#### Exact Holography of Massive M2-brane Theories and Entanglement Entropy

EPJ Web of Conferences
Exact Holography of Massive M2-brane Theories and Entangle- ment Entropy
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 gauge/gravity duality between the N = 6 mass-deformed ABJM theory with Uk(N) × U−k(N) gauge symmetry and the 11-dimensional supergravity on LLM geometries with S O(4)/Zk × S O(4)/Zk isometry. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of mass-deformed ABJM theory and from the implementation of Kaluza-Klein (KK) holography to the LLM geometries. We focus on the chiral primary operator (CPO) with conformal dimension Δ = 1. The non-vanishing vacuum expectation value (vev) implies the breaking of conformal symmetry. In that case, we show that the variation of the holographic entanglement entropy (HEE) from it's value in the CFT, is related to the non-vanishing one-point function due to the relevant deformation as well as the source field. Applying Ryu Takayanagi's HEE conjecture to the 4-dimensional gravity solutions, which are obtained from the KK reduction of the 11-dimensional LLM solutions, we calculate the variation of the HEE. We show how the vev and the value of the source field determine the HEE.
1 Introduction
We consider a non conformal quantum field theory and test the gauge/gravity duality away from the
conformal fixed point in the large N limit. We start with the N = 6 Uk(N)×U−k(N)
Aharony-BergmanJafferis-Maldacena (ABJM) theory with Chern-Simons level k [
1
], as the CFT at the UV fixed point.
The ABJM theory allows the supersymmetry preserving mass deformation and the deformed theory
(mABJM) [
2, 3
] has discrete Higgs vacua presented by the Gomis, Rodriguez-Gomez, Van
Raamsdonk, Verlinde (GRVV) matrices [3]. It was known that the vacua of the mABJM theory have
one-toone correspondence [
4, 5
] with the half BPS Lin-Lunin-Maldacena (LLM) geometries [
6, 7
] having
SO(2,1)×SO(4)/Zk×SO(4)/Zk isometry in 11-dimensions [
5, 8
]. 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 × S 7/Zk. We test this gauge/gravity duality, by calculating the the vacuum
expectation values (vevs) of the chiral primary operator with conformal dimension Δ = 1 using both
sides of the duality.
The novelty of gauge/gravity duality is that, quantities which are difficult to calculate on one side
of the duality can easily be obtained on the other side. One such case is the calculation of
entanglement entropy in dimension higher than two, which is very difficult to obtain from the quantum field
theory. However, Ryu and Takanayagi (RT) suggested a holographic method of entanglement entropy
calculation, for d-dimensional conformal field theory from the AdSd+1 solution of the dual gravity [
9
].
In this letter, we extend RT conjecture to non-conformal case and calculate the entanglement entropy
for mABJM from an asymptotically AdS4 solution of 4-dimensional supergravity, which is obtained
by the KK reduction of the LLM solutions.
2 Construction of 4-dimensional Gravity
A 4-dimensional gravity theory on an asymptotically AdS4 background can be constructed from the
KK reduction of 11-dimensional gravity on the LLM geometries. The KK reduction procedure
involves, expansion of the 11-dimensional fields in terms of the spherical harmonics on S 7 and then
projecting the 11-dimensional equations of motion on the appropriate spherical harmonic elements.
The resulting equations are diagonalized to obtain the equations for infinite towers of gauge invariant
KK modes. In general, this procedure results in the field equations which contain higher derivative
terms. In order to absorb the higher derivative terms and write the canonical 4-dimensional equations
of motion for those KK modes, we need to introduce some field redefinitions. Such field redefinitions
are some times called the KK maps. See [
11, 12
] for details.
Following the above procedure and considering the small mass parameter expansion, only the
equations of motion of few KK modes, which includes the 4-dimensional graviton mode Hμν, one
scalar mode Ψ, and one pseudo scalar mode T , are non trivial at quadratic order in the mass parameter.
These and all the other 4-dimensional fields are related to the 11-dimensional fields by the KK maps.
The equations of motion of those three modes are
− MT2 T = 0,
− MΨ2 Ψ = 0,
8
with MT2 = MΨ2 = − L2
+ 8πGN AT ∇μT ∇νT +
2
M2
T g(μ0ν)T 2 + 8πGN AΨ ∇μΨ∇νΨ +
2
M2
Ψ g(μ0ν)Ψ2 = 0,
(1)
where g(μ0ν) is metric of the pure AdS4, L is the radius of S 7 in 11-dimensions, and GN is the Newtion’s
constant in 4-dimensions. For the detailed definitions of parameters, see [
11, 12
].
Asymptotically AdS4 solutions of the 4-dimensional equations of motion are obtained from the
asymptotic expansion of the LLM solutions in 11-dimensions. Performing the expansion we obtain
the following results
(Lμ0)2
Hi j = − 180
Ψ = −24β3μ0z + O(μ03),
30 + β23 + O μ40 ηi j
where μ0 is the mass parameter of the LLM solutions and β3 parametrizes all LLM solutions. We
have verified that Ψ is dual to CPO of conformal dimension one, while T is dual to a gauge invariant
operator of conformal dimension two. Using these 4-dimensional gravity results, we examine the vev
of chiral primary operators and holographic entanglement entropy in mABJM theory.
3 Exact Holography
In mABJM theory, there is a unique CPO with conformal dimension one O(1) that have a
nonvanishing vev [
11, 12
]. Following the standard perturbative expansion in mABJM theory, we can
calculate the one point function for such CPO and express the result as an expansion in 1/N. In the
large N limit, only the leading terms which is completely determined by the discrete Higgs vacua of
the mABJM theory is relevant. See [
11, 12
] for details. The result is
O
(1) m =
kμ
√
4 2 π n=0
2NB+1
n(n + 1)(Nn − Nn),
where {Nn, Nn} are the occupation numbers of the discrete Higgs vacua, and μ = 4μ0 is the mass
parameter in the mABJM theory. This result is identified with the coefficient of the asymptotic
expansion of the dual scalar field Ψ in (2), by using the one-to-one correspondence between the discrete
Higgs vacua and the LLM solutions. Then, the vev is given by
O(1) =
According to the RT conjecture, the HEE with a subsystem A on the boundary of (d+1)-dimensional
AdS geometry is given by S A = Min(γA) , where GN is the Newton constant in the (d+1)-dimensional
4GN
gravity theory and γA is an area of the surface stretched to the bulk direction, which has the same
boundary with the subsystem A [
9
]. The 4-dimensional metric obtained in section 2 is a perturbation
of pure AdS4, gμν = g(μ0ν) + Hμν. Then the induced metric is given by
∂xμ ∂xν ∂xμ ∂xν
g˜i j = ∂σi ∂σ j gμν = ∂σi ∂σ j gμν + Hμν = g˜i j + H˜ i j.
The deviation H˜ i j of the induced metric from its AdS4 value results in the variation of the area for the
surface whose boundary is a disk of radius l,
δγA =
Inserting the results in (2) into (5), we obtain the expansions of H˜ i j up to μ20-order. Using those
expansions in (6), we obtain the variation of the minimum area,
πL2μ2
0
δγA = − 1440
0
l
dρ √
ρ
Therefore, the HEE up to μ20-order is given by
S A = S (A0) + δS A = 8πGLN2 l − 1 − 43 1 + 3β232 (μ0l)2 .
(7)
We see that the solutions of the 4-dimensional metric (2) for all possible droplets exactly reproduce
the HEE which was calculated from the 11-dimensional LLM solutions [
10
].
Since the metric deformation in (2) is due to the presence of the matter fields Ψ and T , we conclude
that the β3-term in δS A in (7) is originated from the vevs of the CPO O(1), which is dual to the scalar
field Ψ, while the constant term in δS A is originated from the source of a gauge invariant operator
O˜(2), which is dual to the pseudo scalar field T . Therefore, we see the HEE up to μ20-order in the large
N limit is the function of the source and the vevs, δS A = δS A(JO˜(2) , O(1) ), where JO˜(2) denotes the
source of O˜(2).
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 Δ = 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 [
13, 14
]. Though our quantitative results
for the gauge/gravity correspondence involve infinite examples, we need to accumulate more analytic
evidences for CPOs with Δ (≥ 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.
Acknowledgements
This work was supported by the National Research Foundation of Korea(NRF) grant with the
grant number NRF-2016R1D1A1B03931090 (Y.K.), NRF-2017R1D1A1A09000951 (O.K.),
NRF2017R1D1A1B03032523 (D.T.).
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