Flat Space Holography

Transcription

1 Flat Space Holography Daniel Grumiller Institute for Theoretical Physics TU Wien IFT, UNESP Brazil, September 2015

2 Some of our papers on flat space holography A. Bagchi, D. Grumiller and W. Merbis, Stress tensor correlators in three-dimensional gravity, arxiv: A. Bagchi, R. Basu, D. Grumiller and M. Riegler, Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett. 114 (2015) 11, [arxiv: ]. H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Spin-3 Gravity in Three-Dimensional Flat Space, Phys. Rev. Lett. 111 (2013) 12, [arxiv: ]. A. Bagchi, S. Detournay, D. Grumiller and J. Simon, Cosmic Evolution from Phase Transition of Three-Dimensional Flat Space, Phys. Rev. Lett. 111 (2013) 18, [arxiv: ]. A. Bagchi, S. Detournay and D. Grumiller, Flat-Space Chiral Gravity, Phys. Rev. Lett. 109 (2012) [arxiv: ]. Thanks to Bob McNees for providing the LATEX beamerclass! Daniel Grumiller Flat Space Holography 2/31

3 Outline Motivations Assumptions Flat space holography basics Recent results Generalizations & open issues Daniel Grumiller Flat Space Holography 3/31

4 Outline Motivations Assumptions Flat space holography basics Recent results Generalizations & open issues Daniel Grumiller Flat Space Holography Motivations 4/31

5 Holography in our Universe? This talk focuses on holography (in the quantum gravity sense). Daniel Grumiller Flat Space Holography Motivations 5/31

6 Holography in our Universe? This talk focuses on holography (in the quantum gravity sense). Main question: how general is holography? Daniel Grumiller Flat Space Holography Motivations 5/31

7 In memoriam Jakob Bekenstein (May 1, 1947 August 16, 2015) Daniel Grumiller Flat Space Holography Motivations 6/31

8 Testing the holographic principle How general is holography? Holographic principle realized in AdS/CFT correspondence Special case or generic lesson for quantum gravity? AdS d+1 CFT d Use (classical) gravity to learn more about CFTs Strong coupling large N limit: classical gravity Useful tool to calculate correlation functions Useful tool to calculate entanglement entropy CFT d AdS d+1 Use CFTs to learn more about (quantum) gravity Gravity in ultra-quantum limit: simple CFT? Useful tool to address black hole microstates Useful tool for qu-gr puzzles (information paradox) Daniel Grumiller Flat Space Holography Motivations 7/31

9 Testing the holographic principle How general is holography? To what extent do (previous) lessons rely on the particular constructions used to date? Are they tied to stringy effects and to string theory in particular, or are they general lessons for quantum gravity? see numerous talks at KITP workshop Bits, Branes, Black Holes 2012 and at ESI workshop Higher Spin Gravity 2012 Daniel Grumiller Flat Space Holography Motivations 7/31

10 Testing the holographic principle How general is holography? To what extent do (previous) lessons rely on the particular constructions used to date? Are they tied to stringy effects and to string theory in particular, or are they general lessons for quantum gravity? Does holography apply only to unitary theories? originally holography motivated by unitarity Daniel Grumiller Flat Space Holography Motivations 7/31

11 Testing the holographic principle How general is holography? To what extent do (previous) lessons rely on the particular constructions used to date? Are they tied to stringy effects and to string theory in particular, or are they general lessons for quantum gravity? Does holography apply only to unitary theories? originally holography motivated by unitarity plausible AdS/CFT-like correspondence could work non-unitarily AdS/log CFT first example of non-unitary holography DG, (Jackiw), Johansson 08; Skenderis, Taylor, van Rees 09; Henneaux, Martinez, Troncoso 09; Maloney, Song, Strominger 09; DG, Sachs/Hohm 09; Gaberdiel, DG, Vassilevich 10;... DG, Riedler, Rosseel, Zojer 13 recent proposal by Vafa 14 Daniel Grumiller Flat Space Holography Motivations 7/31

12 Testing the holographic principle How general is holography? To what extent do (previous) lessons rely on the particular constructions used to date? Are they tied to stringy effects and to string theory in particular, or are they general lessons for quantum gravity? Does holography apply only to unitary theories? Can we establish a flat space holographic dictionary? the answer appears to be yes see my current talk and recent papers by Bagchi et al., Barnich et al., Strominger et al., Daniel Grumiller Flat Space Holography Motivations 7/31

13 Testing the holographic principle How general is holography? To what extent do (previous) lessons rely on the particular constructions used to date? Are they tied to stringy effects and to string theory in particular, or are they general lessons for quantum gravity? Does holography apply only to unitary theories? Can we establish a flat space holographic dictionary? Generic non-ads holography/higher spin holography? non-trivial hints that it might work at least in 2+1 dimensions Gary, DG Rashkov 12; Afshar et al 12; Gutperle et al 14-15; Gary, DG, Prohazka, Rey 14;... Daniel Grumiller Flat Space Holography Motivations 7/31

14 Testing the holographic principle How general is holography? To what extent do (previous) lessons rely on the particular constructions used to date? Are they tied to stringy effects and to string theory in particular, or are they general lessons for quantum gravity? Does holography apply only to unitary theories? Can we establish a flat space holographic dictionary? Generic non-ads holography/higher spin holography? Address questions above in simple class of 3D toy models Exploit gauge theoretic Chern Simons formulation Restrict to kinematic questions, like (asymptotic) symmetries Daniel Grumiller Flat Space Holography Motivations 7/31

15 Goals of this talk 1. Review general aspects of holography in 3D Daniel Grumiller Flat Space Holography Motivations 8/31

16 Goals of this talk 1. Review general aspects of holography in 3D 2. Discuss flat space holography Daniel Grumiller Flat Space Holography Motivations 8/31

17 Goals of this talk 1. Review general aspects of holography in 3D 2. Discuss flat space holography 3. List selected open issues Daniel Grumiller Flat Space Holography Motivations 8/31

18 Goals of this talk 1. Review general aspects of holography in 3D 2. Discuss flat space holography 3. List selected open issues Address these issues in 3D! Daniel Grumiller Flat Space Holography Motivations 8/31

19 Outline Motivations Assumptions Flat space holography basics Recent results Generalizations & open issues Daniel Grumiller Flat Space Holography Assumptions 9/31

20 Assumptions Working assumptions: 3D Daniel Grumiller Flat Space Holography Assumptions 10/31

21 Assumptions Working assumptions: 3D Restrict to pure gravity theories Daniel Grumiller Flat Space Holography Assumptions 10/31

22 Assumptions Working assumptions: 3D Restrict to pure gravity theories Define quantum gravity by its dual field theory Interesting dichotomy: Either dual field theory exists useful toy model for quantum gravity Or gravitational theory needs UV completion (within string theory) indication of inevitability of string theory Daniel Grumiller Flat Space Holography Assumptions 10/31

23 Assumptions Working assumptions: 3D Restrict to pure gravity theories Define quantum gravity by its dual field theory Interesting dichotomy: Either dual field theory exists useful toy model for quantum gravity Or gravitational theory needs UV completion (within string theory) indication of inevitability of string theory This talk: Remain agnostic about dichotomy Focus on generic features of dual field theories that do not require string theory embedding Daniel Grumiller Flat Space Holography Assumptions 10/31

24 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Daniel Grumiller Flat Space Holography Assumptions 11/31

25 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Daniel Grumiller Flat Space Holography Assumptions 11/31

26 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Daniel Grumiller Flat Space Holography Assumptions 11/31

27 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Formulation as topological gauge theory (Chern Simons) Daniel Grumiller Flat Space Holography Assumptions 11/31

28 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Formulation as topological gauge theory (Chern Simons) Dual field theory (if it exists): 2D Daniel Grumiller Flat Space Holography Assumptions 11/31

29 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Formulation as topological gauge theory (Chern Simons) Dual field theory (if it exists): 2D Infinite dimensional asymptotic symmetries (Brown Henneaux) Daniel Grumiller Flat Space Holography Assumptions 11/31

30 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Formulation as topological gauge theory (Chern Simons) Dual field theory (if it exists): 2D Infinite dimensional asymptotic symmetries (Brown Henneaux) Black holes as orbifolds of AdS 3 (BTZ) Daniel Grumiller Flat Space Holography Assumptions 11/31

31 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Formulation as topological gauge theory (Chern Simons) Dual field theory (if it exists): 2D Infinite dimensional asymptotic symmetries (Brown Henneaux) Black holes as orbifolds of AdS 3 (BTZ) Simple microstate counting from AdS 3 /CFT 2 Daniel Grumiller Flat Space Holography Assumptions 11/31

32 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Formulation as topological gauge theory (Chern Simons) Dual field theory (if it exists): 2D Infinite dimensional asymptotic symmetries (Brown Henneaux) Black holes as orbifolds of AdS 3 (BTZ) Simple microstate counting from AdS 3 /CFT 2 Hawking Page phase transition hot AdS BTZ Daniel Grumiller Flat Space Holography Assumptions 11/31

33 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Formulation as topological gauge theory (Chern Simons) Dual field theory (if it exists): 2D Infinite dimensional asymptotic symmetries (Brown Henneaux) Black holes as orbifolds of AdS 3 (BTZ) Simple microstate counting from AdS 3 /CFT 2 Hawking Page phase transition hot AdS BTZ Simple checks of Ryu Takayanagi proposal Daniel Grumiller Flat Space Holography Assumptions 11/31

34 Gravity in 3D AdS 3 gravity Lowest dimension with black holes and (off-shell) gravitons Weyl = 0, thus Riemann = Ricci Einstein gravity: no on-shell gravitons Formulation as topological gauge theory (Chern Simons) Dual field theory (if it exists): 2D Infinite dimensional asymptotic symmetries (Brown Henneaux) Black holes as orbifolds of AdS 3 (BTZ) Simple microstate counting from AdS 3 /CFT 2 Hawking Page phase transition hot AdS BTZ Simple checks of Ryu Takayanagi proposal Caveat: while there are many string compactifications with AdS 3 factor, applying holography just to AdS 3 factor does not capture everything! Daniel Grumiller Flat Space Holography Assumptions 11/31

35 Picturesque analogy: soap films Both soap films and Chern Simons theories have essentially no bulk dynamics highly non-trivial boundary dynamics most of the physics determined by boundary conditions esthetic appeal (at least for me) Daniel Grumiller Flat Space Holography Assumptions 12/31

36 Outline Motivations Assumptions Flat space holography basics Recent results Generalizations & open issues Daniel Grumiller Flat Space Holography Flat space holography basics 13/31

37 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

38 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Works straightforwardly sometimes, otherwise not Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

39 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Works straightforwardly sometimes, otherwise not Example where it works nicely: asymptotic symmetry algebra Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

40 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Works straightforwardly sometimes, otherwise not Example where it works nicely: asymptotic symmetry algebra Take linear combinations of Virasoro generators L n, Ln L n = L n L n M n = 1 ( Ln + l L ) n Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

41 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Works straightforwardly sometimes, otherwise not Example where it works nicely: asymptotic symmetry algebra Take linear combinations of Virasoro generators L n, Ln L n = L n L n M n = 1 ( Ln + l L ) n Make Inönü Wigner contraction l on ASA [L n, L m] = (n m) L n+m + cl 12 (n3 n) δ n+m, 0 [L n, M m] = (n m) M n+m + cm 12 (n3 n) δ n+m, 0 [M n, M m] = 0 Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

42 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Works straightforwardly sometimes, otherwise not Example where it works nicely: asymptotic symmetry algebra Take linear combinations of Virasoro generators L n, Ln L n = L n L n M n = 1 ( Ln + l L ) n Make Inönü Wigner contraction l on ASA [L n, L m] = (n m) L n+m + cl 12 (n3 n) δ n+m, 0 [L n, M m] = (n m) M n+m + cm 12 (n3 n) δ n+m, 0 [M n, M m] = 0 This is nothing but the BMS 3 algebra (or GCA 2, URCA 2, CCA 2 )! Ashtekar, Bicak, Schmidt, 96; Barnich, Compere 06 L n : diffeos of circle, M n : supertranslations, c L/M : central extensions Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

43 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Works straightforwardly sometimes, otherwise not Example where it works nicely: asymptotic symmetry algebra Take linear combinations of Virasoro generators L n, Ln L n = L n L n M n = 1 ( Ln + l L ) n Make Inönü Wigner contraction l on ASA [L n, L m] = (n m) L n+m + cl 12 (n3 n) δ n+m, 0 [L n, M m] = (n m) M n+m + cm 12 (n3 n) δ n+m, 0 [M n, M m] = 0 This is nothing but the BMS 3 algebra (or GCA 2, URCA 2, CCA 2 )! If dual field theory exists it must be a 2D Galilean CFT! Bagchi et al., Barnich et al. Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

44 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Works straightforwardly sometimes, otherwise not Example where it works nicely: asymptotic symmetry algebra Take linear combinations of Virasoro generators L n, Ln L n = L n L n M n = 1 ( Ln + l L ) n Make Inönü Wigner contraction l on ASA [L n, L m] = (n m) L n+m + cl 12 (n3 n) δ n+m, 0 [L n, M m] = (n m) M n+m + cm 12 (n3 n) δ n+m, 0 [M n, M m] = 0 This is nothing but the BMS 3 algebra (or GCA 2, URCA 2, CCA 2 )! Example where it does not work easily: boundary conditions Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

45 Flat space holography (Barnich et al, Bagchi et al, Strominger et al,...) if holography is true must work in flat space Just take large AdS radius limit of 10 4 AdS/CFT papers? Works straightforwardly sometimes, otherwise not Example where it works nicely: asymptotic symmetry algebra Take linear combinations of Virasoro generators L n, Ln L n = L n L n M n = 1 ( Ln + l L ) n Make Inönü Wigner contraction l on ASA [L n, L m] = (n m) L n+m + cl 12 (n3 n) δ n+m, 0 [L n, M m] = (n m) M n+m + cm 12 (n3 n) δ n+m, 0 [M n, M m] = 0 This is nothing but the BMS 3 algebra (or GCA 2, URCA 2, CCA 2 )! Example where it does not work easily: boundary conditions Example where it does not work: highest weight conditions Daniel Grumiller Flat Space Holography Flat space holography basics 14/31

46 Flat space Einstein gravity as isl(2) Chern Simons theory For details, references and spin-3 generalization see Gary, DG, Riegler, Rosseel 14 Daniel Grumiller Flat Space Holography Flat space holography basics 15/31

47 Flat space Einstein gravity as isl(2) Chern Simons theory For details, references and spin-3 generalization see Gary, DG, Riegler, Rosseel 14 AdS gravity in CS formulation: sl(2) sl(2) gauge algebra Achucarro, Townsend 86; Witten 88 Daniel Grumiller Flat Space Holography Flat space holography basics 15/31

48 Flat space Einstein gravity as isl(2) Chern Simons theory For details, references and spin-3 generalization see Gary, DG, Riegler, Rosseel 14 AdS gravity in CS formulation: sl(2) sl(2) gauge algebra Flat space: isl(2) gauge algebra S flat CS = k A da + 2 4π 3 A A A with isl(2) connection (a = 0, ±1) A = e a M a + ω a L a isl(2) algebra (global part of BMS/GCA) [L a, L b ] = (a b)l a+b [L a, M b ] = (a b)m a+b [M a, M b ] = 0 Note: e a dreibein, ω a (dualized) spin-connection Bulk EOM: gauge flatness Einstein equations F = da + A A = 0 Daniel Grumiller Flat Space Holography Flat space holography basics 15/31

49 Flat space Einstein gravity as isl(2) Chern Simons theory For details, references and spin-3 generalization see Gary, DG, Riegler, Rosseel 14 AdS gravity in CS formulation: sl(2) sl(2) gauge algebra Flat space: isl(2) gauge algebra S flat CS = k A da + 2 4π 3 A A A with isl(2) connection (a = 0, ±1) A = e a M a + ω a L a Boundary conditions in CS formulation: A(r, u, ϕ) = b 1 (r) ( d+a(u, ϕ) + o(1) ) b(r) Daniel Grumiller Flat Space Holography Flat space holography basics 15/31

50 Flat space Einstein gravity as isl(2) Chern Simons theory For details, references and spin-3 generalization see Gary, DG, Riegler, Rosseel 14 AdS gravity in CS formulation: sl(2) sl(2) gauge algebra Flat space: isl(2) gauge algebra S flat CS = k A da + 2 4π 3 A A A with isl(2) connection (a = 0, ±1) A = e a M a + ω a L a Boundary conditions in CS formulation: A(r, u, ϕ) = b 1 (r) ( d+a(u, ϕ) + o(1) ) b(r) Flat space boundary conditions: b(r) = exp ( 1 2 rm 1) and a(u, ϕ) = ( M 1 M(ϕ)M 1 ) du + ( L1 M(ϕ)L 1 N(u, ϕ)m 1 ) dϕ with N(u, ϕ) = L(ϕ) + u 2 M (ϕ) Daniel Grumiller Flat Space Holography Flat space holography basics 15/31

51 Flat space Einstein gravity as isl(2) Chern Simons theory For details, references and spin-3 generalization see Gary, DG, Riegler, Rosseel 14 AdS gravity in CS formulation: sl(2) sl(2) gauge algebra Flat space: isl(2) gauge algebra S flat CS = k A da + 2 4π 3 A A A with isl(2) connection (a = 0, ±1) A = e a M a + ω a L a Boundary conditions in CS formulation: A(r, u, ϕ) = b 1 (r) ( d+a(u, ϕ) + o(1) ) b(r) Flat space boundary conditions: b(r) = exp ( 1 2 rm 1) and a(u, ϕ) = ( M 1 M(ϕ)M 1 ) du + ( L1 M(ϕ)L 1 N(u, ϕ)m 1 ) dϕ with N(u, ϕ) = L(ϕ) + u 2 M (ϕ) metric g µν 1 2 tr A µ A ν ds 2 = M du 2 2 du dr+2n du dϕ+r 2 dϕ 2 Daniel Grumiller Flat Space Holography Flat space holography basics 15/31

52 Outline Motivations Assumptions Flat space holography basics Recent results Generalizations & open issues Daniel Grumiller Flat Space Holography Recent results 16/31

53 Correlation functions in flat space holography AdS/CFT good tool for calculating correlators What about flat space/galilean CFT correspondence? Daniel Grumiller Flat Space Holography Recent results 17/31

54 Correlation functions in flat space holography AdS/CFT good tool for calculating correlators What about flat space/galilean CFT correspondence? What is flat space analogue of T (z 1 )T (z 2 )... T (z 42 ) CFT δ42 δg 42 Γ EH-AdS? EOM Daniel Grumiller Flat Space Holography Recent results 17/31

55 Correlation functions in flat space holography AdS/CFT good tool for calculating correlators What about flat space/galilean CFT correspondence? What is flat space analogue of T (z 1 )T (z 2 )... T (z 42 ) CFT δ42 δg 42 Γ EH-AdS? Does it work? EOM Daniel Grumiller Flat Space Holography Recent results 17/31

56 Correlation functions in flat space holography AdS/CFT good tool for calculating correlators What about flat space/galilean CFT correspondence? What is flat space analogue of T (z 1 )T (z 2 )... T (z 42 ) CFT δ42 δg 42 Γ EH-AdS? Does it work? What is the left hand side in a Galilean CFT? EOM Daniel Grumiller Flat Space Holography Recent results 17/31

57 Correlation functions in flat space holography AdS/CFT good tool for calculating correlators What about flat space/galilean CFT correspondence? What is flat space analogue of T (z 1 )T (z 2 )... T (z 42 ) CFT δ42 δg 42 Γ EH-AdS EOM? Does it work? What is the left hand side in a Galilean CFT? Shortcut to right hand side other than varying EH-action 42 times? Daniel Grumiller Flat Space Holography Recent results 17/31

58 Correlation functions in flat space holography AdS/CFT good tool for calculating correlators What about flat space/galilean CFT correspondence? What is flat space analogue of T (z 1 )T (z 2 )... T (z 42 ) CFT δ42 δg 42 Γ EH-AdS EOM? Does it work? What is the left hand side in a Galilean CFT? Shortcut to right hand side other than varying EH-action 42 times? Start slowly with 0-point function Daniel Grumiller Flat Space Holography Recent results 17/31

59 0-point function (on-shell action) Not check of flat space holography but interesting in its own right Calculate the full on-shell action Γ Daniel Grumiller Flat Space Holography Recent results 18/31

60 0-point function (on-shell action) Not check of flat space holography but interesting in its own right Calculate the full on-shell action Γ Variational principle? Γ = 1 d 3 x g R 1 16πG N 8πG N with I counter-term chosen such that δγ EOM = 0 for all δg that preserve flat space bc s d 2 x γ K I counter-term Daniel Grumiller Flat Space Holography Recent results 18/31

61 0-point function (on-shell action) Not check of flat space holography but interesting in its own right Calculate the full on-shell action Γ Variational principle? Γ = 1 d 3 x g R 1 16πG N 8πG N with I counter-term chosen such that δγ EOM = 0 d 2 x γ K I counter-term for all δg that preserve flat space bc s Result (Detournay, DG, Schöller, Simon 14): Γ = 1 d 3 x 1 g R d 2 x γ K 16πG N 16πG }{{ N } 1 2 GHY! follows also as limit from AdS using Mora, Olea, Troncoso, Zanelli 04 independently confirmed by Barnich, Gonzalez, Maloney, Oblak 15 Daniel Grumiller Flat Space Holography Recent results 18/31

62 0-point function (on-shell action) Not check of flat space holography but interesting in its own right Calculate the full on-shell action Γ Variational principle? Phase transitions? Standard procedure (Gibbons, Hawking 77; Hawking, Page 83) Evaluate Euclidean partition function in semi-classical limit Z(T, Ω) = Dg e Γ[g] = e Γ[gc(T, Ω)] Z fluct. g c path integral bc s specified by temperature T and angular velocity Ω Two Euclidean saddle points in same ensemble if same temperature T = 1/β and angular velocity Ω obey flat space boundary conditions solutions without conical singularities Periodicities fixed: (τ E, ϕ) (τ E + β, ϕ + βω) (τ E, ϕ + 2π) Daniel Grumiller Flat Space Holography Recent results 18/31

63 0-point function (on-shell action) Not check of flat space holography but interesting in its own right Calculate the full on-shell action Γ Variational principle? Phase transitions? 3D Euclidean Einstein gravity: for each T, Ω two saddle points: Hot flat space ds 2 = dτe 2 + dr 2 + r 2 dϕ 2 Flat space cosmology ds 2 = r+ 2 ( r 2 ) 1 0 dτ 2 r 2 E + r2 dr 2 r+ 2 (r 2 r0 2) + ( r2 dϕ r +r 0 ) 2 r 2 dτ E shifted-boost orbifold, see Cornalba, Costa 02 Daniel Grumiller Flat Space Holography Recent results 18/31

64 0-point function (on-shell action) Not check of flat space holography but interesting in its own right Calculate the full on-shell action Γ Variational principle? Phase transitions? Plug two Euclidean saddles in on-shell action and compare free energies F HFS = 1 F FSC = r + 8G N 8G N Daniel Grumiller Flat Space Holography Recent results 18/31

65 0-point function (on-shell action) Not check of flat space holography but interesting in its own right Calculate the full on-shell action Γ Variational principle? Phase transitions? Plug two Euclidean saddles in on-shell action and compare free energies F HFS = 1 F FSC = r + 8G N 8G N Result of this comparison r+ > 1: FSC dominant saddle r+ < 1: HFS dominant saddle Critical temperature: HFS melts into FSC at T > T c Bagchi, Detournay, DG, Simon 13 T c = 1 2πr 0 = Ω 2π Daniel Grumiller Flat Space Holography Recent results 18/31

66 1-point functions (conserved charges) First check of entries in holographic dictionary: identification of sources and vevs In AdS 3 : Note that T µν BY δγ vev δ source T µν EOM BY δgµν NN M M follows from canonical analysis as well (conserved charges) Daniel Grumiller Flat Space Holography Recent results 19/31

67 1-point functions (conserved charges) First check of entries in holographic dictionary: identification of sources and vevs In AdS 3 : Note that T µν BY In flat space: δγ vev δ source T µν EOM BY δgµν NN M M follows from canonical analysis as well (conserved charges) non-normalizable solutions to linearized EOM? analogue of Brown York stress tensor? comparison with canonical results Daniel Grumiller Flat Space Holography Recent results 19/31

68 1-point functions (conserved charges) First check of entries in holographic dictionary: identification of sources and vevs In AdS 3 : Note that T µν BY In flat space: δγ vev δ source T µν EOM BY δgµν NN M M follows from canonical analysis as well (conserved charges) non-normalizable solutions to linearized EOM? analogue of Brown York stress tensor? comparison with canonical results everything works (Detournay, DG, Schöller, Simon, 14) mass and angular momentum: M = g tt 8G N = g tϕ 4G full tower of canonical charges: see Barnich, Compere 06 Daniel Grumiller Flat Space Holography Recent results 19/31

69 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder (ϕ ϕ + 2π): M(u 1, ϕ 1 ) M(u 2, ϕ 2 ) = 0 M(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c M 2s 4 12 N(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c L 2c M τ 12 2s 4 12 with s ij = 2 sin[(ϕ i ϕ j )/2], τ ij = (u i u j ) cot[(ϕ i ϕ j )/2] Fourier modes of Galilean CFT stress tensor on cylinder: M := M n e inϕ c M 24 n N := ( ) Ln inum n e inϕ c L 24 n Conservation equations: u M = 0, u N = ϕ M Daniel Grumiller Flat Space Holography Recent results 20/31

70 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder (ϕ ϕ + 2π): M(u 1, ϕ 1 ) M(u 2, ϕ 2 ) = 0 M(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c M 2s 4 12 N(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c L 2c M τ 12 2s 4 12 with s ij = 2 sin[(ϕ i ϕ j )/2], τ ij = (u i u j ) cot[(ϕ i ϕ j )/2] Short-cut on gravity side: Do not calculate second variation of action Daniel Grumiller Flat Space Holography Recent results 20/31

71 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder (ϕ ϕ + 2π): M(u 1, ϕ 1 ) M(u 2, ϕ 2 ) = 0 M(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c M 2s 4 12 N(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c L 2c M τ 12 2s 4 12 with s ij = 2 sin[(ϕ i ϕ j )/2], τ ij = (u i u j ) cot[(ϕ i ϕ j )/2] Short-cut on gravity side: Do not calculate second variation of action Calculate first variation of action on non-trivial background Daniel Grumiller Flat Space Holography Recent results 20/31

72 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder (ϕ ϕ + 2π): M(u 1, ϕ 1 ) M(u 2, ϕ 2 ) = 0 M(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c M 2s 4 12 N(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c L 2c M τ 12 2s 4 12 with s ij = 2 sin[(ϕ i ϕ j )/2], τ ij = (u i u j ) cot[(ϕ i ϕ j )/2] Short-cut on gravity side: Do not calculate second variation of action Calculate first variation of action on non-trivial background Can iterate this procedure to higher n-point functions Daniel Grumiller Flat Space Holography Recent results 20/31

73 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder (ϕ ϕ + 2π): M(u 1, ϕ 1 ) M(u 2, ϕ 2 ) = 0 M(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c M 2s 4 12 N(u 1, ϕ 1 ) N(u 2, ϕ 2 ) = c L 2c M τ 12 2s 4 12 with s ij = 2 sin[(ϕ i ϕ j )/2], τ ij = (u i u j ) cot[(ϕ i ϕ j )/2] Short-cut on gravity side: Do not calculate second variation of action Calculate first variation of action on non-trivial background Can iterate this procedure to higher n-point functions Summarize first how this works in the AdS case Daniel Grumiller Flat Space Holography Recent results 20/31

74 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On CFT side deform free action S 0 by source term µ for stress tensor S µ = S 0 + d 2 z µ(z, z)t (z) Daniel Grumiller Flat Space Holography Recent results 21/31

75 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On CFT side deform free action S 0 by source term µ for stress tensor S µ = S 0 + d 2 z µ(z, z)t (z) Localize source µ(z, z) = ɛ δ (2) (z z 2, z z 2 ) Daniel Grumiller Flat Space Holography Recent results 21/31

76 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On CFT side deform free action S 0 by source term µ for stress tensor S µ = S 0 + d 2 z µ(z, z)t (z) Localize source µ(z, z) = ɛ δ (2) (z z 2, z z 2 ) 1-point function in µ-vacuum 2-point function in 0-vacuum T 1 µ = T ɛ T 1 T O(ɛ 2 ) Daniel Grumiller Flat Space Holography Recent results 21/31

77 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On CFT side deform free action S 0 by source term µ for stress tensor S µ = S 0 + d 2 z µ(z, z)t (z) Localize source µ(z, z) = ɛ δ (2) (z z 2, z z 2 ) 1-point function in µ-vacuum 2-point function in 0-vacuum T 1 µ = T ɛ T 1 T O(ɛ 2 ) On gravity side exploit sl(2) CS formulation with chemical potentials A = b 1 (d+a)b b = e ρl 0 a z = L + L k L a z = µl Drinfeld, Sokolov 84, Polyakov 87, H. Verlinde 90 Bañados, Caro 04 Daniel Grumiller Flat Space Holography Recent results 21/31

78 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On CFT side deform free action S 0 by source term µ for stress tensor S µ = S 0 + d 2 z µ(z, z)t (z) Localize source µ(z, z) = ɛ δ (2) (z z 2, z z 2 ) 1-point function in µ-vacuum 2-point function in 0-vacuum T 1 µ = T ɛ T 1 T O(ɛ 2 ) On gravity side exploit sl(2) CS formulation with chemical potentials A = b 1 (d+a)b b = e ρl 0 a z = L + L k L a z = µl Expand L(z) = L (0) (z) + ɛl (1) (z) + O(ɛ 2 ) Daniel Grumiller Flat Space Holography Recent results 21/31

79 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On gravity side exploit CS formulation with chemical potentials A = b 1 (d + a)b b = e ρl 0 a z = L + L k L a z = µl Expand L(z) = L (0) (z) + ɛl (1) (z) + O(ɛ 2 ) Write EOM to first subleading order in ɛ L (1) (z) = k 2 3 δ (2) (z z 2 ) Daniel Grumiller Flat Space Holography Recent results 21/31

80 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On gravity side exploit CS formulation with chemical potentials A = b 1 (d + a)b b = e ρl 0 a z = L + L k L a z = µl Expand L(z) = L (0) (z) + ɛl (1) (z) + O(ɛ 2 ) Write EOM to first subleading order in ɛ L (1) (z) = k 2 3 δ (2) (z z 2 ) Solve them using the Green function on the plane G = ln (z 12 z 12 ) L (1) (z) = k 2 4 z 1 G(z 12 ) = 3k z 4 12 Daniel Grumiller Flat Space Holography Recent results 21/31

81 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On gravity side exploit CS formulation with chemical potentials A = b 1 (d + a)b b = e ρl 0 a z = L + L k L a z = µl Expand L(z) = L (0) (z) + ɛl (1) (z) + O(ɛ 2 ) Write EOM to first subleading order in ɛ L (1) (z) = k 2 3 δ (2) (z z 2 ) Solve them using the Green function on the plane G = ln (z 12 z 12 ) L (1) (z) = k 2 4 z 1 G(z 12 ) = 3k z 4 12 This is the correct CFT 2-point function on the plane with c = 6k Daniel Grumiller Flat Space Holography Recent results 21/31

82 2-point functions (anomalous terms) Illustrate shortcut in AdS 3/CFT 2 (restrict to one holomorphic sector) On gravity side exploit CS formulation with chemical potentials A = b 1 (d + a)b b = e ρl 0 a z = L + L k L a z = µl Expand L(z) = L (0) (z) + ɛl (1) (z) + O(ɛ 2 ) Write EOM to first subleading order in ɛ L (1) (z) = k 2 3 δ (2) (z z 2 ) Solve them using the Green function on the plane G = ln (z 12 z 12 ) L (1) (z) = k 2 4 z 1 G(z 12 ) = 3k z 4 12 This is the correct CFT 2-point function on the plane with c = 6k Generalize to cylinder Daniel Grumiller Flat Space Holography Recent results 21/31

83 2-point functions (anomalous terms) Apply shortcut to flat space/galilean CFT (Bagchi, DG, Merbis 15) Exploit results for flat space gravity in CS formulation in presence of chemical potentials (Gary, DG, Riegler, Rosseel 14) Daniel Grumiller Flat Space Holography Recent results 22/31

84 2-point functions (anomalous terms) Apply shortcut to flat space/galilean CFT (Bagchi, DG, Merbis 15) Exploit results for flat space gravity in CS formulation in presence of chemical potentials (Gary, DG, Riegler, Rosseel 14) Localize chemical potentials µ M/L = ɛ M/L δ (2) (u u 2, ϕ ϕ 2 ) Daniel Grumiller Flat Space Holography Recent results 22/31

85 2-point functions (anomalous terms) Apply shortcut to flat space/galilean CFT (Bagchi, DG, Merbis 15) Exploit results for flat space gravity in CS formulation in presence of chemical potentials (Gary, DG, Riegler, Rosseel 14) Localize chemical potentials µ M/L = ɛ M/L δ (2) (u u 2, ϕ ϕ 2 ) Expand around global Minkowski space M = k/2 + M (1) N = N (1) Daniel Grumiller Flat Space Holography Recent results 22/31

86 2-point functions (anomalous terms) Apply shortcut to flat space/galilean CFT (Bagchi, DG, Merbis 15) Exploit results for flat space gravity in CS formulation in presence of chemical potentials (Gary, DG, Riegler, Rosseel 14) Localize chemical potentials µ M/L = ɛ M/L δ (2) (u u 2, ϕ ϕ 2 ) Expand around global Minkowski space M = k/2 + M (1) N = N (1) Write EOM to first subleading order in ɛ M/L u M (1) = k ɛ L ( 3 ϕ δ + ϕ δ ) u N (1) = k ɛ M ( 3 ϕ δ + ϕ δ ) + ϕ M (1) Daniel Grumiller Flat Space Holography Recent results 22/31

87 2-point functions (anomalous terms) Apply shortcut to flat space/galilean CFT (Bagchi, DG, Merbis 15) Exploit results for flat space gravity in CS formulation in presence of chemical potentials (Gary, DG, Riegler, Rosseel 14) Localize chemical potentials µ M/L = ɛ M/L δ (2) (u u 2, ϕ ϕ 2 ) Expand around global Minkowski space M = k/2 + M (1) N = N (1) Write EOM to first subleading order in ɛ M/L u M (1) = k ɛ L ( 3 ϕ δ + ϕ δ ) u N (1) = k ɛ M ( 3 ϕ δ + ϕ δ ) + ϕ M (1) Solve with Green function on cylinder M (1) = 6kɛ L s 4 12 N (1) = 6k(ɛ M 2ɛ L τ 12 ) s 4 12 Daniel Grumiller Flat Space Holography Recent results 22/31

88 2-point functions (anomalous terms) Apply shortcut to flat space/galilean CFT (Bagchi, DG, Merbis 15) Exploit results for flat space gravity in CS formulation in presence of chemical potentials (Gary, DG, Riegler, Rosseel 14) Localize chemical potentials µ M/L = ɛ M/L δ (2) (u u 2, ϕ ϕ 2 ) Expand around global Minkowski space M = k/2 + M (1) N = N (1) Write EOM to first subleading order in ɛ M/L u M (1) = k ɛ L ( 3 ϕ δ + ϕ δ ) u N (1) = k ɛ M ( 3 ϕ δ + ϕ δ ) + ϕ M (1) Solve with Green function on cylinder M (1) = 6kɛ L s 4 12 N (1) = 6k(ɛ M 2ɛ L τ 12 ) s 4 12 Correct 2-point functions for Einstein gravity with c L = 0, c M = 12k Daniel Grumiller Flat Space Holography Recent results 22/31

89 3-point functions (check of symmetries) First non-trivial check of consistency with symmetries of dual Galilean CFT Check of 2-point functions works nicely with shortcut; 3-point too? Daniel Grumiller Flat Space Holography Recent results 23/31

90 3-point functions (check of symmetries) First non-trivial check of consistency with symmetries of dual Galilean CFT Check of 2-point functions works nicely with shortcut; 3-point too? Yes: same procedure, but localize chemical potentials at two points 3 µ M/L (u 1, ϕ 1 ) = ɛ i M/L δ(2) (u 1 u i, ϕ 1 ϕ i ) i=2 Daniel Grumiller Flat Space Holography Recent results 23/31

91 3-point functions (check of symmetries) First non-trivial check of consistency with symmetries of dual Galilean CFT Check of 2-point functions works nicely with shortcut; 3-point too? Yes: same procedure, but localize chemical potentials at two points µ M/L (u 1, ϕ 1 ) = Iteratively solve EOM 3 ɛ i M/L δ(2) (u 1 u i, ϕ 1 ϕ i ) i=2 u M = k ϕµ 3 L + µ L ϕ M + 2M ϕ µ L u N = k ϕµ 3 M + (1 + µ M ) ϕ M + 2M ϕ µ M + µ L ϕ N + 2N ϕ µ L Daniel Grumiller Flat Space Holography Recent results 23/31

92 3-point functions (check of symmetries) First non-trivial check of consistency with symmetries of dual Galilean CFT Check of 2-point functions works nicely with shortcut; 3-point too? Yes: same procedure, but localize chemical potentials at two points µ M/L (u 1, ϕ 1 ) = Iteratively solve EOM 3 ɛ i M/L δ(2) (u 1 u i, ϕ 1 ϕ i ) i=2 u M = k 3 ϕµ L + µ L ϕ M + 2M ϕ µ L u N = k 3 ϕµ M + (1 + µ M ) ϕ M + 2M ϕ µ M + µ L ϕ N + 2N ϕ µ L Result on gravity side matches precisely Galilean CFT results c M M 1 N 2 N 3 = s 2 N 1 N 2 N 3 = c L c M τ s2 13 s2 23 s 2 12 s2 13 s2 23 provided we choose again the Einstein values c L = 0 and c M = 12k Daniel Grumiller Flat Space Holography Recent results 23/31

93 4-point functions (enter cross-ratios) First correlators with non-universal function of cross-ratios Repeat this algorithm, localizing the sources at three points Daniel Grumiller Flat Space Holography Recent results 24/31

94 4-point functions (enter cross-ratios) First correlators with non-universal function of cross-ratios Repeat this algorithm, localizing the sources at three points Derive 4-point functions for Galilean CFTs (Bagchi, DG, Merbis 15) with the cross-ratio function and M 1 N 2 N 3 N 4 2c M g 4 (γ) = s 2 14 s2 23 s 12s 13 s 24 s 34 N 1 N 2 N 3 N 4 = 2c L g 4 (γ) + c M 4 s 2 14 s2 23 s 12s 13 s 24 s 34 g 4 (γ) = γ2 γ + 1 γ γ = s 12 s 34 s 13 s 24 4 = 4g 4(γ)η 1234 (τ τ 14 + τ 23 )g 4 (γ) η 1234 = ( 1) 1+i j (u i u j ) sin(ϕ k ϕ l )/(s 2 13s 2 24) Daniel Grumiller Flat Space Holography Recent results 24/31

95 4-point functions (enter cross-ratios) First correlators with non-universal function of cross-ratios Repeat this algorithm, localizing the sources at three points Derive 4-point functions for Galilean CFTs (Bagchi, DG, Merbis 15) with the cross-ratio function and M 1 N 2 N 3 N 4 2c M g 4 (γ) = s 2 14 s2 23 s 12s 13 s 24 s 34 N 1 N 2 N 3 N 4 = 2c L g 4 (γ) + c M 4 s 2 14 s2 23 s 12s 13 s 24 s 34 g 4 (γ) = γ2 γ + 1 γ γ = s 12 s 34 s 13 s 24 4 = 4g 4(γ)η 1234 (τ τ 14 + τ 23 )g 4 (γ) η 1234 = ( 1) 1+i j (u i u j ) sin(ϕ k ϕ l )/(s 2 13s 2 24) Gravity side yields precisely the same result! Daniel Grumiller Flat Space Holography Recent results 24/31

96 5-point functions (further check of consistency of flat space holography) Last new explicit correlators I am showing to you today (I promise) Repeat this algorithm, localizing the sources at four points Daniel Grumiller Flat Space Holography Recent results 25/31

97 5-point functions (further check of consistency of flat space holography) Last new explicit correlators I am showing to you today (I promise) Repeat this algorithm, localizing the sources at four points Derive 5-point functions for Galilean CFTs (Bagchi, DG, Merbis 15) M 1 N 2 N 3 N 4 N 5 = 4c M g 5 (γ, ζ) 1 i<j 5 s ij N 1 N 2 N 3 N 4 N 5 = 4c L g 5 (γ, ζ) + c M 5 1 i<j 5 s ij with the previous definitions and (ζ = s 25 s 34 s 35 s 24 ) g 5(γ, ζ) = γ + ζ 2(γ ζ) (γ 2 γζ + ζ 2 ) γ(γ 1)ζ(ζ 1)(γ ζ) ( [γ(γ 1)+1][ζ(ζ 1)+1] γζ ) 5 = 4 γ g 5 (γ, ζ)η ζ g 5 (γ, ζ)η g 5 (γ, ζ)τ Daniel Grumiller Flat Space Holography Recent results 25/31

98 5-point functions (further check of consistency of flat space holography) Last new explicit correlators I am showing to you today (I promise) Repeat this algorithm, localizing the sources at four points Derive 5-point functions for Galilean CFTs (Bagchi, DG, Merbis 15) M 1 N 2 N 3 N 4 N 5 = 4c M g 5 (γ, ζ) 1 i<j 5 s ij N 1 N 2 N 3 N 4 N 5 = 4c L g 5 (γ, ζ) + c M 5 1 i<j 5 s ij with the previous definitions and (ζ = s 25 s 34 s 35 s 24 ) g 5(γ, ζ) = γ + ζ 2(γ ζ) (γ 2 γζ + ζ 2 ) γ(γ 1)ζ(ζ 1)(γ ζ) ( [γ(γ 1)+1][ζ(ζ 1)+1] γζ ) 5 = 4 γ g 5 (γ, ζ)η ζ g 5 (γ, ζ)η g 5 (γ, ζ)τ Gravity side yields precisely the same result! Daniel Grumiller Flat Space Holography Recent results 25/31

99 n-point functions (holographic Ward identities and recursion relations) Shortcut to 42 (Bagchi, DG, Merbis 15) Smart check of all n-point functions? Idea: calculate n-point function from (n 1)-point function Daniel Grumiller Flat Space Holography Recent results 26/31

100 n-point functions (holographic Ward identities and recursion relations) Shortcut to 42 (Bagchi, DG, Merbis 15) Smart check of all n-point functions? Idea: calculate n-point function from (n 1)-point function Need Galilean CFT analogue of BPZ-recursion relation n ( 2 T 1 T 2... T n = + c ) 1i 2 ϕ i T 2... T n + disconnected i=2 s 2 1i Daniel Grumiller Flat Space Holography Recent results 26/31

101 n-point functions (holographic Ward identities and recursion relations) Shortcut to 42 (Bagchi, DG, Merbis 15) Smart check of all n-point functions? Idea: calculate n-point function from (n 1)-point function Need Galilean CFT analogue of BPZ-recursion relation n ( 2 T 1 T 2... T n = + c ) 1i 2 ϕ i T 2... T n + disconnected i=2 s 2 1i After small derivation we find (c ij := cot[(ϕ i ϕ j )/2]) n ( 2 M 1 N 2... N n = + c ) 1i 2 ϕ i M 2 N 3... N n +disconnected i=2 s 2 1i N 1 N 2... N n = c L c M M 1 N 2... N n + n u i ϕi M 1 N 2... N n i=1 Daniel Grumiller Flat Space Holography Recent results 26/31

102 n-point functions (holographic Ward identities and recursion relations) Shortcut to 42 (Bagchi, DG, Merbis 15) Smart check of all n-point functions? Idea: calculate n-point function from (n 1)-point function Need Galilean CFT analogue of BPZ-recursion relation n ( 2 T 1 T 2... T n = + c ) 1i 2 ϕ i T 2... T n + disconnected i=2 s 2 1i After small derivation we find (c ij := cot[(ϕ i ϕ j )/2]) n ( 2 M 1 N 2... N n = + c ) 1i 2 ϕ i M 2 N 3... N n +disconnected i=2 s 2 1i N 1 N 2... N n = c L c M M 1 N 2... N n + n u i ϕi M 1 N 2... N n We can also derive same recursion relations on gravity side! Daniel Grumiller Flat Space Holography Recent results 26/31 i=1

103 n-point functions in flat space holography Summary EH action has variational principle consistent with flat space bc s (iff we add half the GHY term!) Daniel Grumiller Flat Space Holography Recent results 27/31

104 n-point functions in flat space holography Summary EH action has variational principle consistent with flat space bc s (iff we add half the GHY term!) 0-point function shows phase transition exists between hot flat space and flat space cosmologies Daniel Grumiller Flat Space Holography Recent results 27/31

105 n-point functions in flat space holography Summary EH action has variational principle consistent with flat space bc s (iff we add half the GHY term!) 0-point function shows phase transition exists between hot flat space and flat space cosmologies 1-point functions show consistency with canonical charges and lead to first entries in holographic dictionary Daniel Grumiller Flat Space Holography Recent results 27/31

106 n-point functions in flat space holography Summary EH action has variational principle consistent with flat space bc s (iff we add half the GHY term!) 0-point function shows phase transition exists between hot flat space and flat space cosmologies 1-point functions show consistency with canonical charges and lead to first entries in holographic dictionary 2-point functions consistent with Galilean CFT for c L = 0, c M = 12k = 3/G N Daniel Grumiller Flat Space Holography Recent results 27/31

107 n-point functions in flat space holography Summary EH action has variational principle consistent with flat space bc s (iff we add half the GHY term!) 0-point function shows phase transition exists between hot flat space and flat space cosmologies 1-point functions show consistency with canonical charges and lead to first entries in holographic dictionary 2-point functions consistent with Galilean CFT for c L = 0, c M = 12k = 3/G N 42 nd variation of EH action leads to 42-point Galilean CFT correlators Daniel Grumiller Flat Space Holography Recent results 27/31

108 n-point functions in flat space holography Summary EH action has variational principle consistent with flat space bc s (iff we add half the GHY term!) 0-point function shows phase transition exists between hot flat space and flat space cosmologies 1-point functions show consistency with canonical charges and lead to first entries in holographic dictionary 2-point functions consistent with Galilean CFT for c L = 0, c M = 12k = 3/G N 42 nd variation of EH action leads to 42-point Galilean CFT correlators all n-point correlators of Galilean CFT reproduced precisely on gravity side (recursion relations!) Daniel Grumiller Flat Space Holography Recent results 27/31

109 n-point functions in flat space holography Summary EH action has variational principle consistent with flat space bc s (iff we add half the GHY term!) 0-point function shows phase transition exists between hot flat space and flat space cosmologies 1-point functions show consistency with canonical charges and lead to first entries in holographic dictionary 2-point functions consistent with Galilean CFT for c L = 0, c M = 12k = 3/G N 42 nd variation of EH action leads to 42-point Galilean CFT correlators all n-point correlators of Galilean CFT reproduced precisely on gravity side (recursion relations!) Fairly non-trivial check that 3D flat space holography can work! Daniel Grumiller Flat Space Holography Recent results 27/31

110 Other selected recent results Some further checks that dual field theory is Galilean CFT: Daniel Grumiller Flat Space Holography Recent results 28/31

111 Other selected recent results Some further checks that dual field theory is Galilean CFT: Microstate counting? Daniel Grumiller Flat Space Holography Recent results 28/31

112 Other selected recent results Some further checks that dual field theory is Galilean CFT: Microstate counting? Works! (Bagchi, Detournay, Fareghbal, Simon 13, Barnich 13) S gravity = S BH = Area cm = 2πh L = S GCFT 4G N 2h M Also as limit from Cardy formula (Riegler 14, Fareghbal, Naseh 14) Daniel Grumiller Flat Space Holography Recent results 28/31

113 Other selected recent results Some further checks that dual field theory is Galilean CFT: Microstate counting? Works! (Bagchi, Detournay, Fareghbal, Simon 13, Barnich 13) S gravity = S BH = Area cm = 2πh L = S GCFT 4G N 2h M Also as limit from Cardy formula (Riegler 14, Fareghbal, Naseh 14) (Holographic) entanglement entropy? Daniel Grumiller Flat Space Holography Recent results 28/31

114 Other selected recent results Some further checks that dual field theory is Galilean CFT: Microstate counting? Works! (Bagchi, Detournay, Fareghbal, Simon 13, Barnich 13) S gravity = S BH = Area cm = 2πh L = S GCFT 4G N 2h M Also as limit from Cardy formula (Riegler 14, Fareghbal, Naseh 14) (Holographic) entanglement entropy? Works! (Bagchi, Basu, DG, Riegler 14) S GCFT EE = c L 6 ln l x + }{{ a} like CFT l y l x c M 6 }{{} like grav anomaly Calculation on gravity side confirms result above (using Wilson lines in CS formulation) Daniel Grumiller Flat Space Holography Recent results 28/31

115 Outline Motivations Assumptions Flat space holography basics Recent results Generalizations & open issues Daniel Grumiller Flat Space Holography Generalizations & open issues 29/31

116 Generalizations & open issues Recent generalizations: adding chemical potentials Works! (Gary, DG, Riegler, Rosseel 14) In CS formulation: A 0 A 0 + µ Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

117 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) Conformal CS gravity at level k = 1 with flat space boundary conditions conjectured to be dual to chiral half of monster CFT. Action (gravity side): I CSG = k 4π d 3 x g ε λµν Γ ρ λσ( µ Γ σ νρ Γσ µτ Γ τ ) νρ Partition function (field theory side, see Witten 07): Z(q) = J(q) = 1 q q + O(q2 ) Note: ln = 4π+quantum corrections Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

118 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity Works! (Barnich, Donnay, Matulich, Troncoso 14) Asymptotic symmetry algebra = super-bms 3 Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

119 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity flat space higher spin gravity Remarkably it exists! (Afshar, Bagchi, Fareghbal, DG, Rosseel 13; Gonzalez, Matulich, Pino, Troncoso 13) New type of algebra: W-like BMS ( BMW ) [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 (n m)θ n+m + cl 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 + cm 12 n(n2 1)(n 2 4) δ n+m, 0 [L, L], [L, M],[M, M] as in BMS 3 [L, U], [L, V ], [M, U], [M, V ] as in isl(3) Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

120 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity flat space higher spin gravity Some open issues: Further checks in 3D (n-point correlators, partition function,...) Barnich, Gonzalez, Maloney, Oblak 15: 1-loop partition function matches BMS 3 character Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

121 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity flat space higher spin gravity Some open issues: Further checks in 3D (n-point correlators, partition function,...) Further generalizations in 3D (massive gravity, adding matter,...) Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

122 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity flat space higher spin gravity Some open issues: Further checks in 3D (n-point correlators, partition function,...) Further generalizations in 3D (massive gravity, adding matter,...) Generalization to 4D? (Barnich et al, Strominger et al) Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

123 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity flat space higher spin gravity Some open issues: Further checks in 3D (n-point correlators, partition function,...) Further generalizations in 3D (massive gravity, adding matter,...) Generalization to 4D? (Barnich et al, Strominger et al) Flat space limit of usual AdS 5 /CFT 4 correspondence? Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

124 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity flat space higher spin gravity Some open issues: Further checks in 3D (n-point correlators, partition function,...) Further generalizations in 3D (massive gravity, adding matter,...) Generalization to 4D? (Barnich et al, Strominger et al) Flat space limit of usual AdS 5 /CFT 4 correspondence? holography seems to work in flat space Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

125 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity flat space higher spin gravity Some open issues: Further checks in 3D (n-point correlators, partition function,...) Further generalizations in 3D (massive gravity, adding matter,...) Generalization to 4D? (Barnich et al, Strominger et al) Flat space limit of usual AdS 5 /CFT 4 correspondence? holography seems to work in flat space holography more general than AdS/CFT Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

126 Generalizations & open issues Recent generalizations: adding chemical potentials 3-derivative theory: flat space chiral gravity (Bagchi, Detournay, DG 12) generalization to supergravity flat space higher spin gravity Some open issues: Further checks in 3D (n-point correlators, partition function,...) Further generalizations in 3D (massive gravity, adding matter,...) Generalization to 4D? (Barnich et al, Strominger et al) Flat space limit of usual AdS 5 /CFT 4 correspondence? holography seems to work in flat space holography more general than AdS/CFT (when) does it work even more generally? Daniel Grumiller Flat Space Holography Generalizations & open issues 30/31

127 Thanks for your attention! Vladimir Bulatov, M.C.Escher Circle Limit III in a rectangle Daniel Grumiller Flat Space Holography Generalizations & open issues 31/31