Freezing and BPS jumping

We report a novel BPS jumping phenomenon of 5d $\mathcal{N}=1$ supersymmetric gauge theories whose brane configuration is equipped with an O$7$-plane. The study of the relation between O$7{}^+$-plane and O$7{}^-$-plane reveals that such BPS jumps take place when the Higgsing is triggered near the O7-plane upon a particular parameter tuning of the theories. We propose two types of gauge theories whose BPS spectra jump. One is the SU($2N+8$) gauge theory with a symmetric hypermultiplet converted to the SU($2N$) gauge theory with an antisymmetric hypermultiplet. The other is pure SO($2N+8$) gauge theory jumping to pure Sp($N$) gauge theory. We explicitly confirm our proposal through the (un)refined instanton partition functions. Furthermore, we discuss feasible generalizations involving an O$p$-plane for supersymmetric gauge theories of eight supercharges in four and three dimensions ($p=6, 5$ respectively).