Semi-classical Strings in the AdS/CFT correspondence

1. Semi-classical Strings in the AdS/CFT correspondence Tristan Mc Loughlin Aug 12th 2008

2. Overview • In the first part we discuss the construction of several families of classical string solutions on R x S2. These solutions are the finite-size analogues of two-magnon scattering solutions and magnon “bound” states. We semi-classically quantize these solutions, invert the implicit energy relations and find the finite size corrections to the infinite volume expressions. Based on 0803.2324[hep-th] with T. Klose. • In the second part we study the spinning folded string in the AdS4 x CP3 geometry. We calculate the one-loop large  correction to the string energy which can be compared to the prediction from the recently conjectured all order Bethe ansatz. Based on 0807.3965[hep-th] with R. Roiban.

3. Motivations • Integrability in planar N=4 SYM/AdS5 string theory has provided many interesting insights. In particular the matching of the spectrum for infinite length operators/strings with infinite charges at all orders in coupling. • Spectrum of both theories described for states/operators of large charge by asymptotic Bethe ansatz. • We do know that the asymptotic ansatz must be corrected • Consideration of strings with large angular momentum J found corrections of the form exp(-2  J/) (Schäfer-Nameki, Zamaklar & Zarembo). • In the gauge based on known BFKL behaviour for high energy scattering wrapping effects were shown to occur for twist-two operators (Kotikov, Lipatov, Rej, Staudacher and Velizhanin). • Concrete calculations of four-loop Konishi operator anomalous dimension (Keeler & Mann),(Fiamberti, Santambrogio, Sieg & Zanon).

4. Theory on a Cylinder • There are new difficulties in finite volume • One cannot arbitrarily separate excitations to define an S-matrix. • There may not be additivity of energies. • Methods for solving QFT in finite volume include TBA • Some results for string worldsheet, complicated at least part due to lack of worldsheet Lorentz invariance. Arutyunov&Frolov • Relatedly is the Lüscher formalism • Ambjorn, Janik & Kristjansen/ (Heller)Janik & Lukowskiapplied arguments of Lüscher to the non-relativistic case at hand and showed that generically the corrections for length L states start at order ^L. They also found from the asymptotic S-matrix the leading contributions to the magnon dispersion relation. Recent calculation of 4-loop Konishi by Janik&Bajnok. See also [Hatsuda&Suzuki], [Gromov, Schafer-Nameki&Vieira]. • We would of course like to know the all-order expressions but perhaps as in the infinite volume case it is useful to consider semi-classical string solutions as a first step.

5. Starting with the string in conformal gauge we take the time direction from the AdS space to be proportional to the worldsheet time t =   and we consider an S2  S5 parameterised by the unit vectorn. • Making the identification one can show that the e.o.m and constraints are equiv. to sine-Gordon e.o.m.

6. Giant Magnons • String solitons: What is the string dual of the single spin chain excitation? How can we reproduce spin chain “sine” behaviour from cont. world-sheet theory? Can we find magnon S-matrix? • Hofman/Maldacena found classical string solution dual to magnon with • Opening angle related to magnon momentum. • From two magnon solution can calculate classical • scattering phase. • In light-cone gauge pw.s.= p, can calculate charges. • World sheet has infinite extent, E & J separately infinite. • Finite size version found by AFZ.

7. Via Pohlmeyer reduction the string e.o.m.‘s are equivalent to s-G theory • Solitonic solutions of s-G theory in infinite volume • Kink • Kink scattering • In finite volume the analogue of the kink solution is the one-phase solution the “Kink-train” Corresponds to single magnon Corresponds to magnon-magnon scattering Decompactification as m 1 For this parameter regime it corresponds to AFZ magnon but for m>1 corresponds to so called helical string.

8. Two phase solutions of the Lamb ansatz form • Three types - fluxon, breather and plasmon E.g. Fluxon Lamb ‘71 • Corresponds to solutions in • c.o.m. frame • Involves elliptic functions • Algebraic curve has extra • symmetries Costabile et al ‘78 Elliptic moduli Decompactification m1, L,T.

9. From sine-Gordon to Strings • The string is a patch of the unit sphere parameterized by a vector + two tangent vectors giving an orthonormal triad. The surface is described by the fundamental eqns. • Choosing for our connection: • Integrability conditions are just s-G eqn. thus each solution of s-G theory corresponds to a surface on a sphere and this is our string solution. Hence we “simply” need to integrate the structure equations- actually doable for solutions of the Lamb form.

10. String solutions of Lamb form: ~ string radius & with: and: finally the angular momentum is given by: ; charge density

11. Example: Fluxon Oscillation (Elementary regime) Finite analogue of scattering sol.

12. Example: Fluxon Oscillation (Elementary regime) • Can also find solutions corresponding to fluxon breather and plasmon for all parameter regimes. • One of the important features of the Hofman/Maldacena construction was their ability to identify the magnon momentum but here we can’t do that as easily as we are considering closed strings and we can’t cleanly separate magnons. As we will see it is useful to expand answers near the decompactification limit i.e. large L. Finite analogue of scattering sol.

13. Semi-classical quantization • All our solutions are periodic in time (as well as space) so we can use the usual Bohr-Sommerfeld quantisation • Invert equation for L and J to remove elliptic moduli. • Breather “bound” states: Simply integrate this order by order and invert to find the dispersion relation. Eg. Fluxon breather in near-decompactification limit. • Also has interesting J=0 limit which agrees with pulsating circular string of Minahan in the appropriate limit. Period Energy Quantum number , 

14. Semi-classical quantization • All our solutions are periodic in time (as well as space) so we can use the usual Bohr-Sommerfeld quantisation • Invert equation for L and J to remove elliptic moduli • Scattering states: e.g. Magnon-- anti-magnon in near-decomp. limit,

15. Semi-classical quantization • Scattering states: let us recall first the decompactifiaction limit where we find that ( for v = A-1 ) we can write the period as where T0 TDelay Interpreting soliton c.o.m. motion as a particle in a box of length /2 L, energy , momentum p = 2 pw.s. , interacting with potential. We thus have the boundary conditions: Jackiw & Woo Particle Momentum Phase shift Identifying the terms in orders of L we get equations for both the energy and the phase Twice the single magnon energy

16. Can repeat including corrections from near-decompactification • limit. At given order in exp. we take leading L term to define • energy: Not simply twice the single magnon energy Agrees with two magnon c.o.m. expression from alg. curve Minahan & Ohlsson Sax Easy enough to find higher exponential corrections. Again using: we can find the exp. corrections to the phase Classical phase shift in c.o.m. frame

17. Summary • We considered classical two-phase solutions describing magnon bound states and scattering solutions. We found exponential corrections to the multi-particle dispersion relation and to the phase-shift in the near-decompactification limit. • Further work • Can we use the DHN method to calculate one-loop corrections? • Find boosted solutions to find phase shift in lab. frame. • Find three phase solutions. • Generalise solutions to non-compact AdS2 x S1 (sinh-G) sector. • Solve theory on finite volume, hopefully our results may provide a guide to or at least a check on final answer.

18. We consider closed super-strings on the background: Conjectured to be dual to N=6 susy 3-dim Chern-Simons theory with level k. ABJM Describes the low energy limit of N M2-branes onR1,8 x C/Zk whose near horizon limit is AdS4 x S7/Zk. WritingS7 as an S1 fibered over CP3 the action of the quotient is simply to reduce the a radius of the S1. We consider the compactification with k and N large but N/k fixed.

19. Explicit expressions after compactification: Originates from S7 as S1 fibered over CP3 From spin connection of AdS4 x S7; J Kähler form Ten dim. radius: This form of the metric can be found by iteratively embedding CPn-1 in CPn. eg Hoxha et al

20. One can consider a semi-classical expansion about such solutions. The expansion is organised as: Charge Densities We are additionallyinterested in the limit: Solution is particularly simple in this limit - becomes homogeneous in worldsheet coordinates.

21. Virasoro Constraint Charges are given by: Scaling limit is:

22. Virasoro Constraint Charges are given by: Can rescale  by  so worldsheet length L is 2. L in scaling limit.

23. Virasoro Constraint Classical String energy is: Generalised scaling function Limit of l = 0 gives universal scaling function. Our goal is to compute the one-loop corrections to the string energy and show that the same form persists.

24. Bosonic Fluctuations Analysis of bosonic fluctuations is almost identical to F&T, FTT We can simply use previously known results to find the spectrum of fluctuations. AdS4 General case of J0. Here SO(4)  SO(6) is preserved from CP3. Two massless modes cancel against ghosts. CP3

25. Bosonic Fluctuations Analysis of bosonic fluctuations is almost identical to F&T, FTT We can simply use previously known results to find the spectrum of fluctuations. 4 2 2 2 0 x 2 Simple case of J=0 - Full SO(6) preserved. c.f. AMII - interpret six massless CP fluctuations as Goldstone bosons which should remain massless to all orders in perturbation theory. m2 AdS4  m2 = 6 2 m2 0 x 6 CP3

26. Fermionic Action and Frequencies • Complete, covariant all-order fermionic action is not known for this background. [Arutyunov&Frolov] [Stefanski] • Actions based on • coset and corresponding to partially kappa-gauge fixed actions have been constructed, and successfully used for this calculation. • Also pure spinor action. • Alternatively one can start with 11-dim membrane action deWit et al and doubly dimensionally reduce Duff et al. [Alday,Arutyunov Bykov], [Krishnan] [Fre’ & Grassi.] Actually only need quadratic-in-fermions piece - generic form Cvetic et al

27. Fermionic Action and Frequencies Consider J=0 string: The vielbien pull back involves the worldsheet coordinate explicitly. can be removed by In particular the coord. dependence in rotating the fermions The action becomes with Now we want to fix kappa-gauge, structure of the derivative terms leads to obvious choice: .

28. 0 x 2 2 x 6 Fermionic Action and Frequencies After kappa gauge fixing action is and the mass matrix has eigenvalues Makes sense from symmetries. Susys form a of . m2 - Note two “light” fermions Can repeat analysis for J0: Where as expected the spectrum reflects the SO(6) broken to SO(4).

29. The one-loop correction to the string energies is given by with so Sum of Fluctuation freq. Two cases: a) In the scaling limit  >> 1 the worldsheet is infinitely long and the momenta are continous. Thus to O() the sum can be replaced by an integral. Imposing a cutoff and performing the integrals one find that the quadratic and logarithmic divergences cancel. The finite piece is given by

30. The one-loop correction to the string energies is given by with so Two cases: b) Very similar to AdS5 result except for last term. Reduces to previous answer in u0 limit.

31. Comparison with all-order Bethe ansatz An all-loop conjecture for the string/gauge spectrum in the planar limit has been recently made. Very similar to N=4 SYM case - again the su(2|2) symmetries constrain the tensor structure of S-matrix and the form of magnon dispersion relation, One important difference is that h() is not simply as in N=4 SYM but has different behaviour at weak and strong coupling It was conjectured that up to this function the dressing phase factors are the same in N=6 CS as in N=4 SYM. Given this simple relation it is easy to obtain predictions for anomalous dimensions of various operators. Gromov & Vieira

32. Comparison with all-order Bethe ansatz G&V identified an SL(2) sector of the Bethe equations and identified a solution that corresponds to the spinning folded string. Their prediction for the universal scaling function is The O(1) piece of the h function is fixed by the one-loop correction to the magnon dispersion relation and by general considerations from the spectral curve analysis. Result to one-loop: which obviously does not match the explicit calculation. We can extend this comparison to the generalised scaling function Additional term found in explicit calculation. G&V conjecture

33. Outlook • Possible resolutions: • Break down of integrability: Known that CP3 bosonic sigma model has a quantum anomaly that ruins integrability. Here we expect fermions to cancel such an effect. Perhaps we can see this by considering only “light” d.o.f. about the J=0 solution à la AM - capture higher loop effects. • Reordering and summation of fluctuation modes proposed by Gromov & Mikhaylov. Equivalent to imposing different cutoffs for “light” and “heavy” modes - produces agreement with all-order conjecture. While natural from the curve p.o.v. it is unclear whether this prescription works for other solutions or higher loops, does not respect SO(6) symmetry of J=0 solution. • Precise identification of S-matrix - different dressing phase, modifications to h() function. • Other directions: Other solutions. Scaling solution related to Wilson loops in AdS5 perhaps there is a similar relation here also to scattering ampl. • Higher orders: need higher-in-fermion terms in string action. Can be found from doubly dim. reducing AdS4 x S7 membrane action. Can also analyze semiclassical spinning membranes at one-loop.