Freelance holography, part II: Moving boundary in gauge/gravity correspondence

We continue developing the freelance holography program, formulating gauge/gravity correspondence where the gravity side is formulated on a space bounded by a generic timelike codimension-one surface inside AdS and arbitrary boundary conditions are imposed on the gravity fields on the surface. Our analysis is performed within the Covariant Phase Space Formalism (CPSF). We discuss how a given boundary condition on the bulk fields on a generic boundary evolves as we move the boundary to another boundary inside AdS and work out how this evolution is encoded in deformations of the holographic effective boundary theory. Our analyses here extend the extensively studied \text{T}\bar{\text{T}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">T</mml:mtext> <mml:mover> <mml:mtext mathvariant="normal">T</mml:mtext> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> -deformation by relaxing the boundary conditions at asymptotic AdS or at the cutoff surface to be any arbitrary one (besides Dirichlet). We discuss some of the implications of our general freelance holography setting.