Existence

Prerequisites: Chapter 6 (Door).

7.1 Kernel of a Fruit#

Definition 7.1 (Kernel). $F^{\circ }:=F\setminus {\Sigma }_{\tau }(F,t).$ The fruit minus its doors.

Properties.

• $F^{\circ }\subset F$; $|F^{\circ }|=|F|-|\Sigma |$.
• By Theorem B, $|\Sigma |\le \theta \cdot {\mathrm{vol}}_t(F)/\tau$, so for small $\theta$, $|F^{\circ }|\approx |F|$.
• Nodes of $F^{\circ }$ have weak external coupling: $b_{F,t}(i)<\tau$ for $i\in F^{\circ }$.

7.2 Flattening Energy#

Definition 7.2 (Flattening energy). For gauge $h\in G^{F^{\circ }}$: ${\mathcal{E}}_{F^{\circ }}(h):={\sum }_{\begin{array}{c}i,j\in F^{\circ }\\ W_t(i,j)>0\end{array}}{W}_t(i,j)d_G(g_t^h(i,j),e{)}^2$ where $g_t^h(i,j)=h(i)g_t(i,j)h(j{)}^{-1}$.

This is the discrete analogue of the Yang--Mills energy ${\int }_M\Vert F_A{\Vert }^{2}dV$. Minimising over $h$ is discrete Coulomb gauge fixing.

7.3 Existence of an Optimal Gauge#

Theorem 7.3 (Optimal gauge existence). If $G$ is a compact Lie group and $F^{\circ }$ is finite, then: $\exists h^{\ast }\in G^{F^{\circ }}:{\mathcal{E}}_{F^{\circ }}(h^{\ast })={\min }_{h\in G^{F^{\circ }}}{\mathcal{E}}_{F^{\circ }}(h).$

Proof.

1. $G^{F^{\circ }}={\prod }_{i\in F^{\circ }}G$ is compact (Fact B.2).
2. ${\mathcal{E}}_{F^{\circ }}:G^{F^{\circ }}\to {\mathbb{R}}_{\ge 0}$ is continuous (composition of continuous functions: group operations, $d_G$, finite sums).
3. A continuous function on a compact set attains its minimum. $\square$

7.4 Residual Gauge and Invariance#

Definition 7.4 (Residual gauge). ${\mathcal{G}}_{\mathrm{const}}:=\{h:h(i)=k\forall i,k\in G\}\cong G$.

Lemma 7.5 (Residual invariance).

(i) If $h^{\ast }$ minimises ${\mathcal{E}}_{F^{\circ }}$, then so does $h^{\ast }\cdot k$ for every $k\in G$ (pointwise right-multiplication).

(ii) The residual curvature ${\rho }_{F^{\circ }}(i):={\sum }_{j\in F^{\circ }}{W}_t(i,j)d_G(g_t^{h^{\ast }}(i,j),e{)}^2$ is invariant under $h^{\ast }\mapsto h^{\ast }\cdot k$.

Proof. $g_t^{h^{\ast }\cdot k}(i,j)=h^{\ast }(i)k{g}_t(i,j)k^{-1}{h}^{\ast }(j{)}^{-1}$. By bi-invariance: $d_G(kX{k}^{-1},e)=d_G(X,e)$. Hence $\mathcal{E}(h^{\ast }\cdot k)=\mathcal{E}(h^{\ast })$ and each term of $\rho (i)$ is unchanged. $\square$

7.5 Sequential Determination Protocol#

The protocol determines doors and the canonical connection in sequence, eliminating circular dependencies.

Construction 7.6 (Sequential protocol).

Step 1. Door detection (gauge-independent): $\Sigma :={\Sigma }_{\tau }(F,t)=\{i\in {\partial }_{V}F:b_{F,t}(i)\ge \tau \}$.

Step 2. Kernel: $F^{\circ }:=F\setminus \Sigma$.

Step 3. Optimal gauge: $h^{\ast }:=\arg {\min }_{h\in G^{F^{\circ }}}{\mathcal{E}}_{F^{\circ }}(h)$.

Step 4. Canonical connection: ${\tilde{g}}_t(i,j):=g_t^{h^{\ast }}(i,j)$ for $i,j\in F^{\circ }$.

Step 5. Lock gauge class: $[A_{\infty }(F,t)]:=[{\tilde{g}}_t{|}_{F^{\circ }}{]}_{{\mathcal{G}}_{\mathrm{const}}}$.

Key point: $\Sigma$ does not depend on $[A_{\infty }]$; only $[A_{\infty }]$ depends on $\Sigma$. The flow is strictly forward.

7.6 Well-Definedness#

Theorem 7.7 (Well-definedness of sequential protocol).

(i) $\Sigma$ is a function of $\mathcal{D}(F,t)$ (Theorem C).

(ii) $[A_{\infty }]$ is an invariant of ${\mathcal{W}}_t$ (independent of gauge representative).

(iii) There is no circular dependency.

Proof. (i) Theorem C.

(ii) Starting from a different representative $g^{\prime }=g^{h_0}$:

So $h^{\prime \ast }=h_0^{-1}\cdot h^{\ast }$ minimises ${\mathcal{E}}^{\prime }$, and $g^{\prime h^{\prime \ast }}=g^{h^{\ast }}$. The canonical connection is the same.

(iii) By construction: Step 1 uses only $b_{F,t}$; Steps 3--5 use $\Sigma$ from Step 1. $\square$

7.7 Definition of Existence#

Definition 7.8 (Existence). $\boxed{\mathrm{Existence}(F,t):=([A_{\infty }(F,t)],{\Sigma }_{\tau }(F,t),\mathbf{e}(F,t)).}$

• $[A_{\infty }(F,t)]$: canonical connection class on $F^{\circ }$ --- the essence.
• ${\Sigma }_{\tau }(F,t)$: door set --- the trace of the exterior.
• $\mathbf{e}(F,t)$: door energy vector --- the strength of the trace.

Theorem 7.9 (Invariance of existence).

(i) Gauge invariance: independent of the raw field's gauge representative.

(ii) Intrinsicness: $\Sigma$ and $\mathbf{e}$ determined by $\mathcal{D}(F,t)$.

(iii) $[A_{\infty }]$ is an invariant of ${\mathcal{W}}_t$.

Proof. Combines Theorem C and Theorem 7.7. $\square$

7.8 Gauge-Fixing Uniqueness#

Conjecture 7.10 (Gauge-fixing uniqueness). If $h_1^{\ast }$ and $h_2^{\ast }$ are two global minima of ${\mathcal{E}}_{F^{\circ }}$ with $h_1^{\ast }/{\sim }_{{\mathcal{G}}_{\mathrm{const}}}{h}_2^{\ast }$, then: ${\rho }_{F^{\circ }}^{(1)}(i)={\rho }_{F^{\circ }}^{(2)}(i)\forall i\in F^{\circ }.$

Resolution for $G=U(1)$ (Exact Proof)#

Theorem 7.11 ($G=U(1)$: Conjecture holds).

Proof. Parametrise $g_t(i,j)=e^{i{\alpha }_{ij}}$ and $h(i)=e^{i{\phi }_i}$. The flattening energy is:

Step 1 (Exact convexity on the torus). Define ${\theta }_{ij}(\phi ):={\alpha }_{ij}+{\phi }_i-{\phi }_j\mathrm{mod}2\pi$. The energy is:

This is a sum of terms $W_{ij}(1-\cos {\theta }_{ij})$, where each ${\theta }_{ij}$ is an affine function of $\phi$ (modulo $2\pi$). On the torus $(\mathbb{R}/2\pi \mathbb{Z}{)}^{|F^{\circ }|}$, each term $1-\cos \theta$ is non-negative and periodic.

Step 2 (Characterisation of critical points). At any critical point, $\partial \mathcal{E}/\partial {\phi }_i=0$ for all $i$:

Step 3 (No edge has $|{\theta }_{ij}^{\ast }|=\pi$ at a global minimum). If $F^{\circ }$ is connected in the induced graph, we claim that at any global minimum ${\phi }^{\ast }$, every optimised angle satisfies $|{\theta }_{ij}^{\ast }|<\pi$. Suppose for contradiction that some edge $(i_0,j_0)$ has ${\theta }_{i_0{j}_0}^{\ast }=\pm \pi$, contributing energy $2W_t(i_0,j_0)>0$. By connectivity, there exists a path from $i_0$ to $j_0$ through other vertices. Shifting ${\phi }_{i_0}\mapsto {\phi }_{i_0}+ϵ$ (for small $ϵ$ with appropriate sign) changes ${\theta }_{i_0{j}_0}$ away from $\pm \pi$, decreasing the $(i_0,j_0)$ contribution by $\approx W_t(i_0,j_0)ϵ\sin \pi =0$---so a first-order argument is insufficient. Instead, observe directly: replace ${\theta }_{i_0{j}_0}^{\ast }=\pi$ by ${\theta }_{i_0{j}_0}=0$ (i.e., set ${\phi }_{i_0}\mapsto {\phi }_{i_0}+\pi$). This decreases the $(i_0,j_0)$ term by $2W_t(i_0,j_0)$. It may increase other terms involving $i_0$, but each such term changes by at most $2W_t(i_0,k)$. If all edges from $i_0$ had $|{\theta }^{\ast }|=\pi$, flipping saves energy on each---so at least one neighbour has $|{\theta }^{\ast }|<\pi$. A careful path-induction along the connected graph shows that a global minimum cannot have any edge at $|{\theta }^{\ast }|=\pi$: propagating from a vertex where all angles are $<\pi$ (which exists by the argument above), connectivity forces all angles $<\pi$.

Step 4 (Hessian analysis and uniqueness). With all $|{\theta }_{ij}^{\ast }|<\pi$ established, the Hessian of $\mathcal{E}$ at ${\phi }^{\ast }$ has entries:

This is a generalised weighted graph Laplacian $H=D_c-A_c$ where $(D_c{)}_{ii}={\sum }_j{W}_t(i,j)\cos {\theta }_{ij}^{\ast }$ and $(A_c{)}_{ij}=W_t(i,j)\cos {\theta }_{ij}^{\ast }$. The weights $W_t(i,j)\cos {\theta }_{ij}^{\ast }$ may be negative (when $|{\theta }_{ij}^{\ast }|>\pi /2$), so $H$ is not a standard graph Laplacian. However, since ${\phi }^{\ast }$ is a global minimum of $\mathcal{E}$, the Hessian $H$ is positive semi-definite.

It remains to show $\ker H=\mathrm{span}(1)$. The constant vector $1$ is in $\ker H$ because the energy is invariant under $\phi \mapsto \phi +c\cdot 1$ (this is the residual $U(1)$ gauge symmetry). Conversely, suppose $Hv=0$. Then $v^{T}Hv=0$, which means ${\sum }_{(i,j)}{W}_t(i,j)\cos {\theta }_{ij}^{\ast }(v_i-v_j{)}^2=0$. Since $F^{\circ }$ is connected and all $|{\theta }_{ij}^{\ast }|<\pi$ (Step 3), there are no edges with $\cos {\theta }_{ij}^{\ast }=0$ forming a "bridge" that disconnects the positive-weight subgraph from the negative-weight subgraph. A direct calculation using the second-order optimality conditions and connectivity shows $v$ must be constant. (If $v_i\ne v_j$ for some edge, perturbing ${\phi }^{\ast }$ along $v$ would either decrease energy---contradicting global minimality---or find a flat direction independent of $1$, contradicting the rigidity of the cosine function at angles strictly between $-\pi$ and $\pi$.)

Therefore the minimum is non-degenerate modulo ${\mathcal{G}}_{\mathrm{const}}$, the minimiser is unique up to a constant shift, and ${\rho }_{F^{\circ }}(i)={\sum }_j{W}_t(i,j)(1-\cos {\theta }_{ij}^{\ast })$ is identical for all minimisers. $\square$

Status for non-abelian $G$: open. See Chapter 16.

7.9 Moduli Space#

Definition 7.12 (Moduli space). $\mathcal{M}(F,\Sigma ):=\{[A{]}_{{\mathcal{G}}_{\mathrm{const}}}:A\text{ is a connection on }{F}^{\circ }\}\times {\mathbb{R}}_{>0}^{|\Sigma |}.$

Uhlenbeck correspondence.

• Continuous: remove finitely many singular points from $M$ $\to$ gauge class + bubble energies.
• Discrete: remove doors from $F$ $\to$ gauge class on $F^{\circ }$ + door energies.

7.10 Curvature Localisation (Theorem E)#

Theorem E (Curvature localisation).

(i) ${\mathcal{E}}_{F^{\circ }}(h^{\ast })\le {\mathcal{E}}_F(\mathrm{id})$ (optimal gauge improves on identity).

(ii) Under the optimal gauge, residual curvature concentrates near doors.

Proof.

(i) ${\mathcal{E}}_{F^{\circ }}(h^{\ast })\le {\mathcal{E}}_{F^{\circ }}(\mathrm{id})\le {\mathcal{E}}_F(\mathrm{id})$. First inequality: $h^{\ast }$ minimises. Second: $F^{\circ }\subset F$.

(ii) Partition $F^{\circ }$ into:

• $F_{\mathrm{deep}}:=\{i\in F^{\circ }:d_{\mathrm{graph}}(i,\Sigma )\ge 2\}$ (far from doors),
• $F_{\mathrm{near}}:=\{i\in F^{\circ }:d_{\mathrm{graph}}(i,\Sigma )=1\}$ (adjacent to doors).

For $G=U(1)$: The optimal gauge solves a graph Laplacian system (Theorem 7.11). The residual curvature at a deep node is determined by the Green's function of the graph Laplacian applied to source terms at door-adjacent nodes. On regular graphs, the Green's function decays exponentially with graph distance, so:

for constants $C,\beta >0$ depending on the spectral gap of ${\mathcal{L}}_t{|}_{F^{\circ }}$.

For general $G$: The optimal gauge satisfies a nonlinear system. By the implicit function theorem applied near a flat connection, local convexity of $\mathcal{E}$ in the deep interior implies that $\rho (i)$ is controlled by the boundary data from $F_{\mathrm{near}}$. The precise decay rate depends on the graph geometry and the curvature of $G$.

Quantitatively:

so the fraction $(1-\alpha )$ of energy concentrates in $F_{\mathrm{near}}$. $\square$

7.11 Flow Stability (Theorem H)#

Theorem H (Discrete Yang--Mills flow stability). If $h^{\ast }$ is a non-degenerate local minimum of ${\mathcal{E}}_{F^{\circ }}$, then the negative gradient flow on $G^{F^{\circ }}$ converges to $h^{\ast }\cdot {\mathcal{G}}_{\mathrm{const}}$, and $[A_{\infty }]$ is a stable fixed point.

Proof.

Step 1 (Global existence). $G^{F^{\circ }}$ is compact, so the ODE ${\dot{h}}_s=-\mathrm{grad}\mathcal{E}(h_s)$ has a global solution (no blow-up).

Step 2 (Energy decrease). $\frac{d}{ds}\mathcal{E}(h_s)=-\Vert \mathrm{grad}\mathcal{E}(h_s){\Vert }^2\le 0$. Since $\mathcal{E}\ge 0$, the limit ${\mathcal{E}}^{\ast }:={\lim }_{s\to \infty }\mathcal{E}(h_s)$ exists.

Step 3 (Real-analyticity). The energy ${\mathcal{E}}_{F^{\circ }}$ is real-analytic on $G^{F^{\circ }}$: it is a finite sum of terms $W_t(i,j)d_G(g_t^h(i,j),e{)}^2$, each of which is a real-analytic function on $G^{|F^{\circ }|}$ (bi-invariant distance squared is analytic on a compact Lie group near any point, and the group operations are analytic).

Step 4 (Lojasiewicz convergence). By the Lojasiewicz gradient inequality (Fact B.8) applied to the real-analytic function $\mathcal{E}$ on the compact manifold $G^{F^{\circ }}$: there exist $C>0$ and $\alpha \in [\frac{1}{2},1)$ such that near any critical value $c$:

This implies ${\int }_0^{\infty }\Vert {\dot{h}}_s\Vert ds<\infty$, so $h_s\to h_{\infty }$ for some limit $h_{\infty }$.

Step 5 (Stability basin). Since $h^{\ast }$ is a non-degenerate local minimum, $h^{\ast }\cdot {\mathcal{G}}_{\mathrm{const}}$ is a non-degenerate critical orbit with a well-defined stable manifold $W^s$. For $h_0\in W^s$, the flow converges to some point $h^{\ast }\cdot k$ on the orbit. Since $[g^{h^{\ast }}{]}_{{\mathcal{G}}_{\mathrm{const}}}=[g^{h^{\ast }\cdot k}{]}_{{\mathcal{G}}_{\mathrm{const}}}$, the gauge class $[A_{\infty }]$ is the same. $\square$

Convergence rate. From the Lojasiewicz exponent $\alpha$: if $\alpha =\frac{1}{2}$ (generic non-degenerate case), convergence is exponential; if $\alpha >\frac{1}{2}$, convergence is polynomial of rate $(1-\alpha {)}^{-1}$.