1 / 47

Spectral curves of open strings attached to the Y=0 Brane

Seminar in KIAS. Spectral curves of open strings attached to the Y=0 Brane. Minkyoo Kim (Wigner Research Centre for Physics) 9th, September, 2013. Work done in collaboration with. Zoltan Bajnok - Wigner Research Center for Physics, Hungary Laszlo Palla

les
Télécharger la présentation

Spectral curves of open strings attached to the Y=0 Brane

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Seminar in KIAS Spectral curves of open strings attached to the Y=0 Brane Minkyoo Kim (Wigner Research Centre for Physics) 9th, September, 2013

  2. Work done in collaboration with ZoltanBajnok - Wigner Research Center for Physics, Hungary Laszlo Palla - Eotvos Roland University, Hungary PiotrSurowka - International Solvay Institute, Beligium

  3. Contents • 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

  4. Old aspects of string theory (1) • Quantum & Relativistic theory for 1-dim. Object • Worldsheet conformal invariance • Spacetimesupersymmetry - lead to 5 consistent superstring theories in 10-dim. target spacetime with YM gauge symmetry based on two kinds of Lie algebras & • Compactification of extra dimension - for low energy N=1 SUSY

  5. Old aspects of string theory (2) • 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 - Landscape, anthropic principle? • In old approach, quantum gauge field theories for nature are included in String theory.

  6. AdS/CFT (1) • 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.

  7. AdS/CFT (2) • Maldacena’s conjecture - IIB string on N=4 SYM in 4D - effective tension coupling constant - string coupling number of colors {} = {} , • Planar limit - Large with fixed “ ” “planar diagram”

  8. AdS/CFT (3) • 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-perturbativeregime of string (gauge) theory has dual, perturbative regime of gauge (string) theory. • What idea was helpful for us?

  9. AdS/CFT and Integrability • 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”

  10. String non-linear sigma model • Type IIB superstring on • 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

  11. Reduced model (1) • 3+3 Cartan generators of • Rotating string ansatz • Neumann-Rosochatius model

  12. Reduced model (2) • 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.

  13. Reduced model (3) • String theory on 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

  14. String Integrability (1) • 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

  15. String Integrability (2) • Lax representation of full coset construction • Generally, strings on semisymmetricsuperspace are integrablefrom 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

  16. String Integrability (3) • MT superstring action • Bosonic case • Lax connection

  17. String Integrability (4) • Flat connection -> path independ. -> Monodromy • Collection of quasi-momenta as eigenvalues of Monodromy • Analogy of Mode numbers and Fourier mode amplitudes in flat space

  18. String Integrability(5)

  19. CFT side • 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.

  20. PerturbativeIntegrability (1) • 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.

  21. PerturbativeIntegrability(2) • One-loop dilatation in SO(6) sector

  22. PerturbativeIntegrability (3) • 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.

  23. Exact Integrability • Scaling limit of all-loop BAES

  24. Examples : Circular strings (1) • Circular string solution

  25. Examples : Circular strings (1) • Quasi-momenta

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

  27. Quantum effects from Al. curve (1) • 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

  28. Quantum effects from Al. curve (2) • Fluctuations of quasi-momenta • GM • Circular string

  29. Quantum effects from Al. curve (3) • One-loop energy shifts • From exact dispersion, we know one-loop effects are finite-size piece.

  30. Finite-size effects – Luscher (1) • 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.

  31. Finite-size effects – Luscher (2) • Successes of Luscher’s method - string theory computation (giant magnon) - gauge theory computation (Konishioperator) recently, done up to 7-loop and matched well with FiNLIE up to 6-loop

  32. Exact Finite-size effects (1) • 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.

  33. Exact Finite-size effects (2) • For Lee-Yang model, • For AdS/CFT, • Each Y-funtions are defined on T-hook lattice.

  34. Coupling-Length diagram

  35. Open strings side (1) • Open string integrability • For analyticity of eigenvalues

  36. Open strings side (2) • Double row transfer matrix

  37. Open strings side (3) • Generating function of transfer matrix • Same Y-system with different asymptotic solutions • The scaling limit gives classical strings curves.

  38. Open strings side (4) • Quasi-momenta for open strings • How can be these curves checked? - From strong coupling limit of open BAEs - From any explicit solutions

  39. Open strings - Y=0 brane (1) • Open string boundary conditions • Scaling limit of all-loop Bethe equations

  40. Open strings - Y=0 brane(2) • 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.

  41. Open strings - Y=0 brane(3) • Full sector quasi-momenta from OBAEs

  42. Open strings - Y=0 brane(4) • Full sector quasi-momenta from Y-systems

  43. Open strings - Y=0 brane(5) • Can be shown that they are equivalent under the following the roots decompositions

  44. Open strings - Y=0 brane(6) • Circular strings • From group element g, we can define the currents. But, it’s not easy to solve the linear problem.

  45. Open strings - Y=0 brane(7) • Trial solutions by RomualdJanik (Thanks so much!) • Satisfied with the integrable boundary conditions and resultant quasi-momenta are matched.

  46. Conclusions • Integrable structures in AdS/CFT • Beyond perturbativeintegrability, we can fully use integrabilitythrough 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

  47. Discussions • Generalization to Z=0 branes • Spectral curves in other models - - Beta-deformed theory, Open string theory • Final destination : Quantum algebraic curve (?) - Today, in 1305:1939, P-system (?)

More Related