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:
where
and
is the Laplace operator on
.
Therefore, on harmonic functions:
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:
Solutions of wave equations
Consider the following family of scalar field profiles, parameterized by a real parameter 
:
where
can be determined recursively from:
This solves the wave equation in
Therefore:
Massless scalar in
To solve the wave equation on
, we take
:
Massless scalar in
Let us parametrize
by a unit vector
.
Suppose that the
dependence is a harmonic polynomial
of order
. We must either take
or
. The solution is:
Inhomogeneous equations, appearence of log terms
Consider the equation with nonzero right hand side:
. When
is proportional to
, the logarithmic terms appear. Indeed:
and therefore:
This expression contains
.
A somewhat special case is the equation:
The solution is:
One can think of
as a family, parametrized by
, of field profiles,
taking values in a different representation for each
. (All these representations
are subspaces of one large space.) The value of the Casimir operator
is given
by Eq. (
77). When
it is zero.
From this point of view, Eq. (
78) is a particular case of the following
general construction. Suppose that we have an operator
acting on a representation space
of
,
commuting with
,
and
is a continuous direct sum of subrepresentations
parametrized by a parameter
,
such that the restriction of
on each
is the multiplication by
.
For
, we want to find
such that
. Consider a 1-parameter family
of vectors
such that
. Then
. We only need a 1-jet of
the family. If it is possible to find a map:
commuting with
with the symmetry, then the equation
can be solved in a covariant way:
For example, if
were equipped with a metric, we could pick for each
the path
going through
with the velocity perpendicular to
. But in our context,
there is no invariant metric, and there is no
-covariant invertion of
.
We can construct a sequence of
-independent functions:
They all depend only on
and grow near the boundary of AdS
as powers of
.
Functions participating in the perturbative expansion of nonlinear beta-deformation
We expect that nonlinear beta-deformation (and other finite-dimensional deformations)
is expressed in terms of
functions 
of Eq. (80) and their derivatives w.r.to 
and 
,
multiplied by polynomials of 
and rational functions of 
.
