minkyoo kim (wigner research centre for physics) 9th, september, 2013 seminar in kias
TRANSCRIPT
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SPECTRAL CURVES OF OPEN STRINGS ATTACHED TO THE Y=0 BRANE
Minkyoo Kim (Wigner Research Centre for
Physics) 9th, September, 2013
Seminar in KIAS
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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
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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
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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
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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.
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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.
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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”
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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?
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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”
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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 back-
ground One way to circumvent - Penrose limit and its curvature correction
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Reduced model (1)
3+3 Cartan generators of
Rotating string ansatz
Neumann-Rosochatius model
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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 con-straints
- if we confine to sphere part, then the 1st equation becomes a SG equation.
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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
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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
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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
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String Integrability (3)
MT superstring action
Bosonic case
Lax connection
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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
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String Integrability (5)
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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.
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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.
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Perturbative Integrability (2)
One-loop dilatation in SO(6) sector
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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.
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Exact Integrability Scaling limit of all-loop BAES
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Examples : Circular strings (1)
Circular string solution
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Examples : Circular strings (1)
Quasi-momenta
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Giant magnon solutions Giant magnon solutions - dual to fundamental excitation of spin-chain - Dispersion relation
- Log cut solution
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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
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Quantum effects from Al. curve (2)
Fluctuations of quasi-momenta
GM
Circular string
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Quantum effects from Al. curve (3)
One-loop energy shifts
From exact dispersion, we know one-loop effects are finite-size piece.
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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.
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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
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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.
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Exact Finite-size effects (2) For Lee-Yang model,
For AdS/CFT,
Each Y-funtions are defined on T-hook lattice.
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Coupling-Length diagram
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Open strings side (1) Open string integrability
For analyticity of eigenvalues
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Open strings side (2) Double row transfer matrix
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Open strings side (3) Generating function of transfer matrix
Same Y-system with different asymptotic so-lutions
The scaling limit gives classical strings curves.
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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
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Open strings - Y=0 brane (1) Open string boundary conditions
Scaling limit of all-loop Bethe equations
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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.
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Open strings - Y=0 brane (3) Full sector quasi-momenta from OBAEs
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Open strings - Y=0 brane (4) Full sector quasi-momenta from Y-systems
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Open strings - Y=0 brane (5) Can be shown that they are equivalent under
the following the roots decompositions
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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.
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Open strings - Y=0 brane (7) Trial solutions by Romuald Janik (Thanks so
much!)
Satisfied with the integrable boundary condi-tions and resultant quasi-momenta are matched.
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Conclusions Integrable structures in AdS/CFT
Beyond perturbative integrability, we can fully use integrability through the spectral curves, exact S-matrices, BAEs and Y-sys-tem.
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
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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 (?)