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Twists and Wilson Loops in the String Theory of Two Dimensional QCD

David J. Gross, Washington Taylor

TL;DR

This work completes a geometric, map-based string description of two-dimensional QCD by expressing the partition function on arbitrary manifolds as a sum over covering maps with twist points (Ω-points) and by formulating an open-string framework for Wilson loops. The model naturally splits into two chiral sectors (orientation-preserving/reversing) that are coupled by orientation-reversing tubes and twist points, yielding a complete 1/N expansion with precise gluing rules. It introduces generalized Frobenius relations and a boundary-observable basis Υσ(U) to capture Wilson-loop data, and provides explicit map-based calculations and consistency checks on spheres and tori, including comparisons to established QCD$_2$ results. Dynamical quarks are incorporated through a fermion determinant path integral, suggesting an open+closed string interpretation with dynamical boundaries and outlining future directions toward a full worldsheet action and potential extension to higher dimensions.

Abstract

Many Texo's have been corrected and a reference added.

Twists and Wilson Loops in the String Theory of Two Dimensional QCD

TL;DR

This work completes a geometric, map-based string description of two-dimensional QCD by expressing the partition function on arbitrary manifolds as a sum over covering maps with twist points (Ω-points) and by formulating an open-string framework for Wilson loops. The model naturally splits into two chiral sectors (orientation-preserving/reversing) that are coupled by orientation-reversing tubes and twist points, yielding a complete 1/N expansion with precise gluing rules. It introduces generalized Frobenius relations and a boundary-observable basis Υσ(U) to capture Wilson-loop data, and provides explicit map-based calculations and consistency checks on spheres and tori, including comparisons to established QCD results. Dynamical quarks are incorporated through a fermion determinant path integral, suggesting an open+closed string interpretation with dynamical boundaries and outlining future directions toward a full worldsheet action and potential extension to higher dimensions.

Abstract

Many Texo's have been corrected and a reference added.

Paper Structure

This paper contains 10 sections, 150 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The partition function
  3. The Complete Theory
  4. Gluing manifolds
  5. Wilson loops
  6. Examples of partition functions
  7. Examples of Wilson loops
  8. Dynamical quarks
  9. Conclusions
  10. Generalized Frobenius relations

Figures (14)

  • Figure 1: Gluing manifolds along a circle and a segment
  • Figure 2: A genus 2 surface built from spheres with boundary
  • Figure 3: the inside (I) and outside (O) of an oriented closed curve $\gamma$
  • Figure 4: A graph $\Gamma$ formed by Wilson loops $\gamma_1, \gamma_2$ with edges $E_a$, vertices $V_b$
  • Figure 5: Numbers of $\Omega$-points in each region $M_j (I_j)$
  • ...and 9 more figures