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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

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