# Development of high-rate capable and ultra-low mass Resistive Plate Chamber
with Diamond-Like Carbon

Atsushi Oya1 Kei Ieki2 Atsuhiko Ochi3 Rina Onda1 Wataru Ootani2 and Kensuke
Yamamoto1 1 Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo
113-0033, Japan  2 International Center for Elementary Particle Physics, The
University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  3 Department of
Physics, Kobe University, Kobe 657-8501 Japan<EMAIL_ADDRESS>
###### Abstract

A new type of resistive plate chamber (RPC) is under development using thin-
film resistive electrodes based on diamond-like carbon (DLC). Planned to be
put on the path of high-intensity low-momentum muon beam of the MEG II
experiment, this detector is required to be high-rate capable and ultra-low
mass. Using a prototype detector with $2\,\mathrm{cm}\times 2\,\mathrm{cm}$
size and $0.1\%\,X_{0}$ material budget, performance studies were conducted
for MIP detection efficiency, timing resolution and high rate capability in
low-momentum muon beam. In this paper, the measured performance is presented
including the result with low-momentum muon beam at rate up to
$1\,\mathrm{MHz/cm^{2}}$. Based on the result, the expected performance of the
full-scale detector in the MEG II experiment is also discussed.

##  1 Introduction

A novel background identification detector is under development for the MEG II
experiment, aiming for sensitivity improvement in the $\mu\to e\gamma$ decay
search [1]. This detector is required to detect positrons ($<5\,\mathrm{MeV}$)
from the radiative muon decays with high energy photons observed in the photon
detector. As this detector is planned to be put in a high-intensity low-
momentum muon beam with $4\,\mathrm{MHz/cm^{2}}$ rate at the beam center,
ultra-low mass ($<0.1\,\%X_{0}$) and high rate capability are necessary in
addition to a high efficiency ($>90\,\%$) for minimum ionizing particles (MIP)
and a good timing resolution ($<1\,\mathrm{ns}$).

The detector is based on a new type of Resistive Plate Chamber (RPC) with
Diamond-Like Carbon (DLC) electrodes sputtered on thin Kapton films as shown
in Fig. 2. The overall material thickness can be kept below $0.1\%\,X_{0}$
with four-layer configuration.

Figure 1: Concept of the detector design.

Figure 2: Single-layer prototype.

A single-layer efficiency as high as 40 % is required for the overall
efficiency of 90 %.

##  2 Single layer prototype

A single-layer prototype with $2\,\mathrm{cm}\times 2\,\mathrm{cm}$ size is
constructed as shown in Fig. 2. The gap of $384\,\mathrm{\mu m}$-thick is made
by pillars formed with a photolithography technique. The gap is filled with a
gas mixture of 94 % Freon (R134a), 5 % iso-$\mathrm{C_{4}H_{10}}$ and 1 %
$\mathrm{SF_{6}}$. The high voltage is supplied by conductive tape attached on
the edge of the DLC surface. The readout is implemented with a 1 cm wide
aluminum strip. The signal is readout from both ends of the strip, then
amplified by a 38 dB amplifier and fed into the DRS4 waveform digitizer [2].

##  3 Performance measurement

Three different measurements were performed to evaluate the performance for
the MIP positron and low-momentum muon. Firstly, the MIP detection efficiency
and timing resolution were measured at low rate, as described in Sec. 3.1.
Secondly, the RPC’s response to a low-momentum muon was measured at low rate,
as described in Sec. 3.2 with discussion on the results. Finally, the
performance in a high-rate low-momentum muon beam was measured, as described
in Sec. 3.3.

###  3.1 Efficiency and timing resolution for MIP positrons at low rate

The efficiency and the timing resolution were measured with positrons from
muon decay ($\sim 10\,\mathrm{kHz/cm^{2}}$). The signal height spectra are
shown in Fig. 3 changing the applied voltage between 2.6–2.75 kV.

Figure 3: Signal height distributions obtained for MIP positrons at low rate.

The detection efficiency was measured to be 60 % with a 10 mV signal threshold
at 2.75 kV, which satisfies the requirement ($>40\,\%$). The timing resolution
was evaluated to be 170 ps at 2.75 kV, which also well satisfies the
requirement ($<1\,\mathrm{ns}$).

###  3.2 Response to low-momentum muon at low rate

The response to a low-momentum muon (28 MeV/$c$) was measured with the muon
beam at the $\mathrm{\pi E5}$ beam line of Paul Scherrer Institute. A
collimated muon beam was injected at 3.2 kHz into the prototype as shown in
Fig. 5. To avoid triggering positron hits, events with both muon and
subsequent (within 100–450 ns from muon hit) decay positron emitted toward
downstream were selected with two scintillation counters. The pulse height
spectra for different voltages between 2.6–2.75 kV are shown in Fig. 5, where
the main peak around 0.1–0.2 V corresponds to the muon events.

Figure 4: Schematic view of the setup for the measurement with low-momentum
muon at low rate

Figure 5: Signal height distributions obtained for low-momentum muon at low
rate.

The pedestal peak is likely due to the accidental triggers by background
radiations. This background prevents us from measuring the efficiency for low-
momentum muon although it is found to be higher than 75 % at 2.75 kV from the
event fraction above the pedestal.

Even though the energy deposit by a low-momentum muon is an order of magnitude
larger than that by a MIP positron, Fig. 5 shows that the signal height is
around the same level as that for the MIP positrons. This can be explained by
the space charge effect, which saturates the evolution of avalanche charge
when the total avalanche gain reaches $\mathrm{O(10^{7}\textrm{--}10^{8})}$
[3].

The rate capability is determined by the gain reduction due to the voltage
drop caused by the signal current flowing on the resistive electrodes [4]. The
observed saturation for the low-momentum muon is, therefore, even advantageous
when the RPC is operated in a low-momentum muon beam at high rate as in MEG II
since the voltage drop can be suppressed.

###  3.3 Performance in low-momentum muon beam at high rate

The detection efficiency for MIP positrons in a high-intensity low-momentum
muon beam was measured at 2.75 kV with a setup shown in Fig. 6. In this
measurement, the RPC was exposed to a muon beam with a spread of
$(\sigma_{x},\sigma_{y})=(13\,\mathrm{mm},23\,\mathrm{mm})$ at a rate of
$1\,\mathrm{MHz/cm^{2}}$ at the center. The readout region of the RPC was
aligned to the beam center.

Two scintillation counters were aligned to select positrons from decay of muon
stopped in the upstream counter. The angle between the flight path of the
triggered positron and the beam axis was set to $30^{\circ}$, which is close
to the average incident angle of the positrons from the radiative muon decay
to the RPC in the MEG II experiment.

A typical event display is also shown in Fig. 6. The waveform data were
analyzed both for the on-triggered timing and off-timing windows with 30 ns
width each. In the on-timing window, the positron events are expected and the
signal height spectrum is shown as the red histogram in Fig. 7, where the
efficiency is evaluated to be 35 %.

Figure 6: Schematic view of the setup of measurement with high rate muon beam
and a typical event display. The on-timing (off-timing) window is depicted as
the red hatched (blue shaded) region. Figure 7: The signal height distribution
obtained at 2.75 kV in test with a high rate low-momentum muon beam. The red
(blue) histogram shows the distribution in the on-timing (off-timing) window.

The signal height spectrum in the off-timing window is shown as blue histogram
in Fig. 7, where the bump around 0.1 V comes from the accidental muon events.

The voltage drop due to the high signal current at high rate was estimated by
comparing the signal height spectra in this measurement (at 2.75 kV) with
those measured in the low rate measurements (at different voltages) in Sec.
3.1 and Sec. 3.2. Both the muon and MIP spectra appear similar to those
obtained at low rate at 2.6–2.65 kV, which suggests a voltage drop of 100–150
V. Though $\sim 150\,\mathrm{V}$ drop is too large to achieve 40 % single
layer efficiency, it can be reduced with further design optimization as will
be discussed in Sec. 4.

The expected voltage drop can also be calculated from the geometry and the
resistivity ($\sim 60\,\mathrm{M\Omega/sq}$ for the prototype) of the DLC, the
measured hit rate and the avalanche charge. The calculation also gives a
voltage drop of 100–150 V, which agrees with the estimation from the observed
signal height distribution. This result confirms that the performance in a
high-rate beam can be projected from the voltage drop calculation.

##  4 Expected performance in MEG II experiment

The full-scale RPC for the MEG II experiment should be 20 cm in diameter. It
would suffer from a much larger voltage drop due to the larger area of the
DLC. In addition, the beam profile of $4\,\mathrm{MHz/cm^{2}}$ at the center
and spread of $\sigma_{x}=\sigma_{y}=20\,\mathrm{mm}$ are harsher than that in
the prototype test. The voltage drop can be mitigated with a multi-strip HV
supply as shown in Fig. 8.

Figure 8: Schematic explaining the structure of multi-strip high voltage
supply.

The voltage drop can be reduced by the small strip pitch independently of the
total detector size, although the pitch cannot be too small because of the
increase of the inactive area.

With this concept, the voltage drop can be kept at a reasonable level of 100 V
for the applied voltage of 2.75 kV with the strip pitch of 1 cm and the
surface resistivity of the DLC of $10\,\mathrm{M\Omega/sq}$. This would allow
us to operate the RPC with the required performance in the high-rate muon beam
of the MEG II experiment.

We still, however, have several technical challenges to be solved to realize
this design. Firstly, the implementation of the conductive strips on the DLC
surface is not trivial. Secondly, the control of the surface resistivity of
the DLC is not easy. We currently suffer a large fluctuation by a factor 3–5.
Finally, tests with $10\,\mathrm{M\Omega/sq}$ have not been performed because
of the issue of resistivity control. A stable operation must be achieved with
$10\,\mathrm{M\Omega/sq}$ resistivity while $>30\,\mathrm{M\Omega/sq}$
resistivity has been used in the measurements so far.

##  5 Conclusion

A new type of RPC with DLC electrodes is under development for the MEG II
experiment, where MIP detection in a high-rate low-momentum muon beam is
required. Using $2\,\mathrm{cm}\times 2\,\mathrm{cm}$ size prototype, the
performance of RPC was measured for MIP positron and low-momentum muon beam
with rate up to $1\,\mathrm{MHz/cm^{2}}$. The results show that 90 % overall
efficiency is achievable with 4-layer configuration even in the high-intensity
muon beam of the MEG II experiment, while there still remain some technical
challenges. Our results are encouraging for possible application to other
experiments where the RPCs have not been available due to the limitted rate
capability of the conventional glass-based RPCs.

##  6 Acknowledgments

This work is supported by JSPS Core-to-Core Program, A. Advanced Research
Networks JPJSCCA20180004.

## References

## References

  * [1] Baldini A M et al. 2018 Eur. Phys. J. C 78 380 URL https://doi.org/10.1140/epjc/s10052-018-5845-6
  * [2] Ritt S 2004 Nuclear Instruments and Methods in Physics Research Section A 518 470–471 URL https://www.sciencedirect.com/science/article/pii/S016890020302922X
  * [3] Lippmann C and Riegler W 2004 Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 517 54–76 ISSN 0168-9002 URL https://www.sciencedirect.com/science/article/pii/S0168900203026421
  * [4] Aielli G et al. 2016 Journal of Instrumentation 11 P07014–P07014 URL https://doi.org/10.1088/1748-0221/11/07/p07014


# Fixed-energy harmonic functions

Aaron Abrams Mathematics Department, Washington and Lee University, Lexington
VA 24450; abramsa at wlu.edu. Research supported by Simons Foundation award
281189. Richard Kenyon Department of Mathematics, Brown University,
Providence, RI 02912; rkenyon at math.brown.edu. Research supported by NSF
grant DMS-1208191 and the Simons Foundation award 327929.

###### Abstract

We study the map from conductances to edge energies for harmonic functions on
finite graphs with Dirichlet boundary conditions. We prove that for any
compatible acyclic orientation and choice of energies there is a unique choice
of conductances such that the associated harmonic function realizes those
orientations and energies. We call the associated function _enharmonic_. For
rational energies and boundary data the Galois group of ${\mathbb{Q}}^{tr}$
(the totally real algebraic numbers) over ${\mathbb{Q}}$ permutes the
enharmonic functions, acting on the set of compatible acyclic orientations.

For planar graphs there is an enharmonic conjugate function; together these
form the real and imaginary parts of a “fixed energy” analytic function. In
the planar scaling limit for ${\mathbb{Z}}^{2}$ (and the fixed south/west
orientation), these functions satisfy a nonlinear analog of the Cauchy-Riemann
equations, namely

$\displaystyle u_{x}v_{y}$ $\displaystyle=$ $\displaystyle 1$ $\displaystyle
u_{y}v_{x}$ $\displaystyle=$ $\displaystyle-1.$

We give an analog of the Riemann mapping theorem for these functions, as well
as a variational approach to finding solutions in both the discrete and
continuous settings.

##  1 Overview

###  1.1 The Dirichlet problem

The classical Dirichlet problem is to find a harmonic function on a domain
that takes specified values on the boundary. It is the mathematical
abstraction of physical equilibrium problems arising in electromagnetism,
fluid flow, gravitation, and other areas. Solutions to the Dirichlet problem
describe physical phenomena from the vibrations of a drum to the dissipation
of heat. Dirichlet himself was investigating the stability of the solar system
[7]. In the century and a half since Dirichlet’s work, there have been
countless investigations into various versions, modifications,
generalizations, and special cases of the Dirichlet problem.

We study the discrete Dirichlet problem on a graph, and in particular we are
interested in the connection between the _edge conductances_ and the _edge
energies_ of the resulting solution. In the classical problem one fixes the
conductances, and then minimizing the Dirichlet energy gives a unique solution
to the Dirichlet problem, namely a harmonic function on the vertices called
the _voltage_. The edge energies are then obtained from this solution. Thus
there is a map, soon called $\Psi$, from edge conductances to edge energies.

Here we reverse the problem and fix the desired energies of the harmonic
function, determining which choices of conductances lead to these energies. We
show (Theorem 9) that in suitable coordinates $\Psi$ has a surprisingly simple
Jacobian, which makes analysis of the point preimages very clean. This set of
preimages is parameterized by a certain set of acyclic orientations of the
graph (Theorems 1 and 3), and one can determine the Dirichlet solutions
through a simple equation, the _enharmonic Laplace equation_ (Theorem 2).
Remarkably, the cardinality of the solution set is independent of the boundary
values, as long as these are distinct (Corollary 4).

We begin in Section 1.2 with notation and an overview of our main results
showing existence of solutions and giving variational descriptions of these
solutions. In Sections 1.3, 1.4, 1.5, 1.6 we give consequences of our theorems
that highlight connections between the Dirichlet problem on a graph and other
areas of mathematics, such as number theory (Theorem 6), discrete analytic
functions (Theorem 11), and rectangle tilings (Corollaries 7 and 8). Some
proofs are given; longer ones are deferred to later sections.

Several explicit examples of the main theorems are given in Section 2.

In Section 3 we study the Jacobian of $\Psi$. Sections 4, 5, 6 contain proofs.
We pose a few questions in Section 7.

###  1.2 Main results

Let ${\mathcal{G}}=(V,E)$ be a connected graph and $B\subset V$ a subset of
$\geq 2$ vertices, which we refer to as _boundary vertices_. We assume that
each edge of ${\mathcal{G}}$ lies on a simple path from a vertex of $B$ to
another vertex of $B$. Let $c_{e}>0$ be a _conductance_ for each edge $e$. It
will sometimes be convenient to refer to edges by their endpoints, e.g., we
also write $c_{xy}$ for the same conductance if $e$ is the edge with endpoints
$x,y$.

Given a function $u:B\to{\mathbb{R}}$ the classical Dirichlet problem is to
find a function $f$ equal to $u$ on $B$ and which minimizes the _Dirichlet
energy_

${\mathcal{E}}(f)=\sum_{e=xy}c_{e}(f(x)-f(y))^{2},$

where the sum is over all edges of ${\mathcal{G}}$. There is a unique
minimizing function $h$; it is the unique function harmonic at each vertex of
$V\setminus B$, that is satisfying, for $x\not\in B$,

$0=\Delta h(x):=\sum_{y\sim x}c_{e}(h(x)-h(y))$

where the sum is over neighbors $y$ of $x$. We define the _energy_ of $h$ on
an edge $e=xy$ to be $c_{e}(h(x)-h(y))^{2}$, and

$\Psi=\Psi_{u}:(0,\infty)^{E}\to[0,\infty)^{E}$ (1)

to be the map from the set of edge conductances to the set of energies of the
associated harmonic function $h$. See an example in (6) below.

Our main theorem relates the conductances and edge energies of a harmonic
function. (See the definition of compatible orientation below.)

######  Theorem 1 (Existence).

Let the graph ${\mathcal{G}}$, boundary $B$, and boundary values
$u:B\to{\mathbb{R}}$ be fixed. For each compatible orientation $\sigma$ of
${\mathcal{G}}$ and tuple ${\mathcal{E}}\in(0,\infty)^{E}$ there is a unique
choice of positive conductances $\\{c_{e}\\}$ on edges so that the associated
harmonic function $h$ has $\text{sgn}(dh)=\sigma$ and energies
${\mathcal{E}}$.

Let ${\mathcal{F}}_{u}$ be the set of real-valued functions on ${\mathcal{G}}$
which have no interior extrema, that is, no local maxima or local minima
except on $B$. For any choice of positive conductances, a harmonic function
with boundary values $u$ on $B$ necessarily lies in ${\mathcal{F}}_{u}$, by
the maximum principle for harmonic functions.

Suppose that $f\in{\mathcal{F}}_{u}$ takes distinct values at the two
endpoints of each edge of ${\mathcal{G}}$. Then $f$ induces an orientation of
the edges of ${\mathcal{G}}$, from larger values to smaller values. This
orientation has the following three obvious properties:

  1. 1.

it is acyclic, that is, there are no oriented cycles,

  2. 2.

it has no interior sinks or sources (as these would correspond to interior
extrema of $f$),

  3. 3.

there are no oriented paths from lower boundary values to higher (or equal)
boundary values.

Given $u$, we call an orientation satisfying the above properties _compatible
(with $u$)_, and let $\Sigma_{u}$ be the set of orientations compatible with
$u$. It is not hard to see that for each $\sigma\in\Sigma_{u}$ there is a
function $h\in{\mathcal{F}}_{u}$ inducing $\sigma$, that is, with
$dh(e):=h(x)-h(y)$ nonzero on every edge $e=xy$, and having sign
$\text{sgn}(dh(e))=\sigma(e)$ on each edge. The set
${\mathcal{F}}_{u}(\sigma)$ of functions $h\in{\mathcal{F}}_{u}$ inducing the
orientation $\sigma$ is therefore an _open polytope_
${\mathcal{F}}_{u}(\sigma)$, since it is defined by (strict) linear
inequalities.

The most basic setting of Theorem 1 above is the case where the boundary $B$
consists in just two vertices. Let $B=\\{v_{0},v_{1}\\}$ and
$u(v_{0})=0,u(v_{1})=1$. Then $\Sigma_{u}$ consists of acyclic orientations
which have a unique sink, at $v_{0}$, and a unique source, at $v_{1}$. These
are called _bipolar orientations_. Their cardinality is the _beta invariant_
of the graph $\tilde{\mathcal{G}}$, in which an edge is added to
${\mathcal{G}}$ from $v_{0}$ to $v_{1}$ if there is no such edge already
present. The beta invariant, also known as the _chromatic invariant_ , is the
derivative at $1$ of the chromatic polynomial of $\tilde{\mathcal{G}}$, or
equivalently the coefficient of $x$ (and of $y$) in the Tutte polynomial
$T(x,y)$, see [4].

By Theorem 1 above, if we fix a graph ${\mathcal{G}}$ with boundary $B$, a
function $u:B\to{\mathbb{R}}$, and on each edge $e$ an energy
${\mathcal{E}}_{e}>0$, the set of functions in ${\mathcal{F}}_{u}$ with
energies $\\{{\mathcal{E}}_{e}\\}$ is in bijection with the set $\Sigma_{u}$
of compatible orientations of ${\mathcal{G}}$. We call these functions
_enharmonic_ (short for “energy-harmonic”) with respect to the energies
$\\{{\mathcal{E}}_{e}\\}$. Note that a function that is enharmonic with
respect to certain energies is also harmonic with respect to certain
conductances, and vice versa; the difference in terminology indicates only
which data were used to define the function.

######  Theorem 2 (Characterization).

Let the graph ${\mathcal{G}}$, boundary $B$, and boundary values
$u:B\to{\mathbb{R}}$ be fixed. Enharmonic functions with energies
${\mathcal{E}}\in(0,\infty)^{E}$ are precisely the local maxima on
${\mathcal{F}}_{u}$ of the functional

$M(h)=\prod_{e=xy\in E}|h(x)-h(y)|^{{\mathcal{E}}_{e}}.$ (2)

The enharmonic function $h$ with energies ${\mathcal{E}}$ and satisfying
$dh=\sigma$ is the unique maximizer of $M(h)$ on the polytope
${\mathcal{F}}_{u}(\sigma)$. An enharmonic function is one which satisfies the
(discrete) _enharmonic Laplace equation_ $Lh(x)=0$ at every interior vertex
$x$, where $L$ is the nonlinear operator

$Lh(x)=\sum_{y\sim x}\frac{{\mathcal{E}}_{xy}}{h(x)-h(y)}$ (3)

and the sum is taken over all vertices $y$ adjacent to $x$.

In practice, for given energies $\\{{\mathcal{E}}_{e}\\}$, determining the
function $h$ and the conductances $\\{c_{e}\\}$ is most easily done via the
equation (3). We prove Theorem 2 in Section 5.

###  1.3 The degree of $\Psi$

The map $\Psi_{u}$ of (1) from conductances to energies is a rational map
(i.e. its coordinate functions are ratios of polynomial functions of the
conductances). To see this, note first that to obtain the harmonic function
$f$ as a function of the conductances and boundary data, we have to invert the
Laplacian which has entries which are linear polynomials in the conductances;
the energies are then obtained from ${\mathcal{E}}_{e}=c_{e}(f(x)-f(y))^{2}$.
Thus $\Psi_{u}$ can be extended to a rational map

$\Psi_{u}:{\mathbb{C}}^{E}\to{\mathbb{C}}^{E}.$

The inverse map $\Psi_{u}^{-1}$ is algebraic. Remarkably, for positive real
energies all inverse images are positive real:

######  Theorem 3.

The rational map $\Psi_{u}$ is of degree $|\Sigma_{u}|$ over ${\mathbb{C}}$.

By this we mean that there is an open dense set of ${\mathbb{C}}^{E}$ such
that the cardinality of the preimage of any point in this set is
$|\Sigma_{u}|$. See Section 4 for a proof. Since Theorem 1 provides
$|\Sigma_{u}|$ preimages of any point with positive real coordinates, such a
preimage must consist only of points of ${\mathbb{C}}^{E}$ with positive real
coordinates:

$\Psi_{u}^{-1}((0,\infty)^{E})\subset(0,\infty)^{E}.$

An immediate consequence is

######  Corollary 4.

The number of compatible orientations $|\Sigma_{u}|$ does not depend on the
values of $u$, as long as $u$ takes distinct values at distinct boundary
vertices.

###### Proof.

The degree of a rational family of rational maps is constant for almost every
parameter value. Thus for almost every choice of values of $u$, the degree of
$\Psi_{u}$ is constant. As $u$ varies among a (full-dimensional) set of
functions taking distinct values, the set of compatible orientations does not
change. So the set of $u$’s where the degree of $\Psi_{u}$ is smaller is a
subset of the set $u$’s where at least two boundary values are the same. ∎

###  1.4 Galois action

Suppose the given energies and boundary values lie in ${\mathbb{Q}}$, with
positive energies. Then the enharmonic equation

$\sum_{y\sim x}\frac{{\mathcal{E}}_{xy}}{h(x)-h(y)}=0$ (4)

supplemented with the boundary equations $h|_{B}=u$ which define the
enharmonic functions, form a rational system of equations defined over
${\mathbb{Q}}$. Thus the absolute Galois group permutes the solutions. Theorem
3 shows that the values of $f$ generate a totally real number field. Thus we
have

######  Corollary 5.

If the values of $u$ and the coordinates of ${\mathcal{E}}\in(0,\infty)^{E}$
are rational then the Galois group of ${\mathbb{Q}}^{tr}$ (the totally real
algebraic numbers) over ${\mathbb{Q}}$ permutes the enharmonic functions of
Theorem 1. Equivalently, $\operatorname{Gal}({\mathbb{Q}}^{tr}/{\mathbb{Q}})$
acts on $\Sigma_{u}$.

We conjecture that for 3-connected graphs, the action is transitive111A graph
with boundary is $3$-connected if, upon adding edges connecting every pair of
boundary vertices, it cannot be disconnected by removing two vertices.:

######  Conjecture 1.

If $G$ is $3$-connected the above Galois action is transitive for generic222By
generic we mean on a Zariski open set. rational energies and boundary data.

Considering all acyclic orientations $\cup\Sigma_{u}$ as ${\mathcal{G}}$ and
$u$ vary, this is reminiscent of _dessins d’enfants_ , a family of
combinatorial objects on which the absolute Galois group
$\operatorname{Gal}(\bar{\mathbb{Q}}/{\mathbb{Q}})$ acts. In the current setup
it is the Galois group of the totally real algebraic numbers which acts. We
prove (see Section 6) that the action is faithful, i.e., that every totally
real number field arises in this way:

######  Theorem 6.

For every totally real number field $K$ there is a tuple
$({\mathcal{G}},B,u,\\{{\mathcal{E}}_{e}\\})$ consisting of a graph
${\mathcal{G}}$, boundary $B$, rational boundary values $u$ and rational edge
energies ${\mathcal{E}}_{e}>0$ such that $K$ is the number field generated by
the values of one of the associated enharmonic functions, and all enharmonic
functions generate isomorphic number fields.

We don’t know if Theorem 6 holds when the size of the boundary is constrained,
for example when the boundary consists in only two points. We show this
however for quadratic fields in Section 2.2.

###  1.5 Discrete analytic functions

A _circular planar network_ [5] is a finite embedded planar graph
${\mathcal{G}}$ with conductances $c:E\to{\mathbb{R}}_{>0}$ and boundary $B$
consisting of a subset of vertices on the outer face. We define the _dual
network_ ${\mathcal{G}}^{*}$ as the usual planar dual of the graph obtained
from ${\mathcal{G}}$ by adding disjoint rays from each boundary vertex of
${\mathcal{G}}$ to infinity. Thus ${\mathcal{G}}^{*}$ has one vertex in every
bounded face of ${\mathcal{G}}$ and one vertex in the exterior face between
every pair of “consecutive” boundary vertices of ${\mathcal{G}}$; there is a
dual edge connecting dual vertices for every primal edge separating the
associated faces and there is a dual edge for each of the infinite rays. Given
a harmonic function $f$ on ${\mathcal{G}}$ with boundary values $u$ on $B$, a
function $g$ on ${\mathcal{G}}^{*}$ is said to be a _harmonic conjugate_ if
for any primal edge $e=xy$ we have $f(y)-f(x)=c_{e}(g(t)-g(s))$ where where
$s,t$ are the faces adjacent to edge $xy$, with $t$ being on the right when
the edge is traversed from $x$ to $y$. These equations are the _discrete
Cauchy-Riemann equations_. They imply in fact that $f$ is harmonic on
${\mathcal{G}}$, and $g$ is harmonic on the dual graph with reciprocal
conductances on the dual edges. In this setting such a pair $f,g$ is also said
to form a _discrete analytic function_.

For a circular planar network $({\mathcal{G}},B,u,\\{{\mathcal{E}}_{e}\\})$
with enharmonic function $f$, an _enharmonic conjugate_ $g$ is a function $g$
on ${\mathcal{G}}^{*}$ satisfying

$(g(t)-g(s))(f(y)-f(x))={\mathcal{E}}_{xy},$

where $s,t$ are the faces adjacent to edge $xy$, with $t$ being on the right
when the edge is traversed from $x$ to $y$. Enharmonicity of $f$ implies
enharmonicity of $g$ on the dual graph with the same energies. Together $f$
and $g$ are harmonic conjugates for the associated conductances. We call the
pair $(f,g)$ the _fixed-energy discrete analytic function_ associated to the
boundary data. These functions are used as $x$\- and $y$-coordinates in the
rectangle tilings discussed in the next section.

It is natural to consider scaling limits of enharmonic functions, that is,
continuous limits of enharmonic functions on a sequence of graphs
approximating (in an appropriate sense) a region in ${\mathbb{R}}^{n}$. These
scaling limits depend strongly on both the underlying graph structure and
choice of orientation. We consider here only the simplest nontrivial case
where the approximating graph is a scaled subgraph of ${\mathbb{Z}}^{2}$, with
edges oriented south and west. We prove in Theorem 11 below that the natural
scaling limit of an enharmonic function $f$ and its enharmonic conjugate $g$
for ${\mathcal{G}}=\epsilon{\mathbb{Z}}^{2}$ satisfy a nonlinear analog of the
Cauchy-Riemann equation:

$\displaystyle f_{x}g_{y}$ $\displaystyle=$ $\displaystyle-1$ (5)
$\displaystyle f_{y}g_{x}$ $\displaystyle=$ $\displaystyle 1.$

Moreover $f,g$ both satisfy the “continuous enharmonic Laplace equation”

$\frac{f_{xx}}{f_{x}^{2}}+\frac{f_{yy}}{f_{y}^{2}}=0.$

We prove (see Section 5.3) an analog of the Riemann mapping theorem in this
context, namely that from any region in an appropriate class there is a
“fixed-energy analytic” map to a rectangle.

In [1] a different continuous analog of the fixed-energy problem is
considered.

###  1.6 Tilings and networks

In 1940, Brooks, Smith, Stone, and Tutte [3], building on work of Dehn [6],
gave a correspondence between the Dirichlet problem for circular planar
networks (with two boundary vertices) and tilings of a rectangle by
rectangles; they called this the _Smith diagram_ of the network. See Figure 2
below for an example. Namely, given such a tiling, one can orient the
rectangle so it sits on the $x$-axis, and then form a network from the tiling
according to the following rules. Place a vertex at height $y$ for each
maximal horizontal segment in the tiling at height $y$; then, for each
rectangular tile add an edge to the network connecting the vertices
corresponding to the top and bottom of the tile. On this edge place a
conductance which is the aspect ratio (width over height) of the tile. The
result is a circular planar graph which can be interpreted as a resistor
network.

If we now imagine the top and bottom vertices connected by a battery
generating a voltage differential equal to the height of the rectangle, then
the voltage function on the vertices (i.e., the $y$-values) is a harmonic
function that solves the Dirichlet problem on the network with boundary
conditions enforced at the top and bottom vertex. The rectangle widths are the
current flows $c_{e}dh$, and the areas are the edge energies $c_{e}(dh)^{2}$.

The BSST correspondence is invertible: beginning with a planar network with a
specified source and sink on the outer face and a specified total voltage
drop, one can solve for the harmonic function and the currents in the wires.
The resulting network can then be represented as a rectangle tiling by
reversing the process described above.

A corollary of Theorem 1 applies to Smith diagrams. This was originally proved
in a different manner in [12] and a number of times since, see [8].

######  Corollary 7.

Given a rectangle tiling of a rectangle there is an isotopic tiling in which
the rectangles have prescribed areas.

Here by _isotopic_ we mean one is obtained from the other by displacing
continuously the walls, floors and ceilings of rectangles in the first to form
the second, without degenerating any rectangles along the way (we also need to
add a proviso that at a point where four rectangles meet only one of the two
possible perturbations is allowed; either the horizontal segments to its left
and right can move independently up and down, or the vertical segments above
and below can move independently left and right, but not both.) See Section 2
for some examples.

Such maps are also called _rectangular cartograms_ [9] and _floorplans_ [2]
and are used e.g. to make geographic maps where the area represents the
measure of some quantity associated to each country, such as population.

Another corollary of Theorems 1 and 3 applies to the tilability of polygons by
rectangles of rational areas. Such a tiling arises from a circular planar
network with a boundary vertex for each horizontal segment of the boundary
polygon. Again the Dirichlet boundary values are the $y$-coordinates of the
horizontal boundary segments.

######  Corollary 8.

If a polygon $P$ can be tiled with rectangles of rational areas, then the
horizontal edge lengths are in a totally real extension field of
${\mathbb{Q}}(v_{1},\dots,v_{k})$, the field generated by the vertical edge
lengths.

For example the unit-area polygon of Figure 1 cannot be tiled with rectangles
of rational areas, because $2^{1/3}$ is not a totally real algebraic number.

Figure 1: A polygon with area $1$ which is not tilable with rational-area
rectangles.

###### Proof.

Assume some vertex of $P$ is at $(0,0)$. As discussed above, the
$y$-coordinates of the horizontal edges of $P$ correspond to the boundary
values $u$ of the enharmonic function. The $y$-coordinates of the interior
horizontal edges are the interior values of the enharmonic function, and
therefore in a totally real extension of ${\mathbb{Q}}(v_{1},\dots,v_{k})$.
The $x$-coordinates of all vertical edges are sums of currents
${\mathcal{E}}_{xy}/(f(x)-f(y))$, which are therefore in the same extension
field. ∎

##  2 Examples

We illustrate our theorems and techniques with a few examples.

###  2.1 A small graph

Consider the graph ${\mathcal{G}}$ shown in Figure 2, with conductances
$a,b,c,d,e$ as labeled. With $u(v_{1})=1$ and $u(v_{0})=0$, we have
$x,y\in[0,1]$, and thus ${\mathcal{F}}_{u}\cong[0,1]^{2}$. Solving for $h$
gives $h(x)=\frac{1}{Z}(ab+ac+bc+ae)$ and $h(y)=\frac{1}{Z}(ab+ac+bc+bd)$,
where $Z=ab+ac+ae+bc+bd+cd+ce+de$ is the determinant of the Laplacian. The
energies are then

$\displaystyle\Psi_{u}(a,b,c,d,e)=\frac{1}{Z^{2}}\Bigl{(}$ $\displaystyle
a(bd+cd+ce+de)^{2},\ b(ac+cd+ce+de)^{2},$ (6) $\displaystyle c(ae-bd)^{2},\
d(ab+ac+bc+ae)^{2},\ e(ab+ac+bc+bd)^{2}\Bigr{)}$

$v_{0}$$v_{1}$$y$$x$$a$$b$$c$$d$$e$$0$$1$$\frac{1}{2}-\frac{1}{2\sqrt{5}}$$\frac{1}{2}+\frac{1}{2\sqrt{5}}$
Figure 2: A graph with $|\Sigma|=2$. With $u(v_{0})=0,u(v_{1})=1$ and all
energies equal, the underlying number field is ${\mathbb{Q}}(\sqrt{5})$.

To find solutions with (for instance) all energies equal to $1$, it is much
easier to solve the enharmonic equation $Lh=0$ than it is to set each
coordinate above equal to 1 and solve the system directly. The enharmonic
equation(s) (see (4)) are

$\displaystyle 0$
$\displaystyle=\frac{{\mathcal{E}}_{a}}{h(x)-h(v_{1})}+\frac{{\mathcal{E}}_{c}}{h(x)-h(y)}+\frac{{\mathcal{E}}_{d}}{h(x)-h(v_{0})}=\frac{1}{h(x)-1}+\frac{1}{h(x)-h(y)}+\frac{1}{h(x)}$
$\displaystyle 0$
$\displaystyle=\frac{{\mathcal{E}}_{b}}{h(y)-h(v_{1})}+\frac{{\mathcal{E}}_{c}}{h(y)-h(x)}+\frac{{\mathcal{E}}_{e}}{h(y)-h(v_{0})}=\frac{1}{h(y)-1}+\frac{1}{h(y)-h(x)}+\frac{1}{h(y)}$

and have the solutions $h(x)=\frac{1}{2}\pm\frac{\sqrt{5}}{10}$ and
$h(y)=\frac{1}{2}\mp\frac{\sqrt{5}}{10}$, from which one can quickly produce
the conductances (for example, $a=e=\frac{1}{2}(15\pm 5\sqrt{5})$). The values
of $h$ are roots of the polynomial $5z^{2}-5z+1$. There are two solutions
because current can flow either way along the edge $xy$. That is,
$|\Sigma|=2$. The two polytopes in ${\mathcal{F}}_{u}=[0,1]^{2}$ on which
$M(h)$ is nonzero are the triangles above and below the diagonal.

The equal-area tilings associated to the two solutions are shown in Figure 2;
the $y$-coordinates of the horizontal edges are $h(x)$ and $h(y)$. The tilings
are not isotopic.

###  2.2 Quadratic number fields

Keeping the same graph ${\mathcal{G}}$ as in the previous example, if we
instead look for solutions with arbitrary energies
${\mathcal{E}}_{a},{\mathcal{E}}_{b},{\mathcal{E}}_{c},{\mathcal{E}}_{d},{\mathcal{E}}_{e}$,
we find that the conductances generate the number field
${\mathbb{Q}}(\sqrt{\delta})$, where

$\delta=({\mathcal{E}}_{a}{\mathcal{E}}_{e}-{\mathcal{E}}_{b}{\mathcal{E}}_{d}+{\mathcal{E}}_{c}S)^{2}+4{\mathcal{E}}_{b}{\mathcal{E}}_{c}{\mathcal{E}}_{d}S.$

Here $S$ is the sum of the five energies. Of course for positive real
energies, $\delta>0$ as all roots are real. It turns out that we can get any
real quadratic number field this way. To see this, we first specify
${\mathcal{E}}_{a}={\mathcal{E}}_{b}={\mathcal{E}}_{c}=1$ so that the
discriminant reduces to
$\delta=4{\mathcal{E}}_{d}^{2}+4{\mathcal{E}}_{d}{\mathcal{E}}_{e}+4{\mathcal{E}}_{e}^{2}+12{\mathcal{E}}_{d}+12{\mathcal{E}}_{e}+9$.
To obtain the number field ${\mathbb{Q}}(\sqrt{D})$ for a positive squarefree
integer $D$, it suffices to find positive rational numbers
${\mathcal{E}}_{d},{\mathcal{E}}_{e}$ such that $\delta=Dr^{2}$ for some
rational $r$. Choose a rational $s$ so that $D^{\prime}=Ds^{2}\in(1/3,4/9).$
Then substituting
${\mathcal{E}}_{d}=\frac{1}{6D^{\prime}-2},{\mathcal{E}}_{e}=\frac{4-9D^{\prime}}{6D^{\prime}-2}$
(both positive) gives discriminant
$\delta=\frac{9}{(3D^{\prime}-1)^{2}}D^{\prime}$ which is a rational square
times $D$.

###  2.3 Jacobi polynomials

One way to generalize the previous graph is the following.333Note that $n=2$
is the only member of this family that is planar. Let ${\mathcal{G}}=K_{n+2}$
be the complete graph on $n+2$ vertices $v_{0},\dots,v_{n+1}$, minus the edge
$v_{0}v_{1}$. Fix boundary $B=\\{v_{0},v_{1}\\}$ and boundary values
$u(v_{0})=0$ and $u(v_{1})=1$. Put energy $a$ on all edges from $v_{0}$ to
$v_{i}$, $i\in[2,n+1]$ and energy $b$ on edges from $v_{1}$ to
$v_{i},~{}i\in[2,n+1]$, and energy $2$ on the remaining edges. Then by
symmetry we can order the values $h(i)$ for $i\in[2,n+1]$ in increasing order.
We have

$M(h)=\prod_{i\in[2,n+1]}h(i)^{a}(1-h(i))^{b}\prod_{1<i<j<n+1}(h(j)-h(i))^{2}.$

In [11, Theorem 6.7.1] it is shown that the values $h(i)$ maximizing $M(h)$
are the roots of the Jacobi orthogonal polynomial $P_{a-1,b-1}(z)$, scaled to
the interval $[0,1]$.

###  2.4 A medium graph

For another example, consider the network shown in Figure 3. Again let
$u(v_{1})=1$ and $u(v_{0})=0$. The edges are labeled 1 through 12, and (by
coincidence) also $|\Sigma_{u}|=12$. We have solved the enharmonic equation
(see (4)) for the twelve enharmonic functions that produce all energies equal
to $1$. From these functions we computed the conductances of the networks,
which determine the shapes of the rectangles in the corresponding Smith
diagrams. In Figure 4 we have scaled the Smith diagrams to be (unit) squares,
so each rectangle in each rectangulation has area $1/12$. Any rectangulation
of a rectangle with the same underlying network (Figure 3) is isotopic to one
of these twelve.

These calculations are too complicated to do by hand; for instance, according
to Mathematica, the width of the rectangle labeled “1” in each picture is a
root of the polynomial

$\displaystyle p(z)=$ $\displaystyle
1270080000000z^{12}-5554584000000z^{11}+10776143400000z^{10}$
$\displaystyle-12235337185000z^{9}+9034493949125z^{8}-4560532680000z^{7}$
$\displaystyle+1610724560815z^{6}-400501165895z^{5}+69535433439z^{4}$
$\displaystyle-8223166134z^{3}+629396649z^{2}-28041714z+551124.$

All other edge lengths in all tilings in Figure 4 lie in the splitting field
of $p$, which is contained in ${\mathbb{R}}$.

Figure 3: A planar network with two boundary vertices $v_{0},v_{1}$. Figure 4:
Smith diagrams for the planar network of Figure 3, with energies $1$. Here
$|\Sigma_{u}|=12$, and the Galois group acts transitively: the underlying
number field for each tiling is a degree-$12$ totally real extension field of
${\mathbb{Q}}$. Each tiling corresponds to a different bipolar orientation of
${\mathcal{G}}$, so these tilings are not isotopic.

###  2.5 A large graph

The variational method makes it very easy to find (numerically) enharmonic
functions on large graphs, see for example Figure 5. In fact this figure
illustrates a certain scaling limit convergence theorem, Theorem 11 below.

Figure 5: Smith diagram for the $40\times 40$ square grid
$[0,40]^{2}\cap{\mathbb{Z}}^{2}$, with edges oriented south and west, with all
energies $1$ and boundary values $u(0,0)=0,~{}u(40,40)=1$. The apparent
regularity here can be quantified: see Theorem 11.

##  3 The Jacobian

The work in this paper began with the observation that the Jacobian of $\Psi$
has a surprisingly simple description, which we give here. Our proof of
Theorem 1 uses the nonvanishing of this Jacobian.

First some definitions. On a connected graph ${\mathcal{G}}=(V,E)$ with
boundary vertices $B$, a _$1$ -form_ is a function $\omega$ on oriented edges
which is antisymmetric under changing orientation: $\omega(-e)=-\omega(e)$
where $-e$ represents the edge $e$ with reversed orientation. Let $\Gamma^{1}$
be the space of $1$-forms. Let $\Gamma^{0}$ be the space of functions on $V$
with zero boundary values.

We define the map $d:\Gamma^{0}\to\Gamma^{1}$ by $df(e)=f(y)-f(x)$ when
$e=\vec{xy}$ is directed from $x$ to $y$. A $1$-form is a _coboundary_ if it
is of the form $df$ for some $f\in\Gamma^{0}$. Let $W_{cob}\subset\Gamma^{1}$
be the space of coboundaries. It has a natural basis consisting of the set of
coboundaries $\\{d{\mathbbm{1}}_{v}\\}_{v\in V\setminus B}$ where
${\mathbbm{1}}_{v}$ is the function which is $1$ at $v$ and zero elsewhere.

The orthogonal complement (for the standard inner product on
${\mathbb{R}}^{E}$) of $W_{cob}$ is the space $W_{cyc}$ of _(relative) cycles_
, or _flows_. These are $1$-forms in the kernel of the boundary operator
$d^{*}:\Gamma^{1}\to{\mathbb{R}}^{V}/{\mathbb{R}}^{B}\cong\Gamma^{0}$, the
transpose of $d$. The mapping $d^{*}$ from $1$-forms to functions is

$d^{*}\omega(x)=\sum_{y\sim x}\omega(\vec{xy})$

where the sum is over vertices $y$ neighboring $x$; in other terminology
$d^{*}\omega$ is the _divergence_ of $\omega$. Cycles (or flows) are $1$-forms
which are divergence-free on the non-boundary vertices.

Note that $W_{cob}$ has dimension $|V|-|B|$, the number of internal vertices,
and $W_{cyc}$ has dimension $|E|-|V|+|B|$.

The differential of the map $\Psi_{u}$ has a surprisingly simple form when
considered as a map from the logarithms of the conductances to the logarithms
of the energies. We assume the boundary values $u$ are fixed. If we fix an
element $\sigma\in\Sigma_{u}$, we can identify ${\mathbb{R}}^{E}$ with the
space of $1$-forms.

Let $P_{cyc}:\Gamma^{1}\to\Gamma^{1}$ be the orthogonal projection onto
$W_{cyc}$, and $P_{cob}=I-P_{cyc}$ the projection onto $W_{cob}$.

######  Theorem 9.

When $\Psi_{u}(\\{c\\})$ is nonzero on every edge, let $J_{\log}$ be the
differential of the map
$\log\circ\Psi_{u}\circ\exp:{\mathbb{R}}^{E}\to{\mathbb{R}}^{E}$. Then
$J_{\log}=P_{cyc}-P_{cob}$.

######  Corollary 10.

$\det D\Psi=(-1)^{|V|-|B|}\prod_{e}(h(x)-h(y))^{2}.$

In particular on $(0,\infty)^{E}$, $\Psi$ fails to be locally injective
precisely when some edge has energy $0$.

###### Proof of Corollary 10.

By the Theorem (and the chain rule)

$\det
D\Psi=(-1)^{{\text{d}im}(W_{cob})}\frac{\prod_{e}{\mathcal{E}}_{e}}{\prod_{e}c_{e}}.$

Substitute ${\mathcal{E}}_{e}=c_{e}(h(x)-h(y))^{2}$ for edges $e=xy$. ∎

###### Proof of Theorem 9.

Since ${\mathbb{R}}^{E}=W_{cob}\oplus W_{cyc}$, we study the action of
$J_{\log}$ on $W_{cob}$ and $W_{cyc}$.

Given a conductance function $c$, let $h$ be the associated harmonic function,
and $\omega=c\cdot dh$ the associated current flow. We assume $\omega$ is
nonzero on each edge, which holds for almost all choices of $c$.

Let $v\in V\setminus B$. We define a $1$-parameter family of conductances
$c_{e}(t),~{}t\in[0,1]$ as follows. Let $h_{1}(v)$ be any real number close
enough to $h(v)$ so that, if we redefine $h(v)$ to be $h_{1}(v)$ then the sign
of $dh$ does not change. Fix $h$ at all vertices but $v$ and let

$h_{t}(v)=h(v)+t(h_{1}(v)-h(v)).$

We fix $\omega$ to its initial value, and define the conductance at time $t$
by $c_{e}(t)=\omega(e)/dh_{t}(e)$. Then $h_{t}$ is harmonic for conductances
$c_{e}(t)$. The associated energies are
${\mathcal{E}}_{e}(t)=\omega(e)dh_{t}(e)$ and thus $\frac{\partial}{\partial
t}\log{\mathcal{E}}_{e}(t)=-\frac{\partial}{\partial t}\log c_{e}(t).$
Evaluating at $t=0$, this vector is a multiple of $d{\mathbbm{1}}_{v}$:

$\left.-\frac{\partial}{\partial t}\log
c_{e}(t)\right|_{t=0}=\left.\frac{\partial}{\partial t}\log
dh_{t}(e)\right|_{t=0}=\frac{h_{1}(v)-h(v)}{h(v)}d{\mathbbm{1}}_{v}.$

Thus $d{\mathbbm{1}}_{v}$ is an eigenvector of $J_{\log}$ with eigenvalue
$-1$. As $v$ varies over $V\setminus B$ the coboundaries $d{\mathbbm{1}}_{v}$
span the coboundary space $W_{cob}$, so $J_{\log}$ is minus the identity on
$W_{cob}$.

Now let $\gamma$ be a cycle in $G$. Let $\delta>0$ be small enough so that
$\omega_{1}=\omega+\delta\gamma$ has the same sign as $\omega$. We now fix $h$
but vary $\omega$ according to
$\omega_{t}(e)=\omega(e)+t(\omega_{1}(e)-\omega(e)).$ Define the conductance
at time $t$ by $c_{e}(t)=\omega_{t}(e)/dh(e)$. We then have
$\frac{\partial}{\partial t}\log{\mathcal{E}}_{e}=\frac{\partial}{\partial
t}\log c_{e}(t)$. Evaluating at $t=0$, this vector is a multiple of $\gamma$:

$\left.\frac{\partial}{\partial t}\log
c_{e}(t)\right|_{t=0}=\left.\frac{\partial}{\partial
t}\log\omega_{t}(e)\right|_{t=0}=\frac{\omega_{1}(e)-\omega(e)}{\omega(e)}\gamma.$

Thus $J_{\log}$ is the identity on $W_{cyc}$. ∎

It is worth pointing out the interpretation of this proof in terms of tilings.
If $G$ is planar, the proof starts with a rectangle tiling with combinatorial
structure described by $G$, and then studies the effect of isotopy on the
diagram. Moving a horizontal segment corresponds to changing the value of $h$
at a single point, which is an eigenvalue $-1$ of $J_{\log}$: the relative
change of the conductances (widths over heights) move in the opposite
direction as the relative change of areas (widths times heights). Moving a
vertical segment corresponds to the eigenvalue $+1$: the relative change of
the conductances move in the same direction as the relative change of areas.

##  4 Proofs of Theorems 1 and 3

We assume $u$ is fixed and takes distinct values at distinct boundary
vertices, and we omit it from the notation.

###### Proof of Theorem 1.

We first prove that the map is surjective onto $\Sigma$, that is, every
$\sigma\in\Sigma$ is realized by some choice of conductances. Given
$\sigma\in\Sigma$, let $f$ be any function in ${\mathcal{F}}(\sigma).$ Such a
function exists because $\sigma$ is acyclic: for example one can define $f$
from a linear extension of $\sigma$, thinking of $\sigma$ as a partial order.
There is a choice of conductances for which this $f$ is harmonic: to see this
take any flow $\omega$ with orientation $\sigma$ and divergence $0$ on
$V\setminus B$, and define the conductance $c_{e}$ on edge $e=xy$ by
$c_{e}=\frac{\omega(e)}{f(x)-f(y)}$. As a concrete example of such a flow
$\omega$, let $P$ be the set of all possible oriented paths from a boundary
vertex to a boundary vertex, with orientation compatible with $\sigma$, and
let the value $\omega(e)$ be the probability that $e$ is on a uniformly chosen
path in $P$. By our condition on ${\mathcal{G}}$ that each edge lies on a
simple path from $B$ to $B$, we see that $\omega$ is nonzero. This completes
the proof that the map is surjective onto $\Sigma$.

Now consider the map $\Psi\colon(0,\infty)^{E}\to[0,\infty)^{E}$ sending a
tuple of conductances to the corresponding tuple of energies of the associated
harmonic function. Corollary 10 shows that the Jacobian of $\Psi$ is nonzero
when the energies are nonzero.

We claim that $\Psi$ is proper: the preimage of a compact set is compact. In
other words, $\Psi$ maps the boundary to the boundary. Take a sequence of
conductance functions leaving every compact set of $(0,\infty)^{E}$. By taking
a subsequence, we can assume it converges to a boundary point of
$[0,\infty]^{E}$. We need to show that along this sequence some edge has
energy tending to $0$ or $\infty$.

Let $S_{0}$ be the set of edges whose conductances are going to zero, and
$S_{\infty}$ those whose conductances are going to $\infty$. If $S_{\infty}$
does not connect any two boundary vertices, then there is a limiting network
obtained by contracting all edges in $S_{\infty}$. The resulting network has
conductances bounded above, which implies a finite current flow. The energies
on the contracted edges and any edges of zero conductance have gone to zero.

If $S_{\infty}$ contains a connection between two boundary points (with
different values of $u$, by genericity), the current flowing between these
points will diverge. This implies that the energy (which can be written as
current flow times $dh$) will diverge along some edges of this connection.

Thus $\Psi$ is proper.

The preimage of $(0,\infty)^{E}$ in the space of conductances is a union of
open sets $U_{\sigma}$ where $dh$ has constant sign $\sigma$. At a finite
boundary point of any such $U_{\sigma}$, some energy goes to zero (since zero
current implies zero energy). Hence the boundary of $U_{\sigma}$ maps into the
boundary of the space of energies. Thus $\Psi$ on any of these open sets
$U_{\sigma}$ is proper and a local homeomorphism (a covering map), and
therefore surjective from this set to the space of energies. To get uniqueness
it remains to show that $U_{\sigma}$ is connected.

In fact we show that $U_{\sigma}$ has the structure of (the interior of) a
polytope. As in the first paragraph of the proof, a point in $U_{\sigma}$ is
determined by a flow compatible with $\sigma$ (and of divergence zero on
interior vertices), and an independent choice of function $f$ in the open
polytope ${\mathcal{F}}_{u}(\sigma)$. The set of current flows $\omega$
compatible with $\sigma$ is the subset of $(0,\infty)^{E}$ defined by the
linear equations $\text{div}(\omega)=0$. It is the interior of a convex
polytope (possibly infinite but with finitely many facets). ∎

###### Proof of Theorem 3.

We have constructed $|\Sigma|$ real preimages of each point in
$(0,\infty)^{E}$. It remains to show that there are no non-real preimages. Let
$\\{c_{e}\\}\in\Psi^{-1}(\\{{\mathcal{E}}_{e}\\})$ be a preimage of a set of
positive real energies. Let $h$ be the corresponding solution to the Dirichlet
problem with conductances $\\{c_{e}\\}$. Then for all non-boundary vertices
$x$,

$\sum_{y\sim x}c_{e}(h(x)-h(y))=0$

and so

$\sum_{y\sim x}\frac{{\mathcal{E}}_{xy}}{h(x)-h(y)}=0.$

We can rewrite this as

$\sum_{y\sim
x}\frac{{\mathcal{E}}_{xy}(\overline{h(x)}-\overline{h(y)})}{|h(x)-h(y)|^{2}}=0,$

or

$\left(\sum_{y\sim
x}\frac{{\mathcal{E}}_{xy}}{|h(x)-h(y)|^{2}}\right)\overline{h(x)}=\sum_{y\sim
x}\frac{{\mathcal{E}}_{xy}}{|h(x)-h(y)|^{2}}\overline{h(y)}.$

This represents $\overline{h(x)}$ as a convex combination of its neighboring
values $\overline{h(y)}$, and thus $h(x)$ lies in the convex hull of the
neighboring values $h(y)$. Since the boundary values of $h$ are real, each
$h(x)$ must be real as well. Finally $c_{e}={\mathcal{E}}_{e}/(h(x)-h(y))^{2}$
so each $c_{e}$ is real and positive. ∎

##  5 Enharmonic functions: a variational method

###  5.1 Proof of Theorem 2

###### Proof of Theorem 2.

Fix a compatible orientation $\sigma$. Recall the open polytope
${\mathcal{F}}_{u}(\sigma)\subset{\mathcal{F}}_{u}$ of functions inducing
orientation $\sigma$. Restricted to ${\mathcal{F}}_{u}(\sigma)$, the function
$\log M(h)$ is concave, since it is a sum of concave functions

${\mathcal{E}}(e)\log|h(x)-h(y)|={\mathcal{E}}(e)\log(\sigma(e)(h(x)-h(y))).$

The function $\log M(h)$ tends to $-\infty$ on the boundary of
${\mathcal{F}}_{u}(\sigma)$ (when $h(x)=h(y)$ for some edge $e=xy$). As a
consequence $\log M(h)$, being analytic, is strictly concave on
${\mathcal{F}}_{u}(\sigma)$. It thus has a unique maximizer on the interior of
${\mathcal{F}}_{u}(\sigma)$, for each $\sigma$. The maximizer is determined by
setting the derivative with respect to $h(x)$ equal to zero for each vertex
$x$, that is

$\sum_{y\sim x}\frac{{\mathcal{E}}(e)}{h(x)-h(y)}=0.$ (7)

Given a solution to (7), set $c_{e}={\mathcal{E}}_{e}/(h(x)-h(y))^{2}$, then
(7) becomes

$\sum_{y\sim x}c_{e}(h(x)-h(y))=0,$ (8)

that is, $h$ is harmonic for the conductances $c_{e}$ and has energies
${\mathcal{E}}_{e}$.

∎

###  5.2 Scaling limits

Let $D\subset{\mathbb{R}}^{2}$ be a Jordan domain with piecewise smooth
boundary and let $u:\partial D\to{\mathbb{R}}$ be continuous. We assume $u$ is
_feasible_ in the sense that the space $\Omega(D,u)$ of smooth functions
$f\colon D\to{\mathbb{R}}$ with boundary values $f|_{\partial U}=u$ and
satisfying $f_{x},f_{y}>0$ is nonempty444The issue of feasibility is not
completely straightforward. Such a function $f$ necessarily has level contours
having tangents pointing in the NW and SE quadrants. Conversely, given a
foliation $\mathcal{F}$ of $D$ by SE-directed leaves, in which the value of
$u$ on the endpoints of each leaf are the same, there is a function $f$ whose
level lines are the leaves of $\mathcal{F}$.. Fix such a function
$f_{0}\in\Omega(D,u)$.

For sufficiently small $\epsilon>0$ let
$D_{\epsilon}=\epsilon{\mathbb{Z}}^{2}\cap D$ be the part of the graph
$\epsilon{\mathbb{Z}}^{2}$ contained in $D$. Let $\sigma$ be the south and
west orientation of the edges of $D_{\epsilon}$. Let $\partial D_{\epsilon}$
be the set of boundary vertices of $D_{\epsilon}$, that is, vertices with a
nearest neighbor in $\epsilon{\mathbb{Z}}^{2}$ not in $D_{\epsilon}$. Let
$u_{\epsilon}:\partial D_{\epsilon}\to{\mathbb{R}}$ be the restriction of
$f_{0}$ to $\partial D_{\epsilon}$.

Let $f_{\epsilon}$ be the unique enharmonic function on $D_{\epsilon}$ with
energy $\epsilon^{2}$ per edge, with boundary values $u_{\epsilon}$ and with
$df_{\epsilon}$ having orientation $\sigma$.

######  Theorem 11.

Under the above hypotheses, as $\epsilon\to 0$ the function $f_{\epsilon}$
converges to a continuous function on $D$, real analytic in the interior of
$D$, and satisfying the enharmonic Laplace equation

$\frac{f_{xx}}{f_{x}^{2}}+\frac{f_{yy}}{f_{y}^{2}}=0.$ (9)

There is a unique (up to an additive constant) conjugate function $g$, also
real analytic and continuous up to the boundary, satisfying (5).

We call $f$ satisfying (9) an _enharmonic function_. A function $g$ satisying
(5) is an _enharmonic conjugate_.

###### Proof.

Assuming $f\in C^{2}$ satisfies (9), the $1$-form
$\omega=-\frac{dx}{f_{y}}+\frac{dy}{f_{x}}$ is closed. Let
$g(p)=\int^{p}_{p_{0}}\omega$ for some fixed $p_{0}\in D$. Then $f$ and $g$
satisfy the “fixed-energy” Cauchy Riemann equations (5). The enharmonicity of
$g$ follows from (5) as well.

Let $F\in\Omega(D,u)$ be the (unique) function maximizing the concave
functional

${\mathcal{M}}(F)=\int_{D}\log(F_{x}F_{y})dx\,dy.$ (10)

Existence and uniqueness of $F$ both follow from strict concavity of $\cal M$
(existence follows from the fact that the space of monotone functions with
boundary values $u$ is compact, and ${\mathcal{M}}$ is upper semicontinuous;
uniqueness follows from the fact that ${\cal
M}(\frac{1}{2}(F_{1}+F_{2}))>{\cal M}(F_{1})+{\cal M}(F_{2})$ unless
$F_{1}=F_{2}$. Analyticity follows from analyticity of the function $\log$,
see [10][Theorem 1.10.4 (vi)].

We will show that $f_{\epsilon}\to F$.

The Euler-Lagrange equation for a critical point of ${\cal M}(F)$ is the
enharmonic equation

$\frac{F_{xx}}{F_{x}^{2}}+\frac{F_{yy}}{F_{y}^{2}}=0.$

Let $M_{\epsilon}$ be the functional (2) for the graph $D_{\epsilon}$ with
energies $\epsilon^{2}$. Let $f_{\epsilon}$ be its maximizer. After
triangulating the faces of $D_{\epsilon}$ by adding diagonals, we can extend
$f_{\epsilon}$ to a piecewise linear function $\tilde{f}_{\epsilon}$ on $D$
(except near the boundary). Then $\log M_{\epsilon}(f_{\epsilon})$ is a
Riemann sum for ${\mathcal{M}}(\tilde{f}_{\epsilon})$, so

$\log M_{\epsilon}(f_{\epsilon})={\mathcal{M}}(\tilde{f}_{\epsilon})+o(1).$

If we restrict $F$ to $D_{\epsilon}$ we likewise find by analyticity of $F$
that

$\log M_{\epsilon}(F)={\mathcal{M}}(F)+o(1).$

Consequently

${\mathcal{M}}(F)=\log M_{\epsilon}(F)+o(1)\leq\log
M_{\epsilon}(f_{\epsilon})+o(1)={\mathcal{M}}(\tilde{f}_{\epsilon})+o(1)\leq{\mathcal{M}}(F)+o(1).$

Therefore all these quantities are within $o(1)$ of each other. By upper
semicontinuity of ${\mathcal{M}}$, we find that $f_{\epsilon}$ converges to
$F$.

We prove the second statement of the theorem in the coming section.

###  5.3 Riemann mapping to a rectangle

Let $D\subset{\mathbb{R}}^{2}$ be a piecewise smooth domain of the following
type. There are four points $a,b,c,d$ on the boundary of $D$ in counter-
clockwise order, so that the boundary of $D$ between $a$ and $b$ is monotone
in the NE direction (that is, is monotone in the $+x,+y$ directions), and
similarly between $b$ and $c$, $c$ and $d$, $d$ and $a$ respectively the
boundary is monotone in the NW, SW, and SE directions. Let $u$ be the function
which is $1$ on $\partial D$ between $b$ and $c$, and zero between $d$ and
$a$. Let $D_{\epsilon}$ be as before the subgraph of
$\epsilon{\mathbb{Z}}^{2}$ with vertices in $D$. Let
$a_{\epsilon},\dots,d_{\epsilon}$ be boundary vertices so that the boundary of
$D_{\epsilon}$ is monotone NE, NW, SW, SE between $a_{\epsilon}$ and
$b_{\epsilon}$, $b_{\epsilon}$ and $c_{\epsilon}$, $c_{\epsilon}$ and
$d_{\epsilon}$, and $d_{\epsilon}$ and $a_{\epsilon}$. Let
$D^{\prime}_{\epsilon}$ be the dual graph of $D_{\epsilon}$, with a vertex for
every bounded face of $D_{\epsilon}$ and, instead of one exterior vertex, four
exterior vertices $v_{ab},v_{bc},v_{cd},v_{da}$ outside of the corresponding
boundary edges of $D_{\epsilon}$. Let $\sigma$ be the south/west orientation
of edges of $D_{\epsilon}$.

Let $f\in\Omega(D,u)$ be the unique function on $D$ with boundary values $1$
on $bc$ and $0$ on $da$, free on the other boundaries, and maximizing
${\mathcal{M}}(f)$; this differs from the previous case because $u$ is only
fixed on part of the boundary, but again existence and uniqueness follow from
concavity and analyticity from arguments of [10]. Then $f$ will satisfy (9) on
$D$, with fixed boundary values $0$ on $da$ and $1$ on $bc$ respectively and
along the free boundary $ab\cup cd$, $f$ will satisfy the condition that
$\frac{dx}{f_{y}}-\frac{dy}{f_{x}}=0$ (this is in fact the Euler-Lagrange
condition along the free boundary).

As before the function $f_{\epsilon}$ on $D_{\epsilon}$ maximizing $M$ will
lie within $o(1)$ of $f$. Using edge energies $\epsilon^{2}$, let
$g_{\epsilon}$ be the enharmonic conjugate of $f_{\epsilon}$, defined on the
dual graph $D^{\prime}_{\epsilon}$ and with value zero on the dual vertex
$v_{ab}$. Let $R_{\epsilon}=g_{\epsilon}(v_{cd})$ be the value at the opposite
dual vertex. Since the sum of the energies is the area of the image under the
rectangle tiling, we have

$R_{\epsilon}=\epsilon^{2}(\text{\\# edges of
$D_{\epsilon}$})+o(1)=2\text{Area}(D)+o(1).$

The above convergence argument applied to $g_{\epsilon}$ shows that
$g_{\epsilon}$ converges to $g$, the maximizer of ${\mathcal{M}}$ with
boundary values $A=2\text{Area}(D)$ and $0$ on $cd$ and $ab$ respectively.

It remains to show that $f$ and $g$ are enharmonic conjugates. The enharmonic
conjugate $f^{*}$ of $f$ is the enharmonic function (well defined up to an
additive constant) which is constant along $ab$ and along $cd$, with free
boundary values along $bc$ and $da$. These are precisely the boundary
conditions satisfied by the function $g$, except that the value of the
constant along $cd$ might be different, so we have that $f^{*}=Bg$ for some
constant $B$. The map $(x,y)\mapsto(f^{*},f)$ is area preserving except for a
factor $2$ (the Jacobian is identically $2$, by (5)) so the image of $D$ has
area $A$, which means the boundary value of $f^{*}$ along $bc$ must also be
$A$. Thus $f^{*}=g$.

This proves that $(f_{\epsilon},g_{\epsilon})$ converges to $(f,g)$. In the
limit we have a real-analytic mapping $(g,f)$ from $D$ to the rectangle
$[0,A]\times[0,1]$. ∎

##  6 Realizing number fields

_Proof of Theorem 6_. Let ${\mathcal{G}}$ be a $d+1$-star graph with vertices
$\\{v,v_{0},\dots,v_{d}\\}$, with $v$ being the central vertex. Take boundary
$B$ consisting of all vertices except $v$. Let $e_{i}$ be the energy on edge
$vv_{i}$ and take rational boundary values $a_{i}$ at $v_{i}$, with
$a_{0}<\dots<a_{d}$. The enharmonic equation for the value $x$ at $v$ is

$\sum_{i=0}^{d}\frac{e_{i}}{x-a_{i}}=0.$ (11)

This equation has exactly $d$ roots, and these are real and interlaced with
the $a_{i}$, that is, there is one in each interval $(a_{i},a_{i+1})$ (this
follows from Theorem 1 above or simply from the fact that the roots are roots
of the derivative of $\prod(x-a_{i})^{e_{i}}$).

Conversely let $p(x)$ be a polynomial with rational coefficients of degree $d$
with all real roots, interlaced with the $a_{i}$ as above. Then we claim that
there is a unique choice of rational $e_{i}$ so that the enharmonic equation
(11) has exactly the same roots.

The map from the tuple of energies $\\{e_{i}\\}$ to the tuple of coefficients
$\\{b_{i}\\}$ of the polynomial $p(x)=b_{0}+b_{1}x+\dots+b_{d}x^{d}$ is a
rational linear map from ${\mathbb{R}}^{d+1}$ to itself. Specifically its
matrix consists of elementary functions in the $a_{i}$ (after removing
$a_{i-1}$ in those for the $i$th column) illustrated for the case $d=3$:

$\begin{pmatrix}b_{0}\\\ b_{1}\\\ b_{2}\\\
b_{3}\end{pmatrix}=\begin{pmatrix}-a_{1}a_{2}a_{3}&-a_{0}a_{2}a_{3}&-a_{0}a_{1}a_{3}&-a_{0}a_{1}a_{2}\\\
a_{1}a_{2}+a_{3}a_{2}+a_{1}a_{3}&a_{0}a_{2}+a_{3}a_{2}+a_{0}a_{3}&a_{0}a_{1}+a_{3}a_{1}+a_{0}a_{3}&a_{0}a_{1}+a_{2}a_{1}+a_{0}a_{2}\\\
-a_{1}-a_{2}-a_{3}&-a_{0}-a_{2}-a_{3}&-a_{0}-a_{1}-a_{3}&-a_{0}-a_{1}-a_{2}\\\
1&1&1&1\\\ \end{pmatrix}\begin{pmatrix}e_{0}\\\ e_{1}\\\ e_{2}\\\
e_{3}\end{pmatrix}$

The determinant of this matrix is a polynomial of degree $\binom{d}{2}$ and
has factors $a_{i}-a_{j}$ for each $i<j$, so is the Vandermonde
$\pm\prod_{i<j}(a_{i}-a_{j})$ and is therefore nonzero. Thus the map from
energies to coefficients $b_{i}$ is invertible over ${\mathbb{Q}}$. This
completes the proof of the claim.

Now let $K$ be a totally real number field and $x\in K$ a primitive element
(i.e. one for which $K={\mathbb{Q}}[x]$). Let $p(X)$ be its minimal
polynomial. Let $r_{1},\dots,r_{d}$ be the roots of $p(X)$, which are real.
Replacing $x$ with $x+m$ for a large enough integer $m$ we can assume that all
$r_{i}>0$. Let $a_{1},\dots,a_{d+1}$ be positive rationals interlacing them.
Now do the above construction with parameters $a_{i}$. $\square$

Note that as a simple corollary to this proof, by letting the $e_{i}$ be
arbitrary positive rationals, modulo a global scale, we can parameterize the
polynomials with roots interlaced with the $a_{i}$ as a simplex

$\\{(e_{0},\dots,e_{d})~{}|~{}\sum e_{i}=1\\}.$

##  7 Other questions

  1. 1.

Can one use the volume-preserving character of the map
$\log\circ\Psi\circ\exp$ to compute the cardinality of $\Sigma$? This is NP
hard in general but integrating the Jacobian over projective space one can
approximate the degree of $\Psi$ and therefore $|\Sigma|$.

  2. 2.

The minimal polynomials for geometric quantities (such as values of the
harmonic function) seem to have coefficients which are (fixed-sign) integer-
coefficient polynomials in the edge energies. Can one give a combinatorial
description of these polynomials?

  3. 3.

What can be said about solutions to the enharmonic equation in the case when
the boundary values $u$ are complex? The number of solutions is the same, but
there is no longer the same interpretation in terms of orientations.

Acknowledgements. We thank Michael Bush, Jeff Lagarias, Russell Lyons, Sanjay
Ramassamy, Günther Rote, Raman Sanyal, Bernd Sturmfels and Cynthia Vinzant for
helpful suggestions and conversations.

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