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Interacting Higher Spins on AdS(D)

Interacting Higher Spins on AdS(D). Mirian Tsulaia University of Crete. Plan. Motivation Free Field Equations, Free Lagrangians Fields on AdS and Minkowski Spaces Interactions Based On: A. Fotopoulos and M. T, arXiv:0705.2939

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Interacting Higher Spins on AdS(D)

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

  2. Plan • Motivation • Free Field Equations, Free Lagrangians • Fields on AdS and Minkowski Spaces • Interactions • Based On: • A. Fotopoulos and M. T, arXiv:0705.2939 • I. Buchbinder, A. Fotopoulos, A. Petkou, M.T, Phys. Rev. D 74, 2006, 105018 • Also: • A. Fotopoulos, K.L. Panigrahi, M. T, Phys. Rev. D74, 2006, 085029 • 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, AdS5 X S5 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; ms ~ s/α’ • Symmetric Phase ms→ 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 • E0 = 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 αμk, ck, bk 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 ck has ghost number 1; bk 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 αμαμ

  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(αi+,ci+,bi+,bi0) determined from Q|V>= 0; Q = Q1 + Q2 + Q3

  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 αiμ+; bi+, ci+ 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 • Αμ0 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 βij = cib0j and lij = αipj

  18. Demand BRST invariance: The Closure requires: • Sii = (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

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