Deformed BPS Monopole in Omega-background

We study the BPS condition in the $\mathrm{\Omega }$-deformed $\mathcal{N}=2$ super Yang-Mills theory when one of the $ϵ$-parameters of the background is zero. We obtain the deformed BPS equation for dyons and the formula for their central charge. In particular, we find that the deformed BPS monopole equation has axially-symmetric solution and is equivalent to the Ernst equation. The monopole charge is shown to be undeformed. We construct one-monopole solution explicitly and examine its profile.