Holography and AdS 2 gravity with a dynamical aether
Received: May
AdS2 gravity
Christopher Eling 0 1 2
0 We discuss the
1 1 Keble Road , Oxford OX1 3NP , U.K
2 Rudolf Peierls Centre for Theoretical Physics, University of Oxford
We study twodimensional Einsteinaether (or equivalently HoravaLifshitz) gravity, which has an AdS2 solution. We examine various properties of this solution in the context of holography. We rst show that the asymptotic symmetry group is the full set of time reparametrizations, the onedimensional conformal group. At the same time there are con gurations with nite energy and temperature, which indicate a violation of the Ward identity associated with onedimensional conformal invariance. These solutions are characterized by a universal causal horizon and we show that the associated entropy of the universal horizon scales with the logarithm of the temperature. puzzles associated with this result and argue that the violation of the Ward identity is associated with a type of explicit breaking of time reparametrizations in the hypothetical 0 + 1 dimensional dual system.
2D Gravity; Field Theories in Lower Dimensions; Holography and condensed

HJEP07(21)4
1 Introduction
2
3
4
5
6
1
Einsteinaether theory in two dimensions
Asymptotic symmetry group
Thermodynamics and conserved charges
Algebra of charges
Discussion
Introduction
reparametrizations of time t ! f (t), which are the asymptotic symmetries of AdS2
spacetime [1{3].
While physics in lower dimensions is simpler, in this case it appears to be
too simple. One way of stating the problem is that the conformal Ward identity, which
generically enforces the tracelessness of the stresstensor, implies in onedimension that
the energy is zero since the stress tensor only has one component T tt. Thus the
quantum mechanical theory has no dynamics. On the gravity side this re ected in the fact
that pure Einstein gravity is trivial and the EinsteinHilbert action in two dimensions is a
topological term.
An early model of a nontrivial two dimensional theory of quantum gravity was
studied by Polyakov [4]. The action is given by the nonlocal Polyakov term which generates
the trace anomaly in twodimensions. In the conformal gauge this reduces to Liouville
gravity, which is a special case of a general class of dilaton gravity theories. These
theories have solutions with an AdS2 metric plus a nontrivial dilaton in the bulk. Recently,
these results have been interpreted in terms of a nearly AdS2/CF T1 (N AdS2/N CF T1)
correspondence (see, e.g. [5{9]). On the gravity side, the dilaton explicitly breaks the time
reparametrization asymptotic symmetry. The dual description is in terms of the rst nite
temperature/energy corrections away from the infrared to a type of quantum mechanical
model of interacting fermions, the SachdevYeKitaev model. This model has an emergent
reparametrization invariance in the T ! 0 limit [10, 11].
{ 1 {
In this paper we will consider another theory of gravity in two dimensions,
Einsteinaether theory [12]. In this theory the metric is coupled to a dynamical unit timelike vector
eld, the \aether". One can also recast the theory into a HoravaLifshitz form,1 which
has a preferred foliation of time and therefore is invariant only under foliation preserving
di eomorphisms [13]. In higher dimensions this class of theories has been studied as a
potential holographic dual to strongly coupled nonrelativistic eld theories invariant under
Lifshitz scaling symmetries, where Lorentz invariance is broken, see e.g. [16{19]. In section
II we discuss the properties of the theory in twodimensions and show it has solution with
an AdS2 metric plus a nontrivial aether pro le in the bulk.
As a rst probe of holographic properties, we investigate the asymptotic symmetries of
HJEP07(21)4
this solution in section III. We nd that the time reparametrizations of AdS2 are unbroken
by the aether. Despite this fact, we show that there are con gurations with
nite energy
and temperature. In the usual case of a higher dimensional CFT, the presence of a
nite
temperature introduces a scale which spontaneously breaks the conformal symmetry. One
nds an energy associated with the thermal state, but the Ward identity enforcing the
traceless stress tensor still holds. However, in 0 + 1 dimensions there is a con ict between
spontaneous breaking and the Ward identity. A
nite temperature con guration has nite
energy, but this is inconsistent with required vanishing T tt as described above. One can
think about a
nite temperature con guration in 0 + 1 dimensions as either an explicit
breaking of the time reparametrization symmetry or as being an anomaly, with the
temperature as the background external eld. Indeed, a \central charge" appears in a number
of our results.
We examine the thermodynamics associated with the AdS2 plus aether solutions in
section IV. These are characterized by the presence of a universal horizon, a surface beyond
which even signals of arbitrary speed cannot reach in nity. The universal horizon therefore
serves as a notion of causal boundary in a nonrelativistic theory of gravity. We derive
a thermodynamical relation of the form E
with the global timelike Killing vector
T where E is the Noether charge associated
eld and T is the temperature of the universal
horizon. Using the thermodynamic relations, we nd there is an entropy associated with
the universal horizon S
ln( T ), where
aspects and possible interpretations of this result.
is a new cuto scale. We discuss the puzzling
Finally, motivated by the violation of the time reparametrization Ward identity, in
section V we study the algebra of charges associated with the asymptotic symmetries,
following Brown and Henneaux [21]. Since the boundary geometry is onedimensional (only
time direction), conserved charges are simply evaluated at points and there is no integration
over space. This causes problems when de ning the Poisson bracket and a potential central
charge independent of where it is evaluated on the boundary. If we instead make the ansatz
that the charges are de ned in terms of an integral over time, we nd the potential central
charge vanishes. This is at least consistent with the lore that there is no conformal anomaly
in onedimension. We interpret the violation of the Ward identity as being due to a novel
1Speci cally the nonprojectable, extended version of the theory without an additional U(
1
)
symmetry [14, 15].
{ 2 {
type of explicit breaking of the time reparametrizations, caused by the presence of the
aether. However, perhaps the \central charge" appearing AdS2 aether system can be seen
as an artifact of the null dimensional reduction of an AdS3=CF T2 system.
2
Einsteinaether theory in two dimensions
Einsteinaether theory is a theory of gravity where the metric gAB is coupled to a
dynamical unit timelike (co)vector eld uA [22]. The aether eld acts as a preferred frame at
every point in spacetime, breaking local Lorentz invariance. To construct the Lagrangian
Lae(gAB; uA), one works in e ective theory and writes down all possible terms up to
second order in an expansion in derivatives of the metric and the aether. The result in
where Lae = R + Lvec, with
and
Sae =
1
16 Gae
Z
d x
4 p
gLae ;
Lvec = KAB
CDrAuC rBuD
(u2 + 1) ;
KAB
CD = c1gABgCD + c2 CA DB + c3 DA CB
c4uAuBgCD :
The coupling constants ci are dimensionless.
In [12], Einsteinaether theory in twodimensions was considered. In this
lowerdimensional setting it was shown that the action reduces to the following form
Sae =
Z
d x
2 p
g
1
2
FABF AB + (rAuA)2 + (u2 + 1) ;
where FAB = rAuB
rBuA. In terms of the original ci coupling constants above,
c1+c4 and
= c1+c2+c3. Also note that in two dimensions the EinsteinHilbert term leads
=
to trivial dynamics, since the Ricci scalar is a total derivative. Finally, since the aether
is twistfree and hypersurface orthogonal in twodimensions, it de nes a preferred time
slicing. Therefore twodimensional Einsteinaether theory is equivalent to twodimensional
HoravaLifshitz gravity [20] .
Variation of the Lagrangian with respect to gAB and uA produces the metric equation
(2.1)
(2.2)
(2.3)
(2.4)
of motion
FAC FBC
1
2 gAB
1
2
and the aether eld equation
F CDFCD
(rC u )
C 2
2 uC rC (rDuD)
C + uAuB = 0 (2.5)
rBF BA +
rA(rC uC )
u
A = 0:
(2.6)
{ 3 {
6
=
The Lagrange multiplier
can be found by multiplying the aether eld equation with
uA and using the unit constraint. The solutions to these eld equations were found and
analyzed in [12]. In particular, when
=
there are only
at spacetime solutions. When
there are nonconstant and constant curvature solutions. In the second class an
AdS2 solution with an aether eld was found. In Fe ermanGraham like coordinates for
the Poincare patch, one nds the solution
ds2 =
r2dt2 +
uAdxA = krdt
dr2
r2
r
pk2
1
dr;
where k = p(
) =(
). We take
and
to be positive and
>
. Note that
no cosmological constant term is needed for this con guration to be a solution.2 A plot of
the ow lines of the aether for this solution on the Penrose diagram of AdS can be found
in
gure 4 of [12]. The aether eld is regular in the Poincare patch, but becomes singular
on the Poincare horizon. In twodimensions, the boundary of AdS2 is disconnected into
two separate boundaries. From the holographic point of view this raises the question of
whether the dual description is terms of a single CF T1 or two systems on the boundaries.
In this paper we will consider the theory in the Poincare and smaller subpatches of the
spacetime, which appears to restrict us to only one boundary system.
3
Asymptotic symmetry group
To investigate the potential holographic dual to this solution, we will analyze the
asymptotic symmetries, in the spirit of Brown and Henneaux. To start, we consider the solution
in (2.7). We want to nd an asymptotic Killing vector, i.e. a Killing vector A that preserves
the following asymptotically AdS boundary conditions
gtt =
r2 + O(
1
);
ut = kr + O(
1
);
gtr = O(1=r3);
ur =
(pk2
1)=r + O(1=r2)
grr = O(1=r4)
The result is
t = (t) +
r =
for arbitrary function (t), associated with an in nitesimal t ! t + (t). This is exactly the
asymptotic Killing vector that arises in studies of asymptotic symmetries in pure AdS2,
see e.g. [1] . The aether eld does not explicitly break the asymptotic symmetry group,
which is the in nite dimensional set of onedimensional conformal transformations. These
can be thought of as \onehalf" of the conformal transformations in twodimensions, which
2Note that we can include a cosmological constant term
in the twodimensional Einsteinaether action.
However, this only a ects the solution (2.7) by changing the value of k.
{ 4 {
(2.7)
3
t
1
kr2t +
gtt =
r2 + stt +
ut = kr + pk2
1 t;
ur =
Under in nitesimal di eomorphisms generated by (3.2) one nds
lead to the Virasoro algebra. Here any mapping t ! f (t) takes the metric ds2 =
dt2 into
We now parametrize the rst order corrections to the metric and aether in the
followAsymptotic symmetries are always spontaneously broken. For example, consider the case
where stt =
t = 0, which corresponds to a choice of vacuum state.
This con
guration is only invariant under transformations
= (1; t), which correspond to in nitesimal
time translations and an overall scale transformation. This a ne subgroup A(
1
) is
isomorphic to the Lorentz subgroup of boosts and null rotations (Lorentz transformations
preserving null vectors) in threedimensional Minkowski spacetime. These are the exact
symmetries of the metric and aether con guration. Thus there is a spontaneous breaking
of time reparametrizations down to A(
1
). Usually the AdS2 vacuum is invariant also under
in nitesimal special conformal transformations generated by (t) = t2, and one has the
SL(2) symmetries, but the presence of the aether breaks this down to A(
1
). In the eld
theory we could interpret this as the usual SL(2) invariant vacuum state plus a source
associated with the aether eld.
4
Thermodynamics and conserved charges
Now suppose we consider the case of a nite di eomorphism of (2.7) preserving the gauge
and boundary conditions. One nds
where ff; tg is the Schwarzian derivative
ds2 =
(r2
r02)d 2 +
uAdxA = (kr + pk2
1r0)d +
ff; tg =
dr2
r2
r
2
0
1r0 + pk2
r
2
0
r2
1r + kr0 !
dr;
(4.3)
and the dot represents a time derivative. Taking, for example, f (t) = er0t one can express
the metric to all orders in 1=r as
stt =
t =
f
_
f
;
...
f (t)
f_(t)
2ff; tg
3 f(t)2
2 f_(t)2
;
kr + pk2
{ 5 {
(3.3)
(3.4)
HJEP07(21)4
(4.1)
(4.2)
which is the AdS2 black hole (or AdSRindler coordinates) plus the aether con guration.
One can verify that this is indeed a solution to the eld equations.
We can also express the above metric in a HoravaLifshitz gauge associated with the
time foliation (slices of constant u)
where
vanishes
ds2 =
(r2
r02)du2 + 2Nrdudr +
uAdxA = (kr + r0pk2
1)du;
1
r2
Nr22 dr2
r
0
Nr =
rpk2
kr + r0pk2
1 + kr0
1
is the shift vector. From this form, we see that there is a universal horizon, de ned as the
location where the dot product of the global timelike Killing vector
rUH =
pk2
k
1
r0 =
r
r0:
TUH =
aAsAj j
2
r=rUH
;
TUH =
r0
2
:
The region beyond this horizon is causally disconnected from in nity, even for signals of
arbitrary speed and therefore de nes a notion of black hole. In [23, 24] it has been argued
there is a Hawking temperature associated with universal horizons, which has the form
where aA = uBrBuA and sA is the unit vector orthogonal to uA.3 Evaluating this formula
for our solution, we nd
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
HJEP07(21)4
temperature.
leads to
Note that this is consistent with the exponential relation between the Poincare time t
and the Schwarzschildlike time . There is a periodicity in imaginary time with period
= 2 =r0. This indicates a potential dual con guration at the boundary is at nite
One important question is the nature of the conserved charges corresponding to the
asymptotic Killing vectors. One way to extract these charges is to employ the covariant
phase space approach of Wald [25]. In general, the variation of the Lagrangian density L
3In [24] it was argued that the Hawking temperature in [23], which was obtained by the tunnelling
method, is o by a factor of two. Here we will use the form in [24].
L = Ei
i + rA A
;
{ 6 {
where i are the elds in the problem, Ei are the equations of motion, and A( i;
the symplectic potential current density. By acting on this equation with two variations,
one can show that onshell
where !A = 1 A( 2)
2 A(
1
). The symplectic form ! is de ned as the integral over a
Cauchy slice
rA!A = 0;
! =
Z
d A!A:
For di eomorphisms generated by a vector eld A, the eld variations are Lie derivatives.
From Hamilton's equations of motion, the variation of the Hamiltonian associated with
A is
H
=
Z
d A!A( i; L
i):
This equation can be expressed in rst in terms of the Noether current density J A =
(4.10)
(4.11)
nally in terms of a surface integral and the antisymmetric Noether potential
density QAB
H
=
Z
QAB
[A B] ;
where J A = 2rBQAB. The surface element nAB is 2r[AtB], where rA and tA are the
unit norms to a surfaces of constant r and t respectively. A Hamiltonian exists for the
asymptotic Killing vectors if there is a BA such that
R
1
dnABB[A B] = R
dnAB [A B].
1
For Einsteinaether theory, the form of the symplectic current and Noether potentials
was found generally in [26, 27]. In the twodimensional case we nd
A( i; L
i
)
AL,
H
=
Z
d A
J A
2rB( [A B]) ;
H
= 2p
( t + _)
{ 7 {
A = p
g
and
(rC uC )(uAgBC
2gABuc) gBC + 2 F AB uB + 2 (rC uC )gAB uB
(4.15)
QAB =
p
g
F AB(uC C ) + (rC uC )(uA B
u
B A
) :
Computing the Hamiltonian associated with the asymptotic Killing vector (3.2), yields
The only contribution to the integral at in nity (here just an evaluation at the boundary)
comes from the Noether current density. The last term can be thought of as an integration
constant since it does not depend on the variation of the elds. One can rede ne the H
by a shift such that for the background con guration where t = 0 it vanishes, i.e.
H
We will work with this form from this point forward.
Another useful way to compute the charge is via the holographic (BrownYork) stress
tensor. Here we consider the onshell gravitational action, which is a boundary term. For
Einsteinaether theory, the e ective action should depend on the boundary metric
and
boundary aether v . The variation of the e ective action W ( ; v) can be expressed as
HJEP07(21)4
where E
= p2
W and J
= p1 Wv . Demanding di eomorphism invariance of the action
W ( ; v), one nds the following Ward identity
W =
Z
ddxp
1
2
E
+ J
v
;
Z
W = 0 =
ddxp (E
D
+ J L v ) ;
D (E
+ J v ) =
J D v :
T
= E
+ J v ;
W = 0 =
Z
ddx
E
+ J v ;
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
where D is the covariant derivative associated with the metric
. We can express this
equation as
In the following we will take the natural de nition of the stress tensor to be
Note that this form is equivalent to the (nonsymmetric) stress tensor that is obtained via
a variation of the vielbein instead of the metric as the fundamental eld (see, e.g. [28]).
The Ward identity associated with onedimensional conformal transformations yields
which seems to imply, in onedimension, a vanishing energy T tt = 0. The associated
charge4 is
H =
Z
To compute the stress tensor, we vary the bulk Einsteinaether action and impose the
eld equations. The result is
W = Sae =
Z
dt h
p h
(rC uC )hAB
2r(AuB)
rC u
C gAB
+ 2 (rBrBu
A
gABrC rBuc) + 2 (rBuB)rA
uA
(4.25)
4This charge has the same value on any surface of constant time since the contribution from the right
hand side of (4.21) vanishes at in nity.
{ 8 {
where hAB = gAB
T tt. Using gAB = r2
rArB. In twodimensions the only nonzero part of the stress tensor is
tt +
, uA = r vt +
and htt = r2 tt, we can extract from this
expression Ett and J t and nd the value for T tt for the metric (3.3) in the limit as r ! 1.
The nal result agrees with (4.18).
In the case where 1(t) = 1, the asymptotic Killing vector is a global symmetry and
the corresponding charge corresponds to an energy of the system. If we use the Hawking
temperature at the universal horizon (4.8), we nd the thermodynamic relation
c2dT 2, relating pressure to central charge [29, 30]. One considers
a conformal transformation that maps the plane into the cylinder. Using the formula for
the transformation of the stress tensor under a conformal mapping, one can show that the
vacuum acquires an energy. This can be interpreted as a Casimir energy since the system
now has an e ective nite size. Here, if we act with an asymptotic di eomorphism, we nd
from (3.4)
The rst term has the form of the transformation of a vector current under di eomorphism,
while the last is an anomalous term, with a \central charge" of 2
. When we start from
the E = 0 vacuum and set f = er0t, this yields the above result. Thus the energy here can
be thought of as a Casimir energy arising from the mapping of the system from the line to
p
a circle.
To obtain the entropy of the system we use the thermodynamic relation
Inserting the formula for the energy, we nd
where
is a new \spontaneously generated" scale (integration constant). It acts as a cuto
since formally the number of states in the system is in nite. One can also extract the free
energy of the system using F = E
T S. This yields
F = 4 p
T
1
ln
:
In Einsteinaether theory there is no general Wald formula for the entropy but in
spherically symmetric black holes in higher dimensions it has been argued the entropy
is proportional to the area of the universal horizon [24, 27]. In twodimensions though
the horizon is a point. Therefore our result is a new prediction for horizon entropy in
twodimensional Einsteinaether theory.
E = f_E + 2p
S =
Z dE =
T 0
Z T 1 dE dT 0:
T 0 dT 0
S = 4 p
ln
T
f
_
f
;
T
{ 9 {
(4.27)
(4.28)
(4.29)
(4.30)
agree if we identify the factor 4 p
A logarithmic dependence of entropy on temperature has been found previously in the
AlmheiriPolchinski dilaton model [5], where it is the contribution at oneloop to the
thermodynamical entropy from (conformal) scalar matter elds. Spradlin and Strominger also
found a logarithmic dependence on temperature in the entanglement entropy for conformal
scalar elds outside an AdS2 black hole [31].5 The formulas for twodimensional entropy
as, again, being proportional to a central charge.
However, a direct connection with these past results, which are obtained at oneloop, is
not clear. It may be that one can consider the aether eld as a type of matter eld on the
AdS2 background and the entropy is an entanglement entropy associated with that eld.
There are in principle two puzzling features to the logarithmic dependence. For
> T
the entropy is negative, and as T ! 0 the entropy S !
1. The zero temperature state
is of course the original Poincare vacuum (2.7). One could argue that
< T and that
as T ! 0, we should also e ectively take the cuto
vacuum state vanishes. Essentially T
! 0, such that the entropy in the
is where the theory is strongly coupled and
the semiclassical picture of Hawking radiation breaks down. However, if we were to take
negative entropy seriously, in quantum information theory there is a notion of a conditional
entropy H(SjO), which can be negative and has a thermodynamic interpretation [33, 34].
This entropy depends on the amount of information an observer O has about some quantum
system S
. One could imagine that the entropy associated with the universal horizon is a
measure of the ignorance of an observer in the preferred frame about the dual quantum
system. Note that in the case of the Poincare vacuum, the universal horizon coincides
with the extremal Killing horizon. Here the aether eld becomes singular and in nitely
stretched, which could be linked to the divergence of the entropy.
5
Algebra of charges
We now investigate whether the violation of the time reparametrization Ward identity
could be associated with an anomaly. One way to determine if this is the case is to consider
whether the algebra of the conserved charges actually has a central extension. We will rst
consider the bracket of two asymptotic Killing vectors, [ 1; 2]A. This is de ned as the
potential changes in 2 due to variations L 1 gAB or L 1 uA and visa versa. In
this case, these charges are higher order (O(r 4)), so the standard Lie bracket is suitable.
One nds
One typically expands the function (t) in terms of a basis of polynomials
[ 1; 2]t = 1 _2
2 _1:
(t) =
1
X
m= 1
amtm+1:
5Note that ln(T = ) acts like the dilaton for the spontaneous breaking of conformal symmetry by nite
temperature [32]. Perhaps here, where such a spontaneous breaking is explicit, this factor does indeed
measure the number of states.
(5.1)
(5.2)
Note that the m =
1 corresponds to time translations, m = 0 to scale transformations,
and m = 1 to special conformal transformations. Denoting m =
as usual that the Lie bracket leads to the Witt algebra
amtm+1, one can show
associated with onedimensional di eomorphisms. In this onedimensional case we only
have one copy of the Witt algebra, instead of the two copies in twodimensions. The
generators ( 1; 0; 1) form a subalgebra since for these cases the vector elds are
nite at
zero and in nity. However, in this case, as we noted earlier, one has to be careful because
the generator of special conformal transformations is not an exact symmetry of the system.
Following original work of Brown and Henneaux, which has been elaborated on in for
example, [35{38], one can show that the conserved Noether charges associated with the
asymptotic vectors satisfy the following algebra
i
P
h
H 1 (g; u); H 2 (g; u)
= H[ 1; 2](g; u) + K 1; 2
The bracket on the left hand side represents the Poisson (or Dirac) bracket of the conserved
charges. The term K 1; 2 does not depend on the dynamical elds and therefore acts as
a central term in the algebra. The Poisson bracket for the charges has been typically
de ned as
where
2 1 = 0, meaning that the variation acts only on the elds. As a result,
On the other hand,
h
H 1 (g; u); H 2 (g; u)
i
P
=
These results do not appear to be consistent with (5.4). In, for example, the AdS3 and
BMS cases, one can show that (5.4) holds by evaluating
2
Q 1 and integrating by parts
over spatial direction. In those cases the total charges were integrals over space. This is
not the case in onedimension where no spatial integrals are present and one is evaluating
at a point on the boundary. One should also have an antisymmetry
2
Q 1 =
1
Q 2 ,
which is not obviously true above. One way to proceed is to de ne the onedimensional
Poisson bracket so that antisymmetry is made manifest
i
P
h
H 1 (g; u); H 2 (g; u)
= ( 2 H 1
1 H 2 ) :
Then (5.4) does hold and one nds the centrallike term
However, note that this expression depends on time in that we must evaluate it at some
t = t0. Again, comparing to the AdS3 case, the discrepancy is due to the lack of a spatial
over R02 d , one
of time.
integral. If we expand the analogous AdS3 expression into modes eim(t ) and integrate
nds that the nonvanishing piece of the central term is independent
A possible resolution is to de ne a total time independent charge in terms of an integral
over time (and invoking a periodicity in imaginary time)
Then if (5.8) holds for H , we nd
H = 2p
Z 2
0
dt (t) t
Z 2
0
If we expand (t) into Fourier modes eimt, for integer m and n, we nd that Km;n vanishes
for all (m; n). This indicates this potential charge algebra is without a central term.
6
It is di cult to interpret the nonzero energy via an anomaly in the onedimensional
conformal symmetry. Therefore we instead interpret the violation of the Ward identity as a
type of explicit breaking of the time reparametrization symmetry. A
nite temperature
is a soft breaking, introducing an e ective length scale in T . However, in one dimension
scale invariance implies that the density of states must scale like (E ) = A (E ) + B=E [6].
The rst term is a possible zero temperature entropy, while the second is the T 1 term we
found from the black hole thermodynamics. This leads to the presence of the logarithm
in the entropy and free energy and means there must be another cuto scale
generated
as well. Thus we have a \spontaneous explicit breaking" supported by the presence of the
aether. It would be interesting to understand a potential holographic dual in more detail.
Our results may also be useful for the study of various condensed matter systems via AdS2
holography, e.g. [39, 40].
Finally, it is possible that twodimensional Einsteinaether theory can be realized as a
dimensional reduction of a gravity theory in AdS3, along the lines of the
EinsteinMaxwelldilaton models discussed in [41, 42]. For example, it is known that nonrelativistic theories
are the result of a null reduction of gravity on higher dimensional Lorentzian manifolds [43,
44]. Perhaps the central charge and logarithmic scaling found here have their origins in
the nonrelativistic limit of a twodimensional CFT.
Acknowledgments
I would like to thank T. Andrade, Y. Oz, and A. Starinets for valuable discussions. This
research was supported by the European Research Council under the European Union's
Seventh Framework Programme (ERC Grant agreement 307955).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[INSPIRE].
HJEP07(21)4
014 [arXiv:1402.6334] [INSPIRE].
[arXiv:1605.06098] [INSPIRE].
holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
[9] G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and
twodimensional quantum gravity dual to SYK/tensor models, arXiv:1702.04266 [INSPIRE].
[10] A. Kitaev, A simple model of quantum holography, talk given at the KITP Program:
entanglement in stronglycorrelated quantum matter, April 6{July 2, University of California,
U.S.A. (2015), part 1 and part 2 available online.
[11] J. Maldacena and D. Stanford, Remarks on the SachdevYeKitaev model, Phys. Rev. D 94
[12] C. Eling and T. Jacobson, Twodimensional gravity with a dynamical aether, Phys. Rev. D
(2016) 106002 [arXiv:1604.07818] [INSPIRE].
74 (2006) 084027 [grqc/0608052] [INSPIRE].
[arXiv:0901.3775] [INSPIRE].
Lett. 104 (2010) 181302 [arXiv:0909.3525] [INSPIRE].
[13] P. Horava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008
[14] D. Blas, O. Pujolas and S. Sibiryakov, Consistent extension of Horava gravity, Phys. Rev.
[15] T. Jacobson, Extended Horava gravity and Einsteinaether theory, Phys. Rev. D 81 (2010)
101502 [Erratum ibid. D 82 (2010) 129901] [arXiv:1001.4823] [INSPIRE].
[16] S. Janiszewski and A. Karch, Nonrelativistic holography from Horava gravity, JHEP 02
(2013) 123 [arXiv:1211.0005] [INSPIRE].
[17] T. Gri n, P. Horava and C.M. MelbyThompson, Lifshitz gravity for Lifshitz holography,
Phys. Rev. Lett. 110 (2013) 081602 [arXiv:1211.4872] [INSPIRE].
[18] C. Eling and Y. Oz, HoravaLifshitz black hole hydrodynamics, JHEP 11 (2014) 067
[arXiv:1408.0268] [INSPIRE].
[19] R.A. Davison, S. Grozdanov, S. Janiszewski and M. Kaminski, Momentum and charge
transport in nonrelativistic holographic
uids from Horava gravity, JHEP 11 (2016) 170
[arXiv:1606.06747] [INSPIRE].
Phys. Rev. D 83 (2011) 124021 [arXiv:1103.3013] [INSPIRE].
[21] J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic
symmetries: an example from threedimensional gravity, Commun. Math. Phys. 104 (1986)
[22] T. Jacobson and D. Mattingly, Gravity with a dynamical preferred frame, Phys. Rev. D 64
[23] P. Berglund, J. Bhattacharyya and D. Mattingly, Towards thermodynamics of universal
horizons in Einsteinaether theory, Phys. Rev. Lett. 110 (2013) 071301 [arXiv:1210.4940]
HJEP07(21)4
[INSPIRE].
[arXiv:1401.1463] [INSPIRE].
[grqc/9307038] [INSPIRE].
arXiv:1309.0907 [INSPIRE].
[arXiv:1601.06795] [INSPIRE].
[24] S. Janiszewski, Asymptotically hyperbolic black holes in Horava gravity, JHEP 01 (2015) 018
D 73 (2006) 024005 [grqc/0509121] [INSPIRE].
[27] A. Mohd, On the thermodynamics of universal horizons in Einstein ther theory,
[30] I. A
eck, Universal term in the free energy at a critical point and the conformal anomaly,
Phys. Rev. Lett. 56 (1986) 746 [INSPIRE].
[hepth/9904143] [INSPIRE].
JHEP 05 (2013) 037 [arXiv:1301.3170] [INSPIRE].
(2005) 673.
[31] M. Spradlin and A. Strominger, Vacuum states for AdS2 black holes, JHEP 11 (1999) 021
[32] C. Eling, Y. Oz, S. Theisen and S. Yankielowicz, Conformal anomalies in hydrodynamics,
[33] M. Horodecki, J. Oppenheim and A. Winter, Partial quantum information, Nature 436
[34] L. del Rio et al., The thermodynamic meaning of negative entropy, Nature 474 (2011) 61.
[35] G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws
and central charges, Nucl. Phys. B 633 (2002) 3 [hepth/0111246] [INSPIRE].
[36] J.i. Koga, Asymptotic symmetries on Killing horizons, Phys. Rev. D 64 (2001) 124012
[grqc/0107096] [INSPIRE].
(2002) 3947 [hepth/0204179] [INSPIRE].
062 [arXiv:1001.1541] [INSPIRE].
[37] S. Silva, Black hole entropy and thermodynamics from symmetries, Class. Quant. Grav. 19
[38] G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010)
[39] H. Liu, J. McGreevy and D. Vegh, NonFermi liquids from holography, Phys. Rev. D 83
(2011) 065029 [arXiv:0903.2477] [INSPIRE].
HJEP07(21)4
arXiv:1408.6855 [INSPIRE].
[1] M. Hotta , Asymptotic isometry and twodimensional antide Sitter gravity , grqc/9809035 [2] M. Cadoni and S. Mignemi , Asymptotic symmetries of AdS2 and conformal group in D = 1, Nucl . Phys. B 557 ( 1999 ) 165 [ hep th/9902040] [INSPIRE].
[3] J. NavarroSalas and P. Navarro , AdS2=CF T1 correspondence and near extremal black hole entropy , Nucl. Phys. B 579 ( 2000 ) 250 [ hep th/9910076] [INSPIRE].
[4] A.M. Polyakov , Quantum geometry of bosonic strings , Phys. Lett. B 103 ( 1981 ) 207 .
[5] A. Almheiri and J. Polchinski , Models of AdS2 backreaction and holography , JHEP 11 ( 2015 ) [6] K. Jensen , Chaos in AdS2 holography , Phys. Rev. Lett . 117 ( 2016 ) 111601 [7] J. Maldacena , D. Stanford and Z. Yang , Conformal symmetry and its breaking in two dimensional Nearly AntideSitter space , PTEP 2016 ( 2016 ) 12C104 [arXiv: 1606 . 01857 ] [8] J. Engels oy, T.G. Mertens and H. Verlinde , An investigation of AdS2 backreaction and [25] R.M. Wald , Black hole entropy is the Noether charge , Phys. Rev. D 48 ( 1993 ) R3427 [26] B.Z. Foster , Noether charges and black hole mechanics in Einsteinaether theory , Phys. Rev.
[28] I. Arav , S. Chapman and Y. Oz , Nonrelativistic scale anomalies , JHEP 06 ( 2016 ) 158 [29] H.W.J. Bloete , J.L. Cardy and M.P. Nightingale , Conformal invariance, the central charge and universal nite size amplitudes at criticality , Phys. Rev. Lett . 56 ( 1986 ) 742 [INSPIRE].
[40] T. Faulkner , H. Liu, J. McGreevy and D. Vegh , Emergent quantum criticality, Fermi surfaces and AdS2, Phys. Rev. D 83 ( 2011 ) 125002 [arXiv: 0907 .2694] [INSPIRE].
[41] A. Castro , D. Grumiller , F. Larsen and R. McNees , Holographic description of AdS2 black [42] M. Cvetic and I. Papadimitriou , AdS2 holographic dictionary , JHEP 12 ( 2016 ) 008 [Erratum