Elements of Bi-Local Holography

Transcription

1 Elements of Bi-Local Holography Antal Jevicki Brown University Fifteenth Workshop on Non- Perturbative QCD,-4 June 08

2 Vector Model / Higher Spin Gravity } Large N } d=3 : * L =(@ ~ ) (@ ~ )+ ( ~ ~) N UV : =0 IR : = * Vasiliev HS Theory [Klebanov & Polyakov 0] [Sezgin & Sundell 0] [Giombi & Yin 09] } d= : AdS_3 HS / Minimal Model [Gaberdiel & Gopakumar 0] } d= SYK model: AdS_ HS Gravity

3 Higher Spin Theory : s=0,,,3.. } AdS{d+} HS fields : Fronsdal 80 ( m )h µ µ s + s r (µ r h µ µ s ) s(s ) (d +s 3) g (µ µ r r h µ3 µ s ) =0 } Fradkin, Vasiliev ( ) :Nonlinear Eqs/HS-gauge symmetry } No known Action, Feynman rules, Loops? 3

4 Gauge Reduction for HS Theory } Physical Realization :obtained by solving the tracelessness & de Donder gauge conditions: For AdS3 results z z H(x i,z, ) = Physical space of hs: AdS + S X s= ˆµ ˆµ Hˆµ ˆµ ˆµ s =0 H z ˆµ ˆµ x µ H µˆµ ˆµ s =0 e ±is H (±s) (x i,z) 4

5 Bi-local (Re)Construction (x,x )= N NX i= i(x ) i (x ) CFT_d => AdS_{d+} + Spin (S^{d-}) } Sumit Das,AJ (006),R.de Mello Koch, Kenta Suzuki, J.Rodrigues, J.Yoon(0-) Bi-local Field: (x µ,xµ ), H(xµ,z; S) d + d = (d + ) + (d ) } Map to Higher Spins in AdS_(d+). 5

6 Large N Action } Collective Action: S col = N Z apple d d x r x (x, x 0 ) x 0 =x + (x, x) N Tr log } AJ Sakita 80 matrix theories Recent :Eff action Gurau } Saddle-point: Quadratic Fluctuations: Z 6 (x,x )= 0(x,x )+ p N (x,x ) S () = 4 4Y d d x a (x,x ) L b bi (x 3,x 4 ) a=

7 Quadratic Kernel } Conformal symmetry: Casimirs of SO(,3) bl bi C C 4 x with C L ABL AB C 4 4 L A B L B C L C D L D A C 7

8 Eigenvalue Problem } For the Bi-local Laplacian: bl bi h,s (x,x )= (h, s) h,s (x,x ) } The eigenfunctions are conformal spinning 3-pt funcs: h,s(~x, ~x ; ~x 3 )= (V µ z µ ) s x x 3 h+s x x 3 h+s x x +s h and the eigenvalues are (h, s) = 4 (h s ) 8

9 d= Example } Eigenfunctions :in terms of by Bessel functions: h,s(~x, ~x ; ~p) = e i~p ~x K h+s (ip ) K h s (i p ) Exact eigenvalues (h, s) = (h + s)(h + s )(h s)(h s ) 4 where 9 ~x ~x + ~x, ~ ~x ~x

10 Momentum Space } Fourier Transform of the eigenfunctions (for CFT_) can be written as h,s( ~ k, ~ k ; ~p) = ( ~ k + ~ k ~p)f (k + )F (k ) s k+ F (k + )F (k )= ei (arcsin p + arcsin k ) p e i h k+ (arcsin p + +arcsin k ) p p (k+ ) (p + ) p (k ) (p ) where k ± k ± k ±, k± i (k0 i ± k i ) 0

11 AdS_3 * S^ Wave Functions:Mom Space } The Laplacian: L AdS3 S = In momentum-space, L AdS3 S = } With : wave z } Compare with Bi-local function: z p p p h,s = e ±ih p arccos i e ±is pz p

12 Momentum Map } Comparing the Eigenfunctions (in momentum space) leads to ~p = ~ k + ~ k q q p z = k + k k + k arcsin k+ = In Fourier (Momentum) Space, just change of (momentum) variables: Momentum space in ads/cft ( also by Skenderis, et al,04 p + arcsin k p e (k i,k i )= e H(p i,p z, )

13 Configuration Space Transform } By chain rule } The fields are transformed } Off-shell/ z = p = (x i,x i )= H(x i,z,s)= q q q + q+ (u u )+ q q (u + u + ) q qq + q + q q + q qq q (u + u + ) q + q+ (u u ) Z Z M(x i,x i x i,z,s) H(x i,z,s) M (x i,z,s x i,x i ) (x i,x i ) 3

14 BI-LOCAL (WITTEN) DIAGRAMS } After expanding the bi-local field as The Action leads to vertices S[ ] = N 4 (x,x )= 0(x,x )+ p N (x,x ) Z 4Y a= N 6 Tr + N 8 Tr d d x a (x,x )K(x,x ; x 3,x 4 ) (x 3,x 4 ) 0?? 0?? 0?? 0?? 0? 0?? 0? + 4

15 Bi-local Propagator } Using the Spinning 3-pt funcs for complete basis, bi-local propagator is given by D(x,x ; x 3,x 4 ) / X h,s,~p h,s (x,x ; p) h,s (x 3,x 4 ; p) x x x 3 x 4 h,s } All-order Vertices V_n : by geometric series } Through Momentum space Map : AdS HS -to appear( Kenta Suzuki,R de Mello Koch, J yoon 08 5

16 Propagator } Orthogonality/Completnes of the Bessel basis Z d x d x x 4 h,s (x,x ; p) h0,s 0(x,x ; p 0 ) where = 6 6 (p p 0 ) Leads to the bi-local propagator: D(x,x ; x 3,x 4 )= X h,s,~p h + s (h h 0 ) (s s 0 ) sin( )sin( ), h s sin( )sin( ) 36 6 h,s h,s (x,x ; p) h,s (x 3,x 4 ; p) x x x 3 x 4 6

17 Cubic Vertex } The cubic vertex in bi-local CFT_: V + (3) (k,k 0 ; k,k 0 ; k 3,k 0 3) / (k + + k0+ ) (k+ + k0+ 3 ) (k+ 3 + k0+ ) Through bi-local map : AdS_3 Higher-spin cubic vertex V + (3) (p,, ; p,, ; p 3, 3, 3 ) / (p + + p+ + p+ 3 p ) + + sin( + ) + p + 3 sin( ) p + sin( + ) + p sin( ) 7

18 Diagrams: 4-point Func } We consider bi-local 4-pt function: D E (x,x 0 ) (x,x 0 ) (x 3,x 0 3) (x 4,x 0 4) which generates any higher spin 4-pt functions through bd s (x, x 0 ) X k=0 ( ) k k!(s k)! x) s k x 0) k on external legs// 8

19 Bi-local Diagrams } For tree-level 4-pt func, we have the contact diagram:/n 4 } And s-, t- u-channel exchange diagrams: /N s =

20 = S-channel Contribution } S-channel contribution can be written as: D E (x,x 0 ) (x,x 0 ) (x 3,x 0 3) (x 4,x 0 4) Z d +i d i dh i X s=0 (h, s) h,s Z where A represents a 3-pt bi-local functions, written as s d d ya(x,x 0 ; x,x 0 ; y, h, s)a(y, h,s; x 3,x A(x,x 0 ; x,x 0 ; y, h, s) =g 3 0 (x 0,x ) h,s (x,x 0 ; y) 0 + (3 permutations of external po

21 Conformal Block } The Remaining integral leads to the Conformal Block Expansion: D (x,x 0 ) (x,x 0 ) (x 3,x 0 3) (x 4,x 0 4) eg 3 0(x 0,x 0 ) 0(x 0 3,x 0 4) x x 34 h F h,s (u, v) +F h,s(u, v) Z d +i d i i E dh i s X s=0 (h, s) h,s + (5 permutations) where F h,s (u, v) is conformal partial wave with u x x 34 x 3 x 4, v x 4x 3 x 3 x 4

22 S-channel Contribution } The s-channel contribution to the bi-local 4-pt function: D E (y,y) (y 0,y) (y 0 3,y3) (y 0 4,y4) 0 = eg 3 0(y 0,y ) 0(y 0 3,y 4 ) Z Z d +i d i dh i s X s=0 (h, s) d y h,s (y,y 0 ; y) h,s(y 3,y 0 4; y) + (5 permutations) h,s

23 Evaluation } Introducing (anti-) holomorphic coordinates by y y + iy, ȳ y iy, the intermediate state integral becomes Z d y h,s (y,y ; y) h,s(y 3,y 4 ; y) = (y ) ( h+s ) (y 34 ) ( h+s ) (ȳ ) ( h s ) (ȳ 34 ) ( h s Z Z dy dȳ (y y ) h+s (y y ) h+s (y y 3 ) h +s (y y 4 ) h +s ) (ȳ ȳ ) h s (ȳ ȳ ) h s (ȳ ȳ 3 ) h s (ȳ ȳ 4 ) h s 3

24 Gives:Conformal Block Expansion } The integral is performed in [Dolan & Osborn ] as Z with d y h,s (y,y ; y) h,s(y 3,y 4 ; y) h+s ( = ) ( h+s y y 34 ( h s ) ( h s " K h +s, h s and conf cross ratios ) ) ( h s ) ( h +s ) F h+s, (, ) + h s K h+s, h s # ( h s ) ( h+s ) F h +s, h s (, ) F, (x, x) x F (, ; ; x) x F (, ; ; x) y y 34 y 3 y 4, ȳȳ 34 ȳ 3 ȳ 4 4

25 Result } s,t,u-channel diagrams give D E (y,y) (y 0,y) (y 0 3,y3) (y 0 4,y4) 0 = eg 3 0(y,y 0 ) 0 0(y3,y 0 4) 0 y y 34 " K h +s, h s + (5 permutations) Z d +i d i } Sum over Conformal Partial Waves ( Operator product expansion s dh i X s=0 ( h s ) ( h +s ) F h+s, (, ) + h s (h, s) h,s K h+s, h s h+s ( ) ( h+s ) ( h s ) ( h s ) ( h s ) ( h+s ) F h +s, h s (, ) # } General Spinning Conformal Blocks 5

26 Question of Locality } No-go Result: [Sleight & Taronna 7] [Ponomarev 7] quartic coupling of higher spin theory contains / AdS type non-localities. } In contrast, bi-local theory leads to S = K V 3 + V 4 + Momentum space) vertices and 3Y V 3 = i( ), V 4 = i= K = C C AdS 4Y i= i( ) 6

27 Summary } Described a Constructive Approach to Vectorial Duality } Bilocal (Re)Construction of Bulk AdS } Diagrams : All Order n-point Vertices } Of-shell ( Action) for Higher Spin Gravity } Reproduce : Conf Partial Wave Exp:Conf Blocks 7