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Seminar in KIAS Minkyoo Kim (Wigner Research Centre for Physics) 9th, September, 2013 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 Quantum & Relativistic theory for 1-dim. Object Worldsheet conformal invariance Spacetime supersymmetry - lead to 5 consistent superstring theories in 10-dim. target spacetime with YM gauge symmetry based on two kinds of Lie algebras 𝑆𝑂(32) & 𝐸8 × 𝐸8 Compactification of extra dimension - 𝐶𝑌3 for low energy N=1 SUSY Finding a specific & unique compactification which allows already known particle physics results and predicts new phenomena like supersymmetry etc. - till now, unsuccessful - huge numbers of false vacua ~10500 - Landscape, anthropic principle? In old approach, quantum gauge field theories for nature are included in String theory. 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. Maldacena’s conjecture - IIB string on 𝐴𝑑𝑆5 × 𝑆 5 - effective tension N=4 SYM in 4D 𝑅2 α′ coupling constant λ = 𝑔𝑌𝑀 2 𝑁 - string coupling 𝑔𝑠 number of colors 𝑁 {𝐸𝑛 (λ, 𝑁)} = {𝐷𝑛 (λ, 𝑁)} λ= 𝑅2 α′ Planar limit - Large 𝑁 with fixed λ , λ 𝑁 = 𝑔𝑠 “𝑔𝑠 → 0 ” “planar diagram” 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” Type IIB superstring on 𝐴𝑑𝑆5 × 𝑆 5 Super-isometry group : PSU(2,2|4) ⊃ SO(4,2) x SO(6) Difficult to quantize strings in curved background One way to circumvent - Penrose limit and its curvature correction 3+3 Cartan generators of 𝑆𝑂(4,2) × 𝑆𝑂(6) Rotating string ansatz Neumann-Rosochatius model 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 1st equation becomes a SG equation. String theory on 𝑅 × 𝑆 2 in static gauge could be reduced to SG model from so called Polmeyer reduction. For example, SG kink Giant magnon O(4) model complex SG model 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 Smatrices, 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 Generalization to Z=0 branes Spectral curves in other models - 𝐴𝑑𝑆5 , 𝐴𝑑𝑆4 , 𝐴𝑑𝑆3 , 𝐴𝑑𝑆2 - Beta-deformed theory, Open string theory Final destination : Quantum algebraic curve (?) - Today, in 1305:1939, P-system (?)