Construction of Higher Spin AdS Theory from O(N) Vector Model QFT

Construction of Higher Spin AdS Theory from O(N) Vector Model QFT CQUeST Spring Workshop on Higher Spins and String Geometry, 28-31 March, 2012, Seoul, Korea Antal Jevicki (Brown University) With: Kewang Jin Robert de Mello Koch Qibin Ye Joao P. Rodrigues

SUBJECT! Ongoing Study of the Vector O(N)3/AdS4 Correspondence with Higher Spin Gravity of Vasiliev 90! Originally Suggested by Klebanov + Polyakov 02 after Suggestion by Sundborg 94, Witten 01, Sezgin + Sundell 02! With Crucial Contribution From the 3-point Correlator Comparison due to Giombi + Yin! Related (Parallel) Correspondence in AdS3: Chern Simons 2

TALK:Part I! Direct Construction of AdS4 Higher Spin Gravity from QFT in terms of Bi-Local Fields, Collective Action, 1/N Expansion B.Sakita + A.J. 80! Applied to Higher Spin Case: S. Das + A.J. 03; R. d M Koch, A.J., K. Jin, J. Rodrigues 10,11! Physical Gauge: Timelike, Spacelike Covariant Gauge.! Collective Dipole - First Quantization/ 2nd: Bi-local Field Theory Represents a bulk description of AdS4 and HS Fields 3

Part II:S-matrix and Triviality! Simplest Example of Duality (g=0 UV Fixed Point) is Characterized by Number of Conserved Current/ Charges! The Coleman-Mandula Theorem: which for ordinary QFT states that the S-Matrix: S=1 (Triviality)! Maldacena + Zhiboedov have addressed the Question for the O(N)3/AdS4 Case: Absence of Appropriate S-Matrix (in AdS) Correlators <O1O2O3.On> 0 Are Simple: Given by Free N-Component Field. Is it :trivial:our construction will illuminate this question 4

O(N) CFT d=3!! Fixed Points : g=0 UV CFT g 0 IR CFT! Infinite Sequence of HS Currents Null Plane! Generating Function! 3d 2d Tracelessness 5

! Currents : Boundary Duals to AdS HS Fields Coupling to HS: Bekaert! Three- Point Functions: Giombi-Yin 2010 CFT Propagator 6

Bilocal(Collective) Field Theory Change from the N-component field to the bilocal field: O(N) invariant 3d + 3d! Represents a more general set than the conformal currents : 3d + 2d 7

The action:! Exact :! The collective(effective) action:! Origin of the interaction: Jacobian! The measure: space momentum cutoff 8

Natural Star Product! Definition of trace! Star product! Tr and Det are natural in bi-local theory. 9

Properties of the collective field representation:! Same partition function! More general than the conformal fields 3 + 2 3 + 3 Extra Dimension Bulk of (AdS 4 )! 1/N as coupling constant: HS Gravity coupling constant: 10

Expansion! Equation of motion:! 1/N expansion parameter:! S c -exact: Reproduces all invariant correlators 11

Topology: Witten diagram! Propagator Collective Field Theory Field Theory 12

Four-point function: + 3+3 dimensions Bi-local space-time 4 dimensions + Higher-Spin fields 13

Hamiltonian Representation:! One also has a Hamiltonian representation! Relativistic dipole:! Allows a single time fixing 14! and a Hamiltonian formulation

The Collective Dipole! Two relativistic particle:! Equations of motion:! Center-of-mass frame:! Elimination of relative time: gauge fix 15

Solving the constraints:! Canonical transformation! Constraints lead to.! One time:! Hamiltonian 16

Hamiltonian:! The bi-local fields has a one-time description.! Canonical collective field representation: Equal-time fields Conjugate momenta:! The Hamiltonian is given as 17

! Scaling of N: and the expansion parameter is 1/N.! 1/N expansion! An infinite series is generated by the expansion! The cubic vertex 18

! Quadratic Hamiltonian: is diagonalized in momentum space:! Spectrum 19

The Map! CFT 3 : collective bi-local fields AdS 4 : higher spin fields " Same number of dimensions 1+2+2 = 1+3+1 " Representation of the conformal group SO(2,3) 20 " From the form of two representations it is seen that one does not have a coordinate transformation

Answer: canonical transformation " Identifying the generators of the dipole with the generators of HS: gives 10 equations of 2 4=8 canonical variables 21

Poisson Brackets! We have the AdS 4 canonical variables in terms of the dipole canonical variables (of CFT 3 ).! A consistency check on the correctness of the map: the Poisson brackets take the canonical form.! Assuming we verify that! We have a successful reconstruction of AdS spacetime from the bi-local one. 22

From bi-local field to HS field: 1 p, p,, p + +! In the dual (momentum) space: ( 2 1 2), one observes the transformation takes the form of a point transformation: p! Consequently perform a Fourier transform: 23

Map(transform)between fields! Changing to AdS variables using an inverse transform gives the AdS higher-spin field in terms of the bi-local one: 24

One Check: the z=0 projection! The bilocal field gives a strong off-shell operator re-construction of the HS field in the bulk:! The quadratic bilocal Hamiltonian transforms into Metsaevs HS light cone Hamiltonian! Check on the identification extra AdS coordinate z : evaluate the z=0 limit 25

! Expanding the delta function in Fourier series, one has the binomial expansion! The conformal operators for a fixed spin s is! The expansion coefficients agree up to an overall normalization 26

Finite Temperature (Free Energy)! In the Singlet Sector H2 Reproduces the Singlet Sector O(N^0) Counting of S.S.! Phase Transition: Yin + Shenker 2011 High Temperature:! Reproduced by Stationary Point of Covariant: S.Das + A.J. 02 27

Summary of Results so far:! Given a direct construction of the dual AdS HS theory from the O(N) QFT! A nonlinear Action/Hamiltonian: Built to correctly reproduce arbitrary n-point correlation function! 1/N Expansion through Witten type diagrams! Bulk description of AdS: one to one map

PART II:Coleman-Mandula Theorem in CFT3/AdS4! The simplest case of the correspondence involves CFT UV with g=0, i.e. A free QFT and HS AdS4! In CFT we have an infinite number of conserved charges/ currents: or in the light-cone:! In such a theory, the Coleman-Mandula theorem would imply that the S-Matrix is 0 :Triviality 29

Maldacena +Zhiboedov! have recently adressed the relevance! of the C-M theorem on CFT3/AdS4 arxiv 1112.1016! Not having an S-Matrix (in AdS or CFT) they studied the implication on the correlation functions: Showed that the theory is (trivial) simple i.e. correlators are given by free fields! Still: Cn=<O1O2.On> 0 arbitrary n-point correlators are non-zero! The AdS4 space-time theory can still be nontrivial:example 2d integrable theories 30

S-Matrix! G. Mack has put forward a proposal that CFT correlators themselves have a structure equivalent to an S-Matrix.! He argued that they can be written in an integral form (Mellin Representation): which would imply their S-Matrix interpretation! claim was strengthened by Penedones, Paulos,..! Non-zero! But it is not clear that this is an S-Matrix. 31

Bi-Local Field Representation! Provides a clear answer.! One has an S-Matrix defining scattering of Dipoles: which can be evaluated!! Before that let me point to a simpler example where a simple (free) theory was used with similar questions. 32

The d=1 Matrix Model! Duality with 2D non-critical string 1990 :! Decoupled Harmonic Oscillators With S-J Rey! Infinite sequence of higher charges 33! In Matrix form there is no obvious S-Matrix!

Collective (Fermi-Droplet) Representation! Collective/density field: Invariants with a large N Hamiltonian or Correctly reproduces all S.Das + A.J. 1990 Correlators 34

! Small Fluctuations of this (collective) theory Masless collective boson! Scattering : On shell-relation Gross + Klebanov 91 S=1 for c=1 35

Three-Point S-Amplitude Liouville as time Scattering E1, E2>0 incoming E3<0 outgoing Dirichlet Boundary Condition 36 Nontrivial Boundary conditions:nonzero S-matrix:2DString

Return To Bi-Local Theory:! we are able to define an S-Matrix for scattering of Collective Dipoles! In a time-like gauge (single time), one has the onshell relation Single time Bi-Lo Single time correlators: Reduction Formula 37

! In light-cone gauge, it would correspond to:! Note: Maldacena + Zhiboedov have discussed (reconstructed) correlators of! So here one is not in a position to consider the above defined S-Matrix 38

Continuing with time-like! Cubic Interaction Transformed to momentum space: 39

Result:! For 1+2 3 scattering, the result is! Using the momentum conservation relationships one can see 40! And consequently S3=0! Evaluation of S4=0 is in progress The Associated Vasiliev s Higher Spin gravity: S=1

Conclusion! I have reviewed some elements of a constructive approach to Higher Spin Gravity in the CFT3/AdS4 correspondence:given a reconstruction of the bulk HS theory in terms of collective fields! For the theory based on free O(N) fields it is argued that the there is a natural S-Matrix and that S= 1 in agreement with the Coleman-Mandula theorem:complementing Maldacena and Zhiboedov! Interesting to perform the calculation in HS theory directly/vasilievs, Joung-Taronna s cubic theory 41

! General Theorem: For all duals coming from a free large N theory ( possesing infinitely many conserved charges): S=1! Any change of boundary conditions will result in non-trivial S-Matrix 42