# Pole Dynamics and an Integral of Motion for Multiple SLE(0)

Tom Alberts  Department of Mathematics, University of Utah, Salt Lake City, UT
84112, USA<EMAIL_ADDRESS>,  Nam-Gyu Kang  School of Mathematics,
Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea
<EMAIL_ADDRESS>and  Nikolai G. Makarov  Department of Mathematics,
California Institute of Technology, Pasadena, CA 91125, USA
<EMAIL_ADDRESS>
###### Abstract.

We study a family of real rational functions with prescribed critical points
and the evolution of its poles and critical points under particular Loewner
flows. The recent work by Peltola and Wang shows that the real locus of these
rational functions contains the multiple SLE$(0)$ curves, the deterministic
$\kappa\to 0$ limit of the multiple SLE$(\kappa)$ system. Our main results
highlight the importance of the poles of the rational function in determining
properties of the SLE$(0)$ curves. We show that solutions to the classical
limit of the the null vector equations, which are used in Loewner evolution of
the multiple SLE$(0)$ curves, have simple expressions in terms of the critical
points and the poles of the rational function. We also show that the evolution
of the poles and critical points under the Loewner flow is a particular
Calogero-Moser integrable system. A key step in our analysis is a new integral
of motion for the deterministic Loewner flow.

###### Key words and phrases:

integral of motion, multiple SLEs

###### 2010 Mathematics Subject Classification:

Primary 30C35; Secondary 60J67, 81T40

Tom Alberts was supported by NSF grants DMS-1811087 and DMS-1715680.

Nam-Gyu Kang was partially supported by Samsung Science and Technology
Foundation (SSTF-BA1401-51) and by a KIAS Individual Grant (MG058103) at Korea
Institute for Advanced Study.

Nikolai G. Makarov was supported by NSF grant no. 1500821.

##  1\. Introduction

Recently, Peltola and Wang [PW20] made a detailed study of the multiple
SLE$(0)$ process, a particular deterministic collection of $n$ curves in a
simply connected domain $D$ that connect $2n$ specified boundary points
according to a particular link pattern. While the curves are fully determined
by the domain, the boundary points, and the link pattern, the name multiple
SLE$(0)$ comes from the fact (which Peltola and Wang prove) that it is the
deterministic limit of the random multiple SLE$(\kappa)$ process as
$\kappa\to\to 0$, when the SLE curves have been conditioned to join according
to the particular link pattern. Unsurprisingly then, the multiple SLE$(0)$
process has the same natural properties as multiple SLE$(\kappa)$: conformal
invariance, reversibility, a cascade type relation, and more. We will use the
conformal invariance to restrict our attention solely to the upper half-plane
$\mathbb{H}$, where all of the relevant descriptions are much simpler. In the
upper half-plane each multiple SLE$(0)$ is determined by distinct boundary
points $x_{1},\ldots,x_{2n}$, which we will shorten to
$\boldsymbol{x}=(x_{1},\ldots,x_{2n})$, and a link pattern $\alpha$ that
specifies in which of the $C_{n}$ possible non-crossing ways the $2n$ boundary
points are connected. Here and throughout $C_{n}$ is the Catalan number

$C_{n}=\frac{1}{n+1}\dbinom{2n}{n},$

which enumerates the set of possible link patterns connecting the points in
$\boldsymbol{x}$. We use SLE$(\kappa;\boldsymbol{x};\alpha)$ to denote the law
on $n$ commuting SLE$(\kappa)$ curves in $\mathbb{H}$ that almost surely
connect the boundary points $\boldsymbol{x}$ according to the link pattern
$\alpha$. For $0<\kappa<4$ there are nice existence arguments for these laws
on curves and the process has been well studied in [BBK05, KL07, Dub07, Gra07,
Law09b, Law09a, KP16, BPW18, PW19] among others. We also make an extensive
study of multiple SLE$(\kappa)$ in the forthcoming [AKM], through the lens of
a rigorous conformal field theory introduced in [KM13, KM]. The work in [AKM]
motivates the approach of the present work, and we explain more of our
intuition in the concluding remarks. As for the multiple
SLE$(0;\boldsymbol{x};\alpha)$ process, Peltola and Wang proved its existence
and many of its properties in the recent [PW20]. Using their notation
$\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{n})$ for the ensemble of $n$ curves
that make up the multiple SLE$(0;\boldsymbol{x};\alpha)$, they show that
$\boldsymbol{\eta}$ can be characterized by any one of five equivalent
properties:

  * •

as the deterministic limit of multiple SLE$(\kappa;\boldsymbol{x};\alpha)$ as
$\kappa\to 0$,

  * •

as the unique minimizer of a certain functional (called the multichordal
Loewner energy) on ensembles of $n$ curves that connect the points in
$\boldsymbol{x}$ according to the pattern $\alpha$,

  * •

as a geodesic multichord, meaning that each $\eta_{j}$ is the hyperbolic
geodesic in the (unique) connected component of
$\mathbb{H}\backslash\bigcup_{k\neq j}\eta_{k}$ that contains the endpoints of
$\eta_{j}$,

  * •

as curves generated by a Loewner type evolution with only one non-trivial
$\nu_{j}$ in (2.4) for each $\eta_{j}$, where the driving terms for the
$\eta_{j}$ are determined by special $\alpha$-dependent solutions to the
classical limit of the Belavin-Polyakov-Zamolodchikov (BPZ) equations as
$\kappa\to 0$, and finally,

  * •

as the real locus of a rational function of degree $n+1$, with real
coefficients and critical points at $x_{1},\ldots,x_{2n}$.

In this paper we concentrate on the last two characterizations. One of our
main results (Theorem 2.2) gives explicit solutions to the classical
$\kappa\to 0$ limit of the BPZ equations. We call the $\kappa=0$ versions the
null vector equations, since they correspond to the so-called “null state”
version of the level two BPZ equations. In the $\kappa>0$ case explicit
formulas for solutions to the BPZ equation are difficult to find, particularly
for the so-called pure solutions that correspond to multiple SLEs connecting
according to a particular link pattern. The most well-known expressions for
these pure solutions are as expectations of certain functionals of Brownian
loops measures [KL07, Law09b, Law09a, PW19]; see also [KP16] for an expression
involving quantum groups. Regardless of the method, except for small $n$ it is
difficult to find exact algebraic formulas for these pure solutions. In the
$\kappa=0$ case we express the solution in terms of the poles of the real
rational function whose locus is the multiple SLE$(0)$. For each link pattern
$\alpha$ these poles are algebraic functions of the boundary points
$\boldsymbol{x}$, and although we do not discuss it in this paper they have an
interpretation as the $\kappa\to 0$ limit of extraneous screening variables
that are integrated out along specific contours to produce solutions to the
$\kappa>0$ BPZ equations (see [Dub06] and [FK15a, FK15b, FK15c, FK15d] for
more on the $\kappa>0$ case). Using our explicit expressions for the solutions
to the null vector equations, we also show (Theorem 2.5) that there is a
natural integral of motion for the Loewner flow generated by these solutions.
As a corollary (Corollary 2.8) we are able to show that this Loewner flow
generates the multiple SLE$(0;\boldsymbol{x};\alpha)$ curves. We also show
(Corollary 2.9) that the evolution of the critical points $\boldsymbol{x}$ and
the poles of the corresponding rational function follows a classical system of
ordinary differential equations known as the Calogero-Moser system.

In the next section we recall basic facts about real rational functions and
state our main results. These are primarily on the interplay between the
critical points and poles of the real rational functions and their evolution
under associated Loewner flows. We prove these results in Section 3 along with
a few auxiliary statements. In Section 4 we provide examples in the $n=1$ and
$n=2$ cases (two and four critical points, respectively). In the $n=1$ case
this is relatively trivial, while in the $n=2$ case, in which there are two
ways to connect four boundary points, we use our methods to find explicit
formulas for the solutions to the null vector equations and for the curves
that make up the multiple SLE$(0;\boldsymbol{x};\alpha)$. We also show that
our formulas agree with the $\kappa\to 0$ limits of the pure partition
functions, which are explicitly known in the $n=2$ case. We conclude with some
final remarks in Section 5.

##  2\. Formulation of Main Results

Our motivation for writing this paper comes from the recent work of Peltola
and Wang [PW20], but most of our results are in fact independent of theirs.
Primarily we show that relatively simple properties of real rational functions
have important consequences for the description of the multiple
SLE$(0;\boldsymbol{x};\alpha)$ curves via their Loewner evolution, but we only
rely on [EG02] or [PW20] to guarantee the existence of these real rational
functions. Otherwise our results are self-contained.

We begin by recalling basic facts about real rational functions and the real
locus. Recall that any real rational function
$R:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ of degree $n+1$ can be written
as $R=P/Q$ where $P$ and $Q$ are polynomials with real coefficients and
$\max\\{\deg P,\deg Q\\}=n+1$. For the real locus we write

$\Gamma(R):=\\{z\in\widehat{\mathbb{C}}:R(z)\in\mathbb{R}\\}$

and since we only consider $R$ that are real rational we always have
$\mathbb{R}\subset\Gamma(R)$. The critical points of $R$ are the zero set of

$R^{\prime}=\frac{P^{\prime}Q-Q^{\prime}P}{Q^{2}},$

considered as a multiset in the case of multiplicities (which we do not
consider). For the set of real rational functions with prescribed critical
points we introduce the following notation.

######  Definition.

For $\boldsymbol{x}=\\{x_{1},\ldots,x_{2n}\\}$ with $x_{j}$ distinct, real,
and finite, let $\operatorname{CRR}_{n+1}(\boldsymbol{x})$ be the set of real
rational functions $R:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ of degree
$n+1$ whose critical points are precisely $\boldsymbol{x}$.

Möbius transformations $\phi$ (each of which can be represented by a matrix in
the group $\operatorname{PSL}(2,\mathbb{C})$) act on
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ via post-composition:

$\displaystyle\phi\circ R=\begin{pmatrix}a&b\\\ c&d\\\
\end{pmatrix}R=\frac{aR+b}{cR+d}=\frac{aP+bQ}{cP+dQ}.$

These post-compositions preserve the critical points of the rational function,
but preserve the real locus iff the Möbius transformation maps the real line
to itself, i.e. if $\phi$ is in the subgroup
$\operatorname{PSL}(2,\mathbb{R})$. Such a $\phi$ either maps $\mathbb{H}$ to
itself or to $-\mathbb{H}$. We restrict our attention to post-compositions by
elements of $\operatorname{PSL}(2,\mathbb{R})$ throughout. Since post-
composition preserves the critical points it induces a natural equivalence
relation on $\operatorname{CRR}_{n+1}(\boldsymbol{x})$. Goldberg [Gol91]
showed that the number of equivalence classes is at most $C_{n}$, and Eremenko
and Gabrielov [EG02] proved that it is exactly $C_{n}$. Peltola and Wang
[PW20, Proposition 4.1] presented the matching lower bound by explicitly
constructing $C_{n}$ non-equivalent elements of
$\operatorname{CRR}_{n+1}(\boldsymbol{x})$. Each such element has a real locus
that is precisely one of the ensembles of multiple
SLE$(0;\boldsymbol{x};\alpha)$ curves, and it is natural to index the
equivalence classes by these ensembles. We consider each multiple
SLE$(0;\boldsymbol{x};\alpha)$ as an ensemble
$\boldsymbol{\eta}(\boldsymbol{x};\alpha)=\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{n})$
of $n$ simple, non-crossing curves in $\mathbb{H}$ that connect the boundary
points in $\boldsymbol{x}$ according to the link pattern $\alpha$. We also
identify $\boldsymbol{\eta}$ with the trace of the curves and hence regard it
as a subset of $\mathbb{H}$.

######  Definition.

For each pair $(\boldsymbol{x};\alpha)$ define
$\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ as the equivalence class of
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ (under post-composition by
elements of $\operatorname{PSL}(2,\mathbb{R})$) whose real locus satisfies

$\Gamma(R)=\boldsymbol{\eta}\cup\overline{\boldsymbol{\eta}}\cup\mathbb{R},\quad\boldsymbol{\eta}=\boldsymbol{\eta}(\boldsymbol{x};\alpha),$

where $\overline{\boldsymbol{\eta}}$ is the reflection of $\boldsymbol{\eta}$
across the real axis.

Our main results describe various relations between the critical points of
rational functions $R\in\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ and
their finite poles. We also prove several facts about the evolution of these
relations under certain Loewner flows, which are themselves determined
entirely by the critical points $\boldsymbol{x}$ and the link pattern
$\alpha$. We express our results in terms of a canonical representative
$R_{\alpha}$ of each equivalence class which we now introduce. For
$R=P/Q\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ the (finite) poles of $R$
are nothing more than the zeros of $Q$, and since $Q$ is real these zeros may
be complex but always appear in conjugate pairs. Furthermore, since $R$ is of
degree $n+1$ with $2n$ critical points the number of zeros of $Q$ must always
be $n$ or $n+1$. This follows from

$R^{\prime}=\frac{P^{\prime}Q-Q^{\prime}P}{Q^{2}},$

and the $2n$ distinct points in $\boldsymbol{x}$ can be the critical points of
$R$ iff $\deg P^{\prime}Q-PQ^{\prime}=2n$. Since $\deg P,\deg Q\leq n+1$ this
is possible iff either $\deg P=\deg Q=n+1$, or $\deg Q=n+1$ and $\deg P=n$, or
$\deg P=n+1$ and $\deg Q=n$. We choose (a particular form of) the last case as
our canonical representation so that $R$ always has $n$ finite poles. It also
has a simple pole at infinity that we can typically ignore. Any
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ can be post-composed into a
rational function of this type, and given such an $R$ the only elements of
$\operatorname{PSL}(2,\mathbb{R})$ that preserve the poles of $R$ under post-
composition are the dilations and translations, i.e. elements of
$\operatorname{PSL}(2,\mathbb{R})$ of with $c=0$. We choose the dilation
factor so that the derivative factors in a specific form, which we make in the
following definition.

######  Definition.

An element $R=P/Q\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ is in canonical
form if $\deg Q=n$ and its derivative factors as

$R^{\prime}(z)=\frac{\prod_{j=1}^{2n}(z-x_{j})}{\prod_{k=1}^{n}(z-\zeta_{k})^{2}},$

where $\boldsymbol{\zeta}=\\{\zeta_{1},\ldots,\zeta_{n}\\}$ are the zeros of
$Q$.

Note that the canonical element is only unique up to post-translations by real
values, which we can generally ignore. If one wishes to be concrete it could
be fixed by assuming $R(z)=z+o(1)$ as $z\to\infty$, noting that the asymptotic
behavior $R(z)=z+O(1)$ is automatic from the above factorization of
$R^{\prime}$. A priori this factorization does not rule out that one or more
of the poles overlaps with one of the critical points, or that the poles
appear with multiplicity. If this were to hold it would lead to extra
complications in the statement of our results (although not most of the
proofs). The next definition is designed to explicitly rule out this
degenerate case.

######  Definition.

We say a rational function $R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ with
pole set $\boldsymbol{\zeta}$ is generic if
$\boldsymbol{x}\cap\boldsymbol{\zeta}=\emptyset$.

Our first main result derives simple but important algebraic relations between
the poles and the critical points.

######  Theorem 2.1.

For distinct real points $\boldsymbol{x}$ let
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ be canonical and generic, and
let
$\boldsymbol{\zeta}=\boldsymbol{\zeta}(R)=\\{\zeta_{1},\ldots,\zeta_{n}\\}$ be
its set of finite poles. Then $\zeta_{k}$ satisfy the stationary relation

(2.1) $\sum_{j=1}^{2n}\frac{1}{\zeta_{k}-x_{j}}=\sum_{l\neq
k}\frac{2}{\zeta_{k}-\zeta_{l}},\qquad k=1,\ldots,n$

and the centroid relation

(2.2) $\frac{1}{n}\sum_{k=1}^{n}\zeta_{k}=\frac{1}{2n}\sum_{j=1}^{2n}x_{j}.$

We refer to (2.1) as a stationary relation since we think of the critical
points and poles of $R$ as interacting particles, and for fixed
$\boldsymbol{x}$ a solution to (2.1) may be regarded as an arrangement of the
poles $\zeta_{1},\ldots,\zeta_{n}$ such that interacting forces between the
particles balance out. The generic assumption easily shows (Lemma 3.1) that
the finite poles of $R$ are all simple, hence the right-hand side of (2.1) is
well-defined.

######  Remark.

For a given $\boldsymbol{x}$ there may be canonical elements that are not
generic. See Example 4.4, where the canonical element has both a critical
point and a double pole at zero. The non-generic case is not a serious issue
though, and only complicates the statement of our results rather than the
proofs. For example, in the stationary relation (2.1) if a pole $\zeta_{k}$
coincides with a critical point then the left-hand side is infinite.
Fortuitously, it turns out that if a critical point and a pole coincide then
the pole is necessarily of order at least two (see Lemmas 3.1 and 3.2), and
therefore the right-hand side of (2.1) is also infinite. In that sense the
stationary relation still holds but would need to be explicitly stated as a
special case. Similarly, in equations (2.8) and (2.9) below we see that the
statements require special explanation in the non-generic case, which we
provide in brief remarks.

Both (2.1) and (2.2) are simple consequences of $R^{\prime}$ having no
residues at each of the poles $\zeta_{k}$. The proof makes clear that the
converse statement is true: namely that if $\boldsymbol{x}$ are distinct real
points, $\boldsymbol{\zeta}$ appear in complex conjugate pairs and satisfy the
stationary relation (2.1), and
$\boldsymbol{x}\cap\boldsymbol{\zeta}=\emptyset$, then the function

(2.3)
$\displaystyle\frac{\prod_{j=1}^{2n}(z-x_{j})}{\prod_{k=1}^{n}(z-\zeta_{k})^{2}}$

has a real rational primitive. Using this one can prove that for a fixed set
of distinct, real $\boldsymbol{x}$ there are exactly $C_{n}$ distinct
solutions $\boldsymbol{\zeta}=\\{\zeta_{1},\ldots,\zeta_{n}\\}$ to the
stationary relation (2.1), although at present the proof is indirect. The
argument is as follows: given a solution $\boldsymbol{\zeta}$, the function
(2.3) has a real rational primitive $R$ which is necessarily in
$\operatorname{CRR}_{n+1}(\boldsymbol{x})$. This primitive falls into one of
the $C_{n}$ equivalence classes
$\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$, and only the poles of post-
dilations and post-translations of this primitive solve (2.1). This is because
any other post-composition would increase the number of poles to $n+1$,
thereby ruling out a solution to (2.1), while the poles of post-dilations and
post-translation have the same pole set as the original rational primitive,
hence lead to the same solution $\boldsymbol{\zeta}$. Thus there are at most
$C_{n}$ solutions. Conversely, every equivalence class
$\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ has a canonical
representative that leads to a solution $\boldsymbol{\zeta}$ of (2.1), and the
resulting solutions must be unique else they would correspond to canonical
rational functions in the same equivalence class, which leads to a
contradiction.

For the remainder of the discussion it is useful to codify the dependence of
the poles on the critical points $\boldsymbol{x}$ and the link pattern
$\alpha$, for which we introduce the following pole functions.

######  Definition.

For each set of distinct real points $\boldsymbol{x}$ and link pattern
$\alpha$ connecting them define $\zeta_{\alpha,k}(\boldsymbol{x})$,
$k=1,\ldots,n$, to be the poles of the (any) canonical element of
$\operatorname{CRR}_{n+1}(\boldsymbol{x})$. For each $\alpha$ and $k$ these
extend to smooth, complex-valued functions of $\boldsymbol{x}$ on the open set
of points in $\mathbb{R}^{2n}$ with distinct coordinates; we call these
functions pole functions. We write
$\boldsymbol{\zeta}_{\alpha}=\\{\zeta_{\alpha,1},\ldots,\zeta_{\alpha,n}\\}$
for the set of pole functions determined by $\alpha$.

Smoothness of the pole functions follows from the fact that for each
$\boldsymbol{x}$ they obey the stationary relation (2.1) and the centroid
relation (2.2). Indeed, the stationary relation (2.1) together with the
implicit function theorem implies that there is a neighborhood of
$\boldsymbol{x}$ on which the poles
$\zeta_{\alpha,k}=\zeta_{\alpha,k}(\boldsymbol{x})$ are smooth functions of
the critical points. We only ever use that the first and second order partial
derivatives of $\partial_{x_{j}}\zeta_{l},\partial^{2}_{x_{j}}\zeta_{l}$ exist
and are continuous, and in particular we do not need an explicit expression
for these derivatives. The existence of these partial derivatives is only used
in a minimal way in the proof of Theorem 2.3.

We now recall the evolutionary description of the multiple
SLE$(0;\boldsymbol{x};\alpha)$ curves via the Loewner equation. One may wish
to grow the curves from any of the boundary points in $\boldsymbol{x}$, either
simultaneously or individually, for which the most general form of the Loewner
evolution is

(2.4)
$\dot{g}_{t}(z)=\sum_{j=1}^{2n}\frac{2\nu_{j}(t)}{g_{t}(z)-x_{j}(t)},\qquad
g_{0}(z)=z,\qquad x_{j}(0)=x_{j}.$

Here $\nu_{j}=\nu_{j}(t)$ are non-negative, measurable functions but otherwise
arbitrary. We consider dynamics for the driving vector
$\boldsymbol{x}(t)=(x_{1}(t),\ldots,x_{2n}(t))$ of the form

(2.5) $\dot{x}_{j}(t)=U_{j}(\boldsymbol{x}(t))\nu_{j}(t)+\sum_{k\neq
j}\frac{2\nu_{k}(t)}{x_{j}(t)-x_{k}(t)},\quad j=1,\ldots,2n.$

The functions $U_{j}=U_{j}(\boldsymbol{x})$, $j=1,\ldots,2n$ determine the
interactions between the points $x_{j}$ (other than those interactions coming
from the Loewner flow, which the second term handles). An arbitrary choice of
smooth functions $U_{j}=U_{j}(\boldsymbol{x})$ will produce curves in
$\mathbb{H}$, but the interaction terms $U_{j}$ that generate the multiple
SLE$(0;\boldsymbol{x};\alpha)$ curves are quite specific. Bauer, Bernard, and
Kytölä [BBK05] first showed that the $U_{j}$ should satisfy the system

(2.6) $\displaystyle\frac{1}{2}U_{j}^{2}+\sum_{k\neq
j}\frac{2}{x_{k}-x_{j}}U_{k}-\sum_{k\neq j}\frac{6}{(x_{k}-x_{j})^{2}}=0,\quad
j=1,\ldots,2n,$

although they did not provide solutions. They arrived at these equations by
taking a formal $\kappa\to 0$ limit of the partial differential equations
satisfied by the interaction terms
$\mathcal{Z}=\mathcal{Z}(x_{1},\ldots,x_{2n})$ of the Loewner evolution for
multiple SLE$(\kappa;\boldsymbol{x};\alpha)$:

(2.7) $\displaystyle\left(\frac{\kappa}{2}\partial_{x_{j}}^{2}+\sum_{k\neq
j}\left(\frac{2}{x_{k}-x_{j}}\partial_{x_{k}}-\frac{(6-\kappa)/\kappa}{(x_{k}-x_{j})^{2}}\right)\right)\mathcal{Z}=0,\quad
j=1,\ldots,2n.$

Taking the $\kappa\to 0$ limit of $\kappa\partial_{x_{j}}\log\mathcal{Z}$
leads to the system (2.6). See also [PW20] for a more recent treatment. For
$\kappa>0$ equations (2.7) are the level two null-state BPZ partial
differential equations of conformal field theory, so motivated by this
terminology we refer to (2.6) as the null vector equations. Note that neither
(2.6) nor (2.7) depend on any particular link pattern $\alpha$. The next
theorem gives a simple expression for a set of solutions to the null vector
equations in terms of the critical points $\boldsymbol{x}$ and pole functions
$\boldsymbol{\zeta}_{\alpha}$.

######  Theorem 2.2.

For distinct real points $\boldsymbol{x}$ and a link pattern $\alpha$
connecting them define

(2.8) $\displaystyle U_{\alpha,j}(\boldsymbol{x}):=\sum_{k\neq
j}\frac{2}{x_{j}-x_{k}}+\sum_{k=1}^{n}\frac{4}{\zeta_{\alpha,k}(\boldsymbol{x})-x_{j}},\quad
j=1,\ldots,2n.$

Then $U_{\alpha,j}$ are real-valued, translation invariant, homogeneous of
degree $-1$, and satisfy the system of null vector equations (2.6).

The proof of Theorem 2.2 is purely algebraic, relying on nothing more than the
stationary relation (2.1). This is unsurprising given that we have written the
null vector equations (2.6) in an entirely algebraic form. Based on the
classical limit of the BPZ equations that is used in [BBK05, PW20] and
described above, one also expects that $U_{j}=\partial_{x_{j}}\log
Z(\boldsymbol{x})$ for a suitable function
$Z:\mathbb{R}^{2n}_{d}\to\mathbb{R}_{+}$, where $\mathbb{R}^{2n}_{d}$ is the
set of points in the real $2n$-dimensional Euclidean space whose coordinates
are all distinct. The next result gives an explicit expression for a $Z$
satisfying this property.

######  Theorem 2.3.

For distinct boundary points $\boldsymbol{x}$ and a link pattern $\alpha$
connecting them define

(2.9) $\displaystyle Z_{\alpha}(\boldsymbol{x}):=\\!\\!\prod_{1\leq j<k\leq
2n}\\!\\!(x_{j}-x_{k})^{2}\\!\\!\prod_{1\leq l<m\leq
n}\\!\\!(\zeta_{\alpha,l}(\boldsymbol{x})-\zeta_{\alpha,m}(\boldsymbol{x}))^{8}\prod_{k=1}^{2n}\prod_{l=1}^{n}(x_{k}-\zeta_{\alpha,l}(\boldsymbol{x}))^{-4}.$

Then $Z_{\alpha}$ is strictly positive and $U_{\alpha,j}=\partial_{x_{j}}\log
Z_{\alpha}$ for $j=1,\ldots,2n$.

An immediate corollary of Theorem 2.3 and the second order differentiability
of the pole functions with respect to $\boldsymbol{x}$ is that the solutions
$U_{\alpha,j}$ of Theorem 2.2 satisfy
$\partial_{x_{j}}U_{\alpha,k}=\partial_{x_{k}}U_{\alpha,j}$. The latter is a
condition for the $U_{j}$ that was laid out in [BBK05].

######  Remark.

Note that we do not assume that the canonical rational function is non-generic
in either Theorem 2.2 or 2.3. While the non-generic case may seem to be
problematic for (2.8) (because then a pole and a critical point coincide and
the right-hand side of (2.8) blows up) in fact the singularity is removable
and the solution $U_{\alpha,j}$ is still well-defined. To see this, for
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ define $U_{\alpha,j}$ as

$U_{\alpha,j}(R):=\sum_{k\neq
j}\frac{2}{x_{j}-x_{k}}+\sum_{\zeta_{k}\in\boldsymbol{\zeta}(R)}\frac{4}{\zeta_{k}-x_{j}},$

where $\boldsymbol{\zeta}(R)$ is the multi-set of poles of $R$. For canonical
$R$ this definition agrees with (2.8). For arbitrary $R$ it is easy to see
that the pole set is preserved under post-translations and post-dilations of
$R$, i.e. $\boldsymbol{\zeta}(R+h)=\boldsymbol{\zeta}(R)$ for $h\in\mathbb{R}$
and $\boldsymbol{\zeta}(rR)=\boldsymbol{\zeta}(R)$ for $r>0$. Since the
critical points are also preserved it follows that
$U_{\alpha,j}(R+h)=U_{\alpha,j}(R)$ and $U_{\alpha,j}(rR)=U_{\alpha,j}(R)$.
Under post-inversion we have $\boldsymbol{\zeta}(1/R)=\mathfrak{Z}(R)$, where
$\mathfrak{Z}(R)$ is the zero set of $R$. The zeros, poles, and critical
points satisfy

$\sum_{\zeta_{k}\in\boldsymbol{\zeta}(R)}\frac{1}{x_{j}-\zeta_{k}}=\sum_{\eta_{m}\in\mathfrak{Z}(R)}\frac{1}{x_{j}-\eta_{m}},\quad
j=1,\ldots,2n,$

which follow from $R^{\prime}(x_{j})=0$ and $R^{\prime}/R$ given by

$\frac{R^{\prime}(z)}{R(z)}=\sum_{\eta_{m}\in\mathfrak{Z}(R)}\frac{1}{z-\eta_{m}}-\sum_{\zeta_{k}\in\boldsymbol{\zeta}(R)}\frac{1}{z-\zeta_{k}}.$

Consequently $U_{\alpha,j}(1/R)=U_{\alpha,j}(R)$, proving that $U_{\alpha,j}$
is constant on the entire equivalence class. The non-generic rational
functions correspond to removable singularities, which we remove by extending
the constant value to the non-generic case. This extension also applies to the
functions $Z_{\alpha}$ of (2.9) since $U_{\alpha,j}=\partial_{x_{j}}\log
Z_{\alpha}$.

The next result proves another algebraic relation satisfied by the
$U_{\alpha,j}$, which is related to the Möbius invariance of the multiple
SLE$(0;\boldsymbol{x};\alpha)$ curves. For multiple
SLE$(\kappa;\boldsymbol{x};\alpha)$ the Möbius invariance manifests itself in
terms of a covariant behavior of the interaction terms
$\mathcal{Z}=\mathcal{Z}(\boldsymbol{x})$ with respect to Möbius
transformations acting on $\boldsymbol{x}$, leading $\mathcal{Z}$ to satisfy
the auxiliary system of partial differential equations

$\sum_{j=1}^{2n}\partial_{x_{j}}\mathcal{Z}=0,\quad\sum_{j=1}^{2n}\left(x_{j}\partial_{x_{j}}-\frac{\kappa-6}{2\kappa}\right)\mathcal{Z}=0,\quad\sum_{j=1}^{2n}\left(x_{j}^{2}\partial_{x_{j}}-\frac{\kappa-6}{\kappa}x_{j}\right)\mathcal{Z}=0.$

We call this system the conformal Ward identities. Using again the ansatz that
our $U_{j}$ are the $\kappa\to 0$ limit of the terms
$\kappa\partial_{x_{j}}\log\mathcal{Z}$, this leads to a system of three
auxiliary algebraic equations that the $U_{j}$ should satisfy. We again call
these auxiliary equations the conformal Ward identities, and the next result
proves that our solutions do in fact satisfy them.

######  Theorem 2.4.

For distinct real points $\boldsymbol{x}$ and a link pattern $\alpha$
connecting them the functions $U_{\alpha,j}$ of (2.8) satisfy the conformal
Ward identities

(2.10)
$\sum_{j=1}^{2n}U_{\alpha,j}=0,\quad\sum_{j=1}^{2n}x_{j}U_{\alpha,j}=-6n,\quad\sum_{j=1}^{2n}x_{j}^{2}U_{\alpha,j}=-6\sum_{j=1}^{2n}x_{j}.$

The proof of Theorem 2.4 is once again purely algebraic, based on nothing more
than the stationary relation (2.1) and the centroid relation (2.2). It also
highlights the role played by the canonical rational function having exactly
$n$ finite poles.

Given these properties of the solutions $U_{\alpha,j}$ we now use them to
construct Loewner flows. We do this in two different ways, although we will
eventually see that the two flows are equivalent. A priori though we do not
know this, and it is through the next two theorems that we conclude they are
equal. The first such definition is the following.

######  Definition.

Let $\boldsymbol{x}$ be distinct boundary points and $\alpha$ a link pattern
connecting them. Define the Loewner system associated to
$(\boldsymbol{x};\alpha)$ as (2.4) with driving functions $x_{j}(t)$ that
evolve as

(2.11)
$\displaystyle\dot{x}_{j}(t)=V_{\alpha,j}(\boldsymbol{x}(t),\boldsymbol{\xi}(t))\nu_{j}(t)+\sum_{k\neq
j}\frac{2\nu_{k}(t)}{x_{j}(t)-x_{k}(t)},\quad j=1,\ldots,2n,$

where, for $\boldsymbol{\xi}=\\{\xi_{1},\ldots,\xi_{n}\\}$, we define
$V_{\alpha,j}(\boldsymbol{x},\boldsymbol{\xi})$ as

(2.12) $\displaystyle
V_{\alpha,j}(\boldsymbol{x},\boldsymbol{\xi})=\sum_{k\neq
j}\frac{2}{x_{j}-x_{k}}+\sum_{k=1}^{n}\frac{4}{\xi_{k}-x_{j}},\quad
j=1,\ldots,2n.$

The points $\boldsymbol{\xi}(t)=\\{\xi_{1}(t),\ldots,\xi_{n}(t)\\}$ are
defined as the image of the poles $\zeta_{\alpha,k}(\boldsymbol{x})$ under the
Loewner flow, i.e. $\xi_{k}(t):=g_{t}(\zeta_{\alpha,k}(\boldsymbol{x}))$. (If
$\zeta_{\alpha,k}(\boldsymbol{x})$ lies in the lower half-plane, then we use
the extension of $g_{t},$ which exists by Schwarz reflection principle.) We
consider the system up until the first time $\tau^{*}=\tau\wedge\tau_{0}$,
where

$\tau=\inf\left\\{t>0:x_{j}(t)=x_{k}(t)\textrm{ for }j\neq
k\right\\},\quad\tau_{0}=\inf\left\\{t>0:x_{j}(t)=\xi_{l}(t)\textrm{ for some
}j,l\right\\}.$

For each $z\in\mathbb{H}$ the solution to (2.4) exists up until time $\tau$ or
the first time $\tau(z)$ such that $g_{t}(z)=x_{j}(t)$ for some $j$. The
Loewner hull is, as usual,
$K_{t}:=\\{z\in\overline{\mathbb{H}}:t\geq\tau(z)\\}$.

The more natural dynamics involve the interaction term
$U_{\alpha,j}(\boldsymbol{x}(t))$, and we will soon see that they actually
agree with these dynamics. The reason we avoid them for now is that they use
the processes $\zeta_{\alpha,k}(\boldsymbol{x}(t))$ for which we do not have
expressions for the time derivatives. Instead we use the processes
$\xi_{k}(t)$ for which we can explicitly compute the time derivative.

The next theorem finds a preserved quantity under these particular dynamics.
It comes about as the classical $\kappa\to 0$ limit of a martingale observable
for a system of multiple SLE$(\kappa;\boldsymbol{x};\alpha)$ curves.
Martingale observables can be thought of as the stochastic counterparts of
preserved quantities or integrals of motion, and in [AKM] we will explain how
conformal field theory produces an infinite family of martingale observables
for systems of multiple SLE$(\kappa)$. These observables are correlation
functions derived from formal algebraic operations on the random Gaussian free
field, although multiple SLE requires a version of the GFF that is shifted by
an appropriate deterministic function; see [SS09, Dub09, SS13] (among others)
for more on this idea. The observable below is the $\kappa\to 0$ limit of the
bosonic observable, or in other words the $\kappa\to 0$ limit of the
deterministic shift used in the study of multiple
SLE$(\kappa;\boldsymbol{x};\alpha)$.

######  Theorem 2.5.

Let $\boldsymbol{x}$ be distinct real points and $\alpha$ be a link pattern
connecting them. Then under the Loewner system associated to
$(\boldsymbol{x};\alpha)$ the quantity

(2.13)
$N_{t}(z):=\log\left(g_{t}^{\prime}(z)\frac{\prod_{j=1}^{2n}(g_{t}(z)-x_{j}(t))}{\prod_{k=1}^{n}(g_{t}(z)-g_{t}(\zeta_{\alpha,k}(\boldsymbol{x})))^{2}}\right)$

satisfies $\dot{N}_{t}(z)=0$ for each $z$ and $t<\min(\tau(z),\tau)$.

Even though the integral of motion (2.13) is motivated by conformal field
theory the proof is purely algebraic. It requires little more than the
definition of the Loewner flow associated to $(\boldsymbol{x};\alpha)$. The
next theorem characterizes the evolving points $\boldsymbol{x}(t)$ and
$\boldsymbol{\xi}(t)$ as the critical points and poles of a rational function
built out of $R$ and $g_{t}$.

######  Theorem 2.6.

Let $\boldsymbol{x}$ be distinct boundary points, $\alpha$ be a link pattern
connecting them, and $R\in\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ be
canonical. Then under the Loewner system associated to
$(\boldsymbol{x};\alpha)$ the map $R\circ g_{t}^{-1}$ is, for each $t<\tau$, a
canonical element in $\operatorname{CRR}_{n+1}(\boldsymbol{x}(t);\alpha)$ with
poles $\xi_{k}(t)=g_{t}(\zeta_{\alpha,k}(\boldsymbol{x}))$.

By definition of the pole functions a canonical element in
$\operatorname{CRR}_{n+1}(\boldsymbol{x}(t);\alpha)$ has poles
$\zeta_{\alpha,k}(\boldsymbol{x}(t))$, so from the last theorem we conclude
that there is no ambiguity between the two natural ways of considering the
flow of the poles. We summarize this in the following corollary.

######  Corollary 2.7.

Let $\boldsymbol{x}$ be distinct boundary points, $\alpha$ be a link pattern
connecting them. Then under the Loewner system associated to
$(\boldsymbol{x};\alpha)$ we have
$\zeta_{\alpha,k}(\boldsymbol{x}(t))=g_{t}(\zeta_{\alpha,k}(\boldsymbol{x}))$,
$k=1,\ldots,n$.

A further consequence of $R\circ g_{t}^{-1}$ being real rational is that it
maps $\mathbb{R}$ into $\mathbb{R}$. But the image of $\mathbb{R}$ under
$g_{t}^{-1}$ is the real line plus the trace of the generated curves, which
are subsets of $\mathbb{H}$. Thus $R$ must map these traces back to the real
line, meaning that they are in the real locus of $R$. This proves the
following corollary.

######  Corollary 2.8.

Let $\boldsymbol{x}$ be distinct boundary points and
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ be generic. Let $\gamma[0,t]$
be the trace of the curves generated by the Loewner flow corresponding to $R$,
where $t<\tau^{*}$. Then $\gamma[0,t]$ is a subset of the real locus
$\Gamma(R)$.

At this point Theorem 2.6 and Corollaries 2.7, 2.8 are only valid up until the
time $\tau^{*}$, but we argue next that they can be extended to the time
$\tau$ instead. To begin the argument, first observe that by Corollary 2.7 the
following, more natural definition of the Loewner system associated to
$(\boldsymbol{x};\alpha)$ is equivalent to the original definition.

######  Definition.

Let $\boldsymbol{x}$ be distinct boundary points and $\alpha$ a link pattern
connecting them. Define the Loewner system associated to
$(\boldsymbol{x};\alpha)$ as (2.4) with driving functions $x_{j}(t)$ that
evolve as

(2.14)
$\displaystyle\dot{x}_{j}(t)=U_{\alpha,j}(\boldsymbol{x}(t))\nu_{j}(t)+\sum_{k\neq
j}\frac{2\nu_{k}(t)}{x_{j}(t)-x_{k}(t)},\quad j=1,\ldots,2n,$

for the solutions $U_{\alpha,j}$ of (2.8). The stopping times $\tau(z)$ and
the hull $K_{t}$ are defined in the same way.

The two Loewner systems associated to $(\boldsymbol{x};\alpha)$ are the same
until time $\tau^{*}$, but now we show that we can extend the global existence
time of solutions to the Loewner system (2.14) past the global existence time
$\tau^{*}$ and up to time $\tau$. Observe that $\tau$ is the first time at
which one of the $n$ curves is completed. The reasons for this extension are:

  * •

If a pole $\zeta_{\alpha,k}=\xi_{k}(0)$ starts on the real line $\mathbb{R}$
then the Loewner map $g_{t}$ keeps it on the real line for all $t>0$.

  * •

If $\zeta_{\alpha,k}\in\mathbb{R}$ and is a double pole of a canonical
rational function in $\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$, then
the canonical rational function is non-generic. As explained after Theorem 2.3
the solutions $U_{\alpha,j}$ are still well-defined even in the non-generic
case and the Loewner system (2.14) has a well-defined initial condition for
$\dot{x}_{j}(t)$, hence there is no issue with starting the dynamics.

  * •

If $\zeta_{\alpha,k}=\xi_{k}(0)\in\mathbb{R}$ then it is impossible that
$\xi_{k}(t)=x_{j}(t)$ for some $j$ without also $x_{j}(t)=x_{l}(t)$ for some
$j\neq l$, otherwise one of the curves would intersect the real line at a
point other than one of the critical points in $\boldsymbol{x}$. The latter is
impossible since the real locus can intersect the real line only at one of the
critical points. Thus, to check for the case $\tau_{0}<\tau$ one only has to
consider $\zeta_{\alpha,k}=\xi_{k}(0)$ for which
$\zeta_{\alpha,k}\not\in\mathbb{R}$.

  * •

If $\zeta_{\alpha,k}\not\in\mathbb{R}$ then its complex conjugate is also a
pole, i.e. $\overline{\zeta_{\alpha,k}}=\zeta_{\alpha,l}$ for some $k\neq l$.
This property is preserved for $t>0$, i.e. $\overline{\xi_{k}(t)}=\xi_{l}(t)$.
Furthermore $\zeta_{\alpha,k}$ and $\zeta_{\alpha,l}$ are in the real locus
$\Gamma(R)$ of any $R\in\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$, and
since the curves generated by the system lie in the real locus (Corollary 2.8)
there will be some time $t$ at which $x_{j}(t)=\xi_{k}(t)=\xi_{l}(t)$ for some
$j,k,l$. By definition $\tau_{0}=t$, and this is the only case in which
$\tau_{0}<\tau$. However at time $\tau_{0}$ the configuration of the system is
in the non-generic case, with a double pole at
$x_{j}(t)=\xi_{k}(t)=\xi_{l}(t)$, and as we observed there is no difficulty
with starting the system in the non-generic case. We continue to use this fact
every time the rational function $R\circ g_{t}^{-1}$ becomes non-generic,
thereby extending the global existence time past $\tau_{0}$ and until time
$\tau$.

There should be a natural way to extend the dynamics beyond time $\tau$, when
the first of the $n$ curves is completed, but we do not consider this at
present. One can avoid the issue by choosing a time parameterization
$\boldsymbol{\nu}=(\nu_{1}(t),\ldots,\nu_{2n}(t))$ such that all $n$ curves
are completed at the same time. There are many such time parameterizations
available, each of which can be constructed by appropriate adjustments of the
functions $\nu_{j}(t)$ as the two tips of each curve approach each other.

Rather than exploring this issue further we instead we turn to the connection
with the Calogero-Moser system, which we will expand on in a later work.

######  Corollary 2.9.

Let $\boldsymbol{x}$ be distinct boundary points and $\alpha$ be a link
pattern connecting them. Let $\boldsymbol{x}(t)$ and $\boldsymbol{\zeta}(t)$
be the evolution of the critical points and poles of a canonical element of
$\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ under the Loewner system
associated to $(\boldsymbol{x};\alpha)$. In the case $\nu_{j}\equiv 1/4$ the
pair $(\boldsymbol{x}(t),\boldsymbol{\zeta}(t))$ form a closed dynamical
system whose evolution follows the Calogero-Moser system

(2.15) $\dot{x}_{j}=\phantom{-}\sum_{k\neq
j}\frac{1}{x_{j}-x_{k}}-\sum_{k=1}^{n}\frac{1}{x_{j}-\zeta_{k}},$

and

(2.16) $\dot{\zeta}_{k}=-\sum_{l\neq
k}\frac{1}{\zeta_{k}-\zeta_{l}}+\sum_{j=1}^{2n}\frac{1}{\zeta_{k}-x_{j}}.$

We do not use the explicit dynamics of Corollary 2.9 anywhere in our
arguments, we simply highlight them for their connection to the Calogero-Moser
system. Note that by using the stationary relation (2.1), which is shown to be
preserved under the flow in the proof of Theorem 2.6, from (2.16) one obtains

$\dot{\zeta}_{k}=\sum_{l\neq k}\frac{1}{\zeta_{k}-\zeta_{l}}.$

Thus the poles $\zeta_{k}$, $k=1,\ldots,n$, form a Calogero-Moser system on
their own.

##  3\. Proofs of Main Results

As always, we assume throughout this section that
$\boldsymbol{x}=\\{x_{1},\ldots,x_{2n}\\}$ with all $x_{j}$ distinct, and that
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ is our rational function. We
first record the following simple fact about generic rational functions.

######  Lemma 3.1.

A generic rational function has only simple poles.

###### Proof.

If not then $R=P/Q$ has a double pole (or higher) at a zero $\zeta_{k}$ of
$Q$, i.e. $Q(\zeta_{k})=Q^{\prime}(\zeta_{k})=0$. Then the Wronskian of $P$
and $Q$ vanishes at $\zeta_{k}$, implying that $\zeta_{k}$ is also a critical
point. ∎

We use Lemma 3.1 in the proof of the stationary relations (2.1) and Theorem
2.1.

###### Proof of Theorem 2.1.

By the definition of canonical $R^{\prime}(z)$ factors as

(3.1) $\displaystyle
R^{\prime}(z)=\frac{\prod_{j=1}^{2n}(z-x_{j})}{\prod_{k=1}^{n}(z-\zeta_{k})^{2}}.$

Its partial fraction decomposition is therefore

(3.2) $\displaystyle
R^{\prime}(z)=\sum_{k=1}^{n}\frac{A_{k}}{(z-\zeta_{k})^{2}}+\sum_{k=1}^{n}\frac{B_{k}}{z-\zeta_{k}}.$

We compute the coefficients $A_{k}$ from

$A_{k}=\lim_{z\to\zeta_{k}}R^{\prime}(z)(z-\zeta_{k})^{2}=\frac{\prod_{j=1}^{2n}(\zeta_{k}-x_{j})}{\prod_{l\neq
k}(\zeta_{k}-\zeta_{l})^{2}}.$

Note that we also used that the poles are all simple in this computation,
which follows from Lemma 3.1. The assumption that $R$ is generic also implies
that $A_{k}\neq 0$. For the $B_{k}$ we similarly have

$B_{k}=\lim_{z\to\zeta_{k}}(R^{\prime}(z)(z-\zeta_{k})^{2})^{\prime}=\left(\sum_{j=1}^{2n}\frac{1}{\zeta_{k}-x_{j}}-\sum_{l\neq
k}\frac{2}{\zeta_{k}-\zeta_{l}}\right)A_{k}.$

However since $R^{\prime}(z)$ has a rational primitive ($R$ itself) we have
$B_{k}=0$ for all $k=1,\ldots,n$, and since we already know $A_{k}\neq 0$ this
proves (2.1).

Finally we prove the centroid relation (2.2). Since $B_{k}=0$ the partial
fraction decomposition (3.2) implies $R^{\prime}(z)=1+O(z^{-2})$ near
$z=\infty$, and therefore

$\lim_{z\to\infty}z(R^{\prime}(z)-1)=0.$

On the other hand, we may use the factorization of $R^{\prime}$ to write
$z(R^{\prime}(z)-1)$ as

$\frac{(2\sum_{k=1}^{n}\zeta_{k}-\sum_{j=1}^{2n}x_{j})z^{2n}+\cdots}{\prod_{k=1}^{n}(z-\zeta_{k})^{2}}.$

Therefore, we obtain the relation
$2\sum_{k=1}^{n}\zeta_{k}=\sum_{j=1}^{2n}x_{j}$. ∎

The partial fraction decomposition (3.2) also allows us to prove the converse
to Lemma 3.1.

######  Lemma 3.2.

A rational function $R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ with all
simple poles is generic.

###### Proof.

Suppose not, so that $\zeta_{k}=x_{m}$ for some $m,k$. Without loss of
generality we may assume that $R$ factors as in (3.1). Then the partial
fraction expansion (3.2) of $R^{\prime}$ still holds, but by assumption
$\zeta_{k}=x_{m}$ and therefore $A_{k}=0$. Since $R^{\prime}$ still has a
rational primitive we know $B_{k}=0$ also. On the other hand, if we let
$q(z)=R^{\prime}(z)(z-\zeta_{k})^{2}$ then $q^{\prime}(\zeta_{k})=B_{k}$
follows by the partial fraction expansion, just as in the proof of Theorem
2.1. But also

$q^{\prime}(z)=\left(\sum_{j=1}^{2n}\frac{1}{z-x_{j}}-\sum_{l\neq
k}\frac{2}{z-\zeta_{l}}\right)q(z).$

The left-hand side tends to zero as $z\to\zeta_{k}$, by
$q^{\prime}(\zeta_{k})=B_{k}=0$, while the right-hand side tends to

$\frac{\prod_{j\neq m}(\zeta_{k}-x_{j})}{\prod_{l\neq
k}(\zeta_{k}-\zeta_{l})^{2}}$

by definition of $q(z)$ and the factorization of $R^{\prime}$. But it is a
contradiction for the quantity above to be zero since $\zeta_{k}=x_{m}$ and
the critical points are all distinct. ∎

We now turn to the proof of Theorem 2.2, that the functions
$U_{\alpha,j}=U_{\alpha,j}(\boldsymbol{x})$ of equation (2.8) satisfy the null
vector equations.

###### Proof of Theorem 2.2.

Write $U_{\alpha,j}=U_{j}$ throughout. Real-valuedness of $U_{j}$ follows from
$x_{k}\in\mathbb{R}$ and the $\zeta_{k}$ occurring in complex conjugate pairs.
For the translation invariance, note that the pole functions
$\zeta_{\alpha,k}$ clearly satisfy
$\zeta_{\alpha,k}(\boldsymbol{x}+h)=\zeta_{\alpha,k}(\boldsymbol{x})+h$ for
$h\in\mathbb{R}$, since if
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ is canonical then
$R(z-h)\in\operatorname{CRR}_{n+1}(\boldsymbol{x}+h;\alpha)$ and is also
canonical. Similarly,
$rR(r^{-1}z)\in\operatorname{CRR}_{n+1}(r\boldsymbol{x};\alpha)$ for $r>0$ and
is canonical, hence
$\zeta_{\alpha,k}(r\boldsymbol{x})=r\zeta_{\alpha,k}(\boldsymbol{x})$. The
homogeneity of $U_{\alpha,j}$ follows from these properties of the pole
function and the definition (2.8).

For the proof that the $U_{j}$ satisfy the null vector equations it is
convenient to introduce

(3.3) $u_{j}:=U_{j}-\sum_{k\neq j}\frac{2}{x_{j}-x_{k}}.$

All we need to check is that $u_{j}$’s satisfy the algebraic equations

(3.4) $\frac{1}{2}\,u_{j}^{2}+2\sum_{k\neq
j}\frac{u_{j}-u_{k}}{x_{j}-x_{k}}=0,\quad j=1,\ldots,2n.$

Using the stationary relation (2.1), we have

$\displaystyle\frac{1}{8}\Big{(}\frac{1}{2}\,u_{j}^{2}$
$\displaystyle+2\sum_{k\neq
j}\frac{u_{j}-u_{k}}{x_{j}-x_{k}}\Big{)}=\Big{(}\sum_{l=1}^{n}\frac{1}{\zeta_{l}-x_{j}}\Big{)}^{2}+\sum_{k\neq
j}\sum_{l=1}^{n}\frac{1}{(\zeta_{l}-x_{j})(\zeta_{l}-x_{k})}$
$\displaystyle=\sum_{l=1}^{n}\frac{1}{\zeta_{l}-x_{j}}\sum_{k=1}^{2n}\frac{1}{\zeta_{l}-x_{k}}+2\sum_{k<l}\frac{1}{(\zeta_{k}-x_{j})(\zeta_{l}-x_{j})}$
$\displaystyle=\sum_{l=1}^{n}\frac{1}{\zeta_{l}-x_{j}}\sum_{k\neq
l}\frac{2}{\zeta_{l}-\zeta_{k}}+2\sum_{k<l}\frac{1}{(\zeta_{k}-x_{j})(\zeta_{l}-x_{j})}=0.$

Here we used the identity

$\sum_{k<l}\frac{1}{(\zeta_{k}-x_{j})(\zeta_{l}-x_{j})}=\sum_{k<l}-\frac{1}{(\zeta_{k}-x_{j})(\zeta_{k}-\zeta_{l})}+\frac{1}{(\zeta_{l}-x_{j})(\zeta_{k}-\zeta_{l})}.$

∎

For the proof of Theorem 2.3 we compare the explicit formula (2.8) for the
$U_{\alpha,j}$ with the logarithmic derivative of $Z_{\alpha}$ of equation
(2.9).

###### Proof of Theorem 2.3.

Write $Z=Z_{\alpha}$ and $U_{j}=U_{\alpha,j}$ throughout. Strict positivity of
$Z$ follows from the $x_{j}$ assumed to be real and distinct and the
$\zeta_{k}$ always appearing in complex conjugate pairs. For the derivative,
using expression (2.9) for $Z$ and computing $\partial_{x_{j}}\log Z$ directly
we have

$U_{j}=\sum_{k\neq
j}\frac{2}{x_{j}-x_{k}}+\sum_{l=1}^{n}\frac{4}{\zeta_{l}-x_{j}}+4\sum_{k=1}^{2n}\sum_{l=1}^{n}\frac{\partial_{x_{j}}\zeta_{l}}{x_{k}-\zeta_{l}}+8\\!\\!\\!\\!\sum_{1\leq
l<m\leq
n}\\!\\!\frac{\partial_{x_{j}}\zeta_{l}-\partial_{x_{j}}\zeta_{m}}{\zeta_{l}-\zeta_{m}}.$

For the last two terms use the stationary relation (2.1) to obtain

$\sum_{l=1}^{n}\partial_{x_{j}}\zeta_{l}\sum_{k=1}^{2n}\frac{1}{x_{k}-\zeta_{l}}=-\sum_{l=1}^{n}\partial_{x_{j}}\zeta_{l}\sum_{m\neq
l}\frac{2}{\zeta_{l}-\zeta_{m}}=-2\\!\\!\\!\\!\sum_{1\leq l<m\leq
n}\\!\\!\frac{\partial_{x_{j}}\zeta_{l}-\partial_{x_{j}}\zeta_{m}}{\zeta_{l}-\zeta_{m}}.$

Thus the last two terms in the above expression for $U_{j}$ vanish, thereby
proving equation (2.8). ∎

We next prove that the $U_{j}$ satisfy the conformal Ward identities, using
only simple algebra, the stationary relation (2.1), the centroid relation
(2.2), and the definition (2.8) of $U_{j}$.

###### Proof of Theorem 2.4.

The first conformal Ward identity holds by symmetry:

$\sum_{j=1}^{2n}U_{j}=\sum_{j=1}^{2n}\sum_{k\neq
j}\frac{2}{x_{j}-x_{k}}+\sum_{j=1}^{2n}\sum_{k=1}^{n}\frac{4}{\zeta_{k}-x_{j}}=\sum_{k=1}^{n}\sum_{l\neq
k}\frac{8}{\zeta_{k}-\zeta_{l}}=0.$

To find a sufficient and necessary condition for the second identity
$\sum_{j=1}^{2n}x_{j}U_{j}=-6n$, we compute

$\sum_{j=1}^{2n}\sum_{k\neq
j}\frac{x_{j}}{x_{j}-x_{k}}=2n^{2}-n,\qquad\sum_{j=1}^{2n}\sum_{k=1}^{n}\frac{x_{j}}{\zeta_{k}-x_{j}}=-2n^{2}+\sum_{k=1}^{n}\sum_{l\neq
k}\frac{2\zeta_{k}}{\zeta_{k}-\zeta_{l}}=-2n^{2}+n(n-1).$

Thus we find $\sum_{j=1}^{2n}x_{j}U_{j}=4n^{2}-2n-8n^{2}+4n(n-1)=-6n$, as
claimed. For the last of the conformal Ward identities we compute

$\sum_{j=1}^{2n}\sum_{k\neq
j}\frac{x_{j}^{2}}{x_{j}-x_{k}}=\sum_{j<k}(x_{j}+x_{k})=(2n-1)\sum_{j=1}^{n}x_{j}$

and therefore

$\sum_{j=1}^{2n}\sum_{k=1}^{n}\frac{x_{j}^{2}}{\zeta_{k}-x_{j}}=-n\sum_{j=1}^{2n}x_{j}-2n\sum_{k=1}^{n}\zeta_{k}+\sum_{k=1}^{n}\sum_{l\neq
k}\frac{2\zeta_{k}^{2}}{\zeta_{k}-\zeta_{l}}=-n\sum_{j=1}^{2n}x_{j}-2\sum_{k=1}^{n}\zeta_{k}.$

Now the last conformal Ward identity follows from the centroid relation of
equation (2.2). ∎

For Theorem 2.5 we show that (2.13) is an integral of motion for the Loewner
flow associated to $(\boldsymbol{x};\alpha)$ via explicit computation. Note
that this proof only uses the defining equations (2.4)-(2.11) of the Loewner
system and the defining equation (2.12) of the solutions $V_{\alpha,j}$, along
with the definition $\xi_{k}(t)=g_{t}(\zeta_{\alpha,k}(\boldsymbol{x}))$. The
rest of the argument is simple algebra.

###### Proof of Theorem 2.5.

Throughout the proof we drop the dependence on $\alpha$ and write
$\xi_{k}(t)=g_{t}(\zeta_{\alpha,k}(\boldsymbol{x}))$. By differentiation we
have

$\dot{N}_{t}(z)=\frac{\dot{g}_{t}^{\prime}(z)}{g_{t}^{\prime}(z)}+\sum_{j=1}^{2n}\frac{\dot{g}_{t}(z)-\dot{x}_{j}(t)}{g_{t}(z)-x_{j}(t)}-2\sum_{k=1}^{n}\frac{\dot{g}_{t}(z)-\dot{\xi}_{k}(t)}{g_{t}(z)-\xi_{k}(t)}.$

From the definitions we have (dropping the $t$ dependence in $x_{j}(t)$ and
$\xi_{k}(t)$ and the $z$ dependence in $g_{t}(z)$)

$\frac{\dot{g}_{t}^{\prime}}{g_{t}^{\prime}}=-\sum_{j=1}^{2n}\frac{2\nu_{j}}{(g_{t}-x_{j})^{2}},\quad\dot{\xi}_{k}=\sum_{j=1}^{2n}\frac{2\nu_{j}}{\xi_{k}-x_{j}},\quad\dot{x}_{j}=\sum_{k\neq
j}\frac{2\nu_{j}}{x_{j}-x_{k}}+\sum_{k=1}^{n}\frac{4\nu_{j}}{\xi_{k}-x_{j}}+\sum_{k\neq
j}\frac{2\nu_{k}}{x_{j}-x_{k}}.$

Thus the first two terms of $\dot{N}_{t}(z)$ sum to

$\frac{\dot{g}_{t}^{\prime}}{g_{t}^{\prime}}+\sum_{j=1}^{2n}\frac{\dot{g}_{t}}{g_{t}-x_{j}}=\sum_{k\neq
j}\frac{2\nu_{k}}{(g_{t}-x_{j})(g_{t}-x_{k})}.$

Now in the expression for $\dot{N}_{t}(z)$ expand out $\dot{x}_{j}$ and group
the terms as $\dot{N}_{t}(z)=\mathrm{I}+\mathrm{II}$, where

$\mathrm{I}=\sum_{j\neq
k}\frac{2\nu_{k}}{(g_{t}-x_{j})(g_{t}-x_{k})}-\sum_{j\neq
k}\frac{2\nu_{j}}{(g_{t}-x_{j})(x_{j}-x_{k})}-\sum_{j\neq
k}\frac{2\nu_{k}}{(g_{t}-x_{j})(x_{j}-x_{k})}$

is the sum of the terms not involving $\xi_{k}$ and

$\mathrm{II}=-\sum_{j,k}\frac{4\nu_{j}}{(g_{t}-x_{j})(\xi_{k}-x_{j})}-\sum_{j,k}\frac{4\nu_{j}}{(g_{t}-x_{j})(g_{t}-\xi_{k})}+\sum_{k}\frac{2\dot{\xi}_{k}}{g_{t}-\xi_{k}}$

is the sum of the terms that do involve $\xi_{k}$. The first and last terms of
I sum to

$\sum_{j\neq k}\frac{2\nu_{k}}{(g_{t}-x_{j})(g_{t}-x_{k})}-\sum_{j\neq
k}\frac{2\nu_{k}}{(g_{t}-x_{j})(x_{j}-x_{k})}=\sum_{k\neq
j}\frac{2\nu_{j}}{(g_{t}-x_{j})(x_{j}-x_{k})},$

and therefore $\mathrm{I}=0$. On the other hand, the first two terms of II sum
to

$\sum_{j,k}\frac{4\nu_{j}}{(g_{t}-x_{j})(\xi_{k}-x_{j})}+\sum_{j,k}\frac{4\nu_{j}}{(g_{t}-x_{j})(g_{t}-\xi_{k})}=\sum_{j,k}\frac{4\nu_{j}}{(\xi_{k}-x_{j})(g_{t}-\xi_{k})},$

and then using the expression above for $\dot{\xi}_{k}(t)$ it follows that

$\mathrm{II}=\sum_{k=1}^{n}\frac{2}{g_{t}(z)-\xi_{k}(t)}\Big{(}\dot{\xi}_{k}(t)-\sum_{j=1}^{2n}\frac{2\nu_{j}(t)}{\xi_{k}(t)-x_{j}(t)}\Big{)}=0.$

This completes the proof. ∎

Now we immediately build off of Theorem 2.5 to prove Theorem 2.6 that $R\circ
g_{t}^{-1}$ is a real rational function.

###### Proof of Theorem 2.6.

We write $\zeta_{k}=\zeta_{\alpha,k}(\boldsymbol{x})$ throughout. By Theorem
2.5 we have

(3.5)
$g_{t}^{\prime}(z)\frac{\prod_{j=1}^{2n}(g_{t}(z)-x_{j}(t))}{\prod_{k=1}^{n}(g_{t}(z)-\xi_{k}(t))^{2}}=\frac{\prod_{j=1}^{2n}(z-x_{j})}{\prod_{k=1}^{n}(z-\zeta_{k})^{2}},$

and the expression on the right is simply $R^{\prime}(z)$, since $R$ is a
canonical element of $\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$. Let
$f_{t}=g_{t}^{-1}$, and then rewrite the above as

(3.6)
$\displaystyle\frac{\prod_{j=1}^{2n}(z-x_{j}(t))}{\prod_{k=1}^{n}(z-\xi_{k}(t))^{2}}=f_{t}^{\prime}(z)\frac{\prod_{j=1}^{2n}(f_{t}(z)-x_{j})}{\prod_{k=1}^{n}(f_{t}(z)-\zeta_{k})^{2}}=f_{t}^{\prime}(z)R^{\prime}(f_{t}(z))=(R\circ
f_{t})^{\prime}(z).$

Expand the left-hand side into partial fractions, as in equation (3.2) in the
proof of Theorem 2.1, and take the anti-derivative of both sides with respect
to $z$ to obtain

$\displaystyle z+\sum_{k=1}^{n}$ $\displaystyle
A_{k}(t)\left(\sum_{j=1}^{2n}\frac{1}{\xi_{k}(t)-x_{j}(t)}-\sum_{l\neq
k}\frac{2}{\xi_{k}(t)-\xi_{l}(t)}\right)\log(z-\xi_{k}(t))$
$\displaystyle-\sum_{k=1}^{n}\frac{A_{k}(t)}{z-\xi_{k}(t)}=R(f_{t}(z))+C_{t},$

for some constant $C_{t}$, and where $A_{k}(t)$, $k=1,\ldots,n$, is given by

$A_{k}(t)=\frac{\prod_{j=1}^{2n}(\xi_{k}(t)-x_{j}(t))}{\prod_{l\neq
k}(\xi_{k}(t)-\xi_{l}(t))^{2}}.$

The composition $R\circ f_{t}$ has no logarithmic singularities (by
rationality of $R$ and smoothness of $g_{t}^{-1}$ on $\mathbb{H}$), therefore

(3.7) $\sum_{j=1}^{2n}\frac{1}{\xi_{k}(t)-x_{j}(t)}=\sum_{l\neq
k}\frac{2}{\xi_{k}(t)-\xi_{l}(t)},$

i.e. the stationary relation (2.1) is preserved. Thus we have

(3.8) $\displaystyle
z-\sum_{k=1}^{n}\frac{A_{k}(t)}{z-\xi_{k}(t)}=R(f_{t}(z))+C_{t},$

so $R\circ f_{t}$ is a rational function. Note that the left-hand side behaves
asymptotically like $z+o(1)$ as $z\to\infty$, as does $g_{t}(z)$ and hence
$f_{t}(z)$. Since $R(z)=z+O(1)$ at infinity it follows that

$C_{t}=\lim_{z\to\infty}z-R(z),$

which in particular is real and does not depend on $t$. Therefore $R\circ
f_{t}$ is real by (3.8) and because the Loewner flow corresponding to (2.11)
and (2.12) preserves that the poles $\xi_{k}(t)$ appear in complex conjugate
pairs. The factorization (3.6) of $(R\circ f_{t})^{\prime}$ shows that $R\circ
f_{t}$ is canonical with critical points $x_{j}(t)$ and poles
$\xi_{k}(t)=g_{t}(\zeta_{\alpha,k}(\boldsymbol{x}))$. ∎

As a consequence of Theorem 2.6 we record the following result, which gives an
algebraic equation for the conformal map $g_{t}$. By searching for real-valued
solutions to this equation we can determine the locations of the $n$ tips of
the curves $\gamma_{j}(t)$, $j=1,\ldots,2n$ that are grown by the Loewner
evolution. We use this technique frequently in the examples.

######  Lemma 3.3.

Let $\boldsymbol{x}$ be distinct boundary points and $\alpha$ be a link
pattern connecting them. Let $\boldsymbol{x}(t)$ and $\boldsymbol{\zeta}(t)$
be the evolution of the critical points and poles of a canonical element of
$\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ under the Loewner system
associated to $(\boldsymbol{x};\alpha)$. Then

$g_{t}(z)-\sum_{k=1}^{n}\frac{A_{k}(\boldsymbol{x}(t),\boldsymbol{\zeta}_{\alpha}(t))}{g_{t}(z)-\zeta_{\alpha,k}(t)}=z-\sum_{k=1}^{n}\frac{A_{k}(\boldsymbol{x}(0),\boldsymbol{\zeta}_{\alpha}(0))}{z-\zeta_{\alpha,k}(0)},$

where $A_{k}(\boldsymbol{x},\boldsymbol{\zeta})$ is defined by

$A_{k}(\boldsymbol{x},\boldsymbol{\zeta})=\frac{\prod_{j=1}^{2n}(\zeta_{k}-x_{j})}{\prod_{l\neq
k}(\zeta_{k}-\zeta_{l})^{2}}.$

###### Proof.

Combine $R\circ g_{t}^{-1}\circ g_{t}=R$ with equation (3.8) for $R\circ
g_{t}^{-1}$. ∎

Finally, we prove Corollary 2.9 that
$(\boldsymbol{x}(t),\boldsymbol{\zeta}_{\alpha}(t))$ evolve, under the Loewner
dynamics associated to the rational function $R$, as a Calogero-Moser system.

###### Proof of Corollary 2.9.

If $\nu_{k}\equiv 1/4$, then it follows from the defining dynamics (2.5) of
$x_{j}(t)$ and formula (2.8) for $U_{j}$ that

$\dot{x}_{j}=\frac{1}{4}\Big{(}U_{j}+\sum_{k\neq
j}\frac{2}{x_{j}-x_{k}}\Big{)}=\sum_{k\neq
j}\frac{1}{x_{j}-x_{k}}+\sum_{k=1}^{n}\frac{1}{\zeta_{k}-x_{j}},$

which shows (2.15). On the the hand, by the definition of
$\zeta_{k}(t):=g_{t}(\zeta_{k}(0))$ and the Loewner flow (2.4), we have

$\dot{\zeta}_{k}=\dot{g}_{t}(\zeta_{k}(0))=\frac{1}{2}\sum_{j=1}^{2n}\frac{1}{g_{t}(\zeta_{k}(0))-x_{j}}=\frac{1}{2}\sum_{j=1}^{2n}\frac{1}{\zeta_{k}-x_{j}}.$

The stationary relation (2.1) implies that

$\dot{\zeta}_{k}=\sum_{l\neq k}\frac{1}{\zeta_{k}-\zeta_{l}}=-\sum_{l\neq
k}\frac{1}{\zeta_{k}-\zeta_{l}}+\sum_{j=1}^{2n}\frac{1}{\zeta_{k}-x_{j}},$

which shows (2.16). ∎

##  4\. Examples

In this section we explicitly compute expressions for the solutions $U_{j}$ to
the null vector equations (2.6), in the cases $n=1$ and $n=2$. We find
explicit formulas for the poles $\zeta_{1},\ldots,\zeta_{n}$ in terms of the
critical points $\boldsymbol{x}$, which we then convert into formulas for the
$U_{j}$ that are purely in terms of the $\boldsymbol{x}$. This is in contrast
to (2.8) in Theorem 2.2 that represents the solutions in terms of the critical
points and the poles, but is only possible because we consider small $n$. For
larger $n$ this type of analysis will be much more complicated.

We summarize the results of our examples in Table 1. To keep the expressions
simple we present the results for specific values of the critical points, but
explicit formulas for general critical points are worked out in the text. For
Examples 4.1, 4.3, 4.4 we use the canonical rational element to carry out the
analysis, which we recall is the unique (up to post-composition by a
translation) element of $\operatorname{CRR}_{n+1}(\boldsymbol{x};\alpha)$ with
exactly $n$ finite poles and leading coefficient $1$ on the derivative. In
Example 4.2 we consider the $n=1$ case but for a rational function without a
finite pole. Our main results do not describe but this case but much of our
theory still holds. In general, for $R=P/Q$ with $\deg P=n+1$ and $\deg
Q=d\leq n+1$ much of the theory still holds, and in the concluding remarks we
briefly describe the differences between the cases $d<n$ and
$d\in\\{n,n+1\\}$, the primary one being that for $d<n$ there are several
curves that simultaneously go off to infinity. Example 4.4 is interesting
because $0$ is both a critical point and a double pole of the canonical
rational function, hence the canonical element is not generic. The example
shows that the analysis can be carried out regardless, and emphasizes that the
generic assumption is convenient rather than crucial. We say more on this in
the concluding remarks.

Table 1. Summary of our examples. The examples contain formulas for arbitrary critical points and poles. | Critical Points | Finite Poles | Rational Function | Real Locus in $\mathbb{\mathbb{H}}$
---|---|---|---|---
Example 4.1 | $-1,1$ | $0$ | $z+1/z$ | $x^{2}+y^{2}=1$
Example 4.2 | $-1,1$ | none | $z^{3}-3z$ | $y^{2}=3(x^{2}-1)$
Example 4.3 | $-3,0,1,2$ | $\pm\sqrt{7/3}$ | $(z^{3}-3)/(3z^{2}-7)$ | Figure 1 and (4.10)
Example 4.4 | $-3,0,1,2$ | $0,0$ | $(z^{3}+7z-3)/z^{2}$ | Figure 2 and (4.12)

###  4.1. The $n=1$ case

We consider the $n=1$ case with critical points $x,y\in\mathbb{R}$.

######  Example 4.1.

We describe a single curve connecting $x,y$, which turns out to be a semi-
circle connecting the two in $\mathbb{H}$. Of course this is already clear
from the hyperbolic geodesic property but we show the explicit calculations.
From the Loewner equation point of view the semi-circle is really two curves
that “meet in the middle”, one growing from $x$ and the other from $y$. We
assume there is a single finite pole $\zeta$, and according to the centroid
relation (2.2) we have $\zeta=\frac{1}{2}(x+y)$. Note this also agrees with
the stationary relation (2.1). The canonical rational function is

$R(z)=z-\frac{(\zeta-x)(\zeta-y)}{z-\zeta}.$

By Theorem 2.2 the solutions to the null vector equations are

$U_{1}=-6/(x-y),\quad U_{2}=-6/(y-x).$

The Loewner dynamics for $g_{t}$ and $x(t)=x_{t}$, $y(t)=y_{t}$ are

$\dot{g}_{t}=\frac{2\mu(t)}{g_{t}-x_{t}}+\frac{2\nu(t)}{g_{t}-y_{t}},\qquad\dot{x}_{t}=-\frac{6\mu(t)-2\nu(t)}{x_{t}-y_{t}},\quad\dot{y}_{t}=\frac{6\nu(t)-2\mu(t)}{x_{t}-y_{t}}.$

For concreteness, we set $\mu=\nu\equiv 1/4$ and $x_{0}=-1,\,y_{0}=1$. Then
$x_{t},y_{t}$, and $\zeta_{t}$ satisfy

$x_{t}=-\sqrt{1-t},\qquad y_{t}=\sqrt{1-t},\qquad\zeta_{t}\equiv
0,\qquad(0\leq t\leq 1).$

By Lemma 3.3 we obtain

$g_{t}(z)-\frac{t-1}{g_{t}(z)}=z+\frac{1}{z},\qquad\gamma_{1}(t)=-\sqrt{1-t}+i\sqrt{t},\qquad\gamma_{2}(t)=\sqrt{1-t}+i\sqrt{t},\qquad(0\leq
t\leq 1).$

Thus the curves lie on the unit circle.

######  Example 4.2.

We consider the $n=1$ case again with critical points $x,y$, but this time we
assume there is no finite pole. Thus the corresponding rational function is a
polynomial, and we will see that its real locus consists of two curves that
start from $x$ and $y$ and merge at infinity. According to Theorem 2.2 we set

$U_{1}=2/(x-y),\quad U_{2}=2/(y-x).$

For simplicity, set $\mu(t)=\nu(t)\equiv 1/4$ and assume $x\leq y$. Then the
Loewner dynamics are

$\dot{g}_{t}=\frac{1}{2}\Big{(}\frac{1}{g_{t}-x_{t}}+\frac{1}{g_{t}-y_{t}}\Big{)},\qquad\dot{x}_{t}=\frac{1}{x_{t}-y_{t}},\quad\dot{y}_{t}=-\frac{1}{x_{t}-y_{t}}.$

Solving the system of ODEs, we find the solutions $x_{t}$ and $y_{t}$ are

$x_{t}=\frac{x_{0}+y_{0}-\sqrt{4t+(x_{0}-y_{0})^{2}}}{2},\qquad
y_{t}=\frac{x_{0}+y_{0}+\sqrt{4t+(x_{0}-y_{0})^{2}}}{2}.$

Let us consider the merging case that $x\equiv x_{0}=0=y_{0}\equiv y$ first.
In this case, we have $x_{t}=-\sqrt{t}$, $y_{t}=\sqrt{t}$. Applying Lemma 3.3
we see that $g_{t}=g_{t}(z)$ satisfies $g_{t}^{3}-3tg_{t}=z^{3}$. Therefore
the SLE(0) paths $\gamma_{1},\gamma_{2}$ are described by

$\gamma_{1}(t)=\sqrt[3]{2}\,e^{\frac{2\pi}{3}i}\,\sqrt{t},\qquad\gamma_{2}(t)=\sqrt[3]{2}\,e^{\frac{\pi}{3}i}\sqrt{t}.$

The curves are a pair of symmetric half-lines starting from the origin that
make the angle $\pi/3$ with the real axis.

Next, we consider the case that curves start from different points, say
$x\equiv x_{0}=-1,y\equiv y_{0}=1$. Solving the system of ODEs, we find the
solutions $x_{t}$ and $y_{t}$ as $x_{t}=-\sqrt{1+t},y_{t}=\sqrt{1+t}$. In this
case, $z^{3}-3z$ is an integral of motion and
$g_{t}^{3}-3(t+1)g_{t}=z^{3}-3z$. Furthermore, the SLE(0) paths
$\gamma_{1},\gamma_{2}$ are described by

$\gamma_{1}(t)=-\frac{1}{2}s+i\frac{\sqrt{3}}{2}\sqrt{s^{2}-4},\qquad\gamma_{2}(t)=\frac{1}{2}s+i\frac{\sqrt{3}}{2}\sqrt{s^{2}-4},$

where $s$ is real and $s^{3}-3s=2(1+t)\sqrt{1+t}\,(s\geq 2)$. The curves lie
on the hyperbola $y^{2}=3(x^{2}-1)$.

###  4.2. The $n=2$ case

We consider the $n=2$ situation with $2$ finite poles. Thus we assume
$-\infty<x_{1}<x_{2}<x_{3}<x_{4}<\infty$ are the critical points. We will
describe the rational function and the real locus for both the rainbow pattern
($x_{1}$ connected to $x_{4}$ and $x_{2}$ connected to $x_{3}$) and the
neighbor pattern ($x_{1}$ connected to $x_{2}$ and $x_{3}$ connected to
$x_{4}$). As expected, the stationary relation

$\sum_{j=1}^{4}\frac{1}{\zeta_{1}-x_{j}}=\frac{2}{\zeta_{1}-\zeta_{2}},\quad\sum_{j=1}^{4}\frac{1}{\zeta_{2}-x_{j}}=-\frac{2}{\zeta_{1}-\zeta_{2}}$

has two pairs of solutions
$\\{\zeta_{1}^{+},\zeta_{2}^{+}\\},\\{\zeta_{1}^{-},\zeta_{2}^{-}\\}$ for the
finite poles:

(4.1)
$\displaystyle\zeta_{1}^{\pm}=\frac{1}{4}\Big{(}s_{1}-\sqrt{s_{1}^{2}-\frac{8}{3}s_{2}\pm\frac{8}{3}\sqrt{S}}\Big{)},\qquad\zeta_{2}^{\pm}=\frac{1}{4}\Big{(}s_{1}+\sqrt{s_{1}^{2}-\frac{8}{3}s_{2}\pm\frac{8}{3}\sqrt{S}}\Big{)},$

where

(4.2) $\displaystyle
S=\Big{(}\sum_{j<k}\frac{2}{(x_{j}-x_{k})^{2}}\Big{)}\Big{/}\Big{(}\sum_{j}\prod_{k\neq
j}\frac{1}{(x_{j}-x_{k})^{2}}\Big{)}=s_{2}^{2}-3s_{1}s_{3}+12s_{4}$

and $s_{k}$’s are the elementary symmetric functions in the four variables
$x_{1},x_{2},x_{3},x_{4}:$

$s_{1}=x_{1}+x_{2}+x_{3}+x_{4},\quad s_{2}=\\!\\!\sum_{1\leq j<k\leq
4}\\!\\!x_{j}x_{k},\quad s_{3}=\sum_{j<k<l}x_{j}x_{k}x_{l},\quad
s_{4}=x_{1}x_{2}x_{3}x_{4}.$

The plus case corresponds to the neighbor pattern while the minus case
corresponds to the rainbow pattern, as we will see shortly. Inserting
$\zeta_{1}^{\pm},\zeta_{2}^{\pm}$ into the system (2.8) to obtain
$U_{j}^{\pm}$, $j=1,2,3,4$, we first claim that the solutions $U_{j}^{\pm}$
can be entirely expressed in terms of $x_{j}$’s as

(4.3) $U_{j}^{\pm}=-\sum_{k\neq
j}\frac{2}{x_{j}-x_{k}}\mp\frac{4\sqrt{S}}{\prod_{k\neq j}(x_{j}-x_{k})}.$

To see this, let

$V_{j}=U_{j}+\sum_{k\neq j}\frac{2}{x_{j}-x_{k}}.$

Then the null vector equations (2.6) can be rewritten in terms of $V_{j}$ as

(4.4) $\frac{1}{2}V_{j}^{2}-2\sum_{k\neq
j}\frac{V_{j}+V_{k}}{x_{j}-x_{k}}-\sum_{j<k}\frac{8}{(x_{j}-x_{k})^{2}}=0.$

If $V_{j}$’s satisfy the above null vector equations, then
$(V_{1},\ldots,V_{4})$ lies on a sphere:

(4.5) $\sum_{j}V_{j}^{2}=32\sum_{j<k}\frac{1}{(x_{j}-x_{k})^{2}}.$

On the other hand, the conformal Ward identities (2.10) can be rewritten in
terms of $V_{j}$ as

(4.6) $\sum_{j}V_{j}=\sum_{j}x_{j}V_{j}=\sum_{j}x_{j}^{2}V_{j}=0.$

Due to the Vandermonde determinant formula, the intersection of these three
hyperplanes is the straight line passing through the origin with direction
vector

$\left(\frac{1}{\prod_{k\neq 1}(x_{1}-x_{k})},\frac{1}{\prod_{k\neq
2}(x_{2}-x_{k})},\frac{1}{\prod_{k\neq 3}(x_{3}-x_{k})},\frac{1}{\prod_{k\neq
4}(x_{4}-x_{k})}\right).$

This line intersects the sphere (4.5) at two points, so there are at most two
solutions $U_{j}$ to both the null vector equations (2.6) and the conformal
Ward identities (2.10). On the other hand, by Theorems 2.2 and 2.4 we already
know that (2.8) provides at least two solutions to the null vector equations
and conformal Ward identities. Thus we have exactly two solutions to (2.6) and
(2.10). It is straightforward to verify that for the proposed solutions (4.3)
the corresponding $V_{j}^{\pm}$ satisfy the version (4.4) of the null vector
equations and the conformal Ward identities (4.6), showing that the
$U_{j}^{\pm}$ are in fact equal to the solutions given by Theorem 2.2, up to a
determination of the sign which we leave as an exercise for the reader.

Now we consider the dynamics for $x_{j}$ driven by $U_{j}^{\pm}$. For
simplicity, we set $\nu_{j}\equiv 1/4$. Then using (4.3) for $U_{j}^{\pm}$ we
find the dynamics of $x_{j}$’s to be

(4.7) $\displaystyle\dot{x}_{j}=\mp\frac{\sqrt{S}}{\prod_{k\neq
j}(x_{j}-x_{k})}.$

Now recall from (4.2) that $S$ is expressible in terms of the elementary
symmetric functions $s_{1},s_{2},s_{3},s_{4}$ in the four variables
$x_{1},x_{2},x_{3},x_{4}$. From these formulas and (4.7) we obtain, by
symmetry,

$\displaystyle\dot{s}_{1}$
$\displaystyle=\mp\sqrt{S}\sum_{j}\frac{1}{\prod_{k\neq j}(x_{j}-x_{k})}=0,$
$\displaystyle\dot{s}_{2}$ $\displaystyle=\mp\sqrt{S}\sum_{j}\frac{\sum_{k\neq
j}x_{k}}{\prod_{k\neq j}(x_{j}-x_{k})}=0,$ $\displaystyle\dot{s}_{3}$
$\displaystyle=\mp\sqrt{S}\sum_{j}\frac{(\prod_{k\neq j}x_{k})\sum_{k\neq
j}1/x_{k}}{\prod_{k\neq j}(x_{j}-x_{k})}=0,$

and

$\dot{s}_{4}=\mp\sqrt{S}\sum_{j}\frac{\prod_{k\neq j}x_{k}}{\prod_{k\neq
j}(x_{j}-x_{k})}=\pm\sqrt{S}.$

We now describe $\dot{g}_{t}$ and $R^{\prime}(z;\boldsymbol{x}(t))$ in terms
of $s_{j}(t)$, where $R(z;\boldsymbol{x}(t))$ is a canonical element in
$\operatorname{CRR}_{n+1}(\boldsymbol{x}(t);\alpha)$:

(4.8)
$\dot{g}_{t}\Big{(}=\frac{1}{2}\sum\frac{1}{g_{t}-x_{j}(t)}\Big{)}=\frac{1}{2}\frac{4g_{t}^{3}-3s_{1}(t)+2s_{t}(t)g_{t}-s_{3}(t)}{g_{t}^{4}-s_{1}(t)g_{t}^{3}+s_{2}(t)g_{t}^{2}-s_{3}(t)g_{t}+s_{4}(t)}$

and

(4.9)
$R^{\prime}(z;\boldsymbol{x}(t))=\frac{\prod_{j}(z-x_{j}(t))}{(z-\zeta_{1}^{\pm}(t))^{2}(z-\zeta_{2}^{\pm}(t))^{2}}=\frac{z^{4}-s_{1}(t)z^{3}+s_{2}(t)z^{2}-s_{3}(t)z+s_{4}(t)}{\big{(}z^{2}-s_{1}(t)z/2+(s_{2}(t)\mp\sqrt{S(t)})/6)\big{)}^{2}}.$

We now consider the dynamics of $x_{j}$, $S$, and $s_{j}$ with the specific
initial data:

$x_{1}(0)=-3,\quad x_{2}(0)=0,\quad x_{3}(0)=1,\quad x_{4}(0)=2.$

Then we have $s_{1}\equiv 0$, $s_{2}\equiv-7$, $s_{3}\equiv-6$, and

$\dot{s}_{4}=\pm\sqrt{S}=\pm\sqrt{49+12s_{4}}.$

Solving this ODE, we find $s_{4}=3t^{2}\pm 7t$. We take up the plus case in
the next example and the minus case in Example 4.4.

######  Example 4.3.

We first consider the case $s_{4}=3t^{2}+7t$. By (4.8) and (4.9), $g_{t}$ is
described by the following ODE:

$\dot{g}_{t}=\frac{2g_{t}^{3}-7g_{t}+3}{g_{t}^{4}-7g_{t}^{2}+6g_{t}+3t^{2}+7t}$

and $R^{+}(z)=(z^{3}-3)/(3z^{2}-7)$ is its integral of motion:

$\frac{g_{t}^{3}-3tg_{t}-3}{3g_{t}^{2}-3t-7}=\frac{z^{3}-3}{3z^{2}-7}.$

The nontrivial solutions to $R^{+}(z)=R^{+}(u)$ are found as

$z=\frac{-9+7u\pm\sqrt{3}\sqrt{27+42u-49u^{2}-36u^{3}+28u^{4}}}{2(-7+3u^{2})}.$

Eliminating parameter $u$, the real locus of $R^{+}$ in $\mathbb{H}$ is given
by

(4.10) $\displaystyle
y=\sqrt{\frac{1}{6}(-7-6x^{2}+\sqrt{49+24x(-9+14x)})},\qquad(-3\leq x\leq
0,1\leq x\leq 2).$

See Figure 1 for its graph. We see that the graph is the neighbor pattern
$\\{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\}\\}$.

Figure 1. The real locus of $R^{+}$

######  Example 4.4.

We next consider the case $s_{4}=3t^{2}-7t$. By (4.8) and (4.9), $g_{t}$ is
described by the following ODE:

(4.11)
$\dot{g}_{t}=\frac{2g_{t}^{3}-7g_{t}+3}{g_{t}^{4}-7g_{t}^{2}+6g_{t}+3t^{2}-7t}$

and $R^{-}(z)=(z^{3}+7z-3)/z^{2}$ is its integral of motion:

$\frac{g_{t}^{3}+(7-3t)g_{t}-3}{g_{t}^{2}-t}=\frac{z^{3}+7z-3}{z^{2}}.$

The nontrivial solutions to $R^{-}(z)=R^{-}(u)$ are found as

$z=\frac{-3+7u\pm\sqrt{9-42u+49u^{2}-12u^{3}}}{2u^{2}}.$

Eliminating the parameter $u$, the real locus of $R^{-}$ in $\mathbb{H}$ is
given by

(4.12) $\displaystyle
y=\begin{cases}\sqrt{-x^{2}+\frac{1}{2}(7+\sqrt{49-24x}\,)},\qquad(-3\leq
x\leq 2),\\\ \sqrt{-x^{2}+\frac{1}{2}(7-\sqrt{49-24x}\,)},\qquad(0\leq x\leq
1).\end{cases}$

See Figure 2 for its graph. We see that the graph is the rainbow pattern
$\\{\\{x_{1},x_{4}\\},\\{x_{2},x_{3}\\}\\}$.

Figure 2. The real locus of $R^{-}$

Given the explicit formulas (4.1) $\zeta_{1}^{\pm}$ and $\zeta_{2}^{\pm}$ for
the poles we now analyze the functions
$Z^{\pm}=Z^{\pm}(x_{1},x_{2},x_{3},x_{4})$ of Theorem 2.3. Recall that
definition (2.9) of $Z^{\pm}$ tells us

(4.13)
$Z^{\pm}=\frac{(\zeta_{1}^{\pm}-\zeta_{2}^{\pm})^{8}\prod_{j<k}(x_{j}-x_{k})^{2}}{\prod_{j}(x_{j}-\zeta_{1}^{\pm})^{4}(x_{j}-\zeta_{2}^{\pm})^{4}}.$

We now use (4.1) to simplify $Z^{\pm}$ (up to a multiplicative constant) as

$\displaystyle Z^{+}$
$\displaystyle=\frac{(x_{3}-x_{1})^{2}(x_{4}-x_{1})^{2}(x_{3}-x_{2})^{2}(x_{4}-x_{2})^{2}}{(x_{2}-x_{1})^{6}(x_{4}-x_{3})^{6}}\Bigg{(}\frac{H^{+}_{\\{\\{1,2\\},\\{3,4\\}\\}}}{H^{-}_{\\{\\{1,3\\},\\{2,4\\}\\}}H^{-}_{\\{\\{1,4\\},\\{2,3\\}\\}}}\Bigg{)}^{4},$
$\displaystyle Z^{-}$
$\displaystyle=\frac{(x_{2}-x_{1})^{2}(x_{3}-x_{1})^{2}(x_{4}-x_{2})^{2}(x_{4}-x_{3})^{2}}{(x_{4}-x_{1})^{6}(x_{3}-x_{2})^{6}}\Bigg{(}\frac{H^{-}_{\\{\\{1,4\\},\\{2,3\\}\\}}}{H^{+}_{\\{\\{1,3\\},\\{2,4\\}\\}}H^{+}_{\\{\\{1,2\\},\\{3,4\\}\\}}}\Bigg{)}^{4},$

where
$H_{\\{\\{j,k\\},\\{l,m\\}\\}}^{\pm}=H_{\\{\\{j,k\\},\\{l,m\\}\\}}^{\pm}(x_{1},x_{2},x_{3},x_{4})$
is a homogeneous function of degree 2 given by

$H_{\\{\\{j,k\\},\\{l,m\\}\\}}^{\pm}=\pm
2\sqrt{S}-(x_{j}+x_{k})(x_{l}+x_{m})+2(x_{j}x_{k}+x_{l}x_{m}).$

For $x_{1}<x_{2}<x_{3}<x_{4}$, $Z^{\pm}$ can be rewritten as

$Z^{+}=\frac{(1-z)^{2}}{(x_{1}-x_{2})^{6}(x_{3}-x_{4})^{6}}F_{0}(1-z),\qquad
Z^{-}=\frac{z^{2}}{(x_{1}-x_{4})^{6}(x_{2}-x_{3})^{6}}F_{0}(z),$

where $z$ is a cross-ratio

$z=\frac{(x_{2}-x_{1})(x_{4}-x_{3})}{(x_{4}-x_{2})(x_{3}-x_{1})}$

and $F_{0}$ is given by

$F_{0}(z)=\frac{(1-z)^{8}}{\big{(}(z+1)(z-1/2)(z-2)+(1-z+z^{2})\sqrt{1-z+z^{2}}\big{)}^{4}}$

up to a multiplicative constant.

Now we compare $Z^{\pm}$ with their analogues in the $\kappa>0$ (but small)
case. In this case the analogues of $Z^{\pm}$ are the so-called pure partition
functions, which we denote by $Z^{\pm}_{\kappa}$. When the logarithmic
derivative of these partition functions is used as an interaction term in the
driving functions of a particular stochastic Loewner system, it is known that
it produces multiple SLE$(\kappa)$ curves that connect
$x_{1},x_{2},x_{3},x_{4}$ almost surely according to the neighbor pattern in
the $Z^{+}_{\kappa}$ case and the rainbow pattern in the $Z^{-}_{\kappa}$
case. Explicit expressions for $Z_{\kappa}^{\pm}$ are known:

$\displaystyle Z_{\kappa}^{+}$
$\displaystyle=(x_{2}-x_{1})^{-2h}(x_{4}-x_{3})^{-2h}(1-z)^{2/\kappa}F(1-z)/F(1)$
$\displaystyle Z_{\kappa}^{-}$
$\displaystyle=(x_{4}-x_{1})^{-2h}(x_{3}-x_{2})^{-2h}z^{2/\kappa}F(z)/F(1),$

where $h=(6-\kappa)/(2\kappa)$ and $F$ is a hypergeometric function:

$F(z)=\,_{2}F_{1}\Big{(}\frac{4}{\kappa},1-\frac{4}{\kappa},\frac{8}{\kappa};z\Big{)},$

see e.g., [PW19, KL07, Law09b, Law09a]. Peltola and Wang [PW20, Corollary
5.12] showed that $-\frac{1}{12}\kappa\log Z_{\kappa,\alpha}$ converges as
$\kappa\to 0$ to what they call the minimal potential, the minimum value of a
certain functional on ensembles of curves. From the relation between the
minimal potential and $U_{j}$’s (see [PW20, (1.8)-(1.10)]), we have

$\lim_{\kappa\to 0}F(z)^{\kappa}=F_{0}(z),\qquad\lim_{\kappa\to
0}(Z_{\kappa}^{\pm})^{\kappa}=Z^{\pm}$

up to a common multiplicative constant. This can be verified directly using
the expressions above together with properties of hypergeometric functions.

##  5\. Concluding Remarks

The approach of this paper was informed by our work in the forthcoming [AKM],
in which we make a detailed study of the multiple SLE$(\kappa)$ process in the
framework of the conformal field theory of [KM13]. In [AKM] the idea is to use
the Gaussian free field (modified with appropriate Dirichlet boundary
conditions) and the calculus induced on Gaussian random fields by Wick’s
formula and the so-called operator product expansion to represent algebras of
correlation functions. These correlation functions describe statistics of the
multiple SLE$(\kappa)$ curves and one of the main results of [AKM] is that
they are martingale observables under the random Loewner evolution that
generates the curves. As mentioned earlier, the integral of motion in Theorem
2.5 is nothing more than the $\kappa\to 0$ limit of the bosonic observable,
i.e. the free field itself (with appropriate boundary conditions) evaluated at
an interior point $z$. It is through this approach that we found it. We point
out that the idea of studying integrals of motion for deterministic systems
via limits of field theories goes back to at least Feynman [Pol19], who used
it to study properties of the Calogero-Moser system.

The idea of studying the poles of the rational function is essentially the
$\kappa\to 0$ limit of the method of screening, which also goes by the name of
the Coulomb gas or Dotsenko-Fateev integrals. Under this method auxiliary
“screening charges” are introduced into the Coulomb gas but are then
integrated out along appropriate contours, which produces solutions
$\mathcal{Z}=\mathcal{Z}(x_{1},\ldots,x_{2n})$ to the positive $\kappa$ system
of BPZ equations (2.7). One can formally apply Laplace’s method/the method of
stationary phase to the integrals in these screened solutions, and under this
method the screening variables become the poles of the rational function. This
use of stationary phase is another reason why we refer to (2.1) as a
stationary relation. One can also view the functions $Z$ of Theorem 2.3 as the
appropriate $\kappa\to 0$ limit of the screened solutions $\mathcal{Z}$ of the
BPZ equations.

The statements of our results are simpler in the case that
$R\in\operatorname{CRR}_{n+1}(\boldsymbol{x})$ is generic, but as we pointed
out it is not strictly required. By Lemmas 3.1 and 3.2 a non-generic function
has double poles at the point where it overlaps with a critical point, and
therefore the stationary relation (2.1) holds in the sense that both sides are
infinite (the centroid relation (2.2) still holds). Even though two poles and
a critical point start at the same point in the non-generic case they
immediately split into three distinct points, and then remain separated for
all but finitely many $t>0$. This is the case in Example 4.4. The integral of
motion $N_{t}(z)$ in Theorem 2.5 is preserved, and $R\circ g_{t}^{-1}$ remains
a real rational function as in Theorem 2.6.

We can also modify our arguments to deal with $\deg Q=d\not\in\\{n,n+1\\}$.
Indeed, the starting point for most of our arguments is the observation that
the derivative of a real rational function with $2n$ distinct and real
critical points has the factorization (3.1). In the context of this paper we
used $d\in\\{n,n+1\\}$ throughout, since that is always the case for multiple
SLE$(0)$, but most of the arguments go through unchanged if $0\leq d<n$.
Assuming that a function $R^{\prime}$ factors as

$R^{\prime}(z)=\frac{\prod_{j=1}^{2n}(z-x_{j})}{\prod_{k=1}^{d}(z-\zeta_{k})^{2}}$

for real and distinct $x_{j}$ and all $\zeta_{k}$ appearing in complex
conjugate pairs, the arguments can be modified to show that

  * •

$R^{\prime}$ has a rational primitive $\iff$ the stationary relation (2.1)
holds (for $k=1,\ldots,d$),

  * •

the centroid relation (2.2) holds $\iff d\in\\{n,n+1\\}$,

  * •

the interaction terms $U_{j}$ of (2.8) are still well-defined, satisfy the
null vector equations, and are still the logarithmic derivatives of the
function $Z$ introduced in (2.9), but only satisfy the conformal Ward
identities if $d\in\\{n,n+1\\}$.

Based on the first point one simply assumes an $R^{\prime}$ that factors as
above and satisfies the stationary relations (2.1). Under these assumptions
Theorems 2.5 and 2.6 still hold, as do Corollaries 2.8 and 2.9. The main
difference is that for $d<n$ the real locus $\Gamma(R)$ contains multiple
curves going off to infinity and meeting there, even for $x_{j}$ all distinct.
This situation will be considered in a forthcoming paper, which also considers
the degenerate cases in which the critical points occur with multiplicities.
These correspond to “fusion rule” type degenerations of the multiple
SLE$(0;\boldsymbol{x};\alpha)$ curves.

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