Bonora L. Fermions and Anomalies in Quantum Field Theories 2023

Textbook in PDF format

The book presents a modern view of anomalies in quantum field theories. It is divided into six parts. The first part is preparatory covering an introduction to fermions, a description of the classical symmetries, and a short introduction to conformal symmetry. The second part of the book is devoted to the relation between anomalies and cohomology. The third part deals with perturbative methods to compute gauge, diffeomorphism and trace anomalies. In the fourth part the same anomalies are calculated with non-perturbative heat-kernel-like methods. Part five is devoted to the family's index theorem and its application to chiral anomalies, and to the differential characters and their applications to global anomalies. Part six is devoted to special topics including a complete calculation of trace and diffeomorphism anomalies of a Dirac fermion in a MAT background in two dimensions, Wess-Zumino terms in field theories, sigma models, their local and global anomalies and their cancelation, and finally the analysis of the worldsheet, sigma model, and target space anomalies of string and superstring theories.
The book is targeted to researchers and graduate students.
Quantum field theory is a theory under construction. It is not an axiom-based theory, where everything can be derived, at least potentially, from first principles (although attempts in this direction have been made). It is rather a mixture of heuristic intuitions and pieces of mathematics, sometimes of very rigorous mathematics; fantastically successful, it must be stressed, in describing quantum physics. The path integral, which is at the basis of modern quantum field theory, is the typical example in this sense: it is not (yet) mathematically rigorous, but it is nevertheless an extremely efficient and powerful tool. And it is an incredible alloy of mathematics and physical intuition. There is a sector of quantum field theory, where this composite texture is particularly visible: anomalies. The anomaly problem is in principle very simple: there are classically conserved (divergenceless, traceless) quantities that are not conserved anymore upon quantization. But, in practice, this simplicity quickly disappears when faced with the ambiguities and variety of approaches the (still) incomplete nature of the quantum field theory formulation requires. The main focus of this book is on how an acceptable degree of confidence about our knowledge of anomalies can be gathered by comparing various methods and techniques of calculation.
The purpose of this book is not a complete historical review of all the methods used to analyze anomalies and their results, nor a description of all the anomalies discovered in the last fifty or so years. This would be impossible in a single volume due to the vastness of the relevant literature. Our purpose is rather to provide a coherent view (as opposed to a sparse set of notions) of the anomaly problem. This implies a selective organization of the material (so, for instance, not all the regularization prescriptions in the perturbative and non-perturbative approaches will be analyzed, and not all the interpretations of the index theorem that have appeared in the literature will be considered).
Selective choices of material accompanied by detailed and explicit calculations and an accurate comparison of the three methods (perturbative, heat kernel like and family’s index theorem) are organized in order to produce a coherent and logical design.
Fermions
Classical and BRST Symmetries
Conformal Symmetry
Effective Actions and Anomalies
Cohomological Analysis of Anomalies
Feynman Diagrams and Regularizations
Perturbative Diffeomorphism and Trace Anomalies
Functional Non-perturbative Methods
Explicit Non-perturbative Derivations
Metric-Axial-Tensor (MAT) Background
Geometry of Anomalies
Anomalies as Obstructions: The Atiyah-Singer Family’s Index Theorem
Global Anomalies
MAT in 2d
Wess-Zumino Terms
Sigma Model Anomalies
Anomalies and (Super)String Theories
Literature and Further Readings