Higher Spin Gauge Theories Lecture II Lecture II a 1. 4 d HS fields - PowerPoint PPT Presentation

Higher Spin Gauge Theories Lecture II Lecture II a 1. 4 d HS fields in spinor notation 2. Weyl algebra 3. Star product 4. Simplest HS algebra 5. Properties of HS algebras 6. Singletons and AdS/CFT Lecture II b 1. Cubic HS action 2. Unfolded dynamics 3. Equations of motion in all orders 4. 4 d HS fields in ten-dimensional space-time 1

Spinorial and tensorial HS models Tensorial HS models in any dimension: HS fields are realized as forms carrying tensor indices. Spinorial 3 d and 4 d HS models: HS fields are realized as forms carrying spinor indices. 2

The case of four dimensions Key fact 2 × 2 = 4 Minkowski coordinates as 2 × 2 hermitian matrices 3 � x n ⇒ x α ˙ α = α , − → x n σ α ˙ α σ α ˙ α = ( I α ˙ σ α ˙ α , k ) n n n =0 I α ˙ α : unit matrix − → σ α ˙ α k , k = 1 , 2 , 3: Pauli matrices α, ˙ α, β, . . . = 1 , 2, ˙ β, . . . = 1 , 2 two-component spinor indices α | = ( x 0 ) 2 − ( x 1 ) 2 − ( x 2 ) 2 − ( x 3 ) 2 det | x α ˙ Lorentz symmetry: sl (2 , C ) ∼ o (3 , 1). 3

Two-component spinors Two-component indices are contracted by the antisymmetric 2 × 2 matrix ǫ 12 = ǫ 12 = 1 , ǫ αγ ǫ βγ = δ β ψ α = ǫ αβ ψ β , ψ α = ψ β ǫ βα ǫ αβ : α , Lorentz invariants ψ α χ α : Lorentz Symmetry: sl 2 ( C ) ∼ o (3 , 1). Dictionary between tensors and multispinors by: αβ = σ [ a α σ b ] β , ˙ β = σ [ a α σ b ] α ˙ σ a σ ab σ ab α , ¯ α ˙ α ˙ α ˙ β α ˙ β ˙ Pair of dotted and undotted indices: vector Pairs of symmetrized indices of the same type: antisymmetric tensors Irreducible representations of the Lorentz group: symmetric multispinors A α 1 ...α n , ˙ β m ⊕ A β 1 ...β m , ˙ α n ∼ ω a 1 ...a p ,b 1 ...b q , p = | n + m | / 2 , q = | n − m | / 2 β 1 ... ˙ α 1 ... ˙ p A a 1 ...a p ,b 1 ...b q η a 1 a 2 = 0 . Irreducibility: A ( a 1 ...a p ,a p +1 ) b 2 ...b q = 0 : , q 4

Gauge connections Gauge 1-forms ω α 1 ...α n , ˙ β m , n + m = 2( s − 1) β 1 ... ˙ ω ( x ) = dx n ω n ( x ) s = 1 : s = 2 : ω α ˙ β ( x ) , ω αβ ( x ) , ¯ ω ˙ β ( x ) α ˙ s = 3 / 2 : ω α ( x ) , ¯ ω ˙ α ( x ) 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 5

Gauge invariant field strengths 0-forms | n − m | = 2 s C α 1 ...α n , ˙ β m , 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 6

HS multiplets Infinite set of spins s = 0 , 1 / 2 , 1 , 3 / 2 , 2 . . . ω α 1 ...α n , ˙ β m and C α 1 ...α n , ˙ β m with all n ≥ 0 and m ≥ 0. β 1 ... ˙ β 1 ... ˙ Generating functions ω ( Y | x ) and C ( Y | x ): Unrestricted functions of com- muting spinor (twistor) variables Y = ( y α , ¯ y ˙ α ) ∞ � 1 α m y α 1 . . . y α n ¯ y ˙ α 1 . . . ¯ y ˙ α m A ( Y | x ) = 2 n ! m ! A α 1 ...α n , ˙ α 1 ... ˙ n,m =0 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 ) . 7

Twistor Central On-shell theorem The full unfolded system for the doubled sets of free fields is ∂ 2 ∂ 2 α ˙ 1 ( y, y | x ) = H ˙ β R ii β C 1 − i i (0 , y | x ) + 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 , ˜ where α ˙ β = h α ˙ α ∧ h α ˙ H αβ = h α ˙ α , H ˙ β , α ∧ h β ˙ y | x ) = D ad ω ( y, ¯ R 1 ( y, ¯ y | x ) � � � � ∂ 2 ∂ ∂ D ad ω = D L − λh α ˙ D = D L + λh α ˙ β β ˜ β + ∂y α ¯ y α ¯ β + y α y ˙ , y ˙ , β y ˙ y ˙ ∂y α ∂ ¯ β ∂ ¯ � � ∂ ∂ α ˙ D L A = d x − ω αβ y α ω ˙ β ¯ ∂y β + ¯ y ˙ . α y ˙ β ∂ ¯ NonAbelian generalization via star-product algebra 8

Weyl algebra associative algebra of functions f (ˆ Weyl algebra A n : Y ) of n pairs of oscillators [ˆ Y µ , ˆ Y ν ] = 2 iC µν , µ, ν = 1 , . . . 2 n . Different types of orderings are equivalent for polynomial f (ˆ Y ) because commutators of oscillators decrease an order of polynomial. Weyl ordering: totally symmetric ∞ � f µ 1 ...µ p ˆ f (ˆ Y µ 1 . . . ˆ Y ) = Y µ p , p =0 f µ 1 ...µ p totally symmetric a + j ] = δ j a − Wick (normal) ordering [ˆ i , ˆ i ∞ � χ i 1 ...i p a ± ) = a + j 1 . . . ˆ a + j q ˆ a − a − f (ˆ j 1 ...j q ˆ i 1 . . . ˆ i q p,q =0 9

Star Product Weyl symbol f ( Y ) of the operator ˆ f (ˆ Y ) is a function of commuting variables Y µ that has the same expansion ∞ � f µ 1 ...µ p Y µ 1 . . . Y µ p f ( Y ) = p =0 Y ν is the Weyl symbol of ˆ Y ν . f ( ˆ Wick symbol f ( a ± ) of the operator ˆ a ± ) is a function of commuting variables a ± that has the same expansion ∞ � χ i 1 ...i p f ( a ± ) = j 1 ...j q a + j 1 . . . a + j q a − i 1 . . . a − i q p,q =0 Star–product algebra is defined by the rule f (ˆ ˆ g (ˆ Weyl star-product ( f ∗ g )( Y ) is a symbol of Y )ˆ Y ) . In particular, [ Y ν , Y µ ] ∗ = 2 iC νµ , [ a , b ] ∗ = a ∗ b − b ∗ a Wick star-product ( f ⋆ g )( a ± ) a ± )ˆ a ± ) . ˆ is a symbol of f (ˆ g (ˆ 10

Examples Y µ ∗ Y ν = Y ( µ Y ν ) + iC µν i ⋆ a + j = a + j a − a + j ⋆ a − i + δ j i = a + j a − a − i , i Problem 2.1. Prove [ Y ν , f ( Y )] ∗ = 2 i ∂ Y ν = C νµ Y µ ∂Y ν f ( Y ) , { Y ν , f ( Y ) } ∗ = 2 Y ν f ( Y ) a + i ⋆ f ( a ± ) = a + i f ( a ± ) , f ( a ± ) ⋆ a − j = f ( a ± ) a − j � � � � ∂ ∂ f ( a ± ) ⋆ a + = a + j + a − i ⋆ f ( a ± ) = a − f ( a ± ) , f ( a ± ) i + ∂a − ∂a + i j 11

Weyl-Moyal star-product For the Weyl ordering, star–product is given by the Weyl-Moyal formula ( f 1 ∗ f 2 )( Y ) = f 1 ( Y ) exp [ i ← ∂ ν − − → ∂ ∂ µ ≡ ∂ µ C νµ ] f 2 ( Y ) , ∂Y µ Problem 2.2. Prove using Campbell-Hausdorf formula for exponentials exp J ν ˆ Y ν Important properties • associativity: ( f ∗ g ) ∗ h = f ∗ ( g ∗ h ) • regularity: star product of any two polynomials of Y is a polynomial The Weyl-Moyal star product has integral representation � 1 dSdT exp( − iS µ T ν C µν ) f 1 ( Y + S ) f 2 ( Y + T ) ( f 1 ∗ f 2 )( Y ) = π 2 M 12

Supertrace str ( f ( Y )) = f (0) Boson-fermion parity for spinorial Y ν f ( Y ) = ( − 1) π ( f ) f ( − Y ) str ( f ( Y ) ∗ g ( Y )) = ( − 1) π ( f ) str ( g ( Y ) ∗ f ( Y )) = ( − 1) π ( g ) str ( g ( Y ) ∗ f ( Y )) Bilinear form str ( f ∗ g ) is invariant under δf = [ ǫ , f ] ∗ provided that fermion fields carry additional Grassmann parity In components ∞ i n + m − 1 � β m ∧ B α 1 ...α n , ˙ β 1 ... ˙ β m , str ( A ∗ B ) = A α 1 ...α n , ˙ β 1 ... ˙ n ! m ! n,m =0 for ∞ � 1 α m y α 1 . . . y α n ¯ y ˙ α 1 . . . ¯ y ˙ α m A ( Y ) = 2 n ! m ! A α 1 ...α n , ˙ α 1 ... ˙ n,m =0 13

NonAbelian HS Algebra R ( Y | x ) = dω ( Y | x ) + ω ( Y | x ) ∗ ∧ ω ( Y | x ) ω 0 = 1 α ˙ β + 2 λh α ˙ ω ˙ 4 i ( ω αβ β β y α ¯ ω = ω 0 + ω 1 , 0 y α y β + ¯ 0 ¯ y ˙ α ¯ y ˙ y ˙ β ) R 0 = 0 , R 1 = D 0 ω 1 = dω 1 + [ ω 0 , ω 1 ] ∗ HS gauge transformation δω ( Y | x ) = Dǫ ( Y | x ) = dǫ ( Y | x ) + [ ω ( Y | x ) , ǫ ( Y | x )] ∗ • The simplest 4 d HS algebra hu (1 , 0 | 4) is the infinite-dimensional Lie algebra of even polynomials f ( − Y ) = f ( Y ) with star-commutator [ f , g ] ∗ as Lie product 14

• T νµ - generators of sp (4) ∼ (3 , 2) ⊂ hu (1 , 0 | 4) : bilinears of Y . Y µ independent generators correspond to spin one spin s generators are homogeneous Weyl symbols ω s ( νY | x ) = ν 2( s − 1) ω ( Y | x ) . hu (1 , 0 | 4) is a global symmetry algebra of the most symmetric vacuum solution of the nonlinear bosonic HS theory • HS algebras possess extensions to superalgebras hu ( n, m | 2 M ), ho ( n, m | 2 M husp (2 n, 2 m | 2 M ) with fermions and non-Abelian spin one YM gauge al- gebras u ( n ) ⊕ u ( m ), o ( n ) ⊕ o ( m ), usp (2 n ) ⊕ usp (2 m ) The construction of HS gauge symmetries is analogous Chan-Paton construction in String Theory Orthogonal and symplectic gauge symmetry result from the construc- tion analogous to orientifolds (Pradisi, Sagnotti) but in the space of auxiliary oscillators rather than in space-time 15

Properties of HS algebras Let T s 1 be homogeneous polynomial of degree 2( s − 1) [ T s 1 , T s 2 ] = T s 1 + s 2 − 2 m = T s 1 + s 2 − 2 + T s 1 + s 2 − 4 + . . . + T | s 1 − s 2 | +2 . Once a gauge field of spin s > 2 appears, the HS symmetry algebra requires an infinite tower of HS gauge fields together with gravity: [ T s , T s ] gives rise to generators T 2 s − 2 , of a gauge field of spin s ′ = 2 s − 2 > s and also gives rise to generators T 2 of o (3 , 2) ∼ sp (4) . The spin- 2 barrier separates theories with usual finite-dimensional lower- spin symmetries from those with infinite-dimensional HS symmetries. The maximal finite-dimensional subalgebra of hu (1 , 0 | 4) is: u (1) ⊕ o (3 , 2), where u (1) is associated with the unit element. Even spin generators T 2 p span a proper subalgebra ho (1 , 0 | 4). 16