NÂ =Â 2 gauge theories on the hemisphere HS4

Using localization techniques, we compute the path integral of N=2 SUSY gauge theory coupled to matter on the hemisphere HS4, 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 determinants 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 QSUSY2 eigenvalues and its multiplicities. In the Dirichlet case, we show how to glue two wave-functions to get back the partition function of round S4. We will also describe how to obtain the same results using SO(5) harmonics basis.