resume.html

Summaries, lecture notes and advised reading

Yang Mills and fiber bundles
Twistor Geometry and Gauge Theory
Complex Geometry, Calabi-Yau Manifolds and Toric Geometry
Lie Algebras: representation and applications
Conformal Field Theory
Beta Function, Renormalization and Quantum Field Theory
Moyal's Star Product
Noncommutative Spaces and Gravity (informal talk)

The complete lecture notes will be available here. (soon ;-))

Yang Mills and fiber bundles (Laurent Claessens):

Plan of the lectures:
I. General manifold theory
Definition of a real manifold. Example : z=f(x,y) and the sphere (I want to explain why it doesn't works with only one chart)
Definition of a tangent vector. Why is it usually denoted by $v^i\partial_i$ ? The differential of a map from a manifold to another. How the matrix $\frac{\partial f^i}{\partial x^j}$ of the differential comes back in the manifold formalism ?
II. Fiber Bundle
The talk will extensively be concerned by the different ways to express local sections of bundles.
-Definition of a vector bundle, example : the tangent bundle; transitions functions
Sections of vector bundle; how to trivialise a section when we know a trivialisation of the vector bundle ?
Connection of a vector bundle; local expression with respect to a trivialisation of the bundle. Proof :-) of the famous transformation law $A'=g^{-1}dg+g^{-1}Ag$.
A few word about the curvature of a connection.
-Definition of a principal bundle. Example 1 : the frame bundle; example 2 : a vector bundle is NOT a GL(V)-principal bundle.
Duality between sections and trivialisation.
Definition of a gauge transformation and a local expression.
-Definition of an associated bundle. How to trivialise an associated bundle when we can trivialize the principal bundle ? Sections, local expression (two forms : an equivariant function with values in the principal bunbdle and an expression with values in the vector space).
Sketch of an answer to the most FAQed question in the world : what is a spinor ?
Action of a gauge transformation on a section.
-Connection on principal bundle : vertical space and associated 1-form. Local expressions : the so-called "gauge potential" and its curvature F. Why F=dA and dF=0 (famous equations !) when the group is abelian ?
Definition of a covariant derivative on any associated bundle from a connection on the principal bundle. The equation $D\psi=d\psi+A\psi$.
Transformation law of a connection under a gauge transformation, transformation of a section under a gauge transformation (with a proof :-) )
Why is the covariant derivative "covariant" ?
III. Electromagnetism
Well known children formalism for the Maxwell's equations.
Why it is better to write $A=A_{\mu}dx^{\mu}$ ?
The Yang-Mills's trick as seen at the university
The Yang-Mills's trick in the bundle formalism
Spin 0 electromagnetism in a U(1)-principal bundle : gauge invariance of the square norm of $D\phi$.
Lecture notes: notes (pdf),
Advised readings: text1 (pdf), text2 (pdf), dgbook.ps

Complex Geometry, Calabi-Yau Manifolds and Toric Geometry (Vincent Bouchard):

Plan of the lectures:
Lecture 1. Complex geometry
1.1 Complex manifolds
1.2 Tensors on complex manifolds
1.3 Exterior forms on complex manifolds
1.4 Cohomology
1.5 Chern classes
    1.5.1 Chern character
1.6 Holomorphic vector bundles
1.7 Divisors and line bundles
Lecture 2. Kahler geometry
2.1 Kahler manifolds
2.2 Forms on Kahler manifolds
2.3 Cohomology
2.4 Holonomy
Lecture 3. Calabi-Yau geometry
3.1 Calabi-Yau manifolds
3.2 Cohomology
3.3 Examples
    3.3.1 Chern class of CP^m
    3.3.2 Calabi-Yau condition for complete intersection manifolds
    3.3.3 The quintic in CP⁴
    3.3.4 The Tian-Yau manifold
3.4 'Local' Calabi-Yau manifolds
Lecture 4. Toric geometry
4.1 Homogeneous coordinates
4.2 Toric Calabi-Yau threefolds
4.3 Toric diagrams
    4.3.1 Toric manifolds as symplectic quotients
    4.3.2 T³ fibrations
    4.3.3 T² x R fibrations
4.4 Example
    4.4.1 O(-1) + O(-1) -> CP¹
    4.4.2 Two dP_2's connected by a CP¹
4.5 Hypersurfaces in toric varieties
    4.5.1 Reflexive polytopes
    4.5.2 Toric interpretation
Lecture notes: notes (tar.gz) or notes (ps)
Main references (see the lecture notes for a full list of references):
-P. Candelas, "Lectures on complex geometry", in Trieste 1987, Proceedings, Superstrings '87. 1987.
-K. Hori et al, Mirror Symmetry, vol.1 of Clay Mathematics Monographs. American Mathematical Society, 2003. 929 p.
-T. Hubsch, Calabi-Yau manifolds: a bestiary for physicists. World Scientific, 1992. 374 p.
-D. D. Joyce, Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford University Press, 2000. 436 p.
-M. Nakahara, Geometry, topology and physics. The Institute of Physics, 2002. 520 p.
-H. Skarke, "String dualities and toric geometry: an introduction," arXiv:hep-th/9806059.

Lie Algebras: representation and applications (Xavier Bekaert, Nicolas Boulanger, Sophie de Buyl, Francis Dolan, Jeong-Hyuck Park):

Plan of the lectures:
part I: Structure of semi-simple complex Lie algebras
- basic definitions
- The Cartan-Weyl basis
- The Killing form
- Simple roots, Cartan matrix, Dynkin diagrams
- The Chevalley basis
- Weyl group
part II: Representation theory
- Fondamental and highest weights
- Freudenthal recursion formula
- Racah-Speiser algorithm
- Weyl's character formula and dimensionality formula
part III: Applications to A_2, the complexification of su(3)
part IV: Clifford algebra
-Gamma matrices: in even dimension, in odd dimension, Lorentz transformations, crucial identities for SYM
-Spinors: Weyl spinor, Majorana spinor, Majorana-Weyl spinor
-Majorana representation and SO(8) triolity
part V:Universal enveloping algebras and higher symmetries
- Some taste of abstract Algebra
    * Free, tensor, symmetric
    * Associative vs Lie : Universal enveloping
- Physical applications
    * Casimir operators
    * Higher symmetries
Advised references:
-J.E.Humphreys, "Introduction to Lie algebras and representation theory", Graduate texts in Mathematics 9, Springer Verlag (2000).
-J.F.Cornwell, "Group theory in physics: An introduction", San Diego, USA: Academics (1997).
-J. Fuchs and C. Schweigert, "Symmetries, Lie Algebras and Representations:A graduate course for physicists", Cambridge, UK, University Press (1997).
-W.Fulton and J.Harris, "Representation Theory: A First Course", Springer Verlag (1991) 3rd Edition.
-Kugo and Townsend, Nucl. Phys. B221: 357, 1983
Lecture notes: partI-II (ps), partIV (pdf), partV (pdf).

Conformal Field Theory (Stéphane Detournay):

Plan of the lectures:
Conformal field theory in two dimensions
- local/global conformal transformations, primary/quasi-primary fields
- conformal Ward identities, operator product expansion
- Energy momentum tensor as generator of conformal symmetry
- radial and conformal normal orderings
- Field/state correspondance
- Virasoro algebra, Verma modules
Lecture notes: notes (pdf).
Advised references:
- Conformal field theory (P Di Francesco, P Mathieu, D Senechal,New York, USA: Springer (1997) ), Chap. 4,5,6,15 essentially
- Introduction to conformal field theory, AN Schellekens ,Fortsch. Phys, 1996,
http://staff.science.uva.nl/~jdeboer/stringtheory/CFT.ps
- Some reviews on the arXiv, see e.g. Ginsparg (hep-th/9108028), Gaberdiel (hep-th/9910156), Walton (hep-th/9911187) and more...
Background material:
Some basic knowledge of Quantum Field Theory (Poincare invariance, Noether's theorem, correlation functions, ...). Chapter 2 of the first reference provides a good idea of what we will need!