Gauging the Standard Model 1-form symmetry via gravitational instantons

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.

💡 Research Summary

In this work the author investigates the fate of the Standard Model’s electric ℤ₆ 1‑form global symmetry when the theory is placed in the background of a Euclidean gravitational instanton, focusing on the Eguchi‑Hanson (EH) space. The EH manifold is an asymptotically locally Euclidean (ALE) self‑dual solution of the vacuum Einstein equations. Its second cohomology group is ℤ, reflecting the existence of a unique normalizable harmonic self‑dual two‑form K that is localized on the bolt S² at radius r = a. Because K is normalizable and self‑dual, turning on a U(1) field strength F = −2π C K does not back‑react on the metric; the energy‑momentum tensor vanishes identically. Flux quantization requires (1/2π)∮_{S²} F = −C ∈ ℤ, so the integer C labels distinct flux sectors.

The presence of such a flux induces a fractional topological charge
 Q = (1/8π²)∫{EH} F∧F = C²/4,
which is a hallmark of a higher‑form background rather than an ordinary instanton (which would have integer Q). The Standard Model gauge group SU(3)×SU(2)×U(1) possesses an electric ℤ₆ 1‑form symmetry. Gauging this symmetry amounts to turning on background fluxes in the Cartan subalgebras of SU(3) and SU(2) and a compensating flux in the hypercharge U(1) such that the combined configuration corresponds to an element of ℤ₆. Explicitly, the fluxes are parametrized by two integers m^{(2)} and m^{(3)} (associated with SU(2) and SU(3) respectively) and a hypercharge flux coefficient C = −3 m^{(2)} − 2 m^{(3)} (mod 6). The resulting topological charges are fractional:  Q{(3)} = m^{(3)2}/6, Q_{(2)} = m^{(2)2}/8, Q_{(1)} =