SPECTRAL CURVES OF OPEN STRINGS ATTACHED TO THE Y=0 BRANE

Minkyoo Kim (Wigner Research Centre for

Physics) 9th, September, 2013

Seminar in KIAS

Work done in collaboration with

Zoltan Bajnok - Wigner Research Center for Physics, Hungary

Laszlo Palla - Eotvos Roland University, Hungary

Piotr Surowka - International Solvay Institute, Beligium

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 so-lution Discussion

Old aspects of string theory (1) 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 &

Compactification of extra dimension - for low energy N=1 SUSY

Old aspects of string theory (2) Finding a specific & unique compactification

which allows already known particle physics results and predicts new phenomena like su-persymmetry 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.

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.

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”

AdS/CFT (3) In this duality, QFT is not included in but equiva-

lent 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?

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”

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

Reduced model (1) 3+3 Cartan generators of

Rotating string ansatz

Neumann-Rosochatius model

Reduced model (2) NR integrable system – particle on sphere with

the potential Infinite conserved charges construced from in-

tegrals 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.

Reduced model (3) String theory on in static gauge could be re-

duced to SG model from so called Polmeyer reduction.

For example, SG kink Giant magnon O(4) model complex SG model

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-momen-tum.

Resolvent and integral equation

String Integrability (2) Lax representation of full coset construction Generally, strings on semisymmetric super-

space 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

String Integrability (3) MT superstring action

Bosonic case

Lax connection

String Integrability (4) Flat connection -> path independ. -> Mon-

odromy

Collection of quasi-momenta as eigenvalues of Monodromy

Analogy of Mode numbers and Fourier mode amplitudes in flat space

String Integrability (5)

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, Addi-tional superconformal generators

-> SCFT, full symmetric group PSU(2,2|4) Knowing conformal dimensions and also struc-

ture constants : in principle, we can determine higher point correlation functions.

Perturbative Integrability (1) Two point function -> conformal dimension Dilatation operator

Chiral primary operator (BPS) It doesn’t have any quantum correc-tions. For non-BPS operators, “Anomalous dimension” In SU(2) sector, there are two fields : X, Z. Heisenberg spin-chain Hamiltonian appears.

Perturbative Integrability (2)

One-loop dilatation in SO(6) sector

Perturbative Integrability (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.

Exact Integrability Scaling limit of all-loop BAES

Examples : Circular strings (1)

Circular string solution

Examples : Circular strings (1)

Quasi-momenta

Giant magnon solutions Giant magnon solutions - dual to fundamental excitation of spin-chain - Dispersion relation

- Log cut solution

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

Quantum effects from Al. curve (2)

Fluctuations of quasi-momenta

GM

Circular string

Quantum effects from Al. curve (3) One-loop energy shifts

From exact dispersion, we know one-loop effects are finite-size piece.

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 de-termine the energy correction – Luscher’s method.

Finite-size effects – Luscher (2) Successes of Luscher’s method - string theory computation (giant magnon)

- gauge theory computation (Konishi opera-tor) recently, done up to 7-loop and matched well with FiNLIE up to 6-loop

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 rota-tions -> Finite size energy on (L,R) = free energy on (R,L) -> Full expression also has entropy density for particles and holes.

Exact Finite-size effects (2) For Lee-Yang model,

For AdS/CFT,

Each Y-funtions are defined on T-hook lattice.

Coupling-Length diagram

Open strings side (1) Open string integrability

For analyticity of eigenvalues

Open strings side (2) Double row transfer matrix

Open strings side (3) Generating function of transfer matrix

Same Y-system with different asymptotic so-lutions

The scaling limit gives classical strings curves.

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

Open strings - Y=0 brane (1) Open string boundary conditions

Scaling limit of all-loop Bethe equations

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.

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

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

Open strings - Y=0 brane (5) Can be shown that they are equivalent under

the following the roots decompositions

Open strings - Y=0 brane (6) Circular strings

From group element g, we can define the cur-rents.

But, it’s not easy to solve the linear problem.

Open strings - Y=0 brane (7) Trial solutions by Romuald Janik (Thanks so

much!)

Satisfied with the integrable boundary conditions and resultant quasi-momenta are matched.

Conclusions 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

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 (?)