The Holography of the 2D inhomogeneously deformed CFT

We systematically study inhomogeneous Hamiltonians in two-dimensional conformal field theories within the framework of the AdS/CFT correspondence by relating them to two-dimensional curved backgrounds. We propose a classification of inhomogeneous Hamiltonians based on the Virasoro coadjoint orbit. The corresponding bulk dual geometries are described by the generalized Ba$\tilde{\text{n}}$ados solutions, for which we introduce a generalized Roberts mapping to facilitate their study. Our classification provides previously underexplored classes of deformations, offering fresh insights into their holographic properties. Revisiting the well-known example of the M$\ddot{\text{o}}$bius Hamiltonian, we establish a connection to the 3D C-metric, which describes three-dimensional accelerating solutions. Furthermore, we extend our analysis to KdV-type asymptotic boundary conditions, revealing a broader class of solvable inhomogeneous Hamiltonians that are not linear combinations of Virasoro charges but instead involve KdV charges.