resumesss.html

Summaries, lecture notes
and advised reading

Integrable models
Algebraic topology
Kac-Moody algebras
Poisson and symplectic geometry
Background field method and pinch technique
D-branes, sheaves and categories
Topological strings
Modular forms
Special geometry

Integrable models (C. Sieg, 3h) :

Plan of the lecture:

                   Integrable structures play an important role in many fields of theoretical
                   physics. These lectures give a pedagogical introduction to integrability in
                   classical as well as quantized models. I will also give an overview on their
                   recent appearance in AdS/CFT.

Lecture notes: notes (pdf)
Advised references:

Algebraic topology (J. Evslin, A. Wijns, 4h) :

Plan of the lecture:
- Aspects of homotopy and homology theory, spectral sequences, ...
- TBA (Jara's part)
Lecture notes: notes (pdf)
Advised references:

Kac-Moody algebras (D. Persson, N. Tabti, 4h) :

Plan of the lecture:
- Some brief recap about finite Lie algebras. Cartan matrices, Dynkin diagrams, roots, weights, Weyl group, representations....
- Affine Lie algebras and their realization in terms of centrally extended loop algebras. Perhaps some mentioning of current algebras in string theory.
- Lorentzian Kac-Moody algebras with focus on their hyperbolic subset. The invariant bilinear form, generalized Casimir operator, real vs. imaginary roots, the fundamental Weyl chamber and integrable representations, level decomposition, character formulas...
- Some applications in physics. E.g. some billiards, E10 and M-theory... (Nassiba's part)
Lecture notes: notes (pdf)
Advised references:

Poisson and symplectic geometry (S. de Buyl, S. Detournay, 4h) :

Plan of the lecture:
- Definition of symplectic and Poisson manifolds, basic tools
- Examples : cotangent bundles, coadjoint orbits
- Geometric formulation of Hamiltonian and Lagrangian mechanics
- Symplectic and Hamiltonian group actions, applications (e.g.
canonical transformations, lagrangian symmetries)

                   There are other, more advanced things we could discuss depending on time
                   (symplectic reduction, (hyper-)Kahler structures, deformation quantization,
                   geometric quantization).

                   Background: classical mechanics and basic differential geometry (e.g.
                   tensor fields, connections, exterior calculus), but could be provided if
                   necessary.

Lecture notes: notes (pdf)
Advised references:

Background field method and pinch technique (V. Mathieu, 3h) :

Plan of the lecture:
- Motivations, why we need physical off-shell Green's functions.
- The S-Matrix pinch technique and the intrinsic pinch technique on the example of the gluon self-energy.
- The background field method and its relation with the pinch technique.
- The absorptive pinch technique construction.
Lecture notes: notes (pdf)
Advised references:

D-branes, sheaves and categories (C. Krishnan, 3h) :

Plan of the lecture:
1. Sigma models, toplogical strings, A/B model (quick overview focussing on things I'll need).
2. Calabi-Yau, and other consistency conditions for B-model. D-branes in the B-model. Intro to sheaves, coherent sheaves, sheaf-cohomology, categories. Derived categories of coherent sheaves as a model for B-branes, Ext groups and open strings.
3. Stability and the lift to physical string theory. Pi-stability.
4. Back to topological strings, this time A-model branes, Fukaya category.
5. Duality between A and B branes, Homological Mirror symmetry conjecture of Kontsevich.

Lecture notes: notes (pdf)
Advised references:

Topological strings (A. Collinucci, T. Wyder, 4h) :

Plan of the lecture:
1. Schwartz type TFT
2. Witten type TFT
3. Topological twisting procedure
4. Topological A model
5. Topological B model
6. Topological String Field Theories

If there is time! (Kahler gravity for A model, Kodaira-spencer theory for B model, maybe also examples of OTSFT;
depending on how much time we have and how detailled we decide to discuss the former topics)

Lecture notes: notes (pdf)
Advised references:

Modular forms (D. Van den Bleeken, 3h) :

Plan of the lecture:
- TBA (Modular, Automorphic forms)
Lecture notes:
Advised references:

Special geometry (J. Rosseel, 2h) :

Plan of the lecture:
- General intro to symplectic geometry
- Applications to CY moduli spaces, relation to e-m duality etc..
Lecture notes: notes (pdf)
Advised references: