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lim AdS 3 / CFT 2 Flat Space Holography Daniel Grumiller Institute for Theoretical Physics TU Wien All about AdS3 ETH Zurich, November 2015 Some of our papers on flat space holography A. Bagchi, D. Grumiller and W.

  1. 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 , ¯ L n M n = 1 L n = L n − ¯ L n + ¯ � � L − n L − n ℓ ◮ Make In¨ on¨ u–Wigner contraction ℓ → ∞ on ASA [ L n , L m ] = ( n − m ) L n + m + c L 12 ( n 3 − n ) δ n + m, 0 [ L n , M m ] = ( n − m ) M n + m + c M 12 ( n 3 − 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 8/25

  2. 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 , ¯ L n M n = 1 L n = L n − ¯ L n + ¯ � � L − n L − n ℓ ◮ Make In¨ on¨ u–Wigner contraction ℓ → ∞ on ASA [ L n , L m ] = ( n − m ) L n + m + c L 12 ( n 3 − n ) δ n + m, 0 [ L n , M m ] = ( n − m ) M n + m + c M 12 ( n 3 − 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 8/25

  3. 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 , ¯ L n M n = 1 L n = L n − ¯ L n + ¯ � � L − n L − n ℓ ◮ Make In¨ on¨ u–Wigner contraction ℓ → ∞ on ASA [ L n , L m ] = ( n − m ) L n + m + c L 12 ( n 3 − n ) δ n + m, 0 [ L n , M m ] = ( n − m ) M n + m + c M 12 ( n 3 − 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 8/25

  4. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 9/25

  5. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 9/25

  6. 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 � CS = k �A ∧ d A + 2 S flat 3 A ∧ A ∧ A� 4 π 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 = d A + A ∧ A = 0 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 9/25

  7. 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 � CS = k �A ∧ d A + 2 S flat 3 A ∧ A ∧ A� 4 π 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 9/25

  8. 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 � CS = k �A ∧ d A + 2 S flat 3 A ∧ A ∧ A� 4 π 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 d u + L 1 − M ( ϕ ) L − 1 − N ( u, ϕ ) M − 1 d ϕ with N ( u, ϕ ) = L ( ϕ ) + u 2 M ′ ( ϕ ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 9/25

  9. 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 � CS = k �A ∧ d A + 2 S flat 3 A ∧ A ∧ A� 4 π 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 d u + L 1 − M ( ϕ ) L − 1 − N ( u, ϕ ) M − 1 d ϕ with N ( u, ϕ ) = L ( ϕ ) + u 2 M ′ ( ϕ ) ◮ metric d s 2 = M d u 2 − 2 d u d r +2 N d u d ϕ + r 2 d ϕ 2 g µν ∼ 1 2 � tr �A µ A ν � → Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Flat space holography basics 9/25

  10. Outline Motivations Flat space holography basics Recent results Generalizations & open issues Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 10/25

  11. Correlation functions in flat space holography AdS/CFT good tool for calculating correlators What about flat space/Galilean CFT correspondence? Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 11/25

  12. 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 ∼ δ 42 � δg 42 Γ EH-AdS � EOM ? Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 11/25

  13. 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 ∼ δ 42 � δg 42 Γ EH-AdS � EOM ? ◮ Does it work? Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 11/25

  14. 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 ∼ δ 42 � δg 42 Γ EH-AdS � EOM ? ◮ Does it work? ◮ What is the left hand side in a Galilean CFT? Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 11/25

  15. 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 ∼ δ 42 � 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 11/25

  16. 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 ∼ δ 42 � 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 11/25

  17. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 12/25

  18. 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? � � 1 d 3 x √ g R − 1 d 2 x √ γ K − I counter-term Γ = − 16 πG N 8 πG N with I counter-term chosen such that � � δ Γ EOM = 0 for all δg that preserve flat space bc’s Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 12/25

  19. 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? � � 1 d 3 x √ g R − 1 d 2 x √ γ K − I counter-term Γ = − 16 πG N 8 πG N with I counter-term chosen such that � � δ Γ EOM = 0 for all δg that preserve flat space bc’s Result (Detournay, DG, Sch¨ oller, Simon ’14): � � 1 d 3 x √ g R − 1 d 2 x √ γ K Γ = − 16 πG N 16 πG 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 12/25

  20. 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 � � D g e − Γ[ g ] = e − Γ[ g c ( T, Ω)] × Z fluct. Z ( T, Ω) = 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 + β Ω) ∼ ( τ E , ϕ + 2 π ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 12/25

  21. 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 d s 2 = d τ 2 E + d r 2 + r 2 d ϕ 2 ◮ Flat space cosmology r 2 d r 2 � � 0 ) + r 2 � � 2 1 − r 2 d ϕ − r + r 0 d s 2 = r 2 0 d τ 2 E + d τ E + ( r 2 − r 2 + r 2 r 2 r 2 shifted-boost orbifold, see Cornalba, Costa ’02 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 12/25

  22. 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 1 F FSC = − r + F HFS = − 8 G N 8 G N Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 12/25

  23. 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 1 F FSC = − r + F HFS = − 8 G N 8 G N ◮ Result of this comparison ◮ r + > 1 : FSC dominant saddle ◮ r + < 1 : HFS dominant saddle Critical temperature: 1 = Ω T c = 2 πr 0 2 π HFS “melts” into FSC at T > T c Bagchi, Detournay, DG, Simon ’13 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 12/25

  24. 1-point functions (conserved charges) First check of entries in holographic dictionary: identification of sources and vevs In AdS 3 : � � � T µν � δ Γ EOM ∼ vev × δ source ∼ BY × δg NN µν ∂ M ∂ M Note that T µν BY follows from canonical analysis as well (conserved charges) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 13/25

  25. 1-point functions (conserved charges) First check of entries in holographic dictionary: identification of sources and vevs In AdS 3 : � � � T µν � δ Γ EOM ∼ vev × δ source ∼ BY × δg NN µν ∂ M ∂ M Note that T µν BY follows from canonical analysis as well (conserved charges) In flat space: ◮ non-normalizable solutions to linearized EOM? ◮ analogue of Brown–York stress tensor? ◮ comparison with canonical results Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 13/25

  26. 1-point functions (conserved charges) First check of entries in holographic dictionary: identification of sources and vevs In AdS 3 : � � � T µν � δ Γ EOM ∼ vev × δ source ∼ BY × δg NN µν ∂ M ∂ M Note that T µν BY follows from canonical analysis as well (conserved charges) In flat space: ◮ non-normalizable solutions to linearized EOM? ◮ analogue of Brown–York stress tensor? ◮ comparison with canonical results everything works (Detournay, DG, Sch¨ oller, Simon, ’14) mass and angular momentum: M = g tt N = g tϕ 8 G 4 G full tower of canonical charges: see Barnich, Compere ’06 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 13/25

  27. 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder ( ϕ ∼ ϕ + 2 π ): � M ( u 1 , ϕ 1 ) M ( u 2 , ϕ 2 ) � = 0 � M ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c M 2 s 4 12 � N ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c L − 2 c M τ 12 2 s 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 n e − inϕ − c M M := 24 n � � � e − inϕ − c L N := L n − inuM n 24 n Conservation equations: ∂ u M = 0 , ∂ u N = ∂ ϕ M Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 14/25

  28. 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder ( ϕ ∼ ϕ + 2 π ): � M ( u 1 , ϕ 1 ) M ( u 2 , ϕ 2 ) � = 0 � M ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c M 2 s 4 12 � N ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c L − 2 c M τ 12 2 s 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 14/25

  29. 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder ( ϕ ∼ ϕ + 2 π ): � M ( u 1 , ϕ 1 ) M ( u 2 , ϕ 2 ) � = 0 � M ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c M 2 s 4 12 � N ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c L − 2 c M τ 12 2 s 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 14/25

  30. 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder ( ϕ ∼ ϕ + 2 π ): � M ( u 1 , ϕ 1 ) M ( u 2 , ϕ 2 ) � = 0 � M ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c M 2 s 4 12 � N ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c L − 2 c M τ 12 2 s 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 14/25

  31. 2-point functions (anomalous terms) First check sensitive to central charges in symmetry algebra Galilean CFT on cylinder ( ϕ ∼ ϕ + 2 π ): � M ( u 1 , ϕ 1 ) M ( u 2 , ϕ 2 ) � = 0 � M ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c M 2 s 4 12 � N ( u 1 , ϕ 1 ) N ( u 2 , ϕ 2 ) � = c L − 2 c M τ 12 2 s 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 14/25

  32. 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 � d 2 z µ ( z, ¯ S µ = S 0 + z ) T ( z ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  33. 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 � d 2 z µ ( z, ¯ S µ = S 0 + z ) T ( z ) ◮ Localize source z ) = ǫ δ (2) ( z − z 2 , ¯ µ ( z, ¯ z − ¯ z 2 ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  34. 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 � d 2 z µ ( z, ¯ S µ = S 0 + z ) T ( z ) ◮ Localize source z ) = ǫ δ (2) ( z − z 2 , ¯ µ ( z, ¯ z − ¯ z 2 ) ◮ 1-point function in µ -vacuum → 2-point function in 0-vacuum � T 1 � µ = � T 1 � 0 + ǫ � T 1 T 2 � 0 + O ( ǫ 2 ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  35. 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 � d 2 z µ ( z, ¯ S µ = S 0 + z ) T ( z ) ◮ Localize source z ) = ǫ δ (2) ( z − z 2 , ¯ µ ( z, ¯ z − ¯ z 2 ) ◮ 1-point function in µ -vacuum → 2-point function in 0-vacuum � T 1 � µ = � T 1 � 0 + ǫ � T 1 T 2 � 0 + O ( ǫ 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 Ba˜ nados, Caro ’04 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  36. 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 � d 2 z µ ( z, ¯ S µ = S 0 + z ) T ( z ) ◮ Localize source z ) = ǫ δ (2) ( z − z 2 , ¯ µ ( z, ¯ z − ¯ z 2 ) ◮ 1-point function in µ -vacuum → 2-point function in 0-vacuum � T 1 � µ = � T 1 � 0 + ǫ � T 1 T 2 � 0 + O ( ǫ 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 ) + ǫ L (1) ( z ) + O ( ǫ 2 ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  37. 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 ) + ǫ L (1) ( z ) + O ( ǫ 2 ) ◮ Write EOM to first subleading order in ǫ ∂ L (1) ( z ) = − k ¯ 2 ∂ 3 δ (2) ( z − z 2 ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  38. 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 ) + ǫ L (1) ( z ) + O ( ǫ 2 ) ◮ Write EOM to first subleading order in ǫ ∂ 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 z 1 G ( z 12 ) = 3 k 2 ∂ 4 z 4 12 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  39. 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 ) + ǫ L (1) ( z ) + O ( ǫ 2 ) ◮ Write EOM to first subleading order in ǫ ∂ 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 z 1 G ( z 12 ) = 3 k 2 ∂ 4 z 4 12 ◮ This is the correct CFT 2-point function on the plane with c = 6 k Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  40. 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 ) + ǫ L (1) ( z ) + O ( ǫ 2 ) ◮ Write EOM to first subleading order in ǫ ∂ 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 z 1 G ( z 12 ) = 3 k 2 ∂ 4 z 4 12 ◮ This is the correct CFT 2-point function on the plane with c = 6 k ◮ Generalize to cylinder Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 15/25

  41. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 16/25

  42. 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 = ǫ M/L δ (2) ( u − u 2 , ϕ − ϕ 2 ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 16/25

  43. 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 = ǫ 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 16/25

  44. 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 = ǫ 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 ǫ M/L � � ∂ u M (1) = − k ǫ L ∂ 3 ϕ δ + ∂ ϕ δ � � ∂ u N (1) = − k ǫ M ∂ 3 + ∂ ϕ M (1) ϕ δ + ∂ ϕ δ Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 16/25

  45. 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 = ǫ 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 ǫ M/L � � ∂ u M (1) = − k ǫ L ∂ 3 ϕ δ + ∂ ϕ δ � � ∂ u N (1) = − k ǫ M ∂ 3 + ∂ ϕ M (1) ϕ δ + ∂ ϕ δ ◮ Solve with Green function on cylinder M (1) = 6 kǫ L N (1) = 6 k ( ǫ M − 2 ǫ L τ 12 ) s 4 s 4 12 12 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 16/25

  46. 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 = ǫ 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 ǫ M/L � � ∂ u M (1) = − k ǫ L ∂ 3 ϕ δ + ∂ ϕ δ � � ∂ u N (1) = − k ǫ M ∂ 3 + ∂ ϕ M (1) ϕ δ + ∂ ϕ δ ◮ Solve with Green function on cylinder M (1) = 6 kǫ L N (1) = 6 k ( ǫ M − 2 ǫ L τ 12 ) s 4 s 4 12 12 ◮ Correct 2-point functions for Einstein gravity with c L = 0 , c M = 12 k Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 16/25

  47. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 17/25

  48. 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 � ǫ i M/L δ (2) ( u 1 − u i , ϕ 1 − ϕ i ) µ M/L ( u 1 , ϕ 1 ) = i =2 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 17/25

  49. 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 � ǫ i M/L δ (2) ( u 1 − u i , ϕ 1 − ϕ i ) µ M/L ( u 1 , ϕ 1 ) = i =2 ◮ Iteratively solve EOM ∂ u M = − k∂ 3 ϕ µ L + µ L ∂ ϕ M + 2 M∂ ϕ µ L ∂ u N = − k∂ 3 ϕ µ M + (1 + µ M ) ∂ ϕ M + 2 M∂ ϕ µ M + µ L ∂ ϕ N + 2 N∂ ϕ µ L Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 17/25

  50. 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 � ǫ i M/L δ (2) ( u 1 − u i , ϕ 1 − ϕ i ) µ M/L ( u 1 , ϕ 1 ) = i =2 ◮ Iteratively solve EOM ∂ u M = − k∂ 3 ϕ µ L + µ L ∂ ϕ M + 2 M∂ ϕ µ L ∂ u N = − k∂ 3 ϕ µ M + (1 + µ M ) ∂ ϕ M + 2 M∂ ϕ µ M + µ L ∂ ϕ N + 2 N∂ ϕ µ L ◮ Result on gravity side matches precisely Galilean CFT results c M � N 1 N 2 N 3 � = c L − c M τ 123 � M 1 N 2 N 3 � = s 2 12 s 2 13 s 2 s 2 12 s 2 13 s 2 23 23 provided we choose again the Einstein values c L = 0 and c M = 12 k Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 17/25

  51. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 18/25

  52. 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) 2 c M g 4 ( γ ) � M 1 N 2 N 3 N 4 � = s 2 14 s 2 23 s 12 s 13 s 24 s 34 � N 1 N 2 N 3 N 4 � = 2 c L g 4 ( γ ) + c M ∆ 4 s 2 14 s 2 23 s 12 s 13 s 24 s 34 with the cross-ratio function g 4 ( γ ) = γ 2 − γ + 1 γ = s 12 s 34 γ s 13 s 24 and ∆ 4 = 4 g ′ 4 ( γ ) η 1234 − ( τ 1234 + τ 14 + τ 23 ) g 4 ( γ ) � ( − 1) 1+ i − j ( u i − u j ) sin( ϕ k − ϕ l ) / ( s 2 13 s 2 η 1234 = 24 ) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 18/25

  53. 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) 2 c M g 4 ( γ ) � M 1 N 2 N 3 N 4 � = s 2 14 s 2 23 s 12 s 13 s 24 s 34 � N 1 N 2 N 3 N 4 � = 2 c L g 4 ( γ ) + c M ∆ 4 s 2 14 s 2 23 s 12 s 13 s 24 s 34 with the cross-ratio function g 4 ( γ ) = γ 2 − γ + 1 γ = s 12 s 34 γ s 13 s 24 and ∆ 4 = 4 g ′ 4 ( γ ) η 1234 − ( τ 1234 + τ 14 + τ 23 ) g 4 ( γ ) � ( − 1) 1+ i − j ( u i − u j ) sin( ϕ k − ϕ l ) / ( s 2 13 s 2 η 1234 = 24 ) ◮ Gravity side yields precisely the same result! Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 18/25

  54. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 19/25

  55. 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 � = 4 c M g 5 ( γ, ζ ) � 1 ≤ i<j ≤ 5 s ij � N 1 N 2 N 3 N 4 N 5 � = 4 c 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 ) ( γ 2 − γζ + ζ 2 ) γ + ζ � � g 5 ( γ, ζ ) = [ γ ( γ − 1)+1][ ζ ( ζ − 1)+1] − γζ 2( γ − ζ ) − γ ( γ − 1) ζ ( ζ − 1)( γ − ζ ) × ∆ 5 = 4 ∂ γ g 5 ( γ, ζ ) η 1234 + 4 ∂ ζ g 5 ( γ, ζ ) η 2345 − 2 g 5 ( γ, ζ ) τ 12345 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 19/25

  56. 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 � = 4 c M g 5 ( γ, ζ ) � 1 ≤ i<j ≤ 5 s ij � N 1 N 2 N 3 N 4 N 5 � = 4 c 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 ) ( γ 2 − γζ + ζ 2 ) γ + ζ � � g 5 ( γ, ζ ) = [ γ ( γ − 1)+1][ ζ ( ζ − 1)+1] − γζ 2( γ − ζ ) − γ ( γ − 1) ζ ( ζ − 1)( γ − ζ ) × ∆ 5 = 4 ∂ γ g 5 ( γ, ζ ) η 1234 + 4 ∂ ζ g 5 ( γ, ζ ) η 2345 − 2 g 5 ( γ, ζ ) τ 12345 ◮ Gravity side yields precisely the same result! Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 19/25

  57. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 20/25

  58. 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 � 2 n � � + c 1 i � T 1 T 2 . . . T n � = � T 2 . . . T n � + disconnected 2 ∂ ϕ i s 2 1 i i =2 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 20/25

  59. 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 � 2 n � � + c 1 i � T 1 T 2 . . . T n � = � T 2 . . . T n � + disconnected 2 ∂ ϕ i s 2 1 i i =2 ◮ After small derivation we find ( c ij := cot[( ϕ i − ϕ j ) / 2] ) � 2 � n � + c 1 i � M 1 N 2 . . . N n � = � M 2 N 3 . . . N n � +disconnected 2 ∂ ϕ i s 2 1 i i =2 n � � N 1 N 2 . . . N n � = c L � M 1 N 2 . . . N n � + u i ∂ ϕ i � M 1 N 2 . . . N n � c M i =1 Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 20/25

  60. 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 � 2 n � � + c 1 i � T 1 T 2 . . . T n � = � T 2 . . . T n � + disconnected 2 ∂ ϕ i s 2 1 i i =2 ◮ After small derivation we find ( c ij := cot[( ϕ i − ϕ j ) / 2] ) � 2 � n � + c 1 i � M 1 N 2 . . . N n � = � M 2 N 3 . . . N n � +disconnected 2 ∂ ϕ i s 2 1 i i =2 n � � N 1 N 2 . . . N n � = c L � M 1 N 2 . . . N n � + u i ∂ ϕ i � M 1 N 2 . . . N n � c M i =1 ◮ We can also derive same recursion relations on gravity side! Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 20/25

  61. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 21/25

  62. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 21/25

  63. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 21/25

  64. 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 = 12 k = 3 /G N Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 21/25

  65. 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 = 12 k = 3 /G N ◮ 42 nd variation of EH action leads to 42-point Galilean CFT correlators Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 21/25

  66. 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 = 12 k = 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 21/25

  67. 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 = 12 k = 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 21/25

  68. Other selected recent results Some further checks that dual field theory is Galilean CFT: Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 22/25

  69. Other selected recent results Some further checks that dual field theory is Galilean CFT: ◮ Microstate counting? Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 22/25

  70. Other selected recent results Some further checks that dual field theory is Galilean CFT: ◮ Microstate counting? Works! (Bagchi, Detournay, Fareghbal, Simon ’13, Barnich ’13) � c M S gravity = S BH = Area = 2 πh L = S GCFT 4 G N 2 h M Also as limit from Cardy formula (Riegler ’14, Fareghbal, Naseh ’14) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 22/25

  71. Other selected recent results Some further checks that dual field theory is Galilean CFT: ◮ Microstate counting? Works! (Bagchi, Detournay, Fareghbal, Simon ’13, Barnich ’13) � c M S gravity = S BH = Area = 2 πh L = S GCFT 4 G N 2 h M Also as limit from Cardy formula (Riegler ’14, Fareghbal, Naseh ’14) ◮ (Holographic) entanglement entropy? Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 22/25

  72. Other selected recent results Some further checks that dual field theory is Galilean CFT: ◮ Microstate counting? Works! (Bagchi, Detournay, Fareghbal, Simon ’13, Barnich ’13) � c M S gravity = S BH = Area = 2 πh L = S GCFT 4 G N 2 h M Also as limit from Cardy formula (Riegler ’14, Fareghbal, Naseh ’14) ◮ (Holographic) entanglement entropy? Works! (Bagchi, Basu, DG, Riegler ’14) = c L 6 ln ℓ x c M ℓ y S GCFT + EE a 6 ℓ x � �� � � �� � like CFT like grav anomaly Calculation on gravity side confirms result above (using Wilson lines in CS formulation) Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Recent results 22/25

  73. Outline Motivations Flat space holography basics Recent results Generalizations & open issues Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 23/25

  74. Generalizations & open issues Recent generalizations: ◮ adding chemical potentials Works! (Gary, DG, Riegler, Rosseel ’14) In CS formulation: A 0 → A 0 + µ Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 24/25

  75. 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): � d 3 x √− g ε λµν Γ ρλσ � � I CSG = k ∂ µ Γ σνρ + 2 3 Γ σµτ Γ τ νρ 4 π Partition function (field theory side, see Witten ’07): Z ( q ) = J ( q ) = 1 q + 196884 q + O ( q 2 ) Note: ln 196883 ≈ 12 . 2 = 4 π + quantum corrections Daniel Grumiller — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 24/25

  76. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 24/25

  77. 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 )(2 n 2 + 2 m 2 − nm − 8) L n + m + 192 c M ( n − m )Λ n + m � c L + 44 � − 96 ( n − m )Θ n + m + c L 12 n ( n 2 − 1)( n 2 − 4) δ n + m, 0 5 c 2 M [ U n , V m ] = ( n − m )(2 n 2 + 2 m 2 − nm − 8) M n + m + 96 c M ( n − m )Θ n + m + c M 12 n ( n 2 − 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 24/25

  78. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 24/25

  79. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 24/25

  80. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 24/25

  81. 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 — lim ℓ →∞ � AdS 3 / CFT 2 � Generalizations & open issues 24/25

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