Stefan Prohazka [email protected]/~grumil/pdf/prohazka2016.pdf · Non-AdS Higher Spin Gravity Stefan Prohazka [email protected] Institute

Non-AdS Higher Spin Gravity

Stefan [email protected]

Institute for Theoretical Physics

AEI Potsdam-Golm13 February 2017

mailto:[email protected]

Outline

How General is Holography?

Higher Spin Theories

AdS Higher Spin Gravity in D = 3

Boundary Conditions for AdS HS Gravity

Non-AdS Spacetimes in AdS Higher Spin Gravity


Summary & Outlook

Outline







Summary & Outlook

Holographic principle

I (Non-gravitational) Boundarytheory (D − 1)could be sufficient to describe a(gravitational) bulk theory (D)[’t Hooft ’93, Susskind ’95]

I Realized:AdS/CFT correspondence[Maldacena ’98]

Bulk(Conformal)Boundary


I Conceptual: How general is holography?I Holographic principle independent of AdS/CFTI More general than AdS and CFT

Non-AdS Holography

I (Non-)AdS/(Non-)CFT new way analyze strongly coupledfield theories

I Strongly coupled systems in laboratoriesI Technological interestI Conventional techniques failI Duality

I Strong-weak: Either boundary or bulk provides weakly coupleddescription

I Lectures in both directions

I Could help computationally and conceptually

Non-relativistic

I Non-AdS necessary to realize specific dual systems [Son ’08,

Balasubramanian-McGreevy ’08, Kachru et al ’08, ...]

I Lifshitz spacetime, anisotropic scaling

t→ λzt ~x→ λ~x (r → r/λ) z 6= 1

I Schrodinger spacetimeI Null-Warped spacetime

Outline







Summary & Outlook

Higher spin theories D = 4: Non-interacting

I Higher spin D = 4I Massless bosonic fields s ≥ 2I Non-interacting EOM, Lagrangian known [Fronsdal ’78,

Fang-Fronsdal ’78]

I φ symmetric in all indices

s = 0 �φ = 0 δφ = 0

s = 1 �φµ − ∂µ∂ρφρ = 0 δφµ = ∂µΛ

s = 2 �φµν − (∂µ∂ρφρν + ∂ν∂

ρφρµ) + ∂µ∂νφρρ = 0 δφµν = ∂(µΛν)

s = 3 �φµνρ − · · · = 0 . . .

. . . . . . . . .

Higher spin theories D = 4: Interacting

I Higher spin D = 4I Interacting?

I Flat space + Higher Spin fields → No-go theorems [Weinberg

’64, Weinberg-Witten ’80, ...]

I One way to circumvent [Fradkin-Vasiliev ’87, Vasiliev ’92]

I (A)dS backgroundI Infinite tower of massless higher spins

Relation to string theory in tensionless limit

I String theory:I Excited states of string

M2 ∝ T

I Take tensionless limit T → 0I All string excited states get massless M → 0

I Leading Regge trajectory

Relation to String theory in tensionless limit

I Limit hints of enlarged gauge symmetry of string theory [Gross

’88, Witten ’88, Moore ’93, ...] → effective description in terms ofVasiliev theory

I Unbroken phase of string theory

I Other direction: String theory from symmetry breaking ofmassless higher spins

I “La Grande Bouffe”: massless HS particles eat Goldstoneparticles [Bianchi et al. ’03]

Holography

I HolographyI Critical O(N) vector model in 3 dimensions ⇐⇒ bosonic

higher spin theory in AdS4 [Klebanov-Polyakov ’02, Sezgin-Sundell

’02]

I Simple modelI Various checks support conjecture [Giombi-Yin ’09, ’10, ...]

Outline







Summary & Outlook

Higher spin theories D = 3

I Fradkin and Vasiliev (D = 3):I Some of the technical difficulties of D = 4 goneI Chern-Simons action [Blencowe ’88]

I Infinite tower of higher spins hs[λ] → sl(N,R) spins ofs = 2, 3, ..., N [Henneaux-Rey ’10, Campoleoni et al. ’10,

Gaberdiel-Hartman ’11]

I Bosonic massless s ≥ 2 HS Fields have no propagating d.o.f.in the bulk

Why D = 3?

I D = 3 → good balance between complexity and tractabilityI Holography

I 2D WN minimal model CFTs in large-N ’t Hooft limit ⇐⇒HS gravitational theories in AdS3 [Gaberdiel-Gopakumar ’11, ...]

I Higher Spins & Strings [Gaberdiel-Gopakumar ’14, ...]

I CFT2 high degree of analytical control

I HS Black holes [Gutperle-Kraus ’11, Ammon et al. ’11, ...]

I Lifshitz black holes [Gutperle et al. ’13]

I Entanglement entropy [Ammon et al. ’13, de Boer-Jottar ’13]

I . . .

sl(3,R)⊕ sl(3,R) AdS HS Gravity

I Spin-3 gravity is sl(3,R)⊕ sl(3,R) Chern-Simons theory

I[A,A] = ICS [A]− ICS [A]

where

ICS [A] =k

4π

∫M

Tr(A ∧ dA+2

3A ∧A ∧A) +B[A].

I Interpretation: Spin-3 field coupled to gravity

gµν =1

2Tr(eµeν) φλµν =

1

3!Tr(e(λeµeν))

eµ =`

2

(Aµ −Aµ

)

Back to GR

I sl(3,R)-Algebra

[Ln, Lm] = (n−m) Ln+m

[Ln, Wm] = (2n−m) Wn+m

[Wn, Wm] = − (n−m)(2n2 + 2m2 − nm− 8) Ln+m

I Restriction to subalgebra sl(2,R):

ICS [A]− ICS [A] ∝∫ √

|g|(R+

2

`2

)d3x

Metric Like Action in D = 3

I sl(2,R)⊕ sl(2,R):

ICS [A]− ICS [A] ∝∫ √

|g|(R+

2

`2

)d3x

I sl(3,R)⊕ sl(3,R) [Campoleoni et al. ’13, ...]

ICS [A]− ICS [A] ∝∫d3x√|g|[(R+

2

`2

)+ ϕµνρ

(Fµνρ −

3

2g(µνFρ)

)− 3

2Rϕµνρϕ

µνρ +9

4Rρσ (2ϕρµνϕ

σµν − ϕρϕσ)

− 1

`2(6ϕµνρϕ

µνρ − 9ϕµϕµ)

]+O(ϕ4)

Enhanced Symmetries

A A

gµν © gµν ?§?

A=g−1Ag+g−1dg

proper gauge transformation

gµν=12Tr(eµeν) gµν=

12Tr(eµeν)

I Singularity resolution [Castro et al. ’12, Krishnan-Roy ’13,...]

Outline







Summary & Outlook

Theory and Boundary Conditions

I Action: sl(3,R)⊕ sl(3,R) CS + Bilinear form = AdS HSGravity

I One action can have different boundary conditions

I Different boundary conditions → (possibly) different physicaltheories

I “The field equations and the boundary conditions areinextricably connected and the latter can in no way beconsidered less important than the former.” [Fock ’55 via Bunster

et al. ’14]

I Especially dramatic in 2 + 1 Einstein gravity (and HSgeneralization)

I No local bulk dof → Information in holonomies and boundarydegrees of freedom

Boundary conditions

I Define theory (Action + BC)

I Find gauge transformations that leave BC invariant →Asymptotic symmetry algebra

I Gives information about possible boundary theory

W3 Boundary Conditions

I aφ = L+1 − 2πk L+(φ)L−1 − π

2kW+(φ)W−2 [Henneaux-Rey ’10,

Campoleoni et al. ’10]

I After a proper gauge transformation

I Generalization of Brown-Henneaux BC

I Lead to generalization of BTZ BH [Gutperle-Kraus ’11, Ammon et

al. ’11, ...]

I ASA is W3 (generalization of Virasoro)

I One can calculate Entropy [Bunster et al. ’14]

SW3 = 2π√

2πk

(√L+ cos

[1

3arcsin

(3

8

√3k

2πL3+W+

)]

+√L− cos

[1

3arcsin

(3

8

√3k

2πL3−W−

)])

u(1) Boundary Conditions

I Motivated by near horizon limit of BH [Afshar et al. 16]

I aφ = J +L0 + J +(3)W0 [Grumiller et al. ’16]

I ASA u(1)⊕ u(1)

I Calculate entropy (easy)

Su(1) = 2π(J+0 + J−0

)I Can be mapped on other boundary conditions

J±0 =√

2πkL± cos

[1

3arcsin

(3

8

√3k

2πL3±W±

)]I One can construct W3 out of u(1)

Su(1) = SW3

I Higher spin soft hair [Hawking, Perry, Strominger ’16]

Outline







Summary & Outlook

Higher Spin Lifshitz Holography

I Lifshitz background is the proposed gravity dual to condensedmatter systems at fixed point [Kachru et al. ’08]

I Theory at this fixed point: anisotropic scale invariancebetween spatial and temporal scaling

t→ λzt ~x→ λ~x (r → r/λ) z 6= 1

I z = 2

ds2 = `2(−e4ρdt2 + dρ2 + e2ρdx2

)φµνλ dx

µ dxν dxλ = −5`3

4e4ρ dt(dx)2

I Spin-3 field needed since Lifshitz is no solution of vacuumEinstein

Higher Spin Lifshitz Holography

I ASA W3 ⊕W3 with c = 3`2GN

I Same ASA as higher spin AdS W3 BC [Henneaux-Rey ’10

,Campoleoni et al. ’10] but topologically different

I Lifshitz in HS theory gets enhanced to full relativisticW3 ⊕W3 [Gary et al. ’14]

I Null warped: Proposed dual to non-relativistic CFTsdescribing cold atoms [Son ’08, Balasubramanian-McGreevy ’08] →HS similar story [Breunholder et al. ’15]

Outline







Summary & Outlook


I So far: AdS higher spin gravity (sl(3,R)⊕ sl(3,R)) withdifferent BC

I Based on generalization of AdS algebra (sl(2,R)⊕ sl(2,R))

I What about other algebras (Poincare, Galilei)?

I Make Non-AdS manifest?


I RelativisticI Poincare iso(2, 1) [Achucarro-Townsend ’87, Witten ’88]

I Higher spin Poincare [Afshar et al. ’13, Gonzalez et al. ’13]

I Non-relativisticI Galilei gravity based on gal with 2 central extension

[Papageorgiou, Schroers ’09, Bergshoeff-Rosseel ’16, Hartong et al. ’16]

→ rememberI Non-relativistic geometry interest for non-AdS holography

[Christensen et al. ’14, Hartong et al. ’14,’15]

I Systematic way to find interesting (higher spin) algebras?

Possible Kinematics D = 4

I Under the assumptions that [Bacry, Levy Leblond ’67]

I Space is isotropicI Parity and time-reversal are automorphismsI Internal transformations form a noncompact subgroup

there are eight types of Lie algebras for kinematical groups.

I All of them given by sequential Inonu-Wigner contractions of(A)dS

(A)dS

poi

nh

ppoi

gal

pgal

car

st

Space-time

Sp

eed-sp

ace

Speed-time

Genera

l

How to Generalize to Higher Spins in D = 3

I Start with higher spin AdS, sl(3,R)⊕ sl(3,R)

I Restricted to spin-2 part → contraction should be that ofkinematical algebras

I Resulting contraction not all commutators of the spin-3 partare vanishing.

IW contractions

I g = h+i and rescale i→ ε i

[ h , h ] ⊆ h + ε i

[ h , i ] ⊆ 1

εh + i

[ i , i ] ⊆ 1

ε2h +

1

εi

I [ h , h ] needs to be a subalgebra

I Contraction leads to

[ h , h ] ⊆ h

[ h , i ] ⊆ i

[ i , i ] ⊆ 0

Example: (A)dS → Poincare, h = JA

[JA , JB

]= εABC JC[

JA , PB]

= εABC PC[PA , PB

]= ±εABC JC

[ h , h ] ⊆ h

[ h , i ] ⊆ i

[ i , i ] ⊆ 0

Example: (A)dS → Poincare, h = JA

[JA , JB

]= εABC JC[

JA , PB]

= εABC PC[PA , PB

]= 0

[ h , h ] ⊆ h

[ h , i ] ⊆ i

[ i , i ] ⊆ 0

Example: Poincare, h = JA

[JA , JB

]= εABC JC[

JA , PB]

= εABC PC[PA , PB

]= ± εABC JC ,

[JA , JBC

]= εMA(B JC)M

[PA , PBC

]= ± εMA(B JC)M .[

JA , PBC]

= εMA(B PC)M

[PA , JBC

]= εMA(B PC)M

[JAB , JCD

]= − η(A(CεD)B)M JM[

JAB , PCD]

= − η(A(CεD)B)M PM[PAB , PCD

]= ∓η(A(CεD)B)M JM .

Example: Poincare, h = JA add JAB

[JA , JB

]= εABC JC[

JA , PB]

= εABC PC[PA , PB

]= ± εABC JC ,

[JA , JBC

]= εMA(B JC)M

[PA , PBC

]= ± εMA(B JC)M .[

JA , PBC]

= εMA(B PC)M

[PA , JBC

]= εMA(B PC)M

[JAB , JCD


JAB , PCD]



Example: Poincare, h = JA add PAB

[JA , JB

]= εABC JC[

JA , PB]

= εABC PC[PA , PB

]= ± εABC JC ,

[JA , JBC

]= εMA(B JC)M

[PA , PBC

]= ± εMA(B JC)M .[

JA , PBC]

= εMA(B PC)M

[PA , JBC

]= εMA(B PC)M

[JAB , JCD


JAB , PCD]



Example: Poincare, h = JA add JAB and PAB

[JA , JB

]= εABC JC[

JA , PB]

= εABC PC[PA , PB

]= ± εABC JC ,

[JA , JBC

]= εMA(B JC)M

[PA , PBC

]= ± εMA(B JC)M .[

JA , PBC]

= εMA(B PC)M

[PA , JBC

]= εMA(B PC)M

[JAB , JCD


JAB , PCD]



All contractions

I Exactly 10 different IW contractions (+7 “traceless”)I 2 space-timeI 2 speed-spaceI 2 speed-timeI 4 general (can be reached by sequential of others)

I Nice because: Could have been a lot more or none

hs3(A)dS

hs3poi

hs3nh

hs3ppoi

hs3gal

hs3pgal

hs3car

hs3st

#3#4

#5

#6

#1 #2

Non-AdS Higher Spin Gravity?

I Non-AdS Higher Spin Algebras 3

I CS action would be nice

ICS [A] =k

4π

∫M

Tr(A ∧ dA+2

3A ∧A ∧A)

I Do we have a Tr?I HS AdS= sl(3,R)⊕ sl(3,R)→ semi-simple → Killing form 3I Contractions have abelian ideal → not semi-simple → Killing

form degenerateI We do not have a trace in the “usual” sense (Killing form)

I What do we actually need?I Symmetric, non-degenerate, (ad-)invariant bilinear formI HS Poincare, HS Carroll 3I HS Galilei 7

HS Galilei: What can we do?

I Same problem already spin-2

I Double extension

I 4 HS Galilei

I 2 HS Galilei of usual “HS Poincare”

I 2 HS Extended Bargmann!

Extended Bargmann 1 (Part I)

[ J , Ga ] = εamGm [ J , Pa ] = εamPm

[ Ga , H ] = −εamPm[ Ga , Gb ] = εabH

∗ [ Pa , Gb ] = εabJ∗

[ J , Ja ] = εamJm [ J , Gab ] = −εm(aGb)m

[ J , Ha ] = εamHm [ J , Pab ] = −εm(aPb)m

[ Ga , Jb ] = −(εamGbm + εabGmm) [ Ga , Hb ] = −(εamPbm + εabPmm)

[ H , Ja ] = εamHm [ H , Gab ] = −εm(aPb)m

[ Pa , Jb ] = −(εamPbm + εabPmm)

Extended Bargmann 1 (Part II)

[ Ja , Jb ] = εabJ

[ Ja , Gbc ] = δa(bεc)mGm [ Ja , Hb ] = εabH

[ Ja , Pbc ] = δa(bεc)mPm [ Gab , Hc ] = −δc(aεb)mPm[ Gab , Gcd ] = ε(a(cδd)b)H

∗ [ Pab , Gcd ] = ε(a(cδd)b)J∗

[ Pa , Gbc ] = εa(bJ∗c) [ Ga , Gbc ] = εa(bH

∗c)

[ Ga , Pbc ] = εa(bJ∗c) [ J , H∗a ] = εamH

∗m

[ J , J∗a ] = εamJ∗m [ H , H∗a ] = εamJ

∗m

[ Ja , J∗ ] = −εamJ∗m [ Ja , H

∗ ] = −εamH∗m[ Ja , J

∗b ] = εabJ

∗ [ Ja , H∗b ] = εabH

∗

[ Ha , H∗ ] = −εamJ∗m [ Ha , H

∗b ] = εabJ

∗

Outline







Summary & Outlook

Summary

I Higher spin theories interesting properties

I Different BC (can) lead to different theories

I Non relativistic HS interesting to look for non relativistic HSalgebras

I Possible + full classification via contractions for kinematicalHS algebras

I HS Extended Bargmann and Carroll

Outlook

I Solution space

I Boundary conditions (done for Carroll Spin-2)

I Boundary theories

I Fronsdal-like equations

I Lie superalgebras

I Spin-2 Extended Bargmann related to Horava-Lifshitz gravity[Lei et al. ’15] → Spin-3 Horava Gravity?

Thank you very much for your attention.

Documents

Stefan Prohazka [email protected]/~grumil/pdf/prohazka2016.pdf · Non-AdS Higher Spin Gravity Stefan Prohazka [email protected] Institute