Summaries,
lecture notes and advised reading

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

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

**Twistor Geometry and Gauge Theory**(Martin Wolf):**Lecture notes:**notes (ps)**Advised references:**

- S. A. Hugget and K. P. Tod "An Introduction to Twistor Geometry", Cambridge University Press

- R. S. Ward and R. O. Wells "Twistor Geometry and Field Theory", Cambridge University Press

**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 CP^{4}

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 T^{3}fibrations

4.3.3 T^{2}x R fibrations

4.4 Example

4.4.1 O(-1) + O(-1) -> CP^{1}

4.4.2 Two dP_2's connected by a CP^{1}

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!

**Beta function, Renormalisation and Quantum Field Theory**(Johanna Erdmenger, Glenn Barnich):**Plan of the lectures:**

*I.*Brief review of field quantization and perturbation theory

*II.*One loop calculations in phi^{4}theory and QED including renormalization.

*III.*Renormalization and Symmetry: The axial anomaly.

*IV.*Renormalization group: i) Callan-Symanzik beta and gamma functions

ii) Evolution of couplings

iii) Wilson approach.**Lecture notes:**notes (pdf)**Recommended reading:**M. Peskin/D. Schroeder:An introduction to quantum field theory, chapters 1-4, 9-10, 12, 19.

**Moyal's Star Product**(Sandrine Cnockaert):**Plan of the lectures:**

Moyal's Star Product and some of its representations.

**Lecture notes:**notes (ps)**References:**see notes.

**Noncommutative Spaces and Gravity**(informal talk by Frank Meyer):**Notes:**notes (ps)