BCFT One-point Functions of Coulomb Branch Operators

We show that supersymmetry can be used to compute the BCFT one-point functioncoefficients for chiral primary operators, in 4d $\mathcal{N}=2$ SCFTs with$\frac{1}{2}$-BPS boundary conditions. The main ingredient is the hemispherepartition function, with the boundary condition on the equatorial $S^3$. Asupersymmetric Ward identity relates derivatives with respect to the chiralcoupling constants to the insertion of the primaries at the pole of thehemisphere. Exact results for the one-point functions can be then obtained interms of the localization matrix model. We discuss in detail the example of thesuper Maxwell theory in the bulk, interacting with 3d $\mathcal{N}=2$ SCFTs onthe boundary. In particular we derive the action of the SL(2,$\mathbb{Z}$)duality on the one-point functions.