Solving $\mathcal{N}=4$ SYM BCFT matrix models at large $N$

Many observables in 4d $\mathcal{N}=4$ SYM with Gaiotto-Witten boundary conditions can be described exactly by matrix models via supersymmetric localization. The boundaries typically introduce new degrees of freedom, through a reduction of the gauge symmetry on the boundary or as explicit boundary degrees of freedom, leading to non-trivial matrix models. We derive the saddle points dominating these matrix models at large $N$, expressed in terms of generalized Lambert W-functions. In string theory the BCFTs are realized by D3-branes ending on D5 and NS5 branes. We independently derive the saddle points from the holographic duals with $\mathrm{A}\mathrm{d}{\mathrm{S}}_4\times {\mathrm{S}}^2\times {\mathrm{S}}^2\times \mathrm{\Sigma }$ geometry and provide precision tests of the dualities.