Integrable spin chains with U(1) 3 symmetry Lisa Freyhult Helsinki - PowerPoint PPT Presentation

Deformations in AdS/CFT Integrable spin chains with U(1) 3 symmetry Lisa Freyhult Helsinki 28/10 2005 freyhult@nordita.dk Deformations in AdS/CFT – p.1/30

Plan • Introduction • β -deformation and generalisations in gauge theory • Corresponding deformations in string theory • Integrability, factorized scattering and the coordinate Bethe ansatz • Yang-Baxter equation and results for integrability • Conclusion and Outlook Based on work with C. Kristjansen and T. Månsson [hep-th/0510221] [Staudacher hep-th/0412188] [Berenstein, Cherkis hep-th/0405215] [Lunin, Maldacena hep-th/0502086] [Frolov hep-th/0503201] [Frolov, Tseytlin, Roiban hep-th/0503192,0507021] [Beisert, Staudacher hep-th/0504190] [Beisert, Roiban hep-th/0505187] Deformations in AdS/CFT – p.2/30

Introduction Using integrability to study the AdS/CFT duality has been a very succesful approach • Scaling dimension of long operators found by diagonalising the Dilatation operator using the Bethe ansatz. • Agrees with the energy of semiclassical spinning strings up to 3 loops. • Agreement on the level of actions, etc. • Succes largely due to integrability Deformations in AdS/CFT – p.3/30

Introduction Gauge-string duality for less supersymmetry? Marginal deformations of N = 4 with deformation parameter β , also called β -deformations [Leigh, Strassler] ⇔ Strings in the Lunin-Maldacena background AdS 5 × S 5 [Lunin, Maldacena] β Possible to define semiclassical strings on this background, string energies typically of the form λ ′ = λ 1 + λ ′ ( e 1 + e 2 ( βJ ) + e 3 ( βJ ) 2 ) + O ( λ ′ 2 ) ` ´ E = J J 2 Also: Extension to three deformation parameters β 1 , β 2 and β 3 . [Frolov] [Beisert, Roiban] Parameters are allowed to be complex. Deformations in AdS/CFT – p.4/30

The Lunin-Maldacena background Obtained by deforming the string sigma model in AdS 5 × S 5 by making a TsT transformation. Sigma model on S 5 : √ Z dσ Z λ “ ” γ αβ ∂ α r i ∂ β r i + r 2 i ∂ α φ i ∂ β φ i + Λ( r 2 S S 5 = − dτ i − 1) 2 2 π Original proposal: Change of variables φ 1 = ϕ 3 − ϕ 2 , φ 2 = ϕ 1 + ϕ 2 + ϕ 3 , φ 3 = ϕ 3 − ϕ 1 1) T-duality on circle parametrised by ϕ 1 2) Shift ϕ 2 → ϕ 2 + ˆ γϕ 1 3) T-duality on circle parametrised by ϕ 1 √ Z dσ »„ « Z λ X X ∂ α r i ∂ β r i + Gr 2 γ 2 Gr 2 1 r 2 2 r 2 2 π γ αβ S = − dτ i ∂ α φ i ∂ β φ i + ˆ ∂ α φ i ∂ β φ j 3 2 i j – γGǫ αβ ( r 2 1 r 2 2 ∂ α φ 1 ∂ β φ 2 + r 2 2 r 2 3 ∂ α φ 2 ∂ β φ 3 + r 2 3 r 2 1 ∂ α φ 3 ∂ β φ 1 ) + Λ( r 2 − 2ˆ i − 1) G − 1 = 1 + ˆ γ 2 ( r 2 1 r 2 2 + r 2 1 r 2 3 + r 2 2 r 2 3 ) Deformations in AdS/CFT – p.5/30

The Lunin-Maldacena background This can be generalized to generate a three parameter deformation. Apply a sequence of TsT dualities: • TsT on ( φ 1 , φ 2 ) T-duality on φ 1 and shift by ˆ γ 3 on φ 2 • TsT on ( φ 2 , φ 3 ) T-duality on φ 2 and shift by ˆ γ 1 on φ 3 • TsT on ( φ 3 , φ 1 ) T-duality on φ 3 and shift by ˆ γ 2 on φ 1 The dual background for complex parameters, β i = ˆ γ i + i ˆ σ i , is found by per- forming SL (2 , R ) transformations. I.e. a sequence of S σ Ts γ TS − 1 gives the σ three complex parameter background Deformations in AdS/CFT – p.6/30

β -deformed N = 4 SYM Superpotential in N = 4 W N =4 = Tr (Φ 1 Φ 2 Φ 3 − Φ 1 Φ 3 Φ 3 ) Two exactly marginal deformations in N = 4 W def = Tr ( e iπβ Φ 1 Φ 2 Φ 3 − e − iπβ Φ 1 Φ 3 Φ 3 ) + h ′ Tr (Φ 3 1 + Φ 3 2 + Φ 3 3 ) The resulting theory is N = 1 supersymmetric and conformal. Set h ′ = 0 . Deformations in AdS/CFT – p.7/30

β -deformed N = 4 SYM In terms of component fields „ | e iπβ Φ 1 Φ 2 − e − iπβ Φ 2 Φ 1 | 2 + | e iπβ Φ 2 Φ 3 − e − iπβ Φ 3 Φ 2 | 2 V = Tr « Φ 1 ] 2 + [Φ 2 , ¯ Φ 2 ] 2 + [Φ 3 , ¯ + | e iπβ Φ 3 Φ 1 − e − iπβ Φ 1 Φ 3 | 2 [Φ 1 , ¯ Φ 3 ] 2 ´ ` + Tr Introduce a more general deformation 3 (6) parameter deformation „ Φ 1 Φ 2 − e − iπβ 1 Φ 2 Φ 1 | 2 + | e iπβ 2 Φ 2 Φ 3 − e − iπβ 2 Φ 3 Φ 2 | 2 | e iπβ 1 V = Tr 1 « Φ 1 ] 2 + [Φ 2 , ¯ Φ 2 ] 2 + [Φ 3 , ¯ + | e iπβ 3 Φ 3 Φ 1 − e − iπβ 3 Φ 1 Φ 3 | 2 [Φ 1 , ¯ Φ 3 ] 2 ´ ` + Tr β i ∈ C This deformation is not supersymmetric but conformal. Deformations in AdS/CFT – p.8/30

Dilatation operator in the deformed theory Consider operators in N = 4 of the form O ( x ) = Tr ( X J 1 Y J 2 Z J 3 + . . . ) X , Y , Z chiral scalars Dilatation operator associated with su (3) nearest neighbour ferromagnetic spin chain. J J λ λ X X D = (1 k,k +1 − P k,k +1 ) H k,k +1 = 8 π 2 8 π 2 k =1 k =1 [Minahan, Zarembo]. This is generalized to the full theory giving the dilatation operator in psu (2 , 2 | 4) . Higher loops introduce interactions beyond nearest neighbours. We can write the su (3) hamiltonian in terms of the generators E ij | k � = δ jk | i � Deformations in AdS/CFT – p.9/30

Dilatation operator The su (3) sector in N = 4 H su (3) 00 E k +1 11 E k +1 00 E k +1 22 E k +1 11 E k +1 22 E k +1 k,k +1 = E k + E k + E k + E k + E k + E k 11 00 22 00 22 11 − E k 12 E k +1 − E k 21 E k +1 − E k 10 E k +1 − E k 01 E k +1 − E k 20 E k +1 − E k 02 E k +1 21 12 01 10 02 20 On matrix form 0 1 0 0 0 0 0 0 0 0 0 B C − 1 0 1 0 0 0 0 0 0 B C B C B C − 1 0 0 1 0 0 0 0 0 B C B C B C 0 − 1 0 1 0 0 0 0 0 B C B C H su (3) = B C 0 0 0 0 0 0 0 0 0 B C B C B C − 1 0 0 0 0 0 1 0 0 B C B C B C − 1 0 0 0 0 0 1 0 0 B C B C B − 1 C 0 0 0 0 0 0 1 0 B C @ A 0 0 0 0 0 0 0 0 0 Deformations in AdS/CFT – p.10/30

The deformed dilatation operator Use the notation q i = e iπβ i = r i e iγ i . H su (3) 00 E k +1 + r 2 11 E k +1 + r 2 00 E k +1 22 E k +1 11 E k +1 + r 2 22 E k +1 k,k +1 = E k 2 E k 3 E k + E k + E k 1 E k 11 00 22 00 22 11 − r 1 e − iγ 1 E k 12 E k +1 21 E k +1 10 E k +1 − r 2 e − iγ 2 E k 01 E k +1 − r 1 e iγ 1 E k − r 2 e iγ 2 E k 21 12 01 10 − r 3 e − iγ 3 E k 20 E k +1 02 E k +1 − r 3 e iγ 3 E k 02 20 0 1 0 0 0 0 0 0 0 0 0 B C r 3 e − iγ 3 B C 0 1 0 0 0 0 0 0 B C B C r 2 r 2 e iγ 2 0 0 0 0 0 0 0 B C 3 B C B C r 2 r 3 e iγ 3 0 0 0 0 0 0 0 B C 2 B C H su (3) = B C 0 0 0 0 0 0 0 0 0 B C B C B r 1 e − iγ 1 C 0 0 0 0 0 1 0 0 B C B C r 2 e − iγ 2 B 0 0 0 0 0 1 0 0 C B C B C r 1 e iγ 1 r 2 0 0 0 0 0 0 0 B C 1 @ A 0 0 0 0 0 0 0 0 0 Deformations in AdS/CFT – p.11/30

Integrability? Is the model integrable? • su (2) : Yes, always! [Berenstein, Cherkis] • su (3) : Yes, when r i = 1 [Beisert, Roiban] No, when r 1 = r 2 = r 3 = r , γ 1 = γ 2 = γ 3 = γ [Berenstein, Cherkis] Maybe not when r i � = 1 , γ 1 � = γ 2 � = γ 3 . . . Investigate this! More general: Any Hamiltonian with U (1) 3 symmetry H k,k +1 H 00 00 E k 00 E k +1 + H 11 11 E k 11 E k +1 + H 22 22 E k 22 E k +1 + H 12 12 E k 11 E k +1 + H 21 12 E k 12 E k +1 = 00 11 22 22 21 H 12 21 E k 21 E k +1 + H 21 21 E k 22 E k +1 + H 01 10 E k 10 E k +1 + H 10 10 E k 11 E k +1 + H 01 01 E k 00 E k +1 + 12 11 01 00 11 H 10 01 E k 01 E k +1 + H 02 20 E k 20 E k +1 + H 20 20 E 22 E 00 + H 02 02 E 00 E 22 + H 20 + 02 E 02 E 20 , 10 02 Deformations in AdS/CFT – p.12/30

A general Hamiltonian 0 1 H 00 0 0 0 0 0 0 0 0 00 B C H 01 H 01 B 0 0 0 0 0 0 0 C 01 10 B C B C H 02 H 02 0 0 0 0 0 0 0 B C 02 20 B C B C H 10 H 10 0 0 0 0 0 0 0 B C 01 10 B C B C H 11 H = . 0 0 0 0 0 0 0 0 B C 11 B C H 12 H 12 B C 0 0 0 0 0 0 0 B C 12 21 B C H 20 H 20 B C 0 0 0 0 0 0 0 02 20 B C B C H 21 H 21 0 0 0 0 0 0 0 B C 12 21 @ A H 22 0 0 0 0 0 0 0 0 22 21 ) ∗ = r 1 e iγ 1 , H 02 02 ) ∗ = r 2 e iγ 2 , H 10 Require hermiticity: H 21 12 = ( H 12 20 = ( H 20 01 = 10 ) ∗ = r 3 e iγ 3 , diagonal terms real. Not all parameters are physical, we are ( H 01 allowed to rescale and add/subtract number operators. ⇒ 9 physical parameters When is this model integrable? Deformations in AdS/CFT – p.13/30

Investigating integrability Integrability ⇔ Factorized scattering Consider an N particle process: scattering occurs as a sequence of two-particle scatterings, in the case of 3 particles: k2 k3 k1 j3 j2 j1 i3 i1 i2 Alternative more technical definition: Existence of an R-matrix that satisfies the Yang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 leads to an infinite number of commuting charges. Deformations in AdS/CFT – p.14/30

Investigating integrability Yang-Baxter equation represented graphically k2 k2 k3 k1 k1 j3 k3 j1 j2 X X = j2 j1 i3 j 1 ,j 2 ,j 3 j 1 ,j 2 ,j 3 j3 i1 i3 i1 i2 i2 R j 1 j 2 i 1 i 2 ( u − v ) R k 1 j 3 j 1 i 3 ( u ) R k 2 k 3 j 2 j 3 ( v ) = R j 2 j 3 i 2 i 3 ( v ) R j 1 k 3 i 1 j 3 ( u ) R k 1 k 2 j 1 j 2 ( u − v ) Factorized scattering leads to a similar relation for the S-matrix S j 1 j 2 i 1 i 2 ( p i 1 , p i 2 ) S k 1 j 3 j 1 i 3 ( p i 1 , p i 3 ) S k 2 k 3 j 2 j 3 ( p i 2 , p i 3 ) = S j 2 j 3 i 2 i 3 ( p i 1 , p i 2 ) S j 1 k 3 i 1 j 3 ( p i 1 , p i 3 ) S k 1 k 2 j 1 j 2 ( p i 2 , p i 3 ) Deformations in AdS/CFT – p.15/30