Gravitational Quantum Foam and Supersymmetric Gauge Theories

TL;DR

This paper explores the connection between gravitational quanta in local SU(N) geometries, supersymmetric gauge theories, and random plane partitions, revealing a discretized foam model that links geometry, quantum counting, and statistical models.

Contribution

It establishes a precise correspondence between Kähler gravity on local SU(N) geometries and supersymmetric gauge theories using geometric quantization and plane partition models.

Findings

Gravitational quanta are counted via a regularized geometric quantization.

The number of quanta matches the perturbative prepotential of 5D N=1 SU(N) Yang-Mills.

Each quantum corresponds to N unit cubes in a plane partition.

Abstract

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.