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Real forms of complex HS field equations and new exact solutions

Real forms of complex HS field equations and new exact solutions. Carlo IAZEOLLA Scuola Normale Superiore, Pisa. ( C.I., E.Sezgin, P.Sundell – Nuclear Physics B, 791 (2008)). Sestri Levante, June 04 2008. Why Higher Spins?. Crucial (open) problem in Field Theory Key role in String Theory

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Real forms of complex HS field equations and new exact solutions

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  1. Real forms of complex HS field equations and new exact solutions Carlo IAZEOLLA Scuola Normale Superiore, Pisa (C.I.,E.Sezgin, P.Sundell – Nuclear Physics B, 791 (2008)) Sestri Levante, June 04 2008

  2. Why Higher Spins? • Crucial (open) problem in Field Theory • Key role in String Theory • Strings beyond low-energy SUGRA • HSGT as symmetric phase of String Theory? • 3. Positive results from AdS/CFT

  3. The Vasiliev Equations • Interactions? Consistent!, in presence of: • Infinitely many gauge fields • Cosmological constant L 0 • Higher-derivative vertices • Consistent non-linear equations for all spins (all symm tensors): • Diff invariant • so(D+1;ℂ)-invariant natural vacuum solutions (SD, HD, (A)dSD) • Infinite-dimensional (tangent-space) algebra • Correct free field limit  Fronsdal or Francia-Sagnotti eqs • Arguments for uniqueness Focus on D=4 AdS bosonic model

  4. The Vasiliev Equations -dim. extension of AdS-gravity with gauge fields valued in HS tangent-space algebra ho(3,2)  Env(so(3,2))/I(D) so(3,2) : Generators of ho(3,2): (symm. and TRACELESS!) Gauge field Adj(ho(3,2)) (master 1-form): But: representation theory of ho(3,2) needs more! • Massless UIRs of all spins in AdS include a scalar! • “Unfolded”eq.ns require a “twisted adjoint” rep. 

  5. The Vasiliev Equations Introduce a master 0-form (contains a scalar, Weyl, HS Weyl and derivatives) (upon constraints, all on-shell-nontrivial covariant derivatives of the physical fields, i.e., all the dynamical information is in the 0-form at a point) e.g. s=2: Ricci=0  Riemann = Weyl [tracelessness  dynamics !] [Bianchi  infinite chain of ids.] Unfolded full eqs: (M.A. Vasiliev, 1990) • Manifest HS-covariance • Consistency (d2= 0)  gauge invariance • NOTE: covariant constancy conditions, but infinitely many fields • + trace constraints DYNAMICS

  6. The Vasiliev Equations Osc. realization: NC extension, x (x,Z): Solving for Z-dependence yields consistent nonlinear corrections as an expansion in Φ. For space-time components, projecting on phys. space {Z=0} 

  7. Exact Solutions: strategy Full eqns: Also the other way around! (base  fiber evolution) Locally give x-dep. via gauge functions (space-time  pure gauge!) Z-eq.ns can be solved exactly: 1) imposing symmetries on primed fields 2) via projectors A general way of solving the homogeneous (=0) eqn.:

  8. Type-1 Solutions SO(3,1)-invariance:  Inserting in the last three constraints: Remain: Integral rep.: gives manageable algebraic equations for n(s)  particular solution, -dependent. Homogeneous (=0) eqn. admits the projector solution:

  9. Type-1 Solutions Remaining constraints yield: Sol.ns depend on one continuous & infinitely many discrete parameters • Physical fields (Z=0): • 1) k = 0 , k • 0-forms: only scalar field • 1-forms: only Weyl-flat metric, • asympt. max. sym space-time • 2)  = 0, (k -k+1)² = 1 • 1-forms: degenerate metric

  10. Conclusions & Outlook • HS algebras and 4D Vasiliev equations generalized to various • space-time signatures. • New exact solutions found, by exploiting the “simple” structure of • HS field equations in the extended (x,Z)-space. Among them, the • first one with HS fields turned on. • “Lorentz-invariant” solution (Type I) • “Projector” solutions & new vacua (Type II) • Solutions to chiral models with HS fields  0 (Type III) • Other interesting solutions, in particular black hole solutions: BTZ • in D=3 [Didenko, Matveev, Vasiliev, 2006]  interesting to elevate it to D=4. • Hints towards 4D Kerr b.h. solution [Didenko, Matveev, Vasiliev, 2008].

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