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The absence of global anomalies of CP symmetry

Kazuya Yonekura

TL;DR

This work addresses whether gauging CP symmetry as a potential solution to the strong CP problem introduces new global anomalies. By extending the analysis of Witten to four dimensions and requiring the gauge group $G$ to be connected and simply connected with no preexisting anomalies, the authors prove that gauging CP does not generate additional global anomalies in $4$D, so the Standard Model content remains anomaly-free under the combined $\mathrm{Pin}^+(4)\ltimes G$ symmetry. The approach relies on anomaly inflow, perturbative and global anomaly formalisms, and bordism invariants, with a reduction to the $SU(2)$ case and obstruction theory to general $G$ under $\pi_0(G)=\pi_1(G)=0$. The results have implications for GUT embeddings like $SU(5)$ and Spin(10), and are connected to string theory realizations via heterotic compactifications, where CP symmetry can be realized consistently in higher dimensions. Overall, the paper solidifies the viability of gauged CP as a mechanism compatible with known anomaly constraints and provides a concrete framework for analyzing CP-related global anomalies in beyond-Standard-Model setups.

Abstract

Some solutions to the strong CP problem assume that CP symmetry is a gauge symmetry, which is then spontaneously broken. For this scenario to be possible, the CP symmetry should not have any nonperturbative (global) anomalies. In this paper, we study anomalies of CP symmetry of fermions which are coupled to gravity and gauge fields with a gauge group $G$. When $G$ is connected and simply connected, we show that gauging a CP symmetry does not produce any new anomaly beyond the one before gauging it. In particular, the standard model matter content does not have anomalies.

The absence of global anomalies of CP symmetry

TL;DR

This work addresses whether gauging CP symmetry as a potential solution to the strong CP problem introduces new global anomalies. By extending the analysis of Witten to four dimensions and requiring the gauge group to be connected and simply connected with no preexisting anomalies, the authors prove that gauging CP does not generate additional global anomalies in D, so the Standard Model content remains anomaly-free under the combined symmetry. The approach relies on anomaly inflow, perturbative and global anomaly formalisms, and bordism invariants, with a reduction to the case and obstruction theory to general under . The results have implications for GUT embeddings like and Spin(10), and are connected to string theory realizations via heterotic compactifications, where CP symmetry can be realized consistently in higher dimensions. Overall, the paper solidifies the viability of gauged CP as a mechanism compatible with known anomaly constraints and provides a concrete framework for analyzing CP-related global anomalies in beyond-Standard-Model setups.

Abstract

Some solutions to the strong CP problem assume that CP symmetry is a gauge symmetry, which is then spontaneously broken. For this scenario to be possible, the CP symmetry should not have any nonperturbative (global) anomalies. In this paper, we study anomalies of CP symmetry of fermions which are coupled to gravity and gauge fields with a gauge group . When is connected and simply connected, we show that gauging a CP symmetry does not produce any new anomaly beyond the one before gauging it. In particular, the standard model matter content does not have anomalies.
Paper Structure (19 sections, 62 equations, 3 figures)

This paper contains 19 sections, 62 equations, 3 figures.

Table of Contents

  1. Introduction and summary
  2. The main result.
  3. CP symmetry
  4. The CP symmetry of a single neutral Weyl fermion
  5. The Pin groups
  6. Inclusion of gauge group
  7. Summary of the group and representation
  8. Gauging CP symmetry
  9. Anomalies
  10. Anomaly inflow
  11. Perturbative anomaly
  12. Global anomalies
  13. The absence of anomalies of the CP symmetry
  14. More details
  15. The case of $\mathrm{SU}(N)$.
  16. ...and 4 more sections

Figures (3)

  • Figure 1: An example of a $(d+1)$-dimensional manifold $Y$ and its $d$-dimensional boundary $X$. In this example, $d=1$, $X=S^1$, and $Y$ is a torus with a disk removed.
  • Figure 2: Gluing of $Y$ and (the orientation reversal $\overline{Y}'$ of) $Y'$ along the common boundary $\partial Y = \partial Y' =X$ to obtain the closed manifold $Y"=Y \cup \overline{Y}'$.
  • Figure 3: A $(d+2)$-dimensional manifold $Z=Z$ whose boundary is $Y$ and $Y'$.