  The question of the dimensional reduction of two-dimensional (2d) quantum
models on a sphere to one-dimensional (1d) models on a circle is adressed. A
possible application is to look at a relation between the 2d anyon model and
the 1d Calogero-Sutherland model, which would allow for a better understanding
of the connection between 2d anyon exchange statistics and Haldane exclusion
statistics. The latter is realized microscopically in the 2d LLL anyon model
and in the 1d Calogero model. In a harmonic well of strength \omega or on a
circle of radius R - both parameters \omega and R have to be viewed as long
distance regulators - the Calogero spectrum is discrete. It is well known that
by confining the anyon model in a 2d harmonic well and projecting it on a
particular basis of the harmonic well eigenstates, one obtains the
Calogero-Moser model. It is then natural to consider the anyon model on a
sphere of radius R and look for a possible dimensional reduction to the
Calogero-Sutherland model on a circle of the same radius. First, the free
one-body case is considered, where a mapping from the 2d sphere to the 1d
chiral circle is established by projection on a special class of spherical
harmonics. Second, the N-body interacting anyon model is considered : it
happens that the standard anyon model on the sphere is not adequate for
dimensional reduction. One is thus lead to define a new spherical anyon-like
model deduced from the Aharonov-Bohm problem on the sphere where each flux line
pierces the sphere at one point and exits it at its antipode.
