Supersymmetric gauge theory with space-time-dependent couplings

We study deformations of N = 4 supersymmetricYang-Mills theory with couplings and masses depending on space-time. The conditions to preserve part of the supersymmetry are derived and a lot of solutions of these conditions are found. The main example is the case with ISO(1, 1) x SO(3) x SO(3) symmetry, in which couplings, as well as masses and the theta parameter, can depend on two spatial coordinates. In the case in which ISO(1, 1) is enhanced to ISO(1, 2), it reproduces the supersymmetric Janus configuration found by Gaiotto and Witten [J. High Energy Phys. 06, 097 (2010)]. When SO(3) x SO(3) is enhanced to SO(6), it agrees with the world-volume theory of D3-branes embedded in F-theory (a background with 7-branes in type IIB string theory). We have also found the general solution of the supersymmetry conditions for the cases with ISO(1, 1) x SO(2) x SO(4) symmetry. Cases with time-dependent couplings and/or masses are also considered.