Chapter 13 — Yang–Mills Flow

Prerequisites: Chapters 2--8 (Part I), Chapter 9 (Theorem H).

13.1 Motivation#

In Chapters 2--8, the world ${\mathfrak{W}}_t$ was analysed at a frozen instant. The question now is: how does the gauge connection evolve over time?

The discrete Yang--Mills gradient flow provides the answer. Starting from an arbitrary connection, the flow minimises curvature energy, automatically stabilising the world's essential structure.

13.2 Discrete Yang--Mills Energy#

Definition 13.1 (Discrete YM energy)#

Let $(V,E,\mathcal{T})$ be a graph with triangle set $\mathcal{T}$ (all 3-cliques). For a $G$-valued connection $A=\{g_{ij}{\}}_{(i,j)\in E}$, the holonomy of triangle $△=(i,j,k)$ is:

The discrete Yang--Mills energy:

Properties.

• $\mathcal{E}(A)=0⟺$ all holonomies are trivial $⟺$ the connection is flat.
• $\mathcal{E}(A)>0⟺$ non-trivial curvature $⟺$ topological obstruction.
• $\mathcal{E}$ is gauge-invariant (Theorem 13.4 below).

13.3 Gradient Flow#

Definition 13.2 (Discrete YM gradient flow)#

On $G^E$ with the product Riemannian structure ${\sum }_{(i,j)}⟨\cdot ,\cdot {⟩}_{\mathfrak{g}}$:

i.e., for each edge $(i,j)$:

Theorem 13.1 (Monotone energy decrease)#

Along the gradient flow: $\frac{d}{ds}\mathcal{E}(A_s)=-\Vert \mathrm{grad}\mathcal{E}(A_s){\Vert }^2\le 0$.

Proof. Chain rule: $\frac{d}{ds}\mathcal{E}=⟨\mathrm{grad}\mathcal{E},{\dot{A}}_s⟩=-\Vert \mathrm{grad}\mathcal{E}{\Vert }^2$. $\square$

Since $\mathcal{E}\ge 0$, the limit ${\mathcal{E}}^{\ast }:={\lim }_{s\to \infty }\mathcal{E}(A_s)$ exists.

13.4 Fixed Points and Stability#

Theorem 13.2 (Fixed-point characterisation)#

A fixed point $A^{\ast }$ of the flow satisfies $\mathrm{grad}\mathcal{E}(A^{\ast })=0$ (critical point). If additionally $\mathrm{Hess}(\mathcal{E}){|}_{A^{\ast }}$ is positive-definite on the normal space to the gauge orbit, then $A^{\ast }$ is Lyapunov stable.

Theorem 13.3 (Connection to $[A_{\infty }]$)#

The gauge class $[A_{\infty }]$ from Construction 7.6 is a stable fixed point of the gradient flow of ${\mathcal{E}}_{F^{\circ }}$ (flattening energy on the fruit kernel). This follows from Theorem H (Chapter 9).

13.5 Gauge Invariance#

Theorem 13.4 (Gauge invariance of YM energy)#

For any gauge transformation $h:V\to G$: $\mathcal{E}(A^h)=\mathcal{E}(A)$.

Proof. ${\Omega }_{△}^h=h(i){\Omega }_{△}h(i{)}^{-1}$. By bi-invariance of $d_G$: $d_G({\Omega }^h,e)=d_G(\Omega ,e)$. $\square$

Corollary. The gradient flow is gauge-equivariant: $\frac{d}{ds}(A^h{)}_s=(\frac{d}{ds}{A}_s{)}^h$.

13.6 The $U(1)$ Case#

Theorem 13.5 ($U(1)$ convergence and uniqueness)#

For $G=U(1)$:

1. The energy $\mathcal{E}$ is convex on the torus $(S^1{)}^{|E|}$ (each term $2(1-\cos \theta )$ is convex in $\theta \in (-\pi ,\pi )$).
2. The global minimum is unique (up to gauge).
3. The flow converges exponentially to $[A_{\infty }]$ from any initial condition.

Proof. Convexity follows from $\frac{d^2}{d{\theta }^2}(1-\cos \theta )=\cos \theta >0$ for $|\theta |<\pi /2$ (near the minimum). The strict convexity of the total energy on the connected quotient space gives uniqueness. Exponential convergence follows from the positive spectral gap of the Hessian (Theorem 7.11). $\square$

13.7 Non-Abelian Difficulties#

$SU(2)$ non-convexity#

For $G=SU(2)$, the geodesic distance $d_{SU(2)}(g,h)=\arccos (|\mathrm{tr}(g^{†}h)|/2)$ is non-convex on $SU(2{)}^{|E|}$:

• Multiple local minima may exist.
• Not every critical point is stable.
• Initial conditions may lead to different limit points.

This is the core reason Conjecture 7.10 (gauge-fixing uniqueness) remains open for non-abelian $G$.

Gribov ambiguity#

For $G=SU(2)$, the space of gauge-inequivalent connections modulo gauge may have non-trivial topology (Gribov copies). This complicates:

• Gauge fixing: no single global gauge choice may work.
• Counting fixed points.
• Global properties of the moduli space.

Workaround: compute gauge-invariant quantities (Axes 1--3) directly, bypassing gauge fixing.

13.8 Integration with the Three Axes#

After the flow converges to $A_{\infty }$:

• Axis 1: $H^{\ast }(K_{\alpha };A)$ computed on the stabilised connection.
• Axis 2: $H^{\ast }(K_{\alpha },L_{\alpha };A)$ reflects the role of doors in the equilibrium state.
• Axis 3: Barcode of $\Sigma$ reflects the final fractal structure.

The three axes read the world's stable properties---those that survive the dynamical relaxation.