Holography, Unfolding and Higher-Spin Theories M.A.Vasiliev Lebedev Institute, Moscow ESI Workshop on “Higher Spin Gravity” Vienna, April 17, 2012
HS theory Higher derivatives in interactions (1983) , Berends, Burgers, van Dam A.Bengtsson, I.Bengtsson, Brink (1984) � ( D p ϕ )( D q ϕ )( D r ϕ ) ρ p + q + r + 1 S = S 2 + S 3 + . . . , S 3 = 2 d − 3 p,q,r HS Gauge Theories ( m = 0): Fradkin, M.V. (1987) [ D n , D m ] ∼ ρ − 2 = λ 2 AdS d : AdS/CFT: (3 d, m = 0) ⊗ (3 d, m = 0) = � ∞ s =0 (4 d, m = 0) Flato, Fronsdal (1978); Sundborg (2001), Sezgin, Sundell (2002,2003), Klebanov, Polyakov (2002), Giombi, Yin (2009). . . Maldacena, Zhiboedov (2011) Thm: Unitary, conformal, local theory conserved current of spin s > 2 is free Is a boundary dual of AdS 4 HS theory free?
Main results CFT 3 dual of AdS 4 HS theory: 3d superconformal HS theory Holography: Unfolding
Plan I Unfolded dynamics II Unfolding and holographic duality III Free massless HS fields in AdS 4 IV Conserved currents and massless equations V AdS 4 HS theory as 3 d conformal HS theory VI Holographic locality at infinity VII Towards nonlinear 3 d conformal HS theory IIX Higher-spin theory and quantum mechanics IX Conclusion
Unfolded dynamics First-order form of differential equations q i ( t ) = ϕ i ( q ( t )) q i ( t 0 ) ˙ initial values: # degrees of freedom = # of dynamical variables Field theory: infinite # of degrees of freedom = spaces of functions= infinite # of undetermined derivatives (generalized momenta) Dirac approach is nice and efficient but noncovariant. Covariant extension t → x n ? Unfolded dynamics: multidimensional generalization ∂ q i ( t ) → W Ω ( x ) = dx n 1 ∧ . . . ∧ dx n p W Ω ∂t → d , n 1 ...n p ( x ) a set of differential forms
Unfolded equations dW Ω ( x ) = G Ω ( W ( x )) , d = dx n ∂ n G Ω ( W ) : function of “supercoordinates” W α ∞ � G Ω ( W ) = f ΩΛ 1 ... Λ n W Λ 1 ∧ . . . ∧ W Λ n n =1 d > 1: Nontrivial compatibility conditions G Λ ( W ) ∧ ∂G Ω ( W ) ≡ 0 ∂W Λ Any solution to generalized Jacobi identities: FDA Sullivan (1968); D’Auria and Fre (1982) The unfolded equation is invariant under the gauge transformation δW Ω = dε Ω + ε Λ ∂G Ω ( W ) , ∂W Λ where the gauge parameter ε Ω ( x ) is a ( p Ω − 1)- form. (No gauge parameters for 0-forms W Ω )
Vacuum geometry h : a Lie algebra. ω = ω α T α : a 1-form taking values in h . G ( ω ) = − ω ∧ ω ≡ − 1 2 ω α ∧ ω β [ T α , T β ] the unfolded equation with W = ω has the zero-curvature form dω + ω ∧ ω = 0 . Compatibility condition: Jacobi identity for h . The FDA gauge transformation is the usual gauge transformation of the connection ω . The zero-curvature equations: background geometry in a coordinate independent way. If h is Poincare or anti-de Sitter algebra it describes Minkowski or AdS d space-time
Free fields unfolded Let W Ω contain p - forms C i (e.g. 0- forms) and G i be linear in ω and C G i = − ω α ( T α ) ij ∧ C j . The compatibility condition implies that ( T α ) ij form some representation T of h , acting in a carrier space V of C i . The unfolded equation is D ω C = 0 D ω ≡ d + ω : covariant derivative in the h - module V . Covariant constancy equation : linear equations in a chosen background h : global symmetry
Scalar field example s = 0: infinite set of totally symmetric 0 -forms C m 1 ...m n ( x ) ( n = 0 , 1 , 2 , . . . ). Off-shell unfolded equations dC m 1 ...m n = e k C m 1 ...m n k ( n = 0 , 1 , . . . ) , Cartesian coordinates: D L = d . The space V of C m 1 ...m n forms an (infinite dimensional) iso ( d − 1 , 1) –module. First two equations ∂ n C = C n , ∂ n C m = C mn All other equations express highest tensors in terms of higher-order derivatives C m 1 ...m n = ∂ m 1 . . . ∂ m n C . C n 1 ...n n describe all derivatives of C ( x ). The system is off-shell: it is equivalent to an infinite set of constraints On-shell system: C kkm 3 ...m n ( x ) = 0
Invariant functionals via Q –cohomology Equivalent form of compatibility condition ∂ Q 2 = 0 , Q = G Ω ( W ) ∂W Ω Q -manifolds Hamiltonian-like form of the unfolded equations dF ( W ( x )) = Q ( F ( W ( x )) , ∀ F ( W ) . Invariant functionals � S = L ( W ( x )) , QL = 0 (2005) L = QM : total derivatives Actions and conserved charges: Q cohomology for off-shell and on-shell unfolded systems, respectively
Properties • General applicability • Manifest (HS) gauge invariance • Invariance under diffeomorphisms Exterior algebra formalism • Interactions: nonlinear deformation of G Ω ( W ) • Local degrees of freedom are in 0-forms C i ( x 0 ) at any x = x 0 (as q ( t 0 ) ) infinite-dimensional module dual to the space of single-particle states • Independence of ambient space-time G Ω ( W ) Geometry is encoded by
Unfolding and holographic duality Unfolded formulation unifies various dual versions of the same system. Duality in the same space-time: ambiguity in what is chosen to be dynamical or auxiliary fields. Holographic duality between theories in different dimensions: universal unfolded system admits different space-time interpretations. Extension of space-time without changing dynamics by letting the dif- ferential d and differential forms W to live in a larger space ∂ ∂ ∂ n ˆ X ˆ X ˆ d = dX n d = dX n n dX n W n → dX n W n + d ˆ ∂X n → ˜ ∂X n + d ˆ n , W ˆ n , ∂ ˆ X ˆ n are some additional coordinates. X ˆ ˆ dW Ω ( X, ˆ X ) = G Ω ( W ( X, ˆ ˜ X ))
A particular space-time interpretation of a universal unfolded system, e.g, whether a system is on-shell or off-shell, depends not only on G Ω ( W ) but, in the first place, on a space-time M d and chosen vacuum solution W 0 ( X ). Two unfolded systems in different space-times are equivalent (dual) if they have the same unfolded form. Most direct way to establish holographic duality between two theories: unfold both to see whether the operators Q of their unfolded formulations coincide. Given unfolded system generates a class of holographically dual theories in different dimensions.
HS gauge connections in AdS 4 Gauge 1-forms n + m = 2( s − 1) ω α 1 ...α n , ˙ β m , β 1 ... ˙ ω ( x ) = dx n ω n ( x ) s = 1 : s = 2 : β ( x ) , ω αβ ( x ) , ¯ β ( x ) ω α ˙ ω ˙ α ˙ s = 3 / 2 : ω α ( x ) , ¯ α ( x ) ω ˙ Frame-like fields: | n − m | = 0 (bosons) or | n − m | = 1 fermions Auxiliary Lorentz-like fields: | n − m | = 2 (bosons) Extra fields: | n − m | > 2 1987
Gauge invariant field strengths 0-forms C α 1 ...α n , ˙ β m , | n − m | = 2 s β 1 ... ˙ (Anti)selfdual Weyl tensors carry only (dotted)undotted spinor indices s = 0 : C ( x ) ¯ s = 1 / 2 : C α ( x ) , C ˙ α ( x ) ¯ s = 1 : C αβ , C ˙ α ˙ β ¯ s = 3 / 2 : C αβγ , C ˙ α ˙ β ˙ γ ¯ s = 2 : C α 1 ...α 4 , C ˙ α 1 ... ˙ α 4 Formulae simplify in terms of generating functions ω ( y, ¯ y | x ), C ( y, ¯ y | x ) ∞ � 1 β 1 ... ˙ ˙ β m A α 1 ...α n , β m ( x ) A ( y, ¯ y | x ) = i n ! m ! y α 1 . . . y α n ¯ y ˙ β 1 . . . ¯ y ˙ n,m =0 Traceless tensors by virtue of Penrose formula: p α ˙ y ˙ p α ˙ β = y α ¯ β β p α ˙ p n p n = 0 . ⇒ β = 0 ⇔ α put the system on-shell y ˙ Twistor auxiliary variables y α , ¯
Central on-shell theorem Infinite set of spins s = 0 , 1 / 2 , 1 , 3 / 2 , 2 . . . Fermions require doubling of fields ω ii ( y, ¯ C i 1 − i ( y, ¯ y | x ) , y | x ) , i = 0 , 1 , ω ii ( y, ¯ y | x ) = ω ii (¯ C i 1 − i ( y, ¯ y | x ) = C 1 − i i (¯ ¯ ¯ y, y | x ) , y, y | x ) . The full unfolded system for the doubled sets of free fields is ∂ 2 ∂ 2 α ˙ 1 ( y, y | x ) = η H ˙ β ⋆ R ii β C 1 − i i (0 , y | x ) + ¯ η H αβ ∂y α ∂y β C i 1 − i ( y, 0 | x ) α ∂y ˙ ∂y ˙ ⋆ D 0 C i 1 − i ( y, y | x ) = 0 ˜ α ˙ β = e α ˙ α ∧ e α ˙ H αβ = e α ˙ α , H ˙ β , y | x ) = D ad α ∧ e β ˙ R 1 ( y, ¯ 0 ω ( y, ¯ y | x ) � � � � ∂ 2 ∂ ∂ 0 ω = D L − λe α ˙ D 0 = D L + λe α ˙ D ad β β ˜ y α β + ∂y α ¯ y ˙ , y α ¯ y ˙ β + , β y ˙ y ˙ ∂y α ∂ ¯ β ∂ ¯ � � ∂ ∂ α ˙ D L A = d x − ω αβ y α ω ˙ β ¯ ∂y β + ¯ y ˙ . α y ˙ β ∂ ¯
Non-Abelian HS algebra Star product � dSdTf ( Y + S ) g ( Y + T ) exp − iS A T A ( f ∗ g )( Y ) = [ Y A , Y B ] ∗ = 2 iC AB , C αβ = ǫ αβ , C ˙ β = ǫ ˙ α ˙ α ˙ β Non-Abelian HS curvature R 1 ( y, ¯ y | x ) → R ( y, ¯ y | x ) = dω ( y, ¯ y | x ) + ω ( y, ¯ y | x ) ∗ ω ( y, ¯ y | x ) ˜ y | x ) → ˜ D 0 C ( y, ¯ DC ( y, ¯ y | x ) = dC ( y, ¯ y | x )+ ω ( y, ¯ y | x ) ∗ C ( y, ¯ y | x ) − C ( y, ¯ y | x ) ∗ ω ( y, − ¯ y | x )
Unfolding as twistor transform Twistor transform C ( Y | x ) � ❅ η ν � ❅ � ❅ � ✠ ❅ ❘ M ( x ) T ( Y ) . W Ω ( Y | x ) are functions on the “correspondence space” C . Space-time M : coordinates x . Twistor space T : coordinates Y . Unfolded equations describe the Penrose transform by mapping functions on T to solutions of field equations in M . Being simple in terms of unfolded dynamics and the corresponding twistor space T , holographic duality in terms of usual space-time may be complicated requiring solution of at least one of the two unfolded systems: a nontrivial nonlinear integral map.
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