Theorem. The finite surface character formula [fdip-006W]
AGENTDRAFTED
For the surface data and coefficients,
\[Z_\Sigma (f)=\sum _{\rho \in \widehat G}
d_\rho ^{\,2-2g}\left (\frac {c_\rho }{d_\rho }\right )^P.\]
The formula uses normalized counting on every edge. In particular,
\(f\equiv 1\) gives \(Z_\Sigma (1)=1\); the unnormalized sum is
\(|G|^{|E|}\).
The exceptional CW description of a sphere with one vertex, no
edges and one face attached by a constant map lies outside the
nonempty polygon convention. Its partition sum is directly \(f(1)\).
The same expression still holds, since \(g=0\), \(P=1\) and
\(f(1)=\sum _\rho d_\rho c_\rho \).
For \(G=\{1,-1\}\) and \(f(h)=e^{-\beta (1-h)}\), the two
coefficients are \(c_+=e^{-\beta }\cosh \beta \) and
\(c_-=e^{-\beta }\sinh \beta \). On the torus the normalized sum is
\(c_+^P+c_-^P\). Multiplication by \(2^{|E|}\) recovers the
exact square-torus partition function, since
\(|E|=2P\). For a disk of \(A\) faces, insertion of its boundary
Wilson sign multiplies the face weight by the sign on each interior
face. This swaps \(c_+\) and \(c_-\) there. The same edge contraction
then gives numerator
\[c_-^A c_+^{P-A}+c_+^A c_-^{P-A}.\]
Division by the partition sum recovers
\((t^A+t^{P-A})/(1+t^P)\), with \(t=\tanh \beta \), under either
counting convention.
For character convolution with normalized group integration, see
equation (9) in [migdal1975recursion, p. 414].
Witten's Section 2.3 uses normalized Haar measure and character
contractions in equations (2.44), (2.48), (2.50)–(2.51)
[witten1991quantum, pp. 163–166]. His continuum construction uses
heat-kernel face weights with area-dependent Casimir factors.
The proof here is for a finite group and an arbitrary fixed central
face weight. It does not assert subdivision invariance for the Wilson
weight or transfer continuum conclusions to these finite models.