Example. A normalized complex holonomy action on the circle [fdip-006I]

✍️sourceAGENTDRAFTED

Use the supplied unit-circle mesh, now with complex fields and total holonomy \(e^{i\theta }\). On \(\mathbb C^n\), take \[\langle z,w\rangle _n=\frac 1n\sum _j\overline {z_j}w_j, \qquad Q_{n,\theta }=n^2L_\theta +\mu I,\qquad \mu >0.\] Here \(L_\theta \) is the unit-weight twisted cycle, \(n\ge 3\), and the same factor \(n^2\) fixes the circle length scale. Let \(d\operatorname {vol}_n\) be the real volume induced by this Hermitian pairing. If \(z_j=x_j+iy_j\), then \[d\operatorname {vol}_n=n^{-n}\prod _jdx_j\,dy_j.\] Indeed \(z\mapsto z/\sqrt n\) is an orthonormal coordinate change in complex dimension \(n\), hence real dimension \(2n\).

Choose the complex Gaussian convention \[\begin {aligned} Z_n(\theta )&=\pi ^{-n}\int _{\mathbb C^n} e^{-\langle z,Q_{n,\theta }z\rangle _n} \,d\operatorname {vol}_n(z),\\ W_n(\theta )&=-\log Z_n(\theta ). \end {aligned}\] Unitary diagonalization gives one factor per eigenvalue \(\lambda >0\). Polar coordinates evaluate it: \[\begin {aligned} \pi ^{-1}\int _{\mathbb R^2}e^{-\lambda (x^2+y^2)}dx\,dy &=2\int _0^\infty e^{-\lambda r^2}r\,dr\\ &=\lambda ^{-1}. \end {aligned}\] Therefore \[Z_n(\theta )=(\det _{\mathbb C}Q_{n,\theta })^{-1}, \qquad W_n(\theta )=\log \det _{\mathbb C}Q_{n,\theta }.\] This complex convention has no factor \(1/2\) in the determinant action; the earlier real Gaussian action uses a different exponent and real field dimension.

Since \(Q_{n,\theta }=n^2(L_\theta +\mu n^{-2}I)\), define the explicitly subtracted finite action by \[\widetilde W_n(\theta )=W_n(\theta )-2n\log n.\] The cycle determinant gives \[\begin {aligned} \widetilde W_n(\theta ) &=\log (2\cosh (n\omega _n)-2\cos \theta ),\\ 2\cosh \omega _n&=2+\mu /n^2,\qquad \omega _n\ge 0. \end {aligned}\] Thus the removed term is independent of holonomy; it is part of the stated normalization, not a finite determinant omitted from the formula.

The identity \(\cosh \omega -1=2\sinh ^2(\omega /2)\) gives \(\omega _n=2\operatorname {arsinh}(\sqrt {\mu }/(2n))\). For \(t>0\), integrate the derivative \((\operatorname {arsinh}t)'=(1+t^2)^{-1/2}\) to obtain \[\frac 1{\sqrt {1+t^2}} \le \frac {\operatorname {arsinh}t}{t}\le 1.\] Consequently \(n\omega _n\to \sqrt \mu \). The positive limiting argument of the logarithm is at least \(2\cosh \sqrt \mu -2>0\), so \[\widetilde W_n(\theta )\longrightarrow W(\theta )=\log (2\cosh \sqrt \mu -2\cos \theta ).\] This is a direct limit of the normalized finite determinants, beyond the separate fixed-mode comparison; no interchange of an infinite product is used.

For \(\mu >0\), \(W(\theta )\) is minimized exactly at trivial holonomy \(\theta =0\) modulo \(2\pi \), since \(\cos \theta \le 1\). Equivalently the integrated matter factor \(e^{-W}\) is largest there. At \(\mu =0\) and nontrivial holonomy the same finite calculation gives \(W(\theta )=\log (2-2\cos \theta )\). At trivial holonomy there is a zero mode: the unpinned Gaussian integral diverges and this determinant action is undefined. Its formal logarithm of zero is not a normalized Gaussian action. The circle, its scale and the connection remain supplied data.