Nonperturbative effects in the Standard Model with gauged 1-form symmetry

We study the Standard Model with gauged ${\mathbb{Z}}_{N=2,3,6}^{(1)}$ subgroups of its ${\mathbb{Z}}_6^{(1)}$ 1-form global symmetry, making the gauge group $\frac{SU(3)\times SU(2)\times U(1)}{{\mathbb{Z}}_N}$. We show that, on a finite ${\mathbb{T}}^3$, there are self-dual instantons of fractional topological charge. They mediate baryon- and lepton-number violating processes. We compare their amplitudes to the ones due to the usual BPST-instantons. We find that the small hypercharge coupling suppresses the fractional-instanton contribution, unless the torus size is taken sub-Planckian, or extra matter is added above the weak scale. We also discuss these results in light of the cosmological bounds on the torus size.