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

2 Why Higher Spins? 1. Crucial (open) problem in Field Theory 2.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 Consistent non-linear equations for all spins (all symm tensors): Diff invariant so(D+1; ℂ )-invariant natural vacuum solutions (S D, H D, (A)dS D ) Infinite-dimensional (tangent-space) algebra Correct free field limit  Fronsdal or Francia-Sagnotti eqs Arguments for uniqueness Focus on D=4 AdS bosonic model Interactions? Consistent!, in presence of: Infinitely many gauge fields Cosmological constant   0 Higher-derivative vertices

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) Gauge field  Adj(ho(3,2)) ( master 1-form ): Generators of ho(3,2): (symm. and TRACELESS!) s o(3,2) : 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 e.g. s=2: Ricci=0  Riemann = Weyl [tracelessness  dynamics !] [Bianchi  infinite chain of ids.] Unfolded full eqs: Manifest HS-covariance Consistency (d 2 = 0)  gauge invariance NOTE: covariant constancy conditions, but infinitely many fields DYNAMICS + trace constraints  DYNAMICS 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) (M.A. Vasiliev, 1990)

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

7 Exact Solutions: strategy Also the other way around! (base  fiber evolution) Locally give x-dep. via gauge functions (space-time  pure gauge!) Full eqns: 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-0 Solutions Vacuum sol.: The gauge function gives

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

10 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

11 Type-2 Solutions Projector vacuum sol. of the nonminimal (s=0,1,2,...) model: K  = 0  the solution describes a maximally symmetric space-time. But! the internal, non-commutative connection is  0 ! No space-time fields turned on, but affects the interactions!

12 Type-3 Solutions Projector sol. of the nonminimal chiral (c 2 =0) model (Euclidean/Kleinian sign.) : the spacetime physical fields (Z=0) are computed from: 0-forms: constant scalar field, anti-selfdual Weyl tensors  0 ! With the simplest projector Weyl tensors blow up at 2 x 2 = 1.  no asympt.lly H 4 for 2 = 1 (0  2 x 2 < 1), (regular on S 4, 2 = -1 ({0  - 2 x 2  1}  {0  - -2 x -2  1})

13 Type-3 Solutions 1-forms: metric & HS fields  0 ! Metric blows up at 2 x 2 = 1, -1, 1/3 ! 2 x 2 = 1  bdry of H 4, H 3,2, g  ~ (1- 2 x 2 ) - 4 x  x degenerate! 2 x 2 = 1/3  inside H 4, g  ~ (1/3- 2 x 2 ) – 4 ( Jx)  ( Jx) degenerate!

14 HS Invariants Ciclicity: Define: Conserved on the field equations: Ciclicity + A  even function of oscilllators

15 Conclusions & Outlook HS algebras and 4D Vasiliev equations generalized to various space-time signatures. 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]. 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. 1.“Lorentz-invariant” solution (Type I) 2.“Projector” solutions & new vacua (Type II) 3.Solutions to chiral models with HS fields  0 (Type III)

16 Real Forms Reality conditions: Tangent-space signature:  ab = (4,0); (3,1); (2,2). Ambient-sp. signature:  AB = (  ab,- 2 ) = (5,0), (4,1); (4,1)’, (3,2); (3,2)’  ho(5;C)  ho(5), ho(4,1), ho(3,2).