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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
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Introduction
- String theory, AdS/CFT and its integrability
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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
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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
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Compactification of extra dimension
- 𝐶𝑌3 for low energy N=1 SUSY
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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?
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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
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In AdS/CFT, specific string configurations on AdS
backgrounds have corresponding dual descriptions
as composite operators in CFT.
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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”
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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.
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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
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“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
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3+3 Cartan generators of 𝑆𝑂(4,2) × 𝑆𝑂(6)
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Rotating string ansatz
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Neumann-Rosochatius model
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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.
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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
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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
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MT superstring action
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Bosonic case
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Lax connection
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Flat connection -> path independ. -> Monodromy
Collection of quasi-momenta as eigenvalues of
Monodromy
Analogy of Mode numbers and Fourier mode
amplitudes in flat space
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N=4 SYM in 4-dimension
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Field contents - All are in the adjoint rep. of SU(N).
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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.
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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.
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One-loop dilatation in SO(6) sector
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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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Scaling limit of all-loop BAES
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Circular string solution
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Quasi-momenta
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Giant magnon solutions
- dual to fundamental
excitation of spin-chain
- Dispersion relation
- Log cut solution
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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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Fluctuations of quasi-momenta
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GM
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Circular string
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One-loop energy shifts
From exact dispersion,
we know one-loop effects
are finite-size piece.
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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.
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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.
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For Lee-Yang model,
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For AdS/CFT,
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Each Y-funtions are defined
on T-hook lattice.
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Open string integrability
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For analyticity of eigenvalues
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Double row transfer matrix
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Generating function of transfer matrix
Same Y-system with different asymptotic solutions
The scaling limit gives classical strings curves.
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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 string boundary conditions
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Scaling limit of all-loop Bethe equations
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SU(2) sector
- (a) Bulk and boundary S-matrices part
- (b), (c) Dressing factors
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Quasi-momentum can be constructed and
compared with Y-system result.
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Full sector quasi-momenta from OBAEs
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Full sector quasi-momenta from Y-systems
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Can be shown that they are equivalent under the
following the roots decompositions
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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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Trial solutions by Romuald Janik (Thanks so much!)
Satisfied with the integrable boundary conditions
and resultant quasi-momenta are matched.
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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
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Generalization to Z=0 branes
Spectral curves in other models
- 𝐴𝑑𝑆5 , 𝐴𝑑𝑆4 , 𝐴𝑑𝑆3 , 𝐴𝑑𝑆2
- Beta-deformed theory, Open string theory
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Final destination : Quantum algebraic curve (?)
- Today, in 1305:1939, P-system (?)