On Liouville integrable defects Anastasia Doikou University of - PowerPoint PPT Presentation

On Liouville integrable defects Anastasia Doikou University of Patras Quantum Integrable Systems and Geometry Olhao, September 2012 Work in collaboration with J. Avan: arXiv:1110.4728 (JHEP 01 (2012) 040) and arXiv:1205.1661 Anastasia Doikou On Liouville integrable defects

General frame Integrable defects (quantum level) impose severe constraints on relevant algebraic and physical quantities (e.g. scattering amplitudes) ( Delfino, Mussardo, Simonetti, Konic, LeClair, ....) In discrete integrable systems there is a systematic description of local defects based on QISM In integrable field theories a defect is introduced as discontinuity plus gluing conditions ( Bowcock, Corrigan, Zambon,... ), integrability issue not systematically addressed; other attempts ( Caudrelier, Kundu, Habibulin,... ) We developed a systematic algebraic means to investigate integrable filed theories with point like defects. Integrability is ensured by construction Anastasia Doikou On Liouville integrable defects

Outline The general frame 1 The L matrix The classical quadratic algebra Local integrals of motion, and relevant Lax pairs for NLS and 2 sine-Gordon models Discrete theories and consistent continuum limits 3 Discussion and future perspectives 4 Anastasia Doikou On Liouville integrable defects

Classical Integrability The Lax pair U , V ; the linear auxiliary problem ( e.g. Faddeev-Takhtajan ): ∂ Ψ( x , t ) = U ( x , t ) Ψ( x , t ) ∂ x ∂ Ψ( x , t ) = V ( x , t ) Ψ( x , t ) ∂ t Compatibility condition leads to Zero curvature condition � � ˙ U ( x , t ) − V ′ ( x , t ) + U ( x , t ) , V ( x , t ) = 0 Gives rise to the equations of motion of the system. Anastasia Doikou On Liouville integrable defects

The monodromy matrix The continuum monodromy matrix � � y � T ( x , y , λ ) = P exp dx U ( x ) x Solution of the differential equation ∂ T ( x , y ) = U ( x , t ) T ( x , y ) ∂ x Anastasia Doikou On Liouville integrable defects

The monodromy matrix The continuum monodromy matrix � � y � T ( x , y , λ ) = P exp dx U ( x ) x Solution of the differential equation ∂ T ( x , y ) = U ( x , t ) T ( x , y ) ∂ x U obeys linear classical algebra, T satisfies the: Classical algebra � � � � T a ( λ ) , T b ( µ ) = r ab ( λ − µ ) , T a ( λ ) T b ( µ ) The classical r -matrix satisfies the CYBE ( Sklyanin, Semenov-Tian-Shansky ) [ r 12 , r 13 ] + [ r 12 , r 23 ] + [ r 13 , r 23 ] = 0 . Anastasia Doikou On Liouville integrable defects

Classical integrability The monodromy matrix T satisfies the classical algebra, thus The transfer matrix t ( λ ) = Tr T ( λ ) provides the charges in involution; � � t ( λ ) , t ( µ ) = 0 integrability ensured by construction. ln t ( λ ) → local integrals of motion Anastasia Doikou On Liouville integrable defects

The defect frame The key object , modified monodromy: Defect monodromy matrix T ( L , − L , λ ) = T + ( L , x 0 , λ ) ˜ L ( x 0 , λ ) T − ( x 0 , − L , λ ) where we define � � T ± = P exp � dx U ± ( x ) The defect ˜ L matrix obeys � � � � L a ( λ 1 ) , ˜ ˜ L b ( λ 2 ) = r ab ( λ 1 − λ 2 ) , L a ( λ 1 ) L b ( λ 2 ) T ± satisfy the classical algebra, thus T obeys the same algebra, integrability also ensured Anastasia Doikou On Liouville integrable defects

The defect frame Auxiliary linear problem for U ± , V ± for the defect theory: ∂ Ψ( x , t ) = U ± Ψ( x , t ) ∂ x ∂ Ψ( x , t ) = V ± Ψ( x , t ) ∂ t The corresponding Zero curvature condition � � U ± ( x , t ) − V ± ′ ( x , t ) + ˙ U ± ( x , t ) , V ± ( x , t ) = 0 x � = x 0 Anastasia Doikou On Liouville integrable defects

The defect frame Auxiliary linear problem for U ± , V ± for the defect theory: ∂ Ψ( x , t ) = U ± Ψ( x , t ) ∂ x ∂ Ψ( x , t ) = V ± Ψ( x , t ) ∂ t The corresponding Zero curvature condition � � U ± ( x , t ) − V ± ′ ( x , t ) + ˙ U ± ( x , t ) , V ± ( x , t ) = 0 x � = x 0 On the defect point Defect zero curvature condition d ˜ L ( x 0 ) = ˜ V + ( x 0 )˜ L ( x 0 ) − ˜ L ( x 0 )˜ V − ( x 0 ) dt Anastasia Doikou On Liouville integrable defects

The NLS model with defect The U ± -operator for the NLS model: ¯ � � � ψ ± � U ± = λ 1 0 0 + . ψ ± 0 − 1 0 2 From the classical algebra for U : Poisson structure � � � � ψ ± ( x ) , ¯ ψ ∓ ( x ) , ¯ ψ ± ( y ) ψ ± ( y ) = δ ( x − y ) , = 0 . The classical r -matrix is the Yangian: r ( λ ) = 1 λ P ( Yang ) P ( a ⊗ b ) = b ⊗ a . Anastasia Doikou On Liouville integrable defects

The NLS model with defect The generic defect ˜ L operator � α ( x 0 ) � β ( x 0 ) ˜ L ( x 0 ) = λ I + . γ ( x 0 ) δ ( x 0 ) From the quadratic classical algebra for ˜ L ( sl 2 algebra): � � α ( x 0 ) , β ( x 0 ) = β ( x 0 ) � � α ( x 0 ) , γ ( x 0 ) = − γ ( x 0 ) � � β ( x 0 ) , γ ( x 0 ) = 2 α ( x 0 ) Establish the Poisson structure! Relevant studies: ( Corrigan-Zambon ) Anastasia Doikou On Liouville integrable defects

The NLS model: local IM First recall that: ∂ T ± ( x , y , t ) = U ± ( x , t ) T ± ( x , y , t ) ∂ x Based on the latter consider the decomposition ansatz: T ± ( x , y ; λ ) = (1 + W ± ( x )) e Z ± ( x , y ) (1 + W ± ( y )) − 1 W ± anti-diagonal, Z ± diagonal. Also, ∞ ∞ W ± ( n ) Z ± ( n ) W ± = Z ± = � � , λ n λ n n =0 n = − 1 Substituting the ansatz to the differential equation above identify W ± ( n ) , Z ± ( n ) matrices. Anastasia Doikou On Liouville integrable defects

The NLS model: local IM Substitution leads to Riccati-type : Differential equations dW ± + W ± U d − U d W ± + W ± U ± a W ± − U ± a = 0 dx dZ ± = U d + U ± a W ± dx Solving the latter one identifies the W ± ( n ) , Z ± ( n ) , hence the charges in involution. Similar differential equations arise within the inverse scattering frame. Anastasia Doikou On Liouville integrable defects

The NLS model: local IM The generating function G ( λ ) = ln tr ( T + ( λ ) ˜ L ( λ, x 0 ) T − ( λ )) which turns to, via the decomposition: Generating function 11 ( λ ) + ln[(1 + W + ( x 0 )) − 1 ˜ G ( λ ) = Z + 11 ( λ ) + Z − L ( x 0 )(1 + W − ( x 0 ))] 11 Also, ∞ H ( n ) � G ( λ ) = λ n n =0 Anastasia Doikou On Liouville integrable defects

The NLS model: local IM The first three integrals of motion: The number of particles � x − � L 0 H (1) = dx ψ − ( x ) ¯ dx ψ + ( x ) ¯ ψ − ( x ) + ψ + ( x ) + α ( x 0 ) x + − L 0 The momentum � x − � L 0 ψ − ( x ) ψ − ′ ( x ) − ψ + ( x ) ψ + ′ ( x ) dx ¯ dx ¯ H (2) = − x + − L 0 ψ + + βψ − − α 2 ψ + ψ + + ¯ ψ + ψ − + γ ¯ ¯ − 2 Anastasia Doikou On Liouville integrable defects

The NLS model: local IM The Hamiltonian � L � x − ψ + ψ + ′′ + | ψ + | 4 � ψ − ψ − ′′ + | ψ − | 4 � 0 � � ¯ ¯ H (3) = dx + dx x + − L 0 ψ + ′ ψ − + α 3 ψ + ′ − βψ − ′ + ¯ ψ + ψ + ) ′ + γ ¯ ( ¯ + 3 ψ + ψ − ′ − α � ψ + + βψ − + 2 ¯ ψ + ψ − � ¯ γ ¯ − Anastasia Doikou On Liouville integrable defects

The NLS model: local IM The Hamiltonian � L � x − ψ + ψ + ′′ + | ψ + | 4 � ψ − ψ − ′′ + | ψ − | 4 � 0 � � ¯ ¯ H (3) = dx + dx x + − L 0 ψ + ′ ψ − + α 3 ψ + ′ − βψ − ′ + ¯ ψ + ψ + ) ′ + γ ¯ ( ¯ + 3 ψ + ψ − ′ − α � ψ + + βψ − + 2 ¯ ψ + ψ − � ¯ γ ¯ − By construction (formally), and also explicitly checked: Involution � � � � � � H 1 , H 2 = H 1 , H 3 = H 2 , H 3 = 0 No sewing constraints arise or used so far, off-shell integrability. Anastasia Doikou On Liouville integrable defects

The NLS model: Lax pair Next step, derive time component of the Lax pair V , and sewing conditions. Explicit expressions ( Faddeev-Takhtajan, Avan-Doikou ): � � a ( x , x 0 )˜ V + ( x , λ, µ ) = t − 1 tr a T + a ( L , x ) r ab ( λ − µ ) T + L a ( x 0 ) T − a ( x 0 , − L ) � � a ( L , x 0 )˜ V − ( x , λ, µ ) = t − 1 tr a T + L a ( x 0 ) T − a ( x 0 , x ) r ab ( λ − µ ) T − a ( x , − L ) � � ˜ a ( L , x 0 ) r ab ( λ − µ )˜ V + ( x 0 , λ, µ ) = t − 1 tr a T + L a ( x 0 ) T − a ( x 0 , − L ) � � ˜ a ( L , x 0 )˜ V − ( x 0 , λ, µ ) = t − 1 tr a T + L a ( x 0 ) r ab ( λ − µ ) T − a ( x 0 , − L ) . Anastasia Doikou On Liouville integrable defects

The NLS model: Lax pair For the left and right bulk theories, and the defect point: � 1 � 0 V ± (1) ( µ, x ) = 0 0 ¯ � ψ − ( x ) � µ V − (2) ( µ, x ) = ψ − ( x ) 0 ¯ � ψ + ( x ) � µ V +(2) ( µ, x ) = ψ + ( x ) 0 ¯ � ψ + ( x 0 ) + β ( x 0 ) � µ ˜ V − (2) ( µ, x 0 ) = ψ − ( x 0 ) 0 ¯ � ψ + ( x 0 ) � µ ˜ V +(2) ( µ, x 0 ) = γ ( x 0 ) + ψ − ( x 0 ) 0 Anastasia Doikou On Liouville integrable defects