  In this paper we show that the Ray-Singer complex analytic torsion is trivial
for even dimensional Calabi-Yau manifolds.
  Then we define the quaternionic analytic torsion for quaternionic manifolds
and prove that they are metric independent.
  In dimension four, the quaternionic analytic torsion equals to the self-dual
analytic torsion. For higher dimensional manifolds, the self-dual analytic
torsion is a conformal invariant.
