Deformation of $N=4$ SYM with varying couplings via fluxes and intersecting branes

We study deformations of $N=4$ supersymmetric Yang-Mills theory with space-time dependent couplings by embedding probe D3-branes in supergravity backgrounds with non-trivial fluxes. The effective action on the world-volume of the D3-branes is analyzed and a map between the deformation parameters and the fluxes is obtained. As an explicit example, we consider D3-branes in a background corresponding to $(p,q)$ 5-branes intersecting them and show that the effective theory on the D3-branes precisely agrees with the supersymmetric Janus configuration found by Gaiotto and Witten in https://doi.org/10.1007/JHEP06(2010)097. D3-branes in an intersecting D3-brane background is also analyzed and the D3-brane effective action reproduces one of the supersymmetric configurations with ISO$(1,1)\times SO\left(2\right)\times SO\left(4\right)$ symmetry found in our previous paper https://arxiv.org/abs/1710.09792.