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