Triangulations of singular constant curvature spheres via Belyi functions and determinants of Laplacians

Technical Summary: Triangulations of Singular Constant Curvature Spheres via Belyi Functions and Determinants of Laplacians

Problem Statement
The paper addresses the problem of computing the zeta-regularized spectral determinant of the Friedrichs Laplacian on singular constant curvature spheres. These surfaces are constructed by cutting and gluing copies of a constant curvature (hyperbolic, spherical, or flat) "double triangle" (an $S^2$-like surface formed by two geodesic triangles). The resulting surfaces possess conical singularities. A primary challenge is that standard gluing formulae for spectral determinants (such as BFK-type formulae) are inapplicable due to the presence of conical singularities along the cuts. Furthermore, while the spectral determinant for smooth metrics on Riemann surfaces has been extensively studied (e.g., by Osgood, Phillips, and Sarnak), closed explicit expressions for singular surfaces, particularly those corresponding to highly symmetric configurations like Platonic solids, have been lacking.

Methodology
The author employs a geometric and analytic approach rooted in the theory of Belyi functions and dessins d'enfants:

1. Geometric Construction: The surfaces are viewed as ramified coverings of the Riemann sphere. A Belyi function $f: \mathbb{C}_x \to \mathbb{C}_z$ (ramified only over $\{0, 1, \infty\}$) defines a combinatorial triangulation of the source sphere $\mathbb{C}_x$.
2. Metric Pullback: A constant curvature metric $m_\beta$ with conical singularities of orders $\beta_0, \beta_1, \beta_\infty$ at $0, 1, \infty$ is defined on the target sphere. The metric on the source surface is the pullback $f^*m_\beta$. This creates a singular metric of constant Gaussian curvature on $\mathbb{C}_x$, isometric to the glued double triangles.
3. Anomaly Formulae: Instead of gluing formulae, the author utilizes anomaly formulae for determinants of Laplacians under conformal changes of metric. The proof relies on relating the determinant on the ramified covering to the determinant on the base sphere via the Liouville action and the stress-energy tensor.
4. Liouville Action and Accessory Parameters: The paper explicitly connects the spectral determinant to the Liouville action. It derives explicit expressions for the accessory parameters (coefficients in the stress-energy tensor) of the triangulated surfaces in terms of the Belyi function and the orders of singularities.
5. Euclidean Reference: To resolve integration constants arising in the anomaly formula, the author evaluates the determinant for a specific case: the Euclidean equilateral triangulation (where the base metric is flat with specific singularities). This allows for the explicit calculation of the constant term $C_f$ depending only on the Belyi function.

Key Contributions and Results
The central result is Theorem 2.1, which provides a closed, explicit formula for the zeta-regularized spectral determinant $\det \Delta_{f^*\beta}$ of the Friedrichs Laplacian on the triangulated singular sphere $(\mathbb{C}_x, f^*m_\beta)$. The formula expresses the determinant in terms of:

• The spectral determinant of the base $S^2$-like double triangle ($\det \Delta_\beta$), which was previously found in closed form in [18].
• The Belyi function $f$ and its ramification data (orders of zeros, poles, and critical points).
• Explicit constants involving the Barnes double zeta function and the Riemann zeta function.

The paper further establishes:

• Explicit Accessory Parameters: Lemma 4.1 provides explicit formulas for the accessory parameters $h_k$ of the uniformizing projective connection in terms of the Belyi function's Taylor/Laurent coefficients and the singularity orders.
• Liouville Action: An explicit expression for the Liouville action is derived as a byproduct, linking it to the spectral determinant.
• Stationary Points: Theorem 4.6 demonstrates that the surfaces corresponding to the five Platonic solids (tetrahedron, octahedron, cube, icosahedron, dodecahedron) and regular dihedra are stationary points of the spectral determinant functional on the space of conical metrics with fixed area and curvature. This is proven by exploiting the high symmetry (automorphisms) of these configurations.
• Specific Examples: Section 5 provides explicit determinant formulas for various triangulations, including:
  • Cyclic triangulations (Shabat polynomials $x^\ell$).
  • Dihedral triangulations.
  • Platonic surfaces (Tetrahedron, Octahedron, Cube, Icosahedron, Dodecahedron) for spherical, Euclidean, and hyperbolic cases.
  • Specific evaluations for the Bolza curve (as a double cover of a hyperbolic octahedron) and surfaces related to Klein's quartic.

Significance and Claims
The paper claims to provide the first closed explicit expressions for the spectral determinants of these specific singular surfaces. The significance lies in:

1. Solving a Specific Spectral Problem: It overcomes the limitation of existing gluing formulae by using Belyi functions and anomaly formulae to handle conical singularities explicitly.
2. Symmetry and Extrema: It confirms that the Platonic surfaces, known for their maximal symmetry groups, correspond to stationary points of the spectral determinant, extending previous results on smooth metrics to the singular setting.
3. Computational Utility: The derived formulas allow for the explicit evaluation of spectral invariants for complex surfaces (like the Bolza curve and Klein's quartic) that were previously only accessible via numerical methods or remained unknown in closed form.
4. Connection to Conformal Field Theory: The explicit evaluation of the Liouville action and accessory parameters offers new data points for checking results derived from conformal blocks.

The author notes that while the results are explicit for the cases studied, the general problem of uniformization for arbitrary moduli remains open, and the current approach relies on the specific structure provided by Belyi functions. The paper does not propose new experimental applications but rather advances the theoretical understanding of spectral geometry on singular surfaces.