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$\mathrm{U}(N)$ lattice Yang-Mills in the 't Hooft regime

Ron Nissim

TL;DR

This work establishes a mass gap, a unique infinite-volume limit, and a large-$N$ limit for $\,\mathrm{U}(N)\,$ lattice Yang–Mills in the 't Hooft regime by overcoming the failure of the Bakry–Émery approach for $\,\mathrm{U}(N)\$ through a random-environment decomposition into $\,\mathrm{U}(1)\times\mathrm{SU}(N)\$. It combines cluster expansions with Langevin-dynamics arguments to prove a mass gap and decorrelation bounds, and uses a marginal SU(N) analysis to derive a mass gap for the marginal field, enabling a robust infinite-volume limit. The Large-$N$ analysis then shows Wilson loops become deterministic and factorize in probability, extending the Wilson-loop factorization property to $\,\mathrm{U}(N)\$. The results provide a rigorous foundation for the strong-coupling, 't Hooft scaling regime of $\ ext{U}(N)$ lattice YM and highlight a productive synthesis of cluster expansions and stochastic dynamics in non-Abelian gauge theories. Overall, the paper broadens the scope of rigorous lattice YM results to include $ ext{U}(N)$ and strengthens connections to large-$N$ and string-theoretic viewpoints.

Abstract

We establish a mass gap, prove the existence of a unique infinite volume limit, and give a new proof of the large $N$ limit for $\mathrm{U}(N)$ lattice Yang-Mills theory in the 't Hooft regime. These results were previously obtained for $\mathrm{SU}(N)$ and $\mathrm{SO}(N)$ lattice Yang-Mills theories as applications of the mixing of the associated Langevin dynamics, which is verified via the Bakry-Émery criterion [SZZ23]. For $\mathrm{U}(N)$, however, this approach fails because its Ricci curvature is not uniformly positive, and as a result the Bakry-Émery condition cannot be easily verified. To overcome this obstacle, we recast the $\mathrm{U}(N)$ theory as a random-environment $\mathrm{SU}(N)$ model, where the randomness arises from a $\mathrm{U}(1)$ field, and combine cluster-expansion and Langevin-dynamics techniques to analyze the resulting $\mathrm{U}(1)\times\mathrm{SU}(N)$ model.

$\mathrm{U}(N)$ lattice Yang-Mills in the 't Hooft regime

TL;DR

This work establishes a mass gap, a unique infinite-volume limit, and a large- limit for lattice Yang–Mills in the 't Hooft regime by overcoming the failure of the Bakry–Émery approach for through a random-environment decomposition into . It combines cluster expansions with Langevin-dynamics arguments to prove a mass gap and decorrelation bounds, and uses a marginal SU(N) analysis to derive a mass gap for the marginal field, enabling a robust infinite-volume limit. The Large- analysis then shows Wilson loops become deterministic and factorize in probability, extending the Wilson-loop factorization property to . The results provide a rigorous foundation for the strong-coupling, 't Hooft scaling regime of lattice YM and highlight a productive synthesis of cluster expansions and stochastic dynamics in non-Abelian gauge theories. Overall, the paper broadens the scope of rigorous lattice YM results to include and strengthens connections to large- and string-theoretic viewpoints.

Abstract

We establish a mass gap, prove the existence of a unique infinite volume limit, and give a new proof of the large limit for lattice Yang-Mills theory in the 't Hooft regime. These results were previously obtained for and lattice Yang-Mills theories as applications of the mixing of the associated Langevin dynamics, which is verified via the Bakry-Émery criterion [SZZ23]. For , however, this approach fails because its Ricci curvature is not uniformly positive, and as a result the Bakry-Émery condition cannot be easily verified. To overcome this obstacle, we recast the theory as a random-environment model, where the randomness arises from a field, and combine cluster-expansion and Langevin-dynamics techniques to analyze the resulting model.
Paper Structure (15 sections, 24 theorems, 142 equations)

This paper contains 15 sections, 24 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattice Yang--Mills
  4. Riemannian Geometry and Lie Groups
  5. Ricci Curvature and Hessian
  6. Lie Groups and Lie Algebras
  7. Products of Riemannian Manifolds
  8. Lie Group Brownian Motion
  9. $\mathrm{U}(N)$ Yang-Mills as random environment $\mathrm{SU}(N)$ Field
  10. Mass Gap
  11. Step 1: Mass gap conditioned on $\mathrm{SU}(N)$ field
  12. Step 2: Localizing conditional observables
  13. Step 3: Mass gap for $\mathrm{SU}(N)$ Marginal Field
  14. Infinite Volume Limit
  15. Large $N$ Limit: Proof of Theorem \ref{['thm:main-large-N-Limit']}

Key Result

Theorem 1.1

Let $d \geq 2$, $N \geq 2$. Then for some fixed $\tilde{\beta}=\tilde{\beta}(d)$, and all $\beta < \tilde{\beta}$, there exists a probability measure $\mu_{\mathrm{U}(N),\beta}$ on $\mathrm{U}(N)^{E_{\Lambda_{\infty}}^+}$ (with $\Lambda_{\infty}=\mathbb{Z}^d$) such that weakly as $L \to \infty$. Additionally, for any smooth local observables $f, g$ with $\Lambda_f, \Lambda_g \subseteq \Lambda$ (s

Theorems & Definitions (63)

  • Theorem 1.1: Mass gap and infinite volume limit
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Large $N$ Limit for $\mathrm{U}(N)$
  • Remark 1.6
  • Definition 2.1: Lattice gauge configuration
  • Remark 2.2
  • Definition 2.3: Plaquette variable
  • Remark 2.4
  • ...and 53 more