Example. An exact matter-weighted sum on the three-by-three torus [fdip-0078]

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On a connected graph, take \(G=\{1,-1\}\) acting by multiplication on the real line. Two gauges induce the same action precisely when their ratio is constant, so each orbit contains \(2^{|V|-1}\) connections. A spanning tree meets every vertex and contains no cycle. Requiring its edges to be positive selects one connection from each orbit. With \(b_1=|E|-|V|+1\) and with zero bare coupling, gauge invariance of the determinant converts the marginal into \[Z_U=2^{-b_1}\sum _s(\det Q_s)^{-1/2},\] where \(s\) runs through the sign patterns on the complementary edges.

Take the square torus with both periods three, \(G=\{1,-1\}\), real scalar matter, unit edge weights, \(m^2=1\) and bare coupling \(\beta =0\). Vertices are \((x,y)\) with coordinates in \(\mathbb Z/3\mathbb Z\); use horizontal edges \(a_{x,y}\) and vertical edges \(b_{x,y}\) in the orientations of that definition. There are 9 matter coordinates and 18 links.

Fix a spanning tree by requiring \[a_{0,y}=a_{1,y}=1\quad (y=0,1,2),\qquad b_{0,0}=b_{0,1}=1.\] These 8 edges connect every vertex without a cycle. Parameterize the 10 remaining links by \(s=(s_1,\ldots ,s_{10})\in \{1,-1\}^{10}\): \[\begin {aligned} (a_{2,0},a_{2,1},a_{2,2})&=(s_1,s_2,s_3),\\ (b_{1,0},b_{2,0},b_{1,1},b_{2,1})&=(s_4,s_5,s_6,s_7),\\ (b_{0,2},b_{1,2},b_{2,2})&=(s_8,s_9,s_{10}). \end {aligned}\] Let \(U_e(s)\) denote these assigned signs.

In the vertex order \((0,0),(1,0),(2,0),(0,1),\ldots ,(2,2)\), let \(E_{v,w}\) be a matrix unit and define \[\begin {aligned} H_s&=\sum _{e\in E}U_e(s)(E_{h(e),s(e)}+E_{s(e),h(e)}),\\ d_s&=\det (5I-H_s),\\ P_{x,y}(s)&=a_{x,y}b_{x+1,y}a_{x,y+1}b_{x,y},\\ k(s)&=\#\{(x,y):P_{x,y}(s)=-1\}. \end {aligned}\] The symbol \(s(e)\) in a matrix index is the edge's source vertex; the tuple \(s\) labels the 10 free signs. All subscripts on links are read modulo three. Every \(d_s\) is a positive integer.

For each connection there is a unique gauge transformation with value one at \((0,0)\) that makes the tree links positive: propagate its vertex values along the tree. The only remaining transformations fixing a connection are the two constant signs, since the graph is connected. Hence every gauge orbit has \(2^8\) elements and exactly one displayed representative. Gauge-invariant normalized edge sums therefore reduce to the following exact formulas: \[\begin {aligned} R&=\sum _{s\in \{1,-1\}^{10}}d_s^{-1/2},& Z_U&=2^{-10}R,\\ \langle P_{0,0}\rangle &=\frac {\sum _sP_{0,0}(s)d_s^{-1/2}}{R},& \mathfrak Z&=(2\pi )^{9/2}Z_U. \end {aligned}\] Here \(Z_U\) is the determinant-weighted edge integral, whereas \(\mathfrak Z\) includes ordinary matter volume.

The finite sums contain 1024 terms. They can be evaluated by forming the displayed integer matrix for each sign tuple and taking its determinant. At the all-positive tuple, the eigenvalues of \(5I-H_s\) are \(1\), 4 copies of \(4\), and 4 copies of \(7\), so \(d_s=4^4\cdot 7^4=614656\). This is the lower bound of the matter-determinant comparison. Grouping by the number of negative plaquettes gives the table. The matter probability in row \(k\) is \(R^{-1}\sum _{s:k(s)=k}d_s^{-1/2}\), and the probability without matter is the orbit count divided by \(1024\).

Translation invariance gives the same expectation at every plaquette. The pure-gauge torus law at \(\beta =0\) instead has \(\langle P_{0,0}\rangle =0\). No large-mass truncation was used in these finite sums.