[Paper Review] Large dynamics of Yang--Mills theory: mean dimension formula

Masaki Tsukamoto|arXiv (Cornell University)|Jul 8, 2014

Mathematical Dynamics and Fractals22 references3 citations

TL;DR

This paper establishes the exact mean dimension formula for the infinite-dimensional dynamical system arising from the anti-self-dual (ASD) Yang–Mills equation on ℝ×S³, using a novel technique in metric mean dimension theory. It proves that the mean dimension equals 8c₂(A) + 3, where c₂(A) is the second Chern class of the connection A, revealing a precise topological invariant for this chaotic, infinite-entropy system.

ABSTRACT

This paper studies the Yang--Mills ASD equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover it becomes an infinite dimensional and infinite entropy system. We study the mean dimension of this huge dynamical system. Mean dimension is a topological invariant of dynamical systems introduced by Gromov. We prove the exact formula of the mean dimension by developing a new technique based on the metric mean dimension theory of Lindenstrauss--Weiss.

Motivation & Objective

• To investigate the large-scale chaotic dynamics of the Yang–Mills ASD equation on ℝ×S³ as an infinite-dimensional, infinite-entropy dynamical system.
• To compute the mean dimension of the space of bounded orbits of the ASD evolution equation.
• To develop a new analytical technique based on metric mean dimension theory to derive exact formulas in gauge-theoretic dynamics.
• To extend Gromov’s mean dimension program to Yang–Mills theory, revealing hidden topological invariants in non-compact geometric PDEs.

Proposed method

• The study models the ASD equation as a non-linear evolution equation on ℝ×S³, using the temporal gauge to express it as ∂A/∂t = −*₃F(A(t)).
• It considers the dynamical system of bounded orbits, which include chaotic solutions analogous to the Bernoulli shift and the Hilbert cube [0,1]^ℤ.
• The paper applies the metric mean dimension theory of Lindenstrauss–Weiss to analyze the complexity of the system, focusing on covering and separation numbers in the L∞-topology.
• A key technical step involves constructing a local coordinate system near a reference ASD connection A via the exponential map and harmonic projection, reducing the problem to a finite-dimensional perturbation space.
• The proof uses a contraction mapping argument on weighted Sobolev spaces L²,W₃ ⊕ H¹,Wₐ ⊕ L²,W₁ to show openness and closedness of solution sets, ensuring regularity of the parameter space.
• The final formula is derived by bounding the covering number of a neighborhood of A using the dimension of the L²-cohomology space H¹,Wₐ, which is shown to be 8c₂(A) + 3.

Experimental results

Research questions

• RQ1 What is the mean dimension of the dynamical system generated by bounded solutions of the ASD Yang–Mills equation on ℝ×S³?
• RQ2 How does the mean dimension relate to topological invariants of the underlying connection, such as the second Chern class c₂(A)?
• RQ3 Can metric mean dimension theory be effectively applied to infinite-dimensional gauge-theoretic PDEs with chaotic dynamics?
• RQ4 Is the mean dimension finite and computable for such systems, despite their infinite entropy and dimensionality?
• RQ5 What is the role of deformation parameters in the dynamics, and how do they affect the system’s complexity?

Key findings

• The mean dimension of the dynamical system of bounded ASD connections on ℝ×S³ is exactly 8c₂(A) + 3, where c₂(A) is the second Chern class of the connection A.
• The mean dimension is finite and explicitly computable, providing a topological invariant for the infinite-dimensional dynamics of Yang–Mills theory.
• The system exhibits infinite entropy and infinite dimensionality, yet its mean dimension is finite and precisely quantified.
• The proof establishes a sharp upper bound on the covering number of a neighborhood of a connection A, showing it grows as (C₀/ε)^{8c₂(A)+3} for small ε.
• The method successfully applies metric mean dimension theory to a non-compact geometric PDE, extending Gromov’s program to gauge theory.
• The result holds for any ASD connection A over ℝ×S³ with finite energy, and the bound is sharp in the sense of topological complexity.