  We study K\"{a}hler gravity on local SU(N) geometry and describe precise
correspondence with certain supersymmetric gauge theories and random plane
partitions. The local geometry is discretized, via the geometric quantization,
to a foam of an infinite number of gravitational quanta. We count these quanta
in a relative manner by measuring a deviation of the local geometry from a
singular Calabi-Yau threefold, that is a A_{N-1} singularity fibred over
\mathbb{P}^1. With such a regularization prescription, the number of the
gravitational quanta becomes finite and turns to be the perturbative
prepotential for five-dimensional \mathcal{N}=1 supersymmetric SU(N)
Yang-Mills. These quanta are labelled by lattice points in a certain convex
polyhedron on \mathbb{R}^3. The polyhedron becomes obtainable from a plane
partition which is the ground state of a statistical model of random plane
partition that describes the exact partition function for the gauge theory.
Each gravitational quantum of the local geometry is shown to consist of N unit
cubes of plane partitions.
