Holographic counterterms from local supersymmetry without boundary conditions

TL;DR

The paper tackles the problem that holographic renormalization traditionally relies on fixing boundary data to obtain finite, well-defined variational problems. By enforcing local, off-shell supersymmetry without any boundary conditions, the authors derive the requisite boundary terms directly from the bulk action, first in 3D supergravity with negative cosmological constant and then in 2D dilaton gravity as a nontrivial test. They show that the Gibbons–Hawking–York boundary term and the holographic counterterm emerge naturally from SUSY without boundary conditions, yielding the standard AdS boundary structure $I_{ extrm{EH}}$ with $ ext{∂M}$-dependent terms like $(K - 1/ extell)$, and similarly in 2D, boundary contributions of the form $(XK - u(X))$. This SUSY-driven approach provides a principled route to holographic renormalization that does not rely on pre-specified boundary data and suggests potential extensions to higher dimensions and broader SUSY frameworks, with implications for the finiteness of boundary stress tensors and AdS/CFT consistency.

Abstract

We show in some lower-dimensional supergravity models that the holographic counterterms which are needed in the AdS/CFT correspondence to make the theory finite, coincide with the counterterms that are needed to make the action supersymmetric without imposing any boundary conditions on the fields.