Three-dimensional gravity and logarithmic CFT Nordic Network Meeting, G¨oteborg

Three-dimensional gravity and logarithmic CFT

Nordic Network Meeting, G¨oteborg

Niklas Johansson

Vienna University of Technology

October 22 2010

Outline

Logarithmic CFT Lightning review LCFTs as limits Gravity duals

Early hints

Candidate theories Tests of the conjectures

TMG NMG GMG

Logarithmic CFT

• LCFTs ([Gurarie ’93]) are non-unitary CFTs.

• Useful for describing: systems with quenched disorder.

• (E.g., spin glasses, quenced random magnets, self-avoiding polymers and percolation.) [Cardy, Gurarie, Ludwig, Tabar...]

• Challenging to describe!

• Defining feature: Two operators O andO^log, have degenerate conformal weights and form alogarithmic pair:

H O^log

O

=

E 1 0 E

O^log O

J

O^log O

= j 0

0 j

O^log O

• Reviews: ^[Flohr] hep-th/0111228,[Gaberdiel]hep-th/0111260

Logarithmic CFT

• LCFTs ([Gurarie ’93]) are non-unitary CFTs.

• Useful for describing: systems with quenched disorder.

• (E.g., spin glasses, quenced random magnets, self-avoiding polymers and percolation.) [Cardy, Gurarie, Ludwig, Tabar...]

• Challenging to describe!

• Defining feature: Two operators O andO^log, have degenerate conformal weights and form alogarithmic pair:

H O^log

O

=

E 1 0 E

O^log O

J

O^log O

= j 0

0 j

O^log O

• Reviews: ^[Flohr] hep-th/0111228,[Gaberdiel]hep-th/0111260

Logarithmic CFT

• LCFTs ([Gurarie ’93]) are non-unitary CFTs.

• Useful for describing: systems with quenched disorder.

• (E.g., spin glasses, quenced random magnets, self-avoiding polymers and percolation.) [Cardy, Gurarie, Ludwig, Tabar...]

• Challenging to describe!

• Defining feature: Two operators O andO^log, have degenerate conformal weights and form alogarithmic pair:

H O^log

O

=

E 1 0 E

O^log O

J

O^log O

= j 0

0 j

O^log O

• Reviews: ^[Flohr] hep-th/0111228,[Gaberdiel]hep-th/0111260

Logarithmic CFT

• LCFTs ([Gurarie ’93]) are non-unitary CFTs.

• Useful for describing: systems with quenched disorder.

• (E.g., spin glasses, quenced random magnets, self-avoiding polymers and percolation.) [Cardy, Gurarie, Ludwig, Tabar...]

• Challenging to describe!

• Defining feature: Two operators O andO^log, have degenerate conformal weights and form alogarithmic pair:

H O^log

O

=

E 1 0 E

O^log O

J

O^log O

= j 0

0 j

O^log O

• Reviews: ^[Flohr] hep-th/0111228,[Gaberdiel]hep-th/0111260

Logarithmic CFT

• LCFTs ([Gurarie ’93]) are non-unitary CFTs.

• Useful for describing: systems with quenched disorder.

• (E.g., spin glasses, quenced random magnets, self-avoiding polymers and percolation.) [Cardy, Gurarie, Ludwig, Tabar...]

• Challenging to describe!

• Defining feature: Two operators O andO^log, have degenerate conformal weights and form alogarithmic pair:

H O^log

O

=

E 1 0 E

O^log O

J

O^log O

= j 0

0 j

O^log O

• Reviews: ^[Flohr] hep-th/0111228,[Gaberdiel]hep-th/0111260

Logarithmic CFT

• LCFTs ([Gurarie ’93]) are non-unitary CFTs.

• Useful for describing: systems with quenched disorder.

• (E.g., spin glasses, quenced random magnets, self-avoiding polymers and percolation.) [Cardy, Gurarie, Ludwig, Tabar...]

• Challenging to describe!

• Defining feature: Two operators O andO^log, have degenerate conformal weights and form alogarithmic pair:

H O^log

O

=

E 1 0 E

O^log O

J

O^log O

= j 0

0 j

O^log O

• Reviews: ^[Flohr] hep-th/0111228,[Gaberdiel]hep-th/0111260

Logarithmic CFT

• LCFTs ([Gurarie ’93]) are non-unitary CFTs.

• Useful for describing: systems with quenched disorder.

• (E.g., spin glasses, quenced random magnets, self-avoiding polymers and percolation.) [Cardy, Gurarie, Ludwig, Tabar...]

• Challenging to describe!

• Defining feature: Two operators O andO^log, have degenerate conformal weights and form alogarithmic pair:

H O^log

O

=

E 1 0 E

O^log O

J

O^log O

= j 0

0 j

O^log O

• Reviews: ^[Flohr] hep-th/0111228,[Gaberdiel]hep-th/0111260

Logarithmic CFTs

...correlators

• There are logs in the correlators!

• One state is zero norm!

• O can be O^L ≡Tzz(z). Then c_L= 0.

hO(z)O(0)i= 0 2z^2h hO(z)O^log(0,0)i= b

2z^2h

hO^log(z,z¯)O^log(0,0)i=−bln (m_L²|z|²) z^2h

Logarithmic CFTs

...correlators

• There are logs in the correlators!

• One state is zero norm!

• O can be O^L ≡Tzz(z). Then c_L= 0.

hO(z)O(0)i= 0 2z^2h hO(z)O^log(0,0)i= b

2z^2h

hO^log(z,z¯)O^log(0,0)i=−bln (m²_L|z|²) z^2h

Logarithmic CFTs

...correlators

• There are logs in the correlators!

• One state is zeronorm!

• O can be O^L ≡Tzz(z). Then c_L= 0.

hO(z)O(0)i= 0 2z^2h hO(z)O^log(0,0)i= b

2z^2h

hO^log(z,z¯)O^log(0,0)i=−bln (m²_L|z|²) z^2h

Logarithmic CFTs

...correlators

• There are logs in the correlators!

• One state is zeronorm!

• O can be O^L ≡Tzz(z). Then c_L= 0.

hO(z)O(0)i= 0 2z^2h hO(z)O^log(0,0)i= b

2z^2h

hO^log(z,z¯)O^log(0,0)i=−bln (m²_L|z|²) z^2h

Logarithmic CFTs

...as limits

• Consider a 1-parameter () family of CFTs.

• Suppose two operators O^A/B() coincide at = 0: O^A(0) =O^B(0).

• The weights satisfy ∆_AB =h^A()−h^B() =.

hO^I(z)O^J(0,0)i= δ^IJc_I() 2z^2hI()

• If thec_I vanish as c_A ∼Bandc_B ∼ −Bwe can define O^log ≡ lim

∆AB→0

O^A− O^B

∆_AB

Logarithmic CFTs

...as limits

• Consider a 1-parameter () family of CFTs.

• Suppose two operators O^A/B() coincide at = 0:

O^A(0) =O^B(0).

• The weights satisfy ∆_AB =h^A()−h^B() =.

hO^I(z)O^J(0,0)i= δ^IJc_I() 2z^2hI()

• If thec_I vanish as c_A ∼Bandc_B ∼ −Bwe can define O^log ≡ lim

∆AB→0

O^A− O^B

∆_AB

Logarithmic CFTs

...as limits

• Consider a 1-parameter () family of CFTs.

• Suppose two operators O^A/B() coincide at = 0:

O^A(0) =O^B(0).

• The weights satisfy ∆_AB =h^A()−h^B() =.

hO^I(z)O^J(0,0)i= δ^IJc_I() 2z^2hI()

• If thec_I vanish as c_A ∼Bandc_B ∼ −Bwe can define O^log ≡ lim

∆AB→0

O^A− O^B

∆_AB

Logarithmic CFTs

...as limits

• Consider a 1-parameter () family of CFTs.

• Suppose two operators O^A/B() coincide at = 0:

O^A(0) =O^B(0).

• The weights satisfy ∆_AB =h^A()−h^B() =.

hO^I(z)O^J(0,0)i= δ^IJc_I() 2z^2hI()

• If thec_I vanish as c_A ∼Bandc_B ∼ −Bwe can define O^log ≡ lim

∆AB→0

O^A− O^B

∆_AB

Logarithmic CFTs

...as limits

• Consider a 1-parameter () family of CFTs.

• Suppose two operators O^A/B() coincide at = 0:

O^A(0) =O^B(0).

• The weights satisfy ∆_AB =h^A()−h^B() =.

hO^I(z)O^J(0,0)i= δ^IJc_I() 2z^2hI()

• If thec_I vanish asc_A ∼Bandc_B ∼ −Bwe can define

O^log ≡ lim

∆AB→0

O^A− O^B

∆_AB

Logarithmic CFTs

...as limits

• Consider a 1-parameter () family of CFTs.

• Suppose two operators O^A/B() coincide at = 0:

O^A(0) =O^B(0).

• The weights satisfy ∆_AB =h^A()−h^B() =.

hO^I(z)O^J(0,0)i= δ^IJc_I() 2z^2hI()

• If thec_I vanish asc_A ∼Bandc_B ∼ −Bwe can define O^log ≡ lim

∆AB→0

O^A− O^B

∆_AB

Logarithmic CFTs

...as limits

• O^A|₌₀=O^B|₌₀ and ∆_AB =h^A()−h^B().

• The c_I vanish as c_A ∼B∆_AB andc_B ∼ −B∆_AB

O^log ≡ lim

∆AB→0

O^A− O^B

∆_AB

• In the limit we get the non-zero correlators hO^A(z)O^log(0,0)i= B

2z^2h(0)

hO^log(z,¯z)O^log(0,0)i=−Bln (m²_L|z|²) z^2h(0)

• Like an LCFT!

Logarithmic CFTs

...as limits

• O^A|₌₀=O^B|₌₀ and ∆_AB =h^A()−h^B().

• The c_I vanish as c_A ∼B∆_AB andc_B ∼ −B∆_AB

O^log ≡ lim

∆AB→0

O^A− O^B

∆_AB

• In the limit we get the non-zero correlators hO^A(z)O^log(0,0)i= B

2z^2h(0)

hO^log(z,¯z)O^log(0,0)i=−Bln (m²_L|z|²) z^2h(0)

• Like an LCFT!

Gravity duals to LCFTs

...early hints

• Early ’08, Li, Song and Stromingerstudied a 3-dimensional gravity theory ([Deser, Jackiw, Templeton ’82])

S_TMG =S_EH+ 1

`²S_CC+ 1 µS_CS

• Linearized around AdS3: gµν =g_µν^AdS+ Re(ψµν) (D^LD^RD^mψ)_µν = 0 with

D^L/R =∂/±1

` D^m=∂/+µ

D^mψ^m = 0 is a massive graviton. ψ^L/R are “pure gauge” ∼Tzz/T¯z¯z.

Gravity duals to LCFTs

...early hints

• Early ’08, Li, Song and Stromingerstudied a 3-dimensional gravity theory ([Deser, Jackiw, Templeton ’82])

S_TMG=S_EH+ 1

`²S_CC+ 1 µS_CS

• Linearized around AdS3: gµν =g_µν^AdS+ Re(ψµν) (D^LD^RD^mψ)_µν = 0 with

D^L/R =∂/±1

` D^m=∂/+µ

D^mψ^m = 0 is a massive graviton. ψ^L/R are “pure gauge” ∼Tzz/T¯z¯z.

Gravity duals to LCFTs

...early hints

• Early ’08, Li, Song and Stromingerstudied a 3-dimensional gravity theory ([Deser, Jackiw, Templeton ’82])

S_TMG=S_EH+ 1

`²S_CC+ 1 µS_CS

• Linearized around AdS3: gµν=g_µν^AdS+ Re(ψµν)

(D^LD^RD^mψ)_µν = 0 with

D^L/R =∂/±1

` D^m=∂/+µ

D^mψ^m = 0 is a massive graviton. ψ^L/R are “pure gauge” ∼Tzz/T¯z¯z.

Gravity duals to LCFTs

...early hints

• Early ’08, Li, Song and Stromingerstudied a 3-dimensional gravity theory ([Deser, Jackiw, Templeton ’82])

S_TMG=S_EH+ 1

`²S_CC+ 1 µS_CS

• Linearized around AdS3: gµν=g_µν^AdS+ Re(ψµν) (D^LD^RD^mψ)_µν = 0

with

D^L/R =∂/±1

` D^m=∂/+µ

D^mψ^m = 0 is a massive graviton. ψ^L/R are “pure gauge” ∼Tzz/T¯z¯z.

Gravity duals to LCFTs

...early hints

• Early ’08, Li, Song and Stromingerstudied a 3-dimensional gravity theory ([Deser, Jackiw, Templeton ’82])

S_TMG=S_EH+ 1

`²S_CC+ 1 µS_CS

• Linearized around AdS3: gµν=g_µν^AdS+ Re(ψµν) (D^LD^RD^mψ)_µν = 0 with

D^L/R =∂/±1

` D^m=∂/+µ

D^mψ^m = 0 is a massive graviton. ψ^L/R are “pure gauge” ∼T /T .

Gravity duals to LCFTs

...early hints

• At the tuning µ`= 1

◦ D^L=D^m =⇒ ψ^L=ψ^m

◦ hψ^Lψ^Li ∼cL→0

• LSS conjectured: ψ^L =ψ^m pure gauge! Chiral, unitary CFT dual!

• The construction [Grumiller, NJ ’08]

ψ^log = lim

µ`→1

ψ^m−ψ^L µ`−1 =⇒

H ψ^log

O

= 2 1

0 2

ψ^log ψ^L

J

ψ^log ψ^L

= 2 0

0 2

ψ^log ψ^L

• Conjecture: µ`= 1 is dual to an LCFT!

Gravity duals to LCFTs

...early hints

• At the tuning µ`= 1

◦ D^L=D^m =⇒ ψ^L=ψ^m

◦ hψ^Lψ^Li ∼cL→0

• LSS conjectured: ψ^L =ψ^m pure gauge! Chiral, unitary CFT dual!

• The construction [Grumiller, NJ ’08]

ψ^log = lim

µ`→1

ψ^m−ψ^L µ`−1 =⇒

H ψ^log

O

= 2 1

0 2

ψ^log ψ^L

J

ψ^log ψ^L

= 2 0

0 2

ψ^log ψ^L

• Conjecture: µ`= 1 is dual to an LCFT!

Gravity duals to LCFTs

...early hints

• At the tuning µ`= 1

◦ D^L=D^m =⇒ ψ^L=ψ^m

◦ hψ^Lψ^Li ∼cL→0

• LSS conjectured: ψ^L =ψ^m pure gauge! Chiral, unitary CFT dual!

• The construction [Grumiller, NJ ’08]

ψ^log = lim

µ`→1

ψ^m−ψ^L µ`−1 =⇒

H ψ^log

O

= 2 1

0 2

ψ^log ψ^L

J

ψ^log ψ^L

= 2 0

0 2

ψ^log ψ^L

• Conjecture: µ`= 1 is dual to an LCFT!

Gravity duals to LCFTs

...early hints

• At the tuning µ`= 1

◦ D^L=D^m =⇒ ψ^L=ψ^m

◦ hψ^Lψ^Li ∼cL→0

• LSS conjectured: ψ^L =ψ^m pure gauge! Chiral, unitary CFT dual!

• The construction [Grumiller, NJ ’08]

ψ^log = lim

µ`→1

ψ^m−ψ^L µ`−1 =⇒

H ψ^log

O

= 2 1

0 2

ψ^log ψ^L

J

ψ^log ψ^L

= 2 0

0 2

ψ^log ψ^L

• Conjecture: µ`= 1 is dual to an LCFT!

Gravity duals to LCFTs

...early hints

• At the tuning µ`= 1

◦ D^L=D^m =⇒ ψ^L=ψ^m

◦ hψ^Lψ^Li ∼cL→0

• LSS conjectured: ψ^L =ψ^m pure gauge! Chiral, unitary CFT dual!

• The construction [Grumiller, NJ ’08]

ψ^log = lim

µ`→1

ψ^m−ψ^L µ`−1 =⇒

H ψ^log

O

= 2 1

0 2

ψ^log ψ^L

J

ψ^log ψ^L

= 2 0

0 2

ψ^log ψ^L

• Conjecture: µ`= 1 is dual to an LCFT!

Gravity duals to LCFTs

...early hints

• At the tuning µ`= 1

◦ D^L=D^m =⇒ ψ^L=ψ^m

◦ hψ^Lψ^Li ∼cL→0

• LSS conjectured: ψ^L =ψ^m pure gauge! Chiral, unitary CFT dual!

• The construction [Grumiller, NJ ’08]

ψ^log = lim

µ`→1

ψ^m−ψ^L µ`−1 =⇒

H ψ^log

O

= 2 1

0 2

ψ^log ψ^L

J

ψ^log ψ^L

= 2 0

0 2

ψ^log ψ^L

• Conjecture: µ`= 1 is dual to an LCFT!

Gravity duals to LCFTs

...early hints

• At the tuning µ`= 1

◦ D^L=D^m =⇒ ψ^L=ψ^m

◦ hψ^Lψ^Li ∼cL→0

• LSS conjectured: ψ^L =ψ^m pure gauge! Chiral, unitary CFT dual!

• The construction [Grumiller, NJ ’08]

ψ^log = lim

µ`→1

ψ^m−ψ^L µ`−1 =⇒

H ψ^log

O

= 2 1

0 2

ψ^log ψ^L

J

ψ^log ψ^L

= 2 0

0 2

ψ^log ψ^L

• Conjecture: µ`= 1 is dual to an LCFT!

Gravity duals to LCFTs

...early hints

• At the tuning µ`= 1

◦ D^L=D^m =⇒ ψ^L=ψ^m

◦ hψ^Lψ^Li ∼cL→0

• LSS conjectured: ψ^L =ψ^m pure gauge! Chiral, unitary CFT dual!

• The construction [Grumiller, NJ ’08]

ψ^log = lim

µ`→1

ψ^m−ψ^L µ`−1 =⇒

H ψ^log

O

= 2 1

0 2

ψ^log ψ^L

J

ψ^log ψ^L

= 2 0

0 2

ψ^log ψ^L

Gravity duals to LCFTs

...candidate theories

• Note that to have a log partner to T_zz (many applications), a metric mode must degenerate. Higher curvature theories!

• Many interesting in 3d!

• New Massive Gravity,[Bergshoeff, Hohm, Townsend ’09]

SNMG = 1 κ²

Z d³x√

−gh

σR+ 1

m² R^µνRµν−3 8R²

−2λm² i

• Add _µ¹S_CS: GeneralisedMassiveGravity. Around AdS3:

(D^LD^RD^m1D^m2ψ)_µν = 0 m_1,2`= m<sup>2</sup>`² 2µ` ±

s 1

2 −σm²`<sup>2</sup>+ m<sup>4</sup>`⁴ 4µ²`².

Gravity duals to LCFTs

...candidate theories

• Note that to have a log partner to T_zz (many applications), a metric mode must degenerate. Higher curvature theories!

• Many interesting in 3d!

• New Massive Gravity,[Bergshoeff, Hohm, Townsend ’09]

SNMG = 1 κ²

Z d³x√

−gh

σR+ 1

m² R^µνRµν−3 8R²

−2λm² i

• Add _µ¹S_CS: GeneralisedMassiveGravity. Around AdS3:

(D^LD^RD^m1D^m2ψ)_µν = 0 m_1,2`= m<sup>2</sup>`² 2µ` ±

s 1

2 −σm²`<sup>2</sup>+ m<sup>4</sup>`⁴ 4µ²`².

Gravity duals to LCFTs

...candidate theories

• Note that to have a log partner to T_zz (many applications), a metric mode must degenerate. Higher curvature theories!

• Many interesting in 3d!

• New Massive Gravity,[Bergshoeff, Hohm, Townsend ’09]

SNMG = 1 κ²

Z d³x√

−gh

σR+ 1

m² R^µνRµν−3 8R²

−2λm² i

• Add _µ¹S_CS: GeneralisedMassiveGravity. Around AdS3:

(D^LD^RD^m1D^m2ψ)_µν = 0 m_1,2`= m<sup>2</sup>`² 2µ` ±

s 1

2 −σm²`<sup>2</sup>+ m<sup>4</sup>`⁴ 4µ²`².

GMG parameter space

• D^m1 =D^L:

Tzz has log-partner!

• D^m1 =D^m2:

O^M has log-partner!

• D^m1 =D^m2=D^L: Rank 3 Jordan cell!

• c_L=c_R = 0: log-NMG Both Tzz andT¯z¯z logged!

• PMG! c_L=c_R 6= 0

Enhanced gauge symmetry!

6 m₂`

-m₁` c_L= 0 c_L = 0

c_R = 0

c_R = 0 m1=m2

es

es v

v u

GMG parameter space

• D^m1 =D^L:

Tzz has log-partner!

• D^m1 =D^m2:

O^M has log-partner!

• D^m1 =D^m2=D^L: Rank 3 Jordan cell!

• c_L=c_R = 0: log-NMG Both Tzz andT¯z¯z logged!

• PMG! c_L=c_R 6= 0

Enhanced gauge symmetry!

6 m₂`

-m₁` c_L= 0 c_L = 0

c_R = 0

c_R = 0 m1=m2

NMG es

es v

v u

GMG parameter space

• D^m1 =D^L:

Tzz has log-partner!

• D^m1 =D^m2:

O^M has log-partner!

• D^m1 =D^m2=D^L: Rank 3 Jordan cell!

• c_L=c_R = 0: log-NMG Both Tzz andT¯z¯z logged!

• PMG! c_L=c_R 6= 0

Enhanced gauge symmetry!

6 m₂`

-m₁` c_L= 0 c_L = 0

c_R = 0

c_R = 0 m1=m2

es

es v

v u

GMG parameter space

• D^m1 =D^L:

Tzz has log-partner!

• D^m1 =D^m2:

O^M has log-partner!

• D^m1 =D^m2=D^L: Rank 3 Jordan cell!

• c_L=c_R = 0: log-NMG Both Tzz andT¯z¯z logged!

• PMG! c_L=c_R 6= 0

Enhanced gauge symmetry!

6 m₂`

-m₁` c_L= 0 c_L = 0

c_R = 0

c_R = 0 m1=m2

NMG es

es v

v u

GMG parameter space

• D^m1 =D^L:

Tzz has log-partner!

• D^m1 =D^m2:

O^M has log-partner!

• D^m1 =D^m2=D^L: Rank 3 Jordan cell!

• c_L=c_R = 0: log-NMG Both Tzz andT¯z¯z logged!

• PMG! c_L=c_R 6= 0

Enhanced gauge symmetry!

6 m₂`

-m₁` c_L= 0 c_L = 0

c_R = 0

c_R = 0 m1=m2

es

es v

v u

Test of the conjectures

...TMG

• c_L= _2G<sup>3`</sup>(1−1/µ`)→0 and ψ^m →ψ^L [Li, Song, Strominger ’08]

• ψ^log is a valid solution. Jordan block! [Grumiller, NJ ’08]

• ψ^log = propagating dof also nonlinearly. [Grumiller, Jackiw, NJ ’08] [Carlip ’08]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Grumiller, NJ ’08]

• Correlators!

hψ^logψ^logi=δ⁽²⁾Sgrav(ψ^log, ψ^log) =−bln (m_L²|z|²) z^2h The ‘new anomaly’b =−^3`_G. [Skenderis, Taylor, v Rees ’09] [Grumiller, Sachs ’09]

• 3-point correlator also match! [Grumiller, Sachs ’09]

• 1-loop partition function consistent with LCFT. [Gaberdiel, Grumiller, Sachs ’10]

Test of the conjectures

...TMG

• c_L= _2G<sup>3`</sup>(1−1/µ`)→0 and ψ^m →ψ^L [Li, Song, Strominger ’08]

• ψ^log is a valid solution. Jordan block! [Grumiller, NJ ’08]

• ψ^log = propagating dof also nonlinearly. [Grumiller, Jackiw, NJ ’08] [Carlip ’08]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Grumiller, NJ ’08]

• Correlators!

hψ^logψ^logi=δ⁽²⁾Sgrav(ψ^log, ψ^log) =−bln (m_L²|z|²) z^2h The ‘new anomaly’b =−^3`_G. [Skenderis, Taylor, v Rees ’09] [Grumiller, Sachs ’09]

• 3-point correlator also match! [Grumiller, Sachs ’09]

• 1-loop partition function consistent with LCFT. [Gaberdiel, Grumiller, Sachs ’10]

Test of the conjectures

...TMG

• c_L= _2G<sup>3`</sup>(1−1/µ`)→0 and ψ^m →ψ^L [Li, Song, Strominger ’08]

• ψ^log is a valid solution. Jordan block! [Grumiller, NJ ’08]

• ψ^log = propagating dof also nonlinearly. [Grumiller, Jackiw, NJ ’08] [Carlip ’08]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Grumiller, NJ ’08]

• Correlators!

hψ^logψ^logi=δ⁽²⁾Sgrav(ψ^log, ψ^log) =−bln (m_L²|z|²) z^2h The ‘new anomaly’b =−^3`_G. [Skenderis, Taylor, v Rees ’09] [Grumiller, Sachs ’09]

• 3-point correlator also match! [Grumiller, Sachs ’09]

• 1-loop partition function consistent with LCFT. [Gaberdiel, Grumiller, Sachs ’10]

Test of the conjectures

...TMG

• c_L= _2G<sup>3`</sup>(1−1/µ`)→0 and ψ^m →ψ^L [Li, Song, Strominger ’08]

• ψ^log is a valid solution. Jordan block! [Grumiller, NJ ’08]

• ψ^log = propagating dof also nonlinearly. [Grumiller, Jackiw, NJ ’08] [Carlip ’08]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Grumiller, NJ ’08]

• Correlators!

hψ^logψ^logi=δ⁽²⁾Sgrav(ψ^log, ψ^log) =−bln (m_L²|z|²) z^2h The ‘new anomaly’b =−^3`_G. [Skenderis, Taylor, v Rees ’09] [Grumiller, Sachs ’09]

• 3-point correlator also match! [Grumiller, Sachs ’09]

• 1-loop partition function consistent with LCFT. [Gaberdiel, Grumiller, Sachs ’10]

Test of the conjectures

...TMG

• c_L= _2G<sup>3`</sup>(1−1/µ`)→0 and ψ^m →ψ^L [Li, Song, Strominger ’08]

• ψ^log is a valid solution. Jordan block! [Grumiller, NJ ’08]

• ψ^log = propagating dof also nonlinearly. [Grumiller, Jackiw, NJ ’08] [Carlip ’08]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Grumiller, NJ ’08]

• Correlators!

hψ^logψ^logi=δ⁽²⁾Sgrav(ψ^log, ψ^log) =−bln (m_L²|z|²) z^2h The ‘new anomaly’b =−^3`_G. [Skenderis, Taylor, v Rees ’09] [Grumiller, Sachs ’09]

• 3-point correlator also match! [Grumiller, Sachs ’09]

• 1-loop partition function consistent with LCFT. [Gaberdiel, Grumiller, Sachs ’10]

Test of the conjectures

...TMG

• c_L= _2G<sup>3`</sup>(1−1/µ`)→0 and ψ^m →ψ^L [Li, Song, Strominger ’08]

• ψ^log is a valid solution. Jordan block! [Grumiller, NJ ’08]

• ψ^log = propagating dof also nonlinearly. [Grumiller, Jackiw, NJ ’08] [Carlip ’08]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Grumiller, NJ ’08]

• Correlators!

hψ^logψ^logi=δ⁽²⁾Sgrav(ψ^log, ψ^log) =−bln (m_L²|z|²) z^2h The ‘new anomaly’b =−^3`_G. [Skenderis, Taylor, v Rees ’09] [Grumiller, Sachs ’09]

• 3-point correlator also match! [Grumiller, Sachs ’09]

• 1-loop partition function consistent with LCFT. [Gaberdiel, Grumiller, Sachs ’10]

Test of the conjectures

...TMG

• c_L= _2G<sup>3`</sup>(1−1/µ`)→0 and ψ^m →ψ^L [Li, Song, Strominger ’08]

• ψ^log is a valid solution. Jordan block! [Grumiller, NJ ’08]

• ψ^log = propagating dof also nonlinearly. [Grumiller, Jackiw, NJ ’08] [Carlip ’08]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Grumiller, NJ ’08]

• Correlators!

hψ^logψ^logi=δ⁽²⁾Sgrav(ψ^log, ψ^log) =−bln (m_L²|z|²) z^2h The ‘new anomaly’b =−^3`_G. [Skenderis, Taylor, v Rees ’09] [Grumiller, Sachs ’09]

• 3-point correlator also match! [Grumiller, Sachs ’09]

• 1-loop partition function consistent with LCFT. [Gaberdiel, Grumiller, Sachs ’10]

GMG parameter space

• D^m1 =D^L:

Tzz has log-partner!

• D^m1 =D^m2:

O^M has log-partner!

• D^m1 =D^m2=D^L: Rank 3 Jordan cell!

• c_L=c_R = 0: log-NMG Both Tzz andT¯z¯z logged!

• PMG! c_L=c_R 6= 0

Enhanced gauge symmetry!

6 m₂`

-m₁` c_L= 0 c_L = 0

c_R = 0

c_R = 0 m1=m2

es

es v

v u

GMG parameter space

• D^m1 =D^L:

Tzz has log-partner!

• D^m1 =D^m2:

O^M has log-partner!

• D^m1 =D^m2=D^L: Rank 3 Jordan cell!

• c_L=c_R = 0: log-NMG Both Tzz andT¯z¯z logged!

• PMG! c_L=c_R 6= 0

Enhanced gauge symmetry!

6 m₂`

-m₁` c_L= 0 c_L = 0

c_R = 0

c_R = 0 m1=m2

NMG es

es v

v u

Test of the conjectures

...NMG

• Two limits: ψ^m1/m2 →ψ^L/R or ψ^m1 →ψ^m2→ψ⁰

• c_L/R = _2G<sup>3`</sup>(σ+<sub>2m</sub><sup>1</sup>2`²)→0. [Bergshoeff, Hohm, Townsend ’09]

• ψ^m1,2 →ψ^L/R,ψ^log is a valid solution. Jordan block! [Liu, Sun ’09]

• ψ^log = propagating dof also nonlinearly. [Blagojevi´c, Cvetkovi´c ’10]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Liu, Sun ’09]

• 2-point correlators match! [Grumiller, Hohm ’09],[Alishahiha, Naseh ’10]

• c_L/R 6= 0! hψ^m1ψ^m1i ∼ −hψ^m2ψ^m2i ∼ ±c. Jordan block, new (consistent) BCs! [Oliva, Tempo, Troncoso ’09]

• Actually this theory has many interesting features already logless.

[Bergshoeff, Hohm, Townsend ’09]

Test of the conjectures

...NMG

• Two limits: ψ^m1/m2 →ψ^L/R or ψ^m1 →ψ^m2→ψ⁰

• c_L/R = _2G<sup>3`</sup>(σ+<sub>2m</sub><sup>1</sup>2`²)→0. [Bergshoeff, Hohm, Townsend ’09]

• ψ^m1,2 →ψ^L/R,ψ^log is a valid solution. Jordan block! [Liu, Sun ’09]

• ψ^log = propagating dof also nonlinearly. [Blagojevi´c, Cvetkovi´c ’10]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Liu, Sun ’09]

• 2-point correlators match! [Grumiller, Hohm ’09],[Alishahiha, Naseh ’10]

• c_L/R 6= 0! hψ^m1ψ^m1i ∼ −hψ^m2ψ^m2i ∼ ±c. Jordan block, new (consistent) BCs! [Oliva, Tempo, Troncoso ’09]

• Actually this theory has many interesting features already logless.

[Bergshoeff, Hohm, Townsend ’09]

Test of the conjectures

...NMG

• Two limits: ψ^m1/m2 →ψ^L/R or ψ^m1 →ψ^m2→ψ⁰

• c_L/R = _2G<sup>3`</sup>(σ+<sub>2m</sub><sup>1</sup>2`²)→0. [Bergshoeff, Hohm, Townsend ’09]

• ψ^m1,2 →ψ^L/R,ψ^log is a valid solution. Jordan block! [Liu, Sun ’09]

• ψ^log = propagating dof also nonlinearly. [Blagojevi´c, Cvetkovi´c ’10]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Liu, Sun ’09]

• 2-point correlators match! [Grumiller, Hohm ’09],[Alishahiha, Naseh ’10]

• c_L/R 6= 0! hψ^m1ψ^m1i ∼ −hψ^m2ψ^m2i ∼ ±c. Jordan block, new (consistent) BCs! [Oliva, Tempo, Troncoso ’09]

• Actually this theory has many interesting features already logless.

[Bergshoeff, Hohm, Townsend ’09]

Test of the conjectures

...NMG

• Two limits: ψ^m1/m2 →ψ^L/R or ψ^m1 →ψ^m2→ψ⁰

• c_L/R = _2G<sup>3`</sup>(σ+<sub>2m</sub><sup>1</sup>2`²)→0. [Bergshoeff, Hohm, Townsend ’09]

• ψ^m1,2 →ψ^L/R,ψ^log is a valid solution. Jordan block! [Liu, Sun ’09]

• ψ^log = propagating dof also nonlinearly. [Blagojevi´c, Cvetkovi´c ’10]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Liu, Sun ’09]

• 2-point correlators match! [Grumiller, Hohm ’09],[Alishahiha, Naseh ’10]

• c_L/R 6= 0! hψ^m1ψ^m1i ∼ −hψ^m2ψ^m2i ∼ ±c. Jordan block, new (consistent) BCs! [Oliva, Tempo, Troncoso ’09]

• Actually this theory has many interesting features already logless.

[Bergshoeff, Hohm, Townsend ’09]

Test of the conjectures

...NMG

• Two limits: ψ^m1/m2 →ψ^L/R or ψ^m1 →ψ^m2→ψ⁰

• c_L/R = _2G<sup>3`</sup>(σ+<sub>2m</sub><sup>1</sup>2`²)→0. [Bergshoeff, Hohm, Townsend ’09]

• ψ^m1,2 →ψ^L/R,ψ^log is a valid solution. Jordan block! [Liu, Sun ’09]

• ψ^log = propagating dof also nonlinearly. [Blagojevi´c, Cvetkovi´c ’10]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Liu, Sun ’09]

• 2-point correlators match! [Grumiller, Hohm ’09],[Alishahiha, Naseh ’10]

• c_L/R 6= 0! hψ^m1ψ^m1i ∼ −hψ^m2ψ^m2i ∼ ±c. Jordan block, new (consistent) BCs! [Oliva, Tempo, Troncoso ’09]

• Actually this theory has many interesting features already logless.

[Bergshoeff, Hohm, Townsend ’09]

Test of the conjectures

...NMG

• Two limits: ψ^m1/m2 →ψ^L/R or ψ^m1 →ψ^m2→ψ⁰

• c_L/R = _2G<sup>3`</sup>(σ+<sub>2m</sub><sup>1</sup>2`²)→0. [Bergshoeff, Hohm, Townsend ’09]

• ψ^m1,2 →ψ^L/R,ψ^log is a valid solution. Jordan block! [Liu, Sun ’09]

• ψ^log = propagating dof also nonlinearly. [Blagojevi´c, Cvetkovi´c ’10]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Liu, Sun ’09]

• 2-point correlators match! [Grumiller, Hohm ’09],[Alishahiha, Naseh ’10]

• c_L/R 6= 0! hψ^m1ψ^m1i ∼ −hψ^m2ψ^m2i ∼ ±c. Jordan block, new (consistent) BCs! [Oliva, Tempo, Troncoso ’09]

• Actually this theory has many interesting features already logless.

[Bergshoeff, Hohm, Townsend ’09]

Test of the conjectures

...NMG

• Two limits: ψ^m1/m2 →ψ^L/R or ψ^m1 →ψ^m2→ψ⁰

• c_L/R = _2G<sup>3`</sup>(σ+<sub>2m</sub><sup>1</sup>2`²)→0. [Bergshoeff, Hohm, Townsend ’09]

• ψ^m1,2 →ψ^L/R,ψ^log is a valid solution. Jordan block! [Liu, Sun ’09]

• ψ^log = propagating dof also nonlinearly. [Blagojevi´c, Cvetkovi´c ’10]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Liu, Sun ’09]

• 2-point correlators match! [Grumiller, Hohm ’09],[Alishahiha, Naseh ’10]

• c_L/R 6= 0! hψ^m1ψ^m1i ∼ −hψ^m2ψ^m2i ∼ ±c. Jordan block, new (consistent) BCs! [Oliva, Tempo, Troncoso ’09]

• Actually this theory has many interesting features already logless.

Test of the conjectures

...NMG

• Two limits: ψ^m1/m2 →ψ^L/R or ψ^m1 →ψ^m2→ψ⁰

• c_L/R = _2G<sup>3`</sup>(σ+<sub>2m</sub><sup>1</sup>2`²)→0. [Bergshoeff, Hohm, Townsend ’09]

• ψ^m1,2 →ψ^L/R,ψ^log is a valid solution. Jordan block! [Liu, Sun ’09]

• ψ^log = propagating dof also nonlinearly. [Blagojevi´c, Cvetkovi´c ’10]

• ∃ consistent BCs forψ^log. ASG = Virasoro². [Liu, Sun ’09]

• 2-point correlators match! [Grumiller, Hohm ’09],[Alishahiha, Naseh ’10]

• c_L/R 6= 0! hψ^m1ψ^m1i ∼ −hψ^m2ψ^m2i ∼ ±c. Jordan block, new (consistent) BCs! [Oliva, Tempo, Troncoso ’09]

• Actually this theory has many interesting features already logless.

[Bergshoeff, Hohm, Townsend ’09]