  We derive the relation between the Hilbert space of certain geometries under
the Bohr-Sommerfeld quantization and the perturbative prepotentials for the
supersymmetric five-dimensional SU(N) gauge theories with massive fundamental
matters and with one massive adjoint matter. The gauge theory with one adjoint
matter shows interesting features. A five-dimensional generalization of
Nekrasov's partition function can be written as a correlation function of
two-dimensional chiral bosons and as a partition function of a statistical
model of partitions. From a ground state of the statistical model we reproduce
the polyhedron which characterizes the Hilbert space.
