Holographic Duals and Operator Dimensions in Dilaton Black Holes with Geometric Deformation

We investigate the holographic dual of a four-dimensional dilaton black hole with a deformed angular geometry r 2 R(r) 2 , where R(r) = (r/r0)N. Using the Brown-York formalism and holographic renormalization, we compute the renormalized boundary stress-energy tensor and analyze its dependence on the geometric parameter N and the dilaton coupling α. The deformation modifies the intrinsic curvature of the boundary and influences the structure of the dual CFT. We identify the boundary operator dual to the geometric deformation and show that its scaling dimension depends linearly on N , making it relevant for negative N and irrelevant for positive N . The dilaton field is found to be dual to an operator of approximate dimension two, consistent with its role in modulating the gauge coupling. Crucially, we verify that in the limit N → 0, α → 0, the solution reduces exactly to the Reissner-Nordström-AdS black hole, with all holographic quantities recovering their standard forms. This confirms the model as a consistent generalization of classical black holes, where the geometric deformation and dilaton field act as physical parameters that smoothly modify the spacetime structure and the dual quantum field theory.