Yerevan State University / Leibniz Universität Hannover

Supersymmetry in Integrable Systems - SIS'10International Workshop, 24-28 August 2010, Yerevan, Armenia

Bum-Hoon Lee

Center for Quantum SpacetimeSogang 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: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, JHEP 0811:066,2008 , 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,

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)

ijIX

ijA

(i) (j) (Nc)

FermionsDDFFTrg

LJI

JIIuu

YM

,

22 )],[

21

41(

41

:,, IA

0, 1, …, 31, …, 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 Reliable 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:0806.1218

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/0202021 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 0212208

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/0305116

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

2 2 2

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 upGround stateExcited state ( : # of Z and W, J1 + J2 = 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

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

• 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 HamiltonianEx)

• 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/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,

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 sheetmomentum 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 thedifference 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 S2 ⊂ S5

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 = 3 n = 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/0604135) J1, E infinity, E-J1, J2, lambda = finiteString 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.