Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

A Gravity Realization of LCFT’s in TwoDimensions and Beyond

Eric Bergshoeff

Groningen University

based on a collaboration with Sjoerd de Haan, Wout Merbis,

Massimo Porrati, Jan Rosseel and Thomas Zojer

College Station, March 13 2012

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Critical Gravity

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Critical Gravity

“Tricritical” Gravity

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Critical Gravity

“Tricritical” Gravity

Conclusions

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Critical Gravity

“Tricritical” Gravity

Conclusions

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Why Higher-Curvature Gravity ?

• Obtain better quantum behaviour

• Test limits of the AdS/CFT correspondence

Problem: Gravity is perturbative non-renormalizable

L ∼ R + a(

Rµνab)2

+ b (Rµν)2 + c R2 :

renormalizable but not unitaryStelle (1977)

massless spin 2 and massive spin 2 have opposite sign !

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Special Cases

• In three dimensions there is no massless spin 2 !

⇒ “3D Topological Massive Gravity”Deser, Jackiw, Templeton (1982)

. “3D New Massive Gravity”Hohm, Townsend + E.B. (2009)

• Degeneracy for special point in parameter space

⇒ “3D Chiral Gravity”Li, Song, Strominger (2008)

. “D ≥ 3 Critical Gravity”Lu and Pope (2011)

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Logarithmic Conformal Field Theories (LCFT’s)

• Einstein mode gets replaced by log mode

• At the same time CFT becomes a rank 2 LCFTGurarie (1993), Flohr (2001), Gaberdiel (2001)

• This talk : Higher degeneracies give rise to higher-rank LCFT’s

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

• Higher-Curvature Gravity theories can be constructed

starting from FP equations and solving for differential

subsidiary conditions

• This requires fields with zero massless degrees of freedom

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Massless Degrees of Freedom

field S ∼

gauge parameters λ1 ∼ λ2 ∼

gauge transformation δ = ∂ +∂

curvature R(S) ∼ ∂

∂

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Zero Massless D.O.F.

“Einstein tensor” G (S) ∼ ⋆ ⋆∂

∂

Requirement : G (S) ∼ ⇒ E.O.M. : G (S) = 0

s = 2 : p + q = D − 1

Example : p = q = 1 ,D = 3 , S ∼

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

“Boosting Up the Derivatives”

Second-order Derivative Generalized FP

Curtright (1980)

(

�−m2)

S = 0 , S tr = 0 , ∂ · S = 0

∂ · S = 0 ⇒ S = G (T )

(

�−m2)

G (T ) = 0 , G (T )tr = 0

Higher-derivative Gauge Theory

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

“Taking the Square Root”

Consider 1-forms in 3D , 3-forms in 7D , etc.

The KG-operator factorizes :(

�−m2)

= D(m)D(−m)

Take the “square root” D(m)S = 0

D(m)G (T ) = 0

Higher-derivative Gauge Theory

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Critical Gravity

“Tricritical” Gravity

Conclusions

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

3D Einstein-Hilbert Gravity

Deser, Jackiw, ’t Hooft (1984)

There are no massless gravitons

Adding higher-derivative terms leads to “massive gravitons”

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Fierz-Pauli

•(

�−m2)

hµν = 0 , ηµν hµν = 0 , ∂µhµν = 0

• LFP = 12 h

µνGµν(h) +12m

2(

hµν hµν − h2)

, h ≡ ηµν hµν

no non-linear extension !

number of propagating modes is 12D(D + 1)− 1− D =

{

5 for 4D2 for 3D

Note : the numbers become 2 (4D) and 0 (3D) for m = 0

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Higher-derivative Extension in 3D

∂µhµν = 0 ⇒ hµν = ǫµαβǫν

γδ∂α∂γhβδ ≡ Gµν(h)

(

�−m2)

G linµν(h) = 0 , R lin(h) = 0

Non-linear generalization : gµν = ηµν + hµν ⇒

L =√−g

[

−R − 1

2m2

(

RµνRµν −3

8R2

)]

“New Massive Gravity” : unitary !

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Mode Analysis

• Take NMG with metric gµν , cosmological constant Λ andcoefficient σ = ±1 in front of R

• lower number of derivatives from 4 to 2 by introducing anauxiliary field fµν

• after linearization and diagonalization the two fields describe amassless spin 2 with coefficient σ = σ − Λ

2m2 and a massivespin 2 with mass M2 = −m2σ

• special cases: 3D NMG and D ≥ 3 “critical gravity” forspecial value of Λ

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Unitary Bulk Region

σ = −1 λ 6= Λ !

c = 0 LCFT at cross-over point λ = 3

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Topological Massive Gravity

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Taking the Square Root

•(

�−m2)

hµν =[

O(m)O(−m)]

µρhρν with

[

O(±m)]

µρ = ǫµ

τρ∂τ ±mδρµ

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Taking the Square Root

•(

�−m2)

hµν =[

O(m)O(−m)]

µρhρν with

[

O(±m)]

µρ = ǫµ

τρ∂τ ±mδρµ

• mhµν = ǫµρσ∂ρhσν :

√FP

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Taking the Square Root

•(

�−m2)

hµν =[

O(m)O(−m)]

µρhρν with

[

O(±m)]

µρ = ǫµ

τρ∂τ ±mδρµ

• mhµν = ǫµρσ∂ρhσν :

√FP

• S = 12

∫

d3x(

ǫµνρhµσ∂ν hρσ +m(hνµhµν − h2)

)

Aragone, Khoudeir (1986)

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

From√FP to Topological Massive Gravity

•√FP : mhµν = ǫµ

ρσ∂ρhσν

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

From√FP to Topological Massive Gravity

•√FP : mhµν = ǫµ

ρσ∂ρhσν

• ∂µhµν = 0 ⇒ hµν = G linµν (h)

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

From√FP to Topological Massive Gravity

•√FP : mhµν = ǫµ

ρσ∂ρhσν

• ∂µhµν = 0 ⇒ hµν = G linµν (h)

• Non-linear generalization : gµν = ηµν + hµν ⇒

L ∼ −√−g R +1

mLLCS : TMG

Deser, Jackiw, Templeton (1982)

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Generalization

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Generalized Massive Gravity (GMG)

S [g ] =1

κ2

∫

d3x√−g

[

σR − 2λm2 +1

m2K

]

+1

µLLCS

m2 = m+m− , µ = − m+m−

m+ −m−

special cases:

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Generalized Massive Gravity (GMG)

S [g ] =1

κ2

∫

d3x√−g

[

σR − 2λm2 +1

m2K

]

+1

µLLCS

m2 = m+m− , µ = − m+m−

m+ −m−

special cases:

• m+ = m− : NMG

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Generalized Massive Gravity (GMG)

S [g ] =1

κ2

∫

d3x√−g

[

σR − 2λm2 +1

m2K

]

+1

µLLCS

m2 = m+m− , µ = − m+m−

m+ −m−

special cases:

• m+ = m− : NMG

• m+ → ∞ : TMG

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Critical Gravity

“Tricritical” Gravity

Conclusions

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

3D critical gravity

At critical point : D(±m) → DL,R or, shortly, L,R

• TMG : Okin ∼ LLR ↔ cL = 0 rank 2 LCFT for left-movers

but cR 6= 0 CFT for right-movers

• NMG : Okin ∼ LLRR ↔ cL = cR = 0 rank 2 LCFT

• GMG : Okin ∼ LLLR ↔ cL = 0 rank 3 LCFT for left-movers

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Critical Gravity beyond 3D

S =1

κ2

∫

dDx√−g

[

σR − 2λm2 +1

m2GµνSµν +

1

m′2LGB

]

Lu and Pope (2011); Deser, Liu, Lu, Pope, Sisman and Tekin (2011)

RµνρσRµνρσ − 4RµνRµν + R2 = W µνρσWµνρσ − 4(D − 3)GµνSµν

after lowering the derivatives and linearization around AdS we find

σ(Λ) ≡ σ − Λ

m2

1

D − 1+ 4

Λ

m′2

(D − 3)(D − 4)

(D − 1)(D − 2)

The critical point is defined by σ(Λcrit) = 0

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Massless and Log Modes

D = 4 : massive − 2 − 1 0 + 1 + 2 ⇒ massless − 2 + 2

Gµν (G(h)) = 0 kµν ≡ Gµν(h) ⇒ Gµν(k) = 0

kµν = 0 trivial solution massless modes

kµν = ∇(µAν) “gauge solutions” “Proca log modes”

k⊥µν 6= ∇(µAν) “non-gauge solutions” “spin 2 log modes”

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Unitarity

〈OiOj〉 ∼(

0 CFT

CFT L

)

i = Einstein, log

< Einstein | Log > 6= 0 ⇒

|S > ≡ | Log > +α| massless > can have either sign ! →

LCFT is non-unitary : non-trivial unitary truncations ?

• away from critical pointLu, Pang, Pope (2011)

• multi-critical points

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Critical Gravity

“Tricritical” Gravity

Conclusions

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Toy Model

Consider r scalars on a fixed AdSD+1 backgound :

S = −1

2

∫

dD+1x√g

r∑

i ,j=1

(Aij∂µφi∂µφj + Bijφiφj )

The field equations are given by

(�−m2)rφr = 0 with r − 1 auxiliary fields φ1 , · · · , φr−1

r = 2 :(

�−m2)

φ2 = φ1 and(

�−m2)

φ1 = 0

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Boundary Conformal Field Theory

rank 3 LCFT : 〈OiOj〉 ∼

0 0 CFT

0 CFT L

CFT L L2

i = Einstein, log, log2

Truncating the log2 modes leads to

〈OiOj〉 ∼(

0 00 CFT

)

i = Einstein, log

with a non-negative scalar product !

• Interactions ?

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

A Gravity Realization ?

Three dimensions : L ∼ Λ + R + R2 + R�R + Rµν�Rµν

Spectrum reduces to one massless and two massive spin two modes

At critical point: Gµν

(

G(G(h)))

= 0

• one tri-critical and three bi-critical points

• positive mass log black holes ?

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Outline

Introduction

Higher-Curvature Gravity

Critical Gravity

“Tricritical” Gravity

Conclusions

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Summary

Critical Gravity ⇔ LCFT

⇓

Higher-Derivative Critical Gravity ⇔ Higher-Rank LCFT

• LCFT is in general non-unitary

Introduction Higher-Curvature Gravity Critical Gravity “Tricritical” Gravity Conclusions

Open Issues

• are non-trivial unitary truncations possible ?

• extend to most general 4D tri-critical gravity

cp. to Nutma (2012), to appear