Example. Binary tetrahedral spinor gauge fields [fdip-006Y]

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Let \(G=2T\subset SU(2)\) be the 24 Hurwitz units. Use its defining complex spinor representation \(\sigma \), with \(\chi _\sigma (h)=2\operatorname {Re}h\), for both the Wilson action and the normalized Wilson loops of Definition [fdip-006V]. The complete character table has irreducibles \(1,\lambda ,\lambda ^2,\sigma ,\sigma \lambda , \sigma \lambda ^2,\tau \), with degrees \(1,1,1,2,2,2,3\). Their squared degrees sum to \(24\). The nonreal characters are kept separately as complex irreducibles here; no real isotypic multiplicity is substituted into the surface formula.

Put \(r=e^{-\beta /2}\) and \(\omega =(-1+i+j+k)/2\). In the class order of that table, \[\begin {array}{c|rrrrrrr} h&1&-1&i&\omega &-\omega &\omega ^2&-\omega ^2\\\hline |[h]|&1&1&6&4&4&4&4\\ f_\beta (h)&1&r^4&r^2&r^3&r&r^3&r \end {array}\] The resulting character coefficients are \[\begin {aligned} a&=\frac {1+r^4+6r^2+8r+8r^3}{24} &&(1),\\ b&=\frac {(1-r)^4}{24} &&(\lambda ,\lambda ^2),\\ c&=\frac {(1-r^2)(1+r^2+4r)}{12} &&(\sigma ),\\ d&=\frac {(1-r^2)(1-r)^2}{12} &&(\sigma \lambda ,\sigma \lambda ^2),\\ e&=\frac {(1-r^2)^2}{8} &&(\tau ). \end {aligned}\] To verify them, let \(s_j=\zeta ^j+\zeta ^{2j}\), where \(\zeta =e^{2\pi i/3}\). Thus \(s_0=2\) and \(s_1=s_2=-1\). The class-weighted inner products give \[\begin {aligned} c_{\lambda ^j}&=\frac {1+r^4+6r^2+4s_j(r+r^3)}{24},\\ c_{\sigma \lambda ^j}&=\frac {1-r^4+2s_j(r-r^3)}{12},\\ c_\tau &=\frac {3+3r^4-6r^2}{24}. \end {aligned}\] Factoring these expressions gives the displayed coefficients. They are nonnegative for \(\beta \ge 0\), and \(a+2b+2c+4d+3e=1\). At \(\beta =0\), only \(a=1\) remains; as \(\beta \to \infty \), each coefficient tends to its degree divided by \(24\), the character expansion of the identity indicator.

For the closed oriented surface, \[\begin {aligned} Z_\Sigma (f_\beta )={}&a^P+2b^P\\ &+2^{2-2g}\left [\left (\frac c2\right )^P +2\left (\frac d2\right )^P\right ]\\ &+3^{2-2g}\left (\frac e3\right )^P. \end {aligned}\] Unnormalized counting multiplies this by \(24^{|E|}\). These powers use complex irreducible dimensions; the quaternionic nature of the spinor does not change its complex degree two.

On the one-face torus, set \(g=1\), \(P=1\). Substitution gives \[Z_{\mathbb T^2}(f_\beta ) =a+2b+\frac c2+d+\frac e3 =\frac {7+16r^2+r^4}{24}.\] For its face loop, \(W_\sigma =\operatorname {Re}H\) and \(f_\beta (H)=e^{-\beta (1-W_\sigma )}\). Differentiating the finite sum therefore gives \[\langle W_\sigma \rangle =1+\frac {d}{d\beta }\log Z_{\mathbb T^2}(f_\beta ) =\frac {7-r^4}{7+16r^2+r^4}.\] This differs from the quaternion-group answer, despite using the same spinor trace convention. The calculation concerns the specified finite gauge group and plaquette; it does not identify the finite model with an \(SU(2)\) continuum theory.