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The algebra of anomaly interplay

Joe Davighi, Nakarin Lohitsiri

TL;DR

The paper develops a bordism-based, non-canonically split framework to analyze the interplay between local and global anomalies via invertible field theories. It formalizes anomaly data as a pullback along symmetry maps between Madsen–Tillman spectra, linking a torsion global anomaly group to a non-torsion local anomaly data through short exact sequences and their noncanonical splitting. By working through concrete 2d, 4d, and 6d examples—including 2d U(1) vs Z/2, 4d U(2) vs SU(2), discrete gauge groups, and 6d U(1) vs Z/2—the authors derive explicit criteria for anomaly cancellation and demonstrate how global anomalies can be computed from perturbative local anomalies in larger symmetry groups (Elitzur–Nair style). The results yield a rigorous, bordism-centric method for both deriving global anomalies from local data and understanding RG-flow–driven anomaly matching, with broad implications for high-energy model building and topological phases of matter.

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.

The algebra of anomaly interplay

TL;DR

The paper develops a bordism-based, non-canonically split framework to analyze the interplay between local and global anomalies via invertible field theories. It formalizes anomaly data as a pullback along symmetry maps between Madsen–Tillman spectra, linking a torsion global anomaly group to a non-torsion local anomaly data through short exact sequences and their noncanonical splitting. By working through concrete 2d, 4d, and 6d examples—including 2d U(1) vs Z/2, 4d U(2) vs SU(2), discrete gauge groups, and 6d U(1) vs Z/2—the authors derive explicit criteria for anomaly cancellation and demonstrate how global anomalies can be computed from perturbative local anomalies in larger symmetry groups (Elitzur–Nair style). The results yield a rigorous, bordism-centric method for both deriving global anomalies from local data and understanding RG-flow–driven anomaly matching, with broad implications for high-energy model building and topological phases of matter.

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.

Paper Structure

This paper contains 21 sections, 111 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The generalities of anomaly interplay
  3. Physics applications
  4. Deriving global anomalies
  5. Anomaly matching
  6. 2d anomalies in $U(1)$ vs. $\mathbb{Z}/2$
  7. 4d anomalies in $U(2)$ vs. $SU(2)$
  8. The spin case
  9. Non-spin generalization: the '$w_2 w_3$ anomaly'
  10. A remark concerning the 5d $SU(2)$ anomaly
  11. 4d $SU(2)$ anomaly from $SU(3)$
  12. 4d discrete gauge anomalies
  13. $\text{Spin} \times \mathbb{Z}/2^n$
  14. $(\text{Spin}\times\mathbb{Z}/2^n)/(\mathbb{Z}/2)$
  15. Going between discrete groups
  16. ...and 6 more sections

Figures (10)

  • Figure 1: Schematic illustration of the extension of the mapping torus $S^4\times S^1$ to the 6-manifold $S^4\times D^2$, made possible by embedding $SU(2)$ as a subgroup of $U(2)$.
  • Figure 2: The $\mathcal{A}_1$-module structure for $\mathbb{Z}/2[w_2^\prime,w_3^\prime,w_2^{\prime\prime}]\{UV\}$, up to degree $12$.
  • Figure 3: The Adams chart for $\Sigma^{-5}H^\bullet(MSO(3)\wedge MSO(2))$ with $t-s\leq 7$.
  • Figure 4: The $\mathcal{A}(1)$-module $\mathcal{M}$.
  • Figure 5: The Adams chart for $\mathbb{Z}/2$.
  • ...and 5 more figures