12 An exact quantum comparison and the boundary measure

12.1 A first-moment identity at the flat weights

Fix the weights (7.1), all marked points finite in the chosen coordinate, and a Liouville parameter $0<b<1$. Set $$\gamma=2b,\qquad Q=b+b^{-1},\qquad \beta_i=\frac{\mu_i}{b},\qquad s=\frac{\sum_i\beta_i-2Q}{\gamma}=-1.$$(12.1) Here $\beta_i$ are the charge conventions of Kupiainen–Rhodes–Vargas; in the physics convention $V_{\alpha_i}=e^{2\alpha_i\phi}$, one has $\beta_i=2\alpha_i$. Write $\Lambda_L>0$ for the cosmological constant, to distinguish it from both the deficit weights and the geometric area.

The exact correlation formula of [68, eqs. (2.11)–(2.15)] is $$\begin{aligned} C_b(\boldsymbol\beta;\boldsymbol p) &=\frac1b\Lambda_L^{-s}\Gamma(s) P_b(\boldsymbol p)\,\mathbb E[Z_{\boldsymbol\beta}^{-s}], \\ P_b(\boldsymbol p) &=\prod_{i<j}|p_i-p_j|^{-\beta_i\beta_j},\\ Z_{\boldsymbol\beta} &=\int_{\mathbb C}\prod_i \left(\frac{|w|_+}{|w-p_i|}\right)^{\gamma\beta_i} M_\gamma(\mathop{}\!\mathrm d^2w),\qquad |w|_+=\max(1,|w|), \end{aligned}$$ where $M_\gamma$ is the normalized Gaussian multiplicative-chaos measure in their reference metric. The only probabilistic property needed for the following calculation is its first moment, $$\mathbb E[M_\gamma(\mathop{}\!\mathrm d^2w)]=|w|_+^{-4}\mathop{}\!\mathrm d^2w.$$(12.3) The standard unit-volume normalization removes the zero-mode Gamma factor: $$C_b^{\mathrm{uv}} :=\frac{\Lambda_L^s}{\Gamma(s)}C_b =\frac1b P_b\,\mathbb E[Z_{\boldsymbol\beta}^{-s}].$$(12.4) The right side defines the continuation at $s=-1$.

This is a deduction from the established correlation formula and the area interpretation of the period pairing; it does not use the main DOZZ structure-constant theorem. It is also an exact finite-$b$ identity, rather than an interchange of a semiclassical limit with two moduli derivatives. The raw expression $C_b^{\mathrm{uv}}\mathop{}\!\mathrm d^{2d}p$ is not the moduli-space volume density: the density is the top exterior power of its logarithmic curvature.

12.2 Analytic continuation and the equivalent residue

The original unconstrained zero-mode integral is not convergent at $s=-1$. The unit-volume function, however, is positive and finite there. Indeed, the extended Seiberg bounds of [68, eq. (2.14)] require $$-s<\min\left\{\frac4{\gamma^2}, \min_i\frac2\gamma(Q-\beta_i)\right\},\qquad \beta_i<Q.$$ At our parameters these inequalities reduce to $$1<b^{-2},\qquad 1<1+\frac{1-\mu_i}{b^2},\qquad \beta_i<Q,$$(12.7) and are strict. In particular, the positive moment has some integrability beyond order one. Theorem 6.1 of [68] gives holomorphic continuation in the charges of the unit-volume $n$-point correlation to a complex neighborhood of this real domain. Thus the pole in (12.2) is an actual meromorphic pole, not just a formal screening calculation.

Since $\operatorname{Res}_{s=-1}\Gamma(s)=-1$, $$\operatorname*{Res}_{s=-1}C_b =-\frac{\Lambda_L}{b}P_b\mathcal A_\mu.$$(12.8) This residue means the coefficient of $(s+1)^{-1}$ with the other charge coordinates fixed. With the transverse physics coordinate $t=\sum_i\alpha_i-Q+b=b(s+1)$, it becomes $\operatorname{Res}_{t=0}C_b=-\Lambda_LP_b\mathcal A_\mu$. Consequently the logarithmic curvature of the absolute residue also equals (12.6).

In Coulomb-gas terminology this is the one-screening resonance $\sum_i\alpha_i=Q-b$; see also [95, Section 3.4]. Its chiral contour integrals are $$\int_\gamma\prod_i(w-p_i)^{-\mu_i}\mathop{}\!\mathrm dw,$$(12.9) precisely the Deligne–Mostow periods (7.4). Their Hermitian pairing is the area. Section 13 gives an independent finite period calculation of the four-point residue and fusion matrix, and states the additional continuation and tail-control input for the spectral-contour derivation. A pointwise identification with an unnormalized modular-double $6j$ function would still require its normalization and contour to be specified. Section 14 fixes these normalizations for the induced finite operator and examines the geometric meaning of that specialization.

12.3 Four points: the exact quantum potential and its measure

Put the marks at $(0,1,z,\infty)$, with the standard primary-field normalization at infinity. Then (12.5) reads $$\begin{aligned} C_b^{\mathrm{uv}}(z) &=\frac1b |z|^{-\mu_1\mu_3/b^2} |1-z|^{-\mu_2\mu_3/b^2}\mathcal A_\mu(z), \\ \mathcal A_\mu(z) &=\int_{\mathbb C}|w|^{-2\mu_1}|w-1|^{-2\mu_2} |w-z|^{-2\mu_3}\mathop{}\!\mathrm d^2w. \end{aligned}$$ The condition at infinity is now $\mu_1+\mu_2+\mu_3=2-\mu_4>1$. The ordinary field normalization at infinity cancels the corresponding area scaling; there is no additional moduli-dependent factor.

For equal weights $\mu_i=1/2$, the area is $$\mathcal A(z)=\int_{\mathbb C}\frac{\mathop{}\!\mathrm d^2w}{|w(w-1)(w-z)|}.$$(12.11) Let $\Omega_1,\Omega_2$ be oriented lattice periods of $\mathop{}\!\mathrm dw/y$ on the elliptic curve (7.9), and put $\tau=\Omega_2/\Omega_1=u+iv\in\mathbb H$. The pillowcase has half the area of its double cover, so $$\mathcal A(z)=\frac12\operatorname{Im} (\overline{\Omega_1}\Omega_2) =\frac12|\Omega_1|^2v.$$ Locally $\Omega_1$ is nonzero and holomorphic. Hence $$\boxed{\quad -2i\partial\bar\partial\log C_b^{\mathrm{uv}} =\frac{\mathop{}\!\mathrm du\wedge\mathop{}\!\mathrm dv}{v^2} =\frac{|\tau'(z)|^2}{(\operatorname{Im}\tau(z))^2} \mathop{}\!\mathrm d(\operatorname{Re}z)\wedge \mathop{}\!\mathrm d(\operatorname{Im}z). \quad}$$(12.12) The elliptic-integral expression for $\tau$ is (7.10). Integrating this form over the labeled moduli space gives $2\pi$, as in Section 7. This is the flat equal-deficit Deligne–Mostow area; the cusped hyperbolic WP area is the different quantity $V_{0,4}(0)=2\pi^2$.

12.4 An actual boundary torsion measure

There is also a measure comparison on the representation side. For a good, boundary-regular $\mathrm{SL}_2(\mathbb C)$ representation of an $n$-holed sphere, Heusener–Porti prove $$\Omega_S=\pm\frac{\omega_G^d}{d!} \wedge\nu_1\wedge\cdots\wedge\nu_n, \qquad d=n-3,$$(12.13) where $\Omega_S$ is the holomorphic Reidemeister-torsion volume form, $\omega_G$ is the Goldman form on the fixed-conjugacy leaf, and the $\nu_i$ are the peripheral volume forms [61, Theorem 3]. With invariant pairing $B(U,V)=-\mathop{\mathrm{tr}}(UV)$ their peripheral formula is $\nu_i=\pm\sqrt2\,\mathop{}\!\mathrm d\mathop{\mathrm{tr}}\rho(\partial_i)$ [61, Proposition 2]. The pairing must be the same in all three kinds of form. For $B\mapsto\lambda^2B$, with $\lambda\in\mathbb C^*$, the homogeneity stated after that theorem is, in our notation, $$\omega_G\mapsto\lambda^2\omega_G,\qquad \nu_i\mapsto\lambda\nu_i,\qquad \Omega_S\mapsto\lambda^{2d+n}\Omega_S.$$ Thus the displayed $\sqrt2$ belongs specifically to $B=-\mathop{\mathrm{tr}}$; it is not retained unchanged when the Goldman form is recalibrated. If the scale of $B$ is negative on the real form, choose its complex square root consistently in the holomorphic forms, then take the positive density on the real leaf.

One must disintegrate by the peripheral forms before fixing the conjugacy classes. Simply pulling the unrestricted top-degree form back to a fixed-trace leaf gives zero. After this disintegration, and on the relevant real component, the torsion density is the Goldman Liouville density, with the chosen normalization of $B$.

Mondello identifies the cone WP and Goldman Poisson structures up to a specified constant for openings in $(0,2\pi)$ [80, Theorems 1.1–1.2]. We calibrate $B$ and the torsion and peripheral forms together so that the symplectic form on our real leaf is $\omega_{\mathrm{WP}}$, with the positive trace-coordinate density fixed in Appendix A. The holonomy statement is local in general; it is globally injective on marked cone metrics when every opening is at most $\pi$. Combining (12.13) with (8.5) gives, locally on a controlled family, $$\boxed{\quad \left(\frac2A\right)^d \mathop{}\!\mathrm dV_{\mathrm{torsion,relative}} \longrightarrow \mathop{}\!\mathrm dV_{\mathrm{CH},\mu}. \quad}$$(12.14) Apply the torsion theorem for $A>0$ and then take the rescaled limit. At the Euclidean endpoint the affine holonomies become reducible, and their ordinary characters discard translational shape data. The limiting twisted periods or a blowup retaining their scale are indispensable. In particular, an unqualified ordinary Goldman construction for the Euclidean isometry group would miss the needed nondegenerate pairing.

12.5 The four-point character surface, including a correction

The preceding measure limit can be calculated directly. Give the four cone points the common opening $\theta=\pi-A/4$, so their total hyperbolic area is $A$. Choose equal peripheral traces $a=-2\cos(\theta/2)=-2\sin(A/8)$. The Fricke relation of Appendix A becomes $$\Phi=x^2+y^2+z^2+xyz-2a^2(x+y+z)+4a^2+a^4-4=0.$$(12.15) Blow up the limiting character by $$x=-2-AX,\qquad y=-2-AY,\qquad z=-2-AZ.$$(12.16) Direct substitution gives $$\begin{aligned} \frac\Phi{A^2} ={}&X^2+Y^2+Z^2-2XY-2XZ-2YZ+1\\ &+A\left(-XYZ+\frac{X+Y+Z}{8}\right) -\frac{A^2}{768}+O(A^3). \end{aligned}$$ On a chart with $u>0$, $v>0$, parametrize the limiting quadric by $$X=\frac1{2v},\qquad Y=\frac{u^2+v^2}{2v},\qquad Z_0=\frac{(u-1)^2+v^2}{2v}.$$(12.18) These are squared side lengths of a Euclidean triangle of area $1/4$; the quadric equation is Heron’s identity. Since $$X+Y-Z_0=\frac uv,\qquad \mathop{}\!\mathrm dX\wedge\mathop{}\!\mathrm dY=\frac{u}{2v^3}\mathop{}\!\mathrm du\wedge\mathop{}\!\mathrm dv,$$ the calibrated density (A.2) yields $A^{-1}\omega_{\mathrm{WP}}\to\frac12\mathop{}\!\mathrm du\wedge\mathop{}\!\mathrm dv/v^2$, with the positive orientation on this chart.

In fact, the implicit-function expansion is $$Z=Z_0+AZ_1+O(A^2),\qquad Z_1=\frac{(1-u)[v^2-u(1-u)]}{16v^2}.$$ Using $\Phi_z=2A(X+Y-Z)+A^2(XY-1/8)+O(A^4)$ gives $$\boxed{\quad \frac{\omega_{\mathrm{WP}}}A =\frac1{2v^2}\left[ 1-\frac{A(1+u^2+v^2)}{16v}+O(A^2)\right] \mathop{}\!\mathrm du\wedge\mathop{}\!\mathrm dv. \quad}$$(12.19) This is an explicit first correction in a fixed blown-up character chart. Here $\tau=u+iv$ is the limiting flat period coordinate, not the exact period ratio or conformal cross-ratio of the finite-$A$ surface. The error is local on compact subsets of the chart. In particular, the displayed correction is not termwise integrable at $v=\infty$; it cannot be integrated naively to recover the global polynomial of (8.10).

12.6 What a normalized quantum kernel would have to produce

The phase-space measure of a unitary change of polarization is more constrained than a bare scalar $6j$ prefactor. In one degree of freedom, let $S(q,Q)$ generate a canonical relation, $p=\partial_qS$, away from its caustic. A branchwise normalized semiclassical kernel has the leading form $$K_\hbar(q,Q)\sim \frac{|\partial_q\partial_QS|^{1/2}}{\sqrt{2\pi\hbar}} \exp\left(\frac{iS}{\hbar}+\frac{i\pi m}{4}\right).$$(12.20) The amplitude follows by applying unitarity to the leading phase difference $(q-q')p$ and changing variables from $Q$ to $p$. Thus $$2\pi\hbar|K_\hbar|^2\mathop{}\!\mathrm dq\mathop{}\!\mathrm dQ \sim |\partial_q\partial_QS|\mathop{}\!\mathrm dq\mathop{}\!\mathrm dQ =|\mathop{}\!\mathrm dq\wedge\mathop{}\!\mathrm dp|.$$(12.21) This is a branchwise assertion. Several stationary branches produce interference; a pointwise square of an oscillatory cosine need not converge without averaging.

Teschner–Vartanov’s Fenchel–Nielsen transition kernel contains the factors $\sqrt{M(\alpha_s)M(\alpha_t)}$ multiplying the $b$-$6j$ symbol [96, eq. (5.14)]. Consequently a proposed measure comparison must include the spectral measure, label Jacobians, and polarization. If a normalized kernel quantizes our cone WP leaf, its classical measure is constrained by (12.21), and its rescaled flat measure by (12.14). This conditional argument does not by itself establish the elliptic continuation of that kernel.

12.7 The auxiliary tetrahedron and the order of limits

There is a concrete reason not to identify the auxiliary tetrahedron in a fusion kernel with the convex realization of our cone sphere without further work. In the conventions of [96, Sections 3–5], the parameters satisfy $$\alpha_k=\frac Q2+\frac{i\ell_k}{4\pi b},\qquad A_k=e^{-2\pi ib\alpha_k+\pi i} =e^{\ell_k/2-\pi ib^2}.$$(12.22) Here $A_k$ is the paper’s exponentiated tetrahedral parameter, not our geometric area $A$. Continuing a boundary length by $\ell_i=i\theta_i$ gives an auxiliary dihedral parameter $\eta_i=\theta_i/2-\pi b^2$, modulo the chosen branch. Thus at the equal-angle Euclidean endpoint, and as $b\to0$, the four external auxiliary dihedrals tend to $\pi/2$. The internal $s,t$ edges are opposite. After reordering faces, the limiting angle Gram matrix would be $$G=\begin{pmatrix} 1&-\cos\eta_s&0&0\\ -\cos\eta_s&1&0&0\\ 0&0&1&-\cos\eta_t\\ 0&0&-\cos\eta_t&1 \end{pmatrix},\qquad \det G=\sin^2\eta_s\sin^2\eta_t.$$(12.23) For interior angles $\eta_s,\eta_t\in(0,\pi)$ this is nonsingular. A nondegenerate Euclidean tetrahedron has a rank-three angle Gram matrix. Hence this real slice cannot converge to the generic Euclidean tetrahedron realizing our flat cone metric. An additional degeneration, a different real slice, or a different geometric identification is needed. This is an obstruction to the proposed identification, not to the existence of Liouville theory with cone charges.

There is also a finite-$b$ shift in the charge convention. At a fixed classical opening $\theta_i=2\pi(1-\mu_i)$, the exact dictionary (12.22) gives $$\alpha_i=\frac{\mu_i}{2b}+\frac b2,$$ whereas (12.1) uses $\alpha_i=\mu_i/(2b)$. The latter corresponds in that dictionary to the shifted opening $\theta_i+2\pi b^2$. For unshifted flat openings the zero-mode parameter would be $s=n/2-1$, not $-1$. Thus the exact correlation identity is a specified charge specialization; it is not an identification with the finite-$b$ kernel at unshifted classical cone angles. Indeed, treating the shifted openings as hyperbolic cone data would give the negative Gauss–Bonnet area $-2\pi n b^2$.

There is a second, independent distinction. Along a heavy-charge path with deficit weights $a_i(A)$, set $\alpha_i(A)=a_i(A)/(2b)$ and $A=2\pi(\sum_i a_i(A)-2)$. Then exactly $$\boxed{\quad s=\frac{A}{4\pi b^2}-1.\quad}$$(12.24) The finite-$b$ flat endpoint of this pure-heavy charge path is $s=-1$, the resonance used above. The original zero-mode convergence region $s>0$ requires $A>4\pi b^2$. In contrast, under the proposed joint scaling $A\asymp\hbar^{2/3}$ with $\hbar\asymp b^2$, one has $A/b^2\to\infty$ and hence $s\to+\infty$. The exact residue identity therefore establishes a quantum origin for the Deligne–Mostow curvature without proving the uniform quantum degeneration proposed in Section 11.