Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory

0.21 MB·English

by  Yevgeny Kats

PreprinttypesetinJHEPstyle-HYPERVERSION
Effect of curvature squared corrections in AdS
9
0
0 on the viscosity of the dual gauge theory
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n
a
J

0
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] Yevgeny Kats and Pavel Petrov
h
t
- Jefferson Physical Laboratory, Harvard University,
p
e Cambridge, MA 02138, U.S.A.
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[

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v Abstract:We use thereal-time finite-temperature AdS/CFT correspondence to com-
3
pute the effect of general R2 corrections to the gravitational action in AdS space on
4
7 the shear viscosity of the dual gauge theory. The R2 terms in AdS are determined by
5
0
. the central charges of the CFT. We present an example of a four-dimensional gauge
2
1 theory in which the conjectured lower bound of 1/4π on the viscosity-to-entropy ratio
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is violated for finite N.
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:
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i
X
r
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Contents
1. Introduction 1
2. AdS black brane metric 4
3. Waves in the AdS black brane background 5
4. Viscosity via AdS/CFT 7
5. Viscosity-to-entropy ratio 10
A. Field redefinitions 11
B. Computing the corrected black brane metric 12
C. Coefficients in the action 14
1. Introduction
It has been conjectured [1, 2] that there exists a lower bound η/s 1/4π on the ratio
≥
between the shear viscosity η and the entropy density s for a field theory at a finite
temperature. Much of the evidence for this conjecture comes from methods based on
the AdS/CFT correspondence [3, 4, 5] that relates a theory of gravity in D-dimensional
anti-deSitter(AdS) space toaconformalfieldtheory(CFT) onthe(D 1)-dimensional
−
boundary. In particular, an AdS space with a black brane is dual to a field theory at
a finite temperature [6]. The real-time (i.e., Lorentzian signature) finite-temperature
AdS/CFTcorrespondence[7,8]allowsonetocomputevarioushydrodynamicquantities
of the boundary theory in its strongly coupled regime by doing a supergravity calcu-
lation in the AdS space. It has been shown, first for particular examples [9, 10, 11, 1]
and later in general [12], that gauge theories that are dual to supergravity theories
in AdS (without higher order terms) actually saturate the bound. The bound is still
saturated to leading order when fundamental matter is included [13]. It was also found
that in the AdS S5 setup dual to = 4 SYM, the leading α′ corrections from string
5
× N
– 1 –
theory (which are quartic in the curvature) increase the ratio η/s [14]. In the weakly
coupled regime, direct calculations using the Boltzmann equation give η/s 1 [15].
≫
There has been a debate [1, 2, 16, 17] over whether the bound may apply to all physical
fluids, including non-relativistic ones. It is desirable to consider the effect of additional
types of corrections to the strongly coupled limit in order to obtain more evidence or a
counterexample to the bound conjecture and to gain more insight into the meaning of
the ratio η/s. In addition to these fundamental questions, the results may turn out to
be relevant to the quark-gluon plasma, whose viscosity is measured to be close to the
bound [18] (and possibly violates the bound [19, 20]).
Hereweconsider theeffect ofcurvaturesquaredcorrections, described bytheaction
R
S = dDx√ g Λ+c R2 +c R Rµν +c R Rµνρσ (1.1)
1 2 µν 3 µνρσ
− 2κ −
Z (cid:18) (cid:19)
in D spacetime dimensions, where c are arbitrary small coefficients, and the negative
i
cosmological constant Λ creates an AdS space with radius
(D 1)(D 2)
L2 = − − (1.2)
2κ( Λ)
−
We take a black brane solution and use the real-time AdS/CFT method [8] and the
Kubo formula to compute the viscosity of the dual boundary theory. The corrected
viscosity-to-entropy ratio, to first order in c , is found to be
i
η 1 c
3
= 1 4(D 4)(D 1) (1.3)
s 4π − − − L2/κ
(cid:20) (cid:21)
While c andc affect the viscosity η and the entropy s, they do not affect the ratioη/s,
1 2
which is indeed expected because of the possibility of a field redefinition, as explained
in appendix A. More interesting is the non-zero effect on η/s due to c . The conjecture
3
that 1/4π is the lower bound for η/s would require c to be negative.
3
It is interesting to ask whether there exist any general consistency conditions on a
gravitational theory that fix the sign of the coefficient c . For example, it is known that
3
many classes of low-energy effective field theories are inconsistent with string theory
(and some are inconsistent with any theory of quantum gravity) [21, 22, 23, 24, 25].
In particular, the coefficients of the low-energy effective action must satisfy certain
inequalities because of causality (or analyticity in the UV) [23] or the requirement that
all charged black holes can decay [22, 25]. However, these are not restrictive enough to
fix the sign of c .
3
On the other hand, for 4-dimensional CFTs that have AdS gravity duals, it was
5
shown [26, 27] that in the limit of large N and large ’t Hooft coupling λ g2 N =
≡ YM
– 2 –
L4/α′2 (while the string coupling g = λ/(4πN) is kept small), c is related to the
s 3
central charges of the CFT as1
c c a
3 = − + 1/N2 (1.4)
L2/κ 16c O
(cid:0) (cid:1)
This gives us
η 1 a
= + 1/N2 (1.5)
s 4π c O
(cid:0) (cid:1)
For the well-known = 4 SU(N) SYM, a = c, so the c term does not appear, but
3
N
in general this does not need to be the case. For example, consider the = 2 Sp(N)
N
gauge theory with 4 fundamental and 1 antisymmetric traceless hypermultiplets. This
superconformal theory arises in string theory in the setup of N D3-branes sitting inside
8 D7-branes coincident on an orientifold 7-plane, and its gravity dual is type IIB string
theory on AdS X5, where X5 S5/Z [30, 31]. The c term in this case comes from
5 2 3
× ≃
the effective action on the worldvolume of the D7-branes/O7-plane system [32, 26].
This theory has a = 1 (12N2 +12N 1), c = 1 (12N2 +18N 2) [26], and then for
24 − 24 −
N 1 eq. (1.5) gives
≫
η 1 1
= 1 (1.6)
s 4π − 2N
(cid:18) (cid:19)
Thus the conjectured bound on η/s is violated.
Note that R4 terms from the 10-dimensional bulk contribute an 1/λ3/2 cor-
O
rection to η/s [14]. However, that contribution is suppressed relative to the (1/N)
(cid:0) O (cid:1)
result in (1.6) in the range 1 λ N λ3/2. Contributions from even higher powers
≪ ≪ ≪
of R in the bulk and powers higher than R2 in the brane effective action are suppressed
by additional factors of 1/λ1/2. Note also that since the contribution of our R2 term
to the action is suppressed by 1/N, the contribution from any new field that could be
sourced as a result (the Kaluza-Klein modes) would be suppressed by 1/N2, and thus
can be neglected (we keep g fixed at a small value and do a large N expansion).
s
Note added: A more detailed justification has been provided later in [33], along with
further illuminating discussions.
1The central charges a and c of a four-dimensional CFT are defined in terms of the trace anomaly
as Tµ = 1 (cW aG) (when we consider the CFT coupled to an external metric) where G =
R h Rµµiνρσ164πR2 Rµ−ν+R2istheGauss-BonnetcombinationandW =R Rµνρσ 2R Rµν+1R2
µνρσ − µν µνρσ − µν 3
is the square of the Weyl tensor. Both a and c are (N2) and in order for a supergravity dual to
O
existthey mustbe equalatleading orderinN [28] andgivenbya=c=π3N2/(4V5), where V5 is the
volume of the compact manifold (without the L5 factor) [29]. However,they may differ at subleading
order: c a (N).
− ∼O
– 3 –
The structure of the paper is as follows. In section 2 we discuss the corrections to
the metric of the AdS black brane. Section 3 deals with the propagation of waves in
the background of the black brane, which then allows us to calculate the viscosity of
the boundary theory in section 4 using AdS/CFT. In section 5 we compute the entropy
and find the correction to the viscosity-to-entropy ratio.
2. AdS black brane metric
The Einstein-Hilbert action in a D-dimensional AdS space
R
S = dDx√ g Λ (2.1)
− 2κ −
Z (cid:18) (cid:19)
possesses the well-known black brane solution in Poincar´e coordinates
1 f(z)dt2 +d~x2 L2dz2
ds2 = − + (2.2)
z2 L2 f(z)
(cid:18) (cid:19)
where
D−1
z
f(z) = 1 (2.3)
− z
(cid:18) 0(cid:19)
For later convenience, we will change the variable z to u as
D−1
z 2
u = (2.4)
z
(cid:18) 0(cid:19)
and then the metric is
f(u)dt2 +d~x2 4L2 du2
ds2 = − + , f(u) = 1 u2 (2.5)
L2z2uD4−1 (D 1)2u2f(u) −
0 −
The curvature squared corrections (1.1) modify the function f(u) to
f(u) = 1 u2 +α+γu4 (2.6)
−
where
2(D 4)κ κ
α = − [(D 1)(Dc +c )+2c ] , γ = 2 (D 3)(D 4)c (2.7)
(D 2)L2 − 1 2 3 L2 − − 3
−
(for details, see appendix B).
– 4 –
In these coordinates, u = 0 is the boundary of the AdS space, and the horizon of
the black brane is at
α+γ
u 1+ (2.8)
H
≃ 2
Note that t is not the natural time coordinate in the boundary theory, because for
u 0 the metric is proportional to
→
ds2 = f(0)dt2 +d~x2 (2.9)
−
rather than the Minkowski metric. Instead, the boundary time coordinate is t =
b
f(0)t.
The parameter z is related to the temperature of the black brane. Consider the
p 0
near-horizon geometry, and change the coordinate u to ρ as
16L2
ρ2 = f(u) (2.10)
(D 1)2f′(u )2u2
− H H
Then the u-t part of the metric (2.5) takes the Rindler space form
ds2 = (2πT)2ρ2dt2 +dρ2 (2.11)
− b
where
D 1 f′(u ) D−3 D 1 D 3 D
T = − | H |uD−1 = − 1+ − α γ (2.12)
8πL2z f(0) H 4πL2z 2(D 1) − D 1
0 0 (cid:20) − − (cid:21)
p
is the temperature.
3. Waves in the AdS black brane background
We now add a small fluctuation φ(t,u):
f(u)dt2 +d~x2 +φ(t,u)(dx dx +dx dx ) 4L2 du2
ds2 = − 1 2 2 1 + (3.1)
L2z2uD4−1 (D 1)2u2f(u)
0 −
In the framework of the AdS/CFT correspondence, the field φ corresponds to the com-
ponent T of the energy-momentum tensor of the boundary theory, whose correlators
12
will be used to calculate the viscosity. We take
φ(t,u) = φ(u)e−iωtb = φ(u)e−i√f(0)ωt (3.2)
– 5 –
Treating the corrections to Einstein equations via Teff from (B.1), we obtain the equa-
µν
tion of motion for φ(u):2
1 2u 8γ u D 1 ω¯2
φ′′ + + 1 − u2 φ′ + φ = 0 (3.3)
− u f(u) D 3f(u) − 2 2(D−3)
(cid:20) − (cid:18) (cid:19)(cid:21) f2(u)u D−1
where we denoted
2 f(0)L2z
0
ω¯ ω
≡ D 1
p −
To solve (3.3), we first consider the singular behavior at the horizon where f(u) = 0.
There the equation reduces to
2 α 3γ ω¯2 D 3
′′ ′
φ 1+ − φ + 1 − (α+γ) φ = 0 (3.4)
− f(u) 2 f2(u) − D 1
(cid:18) (cid:19) (cid:18) − (cid:19)
Writing φ(u) = f(u)νF(u) with F(u) regular at the horizon, and leaving only the most
divergent terms, we obtain two possible values
ω¯ D 2 D ω
ν± = i 1 − α+ γ = i (3.5)
± 2 − D 1 D 1 ± 4πT
(cid:18) − − (cid:19)
The full eq. (3.3) then gives the following equation for F(u):
α+γu4 4γ 2u D 1 u3 1
′′ ′
F (1+2ν) 1 + 4 2ν + − γ + F
− − 1 u2 D 3 1 u2 − D 3 1 u2 u
(cid:20)(cid:18) (cid:18) − (cid:19) − (cid:19) − (cid:18) − (cid:19) − (cid:21)
D 1 u2
+8ν − γF = 0 (3.6)
D 31 u2
− −
We dropped ω¯2 terms, since it will be sufficient for us to keep in φ(u) only terms that
are linear in ω. If we ignore terms proportional to c , we have a solution
i
F(u) = 1 (3.7)
Substituting this solution into terms that are proportional to c in (3.6), we are left
i
with
2u 1 D 1 u2
′′ ′
F (1+2ν) + F +8ν − γ = 0 (3.8)
− 1 u2 u D 31 u2
(cid:20) − (cid:21) − −
2In (3.3), f(u) is just a shorthand for our particular expression (2.6). Also, here and throughout
the paper we work to leading order in the coefficients c .
i
– 6 –
The solution is
D 1
F(u) = k +k (1 u2)−2ν +2νγ − u2 (3.9)
1 2
− D 3
−
where k and k are integration constants. We take k = 1, consistently with our
1 2 1
unperturbed solution F(u) = 1, and we must take k = 0 to have F(u) regular at the
2
horizon, so
D 1
F(u) = 1+2νγ − u2 (3.10)
D 3
−
Then the full φ(u), to first order in ω, and normalized to φ(0) = 1, is
ω α+γu4 D 1
φ±(u) = 1 i ln(1 u2)+ α+2γ − u2 (3.11)
± 4πT − 1 u2 − D 3
(cid:20) − − (cid:21)
and the general solution can be written as
φ(u) = aφ+(u)+bφ−(u) (3.12)
where a and b are constants.
4. Viscosity via AdS/CFT
We will calculate the shear viscosity of the boundary theory using the Kubo formula:
1 1
η = lim dtd~xeiωt [T (x),T (0)] = lim ImGR(ω,0) (4.1)
12 12
ω→0 2ω h i −ω→0 ω
Z
where GR is the retarded Green’s function for T :
12
GR(ω,~q) = i dtd~xe−iq~·~x+iωtθ(t) [T (x),T (0)]
12 12
− h i
Z
To find GR(ω,0) we will use the AdS/CFT correspondence, where the bulk field
corresponding to the boundary operator T is our field φ(t,u). In Euclidean space, the
12
correspondence says [5]
exp dD−1xφ (x)T (x) = e−SSG[φ] (4.2)
0 12 AdS
(cid:28) (cid:18)Z (cid:19)(cid:29)CFT
(cid:12)
Here φ is the classical solution to the supergravity equations of(cid:12)motion, subject to the
condition that the boundary value of φ is φ (x), and S [φ] is the corresponding value
0 SG
of the supergravity action. Taking two functional derivatives with respect to φ and
0
– 7 –
setting φ = 0, we see that the part of S that is quadratic in φ can be written in
0 SG 0
′ ′
terms of the Euclidean Green’s function G(x x) T (x)T (x) as
12 12
− ≡ h i
1 dD−1q
(2)
S = φ (q)G(q)φ ( q) (4.3)
SG −2 (2π)D−1 0 0 −
Z
where we switched to momentum space. The possibility to adapt this relation to the
case of a Lorentzian theory at a finite temperature has been discussed by Herzog and
Son [8]. According to their argument, one needs to consider the fully extended Penrose
diagram for the AdS black brane and set the boundary condition that the solution
includes only positive-frequency modes leaving into the future horizon and negative-
frequency modes emerging from the past horizon, where the notion of positive/negative
frequency should be defined with respect to Kruskal coordinates. The analog of (4.3)
is
1 dD−1q
(2)
S = φ (q)G (q)φ ( q) (4.4)
SG 2 (2π)D−1 i ij j −
Z
where G (q) (with i,j = 1,2) are the Schwinger-Keldysh propagators for the operators
ij
of the thermal theory and their doubler operators (see ref. [8] for a review), and φ and
1
φ are sources for these fields. In the AdS/CFT setup, φ corresponds to the boundary
2 1
value of our field φ, while φ corresponds to the boundary value of a similar field that
2
lives on the left Rindler-like patch of spacetime, whose behavior is also described by
(3.11). We will call these fields φ and φ , respectively. Unruh [34] has shown that in
R L
order to construct purely positive or negative frequency modes one needs to take the
following combinations:
φ+ Right φ− Right
φ = φ = (4.5)
neg pos
 
eω/2Tφ+ Left e−ω/2Tφ− Left
This condition relates the coefficients a and b from (3.12) of the left patch to those
of the right patch. Further, we can set the boundary condition φ = 0 on the AdS
L
boundary of the left patch. Then the full solution in the right and the left patches is
given by
φR(u) = a(φ+(u) eω/Tφ−(u))
−
i α+γu4 D 1
= 1 ln(1 u2)+ α+2γ − u2 + (ω) (4.6)
− 2π − 1 u2 − D 3 O
(cid:20) − − (cid:21)
φL(u) = aeω/2T [φ+(u) φ−(u)]
−
i α+γu4 D 1
= ln(1 u2)+ α+2γ − u2 + (ω) (4.7)
−2π − 1 u2 − D 3 O
(cid:20) − − (cid:21)
– 8 –
where in the last step we picked a convenient value for a. Eq. (4.4) becomes
(2) VD−2 dω
SSG = 2 2π φR,ω(u)G11(ω)φR,−ω(u) (4.8)
Z (cid:12)boundary
(cid:12)
(cid:12)
Our supergravity action, keeping only terms quadratic i(cid:12)n φ, is given by
R
S(2) = dudD−1x√ g Λ+c R2 +c R Rµν +c R Rµνρσ
SG − 2κ − 1 2 µν 3 µνρσ
Z (cid:18) (cid:19)
= VD−2 dudt[A(u)φφ′′+B(u)φ′2 +C(u)φφ′
16κLDzD−1
0 Z
+D(u)φ2 +E(u)φ′′2 +F(u)φ′φ′′] (4.9)
where VD−2 = dD−2x, and the functions A(u), B(u), etc. are given in appendix C.
Identically to ref. [14], we add the appropriate Gibbons-Hawking boundary terms, and
R
obtain the total action as the boundary term
(2) VD−2 dω
S = (u) (4.10)
SG 16κLDzD−1 f(0) 2πFω
0 Z (cid:12)boundary
(cid:12)
where p (cid:12)(cid:12)
′
1 F
′ ′
Fω(u) = 2(C −A)φωφ−ω + B −A− 2 φωφ−ω +
(cid:18) (cid:19)
1+u2
′′ ′ ′′′ ′ ′′ ′ ′
+E(φωφ−ω −φωφ−ω)−E φωφ−ω −Eu(1 u2)φωφ−ω (4.11)
−
Here φ (u) is the Fourier transform of φ(t,u) with respect to t , as in (3.2); (u)
ω b ω
F
should be evaluated for both φ and φ (however, the latter happens not to contribute
R L
to the final result for η). Based on (4.8), we can read off G (ω) from (4.10):
11
1 (u)
ω
G (ω) = F (4.12)
11 8κLDz0D−1 f(0)φR,ω(u)φR,−ω(u)(cid:12)(cid:12)boundary
(cid:12)
p (cid:12)
Using the general relation (cid:12)
ω
G (q) = ReGR(q)+icoth ImGR(q) (4.13)
11
2T
in the limit ω 0, eq. (4.1) gives the viscosity as
→
1 1 (u)
0
η = ImG (0) = Im F (4.14)
−2T 11 − 16κLDzD−1 f(0)T φ2 (u)(cid:12)
0 R,0 (cid:12)boundary
(cid:12)
p (cid:12)
(cid:12)
– 9 –