Semi-Classical strings as probes of AdS/CFT M. Kruczenski Purdue University Based on: arXiv: R. Roiban, A. Tirziu, A. Tseytlin, M.K. arXiv: R. Ishizeki, M.K. arXiv: R. Ishizeki, M. Spradlin, A. Volovich, M.K.

Summary ● Introduction String / gauge theory duality (AdS/CFT) Semi-Classical strings and field theory operators Semi-Classical strings and Wilson loops ● Rotating strings and twist 2 ops.  (S. Gubser, I. Klebanov, A. Polyakov) ● Light-like cusp W.L. and twist 2 ops. (M.K., Y. Makeenko) ● Light like-cusp at one-loop. (R.Roiban, A.Tirziu, A.Tseytlin, M.K.)

● Spiky strings and higher twist operators (M.K.) ● Spiky strings on a sphere (Ryang) ● Giant magnons (Hofman-Maldacena) ● Single spike solutions on S 2 and S 3 ( R.Ishizeki, M.K. ) ● Multiple spikes solutions on S 2 and S 3 ( R. Ishizeki, M. Spradlin, A. Volovich, M.K. ) ● Conclusions

String/gauge theory duality: Large N limit (‘t Hooft) mesons String picture π, ρ,... Quark model Fund. strings ( Susy, 10d, Q.G. ) QCD [ SU(3) ] Large N-limit [SU(N)] Effective strings qq Strong coupling q Lowest order: sum of planar diagrams (infinite number) More precisely: (‘t Hooft coupl.)

AdS/CFT correspondence (Maldacena) Gives a precise example of the relation between strings and gauge theory. Gauge theory N = 4 SYM SU(N) on R 4 A μ, Φ i, Ψ a Operators w/ conf. dim. String theory IIB on AdS 5 xS 5 radius R String states w/ fixed λ large → string th. λ small → field th.

Semi-classical strings Why? We cannot quantize exactly S (+fermions etc.) Semi-classical regime is strong coupling limit  Two important cases of classical solutions Surface ending on the boundary Wilson loops (Maldacena; Rey, Yee) (= bdy. curve) String inside AdS 5 xS 5 Operator in the field theory (Berenstein, Maldacena, Nastase; Gubser, Klebanov, Polyakov)

Rotating Strings and twist 2 operators θ = ω t (Gubser, Klebanov, Polyakov)

Light-like cusps and twist 2 ops. (MK, Makeenko) The anomalous dimensions of twist two operators can also be computed by using the cusp anomaly of light-like Wilson loops (Korchemsky and Marchesini). In AdS/CFT Wilson loops can be computed using surfaces of minimal area in AdS 5 (Maldacena, Rey, Yee) z The result agrees with the rotating string calculation.

One loop correction to light-like cusp. (w/ R. Roiban, A. Tirziu, A. Tseytlin) By doing a quadratic expansion around the classical solution we can compute the one-loop correction to the partition function. The result is finite and corrects the cusp anomaly: It agrees with the previously known result from the rotating string. In fact one can see that both classical solutions are related. cusp rot. string for infinite S limit. String version of the Korchemsky-Marchesini argument

Generalization to higher twist operators (MK) In flat space such solutions are easily found in conf. gauge:

Spiky strings in AdS:

Spiky strings on a sphere: (Ryang ) Similar solutions exist for strings rotating on a sphere: (top view) The metric is: We use the ansatz: And solve for. Field theory interpretation?

Giant magnon: (Hofman-Maldacena) (top view)giant magnon

The energy and angular momentum of the giant magnon solution diverge. However their difference is finite: Angular distance between spikes Interpolating expression:

One more angular momentum: (Dorey, Chen-Dorey-Okamura) There are solutions with another angular momentum J 2. The energy is given by: Justifies interpolating formula for J 2 =1

More general solutions: (Russo, Tseytlin, MK) Strategy: We generalize the spiky string solution and then take the giant magnon limit. In flat space: Consider txS 5 : namely: Use similar ansatz:

The reduced e.o.m. follow from the lagrangian: If we interpret  as time this is particle in a sphere subject to a quadratic potential and a magnetic field. The trajectory is the shape of the string The particle is attracted to the axis but the magnetic field curves the trajectory L describes an integrable system

Single spike solutions on S 2 : (w/ R. Ishizeki; Mosaffa, Safarzadeh) It turns out that looking at rigid strings rotating on a two-sphere one can find other class of solutions and in particular another limiting solution: Antiferromagnetic magnon? (see Roiban,Tirziu,Tseytlin) [Hayashi, Okamura, Suzuki, Vicedo] (Goes around infinite times)

Now the energy and winding number diverge but the difference is finite. The angular momentum is finite This describes excitations over an infinitely wound string. Adding one ang. momentum:

Multiple spike solutions (w/Ishizeki, Spradlin, Volovich) Using methods from integrable systems which are usually used to find soliton solutions in two dimensional field theories we can construct more general solutions. We use dressing method (Zakharov, Mikhailov) This method gives new solutions by dressing a known one. In particular dressing the infinitely wrapped string one gets the single spike. By dressing again one gets two spikes etc. This is a confirmation of the idea of these solutions as excitations of the infinitely wrapped string.

Example of spike scattering Phase shift is the same as the giant magnon (in terms of the dressing param.)

Conclusions: Classical string solutions are a powerful tool to study the duality between string and gauge theory. Both through Wilson loop computations and by matching closed strings with gauge invariant f.t. operators We saw several examples: ● folded strings rotating on AdS 5 and light-like cusps. ● light-like cusps at one-loop ● spiky strings rotating in AdS 5 and S 5 ● giant magnons on S 2 and S 3 ● single spike solutions on S 2 ● multiple spike solutions on S 2