Supersymmetric Gauge Theories with Matters, Toric Geometries and Random Partitions

TL;DR

This paper explores the connection between geometrical quantization, supersymmetric 5D gauge theories with matter, and statistical models of partitions, revealing new insights into their mathematical structure and physical implications.

Contribution

It establishes a novel relation between the Hilbert space of certain geometries and perturbative prepotentials in 5D supersymmetric gauge theories with matter, including a generalization of Nekrasov's partition function.

Findings

Derived the relation between Hilbert space and prepotentials for 5D gauge theories with matter.

Expressed Nekrasov's partition function as a correlation function of chiral bosons.

Reproduced the polyhedron characterizing the Hilbert space from a statistical model ground state.

Abstract

We derive the relation between the Hilbert space of certain geometries under the Bohr-Sommerfeld quantization and the perturbative prepotentials for the supersymmetric five-dimensional SU(N) gauge theories with massive fundamental matters and with one massive adjoint matter. The gauge theory with one adjoint matter shows interesting features. A five-dimensional generalization of Nekrasov's partition function can be written as a correlation function of two-dimensional chiral bosons and as a partition function of a statistical model of partitions. From a ground state of the statistical model we reproduce the polyhedron which characterizes the Hilbert space.