The Graviton in the AdS-CFT correspondence: Solution via the Dirichlet Boundary value problem

TL;DR

The paper addresses computing the two-point function of CFT energy-momentum tensors via the AdS/CFT correspondence by solving a regulated Dirichlet boundary value problem for AdS gravitons with boundary at $x_0=\epsilon$. Using a radiation gauge and time-slicing, it derives the linearized graviton equations, constructs the boundary-to-bulk solution from boundary data, and evaluates the on-shell action to extract the non-local part of $\langle T^i_j(x) T^s_r(y) \rangle$, finding agreement with the Liu-Tseytlin result: $\langle T^i_j(x) T^s_r(y) \rangle = \frac{\kappa d}{2|x-y|^{2d}} [ \frac{1}{2}(J^i_r J^s_j + J^{is} J_{jr}) - \frac{1}{d} \delta^i_j \delta^s_r ]$. The work also demonstrates a gauge transformation to de Donder gauge and shows that, while boundary data constraints prevent arbitrary boundary values, the bulk TT structure is preserved, reinforcing the consistency of the regularization scheme and its relevance for interacting graviton analyses in AdS/CFT.

Abstract

Using the AdS-CFT correspondence we calculate the two point function of CFT energy momentum tensors. The AdS gravitons are considered by explicitly solving the Dirichlet boundary value problem for $x_0=ε$. We consider this treatment as complementary to existing work, with which we make contact.