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

TL;DR

This paper proposes a Hamiltonian-based reformulation of the AdS/CFT correspondence that eliminates the need for unknown boundary terms, clarifying their origin and relation to boundary conformal transformations.

Contribution

It introduces a Hamiltonian approach to supergravity in AdS/CFT, removing the necessity of ad hoc boundary terms and explaining their emergence from the Hamiltonian to Lagrangian transition.

Findings

Boundary terms naturally arise from the Hamiltonian formulation.

The approach clarifies the origin of boundary terms in supergravity.

Conformal transformations on the boundary are induced by AdS isometries.

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 $AdS$.