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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) 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 • 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.
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)
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)?
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”
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. →
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
Highest weight state • Representations • E0 = S + D-3 (for totally symmetric fields) → zero norm → zero mass • Leads to Klein Gordon Equation:
B:The equations must have enough gauge invariance to remove negative norm states, ghosts from the spectrum • Totally symmetric field • Mixed symmetry fields
How to derive: A possible way to take a BRST Charge for a bosonic string Rescale And take formally α’ → ∞ With Commutation Relations
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
3. Lagrangian • Free Lagrangian • Gauge Transformation • Equation of Motion
AdS Deformation for Leading Regge Trajectory + Where γ = cb and M = ½ αμαμ
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
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
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.
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
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
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
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