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

Transcription

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

2 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

3 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

4 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

5 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

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

7 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

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

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

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

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

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

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

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

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

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

17 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

18 ! 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

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

20 The Map! CFT 3 : collective bi-local fields AdS 4 : higher spin fields " Same number of dimensions = " 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

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

22 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

23 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

24 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

25 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

26 ! 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

27 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

28 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

29 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

30 Maldacena +Zhiboedov! have recently adressed the relevance! of the C-M theorem on CFT3/AdS4 arxiv ! 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

31 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

32 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

33 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!

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

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

36 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

37 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

38 ! 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

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

40 Result:! For 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

41 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

42 ! 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

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