1 Interacting Higher Spins on AdS(D) Mirian Tsulaia University of Crete

2 Plan 1. 1.Motivation 2. 2.Free Field Equations, Free Lagrangians 3. 3.Fields on AdS and Minkowski Spaces 4. 4.Interactions Based On: A. Fotopoulos and M. T, arXiv: I. Buchbinder, A. Fotopoulos, A. Petkou, M.T, Phys. Rev. D 74, 2006, Also: A. Fotopoulos, K.L. Panigrahi, M. T, Phys. Rev. D74, 2006, A. Sagnotti, M. T, Nucl. Phys. B 682 (2004), 83.

3 1. Motivation There are two consistent theories which contain fields with arbitrary spin (all spins are required for consistency): A:(Super)string theory – Consistent both Classically and Quantum Mechanically (in D = 26, D = 10) Backgrounds: Flat for bosonic. Flat, pp-wave, AdS 5 X S 5 for Superstring B: Higher Spin Gauge Theory (M.Vasiliev, E.Fradkin- M.Vasiliev) – Consistent Classically. Quantum Mechanically – No S Matrix on AdS Backgrounds: AdS(D)

4   HS Theory is an “analog” of SUGRA, but classical consistency requires an infinite tower of massless HS fields   SUGRA’s are low energy limits of superstring theories   Is HS gauge theory any limit (an effective theory) of String Theory???   Holography: M. Bianchi, J. Morales, H. Samtleben ‘03, E. Sezgin and P. Sundell ‘02   Is HS gauge theory (Massless Fields) a symmetric phase of String Theory (Massive Fields)?

5 A way to look at it:   In string theory masses; m s ~ s/α’   Symmetric Phase m s → 0. α’ → ∞ (High Energy)   Note: In a high energy limit a string might curve only about a highly curved background, e.g. AdS with a small radius   Recently considered by N. Moeller and P. West ’04, also P. Horava, C. Keeler, ’07, previously D. Gross, P. Mende, ’88.   Ways to describe HS fields: M. Vasiliev, “Frame-like”; C. Fronsdal, D. Francia and A. Sagnotti, “Metric-like”

6 2. Fields on AdS(D) and Minkowski Space Strategy: First derive proper field equations for massless fields What is “proper”? A: Mass Shell Condition   Massless Klein-Gordon Equation for HS bosons.   Massless Klein-Gordon Equation for fields with mixed symmetry   Mixed symmetry fields are described by different Young tableaux: eg. →

7 On AdS(D)   Group of Isometries SO(D-1,2)   Maximal Compact Subgroup is SO(2) X SO(D-1)   Build Representations in the Cartan-Weyl Basis

8   Highest weight state   Representations   E 0 = S + D-3 (for totally symmetric fields) → zero norm → zero mass   Leads to Klein Gordon Equation:

9 B: The equations must have enough gauge invariance to remove negative norm states, ghosts from the spectrum   Totally symmetric field   Mixed symmetry fields

10 How to derive: A possible way to take a BRST Charge for a bosonic string Rescale And take formally α’ → ∞ With Commutation Relations

11   BRST Charge is nilpotent in any Dimension Q ~ = 0   One can truncate the number if oscillators α μ k, c k, b k to any finite k without affecting the nilpotency property   To get a free lagrangian for a leading Regge trajectory expand   Analogous for mixed symmetry fields   Fock vacuum   The ghost c k has ghost number 1; b k has ghost number -1

12 3. Lagrangian   Free Lagrangian   Gauge Transformation   Equation of Motion

13 AdS Deformation for Leading Regge Trajectory + Where γ = cb and M = ½ α μ α μ

14 Coming back to α’ → ∞   Taking the point that we have massless fields of any spin, our system (BRST charge, Lagrangian, Gauge Transformations) correctly describes these fields (totally symmetric, mixed symmetry)   Fixing gauge one can see that only physical degrees of freedom propagate

15 4. Cubic Interactions   In order to have nontrivial interactions one has to take three copies of Hilbert space considered before and deform gauge transformations   The Lagrangian   Interaction Vertex   V(α i +,c i +,b i +,b i0 ) determined from Q|V>= 0; Q = Q 1 + Q 2 + Q 3

16   The BRST invariance of |V> ensures the gauge invariance of the action and closure of the nonabelian algebra up to first order in g.   One can solve this equation, taking V to be an arbitrary polynomial in α i μ+ ; b i +, c i + with ghost number zero.   It can be done both for AdS (leading trajectory) and flat space-time.   The solution should not be BRST trivial: |V> ≠ Q|W>. The trivial one can be obtained from the free action by field redefinitions.

17 Finally the solution from String Field Theory vertex   In string field theory V is   Α μ 0 is proportional to (α’) 1/2, but one has to also keep terms which do not contain (α’) 1/2, since they are in the exponential   Put the anzatz   Where β ij = c i b 0 j and l ij = α i p j

18 Demand BRST invariance: The Closure requires:   S ii = (1,1,1)   To consider the whole tower together   One can do this for totally symmetric or mixed symmetry fields

19   We have actually done a field dependent deformation of the initial BRST charge ;   Does not work for AdS: there is no such solution in the Bosonic string

20 Summary:   From Bosonic string theory one can get information about interacting massless bosonic fields on a flat background   To do AdS along these lines one probably has to do super   Even a free BRST on AdS for fermions and mixed symmetry has not yet been done