# A Comprehensive Approach to Characterize Navigation Instruments for Magnetic
Guidance in Biological Systems

Peter Blümler  University of Mainz, Institute of Physics, 55128 Mainz, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>Fabian Raudzus
Kyoto University, Department of Clinical Application, Center for iPS Cell
Research and Application (CiRA), Kyoto, Japan  Kyoto University, Neuronal
Signaling and Regeneration Unit, Graduate School of Medicine, Kyoto, Japan
Friederike Schmid  University of Mainz, Institute of Physics, 55128 Mainz,
Germany

###### Abstract

The non-invasive spatiotemporal control of cellular functions, organization of
tissues, and even the behavior of small animals has become paramount for
advanced therapies. As magnetic fields do not interact with biological matter,
their application is not only suitable for in vitro experiments but also for
in vivo applications, even in deep tissues. Particularly, the remote
manipulation of paramagnetic entities through magnetic instruments has emerged
as a promising approach across various biological contexts.
Despite similarities in basic experimental concepts, variations in the
properties and descriptions of those magnetic instruments among the authors
and studies resulted in a lack of reproducibility and comparability.
Therefore, this article addresses the question of how to standardize the
characterization of magnetic instruments.
Our emphasis lies on the ability of magnetic systems to control the movement
of paramagnetic objects such as ferro- or superparamagnetic particles, within
organisms. This movement is achieved by exerting a force on magnetic particles
by exposing them to a locally varying magnetic field. While it is well-known
that the exerted force depends on the spatial variation (i.e. the gradient) of
the magnetic field, the magnitude of the field is equally important. However,
this second factor is often neglected in the literature. Therefore, we conduct
a comprehensive analysis and discussion of both factors. Furthermore, we
propose a novel descriptor, termed ‘effective gradient’, which combines both
dependencies. To illustrate, we characterize different magnet systems by
calculating and comparing the different quantities and relating them to two
experiments with different superparamagnetic nanoparticles.

###### keywords:

nanoparticle, superparamagnetic, SPIO, ferrofluid, magnetic field, magnetic
flux density, gradient, force, steering, motion

## Introduction

For research, diagnosis, and the application of novel therapies it is often
necessary to spatiotemporally control cellular functions, the organization of
tissues, or the location of substances remotely. Within living organisms,
achieving these goals with minimal interaction with the biological system
presents a significant challenge. Unfortunately, common techniques utilizing
light or ultrasound face limitations due to the energy transfer to the cells,
and depth of tissue penetration. Conversely, techniques based on magnetic
fields present an attractive alternative that is gaining popularity. As most
biological substances such as organic materials and tissues are diamagnetic or
only slightly paramagnetic, they interact very weakly with magnetic fields.
Consequently, biological substances can be penetrated non-invasively, and the
depth of penetration is just limited by the decay of the magnetic field over
distance. On the other hand, objects made from strongly paramagnetic (i.e.
superpara-, ferro- or ferrimagnetic) materials, undergo significant
magnetization in the presence of magnetic fields. This enables their
manipulation by magnetic forces, facilitating remote movement or actuation
within larger diamagnetic environments.
This fundamental principle is currently harnessed for various biological
applications. One notable application is the magnetic guidance of objects
through living organisms for diagnostic or therapeutic purposes. This concept
can be applied to objects spanning a wide range of sizes, from millimeters
(e.g. endoscopes and capsules[1, 2]), micrometers (e.g. microrobots[3, 4, 5,
6]) to nanometers (nanoparticles). The latter are employed in drug delivery
(magnetic drug delivery or targeting [7, 8, 9]), tissue manipulation (by
actuation[10], hyperthermia[11, 12], or other localized therapies[13, 14]) at
target sites. Furthermore, magnetic guidance can be utilized to move cells[15,
16], or cellular extensions such as neurites[17, 18], for applications in
tissue engineering and regenerative medicine[19, 20, 21, 22]. In this context,
the strategy involves incorporating nanoparticles into the cells and
subsequently applying force through the particles within the cells. Typically,
superparamagnetic nanoparticles are used, often coated to improve their
biocompatibility, and may have other functional groups or payloads[23].
Moreover, protein-functionalized superparamagnetic nanoparticles demonstrate
the capacity to modulate intracellular signaling[24, 25], or activate cell
receptors and ion channels[26]. Magnetic guiding can also be used in
microfluidics and nanomechanics[27]. For recent reviews over this very active
and interdisciplinary field, we refer to references[28, 29, 30].
In all of these applications, the generation of a suitable magnetic field
either by resistive or superconducting electro-, or permanent magnets, is a
prerequisite for precise steering, guiding, or actuation. Various guiding
concepts[31, 32, 33] and instruments[34, 35] have been proposed, leading to a
plethora of designs. Comparing such instruments is challenging due to the fact
that two distinct factors contribute to the efficiency of magnetic guidance.
The first is the spatial change of the magnetic field, the so-called gradient,
which predominantly determines the force exerted on a particle carrying an
oriented magnetic dipole. The second is the absolute (local) strength of the
magnetic field, required to induce a dipole in the particle or, in the case
particles with a permanent dipole, to orient them. Both factors must be
jointly considered to predict the resulting magnetic force. Unfortunately, the
second factor tends to be overlooked in literature, which results in confusing
and misleading discussions of magnetic designs. The publication aims to
quantitatively elucidate the interplay between strength and gradient of
magnetic fields in generating the magnetic force on particles. The full
mathematical formalism involves a tensorial description of magnetic gradients,
which complicates the direct comparison of designs. As a practical tool to
compare the efficiency of different magnet types for magnetic guiding, we
propose a modified physical quantity. This quantity can be conceptualized as
an effective magnetic field gradient, presented as a simple three- dimensional
vector instead of a tensor. We discuss the suitability of this effective
gradient is discussed for two commonly used magnet geometries in magnetic
guiding.

symbol | terminology | units | conversion
---|---|---|---
$\vec{B}$ |  magnetic flux density, magnetic induction |  T = Vs/m = kg/(As2) | $\vec{B}=\mu_{0}(\vec{H}+\vec{M})=\mu\vec{H}$
$\chi$ | magnetic susceptibility |  | $\chi=M/H$
$\vec{H}$ | magnetic field, magnetizing field | A/m |
$k_{\text{B}}$ | Boltzmann constant | J/K |  $k_{\text{B}}\equiv 1.380649\cdot 10^{-23}$ kg m2/(K s2)
$\mu$ | permeability | Vs/(Am) | $\mu=\mu_{0}(1+\chi)$
$\mu_{0}$ | permeability of vacuum | Vs/(Am) |  $\mu_{0}\equiv 1.25663706212\cdot 10^{-6}$ kg m/(As)2 $\approx 4\,\pi\cdot 10^{-7}$ kg m/(As)2
$\vec{m}$ | magnetic moment | Am2 | $\vec{m}=\iiint\vec{M}\text{d}V\approx\;M_{\text{s}}V$
$\vec{M}$ | magnetization | A/m | $\vec{M}=\text{d}\vec{m}/\text{d}V\approx\;\vec{m}/V$
$\vec{\nabla}$ | spatial gradient | 1/m | $\vec{\nabla}=(\partial/\partial x,\;\partial/\partial y,\;\partial/\partial z)$
$r$ | radius (of a particle) | m |
$T$ | absolute temperature | K |
$V$ | volume | m3 | sphere: $V=4/3\pi r^{3}$
Table 1: Used symbols and corresponding SI unit in this manuscript. It has to
be mentioned that magnetic field, $H$, and magnetic flux density, $B$, are
used synonymously in the text to adapt to common nomenclature of most
publications. When not specifically mentioned this refers to magnetic flux
density.

## Theory

### Average magnetization of superparamagnetic nanoparticles

At room temperature, a ferro- or ferrimagnetic material consists of small
volume regions known as Weiss domains, wherein the magnetic dipole moments of
strength $\vec{m}_{i}$, are mutually aligned in the same direction. The
density of the sum of all magnetic moments yields the magnetization (see Tab.
1)

$\vec{M}=\frac{1}{V}\sum_{i}\vec{m}_{i}.\\\ $ (1)

These domains are microscopic in size and separated by (Bloch-) walls. Larger
objects consist of numerous such domains (see Fig. 1) with different
magnetization directions. The total magnetization $\langle\vec{M}\rangle$ of
the object is the vector sum of the magnetizations of all domains weighted by
their volume. Figure 1 shows how an external magnetic field induces shifts of
the domain walls, thereby enlarging domains with magnetization components in
the same direction as the magnetic field.
This process leads to an increase in total magnetization from zero
($\langle\vec{M}(\vec{B}=0)\rangle$) in the absence of a magnetic field –
where all subdomain magnetizations cancel each other – to maximum (saturation)
magnetization ($\langle|\vec{M}|\rangle=M_{\text{s}}$) at high magnetic
fields. Each individual domain is fully magnetized ($m_{i}=M_{\text{s}}V_{i}$
where $V_{i}$ is the size of the domain), but their directional average
$\langle\vec{M}(\vec{B})\rangle$ depends on the magnetic field.
When the size of the object is of the order of such a domain (typically $r<30$
nm), the object eventually consists of only one single, fully magnetized
domain. However, due to their small size, their rotational Brownian motion
will have a similar effect on the thermally averaged magnetization,
$\langle\vec{M}(\vec{B})\rangle$, as illustrated on the right side of Fig. 1.
The resulting value can be approximated by a Langevin-function,
$\mathfrak{L}(x)$ [36, 37]: The average magnetization $\langle\vec{M}\rangle$
points in the direction of $\vec{B}$, and its amplitude is given by

$\begin{split}&\big{|}\langle\vec{M}(\vec{B})\rangle\big{|}\;\;=\;\;\frac{m}{V}\,\mathfrak{L}\left(\frac{mB}{k_{\text{B}}T}\right)\;\;\approx\;\;M_{\text{s}}\mathfrak{L}\left(\frac{M_{\text{s}}VB}{k_{\text{B}}T}\right)\\\\[8.61108pt]
&\text{where}\quad\mathfrak{L}(\xi)\approx\coth{\xi}-\frac{1}{\xi}\quad\text{and}\quad\xi=\frac{M_{\text{s}}V}{k_{\text{B}}T}B\quad\text{with}\quad
B=\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}.\end{split}$ (2)

While each particle is fully magnetized, its time-averaged magnetization
follows eq. (2). This behavior is known as superparamagnetism[38]. Chemical
stabilization is required to avoid agglomeration due to mutual dipolar
attraction of the particles[39]. Superparamagnetic particles form the basis of
ferrofluids[40] and find numerous applications in theranostics[41].
Independent of their size and ignoring hysteresis effects in the case of
multi-domain particles, the macroscopic magnetization
$\langle\vec{M}(\vec{B})\rangle$ depends on the field according to eq. (2) and
is also strongly dependent on $M_{\text{s}}$ and the size/volume $V$ of the
magnetic particle (MP).
This is illustrated by Fig. 2 which demonstrates that the field dependence of
$\langle\vec{M}(\vec{B})\rangle$ or the Langevin-function is linear for very
small fields and its steepness strongly depends on the particle size. The
constant saturated region where $M\approx M_{\text{s}}$ is reached at lower
magnetic fields for larger particles. At higher magnetic fields, the behavior
of small particles varies more strongly as a function of field strength than
that of larger particles. This also implies that in the same magnetic
instruments, smaller particles exhibit a stronger field dependence. We
emphasize that eq. (2) relates the amplitudes of $\vec{M},\vec{m},\text{and
}\vec{B}$ to each other, despite their status as spatial vectors. As noted
above, $\langle\vec{M}\rangle$ is parallel to $\vec{B}$. Given that MPs carry
a permanent magnetization, the magnetic field induces a torque, causing a
rotation of the whole particles to align their magnetic moments with
$\vec{B}$. As a consequence, it can be assumed that (with $\cdot$ being the
dot or scalar product of two vectors)

Figure 1: Schematic illustration of the magnetization process of large
particles with several domains (gray circles left of curve) and
superparamagnetic nanoparticles (light blue squares right of curve). The thick
blue line shows the average magnetization as a function of applied magnetic
field, which points to the right. The curve corresponds to the positive
quadrant of Fig. 2. Six snapshots illustrate the growth of domains or particle
orientation along the field direction. The resulting averaged magnetization
$\langle\vec{M}(\vec{B})\rangle$ is sketched by the blue arrows, their lengths
correspond to the value on the ordinate. The red arrows indicate the
magnetization of each domain or particle.

$\langle\vec{M}\rangle\parallel\vec{B}\quad\text{or}\quad\langle\vec{M}\rangle\cdot\vec{B}\;=\;\big{|}\langle\vec{M}\rangle\big{|}\,\big{|}\vec{B}\big{|}\;\;\approx\;\;M_{\text{s}}\,\mathfrak{L}(\xi)\,B,\\\
$ (3)

where $\xi$ is defined as in eq. (2).

Figure 2: Field dependence of the magnetization
$\langle\vec{M}(\vec{B})\rangle$ of superparamagnetic iron-oxide MPs with two
different sizes $r=15$ nm (blue) and $r=25$ nm (red) with $M_{\text{s}}=480$
kA/m at $T=300$ K (values from[42]). The solid lines show
$\langle\vec{M}(\vec{B})\rangle$ as approximated by eq. (2) or
$\mathfrak{L}(\xi)$. The dashed lines show $\mathfrak{P}(\xi)$ in comparison.
($\mathfrak{L}(\xi)$ is defined in eq. (9) and describes the amplitude of the
magnetic force as a function of applied field for a given magnetic gradient).

### The magnetic field and its gradient

Another ingredient for calculating the force is the spatial variation of the
external magnetic field. It is quite intuitive that magnetized MPs experience
no drag in a perfectly homogeneous magnetic field, much like a ball on a
perfectly flat plate, as moving the particle to different position changes
nothing. Drag forces only arise when there is a change in potential energy
with respect to space. In the example of the ball, the potential energy is
proportional to its height, and a force can be induced by tilting the plane,
such that the height depends on the lateral position. Analogously, the theory
of electromagnetism tells us that we can assign a ‘magnetic potential energy’
to a particle with magnetic moment $\vec{m}$ in a magnetic field $\vec{B}$,
given by $\vec{E}=\vec{B}\vec{m}$ [43]. This allows us to induce a magnetic
force on the MPs by spatial variations in the magnetic field, $\vec{B}$.
However, the situation is complicated by the fact that the magnetic field has
three components that can all change with space. The spatial variation must be
described in terms of a second rank tensor (with $\otimes$ being the tensor
product)

$\overline{\overline{G}}\;\;=\;\;\mathbf{grad}\,\vec{B}\;\;=\;\;\vec{\nabla}\,\otimes\,\vec{B}\;\;=\;\;\left(\begin{array}[]{c
c
c}\frac{\partial{B_{x}}}{\partial{x}}&\frac{\partial{B_{x}}}{\partial{y}}&\frac{\partial{B_{x}}}{\partial{z}}\\\\[4.30554pt]
\frac{\partial{B_{y}}}{\partial{x}}&\frac{\partial{B_{y}}}{\partial{y}}&\frac{\partial{B_{y}}}{\partial{z}}\\\\[4.30554pt]
\frac{\partial{B_{z}}}{\partial{x}}&\frac{\partial{B_{z}}}{\partial{y}}&\frac{\partial{B_{z}}}{\partial{z}}\\\
\end{array}\right)$ (4)

where $\partial{B_{i}}/\partial{x}$ etc. are partial derivatives. Fortunately,
as we will soon see, the full knowledge of the tensor is not necessary when
calculating the magnetic force on MPs. Nevertheless, eq. (4) demonstrates that
a magnetic system cannot be described by a single value for a magnetic field
gradient. Indeed, it is not even possible to generate a single gradient
outside magnets (such as, e.g., ($\partial{B_{x}}/\partial{x}\neq 0$ , with
all other partial derivatives in $\mathbf{grad}\,\vec{B}$ being zero). This is
forbidden by the Maxwell equations for electromagnetism, i.e. (with $\times$
as the cross product of two vectors)

$\begin{split}\mathbf{div}\,\vec{B}\;\;&=\;\;\vec{\nabla}\,\cdot\,\vec{B}\;\;=\;\;\frac{\partial{B_{x}}}{\partial{x}}+\frac{\partial{B_{y}}}{\partial{y}}+\frac{\partial{B_{z}}}{\partial{z}}\;\;&=\;\;0,\\\\[12.91663pt]
\mathbf{rot}\,\vec{B}\;\;&=\;\;\vec{\nabla}\,\times\,\vec{B}\;\;=\;\;\left(\begin{array}[]{c}\frac{\partial{B_{z}}}{\partial{y}}-\frac{\partial{B_{y}}}{\partial{z}}\\\\[4.30554pt]
\frac{\partial{B_{x}}}{\partial{z}}-\frac{\partial{B_{z}}}{\partial{x}}\\\\[4.30554pt]
\frac{\partial{B_{y}}}{\partial{x}}-\frac{\partial{B_{x}}}{\partial{y}}\\\
\end{array}\right)\;\;&=\;\;0.\end{split}$ (5)

Thus, each gradient must be accompanied by at least another gradient of equal
strength and opposite direction or sign. A single magnetic field gradient
cannot exist!

### Magnetic force

As discussed in the previous section, the force on a MP is given by the
gradient of the magnetic potential energy. In the context of an MP with
magnetic moment $\vec{m}$ subjected to a magnetic field of strength $\vec{B}$
, this potential energy is given by[43].

$\vec{F}_{\text{m}}\;\;=\;\;\vec{\nabla}(\vec{m}\,\cdot\,\vec{B}).\\\ $ (6)

Averaging over the distribution of $\vec{m}$ and using eq. (2), one obtains

$\langle\vec{F}_{\text{m}}\rangle\;\;=\;\;\vec{\nabla}(V\langle\vec{M}\rangle\,\cdot\,\vec{B}).\\\
$ (7)

From eq. (3) it is known that $\langle\vec{M}\rangle$ and $\vec{B}$ are
parallel. This presents a challenge in magnetic guiding, as the dot product
cancels the sign of the magnetic field, causing MPs to be constantly driven in
the direction of regions with higher magnetic fields. This is completely
analogous to paper clips being equally attracted by the south and north pole
of a permanent magnet (never repelled!). However, it allows us to re-express
eq. (7) as (here $\xi=\frac{M_{\text{s}}V}{k_{\text{B}}T}\,B$ is defined as in
eq. (2))

$\begin{split}\langle\vec{F}_{\text{m}}\rangle\;\;&=\;\;\vec{\nabla}\left(M_{\text{s}}V\,\mathfrak{L}(\xi)\,B\right)\;\;=\;\;M_{\text{s}}V\,\mathfrak{L}(\xi)\,\vec{\nabla}B\;+\;BM_{\text{s}}V\;\vec{\nabla}\mathfrak{L}(\xi)\\\\[8.61108pt]
&=\;\;\left(M_{\text{s}}V\,\mathfrak{L}(\xi)\;+\;B\,\frac{\text{d}\mathfrak{L}}{\text{d}\xi}\,k_{\text{B}}T\right)\vec{\nabla}B\;\;=\;\;M_{\text{s}}V\left(\mathfrak{L}(\xi)\;+\;\xi\,\frac{\text{d}\mathfrak{L}}{\text{d}\xi}\right)\vec{\nabla}B\\\
\end{split}$ (8)
$\begin{split}\quad\quad\;&=:\;\;M_{\text{s}}V\;\mathfrak{P}(\xi)\left(\tfrac{\partial{B}}{\partial{x}},\;\tfrac{\partial{B}}{\partial{y}},\;\tfrac{\partial{B}}{\partial{z}}\right)\;\;=\;\;k_{\text{B}}T\;\mathfrak{P}(\xi)\left(\tfrac{\partial{\xi}}{\partial{x}},\;\tfrac{\partial{\xi}}{\partial{y}},\;\tfrac{\partial{\xi}}{\partial{z}}\right)\\\\[8.61108pt]
\text{with}&\quad\mathfrak{P}(\xi)\;:=\;\coth(\xi)\;-\;\xi\text{csch}^{2}(\xi)\quad\text{and}\quad
B\;=\;|\vec{B}|\;=\;\sqrt{B^{2}_{x}+B^{2}_{y}+B^{2}_{z}},\\\ \end{split}$ (9)

Compared to eq. (6), this equation is much simpler, as it no longer requires
the calculation of the complicated gradient tensor (as seen in eq. (4)). Due
to the loss of direction in the dot product of eq. (3), the magnetic force now
only depends on the rescaled strength of the magnetic field, $\xi$, and its
gradient $\vec{\nabla}\xi$, a simple vector. However, we have introduced a new
non-linear, material- and field-dependent function $\mathfrak{P}(\xi)$, which
accounts for the fact that the average magnetization depends on the magnetic
field. It is often neglected by assuming
$\langle\vec{M}(\vec{B})\rangle=\vec{M}_{\text{s}}$, i.e.,
$\mathfrak{P}(\xi)=1$. We emphasize that $\mathfrak{P}(\xi)$ differs from
$\mathfrak{L}(\xi)$ in eq. (2). The reason is that the total force on the MPs
in a magnetic field gradient has two contributions: The first is the direct
averaged magnetic force on MPs with average magnetization
$\langle\vec{M}(\vec{B})\rangle=\vec{M}_{\text{s}}\mathfrak{L}(\xi)$ (the
first term in eq. (8)), and the second is a thermodynamic force accounting for
their drift to regions in space where their average magnetization would be
energetically more favorable (the second term in eq. (8)). A comparison of
$\mathfrak{L}(\xi)$ and $\mathfrak{P}(\xi)$ is shown in Fig. 2. Both functions
have a similar shape, but $\mathfrak{P}(\xi)$ reaches saturation much more
rapidly and is twice as steep at the origin.

$\frac{\text{d}\mathfrak{P}(0)}{\text{d}B}\;\;=\;\;2\;\frac{\text{d}\mathfrak{L}(0)}{\text{d}B}\;\;=\;\;\frac{2}{3}\;\frac{M_{\text{s}}V}{k_{\text{B}}T}.\\\
$ (10)

Equation (9) shows that two conditions, $|\vec{B}|\neq 0$ and
$\vec{\nabla}|\vec{B}|\neq 0$, must be fulfilled to enable magnetic steering
of nanoparticles. Specifically, both the gradient of the field
$\vec{\nabla}|\vec{B}|$ and the field-induced average magnetization
$\langle\vec{M}(\vec{B})\rangle$ must be non-zero. This complicates the
characterization of magnetic systems in their ability to move MPs, because
$\langle\vec{M}(\vec{B})\rangle$ depends on the nature of the used MP. The
fact that the magnetic force is governed by $\mathfrak{P}(\xi)$ rather than
$\mathfrak{L}(\xi)$, the function that describes the average magnetization of
the particles, somewhat eases the problem. The average magnetization in eq.
(2) will only reach a constant for $B$-values close to saturation. For iron
oxide, this saturation is reached for magnetic fields with local strengths of
around 1 T. However, the curve in Fig. 2 shows that concerning the resulting
magnetic force, one may approximate almost constant behavior
($\mathfrak{P}(\xi)=1$) already at approximately 5 mT for MPs with radius of
ca. $r\approx 20$ nm.
Nevertheless, a local magnetic field must be present to magnetize the MPs, and
it must have a strong gradient to exert a force on them.

### Effective gradient and directionality

Given that magnetic guiding instruments need to be evaluated for their
effectiveness, a comprehensive characterization would involve a three-
dimensional description or dataset of their magnetic flux density. However,
comparing different designs on that ground is rather impractical, and some
simple descriptors are needed. Obviously, summarizing the information of a
general vector field in just a few numbers is impossible. Even if it were
possible, an exact prediction of the force amplitude generated by this
magnetic field is not feasible without knowledge of the characteristics of the
MPs to be used in the device. This is due to the non-linearity of eq. (2) and
the size-dependence of the magnetization, as illustrated in Fig. 2.
Nonetheless, it is highly desirable to extract some more meaningful quantities
than, for instance, the maximal magnetic flux density and its maximal
gradient. Therefore, we propose here to use a quantity that contains
information on both the strength and the gradient of the field. This quantity,
which we refer to as “effective gradient”, $K$, is defined by the expression:

$\vec{K}(x,y,z)\;\;:=\;\;\frac{|\vec{B}(x,y,z)|}{|\vec{B}(x,y,z)|}_{\text{max}}\;\vec{\nabla}|\vec{B}(x,y,z)|\;\;=\;\;\frac{B}{\text{max}(B)}\;\vec{\nabla}B,\\\
$ (11)

where $|\vec{B}|_{\text{max}}=\text{max}(B)$ is the maximal field strength in
the sample volume. Equation (11) combines amplitude and gradient of the
magnitude of the magnetic field. The normalization with
$|\vec{B}|_{\text{max}}$ has the advantage that $K$ has the units of a field
gradient [T/m] and multiplication with $M_{\text{s}}V$ will give a rough
estimate of the force that can be expected for a certain MP (assuming that
$|\vec{B}|_{\text{max}}$ is close to the saturation field strength). As
discussed earlier, this does not replace a detailed analysis, but the average,
minimal, and maximal value (and possibly its standard deviation) of
$|\vec{K}|$ in a specified sample volume should offer a good estimate of force
distribution that can be expected in a magnetic system.
A second potentially important information on this magnetic vector field is
the uniformity, $\vec{U}$, of the direction of the generated magnetic force.
We define it as

$\vec{U}\;\;:=\;\;\frac{\iiint\vec{K}\;\text{d}V}{V\,|K_{xyz}|^{(V)}_{\text{max}}}\;\;\approx\;\;\frac{\sum_{n}\vec{K}}{n\,|K_{xyz}|^{V}_{\text{max}}}\quad\text{with}\quad|K_{xyz}|^{(V)}_{\text{max}}\;\;:=\;\;\text{max}\left(|K_{x}|,\,|K_{y}|,\,|K_{z}|\right)_{V},\\\
$ (12)

i.e., the integral (or, in case of discrete measurements, the sum) over all
components of the effective gradient in eq. (11), normalized by the volume (or
the total number of discrete entries) and $|K_{xyz}|^{(V)}_{\text{max}}$ which
represents the maximal magnitude of all entries, $n$, of all components of
$|\vec{K}|$ in that volume. For instance, $\vec{U}=(1,0,0)$ would indicate
that the effective gradient points exclusively in the $x$-direction. Values
less than one indicate non-uniformities in the directionality.

Figure 3: Magnets used in the experiment: (A) photograph of all components
used: (1) Petri dish (70 mm diameter), (2) 3D-printed distance ring, (3)
Halbach-quadrupole, and (4) Halbach-dipole. All in front of transparent scale
paper (small boxes: 1 mm2, larger boxes. (5 mm)2). (B) Setup of Petri dish
inside the Halbach quadrupole, which provides gradients of ca. 200 mT/m. (C)
Same setup as in B) but enclosed by the additional Halbach dipole, which
generates a relatively homogeneous field of 7.7 mT across the sample. The
combination of di- and quadrupole is named a MagGuider[44] magnet. For more
details about the magnetic field, see Figs. 4 and 5, and Tab. 2.

## Experiment

To illustrate the key points of the prior discussion, we compare the magnetic
fields of two distinct magnet designs acting on MPs like those discussed in
Fig. 2. The first magnetic field is that of a Halbach quadrupole (cf. Fig. 3B)
and the second is the same quadrupole combined with a Halbach dipole, forming
an instrument called MagGuider[44] (cf. Fig. 3C). The pure quadrupole design
was chosen for its provision of strong and homogeneous gradients, commonly
used in experiments involving magnetically manipulated MPs or MP-loaded
cells[45, 46, 47]. However, the magnetic field at its center is weak
(theoretically zero). Therefore, it is an ideal magnetic pattern to illustrate
the difference between magnetic field gradients and force.

Figure 4: Magnetic properties of the quadrupole in Fig. 3B. The magnetic field
components, $B_{x}$ in (B) and $B_{y}$ in (C) were measured for a field of
view of $72\times 72\;\text{mm}^{2}$, for all graphs with $x$-axis in
horizontal and $y$-axis in vertical direction. All the other graphs were
calculated from these input data to display the features for the different
entities discussed in the theoretical section. The first column shows a vector
plot (direction given by arrows, amplitude by their length) of the field with
reduced resolution, followed by the $x$ and $y$ component and their magnitude
in the last column. The first row shows the magnetic flux density, $\vec{B}$,
in mT, followed by the gradient, $\vec{G}$ [mT/m], of its magnitude in (D).
The third row shows the force, $\vec{F}$, [N], exerted on MPs with the
properties of Fig. 7 given in atto-Newton (i.e. 10-18 N) and calculated from
eq. (9). The last row shows the effective gradient, $\vec{K}$ [mT/m], as
defined in eq. (11).

The MagGuider-system is a combination of a quadrupole with a dipolar magnet,
which additionally provides a homogeneous background magnetic field. It is
designed to overcome the problem of low magnetic forces in certain regions.
Since the homogeneous field ideally has no gradients (and only small gradients
in reality), its combination with the quadrupole leads to almost identical
field gradients for both systems. If the geometry is properly chosen (i.e.,
the condition $B>Gr$ must be fulfilled at a distance $r$ from the center, with
$B$ being the strength of the homogeneous dipolar field and $G$ the constant
gradient of the quadrupole), it produces a magnetic force field which is
homogeneous both in amplitude and direction. The principle is explained
elsewhere[31, 44]. Both magnets are shown in Fig. 3. They were designed to
generate rather low magnetic fields, to illustrate the problems associated
with them.

Figure 5: Magnetic properties of the MagGuider system in Fig. 3C. Identical
presentation as in Fig. 4.

Figures 4 and 5 show the magnetic properties of both magnet systems, derived
from two dimensional measurements of the $B_{x}$ and $B_{y}$ component over
the size of the utilized Petri dish. In both figures, the first three columns
help to understand the spatial features of the vector field and its two
components, however, for the discussion the last column of the magnitude is
sufficient. In the case of the pure quadrupole system, Fig. 4H shows the
gradient of the magnetic field magnitude in Fig. 4D (cf. discussion to eq.
(3)). This gradient is nearly constant, representing a high degree of
homogeneity. However, the magnitude of the magnetic field, has radial
dependence and is zero at the center as shown in Fig. 4D. As a consequence,
the magnetization of the MPs and hence the magnetic force acting on them is
very weak in the center with a strongly non-linear increase radially outwards
(see Fig. 4L). This can be alleviated if larger MPs are used as discussed in
the context of Fig. 2. The effective gradient is defined by eq. (11) and shown
in Fig. 4P. This figure shows similar features to $|\vec{B}|$ in Fig. 4D
(because its gradient is constant) but has units of a gradient. Hence, it
represents the influence of the magnetic field on the force on any MP better
than the field gradient (Fig. 4H) but overemphasizes the narrow funnel shape
in Fig. 4L for these particular MPs.
The magnetic properties of the MagGuider system (Fig. 3C) are portrayed in
Fig. 5. Due to the superposition of the homogeneous gradient field of the
quadrupole with the homogeneous magnetic field of the dipole, the negative
field, the gradient, and hence force components in the $y$-direction are
eliminated. Consequently, the MPs are guided exclusively in this direction
(here upwards) of the magnet system. For this particular system, the "guiding
condition" ($B(\approx 8\;\text{mT})\gg Gr(\approx 5.7\;\text{mT}))$[44] is
just barely fulfilled. One can see stronger directional deviations to the
sides for $x\rightarrow\pm\;35$ mm, e.g. in Fig. 5I. Nevertheless, it has
relatively high uniformity along the $y$-direction, $\vec{U}=(-0.07,0.38)$, in
particular in comparison to the quadrupole, where the uniformity is
essentially zero. The presence of the homogeneous magnetic background field
throughout the sample volume results in much less variation of the spatial
features of G, K, and F than in the pure quadrupole system (cf. second to
forth row in Fig. 5), as intended by the MagGuider design principle.

Figure 6: (A) Initial state of a thin layer of oily ferrofluid of 10 nm iron
particles on top of water inside the Petri dish. (B) Exposed to the magnetic
field of the MagGuider magnet (cf. Figs. 3C and 5). Shown is the equilibrium
state after ten seconds. (C/D) Exposed to the quadrupole only (cf. Fig. 3B and
4). (C) Photograph taken ca. 1 second after application of the magnetic field.
(D) Equilibrium state after a minute. Figure 7: Behavior of a dispersion of
water soluble, superparamagnetic nano-particles (A) in the quadrupole, (B) in
the MagGuider system. Shown are photographs at different times (hour:min) as
indicated. Only central slices are concatenated except for the first image
where half the Petri dish is shown. The full Petri dish is shown for the final
(equilibrium) state on the right. In B) the particle concentration is
indicated by a blue arrow. (C) Integration over the image intensities in a
central area of ca. 2.6 cm2 and normalization to a range between 0 and 1
(black: MagGuider, red: quadrupole). Here the intensity = 1 corresponds to the
initial particle concentration (cf. A/B image at 0:00) and 0 to the value
obtained from an empty Petri dish. The lines correspond to a biexponential fit
with the formula and the fit results given in the insert.

Figure 6 provides examples of the distribution of 10 nm superparamagnetic
iron-particles dispersed in light mineral oil and suspended on water at
different times after exposure to the magnetic field of the quadrupole (Fig.
33B and 4) and the MagGuider (Figs. 3C and 5). This qualitative representation
illustrates the interaction of the particles with the magnetic field of the
two devices. In the MagGuider system, the initially uniformly distributed MPs
in Fig. 6A concentrate within one second in the direction of the magnetic
force (Fig. 6B). Conversely, in the pure quadrupole, a fraction of particles
initially moves (cf. Fig. 6C) towards the four poles, while a substantial
portion remains in the center (“cross- shaped” feature in Fig. 6C), since the
magnetic field (not the bare gradient) is much smaller there. In equilibrium
(Fig. 6D), the ferrofluid concentrates at large radii, with a distribution
that is mostly independent of the angle. This state is reached after
approximately ten seconds. Note that, just from inspecting the forces acting
on the individual MPs (cf. Tab. 2), one would not expect such a rapid motion.
However, the velocity is not only determined by the force, but also by the
hydrodynamics and the morphology. Due to the phase separation of water and
ferrofluid, larger drops behave collectively (like a big particle) and slide
with very low drag on top of the water surface.
To conduct a more quantitative analysis, water-soluble superparamagnetic
nanoparticles[48] were used. The results are summarized in Fig. 7. Images of
these particles dispersed in water were captured over a period of more than 70
hours inside the quadrupole of Fig. 3B. The first image on the left shows half
of the Petri dish at time $t=0$ (initial distribution), the subsequent shows a
central slice at different times as indicated, and the last one the final
distribution in the entire Petri dish. From the slices it can be recognized
that the initial brownish color of the dispersion becomes brighter with time
as more and more particles are dragged towards the walls. This happens much
faster in the case of the MagGuider systems, as shown in Fig. 7B (same mode of
presentation). Please note the very different time scale in both figures. In
the final state the particles end up at all walls in the case of the pure
quadrupole, and at the top rim (indicated by the blue arrow) in the case of
the MagGuider as already shown in Fig. 6.
In order to analyze this more quantitatively, the intensity of a central
region of each image was integrated and normalized with respect to an image of
the empty Petri dish and the intensity of the initial particle concentration.
Fig. 7C shows the results, which were fitted by a double exponential decay.
The fast decay is attributed to the drag exerted on the nanoparticles by
magnetic forces. The slower decay reflects the motion of some remaining slower
particles, which might be slowed down by interactions with the bottom of the
dish, or -– in the case of the quadrupole, an unfavorable trajectory:
Particles that initially stay in the low field in the center first have to
diffuse into a regions with stronger magnetic forces (diffusing water needs
ca. 2200 min or 37 h to cover a distance of 35 mm). Both the time constant of
the initial fast intensity decay and the subsequent slower decay are nearly an
order of magnitude smaller for the MagGuider system, even though the gradients
are almost identical in both systems (cf. Tab. 2).
In Tab. 2, we attempt to characterize the relevant properties of both magnetic
systems by the descriptive numbers discussed above, namely, $|\vec{B}|=B$,
$|\vec{G}|=G$, and $|\vec{K}|=K$, and test their use for predicting the
magnetic force on the two different particles used in Figs. 6 and 7. The most
obvious discrepancy arises from the much lower force measured for the smaller
particles ($r=5$ nm) used in Fig. 6. Here the predictions via
$M_{\text{s}}V\;|\vec{G}|$ or $M_{\text{s}}V\;|\vec{K}|$ deviate by one or two
orders of magnitude from the calculated force for both magnet systems. This is
because the magnetic field in both systems is too low to fully magnetize the
MPs, and resulting in $\overline{\mathfrak{P}}(\xi)=10^{-3}$ for the
quadrupole and $\overline{\mathfrak{P}}(\xi)=0.02$ for the MagGuider (where
$\overline{\mathfrak{P}}(\xi)$ is the averaged value in Tab. 2). This
illustrates the importance of knowing the local magnetic fields. The
predictions using $M_{\text{s}}V\;|\vec{K}|$ instead of
$M_{\text{s}}V\;|\vec{G}|$ are slightly better, but still differ from
$|\vec{F}|=F$, in particular when looking at the minimum value. The same holds
for the larger particles ($\overline{\mathfrak{P}}(\xi)\approx 1$), where the
value of $M_{\text{s}}V\;|\vec{G}|$ is identical with the force for the
MagGuider, but again overestimate the minimal force by an order of magnitude
for the quadrupole. The best description is realized by giving the range of
all three quantities $B$, $G$, and $K$, the latter because it connects the
local information on the other two. Without this information (e.g., by only
giving the gradient strengths of the systems, which are close to identical for
both of them), the very different behavior of MPs as in Fig. 7C can neither be
explained nor predicted.

magnet | quantity | average | std. dev. | max. | min.
---|---|---|---|---|---
quadrupole |  $|\vec{B}|$ [mT] | 2.08 | 0.77 | 3.34 | 0.01*
$|\vec{G}|$ [mT/m] | 168.1 | 14.9 | 207.2 | 8.4
$|\vec{K}|$ [mT/m] | 107.7 | 46.6 | 207.2 | 0.02*
$|\vec{F}|\;\;[10^{-22}\;\text{N}]\;(r=5\;\text{nm})$ | 0.16 | 0.07 | 0.31 | $3\,\cdot\,10^{-5}\text{*}$
$M_{\text{s}}V|\vec{G}|\;\;[10^{-22}\;\text{N}]\;(r=5\;\text{nm})$ | 28.1 | 2.5 | 34.7 | 1.4
$M_{\text{s}}V|\vec{K}|\;\;[10^{-22}\;\text{N}]\;(r=5\;\text{nm})$ | 18.0 | 7.8 | 34.7 | 0.003*
$|\vec{F}|\;\;[10^{-18}\;\text{N}]\;(r=65\;\text{nm})$ | 23.2 | 2.1 | 28.6 | 0.2*
$M_{\text{s}}V|\vec{G}|\;\;[10^{-18}\;\text{N}]\;(r=65\;\text{nm})$ | 23.2 | 2.1 | 28.6 | 1.1
$M_{\text{s}}V|\vec{K}|\;\;[10^{-18}\;\text{N}]\;(r=65\;\text{nm})$ | 14.9 | 6.4 | 28.6 | 0.003*
MagGuider |  $|\vec{B}|$ [mT] | 7.96 | 1.54 | 11.59 | 5.09
$|\vec{G}|$ [mT/m] | 163.6 | 39.9 | 287.2 | 108.9
$|\vec{K}|$ [mT/m] | 117.6 | 52.1 | 287.2 | 49.5
$|\vec{F}|\;\;[10^{-22}\;\text{N}]\;(r=5\;\text{nm})$ | 0.62 | 0.27 | 1.50 | 0.26
$M_{\text{s}}V|\vec{G}|\;\;[10^{-22}\;\text{N}]\;(r=5\;\text{nm})$ | 27.4 | 6.7 | 48.1 | 18.2
$M_{\text{s}}V|\vec{K}|\;\;[10^{-22}\;\text{N}]\;(r=5\;\text{nm})$ | 19.7 | 8.7 | 48.1 | 8.3
$|\vec{F}|\;\;[10^{-18}\;\text{N}]\;(r=65\;\text{nm})$ | 22.6 | 5.5 | 39.6 | 15.0
$M_{\text{s}}V|\vec{G}|\;\;[10^{-18}\;\text{N}]\;(r=65\;\text{nm})$ | 22.6 | 5.5 | 39.6 | 15.0
$M_{\text{s}}V|\vec{K}|\;\;[10^{-18}\;\text{N}]\;(r=65\;\text{nm})$ | 16.2 | 7.2 | 39.6 | 6.8
Table 2: Characterization of the two magnet systems (quadrupole and MagGuider)
by a few descriptive values representing the full distributions shown in Figs.
4 and 5. From those the arithmetic average, standard deviation, maximal and
minimal value are determined over a central circular area of 40 mm in
diameter. This was done for magnitude of the magnetic flux density
$|\vec{B}|$, its gradient, $|\vec{G}|$, effective gradient $|\vec{K}|$. For
the two particles used in Figs. 6 and 7 the magnetic force $|\vec{B}|$ (using
eq. (9)) and two simplified (scalar) values ($M_{\text{s}}V\,|\vec{G}|$ and
$M_{\text{s}}V\,|\vec{K}|$ ) are compared. The directionality is
$\vec{U}=(2,0.3)\,\cdot\,10^{-3}$ for the quadrupole and
$\vec{U}=(-0.07,0.38)$ for the MagGuider magnet. (*the value of the minimum
must be zero, because $|\vec{B}(0,0)|=0$. The non-zero values are due to
insufficient discretization of the field and/or imprecise measurements.)

## Discussion

In the application of magnetic systems for magnetic guiding, it is crucial to
have a standardized characterization and description. This not only aids in
comprehending, interpreting, comparing, and replicating results, but also
provides a more comprehensive assessment than relying on the maximal field
gradient alone. This is because both the thermally averaged magnetization and
the field gradient contribute to the overall force. While the average
magnetization is an inherent material property of the used MPs, it is also
dependent on the magnetic field strength. Consequently, both the magnitude of
the magnetic flux and its gradient are essential in evaluating a device’s
capacity to exert a force on MPs. This is illustrated by two magnet systems,
that possess similar gradient strengths but exhibit a tenfold difference in
their ability to move MPs. Therefore, we propose to introduce a different
quantity termed ‘effective gradient’, which describes the joint spatial
characteristics of $|\vec{B}|$ and $\vec{\nabla}|\vec{B}|$. The spatial
distribution of this "effective gradient" can provide a rough estimate of the
generated magnetic force and can be approximated using typical descriptors of
distributions like average, standard deviation, minimum, and maximum over the
volume of interest. Additionally, providing information about the magnetic
field and gradient separately can further enhance characterization.
Furthermore, the orientational homogeneity of the force can be described by a
directionality vector. We hope that these suggestions will facilitate a better
comparison of instruments designed for magnetic guiding.
Although magnetic forces acting on single nanoparticles are in the sub-
attonewton regime and might appear extremely low, they are sufficient to
induce aggregate formation depending on the coating and concentrations of MPs
due the attraction via dipole forces. The force on such an aggregate is then
the multiple of the force on a single MP, scaled by the number of particles in
the aggregate (cf. the ferrofluid drops in Fig. 6).
This underscores that even knowing the magnetic force alone is insufficient to
fully describe the ability of a system for magnetic guiding, because it is the
motion of the MPs, which is of interest. The relevant quantities are the
velocity and acceleration of the MPs. To determine them becomes a
hydrodynamical problem, which can be described by Stokes’ law for MPs or MP
aggregates in liquids (for details and the hydrodynamics of aggregates
see[31]. In addition, Hooke’s law applies in elastic, isotropic solids, and
theories of viscoelasticity come into play in complex fluids like polymer or
protein solutions. Consequently, local factors such as MP concentration, and
rheological characteristics of the surrounding medium such as viscosities or
moduli are needed to predict the particle motion from the force field. In the
complex and heterogeneous environment of biological tissues, simulations,
ideally assisted by experiments, are required to predict the forces needed to
induce the intended motion.

## Materials and Methods

The magnets depicted in Fig. 3 are constructed using cubic permanent neodymium
magnets (Nd2Fe14B) affixed into 3D printed supports (polylactic acid).
Specifically, the quadrupole is assembled from 16 cubic magnets with 8 mm side
length (N45, $B_{\text{R}}=1.35$ T) obtained from magnets4you GmbH, 97816
Lohr, Germany. Their centers are mounted in a circle with a radius $r=67.5$
mm.
The dipole consists of 32 magnets (10 mm cubes, N45, $B_{\text{R}}=1.35$ T)
purchased from dogeo GmbH, 55444 Waldlaubersheim, Germany, and mounted in a
circular arrangement with $r=87$ mm.
The magnetic field components in Figs. 4 and 5 were measured with a Hall-probe
sensor (HMNT-4E04-VR, Lake Shore Cryotronics Inc., Westerville, OH) mounted on
a computer-controlled 3D linear table (OWIS GmbH, Staufen im Breisgau,
Germany) and controlled by custom-written software. A circular area with a
radius of 36 mm was scanned with a step width of 1 mm in $x$ and $y$
direction.
The MPs used in the qualitative display shown in Fig. 6 are 10 nm are iron
particles (EFH1, $M_{\text{s}}=32$ kA/m) dispersed in light mineral oil,
acquired from FerroTec, Santa Clara, CA.
The MPs used in the experiments for Fig. 7 are 130 nm nanomag-D with plain
surfaces (product no. 09-00-132) from micromod, Rostock, Germany. They are
multicore iron oxide particles ($M_{\text{s}}=120$ kA/m) consisting of several
smaller magnetic particles[48].

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## Acknowledgements

We wish to thank Rudolf Zentel (University of Mainz) and Matthias Barz
(University of Leiden) for inspiring this work. Rolf Heumann (University of
Bochum) has to be acknowledged for helpful discussions and suggestions
regarding this manuscript.
This work has been supported by the Deutsche Forschungsgemeinschaft funded
Collaborative Research Center “Nanodimensional polymer therapeutics for tumor
therapy” – SFB 1066 , (DFG grant number 213555243). Further, we thank JSPS-
KAKENHI (grant number 22K06430, F.R.) for the support.

## Author contributions statement

P.B. build the magnets and performed the experiments, otherwise all authors
equally contributed to the manuscript and its revisions.


# Integral Regulators on Mirror Curves With Mass Parameter

Soumya Sinha Babu  University of Georgia, Department of Mathematics and
Statistics, Athens, GA<EMAIL_ADDRESS>
###### Abstract.

In 2015, Codesido-Grassi-Marino laid the foundation of a connection between
local CY 3-folds and the spectra of operators attached to their mirror curves
in the context of Topological String Theory. In a 2024 paper with C. Doran and
M. Kerr, the author previously deduced two consequences of this conjecture;
one relating zeroes of higher normal function to the spectra of operators of
genus one curves, the other connecting integral regulators of $K_{2}$-classes
on mirror curves to dilogarithm values at algebraic arguments. We now show
that the latter continues to hold in the presence of a mass parameter, thus
expanding the range of the conjecture.

###### 2000 Mathematics Subject Classification:

14D07, 14J33, 19E15, 32G20, 34K08

##  1\. Introduction

We consider the lattice triangle,
$\Delta_{m,n}=\text{Conv}((1,0),(0,1),(-m,-n))$ where $m,n\in\mathbb{Z}_{>0}$.
A fan on a triangulation of $\\{1\\}\times\Delta_{m,n}\subset\mathbb{R}^{3}$
determines a local (toric, hence, non-compact) CY 3-fold, $X$. Corresponding
to $X$ one can construct a family of _mirror curves_
$\mathcal{C}_{m,n}\subset\mathbb{C}^{*}\times\mathbb{C}^{*}$, of genus $g$
equal to the number of interior integer points of $\Delta_{m,n}$, cut out by
the Laurent polynomial $F_{m,n}(x,y)$ with Newton polygon $\Delta_{m,n}$. We
assume that $F_{m,n}(x,y)$ is tempered, that is to say it takes the form,

(1.1) $\begin{split}\textstyle
F_{m,n}(x,y)&:=x\textstyle+y+x^{-m}y^{-n}+\sum_{j=1}^{g}a_{j}x^{m^{(j)}_{1}}\mspace{-10.0mu}y^{m^{(j)}_{2}}\\\
&\textstyle+\sum_{\ell=1}^{g_{1}-1}\binom{g_{1}}{\ell}x^{1-\ell\frac{m+1}{g_{1}}}y^{-\ell\frac{n}{g_{1}}}+\sum_{\ell=1}^{g_{2}-1}\binom{g_{2}}{\ell}x^{-\ell\frac{m}{g_{2}}}y^{1-\ell\frac{n+1}{g_{2}}}.\end{split}$

where $g_{1}:=\gcd(m+1,n)$ and $g_{2}=\gcd(m,n+1)$. To keep track of moduli,
we rename $\mathcal{C}_{m,n}$ and $F_{m,n}$ to
$\mathcal{C}^{\underline{a}}_{m,n},F^{\underline{a}}_{m,n}$ respectively. Of
particular importance is the so-called maximal conifold point,
$\hat{\underline{a}}\in(\mathbb{C}^{*})^{g}$, a point in moduli at which the
family degenerates to a rational nodal curve.
We now focus specifically on integral regulators111For a complete discussion
see [DKSB, §2.2].. Temperedness of $F^{\underline{a}}_{m,n}$ implies that the
Milnor symbol $\\{-x,-y\\}\in
K_{2}(\mathbb{C}(\mathcal{C}^{\underline{a}}_{m,n}))$ extends to motivic
cohomology classes on the compactifications
$\overline{\mathcal{C}^{\underline{a}}_{m,n}}\subset\mathbb{P}_{\Delta}$.
Writing $R\\{f,g\\}:=\log(f)\tfrac{dg}{g}-2\pi\mathbf{i}\log(g)\delta_{T_{f}}$
for the standard regulator current for Milnor $K_{2}$-symbols
($T_{f}:=f^{-1}(\mathbb{R}_{<0})$ the cut in branch of $\log$), there is a
symplectic basis $\\{\gamma_{j},\beta_{j}\\}_{j=1}^{g}$ of
$H_{1}(\overline{\mathcal{C}^{\underline{a}}_{m,n}},\mathbb{Z})$ with
regulator periods
$R_{\gamma_{j}}:=\int_{\gamma}R\\{-x,-y\\}|_{\mathcal{C}^{\underline{a}}_{m,n}}\sim-2\pi\mathbf{i}\log(a_{j})$,
and the regulator class $\mathcal{R}=R\\{-x,-y\\}\in
H^{1}(\overline{\mathcal{C}^{\underline{a}}_{m,n}},\mathbb{C}/\mathbb{Z}(2))$
then has a local lift to
$H^{1}(\overline{\mathcal{C}^{\underline{a}}_{m,n}},\mathbb{C})$ given by

$\tilde{\mathcal{R}}=\sum_{\ell=1}^{g}\left(R_{\gamma_{\ell}}\gamma_{\ell}^{*}+R_{\beta_{\ell}}\beta_{\ell}^{*}\right),$

whose Gauss-Manin derivatives are given by,

$\textstyle\omega_{j}:=\nabla_{{\partial}/{\partial
R_{\gamma_{j}}}}\tilde{\mathcal{R}}=\dfrac{a_{j}}{2\pi\mathbf{i}}\mathrm{Res}_{\mathcal{C}^{\underline{a}}_{m,n}}\dfrac{dx\wedge
dy}{x^{2j}yF^{\underline{a}}_{m,n}}.$

Denote by $\hat{\mathcal{C}}_{m,n}$ the fiber of the family over the maximal
conifold point $\underline{\hat{a}}$. It has $g$ nodes $\\{\hat{p}_{j}\\}$,
and the cycles $\\{\hat{\gamma}_{j}\\}_{j=1}^{g}$ passing through each node
generate $H_{1}(\hat{\mathcal{C}}_{m,n})$; we set
$R_{\hat{\gamma}_{j}}:=\int_{\hat{\gamma}_{j}}R\\{x,y\\}$. Let
$\bm{\kappa}=\,_{\underline{\hat{\gamma}}}[\mathrm{Id}]_{\underline{\gamma}(\hat{a})}$
be the change-of-basis matrix.

Recent work of [GHM, Ma, CGM] connects the enumerative geometry of $X$ to the
spectral theory of certain operators $\hat{F}$ on $L^{2}(\mathbb{R})$ attached
to $\mathcal{C}^{\underline{a}}_{m,n}$. In [DKSB], Using tools derived from
asymptotic Hodge Theory and local mirror symmetry, we were able to recast this
relationship in terms of regulator periods, resulting in the following

######  Conjecture 1.1.

For the families $\mathcal{C}^{\underline{a}}_{m,n}$ cut out by
$F^{\underline{a}}_{m,n}$, the regulator period
$R_{\gamma_{1}}({\underline{a}})$ asymptotic to $-2\pi\mathbf{i}\log(a_{1})$
at the origin has value

(1.2)
$\tfrac{1}{2\pi\mathbf{i}}R_{\gamma_{1}}(\hat{\underline{a}})\equiv\tfrac{m+n+1}{\pi}D_{2}(1+\zeta_{m+n+1})\;\;\;\textup{mod
$\mathbb{Q}(1)$}$

at the maximal conifold point $\hat{\underline{a}}$.

######  Acknowledgments.

The author is indebted to Dr. M. Kerr and Dr. C. Doran for their help in
understanding and settling the difficult parts of this problem. The author
also thanks Dr. P. Bousseau for countless discussions on this subject bringing
forward deeper insights into the conjecture.

##  2\. The main result

In [DKSB, §4], we proved Conjecture 1.1 for two infinite families of genus-$g$
curves, namely those cut out by $F^{\underline{a}}_{g,g}$ and
$F^{\underline{a}}_{2g-1,1}$. Note that neither $\Delta_{g,g}$ nor
$\Delta_{2g-1,1}$ contain any interior integer points, which significantly
reduces calculations. In present situation we wish to consider genus-$g$
families cut out by $F^{\underline{a}}_{2g,1}$. There is precisely one
interior integer point $(0,-g)$, with corresponding moduli $a_{g+1}$. In
physics literature such points are called mass parameters. However due to the
temperedness condition, we have

$a_{g+1}=-2.$

######  Theorem 2.1.

Conjecture 1.1 holds for the family $\mathcal{C}^{\underline{a}}_{2g,1}$; that
is,

(2.1)
$\displaystyle\tfrac{1}{2\pi\mathbf{i}}R_{\gamma_{1}}(\underline{\hat{a}})\underset{\mathbb{Q}(1)}{\equiv}$
$\displaystyle\dfrac{2g+2}{\pi}D_{2}(1+\zeta_{2g+2})$

######  Remark 2.2.

Note that the Milnor symbol $\\{x,y\\}$ on the curve defined by substituting
$-x,-y$ for $x,y$ resp. then multiplying the equation by $-1$, being a
pullback, is integrally tempered with the same integral regulator as
$\\{-x,-y\\}$. The new equation replaces $a_{g+1}$ by $-a_{g+1}$, and also
changes the sign of $a_{1},a_{3},a_{5},\dots$ etc.; it is this new equation
which we will use going forward. Note also that Conjecture 1.1 is stated in
terms of the regulator period asymptotic to $-2\pi\mathbf{i}\log(a_{n})$; it
is convenient in this section to drop the negative sign and work with one
asymptotic to $2\pi\mathbf{i}\log(a_{n})$. Thus from now on

$R_{{\gamma}_{n}}\sim 2\pi\mathbf{i}\log(a_{n}).$

We prove Theorem 2.1 by essentially adopting the same techniques discussed in
[DKSB, §4] \- monodromies at $\underline{\hat{a}}$ can be calculated using
power series coming from classical periods, following which the limiting
regulator periods can be found by parameterizing the spectral curve and
attaching specially constructed divisors to it.

######  Proposition 2.3.

Let $\kappa_{j}:=\mathrm{gcd}(j,g+1)$. Then

(2.2) $\bm{\kappa}=\mathrm{diag}(\kappa_{1},\dots,\kappa_{g}).$

It then follows from temperedness that

(2.3)
$\tfrac{1}{2\pi\mathbf{i}}R_{\gamma_{j}}(\underline{\hat{a}})\underset{\mathbb{Q}(1)}{\equiv}\tfrac{\kappa_{j}}{2\pi\mathbf{i}}R_{\hat{\gamma}_{j}}.$

###  2.1. Preliminary results

It is necessary to understand growth of the power series corresponding to
regulator periods and singularities of
$\mathcal{C}^{\hat{\underline{a}}}_{2g,1}$.

######  Lemma 2.4.

The following identity holds,

(2.4)
$\sum_{k=0}^{m/2}\dfrac{2^{-2k}}{\Gamma(1+k)^{2}\Gamma(1+m-2k)}=\dfrac{\Gamma\bigg{(}\dfrac{1+2m}{2}\bigg{)}}{\Gamma(1+m)\Gamma\bigg{(}\dfrac{2+m}{2}\bigg{)}\Gamma\bigg{(}\dfrac{1+m}{2}\bigg{)}}.$

###### Proof.

We reduce the given series into a hypergeometric series, and apply Gauss’
summation theorem as follows,

$\displaystyle\sum_{k=0}^{m/2}\dfrac{2^{-2k}}{\Gamma(1+k)^{2}\Gamma(1+m-2k)}$
$\displaystyle=\dfrac{1}{m!}\leavevmode\nobreak\
{}_{2}F_{1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\dfrac{1-m}{2}\mskip
8.0mu-\dfrac{m}{2}}{1};1\biggr{]}$ (2.5)
$\displaystyle=\dfrac{\Gamma\bigg{(}\dfrac{1+2m}{2}\bigg{)}}{\Gamma(1+m)\Gamma\bigg{(}\dfrac{2+m}{2}\bigg{)}\Gamma\bigg{(}\dfrac{1+m}{2}\bigg{)}}.$

∎

######  Lemma 2.5.

If $a,b,c\in\mathbb{R}_{>>>0}$ are such that $a=b+c$, then

(2.6)
$\underbrace{\dfrac{2^{c}\Gamma(1+a)\Gamma\bigg{(}\dfrac{1+2c}{2}\bigg{)}}{\Gamma(1+b)\Gamma(1+c)\Gamma\bigg{(}\dfrac{1+c}{2}\bigg{)}\Gamma\bigg{(}\dfrac{2+c}{2}\bigg{)}}}_{=:\mathcal{A}_{a,b,c}}\approx\dfrac{1}{\sqrt{2}\pi
c}\sqrt{\dfrac{a}{b}}\Bigg{(}\dfrac{a}{b}\bigg{(}\dfrac{4b}{c}\bigg{)}^{c/a}\Bigg{)}^{a}.$

###### Proof.

Using Duplication formula,

(2.7)
$\dfrac{1}{\Gamma\bigg{(}\dfrac{1+c}{2}\bigg{)}\Gamma\bigg{(}\dfrac{2+c}{2}\bigg{)}}=\dfrac{2^{c}}{\sqrt{\pi}\Gamma(1+c)}$

Thus

(2.8)
$\dfrac{\Gamma\bigg{(}\dfrac{1+2c}{2}\bigg{)}}{\Gamma\bigg{(}\dfrac{1+c}{2}\bigg{)}\Gamma\bigg{(}\dfrac{2+c}{2}\bigg{)}}=\dfrac{2^{c}\Gamma\bigg{(}c+\dfrac{1}{2}\bigg{)}}{\Gamma(c+1)}\approx
2^{c}\sqrt{\dfrac{1}{c}}.$

wherein we have used a modified Sterling’s approximation which says that for
large $x\in\mathbb{R}_{\geq 0}$ and $\alpha,\beta\in\mathbb{R}_{>0}$,

(2.9) $\dfrac{\Gamma(x+\alpha)}{\Gamma(x+\beta)}\approx x^{\alpha-\beta}.$

It follows that

$\displaystyle\mathcal{A}_{a,b,c}\approx\dfrac{4^{c}\Gamma(1+a)}{\sqrt{\pi}\Gamma(1+b)\Gamma(1+c)\sqrt{c}}$
$\displaystyle\approx\dfrac{1}{\sqrt{2}\pi
c}\sqrt{\dfrac{a}{b}}\dfrac{4^{c}a^{a}}{b^{b}c^{c}}e^{-a+b+c}$ (2.10)
$\displaystyle=\dfrac{1}{\sqrt{2}\pi
c}\sqrt{\dfrac{a}{b}}\dfrac{4^{c}a^{a}}{b^{a-c}c^{c}}$
$\displaystyle=\dfrac{1}{\sqrt{2}\pi
c}\sqrt{\dfrac{a}{b}}\Bigg{(}\dfrac{a}{b}\bigg{(}\dfrac{4b}{c}\bigg{)}^{c/a}\Bigg{)}^{a}$

as was to be shown.

∎

######  Lemma 2.6.

Suppose that the fiber over
$\underline{\tilde{a}}=(\tilde{a}_{1},\dots,\tilde{a}_{g+1})$ has $g$-many
singularities, say $\tilde{p}_{j}:=(\tilde{x}_{j},\tilde{y}_{j}),n=1,\dots,g$.
Then for each $j$, $\tilde{p}_{j}$ is a node.

###### Proof.

We argue in the same vein as [DKSB]. The result becomes immediate modulo the
hessian calculation, which in this case boils down to the following - we begin
by defining

(2.11) $\textstyle\tilde{P}(x):=g+1+\sum_{j=1}^{g}(g+1-j)\tilde{a}_{j}x^{-j},$

and observing that

(2.12)
$\textstyle\tilde{P}(\tilde{p}_{j})=\tfrac{g}{\tilde{x}_{j}}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})+\partial_{x}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})=0.$

Thus $\textbf{Z}(\tilde{P})=\\{\tilde{p}_{1},\ldots,\tilde{p}_{g}\\}$, i.e.,
$\tilde{P}$ has no repeated roots; that is,
$\tilde{P}^{\prime}(\tilde{p}_{j})\neq 0$ ($\forall j$). To compute the
Hessians, write

$\displaystyle\textstyle\partial_{xx}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})$
$\displaystyle=\textstyle\sum_{\ell=1}^{g+1}\ell(\ell-1)\tilde{a}_{\ell}\tilde{x}_{j}^{-\ell-1}+2g(2g+1)\tilde{x}_{j}^{-2g-2}\tilde{y}^{-1}_{j}$
(2.13)
$\displaystyle\textstyle=\sum_{\ell=1}^{g+1}\ell(\ell-1)\tilde{a}_{\ell}\tilde{x}_{j}^{-\ell-1}+\tfrac{2g(2g+1)\tilde{y}_{j}}{\tilde{x}_{j}^{2}},$
(2.14)
$\displaystyle\partial_{xy}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})$
$\displaystyle=2g\tilde{x}_{j}^{-2g-1}\tilde{y}_{j}^{-2}=\tfrac{2g}{\tilde{y}_{j}},\;\text{and}$
(2.15)
$\displaystyle\partial_{yy}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})$
$\displaystyle=2\tilde{x}_{j}^{2g}\tilde{y}_{j}^{-3}=\tfrac{2}{\tilde{y}_{j}}.$

It can be shown that

(2.16)
$\partial_{xx}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})=\tfrac{2g^{2}\tilde{y}_{j}}{2\tilde{x}_{j}^{2}}+\tfrac{\tilde{P}^{\prime}(\tilde{x}_{j})}{2},$

therefore,

$\displaystyle H_{F^{\underline{\tilde{a}}}_{2g,1}}(\tilde{p}_{j})$
$\displaystyle=\left(\partial_{xy}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})\right)^{2}-\partial_{xx}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})\partial_{yy}F^{\underline{\tilde{a}}}_{2g,1}(\tilde{p}_{j})$
$\displaystyle\textstyle=\tfrac{4g^{2}}{\tilde{x}_{j}^{2}}-\tfrac{4g^{2}}{\tilde{x}_{j}^{2}}-\tfrac{\tilde{P}^{\prime}(\tilde{x}_{j})}{\tilde{y}_{j}}\;=\;-\tfrac{\tilde{P}^{\prime}(\tilde{x}_{j})}{\tilde{y}_{j}}\neq
0$

as was to be shown. ∎

###  2.2. Monodromy calculations via power series

Let us recall the key result of [DKSB] for monodromy calculations - consider a
$1$-parameter family of curves $\mathcal{C}\to\mathbb{P}^{1}$ with coordinate
$t$, endowed with a section $\omega$ of the relative dualizing sheaf; on
smooth fibers $\mathcal{C}_{t}$, $\omega_{1}$ is a holomorphic $1$-form.
Assume that $\mathcal{C}_{c}$ has a single node $p_{c}$ (i.e. is a “conifold
fiber”), and let $\delta_{0}$ be the “conifold” vanishing cycle pinched at
$p_{c}$. Writing $\varepsilon_{0}$ for a cycle invariant about $t=0$, its
monodromy about $t=c$ is a multiple of $\delta_{0}$, say $\bm{k}\delta_{0}$
for some $\bm{k}\in\mathbb{Z}_{\geq 0}$.

######  Lemma 2.7.

The conifold multiple $\bm{k}$ is computed by

(2.17) $\bm{k}=\dfrac{\underset{m\to\infty}{\lim}b_{m}\cdot c^{m}\cdot
m}{\mathrm{Res}_{p_{c}}\leavevmode\nobreak\ \omega_{c}}.$

For the proof of Proposition 2.3, we need to compute the Picard-Lefschetz
matrix $\bm{\kappa}$, whose entries $\bm{\kappa}_{ij}$ tell how many times the
specialization $\gamma_{i}(\hat{\underline{a}})$ passes through $\hat{p}_{j}$.
In order to invoke Lemma 2.7 for this purpose, we should reinterpret these
numbers as (roughly speaking) conifold multiples for 1-parameter subfamilies
of $\mathcal{C}_{\underline{a}}$ acquiring a _single_ node. The idea is that
$\hat{\underline{a}}$ is a normal-crossing point of the discriminant locus,
whose $g$ local-analytic irreducible components each parametrize fibers
carrying a single node $p_{j}$. These are labeled in such a way that the
$j^{\text{th}}$ component can be followed out to where it meets the
$a_{j}$-axis at $a_{j}=\mathring{a}_{j}$. Call this fiber
$\mathcal{C}_{2g,1}^{\mathring{\underline{a}}_{j}}$, and
$\mathring{p}_{j}=(\mathring{x}_{j},\mathring{y}_{j})$ for the limit of the
node to it.

We have the 1-forms

(2.18)
$\varpi_{j}=\tfrac{1}{2\pi\mathbf{i}}\nabla_{\delta_{a_{j}}}R\\{x,y\\}=\frac{-a_{j}}{2\pi\mathbf{i}}\mathrm{Res}_{\mathcal{C}_{2g,1}}\left(\frac{dx\wedge
dy}{x^{2j}yF_{2g,1}(x,y)}\right)$

and 1-cycles $\gamma_{j}$ ($j=1,\ldots,g$). The computation that follows will
consider periods $\Pi_{jj}=\int_{\gamma_{j}}\varpi_{j}$ on the 1-parameter
families over the $a_{j}$-axes (acquiring a single node at
$a_{j}=\mathring{a}_{j}$), which will suffice to determine the diagonal terms
$\bm{\kappa}_{jj}$. That the remaining, off-diagonal terms are actually zero
follows from the fact that each $\gamma_{j}$ is well-defined on a tubular
neighborhood of the hyperplane in (compactified) moduli defined by $z_{j}=0$,
which is cut by the conifold components carrying $p_{i}$ for every $i\neq j$.

Now $\mathcal{C}_{2g,1}^{\mathring{\underline{a}}_{j}}$ is defined by

(2.19)
$f^{(j)}_{2g,1}:=F^{\underline{\mathring{a}}_{j}}_{2g,1}(x,y)=x+y+\mathring{a}_{j}x^{1-j}+a_{g+1}x^{-g}+x^{-2g}y^{-1},$

and to find the node we set
$f_{2g,1}^{(j)}(\mathring{x}_{j},\mathring{y}_{j})=\mathring{x}_{j}\partial_{x}f_{2g,1}^{(j)}(\mathring{x}_{j},\mathring{y}_{j})$
which gives rise to equations of the form,

(2.20) $\displaystyle
2\mathring{y}_{j}+\mathring{x}_{j}+\mathring{a}_{j}\mathring{x_{j}}^{1-j}+a_{g+1}\mathring{x}_{j}^{-g}$
$\displaystyle=0,$ (2.21)
$\displaystyle\mathring{x}_{j}+(1-j)\mathring{x}_{j}^{-j}-g\mathring{a}_{j}\mathring{x}_{j}^{-g-1}-2g\mathring{x}_{j}^{-2g-1}\mathring{y}_{j}^{-1}$
$\displaystyle=0.$

This yields

(2.22) $\displaystyle\mathring{x}_{j}$
$\displaystyle=\sqrt[g+1]{\dfrac{4(g-j+1)}{j}},$ (2.23)
$\displaystyle\mathring{a}_{j}$
$\displaystyle=-\dfrac{g+1}{g-j+1}\bigg{(}\dfrac{4(g-j+1)}{j}\bigg{)}^{\tfrac{j}{g+1}}.$

In particular, we have the relation

(2.24) $\mathring{a}_{j}\mathring{x}_{j}^{g-j+1}=-\dfrac{4(g+1)}{j}.$

In order to calculate the residue of $\varpi_{j}$ at $\mathring{p}_{j}$,
recall that for any $f(x,y)=Ax^{2}+Bxy+Cy^{2}+\text{higher order
terms}\in\mathbb{C}[x,y]$, we have

(2.25) $\mathrm{Res}^{2}_{\underline{0}}\frac{dx\wedge
dy}{f}:={\mathrm{Res}}_{\underline{0}}\left(\mathrm{Res}_{\bm{Z}(f)}\dfrac{dx\wedge
dy}{f}\right)=\dfrac{1}{\sqrt{B^{2}-4AC}}.$

Changing variables to $X:=x-\mathring{x}_{j}$, $Y:=y-\mathring{x}_{j}$ in
$f^{(j)}_{2g,1}(x,y)$ leads to the equation

$\displaystyle
x^{2g}yf^{(j)}_{2g,1}=\tfrac{(6g^{2}-4(j+1)-4g(j-1)}{\mathring{x_{j}}^{2}}X^{2}-2g\mathring{x}_{j}^{g-1}XY$
$\displaystyle+\mathring{x}_{j}^{2g}Y^{2}$ (2.26) $\displaystyle+\text{higher
order terms}.$

Therefore

$\displaystyle{\mathrm{Res}}^{2}_{\mathring{p}_{j}}\dfrac{dx\wedge
dy}{x^{2g}yf^{(j)}_{2g,1}}$
$\displaystyle=\tfrac{1}{\mathring{x}_{j}^{g-1}\sqrt{4g^{2}-2\big{(}6g^{2}-4(j+1)-4g(j-1)\big{)}^{2}}}$
(2.27)
$\displaystyle=\tfrac{(-1)^{g+1}}{\mathbf{i}\mathring{x}_{j}^{g-1}\sqrt{8(g-j+1)(g+1)}}$

Consequently the residue of $\varpi_{j}$ may now be found:

$\displaystyle\mathrm{Res}_{\mathring{p}_{j}}\varpi_{j}$
$\displaystyle=\dfrac{-\mathring{a}_{j}}{2\pi\mathbf{i}}{\mathrm{Res}}^{2}_{\mathring{p}_{j}}\dfrac{dx\wedge
dy}{x^{2j}yf^{(j)}_{2g,1}}$
$\displaystyle=\dfrac{-\mathring{a}_{j}}{2\pi\mathbf{i}}\cdot\mathring{x}_{j}^{2g-j}\cdot{\mathrm{Res}}^{2}_{\mathring{p}_{j}}\dfrac{dx\wedge
dy}{x^{2g}yf^{(j)}_{2g,1}}$ (2.28)
$\displaystyle=\dfrac{-1}{2\pi}\cdot(\mathring{a}_{j}\mathring{x}_{j}^{2(g-j+1)})\cdot\dfrac{1}{\mathbf{i}\mathring{x}_{j}^{g-1}\sqrt{8(g-j+1)(g+1)}}$
$\displaystyle=\dfrac{\sqrt{g+1}}{\pi j\sqrt{2(g-j+1)}}.$

For the periods of $\varpi_{j}$, we start with those of the regulator class.
Writing $\varphi_{j}:=x^{j-1}F_{2g,1}^{\underline{a}}(x,y)-a_{j}$, (with the
sign flip from our choice of $\gamma_{j}$) yields

$\displaystyle\dfrac{1}{2\pi\mathbf{i}}R_{\gamma_{j}}(\underline{a})$
$\displaystyle\underset{\mathbb{Q}(1)}{\equiv}\log(a_{j})-\sum_{m>0}\dfrac{(-a_{j})^{-m}}{m}[\varphi_{j}^{m}]_{\underline{0}}$
(2.29) $\displaystyle=\log(a_{j})-\sum_{m>0}\dfrac{(-a_{j})^{-m}}{m}\times$
$\displaystyle\textstyle\bm{[(}\underbrace{x^{j}}_{=:A_{j}}+\underbrace{x^{j-1}y}_{=:B_{j}}+\textstyle\sum_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{g+1}a_{k}\underbrace{x^{j-k}}_{=:C^{k}_{j}}+\underbrace{x^{j-2g-1}y^{-1}}_{=:D_{j}}\bm{)}^{m}\bm{]}_{\underline{0}}$

where $[L]_{\underline{0}}$ stands for the constant term (in $x,y$) appearing
in the Laurent polynomial $L$. Now, given
$l_{1},l_{2},\cdots,l_{g}\in\mathbb{Z}$, we define

(2.30) $\displaystyle\mathfrak{l}_{j}$
$\displaystyle:=\dfrac{1}{j}\bigg{(}(g+1)(2l_{j}+l_{g+1})+\sum\limits_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{g}kl_{k}\bigg{)}$ (2.31)
$\displaystyle\mathfrak{l^{\prime}}_{j}$
$\displaystyle:=\dfrac{1}{j}\bigg{(}(g-j+1)(2l_{j}+l_{g+1})+\sum\limits_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{g}(k-j)l_{k}\bigg{)},\text{ and put}$ (2.32)
$\displaystyle\mathcal{L}_{j}$
$\displaystyle:=\\{(l_{1},l_{2},\cdots,l_{g})\in\mathbb{Z}^{g}_{\geq
0}\mid\mathfrak{l}^{\prime}_{j}\in\mathbb{Z}_{\geq
0}\\}\setminus\\{(0,\cdots,0)\\}$

Note that $\mathfrak{l}^{\prime}_{j}\in\mathbb{Z}_{\geq
0}\implies\mathfrak{l}_{j}\in\mathbb{Z}_{\geq 0}$. The upshot of this
construction is if $L_{j},L^{\prime}_{j}\in\mathbb{Z}_{\geq 0}$ are such that

(2.33) $\displaystyle
A_{j}^{L_{j}}B_{j}^{L^{\prime}_{j}}\prod\limits_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{g+1}(C^{k}_{j})^{l_{k}}D_{j}^{l_{j}}$
$\displaystyle=1\text{ and}$ (2.34) $\displaystyle
L_{j}+L^{\prime}_{j}+\sum\limits_{k=1}^{g+1}l_{k}$ $\displaystyle=m$

then $L_{j}=\mathfrak{l}^{\prime}_{j}$, $L^{\prime}_{j}=l_{j}$ and
$m=\mathfrak{l}_{j}$. Thus the lattice $\mathcal{L}_{j}\subset\mathbb{Z}^{g}$
encodes all possible constant terms appearing in (2.2), giving

(2.35)
$\dfrac{1}{2\pi\mathbf{i}}R_{\gamma_{j}}(\underline{a})\underset{\mathbb{Q}(1)}{\equiv}\log(a_{j})\leavevmode\nobreak\
-\sum_{\mathcal{L}_{j}}\dfrac{\Gamma(\mathfrak{l}_{j})}{\Gamma(1+\mathfrak{l}^{\prime}_{j})\Gamma^{2}(1+l_{j})\prod\limits_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{g+1}\Gamma(1+l_{k})}\frac{(-a_{j})^{-\mathfrak{l}_{j}}}{\mathfrak{l}_{j}}\prod\limits_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{g+1}a_{k}^{l_{k}}.$

For the classical periods
$\Pi_{j\ell}=\int_{\gamma_{j}}\varpi_{\ell}=\tfrac{1}{2\pi\mathbf{i}}\delta_{a_{\ell}}R_{\gamma_{j}}$,
it is clear from (2.35) that $\Pi_{j\ell}$ vanishes on the $a_{j}$-axis for
$\ell\neq j$. Focusing then on

(2.36)
$\Pi_{jj}(\underline{a})=\int_{\gamma_{j}}\varpi_{j}=1+\sum_{\mathcal{L}_{j}}\dfrac{\Gamma(1+\mathfrak{l}_{j})}{\Gamma(1+\mathfrak{l}^{\prime}_{j})\Gamma^{2}(1+l_{j})\prod\limits_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{g+1}\Gamma(1+l_{k})}(-a_{j})^{-\mathfrak{l}_{j}}\prod\limits_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{g+1}a_{k}^{l_{k}},$

we set $\underline{a}_{i}=0$ for $i\neq j,g+1$ to obtain

(2.37)
$\displaystyle\mathcal{S}:=1+\sum_{\tiny{l_{j},l_{g+1}}\in\mathbb{Z}_{>0}}\dfrac{\Gamma(1+\tfrac{g+1}{j}l_{j})}{\Gamma(1+\tfrac{g+1}{j}l_{j})\Gamma^{2}(1+l_{j})\Gamma(1+l_{g+1})}(-{a}_{j})^{-\tfrac{g+1}{j}l_{j}}a_{g+1}^{l_{g+1}}.$

Recall that $\kappa_{j}:=\mathrm{gcd}(j,g+1)$. Let us shift indices by
renaming $l_{n}\to l_{g+1}+2l_{n}$ and define,

(2.38)
$\begin{split}n_{j}:&=\dfrac{j}{\kappa_{j}},\mspace{50.0mu}m_{j}:=\dfrac{g+1}{\kappa_{j}}=\dfrac{(g+1)n_{j}}{j},\\\
r_{j}:&=\dfrac{l_{j}}{n_{j}},\mspace{50.0mu}\text{and}\mspace{50.0mu}s_{j}:=a_{j}^{-m_{j}}.\end{split}$

Clearly $n_{j},m_{j},r_{j}\in\mathbb{Z}_{>0}$. Now we have a power series of
the form

(2.39)
$\mathcal{S}=1+\sum_{r_{j}\in\mathbb{N}}{\dfrac{(-1)^{m_{j}r_{j}}\Gamma(1+m_{j}r_{j})a_{g+1}^{n_{j}r_{j}-2l_{j}}}{\Gamma^{2}(1+\tfrac{m_{j}-n_{j}}{2}r_{j})\Gamma^{2}(1+l_{j})\Gamma(1+n_{j}r_{j}-2l_{j})}}s_{j}^{r_{j}}=:\sum_{r_{j}}b_{r_{j}}s_{j}^{r_{j}}.$

Let $\mathring{s}_{j}:=\mathring{a}_{j}^{-m_{j}}$. Setting $a_{g+1}=2$ and
applying Lemma A.1,

(2.40)
$\dfrac{\Gamma(1+m_{j}r_{j})}{\Gamma^{2}(1+\frac{m_{j}-n_{j}}{2}r_{j})\Gamma(1+n_{j}r_{j})}\approx\dfrac{(-1)^{m_{j}r_{j}}\sqrt{2m_{j}}}{2\pi
r_{j}n_{j}\sqrt{m_{j}-n_{j}}}\mathring{s}_{j}^{r_{j}}$

from which we may conclude that

(2.41) $\lim_{r_{j}\to\infty}b_{r_{j}}\cdot
r_{j}\cdot\mathring{s}_{j}^{-r_{j}}=\dfrac{\sqrt{2m_{j}}}{2\pi
n_{j}\sqrt{m_{j}-n_{j}}}.$

Observing that

(2.42)
$\displaystyle\mathrm{Res}_{\mathring{p}_{j}}\varpi_{j}=\dfrac{\sqrt{g+1}}{\pi
j\sqrt{2(g-j+1)}}=\dfrac{\sqrt{2m_{j}}}{2\pi j\sqrt{m_{j}-n_{j}}},$

we apply (2.17) to obtain

(2.43) $\bm{\kappa}_{jj}=\dfrac{\underset{r_{j}\to\infty}{\lim}b_{r_{j}}\cdot
r_{j}\cdot\mathring{s}_{j}^{r_{j}}}{\mathrm{Res}_{\mathring{p}_{j}}\varpi_{j}}=\dfrac{j}{n_{j}}=\kappa_{j}.$

This concludes the proof of Theorem 2.3.

######  Remark 2.8.

Notice that $\kappa_{1}=\kappa_{g}=1$. We document
$\underline{\kappa}:=(\kappa_{1},\ldots,\kappa_{n})$ for $g=2,\dots,10$ in
Table 1.

$g$ | $\underline{\kappa}$
---|---
2 | (1,1)
3 | (1,2,1)
4 | (1,1,1,1)
5 | (1,2,3,2,1)
6 | (1,1,1,1,1,1)
7 | (1,2,1,4,1,2,1)
8 | (1,1,3,1,1,3,1,1)
9 | (1,2,1,2,5,2,1,2,1)
10 | (1,1,1,1,1,1,1,1,1,1)
Table 1. Conifold multiples for small genera.

###  2.3. Normalization of the conifold fibers

Recall that for the family $\mathcal{C}^{\underline{a}}_{m,n}$ attached to
$F^{\underline{a}}_{m,n}$, the maximal conifold point
$\underline{\hat{a}}\in(\mathbb{C}^{*})^{g}$ is defined to be the unique point
(if it exists) on the boundary of the region of convergence of the series
(2.35) where $\mathcal{C}_{m,n}^{\hat{\underline{a}}}$ acquires $g$ nodes
(labeled by $\hat{p}_{j}:=(\hat{x}_{j},\hat{y}_{j})$).

######  Remark 2.9.

Note that uniqueness and existence of $\hat{\underline{a}}$ were essentially
proven in [DKSB, Remark 4.11]. We provide a different, albeit short
explanation - the lattice triangle $\Delta_{m,n}$ gives rise to a unique,
isoradial dimer model222 See [UY, Theorem 1.2].. This phenomenon will be
discussed in full generality in an upcoming work.

We furnish a concrete example locating $\hat{\underline{a}}$ in low-genera
case.

######  Example 2.10.

Consider the (untempered) family $\mathcal{C}_{4,1}$ corresponding to a local
$\mathbb{C}^{3}/\mathbb{Z}_{6}$ geometry cut out by

(2.44) $F_{4,1}(x,y)=x+y+a_{1}+a_{2}x^{-1}+a_{3}x^{-2}+x^{-4}y^{-1}=0.$

$a_{3}$ is a mass parameter, and

(2.45) $z_{1}=\dfrac{a_{1}a_{3}}{a_{2}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ z_{2}=\dfrac{a_{2}}{a_{1}^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ z_{3}=\dfrac{1}{a_{3}^{2}}$

are the large complex structure parameters. The discriminant locus is obtained
by setting

$\displaystyle
729z_{1}^{4}(1-4z_{3})^{2}z_{2}^{4}+108z_{1}^{3}(9z_{2}-2)(4z_{3}-1)z_{2}^{2}+(1-4z_{2})^{2}$
(2.46)
$\displaystyle-4z_{1}(36z_{2}^{2}-17z_{2}+2)+2z_{1}^{2}(108(4z_{3}+1)z_{2}^{3}-27(28z_{3}-5)z_{2}^{2})$
$\displaystyle+72((4z_{3}-1)z_{2}-32z_{3}+8)=0.$

Temperedness amounts to letting $z_{3}=\dfrac{1}{4}$, and the maximal conifold
point

(2.47) $\underline{\hat{z}}=\bigg{(}\dfrac{4}{27},\dfrac{1}{4}\bigg{)}$

can once again be recovered from transverse intersection.

The ansatz that allows us to bypass such considerations is described by

######  Proposition 2.11.

Let $\mathcal{T}_{m}$ denote the $m^{\text{th}}$ Chebyshev polynomial of the
first kind; this is a degree-$m$ polynomial characterized by
$\mathcal{T}_{m}(\cos\theta)=\cos m\theta$. Then we have

(2.48)
$F^{\underline{\hat{a}}}_{g,g}(x,(-1)^{j}x^{-g})=\dfrac{2}{x^{g+1}}(\mathcal{T}_{2g+2}(\tfrac{\sqrt{x}}{2})+(-1)^{j}).$

It follows that

(2.49) $\displaystyle\hat{a}_{j}$
$\displaystyle=(-1)^{j}\dfrac{(2g+2)(2g-j+1)!}{j!(2g-2j+2)!}\;\;\;\text{and}$
(2.50) $\displaystyle\hat{x}_{j}$
$\displaystyle=(-1)^{j/g}\hat{y}_{j}^{-1/g}=(-1)^{j}4\cos^{2}\bigg{(}\dfrac{\pi
j}{2g+2}\bigg{)}$

for $j=1,\ldots,g$. In particular, $\underline{\hat{a}}\in\mathbb{Z}^{g}$.

###### Proof.

That $\hat{x}_{j}\in\mathbf{Z}(\text{RHS}\eqref{ep04})$ is immediate from the
defining property of $\mathcal{T}_{2g+1}$, and the $\hat{x}_{j}$ are distinct
and different from $4$. Moreover, writing $\mathcal{U}_{m}$ for the
$m^{\text{th}}$ Chebyshev polynomial of the second kind, the relation
$(\mathcal{T}_{2g+2}(w)-1)(\mathcal{T}_{2g+2}(w)+1)=(w^{2}-1)U_{2g+1}(w)$
guarantees that all roots other than $4$ of
$(\mathcal{T}_{2g+2}(\tfrac{1}{2x})+1)$ have even multiplicity. So they all
have multiplicity $2$ and are precisely the $\\{\hat{x}_{j}\\}$.

The polynomial
$\hat{F}(x,y):=x+y+\sum_{j=1}^{g}\hat{a}_{j}x^{1-j}+a_{g+1}x^{-g}+x^{-2g}y^{-1}$,
with $\hat{a}_{j}$ as in (2.49), satisfies
$\hat{F}(x,(-1)^{j}x^{-g})=\text{RHS}\eqref{ep04}$ by standard results on
coefficients of $\mathcal{T}_{m}$. Clearly $\hat{F}(\hat{p}_{j})=0$, and the
$\\{\hat{p}_{j}\\}$ are in fact singularities of $\mathbf{Z}(\hat{F})$ since
$\tfrac{\partial\hat{F}}{\partial
x}(x,(-1)^{j}x^{-g})=2\tfrac{d}{dx}(\hat{F}(x,(-1)^{j}x^{-g}))$ and they are
double roots of $\hat{F}(x,(-1)^{j}x^{-g})$. Therefore, by Proposition A.2,
they are all nodes. Since one can also check that (2.35) converges at
$\hat{p}_{j}$, $\mathbf{Z}(\hat{F})$ is the maximal conifold curve. ∎

######  Remark 2.12.

Of course, Proposition 2.11 recovers the predicted maximal conifold point for
the $g=2$ family $\mathcal{C}_{4,1}$, namely $\hat{a}_{1}=-6,\hat{a}_{2}=9$.
Table 2 gathers $\mathcal{T}_{2g+2}$ and $\underline{\hat{a}}$ for a few low
genus cases.

$g$ | $\mathcal{T}_{2g+2}(x)$ | $\underline{\hat{a}}$
---|---|---
1 |  $8x^{4}-8x^{2}$+1 | -4
2 | $32x^{6}-48x^{4}+18x^{2}-1$ | (-6,9)
3 | $128x^{8}-256x^{6}+160x^{4}-32x^{2}+1$ | (-8,20,-16)
4 | $512x^{10}-1280x^{8}+1120x^{6}-400x^{4}+50x^{2}-1$ | (-10,35,-50,25)
Table 2. Maximal conifold points for low genera.

$\hat{\mathcal{C}}_{2g,1}$ admits $g$ distinct uniformizations by
$\mathbb{P}^{1}$(proven in appendix), given by
$z\mapsto(\hat{X}_{j}(z),\hat{Y}_{j}(z))$, with

(2.51) $\displaystyle\hat{X}_{j}(z)$
$\displaystyle=\frac{\hat{x}_{j}\left(1-\tfrac{\zeta_{2g+2}^{j}}{z}\right)^{2}}{\left(1-\tfrac{{\zeta^{2j}_{2g+2}}}{z}\right)\left(1-\tfrac{1}{z}\right)}\;\;\;\text{and}$
(2.52) $\displaystyle\hat{Y}_{j}(z)$
$\displaystyle=\frac{\hat{y}_{j}\left(1-z\right)^{2g+1}}{\left(1-\tfrac{z}{\zeta^{j}_{2g+2}}\right)^{2g}\left(1-\tfrac{z}{\zeta_{2g+2}^{2j}}\right)},$

all of whom map $z=0,\infty$ to $\hat{p}_{j}$, implying that the path joining
$z=0$ to $z=\infty$ on $\mathbb{P}^{1}$ is sent (by the $j^{\text{th}}$ map)
to $\hat{\gamma}_{j}$. As per[DK, §6.2], a formal divisor
$\hat{\mathcal{N}}_{j}$ on $\mathbb{P}^{1}\setminus\\{0,\infty\\}$ is
introduced to each uniformization: for
$X(z)=c_{1}\prod_{j}(1-\tfrac{\alpha_{j}}{z})^{d_{j}}$ and
$Y(z)=c_{2}\prod_{k}(1-\tfrac{z}{\beta_{k}})^{e_{k}}$, this divisor is
$\mathcal{N}:=\sum_{j,k}d_{j}e_{k}[\tfrac{\alpha_{j}}{\beta_{k}}]$. According
to [loc. cit.], the imaginary part of $\int_{0}^{\infty}R\\{X(z),Y(z)\\}$ is
then given by
$D_{2}(\mathcal{N}):=\sum_{j,k}d_{j}e_{k}D_{2}(\tfrac{\alpha_{j}}{\beta_{k}})$.

Working in $\mathcal{B}_{2}(\mathbb{C})$, we invoke scissors congruence
relations

(2.53) $\displaystyle[\xi]+[\tfrac{1}{\xi}]=0,\leavevmode\nobreak\
[\xi]+[\overline{\xi}]=0,\leavevmode\nobreak\ [\xi]+[1-\xi]$
$\displaystyle=0\leavevmode\nobreak\ \text{and}$ (2.54)
$\displaystyle[\xi_{1}]+[\xi_{2}]+[\tfrac{1-\xi_{1}}{1-\xi_{1}\xi_{2}}]+[\tfrac{1-\xi_{2}}{1-\xi_{1}\xi_{2}}]+[1-\xi_{1}\xi_{2}]$
$\displaystyle=0$

to obtain,

$\displaystyle\hat{\mathcal{N}}_{j}=2(2g+2)[1+\zeta^{j}_{2g+2}].$

Consequently we have the identity

(2.55) $\displaystyle D_{2}(\hat{\mathcal{N}}_{j})$
$\displaystyle=2(2g+2)D_{2}(1+\zeta^{j}_{2g+2}).$

We are now ready to prove Theorem 2.1. By the previously mentioned result of
[DK, §6.2], we know that
$\Im(R_{\hat{\gamma}_{j}})=D_{2}(\hat{\mathcal{N}}_{j})$ or

(2.56)
$\Re(\tfrac{1}{2\pi\mathbf{i}}R_{\hat{\gamma}_{j}})=\tfrac{1}{2\pi}D_{2}(\hat{\mathcal{N}}_{j}).$

Next, Proposition 2.3 tells us that
$R_{\gamma_{j}}(\hat{\underline{a}})=\kappa_{j}R_{\hat{\gamma}_{j}}$, while
(2.49) and (2.35) ensure that (mod $\mathbb{Q}(1)$)
$\tfrac{1}{2\pi\mathbf{i}}R_{\gamma_{j}}(\hat{\underline{a}})$ hence
$\tfrac{1}{2\pi\mathbf{i}}R_{\hat{\gamma}_{j}}$ is real. Combining this with
(2.55) gives

(2.57)
$\displaystyle\dfrac{1}{2\pi\mathbf{i}}R_{\gamma_{j}}(\underline{\hat{a}})=\dfrac{1}{2\pi\mathbf{i}}\kappa_{j}R_{\hat{\gamma}_{j}}\underset{\mathbb{Q}(1)}{\equiv}\dfrac{(2g+2)\kappa_{j}}{\pi}D_{2}(1+\zeta^{j}_{2g+2}),$

whence (2.1) follows by setting $j=1$ in (2.57).

##  3\. Explicit series identities

Any torsion modulo $\mathbb{Q}(1)$ in (2.57) is eliminated with regards to
(2.35) as both sides are real,333after changing $\log(\hat{a}_{j})$ to
$\log(|\hat{a}_{j}|)$ and brings forth the relationship

$\displaystyle\dfrac{(2g+2)\cdot\mathrm{gcd}(j,g+1)}{\pi}$ $\displaystyle
D_{2}(1+\zeta^{j}_{2g+2})=\log(|\hat{a}_{j}|)-$ (3.1)
$\displaystyle\sum_{\mathcal{L}_{j}}\dfrac{\Gamma(\mathfrak{l}_{j})}{\Gamma(1+\mathfrak{l}^{\prime}_{j})\Gamma^{2}(1+l_{j})\prod\limits_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{g+1}\Gamma(1+l_{k})}\frac{(-\hat{a}_{j})^{-\mathfrak{l}_{j}}}{\mathfrak{l}_{j}}\sum\limits_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{g+1}\hat{a}_{k}^{l_{k}}$

valid for $j=1,\dots,g$. For the family $\mathcal{C}_{4,1}$, Table 1 and Table
2 say that $\underline{\kappa}=(1,1)$ and $\underline{\hat{a}}=(-6,9)$, thus
proving,

$\displaystyle\dfrac{6}{\pi}D_{2}(1+e^{\pi\mathbf{i}/6})$ $\displaystyle=\log
6-\mspace{-15.0mu}\sum_{l_{1},l_{2},l_{3}\in\mathbb{Z}_{\geq
0}}{}^{\mspace{-25.0mu}\prime}\dfrac{\Gamma(6l_{1}+2l_{2}+3l_{3})(-6)^{-6l_{1}-2l_{2}-3l_{3}}9^{l_{2}}2^{l_{3}}}{\Gamma(1+4l_{1}+2l_{3})\Gamma^{2}(1+l_{1})\Gamma(1+l_{2})\Gamma(1+l_{3})}$
$\displaystyle=\log 6-\mspace{-15.0mu}\sum_{m,r,s\in\mathbb{Z}_{\geq
0}}{}^{\mspace{-25.0mu}\prime}\dfrac{(-1)^{s}\Gamma(6m+2r+3s)3^{-6m-3s}2^{s}}{\Gamma(1+4m+r+2s)\Gamma^{2}(1+m)\Gamma(1+r)\Gamma(1+s)},$

an identity conjectured in [CGM].

##  Appendix A Proof of Parameterizations

We tie up the remaining loose end, namely proof of parameterizations 2.51,
2.52 by slightly altering the proof presented in [DKSB, §4.3] \- that these
maps are of degree 1 is straightforward from considering the intersection of
the image and the boundary divisors of $\mathbb{P}_{\Delta_{2g,1}}$. Now,
$F^{\underline{a}}_{2g,1}$ is irreducible - this is also easy, a rather direct
application of [Gao, Prop. 2.6]. The difficult part is to show that the image
is contained in $\mathcal{C}^{\hat{\underline{a}}}_{2g,1}$. We begin by
observing that444This factorization is due to the dimer model phenomenon, and
will be explained in an upcoming work.,

$\displaystyle F^{\hat{\underline{a}}}_{2g,1}(x,y)$
$\displaystyle=\prod_{i=-\frac{2g+1}{2}}^{\frac{2g+1}{2}}\bigg{(}1+\zeta_{2g+2}^{2gi}x^{\frac{2g}{2g+2}}y^{\frac{2}{2g+2}}+\zeta_{2g+2}^{-i}x^{\frac{2g+1}{2g+2}}y^{\frac{1}{2g+2}}\bigg{)}$
$\displaystyle:=\prod_{i=-\frac{2g+1}{2}}^{\frac{2g+1}{2}}P_{i}(x,y)$

Let $(\hat{x},\hat{y})$ be the node of $F^{\hat{\underline{a}}}_{2g,1}(x,y)$
corresponding to $i=\pm\frac{2g+1}{2}$, then the map from
$\phi:\mathbb{P}^{1}_{z}\to(\mathbb{C}^{*})^{2}_{X,Y}$ parametrizing
$F^{\hat{\underline{a}}}_{2g,1}(x,y)$ such that the fiber over
$(\hat{x},\hat{y})$ is $\\{0,\infty\\}$ can be written as follows,

$\displaystyle X$
$\displaystyle=\hat{x}\dfrac{(z-1)^{2}}{(z-\zeta_{2g+2}^{2g+1})(z-\zeta_{2g+2})}$
$\displaystyle Y$
$\displaystyle=\hat{y}\dfrac{(z-\zeta_{2g+2}^{2g+1})^{2g+1}}{(z-1)^{2g}(z-\zeta_{2g+2})}$

######  Lemma A.1.

$a=\dfrac{\alpha^{n+1}}{\beta^{n}\theta},\leavevmode\nobreak\
\leavevmode\nobreak\ b=\dfrac{\beta^{m+1}}{\alpha^{m}\theta}$

Then the following identity holds,

$a^{\frac{m+1}{N}}b^{\frac{n}{N}}=\dfrac{\alpha}{\theta},\leavevmode\nobreak\
\leavevmode\nobreak\ a^{\frac{m}{N}}b^{\frac{n+1}{N}}=\dfrac{\beta}{\theta}$

###### Proof.

Clearly we have

$a^{m+1}b^{n}=\alpha^{(m+1)(n+1)-mn}\beta^{-n(m+1)+n(m+1)}\theta^{-m-1-n}=\dfrac{\alpha^{N}}{\theta^{N}}$

as was claimed. The other identity follows the same way. ∎

######  Proposition A.2.

$\phi$ is indeed the desired parametrization, that is,

$F^{\hat{\underline{a}}}_{2g,1}(X,Y)=0$

###### Proof.

Using Lemma A.1 with

$m=2g,\leavevmode\nobreak\ n=1,\leavevmode\nobreak\
\alpha=z-1,\leavevmode\nobreak\
\beta=z-\zeta_{2g+2}^{2g+1},\leavevmode\nobreak\ \theta=z-\zeta_{2g+2}$

we have,

$\displaystyle F^{\hat{\underline{a}}}_{2g,1}(X,Y)$
$\displaystyle=\prod_{i=-\frac{2g+1}{2}}^{\frac{2g+1}{2}}\bigg{(}1+\zeta_{2g+2}^{2gi}\hat{x}^{\frac{2g}{2g+2}}\hat{y}^{\frac{2}{2g+2}}\dfrac{z-\zeta_{2g+2}^{2g+1}}{z-\zeta_{2g+2}}+\zeta_{2g+2}^{-i}\hat{x}^{\frac{2g+1}{2g+2}}\hat{y}^{\frac{1}{2g+2}}\dfrac{z-1}{z-\zeta_{2g+2}}\bigg{)}$
$\displaystyle=\prod_{i=-\frac{2g+1}{2}}^{\frac{2g+1}{2}}(z-\zeta_{2g+2})\bigg{(}zP_{i}(\hat{x},\hat{y})-\zeta_{2g+2}P_{i+1}(\hat{x},\hat{y})\bigg{)}$

However as

$P_{\frac{2g+1}{2}+1}(x,y)=P_{\frac{2g+3}{2}}(x,y)=P_{-\frac{2g+1}{2}}(x,y)$

and by assumption

$P_{\frac{N-1}{2}}(\hat{x},\hat{y})=P_{-\frac{N-1}{2}}(\hat{x},\hat{y})=0$

we can conclude that

$F^{\hat{\underline{a}}}_{2g,1}(X,Y)=0$

∎

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