Geometric Quantization and Two Dimensional QCD

TL;DR

This paper explores the geometric quantization of two-dimensional QCD with matter fields, identifying phase spaces as infinite-dimensional manifolds and deriving related equations of motion, including linearized forms related to the `t Hooft equation.

Contribution

It extends geometric quantization methods to 2D QCD with fermionic and bosonic matter, linking classical phase spaces to known quantum equations.

Findings

Identified phase spaces as infinite-dimensional Grassmannian and Disc.

Derived nonlinear equations of motion for these classical systems.

Linearized equations reproduce the `t Hooft equation and its scalar analog.

Abstract

In this article, we will discuss geometric quantization of 2d QCD with fermionic and bosonic matter fields. We identify the respective large-N_c phase spaces as the infinite dimensional Grassmannian and the infinite dimensional Disc. The Hamiltonians are quadratic functions, and the resulting equations of motion for these classical systems are nonlinear. In a previous publication, the first author has shown that the linearization of the equations of motion for the Grassmannian gave the `t Hooft equation. We will see that the linearization in the bosonic case leads to the scalar analog of the `t Hooft equation found by Tomaras.