alx-ai/arxiv_papers · Datasets at Hugging Face

text stringlengths 0 44.4k
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

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