2 Ideal polyhedra, angle cells, and a partition function

2.1 The geometric parametrization

In the polyhedral cell discussion take $n\ge4$; the three-ended sphere is the zero-dimensional pants base case.

Rivin’s intrinsic realization theorem identifies a complete finite-area hyperbolic metric on an $n$-punctured sphere with the boundary metric of a convex ideal polyhedron, uniquely up to ambient isometry. Flat doubles of ideal polygons occur as limiting configurations [87]. For oriented moduli one retains the orientation of the boundary and uses orientation-preserving equivalences.

Fix a combinatorial polyhedron and write $\theta_e$ for its exterior dihedral angles. In the dual graph, the ideal conditions are $$\begin{aligned} &0<\theta_e<\pi,\\ &\sum_{e\in\partial f}\theta_e=2\pi &&\text{for every dual face }f,\\ &\sum_{e\in C}\theta_e>2\pi &&\text{for every simple dual circuit not bounding a face.} \end{aligned}$$ Here the dual graph is the planar, $3$-connected graph of a convex polyhedron, with its oriented embedding fixed. With these hypotheses, these are Rivin’s characterization [88]. Thus the admissible angle assignments form an open convex polytope in an affine subspace. Generic polyhedra have triangular faces; their dual maps are trivalent and have $3n-6$ edges. The $n$ perimeter conditions are independent: their matrix is the unsigned vertex–edge incidence matrix of the primal triangulation. A row relation assigns numbers $c_v$ with $c_v+c_w=0$ across every edge; a triangular face forces its three numbers to vanish, and connectedness forces all of them to vanish. The dimension is therefore $2n-6$.

An edge of the dual graph has metric length $\theta_e$. The resulting metric map has face perimeters $2\pi$ and no short nonfacial circuit. This is the genus-zero instance of a $2\pi$-irreducible metric map. It gives the precise combinatorial object behind the proposed sum over polyhedral types.

2.2 A normalized cell integral

For a dual ribbon graph $G$ with labeled faces, define its constrained Lebesgue measure by disintegrating $\prod_e\mathop{}\!\mathrm d\theta_e$ with respect to the face-perimeter map. Symbolically, at perimeters $\boldsymbol\alpha$, $$\mathop{}\!\mathrm d\nu_G(\boldsymbol\alpha)= \left(\prod_{e\in E(G)}\mathop{}\!\mathrm d\theta_e\right) \left(\prod_{i=1}^{n} \delta\left(\sum_{e\in\partial f_i}\theta_e-\alpha_i\right)\right).$$(2.4) An edge is counted with its boundary multiplicity. This notation specifies a disintegrated measure, rather than Euclidean hypersurface measure with an unstated coarea factor. Concretely, let $P$ be the perimeter matrix and split the edge coordinates as $(u,v)$ so that the square $n\times n$ block $P_v$ is invertible. On the fiber $P_u u+P_vv=\boldsymbol\alpha$, the measure is $$\mathop{}\!\mathrm d\nu_G=\frac{|\mathop{}\!\mathrm du_1\cdots\mathop{}\!\mathrm du_{2n-6}|}{|\det P_v|}, \qquad v=P_v^{-1}(\boldsymbol\alpha-P_u u).$$ Thus deleting $n$ dependent edge variables without this determinant would change the normalization.

The ideal cell description is known to give a volume calculation, beyond the possibility suggested by the realization theorem. Charbonnier–David–Eynard compare the angle measure with the WP form through lambda lengths [50, Section 3.2]. In Budd’s normalization of metric-map volume, $$\boxed{ V_{0,n}(0,\ldots,0)= 2^{n-2}\sum_{[G]}\frac{1}{|\mathop{\mathrm{Aut}}(G)|} \int_{\mathcal P_G}\mathop{}\!\mathrm d\nu_G(2\pi,\ldots,2\pi).}$$(2.5) Here $\mathcal P_G$ imposes positivity and irreducibility, $G$ ranges over oriented trivalent maps with labeled faces, and automorphisms preserve this data. Nontrivalent strata have zero volume. Budd gives this interpretation and the normalization explicitly [48, Section 1.3]. Equivalently, his rooted-map measure gives each root weight $1/(2|E(G)|)$; summing the $2|E(G)|/|\mathop{\mathrm{Aut}}(G)|$ rooted versions gives the displayed unrooted weight. Reversing the cyclic orders is not an allowed equivalence of oriented maps.

Thus there is a genuine partition function over polyhedral combinatorial types. Formula (2.5) is a finite cellwise integral at fixed $n$; it is not a sum over geodesics. The factors in it encode three choices that should never be suppressed: labels, orientation, and the normalization of the constrained measure. Section 3 checks them directly for $n=4$.

2.3 The string-theoretic connection

Saadi–Zwiebach’s closed-string construction uses equal-perimeter polyhedral diagrams and quadratic differentials [92]. Zwiebach’s minimal-area formulation imposes a lower bound $2\pi$ on nontrivial closed curves and uses uniqueness to control the covering of moduli space [100]. These give a concrete historical counterpart to the suggestion that such cells should be used for integration.

Several identifications nevertheless require care. A metric ribbon graph can be thickened into a flat surface using a Jenkins–Strebel differential. A hyperbolic polyhedron has an intrinsic hyperbolic boundary metric and also extrinsic dual data. These constructions may share the same combinatorial and length description without giving the same metric or the same conformal coordinate on moduli. Likewise, a string amplitude includes its specified integrand; a decomposition used for string integration does not automatically select the WP measure. Equation (2.5) uses an explicit measure comparison, not just the common vocabulary of polyhedra.