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The AdS/CFT correspondence maps the vacuum to empty anti-de Sitter spacetime and the stress-tensor conformal family to one-graviton normal modes. ThereforeThe distinction between $E_1$ and $E_2$ is a different excitation within the same one-particle conformal family, not an additional graviton.

Under the AdS/CFT correspondence, each single-trace operator is represented by one bulk field. The leading two-point Witten diagram is a bulk propagator joining the two boundary insertions, equivalently two bulk-to-boundary legs contracted through the quadratic bulk action. The leading connected three-point diagram has one bulk cubic vertex integrated over AdS and three bulk-to-boundary propagators ending at $x_1,x_2,x_3$. The bulk kinetic normalization fixes $C_2$, while the cubic coupling fixes $C_{123}$.