Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Direct and Indirect Loop Equations in Lattice Yang-Mills Theory

Xizhe Liu, Gang Yang

TL;DR

The paper tackles the challenge of building a dynamically independent basis of Wilson-loop observables in SU(2) lattice Yang–Mills within a lattice positivity bootstrap framework. It introduces a geometric plaquette-cut and subloop-cut methodology to generate a complete set of direct SD and trace-reduction equations, and couples this with a vertex-filtering strategy to reveal indirect equations that emerge after eliminating higher-length intermediate loops. Applying the framework to SU(2) in 2–4 dimensions, the authors count independent canonical loops and direct equations, analyze their exponential growth, and demonstrate how indirect equations can tighten bootstrap bounds, albeit with significant computational complexity. The work provides a scalable, generalizable approach for constructing loop equations beyond 2D, linking operator-reduction ideas to the lattice positivity program and offering pathways to extend to higher-rank gauge groups and more complex theories.

Abstract

The dynamics of Wilson loops is governed by an infinite set of Schwinger-Dyson equations and trace relations. In the context of the lattice positivity bootstrap, a central challenge is determining a dynamically independent basis of these operators within a truncated space. We present a systematic framework to solve this problem, utilizing a geometric plaquette-cut and subloop-cut strategy to efficiently generate all (local) direct equations. Furthermore, we identify and analyze ``indirect equations", which arise from the elimination of higher-length intermediate loops. We elucidate the origin of these subtle relations and propose a vertex-filtering strategy to construct them. Applying the above framework to SU(2) lattice Yang-Mills theory, we provide explicit counting of independent canonical loops and equations in 2, 3, and 4 dimensions, along with a statistical analysis of their asymptotic growth.

Direct and Indirect Loop Equations in Lattice Yang-Mills Theory

TL;DR

The paper tackles the challenge of building a dynamically independent basis of Wilson-loop observables in SU(2) lattice Yang–Mills within a lattice positivity bootstrap framework. It introduces a geometric plaquette-cut and subloop-cut methodology to generate a complete set of direct SD and trace-reduction equations, and couples this with a vertex-filtering strategy to reveal indirect equations that emerge after eliminating higher-length intermediate loops. Applying the framework to SU(2) in 2–4 dimensions, the authors count independent canonical loops and direct equations, analyze their exponential growth, and demonstrate how indirect equations can tighten bootstrap bounds, albeit with significant computational complexity. The work provides a scalable, generalizable approach for constructing loop equations beyond 2D, linking operator-reduction ideas to the lattice positivity program and offering pathways to extend to higher-rank gauge groups and more complex theories.

Abstract

The dynamics of Wilson loops is governed by an infinite set of Schwinger-Dyson equations and trace relations. In the context of the lattice positivity bootstrap, a central challenge is determining a dynamically independent basis of these operators within a truncated space. We present a systematic framework to solve this problem, utilizing a geometric plaquette-cut and subloop-cut strategy to efficiently generate all (local) direct equations. Furthermore, we identify and analyze ``indirect equations", which arise from the elimination of higher-length intermediate loops. We elucidate the origin of these subtle relations and propose a vertex-filtering strategy to construct them. Applying the above framework to SU(2) lattice Yang-Mills theory, we provide explicit counting of independent canonical loops and equations in 2, 3, and 4 dimensions, along with a statistical analysis of their asymptotic growth.
Paper Structure (22 sections, 35 equations, 9 figures, 11 tables)

This paper contains 22 sections, 35 equations, 9 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Setup of loops and equations
  3. Wilson loops
  4. Lattice theory.
  5. Loops and letter representation.
  6. Canonical loops.
  7. Loop equations
  8. SD equations.
  9. Trace relations.
  10. Remark.
  11. Constructing loop equations: direct ones
  12. Plaquette-cut for SD equations
  13. Subloop-cut for trace relations
  14. Growth of loops and direct equations
  15. Loop equations: indirect ones
  16. ...and 7 more sections

Figures (9)

  • Figure 1: 2D lattice showing a large Wilson loop and a single plaquette.
  • Figure 2: SU(2) trace relations.
  • Figure 3: Plaquette-cut for direct SD equations. The red arrows represent the links for variation.
  • Figure 4: An example of overlapping plaquette-cut. The red arrows represent the links for variation.
  • Figure 5: Subloop-cut strategy for trace relation.
  • ...and 4 more figures