(In)Formal Higher Spin Theories Best Higher Spin Conference, UMONS Based on a paper with Alexei Sharapov Zhenya Skvortsov LMU, Munich and Lebedev Institute, Moscow April, 27, 2017 E.Skvortsov
Intro Comments Higher-spin theories contain ∞ -many of massless fields, whose AdS/CFT duals are higher-rank conserved tensors. The latter implies that the dual theory is free. Therefore, intuitively, knowledge of some free conformal field theory should be necessary and sufficient in order to build the dual higher-spin theory. But how? Historically, the first idea was to set up some deformation problem in the bulk (Noether procedure) and find interaction terms together with the deformation that respects them Another idea is reconstruction: write down the AdS vertices such that the CFT correlators computed via Witten diagrams coincide with the correlators of the required free dual CFT E.Skvortsov
Main Messages There is a deformation procedure that is orthogonal to the usual weak-field expansion: unfolding. It treats higher-spin symmetries in the exact way. However, locality is not under control. The initial data is some higher-spin algebra. We show that the first vertex is determined by a Hochschild cocycle of the higher-spin algebra. There are no higher-order obstructions and other vertices go for a ride The cocycle can be generated by embedding everything into a bigger algebra, which mathematically provides a resolution. The first vertex gives description of fluctuations over higher-spin background given by any flat connection; Equations for such fluctuations can be easily completed to full nonlinear equations (e.g., Vasiliev equations) E.Skvortsov
Unfolding Unfolded equations are ’de Rham=wedge product of fields’: dW A = Q A ( W ) , � B 1 ... B k W B 1 ∧ ... ∧ W B k Q A ( W ) = Q A k Formal consistency d 2 = 0, d = dx m ∂ m leads to → − ∂ Q A ( W ) 0 ≡ ddW A = dQ A ( W ) = dW B ∧ ∂ W B and we find Jacobi-like quadratic contraints − → ∂ Q A ( W ) Q B ∧ ≡ 0 ∂ W B Example: flat connection of some Lie algebra d ω = 1 2 [ ω, ω ] E.Skvortsov
Q-manifold Unfolding is equivalent to Q-manifold: super-manifold with an odd Q vector-field that is nilpotent, Q 2 = 0. W A coordinates : − → ∂ Q = Q A ( W ) vector-field : ∂ W A . It is important to remember that formal consistency Q 2 = 0 leads to formal gauge invariance of the equations; any PDE’s can be written in the unfolded form; it is unclear which unfolded equations correspond to well-defined PDE’s (very easy to give examples of unfolded equations that are meaningless once d is taken seriously); .... E.Skvortsov
Initial Data HS algebras can be defined in many ways: directly in AdS by trying to build an algebra on Killing tensors ∇ m ξ m ( s − 1) = 0 or HS connection (Fradkin-Vasiliev) ; symmetries of free conformally-invariant equations, for example, � φ = 0 (Eastwood) ; symmetries generated by higher-rank conserved tensors φ∂...∂φ that exist in free theories; U ( so ( d , 2)) / I where I some ideal, e.g. the one that is annihilated by Singleton, � φ = 0; quantization of the algebra of functions on the co-adjoint orbit that corresponds to the free field as a representation of so ( d , 2); In any case it is ∞ -dim; associative; so ( d , 2) is a subalgebra; there is a trace; there is an automorphism π ( f ) that flips the sign of AdS-transvections E.Skvortsov
Initial Data The minimalistic data is ∞ -dimensional associative algebra hs ; equipped with an automorphism π (can be trivial); Physics/reality can add few more details: free conformal field origin of the algebra; π ( f ) = f ( L , − P ), where P are AdS-transvections; oscillator realization, for example the 4 d bosonic algebra is the (even subalgebra) of Weyl algebra A 2 , [ q i , p j ] = 1, i , j = 1 , 2, i.e. even functions f ( q , p ) or differential operators in two variables; Disclaimer: locality is not under control, but something will still survive (what?) — we call it formal higher-spin theories. All Vasiliev equations lead to formal theories. E.Skvortsov
Deformation Problem Natural object is a connection ω of hs Just the knowledge of δω = d ξ − [ ω, ξ ] ⋆ gives (some) cubic interactions (Fradkin, Vasiliev) The only unfolded equations that we can write down are d ω = ω ⋆ ω If the HS algebra admits a deformation (for example, deformed oscillators viewed from the vantage point of undeformed ones), then we can add d ω = ω ⋆ ω + Ψ( ω, ω ) The consistency tells exactly that Ψ is a two-cocycle responsible for a deformation of hs . Full theory: Prokushkin-Vasiliev theory (later) E.Skvortsov
Deformation Problem The study of free HS fields suggests that one should start with d ω = ω ⋆ ω dC = ω ⋆ C − C ⋆ π ( ω ) On AdS (any flat HS background) the equation describes pure gauge Fronsdal fields plus degrees of freedom hidden in C (Bargmann-Wigner equations, Weyl tensors, etc.) Now we can have other deformations d ω = ω ⋆ ω + V ( ω, ω, C ) + O ( C 2 ) dC = ω ⋆ C − C ⋆ π ( ω ) + O ( C 2 ) E.Skvortsov
Weak-field vs. Unfolding AdS is a particular exact solution with ω = Ω ∈ so ( d , 2) and C = 0, i.e. d Ω = Ω ⋆ Ω. ω = Ω + ω (1) + ω (2) + ... , C = 0 + C (1) + C (2) + ... , Free equations see a part of the first vertex D ω (1) = V (Ω , Ω , C (1) ) , D C (1) = 0 � Each unfolded vertex contributes to ∞ -many orders in the weak-field expansion: D ω (2) − V (Ω , Ω , C (2) ) = ω (1) ⋆ ω (1) + V (Ω , ω (1) , C (1) ) + V (Ω , Ω , C (1) , C (1) ) , D C (2) = ω (1) ⋆ C (1) − C (1) ⋆ π ( ω (1) ) + V (Ω , C (1) , C (1) ) . � Looks like expanding Riemann tensor near g (0) , which generates ∞ -many terms via ( g (0) + g (1) + ... ) − 1 E.Skvortsov
Hochschild We need to find at least the first vertex in d ω = ω ⋆ ω + V ( ω, ω, C ) + O ( C 2 ) dC = ω ⋆ C − C ⋆ π ( ω ) + O ( C 2 ) Matrix extensions (global symmetry on the CFT side) allows one to reduce wild Chevalley-Eilenberg problem to the Hochschild one. The answer is V ( ω, ω, C ) = Φ( ω, ω ) ⋆ π ( C ) Φ( a , b ) is a Hochschild two-cocycle of the HS algebra: − a ⋆ Φ( b , c ) + Φ( a ⋆ b , c ) − Φ( a , b ⋆ c ) + Φ( a , b ) ⋆ π ( c ) = 0 E.Skvortsov
Hochschild Complex Hochschild differential acts on functions of n -arguments. In our case it takes values in the twisted-adjoint representation of the higher-spin algebra. Z : δ H Z = a ⋆ Z − Z ⋆ π ( a ) δ H Z = 0 would give (anti)-center of the HS algebra. Possible redefinitions ω → ω + ω C are coboundaries: f ( a ) : δ H f = a ⋆ f ( b ) − f ( a ⋆ b ) + f ⋆ π ( c ) The vertex is the two-cocycle: − a ⋆ Φ( b , c ) + Φ( a ⋆ b , c ) − Φ( a , b ⋆ c ) + Φ( a , b ) ⋆ π ( c ) = 0 The two-cocycle valued in the adjoint of the algebra is responsible for a deformation of ⋆ -product, e.g. hs ( λ ): − a ⋆ Ψ( b , c ) + Ψ( a ⋆ b , c ) − Ψ( a , b ⋆ c ) + Ψ( a , b ) ⋆ c = 0 E.Skvortsov
Higher-Spins on Background of Their Own The first vertex gives for free equations that describe free HS fields over any HS-flat background: d Ω = Ω ⋆ Ω , d ω = Ω ⋆ ω + ω ⋆ Ω + Φ(Ω , Ω) ⋆ π ( C ) , dC = Ω ⋆ C − C ⋆ π (Ω) , The Hochschild term is missing in 3 d , but flat connections give back holes, for example. Fronsdal fields transform under global HS symmetries as δω = ξ ⋆ ω − ω ⋆ ξ + Φ( ξ, Ω) ⋆ π ( C ) − Φ(Ω , ξ ) ⋆ π ( C ) , δ C = ξ ⋆ C + C ⋆ π ( ξ ) , E.Skvortsov
Formal HS Theories aka Vasiliev Equations d ω = ω ⋆ ω + Φ( ω, ω ) ⋆ π ( C ) + O ( C 2 ) dC = ω ⋆ C − C ⋆ π ( ω ) + O ( C 2 ) Given a HS algebra, the first vertex is given by a Hochschild cocycle; It can be shown at least in some of the cases that there are no obstructions at higher orders: the first vertex should be completed by higher-order terms; f ∗ g = f · g + { f , g } + ... Still, how to construct higher-order terms? Resolution: bigger complex where the initial complex is a cohomology itself. Now the cohomology can be computed in different ways and some of them can be simpler. E.Skvortsov
Vasiliev Resolution Having the Hochschild cocycle in front of your eyes, it is easy to see that it can be generated as � � Φ( f , g ) = f ( y ) ♥ δ − 1 g ( y ) ♥ δ − 1 Z � z =0 where Z = κ = exp i [ z · y ] dz ∧ dz and f ( y ) ♥ V ( y , z ) = f ( y ) exp i [ ← − ∂ y · ( − → ∂ y − − → ∂ z )] V ( y , z ) Homotopy operator of the de Rham complex δ − 1 � 1 δ − 1 f α 1 ...α k ( z ) = t k − 1 dt z ν f να 2 ...α k ( zt ) 0 There is a deeper reason: bi-complex with differential= Hochschild + de Rham in z -space E.Skvortsov
Vasiliev Resolution Having the resolution in front of your eyes � � Φ( f , g ) = f ( y ) ♥ δ − 1 g ( y ) ♥ δ − 1 κ � z =0 it is easy to see that the equations come from δ S = C ⋆ κ dz ∧ dz δ W = D S D W = 0 where D = d + [ ω, • ] replaces the arguments of Φ. Here d + W + S is the connection in x - z -space Moral: Vasiliev equations provide the field-theoretical realization of the resolution of the Hochschild complex E.Skvortsov
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