Read-back: YangMills.wilsonLoop_gaugeTransform

Binders and assumptions

The statement is universally quantified over all of the following (the first four are implicit arguments, inferred from the later ones):

• $d \in \mathbb{N}$, the number of lattice directions (allowed to be $0$);
• $R \in \mathbb{N}$, the half-width of the box (allowed to be $0$);
• $N \in \mathbb{N}$, the matrix size (allowed to be $0$);
• a type $G$ equipped with a group structure. No topology, compactness, measurability, or any other structure on $G$ is assumed, and $G$ is not required to be non-trivial;
• $\rho : G \to \mathrm{U}(N)$, a monoid homomorphism (hence a group homomorphism) from $G$ into the unitary group $\mathrm{U}(N)$, meaning the set of $N \times N$ complex matrices $A$ with $A^* A = I$ and $A A^* = I$ (here $A^*$ is the conjugate transpose). Every such element is regarded as an $N\times N$ complex matrix when a trace is taken;
• $g : \mathbb{Z}^d \to G$, an arbitrary function defined on all lattice sites $\mathbb{Z}^d$ (not only on the box), with no constraints;
• $U$, a configuration on the box of half-width $R$ (defined below);
• $k \in \mathbb{N}$, a scale parameter;
• $\gamma$, a loop in dimension $d$ (defined below);
• a hypothesis $h_\gamma$ asserting that $\gamma$ is "in the box at scale $k$" (defined below).

The custom definitions, unfolded

Sites and unit vectors. A site is an element $x \in \mathbb{Z}^d$, i.e. a function from the $d$ directions $\{0,\dots,d-1\}$ to $\mathbb{Z}$. For a direction $\mu$, $e_\mu \in \mathbb{Z}^d$ is the vector with a $1$ in coordinate $\mu$ and $0$ elsewhere.

Box. The box is the finite set

$$B_R = \{\, x \in \mathbb{Z}^d \;:\; -R \le x_\mu \le R \text{ for every direction } \mu \,\}.$$

(When $d = 0$, $\mathbb{Z}^0$ has exactly one point and $B_R$ is that single point; when $R = 0$, $B_0 = \{0\}$.)

Edges and box edges. An edge is a pair $(x,\mu)$ with $x\in\mathbb{Z}^d$ a site and $\mu$ a direction. The set of box edges is

$$E_R = \{\, (x,\mu) \;:\; x \in B_R \text{ and } x + e_\mu \in B_R \,\}.$$

Configurations. A configuration $U$ is a function $U : E_R \to G$, assigning a group element to each box edge (and to nothing else).

Link value (total extension). For an arbitrary edge $e=(x,\mu)$, not necessarily in the box,

$$\operatorname{link}_U(x,\mu) = \begin{cases} U(x,\mu) & \text{if } (x,\mu)\in E_R,\\ 1_G & \text{otherwise.}\end{cases}$$

So every edge outside $E_R$ carries the junk value $1_G$.

Gauge transform. For $g:\mathbb{Z}^d\to G$ and a configuration $U$, the transformed configuration $U^{g}$ is the function on $E_R$ given by

$$U^{g}(x,\mu) = g(x)\cdot U(x,\mu)\cdot g(x+e_\mu)^{-1} \qquad \text{for } (x,\mu)\in E_R.$$

Consequently $\operatorname{link}_{U^g}(x,\mu) = g(x)\,U(x,\mu)\,g(x+e_\mu)^{-1}$ when $(x,\mu)\in E_R$, and $\operatorname{link}_{U^g}(x,\mu) = 1_G$ (not $g(x)g(x+e_\mu)^{-1}$) when $(x,\mu)\notin E_R$.

Steps and step vectors. A step is a pair $s=(\mu, b)$ with $\mu$ a direction and $b$ a Boolean. Its step vector is

$$\vec{s} = \begin{cases} e_\mu & b = \mathrm{true} \ (\text{forward}),\\ -e_\mu & b = \mathrm{false} \ (\text{backward}).\end{cases}$$

When $d = 0$ there are no directions and hence no steps at all.

Path vertices. Given a starting site $x$ and a finite list of steps $s_1,\dots,s_n$, put $x_0 = x$ and $x_{i} = x_{i-1} + \vec{s}_i$. The vertex list is

$$\operatorname{pathVertices}(x; s_1,\dots,s_n) = [\,x_0, x_1, \dots, x_{n-1}\,],$$

i.e. it has exactly $n$ entries and omits the final point $x_n$. For the empty step list it is the empty list.

Path holonomy. With the same notation, the holonomy is the ordered product in $G$

$$\operatorname{hol}_U(x; s_1,\dots,s_n) = \prod_{i=1}^{n} h_i, \qquad h_i = \begin{cases} \operatorname{link}_U(x_{i-1},\mu_i) & s_i = (\mu_i,\mathrm{true}),\\[2pt] \operatorname{link}_U(x_{i-1}-e_{\mu_i},\mu_i)^{-1} & s_i = (\mu_i,\mathrm{false}),\end{cases}$$

multiplied left to right ($h_1 h_2\cdots h_n$; for a backward step, $x_{i-1} - e_{\mu_i} = x_i$). For the empty step list the holonomy is $1_G$. Each factor is $1_G$ whenever the corresponding edge lies outside $E_R$.

Loops. A loop $\gamma$ consists of

• a natural number $\gamma.\mathrm{scale}$,
• a base site $\gamma.\mathrm{base}\in\mathbb{Z}^d$,
• a finite (possibly empty) list of steps $\gamma.\mathrm{steps} = [s_1,\dots,s_n]$,
• a proof that $\sum_{i=1}^n \vec{s}_i = 0$ in $\mathbb{Z}^d$ (closedness). For the empty list this holds automatically, and when $d=0$ it holds automatically as well.

Nothing requires the steps to be non-empty, the vertices to be distinct, or the loop to be non-degenerate (e.g. a forward step immediately followed by the reverse step is allowed).

Refinement. For $j\in\mathbb{N}$, the refined loop $\gamma^{(j)}$ has scale $\gamma.\mathrm{scale}+j$, base $2^{j}\cdot\gamma.\mathrm{base}$ (scalar multiplication in $\mathbb{Z}^d$), and step list obtained by replacing each step $s_i$ by $2^j$ consecutive copies of itself, in order:

$$[\,\underbrace{s_1,\dots,s_1}_{2^j},\ \underbrace{s_2,\dots,s_2}_{2^j},\ \dots,\ \underbrace{s_n,\dots,s_n}_{2^j}\,].$$

The loop at scale $k$. Define $j = k \mathbin{\dot{-}} \gamma.\mathrm{scale}$ using truncated natural-number subtraction: $j = k - \gamma.\mathrm{scale}$ if $k \ge \gamma.\mathrm{scale}$, and $j = 0$ if $k < \gamma.\mathrm{scale}$. Then

$$\gamma@k = \bigl(b_k,\ S_k\bigr) := \bigl(\gamma^{(j)}.\mathrm{base},\ \gamma^{(j)}.\mathrm{steps}\bigr) = \bigl(2^{j}\gamma.\mathrm{base},\ \text{each step repeated } 2^{j} \text{ times}\bigr).$$

In particular, when $k \le \gamma.\mathrm{scale}$ (including $k < \gamma.\mathrm{scale}$) one has $j=0$, and $\gamma@k$ is just $(\gamma.\mathrm{base},\gamma.\mathrm{steps})$ unchanged: there is no coarsening, and the scale $k$ plays no role.

Vertices at scale $k$. $\operatorname{verticesAt}(\gamma,k) = \operatorname{pathVertices}(b_k; S_k)$, the list $[x_0,\dots,x_{m-1}]$ with $x_0 = b_k$, $m = 2^j n$ the length of $S_k$, and final point $x_m$ omitted. (By closedness of $\gamma$, $\sum S_k = 2^j\sum_i \vec s_i = 0$, so $x_m = x_0$.)

In-box hypothesis. $h_\gamma$ asserts

$$\gamma.\mathrm{InBox}(k,R) \;:\Longleftrightarrow\; \forall\, x \in \operatorname{verticesAt}(\gamma,k),\ x\in B_R,$$

i.e. every one of the listed vertices $x_0,\dots,x_{m-1}$ of the scale-$k$ version of $\gamma$ has all coordinates in $[-R,R]$. This is a condition on the listed vertices only; it says nothing directly about edges being in $E_R$. If $\gamma.\mathrm{steps}$ is empty, the vertex list is empty and $h_\gamma$ holds vacuously for every $k$ and $R$.

Wilson loop. For a configuration $U$, scale $k$ and loop $\gamma$,

$$W_\rho(U,k,\gamma) = \operatorname{tr}\Bigl( \rho\bigl(\operatorname{hol}_U(b_k; S_k)\bigr) \Bigr) \in \mathbb{C},$$

the trace (sum of the $N$ diagonal entries) of the $N\times N$ complex unitary matrix $\rho$ of the holonomy of the scale-$k$ path $(b_k,S_k)$ with respect to the link values of $U$.

The assertion

Under all the binders and the hypothesis $h_\gamma$ above, the theorem asserts the equality of complex numbers

$$\boxed{\; W_\rho\bigl(U^{g},k,\gamma\bigr) \;=\; W_\rho\bigl(U,k,\gamma\bigr) \;}$$

that is,

$$\operatorname{tr}\Bigl(\rho\bigl(\operatorname{hol}_{U^{g}}(b_k;S_k)\bigr)\Bigr) = \operatorname{tr}\Bigl(\rho\bigl(\operatorname{hol}_{U}(b_k;S_k)\bigr)\Bigr),$$

where $\operatorname{hol}_{U^g}$ is computed with the link values $\operatorname{link}_{U^g}(x,\mu) = g(x)U(x,\mu)g(x+e_\mu)^{-1}$ on box edges and $1_G$ off the box, and $\operatorname{hol}_U$ with $\operatorname{link}_U(x,\mu) = U(x,\mu)$ on box edges and $1_G$ off the box.

It is an equality of traces only; it does not assert equality of the holonomies in $G$, nor of the matrices $\rho(\cdot)$, nor any conjugacy relation between them.

Degenerate cases included by the quantifiers

• $N = 0$: all matrices are $0\times 0$, every trace is $0\in\mathbb{C}$, and both sides are $0$ regardless of anything else.
• Empty step list: $S_k$ is empty, both holonomies are $1_G$, $\rho(1_G) = I_N$, and both sides equal $\operatorname{tr}(I_N) = N$; the hypothesis $h_\gamma$ is vacuous in this case.
• $d = 0$: there are no directions, so the only loops have empty step lists, and the previous case applies; $B_R$ is a single point.
• $k < \gamma.\mathrm{scale}$: by truncated subtraction $j = 0$, and the theorem concerns $\gamma$ itself (base and steps unchanged), with $h_\gamma$ requiring the unrefined vertices to lie in $B_R$.
• Off-box edges: any step of $S_k$ whose associated edge (as specified in the holonomy formula) is not in $E_R$ contributes the factor $1_G$ (or $1_G^{-1}=1_G$) to both holonomies, irrespective of $g$.
• $R = 0$: $B_0 = \{0\}$ and $E_0 = \varnothing$ (since $0 + e_\mu \notin B_0$ for every $\mu$), so $U$ is the unique function on the empty set, every link value is $1_G$, and $h_\gamma$ requires every listed vertex to be $0$.
• $G$ trivial or $\rho$ trivial: allowed; no non-degeneracy of $G$, $g$, $\rho$ or $U$ is assumed.