N=2 gauge theories on the hemisphere $HS^4$
Using localization techniques, we compute the path integral of $N=2$ SUSY gauge theory coupled to matter on the hemisphere $HS^4$, with either Dirichlet or Neumann supersymmetric boundary conditions. The resulting quantities are wave-functions of the theory depending on the boundary data. The one-loop determinant are computed using $SO(4)$ harmonics basis. We solve kernel and co-kernel equations for the relevant differential operators arising from gauge and matter localizing actions. The second method utilizes full $SO(5)$ harmonics to reduce the computation to evaluating $Q_{SUSY}^2$ eigenvalues and its multiplicities. In the Dirichlet case, we show how to glue two wave-functions to get back the partition function of round $S^4$. We will also describe how to obtain the same results using $SO(5)$ harmonics basis.