Plan of the course (tentative)

1. Simple and extended Poincaré supersymmetry: motivations, definitions, notation. The no-go Coleman-Mandula theorem and how it is evaded in extended susy. 4D-susy via dimensional reduction from D > 4; central charges. Examples of susy realizations on fields: physical and auxiliary fields, on-shell and off-shell realizations. Non-linear realizations of spontaneously broken supersymmetry.

2.  Superspace as a natural framework for supersymmetry. Coset approach. Merits of off-shell superfield formulations. Superspin as a generalization of the square of Pauli-Lubanski vector. Chirality as the simplest Grassmann analyticity. N=1 theories in superspace: matter and super Yang-Mills. Kähler geometry of N=1 sigma models from their superspace formulation.

3. N=2 supersymmetry and harmonic superspace. Grassmann harmonic analyticity as the true N=2 analog of N=1 chirality.  N=2 hypermultiplet on shell and off shell, infinite number of auxiliary fields as the price for off-shell unconstrained formulation. Most general hypermultiplet action and hyper Kähler sigma models.

4.  Geometric principles underlying N=0 and N=1 gauge theories. N=2 SYM theory from preserving harmonic analyticity. Wess-Zumino gauge in N=1 and N=2 supersymmetric Yang Mills (SYM) theories. Superfield and component actions. Maximally extended N=4 SYM theory in N=2 superfield formulation.

5.  Elements of quantum superfield theory. Fixing the gauge, superfield FP ghosts, vertices and propagators. Simplest Feynman supergraphs. Basic idea of non-renormalization theorems.

6. Supergravity as supersymmetrization of Einstein theory. Questions to be answered. N=1 supergravity, the method of conformal compensators. From conformal N=2 SG to N=2 Poincaré SG. Maximally extended N=8 SG.

7.  N=3 harmonic analyticity. The N=3 SYM off-shell action as  the harmonic analogue of Chern-Simons action. Ultraviolet finiteness made manifest.

8. Some instructive toy examples. N=2, N=4 and N=8 supersymmetric mechanics. N=4, 1D harmonic superspace. Gauging supersymmetric quantum mechanics as a tool to relate its various models (based on some recent work).

9. Some uses of superconformal symmetries in diverse dimensions. Conformal/AdS superalgebras. A simple example: conformal and superconformal mechanics. Tensorial superspace and its generalized superconformal symmetry OSp(1|8). Higher spins from non-linear realizations of OSp(1|8).

[Note. This course forms part of the Programa de doctorado de calidad del Tercer Ciclo del Departamento de Física Teórica de la Universitat de València, granted by the Ministerio de Educación y Ciencia. Third cycle students wishing to find reasonably priced accomodation may contact the Dept. secretary alicia.salva@uv.es for information (the actual reservation will have to be made by the participants themselves) ]