Holographic RG flows for fourdimensional \( \mathcal{N}=2 \) SCFTs
Accepted: June
Holographic RG ows for fourdimensional SCFTs
Nikolay Bobev 0 1 4
Davide Cassani 0 1 2
Hagen Triendl 0 1 3
Prince Consort Road 0 1
London SW 0 1
0 Via Marzolo 8 , 35131 Padova , Italy
1 Celestijnenlaan 200D , B3001 Leuven , Belgium
2 INFN , Sezione di Padova
3 Department of Physics, Imperial College London
4 Instituut voor Theoretische Fysica, KU Leuven
We study holographic renormalization group ows from fourdimensional N = 2 SCFTs to either N = 2 or N = 1 SCFTs. Our approach is based on the framework of vedimensional halfmaximal supergravity with general gauging, which we use to study domain wall solutions interpolating between di erent supersymmetric AdS5 vacua. We
AdSCFT Correspondence; Supergravity Models; Extended Supersymmetry

show that a holographic RG
ow connecting two N = 2 SCFTs is only possible if the
avor symmetry of the UV theory admits an SO(3) subgroup. In this case the ratio of
the IR and UV central charges satis es a universal relation which we also establish in eld
theory. In addition we provide several general examples of holographic ows from N = 2
to N = 1 SCFTs and relate the ratio of the UV and IR central charges to the conformal
dimension of the operator triggering the ow. Instrumental to our analysis is a derivation
of the general conditions for AdS vacua preserving eight supercharges as well as for domain
wall solutions preserving eight Poincare supercharges in halfmaximal supergravity.
1 Introduction
2 Gauged halfmaximal supergravity
2.1 Supersymmetric domain walls
3 Holographic
ows between N = 2 SCFTs
3.1
3.2
3.3
3.4
3.5
4.1
5.1
5.2
Review of conditions for N = 4 AdS5 vacua
Uniqueness in the absence of avor symmetries
Two distinct N = 4 AdS5 vacua
Conditions for ows with eight Poincare supercharges
Flow between two N = 4 AdS5 vacua and its holographic dual
4 Field theory derivation of ratio between central charges
Anomalies in fourdimensional N = 2 SCFTs
4.2 RG ow between N = 2 SCFTs
5 Holographic
ows from N = 2 to N = 1 SCFTs
Conditions for N = 2 AdS5 vacua
Review of conditions for minimally supersymmetric ows
5.3 A model with one N = 2 vacuum
5.4 A model with two N = 2 vacua
6 Discussion A Uniqueness of halfmaximal AdS solutions in various dimensions
A.1 Four dimensions
A.2 Six dimensions
A.3 Seven dimensions
B The generator of the IR U(1)R symmetry
was used to obtain a holographic proof of the atheorem. There have been numerous
deformations on the other hand are less accessible with eld theory tools and thus the
supergravity results derived here should teach us important general lessons for the structure
of supersymmetric RG
ows.
To understand the general constraints for the existence of distinct supersymmetric
AdS5 vacua and the ow connecting them we present a detailed analysis of the
supersymmetry conditions in halfmaximal gauged supergravity. The results depend on the number,
n, of vector multiplets in the theory and on the type of gauging performed. The existence
of at least one AdS5 vacuum with 16 supercharges implies that an U(1)
SU(2)
Hc
subgroup of the SO(5; n) global symmetry of the supergravity theory should be gauged [8].
The U(1)
SU(2) gauge eld is dual to the Rsymmetry of the fourdimensional N = 2
SCFT dual to this AdS5 while Hc represents the continuous
avor symmetry. If Hc is
trivial we nd that there is a unique AdS5 vacuum with 16 supercharges in the
supergravity theory.1 However when Hc is nontrivial then it must contain an SO(3) subgroup and
there can be another AdS5 vacuum in the supergravity theory with a di erent value of the
cosmological constant. Moreover these two distinct AdS5 vacua are connected by a regular
supersymmetric domain wall solution in the gauged supergravity theory which we construct
analytically. In addition we establish that the RG
ow in the dual QFT should be triggered
by vacuum expectation values (vevs) for two scalar operators of dimension
= 2 and the
ratio of these vevs has to be a xed constant. One of the two scalar operators belongs to the
energy momentum multiplet of the SCFT and the other one sits in the SO(3)
Hc avor
current multiplet. The di erent values of the cosmological constants of the two AdS5 vacua
translate into di erent values for the conformal anomalies of the dual UV and IR N = 2
SCFTs. We compute this ratio of central charges using our supergravity results and are
able to reproduce it by an anomaly calculation in the dual SCFT. The result is a universal
expression for the IR conformal anomalies in terms of the UV conformal anomalies as well
as the central charges of the SO(3) avor current. We also
nd that these anomalies are
related to the constant that controls the relation between the scalar vevs triggering the ow.
1This result can also be established for AdS vacua with 16 supercharges in four, six, and
sevendimensional halfmaximal gauged supergravity.
{ 2 {
HJEP06(218)
Having described the conditions for the existence of N
= 4 AdS vacua in
vedimensional gauged supergravity it is natural to ask whether there are other AdS vacua
which preserve less supersymmetry. To answer this we analyze the general conditions for
N = 2 AdS5 vacua and then we focus on theories that admit both an N = 4 and one or
more N = 2 vacua. Perhaps not surprisingly we
nd that as we increase the number of
vector multiplets in the supergravity theory we can have an increasing number of distinct
N = 2 AdS5 vacua. The details of this structure depend on the matter content and the
choice of gauging in the supergravity theory. To illustrate our general approach we focus on
two particular examples. We rst establish a holographic analog of the QFT result in [14]
in which it was shown that every fourdimensional N = 2 SCFT with an exactly marginal
ow was in fact rst constructed and discussed in some particular holographic
examples, see [2, 15, 16], but here we o er a more general treatment. Our general setup
should capture the RG
ow relating the N = 2 and N = 1 MaldacenaNun~ez SCFTs [17]
arising from M5branes wrapped on a Riemann surface. While it is widely believed that
this RG
ow exists, and is of the class discussed in [14], its explicit holographic
construction is still elusive. Our results should o er some insight into this problem. Moreover, if
our setup can be embedded in elevendimensional supergravity it can potentially capture
holographic RG
ows connecting the N = 2 MaldacenaNun~ez SCFT [17] and one of the
N = 1 SCFTs studied in [18, 19]. In addition to this we study a setup with one N = 4
and two distinct N = 2 AdS5 vacua and discuss the supersymmetric domain wall solutions
which interpolate between them. This may capture holographic RG
ows which relate the
N = 2 MaldacenaNun~ez SCFT and two of the N = 1 SCFTs of [18, 19].
Finally we would like to note that we do not study a speci c embedding of the gauged
supergravity theories we work with in string or Mtheory. Thus our results are universal
and apply to all supersymmetric AdS vacua which admit a lowerdimensional e ective
description in terms of halfmaximal supergravity. This universality is somewhat similar
in spirit to the results for holographic RG
ows across dimensions discussed in [20].
We begin our presentation in the next section with a brief general introduction to
vedimensional N = 4 gauged supergravity. In section 3 we identify under what conditions
there can be two distinct AdS vacua of such a supergravity theory which preserve all
16 supercharges and construct gravitational domain wall solutions interpolating between
these vacua. Whenever such a ow is possible it exhibits a universal relation between the
UV and IR central charges which we establish by eld theory methods in section 4. We
continue in section 5 with a study of the conditions for the existence of AdS5 vacua with 8
supercharges and a discussion on domain wall solutions connecting such vacua. Section 6
is devoted to a short discussion on our results and their implications for holography. In
appendix A we present the extension of some of the results in section 3 to halfmaximal
gauged supergravity in four, six and seven dimensions. In appendix B we give some more
details on the ow in section 5.
{ 3 {
Gauged halfmaximal supergravity
In this section we review the basic properties of vedimensional gauged N = 4
(halfmaximal) supergravity [21{24] that are relevant for our analysis, mainly following [24].
Ungauged N = 4 supergravity has USp(4) Rsymmetry and consists of a gravity
multiplet and n vector multiplets. The gravity multiplet contains the metric g , four gravitini
i ; i = 1; : : : ; 4 transforming in the 4 of USp(4), six vectors (dubbed the graviphotons)
A0 ; Am, with Am, m = 1; : : : ; 5 transforming in the 5 of USp(4) and A0 being neutral,
four spin1/2 fermions i in the 4 of USp(4), and one neutral real scalar . We will label
the vector multiplets with the index a = 1; : : : ; n. Each vector multiplet contains a vector
Aa , four spin1/2 gaugini ai, and
parametrize the coset space
ve real scalar elds. All together the scalar elds
Mscal = SO(1; 1)
SO(5; n)
SO(5)
SO(n)
;
ds2(Mscal) = 3
2
d 2
dMMN dM MN :
1
8
The isometry group of the scalar manifold, SO(1; 1) SO(5; n), is the global symmetry
group of the ungauged supergravity action. In addition, the scalar eld space admits a
local invariance under SO(5)
SO(n). The group SO(5) is promoted to Spin(5) ' USp(4)
when discussing the couplings to the fermions. It is then convenient to convert the SO(5)
index m of the scalar vielbeine VM
m into USp(4) indices i; j via SO(5) gamma matrices,
This satis es VM ij = VM [ij] and
ij VM ij = 0 and hence transforms in the 5 of USp(4).
Here ij is a 4
4 real symplectic matrix.
where the rst factor is spanned by
while the second factor is spanned by the scalars
in the vector multiplet, which we denote by
x, x = 1 : : : ; 5n.
We indicate the coset
representative of the second factor by V = (VM m; VM a), where M = 1; : : : ; n + 5 labels the
fundamental representation of SO(5; n). Being an element of SO(5; n) this obeys
MN =
VM
m
VN
m + VM aVN a ;
where MN = diag( 1; 1; 1; 1; 1; +1; : : : ; +1) is the at SO(5; n) metric, which is also
used to raise and lower the M; N indices (while the m; n and a; b indices are contracted
with the SO(5) and SO(n) Kronecker delta, respectively). Alternatively, the coset can be
represented by the positive de nite scalar metric
MMN = VM
m
VN
m + VM aVN a ;
which also plays the role of the gauge kinetic matrix for the (5 + n) vector elds AM =
(Am; Aa ). The metric on the scalar manifold, which determines the scalar kinetic terms, is
1
2
VM
ij =
VM
m imj :
{ 4 {
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
In gauged supergravity a subgroup of the global symmetry group SO(1; 1) SO(5; n) is
promoted to a local gauge symmetry by introducing minimal couplings to the gauge elds
and their supersymmetric counterparts. In this way part of the global symmetry group is
broken. When some vector
elds transform in nontrivial nonadjoint representations of
the gauge group, additional Stuckelberglike couplings to antisymmetric ranktwo tensor
elds may be required in order to ensure closure of the gauge symmetry algebra. Such
vector elds can then be gauged away, leaving just massive tensor elds together with the
other vectors [21, 23, 24].
The possible gaugings are classi ed by the embedding tensor formalism [25{27]. This
introduces the gauge couplings via a spurionic object  the embedding tensor  and
elds that consist of a tensor eld for each of the original vector
elds. In N = 4 supergravity, the embedding tensor splits into three di erent
representations of SO(1; 1)
SO(5; n), denoted by M ; MN = [MN] and fMNP = f[MNP ]. Their
transformation under SO(5; n) follows from the indicated index structure. With respect
to SO(1; 1), M and fMNP carry charge
1=2, while MN has charge 1. Supersymmetry
of the Lagrangian imposes a set of quadratic constraints on the embedding tensor, whose
possible solutions parametrize the di erent consistent gauged N = 4 supergravity theories.
In this paper we are interested in theories admitting at least one fully supersymmetric
AdS5 vacuum. In [6] it was shown that a necessary condition for this is M = 0. This
means that the SO(1; 1) part of the global symmetry is not involved in the gauging and
the gauge group is entirely contained in SO(5; n). We therefore take M = 0 from now on.
In this case, the quadratic constraints are simply given by
fR[MN fP Q]
R = 0 ;
M QfQNP = 0 :
The fMNP correspond to structure constants for a (nonAbelian) subgroup of SO(5; n),
while the MN assign the charges under the U(1) gauge eld A0 .
The embedding tensor determines the gauge covariant derivatives,
D
= r
AM fM
NP tNP
A
0 NP tNP ;
where tMN = t[MN] generate so(5; n). It also determines the shift matrices that appear in
the fermion supersymmetry variations and specify the scalar potential.
In the following we abbreviate the contraction of the embedding tensor components
f MNP and
MN with the coset representatives VM
m and VM
a by
f^mnp = f MNP
f^mna = f MNP
f^mab = f MNP
f^abc = f MNP
VM
VM
VM
m
m
m
VN nVP p ;
VN nVP a ;
VN aVP b ;
VM aVN bVP c :
^mn =
^ma =
^ab =
MN
MN
MN
VM
VM
m
m
VN n ;
VN a ;
VM aVN b ;
These \dressed" embedding tensor components will always be denoted by a hat symbol.
Since they depend on the scalars, generically they vary along domain wall solutions. Also,
they appear in the conditions for supersymmetric AdS vacua.
{ 5 {
(2.6)
(2.7)
(2.8)
where A(r) is the warp factor which depends only on the radial coordinate r. The
oneand twoform supergravity elds vanish, while the scalars have a radial pro le,
=
(r),
x =
x(r). In particular, when the solution is AdS5, the scalars are constant and we have
A = r=`, where ` is the AdS radius. The latter is related to the cosmological constant,
which in our conventions is the same as the critical value of the scalar potential, V =
6=`2.
The supersymmetry conditions for solutions of this form (and with M = 0) read [28]
i x0vxa ij
5 j
2P a ij j = 0 ;
SO(5;n)
the SO(5) SO(n) scalar manifold, de ned as
where i are the supersymmetry parameters, satisfying the symplecticMajorana condition
i =
ij C( j )T . A prime means derivative with respect to r and vxam are the vielbeins on
d xvam =
x
(
V
1
dV)am :
Moreover we introduced the shift matrices
where "mnpqr is the totally antisymmetric symbol, and
P ij = P mn
mnij ;
with
P mn =
2 ^mn +
1"mnpqrf^pqr ;
1
36
P aij =
1
p
1f^amn
mn ij :
1
p
The shift matrices also determine the scalar potential as
8
2
3 P ij Pij :
The supersymmetry conditions above are obtained by setting to zero the fermion
variations given in [24].2 Eqs. (2.10), (2.11) arise from the gravitino variation, (2.12) arises
from the variation of the spin 1/2 fermion in the N = 4 gravity multiplet, while (2.13)
q 38 @ P ij, A2a ij = p12 P a ji. For the scalar manifold geometry and the Cli ord algebra we use the
same conventions as in [28]. We have reabsorbed the gauge coupling constant g appearing in [24] into the
embedding tensor.
comes from the gaugino variation. The derivation of (2.10), (2.11) assumes that the
supersymmetry parameters depend on the coordinate r but are constant on R1;3; this means
that we are only describing the Poincare supersymmetries. For generic domain walls these
are all the supersymmetries allowed, however in the special case of AdS solutions one also
has the conformal supersymmetries, which depend on the coordinates on R1;3. For this
reason, the case of AdS solutions will be analyzed separately in the next sections.
As we discuss in detail later, the domain wall supersymmetry conditions imply the
existence of a real superpotential function W constructed out of the shift matrix P mn, which
drives the ow of the warp factor and the scalar elds. Introducing an index X = (0; x),
we can denote the scalars as
X = ( ; x) and the scalar kinetic matrix as
Then the ow equations read
gXY =
3
0
However, this is not the full information encoded into supersymmetry. Indeed, one also
nds a set of algebraic constraints restricting the scalar elds that can possibly ow. After
these constraints are satis ed, the scalar potential (2.17) can be expressed in terms of the
superpotential as
This is su cient to ensure that the Einstein and scalar equations of motion are satis ed [29,
30]. When in particular the superpotential is extremized, @X W = 0, we obtain an AdS
solution with radius ` 1 = W .
The speci c form of the superpotential and of the constraints depends on the amount
of supersymmetry being preserved and will be discussed in the next sections.
3
Holographic
ows between N
= 2 SCFTs
In this section, we rst review the conditions for fully supersymmetric AdS5 vacua in
halfmaximal gauged supergravity. Then we show that if there is one such vacuum and the gauge
group does not contain any compact part in addition to the U(1)
SU(2) Rsymmetry of
the vacuum, then the latter is unique, up to moduli. If on the other hand there is one
N = 4 vacuum preserving an SO(3) in addition to the Rsymmetry and a certain condition
on the gauge coupling constants is satis ed, then we show that there exists a second N = 4
AdS vacuum and we construct an explicit ow connecting the two.
3.1
Review of conditions for N = 4 AdS5 vacua
It was shown in [8] that the necessary and su cient conditions for vedimensional
halfmaximal supergravity to admit a fully supersymmetric AdS5 solution amount to a simple
{ 7 {
set of constraints on the dressed components of the embedding tensor. In addition to the
aforementioned
M = 0, these conditions read:
^[mn ^pq] = 0 ;
^ma = 0 ;
f^mna = 0 ;
p
where necessarily ^mn and f^mnp are not identically zero.3 The rst condition arises from
the gravitino equation while (3.2){(3.4) are equivalent to P a ij = @ P ij = 0. The AdS
cosmological constant is read from the scalar potential (2.17) and is
The conditions above imply [8] that the theory has gauge group
G = U(1)
Hnc
Hc
SO(5; n) ;
where Hc
SO(n) is a compact semisimple subgroup, while Hnc is a generically
noncompact group admitting SO(3) as maximal compact subgroup. If Hnc is simple, it can
be either SO(3), SO(3; 1), or SL(3; R). When ^ab = 0, the product of the U(1) factor in
G with the SO(3) subgroup of Hnc embeds blockdiagonally as SO(2)
SO(3) in SO(5).
6
If ^ab = 0, the U(1) factor is a diagonal subgroup of SO(2)
In the vacuum, the gauge vectors of U(1) and of SO(3)
SO(5) and SO(2)
SO(n).
Hnc are graviphotons, with
U(1) being always gauged by the vector A0, while the gauge vectors of Hc and of the
noncompact generators of Hnc belong to vector multiplets. The noncompact part of Hnc is
spontaneously broken and the corresponding gauge vectors are massive. Finally, the vectors
that are charged under the U(1) factor of the gauge group are eaten up by antisymmetric
ranktwo tensor elds via the Stuckelberg mechanism. In total, the AdS vacuum is invariant
under U(1)
SU(2)
Hc. The U(1)
SU(2) corresponds to the Rsymmetry of the dual
N = 2 SCFT, while Hc represents the avor group of that SCFT.
These properties are most easily seen if we perform a global SO(1; 1) SO(5; n)
transformation sending the N = 4 critical point to the origin of the scalar manifold, so that
and (VM
m; VM a) is the identity element of SO(5; n). By further making an SO(5)
= 1
SO(n)
transformation, we can choose
AB, where A; B; C = 6; 7; : : : ; n + 5. Then f 123 are SU(2) structure constants, while the
3Condition (3.4) di ers by a factor of 2 from the one given in [8] because we are including a factor of
1=2 in the map (2.5) and when evaluating the shift matrices of [24] we are taking VP m =
P QVQm. See
footnote 5 in [8].
{ 8 {
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
nonvanishing 45 implies that the vectors A4 , A5 are eaten up by tensor elds. Moreover,
f 1AB; f 2AB; f 3AB complete the SU(2) structure constants to those of Hnc, while f ABC are
the Hc structure constants.
From (3.5) we nd that the cosmological constant is
VM 5 such that 45 is invariant, i.e.
The N = 4 vacuum may admit a set of moduli, namely at directions of the scalar
potential along which full supersymmetry is preserved. These are deformations of VM 4 and
It was proven in [8] that these moduli span the space U(1) SU(m) for some m.
SU(1;m)
3.2
Uniqueness in the absence of avor symmetries
In the absence of any avor symmetries Hc we can prove that there cannot be two N = 4
AdS5 solutions with di erent values of the cosmological constant. We arrive at this result
by showing that in any two such solutions the contractions ^mn ^mn and f^mnpf^mnp must
take the same value. From (3.4) we infer that the SO(1; 1) scalar
is also unchanged.
Then from (3.5) we conclude that the cosmological constant takes the same value in the
two solutions.
^
V
T
V =
We rst consider the MN components of the embedding tensor, in their dressed version
^mn ^mb
^an ^ab . The supersymmetry conditions (3.2), (3.1) and the spectral theory
of real, antisymmetric matrices imply that by a local SO(5)
evaluated on the solution can be put in the canonical blockdiagonal form:
SO(n) transformation, ^
^ = diag ( 0; 0; 0; ; 1 ; 2 ; : : : ; p ; 0; : : : ; 0 ) ;
i
are the only nonvanishing eigenvalues of ^mn and
where
=
0110 , while
^ab = 0 there are no
i 2; : : : ; i p are the nonvanishing eigenvalues of ^ab. It is understood that when
eigenvalues. Let us now assume there are two di erent
eld
(3.8)
(3.9)
(3.10)
i 1,
con gurations corresponding to N = 4 AdS5 solutions. The two corresponding vielbeins V
are related by an SO(5; n) transformation. However the latter cannot change the eigenvalues
of ^, neither can it reshu e the
eigenvalue with the 's, because the former lives in the
timelike eigenspace while the latter live in the spacelike eigenspace. It follows that ^ is the
same in the two vacua up to SO(5)
is the same in the two vacua.
SO(n) transformations. In particular, ^mn ^mn = 2 2
We now turn to the f MNP components of the embedding tensor. We can assume with
admit an SU(2)
no loss of generality that one of the N = 4 AdS5 solutions sits at the origin of the scalar
manifold. In an SO(5) gauge such that (3.7) is true, the other N = 4 AdS vacuum must
Hnc gauge group with structure constants f^123 = VM 1VN 2VP 3f MNP :
The choice of an SU(2) subgroup inside Hnc is described by the coset Hnc=SU(2). Hence
{ 9 {
the rst three components of the coset representative in the two vacua are related as
(3.11)
(3.12)
HJEP06(218)
VM
VM
VM
where (fc)mb are the noncompact generators of Hnc and
c are free real parameters.
These transformations
have been identi ed in [6, 8] as the Goldstone bosons for the
spontaneous breaking Hnc ! SU(2). Since the V's in the two AdS5 vacua are related by a
gauge transformation, the structure constants f^mnp should be preserved. This can easily
be seen at rst order in c recalling that (3.3) holds for the vacuum at the origin:
f^123 = VM 1VN 2VP 3f MNP =
M
1
N
2 P 3f MNP
= f 123 + 3 cfca[1f 23]a + O( 2) = f 123 + O( 2) :
(3.13)
In particular, f^mnpf^mnp takes the same value in the two vacua. This concludes our proof.
We remark that a similar argument of uniqueness for fully supersymmetric AdS vacua
when Hc is trivial can be derived in N = 4 supergravity in dimensions four, six and seven.
We provide this in appendix A.
Two distinct N = 4 AdS5 vacua
3.3
Hc
Now let us assume that the Hc part of the gauge group is nontrivial. Since by de nition
SO(n) and does not contain any U(1) factor, a nontrivial Hc must contain an SO(3)c
subgroup. As we are going to show, in this case one may have multiple fully supersymmetric
vacua by modifying the choice of the SO(3) subgroup corresponding to the SU(2)
U(1)
vacuum Rsymmetry within the full gauge group G given in (3.6).
We will assume in the following that the rst vacuum is set at the origin of the scalar
manifold and is invariant under Hc (hence the dual SCFT has Hc avor symmetry). In
the second vacuum, the U(1) part of the Rsymmetry must also be a diagonal subgroup
of SO(2)
SO(5) and SO(2)
SO(n). Since A0 is the gauge vector of that U(1) globally
over scalar eld space, this can only be if VM
4 and VM
5 di er from their values in the
rst vacuum by moduli, that is ^45 = 45, as discussed in section 3.1. Therefore the two
vacua are only distinguished by the values of VM
then means that in the second vacuum we
m for m = 1; 2; 3. The condition (3.3)
nd an SO(3)2 subgroup of G that is gauged
by A^m = AM
SO(3)1
SO(3)c
VM m; m = 1; 2; 3. Most generally this subgroup can be a subgroup SO(3)2
SO(3)c, where SO(3)1 is part of the Rsymmetry in the original vacuum, while
Hc. We can use SO(5; n) rotations to choose this SO(3)c group to be in the
M = 6; 7; 8 directions at the origin. We will denote the SO(3)c structure constants by
(3.14)
where
is a real constant, while as before we will take
.
.
.
cosh 2
sinh 2
0
0
0
0
0
0
.
.
.
0
0
0
0
0
0
.
.
.
cosh 3 0 0
sinh 3 0 0
With the choice above for the embedding tensor and for the scalar elds, the only nontrivial
N = 4 condition on the scalars
m is given by (3.3), which leads to
tanh m tanh n =
tanh p ;
with (m; n; p) cyclic permutations of (1; 2; 3). Apart for the trivial solution
m = 0, these
equations have the solution 1 = 2 = 3 =
(or 1 =
2 =
3 = , etc.), with
This implies that a second vacuum can only exist for
for the gauge coupling constant of SO(3)1
U(1). The gauge elds of SO(3)1 are thus
A1;2;3, those of SO(3)c are A6;7;8, while A4;5 are eaten up by tensor elds since they are
charged under the U(1) gauged by A0.
As already seen before, the embedding tensor above leads to an N = 4 vacuum at
the origin of the scalar manifold, with cosmological constant V =
assume that in the second vacuum the coset representative VM
2
3 g2. We can also
m has for m = 1; 2; 3 only
More explicitly, its nontrivial part is
N]P are the generators of so(5; n) in the fundamental representation.
In that vacuum, we nd that the coupling constant of SO(3)2 is
Using this and the fact that ^45 =
p12 g, we nd from (3.4) that the scalar
is
tanh
=
:
j j < 1 :
In order to identify which gauge symmetries are spontaneously broken we study the
covariant derivative of the scalar
elds around the second vacuum. Starting from (2.7),
one can see that in general the scalar covariant derivative reads
D am = d am
^amA0 + f^amnA^n
f^ambA^b ;
where A^n = AP VP n, A^a = AP VP a are dressed vectors and we have de ned d am
We expand the covariant derivative at rst order in the eld uctuations around the
second vacuum. In particular the constants
Then (3.5) gives for the cosmological constant
are nonzero and lead to
D( 17
26) = d( 17
while 17 + 26 remains uncharged (here 6; 7; 8 denote the values taken by the a index). One
also has similar expressions for simultaneous cyclic permutations of the indices m = (123)
and a; b = (678). It follows that A^a = 1
2
1=2 Aa
Aa 5 , with a = 6; 7; 8, are all
massive, and the gauge group SO(3)1
SO(3)c is indeed broken to the diagonal subgroup
SO(3)2 with structure constant (3.21), gauged by A^m = (1
2) 1=2(Am
A5+m), for
m = 1; 2; 3. If moreover SO(3)c is part of a larger gauge group Hc, and there are other
generators of Hc that do not commute with SO(3)c, then the constants f MNP ; M = 6; 7; 8
and N; P > 8 are nonzero. In the second vacuum this leads to nonvanishing structure
constants given by f^mab = sinh f (M=m+5)(a=N)(b=P ) that give a mass to the gauge vectors
corresponding to those symmetries. That means that SO(3)1
Hc is spontaneously broken
to the product of SO(3)2 with the maximal commutant of SO(3)c in Hc .
We emphasize that by the procedure above we nd a possible second N = 4 vacuum
for every inequivalent embedding of SO(3)c into Hc such that the condition (3.20) holds.
In section 3.5 we present a domain wall solution between the two N = 4 vacua above
and discuss its holographic interpretation.
3.4
Conditions for ows with eight Poincare supercharges
Domain wall solutions preserving eight of the sixteen supercharges were only partially
discussed in [28]. Here we provide their complete characterization (when M = 0), which
to the best of our knowledge has not appeared in the literature before.4
Starting from the gravitino shift matrix P de ned in (2.15), we introduce the
superpotential
W = p2 PmnP mn :
(3.27)
4The analysis is also similar in spirit to the one performed in N = 2 supergravity in [31].
Then the supersymmetry conditions are equivalent to the ow equations
together with the constraints
A
0 = W ;
P [mnP pq] = 0 ;
W 1P mn = 0 ;
f^amnPmn = 0 ;
3"mnpqrP pq ^ra = Pp[mf^n]pa :
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
(3.36)
(3.37)
(3.38)
(3.39)
When these constraints are satis ed, the scalar potential (2.17) can be written in terms of
the superpotential as
3
2
Clearly, the ow equations and the form of the potential agree with (2.19), (2.20).
One can show that if the constraints (3.31){(3.34) are satis ed and the superpotential
W is extremized, then the N = 4 AdS conditions of section 3.1 are recovered. In other
words, the
xed points of ows preserving eight supercharges are N = 4 AdS solutions.
The converse implication is of course also true, as an N = 4 AdS5 solution can be seen as
a domain wall preserving eight Poincare supercharges and having constant scalars.
Proof.
Let us prove the supersymmetric ow equations above. We start from the
gravitino equation (2.10). Multiplying by P we obtain
P mnP pq( mnpq)ij j = h2PmnP mn
A
In order to solve this equation while preserving eight degrees of freedom in the
supersymmetry parameter i, we need the two sides to vanish separately [28]. In this way we obtain
the constraint (3.31) and the evolution equation (3.28) for the warp factor, where W is
de ned as in (3.27). Since now
we can write
so that I2 =
the form of the projector
which precisely reduces the number of independent components in i by half.
1 is an almost complex structure. Then the gravitino equation (2.10) takes
PikPkj =
W 2 ij ;
P = W I ;
I j
i j + i 5 i = 0 ;
Using the relations just obtained, the di erential equation (2.11) for the spinor is
solved by
i = eA=2^i ;
where ^i is a covariantly constant spinor on R1;3 (with the covariant derivative including
the USp(4) connection).
We now pass to the supersymmetry condition (2.12). Since it has to hold for any
spinor satisfying the projector (3.39), it must be that
x0vxa ij = 2P a ikIkj :
P a ij Iij = 0 ;
P a(ikIj)k = 0 ;
21 vya ij P a ikIkj = gyx x0 ;
which is equivalent to
0 ik
2
because the terms linear in 5 cannot compensate the others and thus have to vanish
separately. Using (3.38) and noting that I2 =
1 implies Tr(I@ I) = 0, gives the ow
equation (3.29) for , together with constraint (3.32).
It remains to discuss the supersymmetry equation (2.13). The same argument used to
manipulate equation (2.12) allows to infer that (2.13) together with the projection (3.39)
is equivalent to
Separating the terms transforming in di erent irreducible representations of USp(4), we get
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
(3.45)
(3.46)
where to obtain the last equation we used vxa ij vyaij = 4gxy. Recalling the de nition of
the gaugino shift matrix (2.16), the rst and the second are easily seen to correspond to
constraints (3.33) and (3.34), respectively. The third instead gives the ow equation (3.30),
because
21 vya ij P a ikIkj =
This can be seen by an explicit computation: evaluating the derivative of (3.27) one nds
p
2 2 ^anInm +
1"mnpqrInpf^qra ;
1
2
where we used DxVM
m
=
VM avxam.
evaluating 12 vya ij P a ikIkj . This concludes our proof.
Exactly the same expression is obtained by
3.5
Flow between two N = 4 AdS5 vacua and its holographic dual
We now construct a ow connecting the two N = 4 AdS5 vacua discussed in section 3.3.
This should correspond to a holographic RG
ow connecting two N = 2 fourdimensional
SCFTs. We preserve all the eight Poincare supersymmetries along this ow and these get
enhanced to sixteen at the AdS xed points by the eight additional conformal supercharges.
We again use the local symmetry on the scalar manifold to choose the relevant
components of the embedding tensor as in (3.14), (3.15). We see from the solution for the
second N = 4 vacuum that besides
the only
owing scalars should be 1
parametrization (3.17) of the coset representative. Since we do not want to break the
diagonal SO(3)2 symmetry along the
ow, we set the three scalars equal to each other,
1 =
2 =
. We can then construct the shift matrix (2.15) and the
superpotential (3.27). We obtain:
with the superpotential being
P mn = W 4[m 5n] ;
(3.47)
where we are assuming g > 0 for simplicity.5 It is easy to check that the constraints (3.31){
(3.34) are satis ed with no further assumptions.
The metric on the subspace spanned by the two scalars is computed from (2.4) and
reads
V =
The scalar potential is
2)
:
The superpotential and the scalar potential are related as in (2.20), namely
4
1 cosh3 sinh3
+
2 sinh4 (cosh(2 ) + 2)
fully supersymmetric AdS vacua, that is the one at the origin,
and the one at nontrivial values of the scalar elds,
=
2 1=6 ;
=
=
log
;
W =
V =
We recall that we should impose
< 1 in order to have a wellde ned vacuum.
It is easy to compute the masses of the scalar elds at these two vacua. They are given
by the eigenvalues of the matrix gXY @X @Y V where gXY is the inverse of the scalar metric.
5Strictly speaking, formula (3.27) for the superpotential yields the absolute value of the right hand side
of (3.48). However assuming g > 0 we see that both in the rst vacuum (
= 1;
= 0) and in the second
vacuum the right hand side of (3.48) is positive; we can thus remove the absolute value.
It is useful to compute the dimensionless scalar mass, i.e. the combination m2`2 where ` is
the scale of AdS. At the UV vacuum one nds
At the IR vacuum one has
We can now employ the holographic identity m2`2 =
(
4) to extract the conformal
dimensions of the operators dual to the two scalars at the UV and IR AdS vacua. At the
UV vacuum we nd that both scalars are dual to operators of dimension
= 2. In the IR
vacuum
is still dual to an operator of dimension
= 2 and is thus relevant, however
the operator dual to
is irrelevant and has dimension
= 6.
Notice that in an N = 2 SCFT the energymomentum multiplet contains the SU(2)
U(1) Rcurrent as well as a real operator of dimension 2 (see for example page 18 in [32]).
We thus nd that the conformal dimensions computed in (3.54) and (3.55) are consistent
with identifying the scalar
as the gravitational dual to the operator of dimension 2 in the
energymomentum multiplet. This is also consistent with the supergravity analysis since
sits in the gravity multiplet of vedimensional halfmaximal supergravity. Through similar
reasoning one nds that the operator dual to the scalar
is the bottom component in the
UV SO(3) avor current multiplet. This operator is sometimes referred to as momentum
map operator. It transforms as a triplet of both the Rsymmetry and the avor SO(3)'s
and we are giving a vev to the component invariant under the diagonal SO(3) subgroup.
The value of the cosmological constants at the two AdS vacua in (3.54) and (3.55)
determines the ratio of the central charges of the dual SCFTs, see for example [2]. We nd
cIR =
cUV
VIR
VUV
3=2
2
:
Since 2 < 1 this result is compatible with the atheorem. Notice that this is also the same
ratio as (gIR=g) 2, where gIR is the gauge coupling of the IR Rsymmetry, given in (3.21).
The ow equations generated by the superpotential (3.48) via (3.28){(3.30) read
(3.54)
(3.55)
(3.56)
(3.57)
(3.58)
It is possible to solve analytically for
one nds that the solution for
is
and A as a function of . After a short calculation
0 =
0 =
1
3
g
A0 = W :
1
3
g 3 +
g cosh3
;
1 sinh cosh
cosh
1 sinh
;
cosh
1 sinh
1=3
( ) =
(cosh(2 ) + c1 sinh(2 ))1=3 ;
1.5
1.0
red/blue lines are the values for the scalars at the IR vacuum in (3.53). We have
xed g = 1 and
1 = 1:1. Right : numerical solution for A(r) for the same values of g and . The IR/UV is at
large negative/positive values of r. The function A(r) is linear in these regions and the scalars
attain their xed point values as expected from (3.52) and (3.53).
HJEP06(218)
in (3.53) we should
x c1 =
solution for the warp factor,
where c1 is an integration constant. In order for the solution to reach the IR AdS vacuum
+
1 . In a similar way one can
nd the following
A( ) =
log
1
6
1 cosh )(cosh
sinh3(2 )
1 sinh )
3
+ c2 ;
(3.59)
where c2 is a trivial integration constant that can be set to any desired value by shifting
the radial coordinate r. The asymptotic behavior of A close to the two AdS vacua is
AUV
1
2
log ;
1
2
AIR
log(
) :
This is the expected divergent behavior of the metric function close to the two AdS vacua.
One can plug the analytic solution for ( ) back into the second equation in (3.57) and
solve for the function (r) in quadratures. Then one can use this solutions to nd also the
functions (r) and A(r). We were not able to solve for (r) analytically, however a typical
numerical plot for the scalars and metric function is not hard to generate, see gure 1.
It is also instructive to analyze the
ow close to the UV AdS vacuum in order to
understand what drives it. We can linearize the ow equations in (3.57) around the vacuum
Using that the AdS scale is ` = 2=g we nd the approximate solution
v e 2r=` ;
1 + v e 2r=` ;
(3.60)
(3.61)
(3.62)
Since the scalars
and
are dual to operators of dimension 2 in the SCFT we can conclude
that the RG
ow is driven by vacuum expectation values for these two operators. If there
were explicit sources for the operators the approximate UV solution should have had an
re 2r=` term in the asymptotic expansion. This is clearly absent in our setup.
10
5
5
10
4
2
2
4
6
8
A
g
2
:
r
`
:
Expanding the explicit analytic solution in (3.58) around the UV AdS vacuum at
We thus conclude that the constants v and v in (3.62) are related by
1 +
3
+ : : : :
v =
v :
3
It would be interesting to understand eldtheoretically the corresponding relation between
the operator vevs.
Field theory derivation of ratio between central charges
(3.63)
(3.64)
(4.1)
(4.2)
(4.3)
(4.4)
We pause here our supergravity analysis and present a eld theory explanation for the ratio
of UV and IR central charges of N = 2 SCFT's found holographically in (3.56).
Anomalies in fourdimensional N = 2 SCFTs
't Hooft anomalies are6
The Rsymmetry of fourdimensional N = 1 SCFTs is U(1)RN=1 . The cubic and linear
Tr(RN3 =1)
and
Tr(RN =1) :
Via N = 1 supersymmetric Ward identities these anomalies are related to the conformal
anomalies by the wellknown relations [33]
a =
9
32
Tr(RN3 =1)
3
32
Tr(RN =1) ;
c =
9
32
Tr(RN3 =1)
5
32
Tr(RN =1) :
For fourdimensional N = 2 SCFTs the Rsymmetry is SU(2)R
U(1)RN=2 . The
generators of SU(2)R are denoted by Ia, a = 1; 2; 3.7 There is a unique N = 1 superconformal
subalgebra of the N = 2 superconformal algebra. This xes how the U(1)RN=1 is embedded
into the Cartan of the SU(2)R
U(1)RN=2 Rsymmetry, see for example [14, 34],
RN =1 =
1
N =1 that is used to compute the conformal anomalies via (4.2). Continuous
avor symmetries in fourdimensional N = 2 SCFTs are characterized by a avor central
charge kF given by the 't Hooft anomaly (see eq. (2.6) of [34])
where Ta are the generators of the avor group.
6The Tr symbol in all equations below should be understood formally. In the presence of a Lagrangian
it indicates a trace over the charges of the chiral fermions in the theory.
7The indices a; b used in the present eld theory section are unrelated to the SO(n) indices used in the
rest of the paper.
4.2
ow between N = 2 SCFTs
We are interested in an RG
ow which connects two distinct fourdimensional N
= 2
SCFTs. In parallel with the supergravity setup, assume that the UV SCFT has SU(2)R
U(1)RN=2 and an SU(2)F
avor symmetry.8 The generators of the avor symmetry algebra
in the UV will be denoted by Ta. In the IR SCFT the symmetry is S^U(2)R
U(1)RN=2
where S^U(2)R is the diagonal subgroup of SU(2)R
are computed by (4.2) using the generator
SU(2)F . The UV conformal anomalies
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
Tr(RN2 =2Ta) = Tr(T3T3T3) = Tr(Ia) = Tr(Ta) = 0 :
With this at hand it is easy to show that
Tr[(RNIR=1)3) = Tr[(RNUV=1)3]
Tr(RNIR=1) = Tr(RNUV=1) :
8
9 kF ;
Using these identities we arrive at the following simple relations between the UV and IR
conformal anomalies
aIR = aUV
1
4 kF ;
cIR = cUV
1
4 kF :
In unitary SCFTs one can show that kF > 0 so the result above is in harmony with the
atheorem.9
For theories with a = c, such as the large N theories described by our holographic
setup, the result (4.9) can be written as
while for the IR conformal anomalies we use the generator
RNUV=1 =
1
where we are assuming that the SU(2)R and SU(2)F generators are normalized in the
same way, so that the respective structure constants are the same. We now note that the
following identities are true due the properties of the generators of SU(2)R and SU(2)F
kF
= 1 + 418 Tr(R3
Tr(RN =2T3T3)
N =2) + Tr(RN =2I3I3)
:
Now we can use the AdS/CFT dictionary to compare this expression with our
supergravity results. The relation between the SCFT symmetry generators and the supergravity
vectors gauging that symmetry is
RN =2 ! sA0 ;
I3 ! g
below applies to an SU(2) subgroup of the
avor group.
and (4.17) of [35].
8This analysis can be generalized to a more general avor symmetry group. In that case the discussion
9Notice that there are stronger unitarity bounds on the avor central charge given in Equations (4.16)
where the 1=g and =g rescalings are introduced because in the conventions of section 3.3
the supergravity vectors A1;2;3 and A6;7;8 are gauging the SO(3)1 and SO(3)c groups with
gauge couplings g and g= , respectively, while we have assumed that Ia and Ta have the
same structure constants. Moreover, s is a real constant that is taking care of any potential
rescaling of the A0 gauge eld in order to match CFT and supergravity conventions. It
turns out that the speci c value of this constant is not important for our analysis.
Using (4.11), the 't Hoof anomalies translate into coe cients of supergravity
topological terms as
Tr(RN3 =2) ! s3d000 ;
Tr(RN =2I3I3) ! g2 d033 ;
s
Tr(RN =2T3T3) !
s 2
where we are omitting a possible overall factor that will not play any role in our calculation.
Therefore in supergravity language the expression in (4.10) reads
VIR
VUV
3=2
= 1 + 2
d088
s24g82 d000 + d033
:
In vedimensional halfmaximal supergravity, the coe cients d000, d033, d088 are
components of a symmetric tensor d
MN P , with M; N ; P = f0; M g = 0; 1; : : : 5 + n, that controls
the topological term. In particular, the gauge variation of the topological term contains
d
MN P HM^HN ^ AP , where HM are covariant eld strengths [24]. Crucially, the only
nonzero components of the d
MN P tensor are d0MN = dM0N = dMN0 =
MN . Plugging d000 =
0 and d088 =
d033 into (4.13) we obtain precisely the relation (3.56) we found in
supergravity. Thus we nd that the ratio of central charges of the UV and IR N = 2 SCFTs which we
found in supergravity is precisely reproduced by the anomaly matching calculation above.
The discussion above also provides a eld theory counterpart of the constant
entering
in the supergravity embedding tensor and controlling the relation between the vevs of the
operators triggering the ow. Comparing (3.56) with (4.9), we obtain
2 =
kF :
4cUV
The existence of the holographic RG
ow imposes that 0 <
2 < 1 and it is important to
understand whether this constraint can be understood from the dual large N
eld theory.
Unitarity of the SCFT immediately implies that
any
eld theory argument for why one should nd
2 > 0, however we are not aware of
2 < 1. It will be most interesting to
understand better this condition and for which N = 2 SCFT it is obeyed.
5
In this section we study holographic ows between an N = 4 AdS5 vacuum and an N = 2
AdS5 vacuum with a di erent cosmological constant. First we will provide the conditions
for the existence of N = 2 AdS vacua, independently of whether there is also an N = 4
vacuum. Then we consider speci c models allowing for an N = 4 AdS vacuum and study
the existence of N = 2 AdS vacua. Finally, we construct domain wall solutions between
such AdS vacua and discuss their holographic interpretation.
(4.12)
(4.13)
(4.14)
with norm
e
Xm =
mnpqrPnpPqr ;
jXe j
q
XmXem =
e
q
Let us focus on the generic case where this does not vanish (we will comment on the special
case Xe = 0 at the end of this section). Then we can introduce a normalized vector
Xm = Xe m = jXe j ;
Xij j = i :
which speci es an SO(4) subgroup of SO(5). On the spinors, this de nes a reduction
Xij = Xm
USp(4) ! SU(2)+
mij . The supersymmetry preserved by our N = 2 AdS vacuum transforms
under either one of these SU(2) factors. Without loss of generality we can choose SU(2)+,
meaning that the supersymmetry parameters satisfy the projection
SU(2) , where the plus and minus refer to the
1 eigenvalues of
We start by providing general conditions for AdS5 solutions of halfmaximal gauged
supergravity preserving eight supercharges, which have not been discussed in the literature so
far. The only assumption we make is M = 0.
The supersymmetries of an N = 2 AdS5 solution transform as a doublet of SU(2) '
USp(2), hence we need to identify the relevant USp(4) ! SU(2) breaking of the
Rsymmetry of halfmaximal supergravity. This was already discussed in [28] and we
summarize it here. The gravitino shift matrix (2.15) de nes the SO(5) vector
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
HJEP06(218)
Having identi ed the USp(4) ! SU(2) breaking by means of the vector Xm, we nd
that the conditions for an N = 2 AdS5 vacuum are:
( pm
2
3 ^ma
p2f^mnaXn = 0 ;
2
1 "mnpqrf^pqaXr = 0 :
The proof is given below. We observe that (5.6) and (5.8) are selfduality conditions on
the fourdimensional space orthogonal to Xm.10 The AdS radius is xed by
10We can derive some other, nonindependent, relations. Contracting (5.6) with Xn and using (5.5) we
obtain f^[mnpXq] = 0, while contracting (5.7) with Xm we nd ^anXn = 0.
` 1 = W ;
where
r
W =
2 P mnPmn
jXe j
=
2 P mnPmn
As we will discuss in the next section, this expression for W de nes the superpotential
driving supersymmetric ows of the scalar elds. This is extremized at the AdS point.
It would be interesting to study the moduli space of the conditions above. This would
most easily be done by exploiting the symmetry of the scalar manifold to set the undeformed
vacuum at the origin and the unit vector Xm to point in a chosen direction. However, this
analysis goes beyond the scope of the present paper and we leave it for future work.
We can also discuss the spontaneous breaking of the gauge group in the N = 2 vacuum
by looking at the scalar covariant derivative (3.24). Working at leading order in the eld
uctuations around the vacuum, separating the term along the vector Xm from those
transverse to it and using the supersymmetry conditions above, we get
XnXm projects on the subspace transverse to Xm. The terms
containing the A^a gauge vectors signal that all noncompact generators of the gauge group are
spontaneously broken in the N = 2 vacuum and their gauge bosons acquire a mass via the
Stuckelberg mechanism. This is analogous to what happens in N = 4 AdS5 vacua. The
remaining terms give generically mass to some of the vectors of the form
combination A0 + p1
3XmA^m. The U(1) generated by the transformation
nmAn and to the
1
2
A
0
! A0 + p
3
d ;
Am
! Am
Xmd ;
is unbroken and corresponds to the Rsymmetry of the N = 2 vacuum. This also
corresponds to the Rsymmetry of the dual N = 1 SCFT.
Proof. Let us derive the N = 2 supersymmetry conditions given above. Using the AdS
conditions A0 = 1` and 0 =
x0 = 0, the supersymmetry equations (2.10){(2.13) reduce to
i Pij j =
0i =
P a ij j = 0 :
1
` 5 i ;
1
2` i ;
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
Xeij j =
1
`2
2PmnP mn
i ;
and one can easily see that, as long as Xe does not vanish, and after making a
harmless sign choice, this is equivalent to the USp(4) ! SU(2) projection (5.4) together with
Eq. (5.14) is trivially solved in terms of a constant spinor ^i as i = e 2` ^i. However we
must recall that (5.13), (5.14) were derived from the gravitino variation assuming that the
supersymmetry parameter i does not depend on the R1;3 domain wall coordinates,
therefore they only capture the Poincare supersymmetry of AdS. When the conformal
supersymmetries are also taken into account, one nds that the gravitino equation does not constrain
the degrees of freedom in i further than (5.4). For this reason, the analysis from now on
di ers from the one in [28], where only the Poincare supersymmetries were considered.
The remaining two equations, namely (5.15) and (5.16), constrain the embedding
tensor and lead to the actual conditions for N = 2 vacua. Since they must hold on any spinor
satisfying the projection (5.4), we infer that
r
P a ik
kj + Xkj = 0 ;
kj + Xkj = 0 :
Recalling the de nition of the shift matrices (2.15), (2.16) and displaying the SO(5) gamma
matrices, these equations can be rewritten as
kj + Xp( p)kj = 0 ;
p
2 3 ^am( m)ik + f^amn( mn)ik
kj + Xp( p)kj = 0 :
Working out the contractions of the USp(4) indices, (5.19) is equivalent to
2
The rst can be combined with the identity P mnXn = 0 (following from the de nition of
Xn and the fact that P m[nP pqP rs] trivially vanishes in ve dimensions) to give (5.5), while
the second is already the same as (5.6). Separating the di erent USp(4) representations, it
is straightforward to see that (5.20) is equivalent to (5.7), (5.8). This concludes our proof.
The derivation above assumed that Xe does not vanish. When Xe = 0 the solution may
preserve eight Poincare supercharges, which is the situation considered in section 3.
However, it may still be possible to have N = 2 AdS5 vacua with vanishing Xe . This still requires
the existence of a unit vector X, however now unrelated to the Xe de ned in (5.1),
projecting out half of the spinor degrees of freedom as in (5.4). For this to be compatible with the
gravitino equation we also need that Xij and Pij commute, which is equivalent to
demanding P mnXn = 0. The rest of the analysis of the supersymmetry equations is unchanged,
hence conditions (5.5){(5.8) still hold and the AdS radius is given by ` 1 = p2P mnPmn.
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
Review of conditions for minimally supersymmetric
ows
After having identi ed models admitting both N = 4 and N = 2 AdS5 vacua, we will be
interested in describing supersymmetric domain walls connecting them. Away from the
xed points, the domain wall should preserve just four Poincare supercharges, namely the
minimal amount of supersymmetry on R1;3. The necessary and su cient conditions for such
domain walls in halfmaximal supergravity were given in [28].11 Here we summarize them.
The conditions use the same vector X and the same superpotential W de ned in
section 5.1, however now the scalars are nonconstant and depend on the radial coordinate.
In addition to solving the ow equations
HJEP06(218)
one has to impose the following constraints along the ow:
(5.22)
(5.23)
(5.24)
(5.25)
(5.26)
(5.27)
(5.28)
(5.29)
(5.30)
W 1P+mn = 0 ;
^amXm = 0 ;
f^+mna
4
W 2 P+pqf^+pqa P+mn = 0 ;
P+mn =
P mn
1
2
2
1 mnpqrPpqXr
where we have introduced
and
f^+mna = ( pm
both living in the fourdimensional space orthogonal to X and being antiselfdual.12
The superpotential (5.10) can also be written as W = 2pP+mnP+mn. One can then
use (5.26) to show that @ P+ is proportional to P+, and is therefore analogous to (5.28).
We are interested in constructing domain wall solutions interpolating between an
N = 4 and an N = 2 AdS5 vacuum. Thus one of the
restrictive N = 4 conditions (3.1){(3.4).13 The other xed point instead has to satisfy the
N = 2 conditions (5.5){(5.8). One can see that the latter are in fact equivalent to the
constraints (5.25){(5.28), together with the condition that the superpotential is extremized.We
xed points has to satisfy the
now proceed to discuss two explicit examples which display all these features.
11The analysis of [28] was restricted to an embedding tensor satisfying M = 0. Recall that we are also
assuming this condition here as it is necessary for a fully supersymmetric AdS5 vacuum. Also note that
in [28] two superpotentials W
were constructed, depending on the preserved supersymmetry; without loss
of generality here we choose W = W+.
12The \+" subscript comes from the original de nitions in [28]. Although expressed in a slightly di erent
form, these constraints are equivalent to eqs. (3.29), (3.31), (3.32) in [28].
13Notice that the vector Xem has to vanish there, so that the four Poincare supersymmetries preserved
along the ow can be enhanced to eight (plus the conformal supersymmetries).
A model with one N = 2 vacuum
An example of a supersymmetric domain wall solution connecting a maximally
supersymmetric AdS5 vacuum to an N = 2 AdS5 vacuum is the wellknown
FreedmanGubserPilchWarner (FGPW) ow [2]. This was originally constructed in the SO(6) maximal
supergravity, where the UV vacuum is the standard SO(6) invariant critical point, while
the IR N = 2 vacuum is the one rst found in [36]. As discussed in [2] this domain wall
solution can also be described in halfmaximal gauged supergravity by a model with two
N = 4 vector multiplets and a gauging determined by the truncation of SO(6) maximal
supergravity. Here we extend the FGPW model allowing for a more general gauging. We
could also allow for an arbitrary number of vector multiplets as done in section 3 when
studying
ows between two N = 4 vacua (see [15] for such an extension of the FGPW
model), however all essential features of the ow are already captured by a model with two
multiplets, so we restrict to that.
We choose the embedding tensor as
p ;
2
67 =
2 g
1
;
(5.31)
where g and
are parameters. The vectors A1; A2; A3 gauge SU(2), A0 gauges U(1), while
A4; A5; A6; A7 are eaten up by tensor elds.
The FGPW model obtained by truncating SO(6) maximal supergravity has
= 2,
so that 67 = 45. In this case the fully superymmetric vacuum has a complex modulus,
parameterizing the space SU(1; 1)=U(1). Since the conditions of section 3.1 are satis ed,
we have a fully supersymmetric solution at the origin of the scalar manifold for any value
of . In order to obtain an N = 2 vacuum at some other point of the scalar manifold, we
break the SO(3) rotations in the 1; 2; 3 directions by mixing the 1; 2 and 6; 7 directions on
SO(5;2)
the scalar manifold. We thus parameterize the SO(5) SO(2) coset representative as14
HJEP06(218)
(5.32)
(5.33)
V = e 2 t16 2 t27 = BB
^12 =
^45 =
^17 =
^67 =
p2g
g
p ;
2
p2g
^26 = p2g
1 cosh2 ;
0 cosh
B
B
B
B
B
B
B
B
sinh
cosh
0
0
0
0
0
sinh
1 sinh cosh ; f^367 = g sinh2 ;
The dressed embedding tensor (2.8) then reads:
14We could introduce two di erent scalars but the N = 2 vacuum conditions would set them equal.
where by 6,7 we are denoting the values taken by the a index. For the unit vector de ning
the USp(4) ! SU(2) projection of the supersymmetries we nd Xm = sign( ) 3m.
The metric on the space spanned by the scalars ; in this case is
while the scalar potential is
V = g2 cosh2
4
2 sinh2
2 (cosh(2 )
3) :
+
1
4
The N = 2 vacuum conditions (5.5), (5.8) are satis ed automatically. Eq. (5.6) gives
In addition to the N = 4 AdS5 solution
we obtain the N = 2 AdS5 solution
cosh2
=
Note that the latter only exists for j j > 1 since only then we have a real scalar . For
j j ! 1 the N = 2 AdS5 vacuum merges with the N = 4 vacuum at the origin.
In the N = 2 vacuum most of the gauge symmetries are broken. From (5.11) we see
that the nontrivial scalar covariant derivatives around this vacuum are:
D 63 = d 63
D 73 = d 73
g
g
r
j j
j j
3
3
1 2
A ;
1 1
A ;
(5.34)
(5.35)
(5.36)
(5.37)
(5.38)
(5.39)
1
p
2
1A0
A3 , corresponding
D( 62
71) = d( 62
71)
3
2 gp 2 + j j
2
p
2
1A0 + A
(5.40)
The vector elds on the right hand side get a mass through the Stuckelberg mechanism.
The vectors in the rst two lines are just two of the gauge vectors of the gauged SU(2).
The N = 2 vacuum is invariant under the combination
to the U(1) Rsymmetry.
N = 2 vacua. The superpotential reads
g
3
We can now move on to study the supersymmetric ow connecting the N = 4 and the
W =
+
;
(5.41)
where we are assuming g > 0. It is easy to check that with the parameterization (5.32) of
the coset representative, the constraints (5.25){(5.28) are satis ed. This means that it is
g
6
consistent to assume that the only
owing scalars are ; . One can also check that the
scalar potential and the superpotential are related as
V =
in agreement with (2.20). The ow equations (3.29), (3.30) for the scalar elds read
From now on we assume without loss of generality that
> 0 so that we can remove the
absolute values.
Let us call the operators dual to the two scalars O
and O . Expanding around the
N = 4 vacuum one nds that the dimensionless masses of the two scalars are
Using the standard AdS/CFT relation m2`2 =
4) this implies that the dimensions
of the dual operators are15
O
= 2 +
2
;
O
= 2 :
0 =
0 =
3
2
3
2 j j
cosh2
+
Along the RG ow there is operator mixing and in the IR SCFT we have two new eigenstates
of the dilatation operator. The corresponding operator dimensions are
O1 = 3 +
25
72
2 +
;
O2 = 1 +
For any
> 1 we have that
O1 > 4 and thus this is always an irrelevant operator. For
O2 one nds
2
4 <
O2
O2
4;
1
6; 2:5 <
2:5 ;
< 1 :
The ratio of central charges (in the planar limit) of the dual SCFTs is
cIR =
cUV
VN =2
VN =4
3=2
=
27
(2 + )3
:
Since the N = 2 vacuum only exists for
of central charges [2, 14] is realized only for
exactly the one where one
marginal coupling in the dual SCFT.
15One could in principle choose the other root of the quadratic equation for
O , i.e.
O = 2 2 . This
however violates the unitarity bound,
> 1, for 1 <
< 2. Moreover for = 2 we know from the FGPW
model that O = 3 which is obeyed for the choice in (5.46).
> 1, we nd that the wellknown 27=32 ratio
= 2. As already noticed, this value is also
nds a modulus for the N = 4 vacuum, corresponding to a
dashed red/blue lines are the values for the scalars at the IR N = 2 AdS5 vacuum at r !
The UV N = 2 AdS5 vacuum is at r ! 1 with
= 2. Right : a contour plot of the superpotential as a function of the scalars p6 log
= 0 there. We have xed g = 1 and
(horizontal
= p6 log
axis) and
(vertical axis) together with a parametric plot of the numerical solution for the scalars
from the left panel.
Let us compare the ratio in (5.49) with the ratio of central charges from equation (2.22)
in [19] where we
x z = 1 for the UV theory (this corresponds to the MaldacenaNun~ez
N = 2 solution) and the goal is to map the parameter z from [19] to the parameter
in (5.49). From [19] we nd
cIR =
cUV
9z2
1 + (1 + 3z2)3=2
16z2
(5.50)
:
z
2
One can now nd a map between z2 and . The explicit expression is not very illuminating
but one
nds that z2 = 0 is mapped to
= 2 and z2 = 1 is mapped to
= 1. Moreover
the map is monotonic, i.e. if we restrict ourselves to 0
1 we have to restrict
to
be in the range 2
1. This suggest that our model with two vector multiplets may
describe holographic RG ows between the N = 2 MaldacenaNun~ez vacuum and some of
the N = 1 vacua with jzj < 1 studied in [19].
The ow equations (5.43) for this model can be integrated numerically. This is
illustrated in
gure 2. It is clear from this gure that there is a smooth domain wall solution
which interpolates between the N = 4 and N = 2 AdS5 vacua.
To understand better what drives the ow we can expand the BPS
ow equations
near the N = 4 AdS5 vacuum in the UV. The linearized expansion of the BPS equations
depends on the value of . For
> 2 we nd
c e (2 2= )r=` ;
1 + c e 2r=` ;
(5.51)
while for 1 <
3(2
(5.52)
Using (5.46) we conclude that the RG
ow is driven by a source term for the operator
O
O
proportional to the constant c . The constant c is related to the vev of the operator
which is dynamically generated along the RG
ow. The expression in (5.51) has the
expected form for scalar elds with masses as in (5.45). The result in (5.52) however is
di erent since for 1 <
< 2 one should keep quadratic (and higher order) terms in
in
the linearized expansion of the di erential equation for the scalar
in (5.43).
The case
= 2 should be treated separately and the linearized expansion of the BPS
equations near the N = 4 AdS5 then reads
c e r=` ;
3`
1 +
4 c2 re 2r=` + c e 2r=` :
This is the behavior of an RG
ow triggered by sources for operators of dimensions 3 and
2. This behavior was also observed in section 5 of [2]. The regular numerical solution
displayed in
gure 2
xes a particular relation between the constants c and c
which
depends on the value of .
Now we turn our attention to reproducing the ratio (5.49) between the central charges
from a eld theory argument. This can be viewed as a generalization of the results in [14]
which is reproduced by selecting
= 2 above. To this end suppose that we have a
deformation of the N = 2 SCFT dual to the N = 4 AdS5 vacuum in the UV which is such that the
resulting RG ow ends in an N = 1 SCFT with a superconformal Rsymmetry given by the
following linear combination of the Cartan generators of the UV SU(2)
U(1) Rsymmetry
RNIR=1 =
1 +
3
RN =2 +
4
3
2
I3 :
Using this superconformal Rsymmetry and the anomaly relations in (4.2) one readily
nds the following relation between the UV and IR central charges
(5.53)
(5.54)
(5.55)
(5.56)
(5.57)
For
= 1=2 the result above reproduces the anomaly calculation in [14]. When the UV
theory has aUV = cUV, such as in SCFTs with a weakly coupled gravity dual, one nds
that the relations in (5.55) reduce to
aIR = cIR =
( + 1)(
2)2 aUV :
This suggests that to reproduce the supergravity result found in (5.49) above we have to
make the identi cation16
16Unitarity and the atheorem imply that 2 >
> 0 which is mapped to the range 1 >
> 1 in the
supergravity analysis.
aIR = (1 +
cIR = ( 2
3)aUV
( + 1)cUV ;
1)aUV +
( + 1)(4
3 2)cUV :
=
This indeed turns out to be the case since as we show in appendix B the combination
of gauge elds which are massless at the IR N = 2 vacuum in (5.39) corresponds to the
generator
which after the identi cation in (5.57) reduces to (5.54). As an additional consistency
check one can show that the charge of the scalar
under the supergravity gauge
eld
corresponding to the UV superconformal Rsymmetry generator, i.e. the one in (5.54)
with
= 0, is 43 (1 +
1). This should correspond to the superconformal Rcharge of the
operator O in the dual SCFT. It is generally expected that operators dual to supergravity
scalar elds belong to chiral multiplets and thus the conformal dimension of O should be
determined by its Rcharge via the relation
(5.58)
(5.59)
3
2
4
3
(1 +
1) = 2 +
2
:
It is reassuring to nd that this result nicely agrees with the one obtained in (5.46) from
an explicit evaluation of the mass of the scalar .
A model with two N = 2 vacua
We now consider a more involved model displaying an N = 4 vacuum and two distinct
= 2 vacua. Since this is similar to the previous example we studied we keep the
presentation short. We take four vector multiplets and choose the embedding tensor as
p ;
2
67 =
p
2 g 1 ;
1
89 =
p
2 g 2 :
1
(5.60)
For simplicity, we assume g > 0, 1 > 0, 2 > 0. We parameterize an SO(5) SO(4) coset
element in terms of the scalar elds ,
as
SO(5;4)
V = e 2 cos (t16+t27) 2 sin (t18+t29)
0
B
B
B
= BBBB sh cos
0
B
ch
0
0
0
0
0
0 0 0 ch cos2 + sin2
sh cos
0
0
0
0
0
0
sh2 2 sin 2
sh cos
0
0
0
0
0
0
ch cos2 + sin2
sh2 2 sin 2
sh sin
0
0
0
0
0
0
sh2 2 sin 2
ch sin2 + cos2
sh sin
0
0
0
0
0
0
sh2 2 sin 2
ch sin2 + cos2
1
C
C
C
C
C
C
C
C
A
ch
0
0
0
0
0
0
sh cos 0 0 0
sh sin 0 0 0
0 0 0
0 0 0
1 0 0
0 1 0
0 0 1
0 0 0
1
3
q 21 + 1
2 ;
VN(1=) 2 =
g
2
6 1
2=3(2 + 1)2 ;
(5.61)
Again we have Xm = 3m. In addition to the usual N = 4 vacuum at the origin with
cosmological constant V =
3 g2, we obtain two N = 2 vacua by solving the
supersymme2
try conditions in a way similar to the example in section 5.3. The rst N = 2 vacuum is
3 = 1 ;
= 0 ;
e
2 =
1 + 2 1 + 2
while the second is
3 = 2 ;
2
1
3
2 =
1 + 2 2 + 2
q 22 + 2
2 ;
VN(2=) 2 =
Note that for the two vacua to be distinct we need 1 6=
2. In addition it is simple to
nd the ratio of central charges of the dual SCFTs. If we assume that 2 > 1 > 1 we nd
the N = 4 vacuum is in the UV, the vacuum in (5.61) is intermediate and the vacuum
in (5.62) is in the deep IR,
6 2
2=3(2 + 2)2 :
(5.62)
cI(R1) =
VN =4
27 1
(2 + 1)3
VN(2=) 2 ! 3=2
VN =4
=
27 2
(2 + 2)3
: (5.63)
The metric on the space spanned by the three scalars ; ; is
The expression for the scalar potential is not particularly illuminating, however it can
easily be recovered using (2.20) and the superpotential
W =
3
6
Let us now discuss possible supersymmetric ows connecting the three
supersymmetric vacua in this model. The constraints (5.25){(5.28) are satis ed, so a
ow involving
; ;
will not require switching on other scalars. The superpotential above generates the
following ow equations for the scalar elds:
(5.64)
(5.65)
(5.66)
0 =
0 =
0 =
3
2
2
cosh2
+ 2 3
1 1 cos2
+ 2 1 sin2
sinh2
3 ;
2
2
1
1 1 cos2
+ 2 1 sin2
1
1
2 sin(2 ) :
1 sinh(2 ) ;
There are ows from the N = 4 vacuum to either one of the N = 2 vacua with
= 0 (5.61)
or
=2 (5.61). These ows have a constant value for the scalar
and can be constructed
numerically in a way very similar to the one described at the end of section 5.3. On the other
hand, in order to ow from the vacuum in (5.61) to the one in (5.62) the scalar
has to
ow. This seems to imply that the numerical integration of the BPS ow equations is nely
tuned and it is more challenging to construct these ows numerically. This is most likely
related to the fact that both vacua in (5.61) and (5.62) are saddle points of the potential V .
6
Discussion
In this paper we studied the general structure of supersymmetric AdS vacua in halfmaximal
vedimensional gauged supergravity as well as possible supersymmetric domainwall
solutions that connect them. Our results have a direct application to holography where they
translate into constraints on the possible conformal vacua and RG ows of fourdimensional
N = 2 SCFTs with a gravity dual.
The approach we took in this work is \bottomup", i.e. we eschewed any reference to
a particular embedding of the gauged supergravity into string or Mtheory and studied the
general structure of the
vedimensional theory. On one hand this allowed us to obtain
very general results that should be applicable to all fourdimensional N = 2 SCFTs with
a holographic dual, but on the other hand leaves the question open to what are concrete
realizations in ten or eleven dimensions. For instance the domain wall connecting two
supersymmetric AdS5 vacua with sixteen supercharges studied in section 3.5 should imply
ow connecting two N = 2 SCFTs with a gravity dual. We provided
further evidence for this claim with the anomaly calculation in section 4, however we are
not aware of an explicit example of such an RG
ow either in a \topdown" model arising
from string or Mtheory or in
eld theory. A potential realization of this N = 2 RG
ow
might be provided by the theories of class S, i.e. N = 2 SCFTs arising from M5branes
compacti ed on a punctured Riemann surface, discussed in [37]. The vev deformation
of the UV N = 2 SCFTs which reduces the SU(2)R Rsymmetry and the SU(2)F
avor
symmetry to the diagonal subgroup (preserved all along the ow) may be provided by an
appropriate \Higgsing of a puncture" on the Riemann surface. It was furthermore shown
in [37] how to describe this class of strongly interacting N = 2 SCFTs holographically in
Mtheory. What is missing to connect this setup to our results is a wellde ned prescription
to assign a given
vedimensional gauged supergravity theory to any of the AdS5
elevendimensional solutions in [37]. It will be interesting to understand how to make such a link.
We should also stress that the results presented in section 4 for the conformal anomalies
of the UV and IR N = 2 SCFTs are valid beyond the supergravity approximation. It may
be useful to emphasize that the IR central charges aIR and cIR in section 4 are those of
the full IR SCFT. As a consequence of the partial spontaneous breaking of the UV global
symmetry, the IR theory will contain a free sector made of Goldstone bosons in addition to
the interacting sector.17 In class S theories it is known how to separate the contributions
of the Goldstone bosons from the rest, see e.g. [38].
We were also able to describe general constraints for the existence of AdS5 vacua
and domainwalls with eight supercharges in a gauged supergravity theory with at least
one AdS5 vacuum with 16 supercharges. These results should be useful to understand
RG
ows between N = 2 and N = 1 SCFTs in four dimensions. The model with two
vector multiplets discussed in section 5.3 is a particularly simple example of our general
results which nevertheless is rich enough to capture interesting physics. For
= 2 this
model provides a holographic realization of the universal eld theory RG
ow discussed
in [14]. A wellknown \topdown" example of this RG
ow is provided by the N = 1 mass
deformation of N = 4 SYM [1, 2], as well as its Zk orbifold [15, 16, 39]. It is widely expected
that this universal RG
ow should connect also the N = 2 and N = 1 MaldacenaNun~ez
SCFTs arising from M5branes wrapping a smooth Riemann surface [17]. These theories
have holographic dual AdS5 vacua but there is no known domain wall solution connecting
17We thank Prarit Agarwal for useful discussions on this.
them. The supergravity solution with
= 2 described in section 5.3 should serve as a
vedimensional e ective description of this holographic RG
ow. It will certainly be very
interesting to embed this
vedimensional model into a consistent truncation of
elevendimensional supergravity. We are not aware of an explicit embedding of the model with
6= 2 in section 5.3 into string or Mtheory. However it is natural to conjecture that it may
be describing holographic RG
ows between the N = 2 MaldacenaNun~ez SCFT and one of
the N = 1 SCFTs with 0 < jzj < 1 studied in [18, 19]. By the same token we can speculate
that the model with one N = 4 and two N = 2 vacua described in section 5.4 may describe
holographic RG ows connecting the N = 2 MaldacenaNun~ez vacuum with two of the N =
1 theories with jzj < 1 in [18, 19]. To establish these conjectures rigorously one has to show
how to construct a consistent truncation for the elevendimensional supergravity solutions
of [18, 19] to
vedimensional gauged supergravity. Partial progress in this direction was
presented in [40], however the solution to the full problem is still out of reach.
Finally we would like to point out that various interesting conjectures about the
structure of RG
ows in quantum
eld theory were presented in [41, 42]. Supersymmetric CFTs
with holographic duals and the RG ows connecting them provide a natural playing ground
to explore these conjectures and we hope that some of our results may be useful in this
context.
Acknowledgments
We would like to thank Prarit Agarwal, Chris Beem, Sergio Benvenuti, Fridrik
Gautason and Parinya Karndumri for useful discussions. We are particularly grateful to Nick
Halmagyi for participating at the initial stages of the development of this project and for
many important discussions. The work of NB is supported in part by an Odysseus grant
G0F9516N from the FWO, by the KU Lueven C1 grant ZKD1118 C16/16/005, and by the
Belgian Federal Science Policy O ce through the InterUniversity Attraction Pole P7/37.
The work of H.T. was supported by the EPSRC Programme Grant EP/K034456/1.
A
Uniqueness of halfmaximal AdS solutions in various dimensions
Halfmaximal gauged supergravity theories in di erent dimensions share a very similar
structure. Their matter content and their couplings are completely
xed by the number
of vector multiplets and the embedding tensor specifying the gauge group. Therefore a
natural question is the possible existence of a nogo result for multiple N = 4 vacua within
halfmaximal supergravity in dimension other than
ve, similar to the one obtained in
section 3.2. Indeed, in this appendix we show that, again under the assumption that the
only compact subgroup of the gauge group is the Rsymmetry of the vacuum, an analogous
proof holds in dimensions four, six and seven. In more general situations it is natural to
expect that there may be two distinct N = 4 vacua in four, six and seven dimensions. This
should be viewed as a generalization of the vedimensional results presented in section 3.3.
It should then be possible to exhibit holographic RG
ows connecting these distinct AdS
vacua analogous to the ones studied in section 3.5. Indeed, examples of such ows in
sixHJEP06(218)
so that H
have the same properties as Hnc in ve dimensions, see (3.6), but with the
novelty that H+ and H
are electrically and magnetically gauged, respectively. In the
AdS4 vacuum we nd again the breaking
H
! SU(2) :
In the holographically dual 3d N = 4 SCFT, SU(2)+
SU(2) is the Rsymmetry group.
Hc is again compact and semisimple and is gauged under vector multiplet gauge bosons.
It corresponds to the group of avor symmetries in the dual SCFT. The embedding tensor
has components f MNP (while M have to vanish in the N = 4 vacuum). If we de ne
fmnp = VM mVN nVP p( f MNP
f+MNP ) ;
where is the SL(2) complex scalar in the gravity multiplet, then the N = 4 supersymmetry
conditions read
and sevendimensional halfmaximal gauged supergravity have been studied in [43, 44]. It
will be interesting to study this further and understand these holographic RG ows from
the point of the dual SCFT.
A.1
Four dimensions
In four dimensions, fully supersymmetric AdS vacua in halfmaximal supergravity have
been discussed in [6]. There it was shown that the gauge group of N = 4 AdS vacua is
G = H+
H
Hc
SO(6; n) ;
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
VM mVN nVP af MNP = 0 ;
6
1 "mnpqrsfqrs =
f123 =
f456 =
i fmnp :
1
3 2
p i :
Using the quadratic constraints and the symmetries of the scalar manifold one can take
The cosmological constant is V =
3 fmnpfmnp = j j2.
2
Let us x one N = 4 AdS4 vacuum to be at the origin, and let us assume that Hc is
trivial. Then we can argue analogously to the vedimensional case that because of (A.4),
the following identities hold (up to SO(6) rotations) in the second vacuum
VM
VM
VM
1 =
2 =
3 =
M
M
M
N N 1 ; VM
N N 2 ; VM
N N 3 ; VM
4 = ~ M N N 4 ;
5 = ~ M N N 5 ;
6 = ~ M N N 6 ;
where
and ~ describe the embedding of SU(2) into H , respectively, which correspond
to Goldstone directions in VM
mix since they are electrically and magnetically gauged.
m, cf. (3.12). Note that the two SU(2) gauge groups cannot
where H
SO(3; m) and H0
m) for some m
n. As in lower dimensions, this
gauge group is spontaneously broken in a supersymmetric vacuum to its maximal compact
subgroup, which turns out to be
where SO(3) is gauged by three of the four graviphotons and corresponds to the
Rsymmetry group of the dual CFT, while Hc
SO(n
m) corresponds to avor symmetries.
The supersymmetry constraints on the embedding tensor re ect the discussion of the
gauge group and are given by
by brie y reviewing [10].
The gauge group is
In six dimensions, halfmaximally supersymmetric AdS vacua are the only allowed
supersymmetric AdS vacua and have been constructed and studied in [10, 45{47]. Let us start
for m = 1; 2; 3. The gauge coupling g and the mass m~ of the twoform in the gravity
multiplet together also determine the cosmological constant via
Again, if we x one halfmaximal AdS6 vacuum to sit at the origin and we assume
that Hc is empty, we can argue from the third equation in (A.10) that the vielbein
(VM 0; VM m; VM a) of any other N = 4 AdS6 vacuum must be related by the following
embedding
G = H
SO(1; n
H0
SO(4; n) ;
SO(3)
Hc ;
VM
VM
VM
VM
m
VN nVP 0f MNP = 0 ;
VN 0VP af MNP = 0 ;
VN nVP af MNP = 0 ;
VN nVP pf MNP = g "mnp ;
V =
20 m~ 2
g
3m~
3=2
:
VM
VM
VM
VM
0 =
1 =
2 =
3 =
M
M
M
M
N N 0 ;
a = ~bfb0a :
where
has the same form as in (3.12) and therefore describes the embedding of SO(3)
into H0. Similarly,
is a transformation in SO(1; n
m) whose nonvanishing components
are 0a and
a0 given by
Again, the transformations
and
are precisely the Goldstone modes of the model, and
thus the N = 4 vacuum is unique.
When Hc contains an SO(3) subgroup, multiple supersymmetric AdS6 solutions
preserving all sixteen supercharges can be found. A supersymmetric ow between two such
solutions was constructed in [43].
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
Supersymmetric AdS vacua of halfmaximal supergravity in seven dimensions have been
discussed in [7]. Analogous to lower dimensions, the gauge group is of the form
G = H
Hc
SO(3; n) ;
(A.14)
where H is spontaneously broken in the AdS vacuum to its maximal compact subgroup
SO(3), which is gauged by graviphotons and corresponds to the Rsymmetry of the dual
CFT. The compact group Hc
SO(n) corresponds to avor symmetries in the CFT. This
result is found by inspecting the supersymmetry conditions imposed on the embedding
tensor components fMNP ; M . These read:
with
given by (3.12), which corresponds to shifts by a Goldstone boson, establishing
uniqueness of the supersymmetric AdS7 vacuum.
Also in this case, when Hc contains an SO(3) subgroup, one can have multiple AdS7
solutions preserving sixteen supercharges, as well as supersymmetric ows connecting them,
see [44] for an example.
where the gauge coupling constant g determines the cosmological constant. Again, if Hc is
trivial the only transformations that leave these conditions invariant are
M = 0 ;
VM
VM
VN nVP af MNP = 0 ;
VN nVP pf MNP = g "mnp ;
VM
VM
VM
1 =
2 =
3 =
M
M
M
The generator of the IR U(1)R symmetry
In this appendix we show that the generator of the U(1) Rsymmetry at the IR xed point
of the holographic ow discussed in section 5.3 is given in eld theory units by
RNIR=1 =
We can extract the information we need from the action of the supergravity gauge
covariant derivative on the spinor parameter i. The general form of the gauge covariant
derivative was given in eq. (2.7). When acting on the spinor parameter, this reads:
D
= r
1
4
( A^mf^mnp np + A^af^anp np + A0 ^np np) ;
where r is the covariant derivative in the ungauged supergravity theory and we are
suppressing the USp(4) indices on the spinor as well as on the SO(5) gamma matrices.
(A.15)
(A.16)
(B.1)
(B.2)
Before coming to the IR vacuum, let us consider the vacuum at the origin of the scalar
manifold, preserving sixteen supercharges. Recalling the form (5.31) of the embedding
tensor, we have at that point:
D
= r
1
4
gAm"mnp np
2 2
g
p A0 45
where in this equation the indices m; n; p run over 1; 2; 3 only. The embedding of the
SU(2)
U(1) Rsymmetry of the N = 4 vacuum in USp(4) is such that we have the
following identi cation:
45 = RN =2 ;
1
4 "mnp np = Im ;
m = 1; 2; 3 :
Therefore the covariant derivative can be written as
D
= r
gAmIm
g
2 2
p A0RN =2
with
= 3 1 , we nd
and 12 is the generator of the IR Rsymmetry, which should be understood as the linear
N =2 and I3 we are after. In addition, when in the main text we discussed
the gauge symmetries being broken, we found that the combination
Now let us consider the supersymmetric ow discussed in section 5.3. Since we have
found there that Xij = ( 3)ij, the supersymmetries being preserved along the ow are
+ = 1+ 3 . This also implies 45 + =
2
12 +. Acting with the projector 1+ 3 on (B.2)
2
to select these supersymmetries and using the expression for the dressed components of
the embedding tensor given in (5.33), we arrive at
D + = r +
g
2
+ p
1
2
A3 cosh2
0
A (
2 sinh2 )
12 + :
At the UV vacuum = 0 and this yields
D + = r +
g
2 2
p A0RN =2) + :
Of the two symmetries generated by RN =2 and I3, one linear combination is preserved
along the ow, while another one is spontaneously broken, with the associated gauge eld
becoming massive. The symmetry that is preserved is manifest by evaluating the covariant
derivative at the IR vacuum. Recalling that the latter is characterized by cosh2
D + = r +
AIR 12 + ;
AIR = (2 + )
g
6
1
2
1A0
A
3 ;
Abroken = g(p2
1A0 + A3)
(B.3)
(B.5)
(B.7)
= +32 ,
(B.8)
(B.9)
(B.10)
is massive (this is determined up to an overall normalization that will not matter). Inverting
the relation between AIR; Abroken and A0; A3 we obtain
Plugging this in (B.7), we nd that the generator multiplying AIR is (B.1), which is what
we wanted to show.
As an additional consistency check of our results, let us retrieve the ratio of central
charges by studying the topological term in supergravity. After ignoring all other vector
elds, the relevant ChernSimons term of halfmaximal supergravity is
A
0 =
A
3 =
2
3g
1
3g
Abroken +
AIR
Abroken
AIR
2 +
2 +
LCS
IR
LCS
32p2
g3(2 + )3
AIR
^ F IR
^ F IR :
(B.11)
(B.12)
(B.13)
(B.14)
If we also discard the vector becoming massive in the IR vacuum, the remaining
Chern
Simons term is
The coe cient of this term in the supergravity Lagrangian is proportional to the cubic
Rsymmetry anomaly of the IR superconformal Rsymmetry which gives the leading
contribution to the aIR = cIR conformal anomaly. The analogous ChernSimons term in the
UV can be obtained by setting
! 1 in (B.13) to nd
UV
LCS
3227pg32 AUV
^ F UV
^ F UV :
Taking the ratio of the two coe cients in (B.13) and (B.14) above we obtain the same
result as the central charge ratio in (5.49) computed by comparing the IR and UV values
of the cosmological constant.
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