Gauge Loop-String-Hadron Formulation on General Graphs and Applications to Fully Gauge Fixed Hamiltonian Lattice Gauge Theory

TL;DR

This paper introduces a gauge-invariant Loop-String-Hadron formulation for SU(2) Yang-Mills theory on arbitrary graphs, enabling a fully gauge-fixed Hamiltonian approach with explicit quantum number relations.

Contribution

It develops a novel LSH-based representation applicable to general graphs and details the gauge fixing process, connecting it to existing magnetic and prepotential descriptions.

Findings

Provides a gauge-invariant LSH formulation for general graphs

Demonstrates a fully gauge-fixed Hamiltonian representation

Clarifies the relation between quantum numbers and magnetic variables

Abstract

We develop a gauge invariant, Loop-String-Hadron (LSH) based representation of SU(2) Yang-Mills theory defined on a general graph consisting of vertices and half-links. Inspired by weak coupling studies, we apply this technique to maximal tree gauge fixing. This allows us to develop a fully gauge fixed representation of the theory in terms of LSH quantum numbers. We explicitly show how the quantum numbers in this formulation directly relate to the variables in the magnetic description. In doing so, we will also explain in detail the way that the Kogut-Susskind formulation, prepotentials, and point splitting, work for general graphs. In the appendix of this work we provide a self-contained exposition of the mathematical details of Hamiltonian pure gauge theories defined on general graphs.