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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)

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)