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text stringlengths 0 44.4k
=/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

=/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