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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

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