The focus of the present book is on the geometric structures underlying all supersymmetric field theories (classical and quantum). The language of geometric structures on smooth manifolds allows us to describe in a uniform and highly unified way all possible situations: rigid supersymmetry as well as local supergravity, in all space–time dimensions D, for all susy extensions N, and all kinds of supersymmetries: superPoincaré, superconformal, and even rigid susy on general curved space–times.

This book evolved out of the lecture notes of a course in supergravity and super-symmetry taught at SISSA. The lectures were aimed at graduate students who already had a knowledge of supersymmetry and supergravity in the standard approaches (superfields, the Noether method, etc.), and the course was meant as an advanced (and perhaps deeper) topic. This explains why this book does not contain many materials that are fundamental tools for a physicist working in the field of supersymmetry but are more than adequately covered by existing books and reviews (see, e.g., the recent book Supergravity by D.Z. Freedman and A. van Proeyen (Cambridge University Press, 2012); our book instead focuses on the geometric aspects, with particular emphasis on the geometric structures that are universal, that is, that are present mutatis mutandis in all possible situations.

The geometric tools introduced in this book allow recovery of all the results obtained from the more classical approaches to susy, and typically more quickly and with less pain (however, for specific problems other viewpoints may be more efficient).

In our tale there are four main characters: (i) the Atiyah–Bott–Shapiro classification of Clifford modules; (ii) Berger's theorem on the Riemannian holonomy groups and the allied results on parallel tensor and spinor fields; (iii) Kostant theorem on the interplay of the holonomy and isometry groups, which describes the gauging of all susy field theories; (iv) Griffiths' theory of variations of Hodge structures, which gives a unifying view on the geometry of electromagnetic dualities. We pay particular attention to their arithmetic aspects, which are crucial for the quantum theory (and have never been discussed previously, to the best of our knowledge).