Higher-Rank Fields, Currents, and Higher-Spin Holography arXiv:1312.6673 O.A.Gelfond, M.V. + work in progress M.A.Vasiliev Lebedev Institute, Moscow Strings 2014 Princeton, June 25, 2014
HS AdS/CFT correspondence General idea of HS duality Sundborg (2001), Witten (2001) AdS 4 HS theory is dual to 3 d vectorial conformal models Klebanov, Polyakov (2002), Petkou, Leigh (2005), Sezgin, Sundell (2005); Giombi and Yin (2009); Maldacena, Zhiboedov (2011,2012); MV (2012); Giombi, Klebanov; Tseytlin (2013,2014) ... AdS 3 /CFT 2 correspondence Gaberdiel and Gopakumar (2010) Analysis of HS holography helps to uncover the origin of AdS/CFT
Unfolded Dynamics Covariant first-order differential equations 1988 ∞ � dW Ω ( x ) = G Ω ( W ( x )) , G Ω ( W ) = f ΩΛ 1 ... Λ n W Λ 1 ∧ . . . ∧ W Λ n n =1 d > 1: Compatibility conditions G Λ ( W ) ∧ ∂G Ω ( W ) ≡ 0 ∂W Λ Manifest (HS) gauge invariance under the gauge transformation δW Ω = dε Ω + ε Λ ∂G Ω ( W ) ε Ω ( x ) : ( p Ω − 1) − form , ∂W Λ G Ω ( W ): unfolded equations make sense in any Geometry is encoded by space-time d z = dz u ∂ dW Ω ( x ) = G Ω ( W ( x )) , x → X = ( x, z ) , d x → d X = d x + d z , ∂z u X -dependence is reconstructed in terms of W ( X 0 ) = W ( x 0 , z 0 ) at any X 0 Classes of holographically dual models: different G
3 d conformal equations Rank-one conformal massless equations Shaynkman, MV (2001) ∂ 2 ∂ ∂y α ∂y β ) C ± ( ∂x αβ ± i j ( y | x ) = 0 , α, β = 1 , 2 , j = 1 , . . . N Bosons (fermions) are even (odd) functions of y : C i ( − y | x ) = ( − 1) p i C i ( y | x ) Rank-two equations: conserved currents � � ∂ 2 ∂ ∂x αβ − J ( u, y | x ) = 0 Gelfond, MV (2003) ∂y ( α ∂u β ) J ( u, y | x ): generalized stress tensor. Rank-two equation is obeyed by N � C − i ( u + y | x ) C + J ( u, y | x ) = i ( y − u | x ) i =1 Primaries : 3 d currents of all integer and half-integer spins ∞ ∞ � � u α 1 . . . u α 2 s J α 1 ...α 2 s ( x ) , y α 1 . . . y α 2 s ˜ ˜ J ( u, 0 | x ) = J (0 , y | x ) = J α 1 ...α 2 s ( x ) 2 s =0 2 s =0 J asym ( u, y | x ) = u α y α J asym ( x ) ∆ J asym ( x ) = 2 ∆ J α 1 ...α 2 s ( x ) = ∆ ˜ J α 1 ...α 2 s ( x ) = s + 1 ∂ 2 ∂ Conservation equation: ∂u α ∂u β J ( u, 0 | x ) = 0 ∂x αβ
Extension to Sp (2 M ) -invariant space Rank-one unfolded equation (2001) � � ∂ 2 ∂ C ± ( Y | X ) = 0 , ξ AB σ − = ξ AB ∂X AB ± iσ − ∂Y A ∂Y B , Y A - auxiliary commuting variables X AB matrix coordinates of M M , X AB = X BA ( A, B = 1 , . . . , M = 2 n ) Fronsdal (1985), Bandos, Lukierski, Sorokin (1999), MV (2001) ξ MN = dX MN are anti-commuting differentials ξ MN ξ AD = − ξ AD ξ MN C ( X ) , C A ( X ) Y A σ − C ( X | Y ) = 0 : Rank-one primary (dynamical) fields : ⇒ dynamical equations Unfolded equations (2001) ∂ ∂ ∂ ∂ ∂X BD C ( X ) − ∂X AD C ( X ) = 0 Klein-Gordon–like , ∂X AE ∂X BE ∂ ∂ ∂X BD C A ( X ) − ∂X AD C B ( X ) = 0 Dirac–like
Extension to higher ranks and higher dimensions Rank- r unfolded equations: r twistor variables Y A i, j, . . . = 1 , . . . , r i � � r ∂ 2 � ∂ ξ AB C ± ( Y | X ) = 0 , − = ξ AB ∂X AB ± iσ r σ r δ ij , − ∂Y A j ∂Y B j =1 i A rank- r field in M M ∼ a rank-one field in M r M with coordinates X AB . ij → Y � A , Y A � A = 1 . . . r M i Embedding of M M into M r M : X AB − X ˜ A ˜ B → ˜ X AB 11 = X AB 22 = . . . = X AB = X AB rr The map M M − → M r M preserves Sp (2 M ) Field-current correspondence: Flato-Fronsdal (1978) for M = 2 Alternative interpretation: multi-particle states (=higher-rank field) in lower dimension=single-particle states in higher dimensions Problem: pattern of the holographic reduction of higher-dimensional models to the lower-dimensional ones
Rank- r fields and equations − = ξ AB � r ∂ 2 σ r σ r − C ( Y | X ) = 0, Rank- r primary fields: δ ij j =1 ∂Y A j ∂Y B i � C i 1 ; ... ; i n A 1 ; ... ; A n ( X ) Y A 1 · · · Y A n ⇒ tracelessness: δ i 1 i 2 C i 1 ; i 2 ; ... C ( Y | X ) = ( X ) = 0 . ... i 1 i n n Since C ... i m ... i k ... ... A m ... A k ... ( X ) = C ... i k ... i m ... ... A k ... A m ... ( X ) , rank- r primary fields are described by -Young diagrams Y 0 [ h 1 , ..., h m ] obeying h 1 + h 2 ≤ r , h 1 ≤ M Rank- r primary fields C Y 0 ( Y | X ) satisfy rank- r dynamical equations ∂ ∂ ∂ ∂ E A 1 [ r − h 2 +1] , A 2 [ r − h 1 +1] , A 3 [ h 3 ] ,...,A n [ h n ] . . . . . . . . . i 1 [ h 1 ] , i 2 [ h 2 ] , i 3 [ h 3 ] ,..., i n [ h n ] ∂Y A 1 ∂Y A 1 ∂Y A h 1 ∂Y A hn n n 1 1 i 1 i hn i 1 i h 1 n n 1 1 � �� � � �� � h n h 1 ∂ ∂ · · · C Y 0 ( Y | X ) = 0 . ∂X A h 1+1 A h 2+1 ∂X A r − h 2+1 A r − h 1+1 1 2 1 2 � �� � r +1 − h 1 − h 2 The parameter E ... projects to Y 0 [ h 1 , h 2 , h 3 , . . . , h n ] and to its ... rank- r two-column dual Y 1 [ r + 1 − h 2 , r + 1 − h 1 , h 3 , . . . , h n ] with respect to the lower and upper indices, respectively
Multi-linear currents For r = 2 κ , a ( κM − κ ( κ − 1) ) -form 2 � ∂ ∂ ∂ ∂ � Ω( J ) = F i 1 [ κ ] ,..., i N [ κ ] D A 1 [ κ ] ,...,A N [ κ ] J ( Y | X ) . . . . . . . . . � Y =0 ∂Y A κ A κ ∂Y A 1 A 1 1 N 1 ∂Y N ∂Y i κ i κ i 1 i 1 1 N 1 N � �� � � �� � κ κ where N = M + 1 − κ F is described by traceless diagram Y [ κ, . . . κ ] , and � �� � N 3 . . . ξ D κ − 1 D 1 1 . . . ξ D M 1 A 1 κ ξ D 1 1 D 1 2 ξ D 2 1 D 1 κ ξ D κ 1 A 1 D A 1 [ κ ] ,...,A N [ κ ] = ǫ D 1 N . . . 1 . . . ǫ D 1 1 1 ...D M κ ...D M n +1 ξ D n +1 ξ D n n D n D n 1 ξ D κ +1 n +2 . . . ξ D κ − 1 κ ξ D κ n A n 1 . . . ξ D M n A n N . . . ξ D κ κ A κ A κ 2 . . . ξ D M κ A κ D n n κ n N is closed provided that J ( Y | X ) obeys the rank- r = 2 κ equations. The current J η ( Y | X ) = η j 1 ,...,j r ( A ) C j 1 ( Y j 1 | X ) . . . C j r ( Y j r | X ) where A 1 ∂ A 2 ∂ B ( Y j | X ) = 2 X AB + Y B j , j C ( Y j | X ) = j ∂Y A ∂Y C j j and C j ( Y | X ) – rank-one fields, generates r -linear charge Q r η ( C ) Multiparticle algebra: string-like HS algebra (2012)
σ − -cohomology analysis Rank- r primary fields and field equations are represented by the coho- mology groups H 0 ( σ r − ) and H 1 ( σ r − ) , respectively. Higher H p ( σ − ) (and their twisted cousins) are responsible for HS gauge fields and their field equations General H p ( σ − ) via homotopy trick: conjugated operators Ω and Ω ∗ ∂ − = T AB ξ AB , Ω ∗ = T AB Ω := σ r ∂ξ CD , ∂ ∂ ∂ T CD = Y C j δ ij , δ ij , i Y D T A B = Y A T AB = = sp (2 M ) j ∂Y A ∂Y B ∂Y B i j j 2 τ mk τ mk + ν A ∆ = { Ω , Ω ∗ } = 1 B ν A B − ( M + 1 − r ) ν A A ∂ ∂ τ mk = Y mA ∂Y kA − Y kA ∂Y mA are o ( r ) − -generators ∂ ν A B = 2 ξ AD ∂ξ BD + T A B are gl tot M - generators that act on Y A and ξ AB i ∆ is semi positive-definite H ( Ω ) ⊂ ker ∆
Young diagrams and South-West principle Y ′ [ B 1 . . . ] ⊂ Y [ h 1 . . . ] ⊗ ( ⊗ n Y δ [1 , 1]) ⊗ Y A [ a 1 , . . . ] where n is a number of o ( r ) metric tensors δ ij , Y A [ a 1 , . . . ] : ξ AB YD Y ′ [ B 1 . . . , B m ] Y [ h 1 . . . , h k ] o ( r ) YD , gl M : YD , � � τ mk τ mk = 2 ν A B ν A h j ( h j − r − 2( i − 1)) , B = − B i ( B i − M − 1 − 2( i − 1)) , j i � � � ⇒ ∆ = − B i ( B i − 2( i − 1)) + h i ( h i − 2( i − 1)) + r ( B i − h i ) . i j i χ a ( S ( i, j )) = i − j + a , a ∈ R , S ( i, j ) – a sell on the intersection of j − th row and i − th column � � χ a ( Y ) = − 1 Y = S ( i, j ) h i ( h i − 2 i + 1 − 2 a ) 2 i S ( i,j ) ∈ Y min(∆) is reached when all cells of Y ′ ∆ semi-positive ⇒ are maximally south-west. This allows us to find H p ( σ r − ) ∀ p Higher-differential forms are relevant to the nonlinear field equations and invariant Lagrangians for multiparticle theory
Invariant functionals Unfolded equations dF ( W ) = QF ( W ) , F ( W ) is an arbitrary function of W ∂ Q 2 = 0 Q = G Ω ∂W Ω , Q -closed p -form functions L p ( W ) are d -closed, giving rise to the gauge invariant functionals represented by Q -cohomology (2005) � S = Σ p L p So defined L p is d -closed in any space-time realization S is gauge invariant in any space-time
Nonlinear HS Equations One-form W = d x + dx ν W ν + dZ A S A and zero-form B : W ⋆ W = i ( dZ A dZ A + F ( B, dz α dz α ⋆ kκ, d ¯ z ˙ α d ¯ α ⋆ ¯ z ˙ k ¯ κ )) W ⋆ B − B ⋆ W = 0 HS star product � dSdTf ( Z + S, Y + S ) g ( Z − T, Y + T ) exp − iS A T A ( f ⋆ g )( Z, Y ) = [ Y A , Y B ] ⋆ = − [ Z A , Z B ] ⋆ = 2 iC AB , Z − Y : Z + Y normal ordering Inner Klein operators: κ = exp iz α y α , α , y ˙ ¯ κ = exp i ¯ α ¯ κ ⋆ f ( y, ¯ y ) = f ( − y, ¯ y ) ⋆ κ , κ ⋆ κ = 1 z ˙ Nontrivial equations are free of the space-time differentials d Action is not known but probably is not needed
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