Presentation on theme: "Yerevan State University / Leibniz Universität Hannover Supersymmetry in Integrable Systems - SIS'10 International Workshop, 24-28 August 2010, Yerevan,"— Presentation transcript:
Yerevan State University / Leibniz Universität Hannover Supersymmetry in Integrable Systems - SIS'10 International Workshop, August 2010, Yerevan, Armenia Bum-Hoon Lee Center for Quantum Spacetime Sogang University Seoul, Korea Integrability in AdS/CFT and finite size effects in dyonic Magnons
Based on 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: J. Kluson, B.-H.L, K. Panigrahi, C. Park, Magnon like solutions for strings in I-brane background. JHEP 0808;032, 2008, arXiv: B.-H.L, K. Panigrahi, C. Park, Spiky Strings on AdS4 x CP3, JHEP 0811:066,2008, arXiv: B.-H.L, C. Park, Unbounded Multi Magnon and Spike, arXiv: C.Ahn, M. Kim, B-H.L., Quantum finite-size effects for dyonic magnons in the AdS_4 x CP^3. arXiv: [hep-th], to appear in JHEP,
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 Contents
3+1 dim N=4 SYM 4+1 dim Gravity Conformal Field Theory in AdS in the 3+1 dim “boundary” 4+1 dim “bulk”, open string close string I. Duality of AdS-CFT * 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) (i) (j) (N c ) 0, 1, …, 3 1, …, 6 : Nc×Nc mtx, adj. repn. of U(Nc) AdS5 x S5 Conformal x R-symm : SO(4, 2) x SO (6) Isometry of (AdS5) x (S5) Perturbative if gN > 1 Maldacena 97 = N - = 후 ㅎ Ex) #Nc D3 branes :
- Dim. of [Operator] 5D mass AdS/CFT Dictionary (for AdS5 x S5) 4D CFT (QCD) 5D AdS Spectrum : - 4D Operator 5D string states Witten 98 Gubser-Klebanov-Polyakov 98 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.
2). AdS –CFT for M2 Branes in M theory Aharony, Bergman, Jafferis &Maldacena, arXiv: Gravity on 2+1 dim. N=6 C.-S. QFT (ABJM Theory) Gauge Field (Chern-Simons) Scalars : ( =1,2) Fermions
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. 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. Operators with large charges J -> additional parameter Ex) BMN, hep-th/ Both and may be evaluated in powers of
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 Integrability plays crucial role Minahan, Zarembo 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.) 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/
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. Large Operators in Gauge Theory and Integrability 222 J = # of Z fields in SYM = ang. Mom. of string rotating along the equator of. Ex) N=4 SYM : Consider the limit
Hence, the one-loop anomalous dimension of operators Minahan and Zarembo (2002) SU(2) sector in 1-loop) 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. Identifying Z with a spin down and W with a spin up Ground state Excited state ( : # of Z and W, J 1 + J 2 = J ) (with Z and W) -energy and R-charge E=1 and J=1 for Z and E=1 and J=O for W
Ex) single magnon In the large ‘t Hooft coupling limit, the dispersion relation becomes Operator Spin chain configuration the dispersion relation for the magnon This is the same as that of the giant magnon in the string sigma model Note : the all loop dispersion relation conjectured for the magnon
Bethe equatioin for the Large operator (scaling) Distribution of the Bethe roots - density or resolvent Scaling limit of Bethe equations Momentum condition Anomalous dimension Scaling limit
Comments Integrability also for N=6 ABJM model ( AdS4 x CP3) - excitations Ai,Bi Two decoupled Heisenberg XXX Hamiltonian Ex)
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) Comments -continued Dilatation operators and Bethe Ansatz in higher orders - in 2-loops – Beisert, Kristjansen & Staudacher, hep-th/ loops - Beisert, Kristjansen & Staudacher, hep-th/ Beisert hep-th/ , Klose & Plefka, hep-th/ Higher loops – Serban & Staudacher, hep-th/ finite size effects wrapping interactions at loop order higher than length,
The giant magnon Ex) magnon in flat space In the light cone gauge, the solution with where In world sheet ( ) In target space 2 Hofman & Maldacena (2006) 3. Classical String Solutions - Giant magnons & spikes The dual description in the string theory side
- (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 2 - the open string case : a single excitation with momentum p along an infinite string. ~ p
Metric on S5 Parametrization Action : 2 Magnon on the AdS5 x S5 - string rotating on S 2 ⊂ S 5 Dispersion Relation Solution Note : Match with the all loop dispersion relation in the gauge theory if take the large tHooft coupling limit
Spike in flat spacetime In conformal gauge in flat Minkowski solution (Eq. of motion ) (constraints ) Dispersion relation
n = 3n = 10 Gauge Theory Operator
Magnon bound states – dyonic giant magnons The dispersion relation - the giant magnon with two angular momenta, J1 and J2 - the string moving on an RxS3 subspace of AdS5 x S5 Note : Operator of the Gauge theory Hofman-Maldacena limit ( Hofman-Maldacena hep-th/ ) J 1, E infinity, E-J 1, J 2, lambda = finite String equations with Virasoro constraints Pohlmeyer ‘76 Chen-Dorey-Okamura ‘06 is equvalent to the complex sine Gordon equation
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
Comments -continued Giant magnon Spike Finite size corrections Dispersion relations for various solutions obtained
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.
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)
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
resolvent is an analytic function on the physical sheet, and can be written as Monodromy matrix Define Integral equation for the density etc.
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
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
5. summary & Discussion Furthermore, the magnon bound state is also described by a giant magnon with two angular momentum The integrability plays an important role and is shown to exist in both sides of the gauge and string sides. AdS/CFT Dimension of an operator in Gauge Theory = Energy of the corresponding string state Ex) the magnon in the spin chain corresponds to the giant magnon solution in string theory, etc. Also mentioned solutions of Spikes on R x S2 with B field Rotating String on Melvin deformed AdS3 x S3 Three spin spiky solutions on AdS3 x S3 -> circular/helical strings on AdS - Multi magnon and spike solutions Classical strings can be reprented by algebraic curve. Ex) Various magnons (small, pair of small, big) in AdS4 x CP3
Summary & Discussion -continued Much of the AdS / CFT still need to be confirmed such as finding the dual integrable model corresponding to the spike solution, etc.