Noncommutative geometry provides a powerful tool for regularizing quantum field theories in the form of fuzzy physics. Fuzzy physics maintains symmetries, has no fermion-doubling problem and represents topological features efficiently. These lecture notes provide a comprehensive introduction to the field. Starting with the construction of fuzzy spaces, using the concrete examples of the fuzzy sphere and fuzzy complex projective spaces, the book moves on to discuss the technology of star products on noncommutative R2d and on the fuzzy sphere. Scalar, spinor and gauge field theories as well as extended objects such as monopoles and nonlinear sigma modes are treated in considerable detail. A detailed treatment of the regularization of supersymmetry is given using the techniques of fuzzy physics.
Fuzzy Spaces; Star Products; Scalar Fields on the Fuzzy Sphere; Instantons, Monopoles and Projective Modules; Fuzzy Nonlinear Sigma Models; Fuzzy Gauge Theories; The Dirac Operator and Axial Anomaly; Fuzzy Supersymmetry; SUSY Anomalies on the Fuzzy Supersphere; Fuzzy Spaces as Hopf Algebras.