Definition. Finite surface gauge data and face weights [fdip-006V]

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Let \(\Sigma \) be a connected closed oriented surface of genus \(g\), with a finite polygonal cell decomposition. Write \(V,E,\mathcal P\) for its vertices, edges and disk faces, and \(P=|\mathcal P|>0\). Each face has a nonempty cyclic boundary word. Every edge occurs exactly twice among all face boundaries, with opposite orientations; the two occurrences may belong to the same face. The link of each vertex is a circle. These surface conditions are stronger than the square-cell data: singular quotients are excluded. Loops and repeated boundary edges are allowed.

Choose one orientation per edge and a finite group \(G\). A connection is \(U\in G^E\), with the inverse assigned to a reversed traversal. Use the ordered-transport convention: if a face boundary successively traverses \(e_1^{\epsilon _1},\ldots , e_k^{\epsilon _k}\), its holonomy is \[U_p=U_{e_k}^{\epsilon _k}\cdots U_{e_1}^{\epsilon _1}.\] Orient all faces by the orientation of \(\Sigma \). Changing a starting vertex conjugates \(U_p\). A central function, or class function, is a function \(f:G\to \mathbb C\) satisfying \(f(khk^{-1})=f(h)\) for all \(h,k\in G\). It therefore defines the normalized partition sum \[Z_\Sigma (f)=\frac 1{|G|^{|E|}}\sum _{U\in G^E} \prod _{p\in \mathcal P}f(U_p).\] This uses uniform probability measure on each edge. Unnormalized counting gives \(|G|^{|E|}Z_\Sigma (f)\). A complex face weight is allowed for the algebraic identity; if \(f>0\), division by the partition sum defines a probability law.

For a fixed unitary representation \(\sigma \) of complex dimension \(d_\sigma >0\) and \(\beta \ge 0\), the Wilson action corresponds to \[f_\beta (h)=\exp \left [-\beta \left ( 1-\frac {\operatorname {Re}\chi _\sigma (h)}{d_\sigma }\right )\right ].\] The constant \(e^{-\beta }\) per face is part of this weight. For an oriented closed edge path \(C\), its normalized real Wilson loop is \[W_\sigma (C)=\frac {\operatorname {Re}\chi _\sigma (U_C)}{d_\sigma }.\] It is gauge invariant, independent of its starting vertex and unchanged by reversal. For a general central \(f\), reversal of the chosen surface orientation need not preserve the weight; the Wilson weight is inversion invariant.

Choose one unitary representative of each complex irreducible representation, indexed by \(\widehat G\), with dimension \(d_\rho \). The character basis gives the finite expansion \[\begin {aligned} f(h)&=\sum _{\rho \in \widehat G}c_\rho \chi _\rho (h),\\ c_\rho &=\frac 1{|G|}\sum _{h\in G} f(h)\overline {\chi _\rho (h)}. \end {aligned}\] These coefficients are not divided by \(d_\rho \) in the expansion. TauCeti's TauCeti.ClassFunction.apply_eq_sum_characterPairing_mul_character formalizes this finite class-function expansion. Its bilinear pairing is \(|G|^{-1}\sum _h\chi _\rho (h)f(h^{-1})\); substitution by inversion and unitarity give the displayed coefficient. This correspondence concerns the expansion, not the surface partition theorem below.