Gravitational Quantum Foam and Supersymmetric Gauge Theories

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We study Kähler 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 Aₙ₋₁ singularity fibred over P¹. With such a regularization prescription, the number of the gravitational quanta becomes finite and turns to be the perturbative prepotential for five-dimensional N=1 supersymmetric SU(N) Yang-Mills. These quanta are labelled by lattice points in a certain convex polyhedron on R³. 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.