Coset sigma models and Green-Schwarz strings Green-Schwarz string Coset sigma model D ( 2, 1; α ) × D ( 2, 1; α ) IIB on AdS 3 × S 3 × S 3 × S 1 ↔ SL ( 2 ) × SU ( 2 ) × SU ( 2 ) × U ( 1 ) PSU ( 1, 1 | 2 ) × PSU ( 1, 1 | 2 ) IIB on AdS 3 × S 3 × T 4 × U ( 1 ) 4 ↔ SL ( 2 ) × SU ( 2 ) Coset backgrounds supported by pure RR flux [Babichenko, Stefański, Zarembo ’09]
Coset sigma models and Green-Schwarz strings Green-Schwarz string Coset sigma model D ( 2, 1; α ) × D ( 2, 1; α ) IIB on AdS 3 × S 3 × S 3 × S 1 ↔ SL ( 2 ) × SU ( 2 ) × SU ( 2 ) × U ( 1 ) PSU ( 1, 1 | 2 ) × PSU ( 1, 1 | 2 ) IIB on AdS 3 × S 3 × T 4 × U ( 1 ) 4 ↔ SL ( 2 ) × SU ( 2 ) Coset backgrounds supported by pure RR flux [Babichenko, Stefański, Zarembo ’09] Include NSNS flux by adding WZ term � � 2 � k Str 3 J 2 ∧ J 2 ∧ J 2 + J 1 ∧ J 3 ∧ J 2 + J 3 ∧ J 1 ∧ J 2 WZ term breaks Z 4 symmetry but a Lax connection can still be constructed [Cagnazzo, Zarembo ’12]
String theory in uniform light-cone gauge
String theory on AdS 3 × S 3 × T 4 × × • Consider strings in AdS 3 × S 3 × T 4 supported by pure RR flux • Fix light-cone gauge • 8 + 8 physical world-sheet excitation • World-sheet integrability: • Dispersion relation for fundamental excitations • Two-particle S matrix • S matrix defined on a non-compact world-sheet
String theory on AdS 3 × S 3 × T 4 × × • Isometries: PSU ( 1, 1 | 2 ) × PSU ( 1, 1 | 2 ) × U ( 1 ) 4 SU ( 2 ) • × SU ( 2 ) ◦ • Bosonic subgroup SO ( 2, 2 ) × SO ( 4 ) × U ( 1 ) 4
String theory on AdS 3 × S 3 × T 4 × × • Isometries in the decompactification limit: PSU ( 1, 1 | 2 ) × PSU ( 1, 1 | 2 ) × U ( 1 ) × SO ( 4 ) SU ( 2 ) • × SU ( 2 ) ◦ • Bosonic subgroup SO ( 2, 2 ) × SO ( 4 ) × U ( 1 ) × SO ( 4 )
Light-cone gauge Equator of S 3 AdS 3 time X + = φ + t = τ Fix light-cone gauge:
Light-cone gauge Equator of S 3 AdS 3 time X + = φ + t = τ Fix light-cone gauge: World-sheet Hamiltonian: H = E − J Angular momentum on S 3 AdS 3 energy
Light-cone gauge X + = φ + t = τ Fix light-cone gauge: World-sheet Hamiltonian: H = E − J BMN-like ground state on AdS 3 × S 3 Not compatible with coset kappa gauge – use GS string
Light-cone gauge X + = φ + t = τ Fix light-cone gauge: World-sheet Hamiltonian: H = E − J BMN-like ground state on AdS 3 × S 3 Not compatible with coset kappa gauge – use GS string Ground state preserves 8 supersymmetries 8+8 fluctuations: m B = 4 × { 0, 1 } m F = 4 × { 0, 1 }
Light-cone gauge X + = φ + t = τ Fix light-cone gauge: World-sheet Hamiltonian: H = E − J BMN-like ground state on AdS 3 × S 3 Not compatible with coset kappa gauge – use GS string Ground state preserves 8 supersymmetries 8+8 fluctuations: m B = 4 × { 0, 1 } m F = 4 × { 0, 1 } Note massless modes
“Off-shell” symmetries • Physical states satisfy level matching: P | p 1 , . . . , p n � = ( p 1 + · · · + p n ) | p 1 , . . . , p n � = 0 • “Off-shell” states have: P | p 1 , . . . , p n � � = 0
“Off-shell” symmetries • Physical states satisfy level matching: P | p 1 , . . . , p n � = ( p 1 + · · · + p n ) | p 1 , . . . , p n � = 0 • “Off-shell” states have: P | p 1 , . . . , p n � � = 0 • Not all isometries are manifest in light-cone gauge • Construct off-shell symmetry algebra A of generators J that 1 Commute with the gauge-fixed Hamiltonian [ H , J ] = 0 2 Act on generic off-shell states • World-sheet supercurrents constructed to quartic order [Borsato, OOS, Sfondrini, Stefański, Torrielli ’14] [Lloyd, OOS, Sfondrini, Stefański ’14] • For on-shell states A ⊂ psu ( 1, 1 | 2 ) 2 × so ( 4 )
Light-cone gauge symmetry algebra Light-cone gauge breaks isometries to psu ( 1 | 1 ) 4 c.e. × so ( 4 ) 8 supercharges
Light-cone gauge symmetry algebra Light-cone gauge breaks isometries to psu ( 1 | 1 ) 4 c.e. × so ( 4 ) The on-shell algebra P = 0 Q L a , Q b = 1 2 δ b Q L a , Q b � � � � � � H + M = 0 L a R a Q a = 1 2 δ a � � � � � � R , Q R b H − M Q L , Q R b = 0 b su ( 2 ) • ⊂ so ( 4 ) indices
Light-cone gauge symmetry algebra Light-cone gauge breaks isometries to psu ( 1 | 1 ) 4 c.e. × so ( 4 ) The off-shell algebra P � = 0 [David, Sahoo ’10] [Borsato, OOS, Sfondrini, Stefański, Torrielli ’13-’14] Q L a , Q b = 1 2 δ b Q L a , Q b = δ b � � � � � � H + M a C L a R a Q a = 1 2 δ a = δ a � � � � � � R , Q R b H − M Q L , Q R b b C b Two additional central charges
Light-cone gauge symmetry algebra Light-cone gauge breaks isometries to psu ( 1 | 1 ) 4 c.e. × so ( 4 ) The off-shell algebra P � = 0 [David, Sahoo ’10] [Borsato, OOS, Sfondrini, Stefański, Torrielli ’13-’14] Q L a , Q b = 1 2 δ b Q L a , Q b = δ b � � � � � � H + M a C L a R a Q a = 1 2 δ a = δ a � � � � � � R , Q R b H − M Q L , Q R b b C b Central charge e i P − 1 C = i � � 2 h ( λ ) Coupling constant √ √ λ h ( λ ) = 2 π + O ( 1 / λ )
Light-cone gauge symmetry algebra Light-cone gauge breaks isometries to psu ( 1 | 1 ) 4 c.e. × so ( 4 ) The off-shell algebra P � = 0 [David, Sahoo ’10] [Borsato, OOS, Sfondrini, Stefański, Torrielli ’13-’14] Q L a , Q b = 1 2 δ b Q L a , Q b = δ b � � � � � � H + M a C L a R a Q a = 1 2 δ a = δ a � � � � � � R , Q R b H − M Q L , Q R b b C b Central charge e i P − 1 C = i � � 2 h ( λ ) Non-trivial coproduct � � � # C ⊗ 1 + # 1 ⊗ C | p 1 p 2 � C | p 1 p 2 � = e i ( p 1 + p 2 ) − 1 ih � � | p 1 p 2 � 2
Light-cone gauge symmetry algebra Light-cone gauge breaks isometries to psu ( 1 | 1 ) 4 c.e. × so ( 4 ) The off-shell algebra P � = 0 [David, Sahoo ’10] [Borsato, OOS, Sfondrini, Stefański, Torrielli ’13-’14] Q L a , Q b = 1 2 δ b Q L a , Q b = δ b � � � � � � H + M a C L a R a Q a = 1 2 δ a = δ a � � � � � � R , Q R b H − M Q L , Q R b b C b Central charge e i P − 1 C = i � � 2 h ( λ ) Non-trivial coproduct � � C ⊗ 1 + e ip 1 1 ⊗ C � | p 1 p 2 � C | p 1 p 2 � = e i ( p 1 + p 2 ) − 1 ih � � | p 1 p 2 � 2
Representations Particles transform in short representations H 2 = M 2 + 4 CC 2 ( e i P − 1 ) gives the dispersion relation Central charge C = ih � m 2 + 4 h 2 sin 2 p E p = 2
Representations Particles transform in short representations H 2 = M 2 + 4 CC 2 ( e i P − 1 ) gives the dispersion relation Central charge C = ih � m 2 + 4 h 2 sin 2 p � sin p � � m → 0 E p = − − − → E p = 2 h � � 2 2 �
Representations Particles transform in short representations H 2 = M 2 + 4 CC 2 ( e i P − 1 ) gives the dispersion relation Central charge C = ih � m 2 + 4 h 2 sin 2 p � sin p � � m → 0 E p = − − − → E p = 2 h � � 2 2 � Two massive + two massless psu ( 1 | 1 ) 4 c.e. multiplets | χ 1 � | χ 2 � | Y L � | Z R � | η L1 � | η L2 � | η R1 � | η R2 � | T 11 � | T 21 � | T 12 � | T 22 � | Z L � | Y R � χ 1 � χ 2 � | ˜ | ˜ m =+ 1 m =− 1 m = 0 m = 0
Representations Particles transform in short representations H 2 = M 2 + 4 CC 2 ( e i P − 1 ) gives the dispersion relation Central charge C = ih � m 2 + 4 h 2 sin 2 p � sin p � � m → 0 E p = − − − → E p = 2 h � � 2 2 � Two massive + two massless psu ( 1 | 1 ) 4 c.e. multiplets | χ 1 � | χ 2 � | Y L � | Z R � | η L1 � | η L2 � | η R1 � | η R2 � | T 11 � | T 21 � | T 12 � | T 22 � | Z L � | Y R � χ 1 � χ 2 � | ˜ | ˜ m =+ 1 m =− 1 Doublet under su ( 2 ) ◦ ⊂ so ( 4 )
Properties of the S matrix ∆ ( J ) S • Symmetry invariance = S ∆ ( J ) S S † S = 1 • Unitarity = S S S • Yang-Baxter equation = S S S S
The two-particle S matrix Find S matrix by imposing off-shell symmetry [ ∆ ( Q ) , S ] = 0 Non-trivial coproduct
The two-particle S matrix Find S matrix by imposing off-shell symmetry [ ∆ ( Q ) , S ] = 0 Non-trivial coproduct Unique matrix for each pair of representations Four undetermined coefficients – “dressing phases” σ 2 σ 2 σ 2 σ 2 ˜ •◦ ◦◦
The two-particle S matrix Find S matrix by imposing off-shell symmetry [ ∆ ( Q ) , S ] = 0 Non-trivial coproduct Unique matrix for each pair of representations Four undetermined coefficients – “dressing phases” σ 2 σ 2 σ 2 σ 2 ˜ •◦ ◦◦ Scattering of excitations with m = + 1 and m = + 1
The two-particle S matrix Find S matrix by imposing off-shell symmetry [ ∆ ( Q ) , S ] = 0 Non-trivial coproduct Unique matrix for each pair of representations Four undetermined coefficients – “dressing phases” σ 2 σ 2 σ 2 σ 2 ˜ •◦ ◦◦ Scattering of excitations with m = + 1 and m = − 1
The two-particle S matrix Find S matrix by imposing off-shell symmetry [ ∆ ( Q ) , S ] = 0 Non-trivial coproduct Unique matrix for each pair of representations Four undetermined coefficients – “dressing phases” σ 2 σ 2 σ 2 σ 2 ˜ •◦ ◦◦ Scattering of excitations with m = + 1 and m = 0
The two-particle S matrix Find S matrix by imposing off-shell symmetry [ ∆ ( Q ) , S ] = 0 Non-trivial coproduct Unique matrix for each pair of representations Four undetermined coefficients – “dressing phases” σ 2 σ 2 σ 2 σ 2 ˜ •◦ ◦◦ Scattering of excitations with m = 0 and m = 0
The two-particle S matrix Find S matrix by imposing off-shell symmetry [ ∆ ( Q ) , S ] = 0 Non-trivial coproduct Unique matrix for each pair of representations Four undetermined coefficients – “dressing phases” σ 2 σ 2 σ 2 σ 2 ˜ •◦ ◦◦ Phases satisfy crossing equations S matrix exact to all orders in h ( λ )
Massless S matrix • In a relativistic theory scattering of massless modes is problematic v = ∂ E ∂ p = ± 1 • Here there is no Lorentz invariance and the “massless” modes have a non-linear dispersion relation v = ∂ E ∂ p = ± h cos p 2
Massless S matrix • In a relativistic theory scattering of massless modes is problematic v = ∂ E ∂ p = ± 1 • Here there is no Lorentz invariance and the “massless” modes have a non-linear dispersion relation v = ∂ E ∂ p = ± h cos p 2 • Massless modes form doublet under su ( 2 ) ◦ – extra su ( 2 ) S matrix S su ( 2 ) = 1 + i ( w p − w q ) Π Unknown function of momentum
Mixed flux background • AdS 3 × S 3 × T 4 supported by RR+NSNS three-form flux � � � � F = ˜ q Vol AdS 3 + Vol S 3 H = q Vol AdS 3 + Vol S 3 q 2 + q 2 = 1 • Coefficients related by ˜ • Quantised WZW level √ Q NS5 = 2 π/ k = q λ ∈ Z • Dispersion relation � q 2 h 2 sin 2 p kp ) 2 + 4˜ ( m + / E p = 2
Mixed flux background • AdS 3 × S 3 × T 4 supported by RR+NSNS three-form flux � � � � F = ˜ q Vol AdS 3 + Vol S 3 H = q Vol AdS 3 + Vol S 3 q 2 + q 2 = 1 • Coefficients related by ˜ • Quantised WZW level √ Q NS5 = 2 π/ k = q λ ∈ Z • Dispersion relation � q 2 h 2 sin 2 p kp ) 2 + 4˜ ( m + / E p = 2 Momentum-dependent “mass” Rescaled coupling k ∼ Q NS5 / ˜ qh ∼ g s Q D5 + · · ·
Mixed flux background • AdS 3 × S 3 × T 4 supported by RR+NSNS three-form flux � � � � F = ˜ q Vol AdS 3 + Vol S 3 H = q Vol AdS 3 + Vol S 3 q 2 + q 2 = 1 • Coefficients related by ˜ • Quantised WZW level √ Q NS5 = 2 π/ k = q λ ∈ Z • Dispersion relation � q 2 h 2 sin 2 p kp ) 2 + 4˜ ( m + / E p = 2 • S matrix takes the same functional form of p and E p for any q [Hoare, Tseytlin ’13] [Lloyd, OOS, Stefański, Sfondrini ’14]
Bethe ansatz and the spin-chain picture
Bethe ansatz equations • Impose periodic boundary conditions e ip k L = � S ( p k , p j ) j � = k • Non-diagonal S matrix − → nested Bethe equations • 3 types of momentum-carrying roots • 3 types of auxiliary roots • Simplifies in the weak coupling limit h ( λ ) → 0
Massive sector At weak coupling • Two decoupled PSU ( 1, 1 | 2 ) spin-chains • The two spin-chains couple through level matching e ip total = 1 ×
Massive sector At weak coupling • Two decoupled PSU ( 1, 1 | 2 ) spin-chains • The two spin-chains couple through level matching e ip total = 1 Higher orders • Sites in the ( 1 2 ; 1 2 ) L ⊗ ( 1 2 ; 1 2 ) R representation
Massive sector At weak coupling • Two decoupled PSU ( 1, 1 | 2 ) spin-chains • The two spin-chains couple through level matching e ip total = 1 Higher orders • Sites in the ( 1 2 ; 1 2 ) L ⊗ ( 1 2 ; 1 2 ) R representation
Massive sector At weak coupling • Two decoupled PSU ( 1, 1 | 2 ) spin-chains • The two spin-chains couple through level matching e ip total = 1 Higher orders • Sites in the ( 1 2 ; 1 2 ) L ⊗ ( 1 2 ; 1 2 ) R representation • Dynamic supersymmetries
Spin-chain representation ( 1 2 ; 1 2 ) Doublet under su ( 2 ) ⊂ psu ( 1, 1 | 2 ) φ ± Dimension 1 Two bosons 2
Spin-chain representation ( 1 2 ; 1 2 ) Doublet under su ( 2 ) ⊂ psu ( 1, 1 | 2 ) φ ± Dimension 1 Two bosons 2 ψ ± Two fermions Dimension 1 Doublet under su ( 2 ) • automorphism
Spin-chain representation ( 1 2 ; 1 2 ) Doublet under su ( 2 ) ⊂ psu ( 1, 1 | 2 ) ∂ n φ ± Dimension 1 Two bosons 2 + n Two fermions ∂ n ψ ± Dimension 1 + n Doublet under su ( 2 ) • automorphism Derivatives generate sl ( 2 ) descendants
Spin-chain representation ( 1 2 ; 1 2 ) Doublet under su ( 2 ) ⊂ psu ( 1, 1 | 2 ) ∂ n φ ± Dimension 1 Two bosons 2 + n Two fermions ∂ n ψ ± Dimension 1 + n Doublet under su ( 2 ) • automorphism Derivatives generate sl ( 2 ) descendants 1/2-BPS representation
Spin-chain representation ( 1 2 ; 1 2 ) Doublet under su ( 2 ) ⊂ psu ( 1, 1 | 2 ) ∂ n φ ± Dimension 1 Two bosons 2 + n Two fermions ∂ n ψ ± Dimension 1 + n Doublet under su ( 2 ) • automorphism Derivatives generate sl ( 2 ) descendants 1/2-BPS representation In the full psu ( 1, 1 | 2 ) × psu ( 1, 1 | 2 ) (massive) spin-chain: Sites make up 8+8 primary fields φ a ˙ � 1 2 , 1 a � 2 1, 1 � 1 ψ α ˙ a ψ a α � � � 2 , 1 L R 2 D αβ � � 1, 1
Massless modes in the spin-chain When we include the massless modes additional chiral representations appear as sites in the spin-chain T α ˙ ( 0, 0 ) Four free scalars β χ a ˙ � 1 α � 2 , 0 L Two ( 1 2 ; 1 2 ) ⊗ 1 multiplets ∂ L T α ˙ α � � 1, 0 χ ˙ a ˙ 0, 1 α � � R 2 Two 1 ⊗ ( 1 2 ; 1 2 ) multiplets ∂ R T α ˙ α � � 0, 1
Massless modes in the spin-chain With massive + massless modes Sites in different representations – “reducible spin-chain” [OOS, Stefański, Torrielli ’12] At weak coupling Two decoupled psu ( 1, 1 | 2 ) spin-chains of different length Extra equations describing scattering between massless modes Level matching condition � � exp ip L + ip R + ip massless = 1
BPS states From psu ( 1, 1 | 2 ) 2 representation theory • Primaries of three types of 1/2-BPS sites φ massive scalar χ ± massless chiral fermion L χ ± massless anti-chiral fermion R • Expect 1/2-BPS states of the form � J M � � J L � � J R � 1 2 ( J M + J L ) , 1 � � φ χ L χ R 2 ( J M + J R ) • Only completely symmetric states protected when interactions are included
BPS states From psu ( 1, 1 | 2 ) 2 representation theory + interactions J massive bosons � J 2 , J � 2
BPS states From psu ( 1, 1 | 2 ) 2 representation theory + interactions J massive bosons Two + two massless fermions, each appearing maximally once � J 2 , J � 2 � J � J � 2 � 2 2 + 1 2 , J 2 , J 2 + 1 2 2 � J � J � J � 4 2 + 1, J � 2 + 1 2 + 1 2 , J � 2 , J 2 + 1 2 2 � J � J � 2 � 2 2 + 1, J 2 + 1 2 + 1 2 , J 2 + 1 2 � J 2 + 1, J � 2 + 1
BPS states From psu ( 1, 1 | 2 ) 2 representation theory + interactions J massive bosons Two + two massless fermions, each appearing maximally once � J 2 , J � 2 � J � J � 2 � 2 2 + 1 2 , J 2 , J 2 + 1 2 2 � J � J � J � 4 2 + 1, J � 2 + 1 2 + 1 2 , J � 2 , J 2 + 1 2 2 � J � J � 2 � 2 2 + 1, J 2 + 1 2 + 1 2 , J 2 + 1 2 � J 2 + 1, J � 2 + 1 Matches supergravity spectrum [de Boer ’98]
String theory on AdS 3 × S 3 × S 3 × S 1
AdS 3 × S 3 × S 3 × S 1 × × × • Supersymmetry realtes the radii: 1 1 + 1 1 1 = 1 − α = α L 2 = R 2 R 2 R 2 L 2 R 2 L 2 + − + −
AdS 3 × S 3 × S 3 × S 1 × × × • Supersymmetry realtes the radii: 1 1 + 1 1 1 = 1 − α = α L 2 = R 2 R 2 R 2 L 2 R 2 L 2 + − + − One parameter family of backgrounds 0 < α < 1
AdS 3 × S 3 × S 3 × S 1 × × × • Supersymmetry realtes the radii: 1 1 + 1 1 1 = 1 − α = α L 2 = R 2 R 2 R 2 L 2 R 2 L 2 + − + − • Isometries: D ( 2, 1; α ) × D ( 2, 1; α ) × U ( 1 ) ⊃ SO ( 2, 2 ) × SO ( 4 ) × SO ( 4 ) × U ( 1 ) • In the α → 0 and α → 1 limits one of the sphere blows up → obtain the AdS 3 × S 3 × T 4 background −
AdS 3 × S 3 × S 3 × S 1 × × × • Unique supersymmetric geodesic on AdS 3 × S 3 × S 3 • Preserves 4 supersymmetries
AdS 3 × S 3 × S 3 × S 1 × × × • Unique supersymmetric geodesic on AdS 3 × S 3 × S 3 • Preserves 4 supersymmetries • Light-cone gauge “off-shell” symmetry algebra psu ( 1 | 1 ) 2 with four central elements c.e. • Fundamental excitations m B = 2 × { 0, α , 1 − α , 1 } m F = 2 × { 0, α , 1 − α , 1 }
AdS 3 × S 3 × S 3 × S 1 × × × • Unique supersymmetric geodesic on AdS 3 × S 3 × S 3 • Preserves 4 supersymmetries • Light-cone gauge “off-shell” symmetry algebra psu ( 1 | 1 ) 2 with four central elements c.e. • Fundamental excitations m B = 2 × { 0, α , 1 − α , 1 } m F = 2 × { 0, α , 1 − α , 1 } Composite?
AdS 3 × S 3 × S 3 × S 1 × × × • Unique supersymmetric geodesic on AdS 3 × S 3 × S 3 • Preserves 4 supersymmetries • Light-cone gauge “off-shell” symmetry algebra psu ( 1 | 1 ) 2 with four central elements c.e. • Fundamental excitations m B = 2 × { 0, α , 1 − α , 1 } m F = 2 × { 0, α , 1 − α , 1 } • Form 1 + 1 dimensional representations of psu ( 1 | 1 ) 2 c.e.
AdS 3 × S 3 × S 3 × S 1 × × × Off-shell symmetry algebra gives • Dispersion relation � q 2 h 2 sin 2 p kp ) 2 + 4˜ E p = ( m + / 2 • Matrix form of S matrix • 9 dressing phases
Summary
Summary Integrability in AdS 3 / CFT 2 Discussed string theory on AdS 3 × S 3 × T 4 • Supported by RR+NSNS flux • Classical theory is integrable • Quantum theory: light-cone gauge • Constructed “off-shell” symmetry algebra • Exact dispersion relation • All-loop S matrix – satisfies Yang-Baxter equation • Spin-chain picture from Bethe equations Results generalise to AdS 3 × S 3 × S 3 × S 1
Outlook Open string theory questions • Dressing phases – solve crossing equations [Work in progress] • Match with perturbation theory [Sundin, Wulff ’12–’15] [Engelund, McKeown, Roiban ’13] [Bianchi, Hoare ’14] • S matrix matches with perturbative results • Two-loop missmatch for massless dispersion relation p 3 p 3 E Exact E Pert = p − 24 h 2 + · · · = p − 4 π 2 h 2 + · · · p p • Massless su ( 2 ) ◦ S matrix • Winding modes on T 4
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