This school is part of the European Network ENIGMA.
Its main theme is quantum integrability. Its program will be
organized
around four main lecturers delivering 3 lectures of 1.30h each:
- AdS-CFT: Niklas Beisert
- Semi-Classical Methods: Yves Colin de Verdière
- Combinatorics: Philippe Di Francesco
- Correlation Functions: Tetsuji Miwa
In addition, seminars by participants will be organized and time for
discussions will be available.
|
9h30-11h |
|
11h30-13h |
|
14h30-16h |
|
16h30-17h30 |
Monday, 15 |
Introduction |
|
Miwa |
|
Beisert |
|
Seminar |
Tuesday, 16 |
Colin de Verdière |
|
Beisert |
|
Miwa |
|
Seminar |
Wednesday, 17 |
Di Francesco |
|
Colin de Verdière |
|
Free |
|
Free |
Thursday, 18 |
Beisert |
|
Miwa |
|
Di Francesco |
|
Seminar |
Friday, 19 |
Colin de Verdière |
|
Di Francesco |
|
Leave |
|
Leave |
Abstracts:
Niklas Beisert
The study of Maldacena's AdS/CFT (string/gauge) correspondence
has
recently advanced a lot by making use of the apparent
integrability
of the models in the planar limit. Using integrable methods like
the
Bethe ansatz allows us to compute observables away from the
natural
(disjoint) perturbative regimes of the constituent gauge and
string
models and thus see the duality of the models explicitly.
In my set of lectures I will start by giving an overview of
the
AdS/CFT duality. I will then introduce and review several aspects
and
tools of integrability in AdS/CFT such as: * The classical
spectrum
of string theory by means of spectral curves. * Spin chains and
the
Bethe ansatz. * Algebraic construction of the
S-matrix.
Lecture notes:
Intro,
AdsCFT,
SU22,
Curve.
Yves Colin de Verdière
An introduction to semi-classical analysis.
I plan to cover some of the following topics:
-- microlocalization, pseudo-differential operators and Fourier
integral operators, star-products
-- Bohr-Sommerfeld quantization rules around critical levels
-- Semi-classical normal forms
-- Wigner measures and equipartition of waves
References:
. Dimassi-Sjöstrand, Spectral asymptotics in the semi-classical
limit. 1999
. Evans-Zworski, Lectures on semi-classical analysis,
London mathematical society lecture note series (268).
http://math.berkeley.edu/~zworski/
. San Vu Ngoc, Symplectic Technics for semi-classical integrable
systemes, http://www-fourier.ujf-grenoble/~svungoc/
Philippe Di Francesco
Borrowing examples from statistical physics, we
address
various enumerative and algebraic combinatorial
problems,
such as counting maps with marked points at fixed
geodesic
distances, or exploring relations between alternating
sign
matrices, plane partitions and the geometry of
orbital
varieties, via a connection to lattice loop gas. We
will
show how classical or quantum
integrability
inherited from the structure of the physical theories at
hand
can be used to derive exact combinatorial
results.
Lecture notes:
Combinatorics.
Tetsuji Miwa
Correlation functions of the XXZ model.
The XXZ model is an exactly solvable model of quantum spin
chain.
It contains a parameter Delta. When Delta=1, the model is
called
the XXX model, and Bethe invented the Bethe Ansatz method
for
diagonalization of the XXX Hamiltonian. In this lecture I give a
summary
of the recent developments by Boos, Jimbo Smirnov, Takeyama and myself
on the computation of the correlation functions of
the
XXZ model.
The basic ingredient is construction of
certain Grassmann
variables which act on the quantum space C^2
otimes ... otimes C^2
in terms of L operators. We obtain an
algebraic formula for the
correlation functions. This is a generalization
of
the classical results that the nearest and the next nearest neighboor
correlation functions for the XXX model is given by using
log2 and zeta(3).