Abstract

We give a general description of the interplay that can occur between local and global anomalies, in terms of (co)bordism. Mathematically, such an interplay is encoded in the non-canonical splitting of short exact sequences known to classify invertible field theories. We study various examples of the phenomenon in 2, 4, and 6 dimensions. We also describe how this understanding of anomaly interplay provides a rigorous bordism-based version of an old method for calculating global anomalies (starting from local anomalies in a related theory) due to Elitzur and Nair.

Highlights

  • Like the universal coefficient theorem in ordinary cohomology, these short exact sequences defining the cobordism groups split, meaning the most general anomaly ‘factors’ into its global and local parts, crucially this splitting is not canonical

  • This last property allows for an interplay between the global and local anomalies of theories with different symmetries

  • Suppose we identify a symmetry type H for which the bordism group ΩHd+1 is non-vanishing torsion, giving the possibility of a global anomaly

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Summary

Introduction

Inflow relates fermionic anomalies to quantum field theories in one dimension higher. Perturbative anomalies due to massless chiral fermions are not of this type because the Chern–Simons anomaly theory is not strictly topological, having a mild dependence on the background metric [15] It is conjectured [11] that a broader class of physically sensible invertible theories (i.e. the reflection positive ones), not necessarily topological, are still classified up to deformation by the homotopy classes of maps between spectra – only non-torsion elements should be included. Like the universal coefficient theorem in ordinary cohomology, these short exact sequences defining the cobordism groups split, meaning the most general anomaly ‘factors’ into its global and local parts, crucially this splitting is not canonical This last property allows for an interplay between the global and local anomalies of theories with different symmetries (for example between two gauge theories whose gauge groups are related by some obvious map, such as inclusion of a subgroup). Other examples from recent [22, 23] and not-so-recent [17, 20] references are discussed from the point of view of our formalism

The generalities of anomaly interplay
Deriving global anomalies
Anomaly matching
The spin case
Non-spin generalization: the ‘w2w3 anomaly’
Going between discrete groups
A Some bordism calculations
Using the Adams spectral sequence

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