  Given a finite index subfactor, we show that the {\em affine morphisms at
zero level} in the affine category over the planar algebra associated to the
subfactor is isomorphic to the fusion algebra of the subfactor as a *-algebra.
This identification paves the way to analyze the structure of affine
$P$-modules with weight zero for any subfactor planar algebra $P$ (possibly
having infinite depth). Further, for irreducible depth two subfactor planar
algebras, we establish an additive equivalence between the category of affine
$P$-modules and the center of the category of $N$-$N$-bimodules generated by
$L^2(M)$; this partially verifies a conjecture of Jones and Walker.
