Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Modern Point of View on Anomalies

Samuel Monnier

TL;DR

The paper presents anomaly field theory as a unifying, geometric framework to package and analyze both local and global anomalies in quantum field theories, recasting anomalous theories as relative to an anomaly field theory. By leveraging the Atiyah–Segal axioms and extended field theory concepts, it provides a systematic way to compute anomaly data via higher-dimensional invariants, with explicit constructions for complex and Majorana fermions, self-dual fields, RCFTs, and 6d (2,0) SCFTs. It then shows how the classical picture emerges within this framework, detailing anomaly polynomials, symmetry actions, and the interpretation as connections on anomaly line bundles, as well as conditions for anomaly cancellation, including the Green–Schwarz mechanism in 6d and global cobordism constraints. The article culminates with a survey of awareness and progress on global anomaly cancellation in string theory backgrounds, outlining verified cases and highlighting open problems that inform consistency of proposed string vacua. Overall, the work provides a modern, higher-categorical, and geometrical lens to diagnose, cancel, and understand anomalies in high-dimensional QFTs and string theory settings.

Abstract

We review the concept of anomaly field theory, namely the fact that the anomalies of a $d$-dimensional field theory can be encoded in a $d+1$-dimensional field theory functor. We give numerous examples of anomaly field theories, explain how classical facts about anomalies are recovered from the anomaly field theory, and review recent work on global anomaly cancellation in 6d supergravity where this concept was instrumental. We also sketch the status of global anomaly cancellation checks in string theory. This paper is based on a talk given at the Durham Symposium `Higher Structures in M-theory' in August 2018.

A Modern Point of View on Anomalies

TL;DR

The paper presents anomaly field theory as a unifying, geometric framework to package and analyze both local and global anomalies in quantum field theories, recasting anomalous theories as relative to an anomaly field theory. By leveraging the Atiyah–Segal axioms and extended field theory concepts, it provides a systematic way to compute anomaly data via higher-dimensional invariants, with explicit constructions for complex and Majorana fermions, self-dual fields, RCFTs, and 6d (2,0) SCFTs. It then shows how the classical picture emerges within this framework, detailing anomaly polynomials, symmetry actions, and the interpretation as connections on anomaly line bundles, as well as conditions for anomaly cancellation, including the Green–Schwarz mechanism in 6d and global cobordism constraints. The article culminates with a survey of awareness and progress on global anomaly cancellation in string theory backgrounds, outlining verified cases and highlighting open problems that inform consistency of proposed string vacua. Overall, the work provides a modern, higher-categorical, and geometrical lens to diagnose, cancel, and understand anomalies in high-dimensional QFTs and string theory settings.

Abstract

We review the concept of anomaly field theory, namely the fact that the anomalies of a -dimensional field theory can be encoded in a -dimensional field theory functor. We give numerous examples of anomaly field theories, explain how classical facts about anomalies are recovered from the anomaly field theory, and review recent work on global anomaly cancellation in 6d supergravity where this concept was instrumental. We also sketch the status of global anomaly cancellation checks in string theory. This paper is based on a talk given at the Durham Symposium `Higher Structures in M-theory' in August 2018.

Paper Structure

This paper contains 34 sections, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The unreasonable power of self-consistency
  3. Anomalies as consistency conditions
  4. Anomalies from geometric invariants
  5. Field theories as functors
  6. Setup and conventions
  7. Bordism categories
  8. Atiyah-Segal axioms
  9. The trivial theory
  10. Invertible theories
  11. Symmetries
  12. Extended field theories
  13. Anomalies and relative field theories
  14. Relative field theories
  15. Anomalous = relative
  16. ...and 19 more sections

Figures (3)

  • Figure 1: The prescription described in the main text to compute anomalous phases.
  • Figure 2: A bordism $M^{d,1}$ from $M^{d-1}_1$ to $M^{d-1}_2$.
  • Figure 3: Top: A "twisted cylinder" $C = M \times S^1$, with the left boundary identified with $M$ through the identity map and the right boundary identified with $M$ through an automorphism $\phi$. The twisted cylinder is a bordism from $M$ to $\phi M$. In the limit where the length of the cylinder tends to zero, we end up with an infinitesimal bordism. The infinitesimal bordisms generate an action of the automorphism group of $M$ on any bordism with a boundary component isomorphic to $M$.