Flat space higher spin gravity

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Flat space higher spin gravity

Daniel Grumiller

Institute for Theoretical Physics TU Wien

Dutch String meeting, Groningen, February 2015

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Outline

Motivations

Holography basics

Flat space gravity

Flat space higher spin gravity

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Outline

Motivations

Holography basics

Flat space gravity

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General motivations

I Quantum gravity

I Address conceptual issues of quantum gravity

I Black holes (thermodynamics, evaporation, information loss, microstate counting, entanglement entropy, firewalls, ...)

I String theory (is it the right theory? can there be any alternative? ...)

I Holography

I Holographic principle realized in Nature? (yes/no)

I Quantum gravity via AdS/CFT? (define quantum gravity in AdS by constructing/postulating dual CFT)

I How general is holography? (non-unitary holography,higher spin holography,flat space holography, non-AdS holography, ...)

I Applications

I Gauge gravity correspondence (plasmas, condensed matter, ...)

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General motivations

I Quantum gravity

I Holography

I Applications

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General motivations

I Quantum gravity

I Holography

I Applications

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General motivations

I Quantum gravity

I Holography

I Applications

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General motivations

I Quantum gravity

I Holography

I Applications

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General motivations

I Quantum gravity

I Holography

I Applications

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General motivations

I Quantum gravity

I Holography

I Applications

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Specific motivation for 3D

Gravity in 3D is simpler than in higher dimensions

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Specific motivation for 3D

Gravity in 3D is simpler than in higher dimensions

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Specific motivation for flat space higher spin gravity

Massless higher spin theories constrained by no-gos!

I Coleman–Mandula ’67

I Aragone–Deser ’79

I Weinberg–Witten ’80

I recent summary: Bekaert, Boulanger, Sundell ’12

Conclusion: there are no consistent interacting massless higher spin theories in 4- (or higher-) dimensional flat space

Circumventing no-gos:

I Vasiliev ’87-’90: higher spin theories in (A)dS

I Afshar, Bagchi, Fareghbal, DG, Rosseel ’13; Gonzalez, Matulich, Pino, Troncoso ’13: flat space higher spin theories in 3d

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I Vasiliev ’87-’90: higher spin theories in (A)dS

I Afshar, Bagchi, Fareghbal, DG, Rosseel ’13; Gonzalez, Matulich, Pino, Troncoso ’13: flat space higher spin theories in 3d

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Goals of this talk

1. Review general aspects of holography in 3D

2. Discuss flat space holography 3. Generalize to higher spin holography 4. List selected open issues

Address these issues in 3D!

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Goals of this talk

1. Review general aspects of holography in 3D 2. Discuss flat space holography

3. Generalize to higher spin holography 4. List selected open issues

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Goals of this talk

3. Generalize to higher spin holography

4. List selected open issues

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Goals of this talk

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Goals of this talk

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Outline

Motivations

Holography basics

Flat space gravity

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Gravity in 3D

AdS3 gravity

I Lowest dimension with black holes and (off-shell) gravitons

I Weyl= 0, thus Riemann= Ricci

I Einstein gravity: no on-shell gravitons

I Formulation as topological gauge theory (Chern–Simons)

I Dual field theory (if it exists): 2D

I Infinite dimensional asymptotic symmetries (Brown–Henneaux)

I Black holes as orbifolds of AdS3 (BTZ)

I Simple microstate counting from AdS₃/CFT₂

I Hawking–Page phase transition hot AdS↔ BTZ

I Simple checks of Ryu–Takayanagi proposal

Caveat: while there are many string compactifications with AdS₃factor, applying holography just to AdS3 factor does not capture everything!

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Gravity in 3D

AdS3 gravity

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Gravity in 3D

AdS3 gravity

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Gravity in 3D

AdS3 gravity

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Gravity in 3D

AdS3 gravity

Interesting generic constraints from CFT₂! e.g.Hellerman ’09, Hartman, Keller, Stoica ’14

I Black holes as orbifolds of AdS₃ (BTZ)

Caveat: while there are many string compactifications with AdS3factor, applying holography just to AdS₃ factor does not capture everything!

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Gravity in 3D

AdS3 gravity

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Gravity in 3D

AdS3 gravity

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Gravity in 3D

AdS3 gravity

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Gravity in 3D

AdS3 gravity

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Gravity in 3D

AdS3 gravity

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Gravity in 3D

AdS3 gravity

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Holographic algorithm from gravity point of view Universal recipe:

1. Identify bulk theory and variational principle

Example: Einstein gravity with Dirichlet boundary conditions I =− 1

16πGN

Z

d³xp

|g| R+ 2

`²

with δg= fixed at the boundary

2. Fix background and impose suitable boundary conditions 3. Perform canonical analysis and check consistency of bc’s

4. Derive (classical) asymptotic symmetry algebra and central charges 5. Improve to quantum ASA

6. Study unitary representations of quantum ASA 7. Identify/constrain dual field theory

8. If unhappy with result go back to previous items and modify

Apply algorithm above to flat space holography in 3D higher spin theories Goal of this talk:

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2. Fix background and impose suitable boundary conditions Example: asymptotically AdS

ds² = dρ²+ e^2ρ/`γ_ij⁽⁰⁾+γ_ij⁽²⁾+. . .

dxⁱdx^j with δγ⁽⁰⁾ = 0 forρ→ ∞

3. Perform canonical analysis and check consistency of bc’s

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I Find and classify all constraints

I Construct canonical gauge generators

I Add boundary terms and get (variation of) canonical charges

I Check integrability of canonical charges

I Check finiteness of canonical charges

I Check conservation (in time) of canonical charges

I Calculate Dirac bracket algebra of canonical charges

Example: Brown–Henneaux analysis for 3D Einstein gravity {Q[ε], Q[η]}=δεQ[η]

Q[ε]∼ I

dϕL(ϕ)ε(ϕ)

δ_εL=Lε+ 2Lε⁰+ ` 16πG_N ε⁰⁰⁰

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I Find and classify all constraints

I Construct canonical gauge generators

I Add boundary terms and get (variation of) canonical charges

I Check integrability of canonical charges

I Check finiteness of canonical charges

I Check conservation (in time) of canonical charges

I Calculate Dirac bracket algebra of canonical charges Example: Brown–Henneaux analysis for 3D Einstein gravity

{Q[ε], Q[η]}=δεQ[η]

Q[ε]∼ I

dϕL(ϕ)ε(ϕ)

δ_εL=Lε+ 2Lε⁰+ ` 16πG ε⁰⁰⁰

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4. Derive (classical) asymptotic symmetry algebra and central charges Example: Two copies of Virasoro algebra

[L_n,L_m] = (n−m)L_n+m+ c

12(n³−n)δ_n+m,0 with Brown–Henneaux central charge

c= 3`

2GN

Reminder: ASA = quotient algebra of asymptotic symmetries by

‘trivial’ asymptotic symmetries with zero canonical charges

5. Improve to quantum ASA

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Example: semi-classical ASA in spin-3 gravity (Henneaux, Rey ’10;

Campoleoni, Pfenninger, Fredenhagen, Theisen ’10) [W_n, W_m] = 16

5c X

p

L_pLn+m−p+. . .

quantum ASA

[W_n, W_m] = 16 5c+ 22

X

p

:L_pLn+m−p : +. . .

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6. Study unitary representations of quantum ASA Example:

15 20 25 c

Afshar et al ’12

Discrete set of Newton constant values compatible with unitarity

7. Identify/constrain dual field theory

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Example: Monster CFT in (flat space) chiral gravity Witten ’07

Li, Song & Strominger ’08 Bagchi, Detournay & DG ’12

Z(q) =J(q) = 1

q + (1 + 196883)q+O(q²) Note: ln 196883≈12.2 = 4π+ quantum corrections

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8. If unhappy with result go back to previous items and modify Examples: too many!

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Outline

Motivations

Holography basics

Flat space gravity

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Flat space holography (Barnich et al, Bagchi et al, Strominger et al, ...) if holography is true⇒ must work in flat space

Just take large AdS radius limit of10⁴ AdS/CFT papers?

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I Works straightforwardly sometimes, otherwise not

I Example where it works nicely: asymptotic symmetry algebra

I Take linear combinations of Virasoro generatorsL_n,L¯_n

Ln=Ln−L¯−n Mn= 1

` Ln+ ¯L−n I Make In¨on¨u–Wigner contraction `→ ∞ on ASA

[Ln, Lm] = (n−m)Ln+m+cL

12(n³−n)δn+m,0

[Ln, Mm] = (n−m)Mn+m+cM

12 (n³−n)δn+m,0

[Mn, Mm] = 0

I This is nothing but the BMS3 algebra (or GCA2, URCA2, CCA2)!

I Example where it does not work easily: boundary conditions!

I Example where it does not work at all: highest weight conditions!

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12(n³−n)δn+m,0

12 (n³−n)δn+m,0

[Mn, Mm] = 0

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` Ln+ ¯L−n

I Make In¨on¨u–Wigner contraction `→ ∞ on ASA

12(n³−n)δn+m,0

12 (n³−n)δn+m,0

[Mn, Mm] = 0

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12(n³−n)δn+m,0

12 (n³−n)δn+m,0

[Mn, Mm] = 0

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12(n³−n)δn+m,0

12 (n³−n)δn+m,0

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12(n³−n)δn+m,0

12 (n³−n)δn+m,0

[Mn, Mm] = 0

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12(n³−n)δn+m,0

12 (n³−n)δn+m,0

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Not in general! Must (also) work intrinsically in flat space!

Interesting example:

I unitarity of flat space quantum gravity

I extrapolate from AdS: should be unitary (?)

I extrapolate from dS: should be non-unitary (?)

I directly in flat space: both options realized, depending on details of model

Many open issues in flat space holography!

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(Higher spin) gravity as Chern–Simons gauge theory...

...with weird boundary conditions (Achucarro & Townsend ’86; Witten ’88; Ba˜nados ’96)

CS action (for AdS: sl(2)⊕sl(2)):

SCS= k 4π

Z

CS(A)− k 4π

Z

CS( ¯A)

Variational principle:

δSCS|EOM= k 4π

Z

Tr A∧δA−A¯∧δA¯

Well-defined for boundary conditions (similarly for A)¯

A₊= 0 or A−= 0 boundary coordinatesx^± Example: asymptotically AdS₃ (Cartan-version of Brown–Henneaux)

A_ρ=L₀ A¯_ρ=−L₀

A₊=e^ρL₁+e^−ρL(x⁺)L−1 A¯₊= 0

A−= 0 A¯−=−e^ρL−1−e^−ρL(x¯ ⁻)L1

Dreibein: e/`∼A−A, spin-connection:¯ ω∼A+ ¯A

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(Higher spin) gravity as Chern–Simons gauge theory...

...with weird boundary conditions (Achucarro & Townsend ’86; Witten ’88; Ba˜nados ’96)

CS action (for AdS: sl(2)⊕sl(2)):

SCS= k 4π

Z

CS(A)− k 4π

Z

CS( ¯A)

Variational principle:

δSCS|EOM= k 4π

Z

Tr A∧δA−A¯∧δA¯

Well-defined for boundary conditions (similarly for A)¯

A₊= 0 or A−= 0 boundary coordinatesx^± Example: asymptotically AdS₃ (Cartan-version of Brown–Henneaux)

A_ρ=L₀ A¯_ρ=−L₀

A₊=e^ρL₁+e^−ρL(x⁺)L−1 A¯₊= 0

A−= 0 A¯−=−e^ρL−1−e^−ρL(x¯ ⁻)L1

Dreibein: e/`∼A−A, spin-connection:¯ ω∼A+ ¯A

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˙Inönü–Wigner contraction of Virasoro (Barnich & Compère ’06)

BMS3 and GCA2 (or rather, URCA2)

I Take two copies of Virasoro, generators L_n,L¯_n, central charges c,¯c

I Define superrotationsLnand supertranslations Mn

Ln:=L_n−L¯_−n Mn:= ¹_` L_n+ ¯L_−n

I Make ultrarelativistic boost,`→ ∞

[L_n, L_m] = (n−m)L_n+m+c_L₁₂¹ δ_n+m,₀ [L_n, M_m] = (n−m)M_n+m+c_M ₁₂¹ δ_n+m,0 [Mn, Mm] = 0

I Is precisely the (centrally extended) BMS3 algebra!

I Central charges:

cL=c−¯c cM = (c+ ¯c)/`

Example TMG (with gravitational CS couplingµand Newton constantG): c_L= 3

µG c_M = 3 G

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Ln:=L_n−L¯_−n Mn:= ¹_` L_n+ ¯L_−n

I Central charges:

cL=c−¯c cM = (c+ ¯c)/`

µG c_M = 3 G

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Ln:=L_n−L¯_−n Mn:= ¹_` L_n+ ¯L_−n

Global part: contracted to isl(2)(generators: L±1, L0,M±1, M0)

I Is precisely the (centrally extended) BMS₃ algebra!

I Central charges:

c_L=c−¯c c_M = (c+ ¯c)/`

µG c_M = 3 G

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Ln:=L_n−L¯_−n Mn:= ¹_` L_n+ ¯L_−n

I Central charges:

cL=c−¯c cM = (c+ ¯c)/`

µG c_M = 3 G

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Ln:=L_n−L¯_−n Mn:= ¹_` L_n+ ¯L_−n

I Central charges:

cL=c−¯c cM = (c+ ¯c)/`

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Consequence of ultrarelativistic boost for AdS boundary

AdS-boundary:

Limit`→ ∞

Flat space boundary:

Null infinity holography!

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Contraction on gravity side AdS metric (ϕ∼ϕ+ 2π):

ds²_AdS = d(`ρ)²−cosh² ^`ρ_`

dt²+`² sinh² ^`ρ_` dϕ²

Limit`→ ∞ (r =`ρ):

ds²_Flat= dr²−dt²+r² dϕ² =−du²−2 dudr+r² dϕ² BTZ metric:

ds²_BTZ =−(^r_`2² −^r_`⁺²₂)(r²−r₋²)

r² dt²+ r² dr² (^r_`²2 −^r²⁺

`²)(r²−r²₋)

+r² dϕ−

r+

` r−

r² dt2

Limit`→ ∞ (ˆr+= ^r_`⁺ = finite): ds²_FSC= ˆr²₊ 1−^r_r⁻²₂

dt²− 1 1−^r_r²⁻₂

dr² ˆ

r²₊ +r² dϕ−ˆr+r−

r² dt2

Shifted-boost orbifold studied by Cornalba & Costa more than decade ago Describes expanding (contracting) Universe in flat space

Cosmological horizon at r =r−, screening CTCs atr <0

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dt²+`² sinh² ^`ρ_` dϕ² Limit`→ ∞ (r =`ρ):

ds²_Flat= dr²−dt²+r² dϕ²=−du²−2 dudr+r² dϕ²

BTZ metric:

ds²_BTZ =−(^r_`2² −^r_`⁺²₂)(r²−r₋²)

r² dt²+ r² dr² (^r_`²2 −^r²⁺

`²)(r²−r²₋)

+r² dϕ−

r+

` r−

r² dt2

dt²− 1 1−^r_r²⁻₂

dr² ˆ

r² dt2

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dt²+`² sinh² ^`ρ_` dϕ² Limit`→ ∞ (r =`ρ):

ds²_Flat= dr²−dt²+r² dϕ²=−du²−2 dudr+r² dϕ² BTZ metric:

ds²_BTZ =−(^r_`2² −^r_`⁺²₂)(r²−r₋²)

r² dt²+ r² dr² (^r_`²2 −^r

2 +

`²)(r²−r²₋)

+r² dϕ−

r+

` r−

r² dt2

dt²− 1 1−^r_r²⁻₂

dr² ˆ

r² dt2