Minkyoo Kim (Wigner Research Centre for Physics) 9th, September, 2013 Seminar in KIAS

Zoltan Bajnok - Wigner Research Center for Physics, Hungary Laszlo Palla - Eotvos Roland University, Hungary Piotr Surowka - International Solvay Institute, Beligium

 Introduction - String theory, AdS/CFT and its integrability  Bethe ansatz, Spectral curves and Y-system  Explicit example : Circular strings  Quantum effects from algebraic curve  Open strings attached to Y=0 brane - Curves from all-loop Bethe ansatz equations - Curves from scaling limit of Y-system - Direct derivation in case of the explicit string solution  Discussion

 Surprising duality between string and gauge theory - String theory on some supergravity background which obtained from near horizon limit of multiple D-branes can be mapped to the conformal gauge field theory defined in lower dimensional spacetime. - Holographic principle  In AdS/CFT, specific string configurations on AdS backgrounds have corresponding dual descriptions as composite operators in CFT.

 In this duality, QFT is not included in but equivalent to String theory!!  Consider string theory as a framework for nature - Use it to study non-perturbative aspects of QFT - Pursue to check more and prove - Non-perturbative regime of string (gauge) theory has dual, perturbative regime of gauge (string) theory.  What idea was helpful for us?

 In both sides of AdS/CFT, integrable structures appeared.  Drastic interpolation method for integrability -> - all-loop Bethe ansatz equations - exact S-matrices - TBA and Y-system  Very successful for spectral problem  “Integrable aspects of String theory”

 NR integrable system – particle on sphere with the potential  Infinite conserved charges construced from integrals of motion of NR integrable system  Coupled NR systems with Virasoro constraints - if we confine to sphere part, then the 1 st equation becomes a SG equation.

 Coset construction of string action : SU(2) sector  Rescaled currents are flat too.  Monodromy & Transfer matrix  Local charges can be obtained from quasi-momentum.  Resolvent and integral equation

 Lax representation of full coset construction  Generally, strings on semisymmetric superspace are integrable from Z4 symmetry. - All known examples  Classical algebraic curve - Fully use the integrability - Simple poles in Monodromy - 8 Riemann surfaces with poles and cuts - Can read analytic properties - Efficient way to obtain charges and leading quantum effects

 MT superstring action  Bosonic case  Lax connection

 Flat connection -> path independ. -> Monodromy  Collection of quasi-momenta as eigenvalues of Monodromy  Analogy of Mode numbers and Fourier mode amplitudes in flat space

 N=4 SYM in 4-dimension  Field contents - All are in the adjoint rep. of SU(N).  The β -function vanishes at all order. Also, Additional superconformal generators -> SCFT, full symmetric group PSU(2,2|4)  Knowing conformal dimensions and also structure constants : in principle, we can determine higher point correlation functions.

 Two point function -> conformal dimension  Dilatation operator  Chiral primary operator (BPS) It doesn’t have any quantum corrections.  For non-BPS operators, “Anomalous dimension”  In SU(2) sector, there are two fields : X, Z.  Heisenberg spin-chain Hamiltonian appears.

 One-loop dilatation in SO(6) sector

 R-matrix construction -> infinite conserved charges  Bethe ansatz can diagonalize the Hamiltonian.  Periodicity of wave function the Bethe equations  We can determine conformal dimension of SYM from solving Bethe ansatz up to planar & large L limit.

 Scaling limit of all-loop BAES

 Circular string solution

 Quasi-momenta

 Giant magnon solutions - dual to fundamental excitation of spin-chain - Dispersion relation - Log cut solution

 Quasi-momenta make multi-sheet algebraic curve.  Giant-magnon -> logarithmic cut solution in complex plane  Deforming quasi-momenta - Finite-size effects - Classical effects : Resolvent deformation - Quantum effects : Adding poles to original curve

 Fluctuations of quasi-momenta  GM  Circular string

 One-loop energy shifts  From exact dispersion, we know one-loop effects are finite-size piece.

 If we move from plane to cylinder, we have to consider the scattering effects with virtual particles.  Finite-size correction can be analyzed from this effects.  Also, we can use the exact S-matrix to determine the energy correction – Luscher’s method.

 Successes of Luscher’s method - string theory computation (giant magnon) - gauge theory computation (Konishi operator) recently, done up to 7-loop and matched well with FiNLIE up to 6-loop

 Luscher’s method is only valid for not infinite but large J. So, for short operator/slow moving strings, we need other way.  TBA and Y-systems - 2D model on cylinder (L,R(infinite)) -> L-R symmetry under double wick rotations -> Finite size energy on (L,R) = free energy on (R,L) -> Full expression also has entropy density for particles and holes.

 For Lee-Yang model,  For AdS/CFT,  Each Y-funtions are defined on T-hook lattice.

 Open string integrability  For analyticity of eigenvalues

 Double row transfer matrix

 Generating function of transfer matrix  Same Y-system with different asymptotic solutions  The scaling limit gives classical strings curves.

 Quasi-momenta for open strings  How can be these curves checked? - From strong coupling limit of open BAEs - From any explicit solutions

 Open string boundary conditions  Scaling limit of all-loop Bethe equations

 SU(2) sector - (a) Bulk and boundary S-matrices part - (b), (c) Dressing factors  Quasi-momentum can be constructed and compared with Y-system result.

 Full sector quasi-momenta from OBAEs

 Full sector quasi-momenta from Y-systems

 Can be shown that they are equivalent under the following the roots decompositions

 Circular strings  From group element g, we can define the currents. But, it’s not easy to solve the linear problem.

 Trial solutions by Romuald Janik (Thanks so much!)  Satisfied with the integrable boundary conditions and resultant quasi-momenta are matched.

 Integrable structures in AdS/CFT  Beyond perturbative integrability, we can fully use integrability through the spectral curves, exact S- matrices, BAEs and Y-system.  We studied the open strings case. - Direct way as solving the linear equation - Scaling limit of BAEs and Y-system - Used the conjecture by BNPS