$\mathcal{N}=2$ Super-Yang Mills in AdS$_4$ and $F_{\text{AdS}}$-maximization

We investigate the dynamics of four-dimensional $\mathcal{N}=2$ $SU(2)$ super Yang--Mills theory on an AdS background. We propose that the boundary conditions that preserve the AdS super-isometries are determined by maximizing the real part of the AdS partition function $F_{\text{AdS}}=-\log Z_{\text{AdS}}$. At weak coupling $\Lambda L \ll 1$ the maximization singles out the Dirichlet boundary condition with an $SU(2)$ boundary global symmetry, corresponding to the classical vacuum at the origin of the Coulomb branch with fully un-higgsed gauge group. We find that for $\Lambda L \sim \mathcal{O}(1)$ new boundary conditions are favored, with gauge-group higgsed down to $U(1)$, matching the expectation from the flat space limit. We use supersymmetric localization to compute $Z_{\text{AdS}}$ nonperturbatively. We further provide evidence for a relation between $F_{\text{AdS}}$ and the $\mathcal{N}=2$ prepotential in AdS background.