Supersymmetric Gauge Theory with Space-time Dependent Couplings

We study deformations of ${\cal N}=4$ supersymmetric Yang-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 $ISO(1,1)\times SO(3)\times SO(3)$ symmetry, in which couplings, as well as masses and the theta parameter, can depend on two spatial coordinates. In the case $ISO(1,1)$ is enhanced to $ISO(1,2)$, it reproduces the supersymmetric Janus configuration found by Gaiotto and Witten. When $SO(3)\times SO(3)$ 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 also find the general solution of the supersymmetry conditions for the cases with $ISO(1,1)\times SO(2)\times SO(4)$ symmetry. The cases with time dependent couplings and/or masses are also considered.