Supergravity Solutions Dual to Holographic Gauge Theory with External Maxwell Electromagnetic Field

We apply the transformation of mixing azimuthal with wrapped coordinate to the 11D Mtheory with a stack N M5-branes to find the spacetime of a stack of N D4-branes with magnetic field in 10D IIA string theory, after the Kaluza-Klein reduction. With (without) performing the T-duality and taking the near-horizon limit the background becomes the Melvin magnetic field deformed AdS5×S (AdS6×S). We also apply a Lorentz boost on the coordinates of N M5-branes and a transformation of mixing time with wrapped coordinate to obtain the background with inhomogeneously Melvin electric field deformation. Although the found supergravity solutions represent the D-branes under the external Melvin field flux of RR one-form we use a simple observation to see that they are also the solution of Dbranes under the external Maxwell field flux. The relations between using above supergravity solutions and those using other backgrounds in recent by many authors to investigate the holographic gauge theory with external Maxwell electric and magnetic fields through D3/D7 (D4/D8) system are discussed. We also investigate the Wilson loop therein and show that Maxwell magnetic field will increase the quark-antiquark potential and that the Maxwell electric field could increase the quark-antiquark potential only if the direction of quarkantiquark pair does not orthogonal to the electric field.