The N=2 Wonderland: From Calabi Yau Manifolds to Topological Field Theories

This book presents, in a unifying perspective, the topics related to N=2 supersymmetry in two dimensions. Beginning with the Kähler structure of D=4 supergravity Lagrangians, through the analysis of string compactifications on Calabi-Yau manifolds, one reaches the heart of the matter with the chiral ring structure of N=2 conformal field theories and its relation to topological field theory models and Landau-Ginzburg models. In addition, mirror symmetry, topological twists and Picard-Fuchs equations are discussed.
I wrote this book in 1995 together with my former Ph.D. student Paolo Soriani in the last year of my affiliation at SISSA, before coming back to Torino University. The topic of the book is indeed a Wonderland since it deals with the incredible mathematical beauty of the following issues: The Art of Quantizing Zero (namely topological field theories) and the interplay between complex algebraic geometry and field theories in the context of supersymmetry.
It was the time when the fertilizing influence of supersymmetry brought together Advanced Geometry (both differential and algebraic) and Quantum Field Theories. Einstein's dream of reducing all laws of Nature to pure geometry found in those years new declinations and new formulations. In particular the possibility of solving exactly quantum field theories, namely calculating correlation functions, in terms of geometric procedures related to moduli spaces of algebraic varieties constitutes a very deep new conception that, indipendently from specific applications and results constitutes an established paradigma and new acquisition of scientific knowledge.