Unitarity in flat space holographyquark.itp.tuwien.ac.at/~grumil/pdf/Brussels2014.pdf · Unitarity in at space holography Daniel Grumiller Institute for Theoretical Physics Vienna

Unitarity in flat space holography

Daniel Grumiller

Institute for Theoretical PhysicsVienna University of Technology

Solvay Workshop on Holography for black holes and cosmology,Brussels, April 2014

based on work withRosseel and Riegler, 1403.5297

and work with Afshar, Bagchi, Detournay, Fareghbal, Simon

Motivation

I Holographic principle, if correct, must work beyond AdS/CFTholographic principle: ’t Hooft ’93; Susskind ’94

AdS/CFT precursor: Brown, Henneaux ’86AdS/CFT: Maldacena ’97; Gubser, Klebanov, Polyakov ’98; Witten’98

I Does it work in flat space?I Can we find models realizing flat space/field theory correspondences?I Are there higher-spin versions of such models?I Does this correspondence emerge as limit of (A)dS/CFT?

II Is there an analog of Hawking–Page phase transition?I Can we relate S-matrix observables to holographic observables?

Strominger ’13He, Lysov, Mitra, Strominger ’14Banks ’14Cachazo, Strominger ’14

I ...

Address unitarity question in two boundary dimensions!Particular interest: unitarity in flat space higher spin gravity?

Daniel Grumiller — Unitarity in flat space holography 2/17

Motivation

I Holographic principle, if correct, must work beyond AdS/CFTI Does it work in flat space?

Polchinski ’99Susskind ’99Giddings ’00Gary, Giddings, Penedones ’09; Gary, Giddings ’09; ...

I Can we find models realizing flat space/field theory correspondences?I Are there higher-spin versions of such models?I Does this correspondence emerge as limit of (A)dS/CFT?



I ...



Motivation

I Holographic principle, if correct, must work beyond AdS/CFTI Does it work in flat space?I Can we find models realizing flat space/field theory correspondences?

Barnich, Compere ’06Barnich et al. ’10-’14Bagchi et al. ’10-’14Strominger et al. ’13-’14...

flat space chiral gravity: Bagchi, Detournay, DG ’12

I Are there higher-spin versions of such models?I Does this correspondence emerge as limit of (A)dS/CFT?



I ...



Motivation

I Holographic principle, if correct, must work beyond AdS/CFTI Does it work in flat space?I Can we find models realizing flat space/field theory correspondences?I Are there higher-spin versions of such models?

Afshar, Bagchi, Fareghbal, DG, Rosseel ’13Gonzalez, Matulich, Pino, Troncoso ’13

part of larger program: non-AdS holography in higher spin gravityGary, DG, Rashkov ’12Afshar, Gary, DG, Rashkov, Riegler ’12Gutperle, Hijano, Samani ’13Gary, DG, Prohazka, Rey (in prep.) ’14

I Does this correspondence emerge as limit of (A)dS/CFT?



I ...



Motivation

I Holographic principle, if correct, must work beyond AdS/CFTI Does it work in flat space?I Can we find models realizing flat space/field theory correspondences?I Are there higher-spin versions of such models?I Does this correspondence emerge as limit of (A)dS/CFT?

some aspects: yes; other aspects: no



I ...



Motivation

I Holographic principle, if correct, must work beyond AdS/CFTI Does it work in flat space?I Can we find models realizing flat space/field theory correspondences?I Are there higher-spin versions of such models?I Does this correspondence emerge as limit of (A)dS/CFT?I (When) are these models unitary?

this talk!

see also Barnich, Oblak ’14 (induced representations of BMS3)

I Is there an analog of Hawking–Page phase transition?I Can we relate S-matrix observables to holographic observables?


I ...



Motivation

I Holographic principle, if correct, must work beyond AdS/CFTI Does it work in flat space?I Can we find models realizing flat space/field theory correspondences?I Are there higher-spin versions of such models?I Does this correspondence emerge as limit of (A)dS/CFT?I (When) are these models unitary?I Is there an analog of Hawking–Page phase transition?

yes: Bagchi, Detournay, DG, Simon ’13Detournay, DG, Scholler, Simon ’14

I Can we relate S-matrix observables to holographic observables?Strominger ’13He, Lysov, Mitra, Strominger ’14Banks ’14Cachazo, Strominger ’14

I ...



Motivation

I Holographic principle, if correct, must work beyond AdS/CFTI Does it work in flat space?I Can we find models realizing flat space/field theory correspondences?I Are there higher-spin versions of such models?I Does this correspondence emerge as limit of (A)dS/CFT?I (When) are these models unitary?I Is there an analog of Hawking–Page phase transition?I Can we relate S-matrix observables to holographic observables?


I ...



Motivation

I Holographic principle, if correct, must work beyond AdS/CFTI Does it work in flat space?I Can we find models realizing flat space/field theory correspondences?I Are there higher-spin versions of such models?I Does this correspondence emerge as limit of (A)dS/CFT?I (When) are these models unitary?I Is there an analog of Hawking–Page phase transition?I Can we relate S-matrix observables to holographic observables?


I ...



Outline of procedure

Work mostly on CFT side and study “landscape” of possible flat spaceasymptotic symmetry algebras:

1. Consider some symmetry algebra arising in a relativistic CFT2 (twocopies of Virasoro or some W-algebra)

2. Inonu–Wigner contract to Galilean/Ultra-relativistic conformal algebra(GCA2/URCA2)

3. Define vacuum and hermitian conjugation

4. Demand non-negativity norm of vacuum descendants

5. Check consequences for central charges and algebra

6. Find realization on gravity side as asymptotic symmetry algebra

In this talk I will address 1.-5., but not 6.!








































































Main results

1. NO–GO:Generically (see later) you can have only two out of three:

I Unitarity

I Flat space

I Non-trivial higher spin states

Compatible with “spirit” of variousno-go results in higher dimensions!

2. YES–GO:There is (at least) one counter-example, namely a Vasiliev-type of theory,where you can have all three properties!

Unitary, non-trivial flat space higher spin algebra exists!Vasiliev-type flat space chiral higher spin gravity?


Main results


I UnitarityI Flat spaceI Non-trivial higher spin states

Example:Flat space chiral gravityBagchi, Detournay, DG, 1208.1658





Main results



Example:Minimal model holographyGaberdiel, Gopakumar, 1011.2986, 1207.6697





Main results



Example:Flat space higher spin gravity (Galilean W3 algebra)Afshar, Bagchi, Fareghbal, DG and Rosseel, 1307.4768Gonzalez, Matulich, Pino and Troncoso, 1307.5651





Main results


I Unitarity

I Flat space






Main results


I Unitarity

I Flat space







Inonu–Wigner contraction of Virasoro (Barnich & Compere ’06)BMS3 and GCA2/URCA2

I Take two copies of Virasoro, generators Ln, Ln, central charges c, c

I Define superrotations Ln and supertranslations Mn

GCA: Ln := Ln + Ln Mn := −ε(Ln − Ln

)URCA: Ln := Ln − L−n Mn := ε

(Ln + L−n

)I Galilean limit/ultrarelativistic boost: ε ∼ 1/`→ 0

[Ln, Lm] = (n−m)Ln+m + cL112 δn+m, 0

[Ln, Mm] = (n−m)Mn+m + cM112 δn+m, 0

[Mn, Mm] = 0

I Is precisely (centrally extended) BMS3 algebra!I Central charges:

GCA: cL = c+ c cM = −ε(c− c)URCA: cL = c− c cM = ε(c+ c)

I Example: URCA TMG: cL = 3/µG and cM = 3/G [dimensionful!]



I Take two copies of Virasoro, generators Ln, Ln, central charges c, cI Define superrotations Ln and supertranslations Mn



(Ln + L−n

)

I Galilean limit/ultrarelativistic boost: ε ∼ 1/`→ 0

[Ln, Lm] = (n−m)Ln+m + cL112 δn+m, 0

[Ln, Mm] = (n−m)Mn+m + cM112 δn+m, 0

[Mn, Mm] = 0









(Ln + L−n


[Ln, Lm] = (n−m)Ln+m + cL112 δn+m, 0

[Ln, Mm] = (n−m)Mn+m + cM112 δn+m, 0

[Mn, Mm] = 0









(Ln + L−n


[Ln, Lm] = (n−m)Ln+m + cL112 δn+m, 0

[Ln, Mm] = (n−m)Mn+m + cM112 δn+m, 0

[Mn, Mm] = 0









(Ln + L−n


[Ln, Lm] = (n−m)Ln+m + cL112 δn+m, 0

[Ln, Mm] = (n−m)Mn+m + cM112 δn+m, 0

[Mn, Mm] = 0



I Example: URCA TMG: cL = 3/µG and cM = 3/G [dimensionful!]Daniel Grumiller — Unitarity in flat space holography 6/17

(Non-)unitarity in GCAs

Assumptions:I Standard highest weight vacuum:

Ln|0〉 = Mn|0〉 = 0 ∀n ≥ −1

I Standard hermitian conjugation:

L†n := L−n M †n := M−n

Conclusions:

I Gram matrix of level 2 descendants:(〈0|L2L−2|0〉 = cL

2 〈0|L2M−2|0〉 = cM2

〈0|M2L−2|0〉 = cM2 〈0|M2M−2|0〉 = 0

)I Determinant: −c2M/4 ≤ 0I cM 6= 0: one positive and one negative norm stateI cM = 0, cL < 0: one negative norm and one null stateI cM = 0, cL = 0: only null states (trivial)I cM = 0, cL > 0: one positive norm and one null state




Ln|0〉 = Mn|0〉 = 0 ∀n ≥ −1


L†n := L−n M †n := M−n

Conclusions:

I Gram matrix of level 2 descendants:(〈0|L2L−2|0〉 = cL

2 〈0|L2M−2|0〉 = cM2

〈0|M2L−2|0〉 = cM2 〈0|M2M−2|0〉 = 0





Ln|0〉 = Mn|0〉 = 0 ∀n ≥ −1


L†n := L−n M †n := M−n

Conclusions:I Gram matrix of level 2 descendants:(

〈0|L2L−2|0〉 = cL2 〈0|L2M−2|0〉 = cM

2〈0|M2L−2|0〉 = cM

2 〈0|M2M−2|0〉 = 0

)

I Determinant: −c2M/4 ≤ 0I cM 6= 0: one positive and one negative norm stateI cM = 0, cL < 0: one negative norm and one null stateI cM = 0, cL = 0: only null states (trivial)I cM = 0, cL > 0: one positive norm and one null state




Ln|0〉 = Mn|0〉 = 0 ∀n ≥ −1


L†n := L−n M †n := M−n


〈0|L2L−2|0〉 = cL2 〈0|L2M−2|0〉 = cM

2〈0|M2L−2|0〉 = cM

2 〈0|M2M−2|0〉 = 0

)I Determinant: −c2M/4 ≤ 0

I cM 6= 0: one positive and one negative norm stateI cM = 0, cL < 0: one negative norm and one null stateI cM = 0, cL = 0: only null states (trivial)I cM = 0, cL > 0: one positive norm and one null state




Ln|0〉 = Mn|0〉 = 0 ∀n ≥ −1


L†n := L−n M †n := M−n


〈0|L2L−2|0〉 = cL2 〈0|L2M−2|0〉 = cM

2〈0|M2L−2|0〉 = cM

2 〈0|M2M−2|0〉 = 0

)I Determinant: −c2M/4 ≤ 0I cM 6= 0: one positive and one negative norm state

I cM = 0, cL < 0: one negative norm and one null stateI cM = 0, cL = 0: only null states (trivial)I cM = 0, cL > 0: one positive norm and one null state




Ln|0〉 = Mn|0〉 = 0 ∀n ≥ −1


L†n := L−n M †n := M−n


〈0|L2L−2|0〉 = cL2 〈0|L2M−2|0〉 = cM

2〈0|M2L−2|0〉 = cM

2 〈0|M2M−2|0〉 = 0

)I Determinant: −c2M/4 ≤ 0I cM 6= 0: one positive and one negative norm stateI cM = 0, cL < 0: one negative norm and one null state

I cM = 0, cL = 0: only null states (trivial)I cM = 0, cL > 0: one positive norm and one null state




Ln|0〉 = Mn|0〉 = 0 ∀n ≥ −1


L†n := L−n M †n := M−n


〈0|L2L−2|0〉 = cL2 〈0|L2M−2|0〉 = cM

2〈0|M2L−2|0〉 = cM

2 〈0|M2M−2|0〉 = 0

)I Determinant: −c2M/4 ≤ 0I cM 6= 0: one positive and one negative norm stateI cM = 0, cL < 0: one negative norm and one null stateI cM = 0, cL = 0: only null states (trivial)

I cM = 0, cL > 0: one positive norm and one null state




Ln|0〉 = Mn|0〉 = 0 ∀n ≥ −1


L†n := L−n M †n := M−n


〈0|L2L−2|0〉 = cL2 〈0|L2M−2|0〉 = cM

2〈0|M2L−2|0〉 = cM

2 〈0|M2M−2|0〉 = 0



Generalization to higher spin asymptotic symmetry algebras

Facts:I Unitarity in GCA requires cM = 0

I Non-triviality requires then cL 6= 0I Generalization to contracted higher spin algebras straightforwardI All of them contain GCA as subalgebraI Again cM = 0 is necessary for unitarity

Example: Galilean W3 algebra: [Ln, Lm] and [Ln, Mm] as before,

[Ln, Um] = (2n−m)Un+m [Ln, Vm] = [Mn, Um] = (2n−m)Vn+m

[Un, Um] = (n−m)(2n2 + 2m2 − nm− 8)Ln+m +192

cM(n−m)Λn+m

−96(cL + 44

5

)c2M

(n−m)Θn+m +cL12n(n2 − 1)(n2 − 4) δn+m, 0

[Un, Vm] = (n−m)(2n2 + 2m2 − nm− 8)Mn+m +96

cM(n−m)Θn+m

+cM12

n(n2 − 1)(n2 − 4) δn+m, 0



Facts:I Unitarity in GCA requires cM = 0I Non-triviality requires then cL 6= 0

I Generalization to contracted higher spin algebras straightforwardI All of them contain GCA as subalgebraI Again cM = 0 is necessary for unitarity



[Un, Um] = (n−m)(2n2 + 2m2 − nm− 8)Ln+m +192

cM(n−m)Λn+m

−96(cL + 44

5

)c2M

(n−m)Θn+m +cL12n(n2 − 1)(n2 − 4) δn+m, 0

[Un, Vm] = (n−m)(2n2 + 2m2 − nm− 8)Mn+m +96

cM(n−m)Θn+m

+cM12

n(n2 − 1)(n2 − 4) δn+m, 0



Facts:I Unitarity in GCA requires cM = 0I Non-triviality requires then cL 6= 0I Generalization to contracted higher spin algebras straightforward

I All of them contain GCA as subalgebraI Again cM = 0 is necessary for unitarity



[Un, Um] = (n−m)(2n2 + 2m2 − nm− 8)Ln+m +192

cM(n−m)Λn+m

−96(cL + 44

5

)c2M

(n−m)Θn+m +cL12n(n2 − 1)(n2 − 4) δn+m, 0

[Un, Vm] = (n−m)(2n2 + 2m2 − nm− 8)Mn+m +96

cM(n−m)Θn+m

+cM12

n(n2 − 1)(n2 − 4) δn+m, 0



Facts:I Unitarity in GCA requires cM = 0I Non-triviality requires then cL 6= 0I Generalization to contracted higher spin algebras straightforwardI All of them contain GCA as subalgebra

I Again cM = 0 is necessary for unitarity



[Un, Um] = (n−m)(2n2 + 2m2 − nm− 8)Ln+m +192

cM(n−m)Λn+m

−96(cL + 44

5

)c2M

(n−m)Θn+m +cL12n(n2 − 1)(n2 − 4) δn+m, 0

[Un, Vm] = (n−m)(2n2 + 2m2 − nm− 8)Mn+m +96

cM(n−m)Θn+m

+cM12

n(n2 − 1)(n2 − 4) δn+m, 0



Facts:I Unitarity in GCA requires cM = 0I Non-triviality requires then cL 6= 0I Generalization to contracted higher spin algebras straightforwardI All of them contain GCA as subalgebraI Again cM = 0 is necessary for unitarity



[Un, Um] = (n−m)(2n2 + 2m2 − nm− 8)Ln+m +192

cM(n−m)Λn+m

−96(cL + 44

5

)c2M

(n−m)Θn+m +cL12n(n2 − 1)(n2 − 4) δn+m, 0

[Un, Vm] = (n−m)(2n2 + 2m2 − nm− 8)Mn+m +96

cM(n−m)Θn+m

+cM12

n(n2 − 1)(n2 − 4) δn+m, 0



Facts:I Unitarity in GCA requires cM = 0I Non-triviality requires then cL 6= 0I Generalization to contracted higher spin algebras straightforwardI All of them contain GCA as subalgebraI Again cM = 0 is necessary for unitarity



[Un, Um] = (n−m)(2n2 + 2m2 − nm− 8)Ln+m +192

cM(n−m)Λn+m

−96(cL + 44

5

)c2M

(n−m)Θn+m +cL12n(n2 − 1)(n2 − 4) δn+m, 0

[Un, Vm] = (n−m)(2n2 + 2m2 − nm− 8)Mn+m +96

cM(n−m)Θn+m

+cM12

n(n2 − 1)(n2 − 4) δn+m, 0


Unitarity leads to further contraction! (Afshar et al. 1307.4768)

Inonu–Wigner contraction of two W-algebras (generators L,W):

Ln :=Ln + Ln Mn := −ε(Ln − Ln

)Un :=Wn + Wn Vn := −ε

(Wn − Wn

)

Limit cM → 0 requires further contraction:

Un → cM Un

Doubly contracted algebra has unitary representations:

[Ln, Lm] = (n−m)Ln+m +cL12

(n3 − n) δn+m, 0

[Ln, Mm] = (n−m)Mn+m

[Ln, Um] = (2n−m)Un+m

[Ln, Vm] = (2n−m)Vn+m

[Un, Um] ∝ [Un, Vm] = 96(n−m)∑p

MpMn−p

Higher spin states decouple and become null states!






(Wn − Wn

)Limit cM → 0 requires further contraction:

Un → cM Un



(n3 − n) δn+m, 0




[Un, Um] ∝ [Un, Vm] = 96(n−m)∑p

MpMn−p







(Wn − Wn


Un → cM Un



(n3 − n) δn+m, 0




[Un, Um] ∝ [Un, Vm] = 96(n−m)∑p

MpMn−p







(Wn − Wn


Un → cM Un



(n3 − n) δn+m, 0




[Un, Um] ∝ [Un, Vm] = 96(n−m)∑p

MpMn−p

Higher spin states decouple and become null states!Daniel Grumiller — Unitarity in flat space holography 9/17

How general is this decoupling mechanism? (DG, Riegler, Rosseel ’14)

Same conclusions — all higher spin states become null states — for

I Contracted principal embedding (Galilean WN )Unitary for positive cL

I Contracted Polyakov–Bershadsky (Galilean W(2)3 )

Unitary for 1 ≤ cL ≤ 32

I Contracted Feigin–Semikhatov (Galilean W(2)N )

Unitary for 1 ≤ cL ≤ 2(N − 1)2(N + 1) ∼ N3 (for large N)

I Contracted W(2−1−1)4

Unitary for 29−√

661 ≤ cL ≤ 42

I Decoupling of higher spin states seems to be a generic feature!But why?

In all cases above direct consequence of non-linearity in W-algebra!











661 ≤ cL ≤ 42













661 ≤ cL ≤ 42













661 ≤ cL ≤ 42





Same conclusions — all higher spin states become null states — forI Contracted principal embedding (Galilean WN )

Unitary for positive cLI Contracted Polyakov–Bershadsky (Galilean W

(2)3 )

Unitary for 1 ≤ cL ≤ 32I Contracted Feigin–Semikhatov (Galilean W

(2)N )




661 ≤ cL ≤ 42







(2)3 )


(2)N )




661 ≤ cL ≤ 42Upper bound from current algebra part:

u(1) : [On, Om] =2(54− cL)

3n δn+m, 0

su(2) : [Qan, Q

bm] = (a− b)Qa+b

n+m +42− cL

6n δa+b, 0 δn+m, 0







(2)3 )


(2)N )




661 ≤ cL ≤ 42Lower bound: non-negativity of cbare in

cL = cu(1) + csu(2) + cbare

with c... = k dim g/(k + h∨) with level k = (42− cL)/6, thus:

cu(1) = 1 csu(2) = 3(42− cL)/(54− cL)

⇒ cbare ≥ 0 implies cL ≥ 29−√

661 ≈ 3.29













661 ≤ cL ≤ 42













661 ≤ cL ≤ 42




General NO–GO resultSee 1403.5297 for details!

Idea of no-go proof:

I Assume non-linearity in W-algebra, e.g. (limc→∞ f(c)→ 0)

[Wn, Wm] = . . .+ f(c) :AB :n+m + . . .+ ω(c)

s−1∏j=−(s−1)

(n+ j) δn+m, 0

I Define Galilean contraction in usual way

I After contraction ε→ 0 use dimensional arguments, e.g.

I Need rescaling Un → cMUn = Un before taking limit cM → 0

I Central terms do not survive this rescaling

I This is why higher spin states become null states and decouple!





[Wn, Wm] = . . .+ f(c) :AB :n+m + . . .+ ω(c)

s−1∏j=−(s−1)

(n+ j) δn+m, 0

I Define Galilean contraction in usual way, e.g.

Un :=Wn + Wn Vn := −ε(Wn − Wn

)Cn :=An + An Dn := −ε

(An − An

)En :=Bn + Bn Fn := −ε

(Bn − Bn

)

I After contraction ε→ 0 use dimensional arguments, e.g.I Need rescaling Un → cMUn = Un before taking limit cM → 0

I Central terms do not survive this rescalingI This is why higher spin states become null states and decouple!





[Wn, Wm] = . . .+ f(c) :AB :n+m + . . .+ ω(c)

s−1∏j=−(s−1)

(n+ j) δn+m, 0


I After contraction ε→ 0 use dimensional arguments, e.g.No central terms in higher spin generators:

[Vn, Vm] = 0 (trivial)

[Un, Vm] = · · ·+ # cM δn+m, 0 (dimensions!)






Idea of no-go proof:I Assume non-linearity in W-algebra, e.g. (limc→∞ f(c)→ 0)

[Wn, Wm] = . . .+ f(c) :AB :n+m + . . .+ ω(c)

s−1∏j=−(s−1)

(n+ j) δn+m, 0

I Define Galilean contraction in usual wayI After contraction ε→ 0 use dimensional arguments, e.g.

[Un, Um] = . . .+O(1/cM ) (:CF :n+m + :DE :n+m)

+ O(1

c2M)︸︷︷︸

6=0

:DF :n+m +ω(cL)

s−1∏j=−(s−1)

(n+ j) δn+m, 0,

I Need rescaling Un → cMUn = Un before taking limit cM → 0I Central terms do not survive this rescalingI This is why higher spin states become null states and decouple!



Idea of no-go proof:I Assume non-linearity in W-algebra, e.g. (limc→∞ f(c)→ 0)

[Wn, Wm] = . . .+ f(c) :AB :n+m + . . .+ ω(c)

s−1∏j=−(s−1)

(n+ j) δn+m, 0

I Define Galilean contraction in usual wayI After contraction ε→ 0 use dimensional arguments, e.g.

[cMUn, cMUm] = . . .+O(cM ) (:CF :n+m + :DE :n+m)

+O(1) :DF :n+m +c2M ω(cL)

s−1∏j=−(s−1)

(n+ j) δn+m, 0,


I Central terms do not survive this rescalingI This is why higher spin states become null states and decouple!





[Wn, Wm] = . . .+ f(c) :AB :n+m + . . .+ ω(c)

s−1∏j=−(s−1)

(n+ j) δn+m, 0



[Un, Um] = . . .+O(1) :DF :n+m (no central terms!)








[Wn, Wm] = . . .+ f(c) :AB :n+m + . . .+ ω(c)

s−1∏j=−(s−1)

(n+ j) δn+m, 0



[Un, Um] = . . .+O(1) :DF :n+m (no central terms!)





YES–GO... every no-go result is only as good as its premises!

Drop assumption of non-linearity!

I Linear higher spin algebra: Pope–Romans–Shen W∞ algebra!

[V im, V

jn

]=

b i+j2 c∑

r=0

gij2r(m,n)V i+j−2rm+n + ci(m) δij δm+n,0

note: wedge algebra is hs(1)

I Ultra-relativistic contraction of generators

V im = V im − V i

−m , W im = ε

(V im + V i

−m)

and central charges

cV = c− c , cW = ε (c+ c)





[V im, V

jn

]=

b i+j2 c∑

r=0




V im = V im − V i

−m , W im = ε

(V im + V i

−m)

and central charges

cV = c− c , cW = ε (c+ c)





[V im, V

jn

]=

b i+j2 c∑

r=0




V im = V im − V i

−m , W im = ε

(V im + V i

−m)

and central charges

cV = c− c , cW = ε (c+ c)


The Treachery of Algebras — Ceci n’est pas une theorie.


YES–GOAsymptotic symmetry algebra of flat space chiral higher spin gravity

I We do not know if flat space chiral higher spin gravity exists...

I ...but its existence is at least not ruled out by the no-go result!I If it exists, this must be its asymptotic symmetry algebra:

[V im,Vjn

]=

b i+j2 c∑

r=0

gij2r(m,n)V i+j−2rm+n + ciV(m) δij δm+n,0

[V im,Wj

n

]=

b i+j2 c∑

r=0

gij2r(m,n)W i+j−2rm+n

[W i

m,Wjn

]= 0

whereciV(m) = #(i, m) × c and c = −c

I Vacuum descendants W im|0〉 are null states for all i and m!

I AdS parent theory: no trace anomaly, but gravitational anomaly(Like in conformal Chern–Simons gravity → Vasiliev type analogue?)



I We do not know if flat space chiral higher spin gravity exists...I ...but its existence is at least not ruled out by the no-go result!

I If it exists, this must be its asymptotic symmetry algebra:

[V im,Vjn

]=

b i+j2 c∑

r=0


[V im,Wj

n

]=

b i+j2 c∑

r=0


[W i

m,Wjn

]= 0






I We do not know if flat space chiral higher spin gravity exists...I ...but its existence is at least not ruled out by the no-go result!I If it exists, this must be its asymptotic symmetry algebra:

[V im,Vjn

]=

b i+j2 c∑

r=0


[V im,Wj

n

]=

b i+j2 c∑

r=0


[W i

m,Wjn

]= 0







[V im,Vjn

]=

b i+j2 c∑

r=0


[V im,Wj

n

]=

b i+j2 c∑

r=0


[W i

m,Wjn

]= 0







[V im,Vjn

]=

b i+j2 c∑

r=0


[V im,Wj

n

]=

b i+j2 c∑

r=0


[W i

m,Wjn

]= 0





Summary and outlook

Main results:I NO–GO: Generically, only two out of three are possible: unitarity, flat

space, non-trivial higher spin statesI Technical reason: non-linearity of algebraI Same argument eliminates also multi-graviton excitations

I YES–GO: Counterexample exists: flat space hs(1) gravityI Technical reason: linear version of asymptotic symmetry algebra exists

Main open issues:

I Other definitions of vacuum/hermitian conjugation that lead tounitarity? (suggestive: Barnich, Oblak, ’14)

I Other linear higher spin algebras? (W1+∞; what else?)I Construction of unitary flat space chiral higher spin gravity

(FSχHSG3)?

Evidence so far for unitary FSχHSG3:constructed its asymptotic symmetry algebra!


Summary and outlook


space, non-trivial higher spin statesI Technical reason: non-linearity of algebraI Same argument eliminates also multi-graviton excitationsI YES–GO: Counterexample exists: flat space hs(1) gravityI Technical reason: linear version of asymptotic symmetry algebra exists

Main open issues:



(FSχHSG3)?



Summary and outlook



Interpretation: Perhaps flat space + unitarity allows only (specific)Vasiliev-type of theories and no truncation to finite spin?

Main open issues:



(FSχHSG3)?



Summary and outlook



Main open issues:I Other definitions of vacuum/hermitian conjugation that lead to

unitarity? (suggestive: Barnich, Oblak, ’14)


(FSχHSG3)?



Summary and outlook




unitarity? (suggestive: Barnich, Oblak, ’14)I Other linear higher spin algebras? (W1+∞; what else?)

I Construction of unitary flat space chiral higher spin gravity(FSχHSG3)?



Summary and outlook




unitarity? (suggestive: Barnich, Oblak, ’14)I Other linear higher spin algebras? (W1+∞; what else?)I Construction of unitary flat space chiral higher spin gravity

(FSχHSG3)?



... hopefully shed Empire of Light on existence of FSχHSG3 in the future!


References

Thanks for your attention!

D. Grumiller, M. Riegler and J. Rosseel, “Unitarity inthree-dimensional flat space higher spin theories,” arXiv:1403.5297.

H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel,Phys. Rev. Lett. 111 (2013) 121603 [arXiv:1307.4768].

A. Bagchi, S. Detournay, D. Grumiller and J. Simon,Phys. Rev. Lett. 111 (2013) 181301 [arXiv:1305.2919].

A. Bagchi, S. Detournay and D. Grumiller,Phys. Rev. Lett. 109 (2012) 151301 [arXiv:1208.1658].

Thanks to Bob McNees for providing the LATEX beamerclass!


Documents

Unitarity in flat space holographyquark.itp.tuwien.ac.at/~grumil/pdf/Brussels2014.pdf · Unitarity in at space holography Daniel Grumiller Institute for Theoretical Physics Vienna