Small, Medium and Giant Magnons Gordon W. Semenoff University of British Columbia D.Astolfi, V.Forini, G.Grignani and G.Semenoff, hep-th/0702043 B.Ramadanovic and G.Semenoff, arXiv:0803.4028 [hep-th] G.Grignani and G.Semenoff, to appear GGI, May 8 , 2008
The AdS/CFT correspondence asserts an exact duality IIB string on AdS 5 × S 5 ↔ N = 4 Yang-Mills N units of 5-form flux on S 5 ↔ SU(N) gauge group radius of curvature R 4 /α ′ 2 g 2 ′ tHooft coupling = Y M N ≡ λ g 2 closed string coupling 4 πg s = Y M Energies of strings ↔ conformal dimensions of operators Free strings on AdS 5 × S 5 limit N → ∞ , λ = g 2 ↔ Y M N fixed Weak coupling sigma model ↔ strong gauge theory √ λ | g | g ab ∂ a X µ G µν ∂ b X ν ↔ S = N � � d 4 x Tr F 2 S = µν 4 π 4 λ GGI, May 8 , 2008
Finding spectrum of planar N = 4 Yang-Mills has a spin-chain analogy : ( J.Minahan, K.Zarembo hep-th/0212208 ) For example: scalar fields of N = 4 super-conformal YM: Φ 1 , ..., Φ 6 Z = Φ 1 + i Φ 2 Φ = Φ 3 + i Φ 4 , Ψ = Φ 5 + i Φ 6 , Large N planar limit ( N → ∞ , λ = g 2 Y M N fixed) : conformal dimensions of composite operators J Z ′ s + M Φ ′ s Tr [ Z (0) Z (0)Φ(0) Z (0)Φ(0) Z (0) ... ] YM interactions: ∆ = J + M + λ (one loop) + λ 2 (two loops) + . . . Resolving degeneracy ∼ solving PSU (2 , 2 | 4) spin chain with long ranged interactions GGI, May 8 , 2008
Ferromagnetic ground state of the spin chain: Tr Z J 1 2 -BPS operator, dimension ∆ = J protected by supersymmetry Symmetry of ground state SU (2 | 2) × SU (2 | 2) × R 1 ⊂ SU (2 , 2 | 4) One flipped spin is a “Magnon” – short multiplet of this residual symmetry algebra Tr Z J − 1 D µ Z , Tr Z J Φ i ˙ β Tr Z J χ β Tr Z J χ α , α ˙ with ∆ = J + 1 Because of cyclicity of the trace, they have zero magnon momentum e ipk Tr Z k Φ Z J − k ∼ δ ( p ) � k GGI, May 8 , 2008
Two magnons J − 1 � e ip 1 k 1 + ip 2 k 2 Tr ZZ... Φ k 1 ... Φ k 2 ...Z ∼ δ ( p 1 + p 2 ) k 1 ,k 2 =0 ∆ − J = 2 + λ (one − loop) + λ 2 (two − loop) + . . . GGI, May 8 , 2008
Two magnons at one loop λ � (1 − P i,i +1 ) H one loop = 8 π 2 i � ψ ( k 1 , k 2 )Tr ZZ... Φ k 1 ... Φ k 2 ...Z L = J + 2 1 ≤ k 1 <k 2 ≤ L ψ ( k 1 , k 2 ) = e ip 1 k 1 + p 2 k 2 + S ( p 1 , p 2 ) e ip 2 k 1 + p 1 k 2 λ sin 2 p 1 2 + sin 2 p 2 � � E = L + + ... 2 π 2 2 S = e ip 1 + ip 2 − 2 e ip 1 + 1 e ip 1 + ip 2 − 2 e ip 2 + 1 Periodic boundary conditions ψ ( k 1 , k 2 ) = ψ ( k 2 , k 1 + L ) → “Bethe equations” e iLp 1 = S ( p 1 , p 2 ) , e iLp 2 = S ( p 2 , p 1 ) Cyclicity of the trace implies p 1 + p 2 = 0 GGI, May 8 , 2008
• The spin chain is thought to be integrable and solvable using a Bethe Ansatz N.Beisert, B.Eden, M.Staudacher hep-th/0610251 • Problem is simpler in the large volume limit. – planar Yang-Mills theory N → ∞ , λ = g 2 YM N fixed – infinite volume J → ∞ with magnon momenta and λ fixed • Bethe Ansatz has distinct quasi-particles. In infinite volume limit, integrability implies scattering with a factorized S-matrix. • quasi-particle is a magnon • 2-body S-matrix almost completely determined by (super-)symmetry: N.Beisert hep-th/0603038,0606214 • once infinite J spectrum is known – reconstruct finite J GGI, May 8 , 2008
In the SU(2) sector, the spin chain Hamiltonian is “known” to four loops ∞ � n � λ � H = H n 16 π 2 n =0 Permutation operator: L � { a, b, c, ... } = P p + a P p + b P p + c ... P k = P k,k +1 , p =1 {} H 1 = 2 {} − 2 { 1 } H 0 = , − 8 {} + 12 { 1 } − 2 ( { 1 , 2 } + { 2 , 1 } ) H 2 = H 3 = 60 {} − 104 { 1 } + 4 { 1 , 3 } + 24 ( { 1 , 2 } + { 2 , 1 } ) − 4 i� 2 ( { 1 , 2 , 3 } + { 2 , 1 , 3 } ) − 4 ( { 1 , 2 , 3 } + { 3 , 2 , 1 } ) GGI, May 8 , 2008
( − 560 − 4 β ) {} + (1072 + 12 β + 8 � 3 a ) { 1 } H 4 = ( − 84 − 6 β − 4 � 3 a ) { 1 , 3 } + − 4 { 1 , 4 } + ( − 302 − 4 β − 8 � 3 a ) ( { 1 , 2 } + { 2 , 1 } ) (4 β + 4 � 3 a + 2 i� 3 c − 4 i� 3 d ) { 1 , 3 , 2 } + + (4 β + 4 � 3 a − 2 i� 3 c + 4 i� 3 d ) { 1 , 1 , 3 } (4 − 2 i� 3 a ) ( { 1 , 2 , 4 } + { 1 , 4 , 3 } ) + (4 + 2 i� 3 a ) ( { 1 , 3 , 4 } + { 2 , 1 , 4 } ) + + (96 + 4 � 3 a ) ( { 1 , 2 , 3 } + { 3 , 2 , 1 } ) ( − 12 − 2 β − 4 � 3 a ) { 2 , 1 , 3 , 2 } + + (18 + 4 � 3 a ) ( { 1 , 3 , 2 , 4 } + { 2 , 1 , 4 , 3 } ) + ( − 8 − 2 � 3 a − 2 i� 3 b ) ( { 1 , 2 , 4 , 3 } + { 1 , 4 , 3 , 2 } ) ( − 8 − 2 � 3 a + 2 i� 3 b ) ( { 2 , 1 , 3 , 4 } + { 3 , 2 , 1 , 4 } ) + − 10 ( { 1 , 2 , 3 , 4 } + { 4 , 3 , 2 , 1 } ) , β = 4 ζ (3) GGI, May 8 , 2008
Recent computations of the spectrum of short operators suggest that the BES Bethe Ansatz is valid only in the J → ∞ limit. F. Fiamberti, A. Santambroggio, C. Seig, D. Zanon, “Wrapping at four loops” ARXIV:0712.3522 � 2 � 3 � � � � λ λ λ ∆ K = 4 + 12 − 48 + 336 16 π 2 16 π 2 16 π 2 � 4 � λ − (2584 − 384 ζ (3) + 1440 ζ (5)) + ... 16 π 2 C. Keeler and N.Mann, “Wrapping interactions and the Konishi Operator”, ARXIV:0801.1661 � 2 � 3 � λ � � λ � λ ∆ K = 4 + 12 − 48 + 336 16 π 2 16 π 2 16 π 2 � 4 � λ − (2607 + 28 ζ (3) + 140 ζ (5)) + ... 16 π 2 GGI, May 8 , 2008
Deviations from the large spin limit are due to “wrapping interactions”. J.Ambjorn, R.Janik, Ch.Kristjansen, hep-th/0510171 GGI, May 8 , 2008
Magnon with p mag � = 0 ... ... � e ipx ...ZZZ Φ ZZZ... x infinitely long spin chain – isolate a single magnon � 1 + λ π 2 sin 2 p mag E = ∆ − J = , p mag = magnon momentum 2 • Compatible with perturbative YM to three loops • all-loops integrability Ans¨ atze at large J • agrees with BMN limit • Beisert: magnon are 1 2 − BPS states of centrally extended superalgebra SU (2 | 2) × SU (2 | 2) × R 3 • Strong coupling limit λ → ∞ from string dual − → GGI, May 8 , 2008
Hofman-Maldacena hep-th/0604135 identified string dual: Giant Magnon : Soliton solution of classical string sigma model on R 1 × S 2 φ ′ , all others periodic angle coordinate open φ ( r ) − φ ( − r ) = p mag J ( = − i∂/∂φ ) → ∞ θ ( ± r ) → π/ 2 √ � � sin p mag π 2 sin 2 p mag λ 1 + λ � � E = ← at large λ � π 2 2 What about corrections to the large J limit? GGI, May 8 , 2008
Finite size corrections? • finite size and strong coupling from string – apparently yes! Arutyunov, Frolov, Zamaklar hep-th/0606126 √ � � λ � sin p mag 1 − 4 e 2 sin 2 p mag � � e −R + ... � · E = � � π 2 2 • Hubbard model matches exponent, √ R = 2 πJ/ λ | sin p mag / 2 | + ap mag cot p mag / 2 but not pefactor • Bethe Ansatz – maybe? – the integrable Hubbard model agrees with perturbation theory to a few loops, then is extrapolated to large λ and large J , √ 2 π 2 √ � � λ � sin p mag � � λ | sin p mag / 2 | + ... λ sin 2 p mag / 2 e − 2 πJ/ � · 1 − E H = � � π 2 • Perturbative gauge theory – none! – at least for J > #loops. GGI, May 8 , 2008
• Finite size classical Giant Magnon found by Arutyunov, Frolov, Zamlakar hep-th/0606126 √ � λ � sin p mag 1 − 4 e 2 sin 2 p mag � � e −R − � · E = � � π 2 2 e 4 sin 2 p mag 4 R 2 (1 + cos p ) + 2 R (2 + 3 cos p mag + � − 2 + ap mag sin p mag ) + 7 + 6 cos p mag + 6 ap mag sin p mag + e − 2 R + ... a 2 p 2 � � mag (1 − cos p mag ) + √ R = 2 πJ/ λ | sin p mag / 2 | + ap mag cot p mag / 2 • but depend on gauge-fixing parameter a • There is no state of N = 4 SYM dual to a single giant magnon with J < ∞ . Gauge theory dual of finite size giant magnon? GGI, May 8 , 2008
Orbifold AdS 5 × S 5 → AdS 5 × S 5 /Z M Identify longitude on 2-sphere by the action of a discrete group Z M : φ → φ + 2 π/M Non-interacting strings: • choose subset of momenta J = integer · M (rather than J =integer in un-orbifold) • Include wrapped strings ∆ φ = 2 πm/M Giant magnon = wrapped closed string Open ends of magnon are identified identified: p mag = 2 πm/M GGI, May 8 , 2008
Giant magnon is a physical state on orbifold D.Astolfi, V.Forini, G.Grignani and G.Semenoff hep-th/0702043 Finite size corrections are computable by asymptotic expansion in J (and identical to Arutyunov, Frolov, Zamlakar hep-th/0606126 in a = 0 gauge) √ λ � sin p mag � 1 − 4 sin 2 p mag � J � � − 2 − 2 π √ λ | sin p mag / 2 | + . . . ∆ − J = e � � π 2 2 � The exponential correction has been reproduced from BES by R.Janik,T.Lukowski, ArXiv:0708:2208 J. Minahan and O.Ohlsson Sax, “Finite size effects for giant magnons on physical strings” arXiv:0801.2064 N. Gromov, S.Shafer-Nameki, P.Viera, “Quantum wrapped giant magnon”, arXiv:0801.3671 GGI, May 8 , 2008
Why orbifold? String on flat space with magnon boundary condition: X 1 ( τ, σ + 2 π ) = X 1 ( τ, σ ) + p mag and all other variables, including ∂ a X 1 ( τ, σ ) periodic. � ∂ 2 ∂τ 2 − ∂ 2 � σ X 1 = 0 → X 1 ( τ, σ ) = x 1 + p 1 τ + p mag 2 π +oscillators ∂σ 2 Virasoro constraints are 2 p µ p µ + p 2 L 0 = α ′ mag L 0 + ˜ 4 π 2 α ′ + N + ˜ 0 = N − 2 L 0 = p 1 p mag L 0 − ˜ + N − ˜ 0 = N 2 π has no solution unless p 1 p mag = 2 π · integer Indistinguishable from string where X 1 ∼ X 1 + p mag = Z -orbifold of flat space GGI, May 8 , 2008
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