Appendix B — Prerequisites

This appendix collects the background facts used throughout the theory. Readers familiar with compact Lie groups, spectral graph theory, and basic gauge theory may skip to the main text.

B.1 Compact Lie Groups#

Definition. A compact Lie group $G$ is a group that is also a compact smooth manifold, with smooth group operations.

Examples.

• $U(1) = \{e^{i\varphi}:\varphi\in\mathbb{R}\}\cong S^1$ (abelian).
• $SU(2) = \{A\in M_{2\times 2}(\mathbb{C}):A^\dagger A=I,\;\det A=1\}\cong S^3$ (non-abelian).
• Finite groups (with discrete topology).

Fact B.1 (Bi-invariant metric). Every compact Lie group $G$ admits an $\mathrm{Ad}$-invariant inner product $\langle\cdot,\cdot\rangle$ on its Lie algebra $\mathfrak{g}$. The corresponding left-invariant Riemannian metric on $G$ is automatically bi-invariant. The geodesic distance $d_G$ satisfies:

Reference: Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293--329.

Fact B.2 (Tychonoff for finite products). If $G$ is compact and $S$ is a finite set, then $G^S=\prod_{i\in S}G$ is compact. Every continuous real-valued function on $G^S$ attains its minimum and maximum.

B.2 Spectral Graph Theory#

Setup. Let $\mathcal{G}=(V,E,W)$ be a weighted graph with symmetric weights $W(i,j)=W(j,i)\ge0$. Set $d(i)=\sum_j W(i,j)$, $D=\mathrm{diag}(d(i))$, $A=(W(i,j))$.

Definition. The normalised Laplacian is $\mathcal{L}=I-D^{-1/2}AD^{-1/2}$, with eigenvalues $0=\lambda_1\le\lambda_2\le\cdots\le\lambda_n$.

Fact B.3 (Discrete Cheeger inequality). For the Cheeger constant $h:=\min_{\emptyset\ne S\subsetneq V}\phi(S)$:

The lower bound follows from the variational characterisation of $\lambda_2$; the upper bound uses the sweep-cut technique of Alon--Milman.

Reference: Chung, Spectral Graph Theory, CBMS Regional Conference Series 92, AMS, 1997.

Fact B.4 (Davis--Kahan $\sin\Theta$ theorem). Let $\mathcal{L}$ have eigenvalues $\lambda_1\le\cdots\le\lambda_n$ and let $U_k$ be the span of the first $k$ eigenvectors. If $f$ satisfies $R_{\mathcal{L}}(f)\le\mu$ with $\mu<\lambda_{k+1}$, then the angle between $f$ and $U_k$ is $O(\mu/(\lambda_{k+1}-\mu))$.

B.3 Cheeger Constants and Conductance#

Definition. For $\emptyset\ne S\subsetneq V$:

A set $S$ with $\phi(S)\le\theta$ is called a low-conductance set or (in our theory) a fruit.

Fact B.5 (Conductance and mixing time). For the lazy random walk $\tilde P=\frac{1}{2}(I+D^{-1}A)$, the mixing time satisfies:

where $\phi$ is the graph conductance.

Reference: Levin, Peres, Wilmer, Markov Chains and Mixing Times, AMS, 2009.

Fact B.6 (Sinclair--Jerrum conductance bound). For a reversible Markov chain with conductance $\Phi$, the spectral gap $\gamma$ satisfies:

Reference: Sinclair and Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chains, Inform. Comput. 82 (1989), 93--133.

B.4 Uhlenbeck Compactness#

The theory's treatment of doors and existence parallels Uhlenbeck's compactness theorem in continuous gauge theory.

Fact B.7 (Uhlenbeck compactness, informal). Let $\{A_k\}$ be a sequence of connections on a principal $G$-bundle over a closed Riemannian 4-manifold $M$ with $\int_M|F_{A_k}|^2\le E$. Then, after passing to a subsequence and applying gauge transformations, $A_k$ converges smoothly on $M\setminus\{p_1,\ldots,p_L\}$ to a smooth connection $A_\infty$, where $L\le E/\epsilon_0$ for a universal constant $\epsilon_0>0$. The bubble energies $e_l=\lim_{r\to0}\lim_{k\to\infty}\int_{B_r(p_l)}|F_{A_k}|^2$ satisfy $\sum_l e_l\le E$.

The discrete analogue:

• Bounded cut energy $\mathrm{cut}(F,\bar F)\le\theta\cdot\mathrm{vol}(F)$ replaces the energy bound.
• The door threshold $\tau$ plays the role of $\epsilon_0$.
• Door count $|\Sigma|\le\theta\cdot\mathrm{vol}(F)/\tau$ parallels $L\le E/\epsilon_0$.

Reference: Uhlenbeck, Connections with $L^p$ bounds on curvature, Comm. Math. Phys. 83 (1982), 31--42.

B.5 Lojasiewicz--Simon Gradient Inequality#

Fact B.8 (Lojasiewicz inequality, finite-dimensional). Let $f:M\to\mathbb{R}$ be a real-analytic function on a compact Riemannian manifold $M$. For every critical value $c$ of $f$, there exist $C>0$ and $\alpha\in[\frac{1}{2},1)$ such that in a neighbourhood of $f^{-1}(c)$:

Consequence. The negative gradient flow $\dot x=-\mathrm{grad}\,f(x)$ converges as $t\to\infty$, and the trajectory has finite length:

This is used in Theorem H to guarantee convergence of the discrete Yang--Mills gradient flow on the compact space $G^{F^\circ}$.

Reference: Lojasiewicz, Sur les trajectoires du gradient d'une fonction analytique, Seminari 1984, Univ. Bologna, 1984.

B.6 Parallel Transport and Discrete Connections#

In continuous differential geometry, a connection on a principal $G$-bundle assigns to each path in the base manifold a parallel transport map in $G$. The curvature $F_A$ measures the failure of parallel transport around infinitesimal loops to be trivial.

In the discrete setting:

• An edge $G$-label $g_t(i,j)\in G$ is the parallel transport from $i$ to $j$.
• The holonomy around a triangle $(i,j,k)$ is $\Omega_t(i,j,k)=g_t(i,j)\,g_t(j,k)\,g_t(k,i)$.
• $\Omega_t=e$ means the triangle is "flat" (transits are compatible).
• A gauge transformation $h:V\to G$ changes the local frame at each node: $g_t^h(i,j)=h(i)\,g_t(i,j)\,h(j)^{-1}$.

This is the standard framework of lattice gauge theory, originating in Wilson's formulation of lattice QCD.

Reference: Wilson, Confinement of quarks, Phys. Rev. D 10 (1974), 2445--2459.