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Makeenko-Migdal equations for lattice Yang-Mills-Higgs

Hao Shen, Scott Andrew Smith, Rongchan Zhu

TL;DR

The paper develops Makeenko-Migdal master loop equations for lattice Yang-Mills-Higgs theories with $G\in\{SO(N),U(N),SU(N)\}$ by extending observables to include open Wilson lines. It introduces a conditional Langevin dynamics framework and applies Itô's formula to derive a comprehensive set of string operations (deformations, expansions, mergers, switches, etc.) that close the system of equations for Wilson line/loop correlations. The authors first establish results for simple strings, then generalize to arbitrary single strings and finally to collections of strings, yielding a closed, structured recursion with group- and representation-dependent coefficients, including Itô corrections. This framework opens avenues for analyzing gauge-string duality, area versus perimeter laws, and surface sums with boundaries in lattice YMH models, and provides a foundation for potential continuum limits and higher-dimensional extensions.

Abstract

We derive a form of master loop equations for the lattice Yang-Mills-Higgs theory with structure group $SO(N)$, $U(N)$ or $SU(N)$. Compared to the pure Yang-Mills setting, several new operations arise. In fact, to obtain a closed recursion we must broaden the class of observables to include open Wilson lines. Our approach is based on the conditional Langevin dynamic and yields a concise proof via Itô's formula.

Makeenko-Migdal equations for lattice Yang-Mills-Higgs

TL;DR

The paper develops Makeenko-Migdal master loop equations for lattice Yang-Mills-Higgs theories with by extending observables to include open Wilson lines. It introduces a conditional Langevin dynamics framework and applies Itô's formula to derive a comprehensive set of string operations (deformations, expansions, mergers, switches, etc.) that close the system of equations for Wilson line/loop correlations. The authors first establish results for simple strings, then generalize to arbitrary single strings and finally to collections of strings, yielding a closed, structured recursion with group- and representation-dependent coefficients, including Itô corrections. This framework opens avenues for analyzing gauge-string duality, area versus perimeter laws, and surface sums with boundaries in lattice YMH models, and provides a foundation for potential continuum limits and higher-dimensional extensions.

Abstract

We derive a form of master loop equations for the lattice Yang-Mills-Higgs theory with structure group , or . Compared to the pure Yang-Mills setting, several new operations arise. In fact, to obtain a closed recursion we must broaden the class of observables to include open Wilson lines. Our approach is based on the conditional Langevin dynamic and yields a concise proof via Itô's formula.

Paper Structure

This paper contains 24 sections, 10 theorems, 59 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Yang--Mills--Higgs SDEs
  3. Conditional SDEs
  4. Explicit form of the SDEs
  5. Strings on the lattice
  6. Loops and lines
  7. Paths, closed paths, cycles
  8. Backtracking
  9. Operations
  10. Splitting
  11. Expansion
  12. Merger
  13. Switching
  14. Deformation
  15. Breaking
  16. ...and 9 more sections

Key Result

Theorem 1.3

Let $s$ be a collection of strings. Fix an edge $e$. One has where the sum is over all operations in the first table, and $C_{G,s,e}$ is a constant depending on $G,s,e$. Fix $x\in \Lambda$. One has where the sum is over all operations in the second table, and $C_{M,s,x}$ is a constant depending on $M,s,x$.

Figures (25)

  • Figure 1: A closed path $e_1e_2\cdots e_{10}$ with a terminal backtrack $e_1=e_{10}^{-1}$.
  • Figure 2: A line and a loop created from the path in Fig. \ref{['fig:path-term-bt']}.
  • Figure 3: Positive splitting of a line $\ell_i=aebec\in {\mathbb L}_1$ as illustration of \ref{['e:splitting1']}.
  • Figure 4: Positive expansion of a line by $p$.
  • Figure 5: Negative expansion of a line by $p$.
  • ...and 20 more figures

Theorems & Definitions (27)

  • Definition 1.1: Wilson loops
  • Definition 1.2: Wilson lines
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 3.1
  • ...and 17 more