AdS notations

Embedding formalism

We here consider massless Laplace equation in .
We realize as a hyperboloid in parametrized by coordinates .
The equation of the hyperboloid is:

The Laplace operator in can be written as

where is the Euler vector field (rescaling of and ) and is the Laplace operator on .
Therefore, on harmonic functions :

Eigenfunctions of the Laplace operator on the hyperboloid can be obtained as restrictions of harmonic functions on

Our space-time is not just , but . The formulas for Laplace operator on the sphere
are completely analogous. To distinguish between AdS and sphere, we use indices and :
, , , . The total Laplace operator
on is:

Therefore, for the scalar function to be harmonic in :

This means:

Massless scalar in

Let us parametrize by a unit vector .
Suppose that the dependence is a harmonic polynomial
of order . The full solution is:

where is a harmonic function of order in
parametrized by coordinates .

Those solutions where is a polynomial form a finite-dimensional representation.

When we allow to be a rational function with denominator powers of , they form an infinite-dimensional representation. That infinite-dimensional representation is the of Eq. (16).