Yerevan State University / Leibniz Universität Hannover

1. Yerevan State University / Leibniz Universität Hannover Supersymmetry in Integrable Systems - SIS'10 International Workshop, 24-28 August 2010, Yerevan, Armenia Integrability inAdS/CFT and finite size effects in dyonic Magnons Bum-Hoon Lee Center for Quantum Spacetime Sogang University Seoul, Korea

2. B.-H.L, R. Nayak, K. Panigrahi, C. Park On the giant magnon and spike solutions for strings on AdS(3) x S**3.JHEP 0806:065,2008. arXiv:0804.2923 J. Kluson, B.-H.L, K. Panigrahi, C. Park, Magnon like solutions for strings in I-brane background. JHEP 0808;032, 2008, arXiv:0806.3879 B.-H.L, K. Panigrahi, C. Park , Spiky Strings on AdS4 x CP3, JHEP0811:066,2008, arXiv:0807.2559 B.-H.L, C. Park , Unbounded Multi Magnon and Spike, arXiv:0812.2727 C.Ahn, M. Kim, B-H.L., Quantum finite-size effects for dyonic magnons in the AdS_4 x CP^3. arXiv:1007.1598 [hep-th], to appear in JHEP, Based on

3. Contents 1. Duality of AdS-CFT Anomalous Dimensions of the Operators in CFT = Energy of the string states 2. Guage Theory Operators and Integrability 3. Classical String solutions (giant magnon and spikes) and Integrability 4. Algebraic curves, Exact S-matrix, and finite size corrections 5. Summary and discussion

4. 1, …, 6 0, 1, …, 3 : Nc×Nc mtx, adj. repn. of U(Nc) I. Duality of AdS-CFT Maldacena 97 * Dp branes carry tension (energy) and charge (source for p+2 form)  Gravity in AdS space (dim = ((p+1)+1) ) * Dp brane’s low energy dynamics by fluctuating open strings  Yang-Mills in (p+1) dim. (CFT) 3+1 dim N=4 SYM4+1 dim Gravity Conformal Field Theory in AdS in the 3+1 dim “boundary” 4+1 dim “bulk”, open stringclose string Ex) #Nc D3 branes : = N - = 후 ㅎ AdS5 x S5 (i) (j) (Nc) Conformal x R-symm : SO(4, 2) x SO (6) Isometry of (AdS5) x (S5) Perturbative if gN << 1 Reliable if gN >> 1

5. AdS/CFT Dictionary (for AdS5 x S5) Witten 98 Gubser-Klebanov-Polyakov 98 • 4D CFT (QCD)  5D AdS • Spectrum : • - 4D Operator  5D string states - Dim. of [Operator]  5D mass • Not easy to confirm AdS/CFT in practice •  string theory side : reliable for large tHooft coupling • quantization in AdS b.g. not known, etc. • YM theory side : reliable perturbation only for small coupling • operator mixing, etc.

6. 2). AdS –CFT for M2 Branes in M theory 2+1 dim. N=6 C.-S. QFT (ABJM Theory) Gauge Field (Chern-Simons) Scalars : ( =1,2) Fermions Gravity on Aharony, Bergman, Jafferis &Maldacena, arXiv:0806.1218

7. Not easy to confirm AdS/CFT  string theory side : reliable for large tHooft coupling quantization in AdS b.g. not known, etc. YM theory side : reliable perturbation only for small coupling operator mixing, etc. • Operators with large charges J -> additional parameter • Ex) BMN, hep-th/0202021 • Both and may be evaluated in powers of • Two sets of operators : • 1) Chiral primary op. & descendents • -> nonrenormalization theorem • 2) op. w/ large charges  classical string • Ex) chiral primary operator w/ large R-charge Tr phi^L • <-> pointlike string rotating on a big circle of S5 with v=c • some “impurities” (BMN operators) <-> almost pointlike • etc.

8. Integrability plays crucial role • Semiclassical strings of string theory with world-sheet sigma model corresponds to large operators with high excitations in gauge theory Bena, Polchinski, Roiban hep-th/0305116 String Theory Side : - Integrability in string theory - string sigma model on AdS5xS5 admit Lax representation - Exists various methods for string solutions. - Algebraic curve methods, - solution through Pohlmeyer reduction, etc. - Computation of is straightforward Gauge Theory Side : - Operator mixing matrix (that grows exponetially with the size) is dentified with the hamiltonian of an integrable spin chain - the anomalous dimension from the integrability (by algebraic Bethe ansatz, exact scattering matrix, etc.) Minahan, Zarembo 0212208

9. 2. Large Operators in Gauge Theory and Integrability Ex) N=4 SYM : Z, W, X : three complex scalar fields of SYM describing coordinates of the internal space with |Z| + |W| + |X| =1. (Z and Z-bar : the plane on which the equator of lies) 2 2 2 J = # of Z fields in SYM = ang. Mom. of string rotating along the equator of . Consider the limit

10. SU(2) sector in 1-loop) (with Z and W) Minahan and Zarembo (2002) -energy and R-charge E=1 and J=1 for Z and E=1 and J=O for W • Identifying Z with a spin down and W with a spin up • Hence, the one-loop anomalous dimension of operators Ground state Excited state ( : # of Z and W, J1 + J2 = J ) • Dilatation operator is related to the Hamiltonian of the • integrable XXX Heisenberg spin chain model • eigenvalue of the spin chain Hamiltonian • which can be solved by the Algebraic Bethe Ansatz, etc.

11. Ex) single magnon Operator Spin chain configuration the dispersion relation for the magnon Note : the all loop dispersion relation conjectured for the magnon In the large ‘t Hooft coupling limit, the dispersion relation becomes This is the same as that of the giant magnon in the string sigma model

12. Algebraic Bethe Ansatz for spin Hamiltonian (SU(2) sector) • Operators with R-charges (J1= L-J, J2=J) • Bethe equations ( : rapidities ) • or • Cyclicity • Anomalous dimension • Scattering matrix

13. Scaling limit • Bethe equatioin for the Large operator (scaling) • Distribution of the Bethe roots - density or resolvent • Scaling limit of Bethe equations • Momentum condition • Anomalous dimension

14. Comments • Integrability also for N=6 ABJM model ( AdS4 x CP3) • - excitations Ai,Bi  Two decoupled Heisenberg XXX Hamiltonian Ex)

15. Comments -continued • There exist many other types of operators Ex) (Single Trace operators, with higher twists) : The anomalous dimension is dominated by the contribution of the derivatives  Dual description in terms of rotating strings with n cusps (Conjecture) • Dilatation operators and Bethe Ansatz in higher orders • - in 2-loops – Beisert, Kristjansen & Staudacher, hep-th/0303060 • - 3-loops - Beisert, Kristjansen & Staudacher, hep-th/0303060 • Beisert hep-th/0308074, 0310252 • Klose & Plefka, hep-th/0310232 • - Higher loops – Serban & Staudacher, hep-th/0401057 • finite size effects • wrapping interactions at loop order higher than length,

16. 2 3. Classical String Solutions - Giant magnons & spikes • The dual description in the string theory side The giant magnon Ex) magnon in flat space In the light cone gauge , the solution with where Hofman & Maldacena (2006) In target space In world sheet ( )

17. 2 - (closed) string excitation : two excitations carrying world sheet momentum p and –p respectively. two trajectories (blue and green) lie in the different values of , The world sheet momentum of the string excitation corresponds to the difference of the target space coordinate ~ p - the open string case : a single excitation with momentum p along an infinite string.

18. 2 Magnon on the AdS5 x S5 - string rotating on S2 ⊂ S5 Metric on S5 Parametrization Action : Solution Dispersion Relation Note : Match with the all loop dispersion relation in the gauge theory if take the large tHooft coupling limit

19. Spike inflat spacetime in flat Minkowski In conformal gauge (Eq. of motion ) (constraints ) solution Dispersion relation

20. n = 3 n = 10 Gauge Theory Operator

21. Magnon bound states – dyonic giant magnons - the giant magnon with two angular momenta, J1 and J2 - the string moving on an RxS3 subspace of AdS5 x S5 Hofman-Maldacena limit (Hofman-Maldacena hep-th/0604135) J1, E  infinity, E-J1, J2, lambda = finite String equations with Virasoro constraints is equvalent to the complex sine Gordon equation Pohlmeyer ‘76 The dispersion relation Chen-Dorey-Okamura ‘06 Note : Operator of the Gauge theory

22. comments • Magnons and Spikes - in S5, AdS5, and AdS5 x S5 - in different background e.g., Melvin background, NS-NS B field, etc. - with 1, 2, and 3 angular momenta - multi magnons and spikes • Solutions in AdS4 x CP3 – three kinds of giant magnons - small magnon : CP1 & CP2 magnon - Pair of small magnon : RP2 and RP3 magnon - Big magnon : dressed solution

23. Comments -continued • Dispersion relations for various solutions obtained • Finite size corrections Giant magnon Spike

24. Classical Integrability of string sigma model • Focus on an SU(2) reduction of the full sigma-model to the subsector of string moving on • The string action in the conformal gauge • Equation of motion with Virasoro constraints • Eq. of motion in the weak coupling limit or where This is the equation of the classical Heisenberg model, which is completely integrable.

25. Equivalentl to the consistency condition [L, M] = 0 • for the following auxiliary linear problem • The monodromy matrix : parallel transport of the flat connection (L, M) with • The trace of the monodromy matrix : independent of tau_0 •  infinite set of integrals of motion • unimodular and unitary when the spectral parameter is real • Eigenvalues determine the quasi-momentum p(x)

26. The string action can be written as • where • The equation of motion • The equation of motion as the zero curvature condition where • Or as the consistency condition for the following linear problem

27. Monodromy matrix • resolvent • is an analytic function on the physical sheet, and can be written as • Define • Integral equation for the density • etc.

28. 4. Algebraic curves, Exact S-matrix & finite size corrections • Integrability in the spectral problem of AdS/CFT - Gauge theory  Integrable spin chain  small g  all loop - String sigma model  Lax representation • All loop Bethe ansatz and exact S-matrix (for L  infinity) • At finite L, there are corrections • We consider the finite size effects at strong coupling regime • Two independent approaches using integrability in both sides - Algebraic curve  semiclassical effects in string theory - Exact S-matrix  Luesher F-term correction

29. All three kinds of giant magnons (small (on CP1 & CP2), Pair of small (on RP2 and RP3) and Big (dressed solution) can be reproduced in algebraic curve