Short description:

This 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.

Long description:

This 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.

?The topics covered are presented at a very high level in all respects. Though most of the material is well established, the book is strongly influenced by the personality of the author, including his scientific positions and preferences. ? the book is a useful and valuable addition to the existing special literature on the formal aspects of modern quantum field theory.? (Ilya Shapiro, Mathematical Reviews, February, 2024)

Table of Contents:

Part I. Basic tools.- Chapter.1 Fermions.- Chapter 2. Classical and BRST Symmetries.- Chapter 3.Conformal symmetry.- Part II. Anomalies and cohomology.- Chapter 4.E?ective actions and anomalies.- Chapter  5.Cohomological analysis of anomalies.- Part III. Perturbative methods for anomalies.- Chapter 6.Feynman diagrams and regularizations.- Chapter 7.Perturbative di?eomorphism and trace anomalies.- Part IV. Nonperturbative methods. (A) heat kernel.- Chapter 8.Functional non-perturbative methods.- Chapter 9.Explicit non-perturbative derivations.- Chapter 10.Metric-axial-tensor (MAT) background.- Part V. Nonperturbative methods. (B) Index theorem.- Chapter 11.Geometry of anomalies.- Chapter 12.Anomalies as obstructions: the Atiyah-Singer family?s index theorem.- Chapter 13.Global anomalies.- Part VI. Special topics.- Chapter 14.MAT in 2d.- Chapter 15.Wess-Zumino terms.- Chapter 16.Sigma model anomalies.- Chapter 17.Anomalies and (super)string theories.- Chapter 18. Literature and further readings.