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Gauging the Standard Model 1-form symmetry via gravitational instantons

Mohamed M. Anber

Abstract

We investigate the fate of the Standard Model (SM) $\mathbb Z_6^{(1)}$ electric $1$-form global symmetry in the background of gravitational instantons, focusing on Eguchi-Hanson (EH) geometries. We show that EH instantons support quantized $\mathbb Z_6^{(1)}$ fluxes localized on their $S^2$ bolt, inducing fractional topological charge without backreacting on the geometry. The requirement that quark and lepton wavefunctions be globally well-defined under parallel transport imposes boundary conditions, removing ill-defined fermion zero modes; the surviving spectrum is confirmed by an explicit solution of the Dirac equation and by the Atiyah-Patodi-Singer index theorem. The Euclidean path integral in the EH background can be interpreted as a transition amplitude from an entangled state between two identical halves of space to the vacuum. Summing over all $\mathbb{Z}_6^{(1)}$ flux sectors in the path integral gauges the SM $1$-form symmetry; thus, it cannot persist as an exact global symmetry in the semiclassical limit of gravity. We further show that these fluxes induce baryon- and lepton-number violating processes, which are exponentially suppressed due to the smallness of the hypercharge coupling constant.

Gauging the Standard Model 1-form symmetry via gravitational instantons

Abstract

We investigate the fate of the Standard Model (SM) electric -form global symmetry in the background of gravitational instantons, focusing on Eguchi-Hanson (EH) geometries. We show that EH instantons support quantized fluxes localized on their bolt, inducing fractional topological charge without backreacting on the geometry. The requirement that quark and lepton wavefunctions be globally well-defined under parallel transport imposes boundary conditions, removing ill-defined fermion zero modes; the surviving spectrum is confirmed by an explicit solution of the Dirac equation and by the Atiyah-Patodi-Singer index theorem. The Euclidean path integral in the EH background can be interpreted as a transition amplitude from an entangled state between two identical halves of space to the vacuum. Summing over all flux sectors in the path integral gauges the SM -form symmetry; thus, it cannot persist as an exact global symmetry in the semiclassical limit of gravity. We further show that these fluxes induce baryon- and lepton-number violating processes, which are exponentially suppressed due to the smallness of the hypercharge coupling constant.

Paper Structure

This paper contains 23 sections, 225 equations.

Table of Contents

  1. Introduction
  2. Eguchi-Hanson instantons, fractional fluxes, and spinors
  3. Turning on a $U(1)$ gauge field
  4. Turning on a background field of $\mathbb Z_N^{(1)}$ $1$-form symmetry
  5. Spinors in EH space in the presence of abelian fields
  6. Dirac fermion zero modes in EH space
  7. Background of $U(N)$ gauge bundle and fermion zero modes
  8. Background of the $\mathbb Z_6^{(1)}$ $1$-form symmetry of the Standard Model: lepton- and baryon-number violation
  9. Background of $\mathbb Z_3^{(1)}$ $1$-form symmetry
  10. Background of the full $\mathbb Z_6^{(1)}$ $1$-form symmetry
  11. EH spaces as saddles in the path integral: gauging the SM $1$-form symmetry
  12. Outlook and future directions
  13. The Eguchi-Hanson geometry
  14. Mathematical preliminaries
  15. The Eguchi-Hanson instanton
  16. ...and 8 more sections