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

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 (?)