Gravitational quantum foam and supersymmetric gauge theories

Abstract

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 A N − 1 singularity fibred over 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 N = 1 supersymmetric SU ( N ) Yang–Mills. These quanta are labelled by lattice points in a certain convex polyhedron on 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.