Real forms of complex HS field equations
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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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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


Real forms of complex hs field equations and new exact solutions

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


Real forms of complex hs field equations and new exact solutions

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


Real forms of complex hs field equations and new exact solutions

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. 


Real forms of complex hs field equations and new exact solutions

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


Real forms of complex hs field equations and new exact solutions

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} 


Real forms of complex hs field equations and new exact solutions

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.:


Real forms of complex hs field equations and new exact solutions

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:


Real forms of complex hs field equations and new exact solutions

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


Real forms of complex hs field equations and new exact solutions

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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