Dai-Freed anomaly in the standard model and topological inflation

When we impose the discrete symmetry in the standard model we have Dai-Freed global anomalies. However, interestingly if we introduce three right-handed neutrinos we can have an anomaly-free discrete $Z_4$ gauge symmetry. This $Z_4$ symmetry should be spontaneously broken down to the $Z_2$ symmetry to generate the heavy Majorana masses for the right-handed neutrinos. We show that this symmetry breaking naturally generates topological inflation, which is consistent with the CMB observations at present and predicts a significant tensor mode with scalar-tensor ratio $r > 0.03$. The right-handed neutrinos play an important role in reheating processes. The reheating temperature is as high as $\sim 10^8$GeV, and non-thermal leptogenesis successfully takes place.