Spiky Strings nd Gint Mgnons on S 5 M. Kruczenski Purdue University Bsed on: hep-th/0607044 (Russo, Tseytlin, M.K.)
Introduction Summry String / guge theory dulity (AdS/CFT) Clssicl strings nd field theory opertors: folded strings nd spin wves in spin chins folded strings nd twist two opertors Spiky strings nd higher twist opertors (M.K.) Clssicl strings moving in AdS nd their field theory interprettion
Spiky strings on sphere nd gint mgnon limit (Ryng, Hofmn-Mldcen) Spin chin interprettion of the gint mgnon (Hofmn-Mldcen) More generic solutions: Spiky strings nd gint mgnons on S 5 (Russo, Tseytlin, M.K.) Other solutions on S (work in prog. w/ R.Ishizeki ) Conclusions
String/guge theory dulity: Lrge N limit ( t Hooft) mesons q,,... String picture Qurk model q q q Fund. strings ( Susy, 0d, Q.G. ) QCD [ SU(3) ] Lrge N-limit [SU(N)] Strong coupling Effective strings More precisely: N, λ = g N fixed Lowest order: sum of plnr digrms (infinite number) YM ( t Hooft coupl.)
AdS/CFT correspondence (Mldcen) Gives precise exmple of the reltion between strings nd guge theory. Guge theory String theory N = 4 SYM SU(N) on R 4 A, i, Opertors w/ conf. dim. IIB on AdS 5 xs 5 rdius R String sttes w/ 4 s YM s YM g = g ; R / l = ( g N ) / E = R YM N, λ = g N fixed lrge smll string th. field th.
Cn we mke the mp between string nd guge theory precise? It cn be done in prticulr cses. Tke two sclrs X = + i ; Y= 3 + i 4 O = Tr(XX Y..Y X), J X s, J Y s, J +J lrge Compute -loop conforml dimension of O, or equiv. compute energy of bound stte of J prticles of type X nd J of type Y (but on three sphere) R 4 S 3 xr E
Lrge number of ops. (or sttes). All permuttions of Xs nd Ys mix so we hve to dig. huge mtrix. Nice ide (Minhn-Zrembo). Relte to phys. system Tr( X X Y X X Y ) opertor conf. of spin chin mixing mtrix op. on spin chin J j+ j= λ r r H = S S 4 j 4π Ferromgnetic Heisenberg model!
Ground stte (s) Tr( X X X X X X ) Tr( Y Y Y Y Y Y ) First excited sttes πn i k l k = e......, k = ; ( J = J + J ) l J λ ε( k) = ( cosk) k 0 4π λn J More generic (low energy) sttes: Spin wves (FT, BFST, MK, ) (BMN)
Other sttes, e.g. with J =J Spin wves of long wve-length hve low energy nd re described by n effective ction in terms of two ngles, : direction in which the spin points. λ Seff. = J dσ dτ cos θ τ φ ( σ ) + sin ( σ ) 3π J [ θ θ φ ] Tking J lrge with /J fixed: clssicl solutions Moreover, this ction grees with the ction of string moving fst on S 5. Wht bout the cse k ~?
Since (, ) is interpreted s the position of the string we get the shpe of the string from S (σ) Exmples point-like (, ) Folded string
Rottion in AdS 5? (Gubser, Klebnov, Polykov) 3 4 Y + Y + Y + Y Y Y = R 5 6 sinh ; [ 3] ρ Ω cosh ρ ; t ds = cosh ρ dt + dρ + sinh ρ dω [ ] 3 = t λ E S + S S π ln, ( ) ( Φ S Φ ) O = Tr, x = z + t + +
Verifiction using Wilson loops (MK, Mkeenko) The nomlous dimensions of twist two opertors cn lso be computed by using the cusp nomly of light-like Wilson loops (Korchemsky nd Mrchesini). In AdS/CFT Wilson loops cn be computed using surfces of miniml re in AdS 5 (Mldcen, Rey, Yee) z The result grees with the rotting string clcultion.
Generliztion to higher twist opertors (MK) O Tr ( S Φ Φ ) [ ] = + O Tr ( S/ n S/ n S/ n S / n = Φ Φ ΦK Φ ) [ n] + + + + In flt spce such solutions re esily found in conf. gug x = Acos[( n ) σ ] + A( n ) cos[ σ ] + y = Asin[( n ) σ ] + A( n ) sin[ σ ] +
Spiky strings in AdS: 0 0 E S n + S S λ π ln, ( ) O ( S/ n S / n S / n S/ n Φ Φ ΦK Φ ) = Tr + + + + λ λ S = dt (cosh ρ j dt + ) θ ρ π & 4 ln sin j 8π j θ j+ θ j
Spiky strings on sphere: (Ryng ) Similr solutions exist for strings rotting on sphere: (top view) The metric is: ds = dt + dθ + sin θ dϕ We use the nstz: t = κτ, ϕ = ωτ + σ, θ = θ( σ) And solve for θ( σ). Field theory interprettion?
Specil limit: (Hofmn-Mldcen) dθ = dσ κ sinθ κ sin θ A A κ ω sin θ ω = κ d θ dσ = sinθ A θ κ sin cos θ A sinθ = A κ sin σ (top view) gint mgnon
The energy nd ngulr momentum of the gint mgnon solution diverge. However their difference is finite: λ A dσ λ ϕ E J = = sin π κ sin σ π ϕ A cos =, ϕ = Angulr distnce between spikes κ Interpolting expression: E λ ϕ J = + sin π λ ϕ sin, λ > > π λ ϕ + < < sin, λ π
Field theory interprettion: (Hofmn-Mldcen) J j+ j= λ r r H = S S 4 j 4π πn i k l k = e......, k = ; ( J = J + J ) J λ λ k ( k) = ( cosk) ε = sin 4π π k ϕ (The is J ) Sttes with one spin flip nd k~ re gint mgnons
More spin flips: (Dorey, Chen-Dorey-Okmur) In the string side there re solutions with nother ngulr momentum J. The energy is given by: λ ϕ λ ϕ E J = J + sin J + sin, λ < < π J π Justifies interpolting formul for J = In the spin chin, if we flip number J of spins there is bound stte with energy: λ ε ( k ) J π sin k = k ϕ (J is bsorbed in J)
More generl solutions: (Russo, Tseytlin, MK) Strtegy: We generlize the spiky string solution nd then tke the gint mgnon limit. In flt spce: x = Acos[( n ) σ ] + A( n ) cos[ σ ] + y = Asin[( n ) σ ] + A( n ) sin[ σ ] + ξ ξ α σ βτ nmely: x + iy = X = x e i ωτ ( ), = + 3 3 = = Consider txs 5 : ds dt dx d X, X X Use similr nstz: = + = X = x ( ξ) e = r ( ξ) e iω τ i µ ( ξ ) + iω τ
The reduced e.o.m. follow from the lgrngin: L = ( α β ) x' x' + iβω ( x' x x' x ) ω x x + Λ ( x x ) If we interpret ξ s time this is prticle in sphere subject to qudrtic potentil nd mgnetic field. The trjectory is the shpe of the string The prticle is ttrcted to the xis but the mgnetic field curves the trjectory
Using the polr prmeteriztion we get: L = ( α β ) r' C α r ( α β ) r ( α β ) ω + Λ ( r ) µ ' = ( α β ) C r + βω, x = r e iµ Constrints: ω C + βκ = 0, H = α α + β β κ Three ng. moment: J = C dξ β α + r α ( α β ) ( α β ) ω Corresponding to phse rottions of x,,3
Solutions: One ngulr momentum: x 3 =0, x rel (µ =0), r +r =, one vrible. Two ngulr moment: x 3 =0, r +r =, one vrible Since only one vrible we solve them using conservtion of H. Reproduced Ryng, Hofmn-Mldcen nd Chen-Dorey-Okmur Three ngulr moment: r +r +r 3 =, r,
Therefore the three ngulr moment cse is the first non-trivil nd requires more effort. It turns tht this system is integrble s shown long go by Neumnn, Rosochtius nd more recently by Moser. Cn be solved by doing chnge of vribles to ζ +, ζ ζ ω ζ ω r = + ( ω ω ) b b ( )( )
In the new vribles, the system seprtes if we use the Hmilton-Jcobi method: Compute the Hmiltonin: H( p, ζ ) ± ± ζ ± W( ) Find such tht H p ± = W ζ ±, ζ ± = E = const. In this cse we try the nstz: nd it works! Vribles seprte!. W = W( ζ ) + W( ζ ) + A lengthy clcultion gives solution for ζ +, ζ which cn then be trnslted into solution for r
The resulting equtions re still complicted but simplify in the gint mgnon limit in which J We get for r : r = ( ω ω3 ) A s ( ω ω ) ( s A s A ) 3 3 r = 3 ( ω ω3 ) A 3 s3 ( ω ω ) ( s A s A ) 3 3 3 with A sξ s3ξ = tnh + B, A3 = coth β β + B 3 We hve r explicitly in terms of ξ nd integrtion const.
We cn compute the energy nd ngulr extension of the gint grviton obtining: λ ϕ λ ϕ 3 E J = E + E3 = J + sin + J3 + sin π π ϕ = ϕ + ϕ 3 We get two superposed mgnons. However there is reltion: sme group velocity. v = v, v = 3 j E ϕ j j
In the spin chin side we need to use more fields to hve more ngulr moment. We consider therefore opertors of the form: Tr( XXXXYYXYYZZZYYXXYZZZXXXXXXXXXXXX ) Where X = + i ; Y= 3 + i 4, Z= 5 + i 6 The J Y s form bound stte nd the J 3 Z s nother, both superposed to bckground of J X s ( J ) E J J + J + λ ϕ J + λ ϕ 3 J < < 3 sin sin, λ π π The condition of equl velocity ppers becuse in the string side we use rigid nstz which does not llow reltive motion of the two lumps. 3
Some exmples of solutions. 0.8 0.8 0.6 0.6 0.4 0.4 0. 0. 30 0 0 0 0 30 40 50 30 0 0 0 0 0 30 40 50 0.8 0.6 0.4 r,,3 (ξ) 0. 30 0 0 0 0 0 30 40 50
Other solutions on S : (Work in progr. w/ R. Ishizeki) It turns out tht looking t rigid strings rotting on two-sphere one cn find other clss of solutions nd in prticulr nother limiting solution: Antiferromgnetic mgnon? (see Roibn,Tirziu,Tseytlin) (Goes round infinite times)
Conclusions: Clssicl string solutions re powerful tool to study the dulity between string nd guge theory. We sw severl exmples: folded strings rotting on S 5 spiky strings rotting in AdS 5 nd S 5 gint mgnons on S nd S 3 gint mgnons with three ngulr moment work in progress on other sol. on S