Wilson Loop Expectations in Lattice Gauge Theories with Finite Gauge Groups

Abstract

Wilson loop expectations at weak coupling are computed to first order, for four dimensional lattice gauge theories with finite gauge groups which satisfy some mild additional conditions. This continues recent work of Chatterjee, which considered the case of gauge group $\mathbb {Z}_2$. The main steps are (1) reducing the first order computation to a problem of Poisson approximation, and (2) using Stein’s method to carry out the Poisson approximation.

Appendices

Appendix A

We prove the representation theory statements which were made in Sect. 1.

Lemma A.1

Let G be a finite group of order at least 3. There exists a faithful unitary representation $\rho$ such that $\Vert {A}\Vert _{op} < 1$.

Proof

Let $n := |G| \ge 3$. First, take ${\tilde{\rho }}$ to be the regular representation of G, with character ${\tilde{\chi }}$. This faithfully represents G as a set of permutation matrices on the vector space $\mathbb {C}^n$, and moreover, we have ${\tilde{\chi }}(1) = n$, and ${\tilde{\chi }}(g) = 0$ for all $g \ne 1$, so that $G_0 = G \backslash \{1\}$. Enumerate $G = \{g_1, \ldots , g_n\}$, with $g_1 = 1$, and let $\pi _1, \ldots , \pi _n$ be the permutations which give $\rho (g_1), \ldots , \rho (g_n)$. Observe that the permutations $\pi _2, \ldots , \pi _n$ (i.e. the permutations which correspond to non-identity elements), have no fixed points. Moreover, for any $2 \le i \ne i' \le n$, and any $1 \le j \le n$, we have $\pi _i(j) \ne \pi _{i'}(j)$. This implies that

$$\begin{aligned} A = \frac{1}{n-1} \sum _{g \ne 1} \rho (g) = \frac{1}{n-1} (\mathbf {1} - I), \end{aligned}$$

where $\mathbf {1}$ is the $n\times n$ matrix with every entry equal to 1, and I is the $n \times n$ identity matrix. Now define the subspace

$$\begin{aligned} V := \bigg \{x \in \mathbb {C}^n : \sum _{i=1}^n x_i = 0\bigg \}. \end{aligned}$$

Observe that on V, the matrix A acts as $-\frac{1}{n-1}$ times the identity. As $n \ge 3$, we have $\frac{1}{n-1} \le \frac{1}{2}$. Thus the desired representation $\rho$ may be obtained by taking the subrepresentation of ${\tilde{\rho }}$ given by the invariant subspace V. $\quad \square$

Lemma A.2

Let G be a finite group of order at least 3, and let $\rho$ be a faithful unitary representation of G. If $\rho$ is irreducible, then $\Vert {A}\Vert _{op} < 1$.

Proof

Observe that for any $h \in G$, we have $h G_0 h^{-1} = G_0$. This implies that for any $h \in G$, we have

$$\begin{aligned} \rho (h) A \rho (h^{-1}) = A, \end{aligned}$$

i.e. A is an intertwiner. Since $\rho$ is irreducible, we may apply Schur’s lemma to obtain that A must be a multiple $\lambda$ of the identity. Since A is Hermitian, and $\Vert {A}\Vert _{op} \le 1$, we have $\lambda \in [-1, 1]$. If $|\lambda | = 1$, then we must have that for all $g \in G_0$, $\rho (g)$ is $\lambda$ times the identity. Because $\rho$ is faithful, we have $\lambda \ne 1$. Now if $\lambda = -1$, then by the definition of $G_0$, we must have $G_0 = G \backslash \{1\}$, which by assumption is of size at least 2. This then contradicts the assumption that $\rho$ is faithful. Thus $|\lambda | \ne 1$, i.e. $\lambda \in (-1, 1)$. $\quad \square$

Appendix B

We prove the topological statements which were needed in Sect. 4.

Proof of Lemma 4.1.10

Suppose without loss of generality that e is the edge between (0, 0, 0, 0) and (1, 0, 0, 0). We start with $S_e$. Observe that $S_e$ is defined by including all plaquettes which are contained in a 3-cell which contains e. Consequently, we may attach all such 3-cells to $S_e$, without changing its fundamental group. Call the resulting space ${\tilde{S}}_e$. After doing this, we may then attach all 4-cells whose boundary 3-cells are in ${\tilde{S}}_e$, without changing the fundamental group. The resulting space is the union of the rectangular prisms $[0, 1] \times [-1, 1]^2 \times \{0\}$, $[0, 1] \times [-1, 1] \times \{0\} \times [-1, 1]$, and $[0, 1] \times \{0\} \times [-1, 1]^2$. The fundamental group of any of these prisms is trivial, and the intersection of these three prisms is the line segment $[0, 1] \times \{0\}^3$, which is path connected. Thus by the Seifert-Van Kampen theorem, we obtain that the fundamental group of the union of these spaces is also trivial, and thus also $\pi _1(S_e)$ is trivial.

Now onto $\partial S_e$. As noted right before Lemma 4.1.10, recall that $\partial S_e$ is the union of the boundaries of three rectangular prisms. Denote the three boundaries by $B_1, B_2, B_3$. Each $B_i$, $1 \le i \le 3$ is simply connected. Recall also that $B_1, B_2$ intersect on the boundary of a rectangle, which is path connected. Thus by Seifert-Van Kampen, $B_1 \cup B_2$ is also simply connected. Now observe that $B_1 \cup B_2$ and $B_3$ intersect on the union of the boundaries of two rectangles, and moreover, these two boundaries have nonempty intersection. Thus the intersection of $B_1 \cup B_2$ and $B_3$ is path connected, and thus again by Seifert-Van Kampen, $\partial S_e = B_1 \cup B_2 \cup B_3$ is simply connected.

Finally, we look at $S_e^c$. Observe $S_e \cap S_e^c = \partial S_e$, while $S_e \cup S_e^c = S_2(\Lambda )$. Thus by Seifert-Van Kampen, we have that $\pi _1(S_2(\Lambda )) = \pi _1(S_e) * \pi _1(S_e^c)$, where $*$ denotes free product of groups. As both $\pi _1(S_2(\Lambda )) = \pi _1(S_e) = \{1\}$, we thus must also have $\pi _1(S_e^c) = 1$. $\quad \square$

Proof of Lemma 4.1.20

First, to see why $S_2(B)$ is simply connected, we may attach all 3-cells whose boundary 2-cells are all contained in $S_2(B)$, without changing the fundamental group. Call the resulting 3-complex $S_3(B)$. We may then attach all 4-cells whose boundary 3-cells are all contained in $S_3(B)$, without changing its fundamental group. The resulting space is a four dimensional cube in $\mathbb {R}^4$, which is simply connected.

Now onto $\partial S_2(B)$. If B is contained in the interior of $\Lambda$, then by attaching 3-cells and 4-cells to $\partial S_2(B)$ as before, we obtain the boundary of a four dimesional cube in $\mathbb {R}^4$, which is simply connected. If instead B intersects the boundary of $\Lambda$, then upon attaching 3-cells and 4-cells, we obtain a space which is homeomorphic to a three dimensional unit ball, which is simply connected. Here we have used the assumption that all side lengths of B are strictly less than the side length of $\Lambda$.

Finally, we look at $S_2^c(B)$. Observe that $S_2(B) \cap S_2^c(B) = \partial S_2(B)$, which is simply connected. Also observe $S_2(B) \cup S_2^c(B) = S_2(\Lambda )$, which is simply connected. Thus by Seifert-Van Kampen, we have $\pi _1(S_2(\Lambda )) = \pi _1(S_2(B)) * \pi _1(S_2^c(B))$. As both $\pi _1(S_2(\Lambda )) = \pi _1(S_2(B)) = \{1\}$, we must also have $\pi _1(S_2^c(B)) = \{1\}$. $\quad \square$

Lemma B.1

Let $P\subseteq {\Lambda _2}$ be such that for every plaquette $p \in P$, every 3-cell which contains p is completely contained in $\Lambda$. If $|P| \le 5$, then $\pi _1(S_2({\Lambda _2}\backslash P)) = \{1\}$. If $|P| = 6$, then $\pi _1(S_2({\Lambda _2}\backslash P)) \ne \{1\}$ if and only if $P= P(e) \subseteq {\Lambda _2}.$

Proof

Fix a vertex $x_0 \in \Lambda _0$, and a spanning tree T of $S_1(\Lambda )$. We have the presentation

$$\begin{aligned} \pi _1(S_2({\Lambda _2}\backslash P), x_0) = \langle a_e, e \in S_1(\Lambda )\backslash T ~|~ C_p, p \in {\Lambda _2}\backslash P\rangle . \end{aligned}$$

If we can show that for all $p \in P$, we have $C_p = 1$, then we would have

$$\begin{aligned} \pi _1(S_2({\Lambda _2}\backslash P), x_0) = \langle a_e, e \in S_1(\Lambda )\backslash T ~|~ C_p, p \in {\Lambda _2}\rangle = \pi _1(S_2({\Lambda _2}\backslash \varnothing ), x_0). \end{aligned}$$

But $S_2({\Lambda _2}\backslash \varnothing ) = S_2(\Lambda )$ is the 2-skeleton of $\Lambda$, which is simply connected. Thus our goal is to show that $C_p = 1$, for all $p \in P$.

This follows from the following observation. For $p \in P$, if there is a 3-cell c which contains p, and such that for every other plaquette $p'$ of c, we have $C_{p'} = 1$, then also $C_p = 1$. Here we’ve used the assumption that since c contains a plaquette of $P$, c is contained in $\Lambda$. Now if $|P| \le 5$, then there always exists a plaquette $p \in P$ for which there is such a 3-cell c. The same holds if $|P| = 6$ but $P$ is not a minimal vortex. $\quad \square$

The proof of the following lemma is due to Ciprian Manolescu.

Lemma B.2

For any plaquette set $P\subseteq {\Lambda _2}$, we have $\mathrm {rk}(\pi _1(S_2({\Lambda _2}\backslash P))) \le |P|$.

Proof

Suppose $\Lambda = ([a_1, b_1] \times \cdots \times [a_4, b_4]) \cap \mathbb {Z}^4$. By going to the dual lattice, we have that the fundamental group of $S_2({\Lambda _2}\backslash P)$ is the same as the fundamental group of the complement of a 2-complex M made of $|P|$ plaquettes in $B := [-(a_1+ 1/2), b_1+1/2] \times \cdots \times [-(a_4 + 1/2), b_4 + 1/2]$. Pick a point $x_0 \in B - M$, and take non intersecting paths in B from $x_0$ to the center of each plaquette in M, such that the paths don’t intersect M anywhere else. This is possible by the transversality theorem [14] (and since $1 + 2 < 4$). These paths form a “star" S. Let N be a small neighborhood of S, such that N intersects the plaquettes of M in small disks around the center points. Let $Q := B \backslash N$, and observe that Q is homeomorphic to $S^3 \times [0, 1]$. Now decompose $B \backslash M = (N \backslash M) \cup (Q \backslash M)$.

Observe that the 2-complex M with the center point of each plaquette removed deformation retracts to its 1-skeleton. Thus $Q \backslash M$ is homotopy equivalent to $S^3 \times [0, 1]$ minus a 1-dimensional space, and thus by the transversality theorem [14] (and since $1 + 2 < 4$), $Q \backslash M$ is simply connected. Moving to $N \backslash M$, observe that this space is homeomorphic to a four dimensional cube with $|P|$ parallel 2-dimensional hyperplanes removed, and thus $\pi _1(N \backslash M)$ is the free group on $|P|$ generators. Now finish by the Seifert-Van Kampen theorem. $\quad \square$

Index of notation

There are some quantities which are defined in both the Abelian case and the general case. This was done to emphasize the analogies between certain aspects of the proofs in the Abelian and general cases. The result is that some entries of the index point to multiple pages; the earlier page contains the definition in the Abelian case, while the later page contains the definition in the general case.

Notation	Description
G	Gauge group
$1$	Identity of G
$\rho$	Unitary representation G
$\chi$	Character of $\rho$
$\Lambda$	Finite 4d cube
p	Plaquette
$\beta$	Inverse coupling constant
$\mu _{\Lambda , \beta }$	Lattice gauge theory
$\gamma$	Closed loop
$W_\gamma$	Wilson loop variable
$\Delta _G$	Strictly positive real number defined in terms of G
$\varphi _\beta$	Function on G
$r_\beta$	Sum of $\varphi _\beta (g)$ over $g \in G, g \ne 1$
$A_\beta$	Weighted average of $\rho (g)$ over $g \in G, g \ne 1$, with weight $\varphi _\beta (g)$
$\ell$	Length of $\gamma$
df	Exterior derivative of f
$\delta f$	Coderivative of f
$\langle f, g \rangle$	Inner product of differential forms
N	Side length of $\Lambda$
$B_\gamma$	Cube of side length $\ell$ which contains $\gamma$
L	$\ell ^\infty$ distance between the boundaries of $B_\gamma$ and $\Lambda$
$P$	Plaquette set
$V$	Vortex
$\Phi$	Function on plaquette sets
$\Sigma$	Random edge configuration with law $\mu _{\Lambda , \beta }$
$P(\Sigma )$	Random plaquette set, support of $d\Sigma$
E	Event which captures the typical behavior
$N_\gamma$	Count of weakly dependent rare events
$\alpha _\beta$	Upper bound on $\Phi$
$\mathcal {P}_\Lambda$	Set of nonempty vortices contained in $\Lambda$
$\Xi _{\mathcal {P}}$	Roughly, a restricted partition function
$N_\Lambda (V_1, \ldots , V_n)$	Vortices which are incompatible with at least one of $V_1, \ldots , V_n$
$\rho _{\mathcal {P}}$	Reduced correlations
$S_1(\Lambda )$	1-skeleton of $\Lambda$
$S_2(\Lambda )$	2-skeleton of $\Lambda$
$x_0$	Base point of $S_1(\Lambda )$
T	Spanning tree of $S_1(\Lambda )$
$a_e$	A closed loop corresponding to e
$\psi _T^{x_0}(\sigma )$	Homomorphism induced by $\sigma$
$C_p$	Closed loop which winds around the boundary of p
$\mathrm {supp}(\sigma )$	The set of plaquettes p for which $\sigma _p \ne 1$
GF(T)	Set of edge configurations which are identity on T
$S_e$	A certain 2-complex corresponding to e
$S_2(B)$	2-skeleton of B
$\partial S_2(B)$	Roughly, the 2-skeleton of the boundary of B
$S_2^c(B)$	Roughly, the 2-skeleton of the complement of B
K	Knot
$\mathcal {K}$	Collection of all knots in $\Lambda$
$\Gamma$	Collection of vortices, i.e. a subset of $\mathcal {P}_\Lambda$
$P(\Gamma )$	Plaquette set which is the union of elements of $\Gamma$
$I(\Gamma )$	Indicator which tracks whether the elements of $\Gamma$ are compatible
$\mathbf {\Gamma }$	Random collection of vortices
$S_2({\Lambda _2}\backslash P)$	2-complex obtained by deleting the plaquettes of P from $S_2(\Lambda )$
$G^s(P)$	A certain graph induced by P
$\mathcal {A}(m, s)$	Plaquette sets for which $G^s(P)$ is connected
$\mathcal {A}(m, s, p)$	Plaquette sets containing p, for which $G^s(P)$ is connected