Unitarity in flat space holography
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1 Unitarity in flat space holography Daniel Grumiller Institute for Theoretical Physics Vienna University of Technology Solvay Workshop on Holography for black holes and cosmology, Brussels, April 2014 based on work with Rosseel and Riegler, and work with Afshar, Bagchi, Detournay, Fareghbal, Simon
2 Motivation Holographic principle, if correct, must work beyond AdS/CFT holographic principle: t Hooft 93; Susskind 94 AdS/CFT precursor: Brown, Henneaux 86 AdS/CFT: Maldacena 97; Gubser, Klebanov, Polyakov 98; Witten 98 Daniel Grumiller Unitarity in flat space holography 2/17
3 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Polchinski 99 Susskind 99 Giddings 00 Gary, Giddings, Penedones 09; Gary, Giddings 09;... Daniel Grumiller Unitarity in flat space holography 2/17
4 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Can we find models realizing flat space/field theory correspondences? Barnich, Compere 06 Barnich et al Bagchi et al Strominger et al flat space chiral gravity: Bagchi, Detournay, DG 12 Daniel Grumiller Unitarity in flat space holography 2/17
5 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Can we find models realizing flat space/field theory correspondences? Are there higher-spin versions of such models? Afshar, Bagchi, Fareghbal, DG, Rosseel 13 Gonzalez, Matulich, Pino, Troncoso 13 part of larger program: non-ads holography in higher spin gravity Gary, DG, Rashkov 12 Afshar, Gary, DG, Rashkov, Riegler 12 Gutperle, Hijano, Samani 13 Gary, DG, Prohazka, Rey (in prep.) 14 Daniel Grumiller Unitarity in flat space holography 2/17
6 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Can we find models realizing flat space/field theory correspondences? Are there higher-spin versions of such models? Does this correspondence emerge as limit of (A)dS/CFT? some aspects: yes; other aspects: no Daniel Grumiller Unitarity in flat space holography 2/17
7 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Can we find models realizing flat space/field theory correspondences? Are there higher-spin versions of such models? Does this correspondence emerge as limit of (A)dS/CFT? (When) are these models unitary? this talk! see also Barnich, Oblak 14 (induced representations of BMS 3 ) Daniel Grumiller Unitarity in flat space holography 2/17
8 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Can we find models realizing flat space/field theory correspondences? Are there higher-spin versions of such models? Does this correspondence emerge as limit of (A)dS/CFT? (When) are these models unitary? Is there an analog of Hawking Page phase transition? yes: Bagchi, Detournay, DG, Simon 13 Detournay, DG, Schöller, Simon 14 Daniel Grumiller Unitarity in flat space holography 2/17
9 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Can we find models realizing flat space/field theory correspondences? Are there higher-spin versions of such models? Does this correspondence emerge as limit of (A)dS/CFT? (When) are these models unitary? Is there an analog of Hawking Page phase transition? Can we relate S-matrix observables to holographic observables? Strominger 13 He, Lysov, Mitra, Strominger 14 Banks 14 Cachazo, Strominger Daniel Grumiller Unitarity in flat space holography 2/17
10 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Can we find models realizing flat space/field theory correspondences? Are there higher-spin versions of such models? Does this correspondence emerge as limit of (A)dS/CFT? (When) are these models unitary? Is there an analog of Hawking Page phase transition? Can we relate S-matrix observables to holographic observables? Strominger 13 He, Lysov, Mitra, Strominger 14 Banks 14 Cachazo, Strominger 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
11 Outline of procedure Work mostly on CFT side and study landscape of possible flat space asymptotic symmetry algebras: Daniel Grumiller Unitarity in flat space holography 3/17
12 Outline of procedure Work mostly on CFT side and study landscape of possible flat space asymptotic symmetry algebras: 1. Consider some symmetry algebra arising in a relativistic CFT 2 (two copies of Virasoro or some W-algebra) Daniel Grumiller Unitarity in flat space holography 3/17
13 Outline of procedure Work mostly on CFT side and study landscape of possible flat space asymptotic symmetry algebras: 1. Consider some symmetry algebra arising in a relativistic CFT 2 (two copies of Virasoro or some W-algebra) 2. İnönü Wigner contract to Galilean/Ultra-relativistic conformal algebra (GCA 2 /URCA 2 ) Daniel Grumiller Unitarity in flat space holography 3/17
14 Outline of procedure Work mostly on CFT side and study landscape of possible flat space asymptotic symmetry algebras: 1. Consider some symmetry algebra arising in a relativistic CFT 2 (two copies of Virasoro or some W-algebra) 2. İnönü Wigner contract to Galilean/Ultra-relativistic conformal algebra (GCA 2 /URCA 2 ) 3. Define vacuum and hermitian conjugation Daniel Grumiller Unitarity in flat space holography 3/17
15 Outline of procedure Work mostly on CFT side and study landscape of possible flat space asymptotic symmetry algebras: 1. Consider some symmetry algebra arising in a relativistic CFT 2 (two copies of Virasoro or some W-algebra) 2. İnönü Wigner contract to Galilean/Ultra-relativistic conformal algebra (GCA 2 /URCA 2 ) 3. Define vacuum and hermitian conjugation 4. Demand non-negativity norm of vacuum descendants Daniel Grumiller Unitarity in flat space holography 3/17
16 Outline of procedure Work mostly on CFT side and study landscape of possible flat space asymptotic symmetry algebras: 1. Consider some symmetry algebra arising in a relativistic CFT 2 (two copies of Virasoro or some W-algebra) 2. İnönü Wigner contract to Galilean/Ultra-relativistic conformal algebra (GCA 2 /URCA 2 ) 3. Define vacuum and hermitian conjugation 4. Demand non-negativity norm of vacuum descendants 5. Check consequences for central charges and algebra Daniel Grumiller Unitarity in flat space holography 3/17
17 Outline of procedure Work mostly on CFT side and study landscape of possible flat space asymptotic symmetry algebras: 1. Consider some symmetry algebra arising in a relativistic CFT 2 (two copies of Virasoro or some W-algebra) 2. İnönü Wigner contract to Galilean/Ultra-relativistic conformal algebra (GCA 2 /URCA 2 ) 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 Daniel Grumiller Unitarity in flat space holography 3/17
18 Outline of procedure Work mostly on CFT side and study landscape of possible flat space asymptotic symmetry algebras: 1. Consider some symmetry algebra arising in a relativistic CFT 2 (two copies of Virasoro or some W-algebra) 2. İnönü Wigner contract to Galilean/Ultra-relativistic conformal algebra (GCA 2 /URCA 2 ) 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.! Daniel Grumiller Unitarity in flat space holography 3/17
19 Main results 1. NO GO: Generically (see later) you can have only two out of three: Unitarity Flat space Non-trivial higher spin states Daniel Grumiller Unitarity in flat space holography 4/17
20 Main results 1. NO GO: Generically (see later) you can have only two out of three: Unitarity Flat space Non-trivial higher spin states Example: Flat space chiral gravity Bagchi, Detournay, DG, Daniel Grumiller Unitarity in flat space holography 4/17
21 Main results 1. NO GO: Generically (see later) you can have only two out of three: Unitarity Flat space Non-trivial higher spin states Example: Minimal model holography Gaberdiel, Gopakumar, , Daniel Grumiller Unitarity in flat space holography 4/17
22 Main results 1. NO GO: Generically (see later) you can have only two out of three: Unitarity Flat space Non-trivial higher spin states Example: Flat space higher spin gravity (Galilean W 3 algebra) Afshar, Bagchi, Fareghbal, DG and Rosseel, Gonzalez, Matulich, Pino and Troncoso, Daniel Grumiller Unitarity in flat space holography 4/17
23 Main results 1. NO GO: Generically (see later) you can have only two out of three: Unitarity Flat space Non-trivial higher spin states Compatible with spirit of various no-go results in higher dimensions! Daniel Grumiller Unitarity in flat space holography 4/17
24 Main results 1. NO GO: Generically (see later) you can have only two out of three: Unitarity Flat space Non-trivial higher spin states Compatible with spirit of various no-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? Daniel Grumiller Unitarity in flat space holography 4/17
25 Daniel Grumiller Unitarity in flat space holography 5/17
26 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2/URCA 2 Take two copies of Virasoro, generators L n, Ln, central charges c, c Daniel Grumiller Unitarity in flat space holography 6/17
27 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2/URCA 2 Take two copies of Virasoro, generators L n, Ln, central charges c, c Define superrotations L n and supertranslations M n GCA: L n := L n + L n M n := ɛ ( L n L n ) URCA: L n := L n L n M n := ɛ ( L n + L n ) Daniel Grumiller Unitarity in flat space holography 6/17
28 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2/URCA 2 Take two copies of Virasoro, generators L n, Ln, central charges c, c Define superrotations L n and supertranslations M n GCA: L n := L n + L n M n := ɛ ( L n L n ) URCA: L n := L n L n M n := ɛ ( L n + L n ) Galilean limit/ultrarelativistic boost: ɛ 1/l 0 [L n, L m ] = (n m) L n+m + c L 1 12 δ n+m, 0 [L n, M m ] = (n m) M n+m + c M 1 12 δ n+m, 0 [M n, M m ] = 0 Daniel Grumiller Unitarity in flat space holography 6/17
29 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2/URCA 2 Take two copies of Virasoro, generators L n, Ln, central charges c, c Define superrotations L n and supertranslations M n GCA: L n := L n + L n M n := ɛ ( L n L n ) URCA: L n := L n L n M n := ɛ ( L n + L n ) Galilean limit/ultrarelativistic boost: ɛ 1/l 0 [L n, L m ] = (n m) L n+m + c L 1 12 δ n+m, 0 [L n, M m ] = (n m) M n+m + c M 1 12 δ n+m, 0 [M n, M m ] = 0 Is precisely (centrally extended) BMS 3 algebra! Central charges: GCA: c L = c + c c M = ɛ(c c) URCA: c L = c c c M = ɛ(c + c) Daniel Grumiller Unitarity in flat space holography 6/17
30 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2/URCA 2 Take two copies of Virasoro, generators L n, Ln, central charges c, c Define superrotations L n and supertranslations M n GCA: L n := L n + L n M n := ɛ ( L n L n ) URCA: L n := L n L n M n := ɛ ( L n + L n ) Galilean limit/ultrarelativistic boost: ɛ 1/l 0 [L n, L m ] = (n m) L n+m + c L 1 12 δ n+m, 0 [L n, M m ] = (n m) M n+m + c M 1 12 δ n+m, 0 [M n, M m ] = 0 Is precisely (centrally extended) BMS 3 algebra! Central charges: GCA: c L = c + c c M = ɛ(c c) URCA: c L = c c c M = ɛ(c + c) Example: URCA TMG: c L = 3/µG and c M = 3/G [dimensionful!] Daniel Grumiller Unitarity in flat space holography 6/17
31 (Non-)unitarity in GCAs Assumptions: Standard highest weight vacuum: L n 0 = M n 0 = 0 n 1 Daniel Grumiller Unitarity in flat space holography 7/17
32 (Non-)unitarity in GCAs Assumptions: Standard highest weight vacuum: L n 0 = M n 0 = 0 Standard hermitian conjugation: n 1 L n := L n M n := M n Daniel Grumiller Unitarity in flat space holography 7/17
33 (Non-)unitarity in GCAs Assumptions: Standard highest weight vacuum: L n 0 = M n 0 = 0 Standard hermitian conjugation: n 1 L n := L n M n := M n Conclusions: Gram matrix of level 2 descendants: ( 0 L2 L 2 0 = c L 2 0 L 2 M 2 0 = c M 2 0 M 2 L 2 0 = c M 2 0 M 2 M 2 0 = 0 ) Daniel Grumiller Unitarity in flat space holography 7/17
34 (Non-)unitarity in GCAs Assumptions: Standard highest weight vacuum: L n 0 = M n 0 = 0 Standard hermitian conjugation: n 1 L n := L n M n := M n Conclusions: Gram matrix of level 2 descendants: ( 0 L2 L 2 0 = c L 2 0 L 2 M 2 0 = c M 2 0 M 2 L 2 0 = c M 2 0 M 2 M 2 0 = 0 Determinant: c 2 M /4 0 ) Daniel Grumiller Unitarity in flat space holography 7/17
35 (Non-)unitarity in GCAs Assumptions: Standard highest weight vacuum: L n 0 = M n 0 = 0 Standard hermitian conjugation: n 1 L n := L n M n := M n Conclusions: Gram matrix of level 2 descendants: ( 0 L2 L 2 0 = c L 2 0 L 2 M 2 0 = c M 2 0 M 2 L 2 0 = c M 2 0 M 2 M 2 0 = 0 Determinant: c 2 M /4 0 c M 0: one positive and one negative norm state ) Daniel Grumiller Unitarity in flat space holography 7/17
36 (Non-)unitarity in GCAs Assumptions: Standard highest weight vacuum: L n 0 = M n 0 = 0 Standard hermitian conjugation: n 1 L n := L n M n := M n Conclusions: Gram matrix of level 2 descendants: ( 0 L2 L 2 0 = c L 2 0 L 2 M 2 0 = c M 2 0 M 2 L 2 0 = c M 2 0 M 2 M 2 0 = 0 Determinant: c 2 M /4 0 c M 0: one positive and one negative norm state c M = 0, c L < 0: one negative norm and one null state ) Daniel Grumiller Unitarity in flat space holography 7/17
37 (Non-)unitarity in GCAs Assumptions: Standard highest weight vacuum: L n 0 = M n 0 = 0 Standard hermitian conjugation: n 1 L n := L n M n := M n Conclusions: Gram matrix of level 2 descendants: ( 0 L2 L 2 0 = c L 2 0 L 2 M 2 0 = c M 2 0 M 2 L 2 0 = c M 2 0 M 2 M 2 0 = 0 Determinant: c 2 M /4 0 c M 0: one positive and one negative norm state c M = 0, c L < 0: one negative norm and one null state c M = 0, c L = 0: only null states (trivial) ) Daniel Grumiller Unitarity in flat space holography 7/17
38 (Non-)unitarity in GCAs Assumptions: Standard highest weight vacuum: L n 0 = M n 0 = 0 Standard hermitian conjugation: n 1 L n := L n M n := M n Conclusions: Gram matrix of level 2 descendants: ( 0 L2 L 2 0 = c L 2 0 L 2 M 2 0 = c M 2 0 M 2 L 2 0 = c M 2 0 M 2 M 2 0 = 0 Determinant: c 2 M /4 0 c M 0: one positive and one negative norm state c M = 0, c L < 0: one negative norm and one null state c M = 0, c L = 0: only null states (trivial) c M = 0, c L > 0: one positive norm and one null state ) Daniel Grumiller Unitarity in flat space holography 7/17
39 Generalization to higher spin asymptotic symmetry algebras Facts: Unitarity in GCA requires c M = 0 Daniel Grumiller Unitarity in flat space holography 8/17
40 Generalization to higher spin asymptotic symmetry algebras Facts: Unitarity in GCA requires c M = 0 Non-triviality requires then c L 0 Daniel Grumiller Unitarity in flat space holography 8/17
41 Generalization to higher spin asymptotic symmetry algebras Facts: Unitarity in GCA requires c M = 0 Non-triviality requires then c L 0 Generalization to contracted higher spin algebras straightforward Daniel Grumiller Unitarity in flat space holography 8/17
42 Generalization to higher spin asymptotic symmetry algebras Facts: Unitarity in GCA requires c M = 0 Non-triviality requires then c L 0 Generalization to contracted higher spin algebras straightforward All of them contain GCA as subalgebra Daniel Grumiller Unitarity in flat space holography 8/17
43 Generalization to higher spin asymptotic symmetry algebras Facts: Unitarity in GCA requires c M = 0 Non-triviality requires then c L 0 Generalization to contracted higher spin algebras straightforward All of them contain GCA as subalgebra Again c M = 0 is necessary for unitarity Daniel Grumiller Unitarity in flat space holography 8/17
44 Generalization to higher spin asymptotic symmetry algebras Facts: Unitarity in GCA requires c M = 0 Non-triviality requires then c L 0 Generalization to contracted higher spin algebras straightforward All of them contain GCA as subalgebra Again c M = 0 is necessary for unitarity Example: Galilean W 3 algebra: [L n, L m ] and [L n, M m ] as before, [L n, U m ] = (2n m)u n+m [L n, V m ] = [M n, U m ] = (2n m)v n+m [U n, U m ] = (n m)(2n 2 + 2m 2 nm 8)L n+m (n m)λ n+m c M 96( c L + 44 ) 5 (n m)θ n+m + c L 12 n(n2 1)(n 2 4) δ n+m, 0 c 2 M [U n, V m ] = (n m)(2n 2 + 2m 2 nm 8)M n+m + 96 c M (n m)θ n+m + c M 12 n(n2 1)(n 2 4) δ n+m, 0 Daniel Grumiller Unitarity in flat space holography 8/17
45 Unitarity leads to further contraction! (Afshar et al ) İnönü Wigner contraction of two W-algebras (generators L, W): L n := L n + L n M n := ɛ ( L n L n ) U n := W n + W n V n := ɛ ( W n W n ) Daniel Grumiller Unitarity in flat space holography 9/17
46 Unitarity leads to further contraction! (Afshar et al ) İnönü Wigner contraction of two W-algebras (generators L, W): L n := L n + L n M n := ɛ ( L n L n ) U n := W n + W n V n := ɛ ( W n W n ) Limit c M 0 requires further contraction: U n c M U n Daniel Grumiller Unitarity in flat space holography 9/17
47 Unitarity leads to further contraction! (Afshar et al ) İnönü Wigner contraction of two W-algebras (generators L, W): L n := L n + L n M n := ɛ ( L n L n ) U n := W n + W n V n := ɛ ( W n W n ) Limit c M 0 requires further contraction: U n c M U n Doubly contracted algebra has unitary representations: [L n, L m ] = (n m)l n+m + c L 12 (n3 n) δ n+m, 0 [L n, M m ] = (n m)m n+m [L n, U m ] = (2n m)u n+m [L n, V m ] = (2n m)v n+m [U n, U m ] [U n, V m ] = 96(n m) p M p M n p Daniel Grumiller Unitarity in flat space holography 9/17
48 Unitarity leads to further contraction! (Afshar et al ) İnönü Wigner contraction of two W-algebras (generators L, W): L n := L n + L n M n := ɛ ( L n L n ) U n := W n + W n V n := ɛ ( W n W n ) Limit c M 0 requires further contraction: U n c M U n Doubly contracted algebra has unitary representations: [L n, L m ] = (n m)l n+m + c L 12 (n3 n) δ n+m, 0 [L n, M m ] = (n m)m n+m [L n, U m ] = (2n m)u n+m [L n, V m ] = (2n m)v n+m [U n, U m ] [U n, V m ] = 96(n m) p M p M n p Higher spin states decouple and become null states! Daniel Grumiller Unitarity in flat space holography 9/17
49 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Daniel Grumiller Unitarity in flat space holography 10/17
50 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Contracted Polyakov Bershadsky (Galilean W (2) 3 ) Unitary for 1 c L 32 Daniel Grumiller Unitarity in flat space holography 10/17
51 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Contracted Polyakov Bershadsky (Galilean W (2) 3 ) Unitary for 1 c L 32 Contracted Feigin Semikhatov (Galilean W (2) N ) Unitary for 1 c L 2(N 1) 2 (N + 1) N 3 (for large N) Daniel Grumiller Unitarity in flat space holography 10/17
52 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Contracted Polyakov Bershadsky (Galilean W (2) 3 ) Unitary for 1 c L 32 Contracted Feigin Semikhatov (Galilean W (2) N ) Unitary for 1 c L 2(N 1) 2 (N + 1) N 3 (for large N) Contracted W (2 1 1) 4 Unitary for c L 42 Daniel Grumiller Unitarity in flat space holography 10/17
53 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Contracted Polyakov Bershadsky (Galilean W (2) 3 ) Unitary for 1 c L 32 Contracted Feigin Semikhatov (Galilean W (2) N ) Unitary for 1 c L 2(N 1) 2 (N + 1) N 3 (for large N) Contracted W (2 1 1) 4 Unitary for c L 42 Daniel Grumiller Unitarity in flat space holography 10/17
54 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Contracted Polyakov Bershadsky (Galilean W (2) 3 ) Unitary for 1 c L 32 Contracted Feigin Semikhatov (Galilean W (2) N ) Unitary for 1 c L 2(N 1) 2 (N + 1) N 3 (for large N) Contracted W (2 1 1) 4 Unitary for c L 42 Upper bound from current algebra part: û(1) : [O n, O m ] = 2(54 c L) 3 su(2) ˆ : n δ n+m, 0 [Q a n, Q b m] = (a b)q a+b n+m + 42 c L n δ a+b, 0 δ n+m, 0 6 Daniel Grumiller Unitarity in flat space holography 10/17
55 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Contracted Polyakov Bershadsky (Galilean W (2) 3 ) Unitary for 1 c L 32 Contracted Feigin Semikhatov (Galilean W (2) N ) Unitary for 1 c L 2(N 1) 2 (N + 1) N 3 (for large N) Contracted W (2 1 1) 4 Unitary for c L 42 Lower bound: non-negativity of c bare in c L = cû(1) + cŝu(2) + c bare with c... = k dim g/(k + h ) with level k = (42 c L )/6, thus: cû(1) = 1 cŝu(2) = 3(42 c L )/(54 c L ) c bare 0 implies c L Daniel Grumiller Unitarity in flat space holography 10/17
56 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Contracted Polyakov Bershadsky (Galilean W (2) 3 ) Unitary for 1 c L 32 Contracted Feigin Semikhatov (Galilean W (2) N ) Unitary for 1 c L 2(N 1) 2 (N + 1) N 3 (for large N) Contracted W (2 1 1) 4 Unitary for c L 42 Decoupling of higher spin states seems to be a generic feature! But why? Daniel Grumiller Unitarity in flat space holography 10/17
57 How general is this decoupling mechanism? (DG, Riegler, Rosseel 14) Same conclusions all higher spin states become null states for Contracted principal embedding (Galilean W N ) Unitary for positive c L Contracted Polyakov Bershadsky (Galilean W (2) 3 ) Unitary for 1 c L 32 Contracted Feigin Semikhatov (Galilean W (2) N ) Unitary for 1 c L 2(N 1) 2 (N + 1) N 3 (for large N) Contracted W (2 1 1) 4 Unitary for c L 42 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! Daniel Grumiller Unitarity in flat space holography 10/17
58 General NO GO result See for details! Idea of no-go proof: Assume non-linearity in W-algebra, e.g. (lim c f(c) 0) [W n, W m ] =... + f(c) :AB : n+m ω(c) s 1 j= (s 1) (n + j) δ n+m, 0 Daniel Grumiller Unitarity in flat space holography 11/17
59 General NO GO result See for details! Idea of no-go proof: Assume non-linearity in W-algebra, e.g. (lim c f(c) 0) [W n, W m ] =... + f(c) :AB : n+m ω(c) s 1 j= (s 1) (n + j) δ n+m, 0 Define Galilean contraction in usual way, e.g. U n := W n + W n V n := ɛ ( W n W ) n C n := A n + Ān D n := ɛ ( ) A n Ān E n := B n + B n F n := ɛ ( B n B ) n Daniel Grumiller Unitarity in flat space holography 11/17
60 General NO GO result See for details! Idea of no-go proof: Assume non-linearity in W-algebra, e.g. (lim c f(c) 0) [W n, W m ] =... + f(c) :AB : n+m ω(c) s 1 j= (s 1) Define Galilean contraction in usual way After contraction ɛ 0 use dimensional arguments, e.g. No central terms in higher spin generators: [V n, V m ] = 0 (trivial) [U n, V m ] = + # c M δ n+m, 0 (dimensions!) (n + j) δ n+m, 0 Daniel Grumiller Unitarity in flat space holography 11/17
61 General NO GO result See for details! Idea of no-go proof: Assume non-linearity in W-algebra, e.g. (lim c f(c) 0) [W n, W m ] =... + f(c) :AB : n+m ω(c) s 1 j= (s 1) Define Galilean contraction in usual way After contraction ɛ 0 use dimensional arguments, e.g. [U n, U m ] =... + O(1/c M ) (:CF : n+m + :DE : n+m ) + O( 1 c 2 ) :DF : n+m + ω(c L ) M }{{} 0 s 1 j= (s 1) (n + j) δ n+m, 0 (n + j) δ n+m, 0, Daniel Grumiller Unitarity in flat space holography 11/17
62 General NO GO result See for details! Idea of no-go proof: Assume non-linearity in W-algebra, e.g. (lim c f(c) 0) [W n, W m ] =... + f(c) :AB : n+m ω(c) s 1 j= (s 1) Define Galilean contraction in usual way After contraction ɛ 0 use dimensional arguments, e.g. [c M U n, c M U m ] =... + O(c M ) (:CF : n+m + :DE : n+m ) + O(1) :DF : n+m +c 2 M ω(c L ) s 1 j= (s 1) (n + j) δ n+m, 0 (n + j) δ n+m, 0, Need rescaling U n c M U n = Ũn before taking limit c M 0 Daniel Grumiller Unitarity in flat space holography 11/17
63 General NO GO result See for details! Idea of no-go proof: Assume non-linearity in W-algebra, e.g. (lim c f(c) 0) [W n, W m ] =... + f(c) :AB : n+m ω(c) Define Galilean contraction in usual way s 1 j= (s 1) After contraction ɛ 0 use dimensional arguments, e.g. (n + j) δ n+m, 0 [Ũn, Ũm] =... + O(1) :DF : n+m (no central terms!) Need rescaling U n c M U n = Ũn before taking limit c M 0 Central terms do not survive this rescaling Daniel Grumiller Unitarity in flat space holography 11/17
64 General NO GO result See for details! Idea of no-go proof: Assume non-linearity in W-algebra, e.g. (lim c f(c) 0) [W n, W m ] =... + f(c) :AB : n+m ω(c) Define Galilean contraction in usual way s 1 j= (s 1) After contraction ɛ 0 use dimensional arguments, e.g. (n + j) δ n+m, 0 [Ũn, Ũm] =... + O(1) :DF : n+m (no central terms!) Need rescaling U n c M U n = Ũn before taking limit c M 0 Central terms do not survive this rescaling This is why higher spin states become null states and decouple! Daniel Grumiller Unitarity in flat space holography 11/17
65 YES GO... every no-go result is only as good as its premises! Drop assumption of non-linearity! Daniel Grumiller Unitarity in flat space holography 12/17
66 YES GO... every no-go result is only as good as its premises! Drop assumption of non-linearity! Linear higher spin algebra: Pope Romans Shen W algebra! [ V i m, V j n ] i+j 2 = r=0 note: wedge algebra is hs(1) g ij i+j 2r 2r (m, n) Vm+n + c i (m) δ ij δ m+n,0 Daniel Grumiller Unitarity in flat space holography 12/17
67 YES GO... every no-go result is only as good as its premises! Drop assumption of non-linearity! Linear higher spin algebra: Pope Romans Shen W algebra! [ V i m, V j n ] i+j 2 = r=0 note: wedge algebra is hs(1) g ij i+j 2r 2r (m, n) Vm+n + c i (m) δ ij δ m+n,0 Ultra-relativistic contraction of generators Vm i = Vm i V m i, Wm i = ɛ ( Vm i + V m i ) and central charges c V = c c, c W = ɛ (c + c) Daniel Grumiller Unitarity in flat space holography 12/17
68 The Treachery of Algebras Ceci n est pas une the orie. Daniel Grumiller Unitarity in flat space holography 13/17
69 YES GO Asymptotic symmetry algebra of flat space chiral higher spin gravity We do not know if flat space chiral higher spin gravity exists... Daniel Grumiller Unitarity in flat space holography 14/17
70 YES GO Asymptotic symmetry algebra of flat space chiral higher spin gravity We do not know if flat space chiral higher spin gravity exists......but its existence is at least not ruled out by the no-go result! Daniel Grumiller Unitarity in flat space holography 14/17
71 YES GO Asymptotic symmetry algebra of flat space chiral higher spin gravity We do not know if flat space chiral higher spin gravity exists......but its existence is at least not ruled out by the no-go result! If it exists, this must be its asymptotic symmetry algebra: where [ V i m, Vn j ] i+j = 2 r=0 [ V i m, Wn j ] i+j = 2 r=0 g ij 2r (m, n) Vi+j 2r m+n + c i V(m) δ ij δ m+n,0 g ij 2r (m, n) Wi+j 2r m+n c i V(m) = #(i, m) c and c = c [ W i m, Wn j ] = 0 Daniel Grumiller Unitarity in flat space holography 14/17
72 YES GO Asymptotic symmetry algebra of flat space chiral higher spin gravity We do not know if flat space chiral higher spin gravity exists......but its existence is at least not ruled out by the no-go result! If it exists, this must be its asymptotic symmetry algebra: where [ V i m, Vn j ] i+j = 2 r=0 [ V i m, Wn j ] i+j = 2 r=0 g ij 2r (m, n) Vi+j 2r m+n + c i V(m) δ ij δ m+n,0 g ij 2r (m, n) Wi+j 2r m+n c i V(m) = #(i, m) c and c = c [ W i m, Wn j ] = 0 Vacuum descendants W i m 0 are null states for all i and m! Daniel Grumiller Unitarity in flat space holography 14/17
73 YES GO Asymptotic symmetry algebra of flat space chiral higher spin gravity We do not know if flat space chiral higher spin gravity exists......but its existence is at least not ruled out by the no-go result! If it exists, this must be its asymptotic symmetry algebra: where [ V i m, Vn j ] i+j = 2 r=0 [ V i m, Wn j ] i+j = 2 r=0 g ij 2r (m, n) Vi+j 2r m+n + c i V(m) δ ij δ m+n,0 g ij 2r (m, n) Wi+j 2r m+n c i V(m) = #(i, m) c and c = c [ W i m, Wn j ] = 0 Vacuum descendants W i m 0 are null states for all i and m! AdS parent theory: no trace anomaly, but gravitational anomaly (Like in conformal Chern Simons gravity Vasiliev type analogue?) Daniel Grumiller Unitarity in flat space holography 14/17
74 Summary and outlook Main results: NO GO: Generically, only two out of three are possible: unitarity, flat space, non-trivial higher spin states Technical reason: non-linearity of algebra Same argument eliminates also multi-graviton excitations Daniel Grumiller Unitarity in flat space holography 15/17
75 Summary and outlook Main results: NO GO: Generically, only two out of three are possible: unitarity, flat space, non-trivial higher spin states Technical reason: non-linearity of algebra Same argument eliminates also multi-graviton excitations YES GO: Counterexample exists: flat space hs(1) gravity Technical reason: linear version of asymptotic symmetry algebra exists Daniel Grumiller Unitarity in flat space holography 15/17
76 Summary and outlook Main results: NO GO: Generically, only two out of three are possible: unitarity, flat space, non-trivial higher spin states Technical reason: non-linearity of algebra Same argument eliminates also multi-graviton excitations YES GO: Counterexample exists: flat space hs(1) gravity Technical reason: linear version of asymptotic symmetry algebra exists Interpretation: Perhaps flat space + unitarity allows only (specific) Vasiliev-type of theories and no truncation to finite spin? Daniel Grumiller Unitarity in flat space holography 15/17
77 Summary and outlook Main results: NO GO: Generically, only two out of three are possible: unitarity, flat space, non-trivial higher spin states Technical reason: non-linearity of algebra Same argument eliminates also multi-graviton excitations YES GO: Counterexample exists: flat space hs(1) gravity Technical reason: linear version of asymptotic symmetry algebra exists Main open issues: Other definitions of vacuum/hermitian conjugation that lead to unitarity? (suggestive: Barnich, Oblak, 14) Daniel Grumiller Unitarity in flat space holography 15/17
78 Summary and outlook Main results: NO GO: Generically, only two out of three are possible: unitarity, flat space, non-trivial higher spin states Technical reason: non-linearity of algebra Same argument eliminates also multi-graviton excitations YES GO: Counterexample exists: flat space hs(1) gravity Technical reason: linear version of asymptotic symmetry algebra exists Main open issues: Other definitions of vacuum/hermitian conjugation that lead to unitarity? (suggestive: Barnich, Oblak, 14) Other linear higher spin algebras? (W 1+ ; what else?) Daniel Grumiller Unitarity in flat space holography 15/17
79 Summary and outlook Main results: NO GO: Generically, only two out of three are possible: unitarity, flat space, non-trivial higher spin states Technical reason: non-linearity of algebra Same argument eliminates also multi-graviton excitations YES GO: Counterexample exists: flat space hs(1) gravity Technical reason: linear version of asymptotic symmetry algebra exists Main open issues: Other definitions of vacuum/hermitian conjugation that lead to unitarity? (suggestive: Barnich, Oblak, 14) Other linear higher spin algebras? (W 1+ ; what else?) Construction of unitary flat space chiral higher spin gravity (FSχHSG 3 )? Evidence so far for unitary FSχHSG 3 : constructed its asymptotic symmetry algebra! Daniel Grumiller Unitarity in flat space holography 15/17
80 ... hopefully shed Empire of Light on existence of FSχHSG 3 in the future! Daniel Grumiller Unitarity in flat space holography 16/17
81 References Thanks for your attention! D. Grumiller, M. Riegler and J. Rosseel, Unitarity in three-dimensional flat space higher spin theories, arxiv: H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Phys. Rev. Lett. 111 (2013) [arxiv: ]. A. Bagchi, S. Detournay, D. Grumiller and J. Simon, Phys. Rev. Lett. 111 (2013) [arxiv: ]. A. Bagchi, S. Detournay and D. Grumiller, Phys. Rev. Lett. 109 (2012) [arxiv: ]. Thanks to Bob McNees for providing the L A TEX beamerclass! Daniel Grumiller Unitarity in flat space holography 17/17
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