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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

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
- Strings beyond low-energy SUGRA
- HSGT as symmetric phase of String Theory?
- 3. Positive results from AdS/CFT

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

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.

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

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}

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.:

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:

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

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].