The classical integrable structure of AdS/CFT Beno t Vicedo DESY, - PowerPoint PPT Presentation

Motivation Constrained IS GS superstring Lie dialgebra Conclusions The classical integrable structure of AdS/CFT Benoˆ ıt Vicedo DESY, Hamburg Cambridge, UK Thursday 24-th February, 2011

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Motivation λ classical AdS strings QISM S -matrix ? TBA perturb. gauge theory L TBA approach: Assumes integrability at finite λ, L . • At L ≫ 1, factorizability of the S -matrix � Fix 2-body S -matrix using Yangian symmetry (universal R -matrix?) • Zamolodchikov’s TBA trick � Ground state energy E 0 ( L ) . Claim: Excited states described by solutions of Y -system (boundary & analyticity conditions?). [Frolov-Arutyunov, Bombardelli-Tateo-Fioravanti, Gromov-Kazakov-Kozak-Vieira ’09] Need to prove integrability ∀ ( λ, L ) � QISM.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Green-Schwarz superstring on AdS -spaces Described by σ -model on semi-symmetric spaces [Zarembo ’10] Σ ≡ R × S 1 = super ( AdS n × Y 10 − n ) ≡ G / H − → Let g : Σ → G , A = − g − 1 dg ∈ Ω 1 (Σ , g ) , g = Lie G and impose • Global left G -action: under g �→ Ug , U ∈ G , have A �→ A . • Local right H -action: under g �→ gh , h : Σ → H have A �→ h − 1 Ah − h − 1 dh , hence A ( n ) ∈ g n . A ( 1 , 2 , 3 ) �→ h − 1 A ( 1 , 2 , 3 ) h , where Lagrangian: 2 � A ( 2 ) ∧ ∗ A ( 2 ) � 2 � A ( 1 ) ∧ A ( 3 ) � + � Λ , dA − A 2 � L GS := − 1 − 1 . � �� � � �� � � �� � Maurer − Cartan kinetic WZW

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Classical integrability (i) Equations of motion for L GS : Can be neatly written as zero-curvature equation dJ BPR ( z ) − J BPR ( z ) ∧ J BPR ( z ) = 0 , for the following Lax connection [Bena-Polchinski-Roiban ’03] − + z − 1 A ( 3 ) + A ( 0 ) + z A ( 1 ) + z 2 A ( 2 ) J BPR ( z ) := z − 2 A ( 2 ) + where z ∈ C and A ( 2 ) ± := A ( 2 ) ± ∗ A ( 2 ) . Integrals of motion: Ω( z , σ, τ ) := P ← − − � [ γ ( σ,τ )] J BPR ( z ) exp γ ( σ,τ ) ( σ ′ , τ ′ ) Ω( z , σ ′ , τ ′ ) = T ˜ γ ( z ) · Ω( z , σ, τ ) · T ˜ γ ( z ) − 1 , γ ˜ γ ( σ ′ ,τ ′ ) ∂ σ str Ω( z ) m = ∂ τ str Ω( z ) m = 0 . ( σ, τ ) ⇒

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Classical integrability (ii) Involution property: Integrability also requires { str Ω( z ) m , str Ω( z ′ ) n } = 0 , ∀ m , n ∈ N , z , z ′ ∈ C . Writing J BPR ( z ) = L BPR ( z ) d σ + M BPR ( z ) d τ , this would follow provided [Maillet ’85] � ? � L BPR ( z , σ ) ⊗ , L BPR ( z ′ , σ ′ ) = � r ( z , z ′ ) , L BPR ( z , σ ) ⊗ 1 + 1 ⊗ L BPR ( z ′ , σ ′ ) � δ ( σ, σ ′ ) � � s ( z , z ′ ) , L BPR ( z , σ ) ⊗ 1 − 1 ⊗ L BPR ( z ′ , σ ′ ) δ ( σ, σ ′ ) − − 2 s ( z , z ′ ) ∂ σ δ ( σ, σ ′ ) , for some r ( z , z ′ ) , s ( z , z ′ ) taking values in g ⊗ g . Problem: J BPR ( z ) does not have this property! � Rederive Lax connection within Hamiltonian formalism.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Constrained Hamiltonian systems Legendre transform: Given Lagrangian L ∈ C ( TM ) , define D L : TM → T ∗ M ≡ P , ( q , ˙ q ) �→ ( q , p = ∂ L /∂ ˙ q ) . If D L ( TM ) = Σ ⊂ P then L ∈ C ( TM ) � H = p ˙ q − L ∈ C (Σ) Σ ≡ { φ A ≈ 0 } ⊂ P Constraint surface: φ A = γ a , χ α {· , γ a } γ a : first class , { γ a , φ A } ≈ 0 χ α : second class φ A ≈ 0 d . o . f . = ( 2 n − m − 2 p ) − m � �� � dim Σ

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Extensions Lemma Let F , G ∈ C ( P ) be such that F ≈ G, i.e. F | Σ = G | Σ , then � f A φ A . F = G + A A f A φ A Extension: Given F ∈ C ( P ) , can ‘extend’ F � F + � If F is first class, i.e. { F , φ A } ≈ 0, then F � F + � a f a γ a e.g. : Extended Hamiltonian � � u a γ a u A , B φ A φ B H � H E ≡ H + + . ���� a A , B dyn. � �� � � �� � gauge tr. unphys.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Symmetry Moment map µ : P → R generates symmetry δ F = { F , µ } . It preserves Σ if δφ A = { φ A , µ } ≈ 0 , ∀ A ( ⋆ ) i.e. provided µ is first class. Can ensure this by µ � µ + � α m α χ α . Remark ( ⋆ ) is preserved by: {· , γ a } � m a γ a µ � µ + a � m A , B φ A φ B {· , µ } + φ A ≈ 0 A , B

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Constrained integrable systems n -dimensional constrained Hamiltonian system ( P , ω, H , φ A ) , has 2 ( n − p − m ) ≡ 2 k indep. phase-space variables. Definition ( P , ω, H , φ A ) is integrable if ∃ µ 1 , . . . , µ k ∈ C ( P ) s.t. { µ i , φ A } ≈ 0 , ∀ i , A ( I ) � � = 0 , ∀ i , j and d µ 1 ∧ . . . ∧ d µ k � = 0 ( II ) µ i , µ j and where H ≈ H ( µ ) , with µ = ( µ 1 , . . . , µ k ) : P → R k . Remark a m i , a γ a + . . . Strong equality forbids extensions µ i � µ i + � � Apply these ideas to the Green-Schwarz superstring

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Lagrangian to Hamiltonian GS superstring Start from the Lagrangian (dynamical variables A 0 , A 1 , Λ , h αβ ) √ − hh αβ str ( A ( 2 ) β ) + ǫ αβ str ( A ( 1 ) α A ( 2 ) α A ( 3 ) L GS = − 1 2 [ β )] + str Λ( ∂ 0 A 1 − ∇ 1 A 0 ) . Dirac’s consistency algorithm: p αβ ≈ 0. Primary constraints: Π 0 ≈ 0 , Π 1 ≈ Λ , Π Λ ≈ 0 , Π 1 − Λ is second class with Π Λ � Dirac bracket. Π 0 ≈ 0 ⇒ C ( 0 ) ≈ C ( 1 ) ≈ C ( 2 ) ≈ C ( 3 ) ≈ 0; Secondary constraints: ˙ and C ( 2 ) second class pair � Eliminate. Π ( 2 ) 0 p αβ ≈ 0 ⇒ T ± ≈ 0 (Virasoro constraints). ˙ C ( 0 , 1 , 3 ) ≈ ˙ T ± ≈ 0 ⇒ no new constraints. � Tertiary constraints: ˙ Partial gauge fixing conditions to eliminate Π ( 0 , 1 , 3 ) , p αβ . 0

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Hamiltonian GS superstring In summary, • Phase-space P parametrised by ( A ( 0 , 1 , 2 , 3 ) , Π ( 0 , 1 , 2 , 3 ) ) with 1 1 { A ( i ) 1 1 ( σ ) , Π ( 4 − i ) ( σ ′ ) } D.B. = C ( i 4 − i ) δ ( σ − σ ′ ) . 1 2 12 • Total set of constraints { Φ A } = { T ± , C ( 0 , 1 , 3 ) } . • First class constraints { Γ a } = {T ± , C ( 0 ) , K ( 1 , 3 ) } , where √ K ( 1 , 3 ) ≡ 2 T ± ≡ T ± ∓ str ( A ( 1 , 3 ) λ [ A ( 2 ) ± , i C ( 1 , 3 ) ] + . C ( 3 , 1 ) ) , 1 • Extended Hamiltonian − str ( k ( 3 ) K ( 1 ) ) − str ( k ( 1 ) K ( 3 ) ) H E = ρ + T + + ρ − T − − str ( µ ( 0 ) C ( 0 ) ) . � �� � � �� � � �� � κ − symmetry coset conformal tr.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Hamiltonian Lax connection Lax connection: Take a general linear combination L = A ( 0 ) + aA ( 1 ) + bA ( 2 ) + cA ( 3 ) 1 1 1 1 + ρ ( ∇ 1 Π 1 ) ( 0 ) + γ ( ∇ 1 Π 1 ) ( 1 ) + β ( ∇ 1 Π 1 ) ( 2 ) + α ( ∇ 1 Π 1 ) ( 3 ) , where a , b , c , α, β, γ, ρ ∈ C . Fix them by imposing: ( I ) str Ω( L ) j are first class, i.e. { str Ω( L ) j , Φ( σ ) } ≈ 0 , ∀ Φ( σ ) ∈ { T ± , C ( 0 , 1 , 3 ) } . ( II ) L is the σ -component of a strongly flat connection, i.e. { L , P 0 } = ∂ σ M + [ M , L ] , for some M , where P 0 is the energy. � Lax connection is then J ( z ) := L ( z ) d σ + M ( z ) d τ .

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Strong zero curvature equation • σ -translation is unambiguously generated on C ( P ) by P 1 ≡ T + − T − − str ( A ( 1 ) 1 C ( 3 ) ) − str ( A ( 3 ) 1 C ( 1 ) ) − str ( A ( 0 ) 1 C ( 0 ) ) , � in the sense that { F ( σ ) , P 1 } = ∂ σ F ( σ ) with P 1 = d σ ′ P 1 ( σ ′ ) . • Generator of τ -translations is defined only on C (Σ) by P 0 ≡ T + + T − − str ( A ( 1 ) 1 C ( 3 ) ) + str ( A ( 3 ) 1 C ( 1 ) ) − str ( A ( 0 ) ˜ 1 C ( 0 ) ) . We are free to extend this definition as follows P 0 � P 0 ≡ ˜ ˜ P 0 + str ( C ( 1 ) C ( 3 ) ) .

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Fixing parameters Total of six independent constraints on seven parameters { a , b , c , α, β, γ, ρ } . Solution depends on one parameter z ∈ C : ( I ) First class monodromy fixes 4 ( z − 3 + 3 z ) , 2 ( z − 1 − z 3 ) a = 1 α = 1 2 ( z − 2 + z 2 ) , 2 ( z − 2 − z 2 ) b = 1 β = 1 4 ( 3 z − 1 + z 3 ) , 2 ( z − 3 − z ) . c = 1 γ = 1 ( II ) Strong zero-curvature fixes 2 ( 1 − z 4 ) . ρ = 1

Motivation Constrained IS GS superstring Lie dialgebra Conclusions Hamiltonian Lax connection Final result reads [BV ’09] � C ( 0 ) + z − 3 C ( 1 ) + z − 1 C ( 3 ) � L ( z ) = L BPR ( z ) + 1 2 ( 1 − z 4 ) . This is an extension of the BPR Lax connection. Relation to pure spinors: Can also write � L ( z ) = L p . s . ( z ) 2 ( 1 − z 4 ) C ( 0 ) . � ghosts = 0 + 1 � Extension of the (matter part of) the pure spinor connection! In fact C ( 1 , 3 ) have second class parts, so L 0 ( z ) := L p . s . ( z ) | ghosts = 0 is a ‘Dirac’ extension of L BPR ( z ) . � Hint at a deeper connection between GS and PS? ...