On the origin of supergravity boundary terms in the AdS/CFT correspondence

TL;DR

The paper addresses the incompleteness of the standard AdS/CFT framework caused by undetermined boundary terms in the supergravity action. It proposes a Hamiltonian formulation of supergravity that eliminates boundary terms, showing that such terms reappear naturally when transitioning to the Lagrangian description; this is illustrated for the graviton sector on AdS_{d+1}×E and for fermions on AdS_{d+1}, including how AdS isometries induce conformal transformations on the boundary. The authors derive explicit metric- and fermion-related boundary terms, demonstrate their compatibility with conformal invariance and the correct boundary two-point functions, and discuss implications for higher-point functions and extensions to full supergravity, including IIB on AdS_5×S^5.

Abstract

The standard formulation of the AdS/CFT correspondence is incomplete since it requires adding to a supergravity action some a priori unknown boundary terms. We suggest a modification of the correspondence principle based on the Hamiltonian formulation of the supergravity action, which does not require any boundary terms. Then all the boundary terms of the standard formulation naturally appear by passing from the Hamiltonian version to the Lagrangian one. As examples the graviton part of the supergravity action on the product of $AdS_{d+1}$ with a compact Einstein manifold $\cal E$ and fermions on $AdS_{d+1}$ are considered. We also discuss conformal transformations of gravity fields on the boundary of $AdS$ and show that they are induced by the isometries of $AdS$.