1001.0002.txt

arXiv:1001.0002v2  [hep-th]  9 Mar 2010Gravity duals for logarithmic conformal field theories
Daniel Grumiller and Niklas Johansson
Institute for Theoretical Physics, Vienna University of Te chnology
Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria
E-mail:[email protected], [email protected]. ac.at
Abstract. Logarithmic conformal fieldtheories with vanishingcentra l charge describe systems
withquencheddisorder, percolation ordiluteself-avoidi ngpolymers. Inthesetheories theenergy
momentum tensor acquires a logarithmic partner. In this tal k we address the construction of
possible gravity duals for these logarithmic conformal fiel d theories and present two viable
candidates for such duals, namely theories of massive gravi ty in three dimensions at a chiral
point.
Outline
Thistalk isorganized asfollows. Insection 1werecall sali ent featuresof2-dimensionalconformal
field theories. In section 2 we review a specific class of logar ithmic conformal field theories where
the energy momentum tensor acquires a logarithmic partner. In section 3 we present a wish-list
for gravity duals to logarithmic conformal field theories. I n section 4 we discuss two examples
of massive gravity theories that comply with all the items on that list. In section 5 we address
possible applications of an Anti-deSitter/logarithmic co nformal field theory correspondence in
condensed matter physics.
1. Conformal field theory distillate
Conformal field theories (CFTs) are quantum field theories th at exhibit invariance under angle
preserving transformations: translations, rotations, bo osts, dilatations and special conformal
transformations. In two dimensions the conformal algebra i s infinite dimensional, and thus
two-dimensional CFTs exhibit a particularly rich structur e. They arise in various contexts in
physics, including string theory, statistical mechanics a nd condensed matter physics, see e.g. [1].
The main observables in any field theory are correlation func tions between gauge invariant
operators. There exist powerful tools to calculate these co rrelators in a CFT. The operator
content of various CFTs may differ, but all CFTs contain at leas t an energy momentum tensor
Tµν. Conformal invariance requires the energy momentum tensor to be traceless, Tµ
µ= 0,
in addition to its conservation, ∂µTµν= 0. In lightcone gauge for the Minkowski metric,
ds2= 2dzd¯z, these equations take a particularly simple form: Tz¯z= 0,Tzz=Tzz(z) :=OL(z)
andT¯z¯z=T¯z¯z(¯z) :=OR(¯z). Conformal Ward identities determine essentially unique ly the form
of 2- and3-point correlators between thefluxcomponents OL/Rof theenergy momentum tensor:∝an}b∇acketle{tOR(¯z)OR(0)∝an}b∇acket∇i}ht=cR
2¯z4(1a)
∝an}b∇acketle{tOL(z)OL(0)∝an}b∇acket∇i}ht=cL
2z4(1b)
∝an}b∇acketle{tOL(z)OR(0)∝an}b∇acket∇i}ht= 0 (1c)
∝an}b∇acketle{tOR(¯z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht=cR
¯z2¯z′2(¯z−¯z′)2(1d)
∝an}b∇acketle{tOL(z)OL(z′)OL(0)∝an}b∇acket∇i}ht=cL
z2z′2(z−z′)2(1e)
∝an}b∇acketle{tOL(z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht= 0 (1f)
∝an}b∇acketle{tOL(z)OL(z′)OR(0)∝an}b∇acket∇i}ht= 0 (1g)
The real numbers cL,cRare the left and right central charges, which determine key p roperties of
the CFT. We have omitted terms that are less divergent in the n ear coincidence limit z,¯z→0 as
well as contact terms, i.e., contributions that are localiz ed (δ-functions and derivatives thereof).
If someone provides us with a traceless energy momentum tens or and gives us a prescription
how to calculate correlators,1but does not reveal whether the underlying field theory is a CF T,
thenwecanperformthefollowing check. Wecalculate all 2- a nd3-point correlators of theenergy
momentum tensor with itself, and if at least one of the correl ators does not match precisely with
the corresponding correlator in (1) then we know that the fiel d theory in question cannot be a
CFT. On the other hand, if all the correlators match with corr esponding ones in (1) we have
non-trivial evidence that the field theory in question might be a CFT. Let us keep this stringent
check in mind for later purposes, but switch gears now and con sider a specific class of CFTs,
namely logarithmic CFTs (LCFTs).
2. Logarithmic CFTs with an energetic partner
LCFTs were introduced in physics by Gurarie [2]. We focus now on some properties of LCFTs
and postpone a physics discussion until the end of the talk, s ee [3,4] for reviews. There are two
conceptually different, but mathematically equivalent, way s to define LCFTs. In both versions
there exists at least one operator that acquires a logarithm ic partner, which we denote by Olog.
We focus in this talk exclusively on theories where one (or bo th) of the energy momentum
tensor flux components is the operator acquiring such a partn er, for instance OL. We discuss
now briefly both ways of defining LCFTs.
According to the first definition “acquiring a logarithmic pa rtner” means that the
Hamiltonian Hcannot be diagonalized. For example
H/parenleftbigg
Olog
OL/parenrightbigg
=/parenleftbigg
2 1
0 2/parenrightbigg/parenleftbigg
Olog
OL/parenrightbigg
(2)
Theangularmomentum operator Jmay ormay not bediagonalizable. Weconsider onlytheories
whereJis diagonalizable:
J/parenleftbigg
Olog
OL/parenrightbigg
=/parenleftbigg
2 0
0 2/parenrightbigg/parenleftbigg
Olog
OL/parenrightbigg
(3)
The eigenvalues 2 arise because the energy momentum tensor a nd its logarithmic partner both
correspond to spin-2 excitations.
1This is exactly what the AdS/CFT correspondence does: given a gravity dual we can calculate the energy
momentum tensor and correlators.The second definition makes it more transparent why these CFT s are called “logarithmic”
in the first place. Suppose that in addition to OL/Rwe have an operator OMwith conformal
weightsh= 2+ε,¯h=ε, meaning that its 2-point correlator with itself is given by
∝an}b∇acketle{tOM(z,¯z)OM(0,0)∝an}b∇acket∇i}ht=ˆB
z4+2ε¯z2ε(4)
The correlator of OMwithOLvanishes since the latter has conformal weights h= 2,¯h= 0, and
operators whose conformal weights do not match lead to vanis hing correlators. Suppose now
that we send the central charge cLand the parameter εto zero, and simultaneously send ˆBto
infinity, such that the following limits exist:
bL:= lim
cL→0−cL
ε∝ne}ationslash= 0B:= lim
cL→0/parenleftbigˆB+2
cL/parenrightbig
(5)
Then we can define a new operator Ologthat linearly combines OL/M.
Olog=bLOL
cL+bL
2OM(6)
Taking the limit cL→0 leads to the following 2-point correlators:
∝an}b∇acketle{tOL(z)OL(0,0)∝an}b∇acket∇i}ht= 0 (7a)
∝an}b∇acketle{tOL(z)Olog(0,0)∝an}b∇acket∇i}ht=bL
2z4(7b)
∝an}b∇acketle{tOlog(z,¯z)Olog(0,0)∝an}b∇acket∇i}ht=−bLln(m2
L|z|2)
z4(7c)
These 2-point correlators exhibit several remarkable feat ures. The flux component OLof the
energy momentum tensor becomes a zero norm state (7a). Never theless, the theory does not
become chiral, because the left-moving sector is not trivia l:OLhas a non-vanishing correlator
(7b) with its logarithmic partner Olog. The 2-point correlator (7c) between two logarithmic
operators Ologmakes it clear why such CFTs have the attribute “logarithmic ”. The constant
bL, sometimes called “new anomaly”, defines crucial propertie s of the LCFT, much like the
central charges do in ordinary CFTs. The mass scale mLappearing in the last correlator above
has no significance, and is determined by the value of Bin (5). It can be changed to any finite
value by the redefinition Olog→ Olog+γOLwith some finite γ. We setmL= 1 for convenience.
Conformal Ward identities determine again essentially uni quely the form of 2- and 3-point
correlators in a LCFT. For the specific case where the energy m omentum tensor acquires a
logarithmic partner the 3-point correlators were calculat ed in [5]. The non-vanishing ones are
given by
∝an}b∇acketle{tOL(z,¯z)OL(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=bL
z2z′2(z−z′)2(8a)
∝an}b∇acketle{tOL(z,¯z)Olog(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=−2bLln|z′|2+bL
2
z2z′2(z−z′)2(8b)
∝an}b∇acketle{tOlog(z,¯z)Olog(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=lengthy
z2z′2(z−z′)2(8c)
If alsoORacquires a logarithmic partner O/tildewiderlogthen the construction above can be repeated,
changing everywhere L→R,z→¯zetc. In that case we have a LCFT with cL=cR= 0 andbL,bR∝ne}ationslash= 0. Alternatively, it may happen that only OLhas a logarithmic partner Olog. In that
case we have a LCFT with cL=bR= 0 andbL,cR∝ne}ationslash= 0. This concludes our brief excursion into
the realm of LCFTs.
Given that LCFTs are interesting in physics (see section 5) a nd that a powerful way to
describe strongly coupled CFTs is to exploit the AdS/CFT cor respondence [6] it is natural to
inquire whether there are any gravity duals to LCFTs.
3. Wish-list for gravity duals to LCFTs
In this section we establish necessary properties required for gravity duals to LCFTs. We
formulate them as a wish-list and explain afterwards each it em on this list.
(i) We wishfora 3-dimensional action Sthat dependsonthemetric gµνandpossiblyonfurther
fields that we summarily denote by φ.
(ii) We wish for the existence of AdS 3vacua with finite AdS radius ℓ.
(iii) We wish for a finite, conserved and traceless Brown–Yor k stress tensor, given by the first
variation of the full on-shell action (including boundary t erms) with respect to the metric.
(iv) We wish that the 2- and 3-point correlators of the Brown– York stress tensor with itself are
given by (1).
(v) We wish for central charges (a la Brown–Henneaux [7]) tha t can be tuned to zero, without
requiring a singular limit of the AdS radius or of Newton’s co nstant. For concreteness we
assumecL= 0 (in addition cRmay also vanish, but it need not).
(vi) We wish for a logarithmic partner to the Brown–York stre ss tensor, so that we obtain a
Jordan-block structure like in (2) and (3).
(vii) We wish that the 2- and non-vanishing 3-point correlat ors of the Brown–York stress tensor
with its logarithmic partner are given by (7) and (8) (and the right-handed analog thereof).
We explain now why each of these items is necessary. (i) is req uired since the AdS/CFT
correspondence relates a gravity theory in d+1 dimensions to a CFT in ddimensions, and we
chosed= 2 on the CFT side. (ii) is required since we are not merely loo king for a gauge/gravity
duality, butreallyforanAdS/CFTcorrespondence,whichre quirestheexistenceofAdSsolutions
on the gravity side. (iii) is required since we desire consis tency with the AdS dictionary, which
relates the vacuum expectation value of the renormalized en ergy momentum tensor in the CFT
∝an}b∇acketle{tTij∝an}b∇acket∇i}htto the Brown–York stress tensor TBY
ij:
∝an}b∇acketle{tTij∝an}b∇acket∇i}ht=TBY
ij=2√−gδS
δgij/vextendsingle/vextendsingle/vextendsingle
EOM(9)
The right hand side of this equation contains the first variat ion of the full on-shell action with
respect to the metric, which by definition yields the Brown–Y ork stress tensor. (iv) is required
since the 2- and 3-point correlators of a CFT are fixed by confo rmal Ward identities to take
the form (1). (v) is required because of the construction pre sented in section 2, where a LCFT
emerges from taking an appropriate limit of vanishing centr al charge, so we need to be able
to tune the central charge without generating parametric si ngularities. Actually, there are
two cases: either left and right central charge vanish and bo th energy momentum tensor flux
components acquire a logarithmic partner, or only one of the m acquires a logarithmic partner,
which for sake of specificity we always choose to be left. (vi) is required, since we consider
exclusively LCFTs where the energy momentum tensor acquire s a logarithmic partner. (vii) is
required since the 2- and 3-point correlators of a LCFT are fix ed by conformal Ward identities to
taketheform(7), (8). Ifanyoftheitemsonthewish-listabo veisnotfulfilleditisimpossiblethat
the gravitational theory under consideration is a gravity d ual to a LCFT of the type discussedin section 2.2On the other hand, if all the wishes are granted by a given grav itational theory
there are excellent chances that this theory is dual to a LCFT . Until recently no good gravity
duals for LCFTs were known [8–12].
Before addressing candidate theories that may comply with a ll wishes we review briefly how
to calculate correlators on the gravity side [6], since we sh all need such calculations for checking
several items on the wish-list. The basic identity of the AdS /CFT dictionary is
∝an}b∇acketle{tO1(z1)O2(z2)...On(zn)∝an}b∇acket∇i}ht=δ(n)S
δj1(z1)δj2(z2)...δjn(zn)/vextendsingle/vextendsingle/vextendsingle
ji=0(10)
The left hand side is the CFT correlator between noperators Oi, whereOiin our case comprise
theleft-andright-moving fluxcomponentsoftheenergymome ntumtensor andtheirlogarithmic
partners. The right hand side contains the gravitational ac tionSdifferentiated with respect to
appropriate sources jifor the corresponding operators. According to the AdS/CFT d ictionary
“appropriate sources” refers to non-normalizable solutio ns of the linearized equations of motion.
We shall be more concrete about the operators, actions, sour ces and non-normalizable solutions
to the linearized equations of motion in the next section. Fo r now we address possible candidate
theories of gravity duals to LCFTs.
The simplest candidate, pure 3-dimensional Einstein gravi ty with a cosmological constant
described by the action
SEH=−1
8πGN/integraldisplay
Md3x√−g/bracketleftig
R+2
ℓ2/bracketrightig
−1
4πGN/integraldisplay
∂Md2x√−γ/bracketleftig
K−1
ℓ/bracketrightig
(11)
does not comply with the whole wish list. Only the first four wi shes are granted: The 3-
dimensional action (12) depends on the metric. The equation s of motion are solved by AdS 3.
ds2
AdS3=gAdS3µνdxµdxν=ℓ2/parenleftbig
dρ2−1
4cosh2ρ(du+dv)2+1
4sinh2ρ(du−dv)2/parenrightbig
(12)
The Brown–York stress tensor (9) is finite, conserved and tra celess. The 2- and 3-point
correlators on the gravity side match precisely with (1). Ho wever, the central charges are given
by [7]
cL=cR=3ℓ
2GN(13)
and therefore allow no tuning to cL= 0 without taking a singular limit. Moreover, there is no
candidate for a logarithmic partner to the Brown–York stres s tensor. Thus, pure 3-dimensional
Einstein gravity cannot be dual to a LCFT.
Adding matter fields to Einstein gravity does not help neithe r. While this may lead to other
kinds of LCFTs, it cannot produce a logarithmic partner for t he energy momentum tensor. This
is so, because the energy momentum tensor corresponds to gra viton (spin-2) excitations in the
bulk, and the only field producing such excitations is the met ric.
Therefore, what we need is a way to provide additional degree s of freedom in the gravity
sector. The most natural way to do this is by considering high er derivative interactions of the
metric. Thefirstgravity modelofthistypewas constructedb yDeser, Jackiw andTempleton [13]
who introduced a Chern–Simons term for the Christoffel connec tion.
SCS=−1
16πGNµ/integraldisplay
d3xǫλµνΓρσλ/bracketleftig
∂µΓσρν+2
3ΓσκµΓκσν/bracketrightig
(14)
2Other types of LCFTs exist, e.g. with non-vanishing central charge or with logarithmic partners to operators
other than the energy momentum tensor. The gravity duals for such LCFTs need not comply with all the items
on our wish list.Hereµis a real coupling constant. Adding this action to the Einste in–Hilbert action (11)
generates massive graviton excitations in the bulk, which i s encouraging for our wish list since
we need these extra degrees of freedom. The model that arises when summing the actions (11)
and (14),
SCTMG=SEH+SCS (15)
is known as “cosmological topologically massive gravity” ( CTMG) [14]. It was demonstrated by
KrausandLarsen[15]that thecentral charges inCTMG areshi ftedfromtheir Brown–Henneaux
values:
cL=3ℓ
2GN/parenleftbig
1−1
µℓ/parenrightbig
cR=3ℓ
2GN/parenleftbig
1+1
µℓ/parenrightbig
(16)
This is again good news concerning our wish list, since cLcan be made vanishing by a (non-
singular) tuning of parameters in the action.
µℓ= 1 (17)
CTMG (15) with the tuning above (17) is known as “cosmologica l topologically massive gravity
at the chiral point” (CCTMG). It complies with the first five it ems on our wish list, but we still
have to prove that also the last two wishes are granted. To thi s end we need to find a suitable
partner for the graviton.
4. Keeping logs in massive gravity
4.1. Login
In this section we discuss the evidence for the existence of s pecific gravity duals to LCFTs that
has accumulated over the past two years. We start with the the ory introduced above, CCTMG,
and we end with a relatively new theory, new massive gravity [ 16].
4.2. Seeds of logs
Given that we want a partner for the graviton we consider now g raviton excitations ψaround
the AdS background (12) in CCTMG.
gµν=gAdS3µν+ψµν (18)
Li,SongandStrominger[17]foundanicewaytoconstructthe m,andwefollowtheirconstruction
here. Imposing transverse gauge ∇µψµν= 0 and defining the mutually commuting first order
operators
/parenleftbig
DM/parenrightbigβ
µ=δβ
µ+1
µεµαβ∇α/parenleftbig
DL/R/parenrightbigβ
µ=δβ
µ±ℓεµαβ∇α (19)
allows to write the linearized equations of motion around th e AdS background (12) as follows.
(DMDLDRψ)µν= 0 (20)
A mode annihilated by DM(DL) [DR]{(DL)2but not by DL}is called massive (left-moving)
[right-moving] {logarithmic }and is denoted by ψM(ψL) [ψR]{ψlog}. Away from the chiral
point,µℓ∝ne}ationslash= 1, the general solution to the linearized equations of moti on (20) is obtained from
linearly combining left, right and massive modes [17]. At th e chiral point DMdegenerates with
DLand the general solution to the linearized equations of moti on (20) is obtained from linearly
combining left, right and logarithmic modes [18]. Interest ingly, we discovered in [18] that the
modesψlogandψLbehave as follows:
(L0+¯L0)/parenleftbigg
ψlog
ψL/parenrightbigg
=/parenleftbigg
2 1
0 2/parenrightbigg/parenleftbigg
ψlog
ψL/parenrightbigg
(21)whereL0=i∂u,¯L0=i∂vand
(L0−¯L0)/parenleftbigg
ψlog
ψL/parenrightbigg
=/parenleftbigg
2 0
0 2/parenrightbigg/parenleftbigg
ψlog
ψL/parenrightbigg
(22)
If we define naturally the Hamiltonian by H=L0+¯L0and the angular momentum by
J=L0−¯L0we recover exactly (2) and (3), which suggests that the CFT du al to CCTMG
(if it exists) is logarithmic, as conjectured in [18]. It was further shown with Jackiw that the
existence of the logarithmic excitations ψlogis not an artifact of the linearized approach, but
persists in the full theory [19].
Thus, also the sixth wish is granted in CCTMG. The rest of this section discusses the last
wish.
4.3. Growing logs
We assume now that there is a standard AdS/CFT dictionary [6] available for LCFTs and check
if CCTMG indeed leads to the correct 2- and 3-point correlato rs. To this end we have to identify
the sources jithat appear on the right hand side of the correlator equation (10). Following the
standard AdS/CFT prescription the sources for the operator sOL(OR) [Olog] are given by left
(right) [logarithmic] non-normalizablesolutions tothel inearized equations of motion (20). Thus,
our first task is to find all solutions of the linearized equati ons of motion and to classify them
into normalizable and non-normalizable ones, where “norma lizable” refers to asymptotic (large
ρ) behavior that is exponentially suppressed as compared to t he AdS background (12).
A construction of all normalizable left and right solutions was provided in [17], and the
normalizable logarithmic solutions were constructed in [1 8].3The non-normalizable solutions
were constructed in [25]. It turned out to be convenient to wo rk in momentum space
ψL/R/log
µν(h,¯h) =e−ih(t+φ)−i¯h(t−φ)FL/R/log
µν(ρ) (23)
The momenta h,¯hare called “weights”. All components of the tensor Fµνare determined
algebraically, except for one that is determined from a seco nd order (hypergeometric) differential
equation. Ingeneral oneofthelinearcombinations of theso lutionsis singularattheorigin ρ= 0,
whiletheother isregular there. We keep onlyregular soluti ons. For each given set ofweights h,¯h
the regular solution is either normalizable or non-normali zable. It turns out that normalizable
solutions exist for integer weights h≥2,¯h≥0 (orh≤ −2,¯h≤0). All other solutions are
non-normalizable.
An example for a normalizable left mode is given by the primar y with weights h= 2,¯h= 0
ψL
µν(2,0) =e−2iu
cosh4ρ
1
4sinh2(2ρ) 0i
2sinh(2ρ)
0 0 0
i
2sinh(2ρ) 0 −1

µν(24)
Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant.
The corresponding logarithmic mode is given by
ψlog
µν(2,0) =−1
2(i(u+v)+lncosh2ρ)ψL
µν(2,0) (25)
Evidently, it behaves asymptotically like its left partner (24), except for overall linear growth in
ρ. It is also worthwhile emphasizing that the logarithmic mod e (25) depends linearly on time
3All these modes are compatible with asymptotic AdS behavior [20,21], and they appear in vacuum expectation
values of 1-point functions. Indeed, the 1-point function /angbracketleftTij/angbracketrightinvolves both ψlogandψR[21–24].t= (u+v)/2. Both features are inherent to all logarithmic modes. All o ther normalizable
modes can be constructed from the primaries (24), (25) algeb raically.
An example for a non-normalizable left mode is given by the mo de with weights h= 1,
¯h=−1
ψL
µν(1,−1) =1
4e−iu+iv
0 0 0
0 cosh(2 ρ)−1−2i/radicalig
cosh(2ρ)−1
cosh(2ρ)+1
0−2i/radicalig
cosh(2ρ)−1
cosh(2ρ)+1−4
cosh(2ρ)+1

µν(26)
Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant,
except for the vv-component, which grows like e2ρ. The corresponding logarithmic mode grows
again faster than its left partner (26) by a factor of ρand depends again linearly on time.
Given a non-normalizable solution ψLobviously also αψLis a non-normalizable solution,
with some constant α. To fix this normalization ambiguity we demand standard coup ling of the
metric to the stress tensor:
S(ψuL
v,Tv
u) =1
2/integraldisplay
dtdφ/radicalig
−g(0)ψuu
LTuu=/integraldisplay
dtdφe−ihu−i¯hvTuu (27)
HereSis either someCFT action withbackgroundmetric g(0)or adualgravitational action with
boundary metric g(0). The non-normalizable mode ψLis the source for the energy-momentum
flux component Tuu. The requirement (27) fixes the normalization. The discussi on above
focussed on left modes. For the right modes essentially the s ame discussion applies, but with
the substitutions L↔R,h↔¯handu↔v.
4.4. Logging correlators
Generically the 2-point correlators on the gravity side bet ween two modes ψ1(h,¯h) andψ2(h′,¯h′)
in momentum space are determined by
∝an}b∇acketle{tψ1(h,¯h)ψ2(h′,¯h′)∝an}b∇acket∇i}ht=1
2/parenleftbig
δ(2)SCCTMG(ψ1,ψ2)+δ(2)SCCTMG(ψ2,ψ1)/parenrightbig
(28)
where∝an}b∇acketle{tψ1ψ2∝an}b∇acket∇i}htstands for the correlation function of the CFT operators dua l to the graviton
modesψ1andψ2. On the right hand side one has to plug the non-normalizable m odesψ1
andψ2into the second variation of the on-shell action and symmetr ize with respect to the two
modes. The second variation of the on-shell action of CCTMG
δ(2)SCCTMG=−1
16πGN/integraldisplay
d3x√−g/parenleftbig
DLψ1∗/parenrightbigµνδGµν(ψ2)+boundary terms (29)
turns out to be very similar to the second variation of the on- shell Einstein–Hilbert action
δ(2)SEH=−1
16πGN/integraldisplay
d3x√−gψ1µν∗δGµν(ψ2)+boundary terms (30)
Thissimilarity allows ustoexploitresultsfromEinsteing ravity forCCTMG,aswenowexplain.4
The bulk term in CCTMG (29) has the same form as in Einstein the ory (30) with ψ1replaced
byDLψ1. Now, consider boundary terms. Possible obstructions to a w ell-defined Dirichlet
boundary value problem can come only from the variation δGµν(ψ2), sinceDLis a first order
operator. Thus any boundary terms appearing in (29) contain ing normal derivatives must be
4Alternatively, one can follow the program of holographic re normalization, as it was done by Skenderis, Taylor
and van Rees [23]. Their results for 2-point correlators agr ee with the results presented here.identical with those in Einstein gravity upon substituting ψ1→ DLψ1. In addition there can be
boundary terms which do not contain normal derivatives of th e metric. However, it turns out
that such terms can at most lead to contact terms in the hologr aphic computation of 2-point
functions. The upshot of this discussion is that we can reduc e the calculation of all possible 2-
point functions in CCTMG to the equivalent calculation in Ei nstein gravity with suitable source
terms. To continue we go on-shell.5
DLψL= 0 DLψR= 2ψRDLψlog=−2ψL(31)
These relations together with the comparison between CCTMG (29) and Einstein gravity (30)
then establish
∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼2∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htEH (32a)
∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32b)
∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32c)
∝an}b∇acketle{tψR(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32d)
∝an}b∇acketle{tψL(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼ −2∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htEH (32e)
Here the sign ∼means equality up to contact terms. Evaluating the right han d sides in Einstein
gravity yields
∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htEH=δh,h′δ¯h,¯h′cBH
24h
¯h(h2−1)t1/integraldisplay
t0dt (33)
and similarly for the right modes, with h↔¯h. The quantity cBHis the Brown–Henneaux
central charge (13). The calculation of the 2-point correla tor between two logarithmic modes
cannot be reduced to a correlator known from Einstein gravit y. The result is given by [25]
∝an}b∇acketle{tψlog(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼ −δh,h′δ¯h,¯h′ℓ
4GNh
¯h(h2−1)/parenleftbig
ψ(h−1)+ψ(−¯h)/parenrightbigt1/integraldisplay
t0dt(34)
whereψis the digamma function. An ambiguity in defining ψlog, viz.,ψlog→ψlog+γψL, was
fixed conveniently in the result (34). This ambiguity corres ponds precisely to the ambiguity of
the LCFT mass scale mLin (7c) (see also the discussion below that equation).
To compare the results (32)-(34) with the Euclidean 2-point correlators in the short-
distance limit (1), (7) we take the limit of large weights h,−¯h→ ∞(e.g. lim h→∞ψ(h) =
lnh+O(1/h)) and Fourier-transform back to coordinate space (e.g. h3/¯his Fourier-transformed
into∂4
z/(∂z∂¯z)δ(2)(z,¯z)∝∂4
zln|z| ∝1/z4). Straightforward calculation establishes perfect
agreement with the LCFT correlators (1), (7), provided we us e the values
cL= 0 cR=3ℓ
GNbL=−3ℓ
GN(35)
These are exactly the values for central charges cL,cR[15] and new anomaly bL[23,25] found
before. Thus, at the level of 2-point correlators CCTMG is in deed a gravity dual for a LCFT.
5Above by “on-shell” we meant that the background metric is Ad S3(12) and therefore a solution of the classical
equations of motion. Here by “on-shell” we mean additionall y that the linearized equations of motion (20) hold.Ψ1
Ψ3Ψ2
Figure 1. Witten diagram for three graviton correlator
We evaluate now the Witten diagram in Fig. 1, which yields the 3-point correlator on the
gravity side between three modes ψ1(h,¯h),ψ2(h′,¯h′) andψ3(h′′,¯h′′) in momentum space.
∝an}b∇acketle{tψ1(h,¯h)ψ2(h′,¯h′)ψ3(h′′,¯h′′)∝an}b∇acket∇i}ht=1
6/parenleftbig
δ(3)SCCTMG(ψ1,ψ2,ψ3)+5 permutations/parenrightbig
(36)
On the right hand side one has to plug the non-normalizable mo desψ1,ψ2andψ3into the third
variation of the on-shell action and symmetrize with respec t to all three modes.
δ(3)SCCTMG∼ −1
16πGN/integraldisplay
d3x√−g/bracketleftig/parenleftbig
DLψ1/parenrightbigµνδ(2)Rµν(ψ2,ψ3)+ψ1µν∆µν(ψ2,ψ3)/bracketrightig
(37)
The quantity δ(2)Rµν(ψ2,ψ3) denotes the second variation of the Ricci-tensor and the te nsor
∆µν(ψ2,ψ3) vanishes if evaluated on left- and/or right-moving soluti ons. All boundary terms
turn out to be contact terms, which is why only bulk terms are p resent in the result (37) for the
third variation of the on-shell action. We compare again wit h Einstein gravity.
δ(3)SEH∼ −1
16πGN/integraldisplay
d3x√−gψ1µνδ(2)Rµν(ψ2,ψ3) (38)
Once more we can exploit some results from Einstein gravity f or CCTMG, and we find the
following results [25] for 3-point correlators without log -insertions:
∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼2∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htEH (39a)
∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39b)
∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39c)
∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψL(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39d)
with one log-insertion:
∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (40a)
∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (40b)
∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼ −2∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψL(h′′,¯h′′)∝an}b∇acket∇i}htEH (40c)and with two or more log-insertions:
lim
|weights|→∞∝an}b∇acketle{tψR(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (41a)
lim
|weights|→∞∝an}b∇acketle{tψL(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼δh′′,−h−h′δ¯h′′,−¯h−¯h′Plog(h,h′,¯h,¯h′)
¯h¯h′(¯h+¯h′)(41b)
lim
|weights|→∞∝an}b∇acketle{tψlog(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼δh′′,−h−h′δ¯h′′,−¯h−¯h′lengthy
¯h¯h′(¯h+¯h′)(41c)
Thelast two correlators so far could becalculated qualitat ively only (Plogis a known polynomial
in the weights and also contains logarithms in the weights, a s expected on general grounds),
and it would be interesting to calculate them exactly. They a re in qualitative agreement with
corresponding LCFT correlators. All other correlators hav e been calculated exactly [25], and
they are in precise agreement with the LCFT correlators (1), (8), provided we use again the
values (35) for central charges and new anomaly.
Inconclusion, also theseventh wishisgranted forCCTMG.6Thus, thereareexcellent chances
that CCTMG is dual to a LCFT with values for central charges an d new anomaly given by (35).
4.5. Logs don’t grow on trees
From the discussion above it is clear that possible gravity d uals for LCFTs are sparse in theory
space: Einstein gravity (11) does not provide a gravity dual for any tuning of parameters and
CTMG (15) does potentially provide a gravity dual only for a s pecific tuning of parameters (17).
Any candidate for a novel gravity dual to a LCFT is therefore w elcomed as a rare entity.
Very recently another plausible candidate for such a gravit ational theory was found [26].
That theory is known as “new massive gravity” [16].
SNMG=1
16πGN/integraldisplay
d3x√−g/bracketleftig
σR+1
m2/parenleftbig
RµνRµν−3
8R2/parenrightbig
−2λm2/bracketrightig
(42)
Heremis a mass parameter, λa dimensionless cosmological parameter and σ=±1 the sign of
the Einstein-Hilbert term. If they are tuned as follows
λ= 3 ⇒m2=−σ
2ℓ2(43)
then essentially the same story unfolds as for CTMG at the chi ral point. The main difference
to CCTMG is that both central charges vanish in new massive gr avity at the chiral point
(CNMG) [27,28].
cL=cR=3ℓ
2GN/parenleftbigg
σ+1
2ℓ2m2/parenrightbigg
= 0 (44)
Therefore, both left and right flux component of the energy mo mentum tensor acquire a
logarithmic partner. It is easy to check that CNMG grants us t he first six wishes from section
3. The seventh wish requires again the calculation of correl ators. The 3-point correlators have
not been calculated so far, but at the level of 2-point correl ators again perfect agreement with
a LCFT was found, provided we use the values [26]
cL=cR= 0bL=bR=−σ12ℓ
GN(45)
6The sole caveat is that two of the ten 3-point correlators wer e calculated only qualitatively. It would be
particularly interesting to calculate the correlator betw een three logarithmic modes (41c), since it contains an
additional parameter independent from the central charges and new anomaly that determines LCFT properties.Itislikely thatasimilarstorycanberepeatedforgeneralm assivegravity [16], whichcombines
new massive gravity (42) with a gravitational Chern–Simons term (14). Thus, even though they
are sparse in theory space we have found a few good candidates for gravity duals to LCFTs:
cosmological topologically massive gravity, new massive g ravity and general massive gravity. In
all cases we have to tune parameters in such a way that a “chira l point” emerges where at least
one of the central charges vanishes.
4.6. Chopping logs?
Sofarwe were exclusively concerned with findinggravitatio nal theories wherelogarithmic modes
can arise. In this subsection we try to get rid of them. The rat ionale behind the desire to
eliminate the logarithmic modes is unitarity of quantum gra vity. Gravity in 2+1 dimensions is
simple and yet relevant, as it contains black holes [29], pos sibly gravity waves [13] and solutions
that are asymptotically AdS. Thus, it could provide an excel lent arena to study quantum gravity
in depth provided one is able to come up with a consistent (uni tary) theory of quantum gravity,
for instance by constructing its dual (unitary) CFT. Indeed , two years ago Witten suggested a
specific CFT dual to 3-dimensional quantum gravity in AdS [30 ]. This proposal engendered a
lot of further research (see [31–37] for some early referenc es), including the suggestion by Li,
Song and Strominger [17] to construct a quantum theory of gra vity that is purely right-moving,
dubbed“chiral gravity”. To make a long story [18,19,24,38– 81] short, “chiral gravity” is nothing
but CCTMG with the logarithmic modes truncated in some consi stent way.
We discuss now two conceptually different possibilities of im plementing such a truncation.
The first option was proposed in [18]. If one imposes periodic ity in time for all modes, t→t+β,
then only the left- and right-moving modes are allowed, whil e the logarithmic modes are
eliminated since they grow linearly in time, see e.g. (25). T he other possibility was pursued
in [22]. It is based upon the observation that logarithmic mo des grow logarithmically faster in
e2ρthan their left partners, see e.g. (25). Thus, imposing boun dary conditions that prohibit this
logarithmic growth eliminates all logarithmic modes.
Currently it is not known whether chiral gravity has its own d ual CFT or if it exists merely
as a zero-charge superselection sector of the logarithmic C FT. In the latter case it is unclear
whether or not the zero-charge superselection sector is a fu lly-fledged CFT. Another alternative
is that neither the LCFT nor its chiral truncation dual to chi ral gravity exists. In that case
CTMG is unlikely to exist as a consistent quantum theory on it s own. Rather, it would require
a UV completion, such as string theory.
4.7. Logout
We summarize now the key results reviewed in this section as w ell as some open issues.
Cosmological topologically massive gravity (15) at the chi ral point (17) is likely to be dual
to a LCFT with a logarithmic partner for one flux component of t he energy momentum tensor
since 2- [23] and 3-point correlators [25] match. The values of central charges and new anomaly
are given by (35). The detailed calculation of the correlato r with three log-insertions (41c)
still needs to be performed and will determine another param eter of the LCFT. New massive
gravity (42) at the chiral point (43) is likely to be dual to a L CFT with a logarithmic partner
for both flux components of the energy momentum tensor since 2 -point correlators match [26].
The central charges vanish and the new anomalies are given by (45). The calculation of 3-
point correlators still needs to be performed and will provi de a more stringent test of the
conjectured duality to a LCFT. A similar story is likely to re peat for general massive gravity
(the combination of topologically and new massive gravity) at a chiral point, and it could be
rewardingtoinvestigate thisissue. Finallyweaddressedp ossibilitiestoeliminatethelogarithmic
modes and their partners, since such an elimination might le ad to a chiral theory of quantum
gravity [17], called “chiral gravity”. The issue of whether chiral gravity exists still remains open.5. Towards condensed matter applications
In this final section we review briefly some condensed matter s ystems where LCFTs do arise,
see [3,4] for more comprehensive reviews. We focus on LCFTs w here the energy-momentum
tensor acquires a logarithmic partner, i.e., the class of LC FTs for which we have found possible
gravity duals.7Condensed matter systems described by such LCFTs are for ins tance systems
at (or near) a critical point with quenched disorder, like sp in glasses [83]/quenched random
magnets [84,85], dilute self-avoiding polymers or percola tion [86]. “Quenched disorder” arises
in a condensed matter system with random variables that do no t evolve with time. If the
amount of disorder is sufficiently large one cannot study the e ffects of disorder by perturbing
around a critical point without disorder — standard mean fiel d methods break down. The
system is then driven towards a random critical point, and it is a challenge to understand its
precise nature. Mathematically, the essence of the problem lies in the infamous denominator
arising in correlation functions of some operator Oaveraged over disordered configurations (see
e.g. chapter VI.7 in [87])
∝an}b∇acketle{tO(z)O(0)∝an}b∇acket∇i}ht=/integraldisplay
DVP[V]/integraltext
Dφexp/parenleftbig
−S[φ]−/integraltext
d2z′V(z′)O(z′)/parenrightbig
O(z)O(0)/integraltext
Dφexp/parenleftbig
−S[φ]−/integraltext
d2z′V(z′)O(z′)/parenrightbig (46)
HereS[φ] is some 2-dimensional8quantum field theory action for some field(s) φandV(z) is a
random potential with some probability distribution. For w hite noise one takes the Gaussian
probability distribution P[V]∝exp/parenleftbig
−/integraltext
d2zV2(z)/(2g2)/parenrightbig
, wheregis a coupling constant that
measuresthestrengthoftheimpurities. Ifit werenot forth edenominatorappearingontheright
hand side of the averaged correlator (46) we could simply per form the Gaussian integral over
the impurities encoded in the random potential V(z). This denominator is therefore the source
of all complications and to deal with it requires suitable me thods, see e.g. [88]. One possibility is
to eliminate the denominator by introducing ghosts. This so -called “supersymmetric method”
works well if the original quantum field theory described by t he actionS[φ] is very simple, like a
free field theory. Another option is the so-called replica tr ick, where one introduces ncopies of
the original quantum field theory, calculates correlators i n this setup and takes the limit n→0
in the end, which formally reproduces the denominator in (46 ). Recently, Fujita, Hikida, Ryu
and Takayanagi combined the replica method with the AdS/CFT correspondence to describe
disordered systems [89] (see [90,91] for related work), ess entially by taking ncopies of the CFT,
exploiting AdS/CFT to calculate correlators and taking for mally the limit n→0 in the end.
Like other replica tricks their approach relies on the exist ence of the limit n→0.
One of the results obtained by the supersymmetric method or r eplica trick is that correlators
like the one in (46) develop a logarithmic behavior, exactly as in a LCFT [84]. In fact, in
then→0 limit prescribed by the replica trick, the conformal dimen sions of certain operators
degenerate. This produces a Jordan block structure for the H amiltonian in precise parallel to
theµℓ→1 limit of CTMG. More concretely, LCFTs can be used to compute correlators of
quenched random systems!
This suggests yet-another route to describe systems with qu enched disorder, and our present
results add to this toolbox. Namely, instead of taking ncopies of an ordinary CFT we may
start directly with a LCFT. If this LCFT is weakly coupled we c an work on the LCFT side
perturbatively, using the results mentioned above [3,4,84 –86]. On the other hand, if the LCFT
becomes strongly coupled, perturbative methods fail. To ge t a handle on these situations we
can exploit the AdS/LCFT correspondence and work on the grav ity side. Of course, to this end
7A well-studied alternative case is a LCFT with c=−2 [2,82]. There is no obvious way to construct a gravity
dual for such LCFTs, even when considering CTMG or new massiv e gravity away from the chiral point. We thank
Ivo Sachs for discussions on this issue.
8Analog constructions work in higher dimensions, but we focu s here on two dimensions.one needs to construct gravity duals for LCFTs. The models re viewed in this talk are simple
and natural examples of such constructions.
Acknowledgments
We thank Matthias Gaberdiel, Gaston Giribet, Olaf Hohm, Rom an Jackiw, David Lowe, Hong
Liu, Alex Maloney, John McGreevy, Ivo Sachs, Kostas Skender is, Wei Song, Andy Strominger
and Marika Taylor for discussions. DG thanks the organizers of the “First Mediterranean
Conference on Classical and Quantum Gravity” for the kind in vitation and for all their efforts to
make the meeting very enjoyable. DG and NJ are supported by th e START project Y435-N16
of the Austrian Science Foundation (FWF). During the final st age NJ has been supported by
project P21927-N16 of FWF. NJ acknowledges financial suppor t from the Erwin-Schr¨ odinger-
Institute (ESI) during the workshop “Gravity in three dimen sions”.
References
[1] Di Francesco P, Mathieu P and Senechal D 1997 Conformal Field Theory (Springer)
[2] Gurarie V 1993 Nucl. Phys. B410535–549 ( Preprint hep-th/9303160 )
[3] Flohr M 2003 Int. J. Mod. Phys. A184497–4592 ( Preprint hep-th/0111228 )
[4] Gaberdiel M R 2003 Int. J. Mod. Phys. A184593–4638 ( Preprint hep-th/0111260 )
[5] Kogan I I and Nichols A 2002 JHEP01029 (Preprint hep-th/0112008 )
[6] Aharony O, Gubser S S, Maldacena J M, Ooguri H and Oz Y 2000 Phys. Rept. 323183–386 ( Preprint
hep-th/9905111 ); for a review focussed on the condensed matter perspective see
McGreevy J, ( Preprint 0909.0518 )
[7] Brown J D and Henneaux M 1986 Commun. Math. Phys. 104207–226
[8] Ghezelbash A M, Khorrami M and Aghamohammadi A 1999 Int. J. Mod. Phys. A142581–2592 ( Preprint
hep-th/9807034 )
[9] Myung Y S and Lee H W 1999 JHEP10009 (Preprint hep-th/9904056 )
[10] Kogan I I 1999 Phys. Lett. B45866–72 (Preprint hep-th/9903162 )
[11] Lewis A 2000 Phys. Lett. B480348–354 ( Preprint hep-th/9911163 )
[12] Moghimi-Araghi S, Rouhani S and Saadat M 2004 Nucl. Phys. B696492–502 ( Preprint hep-th/0403150 )
[13] Deser S, Jackiw R and Templeton S 1982 Phys. Rev. Lett. 48975–978
Deser S, Jackiw R and Templeton S 1982 Ann. Phys. 140372–411; 1988 Erratum-ibid. 185406
[14] Deser S 1984 Quantum Theory Of Gravity ed Christensen S M (Bristol: Adam Hilger) pp 374–381 ( Preprint
Print-82-0692(Brandeis) )
[15] Kraus P and Larsen F 2006 JHEP01022 (Preprint hep-th/0508218 )
[16] Bergshoeff E A, Hohm O and Townsend P K 2009 Phys. Rev. Lett. 102201301 ( Preprint 0901.1766 )
[17] Li W, Song W and Strominger A 2008 JHEP04082 (Preprint 0801.4566 )
[18] Grumiller D and Johansson N 2008 JHEP07134 (Preprint 0805.2610 )
[19] Grumiller D, Jackiw R and Johansson N 2009 Fundamental Interactions - A Memorial Volume for Wolfgang
Kummer (World Scientific) ( Preprint 0806.4185 )
[20] Grumiller D and Johansson N 2009 Int. J. Mod. Phys. D172367–2372 ( Preprint 0808.2575 )
[21] Henneaux M, Martinez C and Troncoso R 2009 Phys. Rev. D79081502R ( Preprint 0901.2874 )
[22] Maloney A, Song W and Strominger A 2009 ( Preprint 0903.4573 )
[23] Skenderis K, Taylor M and van Rees B C 2009 JHEP09045 (Preprint 0906.4926 )
[24] Ertl S, Grumiller D and Johansson N 2009 ( Preprint 0910.1706 )
[25] Grumiller D and Sachs I 2009 ( Preprint 0910.5241 )
[26] Grumiller D and Hohm O 2009 ( Preprint 0911.4274 )
[27] Liu Y and Sun Y W 2009 JHEP04106 (Preprint 0903.0536 )
[28] Bergshoeff E A, Hohm O and Townsend P K 2009 Phys. Rev. D79124042 ( Preprint 0905.1259 )
[29] Banados M, Teitelboim C and Zanelli J, 1992 Phys. Rev. Lett. 691849 (Preprint hep-th/9204099 )
[30] Witten E 2007 ( Preprint 0706.3359 )
[31] Manschot J 2007 JHEP10103 (Preprint 0707.1159 )
[32] Gaiotto D and Yin X 2007 JHEP08029 (Preprint 0707.3437 )
[33] Gaberdiel M R 2007 JHEP11087 (Preprint 0707.4073 )
[34] Yin X 2007 ( Preprint arXiv:0710.2129[hep-th] )
[35] Yin X 2008 JHEP09120 (Preprint 0711.2803 )
[36] Maloney A and Witten E 2007 ( Preprint 0712.0155 )[37] Manschot J and Moore G W 2007 ( Preprint 0712.0573 )
[38] Carlip S, Deser S, Waldron A and Wise D K 2009 Class. Quant. Grav. 26075008 ( Preprint 0803.3998 )
[39] Carlip S 2008 JHEP10078 (Preprint 0807.4152 )
[40] Hotta K, Hyakutake Y, Kubota T and Tanida H 2008 JHEP07066 (Preprint 0805.2005 )
[41] Li W, Song W and Strominger A 2008 ( Preprint 0805.3101 )
[42] Park M I 2008 JHEP09084 (Preprint 0805.4328 )
[43] Sachs I and Solodukhin S N 2008 JHEP08003 (Preprint 0806.1788 )
[44] Lowe D A and Roy S 2008 Phys. Lett. B668159–162 ( Preprint 0806.3070 )
[45] Myung Y S, Lee H W and Kim Y W 2008 ( Preprint 0806.3794 )
[46] Carlip S, Deser S, Waldron A and Wise D K 2008 Phys. Lett. B666272–276 ( Preprint 0807.0486 )
[47] Lee H W, Kim Y W and Myung Y S 2008 ( Preprint 0807.1371 )
[48] Sachs I 2008 JHEP09073 (Preprint 0807.1844 )
[49] Gibbons G W, Pope C N and Sezgin E 2008 Class. Quant. Grav. 25205005 ( Preprint 0807.2613 )
[50] Anninos D, Li W, Padi M, Song W and Strominger A 2009 JHEP03130 (Preprint 0807.3040 )
[51] Giribet G, Kleban M and Porrati M 2008 JHEP10045 (Preprint 0807.4703 )
[52] Strominger A 2008 ( Preprint 0808.0506 )
[53] Compere G and Detournay S 2009 Class. Quant. Grav. 26012001 ( Preprint 0808.1911 )
[54] Myung Y S 2008 Phys. Lett. B670220–223 ( Preprint 0808.1942 )
[55] de Haro S 2009 JHEP01042 (Preprint 0808.2054 )
[56] Stevens K A, Schleich K and Witt D M 2009 Class. Quant. Grav. 26075012 ( Preprint 0809.3022 )
[57] Deser S 2009 Fundamental Interactions - A Memorial Volume for Wolfgang K ummer(World Scientific)
(Preprint 0810.5384 )
[58] Hotta K, Hyakutake Y, Kubota T, Nishinaka T and Tanida H 2 009JHEP01010 (Preprint 0811.0910 )
[59] Quevedo H and Sanchez A 2009 Phys. Rev. D79024012 ( Preprint 0811.2524 )
[60] Oh J J and Kim W 2009 JHEP01067 (Preprint 0811.2632 )
[61] Garbarz A, Giribet G and Vasquez Y 2009 Phys. Rev. D79044036 ( Preprint 0811.4464 )
[62] Kim W and Son E J 2009 Phys. Lett. B67390–94 (Preprint 0812.0876 )
[63] Mann R B, Oh J J and Park M I 2009 Phys. Rev. D79064005 ( Preprint 0812.2297 )
[64] Blagojevic M and Cvetkovic B 2009 JHEP05073 (Preprint 0812.4742 )
[65] Nam S and Park J D 2009 Class. Quant. Grav. 26145015 ( Preprint 0902.0982 )
[66] Hellerman S 2009 ( Preprint 0902.2790 )
[67] Sezgin E and Tanii Y 2009 Class. Quant. Grav. 26235005 ( Preprint 0903.3779 )
[68] Anninos D, Esole M and Guica M 2009 JHEP10083 (Preprint 0905.2612 )
[69] Compere G and Detournay S 2009 JHEP08092 (Preprint 0906.1243 )
[70] Hotta K, Hyakutake Y, Kubota T, Nishinaka T and Tanida H 2 009Phys. Lett. B680279–285 ( Preprint
0906.1255 )
[71] Anninos D 2009 ( Preprint 0906.1819 )
[72] Carlip S 2009 JHEP09083 (Preprint 0906.2384 )
[73] Chow D D K, Pope C N and Sezgin E 2009 ( Preprint 0906.3559 )
[74] Becker M, Bruillard P and Downes S 2009 JHEP10004 (Preprint 0906.4822 )
[75] Blagojevic M and Cvetkovic B 2009 JHEP09006 (Preprint 0907.0950 )
[76] Vasquez Y 2009 ( Preprint 0907.4165 )
[77] Duncan J F and Frenkel I B 2009 ( Preprint 0907.4529 )
[78] Andrade T and Marolf D 2009 ( Preprint 0909.0727 )
[79] Miskovic O and Olea R 2009 JHEP12046 (Preprint 0909.2275 )
[80] Skenderis K, Taylor M and van Rees B C 2009 ( Preprint 0909.5617 )
[81] Afshar H R, Alishahiha M and Naseh A 2009 ( Preprint 0910.4350 )
[82] Gaberdiel M R and Kausch H G 1996 Phys. Lett. B386131–137 ( Preprint hep-th/9606050 )
[83] Binder K and Young A P 1986 Rev. Mod. Phys. 58801–976
[84] Cardy J L 1999 ( Preprint cond-mat/9911024 )
[85] Reza Rahimi Tabar M 2000 Nucl. Phys. B588630–637 ( Preprint cond-mat/0002309 )
[86] Gurarie V and Ludwig A W W 2002 J. Phys. A35L377–L384 ( Preprint cond-mat/9911392 )
[87] Zee A 2003 Quantum field theory in a nutshell (Princeton University Press)
[88] Bernard D 1995 ( Preprint hep-th/9509137 )
[89] Fujita M, Hikida Y, Ryu S and Takayanagi T 2008 JHEP12065 (Preprint 0810.5394 )
[90] Kiritsis E and Niarchos V 2009 Nucl. Phys. B812488–524 ( Preprint 0808.3410 )
[91] Myers R C and Wapler M C 2008 JHEP12115 (Preprint 0811.0480 )