Holography and phase-transition of flat space

Transcription

1 Holography and phase-transition of flat space Daniel Grumiller Institute for Theoretical Physics Vienna University of Technology Workshop on Higher-Spin and Higher-Curvature Gravity, São Paulo, 4. November 2013, 16:00 BRST based on work with Afshar, Bagchi, Detournay, Fareghbal, Rosseel, Schöller, Simon

2 Motivation Holographic principle, if correct, must work beyond AdS/CFT Daniel Grumiller Holography and phase-transition of flat space 2/16

3 Motivation Holographic principle, if correct, must work beyond AdS/CFT Does it work in flat space? Daniel Grumiller Holography and phase-transition of flat space 2/16

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? Daniel Grumiller Holography and phase-transition of flat space 2/16

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? Daniel Grumiller Holography and phase-transition of flat space 2/16

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? Daniel Grumiller Holography and phase-transition of flat space 2/16

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? Daniel Grumiller Holography and phase-transition of flat space 2/16

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? Daniel Grumiller Holography and phase-transition of flat space 2/16

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 related S-matrix observables to holographic observables?... Daniel Grumiller Holography and phase-transition of flat space 2/16

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 related S-matrix observables to holographic observables?... Address these question in 3D! Daniel Grumiller Holography and phase-transition of flat space 2/16

11 Gravity 3D is a spellbinding experience... so let us consider 3D gravity! Daniel Grumiller Holography and phase-transition of flat space 3/16

12 (Higher spin) gravity as Chern Simons gauge theory......with weird boundary conditions (Achucarro & Townsend 86; Witten 88; Bañados 96) CS action: S CS = k 4π Variational principle: δs CS EOM = k 4π CS(A) k 4π CS(Ā) tr ( A δa Ā δā) Well-defined for boundary conditions (similarly for Ā) A + = 0 or A = 0 boundary coordinates x ± Example: asymptotically AdS 3 (Arctan-version of Brown Henneaux) Daniel Grumiller Holography and phase-transition of flat space 4/16

13 (Higher spin) gravity as Chern Simons gauge theory......with weird boundary conditions (Achucarro & Townsend 86; Witten 88; Bañados 96) CS action: S CS = k 4π Variational principle: δs CS EOM = k 4π CS(A) k 4π CS(Ā) tr ( A δa Ā δā) Well-defined for boundary conditions (similarly for Ā) A + = 0 or A = 0 boundary coordinates x ± Example: asymptotically AdS 3 (Cartan-version of Brown Henneaux) A ρ = L 0 Ā ρ = L 0 A + = e ρ L 1 + e ρ L(x + ) L 1 Ā + = 0 A = 0 Ā = e ρ L 1 e ρ L(x ) L 1 Dreibein: e/l A Ā, spin-connection: ω A + Ā Daniel Grumiller Holography and phase-transition of flat space 4/16

14 Non-AdS holography Variational principle for non-vanishing boundary connection (Gary, DG & Rashkov 12) Want to relax A = Ā+ = 0, e.g. to δa = δā+ = 0 Daniel Grumiller Holography and phase-transition of flat space 5/16

15 Non-AdS holography Variational principle for non-vanishing boundary connection (Gary, DG & Rashkov 12) Want to relax A = Ā+ = 0, e.g. to δa = δā+ = 0 Add boundary term to CS action S B = k dx + dx tr ( ) A + A + 4π Ā+Ā Daniel Grumiller Holography and phase-transition of flat space 5/16

16 Non-AdS holography Variational principle for non-vanishing boundary connection (Gary, DG & Rashkov 12) Want to relax A = Ā+ = 0, e.g. to δa = δā+ = 0 Add boundary term to CS action S B = k dx + dx tr ( ) A + A + 4π Ā+Ā Invariant under (anti-)holomorphic boundary diffeos, but not fully boundary diff-invariant! Daniel Grumiller Holography and phase-transition of flat space 5/16

17 Non-AdS holography Variational principle for non-vanishing boundary connection (Gary, DG & Rashkov 12) Want to relax A = Ā+ = 0, e.g. to δa = δā+ = 0 Add boundary term to CS action S B = k dx + dx tr ( ) A + A + 4π Ā+Ā Invariant under (anti-)holomorphic boundary diffeos, but not fully boundary diff-invariant! Variation of full action: δ ( S CS + S B ) = k 2π dx + dx ( ) A + δa + Ā δā+ Daniel Grumiller Holography and phase-transition of flat space 5/16

18 Non-AdS holography Variational principle for non-vanishing boundary connection (Gary, DG & Rashkov 12) Want to relax A = Ā+ = 0, e.g. to δa = δā+ = 0 Add boundary term to CS action S B = k dx + dx tr ( ) A + A + 4π Ā+Ā Invariant under (anti-)holomorphic boundary diffeos, but not fully boundary diff-invariant! Variation of full action: δ ( S CS + S B ) = k 2π dx + dx ( ) A + δa + Ā δā+ Gain: can do non-ads holography in (higher-spin) gravity, including Lobachevsky, warped AdS, Lifshitz and Schrödinger Daniel Grumiller Holography and phase-transition of flat space 5/16

19 Non-AdS holography Variational principle for non-vanishing boundary connection (Gary, DG & Rashkov 12) Want to relax A = Ā+ = 0, e.g. to δa = δā+ = 0 Add boundary term to CS action S B = k dx + dx tr ( ) A + A + 4π Ā+Ā Invariant under (anti-)holomorphic boundary diffeos, but not fully boundary diff-invariant! Variation of full action: δ ( S CS + S B ) = k 2π dx + dx ( ) A + δa + Ā δā+ Gain: can do non-ads holography in (higher-spin) gravity, including Lobachevsky, warped AdS, Lifshitz and Schrödinger Simplest examples: Lobachevsky holography for SO(3, 2) broken to SO(2, 2) U(1) (conformal CS gravity, Bertin, Ertl, Ghorbani, DG, Johansson & Vassilevich 12) and null warped holography for principally embedded spin-3 gravity (in prep. with Gary & Perlmutter) Daniel Grumiller Holography and phase-transition of flat space 5/16

20 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2 (or rather, URCA 2) Take two copies of Virasoro, generators L n, Ln, central charges c, c Daniel Grumiller Holography and phase-transition of flat space 6/16

21 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2 (or rather, URCA 2) Take two copies of Virasoro, generators L n, Ln, central charges c, c Define superrotations L n and supertranslations M n L n := L n L n M n := 1 ( l Ln + L ) n Daniel Grumiller Holography and phase-transition of flat space 6/16

22 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2 (or rather, URCA 2) Take two copies of Virasoro, generators L n, Ln, central charges c, c Define superrotations L n and supertranslations M n L n := L n L n M n := 1 ( l Ln + L ) n Make ultrarelativistic boost, l [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 Holography and phase-transition of flat space 6/16

23 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2 (or rather, URCA 2) Take two copies of Virasoro, generators L n, Ln, central charges c, c Define superrotations L n and supertranslations M n L n := L n L n M n := 1 ( l Ln + L ) n Make ultrarelativistic boost, l [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 the (centrally extended) BMS 3 algebra! Central charges: c L = c c c M = (c + c)/l Daniel Grumiller Holography and phase-transition of flat space 6/16

24 İnönü Wigner contraction of Virasoro (Barnich & Compère 06) BMS 3 and GCA 2 (or rather, URCA 2) Take two copies of Virasoro, generators L n, Ln, central charges c, c Define superrotations L n and supertranslations M n L n := L n L n M n := 1 ( l Ln + L ) n Make ultrarelativistic boost, l [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 the (centrally extended) BMS 3 algebra! Central charges: c L = c c c M = (c + c)/l Example TMG (with gravitational CS coupling µ and Newton constant G): c L = 3 c M = 3 µg G Daniel Grumiller Holography and phase-transition of flat space 6/16

25 Consequence of ultrarelativistic boost for AdS boundary AdS-boundary: Flat space boundary: Limit l Null infinity holography! Daniel Grumiller Holography and phase-transition of flat space 7/16

26 Contraction on gravity side AdS metric (ϕ ϕ + 2π): ds 2 AdS = d(lρ) 2 cosh 2( lρ l ) dt 2 + l 2 sinh 2( lρ) l dϕ 2 Daniel Grumiller Holography and phase-transition of flat space 8/16

27 Contraction on gravity side AdS metric (ϕ ϕ + 2π): ds 2 AdS = d(lρ) 2 cosh 2( lρ l Limit l (r = lρ): ) dt 2 + l 2 sinh 2( lρ) l dϕ 2 ds 2 Flat = dr 2 dt 2 + r 2 dϕ 2 = du 2 2 du dr + r 2 dϕ 2 Daniel Grumiller Holography and phase-transition of flat space 8/16

28 Contraction on gravity side AdS metric (ϕ ϕ + 2π): ds 2 AdS = d(lρ) 2 cosh 2( lρ l Limit l (r = lρ): BTZ metric: ) dt 2 + l 2 sinh 2( lρ) l dϕ 2 ds 2 Flat = dr 2 dt 2 + r 2 dϕ 2 = du 2 2 du dr + r 2 dϕ 2 ds 2 BTZ = ( r2 l 2 r2 + l 2 )(r 2 r 2 ) r 2 dt 2 + r 2 dr 2 ( r2 l 2 r2 + l 2 )(r 2 r 2 ) +r 2 ( dϕ r + l r r 2 dt ) 2 Daniel Grumiller Holography and phase-transition of flat space 8/16

29 Contraction on gravity side AdS metric (ϕ ϕ + 2π): ds 2 AdS = d(lρ) 2 cosh 2( lρ l Limit l (r = lρ): BTZ metric: ) dt 2 + l 2 sinh 2( lρ) l dϕ 2 ds 2 Flat = dr 2 dt 2 + r 2 dϕ 2 = du 2 2 du dr + r 2 dϕ 2 ds 2 BTZ = ( r2 l 2 r2 + l 2 )(r 2 r 2 ) Limit l (ˆr + = r + l r 2 dt 2 + = finite): ds 2 FSC = ˆr 2 + ( 1 r 2 r 2 ) dt 2 1 r 2 dr 2 ( r2 l 2 r2 + l 2 )(r 2 r 2 ) +r 2 ( dϕ 1 r2 r 2 dr 2 ˆr 2 + r + l + r 2 ( dϕ ˆr + r r 2 dt ) 2 r r 2 dt ) 2 Daniel Grumiller Holography and phase-transition of flat space 8/16

30 Contraction on gravity side AdS metric (ϕ ϕ + 2π): ds 2 AdS = d(lρ) 2 cosh 2( lρ l Limit l (r = lρ): BTZ metric: ) dt 2 + l 2 sinh 2( lρ) l dϕ 2 ds 2 Flat = dr 2 dt 2 + r 2 dϕ 2 = du 2 2 du dr + r 2 dϕ 2 ds 2 BTZ = ( r2 l 2 r2 + l 2 )(r 2 r 2 ) Limit l (ˆr + = r + l r 2 dt 2 + = finite): ds 2 FSC = ˆr 2 + ( 1 r 2 r 2 ) dt 2 1 r 2 dr 2 ( r2 l 2 r2 + l 2 )(r 2 r 2 ) +r 2 ( dϕ 1 r2 r 2 dr 2 ˆr 2 + r + l + r 2 ( dϕ ˆr + r r 2 dt ) 2 r r 2 dt ) 2 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 at r < 0 Daniel Grumiller Holography and phase-transition of flat space 8/16

31 Flat-space boundary conditions Barnich & Compere 06; Bagchi, DG, Detournay 12; Barnich & González 13 In metric formulation: g uu = O(1) g ur = 1 + O(1/r) g uϕ = O(1) g rr = O(1/r 2 ) g rϕ = h 0 + O(1/r) g ϕϕ = r 2 + (h 1 u + h 2 ) r + O(1) Functions h i depend on ϕ only Daniel Grumiller Holography and phase-transition of flat space 9/16

32 Flat-space boundary conditions Barnich & Compere 06; Bagchi, DG, Detournay 12; Barnich & González 13 In metric formulation: g uu = O(1) g ur = 1 + O(1/r) g uϕ = O(1) g rr = O(1/r 2 ) g rϕ = h 0 + O(1/r) g ϕϕ = r 2 + (h 1 u + h 2 ) r + O(1) Functions h i depend on ϕ only Fluctuations of g uu to same order as background Daniel Grumiller Holography and phase-transition of flat space 9/16

33 Flat-space boundary conditions Barnich & Compere 06; Bagchi, DG, Detournay 12; Barnich & González 13 In metric formulation: g uu = O(1) g ur = 1 + O(1/r) g uϕ = O(1) g rr = O(1/r 2 ) g rϕ = h 0 + O(1/r) g ϕϕ = r 2 + (h 1 u + h 2 ) r + O(1) Functions h i depend on ϕ only Fluctuations of g uu to same order as background Includes flat background and FSC Daniel Grumiller Holography and phase-transition of flat space 9/16

34 Flat-space boundary conditions Barnich & Compere 06; Bagchi, DG, Detournay 12; Barnich & González 13 In metric formulation: g uu = O(1) g ur = 1 + O(1/r) g uϕ = O(1) g rr = O(1/r 2 ) g rϕ = h 0 + O(1/r) g ϕϕ = r 2 + (h 1 u + h 2 ) r + O(1) Functions h i depend on ϕ only Fluctuations of g uu to same order as background Includes flat background and FSC Canonical boundary charges finite, integrable and conserved in TMG Daniel Grumiller Holography and phase-transition of flat space 9/16

35 Flat-space boundary conditions Barnich & Compere 06; Bagchi, DG, Detournay 12; Barnich & González 13 In metric formulation: g uu = O(1) g ur = 1 + O(1/r) g uϕ = O(1) g rr = O(1/r 2 ) g rϕ = h 0 + O(1/r) g ϕϕ = r 2 + (h 1 u + h 2 ) r + O(1) Functions h i depend on ϕ only Fluctuations of g uu to same order as background Includes flat background and FSC Canonical boundary charges finite, integrable and conserved in TMG Mass tower: M n dϕ e inϕ( ) g uu + h 1 Daniel Grumiller Holography and phase-transition of flat space 9/16

36 Flat-space boundary conditions Barnich & Compere 06; Bagchi, DG, Detournay 12; Barnich & González 13 In metric formulation: g uu = O(1) g ur = 1 + O(1/r) g uϕ = O(1) g rr = O(1/r 2 ) g rϕ = h 0 + O(1/r) g ϕϕ = r 2 + (h 1 u + h 2 ) r + O(1) Functions h i depend on ϕ only Fluctuations of g uu to same order as background Includes flat background and FSC Canonical boundary charges finite, integrable and conserved in TMG Mass tower: M n dϕ e inϕ( ) g uu + h 1 Angular momentum tower: L n 1 µ M n+ dϕ e inϕ( ) inug uu +inr(1+g ur )+2g uϕ +r u g rϕ h 0 h 1 inh 2 Daniel Grumiller Holography and phase-transition of flat space 9/16

37 Asymptotic symmetry algebra Canonical analysis of TMG yields ASA with central charges [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 c L = 3 µg c M = 3 G Daniel Grumiller Holography and phase-transition of flat space 10/16

38 Asymptotic symmetry algebra Canonical analysis of TMG yields ASA with central charges [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 c L = 3 µg c M = 3 G Coincides precisely with results from İnönü Wigner contraction! Daniel Grumiller Holography and phase-transition of flat space 10/16

39 Asymptotic symmetry algebra Canonical analysis of TMG yields ASA with central charges [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 c L = 3 µg c M = 3 G Coincides precisely with results from İnönü Wigner contraction! Einstein gravity (µ, G finite): c L = 0, c M 0 Daniel Grumiller Holography and phase-transition of flat space 10/16

40 Asymptotic symmetry algebra Canonical analysis of TMG yields ASA with central charges [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 c L = 3 µg c M = 3 G Coincides precisely with results from İnönü Wigner contraction! Einstein gravity (µ, G finite): c L = 0, c M 0 Conformal CS gravity (G, 8µG = 1/k finite): c L 0, c M = 0 Daniel Grumiller Holography and phase-transition of flat space 10/16

41 Asymptotic symmetry algebra Canonical analysis of TMG yields ASA with central charges [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 c L = 3 µg c M = 3 G Coincides precisely with results from İnönü Wigner contraction! Einstein gravity (µ, G finite): c L = 0, c M 0 Conformal CS gravity (G, 8µG = 1/k finite): c L 0, c M = 0 Note: generalizations to higher curvature theories like NMG, GMG, PMG,... should be straightforward Daniel Grumiller Holography and phase-transition of flat space 10/16

42 Phase transition ( ) Hot flat space melts into expanding Universe General philosophy: Everything in AdS/CFT could have counterpart in flat space limit Take l of AdS/CFT papers? Daniel Grumiller Holography and phase-transition of flat space 11/16

43 Phase transition ( ) Hot flat space melts into expanding Universe General philosophy: Everything in AdS/CFT could have counterpart in flat space limit Take l of AdS/CFT papers? Does not always work (straightforwardly) Daniel Grumiller Holography and phase-transition of flat space 11/16

44 Phase transition ( ) Hot flat space melts into expanding Universe General philosophy: Everything in AdS/CFT could have counterpart in flat space limit Take l of AdS/CFT papers? Does not always work (straightforwardly) Need often case-by-case analysis Daniel Grumiller Holography and phase-transition of flat space 11/16

45 Phase transition ( ) Hot flat space melts into expanding Universe General philosophy: Everything in AdS/CFT could have counterpart in flat space limit Take l of AdS/CFT papers? Does not always work (straightforwardly) Need often case-by-case analysis Specific question: Is there a flat space analogue of Hawking Page phase transition? Daniel Grumiller Holography and phase-transition of flat space 11/16

46 Phase transition ( ) Hot flat space melts into expanding Universe General philosophy: Everything in AdS/CFT could have counterpart in flat space limit Take l of AdS/CFT papers? Does not always work (straightforwardly) Need often case-by-case analysis Specific question: Is there a flat space analogue of Hawking Page phase transition? Answer: yes! (standard Euclidean path integral methods) Daniel Grumiller Holography and phase-transition of flat space 11/16

47 Phase transition ( ) Hot flat space melts into expanding Universe General philosophy: Everything in AdS/CFT could have counterpart in flat space limit Take l of AdS/CFT papers? Does not always work (straightforwardly) Need often case-by-case analysis Specific question: Is there a flat space analogue of Hawking Page phase transition? Answer: yes! (standard Euclidean path integral methods) Small temperature: hot flat space stable, FSC unstable Large temperature: FSC stable, hot flat space unstable Daniel Grumiller Holography and phase-transition of flat space 11/16

48 Phase transition ( ) Hot flat space melts into expanding Universe General philosophy: Everything in AdS/CFT could have counterpart in flat space limit Take l of AdS/CFT papers? Does not always work (straightforwardly) Need often case-by-case analysis Specific question: Is there a flat space analogue of Hawking Page phase transition? Answer: yes! (standard Euclidean path integral methods) Small temperature: hot flat space stable, FSC unstable Large temperature: FSC stable, hot flat space unstable Melting point of flat space: T critical = 1 2π r Daniel Grumiller Holography and phase-transition of flat space 11/16

49 Hot flat space (at large T ) The Universe Flat The Flat Universe (FSC) Daniel Grumiller Holography and phase-transition of flat space 12/16

50 Unitarity in 3D (higher spin) gravity? General remarks on unitarity (Afshar, Gary, DG, Rashkov, Riegler 12) Wanted: unitary family of topological models with c 1 Daniel Grumiller Holography and phase-transition of flat space 13/16

51 Unitarity in 3D (higher spin) gravity? General remarks on unitarity (Afshar, Gary, DG, Rashkov, Riegler 12) Wanted: unitary family of topological models with c 1 Pure Einstein/principally embedded higher spin gravity: works only for c O(1) (Castro, Gaberdiel, Hartman, Maloney, Volpato) Daniel Grumiller Holography and phase-transition of flat space 13/16

52 Unitarity in 3D (higher spin) gravity? General remarks on unitarity (Afshar, Gary, DG, Rashkov, Riegler 12) Wanted: unitary family of topological models with c 1 Pure Einstein/principally embedded higher spin gravity: works only for c O(1) (Castro, Gaberdiel, Hartman, Maloney, Volpato) Non-principal embeddings (current algebra level k): no-go result (Castro, Hijano & Lepage-Jutier) lim c sign c = lim c sign k Daniel Grumiller Holography and phase-transition of flat space 13/16

53 Unitarity in 3D (higher spin) gravity? General remarks on unitarity (Afshar, Gary, DG, Rashkov, Riegler 12) Wanted: unitary family of topological models with c 1 Pure Einstein/principally embedded higher spin gravity: works only for c O(1) (Castro, Gaberdiel, Hartman, Maloney, Volpato) Non-principal embeddings (current algebra level k): no-go result (Castro, Hijano & Lepage-Jutier) lim c sign c = lim c sign k Circumvented: lim replaced by finite, but arbitrarily large Daniel Grumiller Holography and phase-transition of flat space 13/16

54 Unitarity in 3D (higher spin) gravity? General remarks on unitarity (Afshar, Gary, DG, Rashkov, Riegler 12) Wanted: unitary family of topological models with c 1 Pure Einstein/principally embedded higher spin gravity: works only for c O(1) (Castro, Gaberdiel, Hartman, Maloney, Volpato) Non-principal embeddings (current algebra level k): no-go result (Castro, Hijano & Lepage-Jutier) lim c sign c = lim c sign k Circumvented: lim replaced by finite, but arbitrarily large Example: WN 2 -gravity, with discrete set of unitary values of c: Plot: spin-100 gravity (W101) Plotted: central charge as function of CS level Points in plot correspond to unitary points 0 Α For large N: c N O(1/N) Daniel Grumiller Holography and phase-transition of flat space 13/16

55 Unitarity in flat space? Flat-space chiral gravity ( ) Define standard vacuum: L n 0 = M n 0 = 0, n 1 Daniel Grumiller Holography and phase-transition of flat space 14/16

56 Unitarity in flat space? Flat-space chiral gravity ( ) Define standard vacuum: L n 0 = M n 0 = 0, n 1 Consider level-2 descendants and their inner products: ( 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 Holography and phase-transition of flat space 14/16

57 Unitarity in flat space? Flat-space chiral gravity ( ) Define standard vacuum: L n 0 = M n 0 = 0, n 1 Consider level-2 descendants and their inner products: ( 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 For c M 0: always positive and negative norm state! ) Daniel Grumiller Holography and phase-transition of flat space 14/16

58 Unitarity in flat space? Flat-space chiral gravity ( ) Define standard vacuum: L n 0 = M n 0 = 0, n 1 Consider level-2 descendants and their inner products: ( 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 For c M 0: always positive and negative norm state! Unitarity requires c M = 0 (impossible in Einstein gravity) ) Daniel Grumiller Holography and phase-transition of flat space 14/16

59 Unitarity in flat space? Flat-space chiral gravity ( ) Define standard vacuum: L n 0 = M n 0 = 0, n 1 Consider level-2 descendants and their inner products: ( 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 For c M 0: always positive and negative norm state! Unitarity requires c M = 0 (impossible in Einstein gravity) Within TMG family: only conformal CS gravity can be unitary S CSG [g] = k 4π (Γ dγ Γ Γ Γ) ) Daniel Grumiller Holography and phase-transition of flat space 14/16

60 Unitarity in flat space? Flat-space chiral gravity ( ) Define standard vacuum: L n 0 = M n 0 = 0, n 1 Consider level-2 descendants and their inner products: ( 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 For c M 0: always positive and negative norm state! Unitarity requires c M = 0 (impossible in Einstein gravity) Within TMG family: only conformal CS gravity can be unitary S CSG [g] = k 4π (Γ dγ Γ Γ Γ) Canonical analysis yields c L = 24k (and c M = 0) ) Daniel Grumiller Holography and phase-transition of flat space 14/16

61 Unitarity in flat space? Flat-space chiral gravity ( ) Define standard vacuum: L n 0 = M n 0 = 0, n 1 Consider level-2 descendants and their inner products: ( 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 For c M 0: always positive and negative norm state! Unitarity requires c M = 0 (impossible in Einstein gravity) Within TMG family: only conformal CS gravity can be unitary S CSG [g] = k 4π (Γ dγ Γ Γ Γ) Canonical analysis yields c L = 24k (and c M = 0) Entropy of FSC in flat space chiral gravity (h L = k r+): 2 cl h L S = 4πk ˆr + = 2π 6 Coincides with chiral version of Cardy formula Daniel Grumiller Holography and phase-transition of flat space 14/16 )

62 Outlook to higher spins Towards flat space higher spin gravity in 3D İnönü Wigner contractions of 2 copies of higher spin algebras? Daniel Grumiller Holography and phase-transition of flat space 15/16

63 Outlook to higher spins Towards flat space higher spin gravity in 3D İnönü Wigner contractions of 2 copies of higher spin algebras? BMW W-BMS algebras at finite central charge? Daniel Grumiller Holography and phase-transition of flat space 15/16

64 Outlook to higher spins Towards flat space higher spin gravity in 3D İnönü Wigner contractions of 2 copies of higher spin algebras? BMW W-BMS algebras at finite central charge? Generalizations to non-principal embeddings? Daniel Grumiller Holography and phase-transition of flat space 15/16

65 Outlook to higher spins Towards flat space higher spin gravity in 3D İnönü Wigner contractions of 2 copies of higher spin algebras? BMW W-BMS algebras at finite central charge? Generalizations to non-principal embeddings? Unitary representations of flat-space higher spin algebras? Daniel Grumiller Holography and phase-transition of flat space 15/16

66 Outlook to higher spins Towards flat space higher spin gravity in 3D İnönü Wigner contractions of 2 copies of higher spin algebras? BMW W-BMS algebras at finite central charge? Generalizations to non-principal embeddings? Unitary representations of flat-space higher spin algebras? Answers in talk by Jan Rosseel! Daniel Grumiller Holography and phase-transition of flat space 15/16

67 References H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Phys. Rev. Lett. 111 (2013) [arxiv: [hep-th]]. A. Bagchi, S. Detournay, D. Grumiller and J. Simon, Phys. Rev. Lett. 111 (2013) [arxiv: [hep-th]]. A. Bagchi, S. Detournay and D. Grumiller, Phys. Rev. Lett. 109 (2012) [arxiv: [hep-th]]. Thanks to Bob McNees for providing the L A TEX beamerclass! Daniel Grumiller Holography and phase-transition of flat space 16/16