Boundary Current Algebra and Multiparticle HS Symmetry - PowerPoint PPT Presentation

Boundary Current Algebra and Multiparticle HS Symmetry arXiv:1212.6071, arXiv:1301.3123 O.Gelfond, MV M.A.Vasiliev Lebedev Institute, Moscow Higher Spins, Strings and Holography GGI, Florence, May 9, 2013

Plan I Higher-spin algebra II Free fields and currents III Twistor current operator algebra as multiparticle symmetry IV Multiparticle symmetry as a string-like HS symmetry V Butterfly formulae for n -point functions VI Conclusion

Higher-Spin Theory versus String Theory HS theories: Λ � = 0, m = 0 symmetric fields s = 0 , 1 , 2 , . . . ∞ String Theory: Λ = 0, m � = 0 except for a few zero modes mixed symmetry fields − → s = 0 , 1 , 2 , . . . ∞ String theory has much larger spectrum: HS Theory: first Regge trajectory Pattern of HS gauge theory is determined by HS symmetry What is a string-like extension of a global HS symmetry underlying a string-like extension of HS theory?

Global Higher-Spin Symmetry HS symmetry in AdS d +1 : Maximal symmetry of a d -dimensional free conformal field(s)=singletons usually, scalar (Rac) and/or spinor (Di) Admissibility condition: a set of fields resulting from gauging a global HS symmetry should match some its unitary representation. Example: SUSY algebra admits a UIRREP (2 , N × 3 / 2 , 1 2 N ( N − 1) × 1 , . . . ) There should be a HS-module containing the AdS d +1 module associated with gravity : D (2 , E 0 (2)) D ( s, E 0 ( s )) is a massless module of spin s . E 0 ( s ) for s ≥ 1 is the boundary of the unitarity region

Oscillator realization P a = P a AB { Y A , Y B } , M ab = M ab AB { Y A , Y B } , [ Y A , Y B ] = C AB Tensoring modules: Y A → Y A i , [ Y A , Y B j ] = δ ij C AB , i, j = 1 , . . . N i � � P a = P a M ab = M ab { Y A , Y B { Y A , Y B i } , i } AB i AB i i i If | E 0 (2) � vacuum was a Fock vacuum for Y A E 0 increases as NE 0 . If there was gravity at N = 1: no gravity at N > 1. Incompatibility of AdS extension of Minkowski first quantized string � 1 M ab = nx [ a − n x b ] n + p [ a x b ] , P a = p a n � =0 since [ P a , P b ] = − λ 2 M ab implies that P a should involve all modes and hence lead to the infinite vacuum energy: no graviton What is a symmetry that is able to unify HS gauge theory with String? Current operator algebra

3 d conformal equations and HS symmetry Conformal invariant massless equations in d = 3 ∂ 2 ∂ ∂y α ∂y β ) C ± ( ∂x αβ ± i j ( y | x ) = 0 , α, β = 1 , 2 , j = 1 , . . . N Shaynkman, MV (2001) Generalization to matrix space: α, β = 1 , 2 , . . . M . Bosons and fermions are even (odd) functions of y : C i ( − y | x ) = ( − 1) p i C i ( y | x “Classical” field Φ j ( y | x ) = C + j ( y | x ) + i p j C − Φ j ( y | x ) = C − j ( y | x ) + i p j C + j ( iy | x ) , j ( iy | x ) � � � � ∂ 2 ∂ 2 ∂ ∂ ∂x αβ + i Φ j ( y | x ) = 0 , ∂x αβ − i Φ j ( y | x ) = 0 ∂y α ∂y β ∂y α ∂y β Initial data: C ± j ( y | 0): Maximal symmetry: all operators on the space of functions of y . ∂ Y A = ( y α , [ Y A , Y B ] = C AB . A ( Y A ) : ∂y β ) A = 1 , 2 , 3 , 4 , Algebra of oscillators: 3 d conformal HS algebra = AdS 4 HS algebra sp (4) subalgebra is spanned by bilinears T AB = { Y A , Y B } .

Currents Rank-two equations: conserved currents � � ∂ 2 ∂ ∂x αβ − J ( u, y | x ) = 0 ∂y ( α ∂u β ) J ( u, y | x ): generalized stress tensor. Rank-two equation is obeyed by N � J ( u, y | x ) = Φ i ( u + y | x ) Φ i ( y − u | x ) i =1 Rank-two fields: bilocal fields in the twistor space. 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 Differential equations: conservation condition ∂ 2 ∂ 2 ∂ ∂ ˜ J ( u, 0 | x ) = 0 , J (0 , y | x ) = 0 ∂x αβ ∂x αβ ∂u α ∂u β ∂y α ∂y β

D -functions Unfolded dynamics leads to quantization : Particles and antiparticles: definite frequencies � dξ M c ± ( ξ ) exp ± i [ ξ α ξ β x αβ + y α ξ α ] C ± ( y | x ) = (2 π ) − M/ 2 Time: x αβ = tT αβ with a positive definite T αβ . c ± ( ξ ) = const Solutions with � dξ M exp ± i [ ξ α ξ β x αβ + y α ξ α ] . D ± ( y | x ) = ∓ i (2 π ) − M D ± ( y | x ) = D ± ( x ) exp[ − i 4 x − 1 αβ y α y β ] 2 M π M/ 2 exp ± iπI x i D ± ( x ) = ± 4 | det | x || − 1 / 2 Normalization is such that D ± ( y | 0) = ∓ iδ M ( y ) Rank-one twistor to boundary evolution � d M y ′ D ∓ ( y ′ − y | x ′ − x ) C ± ( y ′ | x ′ ) . C ± ( y | x ) = ∓ i

AdS/CFT from twistors Bulk extesion is trivially achieved by means of twistor-to-bulk D -function D ( y | X ) , X = ( x, z ) D ± ( y | 0) = ∓ iδ M ( y ) D 0 D ( y | X ) = 0 , Twistor-like transforms make the correspondence tautological J ( u, y | 0) � ❅ CFT 3 AdS 4 � ❅ � ❅ � ✠ ❅ ❘ J ( x ) C ( X ) Being simple in terms of unfolded dynamics and twistor space holo- graphic duality in terms of usual space-time may be obscure

Quantization Operator fields obey � � k ( y ′ | x ′ )] = 1 C − C + D − ( y − y ′ | x − x ′ ) + ( − 1) p j p k D − ( y + y ′ | x − x ′ ) [ ˆ j ( y | x ) , ˆ 2 i Commutation relations make sense at x = x ′ � � k ( y ′ | x )] = 1 C − C + δ ( y − y ′ ) + ( − 1) p j p k δ ( y + y ′ ) [ ˆ j ( y | x ) , ˆ 2 δ jk Singularity at ( y, x ) = ( y ′ , x ′ ) does not imply singularity at x = x ′ . Space-time operator algebra is reconstructed by twistor-to-boundary D -functions from the operator algebra in the twistor space. Φ j ( y 1 | x )ˆ J jk ( y 1 , y 2 | x ) =: ˆ Quantum currents: Φ k ( y 2 | x ) : Generating function J 2 g with test-function g � J 2 dw 1 dw 2 g mn ( w 1 , w 2 ) J mn ( w 1 , w 2 | 0) , g = � J 2 dw 1 dw 2 g mn ab ( w 1 , w 2 ) J ab mn ( w 1 , w 2 | x ) = J 2 g ( x ) = g ( x ) g mn x -dependence of ab ( x ) ( a, b = ± ) is reconstructed by D -functions

Twistor current algebra Elementary computation gives J 2 g J 2 g ′ = J 4 g × g ′ + J 2 [ g ,g ′ ] ⋆ + N tr ⋆ ( g ⋆ g ′ ) J 0 Convolution product ⋆ is related to HS star-product via half-Fourier transform � g ( w, v ) = (2 π ) − M/ 2 d M u exp[ iw α u α ] g ( v + u, v − u ) ˜ Star product of AdS 4 HS theory results from OPE of boundary currents Full set of operators J 2 m =: J 2 g . . . J 2 J 0 : g = Id g g � �� � m What is the associative twistor operator algebra?! Since J 2 g 1 J 2 g 2 − J 2 g 2 J 2 g 1 = J 2 [ g 1 ,g 2 ] ⋆ This is universal enveloping algebra U ( h ) of the HS algebra h

Explicit construction of multiparticle algebra Universal enveloping algebra U ( l ( A )) of a Lie algebra l ( A ) associated with an associative algebra A has remarkable properties allowing to obtain very explicit description of the operator product algebra Let { t i } be some basis of A a = a i t i , t i ⋆ t j = f k a ∈ A : ij t k t i ∼ J 2 , a i ∼ g ( w 1 , w 2 ) U ( l ( A )) is algebra of functions of α i (commutative analogue of t i ) Explicit composition law of M ( A ) � ← − − → � ∂ ∂ f n F ( α ) ◦ G ( α ) = F ( α ) exp ij α n G ( α ) ∂α i ∂α j ← − − → ∂ ∂ where derivatives ∂α i and ∂α j act on F and G , respectively. Associativity of ⋆ of A implies associativity of ◦ of M ( A ) As a linear space, A is represented in M ( A ) by linear functions F ( α ) = a i α i a i α i ⇔ a i t i

Operator product algebra Composition law for linear functions F ( α ) ◦ G ( α ) = F ( α ) G ( α ) + F ( α ) ⋆ G ( α ) differs from current operator algebra F ( α ) ⋄ G ( α ) = F ( α ) G ( α ) + 1 2[ F ( α ) , G ( α )] ⋆ + N tr ⋆ ( F ( α ) G ( α )) Uniqueness of the Universal enveloping algebra implies that the two composition laws are related by a basis change ν = ν i α i ∈ A is replaced by Generating function G ν = exp ν G ν = exp[ −N 4 tr ⋆ ln ⋆ ( e ⋆ − 1 4 ν ⋆ ν ) exp[ ν ⋆ ( e ⋆ − 1 2 v ) − 1 ˜ ⋆ ] � ∂ n � T u � ∂ν i 1 . . . ∂ν i n ˜ � i 1 ...i n = G ( ν ) � ν =0 The resulting composition law is   N 4 det ⋆ | e ⋆ − 1 4 ν ⋆ ν | det ⋆ | e ⋆ − 1 4 µ ⋆ µ |   G ν ⋄ � � � G µ = G σ 1 , − 1   ( ν,µ ) det ⋆ | e ⋆ − 1 2 ( ν, µ ) ⋆ σ 1 , − 1 2 ( ν, µ ) | 4 σ 1 , − 1 2

2 ( ν, µ ) = 2( e ⋆ − ( e ⋆ − 1 2 µ ) ⋆ ( e ⋆ + 1 ⋆ ( e ⋆ − 1 4 ν ⋆ µ ) − 1 2 µ ) σ 1 , − 1 ⋆ Generating function for correlators � J 2 n J 2 m � of all currents   N 4 det ⋆ | e ⋆ − 1 4 ν ⋆ ν | det ⋆ | e ⋆ − 1 4 µ ⋆ µ |   � ˜ G ν ˜ G µ � =   det ⋆ | e ⋆ − 1 2 ( ν, µ ) ⋆ σ 1 , − 1 2 ( ν, µ ) | 4 σ 1 , − 1 � ∂ n � J 2 n g 1 ...g n = g i 1 . . . g i n ∂ν i 1 . . . ∂ν i n ˜ � G ν � ν =0 Theories with different N : different frames of the same algebra! U ( h ) possesses different invariants (traces) generating different (inequiv- alent) systems of n -point functions What are models associated with different frame choices?! Infinitely many (conformal?) nonlinear models not respecting Wick theorem!?