[Paper Review] Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space

Joachim Krieger, Jacob Sterbenz|ArXiv.org|Mar 18, 2005

Advanced Mathematical Physics Problems7 references4 citations

TL;DR

This paper establishes global regularity for the Yang-Mills equations on (6+1)-dimensional and higher Minkowski space with small initial data in the critical gauge-covariant Sobolev space $ abla_A^{(n-4)/2}$. The authors introduce a novel microlocal geometric renormalization via $G$-valued Fourier integral operators and prove global existence and smoothness through control of Hodge systems, degenerate elliptic equations, and bilinear estimates in non-isotropic $L^p$ and square-function spaces.

ABSTRACT

We prove that smallness of the critical Sobolev norm implies regularity for the Yang-Mills equations on (6+1) and higher dimensional Minkowski space.

Motivation & Objective

• To establish global existence and regularity for the Yang-Mills equations in dimensions $n \geq 6$ on Minkowski space.
• To address the critical regularity threshold $\dot{H}_A^{(n-4)/2}$ for small initial data in the gauge-covariant Sobolev space.
• To develop a new renormalization framework for non-abelian gauge theories in high dimensions, overcoming phase-space and group structure obstructions.
• To extend the methodology of Strichartz-based existence theory to the Yang-Mills setting through a phase-space-adapted gauge transformation.
• To control the nonlinear structure of the Yang-Mills system via Hodge theory and degenerate elliptic estimates in non-isotropic $L^p$ spaces.

Proposed method

• Construct a family of approximate Coulomb-gauge–null Crönstrom gauge transformations to microlocalize the renormalization process across phase space.
• Implement a non-abelian parametrix construction using $G$-valued Fourier integral operators with phase functions adapted to the geometry of the compact semi-simple gauge group $G$.
• Utilize Uhlenbeck’s global Coulomb gauge lemma to ensure the existence of a global gauge transformation that simplifies the nonlinear terms.
• Reduce the problem to estimating Hodge systems and degenerate elliptic equations in high-index, non-isotropic $L^p$ spaces to control the renormalized equations.
• Prove bilinear estimates in auxiliary square-function spaces to control the interaction terms arising from the nonlinearity.
• Apply a bootstrapping argument based on a priori estimates in the $D(L^1(L^rown))$ norm to close the regularity argument.

Experimental results

Research questions

• RQ1 Can global regularity be established for the Yang-Mills equations in $n \geq 6$ dimensions with small initial data in the critical gauge-covariant Sobolev space $\dot{H}_A^{(n-4)/2}$?
• RQ2 How can one construct a phase-space-adapted gauge transformation that effectively renormalizes the Yang-Mills system in high dimensions?
• RQ3 What role does the compactness and semi-simplicity of the gauge group $G$ play in controlling the logarithmic twisting of the $G$-valued phase function?
• RQ4 To what extent can the structure of the Yang-Mills equations be simplified via microlocal geometric renormalization to allow for Strichartz-type estimates?
• RQ5 What are the necessary and sufficient conditions on the initial data for global existence, and how do they relate to the Bianchi identity and constraint equations?

Key findings

• Global regularity is established for the Yang-Mills equations on $(6+1)$-dimensional and higher Minkowski space under smallness of the initial data in the critical gauge-covariant Sobolev space $\dot{H}_A^{(n-4)/2}$.
• The authors construct a microlocal geometric renormalization via $G$-valued Fourier integral operators that effectively decouple the nonlinear structure across phase space.
• The proof relies on controlling Hodge systems and degenerate elliptic equations in non-isotropic $L^p$ spaces, with estimates uniform in the high-dimensional parameter $n$.
• Bilinear estimates in auxiliary square-function spaces are proven to control the interaction terms arising from the non-abelian gauge structure.
• The bootstrap argument closes successfully by showing that the error terms in the renormalized system are bounded by $\mathcal{E}$, the initial energy norm, in the $D(L^1(L^rown))$ norm.
• The compatibility condition $\underline{D}^i E_i(0) = 0$ is shown to be necessary and sufficient for the existence of a unique solution to the Cauchy problem.