#### Exact holography of the mass-deformed M2-brane theory

Eur. Phys. J. C
Exact holography of the mass-deformed M2-brane theory
Dongmin Jang 1
Yoonbai Kim 1
O-Kab Kwon 1
D. D. Tolla 0 1
0 University College, Sungkyunkwan University , Suwon 440-746 , South Korea
1 Department of Physics, BK21 Physics Research Division, Institute of Basic Science, Sungkyunkwan University , Suwon 440-746 , South Korea
We test the holographic relation between the vacuum expectation values of gauge invariant operators in N = 6 Uk (N ) × U−k (N ) mass-deformed ABJM theory and the LLM geometries with Zk orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza-Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension = 1, which is given by O( =1) = N 23 f( =1), for large N and k = 1. Here the factor f( ) is independent of N . Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of k = 1 for LLM geometries represented by rectangularshaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.
1 Introduction
In the context of the AdS/CFT correspondence [1–4], it was
conjectured that the string/M theory on AdSd+1 × X with a
compact manifold X is dual to d-dimensional conformal field
theory (CFT). The conjecture was soon extended to quantum
field theories (QFTs) which can be obtained from the CFTs at
the Ultraviolet (UV) fixed point by adding relevant operators
to the action or considering vacua where the conformal
symmetry is broken. Then the dual geometries for those QFTs
are asymptotic to AdSd+1 × X . Due to computational
diffia e-mail:
b e-mail:
c e-mail:
d e-mail:
culties on both sides, most of the efforts to test the duality
were focused on the large N limit of the QFT, N being the
rank of the gauge group.
In this letter, we analyze a model which shows a supporting
evidence for an exact dual relation away from the conformal
fixed point in the large N limit. We consider the N = 6
Uk (N ) × U−k (N ) Aharony–Bergman–Jafferis–Maldacena
(ABJM) theory with Chern–Simons level k [5], as the CFT at
the UV fixed point. The ABJM theory allows the
supersymmetry preserving mass deformation and the deformed
theory (mABJM) [6,7] has discrete Higgs vacua as represented
by the Gomis, Rodriguez-Gomez, Van Raamsdonk, Verlinde
(GRVV) matrices [7]. It was known that the vacua of the
mABJM theory have one-to-one correspondence [8,9] with
the half BPS Lin–Lunin–Maldacena (LLM) geometries [10,
11] with Zk orbifold having SO(2,1)×SO(4)/Zk ×SO(4)/Zk
isometry in 11-dimensions [9,12]. Since the mABJM
theory is obtained by a relevant deformation from the ABJM
theory at the UV fixed point, the dual geometry should be
asymptotically AdS4 × S7/Zk .
Here we test the above gauge/gravity duality. On the field
theory side, we calculate the vacuum expectation values
(VEVs) of the chiral primary operator (CPO) with
conformal dimension = 1 for all possible supersymmetric vacua
of the mABJM theory with any k. In gravity side, we
implement the Kaluza–Klein (KK) reduction on S7 and construct
4-dimensional quadratic action from 11-dimensional
supergravity on the AdS4 × S7 background. Applying the
holographic renormalization method [13–15], we obtain an exact
holographic relation for the VEVs of the CPO with = 1,
which is given by O( =1) = N 23 f( =1). Here we consider
k = 1 case and f( ) is a function of the conformal
dimension and also depends on some parameters of LLM solutions
[11]. This result is extended to k > 1 for specific types of
LLM solutions. From our results, we notice that the VEVs of
CPOs obtained from the classical supersymmetric vacuum
solutions at finite N (≥2) are read from classical
supersymmetric solutions in dual gravity theory.
The SU(4) global symmetry of the ABJM theory is broken
to SU(2) × SU(2) × U(1) symmetry under the
supersymmetry preserving mass deformation [6,7]. To make
manifest the broken symmetry, we split the scalar fields into
Y A = (Z a , W †a ), where A = 1, 2, 3, 4 and a = 1, 2.
Then the Higgs vacua of the mABJM theory are represented
as direct sums of irreducible n × (n + 1) GRVV
matri(n) with the occupation number Nn, and their
ces [7], Ma
Hermitian conjugates (n + 1) × n matrices M¯ a(n) with Nn
[9]. (See also [16,17] for the vacuum solutions.) Since Z a
and W †a are N × N matrices, we have two constraints,
n∞=0 n + 21 Nn + Nn = N and n∞=0 Nn = n∞=0 Nn.
In addition, in order to have supersymmetric vacua the range
of the occupation numbers should be 0 ≤ Nn, Nn ≤ k [8,9].
As a result, the supersymmetric vacua of the mABJM theory
are completely classified in terms of the occupation numbers,
{Nn, Nn}.
The LLM geometry with Zk orbifold is determined by the
two functions Z and V ,
Z (x˜, y˜) =
V (x˜, y˜) =
2NB+1 (−1)i+1(x˜ − x˜i )
i=1 2 (x˜ − x˜i )2 + y˜2
(−1)i+1
i=1 2 (x˜ − x˜i )2 + y˜2
where x˜ and y˜ are 11-dimensional coordinates, the x˜i are
the positions of boundaries of black and white strips in the
droplet picture [11], and NB is the number of finite black
droplets. Due to the quantization condition of the 4-form
flux, the difference between the consecutive x˜i is quantized as
x˜i+1 − x˜i = 2πlP3μ0Z with the Planck length lP and the mass
parameter μ0. This implies that all possible LLM
geometries are parametrized by the quantized x˜i . For the
asymptotic expansion of the LLM geometries, it is convenient to
introduce new parameters [18],
C p =
i=1
(−1)i+1
√ A
where A is the area in the Young-diagram picture in the LLM
solution, defined as
n=0
ln(k − ln) + ln(k − ln) .
Here {ln, ln} are set of parameters classifying the LLM
geometries in the droplet picture. See [9] for the details. There
is one-to-one correspondence between {ln, ln} and the
occupation numbers {Nn, Nn} in the vacua of the mABJM theory
[9].
2 Discrete Higgs vacua and dual geometries
3 Kaluza–Klein holography ,
In order to implement the KK holography method [19–21],
we consider asymptotic expansion of the LLM geometries
with Zk orbifold and regard the deviation from AdS4 × S7/Zk
geometry as solutions to perturbed equations of motion in
11dimensional supergravity on such background. According to
the dictionary of the gauge/gravity duality [3,4], the
deviations in asymptotic limit encode the information of VEVs of
CPOs [22] in the mABJM theory.
Our purpose in this letter is to compare quantitatively the
VEVs of CPOs in the mABJM theory with the corresponding
asymptotic coefficients of KK scalar fields, based on the KK
holographic procedure [19–21]. Since the elements of the
GRVV matrices are real numbers, one can compute the VEVs
of CPOs in terms of numerical values and compare them with
the corresponding coefficients of the KK scalars in gravity
side. The number of supersymmetric vacua is numerous for
a given N and thus large number of non-trivial tests can be
carried out.
More precisely, the VEV of CPO with conformal
dimension is proportional to the coefficient of z -term in the
asymptotic expansion of the dual scalar field on the gravity
side [19], where z represents the coordinate in holographic
direction. When we restrict our interest to the CPOs with
low conformal dimensions, it is sufficient to consider the
dual LLM geometry near the asymptotic limit.
In particular, the VEVs of CPO with = 1 are
holographically determined by the solutions of the linearized
supergravity equations of motion on the AdS4 × S7/Zk
background. In this case, diagonalized gauge invariant fields in
11-dimensions can be identified with 4-dimensional
gravity fields without non-trivial field redefinitions. However, for
≥ 2, nonlinear terms in the equations of motion are not
negligible and non-trivial field redefinitions in the
construction of the 4-dimensional gravity theory are necessary [19].
In this letter we focus on the CPO with = 1 and leave our
study of CPOs with ≥ 2 as future work.
3.1 Field theory side The CPO with conformal dimension is O
( ) = C AB11,,......,,AB Tr Y A1 YB†1 · · · Y A YB† ,
where the coefficients, C AB11,,......,,AB , are symmetric in upper
as well as lower indices and traceless over one upper and
one lower indices. The CPO in (3.1) is written by
reflecting the global SU(4) symmetry of the ABJM theory. On the
other hand, in the mABJM theory the CPO should reflect the
SU(2)×SU(2)×U(1) global symmetry, of which the explicit
form will be given later.
in the ABJM theory
For a given vacuum, the complex scalar fields near the
vacuum are written as Y A(x ) = Y0A + Yˆ A, where the Y0A
( A = 1, 2, 3, 4) are the vacuum solutions represented by
GRVV matrices [7], and the Yˆ A are field operators. Then the
VEV of a CPO with dimension for a specific vacuum in
the mABJM theory is given by
where · · · m and · · · 0 denote the VEVs in the mABJM
theory and the ABJM theory, respectively. The N1 -corrections
come from the contributions of multi-trace terms [23–25].
Here we note that quantum corrections of scalar fields are
absent due to the high supersymmetry of the mABJM
theory. The smallest amount of supersymmetry which protects
R-charges of BPS operators in 3-dimensional Chern–Simons
matter theories is N = 3 [26]. In our case, however, the vacua
in the mABJM theory preserve the N = 6 supersymmetries
[8,9]. The second term in the above equation is a one point
function in a conformal field theory and is vanishing.
Therefore, in the large N limit we have
O( )(Y A) m = O( )(Y0A).
The vacua parametrized by the occupation numbers {Nn , Nn}
of the GRVV matrices are composed of N × N matrices
having numerical matrix components. Therefore, the
resulting VEV O( ) 0 is a numerical value for a given N . We
compare the specific value of VEV with the corresponding
asymptotic coefficient in gravity side.
I1 (x ) = 0,
I1 (x ) = 0,
3.2 Gravity side
We start with the k = 1 case and write the fluctuations of
11dimensional supergravity fields on the AdS4×S7 background
as
where g0pq and Fp0qrs represent the background geometry. To
construct the 4-dimensional gravity theory, we implement
KK reduction on S7. This reduction involves the expansion
of the fluctuations in (3.4) in terms of S7 spherical
harmonics. The expansion is generally expressed in terms of scalar,
vector, and tensor spherical harmonics. However, the dual
gravity fields of the CPO with = 1 are built purely from
the coefficients of the scalar spherical harmonics. Here the
truncated expansion involving only the scalar spherical
harmonics is given,
I1 = 14(I(1I1++7)3) 18(I1 − 1)φˆ I1 + 7ψˆ I1 ,
with gauge invariant combinations,
ψˆ I1 = 18g0μν h μI1ν − L μνρσ ∇μsνIρ1σ .
In the subsequent discussion, we will expand the LLM
solution as in (3.4) and read the corresponding values of I1 and
I1 .
4 Exact holography
In mABJM theory, the CPO defined in (3.1) is constrained
by the SU(2) × SU(2) × U(1) global symmetry. In particular
for the = 1 case, we have [30]
hμν (x , y) = h μI1ν (x )Y I1 (y), hμa (x , y) = s μI1 (x )∇a Y I1 (y),
h(ab)(x , y) = s I1 (x )∇(a ∇b)Y I1 (y),
fμνρσ (x , y) = 4∇[μsνIρ1σ ](x )Y I1 (y),
fμνρa (x , y) = −s μI1νρ (x )∇a Y I1 (y),
where I1 is non-negative integer, x denotes the AdS4
coordinates, y the S7 coordinates, and we divide the 11-dimensional
indices p, q, . . . into the indices of AdS4, μ, ν, . . . , and
those of S7, a, b, . . . . The notation (ab) is for symmetrized
traceless combination, while the notation [ab · · · ] is for
anti-symmetrization among the indices, a, b, . . . . The scalar
spherical harmonic Y I1 is determined by the eigenvalue
equation,
Y I1 = 0,
where L = (32π 2k N )1/6lP is the radius of S7. The expansion
(3.5) follows the convention of [19,27]. See also [28,29] for
the linearized equations of motion on AdS4 × S7 background
in the de Donder gauge.
Plugging (3.5) into the linearized equations of motion on
the AdS4 × S7 background and collecting relevant equations
of motion, we obtain two diagonalized equations for KK
scalar fields in 4-dimensions (see [30] for details),
where the overall numerical factor is determined by the
normalization condition, C (AI1),B..1.,,A...,B C B(J1),.A..1,,B...,A + (c.c.) =
δ I J . We have verified that all CPOs with = 1 except for
O(1) in (4.1) have vanishing VEVs for all supersymmetric
vacua of the mABJM theory.
Plugging (4.1) into (3.3) and expressing the vacuum
solutions in terms of the GRVV matrices, we obtain
n(n + 1)(Nn − Nn).
n=0
Here μ is the mass parameter in the mABJM theory and has
the relation μ = 4μ0 with the mass parameter μ0 in the LLM
geometries.
The LLM geometries near the asymptotic limit can be
regarded as AdS4 × S7/Zk plus small fluctuations. Though
the gauge conditions of the LLM solutions in 11-dimensional
supergravity are not clear, the 4-dimensional fields I1 and
I1 in (3.8) are gauge invariant and can be read from the
asymptotic expansions. According to the holographic
dictionary, asymptotic coefficients of I1 and I1 encode the
VEVs of the corresponding CPOs in the mABJM theory.
Warp factors in the LLM geometries [11] are completely
fixed by Z and V in (2.1), which are functions of x˜ and
y˜. To implement the holographic renormalization procedure
[13–15], we should rewrite the LLM solution in terms of the
Fefferman–Graham (FG) coordinate system,
L2
dsF2G = g1(z, τ )(−dt 2 + dw12 + dw22) + 4z2 dz2
+ g2(z, τ )dτ 2 + g3(z, τ )dsS23 + g4(z, τ )dsS23 ,
˜
where z is the holographic direction and τ is one of the S7
coordinates in the asymptotic limit. Here the warp factors
gi (z, τ ), i = 1, 2, 3, 4, are defined as
where Gtt , Gx x , Gθθ , and Gθ˜θ˜ with polar coordinates ρ and
ξ are the warp factors in the LLM solutions [11]. For the
detailed expressions of the LLM solution, see also [30]. For
a general droplet parametrized by the Ci in (2.2), the
asymptotic expansion of these warp factors gives
g1 = 4z2
g3 =
g4 =
1 −
with β3 = 2C13 − 3C1C2 + C3. From the asymptotic
expansion of the warp factors and a similar expansion for the 4-form
flux [30], one can read the fluctuations in (3.4), which will
later be used in the construction of the modes I1 and I1 .
We need to express the LLM geometries in terms of
the spherical harmonics on S7. Since the geometries have
SO(4)×SO(4) isometry, they can be appropriately expressed
in terms of the spherical harmonics having the same
isometry. The scalar spherical harmonics on S7 are defined by the
eigenvalue equation (3.6). In μ0z → 0 limit the warp factors
in (4.3) depend only on the τ coordinate, and thus the
appropriate spherical harmonics are the solutions of (3.6) which
also depends only on τ coordinate. One obtains two kinds of
such solutions represented by the hypergeometric function
2 F1(a, b, c; τ 2), which correspond to those with I1 = 4i
and I1 = 4i + 2 (i = 0, 1, 2, . . .). The first few nonvanishing
Y I1 are given by
Y 0 = 1, Y 2
where we used the normalization π34 Y I1 Y J1 = 2I13−I11!(δII11+J13)! .
According to the dictionary of gauge/gravity duality, the
mass of the scalar mode is related to the conformal dimension
of the corresponding operator. In the AdS4/CFT3
correspondence the relation is
m2 L2AdS4 =
− 3).
4
From (3.7) we see that the masses of the 4-dimensional scalar
modes have the form m2 = n(n − 6)/L2 with n = I1 + 12
for I1 and n = I1 for I1 , respectively. So the relation (4.7)
is rewritten as n(n − 6) = 4 ( − 3). From this relation the
scalar mode I1 satisfying the relation (I1 + 12)(I1 + 6) =
4 ( − 3) cannot be the dual scalar field of the CPO with
= 1. On the other hand, we notice that the field I1
satisfies the relation I1(I1 − 6) = 4 ( − 3), which implies
I1 . We naturally expect that the dual scalar field for the
= 2
CPO with = 1 in (4.1) is nothing but I1 in (3.8) with
I1 = 2.
By writing the asymptotic expansion of the LLM
geometries (4.5) in terms of the scalar spherical harmonics (4.6), we
obtain the asymptotic behavior of the 4-dimensional scalar
modes, I1 and I1 with I1 = 2,
According to the holographic renormalization procedure for
the scalar action on the AdS4 background, we have
where N is a numerical number depending on the
normalization of the scalar I1=2, ψ (1) is the coefficient of the
radial coordinate z in the expansion of the scalar mode, and
λ is the ’t Hooft coupling constant defined as λ = N / k in
ABJM theory. The overall coefficient N 2/√λ in (4.9) comes
from the behavior of the 4-dimensional Newtonian constant,
G4 ∼ √N 2 , in the large N limit. In the case k = 1, the overall
1
λ 3
normalization in (4.9) is reduced to N 2 . Actually the N
3/2dependence in the right-hand side of (4.9) is a peculiar
behavior of the normalization factor in holographic dual relation
for the M2-brane theory [5,31,32]. The numerical factor N in
(4.9) depends on the conventions of fields in the gauge theory
and the gravity theory. By identifying the occupation number
of vacua in the mABJM theory with the discrete torsion in
the LLM geometries [9], i.e.,
the normalization factor N is fixed.
Comparing the values in (4.2) in k = 1 field theory with
the corresponding values of β3 in gravity side, we obtain an
exact holographic relation in the large N limit,
n=0
1
n(n + 1)(ln − ln) = 3 (2C˜ 13 − 3C˜ 1C˜ 2 + C˜ 3),
p
where C˜ p ≡ A 2 C p, A being the area of the Young diagram
in (2.2). For details, see also [30].
We also obtain the normalization factor N for k = 1 with
NB = 1. In the Young-diagram picture, this corresponds to
the rectangular-shaped diagrams. For this case the exact dual
relation is given by
m =
where N˜ = A/ k and λ = N / k is ’t Hooft coupling constant
in the ABJM theory. In the large N limit, N˜ approaches N
and the overall factor N 2/√λ in (4.9) appears. For k = 1,
the holographic relation (4.12) reduces to the result in (4.11).
We obtained Eqs. (4.11) and (4.12) in the large N limit.
As we see in (3.2), to obtain the VEVs of CPOs in finite
N , we need to take into account the 1/N -corrections, which
correspond to quantum corrections in field theory side.
However, we notice that without considering the quantum
corrections in calculations of VEVs of CPO with = 1 the
gauge/gravity relations (4.11) and (4.12) are satisfied even at
the finite N (≥2) exactly. This fact suggests that the VEVs of
CPOs obtained from the classical supersymmetric vacuum
solutions at finite N are read from classical supersymmetric
solutions in dual gravity theory.
5 Conclusion
In this letter, we carried out the KK reduction and the
holographic renormalization procedure for the mABJM theory
and the LLM geometry in 11-dimensional supergravity. By
calculating the VEVs of CPO with = 1 in field theory side
and the corresponding asymptotic coefficients in gravity side,
we found a supporting evidence for an exact gauge/gravity
duality with k = 1 in the large N limit. We could test the
duality since discrete Higgs vacua exist in the mABJM theory
and they correspond one-to-one with the LLM geometries.
We also extended the exact holographic relation to the case of
any k for LLM geometries represented by rectangular-shaped
Young diagrams.
It seems that the Higgs vacua of the mABJM theory are
parametrized by the VEVs of CPOs and those are
nonrenormalizable due to the high supersymmetry. This is similar to
the case of the Coulomb branch in large N limit in N = 4
super Yang–Mills theory [19,20]. Though our quantitative
results for the gauge/gravity correspondence involve
infinite examples, we need to accumulate more analytic
evidence for CPOs with (≥2) and k (≥1) to define
supersymmetric vacua. One should also test the dictionary of the
gauge/gravity duality for one point functions of vector and
tensor fields. For instance, it is important to verify that one
point functions of the energy-momentum tensor vanish for
all possible supersymmetric vacua, since the mABJM theory
is a supersymmetric theory. We leave these issues for future
study.
One necessary condition of the supergravity
approximation in AdS/CFT correspondence is the large N limit.
It was reported recently that the dual gravity limit of the
ABJM theory is broken down at the sub-leading order of
N in one-loop quantum correction in the supergravity
theory [33]. This result indicates that though we can calculate
the 1/N -corrections in (3.2) in the field theory side,
finding the corresponding results in 11-dimensional
supergravity is non-trivial. That is, quantum corrections in
supergravity may not give corresponding 1/N -corrections. Therefore,
obtaining meaningful results in ABJM theory in terms of the
gauge/gravity at finite N is very limited.
On the other hand, we noticed that when we insert the
classical supersymmetric vacuum solutions only into the CPO
with = 1, i.e., without considering the quantum
corrections, the dual relations (4.11) and (4.12) are satisfied at the
finite N (≥2) exactly. Though we need to consider quantum
corrections for finite N seriously, we think that this
observation may give some intuition for the understanding of the
gauge/gravity duality. We need more investigation in this
direction.
Recently, it was reported that the mABJM theory on S3 has
no gravity dual for the mass parameter larger than a critical
value [34] (see also [35–37]). Though the setup is different
from ours, which is the mABJM theory on R2,1, it is also
intriguing to investigate the large mass region for our case. It
seems promising to pursue this issue since the LLM
geometries have no singularity over the whole transverse region.
Acknowledgements We would like to thank Changrim Ahn, Loriano
Bonora, Kyung Kiu Kim, and Chanyong Park for helpful discussions.
This work was supported by the National Research Foundation of Korea
(NRF) grant with the Grant Number NRF-2014R1A1A2057066 and
NRF-2016R1D1A1B03931090 (Y.K.) and NRF-2014R1A1A2059761
(O.K.).
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