Gauging the Standard Model 1-form symmetry via gravitational instantons

A bstract We investigate the fate of the Standard Model (SM) $$ {\mathbb{Z}}_6^{(1)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>6</mml:mn> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:msubsup> </mml:math> electric 1-form global symmetry in the background of gravitational instantons, focusing on Eguchi-Hanson (EH) geometries. We show that EH instantons support quantized $$ {\mathbb{Z}}_6^{(1)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>6</mml:mn> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:msubsup> </mml:math> 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 $$ {\mathbb{Z}}_6^{(1)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>6</mml:mn> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:msubsup> </mml:math> 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.