Real forms of complex HS field equations and new exact solutions

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Real forms of complex HS field equations Real forms of complex HS field equations and new exact solutionsand new exact solutions

Carlo IAZEOLLA

Scuola Normale Superiore, Pisa

Sestri Levante, June 04 2008

(C.I., E.Sezgin, P.Sundell – Nuclear Physics B, 791 (2008))

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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

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The Vasiliev Equations

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

Interactions? Consistent!, in presence of: • Infinitely many gauge fields• Cosmological constant 0 • Higher-derivative vertices

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-dim. extension of AdS-gravity with gauge fields valuedin 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!)

so(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.

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e.g. s=2: Ricci=0 Riemann = Weyl [tracelessness dynamics !] [Bianchi infinite chain of ids.]

Unfolded full eqs:

• Manifest HS-covariance• Consistency (d2 = 0) gauge invariance • NOTE: covariant constancy conditions, but infinitely many fields + trace constraints DYNAMICSDYNAMICS

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)

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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:

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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.:

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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.

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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

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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)

Real forms of complex HS field equations and new exact solutions

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