HIGHER SPIN DUALITY from THERMOFIELD DOUBLE QFT. AJ+Kenta Suzuki+Jung-Gi Yoon Workshop on Double Field Theory ITS, ETH Zurich, Jan 20-23,2016

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1 HIGHER SPIN DUALITY from THERMOFIELD DOUBLE QFT AJ+Kenta Suzuki+Jung-Gi Yoon Workshop on Double Field Theory ITS, ETH Zurich, Jan 20-23,2016

2 Overview } Construction of AdS HS Gravity from CFT } Simplest Example of AdS/CFT Correspondence } Bi-local (Re)Construction of Higher Spins } Bulk Space-Time } At Finite Temperature } From Thermofield Double Field Theory to HS Gravity } Rindler (HS) Gravity } Black Hole: Domain Duality 2

3 Higher Spin Gravity/Duality 3 CFT3 : Vector O(N), U(N) O(N), U(N) symmetry ' i Boson : (i=1,,n) [I.R.Klebanov & A.M. Polyakov, 2002] Fermion : (i=1,,n) [E. Sezgin & P. Sundell, i2005] 2dMinimal CFT[Gaberdiel +Gopakumar] Higher Spin Theory:4D s=0,1,2,3, (Fronsdal,80,Fradkin) Vasiliev (1986~1996) Not a String Theory Other Dimensions

4 Critical CFT s in 3d O(N): Symmetry Fixed Points : g=0 g 0 UV CFT IR CFT Null Plane Sp(2N): Fermion: de Sitter Strominger, Hartman, Anninos 11 4

5 Infinite Sequence of HS Currents Generating Function 3d 2d Tracelessness Currents and Boundary Duals to AdS HS Fields of Fronsdal: Comparison of Correlation Functions 5

6 Collective Fields/Large N 6 CFT 3 Collective Field Higher Spin Theory L = 1 µ µ ~' n g=0 g N (~' ~' )2 UV n g=g c IR ' i (x µ ) (i=1,2,,n) (x µ 1,xµ 2 )=~' (x 1) ~' (x 2 ) Bi-Local O(N) invariant H µ1,µ 2,,µ s (X) X =(x µ,z) AdS 4 Non-Linear Equation H + GH H + =0 1/N Expansion G=1/N Equal Correlation Function

7 Bilocal Field Theory Exact construction Change from field bilocal field: to the O(N) invariant 3d + Represents a more general set than the conformal fields: 3d + 2d 3d 7

8 Collective/BiLocal Action } Bi-Local field : O(N) invariant singlet NX ' i (x µ 1 ) ' i (x µ 2 )= (xµ 1,xµ 2 ) i=1 Z Z col = [ Y Z D ] µ exp (x,y) S col d 3 xl + N 2 Trlog Measure Large N Entropy 8

9 1/N as A Witten Expansion (x 1,x 2 )= p N e (x1,x 2 ) S col = e c e + 1 N hv 3 e E e E e E +V 4 + Recover the correlation functions 9 h~' (x 1 ) ~' (x 0 1) ~' (x 2 ) ~' (x 0 2) ~' (x n ) ~' (x 0 n)i O(N) D = e (x1,x 0 1) e E (x 2,x 0 2) e (x n,x 0 n) col

10 Strong Operator Identification (Bulk HS ) } [S. Das, AJ, 2004] } Need to demonstrate (x µ 1,xµ 2 ) H µ 1 µ 2 µ s (X) Collective Higher Spin Theory (x µ 1,xµ 2 ) X + Spin 3+3=6 AdS 4 Internal AdS4 Issue : Gauges of HS Theory/ and Reductions 10

11 HS Equations } Tensor Fields (Symmetric, traceless) : } 5-D version X A X A = 1 H (X, Y )= X s H A1 A 2 A s (X) Y A 1 Y A2 Y A s 11

12 Fronsdal Gauge } Consider the field H(X,Y) } } } K 2 = P K = P =0 H =0 Traceless Condition r H =0 De Donder Gauge Condition m 2 ' µ =0 Equation of motion 12

13 The Map 13 Collective EAdS 4 (u µ 1,uµ 2 ) H (X, z ) S 3 S 3 EAdS 4 S 2 SO(1,4) J (1) + J (2) = J Addition of Angular Momenta [C. Fronsdal & M. Flato,1978] D 1 2, 0 D 1 2, 0 = 1X D (s +1,s) s=0

14 Propagator: One Loop } Collective } One Loop agreement with[a.giombi,klebanov, Tseytlin 2013,2014] } Sign of Stronger Operator Identities } Bulk col 2 1@ tr log col 1 2 tr = 1 8 log 2 3 (3) 16 2 Trlog col = Trlog scalar+hs gh col scalar+hs 14

15 Light-Cone Gauge CFT 3 : collective bi-local fields AdS 4 : higher spin fields Same number of dimensions = Representation of the conformal group SO(2,3) It is a canonical transformation 15

16 Light Cone Map: } Light-cone (KJin,Aj,dMello,Rodrigues, 2011 /Brodsky at al /Polchinski p + = p p+ 2 x = x 1 p+ 1 + x 2 p+ 2 p p+ 2 : p x = p 1 + p 2 s p z = p + 2 p + 1 p 1 = 2 arctan s s p + 2 p + 1 p + 1 p + 2 p 2 x = x 1p x 2p + 2 p p+ 2q z = (x 1 x 2 ) p + 1 p+ 2 p p+ 2 q p = p + 1 p+ 2 (x 1 x 2 ) + x 1 x 2 2 s p + 2 p + 1 p 1 + s p + 1 p + 2 p 2! 16

17 Covariant Map } Bi-local Map [AJ, RdM Koch, J P Rodrigues and J Yoon, 2014] } From Bi-local Z collective field to HS field in AdS3 A s (~x, z) = d 2 ~pdp z d~p 1 d~p 2 f(~x, z; ~p, p z ; ~p 1, ~p 2 ) ( p 2 ~p 1 ~p 2 2~p 1 ~p 2 p z ) (2) (~p 1 + ~p 2 ~p)a(~p 1, ~p 2 ) ~p = ~p 1 + ~p 2 p z =2 p '1 ' 2 ~p 1 ~p 2 sin 2 2~p2 ~p 1 = arctan ( ~p 2 ~p 1 )p z 17

18 Conclusion Strong Operator Correspondence : Bi-Local and HS Fields CFT Gauge Invariant Data of HS Gravity Holography : A Gauge Phenomena Nonlinear/ 1/N=G 18

19 FINITE TEMPERATURE } O(N) Vector Model : Phases T c p N } at [S. H. Shenker & X. Yin, 2011] } Lower Phase : F low ' 4 (5)T 4 } Higher Phase : 1 T T c T F high ' 4 (3)NT 2 T c Hamiltonian Formalism Collective Field Theory F low = X log 1 e singlet F high = N 2 tr log H [AJ, K. Jin, J. Yoon, 2014] Action Formalism Collective Field Theory [AJ, K. Jin, J. Yoon, 2014] 19

20 Thermo-field Dynamics } [Schwinger-Kelydish,Y. Takahashi and H. Umezawa, 1975] H =!a a H e =!ea ea } Double Hilbert space H TFD H H e } Path Integral along Contour in Complex time plane Im t t i t f Re t t i - iβ/2 t i - iβ t f - iβ/2 20

21 DOMAIN ADS/CFT DUALITY } Space-Times with L-R Wedges: } Maldacena,] } Karczmarek, Nogueira& Raamsdonk 12] Σ p i i i = } Two (L-R) Boundaries:Entangled Hyperbolic CFT s:domain Duality } Connected Bulk Space-Time from Decoupled CFT s? [ Bousso,Mathur,Avery+Chowdury] 21

22 Thermo-field Dynamics } Thermal Vacuum 0( )i } : Expectation value equals Finite Temperature VEV } Inverse Temperature 0( )i e (a ea aea) 0i = 1! log coth hoi h0( ) O 0( )i = 1 Z( ) Tr (e H O) 22

23 Dual HS Theory:AdS SPACE-TIMES:with HORIZONS } D=4 Rindler-AdS: Boundary Metric is given by, 23

24 Hyperbolic Black Holes } D=4 Hyperbolic BH: } BH Temperature: Horizon location from 24

25 FLUCTUATING MODES } Scalar in Pure AdS: } z-component of the Mode satisfies 25

26 Effective Potential of Pure AdS } The Effective Potential } The Frequency is Bounded by } Bulk Radial Momentum is real Propagating Modes 26

27 Effective Potential /Evanescent Mode } AdS BlackHole [ Rey& Rosenhaus 14] V is the tortoise coordinate defined by Near Horizon, } Bulk Radial Momentum can be imaginary Evanescent Modes Π 2 r 27

28 TFD : O(N) Vector Model } O(N) Vector Model } O(N) Symmetry U ab b U 2 O(N) } TFD of O(N) Vector Model : Doubled Vector field: H TFD H H e } O(N) Symmetry of TFD a }? a ea H = NX Z i=1 d~x O(N) O(N) apple 1 2 ( i ) i ) 2 U ab b V ab eb U, V 2 O(N) a ea O(N) }? a ea U ab b U ab eb U 2 O(N) 28

29 LOW TEMPERATURE PHASE:O(N) O(N) GAGING } Collective TFD with } Invariant variables O(N) O(N) } Collective TFD Hamiltonian(Doubled) (t; ~x, ~y) = i (t, ~x) i (t, ~y) e (t; ~x, ~y) = ei (t, ~x) ei (t, ~y) H TFD = H col e Hcol } Classical Solution is not finite temperature two-point function. 0(t; ~x, ~y) 6= h' i (t, ~x)' i (t, ~y)i } Thermal AdS4 background 29

30 High Temperature Phase 30 Singlet Constraint : Diagonal O(N) singlet j ej U jk k U jk ek J ij + e J ij i =0 Invariant Collective Fields i (t, ~x) i (t, ~y) i (t, ~x) ei (t, ~y) ei (t, ~x) ei (t, ~y) Hamiltonian of TFD H TFD = H vec e Hvec

31 HIGH TEMP PHASE:Cntd } Collective TFD with Diagonal O(N) Symmetry } Invariant Variables i (t, ~x) i (t, ~y) i (t, ~x) ei (t, ~y) ei (t, ~x) ei (t, ~y) } Define Collective Field ((~x, i), (~y, j)) a (~x) a (~y) i ea (~x) a (~y)! i a (~x) ea (~y) ea (~x) ea (~y) } Correct Hamiltonian of TFD of O(N) Vector Model H TFD = 2 N Tr [?? ]+N 8 Tr 1 + N 2 Tr r 2? + V 31

32 Cont.:TFD FLUCTUATIONS } Large N, Background : 1 2 0? r 2? 0 = 1 8 I } Solution with one free parameter F 0((~x, i), (~y, j)) = Z d~p (2 ) 2 2 ~p cosh F (~p)e i~p (~x ~y) i~p (~x i sinh F (~p)e ~y) i sinh F (~p)e i~p (~x ~y) i~p (~x cosh F (~p)e ~y) } Agreement with finite temperature two-point function 0((~x, i), (~y, j)) = h ((~x, i), (~y, j))i = 0 D h a (~x) a (~y)i i a (~x) ea (~y) D D E i e a (~x) (~y)e a e a (~x) ea (~y) 1 A F (~p) = 2 tanh 1 e ~p cf. In QM, 2 = 2 tanh 1 e! 32

33 Fluctuation } Expand Collective Field around Background (t;(~x, i), (~y, j)) = 0((~x, i), (~y, j)) + 1 p N (t;(~x, i), (~y, j)) (t;(~x, i), (~y, j)) = p N (t;(~x, i), (~y, j)) } Quadratic Hamiltonian H (2) = 2Tr [? 0? ]+ 1 8 Tr 1 0?? 1 0??

34 Modes } (~p 1, ~p 2 ) create a states with p 0 = ~p 1 + ~p 2 ~p = ~p 1 + ~p 2 } e (~p 1, ~p 2 ) create states with p 0 = ~p 1 ~p 2 ~p = ~p 1 ~p 2 } (p z ) 2 =(p 0 ) 2 ~p 2 = 0 34

35 Evanescent Mode } (~p 1, ~p 2 ) Mixed bi-locals: create states with p 0 = ~p 1 ~p 2 ~p = ~p 1 ~p 2 (p z ) 2 =(p 0 ) 2 ~p } p z : pure imaginary } This mode exponentially decay along z direction } Evanescent mode } [S. Leichenauer and V. Rosenhaus, 2013], [S. J. Rey and V. Rosenhaus, 2014] 35

36 BI-LOCAL MAP FOR RINDLER-ADS } Bi-local Map to to Rindler-AdS: (4.1) 36

37 } Mixed State: Imaginary Value in Radial Direction Momentum Evanescent Modes 37

38 CONCLUSION } Bi-local Construction of Rindler-AdS Spacetime from the Two Entangled O(N) Vector Model CFT s } Entanglement through Diagonal Gauging of O(N) } Extra Degrees of Freedom: Evanescent Modes } Supports: More General Duality Domain Duality } Mixing: LR coupled through Gauss Law } Does not Support :Reconstructions beyond the Horizon: 38