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

Victor Kalvin 111 Department of Mathematics, Dawson College, 3040 Sherbrooke
St. W, Montreal, Quebec H3Z 1A4, Canada; E-mail<EMAIL_ADDRESS>
###### Abstract

We study the zeta-regularized spectral determinant of the Friedrichs
Laplacians on the singular spheres obtained by cutting and glueing copies of
constant curvature (hyperbolic, spherical, or flat) double triangle. The
determinant is explicitly expressed in terms of the corresponding Belyi
functions and the determinant of the Friedrichs Laplacian on the double
triangle. The latter determinant was found in a closed explicit form in
ArXiv:2112.02771 [18]. In examples we consider the cyclic, dihedral,
tetrahedral, octahedral, and icosahedral triangulations, and find the
determinant for the corresponding spherical, Euclidean, and hyperbolic
Platonic surfaces. These surfaces correspond to stationary points of the
determinant.

##  1 Introduction

We study the zeta-regularized spectral determinant of the Friedrichs scalar
Laplacian on the surfaces constructed by cutting and gluing copies of a
constant curvature $S^{2}$-like double triangle. We call these surfaces glued
or triangulated. The geometric (combinatorial) cutting and gluing scheme is
described in terms of the corresponding Belyi function [3]. Or, equivalently,
in terms of dessins d’enfants [12].

The Belyi function maps the (source) Riemann surface to the (target) Riemann
sphere. Any constant curvature double triangle is isometric to the target
Riemann sphere with explicitly constructed conformal metric with three conical
singularities [18]. The glued surface is isometric to the source Riemann
surface equipped with the pullback of the explicit conformal metric by the
Belyi function. This provides us with a geodesic triangulation and an explicit
uniformization of the glued surface. In particular, in the case of flat right
isosceles $S^{2}$-like double triangle, we get the square-tiled flat surfaces,
see e.g. [44, 58] and references therein.

In other words, for studying the spectral determinant we suggest using the
natural decomposition via triangulation based on the celebrated results of
Belyi, Grothendieck, Shabat and Voevodskii. The price for this is that we
first need to study the spectral determinant on surfaces with conical
singularities [18, 19, 20, 21].

The main idea is to explicitly express the spectral determinant of the glued
surface in terms of the corresponding Belyi function and the spectral
determinant of the constant curvature $S^{2}$-like double triangle. The latter
determinant was found in a closed explicit form in [18].

Let us recall that not all Riemann surfaces can be glued/triangulated in this
manner. But for the most interesting surfaces, like the Fermat curve, the
Bolza curve, the Hurwitz surfaces, including Klein’s quartic, the Platonic
surfaces, etc., it is certainly possible, thanks to the famous Belyi theorem
[3, 22, 45]. Due to the highest possible number of authomorphisms [45, 13],
some of the triangulated surfaces (equipped with the natural smooth hyperbolic
metric) are expected to be stationary points of the spectral determinant. To
the best of our knowledge, no closed explicit expression for the spectral
determinant of any of these surfaces is known yet.

In this paper, we study the spectral determinant of the genus zero
triangulated surfaces. For the triangulations of singular constant curvature
(hyperbolic, spherical, or flat) spheres via Belyi functions, we explicitly
express the spectral determinant in terms of the Belyi maps, and the
determinant of the constant curvature $S^{2}$-like double triangle. In
particular, with each bicolored plane tree [4, 5] we can naturally associate
an infinite family of constant curvature singular spheres, and then the
corresponding infinite family of explicit spectral invariants, that is the
family of the corresponding spectral determinants. For some closely related
geometric constructions and invariants see e.g. [14, 38, 39, 47, 51].

Let us stress that due to conical singularities on the cuts, none of the
presently known BFK-type gluing formulae [7] can be used. We rely on a
completely different approach relating the determinants of Laplacians on the
target Riemann sphere and on the ramified covering via anomaly formulae for
the determinants [18, 20].

As a byproduct, our approach allows for explicit evaluation of the Liouville
action [8, 59, 50], which can be of independent interest, cf. [36, 37, 49]. It
would be interesting to check if the explicit expressions can be reproduced by
using conformal blocks [59].

As is known [50], the Liouville action generates the famous accessory
parameters as their common antiderivative. We explicitly express the accessory
parameters of the triangulated singular constant curvature spheres in terms of
the Belyi functions and the orders of conical singularities. We also find the
derivative of the Liouville action with respect to the order of a conical
singularity in the spirit of [59]. As one can expect, this allows us to show
that the five Platonic solids are special also in the context of this paper:
their surfaces correspond to stationary points of the determinant.

Recall that for smooth metrics on Riemann surfaces extrema of spectral
determinants were studied in a series of papers by Osgood, Phillips, and
Sarnak [33, 34, 35]; see also [23] for an extension of their results, and [11]
for recent closely related results in the four-dimensional case.

We illustrate the main results of this paper with a number of examples: We
consider the cyclic, dihedral, tetrahedral, octahedral, and icosahedral
triangulations [25], and find explicit expressions for the spectral
determinant of the corresponding spherical, Euclidean, and hyperbolic Platonic
surfaces. In particular, we explicitly evaluate the determinants of the
regular hyperbolic octahedron with vertices of angle $\pi$ (a double cover is
the Bolza curve) and the regular hyperbolic icosahedron corresponding to the
tessellation by $(2,3,7)$-triangles (related to Klein’s quartic [41, 22, 45]);
see Example 5.4 and Example 5.15.

As the angles of the conical points of a Platonic surface go to zero (and the
area remains fixed), the determinant grows without any bound, cf. Fig 7. In
the limit, the conical points turn into cusps and we obtain an ideal
polyhedron [16, 17, 24]. The spectrum of the corresponding Laplacians is no
longer discrete.

This paper is organized as follows. Section 2 contains preliminaries and the
main results of the paper (Theorem 2.1). Namely, after giving some
preliminaries on constant curvature $S^{2}$-like double triangles and their
spectral determinants in Subsection 2.1, we describe the geometric setting of
this paper and formulate the main results in Theorem 2.1 of Subsection 2.2.

Section 3 is entirely devoted to the proof of Theorem 2.1: In Subsection 3.1
we prove Proposition 3.1, which is a preliminary version of Theorem 2.1. In
Subsection 3.2 we refine the result of Proposition 3.1 by using the natural
Euclidean equilateral triangulation [45, 55]. This completes the proof of
Theorem 2.1.

Section 4 is devoted to the uniformization, the accessory parameters, the
Liouville action, and stationary points of the spectral determinant.

Finally, Section 5 contains illustrating examples and applications.

##  2 Preliminaries and main results

###  2.1 Double triangles

Consider the involutional tessellation of the standard round sphere
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1$ in $\mathbb{R}^{3}$ by two congruent
$(1,1,1)$-triangles; as usual, by a $(k,\ell,m)$-triangle we mean a geodesic
triangle (spherical, Euclidean, or hyperbolic) with internal angles $\pi/k$,
$\pi/\ell$, and $\pi/m$. In particular, the $(1,1,1)$-triangles are
hemispheres. Let the standard sphere in $\mathbb{R}^{3}$ be identified with
the Riemann sphere $\overline{\mathbb{C}}_{z}=\mathbb{C}\cup\infty$ by means
of the stereographic projection. Let the $(1,1,1)$-triangles correspond to the
upper $\Im z\geqslant 0$ and lower $\Im z\leqslant 0$ half-planes. Without
loss of generality, we can assume that three marked points on the sphere (the
vertices of the congruent $(1,1,1)$-triangles) have the coordinates $z=0$,
$z=1$, and $z=\infty$.

Let us endow the Riemann sphere with a unique unit area constant curvature
metric having conical singularities of order $\beta_{0}$, $\beta_{1}$, and
$\beta_{\infty}$ at the marked points $z=0$, $z=1$, and $z=\infty$
correspondingly. For simplicity, we assume that the orders $\beta_{j}$ are in
the interval $(-1,0]$, or, equivalently, that the cone angles
$2\pi(\beta_{j}+1)$ are positive and do not exceed $2\pi$; here
$j\in\\{0,1,\infty\\}$. We denote the metric by $m_{\boldsymbol{\beta}}$ and
say that it represents the divisor

$\boldsymbol{\beta}=\beta_{0}\cdot 0+\beta_{1}\cdot
1+\beta_{\infty}\cdot\infty$

of degree $|\boldsymbol{\beta}|=\beta_{0}+\beta_{1}+\beta_{\infty}$.

The metric $m_{\boldsymbol{\beta}}$ can be explicitly constructed as the
pullback of the Gaussian curvature $2\pi(|\boldsymbol{\beta}|+2)$ model metric

$\frac{4|dw|^{2}}{(1+2\pi(|\boldsymbol{\beta}|+2)|w|^{2})^{2}}$ (2.1)

by an appropriately normalized Schwarz triangle function $z\mapsto
w=w_{\boldsymbol{\beta}}(z)$, see [18, Appendix]. Note that the conditions
$\beta_{j}-|\boldsymbol{\beta}|/2>0$ are necessary and sufficient for the
existence of the metric.

The Schwarz triangle function $w_{\boldsymbol{\beta}}$ maps the upper half-
plane $\Im z>0$ into a geodesic (light-coloured) triangle with internal angle
$\pi(\beta_{0}+1)$ at the vertex $w_{\boldsymbol{\beta}}(0)=0$,
$\pi(\beta_{1}+1)$ at the vertex $w_{\boldsymbol{\beta}}(1)\in\mathbb{R}$, and
$\pi(\beta_{\infty}+1)$ at the vertex $w_{\boldsymbol{\beta}}(i\infty)$. The
analytic continuation of $w_{\boldsymbol{\beta}}$ (from the upper half-plane
$\Im z>0$ through the interval $(0,1)$ of the real axis) maps the lower half-
plane $\Im z<0$ into the (dark-coloured) reflection of the geodesic triangle
in the side joining the points $w_{\boldsymbol{\beta}}(0)$ and
$w_{\boldsymbol{\beta}}(1)$. Thus we obtain a geodesic with respect to the
model metric (2.1) bicolored double triangle, which is hyperbolic for
$|\boldsymbol{\beta}|<-2$ (cf. Fig. 1), spherical for
$|\boldsymbol{\beta}|>-2$ (cf. Fig. 2), and Euclidean for
$|\boldsymbol{\beta}|=-2$ (cf. Fig. 3).

$w_{\boldsymbol{\beta}}(1)$$w_{\boldsymbol{\beta}}(0)$$w_{\boldsymbol{\beta}}(i\infty)$$w_{\boldsymbol{\beta}}(-i\infty)$$\pi(\beta_{0}+1)$$\pi(\beta_{\infty}+1)$$\pi(\beta_{1}+1)$
Figure 1: Hyperbolic double triangle.
$w_{\boldsymbol{\beta}}(i\infty)$$w_{\boldsymbol{\beta}}(0)$$w_{\boldsymbol{\beta}}(1)$$w_{\boldsymbol{\beta}}(-i\infty)$$\pi(\beta_{0}+1)$$\pi(\beta_{\infty}+1)$$\pi(\beta_{1}+1)$
Figure 2: Spherical double triangle.
$w_{\boldsymbol{\beta}}(0)$$w_{\boldsymbol{\beta}}(i\infty)$$w_{\boldsymbol{\beta}}(1)$$w_{\boldsymbol{\beta}}(-i\infty)$$\pi(\beta_{0}+1)$$\pi(\beta_{\infty}+1)$$\pi(\beta_{1}+1)$
Figure 3: Euclidean double triangle.

To make it $S^{2}$-like, the double triangle is folded along the interval
$[w_{\boldsymbol{\beta}}(0),w_{\boldsymbol{\beta}}(1)]\subset\mathbb{R}$, and
then the corresponding sides of the light- and dark-coloured triangles are
glued pairwise. Namely, the side joining $w_{\boldsymbol{\beta}}(0)$ and
$w_{\boldsymbol{\beta}}(i\infty)$ is glued to the side joining
$w_{\boldsymbol{\beta}}(0)$ and $w_{\boldsymbol{\beta}}(-i\infty)$, and then
the side joining $w_{\boldsymbol{\beta}}(1)$ and
$w_{\boldsymbol{\beta}}(i\infty)$ is glued to the side joining
$w_{\boldsymbol{\beta}}(1)$ and $w_{\boldsymbol{\beta}}(-i\infty)$. The
resulting $S^{2}$-like bicolored double triangle is isometric to the genus
zero constant curvature surface
$(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$ with three conical
singularities. The isometry is given by the Schwarz triangle function
$w_{\boldsymbol{\beta}}$.

The potential $\phi$ of the conformal metric
$m_{\boldsymbol{\beta}}=e^{2\phi}|dz|^{2}$ satisfies the Liouville equation

$e^{-2\phi}(-4\partial_{z}\partial_{\bar{z}}\phi)={2\pi(|\boldsymbol{\beta}|+2)},\quad
z\in\mathbb{C}\setminus\\{0,1\\},$ (2.2)

and obeys the asymptotics

$\displaystyle\phi(z)$ $\displaystyle=\beta_{0}\log|z|+\phi_{0}+o(1),\quad
z\to 0,\quad\phi_{0}=\Psi(\beta_{0},\beta_{1},\beta_{\infty}),$ (2.3)
$\displaystyle\phi(z)$ $\displaystyle=\beta_{1}\log|z-1|+\phi_{1}+o(1),\quad
z\to 1,\quad\phi_{1}=\Psi(\beta_{1},\beta_{0},\beta_{\infty}),$
$\displaystyle\phi(z)$
$\displaystyle=-(\beta_{\infty}+2)\log|z|+\phi_{\infty}+o(1),\quad
z\to\infty,\quad\phi_{\infty}=\Psi(\beta_{\infty},\beta_{1},\beta_{0}).$

Here
$\Psi(\beta_{1},\beta_{2},\beta_{3})=\Phi(\beta_{1},\beta_{2},\beta_{3})+\log
2$ with explicit function $\Phi$ from [18, Prop. A.2)].

The Liouville equation indicates that outside of the marked points the
Gaussian curvature of the metric $m_{\boldsymbol{\beta}}$ is
$2\pi(|\boldsymbol{\beta}|+2)$, while the asymptotics (2.3) indicate that at
the marked points $z=0$,$z=1$, and $z=\infty$ the metric has conical
singularities of order $\beta_{0}$, $\beta_{1}$, and $\beta_{\infty}$
correspondingly [54].

On the singular constant curvature surface
$(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$ we consider the Laplace-
Beltrami operator
$\Delta_{\boldsymbol{\beta}}=-e^{-2\phi}4\partial_{z}\partial_{\bar{z}}$ as an
unbounded operator in the usual $L^{2}$-space. The operator is initially
defined on the smooth functions supported outside of the conical singularities
and not essentially selfadjoint. We take the Friedrichs selfadjoint extension,
which we still denote by $\Delta_{\boldsymbol{\beta}}$ and call the Friedrichs
Laplacian or simply Laplacian for short. The spectrum of
$\Delta_{\boldsymbol{\beta}}$ consists of non-negative isolated eigenvalues of
finite multiplicity, and the zeta-regularized spectral determinant
$\det\Delta_{\boldsymbol{\beta}}$ of $\Delta_{\boldsymbol{\beta}}$ can be
introduced in the standard well-known way.

In what follows it is important that the spectral zeta-regularized determinant
of the Friedrichs Laplacian $\Delta_{\boldsymbol{\beta}}$ on the constant
curvature surface $(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$ with
three conical singularities, or, equivalently, on the isometric $S^{2}$-like
bicolored double triangle, is an explicit function

$(-1,0]^{3}\ni(\beta_{0},\beta_{1},\beta_{\infty})\mapsto\det\Delta_{\boldsymbol{\beta}}$
(2.4)

found in [18, Corollary 1.3].

###  2.2 Triangulations and determinants of Laplacians

Recall that a (non-constant) meromorphic function
$f:X\to\overline{\mathbb{C}}_{z}$ on a compact Riemann surface $X$ is called a
Belyi function, if it is ramified at most above three points [3]. Any Belyi
function can alternatively be described via the corresponding dessin d’enfant,
which is usually defined as the graph formed on $X$ by the preimages
$f^{-1}([0,1])$ of the real line segment $[0,1]$ with black points placed at
the preimages $f^{-1}(0)$ of zero and white points placed at the preimages
$f^{-1}(1)$ of $1$, e.g. [10, 12, 30, 40, 57]. Any meromorphic function on the
Riemann sphere is a rational function. If, in addition, $f$ has only a single
pole that is at infinity, then $f$ is a polynomial [4, 5, 30].

Consider a Belyi function
$f:\overline{\mathbb{C}}_{x}\to\overline{\mathbb{C}}_{z}$ ramified at most
above the marked points $z=0$, $z=1$, and $z=\infty$. This defines a ramified
(branched) covering and a bicolored triangulation of the Riemann sphere
$\overline{\mathbb{C}}_{x}$, e.g. [55, 30]. Namely, the function $f$ maps: i)
the sides of the bicolored triangles on $\overline{\mathbb{C}}_{x}$ to the
line segments $(-\infty,0)$, $(0,1)$, and $(1,\infty)$ of the real axis; ii)
the vertices of the triangles to the marked points $0$, $1$, and $\infty$;
iii) the light-colored triangles to the upper half-plane $\Im z>0$, and the
dark-colored triangles to the lower half-plane $\Im z<0$. The number of
bicolored double triangles on $\overline{\mathbb{C}}_{x}$ is exactly the
degree $\deg f=\max\\{\deg P,\deg Q\\}$ of $f$, where $f(x)=P(x)/Q(x)$ with
coprime polynomials $P$ and $Q$.

For example, the Belyi function

$f(x)=1728\frac{x^{5}(x^{10}-11x^{5}-1)^{5}}{(x^{20}+228(x^{15}-x^{5})+494x^{10}+1)^{3}},\quad\deg
f=60,$ (2.5)

defines the icosahedral triangulation [26], which corresponds to the
tessellation of the standard round sphere with bicolored spherical
$(2,3,5)$-triangles in Fig. 4, Fig. 2. As usual, we identify the Riemann
sphere $\overline{\mathbb{C}}_{x}$ with the standard round sphere in
$\mathbb{R}^{3}$ by means of the stereographic projection.

Figure 4: Icosahedral triangulation

Depending on the sign of $|\boldsymbol{\beta}|+2$, the composition
$w_{\boldsymbol{\beta}}\circ f$ (of the Schwarz triangle function
$w_{\boldsymbol{\beta}}$ with a Belyi function
$f:\overline{\mathbb{C}}_{x}\to\overline{\mathbb{C}}_{z}$) maps the light-
coloured (resp. the dark-coloured) triangles defined by $f$ on
$\overline{\mathbb{C}}_{x}$ to the light-coloured (resp. the dark-coloured)
hyperbolic, spherical, or Euclidean triangle in Fig. 1, Fig. 2, and Fig. 3.
The pullback of the model metric (2.1) by this composite function, or,
equivalently, the pullback $f^{*}m_{\boldsymbol{\beta}}$ of the metric
$m_{\boldsymbol{\beta}}$ by $f$, is a singular metric of Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+2)$ on $\overline{\mathbb{C}}_{x}$. The
triangulated surface $(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$
is isometric to the one obtained by cutting and gluing $\deg f$ copies of the
$S^{2}$-like bicolored double triangle in accordance with a combinatorial
scheme prescribed by $f$. In particular, Fuchsian triangle groups are included
into consideration [10].

Let us note that the above construction also works in the opposite
(constructive) direction: the combinatorial cutting and gluing schemes of
bicolored double triangles can be described with the help of a (fixed) base
star in $\overline{\mathbb{C}}_{z}$ with the terminal vertices at
$z=0,1,\infty$ and the constellations. For any constellation there exists a
Belyi function defining the cutting and gluing scheme, e.g. [30].

For instance, one can preserve the gluing scheme of bicolored double triangles
in Fig. 4, and replace each light-coloured (resp. dark-coloured) spherical
$(2,3,5)$-triangle with the image of the upper (resp. lower) complex half-
plane under the Schwarz triangle function $w_{\boldsymbol{\beta}}$ in Fig. 1,
Fig. 2, or Fig. 3. Then, as a result, we obtain a hyperbolic, Euclidean, or
spherical singular surface isometric to
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$, where $f$ is the
Belyi function (2.5). However, in general, it is not easy to find a Belyi
function corresponding to a particular gluing scheme, even though for the
regular dihedra and the surfaces of five Platonic solids they were first found
by Schwarz and Klein [25], see also [30, 31].

The main result of this paper is a closed explicit formula for the zeta-
regularized spectral determinant of the Friedrichs Laplacian on the
triangulated singular constant curvature spheres
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$. We express the
determinant in terms of the Belyi function $f$ and the determinant (2.4) of
the corresponding $S^{2}$-like bicolored double triangle. Recall that the
latter determinant is explicitly evaluated in [18].

Let us list the preimages of the marked points
$\\{0,1,\infty\\}\subset\overline{\mathbb{C}}_{z}$ under $f$ as the marked
points $x_{1},x_{2},\dots,x_{n}$ on the Riemann sphere
$\overline{\mathbb{C}}_{x}$; here $n=\deg f+2$. We shall always assume that
$x_{k}=\infty$ for some $k\leqslant n$: this can always be achieved by
replacing $f$ with equivalent Belyi function $f\circ\mu$, where $\mu$ is a
Möbius transformation satisfying $\mu(\infty)=x_{k}$. The surfaces
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$ and
$(\overline{\mathbb{C}}_{x},(f\circ\mu)^{*}m_{\boldsymbol{\beta}})$ are
isometric, and, as a consequence, the corresponding spectral determinants are
equal.

Introduce the ramification divisor

$\boldsymbol{f}:=\sum_{k=1}^{n}\operatorname{ord}_{k}f\cdot x_{k},$

where $\operatorname{ord}_{k}f$ is the ramification order of the Belyi
function $f$ at $x_{k}$. If $x_{k}$ is a pole of $f$, then its multiplicity
coincides with $\operatorname{ord}_{k}f+1$. If $f^{\prime}(x_{k})=0$, then
$\operatorname{ord}_{k}f+1$ is the order of zero at $x=x_{k}$ of the function
$x\mapsto f(x)-f(x_{k})$. If $x_{k}$ is not a pole of $f$ and
$f^{\prime}(x_{k})\neq 0$, then the ramification order
$\operatorname{ord}_{k}f$ is zero. One can also interpret $x_{k}$ as a vertex
in the dessin d’enfant corresponding to $f$, then the number
$\operatorname{ord}_{k}f+1$ is its graph-theoretic degree, or, equivalently,
the number of edges emanating from the vertex $x_{k}$. By the Riemann-Hurwitz
formula the degree $|\boldsymbol{f}|:=\sum_{k=1}^{n}\operatorname{ord}_{k}f$
of the divisor $\boldsymbol{f}$ satisfies $|\boldsymbol{f}|=2\deg f-2$.

From the Liouville equation (2.2) and the asymptotics (2.3) it follows that
the potential

$f^{*}\phi:=\phi\circ f+\log|f^{\prime}|$

of the pullback metric $f^{*}m_{\boldsymbol{\beta}}=e^{2f^{*}\phi}|dx|^{2}$
satisfies the Liouville equation

$e^{-2f^{*}\phi}\bigl{(}-4\partial_{x}\partial_{\bar{x}}(f^{*}\phi)\bigr{)}={2\pi(|\beta|+2)},\quad
x\in\mathbb{C}\setminus\\{x_{1},\dots,x_{n}\\},$ (2.6)

and obeys the asymptotics

$\displaystyle(f^{*}\phi)(x)$
$\displaystyle=(f^{*}\beta)_{k}\log|x-x_{k}|+O(1),\quad x\to x_{k}\neq\infty,$
(2.7) $\displaystyle(f^{*}\phi)(x)$
$\displaystyle=-((f^{*}\beta)_{k}+2)\log|x|+O(1),\quad x\to x_{k}=\infty,$

where

$\left(f^{*}\boldsymbol{\beta}\right)_{k}:=(\operatorname{ord}_{k}f+1)(\beta_{f(x_{k})}+1)-1.$
(2.8)

Geometrically this means that outside of the points $x_{1},\dots,x_{k}$ the
metric $f^{*}m_{\boldsymbol{\beta}}$ is a regular metric of constant Gaussian
curvature $2\pi(|\beta|+2)$, while at each point $x_{k}$ the metric has a
conical singularity of order $(f^{*}\beta)_{k}$, or, equivalently, of angle
$2\pi((f^{*}\beta)_{k}+1)$, see e.g. [54]. That is because
$\operatorname{ord}_{k}f+1$ vertices of bicolored double triangles meet
together at $x_{k}$ to form the conical singularity. Each of the vertices
makes a contribution of angle $2\pi(\beta_{f(x_{k})}+1)$ into the angle of the
conical singularity, cf. (2.8). Note that it could be so that for a point
$x_{k}$ we have $(f^{*}\beta)_{k}=0$ (for example, this is the case if
$\operatorname{ord}_{k}f=1$ and $\beta_{f(x_{k})}=-1/2$), then $x_{k}$ is also
a regular point of the metric $f^{*}m_{\boldsymbol{\beta}}$. Clearly,
$\beta_{f(x_{k})}=\beta_{\infty}$ if $x_{k}$ is a pole of $f$,
$\beta_{f(x_{k})}=\beta_{0}$ if $x_{k}$ is a zero of $f$, and
$\beta_{f(x_{k})}=\beta_{1}$ if $f(x_{k})=1$, where $\beta_{j}$ is the same as
in (2.3).

Now we are in a position to formulate the main results of this paper.

######  Theorem 2.1 (Spectral determinant of triangulated spheres).

Let $\det\Delta_{\boldsymbol{\beta}}$ stand for the explicit function (2.4),
whose value at a point is the zeta-regularized spectral determinants of the
Friedrichs selfadjoint extension of the Laplace-Beltrami operator on the unit
area $S^{2}$-like double triangle, or, equivalently, on
$(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$; see Sec. 2.1 and [18].

Let $f:\overline{\mathbb{C}}_{x}\to\overline{\mathbb{C}}_{z}$ be a Belyi
function unramified outside of the set $\\{0,1,\infty\\}$ and such that
$f(\infty)\in\\{0,1,\infty\\}$. Denote by $\operatorname{ord}_{k}f$ the
ramification order of $f$ at $x_{k}$, where
$x_{1},x_{2},\dots,x_{n}\in\overline{\mathbb{C}}_{x}$ are the preimages of the
points $0$, $1$, and $\infty$ under $f$.

Consider the surface $(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$
isometric to the one glued from $\deg f$ copies of the double triangle in
accordance with a pattern defined by $f$. Then for the zeta-regularized
spectral determinant $\det\Delta_{f^{*}\boldsymbol{\beta}}$ of the Friedrichs
selfadjoint extension of the Laplace-Beltrami operator on
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$ we have

$\displaystyle\log\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{\deg f}=$
$\displaystyle\deg
f\cdot\log{\det\Delta_{\boldsymbol{\beta}}}+\frac{1}{6}\sum_{k}\left(\operatorname{ord}_{k}f+1-\frac{1}{\operatorname{ord}_{k}f+1}\right)\frac{\phi_{f(x_{k})}}{\beta_{f(x_{k})}+1}$
(2.9)
$\displaystyle-\frac{1}{6}\sum_{k}\left((f^{*}\boldsymbol{\beta})_{k}+1+\frac{1}{(f^{*}\boldsymbol{\beta})_{k}+1}\right)\log(\operatorname{ord}_{k}f+1)$
$\displaystyle-\sum_{k}\Bigl{(}\mathcal{C}\left(\left(f^{*}\boldsymbol{\beta}\right)_{k}\right)-(\operatorname{ord}_{k}f+1)\mathcal{C}(\beta_{f(x_{k})})\Bigr{)}-(\deg
f-1){\bf C}+C_{f},$

where $\phi_{j}$ and $\beta_{j}$ with $j\in\\{0,1,\infty\\}$ is the (explicit)
uniformization data in (2.3), and $(f^{*}\boldsymbol{\beta})_{k}$ is the same
as in (2.8). The real-analytic function
$(-\infty,0]\ni\beta\mapsto\mathcal{C}(\beta)$ is defined by the equality

$\mathcal{C}(\beta)=2\zeta^{\prime}_{B}(0;\beta+1,1,1)-2\zeta_{R}^{\prime}(-1)-\frac{\beta^{2}}{6(\beta+1)}\log
2-\frac{\beta}{12}+\frac{1}{2}\log(\beta+1),$ (2.10)

where $\zeta^{\prime}_{B}$ and $\zeta_{R}^{\prime}$ stand for the derivatives
with respect to $s$ of the Barnes double zeta function
$\zeta_{B}(s;\beta+1,1,1)$ and the Riemann zeta function $\zeta_{R}(s)$
respectively. For the constant ${\bf C}$ in (2.9) we have

${\bf C}=\frac{1}{6}-\frac{4}{3}\log 2-4\zeta_{R}^{\prime}(-1)-\log\pi.$
(2.11)

Finally, the constant $C_{f}$ in (2.9) depends only on the Belyi function $f$.
It is given by

$\displaystyle C_{f}$
$\displaystyle=\frac{1}{18}\sum_{k:x_{k}\neq\infty}\sum_{\ \ \ell:x_{k}\neq
x_{\ell}\neq\infty}\frac{(\operatorname{ord}_{k}f-2)(\operatorname{ord}_{\ell}f-2)}{\operatorname{ord}_{k}f+1}\log|x_{k}-x_{\ell}|$
(2.12)
$\displaystyle\qquad\qquad+\frac{1}{6}\sum_{k}\left(\frac{\operatorname{ord}_{k}f+1}{3}+\frac{3}{\operatorname{ord}_{k}f+1}\right)\log(\operatorname{ord}_{k}f+1)$
$\displaystyle\qquad\qquad+\frac{1}{6}\left(\deg
f-\sum_{k}\frac{3}{\operatorname{ord}_{k}f+1}\right)\log A_{f},$

where

$A_{f}=\frac{|f(x)|^{-2/3}|f(x)-1|^{-2/3}|f^{\prime}(x)|}{\prod_{k:x_{k}\neq\infty}|x-x_{k}|^{\frac{1}{3}(\operatorname{ord}_{k}f-2)}}.$
(2.13)

Note that $A_{f}$ is a scaling coefficient that does not depend on $x$ and can
be easily evaluated for any particular Belyi function $f$.

As we show in the proof of Theorem 2.1, the expression (2.12) for $C_{f}$
comes from the Euclidean equilateral triangulation defined on
$\overline{\mathbb{C}}_{x}$ by $f$, cf. [45, 55]. The readers whose interests
are more in the results and applications than in the proof may wish to skip
Section 3 and proceed directly to Sections 4 and 5.

##  3 Proof of the main results

###  3.1 Equality (2.9) is valid with a constant $C_{f}$

In this subsection we prove that the representation for the spectral
determinant (2.9) is valid with some constant $C_{f}$ that does not depend on
the metric $m_{\boldsymbol{\beta}}$ on the target Riemann sphere
$\overline{\mathbb{C}}_{z}$. As a byproduct we obtain an explicit formula for
$C_{f}$ that is different from the one in (2.12).

######  Proposition 3.1.

The equality (2.9) in Theorem 2.1 is valid with a constant $C_{f}$ that
depends only on the Belyi function $f$. Moreover,

$\displaystyle
C_{f}=\frac{1}{6}\sum_{k:x_{k}\neq\infty,f(x_{k})\neq\infty}\left(\frac{\operatorname{ord}_{k}f}{\operatorname{ord}_{k}f+1}\log|c_{k}|+(\operatorname{ord}_{k}f+2)\log(\operatorname{ord}_{k}f+1)\right)$
(3.1)
$\displaystyle+\frac{1}{6}\sum_{k:x_{k}\neq\infty,f(x_{k})=\infty}\left(\frac{\operatorname{ord}_{k}f+2}{\operatorname{ord}_{k}f+1}\log|c_{k}|-\operatorname{ord}_{k}f\log(\operatorname{ord}_{k}f+1)\right)$
$\displaystyle+\left.\frac{1}{6}\left(-\frac{\operatorname{ord}_{k}f}{\operatorname{ord}_{k}f+1}\log|c_{k}|-(\operatorname{ord}_{k}f+2)\log(\operatorname{ord}_{k}f+1)\right)\right|_{k:x_{k}=\infty,f(\infty)=\infty}$
$\displaystyle+\left.\frac{1}{6}\left(-\frac{\operatorname{ord}_{k}f+2}{\operatorname{ord}_{k}f+1}\log|c_{k}|+\operatorname{ord}_{k}f\log(\operatorname{ord}_{k}f+1)\right)\right|_{k:x_{k}=\infty,f(\infty)\neq\infty},$

where

  * •

$c_{k}$ stands for the first nonzero coefficient in the Taylor series of
$f(x)-f(x_{k})$ at $x_{k}$, if $x_{k}$ is not a pole of $f$;

  * •

$c_{k}$ stands for the first nonzero coefficient of the Laurent series of $f$
at $x_{k}$, if $x_{k}$ is a pole of $f$.

The proof of Proposition 3.1 is preceded by Lemma 3.2 below. To formulate the
lemma we need the following refined version of the asymptotics (2.7) for the
metric potential $f^{*}\phi$:

$\displaystyle(f^{*}\phi)(x)=\left(f^{*}\boldsymbol{\beta}\right)_{k}\log|x-x_{k}|+(f^{*}\phi)_{k}+o(1),\quad
x\to x_{k}\neq\infty,$ (3.2)
$\displaystyle(f^{*}\phi)(x)=-\bigl{(}\left(f^{*}\boldsymbol{\beta}\right)_{k}+2\bigr{)}\log|x|+(f^{*}\phi)_{k}+o(1),\quad
x\to x_{k}=\infty,$

where $\left(f^{*}\boldsymbol{\beta}\right)_{k}$ is the same as in (2.8).
Since $f^{*}\phi=\phi\circ f+\log|f^{\prime}|$, it is not hard to see that the
coefficients $(f^{*}\phi)_{k}$ in (3.2) satisfy

$\displaystyle(f^{*}\phi)_{k}=\phi_{f(x_{k})}+(\beta_{f(x_{k})}+1)\log|c_{k}|+\log(\operatorname{ord}_{k}f+1),\quad\text{
if }\quad f(x_{k})\neq\infty,$ (3.3)
$\displaystyle(f^{*}\phi)_{k}=\phi_{f(x_{k})}-(\beta_{f(x_{k})}+1)\log|c_{k}|+\log(\operatorname{ord}_{k}f+1),\quad\text{
if }\quad f(x_{k})=\infty.$

Here $\beta_{j}$ and $\phi_{j}$ with $j\in\\{0,1,\infty\\}$ are the same as in
(2.3), and $c_{k}$ is the same as in Proposition 3.1.

######  Lemma 3.2.

Recall that $m_{\boldsymbol{\beta}}=e^{2\phi}|dz|^{2}$ and
$f^{*}m_{\boldsymbol{\beta}}=e^{f^{*}\phi}|dx|^{2}$. We have

$\displaystyle(|\boldsymbol{\beta}|+2)\int_{\mathbb{C}}(f^{*}\phi)e^{2f^{*}\phi}\frac{dx\wedge
d\bar{x}}{-2i}=(|\boldsymbol{\beta}|+2)\deg f\int_{\mathbb{C}}\phi
e^{2\phi}\frac{dz\wedge d\bar{z}}{-2i}$
$\displaystyle-\sum_{k:f(x_{k})\neq\infty}(f^{*}\phi)_{k}\operatorname{ord}_{k}f+\sum_{k:f(x_{k})=\infty}(f^{*}\phi)_{k}\operatorname{ord}_{k}f+\sum_{k}(f^{*}\boldsymbol{\beta})_{k}\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|$
$\displaystyle+2\sum_{k:x_{k}\neq\infty,f(x_{k})=\infty}(f^{*}\phi)_{k}+\left.2\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|\right|_{k:x_{k}=\infty}-\left.2(f^{*}\phi)_{k}\right|_{k:x_{k}=\infty,f(\infty)\neq\infty},$

where $\left(f^{*}\boldsymbol{\beta}\right)_{k}$ and $(f^{*}\phi)_{k}$ are the
coefficients in the asymptotics (3.2) satisfying the equalities (2.8) and
(3.3) respectively.

###### Proof.

We begin with the equality

$\int_{\mathbb{C}}(f^{*}\phi)e^{2f^{*}\phi}\frac{dx\wedge d\bar{x}}{-2i}=\deg
f\cdot\int_{\mathbb{C}}\phi e^{2\phi}\frac{dz\wedge
d\bar{z}}{-2i}+\int_{\mathbb{C}}(\log|f^{\prime}|)e^{2f^{*}\phi}\frac{dx\wedge
d\bar{x}}{-2i}$

that easily follows from the relation $f^{*}\phi=\phi\circ
f+\log|f^{\prime}|$. Thus, in order to prove the lemma, we only need to
evaluate the last integral.

Thanks to the Liouville equation (2.6) we have

$\displaystyle(|\boldsymbol{\beta}|+2)\int_{\mathbb{C}}(\log|f^{\prime}|)e^{2f^{*}\phi}\frac{dx\wedge
d\bar{x}}{-2i}=\lim_{\epsilon\to
0}\frac{1}{2\pi}\int_{\mathbb{C}^{\epsilon}}(\log|f^{\prime}|)\bigl{(}-4\partial_{x}\partial_{\bar{x}}(f^{*}\phi)\bigr{)}\frac{dx\wedge
d\bar{x}}{-2i}$ (3.4) $\displaystyle=\lim_{\epsilon\to
0}\frac{1}{2\pi}\oint_{\partial\mathbb{C}^{\epsilon}}\Bigl{(}(f^{*}\phi)\partial_{\vec{n}}(\log|f^{\prime}|)-(\log|f^{\prime}|)\partial_{\vec{n}}(f^{*}\phi)\Bigr{)}\,|dx|,$

where $\mathbb{C}^{\epsilon}$ stands for the large disk $|x|>1/\epsilon$ with
the small disks $|x-x_{k}|<\epsilon$ encircling the marked points $x_{k}$
removed, and ${\vec{n}}$ is the unit outward normal. The last equality in
(3.4) is valid because $x\mapsto\log|f^{\prime}(x)|$ is a harmonic function on
$\mathbb{C}^{\epsilon}$.

Notice that if $x_{k}$ is not a pole of $f$, then we have

$\displaystyle\log|f^{\prime}(x)|=\operatorname{ord}_{k}f\log|x-x_{k}|+\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|+o(1),\
x\to x_{k}\neq\infty,$ (3.5)
$\displaystyle\log|f^{\prime}(x)|=-(\operatorname{ord}_{k}f+2)\log|x|+\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|+o(1),\
x\to x_{k}=\infty.$

Besides, if $x_{k}$ is a pole of $f$, we have

$\displaystyle\log|f^{\prime}(x)|=$
$\displaystyle-(\operatorname{ord}_{k}f+2)\log|x-x_{k}|$ (3.6)
$\displaystyle\qquad\qquad+\log|(\operatorname{ord}_{k}f+1)c_{k}|+o(1),\quad
x\to x_{k}\neq\infty,$ $\displaystyle\log|f^{\prime}(x)|=$
$\displaystyle\operatorname{ord}_{k}f\log|x|+\log|(\operatorname{ord}_{k}f+1)c_{k}|+o(1),\quad
x\to x_{k}=\infty.$

Below, for the derivatives along the unit outward normal vector ${\vec{n}}$ we
use the equalities

$\partial_{\vec{n}}=-\partial_{|x-x_{j}|}\quad\text{on}\quad|x-x_{j}|=\epsilon;\quad\partial_{\vec{n}}=\partial_{|x|}\quad\text{on}\quad|x|=1/\epsilon.$

We calculate the integral in right hand side of (3.4) relying on (3.2)
together with (3.5) and (3.6). As a result, we obtain

$\displaystyle\lim_{\epsilon\to
0}\frac{1}{2\pi}\oint_{\partial\mathbb{C}^{\epsilon}}\Bigl{(}(f^{*}\phi)\partial_{\vec{n}}(\log|f^{\prime}|)-(\log|f^{\prime}|)\partial_{\vec{n}}(f^{*}\phi)\Bigr{)}\,|dx|$
$\displaystyle=\sum_{j=1}^{n}\frac{1}{2\pi}\lim_{\epsilon\to
0}\oint_{|x-x_{j}|=\epsilon}\Bigl{(}(f^{*}\phi)\partial_{\vec{n}}(\log|f^{\prime}|)-(\log|f^{\prime}|)\partial_{\vec{n}}(f^{*}\phi)\Bigr{)}\,|dx|$
$\displaystyle=\sum_{k:x_{k}\neq\infty,f(x_{k})\neq\infty}\Bigl{(}-(f^{*}\phi)_{k}\operatorname{ord}_{k}f+(f^{*}\boldsymbol{\beta})_{k}\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|\Bigr{)}$
$\displaystyle+\sum_{k:x_{k}\neq\infty,f(x_{k})=\infty}\Bigl{(}(f^{*}\phi)_{k}(\operatorname{ord}_{k}f+2)+(f^{*}\boldsymbol{\beta})_{k}\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|\Bigr{)}$
$\displaystyle+\Bigl{(}-(f^{*}\phi)_{k}(\operatorname{ord}_{k}f+2)+(\log\left|(\operatorname{ord}_{k}f+1)c_{n}\right|)\bigl{(}(f^{*}\boldsymbol{\beta})_{k}+2\bigr{)}\Bigr{)}\Bigr{|}_{k:x_{k}=\infty,f(\infty)\neq\infty}$
$\displaystyle+\Bigl{(}(f^{*}\phi)_{k}\operatorname{ord}_{k}f+((f^{*}\boldsymbol{\beta})_{k}+2)\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|\Bigr{)}\Bigr{|}_{k:x_{k}=\infty,f(\infty)=\infty}.$

After regrouping the terms in the right hand side, this implies the assertion
of Lemma 3.2. ∎

######  Lemma 3.3.

For the determinant $\det\Delta_{f^{*}\boldsymbol{\beta}}$ of the Friedrichs
Laplacian on $(\overline{\mathbb{C}}_{x},f^{*}m_{\beta})$ the anomaly formula

$\displaystyle\log\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{\deg f}=-$
$\displaystyle\frac{|\beta|+2}{6}\int_{\mathbb{C}}f^{*}\phi
e^{2f^{*}\phi}\,\frac{dx\wedge
d\bar{x}}{-2i}+\frac{1}{6}\sum_{k:x_{k}\neq\infty}\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}$
(3.7)
$\displaystyle-\left.\frac{1}{6}\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}+2}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}\right|_{k:x_{x}=\infty}-\sum_{k}\mathcal{C}\bigl{(}\left(f^{*}\boldsymbol{\beta}\right)_{k}\bigr{)}+{\bf
C}$

is valid. Here $\mathcal{C}(\beta)$ is the function defined in (2.10), and
$\bf C$ stands for the constant (2.11).

###### Proof.

The main argument of the proof is similar to the one in [18, Proof of Prop.
2.1]. For this reason, we skip the details that can be easily restored from
the references [18, 19, 20].

Let us first obtain an asymptotics for the determinant of the Friedrichs
Dirichlet Laplacian
$\Delta^{D}_{f^{*}\boldsymbol{\beta}}\\!\\!\restriction_{|x|\leqslant
1/\epsilon}$ on the disk $|x|\leqslant 1/\epsilon$ endowed with the metric
$f^{*}m_{\boldsymbol{\beta}}$ as $\epsilon\to 0^{+}$. Denote by
$\Delta^{D}_{0}\\!\\!\restriction_{|x|\leqslant 1/\epsilon}$ the selfadjoint
Dirichlet Laplacian on the disk $|x|\leqslant 1/\epsilon$ equipped with the
flat background metric $|dx|^{2}$. As is known [56], for the determinant of
the Dirichlet Laplacian on the flat disk one has

$\log\det\Delta^{D}_{0}\\!\\!\restriction_{|x|\leqslant
1/\epsilon}=\frac{1}{3}\log\epsilon+\frac{1}{3}\log 2-\frac{1}{2}\log
2\pi-\frac{5}{12}-2\zeta_{R}^{\prime}(-1).$

On the other hand, the Polyakov-Alvarez type formula from [20, Theorem 1.1.2]
reads

$\displaystyle\log\frac{\det\Delta^{D}_{f^{*}\boldsymbol{\beta}}\\!\\!\restriction_{|x|\leqslant
1/\epsilon}}{\det\Delta^{D}_{0}\\!\\!\restriction_{|x|\leqslant 1/\epsilon}}=$
$\displaystyle-\frac{|f^{*}\boldsymbol{\beta}|+2}{6\deg
f}\int_{\mathbb{C}}(f^{*}\phi)e^{2f^{*}\phi}\frac{dx\wedge d\bar{x}}{-2i}$
(3.8)
$\displaystyle-\frac{1}{12\pi}\oint_{|x|=1/\epsilon}(f^{*}\phi)\partial_{\vec{n}}(f^{*}\phi)\,|dx|-\frac{\epsilon}{6\pi}\oint_{|x|=1/\epsilon}(f^{*}\phi)\,|dx|$
$\displaystyle-\frac{1}{4\pi}\oint_{|x|=1/\epsilon}\partial_{\vec{n}}(f^{*}\phi)\,|dx|+\frac{1}{6}\sum_{k:x_{k}\neq\infty}\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}$
$\displaystyle-\sum_{k:x_{k}\neq\infty}\mathcal{C}\bigl{(}\left(f^{*}\boldsymbol{\beta}\right)_{k}\bigr{)},$

where
$|f^{*}\boldsymbol{\beta}|=\sum_{k}\left(f^{*}\boldsymbol{\beta}\right)_{k}$
is the degree of the divisor
$f^{*}\boldsymbol{\beta}=\sum_{k}\left(f^{*}\boldsymbol{\beta}\right)_{k}\cdot
x_{k}$.

By the Gauss-Bonnet theorem [54], the product of the (regularized) Gaussian
curvature by the total area of the singular sphere
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$ equals
$2\pi(|f^{*}\boldsymbol{\beta}|+2)$. Since the (regularized) Gaussian
curvature is the one of $m_{\boldsymbol{\beta}}$, and the total area is $\deg
f$, we conclude that

$\frac{|f^{*}\boldsymbol{\beta}|+2}{\deg f}=|\boldsymbol{\beta}|+2.$

In (3.8) we also replace the integrals along the circle $|x|=1/\epsilon$ with
their asymptotics. As a result, the Polyakov-Alvarez type formula (3.8)
implies

$\displaystyle\log{\det\Delta^{D}_{f^{*}\boldsymbol{\beta}}\\!\\!\restriction_{|x|\leqslant
1/\epsilon}}=-\frac{|\boldsymbol{\beta}|+2}{6}\int_{\mathbb{C}}(f^{*}\phi)e^{2f^{*}\phi}\frac{dx\wedge
d\bar{x}}{-2i}+\frac{1}{6}\sum_{k:x_{k}\neq\infty}\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}$
$\displaystyle-\sum_{k:x_{k}\neq\infty}\mathcal{C}\bigl{(}\left(f^{*}\boldsymbol{\beta}\right)_{k}\bigr{)}+\frac{1}{6}\left(f^{*}\boldsymbol{\beta}\right)_{k}\left((f^{*}\phi)_{k}+3\right)|_{k:x_{k}=\infty}+\frac{1}{3}\log
2-\frac{1}{2}\log 2\pi+\frac{7}{12}$
$\displaystyle-2\zeta_{R}^{\prime}(-1)+\frac{1}{6}\left(\left(\left(f^{*}\boldsymbol{\beta}\right)_{k}+2\right)^{2}-2\left(\left(f^{*}\boldsymbol{\beta}\right)_{k}+1\right)\right)|_{k:x_{k}=\infty}\log\epsilon+o(1),\
\epsilon\to 0^{+}.$

In a similar way, one can also obtain the asymptotics for the determinant of
theFriedrichs Dirichlet Laplacian on the cap $|x|\geqslant 1/\epsilon$ of the
singular sphere $(\overline{\mathbb{C}}_{x},f^{*}m_{\beta})$:

$\displaystyle\log\det\Delta^{D}_{f^{*}\boldsymbol{\beta}}\\!\\!\restriction_{|x|\geqslant
1/\epsilon}=-\frac{1}{6}((\left(f^{*}\boldsymbol{\beta}\right)_{k}+1)^{2}+1)\log\epsilon-\frac{1}{6}\left(\left(f^{*}\boldsymbol{\beta}\right)_{k}+1+\frac{1}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}\right)(f^{*}\phi)_{k}$
$\displaystyle-\mathcal{C}\bigl{(}\left(f^{*}\boldsymbol{\beta}\right)_{k}\bigr{)}-2\zeta_{R}^{\prime}(-1)-\frac{5}{12}+\frac{1}{3}\log
2-\frac{1}{2}\log
2\pi-\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}}{2}+o(1),\quad\epsilon\to
0^{+},$

where $k$ is such that $x_{k}=\infty$. In total, we have

$\displaystyle\log{\det\Delta^{D}_{f^{*}\boldsymbol{\beta}}\\!\\!\restriction_{|x|\leqslant
1/\epsilon}}+\log\det\Delta^{D}_{f^{*}\boldsymbol{\beta}}\\!\\!\restriction_{|x|\geqslant
1/\epsilon}=-\frac{|\boldsymbol{\beta}|+2}{6}\int_{\mathbb{C}}(f^{*}\phi)e^{2f^{*}\phi}\frac{dx\wedge
d\bar{x}}{-2i}$ (3.9)
$\displaystyle+\frac{1}{6}\sum_{k:x_{k}\neq\infty}\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}-\frac{1}{6}\left(1+\frac{1}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}\right)(f^{*}\phi)_{k}$
$\displaystyle-\sum_{k}\mathcal{C}\bigl{(}\left(f^{*}\boldsymbol{\beta}\right)_{k}\bigr{)}+{\bf
C}+\log 2+o(1),\quad\epsilon\to 0^{+}.$

Now we cut the singular sphere
$\left(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}\right)$ along the
circle $|x|=1/\epsilon$ with the help of the BFK formula [7, Theorem B∗] to
obtain

$\displaystyle\log\det\Delta_{f^{*}\boldsymbol{\beta}}=$
$\displaystyle\log\deg f-\log 2$ (3.10) $\displaystyle+\lim_{\epsilon\to
0^{+}}\left(\log\det\Delta^{D}_{f^{*}\boldsymbol{\beta}}\\!\\!\restriction_{|x|\leqslant
1/\epsilon}+\log\det\Delta^{D}_{f^{*}\boldsymbol{\beta}}\\!\\!\restriction_{|x|\geqslant
1/\epsilon}\right).$

Here $\deg f$ is the total area of the singular sphere, and the number $-\log
2$ is the value of the difference

$\log\det\mathcal{N}(\epsilon)-\log\oint_{|x|=1/\epsilon}e^{f^{*}\phi}\,|dx|,$

where $\det\mathcal{N}(\epsilon)$ is the determinant of the Neumann jump
operator on the cut $|x|=1/\epsilon$, see [18, Proof of Prop. 2.1] or [19]. As
demonstrated in [20], the BFK formula (3.10) remains valid in spite of the
fact that the metric $f^{*}m_{\boldsymbol{\beta}}$ is singular.

In the limit, the asymptotics (3.9) together with the BFK formula (3.10)
implies the anomaly formula (3.7). This completes the proof of Lemma 3.3. ∎

###### Proof of Proposition 3.1.

Thanks to Lemma 3.2, the integral in the right-hand side of the anomaly
formula in Lemma 3.3 reduces to an integral on the target sphere
$(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$. Our next purpose is to
express that integral in terms of the determinant of Laplacian
$\Delta_{\boldsymbol{\beta}}$. With this aim in mind, we write down the
anomaly formula

$\displaystyle\log{\det\Delta_{\boldsymbol{\beta}}}=-\frac{|\boldsymbol{\beta}|+2}{6}\int_{\mathbb{C}}\phi
e^{2\phi}\,\frac{dz\wedge d\bar{z}}{-2i}+$
$\displaystyle\frac{1}{6}\sum_{j=0,1}\frac{\beta_{j}}{\beta_{j}+1}\phi_{j}$
(3.11)
$\displaystyle-\frac{1}{6}\frac{\beta_{\infty}+2}{\beta_{\infty}+1}\phi_{\infty}-\sum_{j\in\\{0,1,\infty\\}}\mathcal{C}(\beta_{j})+{\bf
C},$

which is (3.7) in the particular case of $f\equiv 1$. Now we change the index
of summation from $j\in\\{0,1,\infty\\}$ to $k=1,2,\dots,n$ as follows:

$\displaystyle\log{\det\Delta_{\boldsymbol{\beta}}}=-\frac{|\boldsymbol{\beta}|+2}{6}\int_{\mathbb{C}}\phi
e^{2\phi}\,\frac{dz\wedge
d\bar{z}}{-2i}+\frac{1}{6}\sum_{k:f(x_{k})\neq\infty}\frac{\operatorname{ord}_{k}f+1}{\deg
f}\frac{\beta_{f(x_{k})}}{\beta_{f(x_{k})}+1}\phi_{f(x_{k})}$ (3.12)
$\displaystyle-\sum_{k}\frac{\operatorname{ord}_{k}f+1}{\deg
f}\mathcal{C}(\beta_{f(x_{k})})-\frac{1}{6}\sum_{k:f(x_{k})=\infty}\frac{\operatorname{ord}_{k}f+1}{\deg
f}\frac{\beta_{f(x_{k})}+2}{\beta_{f(x_{k})}+1}\phi_{f(x_{k})}+{\bf C}.$

Here we rely on the obvious equalities

$\sum_{k:f(x_{k})=j}(\operatorname{ord}_{k}f+1)=\deg f\ \text{ for any fixed
}j\in\\{0,1,\infty\\}.$ (3.13)

The equality (3.12) together with the result of Lemma 3.2 allows us to express
the integral in the anomaly formula (3.7) in terms of
$\det\Delta_{\boldsymbol{\beta}}$ and the explicit uniformization data in
(2.3). In the remaining part of the proof, we show that as a result, we arrive
at the equality (2.9) with $C_{f}$ given by (3.1).

It is easy to see that proceeding as discussed above, we get the terms

$\deg
f\cdot\log\det\Delta_{\boldsymbol{\beta}}-\sum_{k}\left(\mathcal{C}\left(\left(f^{*}\boldsymbol{\beta}\right)_{k}\right)-(\operatorname{ord}_{k}f+1)\mathcal{C}(\beta_{f(x_{k})})\right)-(\deg
f-1){\bf C}$ (3.14)

in the right hand side of (2.9), we omit the details.

It is considerably harder to show that we also obtain the other terms in the
right-hand sides of (2.9) and (3.1). With this aim in mind, we separately
consider the following four possibilities:

  * •

$x_{k}\neq\infty$ is not a pole of $f$,

  * •

$x_{k}\neq\infty$ is a pole of $f$,

  * •

$x_{k}=\infty$ is a not pole of $f$, and

  * •

$x_{k}=\infty$ is a pole of $f$.

If $x_{k}\neq\infty$ is not a pole of $f$, then, in addition to the terms
listed in (3.14), we get $\frac{1}{6}$ times

$\displaystyle{\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}}-\Bigl{(}-(f^{*}\phi)_{k}\operatorname{ord}_{k}f+(f^{*}\boldsymbol{\beta})_{k}\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|\Bigr{)}$
$\displaystyle-{(\operatorname{ord}_{k}f+1)\frac{\beta_{f(x_{k})}}{\beta_{f(x_{k})}+1}\phi_{f(x_{k})}}$
$\displaystyle=\left(\operatorname{ord}_{k}f+1-\frac{1}{\operatorname{ord}_{k}f+1}\right)\frac{\phi_{f(x_{k})}}{\beta_{f(x_{k})}+1}-\left((f^{*}\boldsymbol{\beta})_{k}+1+\frac{1}{(f^{*}\boldsymbol{\beta})_{k}+1}\right)\log(\operatorname{ord}_{k}f+1)$
$\displaystyle+\left(\frac{\operatorname{ord}_{k}f}{\operatorname{ord}_{k}f+1}\log|c_{k}|+(\operatorname{ord}_{k}f+2)\log(\operatorname{ord}_{k}f+1)\right).$

Here the first two terms in the right-hand side contribute to the first and
the second sums in the right-hand side of (2.9) correspondingly. The last term
goes to $C_{f}$, cf. (3.1).

If $x_{k}\neq\infty$ is a pole of $f$, then we get the terms listed in (3.14)
and also $\frac{1}{6}$ times

$\displaystyle\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}-\Bigl{(}(f^{*}\phi)_{k}(\operatorname{ord}_{k}f+2)+(f^{*}\boldsymbol{\beta})_{k}\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|\Bigr{)}$
$\displaystyle+(\operatorname{ord}_{k}f+1)\frac{\beta_{f(x_{k})}+2}{\beta_{f(x_{k})}+1}\phi_{f(x_{k})}$
$\displaystyle=\left(\operatorname{ord}_{k}f+1-\frac{1}{\operatorname{ord}_{k}f+1}\right)\frac{\phi_{f(x_{k})}}{\beta_{f(x_{k})}+1}-\left((f^{*}\boldsymbol{\beta})_{k}+1+\frac{1}{(f^{*}\boldsymbol{\beta})_{k}+1}\right)\log(\operatorname{ord}_{k}f+1)$
$\displaystyle+\left(\frac{\operatorname{ord}_{k}f+2}{\operatorname{ord}_{k}f+1}\log|c_{k}|-\operatorname{ord}_{k}f\log(\operatorname{ord}_{k}f+1)\right).$

Here again, the first two terms in the right-hand side contribute to the first
and the second sums in the right-hand side of (2.9). The last term goes to
$C_{f}$, cf. (3.1).

Similarly, if $x_{k}=\infty$ is not a pole of $f$, we get the terms listed in
(3.14) and $\frac{1}{6}$ times

$\displaystyle-\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}+2}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}-\Bigl{(}-(f^{*}\phi)_{k}(\operatorname{ord}_{k}f+2)+((f^{*}\boldsymbol{\beta})_{k}+2)(\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|)\Bigr{)}$
$\displaystyle-(\operatorname{ord}_{k}f+1)\frac{\beta_{f(x_{k})}}{\beta_{f(x_{k})}+1}\phi_{f(x_{k})}$
$\displaystyle=\left(\operatorname{ord}_{k}f+1-\frac{1}{\operatorname{ord}_{k}f+1}\right)\frac{\beta_{f(x_{k})}}{\beta_{f(x_{k})}+1}\phi_{f(x_{k})}$
$\displaystyle-\left((f^{*}\boldsymbol{\beta})_{k}+1+\frac{1}{(f^{*}\boldsymbol{\beta})_{k}+1}\right)\log(\operatorname{ord}_{k}f+1)$
$\displaystyle+\left(-\frac{\operatorname{ord}_{k}f+2}{\operatorname{ord}_{k}f+1}\log|c_{k}|+\operatorname{ord}_{k}f\log(\operatorname{ord}_{k}f+1)\right).$

Here, as before, the first two terms in the right-hand side are in agreement
with (2.9), and the last term goes to $C_{f}$, cf. (3.1).

Finally, if $x_{k}=\infty$ is a pole of $f$, we get the terms listed in (3.14)
and $\frac{1}{6}$ times

$\displaystyle-\frac{\left(f^{*}\boldsymbol{\beta}\right)_{k}+2}{\left(f^{*}\boldsymbol{\beta}\right)_{k}+1}(f^{*}\phi)_{k}-\Bigl{(}(f^{*}\phi)_{k}\operatorname{ord}_{k}f+\bigl{(}(f^{*}\boldsymbol{\beta})_{k}+2\bigr{)}\log\left|(\operatorname{ord}_{k}f+1)c_{k}\right|\Bigr{)}$
$\displaystyle+(\operatorname{ord}_{k}f+1)\frac{\beta_{f(x_{k})}+2}{\beta_{f(x_{k})}+1}\phi_{f(x_{k})}$
$\displaystyle=\left(\operatorname{ord}_{k}f+1-\frac{1}{\operatorname{ord}_{k}f+1}\right)\frac{\phi_{f(x_{k})}}{\beta_{f(x_{k})}+1}-\left((f^{*}\boldsymbol{\beta})_{k}+1+\frac{1}{(f^{*}\boldsymbol{\beta})_{k}+1}\right)\log(\operatorname{ord}_{k}f+1)$
$\displaystyle+\left(-\frac{\operatorname{ord}_{k}f}{\operatorname{ord}_{k}f+1}\log|c_{k}|-(\operatorname{ord}_{k}f+2)\log(\operatorname{ord}_{k}f+1)\right),$

which is in agreement with (2.9) and (3.1). This completes the proof of
Theorem 3.1. ∎

###  3.2 Euclidean equilateral triangulation

By Proposition 3.1 the constant $C_{f}$ does not depend on the metric
$m_{\boldsymbol{\beta}}$. In this section, we make a particular choice of
$m_{\boldsymbol{\beta}}$ that significantly simplifies the calculation of the
constant $C_{f}$. As we show in the proof of Proposition 3.4 below, it makes
sense to consider the Euclidean equilateral triangulation naturally associated
with Belyi function $f$, cf. [45, 55].

######  Proposition 3.4.

The constant $C_{f}$ in (3.1) satisfies (2.12) with $A_{f}$ from (2.13); see
also Remark 3.5 at the end of this subsection.

###### Proof.

Here we consider the pull-back by $f$ of the flat singular metric

$m_{\boldsymbol{\beta}}=c^{2}|z|^{-4/3}|z-1|^{-4/3}|dz|^{2},\quad\boldsymbol{\beta}=\left(-\frac{2}{3}\right)\cdot
0+\left(-\frac{2}{3}\right)\cdot 1+\left(-\frac{2}{3}\right)\cdot\infty,$

where $c>0$ is a scaling coefficient that normalizes the area of the metric to
one. Note that the surface
$(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$ is isometric to two
congruent Euclidean equilateral triangles glued together along their sides
[54, 21]. In terms of [45], the corresponding bicolored double triangle in
Fig. 3 (with $\beta_{0}=\beta_{1}=\beta_{\infty}=-2/3$) is a “butterfly” that
puts the wings together to become $S^{2}$-like, see also [55].

Relying on anomaly formulae we can obtain expressions for the determinants of
Laplacians on the base $(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$
and on the ramified covering
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$. These determinants
satisfy the relation (2.9) from Theorem 3.1. As we show below, this leads to
the equalities (2.12) and (2.13) for $C_{f}$ and $A_{f}$ correspondingly, and
thus proves the assertion.

The potential $\phi$ of the metric $m_{\boldsymbol{\beta}}=e^{2\phi}|dz|^{2}$
has the asymptotics

$\phi(x)=-\frac{2}{3}\log|z|+\log c+o(1),\quad|z|\to
0;\quad\phi(x)=-\frac{2}{3}\log|z-1|+\log c+o(1),\quad|z|\to 1;$
$\phi(x)=-\frac{4}{3}\log|z|+\log c+o(1),\quad|z|\to\infty.$

This asymptotics is particularly simple, because for the coefficients
$\phi_{j}$ we have $\phi_{j}=\log c$ (instead of a cumbersome expression with
a lot of Gamma functions for $\phi_{j}$ in (2.3), see (5.2)), and the weights
$\beta_{j}$ of the marked points are $-2/3$. Moreover, thanks to our special
choice of $m_{\boldsymbol{\beta}}$, for the Laplacian
$\Delta_{\boldsymbol{\beta}}$ on
$(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$ the right hand side of
the anomaly formula (3.11) takes the following particularly simple form:

$\log{\det\Delta_{\boldsymbol{\beta}}}=-\frac{4}{3}\log
c-3\mathcal{C}\left(-\frac{2}{3}\right)+{\bf C}.$

This together with the relation (2.9) proved in Proposition 3.1 gives

$\displaystyle\log\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{\deg f}=\deg
f\cdot\left(-\frac{4}{3}\log c+{\bf C}\right)+\frac{1}{2}\left(3\deg
f-\sum_{k}\frac{1}{\operatorname{ord}_{k}f+1}\right){\log c}$ (3.15)
$\displaystyle-\frac{1}{6}\sum_{k}\left(\frac{\operatorname{ord}_{k}f+1}{3}+\frac{3}{\operatorname{ord}_{k}f+1}\right)\log(\operatorname{ord}_{k}f+1)$
$\displaystyle-\sum_{k}\mathcal{C}\left(\frac{\operatorname{ord}_{k}f-2}{3}\right)-(\deg
f-1){\bf C}+C_{f}.$

One can think of the surface
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$ as of the $\deg f$
copies of the flat bicolored double triangle glued along the edges in a way
prescribed by $f$. For the pull-back metric we obtain

$f^{*}m_{\boldsymbol{\beta}}=c^{2}|f(x)|^{-4/3}|f(x)-1|^{-4/3}|f^{\prime}(x)|^{2}\,|dx|^{2}.$
(3.16)

As is well known [54], the flat metric $f^{*}m_{\boldsymbol{\beta}}$ can
equivalently be written in the standard form

$f^{*}m_{\boldsymbol{\beta}}=c^{2}A_{f}^{2}\prod_{k:x_{k}\neq\infty}|x-x_{k}|^{\frac{2}{3}(\operatorname{ord}_{k}f-2)}\,|dx|^{2}.$
(3.17)

Now the representation (2.13) for the scaling coefficient $A_{f}$ is an
immediate consequence of the equalities (3.16) and (3.17).

Clearly, the metric potential $f^{*}\phi$ of
$f^{*}m_{\boldsymbol{\beta}}=e^{2f^{*}\phi}|dx|^{2}$ obeys the asymptotics

$\displaystyle(f^{*}\phi)(x)=$
$\displaystyle\frac{\operatorname{ord}_{k}f-2}{3}\log|x-x_{k}|+\log cA_{f}$
(3.18) $\displaystyle+\sum_{\ell:x_{k}\neq
x_{\ell}\neq\infty}\frac{\operatorname{ord}_{\ell}f-2}{3}\log|x_{k}-x_{\ell}|+o(1),\quad
x\to x_{k}\neq\infty,$

$(f^{*}\phi)(x)=-\left(\frac{\operatorname{ord}_{k}f-2}{3}+2\right)\log|x|+\log
cA_{f}+o(1),\quad x\to x_{k}=\infty.$ (3.19)

Therefore, for the Laplacian $\widetilde{\Delta}_{f^{*}\boldsymbol{\beta}}$
induced by the scaled flat metric

$(cA_{f})^{-2}\cdot
f^{*}m_{\boldsymbol{\beta}}=\prod_{k:x_{k}\neq\infty}|x-x_{k}|^{\frac{2}{3}(\operatorname{ord}_{j}f-2)}\,|dx|^{2}$

of total area $\deg f\cdot(cA_{f})^{-2}$, the anomaly formula from [20, Prop.
3.3] gives

$\displaystyle\log\frac{\det\widetilde{\Delta}_{f^{*}\boldsymbol{\beta}}}{\deg
f\cdot(cA_{f})^{-2}}=\frac{1}{18}\sum_{k:x_{k}\neq\infty}\sum_{\ \
\ell:x_{k}\neq
x_{\ell}\neq\infty}\frac{(\operatorname{ord}_{k}f-2)(\operatorname{ord}_{\ell}f-2)}{\operatorname{ord}_{k}f+1}\log|x_{k}-x_{\ell}|$
(3.20)
$\displaystyle-\sum_{k=1}^{n}\mathcal{C}\left(\frac{\operatorname{ord}_{k}f-2}{3}\right)+{\bf
C}.$

The standard rescaling property of the determinant reads

$\log\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{\deg
f}=\log\frac{\det\widetilde{\Delta}_{f^{*}\boldsymbol{\beta}}}{\deg
f\cdot(cA_{f})^{-2}}-2(\zeta(0)+1)\cdot\log cA_{f},$

where

$\zeta(0)=-\frac{1}{12}\sum_{k}\left(\frac{\operatorname{ord}_{k}f+1}{3}-\frac{3}{\operatorname{ord}_{k}f+1}\right)-1$

is the value of the spectral zeta function of
$\widetilde{\Delta}_{f^{*}\boldsymbol{\beta}}$ at zero; for details we refer
to [20, Section 1.2].

This together with (3.13) allows one to rewrite the equality (3.20) in the
form

$\displaystyle\log\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{\deg
f}=\frac{1}{18}\sum_{k:x_{k}\neq\infty}\sum_{\ \ \ell:x_{k}\neq
x_{\ell}\neq\infty}\frac{(\operatorname{ord}_{k}f-2)(\operatorname{ord}_{\ell}f-2)}{\operatorname{ord}_{k}f+1}\log|x_{k}-x_{\ell}|$
$\displaystyle+\frac{1}{6}\left(\deg
f-\sum_{k=1}^{n}\frac{3}{\operatorname{ord}_{k}f+1}\right)\log
cA_{f}-\sum_{k=1}^{n}\mathcal{C}\left(\frac{\operatorname{ord}_{k}f-2}{3}\right)+{\bf
C}.$

Substituting this into the left-hand side of (3.15), we finally arrive at
(2.12). This completes the proof. ∎

######  Remark 3.5.

Let $c_{k}$ be the coefficient of Taylor or Laurent series from Proposition
3.1. Let $A_{f}$ be the constant defined in (2.13). Then the asymptotics
(3.19) together with (3.2) and (3.3) implies

$A_{f}=\left\\{\begin{array}[]{cc}(\operatorname{ord}_{k}f+1)|c_{k}|^{1/3},&\text{
where }k\text{ is such that }x_{k}=\infty\text{ and }f(\infty)\neq\infty,\\\
(\operatorname{ord}_{k}f+1)|c_{k}|^{-1/3},&\text{ where }k\text{ is such that
}x_{k}=\infty\text{ and }f(\infty)=\infty.\end{array}\right.$

##  4 Uniformization, Liouville action, and stationary points of the
determinant

Consider a constant curvature sphere
$(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$ with conical singularities
of order $\beta_{k}$ located at $x_{k}\in\overline{\mathbb{C}}_{x}$. The
parameters $x_{1},\dots,x_{n}$ are called moduli. By the Gauss-Bonnet theorem
[54], the (regularized) Gaussian curvature $K$ of the singular sphere
$(\overline{\mathbb{C}}_{x},e^{2\phi}|dx|^{2})$ satisfies the equality
$K=2\pi(|\boldsymbol{\beta}|+2)/S_{\varphi}$, where
$|\boldsymbol{\beta}|=\sum_{k}\beta_{k}$ is the degree of the divisor
$\boldsymbol{\beta}=\sum\beta_{k}\cdot x_{k}$, and

$S_{\varphi}=\int_{\mathbb{C}}e^{2\varphi}\frac{dx\wedge d\bar{x}}{-2i}$ (4.1)

is the total surface area of
$(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$.

The potential $\varphi$ of the metric $e^{2\varphi}|dx|^{2}$ is a solution to
the Liouville equation

$e^{-2\varphi}(-4\partial_{x}\partial_{\bar{x}}\varphi)=K,\quad
x\in\mathbb{C}\setminus\\{x_{1},x_{2},\dots,x_{n}\\},$ (4.2)

having the following asymptotics

$\displaystyle\varphi(x)=\beta_{k}\log|x-x_{k}|+\varphi_{k}+o(1),\quad x\to
x_{k}\neq\infty,$ (4.3)
$\displaystyle\varphi(x)=-(\beta_{k}+2)\log|x|+\varphi_{k}+o(1),\quad x\to
x_{k}=\infty.$

The metric potential $\varphi$ and the coefficients $\varphi_{k}$ in the
asymptotics depend on the divisor $\boldsymbol{\beta}$, i.e. on the moduli
$x_{k}$ and the orders $\beta_{k}$ of the conical singularities.

Introduce the classical stress-energy tensor
$T_{\varphi}:=2(\partial_{x}^{2}\varphi-(\partial_{x}\varphi)^{2})$ of the
Liouville field theory. The stress-energy tensor is a meromorphic function on
$\overline{\mathbb{C}}$ satisfying

$\displaystyle T_{\varphi}(x)=$
$\displaystyle\sum_{k:x_{k}\neq\infty}\left(\frac{{s}_{k}}{2(x-x_{k})^{2}}+\frac{h_{k}}{x-x_{k}}\right),$
(4.4) $\displaystyle T_{\varphi}(x)=$
$\displaystyle\frac{{s}_{k}}{2x^{2}}+\frac{h_{k}}{x^{3}}+O(x^{-4})\text{ as
}x\to x_{k}=\infty.$

Here $s_{k}=-\beta_{k}(2+\beta_{k})$ are the weights of the second order
poles, and $h_{k}$ are the famous accessory parameters, e.g. [15, 29, 50] and
references therein. Note that the meromorphic quadratic differential
$T_{\varphi}dx^{2}$ is a uniformizing projective connection compatible with
the divisor $\boldsymbol{\beta}$ [54, 53].

Recall that one of the approaches to the uniformization consists of finding
appropriate values of the accessory parameters $h_{k}$, and two appropriately
normalized linearly independent solutions $u_{1}$ and $u_{2}$ to the Fuchsian
differential equation

$\partial_{x}^{2}u+\frac{1}{2}T_{\varphi}u=0.$

Then the metric potential $\varphi$ can be found in the form

$\varphi=\log 2+\log|\partial_{x}w|-\log(1+K|w|^{2}),$

where $w=u_{1}/u_{2}$ is an analytic in
$\mathbb{C}\setminus\\{x_{1},\dots,x_{n}\\}$ function called the developing
map. It satisfies the Schwarzian differential equation
$\\{w,x\\}=T_{\varphi}(x)$, where
$\\{w,x\\}=\frac{2w^{\prime}w^{\prime\prime\prime}-3w^{\prime\prime
2}}{2w^{\prime 2}}$ is the Schwarzian derivative. However, the accessory
parameters can be determined in some special cases only, and, in general, they
remain elusive.

It turns out that in the geometric setting of this paper, the accessory
parameters can be found explicitly in terms of the Belyi function $f$ and the
orders $\beta_{0}$, $\beta_{1}$, and $\beta_{\infty}$ of the conical
singularities of the metric $m_{\boldsymbol{\beta}}=e^{2\phi}|dz|^{2}$.
Indeed, consider the constant curvature unit area singular sphere
$(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}})$, see Sec. 2.1. For the
corresponding stress-energy tensor we have

$T_{\phi}(z)=\frac{\mathfrak{s}_{0}}{2z^{2}}+\frac{\mathfrak{h}_{0}}{z}+\frac{\mathfrak{s}_{1}}{2(z-1)^{2}}+\frac{\mathfrak{h}_{1}}{z-1},\quad
T_{\phi}(z)=\frac{\mathfrak{s}_{\infty}}{2z^{2}}+\frac{\mathfrak{h}_{\infty}}{z^{3}}+O(z^{-4})\quad\text{as}\quad
z\to\infty,$

where

$\mathfrak{s}_{k}=-\beta_{k}(2+\beta_{k}),\quad k\in\\{0,1,\infty\\}.$ (4.5)

The accessory parameters

$\mathfrak{h}_{0}=-\mathfrak{h}_{1}=\frac{{\mathfrak{s}}_{0}+{\mathfrak{s}}_{1}-{\mathfrak{s}}_{\infty}}{2},\quad\mathfrak{h}_{\infty}=\frac{{\mathfrak{s}}_{1}+{\mathfrak{s}}_{\infty}-{\mathfrak{s}}_{0}}{2}$
(4.6)

were first found by Schwarz. The stress-energy tensors satisfy the relation

$T_{f^{*}\phi}=(T_{\phi}\circ f)(f^{\prime})^{2}+\\{f,x\\}.$ (4.7)

As a consequence, we obtain the following simple result:

######  Lemma 4.1 (Accessory parameters).

Let $f:\overline{\mathbb{C}}_{x}\to\overline{\mathbb{C}}_{z}$ be a Belyi
function unramified outside of the set $\\{0,1,\infty\\}$ and such that
$f(\infty)\in\\{0,1,\infty\\}$. Then the stress-energy tensor $T_{f^{*}\phi}$
of the pull back metric $f^{*}m_{\beta}=e^{f^{*}\phi}|dx|^{2}$ of $m_{\beta}$
by $f$ satisfies the relations (4.4), where $x_{1},\dots,x_{n}$ with $n=\deg
f+2$ are the preimages of the points
$\\{0,1,\infty\\}\subset\overline{\mathbb{C}}_{z}$ under $f$, the weights of
the second order poles in (4.4) are given by the equalities

$s_{k}=-(f^{*}\boldsymbol{\beta})_{k}((f^{*}\boldsymbol{\beta})_{k}+2)$

with orders $(f^{*}\boldsymbol{\beta})_{k}$ from (2.8), and the accessory
parameters $h_{k}$ can be found as follows:

1\. If $x_{k}$ is not a pole of $f$, then the accessory parameter $h_{k}$
satisfies

$h_{k}=\left\\{\begin{array}[]{cc}\frac{d_{k}}{c_{k}}{\mathfrak{s}}_{f(x_{k})}+c_{k}\mathfrak{h}_{f(x_{k})}&\text{
for }\quad\operatorname{ord}_{k}f=0,\\\
-\frac{d_{k}}{c_{k}}\frac{(f^{*}\beta)_{k}((f^{*}\beta)_{k}+2)}{\operatorname{ord}_{k}f+1}&\text{
for }\quad\operatorname{ord}_{k}f>0,\end{array}\right.$ (4.8)

where $\mathfrak{s}_{k}$ and $\mathfrak{h}_{k}$ are the same as in (4.5) and
(4.6), and the coefficients $c_{k}$ and $d_{k}$ are those from the first or
the second expansion

$\displaystyle f(x)-f(x_{k})$
$\displaystyle=c_{k}x^{-\operatorname{ord}_{k}f-1}\left(1+\frac{d_{k}}{c_{k}}\frac{1}{x}+O(x^{-2})\right),\quad
x\to x_{k}=\infty,$ $\displaystyle f(x)-f(x_{k})$
$\displaystyle=c_{k}(x-x_{k})^{\operatorname{ord}_{k}f+1}\left(1+\frac{d_{k}}{c_{k}}(x-x_{k})+O\left((x-x_{k})^{2}\right)\right),\quad
x\to x_{k}\neq\infty,$

depending on whether $x_{k}=\infty$ or $x_{k}\neq\infty$,

2\. If $x_{k}$ is a pole of $f$, then the accessory parameter $h_{k}$
satisfies

$h_{k}=\left\\{\begin{array}[]{cc}-\frac{d_{k}}{c_{k}}{\mathfrak{s}}_{\infty}+\frac{1}{c_{k}}\mathfrak{h}_{\infty}&\text{
for }\quad\operatorname{ord}_{k}f=0,\\\
\frac{d_{k}}{c_{k}}\frac{(f^{*}\beta)_{k}((f^{*}\beta)_{k}+2)}{\operatorname{ord}_{k}f+1}&\text{
for }\quad\operatorname{ord}_{k}f>0,\end{array}\right.$ (4.9)

where $\mathfrak{s}_{\infty}$ and $\mathfrak{h}_{\infty}$ are the same in
(4.5) and (4.6), and the coefficients $c_{k}$ and $d_{k}$ are those from the
first or the second expansion

$\displaystyle f(x)$
$\displaystyle=c_{k}x^{\operatorname{ord}_{k}f+1}\left(1+\frac{d_{k}}{c_{k}}\frac{1}{x}+O(x^{-2})\right),\quad
x\to x_{k}=\infty,$ $\displaystyle f(x)$
$\displaystyle={c_{k}}{(x-x_{k})^{-\operatorname{ord}_{k}f-1}}\left(1+\frac{d_{k}}{c_{k}}(x-x_{k})+O((x-x_{k})^{2})\right),\quad
x\to x_{k}\neq\infty,$

depending on whether $x_{k}=\infty$ or $x_{k}\neq\infty$.

###### Proof.

If $f(x_{k})\neq\infty$ and $x_{k}=\infty$, then

$f(x)-f(x_{k})=c_{k}x^{-\operatorname{ord}_{k}f-1}\left(1+\frac{d_{k}}{c_{k}}\frac{1}{x}+O(x^{-2})\right),\quad
x\to x_{k}=\infty,$

and the asymptotics can be differentiated. As a consequence, for the
contributions into

$(T_{\phi}\circ
f)(f^{\prime})^{2}(x)=\left(\frac{-\beta_{f(x_{k})}(2+\beta_{f(x_{k})})}{2(f(x)-f(x_{k}))^{2}}+\frac{\mathfrak{h}_{f(x_{k})}}{f(x)-f(x_{k})}+O(1)\right)(f^{\prime}(x))^{2}$

we obtain

$\frac{(f^{\prime}(x))^{2}}{(f(x)-f(x_{k}))^{2}}=\frac{(\operatorname{ord}_{k}f+1)^{2}}{x^{2}}+2\frac{d_{k}}{c_{k}}\frac{\operatorname{ord}_{k}f+1}{x^{3}}+O(x^{-4}),$
$\frac{h_{f(x_{k})}}{f(x)-f(x_{k})}(f^{\prime}(x))^{2}=h_{f(x_{k})}{c_{k}}(\operatorname{ord}_{k}f+1)^{2}x^{-\operatorname{ord}_{k}f-3}+O(x^{-4}).$

Besides, for the Schwarzian derivative, we get

$\\{f,x\\}=-\frac{1}{2}(\operatorname{ord}_{k}f)(\operatorname{ord}_{k}f+2)\left(\frac{1}{x^{2}}+\frac{2d_{k}}{c_{k}(\operatorname{ord}_{k}f+1)}\frac{1}{x^{3}}+O(x^{-4})\right),\quad
x\to x_{k}=\infty.$

These together with (4.7) imply

$T_{f^{*}\phi}(x)=-\frac{(f^{*}\beta)_{k}((f^{*}\beta)_{k}+2)}{2x^{2}}+\frac{h_{k}}{x^{3}}+O(x^{-4})\quad\text{as}\quad
x\to x_{k}=\infty,$

where the accessory parameter $h_{k}$ satisfies (4.8).

Similarly, if $f(x_{k})=\infty$ and $x_{k}=\infty$, then starting with the
asymptotics

$f(x)=c_{k}x^{\operatorname{ord}_{k}f+1}\left(1+\frac{d_{k}}{c_{k}}\frac{1}{x}+O(x^{-2})\right),\quad
x\to x_{k}=\infty,$

we arrive at

$T_{f^{*}\phi}(x)=-\frac{(f^{*}\beta)_{k}((f^{*}\beta)_{k}+2)}{2x^{2}}+\frac{h_{k}}{x^{3}}+O(x^{-4})\quad\text{as}\quad
x\to x_{k}=\infty,$

where the accessory parameter $h_{k}$ satisfies (4.9).

The cases $f(x_{k})=\infty$ with $x_{k}\neq\infty$, and $f(x_{k})\neq\infty$
with $x_{k}\neq\infty$ are similar, we omit the details. ∎

######  Remark 4.2.

In particular, by using Lemma 4.1 with $\boldsymbol{\beta}=\beta_{0}\cdot
0+\beta_{1}\cdot 1+\left(-\frac{2}{3}\right)\cdot\infty$ and $f(x)=x^{3}$, we
recover the values of the accessory parameters found in [29, Sec. 4.13] for a
constant curvature Hyperbolic or Euclidean metric with four conical
singularities located at the vertices of a regular tetrahedron.

Introduce the Liouville action

$\mathcal{S}_{\boldsymbol{\beta}}[\varphi]=2\pi(|\boldsymbol{\beta}|+2)\left(\frac{1}{S_{\varphi}}\int_{\mathbb{C}}\varphi
e^{2\varphi}\frac{dx\wedge
d\bar{x}}{-2i}-1\right)+2\pi\sum_{k}\beta_{k}\varphi_{k}+4\pi\varphi_{k}|_{k:x_{k}=\infty},$
(4.10)

where $S_{\varphi}$ is the total area of the singular sphere
$(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$, and $\varphi_{k}$ are the
coefficients in the asymptotics (4.3). As is demonstrated in [18], this new
definition of the Liouville action is in agreement, for instance, with that in
[8, 59, 50]. It is not hard to show that the Liouville equation (4.2) is the
Euler-Lagrange equation for the Liouville action functional
$\psi\mapsto\mathcal{S}_{\boldsymbol{\beta}}[\psi]$.

######  Remark 4.3.

In the geometric setting of this paper we have $\varphi=f^{*}\phi$, and the
anomaly formula from Lemma 3.3 can equivalently be written as follows:

$\log\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{\deg
f}=\frac{|f^{*}\boldsymbol{\beta}|+2}{6}-\frac{1}{12\pi}\left(\mathcal{S}_{f^{*}\boldsymbol{\beta}}[f^{*}\phi]-\pi\log\mathcal{H}_{f^{*}\boldsymbol{\beta}}[f^{*}\phi]\right)-\sum_{k}\mathcal{C}((f^{*}\beta)_{k})+{\bf
C}.$

Here the functional $\mathcal{H}_{f^{*}\boldsymbol{\beta}}[f^{*}\phi]$ is
defined explicitly via the equality

$\mathcal{H}_{f^{*}\boldsymbol{\beta}}[f^{*}\phi]=\exp\left(2\sum_{k}\frac{(f^{*}\beta)_{k}((f^{*}\beta)_{k}+2)}{(f^{*}\beta)_{k}+1}(f^{*}\phi)_{k}\right)$

with $(f^{*}\beta)_{k}$ from (2.8) and $(f^{*}\phi)_{k}$ from (3.3). Thus, as
a consequence of the explicit expression for the determinant of Laplacian in
Theorem 2.1 and the anomaly formula in Lemma 3.3, one also obtains an explicit
expression for the Liouville action
$\mathcal{S}_{f^{*}\boldsymbol{\beta}}[f^{*}\phi]$, cf. [36, 37, 49]. It would
be interesting to check if this result can be reproduced by using conformal
blocks [59].

In the remaining part of this section for simplicity, we assume that the
orders $\beta_{k}$ of the conical singularities meet the condition
$|\boldsymbol{\beta}|\leqslant-2$, i.e. we exclude form consideration the
spherical metrics. This allows us to differentiate the hyperbolic metric
potential and the corresponding Liouville action with respect to $x_{k}$ and
$\beta_{k}$ relying on known regularity results, e.g. [24, 29, 42, 43, 50]. In
the Euclidean case, we have $|\boldsymbol{\beta}|=-2$, and the metrics can be
written explicitly [52], which immediately justifies the differentiation. Let
us also note that there are good grounds to believe [16, 17, 18, 24, 53] that
the potential $\varphi$ of a constant curvature metric is necessarily a real-
analytic function of the orders of conical singularities on the existence and
uniqueness set

$\\{\beta_{k}\in(0,1):\beta_{k}-|\boldsymbol{\beta}|/2>0,k=1,\dots,n\\},$

and the results below remain valid on that set.

Next, we show that the Liouville action
$\mathcal{S}_{\boldsymbol{\beta}}[\varphi]$ generates the accessory parameters
$h_{k}$ as their common antiderivative.

######  Lemma 4.4 (After P. Zograf and L. Takhtajan).

Assume that $\beta_{k}\in(0,1)$, $k=1,\dots,n$, and
$|\boldsymbol{\beta}|\leqslant-2$. Let $\varphi$ be a (unique) solution to the
Liouville equation (4.2) satisfying the area condition (4.1) with some fixed
$S_{\varphi}>0$, and having the asymptotics (4.3). Then the Liouville action
(4.10) meets the equalities

$-\frac{1}{2\pi}{\partial_{x_{k}}\mathcal{S}_{\boldsymbol{\beta}}[\varphi]}=h_{k},\quad
k=1,\dots,n,$ (4.11)

where $x_{1},x_{2},\dots,x_{n}$ are the moduli, and $h_{1},h_{2},\dots,h_{n}$
are the accessory parameters.

Note that in the geometric setting of this paper, we have $\varphi=f^{*}\phi$.
Thus the moduli $x_{k}$, $k=1,\dots,\deg f+2$, are the preimages of the points
$\\{0,1,\infty\\}$ under $f$, and the accessory parameters $h_{k}$ are those
found in Lemma 4.1.

###### Proof.

As is shown in [18], the expression in the right hand side of (4.10) is an
equivalent regularization of the Liouville action introduced in [8, 59, 50].
Hence, in the case of $|\boldsymbol{\beta}|<-2$ and $K=-1$ the assertion of
the lemma is just a reformulation of the result proven in [8, 50], see also
[60] for the first proof of Polyakov’s conjecture (4.11). Next we show that in
the (hyperbolic) case $|\boldsymbol{\beta}|<-2$ the assertion remains valid
for any $S_{\varphi}>0$ and $K=2\pi(|\boldsymbol{\beta}|+2)/S_{\varphi}<0$.

Consider a (unique) metric potential $\varphi$ such that
$|\boldsymbol{\beta}|<-2$ and $K=-1$. Clearly,
$S_{\varphi}=-2\pi(|\boldsymbol{\beta}|+2)$, and for any $C>0$ we have the
following transformation laws for the surface area, the Liouville action, and
the stress-energy tensor:

$S_{\varphi+\log
C}=C^{2}S_{\varphi},\quad\mathcal{S}_{\boldsymbol{\beta}}[\varphi+\log
C]=\mathcal{S}_{\boldsymbol{\beta}}[\varphi]+4\pi(|\boldsymbol{\beta}|+2)\log
C,\quad T_{\varphi+\log C}=T_{\varphi}.$

This implies that the rescaling $\varphi\mapsto\varphi+\log C$ multiplies the
total area by $C^{2}$, but does not affect the equalities (4.11). Thus, in the
case $|\boldsymbol{\beta}|<-2$, the assertion of lemma is valid for any fixed
$S_{\varphi}>0$.

In the case $|\boldsymbol{\beta}|=-2$ the integral term in (4.10) disappears
and the metric $e^{2\varphi}|dx|^{2}$ is flat. As is known [52], up to a
rescaling $\varphi\mapsto\varphi+\log C$, the potential $\varphi$ can be
written explicitly in the form

$\varphi(\boldsymbol{\beta})=\sum_{k:x_{k}\neq\infty}\beta_{k}\log|x-x_{k}|,\quad|\boldsymbol{\beta}|=-2.$
(4.12)

As a result, the equality (4.11) follows from a simple direct computation. ∎

It may also be possible to prove Polyakov’s conjecture (4.11) for the
spherical case $|\boldsymbol{\beta}|>-2$ along the lines of [8, 50], however
this goes out of the scope of this paper.

######  Lemma 4.5 (After A. Zamolodchikov and Al. Zamolodchikov).

Assume that $\beta_{k}\in(-1,0)$ and $|\boldsymbol{\beta}|\leqslant-2$. Let
$\varphi$ stand for a (unique) solution to the Liouville equation (4.2)
satisfying the area condition (4.1) with a fixed $S_{\varphi}>0$, and having
the asymptotics (4.3). Then for any fixed configuration $x_{1},\dots,x_{n}$
the Liouville action (4.10) satisfies

$-\frac{1}{2\pi}{\partial_{\beta_{k}}\mathcal{S}_{\boldsymbol{\beta}}[\varphi]}=1-2\varphi_{k},$
(4.13)

where $\varphi_{k}$ is the coefficient in the corresponding asymptotics (4.3).

In the geometric setting of this paper, we have $\varphi=f^{*}\phi$ and
$\varphi_{k}=(f^{*}\phi)_{k}$, see (3.3).

###### Proof.

In the case $|\boldsymbol{\beta}|<-2$, the proof essentially repeats the one
in [18, Proof of Lemma 3.1], where the differentiation with respect to
$\beta_{k}$ is now justified by the results [24, 42, 43] on the regularity of
$\beta_{k}\mapsto\varphi(\boldsymbol{\beta})$ for the hyperbolic metric
$e^{2\varphi}|dx|^{2}$ on the Riemann sphere. Indeed, one need only notice
that the index $j=k$ in [18, Proof of Lemma 3.1] now runs from $1$ to $n$, and
the region $\mathbb{C}_{R}$ is defined as follows:

$\mathbb{C}_{R}:=\\{x\in\mathbb{C}:|x|\leqslant R,\ |x-x_{k}|\geqslant 1/R,\
k=1,\dots n\\}.$

In the case $|\boldsymbol{\beta}|=-2$ the integral term in (4.10) disappears,
and the equality (4.13) can be verified by a direct computation, cf. (4.12).
We omit the details. ∎

Our choice of examples in Section 5 is partially motivated by the following
result:

######  Theorem 4.6.

The (hyperbolic or flat) surfaces of five Platonic solids and the regular
constant curvature dihedra are critical points of the spectral determinant on
the conical metrics of fixed area and fixed Gaussian curvature.

More precisely: Consider the divisors
$\boldsymbol{\beta}=\sum_{k}\beta_{k}\cdot x_{k}$ of degree
$|\boldsymbol{\beta}|\leqslant-2$ with distinct marked points
$x_{1},\dots,x_{n}$ and weights $\beta_{k}\in(-1,0)$. Then for any fixed $S>0$
and any divisor $\boldsymbol{\beta}$ there exists a unique metric
$e^{2\varphi}|dx|^{2}$ on $\overline{\mathbb{C}}$ of total area $S$, Gaussian
curvature $K=2\pi(|\boldsymbol{\beta}|+2)/S$, and representing the divisor
$\boldsymbol{\beta}$. Consider the spectral determinant
$\det\Delta_{\boldsymbol{\beta}}$ of the surface
$(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$ as a function on the
configuration space

$\mathcal{Z}_{n}(S,K)=\left\\{\boldsymbol{\beta}=\sum_{k\leqslant
n}\beta_{k}\cdot x_{k}:x_{j}\neq x_{k}\in\overline{\mathbb{C}}\text{ for
}j\neq k,\beta_{k}\in(-1,0),2\pi(|\boldsymbol{\beta}|+2)=SK\right\\}$

with some fixed values $S>0$, $K\leqslant 0$, and $n\geqslant 3$.

  1. 1.

If $\boldsymbol{\beta}_{0}\in\mathcal{Z}_{n}(S,K)$ is a divisor such that the
corresponding surface $(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$ is
isometric to the one of a Platonic solid, then $\boldsymbol{\beta}_{0}$ is a
stationary point of the function

$\mathcal{Z}_{n}(S,K)\ni\boldsymbol{\beta}\mapsto\det\Delta_{\boldsymbol{\beta}},$
(4.14)

where $n$ is the number of vertices of the Platonic solid.

  2. 2.

If $\boldsymbol{\beta}_{0}\in\mathcal{Z}_{n}(S,K)$ is a divisor such that the
corresponding surface $(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$ is
isometric to the regular dihedron with $n$ vertices, then
$\boldsymbol{\beta}_{0}$ is a stationary point of the function (4.14) with the
corresponding value of $n$.

###### Proof.

Consider the potential $\varphi(x;\beta_{1},\beta_{2},\beta_{3},\beta_{4})$ of
a (unique) unit area constant curvature metric $e^{2\varphi}|dx|^{2}$
representing the divisor

$\boldsymbol{\beta}=\beta_{1}\cdot 0+\beta_{2}\cdot(-1)+\beta_{3}\cdot
e^{i\frac{\pi}{3}}+\beta_{4}\cdot e^{-i\frac{\pi}{3}}.$

Recall that the Gauss-Bonnet theorem [54] implies that the (regularized)
Gaussian curvature of the surface
$(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$ equals
$2\pi(|\boldsymbol{\beta}|+2)$. The four marked points in the divisor
$\boldsymbol{\beta}$ are in an equi-anharmonic position. In particular, if the
orders of the conical singularities satisfy
$\beta_{k}=|\boldsymbol{\beta}|/4$, then the surface
$(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$ is isometric to the one of
a unit area regular tetrahedron of Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+2)\leqslant 0$.

Notice that $\varphi(\bar{x};\beta_{1},\beta_{2},\beta_{3},\beta_{4})$ is the
potential of a (unique) unit area Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+2)$ metric representing the divisor

$\boldsymbol{\beta}=\beta_{1}\cdot 0+\beta_{2}\cdot(-1)+\beta_{4}\cdot
e^{i\frac{\pi}{3}}+\beta_{3}\cdot e^{-i\frac{\pi}{3}}.$

Similarly, the potential
$\varphi(e^{i\frac{2\pi}{3}}x;\beta_{1},\beta_{2},\beta_{3},\beta_{4})$
corresponds to the divisor

$\boldsymbol{\beta}=\beta_{1}\cdot 0+\beta_{4}\cdot(-1)+\beta_{2}\cdot
e^{i\frac{\pi}{3}}+\beta_{3}\cdot e^{-i\frac{\pi}{3}};$

$\varphi(e^{-i\frac{2\pi}{3}}x;\beta_{1},\beta_{2},\beta_{3},\beta_{4})$
corresponds to the divisor

$\boldsymbol{\beta}=\beta_{1}\cdot 0+\beta_{3}\cdot(-1)+\beta_{4}\cdot
e^{i\frac{\pi}{3}}+\beta_{2}\cdot e^{-i\frac{\pi}{3}};$

and $\varphi(\frac{x+1}{2x-1};\beta_{1},\beta_{2},\beta_{3},\beta_{4})+\log
3-2\log|2x-1|$ corresponds to the divisor

$\boldsymbol{\beta}=\beta_{2}\cdot 0+\beta_{1}\cdot(-1)+\beta_{4}\cdot
e^{i\frac{\pi}{3}}+\beta_{3}\cdot e^{-i\frac{\pi}{3}}.$

As a consequence of these symmetries, we have

$\varphi(x;\beta_{1},\beta_{2},\beta_{3},\beta_{4})=\varphi(\bar{x};\beta_{1},\beta_{2},\beta_{4},\beta_{3})=\varphi(e^{i\frac{2\pi}{3}}x;\beta_{1},\beta_{3},\beta_{4},\beta_{2})=\varphi(e^{-i\frac{2\pi}{3}}x;\beta_{1},\beta_{4},\beta_{2},\beta_{3})$
$=\varphi\left(\frac{x+1}{2x-1};\beta_{2},\beta_{1},\beta_{4},\beta_{3}\right)+\log
3-2\log|2x-1|.$

For the coefficients
$\varphi_{k}=\varphi_{k}(\beta_{1},\beta_{2},\beta_{3},\beta_{4})$ in the
asymptotics (4.3) the latter equalities imply

$\displaystyle(\varphi_{1}-\varphi_{\ell})|_{\beta_{k}=|\boldsymbol{\beta}|/4}=(|\boldsymbol{\beta}|/4+1)\log
3,\quad\ell=2,3,4,$ (4.15)
$\displaystyle\sum_{j=1}^{4}(\partial_{\beta_{1}}\varphi_{j}-\partial_{\beta_{\ell}}\varphi_{j})|_{\beta_{k}=|\boldsymbol{\beta}|/4}=\log
3,\quad\ell=2,3,4.$

Denote the Friedrichs Laplacian on
$(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$ by
$\Delta_{\boldsymbol{\beta}}$. As is proven in [18, Sec. 2], the spectral
determinant $\det\Delta_{\boldsymbol{\beta}}$ satisfies the anomaly formula

$\log\det\Delta_{\boldsymbol{\beta}}=-\frac{|\boldsymbol{\beta}|+1}{6}-\frac{1}{12\pi}\left(\mathcal{S}_{\boldsymbol{\beta}}[\varphi]-\pi\log\mathcal{H}_{\boldsymbol{\beta}}[\varphi]\right)-\sum_{k=1}^{4}\mathcal{C}(\beta_{k})+\bf
C,$ (4.16)

where $\mathcal{S}_{\boldsymbol{\beta}}[\varphi]$ is the Liouville action
(4.10) and

$\mathcal{H}_{\boldsymbol{\beta}}[\varphi]=\exp\left\\{2\sum_{k=1}^{4}\left(\beta_{k}+1-\frac{1}{\beta_{k}+1}\right)\varphi_{k}\right\\}.$
(4.17)

Since $|\boldsymbol{\beta}|$ is fixed, we can set, for example,
$\beta_{1}=|\boldsymbol{\beta}|-\sum_{k>1}\beta_{k}$, and consider the
determinant $\det\Delta_{\boldsymbol{\beta}}$ as a function of
$(\beta_{2},\beta_{3},\beta_{4})$. Then, thanks to the anomaly formula (4.16),
Lemma 4.5, and the equality (4.17), we have

$\displaystyle\partial_{\beta_{\ell}}\left(\log\det\Delta_{\boldsymbol{\beta}}|_{\beta_{1}=|\boldsymbol{\beta}|-\sum_{k>1}\beta_{k}}\right)|_{\beta_{k}=\frac{|\boldsymbol{\beta}|}{4}}$
$\displaystyle=\frac{1}{3}(\varphi_{1}-\varphi_{\ell})|_{\beta_{k}=\frac{|\boldsymbol{\beta}|}{4}}$
$\displaystyle-\frac{1}{6}\left(1+\left(\frac{|\boldsymbol{\beta}|}{4}+1\right)^{-2}\right)(\varphi_{1}-\varphi_{\ell})|_{\beta_{k}=\frac{|\boldsymbol{\beta}|}{4}}$
$\displaystyle-\frac{1}{6}\left(\frac{|\boldsymbol{\beta}|}{4}+1-\frac{1}{\frac{|\boldsymbol{\beta}|}{4}+1}\right)\sum_{j=1}^{4}(\partial_{\beta_{1}}\varphi_{j}-\partial_{\beta_{\ell}}\varphi_{j})|_{\beta_{k}=\frac{|\boldsymbol{\beta}|}{4}}.$

Here the right-hand side is equal to zero because of (4.15).

Now we are in a position to study the determinant under a small perturbation
of the coordinate of a vertex. Let us consider the potential $\varphi(x)$ of a
(unique) unit area constant curvature metric $e^{2\varphi}|dx|^{2}$
representing the divisor

$\boldsymbol{\beta}=\frac{|\boldsymbol{\beta}|}{4}\cdot
h+\frac{|\boldsymbol{\beta}|}{4}\cdot(-1)+\frac{|\boldsymbol{\beta}|}{4}\cdot
e^{i\frac{\pi}{3}}+\frac{|\boldsymbol{\beta}|}{4}\cdot e^{-i\frac{\pi}{3}}.$

Here $h$ is a small complex number. In the case $h=0$ the surface
$(\overline{\mathbb{C}}_{x},e^{2\varphi}|dx|^{2})$ is isometric to the one of
a unit area constant curvature regular tetrahedron.

Consider, for example, the rotation $x\mapsto e^{i\frac{2\pi}{3}}x$. Notice
that $\chi(x):=\varphi(e^{i\frac{2\pi}{3}}x)$ is the potential of a (unique)
unit area Gaussian curvature $2\pi(|\boldsymbol{\beta}|+2)$ metric
representing the divisor

${\boldsymbol{\gamma}}=\frac{|\boldsymbol{\beta}|}{4}\cdot(e^{-i\frac{2\pi}{3}}h)+\frac{|\boldsymbol{\beta}|}{4}\cdot(-1)+\frac{|\boldsymbol{\beta}|}{4}\cdot
e^{i\frac{\pi}{3}}+\frac{|\boldsymbol{\beta}|}{4}\cdot e^{-i\frac{\pi}{3}}.$

The surfaces $({\overline{\mathbb{C}}}_{x},e^{2\varphi}|dx|^{2})$ and
$({\overline{\mathbb{C}}}_{x},e^{2\chi}|dx|^{2})$ are isometric, the isometry
is given by the rotation. As a consequence,
$\det\Delta_{\boldsymbol{\beta}}=\det\Delta_{\boldsymbol{\gamma}}$. Equating
the directional derivative of $\det\Delta_{\boldsymbol{\beta}}$ along $h$ with
the one along $e^{i\frac{2\pi}{3}}h$, we immediately conclude that

$\partial_{\Re h}\det\Delta_{\boldsymbol{\beta}}=\partial_{\Im
h}\det\Delta_{\boldsymbol{\beta}}=0.$

It remains to note that the determinants of the Laplacians
$\Delta_{\boldsymbol{\beta}}$ and
$\Delta_{\boldsymbol{\beta}}^{S_{\varphi}}=\frac{1}{S_{\varphi}}\Delta_{\boldsymbol{\beta}}$
satisfy the standard rescaling property

$\log\det\Delta_{\boldsymbol{\beta}}^{S_{\varphi}}=\log\det\Delta_{\boldsymbol{\beta}}+\zeta_{\boldsymbol{\beta}}(0)\log
S_{\varphi},$

where the value $\zeta_{\boldsymbol{\beta}}(0)$ of the spectral zeta function
at zero [20] does not depend on the moduli $x_{1},\dots,x_{4}$ and satisfies

$\zeta_{\boldsymbol{\beta}}(0)=\frac{|\boldsymbol{\beta}|+2}{6}-\frac{1}{12}\sum_{k}\left(\beta_{k}+1-\frac{1}{\beta_{k}+1}\right)-1,\quad\partial_{\beta_{\ell}}\left(\zeta_{\boldsymbol{\beta}}(0)|_{\beta_{1}=|\boldsymbol{\beta}|-\sum_{k>1}\beta_{k}}\right)|_{\beta_{k}=\frac{|\boldsymbol{\beta}|}{4}}=0.$

Due to the invariance of the spectral determinant under the Möbius
transformations, this completes the proof of the first assertion.

For the octahedron, cube, dodecahedron, icosahedron, and dihedra there are
more symmetries to consider, but the idea and the steps of the proof remain
exactly the same. We omit the details. The case of constant curvature (flat,
spherical, or hyperbolic) metrics with three conical singularities is studied
in [18]. ∎

As is well-known, starting from four punctures on the $2$-sphere, explicit
construction of the general uniformization map is an open long-standing
problem. In this paper, we rely on the uniformization via Belyi functions.
There is another straightforward special case that deserves to be mentioned.

######  Remark 4.7.

In the case of a divisor

$\boldsymbol{\lambda}=\left(-1/2\right)\cdot 0+\left(-1/2\right)\cdot
1+\left(-1/2\right)\cdot\lambda+\left(-1/2\right)\cdot\infty,\quad\lambda\in\mathbb{C},$

the corresponding constant curvature unit area metric
$m_{\boldsymbol{\lambda}}$ with three or four conical singularities of angle
$\pi$ can be written explicitly, e.g. [6, 29, 52].

Recall that by using a suitable Möbius transformation we can always normalize
the marked points so that any three of them are at $0,1,\infty$. As we permute
the marked points $0,1,\lambda,\infty$ by Möbius transformations so that three
of them are still $0,1,\infty$, the fourth point is one of the following six:

$\lambda,\quad\frac{1}{\lambda},\quad
1-\lambda,\quad\frac{1}{1-\lambda},\quad\frac{\lambda}{\lambda-1},\quad\frac{\lambda-1}{\lambda}.$
(4.18)

In general, these six points are distinct. The exceptions are the following
three cases:

  * •

$\lambda=0$ or $\lambda=1$ or $\lambda=\infty$. In this case, the set (4.18)
contains only three distinct numbers: $0,1,\infty$. This case is studied in
[18], we do not discuss it here.

  * •

Harmonic position of four points: $\lambda=-1$ or $\lambda=1/2$ or
$\lambda=2$. The set (4.18) contains only three distinct numbers: $-1,1/2,2$.
The surface $(\overline{\mathbb{C}}_{x},m_{\boldsymbol{\lambda}})$ is
isometric to a unit area flat regular dihedron with four conical singularities
of angle $\pi$.

  * •

Equi-anharmonic position of four points: $\lambda=\frac{1+i\sqrt{3}}{2}$ or
$\lambda=\frac{1-i\sqrt{3}}{2}$. The set (4.18) contains only the numbers
$\frac{1\pm i\sqrt{3}}{2}$. The surface
$(\overline{\mathbb{C}}_{x},m_{\boldsymbol{\lambda}})$ is isometric to the
surface of a unit area regular Euclidean tetrahedron.

In general, for $\lambda\neq 0,1,\infty$, the metric is flat, and we have

$m_{\boldsymbol{\lambda}}=c^{2}_{\lambda}|x|^{-1}|x-1|^{-1}|x-\lambda|^{-1}|dx|^{2},$

where $c^{2}_{\lambda}$ is a scaling factor that guarantees that the surface
$(\overline{\mathbb{C}}_{x},m_{\boldsymbol{\lambda}})$ is of unit area, see
[52]. For the spectral determinant of the Friedrichs Laplacian
$\Delta_{\lambda}$ on the flat surface
$(\overline{\mathbb{C}}_{x},m_{\boldsymbol{\lambda}})$ the anomaly formula
(4.16) gives

$\log\det\Delta_{\lambda}=-\log
c_{\lambda}+\frac{1}{6}(\log|\lambda|+\log|\lambda-1|)-4\mathcal{C}(-1/2)+{\bf
C},$

where ${\bf C}$ is the same as in (2.11). Besides, thanks to [20, Appendix],
we have

$\mathcal{C}\left(-\frac{1}{2}\right)=-\zeta_{R}^{\prime}(-1)-\frac{1}{6}\log
2+\frac{1}{24}.$ (4.19)

As is well-known, e.g. [9, Sec. 2.9], the scaling factor $c^{2}_{\lambda}$
satisfies

$c^{-2}_{\lambda}=\int_{\mathbb{C}}\frac{dx\wedge
d\bar{x}}{-2i|x|\cdot|x-1|\cdot|x-\lambda|}=8|k|\left(K^{\prime}\overline{K}+\overline{K^{\prime}}K\right),\quad\lambda=\frac{(k+1)^{2}}{4k},$

where $K=K(k)$ is the complete elliptic integral of the first kind, and
$K^{\prime}=K(\sqrt{1-k^{2}})$.

In total, in terms of $\tau=iK^{\prime}/K$, we get

$K=\frac{\pi}{2}\vartheta_{3}^{2}(0|\tau),\quad K^{\prime}=-i\tau K,\quad
k=\frac{\vartheta_{2}^{2}(0|\tau)}{\vartheta_{3}^{2}(0|\tau)},$
$\det\Delta_{\lambda}=\frac{2^{2/3}}{\pi}|1-k^{2}|^{1/3}|k|^{1/6}\sqrt{\Im\tau}|K|=\sqrt{\Im\tau}|\eta(\tau/2)|^{2}.$

Here $\eta(\tau/2)$ is the Dedekind eta function. The last equality is a
consequence of the identities

$2\eta^{3}(\tau)=\vartheta_{2}(0|\tau)\vartheta_{3}(0|\tau)\vartheta_{4}(0|\tau),\quad\eta^{2}(\tau/2)=\vartheta_{4}(0|\tau)\eta(\tau),\quad\vartheta^{4}_{3}(0|\tau)=\vartheta^{4}_{2}(0|\tau)+\vartheta^{4}_{4}(0|\tau).$

By analyzing the expression $\sqrt{\Im\tau}|\eta(\tau/2)|^{2}$, it is not hard
to see that there are only two stationary points: $\tau=2i$ is a saddle point,
and $\tau=2e^{2\pi i/3}$ is the unique absolute maximum of the determinant
$\mathbb{C}\setminus\\{0,1\\}\ni\lambda\mapsto\det\Delta_{\lambda}$, cf. [33,
Sec. 4]. The case $\tau=2i$ (resp. $\tau=2e^{i\pi/3}$) corresponds to a
harmonic (resp. to an equi-anharmonic) position of four points in the divisor
$\boldsymbol{\lambda}$, cf. Theorem 4.6.

The reader may also find it interesting that in [28, Sec. 3.5.2.] the authors
demonstrate that the determinant $\det\Delta_{\lambda}$ is maximal for some
(not equi-anharmonic nor harmonic) positions of four points on the unit
circle.

##  5 Examples and applications

###  5.1 Determinant for triangulations by plane trees

By Riemann’s existence theorem, the planar bicolored trees are in one-to-one
correspondence with the (classes of equivalence of) Shabat polynomials [4,
30], see also [5]. Recall that a Shabat polynomial, also known as a
generalized Chebyshev polynomial, is a polynomial with at most two critical
values. Thanks to Theorem 2.1, to each bicolored plane tree we can associate a
family of spectral invariants $\det\Delta_{f^{*}\boldsymbol{\beta}}$. Indeed,
a Belyi function $f:\overline{\mathbb{C}}_{x}\to\overline{\mathbb{C}}_{z}$ (in
this case it is a Shabat polynomial) only prescribes a certain gluing scheme
of the bicolored double triangles. We can still make any suitable choice of
the angles of those triangles, or, equivalently, of the orders $\beta_{0}$,
$\beta_{1}$, and $\beta_{\infty}$ of three conical singularities of the
constant curvature metric $e^{2\phi}|dz|^{2}$ on the target Riemann sphere
$\overline{\mathbb{C}}_{z}$.

As an example, consider the Shabat polynomial

$f(x)=x^{\ell},\quad\ell\in\mathbb{N}.$

The ramification divisor is

$\boldsymbol{f}=(\ell-1)\cdot
0+(\ell-1)\cdot\infty,\quad|\boldsymbol{f}|=2\ell-2,$

where $x=0$ is the only point with $f^{\prime}(x)=0$, and $x=\infty$ is the
only pole of $f$. The corresponding bicolored tree is the inverse image of the
line segment $[0,1]$ under $f$, see Fig. 5.

$1$$e^{\frac{2\pi i}{\ell}}$$e^{-\frac{2\pi i}{\ell}}$$e^{\frac{4\pi
i}{\ell}}$$e^{-\frac{4\pi i}{\ell}}$ Figure 5: Dessin d’enfant representing
the polynomial $f(x)=x^{\ell}$

The black colored point is the preimage of the point $z=0$, and the white
colored points are the $\ell$ preimages $x=\sqrt[\ell]{1}$ of $z=1$. This
describes the cyclic triangulation of the Riemann sphere, or, equivalently,
the tessellation of the standard round sphere with $(\ell,\infty,\ell)$
bicolored double triangles, see Fig. 6.

Clearly, the first non-zero coefficient in the Taylor expansion of $f-f(0)$ at
zero is $c_{1}=1$, and the first non-zero coefficient in the Laurent expansion
of $f$ at infinity is $c_{2}=1$. Hence, the equality (3.1) immediately implies

$C_{f}=\frac{1}{6}\left(\frac{\ell-1}{\ell}\log|c_{1}|+(\ell+1)\log\ell\right)+\frac{1}{6}\left(-\frac{\ell-1}{\ell}\log|c_{2}|-(\ell+1)\log\ell\right)=0,$
(5.1)

where $C_{f}$ is the constant from Theorem 3.1.

The pullback of the divisor $\boldsymbol{\beta}=\beta_{0}\cdot
0+\beta_{1}\cdot 1+\beta_{\infty}\cdot\infty$ by $f(x)=x^{\ell}$ is the
divisor
$f^{*}{\boldsymbol{\beta}}=\sum_{k}\left(f^{*}\boldsymbol{\beta}\right)_{k}\cdot
x_{k}$. For the latter one we have

$f^{*}{\boldsymbol{\beta}}=(\ell(\beta_{0}+1)-1)\cdot
0+\beta_{1}\cdot\\{\sqrt[\ell]{1}\\}+(\ell(\beta_{\infty}+1)-1)\cdot\infty,$

where $\\{\sqrt[\ell]{1}\\}$ stands for the set of $\ell$ radicals
$\sqrt[\ell]{1}$ in $\mathbb{C}_{x}$ (those are the white colored points of
the ”snowflake“ in Fig. 5). The notation $\beta_{1}\cdot\\{\sqrt[\ell]{1}\\}$
means that each element of the set $\\{\sqrt[\ell]{1}\\}$ is a marked point of
weight $\beta_{1}$.

Figure 6: Cyclic triangulation

######  Theorem 5.1 (Cyclic triangulation).

Let $m_{\boldsymbol{\beta}}$ be the unit area Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+1)$ metric of $S^{2}$-like double triangle, see
Section 2.1. Let $f(x)=x^{\ell}$ with $\ell\in\mathbb{N}$, cf. Fig 6. Then for
the zeta-regularized spectral determinant of the Friedrichs Laplacian
$\Delta_{f^{*}\boldsymbol{\beta}}$ corresponding to the area $\ell$ pullback
metric $f^{*}m_{\boldsymbol{\beta}}=e^{2f^{*}\phi}|dx|^{2}$ we have

$\displaystyle\log\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{\ell}=$
$\displaystyle\ell\Bigl{(}\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}(\beta_{0})+\mathcal{C}(\beta_{\infty})-{\bf
C}\Bigr{)}$
$\displaystyle+\frac{1}{6}\left(\ell-\frac{1}{\ell}\right)\left(\frac{\Psi(\beta_{0},\beta_{1},\beta_{\infty})}{\beta_{0}+1}+\frac{\Psi(\beta_{\infty},\beta_{1},\beta_{0})}{\beta_{\infty}+1}\right)$
$\displaystyle-\frac{1}{6}\left(\ell(\beta_{0}+\beta_{\infty}+2)+\frac{1}{\ell(\beta_{0}+1)}+\frac{1}{\ell(\beta_{\infty}+1)}\right)\log\ell$
$\displaystyle-\mathcal{C}\left(\ell(\beta_{0}+1)-1\right)-\mathcal{C}\left(\ell(\beta_{\infty}+1)-1\right)+{\bf
C},$

where the right hand side is an explicit function of $\ell\in\mathbb{N}$ and
$(\beta_{0},\beta_{1},\beta_{\infty})\in(-1,0]^{3}$.

Here $\beta\mapsto\mathcal{C}(\beta)$ is the function (2.10), $\bf C$ is the
constant introduced in (2.11), and the function

$\displaystyle\Psi(\beta_{0},\beta_{1},\beta_{\infty})$
$\displaystyle=\log\frac{\Gamma(-\beta_{0})}{\Gamma(1+\beta_{0})}$ (5.2)
$\displaystyle+\frac{1}{2}\log\frac{\Gamma\left(2+\frac{|\boldsymbol{\beta}|}{2}\right)\Gamma\left(\beta_{0}-\frac{|\boldsymbol{\beta}|}{2}\right)\Gamma\left(1+\frac{|\boldsymbol{\beta}|}{2}-\beta_{1}\right)\Gamma\left(1+\frac{|\boldsymbol{\beta}|}{2}-\beta_{\infty}\right)}{\pi\Gamma\left(-\frac{|\boldsymbol{\beta}|}{2}\right)\Gamma\left(1+\frac{|\boldsymbol{\beta}|}{2}-\beta_{0}\right)\Gamma\left(\beta_{1}-\frac{|\boldsymbol{\beta}|}{2}\right)\Gamma\left(\beta_{\infty}-\frac{|\boldsymbol{\beta}|}{2}\right)}$

is the one from (2.3). Recall that $\det\Delta_{\boldsymbol{\beta}}$ stands
for an explicit function (2.4), whose values are the determinants of the unit
area $S^{2}$-like double triangles of Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+2)$.

###### Proof.

Recall that
$\Psi(\beta_{1},\beta_{2},\beta_{3})=\Phi(\beta_{1},\beta_{2},\beta_{3})+\log
2$ with explicit function $\Phi$ from [18, Prop. A.2)]. This implies (5.2),
where $\Gamma$ stands for the Gamma function.

Now the assertion is an immediate consequence of Theorem 2.1 together with the
asymptotics (2.3), and the calculation of $C_{f}$ in (5.1). ∎

######  Example 5.2 (Dihedra).

For the determinant of the Gaussian curvature $2\pi(\beta+2/\ell)$ area $\ell$
regular dihedron with $\ell$ conical singularities of order $\beta$ we obtain

$\displaystyle\log{\det\Delta^{\ell}_{Dihedron}}=$
$\displaystyle\ell\log\det\Delta_{\boldsymbol{\beta}}+2\ell\mathcal{C}\left(\frac{1}{\ell}-1\right)$
(5.3)
$\displaystyle+\frac{\ell^{2}-1}{3}\Psi\left(\frac{1}{\ell}-1,\beta,\frac{1}{\ell}-1\right)+\frac{1}{3}\log\ell+(1-\ell){\bf
C}.$

This is a direct consequence of Theorem 5.1, where we take

$\boldsymbol{\beta}=\left(\frac{1}{\ell}-1\right)\cdot 0+\beta\cdot
1+\left(\frac{1}{\ell}-1\right)\cdot\infty.$

In particular, when $\beta=-2/\ell$, we obtain the determinant of the flat
regular dihedron of area $\ell$. In the case $\beta=0$, we obtain a surface
isometric to the round sphere in $\mathbb{R}^{3}$ of area $\ell$ and its
determinant. Finally, as $\beta\to-1^{+}$ the determinant increases without
any bound in accordance with the asymptotics

$\displaystyle\log{\det\Delta^{\ell}_{Dihedron}}=\frac{\ell}{12}\left(-2\log(\beta+1)+\log\left(1-\frac{2}{\ell}\right)+\log
2\pi-2+24\zeta_{R}^{\prime}(-1)\right)\frac{1}{\beta+1}$
$\displaystyle-\frac{\ell}{2}\log(\beta+1)+O(1)$

of the right-hand side in (5.3). In the limit $\beta=-1$ we obtain a surface
of Gaussian curvature $2\pi(2/\ell-1)$ with $\ell$ cusps. The spectrum of the
corresponding Laplacian is no longer discrete [16, 17].

######  Example 5.3 (Tetrahedron).

Here we find the spectral determinant of Laplacian for a constant curvature
regular tetrahedron of total area $4\pi$ with (four) conical singularities of
order $\beta$. Or, equivalently, the spectral determinant of the Platonic
surface of Gaussian curvature $2\beta+1$ with four faces. Up to a rescaling,
this is a particular case of Theorem 5.1, that corresponds to the choice
$\ell=3$ and

$\boldsymbol{\beta}=\frac{\beta-2}{3}\cdot 0+\beta\cdot
1+\left(-\frac{2}{3}\right)\cdot\infty.$

Here and in the remaining part of this section we use the standard rescaling
property [20, Sec. 1.2] of the determinant

$\log\det\Delta^{4\pi}_{f^{*}\boldsymbol{\beta}}=\log\det\Delta_{f^{*}\boldsymbol{\beta}}-\zeta_{f^{*}\boldsymbol{\beta}}(0)\log\frac{4\pi}{\deg
f}$

in order to normalize the total area to $4\pi$, where

$\zeta_{f^{*}\boldsymbol{\beta}}(0)=\frac{|f^{*}\boldsymbol{\beta}|+2}{6}-\frac{1}{12}\sum_{k}\left((f^{*}\boldsymbol{\beta})_{k}+1-\frac{1}{(f^{*}\boldsymbol{\beta})_{k}+1}\right)-1.$

As a result, in the case $\beta=0$, when all conical singularities disappear,
we obtain a surface isometric to the standard unit sphere
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1$ in $\mathbb{R}^{3}$ and its determinant

$\log{\det\Delta}=1/2-4\zeta^{\prime}_{R}(-1).$

In general, for the constant curvature regular tetrahedron of total area
$4\pi$ with conical singularities of order $\beta$, Theorem 5.1 gives

$\displaystyle\log{\det\Delta^{4\pi}_{Tetrahedron}}$
$\displaystyle=3\left(\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}\left(\frac{\beta-2}{3}\right)+\mathcal{C}\left(-\frac{2}{3}\right)\right)$
(5.4)
$\displaystyle+\frac{4}{3}\left(\frac{1}{\beta+1}\Psi\left(\frac{\beta-2}{3},\beta,-\frac{2}{3}\right)+{\Psi\left(-\frac{2}{3},\beta,\frac{\beta-2}{3}\right)}\right)$
$\displaystyle-\frac{1}{3}\left(\beta-3+\frac{1}{\beta+1}\right)\left(\log(4\pi)-\frac{1}{2}\log
3\right)-\mathcal{C}\left(\beta\right)-2{\bf C},$

where the right hand side is an explicit function of $\beta$. A graph of this
function is depicted in Fig. 7 as a solid line.

Figure 7: Graphs of the logarithm of the spectral determinant of Laplacian on
the surfaces of Platonic solids of area $4\pi$ as a function of the order
$\beta\in(-1,0]$ of the conical singularities: a. Regular Tetrahedron of
Gaussian curvature $2\beta+1$ (solid line), b. Regular Octahedron of Gaussian
curvature $3\beta+1$ (dotted line), c. Regular Cube of Gaussian curvature
$4\beta+1$ (dashed line), d. Regular Icosahedron of Gaussian curvature
$6\beta+1$ (dash-dotted line), e. Regular Dodecahedron of Gaussian curvature
$10\beta+1$ (long-dashed line). The point
$(0,1/2-4\zeta^{\prime}_{R}(-1))\approx(0,1.16)$ on the graphs corresponds to
the logarithm of the spectral determinant of the unit round sphere in
$\mathbb{R}^{3}$. As $\beta\to-1^{+}$, the determinants increase without any
bound in accordance with the asymptotics (5.5), (5.8), (5.22), (5.27), and
(5.30). In the limit $\beta=-1$, the conical singularities turn into cusps,
and one obtains the ideal Platonic surfaces; the spectrum of the corresponding
Laplacians is no longer discrete.

In particular, in the case $\beta=-1/2$ we obtain a surface
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$ isometric to the
surface of a Euclidean regular tetrahedron. Note that $\mathcal{C}(-1/2)$ can
be evaluated as in (4.19). As a result, the formula (5.4) for the determinant
reduces to

$\log{\det\Delta^{4\pi}_{Tetrahedron}}|_{\beta=-1/2}=\log\frac{4}{3}-3\log\Gamma\left(\frac{2}{3}\right)+\frac{3}{2}\log\pi.$

Alternatively, the latter expression for the determinant can be obtained by
applying the partially heuristic Aurell-Salomonson formula [2] to the
explicitly evaluated pullback of the flat metric

$m_{\boldsymbol{\beta}}=c^{2}|z|^{-1}|z-1|^{-1}{|dz|^{2}}$

by $f(x)=x^{3}$, where $c$ is a scaling coefficient that normalizes the total
area of $m_{\boldsymbol{\beta}}$ to $4\pi/3$; for a rigorous proof of the
Aurell-Salomonson formula we refer to [20, Sec. 3.2].

Finally, as $\beta\to-1^{+}$, the determinant grows without any bound in
accordance with the asymptotics

$\log{\det\Delta^{4\pi}_{Tetrahedron}}=\left(-\frac{2}{3}\log(\beta+1)-\frac{2}{3}+8\zeta_{R}^{\prime}(-1)\right)\frac{1}{\beta+1}-2\log(\beta+1)+O(1)$
(5.5)

of the right hand side in (5.4), cf. Fig 7. In the limit $\beta=-1$ we get a
surface isometric to an ideal tetrahedron: a surface of Gaussian curvature
$-2$ with four cusps; the spectrum of the Laplacian on an ideal tetrahedron is
not discrete [16, 17].

######  Example 5.4 (Octahedron).

Here we find the determinant of Laplacian for a constant curvature regular
octahedron of area $4\pi$ with (six) conical singularities of order $\beta$.
Or, equivalently, the spectral determinant of the Platonic surface of Gaussian
curvature $3\beta+1$ with eight faces. In Theorem 5.1 we substitute $\ell=4$
and

$\boldsymbol{\beta}=\frac{\beta-3}{4}\cdot 0+\beta\cdot
1+\frac{\beta-3}{4}\cdot\infty.$ (5.6)

As a result, after an appropriate rescaling, we obtain

$\displaystyle\log\det\Delta^{4\pi}_{Octahedron}=4\log\det\Delta_{\boldsymbol{\beta}}-\left(\beta+1+\frac{1}{\beta+1}\right)\left(\frac{2}{3}\log
2+\frac{1}{2}\log\pi\right)$ (5.7)
$\displaystyle+\frac{5}{3}\log\pi+\frac{5}{\beta+1}\Psi\left(\frac{\beta-3}{4},\beta,\frac{\beta-3}{4}\right)+2\log
2+8\mathcal{C}\left(\frac{\beta-3}{4}\right)-2\mathcal{C}\left(\beta\right)-3{\bf
C},$

where the right hand side is an explicit function of $\beta$. A graph of this
function is depicted in Fig. 7 as a dotted line.

In the case $\beta=0$ we obtain a surface isometric to the standard unit
sphere in $\mathbb{R}^{3}$ and a representation for its determinant in terms
of the determinant of spherical $(4,\infty,4)$ double triangle, see also
Example 5.5 below.

In the case $\beta=-1/3$ we obtain a surface
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$ isometric to the
(flat) surface of Euclidean regular octahedron. The formula (5.7) for the
determinant reduces to

$\log\det\Delta^{4\pi}_{Octahedron}|_{\beta=-1/3}=6\zeta_{R}^{\prime}(-1)+\frac{35}{24}\log\frac{4}{3}-\frac{13}{2}\log\Gamma\left(\frac{2}{3}\right)+\frac{13}{4}\log\pi.$

Let us also note, that in the case $\beta=-1/2$ we get the tessellation of the
singular sphere
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}|_{\beta=-1/2})$ by the
hyperbolic $(2,3,8)$-triangle. The surface
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}|_{\beta=-1/2})$, where
$\boldsymbol{\beta}$ is the divisor (5.6), is isometric to a regular
hyperbolic octahedron with conical singularities of angle $\pi$. This is
remarkable, as a double of
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}|_{\beta=-1/2})$ is the
Bolza curve, known as the most symmetrical genus two smooth hyperbolic
surface, see e.g. [22]. To the best of our knowledge, the exact value of the
spectral determinant of the Bolza curve, endowed with the smooth Gaussian
curvature $-1$ metric, is not yet known. For a numerical study see [48].

Finally, as $\beta\to-1^{+}$, the determinant grows without any bound in
accordance with the asymptotics

$\log\det\Delta^{4\pi}_{Octahedron}=\left(-\log(\beta+1)+\frac{1}{2}\log
2-1+12\zeta_{R}^{\prime}(-1)\right)\frac{1}{\beta+1}-3\log(\beta+1)+O(1)$
(5.8)

of the right-hand side in (5.7). In the limit $\beta=-1$ we get a surface
isometric to an ideal octahedron: a surface of Gaussian curvature $-2$ with
six cusps, cf. [16, 17]. The spectrum of the corresponding Laplacian is no
longer discrete.

######  Example 5.5 (Spindles).

Let $m_{\boldsymbol{\beta}}^{S}=e^{2\phi}|dz|^{2}$ be the metric of a spindle
with two antipodal singularities [53], where

$\phi(z)=\beta\log|z|+\log 2+\log(\beta+1)-\log(1+|z|^{2\beta+2}).$

The (regularized) Gaussian curvature of $m_{\boldsymbol{\beta}}^{S}$ is $1$,
and the total area is $S=4\pi(\beta+1)$. The metric represents the divisor
$\boldsymbol{\beta}=\beta_{0}\cdot 0+\beta_{1}\cdot
1+\beta_{\infty}\cdot\infty$ with
$\beta_{0}=\beta_{\infty}=:\beta\in(-1,\infty)$ and $\beta_{1}=0$. The spindle
$(\overline{\mathbb{C}}_{z},m_{\boldsymbol{\beta}}^{S})$ is isometric to the
spherical double triangle glued from two copies of a spherical triangle (a
lune) with internal angles $\bigl{(}\pi(\beta+1),\pi,\pi(\beta+1)\bigr{)}$,
cf. Fig. 6.

For the asymptotics of the metric potential we have

$\displaystyle\phi(z)$ $\displaystyle=\beta\log|z|+\phi_{0}+o(1),\quad z\to
0,\quad\phi_{0}=\log 2(\beta+1),$ $\displaystyle\phi(z)$
$\displaystyle=-(\beta+2)\log|z|+\phi_{\infty}+o(1),\quad
z\to\infty,\quad\phi_{\infty}=\log 2(\beta+1).$

Clearly, for the pullback of $m_{\boldsymbol{\beta}}^{S}$ by the Shabat
polynomial $f(x)=x^{\ell}$ we have

$f^{*}m_{\boldsymbol{\beta}}^{S}=\frac{4\ell^{2}(\beta+1)^{2}|x|^{2(\ell(\beta+1)-1)}|dx|^{2}}{(1+|x|^{2\ell(\beta+1)})^{2}},\quad
f^{*}\boldsymbol{\beta}=(\ell(\beta+1)-1)\cdot
0+(\ell(\beta+1)-1)\cdot\infty.$

The surface $(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}^{S})$ is
isometric to the surface glued from $\ell$ copies of the spindle
$(\overline{\mathbb{C}}_{z},m^{S}_{\boldsymbol{\beta}})$ with a cut from the
conical point at $z=0$ to the conical point at $z=\infty$. Or, equivalently,
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}^{S})$ is triangulated
by $2\ell$ bicolored copies of a spherical triangle with internal angles
$\bigl{(}\pi(\beta+1),\pi,\pi(\beta+1)\bigr{)}$ and unit Gaussian curvature,
cf. Fig. 6. The surface
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}^{S})$ is again a
spindle with two antipodal conical singularities: the metric
$m_{\boldsymbol{\beta}}^{S}$ coincides with $f^{*}m_{\boldsymbol{\beta}}^{S}$
after the replacement of $\beta$ by $\ell(\beta+1)-1$ (and $z$ by $x$).

As is known [18, 20, 27, 46], the spectral determinant of the Friedrichs
Laplacian $\Delta^{S}_{\boldsymbol{\beta}}$ on the spindle
$(\overline{\mathbb{C}}_{z},m^{S}_{\boldsymbol{\beta}})$ satisfies

$\log\frac{\Delta^{S}_{\boldsymbol{\beta}}}{S}=\frac{\beta+1}{3}-\frac{1}{3}\left(\beta+1+\frac{1}{\beta+1}\right)\log
2(\beta+1)-2\mathcal{C}(\beta)+{\bf C},\quad\beta>-1.$ (5.9)

For the spectral determinant of the spindle
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}^{S})$ Theorem 3.1
(after an appropriate rescaling) gives

$\displaystyle\log\frac{\det\Delta^{\ell S}_{f^{*}\boldsymbol{\beta}}}{\ell
S}=\ell\Bigl{(}\log\frac{\det\Delta^{S}_{\boldsymbol{\beta}}}{4\pi(\beta+1)}+2\mathcal{C}(\beta)-{\bf
C}\Bigr{)}+\frac{1}{3}\left(\ell-\frac{1}{\ell}\right)\frac{\log
2(\beta+1)}{\beta+1}$ (5.10)
$\displaystyle-\frac{1}{6}\left(2\ell(\beta+1)+\frac{2}{\ell(\beta+1)}\right)\log\ell-2\mathcal{C}\left(\ell(\beta+1)-1\right)+{\bf
C}.$

As expected, after the substitution (5.9) and the replacement of
$\ell(\beta+1)-1$ by $\beta$, the equality (5.10) reduces to the one in (5.9).

In particular, for $\beta=\frac{1}{\ell}-1$ the pullback of the spindle metric
$m_{\boldsymbol{\beta}}^{S}$ by $f$ is the metric of the standard round sphere
of total area $\ell S=4\pi$. In this case, the equality (5.10) expresses the
determinant of Laplacian of the standard sphere

$\det\Delta^{4\pi}_{f^{*}\boldsymbol{\beta}}=\exp(1/2-4\zeta^{\prime}_{R}(-1))$

in terms of the determinants $\det\Delta^{4\pi/\ell}_{\boldsymbol{\beta}}$ of
the spherical double triangle (or, equivalently, of the bicolored double lune)
corresponding to the cyclic triangulation of the sphere via the Shabat
polynomial $f(x)=x^{\ell}$, cf. Fig. 6.

###  5.2 Determinant for dihedral triangulation

Consider the Belyi function

$f(x)=1-\left(\frac{1-x^{\ell}}{1+x^{\ell}}\right)^{2},\quad\ell\in\mathbb{N}.$
(5.11)

This is a ramified covering of degree $2\ell$. For the ramification divisor of
$f$ we have

${\boldsymbol{f}}=(\ell-1)\cdot
0+1\cdot\\{\sqrt[\ell]{1}\\}+1\cdot\\{\sqrt[\ell]{-1}\\}+(\ell-1)\cdot\infty,\quad|{\boldsymbol{f}}|=4\ell-2.$

This Belyi function defines a tessellation of the standard round sphere with
$(2,2,\ell)$-triangles, cf. Fig 8. As we show in the proof of Theorem 5.6
below, to this Belyi map there corresponds the constant

$C_{f}=\frac{2}{3}\left(\ell-\frac{1}{\ell}\right)\log 2.$ (5.12)

The Belyi map $f$ sends the marked points listed in the ramification divisor
$\boldsymbol{f}$ to the points $0,1,\infty$ as follows:

$f(0)=0,\quad f(\infty)=0,\quad f(\sqrt[\ell]{-1})=\infty,\quad
f(\sqrt[\ell]{1})=1.$

For the pullback divisor of $\boldsymbol{\beta}=\beta_{0}\cdot
0+\beta_{1}\cdot 1+\beta_{\infty}\cdot\infty$ by $f$ we obtain

$f^{*}\boldsymbol{\beta}=(\ell(\beta_{0}+1)-1)\cdot
0+(2\beta_{1}+1)\cdot\\{\sqrt[\ell]{1}\\}+(2\beta_{\infty}+1)\cdot\\{\sqrt[\ell]{-1}\\}+(\ell(\beta_{0}+1)-1)\cdot\infty.$

Figure 8: Dihedral triangulation

######  Theorem 5.6 (Dihedral triangulation).

Let $m_{\boldsymbol{\beta}}$ be the unit area Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+1)$ metric of $S^{2}$-like double triangle, see
Section 2.1. For the spectral zeta-regularized determinant of the Friedrichs
Laplacian $\Delta_{f^{*}\boldsymbol{\beta}}$ corresponding to the area $2\ell$
pullback metric $f^{*}m_{\boldsymbol{\beta}}$ on the Riemann sphere
$\overline{\mathbb{C}}_{x}$, where $f$ is the dihedral Belyi map (5.11), we
have the following explicit expression:

$\displaystyle\log$
$\displaystyle\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{2\ell}=2\ell\Bigl{(}\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}_{\boldsymbol{\beta}}-{\bf
C}\Bigr{)}+C_{f}$
$\displaystyle+\frac{1}{3}\left(\ell-\frac{1}{\ell}\right)\frac{\Psi(\beta_{0},\beta_{1},\beta_{\infty})}{\beta_{0}+1}+\frac{\ell}{4}\left(\frac{\Psi(\beta_{1},\beta_{0},\beta_{\infty})}{\beta_{1}+1}+\frac{\Psi(\beta_{\infty},\beta_{1},\beta_{0})}{\beta_{\infty}+1}\right)$
$\displaystyle-\frac{1}{3}\left(\ell(\beta_{0}+1)+\frac{1}{\ell(\beta_{0}+1)}\right)\log\ell-\frac{\ell}{3}\left(\beta_{1}+\beta_{\infty}+2+\frac{1}{4\beta_{1}+4}+\frac{1}{4\beta_{\infty}+4}\right)\log
2$
$\displaystyle-2\mathcal{C}\left(\ell(\beta_{0}+1)-1\right)-\ell\Bigl{(}\mathcal{C}\left(2\beta_{1}+1\right)+\mathcal{C}\left(2\beta_{\infty}+1\right)\Bigr{)}+{\bf
C},$

where
$\mathcal{C}_{\boldsymbol{\beta}}=\mathcal{C}(\beta_{0})+\mathcal{C}(\beta_{1})+\mathcal{C}(\beta_{\infty})$
and $C_{f}$ is the same as in (5.12).

Recall that $\det\Delta_{\boldsymbol{\beta}}$ stands for an explicit function
(2.4), whose values are the determinants of the unit area $S^{2}$-like double
triangles of Gaussian curvature $2\pi(|\boldsymbol{\beta}|+2)$ tessellating
the singular sphere $(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$.
The function $\beta\mapsto\mathcal{C}(\beta)$ is defined in (2.10), the
function $\Psi$ is the same as in (5.2), and $\bf C$ is the constant
introduced in (2.11).

###### Proof.

For the derivative of the Belyi function (5.11) we have

$f^{\prime}(x)=\frac{4\ell x^{\ell-1}(1-x^{\ell})}{(x^{\ell}+1)^{3}}.$

As a result, we immediately obtain the asymptotics in vicinities of the
critical points of the form (3.5). The first non-zero coefficients $c_{k}$ in
the Taylor expansions (cf. Proposition 3.1) satisfy

$|c_{k}|=\left\\{\begin{array}[]{cc}4,&x_{k}=0,\\\
\ell^{2}/4,&x_{k}\in\\{\sqrt[\ell]{1}\\},\\\ 4,&x_{k}=\infty,\quad
k=n=2\ell+2.\end{array}\right.$

Similarly, we obtain the asymptotics in vicinities of the poles of the form
(3.6) with the first non-zero Laurent coefficients $c_{k}$ satisfying

$|c_{k}|=\ell^{2}/4,\quad x_{k}\in\\{\sqrt[\ell]{-1}\\}.$

Now the expression (3.1) for $C_{f}$ implies (5.12). As a result, the
assertion is an immediate consequence of Theorem 2.1 together with the
asymptotics (2.3). ∎

######  Example 5.7 (Dihedra).

Here we deduce an alternative formula for the determinant of the regular
dihedron of Gaussian curvature $K=2\pi(\beta+1/\ell)$ and area $2\ell$ with
$2\ell$ conical singularities of order $\beta$: In Theorem 5.6 we take

$\boldsymbol{\beta}=\left(\frac{1}{\ell}-1\right)\cdot
0+\frac{\beta-1}{2}\cdot 1+\frac{\beta-1}{2}\cdot\infty,$

and obtain

$\displaystyle\log$
$\displaystyle\,{\det\Delta^{2\ell}_{Dihedron}}=2\ell\Bigl{(}\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}_{\boldsymbol{\beta}}-\mathcal{C}\left(\beta\right)\Bigr{)}+\frac{1}{3}\log\ell+(1-2\ell){\bf
C}$
$\displaystyle+\frac{\ell^{2}-1}{3}{\Psi\left(\frac{1}{\ell}-1,\frac{\beta-1}{2},\frac{\beta-1}{2}\right)}+\frac{\ell}{\beta+1}{\Psi\left(\frac{\beta-1}{2},\frac{\beta-1}{2},\frac{1}{\ell}-1\right)}$
$\displaystyle+\left(\frac{2}{3}\left(\ell-\frac{1}{\ell}\right)-\frac{\ell}{3}\left(\beta+1+\frac{1}{\beta+1}\right)+1\right)\log
2.$

######  Example 5.8 (Octahedron).

We obtain an alternative formula for the determinant of Laplacian on a regular
octahedron of Gaussian curvature $3\beta+1$ with (six) conical singularities
of order $\beta$: In Theorem 5.6 we take $\ell=2$ and

$\boldsymbol{\beta}=\frac{\beta-1}{2}\cdot 0+\frac{\beta-1}{2}\cdot
1+\frac{\beta-1}{2}\cdot\infty.$

This together with the standard rescaling property of determinants implies

$\displaystyle\log$
$\displaystyle{\det\Delta^{4\pi}_{Octahedron}}=4\log\det\Delta_{\boldsymbol{\beta}}-\left(\beta+1+\frac{1}{\beta+1}\right)\left(\log
2+\frac{1}{2}\log\pi\right)+\frac{5}{3}\log\pi$
$\displaystyle+\frac{3}{{\beta+1}}{\Psi\left(\frac{\beta-1}{2},\frac{\beta-1}{2},\frac{\beta-1}{2}\right)}+3\log
2+12\mathcal{C}\left(\frac{\beta-1}{2}\right)-6\mathcal{C}\left(\beta\right)-3{\bf
C},$

where the right hand side is an explicit function of $\beta$; this expression
is equivalent to the one in (5.7). A graph of this function is depicted in
Fig. 7 as a dotted line.

###  5.3 Determinant for tetrahedral triangulation

The tetrahedral Belyi map is given by the function

$f(x)=-64\frac{(x^{3}+1)^{3}}{(x^{3}-8)^{3}x^{3}},\quad\deg f=12.$ (5.13)

For the ramification divisor, we obtain

$\boldsymbol{f}=2\cdot\left\\{0,\sqrt[3]{8}\right\\}+1\cdot\left\\{\sqrt[3]{-10\pm
6\sqrt{3}}\right\\}+2\cdot\left\\{\sqrt[3]{-1},\infty\right\\},\quad|\boldsymbol{f}|=22.$

The Belyi map sends the marked points listed in the divisor $\boldsymbol{f}$
to the points $0,1,\infty$ of the target Riemann sphere
$\overline{\mathbb{C}}_{z}$ as follows:

$f(0)=\infty,\quad f(\sqrt[3]{-1})=0,\quad f\left(\sqrt[3]{-10\pm
6\sqrt{3}}\right)=1,\quad f(\sqrt[3]{8})=\infty,\quad f(\infty)=0.$

Here $f(x)=1$ are the edge midpoints of a regular tetrahedron. The poles
$f(x)=\infty$ correspond to its vertices. The zeros $f(x)=0$, i.e. the roots
of the numerator and $x=\infty$, are the centers of the faces. This defines a
tessellation of the standard round sphere with spherical $(2,3,3)$-triangles,
cf. Fig 9. A picture of the corresponding dessin d’enfant can be found e.g. in
[31, Fig. 2]. In the proof of Theorem 5.9 below we show that to the Belyi
function (5.13) there corresponds the constant

$C_{f}=\log 2+\frac{9}{4}\log 3.$ (5.14)

For the pullback of the divisor $\boldsymbol{\beta}=\beta_{0}\cdot
0+\beta_{1}\cdot 1+\beta_{\infty}\cdot\infty$ by $f$ we obtain

$f^{*}\boldsymbol{\beta}=(3\beta_{0}+2)\cdot\left\\{\sqrt[3]{-1},\infty\right\\}+(2\beta_{1}+1)\cdot\left\\{\sqrt[3]{-10\pm
6\sqrt{3}}\right\\}+(3\beta_{\infty}+2)\cdot\left\\{0,\sqrt[3]{8}\right\\}.$

Figure 9: Tetrahedral triangulation

######  Theorem 5.9 (Tetrahedral triangulation).

Let $m_{\boldsymbol{\beta}}$ be the unit area Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+1)$ metric of $S^{2}$-like double triangle, see
Section 2.1. For the determinant of the Friedrichs Laplacian
$\det\Delta_{f^{*}\boldsymbol{\beta}}$ corresponding to the pullback metric
$f^{*}m_{\boldsymbol{\beta}}$ of area $12$, where $f$ is the tetrahedral Belyi
map (5.13), we have

$\displaystyle\log$
$\displaystyle\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{12}=12\Bigl{(}\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}_{\boldsymbol{\beta}}-{\bf
C}\Bigr{)}+C_{f}$
$\displaystyle+\frac{16}{9}\frac{\Psi(\beta_{0},\beta_{1},\beta_{\infty})}{\beta_{0}+1}+\frac{3}{2}\frac{\Psi(\beta_{1},\beta_{0},\beta_{\infty})}{\beta_{1}+1}+\frac{16}{9}\frac{\Psi(\beta_{\infty},\beta_{1},\beta_{0})}{\beta_{\infty}+1}$
$\displaystyle-\frac{2}{3}\left(3(\beta_{0}+\beta_{\infty}+2)+\frac{1}{3(\beta_{\infty}+1)}+\frac{1}{3(\beta_{0}+1)}\right)\log
3-\left(2(\beta_{1}+1)+\frac{1}{2(\beta_{1}+1)}\right)\log 2$
$\displaystyle-4\mathcal{C}\left(3\beta_{0}+2\right)-6\mathcal{C}\left(2\beta_{1}+1\right)-4\mathcal{C}\left(3\beta_{\infty}+2\right)+{\bf
C},$

where
$\mathcal{C}_{\boldsymbol{\beta}}=\mathcal{C}(\beta_{0})+\mathcal{C}(\beta_{1})+\mathcal{C}(\beta_{\infty})$
and $C_{f}$ is the same as in (5.14).

Recall that $\det\Delta_{\boldsymbol{\beta}}$ stands for an explicit function
(2.4), whose values are the determinants of the unit area $S^{2}$-like double
triangles of Gaussian curvature $2\pi(|\boldsymbol{\beta}|+2)$ tessellating
the singular sphere $(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$.
The function $\beta\mapsto\mathcal{C}(\beta)$ is defined in (2.10), the
function $\Psi$ is the same as in (5.2), and $\bf C$ is the constant
introduced in (2.11).

###### Proof.

For the derivative of the Belyi function in (5.13) we have

$f^{\prime}(x)=\frac{192(x^{3}+1)^{2}(x^{6}+20x^{3}-8)}{x^{4}(x^{3}-8)^{4}}.$

In exactly the same way as in the proof of Theorem 5.6 we obtain

$|c_{k}|=\left\\{\begin{array}[]{cc}2^{6}/3^{3},&x_{k}\in\\{\sqrt[3]{-1}\\},\\\
2\sqrt{3}\pm 3,&x_{k}\in\left\\{\sqrt[3]{-10\pm 6\sqrt{3}}\right\\},\\\
2^{6},&x_{k}=\infty,\ k=n,\\\ 2^{3},&x_{k}=0,\\\
2^{3}/3^{3},&x_{k}\in\left\\{\sqrt[3]{8}\right\\}.\end{array}\right.$

This together with the expression (3.1) for $C_{f}$ implies the value stated
in (5.14). The remaining part of the assertion is a direct consequence of
Theorem 2.1. ∎

######  Example 5.10 (Tetrahedron).

Here we deduce an alternative formula for the spectral determinant of a
regular tetrahedron of Gaussian curvature $K=2\beta+1$: in Theorem 5.9 we
substitute

$\boldsymbol{\beta}=\left(-\frac{2}{3}\right)\cdot
0+\left(-\frac{1}{2}\right)\cdot 1+\frac{\beta-2}{3}\cdot\infty.$

As a result, after some rescaling we obtain

$\displaystyle\log$
$\displaystyle\det\Delta^{4\pi}_{Tetrahedron}=12\Bigl{(}\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}_{\boldsymbol{\beta}}\Bigr{)}+\log
2+\frac{7}{12}\log 3+\frac{4}{3}\log\pi$
$\displaystyle+\frac{16}{3}{\Psi\left(-\frac{2}{3},-\frac{1}{2},\frac{\beta-2}{3}\right)+3\Psi\left(-\frac{1}{2},-\frac{2}{3},\frac{\beta-2}{3}\right)}+\frac{16}{3(\beta+1)}\Psi\left(\frac{\beta-2}{3},-\frac{1}{2},-\frac{2}{3}\right)$
$\displaystyle-\frac{1}{3}\left(\beta+1+\frac{1}{\beta+1}\right)\log
3\pi-4\mathcal{C}\left(\beta\right)-11{\bf C},$

where the right hand side is an explicit function of $\beta$. A graph of this
function is depicted in Fig. 7 as a solid line.

In the case $\beta=0$ the surface
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$ is isometric to a
unit sphere in $\mathbb{R}^{3}$. The sphere
$(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}}|_{\beta=0})$ is
tessellated by the double of $(2,3,3)$-triangle, and the above formula for the
determinant is a representation for the determinant
$\det\Delta^{4\pi}_{Tetrahedron}|_{\beta=0}$ of the unit sphere in terms of
the determinant of the double of spherical $(2,3,3)$-triangle.

###  5.4 Determinant for octahedral triangulation

To the octahedral triangulation, there corresponds the Belyi function

$f(x)=-2^{2}\cdot
3^{3}\frac{(x^{4}+1)^{4}x^{4}}{(x^{8}-14x^{4}+1)^{3}},\quad\deg f=24.$ (5.15)

In particular, thanks to the identification of the Riemann sphere
$\overline{\mathbb{C}}_{x}$ with the standard round sphere in $\mathbb{R}^{3}$
via the stereographic projection, $f$ describes the tessellation of the
standard round sphere with spherical $(2,3,4)$-triangles, cf. Fig 10. The
ramification divisor of $f$ is

$\boldsymbol{f}=3\cdot\left\\{0,\infty,\sqrt[4]{-1}\right\\}+1\cdot\left\\{\sqrt[4]{1},\sqrt[4]{-17\pm
3\cdot
2^{5/2}}\right\\}+2\cdot\left\\{\sqrt[4]{(2\pm\sqrt{3})^{2}}\right\\},\quad|\boldsymbol{f}|=46.$
(5.16)

Here $\\{\sqrt[4]{1},\sqrt[4]{-17\pm 3\cdot 2^{5/2}}\\}$ are the edge
midpoints of a cube, for any $x$ in this set we have $f(x)=1$. The poles
$\sqrt[4]{(2\pm\sqrt{3})^{2}}$ of $f$ correspond to the vertices of the cube.
The zeros $\\{0,\infty,\sqrt[4]{-1}\\}$ of $f$ are the centers of the cube
faces.

For the octahedron dual to the cube: the points
$\\{\sqrt[4]{1},\sqrt[4]{-17\pm 3\cdot 2^{5/2}}\\}$ correspond to the edge
midpoints, the poles of $f$ correspond to the centers of the faces, and the
zeros of $f$ are the vertices. A picture of the corresponding dessin d’enfant
can be found e.g. in [31, Fig. 3].

In the proof of Theorem 5.11 below we show that to the Belyi function (5.15)
there corresponds the constant

$C_{f}=\frac{9}{4}\log 3+\frac{119}{18}\log 2.$ (5.17)

For the pullback of the divisor $\boldsymbol{\beta}=\beta_{0}\cdot
0+\beta_{1}\cdot 1+\beta_{\infty}\cdot\infty$ by $f$ we get

$\displaystyle
f^{*}\boldsymbol{\beta}=(4\beta_{0}+3)\cdot\left\\{0,\infty,\sqrt[4]{-1}\right\\}+$
$\displaystyle(2\beta_{1}+1)\cdot\left\\{\sqrt[4]{1},\sqrt[4]{-17\pm 3\cdot
2^{5/2}}\right\\}$
$\displaystyle\qquad\qquad+(3\beta_{\infty}+2)\cdot\left\\{\sqrt[4]{(2\pm\sqrt{3})^{2}}\right\\}.$

Figure 10: Octahedral triangulation

######  Theorem 5.11 (Octahedral triangulation).

Let $m_{\boldsymbol{\beta}}$ be the unit area Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+1)$ metric of $S^{2}$-like double triangle, see
Section 2.1. For the determinant of Laplacian corresponding to the pullback
metric $f^{*}m_{\boldsymbol{\beta}}$ of area $24$, where $f$ is the Belyi map
(5.15), we have

$\displaystyle\log$
$\displaystyle\frac{\det\Delta_{f^{*}\boldsymbol{\beta}}}{24}=24\Bigl{(}\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}_{\boldsymbol{\beta}}-{\bf
C}\Bigr{)}+C_{f}$
$\displaystyle+\frac{15}{4}\frac{\Psi(\beta_{0},\beta_{1},\beta_{\infty})}{\beta_{0}+1}+3\frac{\Psi(\beta_{1},\beta_{0},\beta_{\infty})}{\beta_{1}+1}+\frac{32}{9}\frac{\Psi(\beta_{\infty},\beta_{1},\beta_{0})}{\beta_{\infty}+1}$
$\displaystyle-\left(4(2\beta_{0}+\beta_{1}+3)+\frac{1}{2(\beta_{0}+1)}+\frac{1}{\beta_{1}+1}\right)\log
2-4\left(\beta_{\infty}+1+\frac{1}{9(\beta_{\infty}+1)}\right)\log 3$
$\displaystyle-8\mathcal{C}\left(3\beta_{\infty}+2\right)-6\mathcal{C}\left(4\beta_{0}+3\right)-12\mathcal{C}\left(2\beta_{1}+1\right)+{\bf
C},$

where
$\mathcal{C}_{\boldsymbol{\beta}}=\mathcal{C}(\beta_{0})+\mathcal{C}(\beta_{1})+\mathcal{C}(\beta_{\infty})$
and $C_{f}$ is the same as in (5.17).

Recall that $\det\Delta_{\boldsymbol{\beta}}$ stands for an explicit function
(2.4), whose values are the determinants of the unit area $S^{2}$-like double
triangles of Gaussian curvature $2\pi(|\boldsymbol{\beta}|+2)$ tessellating
the singular sphere $(\overline{\mathbb{C}}_{x},f^{*}m_{\boldsymbol{\beta}})$.
The function $\beta\mapsto\mathcal{C}(\beta)$ is defined in (2.10), the
function $\Psi$ is the same as in (5.2), and $\bf C$ is the constant
introduced in (2.11).

###### Proof.

Here we find $C_{f}$ by using Proposition 3.4. Namely, we first write the
potential of the pullback of
$m_{\boldsymbol{\beta}}=c|z|^{-4/3}|z-1|^{-4/3}|dz|^{2}$ by $f$ in the form

$\bigl{(}f^{*}\phi\bigr{)}(x)=\frac{1}{3}\log|x|+\frac{1}{3}\log|x^{4}+1|-\frac{1}{3}\log|x^{4}-1|-\frac{1}{3}\log|(x^{4}+17)^{2}-9\cdot
2^{5}|+\log cA,$

cf. (3.17). It is easy to see that as $x\to x_{k}\in\\{\sqrt[4]{-1}\\}$ the
metric potential $f^{*}\phi$ satisfies the asymptotics

$\displaystyle\bigl{(}f^{*}\phi\bigr{)}(x)$
$\displaystyle=\frac{1}{3}\log|x-x_{k}|+\frac{1}{3}\log\left|(x^{4}+1)^{\prime}\restriction_{x=\sqrt[4]{-1}}\right|-\frac{1}{3}\log|-2|$
$\displaystyle\qquad-\frac{1}{3}\log|(-1+17)^{2}-9\cdot 2^{5}|+\log cA+o(1)$
$\displaystyle=\frac{1}{3}\log|x-x_{k}|+\log cA-\frac{4}{3}\log 2+o(1).$

This together with (3.18) implies that for any $k$, such that
$x_{k}\in\\{\sqrt[4]{-1}\\}$, we have

$\sum_{\ell:k\neq\ell<n}\frac{\operatorname{ord}_{\ell}f-2}{3}\log|x_{k}-x_{\ell}|=-\frac{4}{3}\log
2.$

Similarly, we obtain

$\sum_{\ell:k\neq\ell<n}\frac{\operatorname{ord}_{\ell}f-2}{3}\log|x_{k}-x_{\ell}|=\left\\{\begin{array}[]{cc}0,&x_{k}=0,\\\
-\log 2-\frac{2}{3}\log 3,&x_{k}\in\\{\sqrt[4]{1}\\},\\\
\frac{1}{3}\log\frac{3\pm 2\sqrt{2}}{144},&x_{k}\in\left\\{\sqrt[4]{-17\pm
3\cdot 2^{5/2}}\right\\}.\end{array}\right.$

As a consequence, by using the expression (5.16) for the ramification divisor
$\boldsymbol{f}$, for the first line in (2.12) we obtain

$\displaystyle\frac{1}{18}\sum_{k:k<n}\sum_{\ell:k\neq\ell<n}\frac{(\operatorname{ord}_{k}f-2)(\operatorname{ord}_{\ell}f-2)}{\operatorname{ord}_{k}f+1}\log|x_{k}-x_{\ell}|=\log
2+\frac{2}{3}\log 3.$ (5.18)

Thanks to (5.16), it is also easy to verify that for the second line in (2.12)
one has

$\frac{1}{6}\sum_{k\leqslant
n}\left(\frac{\operatorname{ord}_{k}f+1}{3}+\frac{3}{\operatorname{ord}_{k}f+1}\right)\log(\operatorname{ord}_{k}f+1)=\frac{17}{2}\log
2+\frac{8}{3}\log 3.$ (5.19)

Finally, as the scaling constant $A$ satisfies (2.13), we get

$A=\frac{|f(x)|^{-2/3}|f(x)-1|^{-2/3}|f^{\prime}(x)|}{|x|^{1/3}|x^{4}+1|^{1/3}|x^{4}-1|^{-1/3}|(x^{4}+17)^{2}-9\cdot
2^{5}|^{-1/3}}=12\cdot 2^{2/3}.$

This allows us to calculate the value of the last line in (2.12):

$\frac{1}{6}\left(n-2-\sum_{k\leqslant
n}\frac{3}{\operatorname{ord}_{k}f+1}\right)\log A=-\frac{13}{12}\log(12\cdot
2^{2/3}).$ (5.20)

Adding the values (5.18), (5.19), (5.20) of the lines in (2.12) together, we
arrive at (5.17). ∎

######  Example 5.12 (Cube).

Here we find the spectral determinant of a cube of Gaussian curvature
$4\beta+1$ with (eight) conical singularities of order $\beta$: in Theorem
5.11 we substitute

$\boldsymbol{\beta}=\left(-\frac{3}{4}\right)\cdot
0+\left(-\frac{1}{2}\right)\cdot 1+\frac{\beta-2}{3}\cdot\infty.$

After rescaling, for the determinant of a cube of (regularized) Gaussian
curvature $4\beta+1$ with eight conical singularities of order $\beta$ we
obtain

$\displaystyle\log$
$\displaystyle\,{\det\Delta^{4\pi}_{Cube}}=24\Bigl{(}\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}_{\boldsymbol{\beta}}\Bigr{)}+\frac{5}{4}\log
3-\frac{7}{18}\log 2+2\log\pi$ (5.21)
$\displaystyle+15{\Psi\left(-\frac{3}{4},-\frac{1}{2},\frac{\beta-2}{3}\right)}+6{\Psi\left(-\frac{1}{2},-\frac{3}{4},\frac{\beta-2}{3}\right)}+\frac{32}{3(\beta+1)}{\Psi\left(\
\frac{\beta-2}{3},-\frac{1}{2},-\frac{3}{4}\right)}$
$\displaystyle-\frac{2}{3}\left(\beta+1+\frac{1}{\beta+1}\right)\log\frac{3\pi}{2}-8\mathcal{C}\left(\beta\right)-23{\bf
C},$

where the right hand side is an explicit function of $\beta$. A graph of this
function is depicted in Fig. 7 as a dashed line. As $\beta\to-1^{+}$, the
determinant of Laplacian grows without any bound in accordance with the
asymptotics

$\log{\det\Delta^{4\pi}_{Cube}}=\left(-\frac{4}{3}\log(\beta+1)+\frac{2}{3}\log
3-\frac{4}{3}+16\zeta^{\prime}_{R}(-1)\right)\frac{1}{\beta+1}-4\log(\beta+1)+O(1)$
(5.22)

of the right hand side in (5.21). In the limit $\beta=-1$ we get a surface
isometric to an ideal cube: a surface of Gaussian curvature $-3$ with eight
cusps, cf. [16, 17]. The spectrum of the corresponding Laplacian is no longer
discrete.

To the case of a Euclidean cube there corresponds the value $\beta=-1/4$. The
formula (5.21) for the determinant reduces to

$\displaystyle\log{\det\Delta^{4\pi}_{Cube}}|_{\beta=-1/4}=$
$\displaystyle\frac{32}{3}\zeta_{R}^{\prime}(-1)-\frac{37}{18}\log
2+\frac{25}{12}\log 3$
$\displaystyle+\frac{16}{3}\log\Gamma\left(\frac{2}{3}\right)-\frac{86}{9}\log\Gamma\left(\frac{3}{4}\right)+\frac{19}{9}\log\pi.$

######  Example 5.13 (Octahedron).

Now we can deduce yet another formula for the determinant of Laplacian on a
regular octahedron of Gaussian curvature $3\beta+1$ with (six) conical
singularities of order $\beta$. In Theorem 5.11 we take

$\boldsymbol{\beta}=\frac{\beta-3}{4}\cdot 0+\left(-\frac{1}{2}\right)\cdot
1+\left(-\frac{2}{3}\right)\cdot\infty.$

As a result, after an appropriate rescaling we obtain

$\displaystyle\log$
$\displaystyle\,{\det\Delta^{4\pi}_{Octahedron}}=24\Bigl{(}\log\det\Delta_{\boldsymbol{\beta}}+\mathcal{C}_{\boldsymbol{\beta}}\Bigr{)}-\frac{13}{12}\log
3+\frac{71}{18}\log 2$
$\displaystyle+\frac{15}{\beta+1}\Psi\left(\frac{\beta-3}{4},-\frac{1}{2},-\frac{2}{3}\right)+6{\Psi\left(-\frac{1}{2},\frac{\beta-3}{4},-\frac{2}{3}\right)}+\frac{32}{3}\Psi\left(-\frac{2}{3},-\frac{1}{2},\frac{\beta-3}{4}\right)$
$\displaystyle-\frac{1}{2}\left(\beta+1+\frac{1}{\beta+1}\right)\log\frac{8\pi}{3}-6\mathcal{C}\left(\beta\right)-23{\bf
C}+\frac{5}{3}\log\pi.$

###  5.5 Determinant for icosahedral triangulation

The icosahedral Belyi function is given by

$f(x)=1728\frac{x^{5}(x^{10}-11x^{5}-1)^{5}}{(x^{20}+228(x^{15}-x^{5})+494x^{10}+1)^{3}},\quad\deg
f=60.$ (5.23)

The ramification divisor of $\boldsymbol{f}$ is

$\boldsymbol{f}=2\cdot\left\\{20\text{ poles of
}f\right\\}+4\cdot\left\\{12\text{ zeros of }f\right\\}+1\cdot\left\\{30\text{
zeros of }f-1\right\\},\quad|\boldsymbol{f}|=118.$

This defines tessellation of a standard round sphere with bicolored spherical
$(2,3,5)$ double triangle, cf. Fig 4. The $20$ poles of $f$ are the
coordinates of the centers of the faces, the $30$ solutions to $f(x)=1$ are
the edge midpoints, and the $12$ zeros of $f$ ($x=\infty$ is also a zero of
$f$) are the vertices of a regular icosahedron inscribed into the sphere. In
terms of the dodecahedron that is dual to the icosahedron: The poles of $f$
are the coordinates of the $20$ vertices, the $12$ zeros of $f$ are the
centers of its faces, the $30$ solutions to the equation $f(x)=1$ correspond
to the edge midpoints. A picture of the corresponding dessin d’enfant can be
found e.g. in [31, Fig. 1].

For the icosahedral Belyi function evaluation of the right hand side in (2.12)
gives

$C_{f}=\frac{139}{15}\log 2+\frac{63}{10}\log 3+\frac{125}{36}\log 5.$ (5.24)

For the pullback of the divisor $\boldsymbol{\beta}$ by $f$ we have

$f^{*}\boldsymbol{\beta}=(5\beta_{0}+4)\cdot\left\\{12\text{ zeros of
}f\right\\}+(2\beta_{1}+1)\cdot\left\\{30\text{ zeros of
}f-1\right\\}+(3\beta_{\infty}+2)\cdot\left\\{20\text{ poles of }f\right\\}.$

######  Theorem 5.14 (Icosahedral triangulation).

Let $m_{\boldsymbol{\beta}}$ be the unit area Gaussian curvature
$2\pi(|\boldsymbol{\beta}|+1)$ metric of $S^{2}$-like double triangle, see
Section 2.1. Then for the determinant of Laplacian corresponding to the area
$60$ pullback metric $f^{*}m_{\boldsymbol{\beta}}$, where $f$ is the