Weil-Petersson Volumes of the Moduli Spaces of CY Manifolds

In this paper it is proved that the volumes of the moduli spaces of polarized CY manifolds with respect to the Weil-Petersson metrics are finite and they are rational numbers.Comment: The paper was accepted for publication in Communication in Analysis and Geometry. The remark of the referee were incorporated. Some typos were correcte

Minimal representations and reductive dual pairs in conformal field theory

A minimal representation of a simple non-compact Lie group is obtained by ``quantizing'' the minimal nilpotent coadjoint orbit of its Lie algebra. It provides context for Roger Howe's notion of a reductive dual pair encountered recently in the description of global gauge symmetry of a (4-dimensional) conformal observable algebra. We give a pedagogical introduction to these notions and point out that physicists have been using both minimal representations and dual pairs without naming them and hence stand a chance to understand their theory and to profit from it.Comment: 21 page