r ~(P )
b1 b2r , due to the
implicit vielbein factors present in it. Taking into account the (L ) 6 from the
determinant factor out front,
D5 retains its form under the rescaling except that
each factor of Fab picks up a corresponding (L ) 2 prefactor. This combines with
the 's already present so that all Fab in
D5 are accompanied by op.
One can check that (M 1)ab ^(P )D^ a
b
gets a net factor of (L ) 1 when expressed in
terms of the rescaled closed string elds, while Fab in Mab acquires an op prefactor.
string
terms of
m via (2.19).
elds around x0m is accompanied by a factor of op when we express Xm in
simply
for the fermion.
Hence we write
Together, these observations show that all open string interaction vertices between
and powers of Fab; @aXm, and Xm are controlled by the expected power of op. The
2
overall prefactor of the leading
2 term is D5(L )5 = ( op gym6 ) 2(L ) 1. We can make
a rescaling5 of ^ analogous to (2.19) such that the coe cient of this leading order term is
i=gy2m6 . We will assume this has been done and continue using the same notation
(2.40)
(2.41)
(2.42)
where
and
Sf =
i
2gy2m6
Z
d x
6 p
g6 ^ (1
(D05)) aD^ a(0) ^
(1 + O( op; )) ;
(D05) :=
012r
1
;
D^ a(0) :=
!MN;a
MN
1
4
12 +
1
(D05) we took the q = 0, r = 3 term in (2.38) and used that ( g6) 1=2"b1 b6 (bP1 ) b6 =
to leading order in open and closed string uctuations. In (2.42), !MN;P are the
components of the spin connection with respect to the background metric GMN , evaluated
5The ^ in (2.36) must have units of (length)1=2. It would be natural to include a factor of 1=2 out in
front of (2.36) so that they are dimensionless. Then the rescaling would be ^~ = o3=p4 ^ .
at xm = x0m, and all gamma matrices with covariant indices are de ned using the vielbeine
of the background metric.
Let us evaluate (2.42) in more detail. It follows from the background (2.3) that
1
where we recall that (r; ; ) are spherical coordinates for the directions spanned by ~r. But
the second term drops out of (2.40) because
Regarding the tendimensional spin connection, there are nonzero components of the type
!bm;a when z0 6= 0. (See appendix B for details.) However, the contribution of these
aD^ a(0) =
a
(i 2) :
(2.45)
The projector in the last term of (2.45) will either give the identity or zero when acting
on
1;2, depending on the 10D chirality of the latter. The two possibilities distinguish
between a D5brane and an antiD5, and only one choice will lead to a supersymmetric
worldvolume theory on the brane. We will see that the supersymmetric theory corresponds to
Thus the coupling to the background F (5) provides a necessary masslike term for the
It is now straightforward to diagonalize the operator 12 (1
(D05)) aD^ a(0) with respect
to the auxiliary doublet structure. Introducing the unitary transformation
1;2 =
one nds
U
1 (1
2
while
of M
U := p
1
2
1
(
(D05)) aD^ a(0) U y =
aDa +
r
2pr2 + z02
y(1 + )
)
2
1 (1 +
3) ;
(2.48)
where Da := @a + 14 !bc;a bc. Thus, setting ( ; 0)T := U ^ , one sees that 0 is projected out
encodes the physical degrees of freedom. Using (2.46), and recalling the de nition
in (2.32), the nal result for (2.40) takes the form
Sf =
i
2gy2m6
Z
d x
6 p
g6
n aDa + M
yo
(1 + O( op; )) :
(2.49)
Note that for a tendimensional MajoranaWeyl spinor, the bilinear
M1 Mp
vanishes
unless p = 3 (or 7), so the gamma matrix structure of the mass term is as it had to be.
contains the degrees of freedom of a single sixdimensional Dirac fermion and we could
write (2.49) in sixdimensional language, but for now it is more convenient to work directly
with the `10D' form.
Finally, we will infer from (2.49) and the bosonic YangMills terms (2.31), the
nonabelian analogs of the leading terms in (2.49) that complete (2.31) into a supersymmetric
invariant. Clearly the covariant derivative Da should be generalized to a gauge covariant
derivative, Da := @a + 14 !bc;a bc + [Aa; ]. We will, for convenience, continue to use the
same notation for this covariant derivative as we did above. A natural ansatz that will yield
the Yukawa couplings is simply to extend this to a tendimensional covariant derivative:
aDa
m[ m; ]. Our ultimate justi cation for this ansatz (detailed below) will be
HJEP07(21)65
that supersymmetry requires it.
Hence we take the fermionic terms of the YangMills action to be
Sym;f :=
i
2gy2m6
Z
d x
6 p
g6 Tr
is now valued in the adjoint representation of su(Nf ).
We've included a boundary action for the fermion,
f
Sbndry :=
i
Z
d5xp
n
y o
:
(2.50)
(2.51)
The analysis of [34] for fermions on antide Sitter space demonstrates that such boundary
terms are necessary in order to have a wellde ned variational principle. We will see that the
boundary action (2.51) is also required for supersymmetry. Without it, the supersymmetry
variation of the action would produce boundary terms that do not vanish on their own.
These points are analyzed in sections 4.3 and 4.4 below. In principle such boundary terms
should have already been present in (2.36), but we are not aware of any previous work on
this issue.
3
Supersymmetry
As noted previously, the intersecting Dbrane system of gure 1 preserves eight
superp
symmetries. In the nearhorizon limit of the D3brane geometry, the symmetry algebra is
enhanced to osp(4j4) with sixteen odd generators, provided the D3 and D5branes have
zero transverse separation. The leading lowenergy e ective description in the regime
Nc
gsNc
Nc= gsNc consists of a sixdimensional YangMills theory on
the rigid background (2.17) in which the transverse separation appears as a parameter,
(along with decoupled supergravity and u(1) sectors). Thus one expects the YangMills
theory to possess eight supersymmetries when z0 6= 0 and sixteen when z0 = 0.
In this section we rst review the Killing spinors of the background geometry [35, 36]
and the induced Killing spinors on the D5brane worldvolume [37]. Then, using the latter
as generators, we exhibit the full set of supersymmetry transformations on the YangMills
elds and establish the invariance of the action, (2.31) plus (2.50), modulo boundary terms.
To analyze the spectrum of modes on the asymptotically AdS4 space we choose an
adapted basis for the sixdimensional gamma matrices:
The next step is then to diagonalize the operator
set of eigenspinors on the twosphere. This is an S2 Dirac operator coupled to a Dirac
2D~ over a complete
monopole background. The eigenvalue equation
;r =
;r
3
;
is equivalent to the dim g equations
~
D
= iM ;
D
2 sin
ips ( 1
cos ) 2
( )s = iM ( )s :
(C.31)
(C.32)
(C.33)
(C.34)
(C.36)
(C.37)
(C.38)
+0 ;
(C.35)
= 0 spinors
HJEP07(21)65
Here
=
speci es the northern or southern patch of the S2 respectively. The two
solutions will be related by a transition function, (+)s = eips ( )s, on the overlap.
This is a classic problem with a completely explicit solution. (See appendix C of [49]
for a recent treatment.)
The eigenspinors are labeled by three indices, ; j; m, where
2 f+; ; 0g and (j; m) are angular momentum quantum numbers. Let
j :=
1
2 (jpsj
1) :
Then the eigenspinors with
=
have jvalues starting at j + 1 and increasing integer
steps, while m runs from
j to j in integer steps as usual. They are given by
1
2
( ;)js;m( ; ) = p N
j
m; 1 2ps ei(m+ ps=2)
djm; 1 2ps ( )12 + i djm; 12 ps ( ) 1
where djm;m0 ( ) is a Wigner little d function38 and
+0 = (1; 0)T . The
correspond to the special value j = j only, and their form depends on the sign of ps:
0(;j)s;m( ; ) = ei(m+ ps=2)
( N mj; j djm; j ( ) +0 ; ps > 0 ;
N mj;j djm;j ( ) 1 +0 ; ps < 0 :
Note these solutions only exist when ps is nonzero; j takes an unphysical value when
j
ps = 0. If ps = 0 then the
=
Nm;m0 are normalization coe cients:
solutions are a complete set with j 2 f 21 ; 23 ; : : :g. The
where the choice of phase will be convenient below. The corresponding eigenvalues are
is m0 Yjm( ; ) =
2j+1 1=2 eim djm;m0 ( ).
4
38These solutions can also be expressed in terms of spinweighted spherical harmonics. The relationship
j
Nm;m0 = ei jm m0j=2
r 2j + 1
4
;
M s;j =
2
p(2j + 1)2
ps2 :
The
= 0 modes are zero modes of
D~ , but this does not mean that they correspond
to massless spinors on AdS4 as there are other terms in the equation (C.30) that must be
taken into account.
To nd the fourdimensional spectrum we insert the mode expansion
into the linearized equation (C.30), using (C.31). Note that
( )s =
X
s
;j;m(x ; r)
( ;j)s;m( ; ) ;
=
3
;
B6 = B4
Here B4 is the product over the imaginary
;r and satis es ( ;r) =
B4
;rB4 1. (We
also used that it is necessarily the product of an odd number of 's, as charge conjugation
reverses chirality for Spin(1; 3).) Hence we'll need the action of 3 and charge conjugation,
on the eigenspinors. These are found to be
3 ( ;j);;sm = ( );s
;j;m ;
=
;
and
3 ( );s
0;j ;m = sgn(ps) 0(;j);s;m ;
ps
2
1( ( ;j);s;m) =
sgn m +
( ); s
;j; m ;
=
;
and
1( 0(;j);s;m) = 0(;j); ; sm :
In order to obtain the latter one requires the property djm;m0 ( ) = ( 1)m m0 dj m; m0 ( ).
The phase of (C.37) was chosen to make the action of charge conjugation as simple as
possible. Remember also that p s =
ps. See the discussion under (4.25).
Using all these facts, we nd that (C.30) splits into two families of coupled systems for
the modes s
;j;m. The coupled system for the
= 0 modes (which exist when ps 6= 0) is
(C.41)
(C.42)
(C.43)
(C.44)
(C.45)
(C.46)
7!
1
and
where
B
0
0
D+ B
D
0
0 !
s
0;j ;m
s
B4( 0;j ; m
)
!
= 0 ;
0 =
=
D4
h
ims;z3
D + rDr is the standard Dirac operator on the asymptotically AdS4 space.
D= 4 = (r2 + z02)1=2 r
1
Inserting (C.45) into (C.43) and dividing through by (r2 + z02)1=2, we have
where
B0 :=
sgn(ps)
3r
+
imz3;s ;
which is a more useful form for studying the large r asymptotics of solutions.
At this point we will content ourselves with understanding the r !
1 behavior of
solutions. Then it is su cient to expand the matrix operator in (C.46) through O(1=r).
To this order it diagonalizes and reduces to
2r
+ sgn(ps)my;s
1 + jpsj 1
2
r
+ O(1=r2)
s
0;j ;m = 0 ;
along with an equivalent equation for the conjugate spinor. The equation diagonalizes with
respect to r. If we decompose
into eigenspinors,
0;j ;m = 0;j ;m + 0s;;j ;m ;
s s;+
with
r s;
0;j ;m =
s;
0;j ;m ;
then the leading behavior of solutions is
0;j ;m / e sgn(ps)my;srr 23 m0 (1 + O(1=r)) ;
s;
m0 :=
1 + jpsj :
2
When my;s 6= 0 we have exponential decay or blowup behavior. When my;s = 0 we have
powerlaw behavior dictated by the mass m0, which we have de ned in such a way that
it can be identi ed with a standard AdS4 mass for the fermion. In other words, the
asymptotic behavior of solutions to (D= 4 + m)
= 0 on AdS4 is
/ r 23 m
. Since the ms0
s;
are all negative, we see that the normalizable modes in the case my;s = 0 are necessarily
associated with
0;j ;m. However the normalizable (exponentially decaying) modes when
my;s 6= 0 could be associated with either
psmy;s. It will be associated with
s;
0;j ;m if this sign is negative. We will comment further
s;
0;j ;m, depending on the sign of the product
Taking similar steps, one nds that the coupled system for the
=
modes can be
on this below.
put in the following form:
where
(C.47)
(C.48)
(C.49)
(C.50)
(C.51)
(C.52)
0
B
BB C
B
0
D
B C
D
3r
ps
2r
;
D
C = my;s
B = sgn m + ps (mz1;s
s
C C B B4( +;j; m
s
s
+;j;m
;j;m
B4(
s
;j; m
)
r
r2 + z02 +
1
C
A
) CCCC = 0 ;
imz3;s ;
This transformation diagonalizes (C.53) to the order we are working. The new variables
; satisfy the asymptotic equations
2r
jC(r)j + O(1=r2) ( (sj;m); (sj;m)) = 0 ;
1
ps 2
2r
where the +( ) is for ( ) respectively, and
jC(r)j =
my;s
+ jM s;j j2 =
my2;s
r
psmy;s +
r
(2j + 1)2
4r2
=
21r (2j + 1) ;
( jmy;sj
ps sgn(my;s) + O(1=r2) ; my;s 6= 0 ;
2r
my;s = 0 :
to r, as in (C.49). Then the asymptotic behavior of solutions to (C.55) is
Let (j;m) and (sj;;m) denote the positive and negative chirality components with respect
;s
Henceforth restrict our analysis to the r ! 1 behavior of solutions. Working through
O(1=r) the B entries can be dropped and the system reduces to
20
ps
2
1
1
ijM s;j j r
my;s
p2s + ijM s;j j r
1 1
A + O(1=r2)5
s
+;j;m
;j;m
!
e
i = C=jCj, and consider the unitary transformation
along with an equivalent equation for the conjugates. Let (r) denote the phase of C,
s
(j;m)
(j;m)
:= U
s
+;j;m
;j;m
e i =2 ei =2 !
e i =2
ei =2
s
+;j;m
;j;m
!
:
HJEP07(21)65
where the AdS4 masses are
s;
(j;m) /
s;
(j;m) /
8
< e jmy;sjrr 23 (1+ p2s sgn(my;s))(1 + O(1=r)) ; my;s 6= 0 ;
: r 23 m( )
j (1 + O(1=r)) ;
8
< e jmy;sjrr 23 (1 p2s sgn(my;s))(1 + O(1=r)) ; my;s 6= 0 ;
: r 23 m( )
j (1 + O(1=r)) ;
my;s = 0 ;
my;s = 0 ;
m
j
( ) = j
1
2
;
j
j +
:
3
2
The normalizable modes for
while the normalizable modes of
are those that have positive r chirality asymptotically,
are those that have negative r chiarlity asymptotically.
In both cases the normalizable modes along Lie algebra directions with my;s 6= 0 are
exponentially decaying while those along directions with my;s = 0 are powerlaw decaying.
(C.54)
(C.55)
(C.56)
(C.57)
(C.58)
(C.59)
s
0;j ;m modes as lling in a lower j = j rung for the
tower in the sense that
Recall that j starts at j + 1 = 12 (jpsj + 1) for these modes. However we can view the
jpsj
2
1
+
3
2
= m0 :
Also the asymptotic rchiralities match provided sgn(ps)my;s < 0 whenever my;s 6= 0.
Assuming this is the case, for the same reasons as discussed under (C.18), we can identify
s
(j ;m)
s
0;j ;m ;
12 =
i r ;
as the lowest rung of the
tower for those s such that ps 6= 0.
Finally we note that the rchirality condition can be translated back to a condition
on the sixdimensional
or on the tendimensional . First, since the action of r
commutes with the rotation U relating ; to the s;j;m, we see that
will be an asymptotic
eigenspinor of
when restricted to normalizable modes of
or
only. We will have
=
totically for the normalizable
type modes and
= +i r
asymptotically for the
i r
normalizable type modes. One can then show from (C.20) and (C.27) that
1
i r
= 0
()
1
r y
= 0 :
Hence positive (negative) r chirality corresponds to negative (positive) r y chirality.
D
Boundary supersymmetry
In this appendix we provide some of the details of the asymptotic analysis that we quoted
in subsection 4.4. We begin with Br and B
bndry, appearing in (4.62). From (3.29),
(C.60)
(C.61)
HJEP07(21)65
(C.62)
asymp(C.63)
+
(D.1)
B
r = " Tr
1
2
+
Tr
Meanwhile B
we infer
B
r y +
1
2 Fab abr + (Da m) m ar
1
2
r
"
:
[ m; n] mn r
M
m
m
y r
bndry is de ned in terms of the supersymmetry variation of the boundary
action, (3.26), according to (4.61). Taking the variation of (3.26) with respect to (3.27),
+
1
r
1
sin
(D
y
)
(D
y
)
y
r
+
Tr
n
y
"
o
:
(D.2)
in the large r expansion of Br + B
bndry.
Since the boundary measure in (4.62) is O(r3) as r ! 1, we must work through O(r 3)
For the moment we set aside the last terms of (D.1) and (D.2) involving the variation
of the fermion, and we focus on the remaining terms. Since " = O(r1=2) and
= O(r 3=2),
we must compute the terms in squarebrackets through O(r 2), utilizing the eld
asymptotics (4.58). All terms can contribute at this order. We expand out, plug in vielbein
factors, and collect terms together as follows:
The rst four sets of terms are proportional to the projector 12 (1
zi term, the relevant spinor bilinear is "
r y) acting to the
= O(1=r2). However,
(nn) is nonzero, then we must set the superconformal generators
0 to zero, which implies "
= 0. Hence, we get an extra order of suppression from the
zi does not contribute, and therefore this term can
be neglected. The same reasoning applies to the
remaining terms in this set involve the spinor bilinear "
1 part of
y in the rst term. The
+, which is O(1=r2). Thus we
need to evaluate them through O(1=r), which corresponds precisely to the contribution
from X~ in
y; A ; A . Speci cally,
2
r y +
r
2(r2 + z0 ) D
2
2(r2 + z02) D
r sin
y
y
1
! r
! r
1
r
r^ X~(n) +
^ X~(n) +
1 ^ X~(n) +
;
:
;
+
+
1
2
+
Tr
2
r y +
1
1
2(r2 + z02)
r 1 +
zi
zi
(nn) part of
+
(D
zi ) zi r +
2(r2 +z02) F
y( y r
r sin
1
2
F
r y
"
F
1
r sin
+ D
o
:
( y
r )
r2 +z02
r sin
) +
(D
zi ) zi r +
1
2
[ zi ; y] ziyr + (D
y) y r+
zi zi + [ zi ; zj ] zizj
r
zi zi (1
r y)+
y( y r +
HJEP07(21)65
+
(D.3)
(D.4)
(D.5)
Hence the relevant combination is
1
r
y
^ +
X~(n)(1 +
r y
)
rr^ +
^ +
+
=
=
2 ry
r
r
2 ry(~ (r) X~(n)) + ;
where we used
r y + =
+ and r^ r + ^
+ ^
= ( r1 ; r2 ; r3 )
~ (r). (See (B.12).)
Next consider the set of six terms inside the large round brackets of (D.3). It follows
from the eld asymptotics (4.58) that all of these terms start at O(1=r2). Furthermore all
of the gamma matrix structures associated with these terms commute with
they involve "+ + = O(1=r) and "
= O(1=r2), and we only need to worry about
the former. The order O(1=r2) terms in the round brackets are all of the form D(nn) or
ad( (znin)) acting on X~ , where D(nn) = @ + ad(a(nn)). Speci cally, the relevant combination
. Hence
of terms is
1
2r2 (
=
r ^
1
2r2
r ^ +
y rr^) D(nn)X~(n) + ( zi r ^
zi r ^
ziyrr^) [ (znin); X~(n)]
The remaining terms in the square brackets of (D.3) start at O(1=r2) and anticommute
with
r y
. Hence they involve the couplings "+
and "
+, and we only need to keep
the former. One simply needs to evaluate (A ; zi ) on their leading behavior, (a(nn); (znin)).
Collecting results, we have
" Tr
1
2
1
1
2r2
2r2
Tr
"+ Tr
"+ Tr
n
n y
D(nn) +
zi ad( (znin)) (~ (r) X~(n)) +o
+
2
r y
+D(nn) zi
(nn)
"
o
+ O(r 7=2) :
zi + 21 [ (znin); (znjn)] zizj
+
(D.7)
(D.8)
(D.9)
Plugging in (3.21) and (4.56) leads to the result in the text, (4.63).
The next step is to analyze the asymptotics of " , as given in (3.27). Our goal will
+ through O(r 3=2) since this is the only order that can contribute
be to compute ( " )
to (D.7), given the asymptotics of
give the supersymmetry variation of the nonnormalizable mode,
to set this eld to zero, its variation need not be zero. The reason is that we are allowing
0
(nn). Even if we choose
certain nonnormalizable modes of the bosonic
be turned on, and they can source the supersymmetry variation of the nonnormalizable
elds  namely (a(nn); (znin); X~(nn))  to
, (4.58). We note that the O(r 3=2) terms of ( " )
+
fermion modes.
where
We expand out (3.27) and collect terms as follows:
"
= fMr + M
+ M
+ Mrestg " ;
Mr = (r2 + z02)1=2Dr y ry +
M
M
(r2 + z02)1=2
r
r sin
D
D
y y +
y
y +
r
r
y
r sin
2(r2 + z02) Fr
2(r2 + z02) Fr
+
r ;
r ;
2(r2 + z02) F
r2 sin
;
and
Mrest =
(r2 + z02)1=2Dr zi rzi +
1
r
+ F r
r +
F
+
+
1
D
1
r sin
F
sin
zi
+
D
1
2
yzi
1
+
(r2 + z02)1=2 D
+ [ y; zi ] yzi +
+ D
zi zi + [ zi ; zj ] zizj :
(D.10)
Let's start with Mrest. It follows from the eld asymptotics that all seven terms in
the big squarebrackets are O(1=r2). Furthermore the gamma matrix structure of each of
these terms is such that it maps the ( )chirality eigenspace of r y to the ( )chirality
eigenspace. Hence, these terms acting on "+ give an O(r 3=2) contribution to ( " ) , while
these terms acting on "
give an O(r 5=2) contribution to ( " )+. Therefore these terms
can be neglected to the order we are working. In contrast the terms in the last line preserve
the chirality and so acting on "+ they give a contribution to ( " )+ that is O(r 3=2) that
must be kept. Finally, consider the rst two terms of Mrest. Using (2.30) one nds
(r2 + z02)1=2Dr zi rzi +
(r2 + z02)1=2
zi
yzi =
The Dr zi term is O(1=r2) and exchanges r y chiralities. If (znin) is nonzero then the
projector annihilates ", so the last term is also e ectively O(1=r2) and exchanges chiralities.
Hence these terms are on the same footing as the squarebracketed terms and can be
neglected. In summary,
(Mrest")+ =
(Mrest") = O(r 3=2) :
1
2r2
2
= Dr zi rzi +
zi zir 1
r y :
(D.11)
+ D(nn) zi
(nn)
zi + 21 [ (znin); (znjn)] zizj "+ + O(r 5=2) ;
Now consider M y. Plugging in (2.30) we have
Mr =
2(r2 + z02)Dr y ry + 2
r y( ry +
) +
2(r2 + z02)Dr y + 2
r y
2
r y1 + O(1=r)
ry 1
ry +
r y
2(r2 + z02) F
2(r2 + z02) Dr y
1
4r2(r2 + z02) P
y
;
where in the last step we recalled the de nition, (4.5). The rst term will drop out of (D.7)
since it involves the opposite projector. The large r expansion of P
in (4.47). Using that result here gives
y was determined
1
Mr = O(1=r)
ry 1
r y
r^ X~(nn) + O(1=r)
:
(D.14)
1
2r2
= O(1=r)
y 1
r y
+
^ X~(nn) + O(1=r)
r ;
(D.15)
= O(1=r)
y 1
r y
^ X~(nn) + O(1=r)
r :
(D.16)
y 1
2r2
2r2
r y
1
2r2
r y
1
2r2
+
+
sin
4r(r2 + z02) P
1
4r(r2 + z02) P
r
^ r + ^ r
"+ + O(r 5=2)
y r^ r + ^
+ ^
X~(nn) "+ + O(r 5=2)
y~ (r) X~(nn) "+ + O(r 5=2) ;
Similar manipulations lead to
M
and
M
r sin
Thus we have
((Mr + M + M )")+ =
((Mr + M + M )") = O(r 3=2) ;
(D.17)
Combining (D.12) and (D.17) leads to the result quoted in the text, (4.65).
Our nal goal is to derive the asymptotics of
due to the massless AdS4 fermions,
as given in (4.55) with (4.67). The leading behavior of these modes as r ! 1 is O(r 3=2)
and the rst subleading behavior is O(r 5=2). They are solutions to the fermion equation
of motion
0 =
D
+ rDr +
D
+ M
y +
zi ad( zi ) +
y ad( y
(D.18)
Our analysis in appendix C.2 shows that the massless modes are in the simultaneous kernel
of ad( y1) and ad(P ). Taking this into account with respect to the eld asymptotics (4.58),
the large r form of the equation of motion is
0 =
3
2r
1
r
r ad(a(rn)) +
~ D~
1 h
^ +
y +
^ +
1 h
r
y i
r^
;
where ~ D~ is (the 10D embedding of) the standard Dirac operator on the twosphere.
The rst three terms give the leading order equation of motion while the remaining terms
give O(1=r) corrections.
Note that this equation only involves the asymptotics of the bosonic modes that we
keep in the truncation, (5.1), and therefore the asymptotics of the solution to the order we
need will be the same as in the truncated theory. Hence we will derive the equations of
motion for the fermion in the truncated theory, which we quoted in (5.7), and then consider
the asymptotics of it.
We rst use results from appendix C.2 to determine the form of the 10D fermion,
,
restricted to the massless AdS4 modes. These are the j = 1=2 doublet ( 12 ;m)(x ; r). They
satisfy (C.55) with the plus sign, and since my;s = ps = 0 for these modes, we have
2
jC(r)j = jM s; 1 j=r = 1=r. Hence
( 12 ;m)(x ; r) = ( r) 3=2 ~( 12 ;m)(x ) (1 + O(1=r)) :
and therefore the corresponding
; 12 ;m modes are
; 12 ;m = p12 e i =4
spinor, (C.39), restricted to these modes, which we will denote by
normalizable modes.
The boundary data ~( 12 ;m) can be decomposed into eigenspinors of r, ~( 12 ;m)
=
and we will see that ~( 12 ;m) corresponds to the normalizable modes and ~( 12 ;m) to the
non+
The phase of C(r) that appears in the unitary transformation of (C.54) is
=
=2,
( 12 ;m). Hence the 6D
( )
j=1=2, takes the form
( )
j=1=2 =
X
m= 1=2
( 12 ;m)(x ; r)
1
p
2
e i =4
+; 12 ;m + ei =4
; 12 ;m
where the
are given by
Hence
; 12 ;m = N m1=;21=2eim
0
A :
( )
j=1=2 =
X
m
( 12 ;m)(x ; r)
N m1=;21=2eim
0 d1m=;21=2( ) 1
A :
Now, using d11==22;1=2 = d1=12=2; 1=2 = cos 2 , d1=12=2;1=2 =
d11==22; 1=2 = sin 2 , and N11==22; 1=2 =
iN11==22;1=2
iN1=2, one nds that this spinor can be expressed in the form
( )
j=1=2 = N1=2
cos 2 sin 2
sin 2 cos 2
ei 3 =2
0
i ( 12 ; 12 )(x ; r) A
1
= exp i
exp
2
2
6D(x ; r) ;
where in the last step we introduced the 6D spinor
i ( 12 ; 12 )(x ; r) A ;
1
and wrote the expression in 6D notation with the de nitions (C.31).
(D.20)
r ~( 12 ;m)
,
(D.21)
(D.22)
(D.23)
(D.24)
(D.25)
6D has a large r expansion starting at O(r 3=2) with the leading behavior given in
terms of the boundary spinors ~, (D.20). If one restricts the 4D spinors to r eigenspaces,
, this corresponds to restricting 6D to 6D de ned by
i r
6D = (
12) 6D =
6D :
We use this to express j(=)1=2 in the form
( );
2
exp
2
6+D(x ; r) + exp
exp
2
6D(x ; r) :
This result is straightforwardly expressed in 10D notation via (C.20). We nd
( )
j=1=2 = hS2( ; ) +(x ; r) + hS2( ; )
(x ; r) ;
where we made use of (3.8),
+ +
is de ned in terms of 6D via (C.27), and
r y
:
y
This is (4.54), which is given in a natural basis with respect to the S2 frame in which
are constant. Indeed, this was assumed throughout the analysis in appendix C.2.
This is to be plugged into the full fermion equation of motion,
E :=
expand the Dirac operator,
with the bosonic elds restricted to (5.1) as well. The basic idea it to pull the factors of
; ) through to the left and collect the terms that are proportional to each. We
satisfy
; ;
identities:
and
(D.26)
(D.27)
(D.28)
(D.29)
(D.30)
(D.32)
(D.33)
(D.34)
aDa =
D + rDr +
D~= S2 +
1
sin
ad(A ) +
ad(A ) ; (D.31)
with D~= S2 the standard Dirac operator on the unit S2. Then we make use of the following
D~= S2h( ; ) =
yh( ; )
h( ; ) r ;
rh( ; ) = h( ; ) r ;
= r r yh( ; )
=
h( ; ) r ;
hS2( ; )
hS2( ; )
hS2( ; )
hS2( ; ) ^ ( ;
= hS2( ; ) ^ ( ;
= hS2( ; ) r^ ( ;
; y)
;
; y)
; y)
;
:
+ (r2 + z02)1=2( ;
+ hS2 (
; )
D
+ rDr
The masslike term
1
r(r2 + z2)
0
h1h2h3
r r y
r
1 ^A +
1
r sin
^A + r^ y =
( ^ ^ X~ + ^^ X~ + r^r^ X~ ) =
1
2r2 X~ :
where we used (5.8) and (5.10), vanishes for the AdS4 background where z0 = 0, and
in general the rdependent mass vanishes asymptotically like O(1=r2). Plugging in the
truncation ansatz (5.1) for the bosonic modes, observe that
Hence the quantities in curly brackets in (D.35) are independent of ;
on this ansatz.
After introducing the triplet notation (5.10) and the metric (5.4), we obtain the result
quoted in the text, (5.6) and (5.7):
; y
1
2r2
1
1 ^A +
1
r
1 ^A +
r sin
r sin
1
1
=
+ zi [ ~ zi ; +
]
^A + r^ y;
r + + zi [ ~ zi ;
^A + r^ y; +
: (D.35)
z
2
0
r(r2 + z02)1=2
Note for these last three we are employing (D.29) as well. Then we nd
E
= hS2 ( ; )
D
+ + rDr
(D.36)
(D.37)
(D.39)
(D.40)
O(1=r) in the operator acting on , one nds
The asymptotics of
0 =
1 h
D(nn) +
hi ad(X(in))i + O(1=r2)
:
e =
D
+ rDr
h1h2h3
+ zi [ zi ; ] + hi [X i; ] = 0 : (D.38)
Now we analyze the large r asymptotics of this equation. Keeping terms through
r(r2 + z02)1=2
z
2
are
2r
1
1
+(x ; r) =
(x ; r) =
( r)3=2 0
( r)3=2
(nn)(x ) +
r (0n)(x ) +
( r)5=2 1+(x ) + O(r 7=2) ;
( r)5=2 1 (x ) + O(r 7=2) :
1
1
This is consistent with (4.56), remembering that ( r3 )cart = ( r)S2 . The
1 are found
by plugging this expansion back into (D.39) and solving it at the rst subleading order.
1+ =
1 =
h
r h
zi ad( (znin))
i (n) + h ad(a(rnn)) +
0
r hi ad(X(in))i (nn) ;
0
D(nn) + zi ad( (znin))
i (nn) + r h ad(a(rnn))+ hi r ad(X(in))i 0(n) : (D.41)
0
This can be expressed in terms of Cartesian frame quantities using ( r)S2 = ( r3 )cart and
( r~ (h))S2
+ = ( r ;
r ; ry)S2
+ = ( y
;
y; ry)S2
+ =
y(~ (r))cart
+ ;
(D.42)
which leads to the results for (D.40) quoted in (4.67).
E
Some details on the truncation
F^ zi = D
ones we have
Here we collect expressions for the components of the nonabelian eldstrength and
covariant derivatives evaluated on the truncation ansatz (5.1). We use a 10D notation A^M for
the gauge eld and Higgs elds in which we identify (A^zi ; A^y)
( zi ; y) and, for example,
zi . There is nothing to say about F ; F r; F^ zi ; F^rzi ; F^zizj . For the remaining
HJEP07(21)65
and
F
F
F^ y
^
Fzi
^
Fzi
F^ziy
trnc sin
trnc
trnc
trnc
trnc
trnc sin
2r
1
2r2
2r
1
2r2
2r
1 ^ D X~ ;
^ D X~ ;
r^ D X~ ;
2r
1 ^ [ zi ; X~ ] ;
^ [ zi ; X~ ] ;
r^ [ zi ; X~ ] ;
Fr
Fr
^
Fry
F
^
F y
^
F y
trnc
trnc
trnc
trnc
trnc
trnc sin
1
2r2
^ ~
X
sin
2r2
P
2r2
P
2
sin
1
2r2
2r2
X
X
2r
1 ^ DrX~ ;
^ X~ +
2
2r3
r^ X~ +
sin
2r
sin
2r r^
^ DrX~ ;
1
2r2
2X~
r^ DrX~ ;
1
2 2r [X~ ;
1
1
2 2r [X~ ;
2 2r [X~ ;
X~ ] ;
(E.1)
(E.2)
(E.3)
We also list some formulae that are used in subsection 6.4 for the reduction of the BPS
equations. From (E.3) one nds that
and converting to the Cartesian coordinate system results in
HJEP07(21)65
Dr y trnc
1
sin
D
Fr + D
y trnc
y trnc
sin
2r2
DrX~
1 ^
2r
1
2 2r2 [X~ ; X~ ] ;
DrX~
1
2 2r2 [X~ ; X~ ] ;
1 ^
2r
DrX~
1
2 2r2 [X~ ; X~ ] ;
Dr3
Dr2
Dr2
y trnc
y trnc
y trnc
1
2r2
2r2
1
2r2
DrX
DrX
DrX
1
1
1
1
sin
1 i
2 p~q~ dxp~ dxq~ :
Likewise, converting from Fpr; Fp ; Fp , to the Cartesian frame Fpri results in
Fpr1
Fpr2
Fpr3
trnc
trnc
trnc
1
1
2r2
2r2 sin cos ( 2r2Fpr) + sin sin DpX
cos DpX
2 ;
sin cos DpX
3 + sin sin ( 2r2Fpr) + cos DpX
1 ;
2r2 sin cos DpX
2
sin sin DpX
1 + cos ( 2r2Fpr) ;
while
Fpy
trnc
1
2r2 sin cos DpX
1 + sin sin DpX
2 + cos DpX
3 :
Identical expressions hold for the F^zpri and F^zpy upon replacing Dp ! ad( zp ).
F
The BPS energy
from (6.30).
In this appendix we show how one obtains (6.65) from (6.62), and as a special case, (6.37)
parameterize R4 with the standard orientation. Introduce a basis of selfdual twoforms,
First we introduce some notation that exposes the structure of 04. Let xp~ = (x1; x2; z^1; z^2)
!1 = dx2 dz^2
dx1 dz^1 ;
!2 = dx2 dz^1 + dx1 dz^2 ;
!3 = dx1 dx2 + dz^1 dz^2 : (F.1)
These can be expressed in terms of 't Hooft matrices,
(E.4)
(E.5)
(E.6)
(E.7)
(F.2)
where our conventions are
0 0 0
1 = BBB 01 00
1 0 1
0 1 C
C ;
A
0
1 0 0
0 0
2 = BBB 00
0 1 0 C
1 0 0 C
C ;
A
1 0 0 0
1 0 0 0 C
3 = BBB 0 0 0 1 C
C :
A
0 0
1 0
Note this is a slightly di erent convention than the standard one given in [100] in that
( 1; 2; 3)here = ( 2; 1; 3)standard :
With our convention matrix multiplication gives the quaternion algebra, i j =
k, with a plus sign in front of the
rather than a minus.
Then, in terms of the twoforms (F.1), one has
(F.3)
(F.4)
ij +
(F.5)
(F.6)
(F.8)
04 =
1
4(r2 + z02)2
dy^ dr1 dr2 dr3 + ( dy^ dr1 + dr2 dr3) ^ !1+
+ ( dy^ dr2 + dr3 dr1) ^ !2 + ( dy^ dr3 + dr1 dr2) ^ !3 :
Dropping the !1 and !2 terms gives !0 .
4
Converting to spherical coordinates, (r; ; ), results in
04 = r2 sin d d r^i + r dy d ^i + r sin dy d ^
i ^ !i + dr terms :
Here we have suppressed terms that have a leg along the radial direction since they will
not contribute to the boundary integral. It follows that
( 04 ^ !CS)12 z^1z^2y^ =
2
1 ( i)p~q~ !CS A^y r2 sin r^i +A^ r ^i A^ r sin
^i ; A^p~ ; A^q~ ;
(F.7)
where we are using the notation !CS(A^A; A^B; A^C )
(!CS)ABC for the components of the
ChernSimons form. If we want !40 ^ !CS instead, then we drop the i = 1; 2 terms.
These expressions integrated against dx1 dx2 d d dz^1 dz^2 dy^ at the boundary r ! 1.
Hence we need the large r limit of (F.7). The leading behavior of the A^p~ is O(1) and given
by the nonnormalizable S2 singlet modes. Therefore the furthest we need to go in the
subleading asymptotics of ( y; A ; ) is the X~(n) terms, which will yield a nite contribution
to (F.7) as r ! 1. In fact, if one restricts to the X~(n) terms, the rst factor in !CS collapses
A^y r2 sin r^ + A^ r ^
A^ r sin
^ !
1
2 sin
X~(n) :
One might worry that the
y1 and 't Hooft charge terms in the asymptotics of ( y; A ; )
will lead to a divergence, but this is not the case. The 't Hooft charge drops out of (F.7).
The
y1 term can contribute, but integration over the twosphere will pick out subleading
behavior in the A^p~ factors such that the result is
nite. (The integration over S2 should
be carried out before the r ! 1 limit is taken.) We thus have
Z
d x
S2
d d sin
~ p~q~
2 !CS(X~(n); A^p~; A^q~) + r2r^ !CS( y ; A^p~; A^q~) :
1
(F.9)
such that
is equivalent to
1
X
m= 1
Here the normalization convention is consistent with the one taken in (4.44). Then (F.9)
ap~;(1;m)(x ; r)Y1m( ; ) =
r^ ~ap~(n)(x ) + O(r 3) :
(F.13)
Both ChernSimons terms are of a similar structure in that they involve an
adjointvalued scalar in one of the factors. When this is the case, one can obtain the following
equivalent expression, starting from the de nition (6.28):
!CS(X~(n); A^p~; A^q~) = 2 Tr nX~(n)F^p~q~o + @q~ h Tr fX~(n)Ap~g
(F.10)
is constant and that any powerlaw modes of A^p~ commute with
Here it should be understood that the total derivative term is only present when p~; q~ = 1; 2.
An analogous expression holds with X~(n) !
1
y . However in this case we can use that
y
1
y1 to observe that the
total derivative terms just subtract o half of the rst term, resulting in:
Z
lim
r!1 S2
d r2r^i !CS( y1; A^p~; A^q~) = lim
Z
r!1 S2
d r2r^i Tr
n y1F^p~q~o :
Now let us recall the mode expansion of A^p~ = (Ap; zp). The terms we need are
(F.11)
ap~;(1;m)(x ; r)Y1m( ; ) +
(F.12)
where ap~(x ; r) = a(p~nn)(x ) + O(r 1) as usual, and we introduce the triplet notation, ~ap~,
1
X
m= 1
p
2r2
1 Z
04 ^ !CS =
Z
(nn)
2 Tr fX~(n)fp~q~ g
p Tr f y1f~p~(q~n)g+
1
3
(nn) i
(nn) + [a(p~nn); a(q~nn)] and f~q~(q~n) = 2@[p~~aq~(]n), and we used the integral
Z
S2
d r^ir^j =
4
3 ij :
(F.14)
(F.15)
This reproduces the magnetic contribution to the energy bound given in (6.65).
Dropping the terms proportional to the rst two 't Hooft symbols will give the result for
04 ! !40. Furthermore there are some simpli cations if we plug in the explicit form of p3~q~:
!40 ^ !CS =
2 Z
d2x 2 Tr nX(3n) f1(2nn) + [ (zn1n); (zn2n)] o
1
3
p Tr f y1f132(n)g+
(nn) i
@1 h Tr fX(3n)a2 g
(nn) i
:
(F.16)
For the second term we can pull y1 out of the integral. Then we are simply computing
the total magnetic ux of the third component of the normalizable mode of the gauge eld
triplet. (See (6.35).) Meanwhile by Stokes' theorem the last two terms give a line integral
around the circle at in nity:
Z
R2
h
2 n
(nn) i
g
h
(nn) io =
g
Tr
nX(3n)a(nn)o : (F.17)
I
S1
1
Taking these facts into account one nds that (F.16) reproduces the magnetic energy
contribution to (6.37).
For the electric energy contribution, we rst note that
(?E)12 z^1z^2y^ = r2 sin grrFr0 :
(F.18)
(grr) 1 and the de nition (6.36), one quickly nds the remaining terms
in (6.37) and (6.65).
Open Access.
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