Finite-dimensional example: view from SYMBy AdS/CFT the non-normalizable excitations correspond to the deformations of the SYM action:whereis a complex combination of the SYM scalars and
is a density of the conformal weight
. We observe that:
for any integer
the space of densities has a finite-dimensional subspace invariant under the conformal group
Acting on this space by the supersymmetries we generate a finite-dimensional representation of the full superconformal group
Point 2 is obvious given 1. To explain 1, we observe that the deformation (16) is conformal-invariant if we transformas a “density” of weight
:
Consider the linear space consisting of the densitiesof the following form:
whereis a homogeneous polynomial in
of degree
, and
a polynomial in
of degree
(not necessarily homogeneous). One can see this space is closed under the action of the conformal transformations. This means that the (infinite-dimensional) representation of the conformal group
in the space of densities of the weight
for
has an invariant finite-dimensional subspace consisting of the densities of the form (17).
There is no invariant complementary subspace, therefore the space of densities is not a semisimple representation of
.