Lectures on Anomalies (0802.0634v1)

Abstract: These lectures on anomalies are relatively self-contained and intended for graduate students who are familiar with the basics of quantum field theory. We begin with several derivations of the abelian anomaly: anomalous transformation of the measure, explicit computation of the triangle Feynman diagram, relation to the index of the Euclidean Dirac operator. The chiral (non-abelian) gauge anomaly is derived by evaluating the anomalous triangle diagram with three non-abelian gauge fields coupled to a chiral fermion. We discuss in detail the relation between anomaly, current non-conservation and non-invariance of the effective action, with special emphasis on the derivation of the anomalous Slavnov-Taylor/Ward identities. We show why anomalies always are finite and local. A general characterization is given of gauge groups and fermion representations which may lead to anomalies in four dimensions, and the issue of anomaly cancellation is discussed, in particular the classical example of the standard model. Then, we move to more formal developments and arbitrary even dimensions. After introducing a few basic notions of differential geometry, in particular characteristic classes, we derive the descent equations. We prove the Wess-Zumino consistency condition and show that relevant anomalies correspond to BRST cohomologies at ghost number one. We discuss why and how anomalies are related to characteristic classes in two more dimensions and outline their computation in terms of the index of an appropriate Dirac operator. Finally we derive the gauge and gravitational anomalies in arbitrary even dimensions from the appropriate index and explain the anomaly cancellations in ten-dimensional IIB supergravity and in type I and heterotic superstrings.

• The paper introduces a comprehensive mathematical treatment of gauge and gravitational anomalies using techniques like differential geometry and index theorems.
• It details how Abelian and non-Abelian anomalies break classical symmetries in quantum field theories and emphasizes the role of BRST cohomology and consistency conditions.
• The work demonstrates practical anomaly cancellation mechanisms in high-energy theories, including applications in type IIB supergravity and heterotic string theory.

An Analysis of Gauge and Gravitational Anomalies in High-Energy Physics

The document under discussion provides an in-depth exploration of anomalies within the context of high-energy theoretical physics, particularly focusing on the lectures delivered by Adel Bilal. These lectures are a comprehensive mathematical treatment aimed at clarifying the concept of anomalies in gauge theories and their implications in various dimensions.

Anomalies in Quantum Field Theory

At its core, an anomaly refers to the breakdown of a symmetry that holds at the classical level when theory transitions to quantum mechanics. This breach manifests as non-conservation of current and is a subtle aspect of quantum field theories, particularly in their application to particle physics and string theory.

• Abelian Anomalies: These arise in the context of U(1) gauge theories when chiral transformations of fermions occur. The lectures illustrate this by deriving the anomaly from path integrals and Feynman diagrams, showing its relation to the abelian current non-conservation.
• Non-Abelian Anomalies: More intricate, these occur in SU(N) gauge theories wherein chiral fermions couple to non-abelian gauge fields. The treatment details how these anomalies affect gauge invariance and can lead to inconsistencies unless they are canceled.

Formal Developments and Differential Geometry

The document navigates through advanced mathematical techniques to describe anomalies:

• Differential Forms and Characteristic Classes: The use of differential geometry simplifies many aspects of gauge field theories. The document systematically reviews differential forms, the exterior derivative, and the concept of closed vs. exact forms. These are foundational to understanding topological aspects of gauge theories.
• Gauge Bundles and Topological Invariants: The text elucidates the significance of gauge bundles in understanding anomalies, emphasizing their topological nature as evident in the paper of instantons and magnetic monopoles.

Consistency Conditions and BRST Cohomology

• Wess-Zumino Consistency Condition: An anomaly must satisfy certain consistency conditions, expressed in this framework as Wess-Zumino conditions. This crucial insight, tied with BRST cohomology, positions the anomaly within a framework where it captures essential topological and group-theoretical information.
• Descent Equations: These are employed to illustrate the relation between gauge variations in different dimensions, stressing their utility in anomaly computations across various dimensional spaces.

Anomalies in Various Dimensions and Their Implications

• Index Theorems and Euclidean Techniques: The comprehensive discussion ties anomalies to index theorems of Dirac operators, bridging anomalies in d dimensions to mathematical structures in d+2. This relation is pivotal for theoretical consistency and computational tractability.
• Gravitational Anomalies: Extending the discourse to gravitational interactions, it highlights anomalies related to local Lorentz and diffeomorphism invariance, crucial for completeness in theories like supergravity.

Specific Cases and Theoretical Applications

The document concludes with applications to ten-dimensional theories such as type IIB supergravity, and heterotic superstring theories, showing the real-world impact of these theoretical considerations:

• Type IIB Supergravity: The lectures demonstrate how anomaly cancellations occur naturally in this framework, a triumph of string theory, illustrating the robustness of theoretical predictions.
• Green-Schwarz Mechanism: A cornerstone method for anomaly cancellation in heterotic string theory, this mechanism underscores the role of additional fields and higher-dimensional inflow in maintaining physical consistency.

Conclusion

This detailed account of anomalies under the lens of high-energy theoretical physics provides essential insights into their origin, manifestation, and resolutions within modern physical theories. The treatment underscores the elegance and necessity of mathematical rigor in resolving these profound phenomena, guiding both theoretical advancements and practical implementations in physic's quest to unify fundamental forces. The interplay between quantum mechanics and gauge theories through these anomalies continues to be a fertile ground for research and introspection in the physics community.