41 research outputs found
Free Fermion and Seiberg-Witten Differential in Random Plane Partitions
A model of random plane partitions which describes five-dimensional
supersymmetric SU(N) Yang-Mills is studied. We compute the wave
functions of fermions in this statistical model and investigate their
thermodynamic limits or the semi-classical behaviors. These become of the WKB
type at the thermodynamic limit. When the fermions are located at the main
diagonal of the plane partition, their semi-classical wave functions are
obtained in a universal form. We further show that by taking the
four-dimensional limit the semi-classical wave functions turn to live on the
Seiberg-Witten curve and that the classical action becomes precisely the
integral of the Seiberg-Witten differential. When the fermions are located away
from the main diagonal, the semi-classical wave functions depend on another
continuous parameter. It is argued that they are related with the wave
functions at the main diagonal by the renormalization group flow of the
underlying gauge theory.Comment: 32 pages, 3 figures, typos correcte
Gravitational Quantum Foam and Supersymmetric Gauge Theories
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.Comment: 43 pages, 12 figures: V2 typos correcte