# Diversities and Conformities111Research funded in part by NSERC.

Andrew Poelstra222Simon Fraser University, Burnaby, British Columbia, V5A 1S6,
Canada<EMAIL_ADDRESS>
###### Abstract

Diversities have recently been developed as multiway metrics admitting clear
and useful notions of hyperconvexity and tight span. In this note we consider
the analytic properties of diversities, in particular the generalizations of
uniform continuity, uniform convergence, Cauchy sequences and completeness to
diversities. We develop conformities, a diversity analogue of uniform spaces,
which abstract these concepts in the metric case. We show that much of the
theory of uniform spaces admits a natural analogue in this new structure; for
example, conformities can be defined either axiomatically or in terms of
uniformly continuous pseudodiversities.

Just as diversities can be restricted to metrics, conformities can be
restricted to uniformities. We find that these two notions of restriction,
which are functors in the appropriate categories, are related by a natural
transformation.

##  1 Introduction

The theory of metric spaces is well-understood and forms the basis of much of
modern analysis. In 1956, Aronszajn and Panitchpakdi developed the notion of
_hyperconvex_ metric spaces [2] in order to apply the Hahn-Banach theorem in a
more general setting. In fact, every metric space can be embedded
isometrically in a minimal hyperconvex space, as discovered by J. R. Isbell
[9] (as the “hyperconvex hull”) and later by A. W. M. Dress [5] (as the
“metric tight span”).

These minimal hyperconvex spaces, or _tight spans_ , proved to be powerful
tools for the analysis of finite metric spaces. The theory of tight spans, or
_T-theory_ , is overviewed in [6]. Its history, as well as applications to
phylogeny, are given in [4].

In light of these applications of T-theory, D. Bryant and P. Tupper developed
the theory of _diversities_ alongside an associated tight span theory in [4].
Diversities are multiway metrics mapping finite subsets of a ground space $X$
to the nonnegative reals. The axioms were chosen based on their specific
applications to phylogeny (where they had already appeared in special cases)
and their ability to admit a tight span theory. This diversity tight span
theory contains the metric tight span theory as a special case (using so-
called diameter diversities), but also allows new behavior which may be useful
in situations such as microbial phylogeny, where the idea of a historical
“phylogenetic tree” does not make sense. Several examples, along with
pictures, of this phenomenon are given in [4].

A classic paper by André Weil [11] developed the theory of _uniform spaces_ ,
which generalize metric spaces. Uniform spaces admit notions of uniform
continuity, uniform convergence and completeness which coincide with the
standard notions when metric spaces are considered as uniform spaces. This
theory has been described in Bourbaki’s _General Toplogy_ [3] as well as John
Kelley’s classic text [10]. The metric topology can be derived purely from
properties of the uniform space (via the so-called _uniform topology_), and in
this sense uniform spaces lie “between” metric spaces and topologies.

In this note, we develop _conformities_ , which generalize diversities in
analogy to Weil’s uniform space generalization of metrics. We will describe
uniform continuity, uniform convergence, Cauchy sequences and completeness for
diversities, and show that these can be characterized in terms of
conformities, giving an abstract framework in which to analyze the uniform
structure of diversities. This is motivated by the observation that while
diversities generalize metric spaces in a straightforward way (in fact they
restrict to metric spaces), they can exhibit very non-smooth behavior with
respect to these spaces (c.f. our Theorem 1). Therefore the existing tools for
metric spaces are insufficient to get a handle on the behavior of diversities.

##  2 Preliminaries

Throughout this paper, we will denote the _finite power set_ of a given set
$X$ by

$\mathcal{P}_{\textnormal{fin}}(X)=\\{A\subseteq X:|A|<\infty\\}.$

We begin with the Bryant-Tupper definition from [4]: a _diversity_ is a pair
$(X,\delta)$ where $X$ is some set and
$\delta:\mathcal{P}_{\textnormal{fin}}(X)\to\mathbb{R}$ is a function
satisfying

  1. D1.

If $A\in\mathcal{P}_{\textnormal{fin}}(X)$, $\delta(A)\geq 0$ and
$\delta(A)=0$ iff $|A|\leq 1$.

  2. D2.

If $A,B,C\in\mathcal{P}_{\textnormal{fin}}(X)$ with $C\neq\varnothing$, then

$\delta(A\cup B)\leq\delta(A\cup C)+\delta(C\cup B)$

If for some $A\in\mathcal{P}_{\textnormal{fin}}(X)$, $\delta(A)=0$ but $|A|>1$
we have the weaker notion of a _pseudodiversity_. It is shown in [4] from
these axioms that if $A\subseteq B$ then $\delta(A)\leq\delta(B)$, i.e.,
(pseudo)diversities are monotonic, and that the restriction of a diversity to
sets of size 2 forms a pseudometric $d(x,y)=\delta(\\{x,y\\})$. We call this
metric the _induced metric_ of the diversity.

For a metric space $(X,d)$, there are two important diversities on $X$ having
$d$ as an induced metric:

  * •

The _diameter diversity_ $(X,\textnormal{diam}_{d})$ defined by

$\textnormal{diam}_{d}(\\{x_{1},\ldots,x_{n}\\})=\sup_{i,j}d(x_{i},x_{j})$

When $X=\mathbb{R}^{n}$ and $d$ is the Euclidean metric we refer to this
diversity simply by diam.

  * •

The _Steiner tree diversity_ $(X,\delta)$ defined for each finite set
$A\subseteq X$ as the infimum of the size of the minimum Steiner tree on $A$.

(Recall that a Steiner tree on $A$ is a tree whose vertex set $V$ satisfies
$A\subseteq V\subseteq X$, with each edge $(x,y)$ weighted by $d(x,y)$. The
size of the tree is the sum of its edge weights.)

In fact, these examples are the extremes of diversity behavior relative to
their induced metrics, in the sense that for any diversity
$(X,\delta^{\prime})$ which induces a metric $d$, we have

$\textnormal{diam}_{d}\leq\delta^{\prime}\leq\delta$

where $\delta$ is the Steiner tree diversity on $(X,d)$. This can be shown by
a straightforward argument333Bryant and Tupper, upcoming..

To demonstrate the difference between the diameter and Steiner tree
diversities, consider the Euclidean metric $(\mathbb{R}^{3},d)$. The induced
metric of both the diameter and Steiner tree diversity is the Euclidean
metric. For any finite set $A$ contained in an $\epsilon$-ball,
$\textnormal{diam}(A)<\epsilon$. To contrast, in any $\epsilon$-ball we can
find finite sets $A$ for which $\delta(A)$ is arbitrarily large:

######  Theorem 1.

The Steiner tree diversity function $\delta$ on $\mathbb{R}^{3}$ is unbounded
on every open set of the Euclidean topology.

###### Proof.

Without loss of generality, we show the result for $\epsilon$-balls about 0.
For each $n\in\mathbb{N}$, define

$G_{n}=\left\\{\left(\frac{i}{n^{2}},\frac{j}{n^{2}},\frac{k}{n^{2}}\right):0\leq
i,j,k<n\right\\}$

which is a grid of points contained in the cube $[0,1/n]^{3}$. Since there are
$n^{3}$ points, a minimum spanning tree connecting the members of $G_{n}$ must
have $n^{3}-1$ edges, each of length $\geq 1/n^{2}$, since that is the least
distance between two points. Therefore the size of the minimum spanning tree
on $G_{n}$ is at least $(n^{3}-1)/n^{2}$, which can be taken as large as we
like by taking $n$ large enough. Since the minimal Steiner tree on $G_{n}$ has
size at least 0.615 times that of the minimal spanning tree [7], we have
$\delta(G_{n})\to\infty$ as $n\to\infty$ even though
$\textnormal{diam}(G_{n})\to 0$. ∎

A similar construction for the Steiner tree diversity on $\mathbb{R}^{2}$
gives sets of diversity $(0.615-\epsilon)$ for every $\epsilon>0$ in every
Euclidean ball. On $\mathbb{R}$ the Steiner tree diversity and diameter
diversity are identical. The dramatic difference between the many-point
behavior of these two diversities in dimension 2 or higher demonstrates that
diversities are not characterized by their induced metrics, even up to a
constant.

In this section and the next, we will define uniform convergence, uniform
continuity and completeness explicitly in terms of an underlying diversity; in
Section 4 we will describe _conformities_ , which abstract these properties
for diversities.This is in analogy to Weil’s uniformities, which abstract the
same concepts for metric spaces.

With this goal in mind, we start with the following definitions: let
$(X,\delta_{X})$ and $(Y,\delta_{Y})$ be diversities. Given $x\in X$, a
sequence $\\{x_{n}\\}\subset X$ _converges to $x$_, denoted $x_{n}\to x$, if

$\lim_{N\to\infty}\sup_{i_{1},i_{2},\ldots,i_{n}\geq
N}\delta_{X}(\\{x,x_{i_{1}},x_{i_{2}},\ldots,x_{i_{n}}\\})=0$

The sequence $\\{x_{n}\\}$ is a _Cauchy sequence_ if

$\lim_{N\to\infty}\sup_{i_{1},i_{2},\ldots,i_{n}\geq
N}\delta_{X}(\\{x_{i_{1}},x_{i_{2}},\ldots,x_{i_{n}}\\})=0$

It is an easy consequence of these definitions and the diversity axioms that
limits are unique and every convergent sequence is Cauchy. If every Cauchy
sequence is convergent, we call the diversity _complete_.

Finally, if $f:X\to Y$ is a function such that for every $\epsilon>0$, there
exists some $d>0$ such that $\delta_{X}(A)<d\implies\delta_{Y}(f(A))<\epsilon$
for every $A\in\mathcal{P}_{\textnormal{fin}}(X)$, we say $f$ is _uniformly
continuous_.

It is not hard to see that for diameter diversities, these definitions
coincide exactly with the standard ones on the induced metric.

For the second half of the paper, we will work extensively with filters, so we
state the definition here: given a ground set $X$, define a _filter_ as a
collection $\mathcal{F}$ of subsets of $X$ satisfying $A\cap B$ whenever $A$,
$B$ are in $\mathcal{F}$, and $B\in\mathcal{F}$ whenever $B\supseteq A$ and
$A\in\mathcal{F}$. A _filter base_ becomes a filter when all supersets of its
elements are added, in which case we say the base _generates_ the filter.

In this paper, we additionally require that $\varnothing\notin\mathcal{F}$.

##  3 Comparison with metrics

In this section, we contrast the uniform convergence of sequences with respect
to diversities and their induced metrics. In particular, we show that although
the Cauchy property for sequences is much stronger for diversities (we
demonstrate a sequence which is not Cauchy with respect to a diversity, even
though it is Cauchy with respect to the induced metric), completeness of a
diversity is equivalent to completeness of its induced metric. This tells us
that every diversity which induces a Euclidean metric (e.g., the Steiner tree
diversity on $\mathbb{R}^{n}$) is complete.

Since the set of Cauchy sequences in a diversity may be smaller than the set
of Cauchy sequences of its induced metric, this may provide a simpler way to
determine completeness of metric spaces.

At the end of the section, we construct the analogue of completion for
diversities.

###  3.1 Completeness in diversities and metric spaces

######  Theorem 2.

Let $(X,\delta)$ be a diversity, $d$ its induced metric. If $(X,d)$ is a
complete metric space, then $(X,\delta)$ is a complete diversity.

###### Proof.

Suppose that $(X,d)$ is complete. Let $\\{x_{n}\\}$ be a Cauchy sequence in
$(X,\delta)$. Then it is also Cauchy in $(X,d)$, and therefore converges to
some element $x$. We claim that $x_{n}\to x$ in $(X,\delta)$. To this end, let
$\epsilon>0$. Then there exists $N$ such that:

  * •

$d(x_{n},x)<\epsilon$ for all $n>N$ (since $x_{n}\to x$ in $(X,d)$)

  * •

$\delta(\\{x_{n_{1}},x_{n_{2}},\ldots,x_{n_{m}}\\})<\epsilon$ for all
$n_{i}>N$ (since $\\{x_{n}\\}$ is Cauchy in $(X,\delta)$).

Therefore, for all $n_{1},\ldots,n_{m}>N$,

$\displaystyle\delta(\\{x,x_{n_{1}},\ldots,x_{n_{m}}\\})$
$\displaystyle\leq\delta(\\{x,x_{n_{1}}\\})+\delta(\\{x_{n_{1}},\ldots,x_{n_{m}}\\})$
$\displaystyle=d(x,x_{n_{1}})+\delta(\\{x_{n_{1}},\ldots,x_{n_{m}}\\})<2\epsilon$

i.e., $x_{n}\to x$ in $(X,\delta)$. ∎

As mentioned, the set of Cauchy sequences in a diversity may be strictly
smaller than the set of Cauchy sequences in the induced metric. For example,
let $(X,\delta)$ be the Steiner tree diversity on $\mathbb{R}^{3}$, and
consider the sets $\\{G_{n}\\}_{n\in\mathbb{N}}$ from Theorem 1.

Order each set $G_{n}$ somehow and define the sequence $\\{x_{i}\\}$ by
concatenating them, i.e.,

$\\{x_{n}\\}=G_{1}G_{2}G_{3}\cdots$

which is Cauchy in the induced metric of $(X,\delta)$ (since eventually every
pair of points is confined to arbitrarily small cubes $[0,\epsilon]^{3}$).
However, it is not Cauchy in $(X,\delta)$, since we saw in the proof of
Theorem 1 that $\delta(G_{n})$ becomes arbitrarily large as $n\to\infty$. In
other words, every tail of $\\{x_{n}\\}$ has arbitrarily large finite sets, so
$\\{x_{n}\\}$ is not Cauchy.

In light of this example, it is interesting to know that every complete
diversity has a complete induced metric, which is proved with the following
Lemma:

######  Lemma 3.

Let $(X,\delta)$ be a diversity, $d$ its induced metric. Let $\\{x_{n}\\}$ be
Cauchy in $(X,d)$. Then it has a subsequence that is Cauchy in $(X,\delta)$.

###### Proof.

Define the subsequence $\\{x_{n_{i}}\\}$ by

$n_{i}=\min\\{n:d(x_{n},x_{m})<2^{-i}\text{ for all $m\geq n$}\\}$

Given $\epsilon>0$, choose $N$ such that $2^{1-N}<\epsilon$. Then for all
$i_{1}\leq i_{2}\leq\cdots\leq i_{m}$ greater than $N$,

$\displaystyle\delta(\\{x_{i_{1}},\ldots,x_{i_{m}}\\})$
$\displaystyle\leq\delta(\\{x_{i_{1}},x_{i_{2}}\\})+\cdots+\delta(\\{x_{i_{m-1}},x_{i_{m}}\\})$
$\displaystyle<1/2^{i_{1}}+\cdots+1/2^{i_{m}}$
$\displaystyle<\sum_{i=N}^{\infty}1/2^{i}$ $\displaystyle=2^{1-N}<\epsilon$

That is, $\\{x_{n_{i}}\\}$ is Cauchy in $(X,\delta)$. ∎

######  Theorem 4.

Let $(X,\delta)$ be a diversity, $d$ its induced metric. If $(X,\delta)$ is a
complete diversity, then $(X,d)$ is a complete metric space.

###### Proof.

Let $\\{x_{n}\\}$ be a Cauchy sequence in $(X,d)$. Then by Lemma 3 it has a
subsequence $\\{x_{i_{n}}\\}$ that is Cauchy in $(X,\delta)$, which converges
to some element $x$ since the diversity is complete. (It converges in both
$(X,\delta)$ and $(X,d)$.)

Then $x_{n}$ converges to $x$ in $(X,d)$, since for any $\epsilon$ we have
$d(x_{n},x)\leq d(x_{n},x_{i_{m}})+d(x_{i_{m}},x)<2\epsilon$ for $m,n$ large
enough. ∎

###  3.2 Completion

In light of the equivalence between metric completeness and diversity
completeness, it is perhaps not so surprising that every diversity can be
completed in a canonical way. To do so, we require two more definitions from
[4]: an _embedding_ $\pi:Y_{1}\to Y_{2}$ is an injective map between
diversities $(Y_{1},\delta_{1})$ and $(Y_{2},\delta_{2})$ such that
$\delta_{1}(A)=\delta_{2}(\pi(A))$ for all
$A\in\mathcal{P}_{\textnormal{fin}}(Y_{1})$. A _isomorphism_ is a surjective
embedding.

######  Theorem 5.

Every diversity $(X,\delta)$ can be embedded in a complete diversity.

###### Proof.

Let $\hat{X}$ be the set of all Cauchy sequences in $X$. Identify any two
sequences $\\{x_{i}\\},\\{y_{i}\\}$ which satisfy
$\lim_{n\to\infty}\delta(\\{x_{n},y_{n}\\})=0$ (so $\hat{X}$ is actually a set
of equivalence classes). Define the function $\hat{\delta}$ from
$\mathcal{P}_{\text{fin}}(\hat{X})\to\mathbb{R}$ by

$\hat{\delta}(\\{\\{x^{1}_{i}\\},\\{x^{2}_{i}\\},\ldots,\\{x^{n}_{i}\\}\\})=\lim_{N\to\infty}\sup_{i_{1},\ldots,i_{n}\geq
N}\delta(\\{x^{1}_{i_{i}},x^{2}_{i_{2}},\ldots,x^{n}_{i_{n}}\\})$

It can then be shown that $(\hat{X},\hat{\delta})$ is a complete diversity,
and that the map $x\mapsto\\{x,x,x,\ldots\\}$ from $(X,\delta)$ is an
embedding. The proof is an exercise in notation. ∎

This completion is dense in the sense that every member $x$ of $\hat{X}$ has a
sequence $\\{x_{i}\\}\subseteq X$ with $x_{i}\to x$ in $\hat{X}$. (Let
$\\{y_{i}\\}$ be a representative of $x$ and define
$x_{i}=\\{y_{1},y_{2},\ldots,y_{i},y_{i},y_{i},\ldots\\}$.) It also satisfies
a universal property analogous to that for metric completion:

######  Theorem 6.

Let $(X,\delta)$ be a diversity, $(\hat{X},\hat{\delta})$ its completion. Then
for any complete diversity $(Y,\gamma)$ and any uniformly continuous function
$f:X\to Y$, there is a unique uniformly continuous function
$\hat{f}:\hat{X}\to Y$ which extends $f$.

###### Proof.

Let $\\{x_{i}\\}$ be a representative sequence of some member of $\hat{X}$,
and define $\hat{f}(\\{x_{i}\\})=\lim_{i\to\infty}f(x_{i})$, which is defined
and independent of representative since $f$ is uniformly continuous and $Y$ is
complete. To show $\hat{f}$ is uniformly continuous, pick $\epsilon>0$ and
$d>0$ such that $\gamma(f(A))<\epsilon$ whenever $\delta(A)<d$ for all
$A\in\mathcal{P}_{\textnormal{fin}}(X)$. Then for all
$B=\\{\\{x_{i}^{1}\\},\\{x_{i}^{2}\\},\ldots,\\{x_{i}^{m}\\}\\}\in\mathcal{P}_{\textnormal{fin}}(\hat{X})$
with $\hat{\delta}(B)<d/2$, we have
$\gamma(\hat{f}(B))=\gamma(\\{\lim_{i\to\infty}f(x^{n}_{i})\\}_{n=1}^{m}))<\epsilon$
since for large enough $N$, $\delta(\\{f(x^{n}_{N})\\}_{n=1}^{m})<3d/4$.

To show uniqueness of $\hat{f}$, let $\hat{g}$ be another uniformly continuous
function extending $f$ to $\hat{X}$. For all $x\in\hat{X}$ we have
$\\{x_{i}\\}\subset X$ with $x_{i}\to x$ in $\hat{X}$, and by uniform
continuity $\hat{g}(x)=\lim_{i\to\infty}f(x_{i})=\hat{f}(x)$. ∎

This is a universal property in the sense that for every complete diversity
$\hat{X}^{\prime}$ extending $X$ and having the property, there is an
isomorphism $\tilde{j}:\hat{X}^{\prime}\to\hat{X}$. (Specifically, let
$\tilde{j}$ be the unique uniformly continuous extension of the identity map
$j:X\to\hat{X}$ to $\hat{X}^{\prime}$.)

##  4 Conformities

In this section we introduce a generalization of diversities analogous to
_uniformities_ , which generalize metric spaces. Uniformities lie between
metric spaces and topologies, in the sense that every metric space defines a
uniformity, and every uniformity defines a topology (which coincides with the
metric topology when the uniformity came from a metric). Uniformities
characterize uniform continuity, uniform convergence and Cauchy sequences,
which are not topological concepts.

The carry-over from the metric case is natural but nontrivial, since
diversities can behave differently on sets of different cardinality. Since
this construction is qualitatively different from metric uniformities, it
requires a different name. We asked ourselves “what would you call a
uniformity that came from a diversity?”, and the answer was clear: a
_conformity_.

Throughout this section, we will give the analogous definitions and results
for uniformities, using the standard treatment from Kelley [10]. We begin by
defining conformities and comparing them to uniformities; we show that just
like uniformities, conformities have a countable base if and only if they are
generated by some pseudodiversity.

We then briefly touch on the problem of completion for conformities.

Finally, we define power conformities: from a conformity defined on a set $X$,
we can construct a conformity on $\mathcal{P}_{\textnormal{fin}}(X)$ from
which pseudodiversities can be considered uniformly continuous functions. We
show that every conformity is generated by exactly the set of
pseudodiversities which are uniformly continuous from its power conformity to
$\mathbb{R}$. This gives an equivalent definition of conformity in terms of
pseudodiversities.

###  4.1 Conformities of diversities

Recall that for $(X,d)$ a metric space, $\\{x_{n}\\}$ a sequence in $X$, that
$\\{x_{n}\\}$ is Cauchy iff for each $\epsilon>0$ there is some $N$ such that
every pair of points $(x_{i},x_{j})$ with $i>N,j>N$ has
$d(x_{i},x_{j})<\epsilon$.

Similarly, let $f:X\to Y$ be a function between metric spaces $(X,d)$ and
$(Y,g)$. Then $f$ is uniformly continuous iff each $\epsilon>0$ has a
$\delta>0$ such that whenever pairs of points $(x,y)\in X\times X$ satisfy
$d(x,y)<\delta$, the pairs $(f(x),f(y))$ satisfy $g(f(x),f(y))<\epsilon$.

A similar characterization of uniform convergence of sequences of functions
can be given in terms of pairs of points. From these observations arises the
theory of uniformities, which is described in any standard text on analysis
(c.f. [3, 10]). We briefly describe the theory here. For any set $X$ define a
_uniformity_ on $X$ as a filter $\mathcal{U}$ on $X\times X$ satisfying

  1. U1.

$(x,x)\in U$ for every $x\in X$, $U\in\mathcal{U}$.

  2. U2.

If $U\in\mathcal{U}$, $(x,y)\in U$, then $(y,x)\in U$.

  3. U3.

For every $U\in\mathcal{U}$, there exists some $V\in\mathcal{U}$ with $V\circ
V\subseteq U$, where in general we define

$U\circ V:=\\{(x,z):(x,y)\in U,(y,z)\in V\text{ for some }y\in X\\}$

In particular, for any pseudometric space $(X,d)$ we can define the _metric
uniformity_ as the filter on $X\times X$ defined by

$U^{\epsilon}=\\{(x,y):d(x,y)<\epsilon\\}$

for each $\epsilon>0$. We see from this example that (U1) expresses the
requirement that $d(x,x)=0$ for all $x\in X$, (U2) expresses symmetry, and
(U3) expresses the triangle inequality.

Uniform structure can be defined entirely with respect to uniformities. For
example, given sets $X,Y$ and uniformities $\mathcal{U},\mathcal{V}$ on $X$
and $Y$ respectively, we can call a function $f:X\to Y$ _uniformly continuous_
if $f^{-1}(V)\in\mathcal{U}$ for every $V\in\mathcal{V}$. (Here $f$ acts on
members of $V$ componentwise.) A sequence $\\{x_{n}\\}\subset X$ is _Cauchy_
if for every $U\in\mathcal{U}$, there is some $N$ such that pairs of elements
$(x_{i},x_{j})$ of $\\{x_{n}\\}$ are in $U$ whenever $i,j>N$. It is not hard
to see that for metric uniformities, these definitions coincide with the
ordinary ones for metric spaces.

To abstract the uniform structure of diversities, uniformities are clearly
insufficient. For one thing, since diversities map finite sets rather than
pairs, we should seek a filter on $\mathcal{P}_{\textnormal{fin}}(X)$ rather
than $X\times X$. Then symmetry is no longer required, but now monotonicity
is. Finally, it is not meaningful to compose finite sets as in (U3), so we
will need a different way to express an analogue of the triangle inequality.

Putting all this together, we define a _conformity_ $\mathcal{C}$ on $X$ as a
filter on $\mathcal{P}_{\textnormal{fin}}(X)$ satisfying

  1. C1.

$\\{x\\}\in C$ for every $x\in X$, $C\in\mathcal{C}$.

  2. C2.

For every $C\in\mathcal{C}$, whenever $A\in C$ and $B\subseteq A$, we have
$B\in C$.

  3. C3.

For every $C\in\mathcal{C}$, there exists some $D\in\mathcal{C}$ with $D\circ
D\subseteq C$, where in general we define

$U\circ V:=\\{u\cup v:u\in U,v\in V\text{ and }u\cap v\neq\varnothing\\}$

Often the term _conformity_ is also used to refer to the pair
$(X,\mathcal{C})$.

An observation that will be necessary later (one which also holds for
uniformities) is that for any $D\in\mathcal{C}$, $(D\circ D)\circ
D=D\circ(D\circ D)$, so that $D\circ D\circ D$ is defined unambiguously. To
estimate the size of this, we also note that $D\circ D\circ D\subseteq(D\circ
D)\circ(D\circ D)$.

As in the metric case, there is a canonical way to generate a conformity from
a diversity; if $\delta$ is a pseudodiversity on $X$, we have the conformity
generated by the sets

$C^{\epsilon}=\\{A:\delta(A)\leq\epsilon\\}=\delta^{-1}[0,\epsilon]$

for each $\epsilon>0$. (This is equivalent to one using strict inequalities,
but typographically nicer.)

As in the metric case, uniform structure can be defined on conformities in a
way that generalizes that of diversities: let $(X,\mathcal{C})$ and
$(Y,\mathcal{D})$ be conformities. Then a function $f$ is _uniformly
continuous_ from $X$ to $Y$ if for all $D\in\mathcal{D}$, the set
$\\{f^{-1}(d):d\in D\\}$ is in $\mathcal{C}$. A sequence $\\{x_{n}\\}$ on $X$
is a _Cauchy sequence_ if for all $C\in\mathcal{C}$,
$\mathcal{P}_{\textnormal{fin}}(\\{x_{n}\\}_{n\geq N})\subseteq C$ for some
integer $N$. For conformities generated from diversities in the above way,
these definitions coincide with those given in the previous section.

More generally, given a collection of pseudodiversities
$\\{\delta_{\alpha}\\}_{\alpha\in\mathcal{A}}$, we can generate a conformity
from the sets
$\left\\{\delta^{-1}_{\alpha}[0,\epsilon]\right\\}_{\alpha\in\mathcal{A},\epsilon>0}$.
We therefore seek a characterization of conformities in terms of the
diversities which generate them. (In a later section, we will see that all
conformities can be described in this way, so that we can _define_
conformities in terms of such sets.) We begin by stating a result from Kelley
[10] along with a summary of his proof:

######  Theorem 7.

A uniformity is generated by a single pseudometric iff it has a countable
base.

The standard proof of this theorem goes as follows: it is obvious that any
uniformity generated by a pseudometric has a countable base. Conversely, if
there exists a countable base for a uniformity on $X$, there exists a
countable base $\\{U_{n}\\}_{n\in\mathbb{N}}$ for which the following argument
holds. Define the function $f(x,y)=2^{-n}$, where $n=\sup\\{i:(x,y)\in
U_{i}\\}$. This generates the uniformity but does not satisfy the triangle
inequality, so define

$d(x,y)=\inf\sum_{i=1}^{m-1}f(x_{i},x_{i+1})$

where the infimum is taken over all sequences $\\{x_{i}\\}_{i=1}^{m}$ with
$x_{1}=x$ and $x_{m}=y$. This clearly satisfies the triangle inequality, so it
just remains to be shown that $d$ generates the uniformity. This is done by
proving that $d(x,y)\leq f(x,y)\leq 2d(x,y)$, which follows from technical
constraints on $\\{U_{n}\\}$.

Given a conformity with a countable base $\\{C_{n}\\}$ on a set $X$, one might
try to translate this proof directly: define a function
$f(A):\mathcal{P}_{\textnormal{fin}}(X)\to\mathbb{R}$ by $f(A)=\sup\\{i:A\in
C_{i}\\}$, then somehow tweak $f$ to (a) satisfy the triangle inequality and
(b) generate the same conformity as $f$. However, it appears that any direct
analogue to the “infimum over all paths” strategy used in the metric case
(there are several) cannot satisfy both (a) and (b) simultaneously.

Nonetheless, the result is true, which is the content of the next theorem.

######  Lemma 8.

Let $(X,\mathcal{C})$ have a countable base. Then it has a countable base
$\\{C_{n}\\}$ satisfying $C_{0}=\mathcal{P}_{\textnormal{fin}}(X)$,
$C_{i}\circ C_{i}\circ C_{i}\subseteq C_{i-1}$ for $i>0$.

###### Proof.

Let $\\{V_{n}\\}$ be a countable base for $\mathcal{C}$. Define
$W_{0}=\mathcal{P}_{\textnormal{fin}}(X)$, $W_{n}=V_{n}\cap W_{n-1}$. Then
$\\{W_{n}\\}$ is a nested countable base. Finally, choose $\\{C_{n}\\}$ as
$C_{i}=W_{n_{i}}$, where $n_{i}$ are chosen inductively as $n_{0}=0$, then
$(W_{n_{i}}\circ W_{n_{i}})\circ(W_{n_{i}}\circ W_{n_{i}})\subseteq
W_{n_{i-1}}$.∎

######  Theorem 9.

Let $(X,\mathcal{C})$ be a conformity. There exists a pseudodiversity $\delta$
which generates $\mathcal{C}$ iff $\mathcal{C}$ has a countable base.

###### Proof.

If $\delta$ exists the sets $\\{C^{1/n}\\}_{n\in\mathbb{N}}$ are our base.

Conversely, let $\\{C_{n}\\}_{1}^{\infty}$ be a base for $\mathcal{C}$
satisfying $C_{0}=\mathcal{P}_{\textnormal{fin}}(X)$ and $C_{i}\circ
C_{i}\circ C_{i}\subseteq C_{i-1}$ for $i>0$. Define $\delta^{\prime}$ on
$\mathcal{P}_{\textnormal{fin}}(X)$ by

$\delta^{\prime}(A)=\left\\{\begin{array}[]{lr}0&A\in C_{n}\text{ for all
}n\\\ 2^{-k}&A\in C_{n}\text{ for }0\leq n\leq k\text{, but }A\notin
C_{k+1}\end{array}\right.$

Notice that for $k\geq 0$,

$\delta^{\prime-1}([0,2^{-k}])=C_{k}$ (1)

and that $\delta^{\prime}$ is monotonic: by (C2), if $A\subseteq B$, then
$A\in C_{n}$ whenever $B$ is.

Define a _chain_ as a sequence $\\{A_{i}\\}_{i=1}^{n}$ in
$\mathcal{P}_{\textnormal{fin}}(X)$ with $A_{i}\cap A_{i-1}\neq\varnothing$
for $i=2,\ldots,n$. Define a _cycle_ as a chain with $A_{1}\cap
A_{n}\neq\varnothing$. Write

$\bar{\delta}(A)=\inf_{\begin{subarray}{c}\text{chains covering
$A$}\end{subarray}}\leavevmode\nobreak\ \sum_{i=1}^{n}\delta^{\prime}(A_{i})$
$\delta(A)=\inf_{\begin{subarray}{c}\text{cycles covering
$A$}\end{subarray}}\leavevmode\nobreak\ \sum_{i=1}^{n}\delta^{\prime}(A_{i})$

Notice $\delta(\varnothing)=\bar{\delta}(\varnothing)=0$.

We claim that $\delta$ is our desired pseudodiversity, since the sets
$(\delta^{\prime})^{-1}[0,\epsilon]$ generate the conformity, and
$\delta\leq\delta^{\prime}\leq 4\delta$. We prove this in three stages.

  1. S1.

First of all, $\delta$ is a pseudodiversity. By (C1), for every $x\in X$,
$n\in\mathcal{N}$, $\\{x\\}\in C_{n}$ so that $\delta^{\prime}(\\{x\\})=0$.
Also $\\{x\\}$ is a cycle covering itself, so $\delta(\\{x\\})=0$.

The triangle equality also holds: let $\epsilon>0$,
$A,C\in\mathcal{P}_{\textnormal{fin}}(X)$ and
$B\in\mathcal{P}_{\textnormal{fin}}(X)$ be nonempty. Choose cycles
$\\{A_{i}\\}_{1}^{n}$ and $\\{B_{i}\\}_{1}^{m}$ covering $A\cup B$ and $B\cup
C$, respectively, and for which

$\sum_{i=1}^{n}\delta^{\prime}(A_{i})\leq\delta(A\cup B)+\epsilon\qquad\text{
and }\qquad\sum_{i=1}^{m}\delta^{\prime}(B_{i})\leq\delta(B\cup C)+\epsilon$

Then $\\{A_{i}\\}_{1}^{n}\cup\\{B_{i}\\}_{1}^{m}$ forms a cycle (after
reordering) covering $A\cup C$, so

$\delta(A\cup
C)\leq\sum_{i=1}^{n}\delta^{\prime}(A_{i})+\sum_{i=1}^{m}\delta^{\prime}(B_{i})\leq\delta(A\cup
B)+\delta(B\cup C)+2\epsilon$

  2. S2.

Next, we notice that

     * •

Every cycle is a chain, so $\delta\leq\bar{\delta}$.

     * •

If $\\{A_{1},\ldots,A_{n-1},A_{n}\\}$ is a chain, then
$\\{A_{1},\ldots,A_{n-1},A_{n},A_{n-1},\ldots,A_{1}\\}$ is a cycle — and the
sum of $\delta^{\prime}$ over this cycle is less than twice the sum of
$\delta^{\prime}$ over the original chain. We conclude that

$\delta\leq\bar{\delta}\leq 2\delta$ (2)

  3. S3.

Finally, we claim that $\bar{\delta}\leq\delta^{\prime}\leq 2\bar{\delta}$.
This combined with (2) will give the main result.

Trivially, $\bar{\delta}\leq\delta^{\prime}$. For the other inequality, choose
$A\in\mathcal{P}_{\textnormal{fin}}(X)$. Our strategy is to induct on the
greatest integer $N$ such that $\bar{\delta}(A)<2^{-N}$.

The case $N=0$ is easy, because then $\delta^{\prime}\leq 1\leq
2\bar{\delta}$. (This also covers the case $\bar{\delta}(A)=1$, which is not
covered by the induction.) When $N>0$, we can choose positive $\epsilon$ less
than $\left(2^{-N}-\bar{\delta}(A)\right)$, and a chain $\\{A_{i}\\}_{1}^{n}$
with

$\sum_{i=1}^{n}\delta^{\prime}(A_{i})<\bar{\delta}(A)+\epsilon<2^{-N}$ (3)

If $n=1$, we have
$\delta^{\prime}(A)\leq\delta^{\prime}(A_{1})<2^{-N}<2\bar{\delta}(A)$.
Otherwise, there is $k<n$ such that

$\sum_{i=1}^{k-1}\delta^{\prime}(A_{i})\leq\frac{\bar{\delta}(A)}{2}\text{ and
}\sum_{i=k+1}^{n}\delta^{\prime}(A_{i})\leq\frac{\bar{\delta}(A)}{2}$ (4)

Since $\\{A_{i}\\}_{1}^{k-1}$ and $\\{A_{i}\\}_{k+1}^{n}$ are chains whose sum
under $\delta^{\prime}$ is less than half that of $\\{A_{i}\\}_{1}^{n}$, the
inductive hypothesis applies to them and we may write

$\displaystyle\delta^{\prime}(A_{1}\cup\cdots\cup A_{k-1})$ $\displaystyle\leq
2\bar{\delta}(A_{1}\cup\cdots\cup A_{k-1})$ inductive hypothesis
$\displaystyle\leq 2\sum_{i=1}^{k-1}\delta^{\prime}(A_{i})$
$\displaystyle\text{definition of }\bar{\delta}$
$\displaystyle\leq\bar{\delta}(A)$ $\displaystyle\text{by
}\eqref{howiwishyouwerehere}$ $\displaystyle<2^{-N}$

Similarly $\delta^{\prime}(A_{k+1}\cup\cdots\cup A_{n})<2^{-N}$, and
$\delta^{\prime}(A_{k})<2^{-N}$ by (3). So

$(A_{1}\cup\cdots\cup A_{k-1})\in C_{N+1}\text{ and }A_{k}\in C_{N+1}\text{
and }(A_{k+1}\cup\cdots\cup A_{n})\in C_{N+1}$

Our double-composition hypothesis gives

$(A_{1}\cup\cdots\cup A_{k-1})\cup A_{k}\cup(A_{k+1}\cup\cdots\cup A_{n})\in
C_{N}$

And by monotonicity of $\delta^{\prime}$,

$\delta^{\prime}(A)\leq\delta^{\prime}(A_{1}\cup\cdots\cup A_{n})\leq
2^{-N}\leq 2\bar{\delta}(A)$

∎

This characterizes the conformities generated by single pseudodiversities.
Later we will describe _every_ conformity in terms of the pseudodiversities
that generate them.

###  4.2 Induced uniformities and completeness

Given a conformity $\mathcal{C}$, we define its _induced uniformity_ as the
uniformity generated by the sets

$U_{C}=\\{(x,y):\\{x,y\\}\in C\\}$

for every $C\in\mathcal{C}$. It is straightforward to show that this is a
uniformity; since every singleton $\\{x\\}$ is in every $C\in\mathcal{C}$, we
have every pair $(x,x)$ in every generator of the induced uniformity, proving
(U1). Since $\\{x,y\\}=\\{y,x\\}$ we have (U2). Finally, (U3) follows from the
observation that whenever $\\{x,y\\}\in C\in\mathcal{C}$ and $\\{y,z\\}\in
D\in\mathcal{C}$, the set $D\circ D\in\mathcal{C}$ contains $\\{x,y,z\\}$.
Then $\\{x,z\\}\in D\circ C$ by (C2). In other words, if $C\circ D\subseteq E$
in the conformity, then $U_{C}\circ U_{D}\subseteq U_{E}$ in the induced
uniformity. Thus (U3) is implied by (C3).

######  Theorem 10.

Let $X$ be a set, $\\{\delta_{A}\\}_{A\in\mathcal{A}}$ a family of diversities
which generate a conformity $\mathcal{C}$. For each $\delta_{A}$, write
$d_{A}$ for its induced metric. Then the uniformity generated by the metrics
$\\{d_{A}\\}_{A\in\mathcal{A}}$ is exactly the induced uniformity of
$\mathcal{C}$.

###### Proof.

Denote by $\mathcal{U}_{d}$ the uniformity generated by
$\\{d_{A}\\}_{A\in\mathcal{A}}$, and by $\mathcal{U}_{c}$ the uniformity
induced by $\mathcal{C}$. A base for $\mathcal{C}$ is

$C_{\epsilon,A}=\\{F:\delta_{A}(F)<\epsilon\\}$

where $\epsilon$ ranges over $\mathbb{R}^{+}$ and $A$ ranges over
$\mathcal{A}$. Then a base for $\mathcal{C}$ is

$U_{C_{\epsilon,A}}=\\{(x,y):\delta_{A}(\\{x,y\\})<\epsilon\\}=\\{(x,y):d_{A}(x,y)<\epsilon\\}$

For the same $\epsilon,A$. But this is just the canonical base for
$\mathcal{U}_{d}$! ∎

######  Corollary 11.

Let $(X,\mathcal{C})$ be a conformity. Then $\mathcal{C}$ has a countable base
iff its induced uniformity does.

###### Proof.

By Theorem 9 $\mathcal{C}$ has a countable base iff it is generated by a
single pseudodiversity; by Theorem 10 this occurs iff the induced uniformity
is generated by a single pseudometric. A standard result [3, 10] gives that
uniformities with countable bases are exactly those generated by single
pseudometrics. ∎

Next, we give some standard definitions. For a uniform space
$(X,\mathcal{U})$, the _uniform topology_ of $\mathcal{U}$ on $X$ is the
smallest topology containing the sets

$N(x,U)=\\{y:(x,y)\in U\\}$

for all $x\in X$, $U\in\mathcal{U}$. Notice that if $\mathcal{U}$ is generated
by a pseudometric, this coincides with the pseudometric topology.

With the same space $(X,\mathcal{U})$, we call a filter $\mathcal{F}$ on $X$
_Cauchy_ if for every $U\in\mathcal{U}$, there is some $F\in\mathcal{F}$ with
$F\times F\subseteq U$. We say that $\mathcal{F}$ _converges_ to some $x\in X$
if every neighborhood of $x$ (in the uniform topology) is in $\mathcal{F}$. We
then call a uniformity _complete_ if every Cauchy filter converges. It can be
shown that a metric space is complete iff its generated uniformity is, and
that every uniformity can be embedded minimally (i.e., satisfying a universal
property with respect to uniformly continuous maps) in a complete uniformity
[3, 10].

The analogous definitions for conformities are as follows.

Let $F$ be a filter on $X$. If for all $C\in\mathcal{C}$, there exists $f\in
F$ with $\mathcal{P}_{\textnormal{fin}}(f)\subseteq C$, then $F$ is a _Cauchy
filter_. If $x\in X$ and for all $C\in\mathcal{C}$ there exist $f\in F$ with
$\mathcal{P}_{\textnormal{fin}}(f)\subseteq\\{A:A\cup\\{x\\}\in C\\}$, then
$F$ _converges to_ $x$. Finally, if every Cauchy filter converges to some
point in $X$, we say $\mathcal{C}$ is _complete_.

######  Theorem 12.

A pseudodiversity $(X,\delta)$ is complete iff its conformity $\mathcal{C}$
is.

###### Proof.

Suppose $(X,\delta)$ is complete and let $F$ be a Cauchy filter on $X$. Then
for every $\epsilon>0$ there is some $f^{\epsilon}\in F$ so that
$\mathcal{P}_{\textnormal{fin}}(f^{\epsilon})\subseteq\\{A:\delta(A)<\epsilon\\}$.
Take some sequence $\epsilon_{n}\to 0$, and define the sets $g^{n}\subseteq X$
by $g^{1}=f^{\epsilon_{1}}$, $g^{n}=f^{\epsilon_{n}}\cap g^{\epsilon_{n-1}}$
for $n>1$.

Choose $x^{n}\in g^{n}$ for each $n$ to form a Cauchy sequence $\\{x_{n}\\}$,
with some limit $x$. For any $\epsilon>0$, find an integer $N$ so that
$\epsilon_{n}<\epsilon$ and $\delta(\\{x_{n},x\\})<\epsilon$ for all $n\geq
N$. Then if $a\in\mathcal{P}_{\textnormal{fin}}(f^{\epsilon_{n}})$, so is
$a\cup\\{x_{n}\\}$, so that
$\delta(a\cup\\{x\\})\leq\delta(a\cup\\{x_{n}\\})+\delta(\\{x_{n},x\\})<2\epsilon$.
We conclude that $F$ converges to $x$.

Conversely, suppose that every Cauchy filter converges in $\mathcal{C}$, and
let $\\{x_{n}\\}$ be a Cauchy sequence in $(X,\delta)$. Choose the sets
$F_{N}=\\{x_{n}\\}_{N}^{\infty}$. These sets generate a Cauchy filter with
some limit $x$. It is clear that $x_{n}\to x$. ∎

For any conformity $(X,\mathcal{C})$ generated by a diversity, the conformity
is complete iff its diversity is iff the diversity’s uniformity is [3, 10];
thus completeness of the conformity is equivalent to completeness of its
induced uniformity. In fact, this is true in general, as the next theorem
shows.

######  Theorem 13.

Let $(X,\mathcal{C})$ be a conformity with complete induced uniformity
$\mathcal{U}$. Then $\mathcal{C}$ is complete.

###### Proof.

Suppose that $\mathcal{U}$ is complete, and let $\mathcal{F}$ be a Cauchy
filter with respect to $\mathcal{C}$. Then $\mathcal{F}$ is also Cauchy with
respect to $\mathcal{U}$, since for all $C\in\mathcal{C}$, we have
$\\{\\{x,y\\}:x,y\in F\\}\subseteq\mathcal{P}_{\textnormal{fin}}(F)\subseteq
C$ for some $F\in\mathcal{F}$; then $F\times F\subseteq U_{C}$. Thus
$\mathcal{F}$ converges in $\mathcal{U}$ to some element $x$, and we claim
that it also converges to $x$ in $\mathcal{C}$. To this end, fix
$C\in\mathcal{C}$. Choose $D\in\mathcal{C}$ so that $D\circ D\subseteq C$. and
$F\in\mathcal{F}$ so that (a) $y\in F$ whenever $(x,y)\in U_{D}$ and (b)
$\mathcal{P}_{\textnormal{fin}}(F)\subseteq D$. Then for all
$A\in\mathcal{P}_{\textnormal{fin}}(F)$, $A\cup\\{x\\}\in C$. (If
$A=\varnothing$, $A\cup\\{x\\}\in C$ trivially. Otherwise pick $y\in A$, and
we will have $A\in D$ and $\\{x,y\\}\in D$, so that
$A\cup\\{x,y\\}=A\cup\\{x\\}\in C$.)∎

We end this section with two open questions:

  1. 1.

Does the converse to Theorem 13 hold; that is, if a conformity
$(X,\mathcal{C})$ is complete, must its induced uniformity be?

  2. 2.

We saw in Section 3.2 that for any diversity $(X,\delta)$ it is possible to
embed $X$ in a complete diversity which was universal, meaning that any
uniformly continuous map from $X$ to a complete diversity factored through the
embedding. It is shown in [10] that every uniformity can be embedded in a
complete uniformity. This embedding is also universal.

Is there a notion of universal completion for conformities?

###  4.3 Diversities of conformities

Not every conformity has a countable base. For example, let $X$ be the space
of functions $f:[0,1]\to[0,1]$, and consider the “pointwise convergence”
conformity generated by the sets

$C^{x}_{\epsilon}=\\{\\{f_{1},\ldots,f_{n}\\}:\textnormal{diam}(\\{f_{1}(x),\ldots,f_{n}(x)\\})<\epsilon\\}$

for every $\epsilon>0$, $x\in[0,1]$. This conformity has no countable base by
Corollary 11, since its induced uniformity does not have a countable base [8].
Thus by Theorem 9 it is not generated by any pseudodiversity.

In this section we will show that every conformity is generated by the
collection of pseudodiversities which are uniformly continuous with respect to
it, in an appropriate sense. In the case of uniformities this is done by
constructing a so-called product uniformity; given a uniformity on a set $X$,
the product uniformity is constructed on $X\times X$. Then a given
pseudometric $d$ may or may not be uniformly continuous from the product
uniformity to the Euclidean uniformity on $\mathbb{R}$. It can be proven [3,
10] that a uniformity $\mathcal{U}$ is exactly the uniformity generated by all
pseudometrics which are uniformly continuous from its product uniformity.

Since pseudodiversities are functions on finite sets rather than pairs, given
a conformity on a set $X$ we seek a conformity on
$\mathcal{P}_{\textnormal{fin}}(X)$ from which to judge uniform continuity of
pseudodiversities.

In fact, such a conformity exists for which we can prove the same result:
given a conformity $(X,\mathcal{C})$, define the _power conformity_
$\mathcal{C}^{P}$ as the conformity on $\mathcal{P}_{\textnormal{fin}}(X)$
generated by the sets

$C_{u}=\left\\{\\{A_{1},\ldots,A_{n}\\}:n\leq 1\text{ or
}\bigcup_{i=1}^{n}A_{i}\in u\right\\}$ (5)

where $u$ ranges over all members of $\mathcal{C}$.

######  Lemma 14.

A power conformity is a conformity.

###### Proof.

First, the $C_{u}$’s form a filter base since $C_{u}\cap C_{v}=C_{u\cap
v}\in\mathcal{C}^{P}$ for any $C_{u}$, $C_{v}\in\mathcal{C}^{P}$. For all
$A\in\mathcal{P}_{\textnormal{fin}}(X)$, $\\{A\\}$ is in every $C_{u}$ by
definition. It is immediate that whenever $\\{A_{i}\\}$ is in $C_{u}$, so is
every subset of $\\{A_{i}\\}$.

Finally, every $C_{u}$ has a $C_{v}$ with $C_{v}\circ C_{v}\subseteq C_{u}$:
choose $v$ with $v\circ v\subseteq u$ in $\mathcal{C}$. If
$\\{A_{i}\\}_{i=1}^{n}$, $\\{B_{i}\\}_{i=1}^{m}$ are in $C_{v}$ with some
$A_{i}$ equal to some $B_{j}$, then (a) $m\leq 1$ and $n\leq 1$, so their
union has at most one element and therefore must lie in $C_{u}$, (b) exactly
one of $m\leq 1$ or $n\leq 1$, in which case one of the sets is a subset of
the other, so their union lies in $v$ (and therefore $u$), or (c) $m>1$ and
$n>1$, so the sets $\bigcup_{i=1}^{n}A_{i}$ and $\bigcup_{i=1}^{m}B_{i}$ are
sets in $v$ with nonempty intersection. Then since $v\circ v\subseteq u$,
their union lies in $u$. In every case we have
$\\{A_{i}\\}_{i=1}^{n}\cup\\{B_{i}\\}_{i=1}^{m}\in C_{u}$. ∎

######  Theorem 15.

Let $(X,\mathcal{C})$ be a conformity. A pseudodiversity $\delta$ is uniformly
continuous from the $\mathcal{C}^{P}$ to $(\mathbb{R},\textnormal{diam})$ iff
the set $V_{\epsilon}=\\{A:\delta(A)<\epsilon\\}$ is in $\mathcal{C}$ for each
$\epsilon>0$.

###### Proof.

First, suppose that every $V_{\epsilon}$ is in $\mathcal{C}$. For each
$\epsilon>0$, the set

$C_{u}=\left\\{\\{A_{1},\ldots,A_{n}\\}:n\leq 1\text{ or
}\delta\left(\bigcup_{i=1}^{n}A_{i}\right)<\epsilon\right\\}$

is in $\mathcal{C}^{P}$. (Notice it has the form of (5) with
$u=V_{\epsilon}$.) Let $\\{A,B\\}\in C_{\epsilon}$; then
$\delta(A)\leq\delta(A\cup B)<\epsilon$ and similarly $\delta(B)<\epsilon$.
Thus $|\delta(A)-\delta(B)|<\epsilon$, so $\delta$ is uniformly continuous.

Conversely, suppose $\delta$ is uniformly continuous. Then for any
$\epsilon>0$, there exists some $u\in\mathcal{C}$, such that every
$\\{A_{1},\ldots,A_{n}\\}\in C_{u}$ satisfies
$\sup_{i,j}|\delta(A_{i})-\delta(A_{j})|<\epsilon$. Since for any $A\in u$,
the set $\\{A,\varnothing\\}$ lies in $C_{u}$, this implies that
$\delta(A)<\epsilon$, which in turn implies that $u\subseteq V_{\epsilon}$,
which finally implies that $V_{\epsilon}$ is in $\mathcal{C}$. ∎

######  Corollary 16.

Every conformity is generated by the pseudodiversities which are uniformly
continuous from its power conformity to $(\mathbb{R},\textnormal{diam})$.

###### Proof.

Let $\mathcal{C}$ be a conformity, $\mathcal{D}$ the conformity generated by
the pseudodiversities which are uniformly continuous from the power conformity
to $(\mathbb{R},\textnormal{diam})$. By Theorem 9 we have
$\mathcal{C}\subseteq\mathcal{D}$, since every member $u$ of $\mathcal{C}$ is
in a countably-based subconformity of $\mathcal{C}$. (Take $u_{0}=u$, $u_{i}$
such that $u_{i}\circ u_{i}\subseteq u_{i-1}$, $i>0$ as a base.)

Then by Theorem 15, every pseudodiversity which is uniformly continuous
generates a subset of $\mathcal{C}$; that is,
$\mathcal{D}\subseteq\mathcal{C}$. ∎

We saw at the beginning of this section that some conformities can be
generated by sets of the form
$\left\\{\delta^{-1}_{\alpha}[0,\epsilon]\right\\}_{\alpha\in\mathcal{A},\epsilon>0}$,
where $\mathcal{A}$ is some collection of pseudodiversities, $\epsilon>0$.
What we have just shown is that all conformities are generated in this way, so
that we may _define_ a conformity as a filter generated in this way by some
collection of diversities.

##  5 Category theory

In [4], Bryant and Tupper introduced the category Dvy whose objects are
diversities and morphisms nonexpansive maps (functions $f$ between diversities
$(X,\delta)$ and $(Y,\rho)$ such that $\rho(f(A))\leq\delta(A)$ for all finite
$A\subseteq X$). Compare with Met [1], whose objects are metric spaces and
morphisms nonexpansive maps (functions $f$ between metric spaces $(X,d)$ and
$(Y,p)$ such that $p(f(x),f(y))\leq d(x,y)$ for all $x,y\in X$).

It is not hard to see that for both metric spaces and diversities,
nonexpansive maps are uniformly continuous. In the metric case, they are also
continuous.

We introduce the category Conf, whose objects are conformities and morphisms
uniformly continuous functions. Compare with Unif [1], whose objects are
uniformities and morphisms uniformly continuous functions.

We also recall Top, whose objects are topological spaces and morphisms
continous maps, and CAT, whose objects are categories and morphisms are
functors (maps between categories which preserve composition).

With these categories in hand, we can summarize the relationships between
diversities, conformities and metric spaces by observing that the maps in the
following diagram in CAT are functors, and that the diagram as a whole
commutes.

MetDivTop
UnifConf$\scriptstyle{t_{m}}$$\scriptstyle{u_{d}}$$\scriptstyle{r_{d}}$$\scriptstyle{u_{\delta}}$$\scriptstyle{t_{u}}$$\scriptstyle{r_{\delta}}$

where:

  * •

$r_{\delta}$ maps conformities to their induced uniform spaces;

  * •

$r_{d}$ maps diversities to their induced metric spaces;

  * •

$u_{\delta}$ maps diversities to the conformities that they generate;

  * •

$u_{d}$ maps metric spaces to the uniform spaces that they generate;

  * •

$t_{m}$ maps metric spaces to their metric topologies;

  * •

and $u_{m}$ maps uniform spaces to their uniform topologies.

Notice that each functor leaves the underlying sets unchanged, e.g. $u_{d}$
maps a metric space $(X,d)$ to a uniform space $(X,\mathcal{U})$. The
morphisms are also unchanged as functions, e.g., a nonexpansive map $f:X\to Y$
in Met is considered a continuous map in Top under $t_{m}$ and a uniformly
continous map in Unif under $u_{d}$, but it is the same function from the set
$X$ to the set $Y$ in all cases.

## References

  * [1] J. Adámek, H. Herrlich, and G.E. Strecker, _Abstract and concrete categories: The joy of cats_ , Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1990.
  * [2] N. Aronszajn and P. Panitchpakdi, _Extension of uniformly continuous transformations and hyperconvex metric spaces_ , Pacific J. Math. 6 (1956), 405–439.
  * [3] Nicolas Bourbaki, _General topology_ , New York: Springer-Verlag, Berlin, 1989, Translation of: Topologie générale.
  * [4] David Bryant and Paul F. Tupper, _Hyperconvexity and tight-span theory for diversities_ , Adv. Math. 231 (2012), no. 6, 3172–3198.
  * [5] A.W.M. Dress, _Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces_ , Adv. Math. 53 (1984), no. 3, 321–402.
  * [6] A.W.M. Dress, V. Moulton, and W. Terhalle, _T-Theory: an overview_ , European J. Combin. 17 (1996), no. 2–3, 161–175.
  * [7] Ding-Zhu Du, _On Steiner ratio conjectures_ , Ann. Opp. Research 33 (1991), no. 6, 437–449.
  * [8] Simon Henryx(mathoverflow.net/users/22131), _Reference: uniformity of pointwise convergence has no countable base_ , MathOverflow, http://mathoverflow.net/questions/134421 (version: 2013-06-21).
  * [9] J.R. Isbell, _Six theorems about injective metric spaces_ , Comment. Math. Helv. 39 (1964), 65–76.
  * [10] John L. Kelley, _General topology_ , van Nostrand, 1975, Reprinted (1975) by Springer-Verlag.
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# Higher order degenerations of Fay’s identities and applications to
integrable equations

Christian Klein  Institut de Mathématiques de Bourgogne, UMR 5584
Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon
Cedex, France
Institut Universitaire de France
E-mail<EMAIL_ADDRESS>and  Jordi Pillet  Institut de
Mathématiques de Bourgogne, UMR 5584
Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon
Cedex, France
E-mail<EMAIL_ADDRESS>Department of Mathematics and Statistics,
Concordia University, 1455 de Maisonneuve W., Montreal, QC H3G 1M8, Canada.

###### Abstract.

Higher order degenerated versions of Fay’s trisecant identity are presented.
It is shown that these lead to solutions for Schwarzian Kadomtsev-Petviashvili
equations.

This work is partially supported by the isite BFC, the EIPHI Graduate School
(contract ANR-17-EURE-0002) and by the European Union Horizon 2020 research
and innovation program under the Marie Sklodowska-Curie RISE 2017 grant
agreement no. 778010 IPaDEGAN. We thank M. Pavlov for helpful discussions and
hints.

##  1\. Introduction

Solutions to integrable partial differential equations (PDEs) in terms of
multi-dimensional theta functions on compact Riemann surfaces appeared in the
1970s in the search for quasi-periodic solutions, see for instance [5, 1] for
a historic account. These solutions were constructed via the _Baker-Akhiezer_
function, a function with an essential singularity on the Riemann surface
first introduced by Clebsch and Gordan. Mumford and coworkers introduced in
[19] a complementary approach based on Fay’s celebrated trisecant identity for
theta functions [7],

$\Theta^{*}_{ad}\Theta^{*}_{cb}\Theta_{ac}\Theta_{bd}+\Theta^{*}_{ca}\Theta^{*}_{db}\Theta_{bc}\Theta_{ad}=\Theta^{*}_{cd}\Theta^{*}_{ab}\Theta\Theta_{a+b,c+d},$

where we have introduced the notation

(1)
$\Theta^{*}_{ab}=\Theta^{*}\left(\int_{a}^{b}\right),\quad\Theta_{ab}=\Theta\left(\mathrm{z}+\int_{a}^{b}\right);$

here $\Theta(\mathrm{z})$, $\mathrm{z}\in\mathbb{C}^{g}$, is the
$g$-dimensional Riemann theta function, $\Theta^{*}(\mathrm{z})$ is a theta
function with an odd non-singular characteristic, see the definitions (6),
(7), $\Theta=\Theta(\mathrm{z})$, $\Theta^{*}=\Theta^{*}(0)=0$, and $a,b,c,d$
are points on a Riemann surface $\mathcal{R}$ with genus $g$. The Abel map
$\int_{a}^{b}$ between two points $a$ and $b$ on $\mathcal{R}$ is defined at
the beginning of section 2. Note that the name trisecant identity refers to
secants on the so-called Kummer variety, see [22] for a comprehensive review.

Since Fay’s identity (9) holds for arbitrary points $a$, $b$, $c$, $d$ on the
Riemann surface $\mathcal{R}$, it is possible to consider the identity in the
limit that two or more points coincide111Note that there are generalizations
of Fay’s identity to more than 4 points and degenerations thereof, see for
instance [8, 6, 2] and references therein.. This leads to identities between
derivatives of theta functions making it possible to identify solutions to
certain PDEs from degenerated identies. In [19] this was done for the Sine-
Gordon equation and the Kadomtsev-Petviasvili (KP) equation. On special
Riemann surfaces (hyperelliptic, trigonal) the latter solutions lead to
algebro-geometric solutions for the Korteweg-de Vries (KdV) [19] and the
Boussinesq equation [1]. In [14] previously known solutions to the Ernst
equation [17] were reconstructed via Fay’s identity, see also [15], in [12]
known solutions to the Camassa-Holm equation [9] were obtained with Mumford’s
approach. In [11] Kalla presented a new degenerated identity allowing to
identify known solutions to the nonlinear Schrödinger [10, 20] and Davey-
Stewartson equations [18] and to construct solutions to vector nonlinear
Schrödinger equations in terms of theta functions. For a recent review on
completely integrable dispersive PDEs, we refer to [16]. In this paper we
generalize Kalla’s approach to higher order in the local parameter near the
point $a$. We obtain with the above notation
Main theorem Part I
Let $a$, $b$ be points on a compact Riemann surface $\mathcal{R}$, and let
$U:=D_{b}\ln\Theta\Theta^{*}_{ba}$. Then $U$ satisfies

(2)
$\begin{split}0&=2(D_{a}U)^{2}D_{a}^{\prime\prime}D_{a}U-2D_{a}UD_{a}^{\prime\prime}UD_{a}^{2}U-(D_{a}U)^{2}D_{a}^{4}U+4D_{a}UD_{a}^{3}UD_{a}^{2}U\\\
&-3(D_{a}^{2}U)^{3}+3(D_{a}^{\prime}U)^{2}D_{a}^{2}U-3(D_{a}U)^{2}(D_{a}^{\prime})^{2}DU.\end{split}$

This identity has similarities to the classical identity (14) by Fay in the
sense that it involves the derivatives $D_{a}^{\prime\prime}$,
$D_{a}^{\prime}$ and $D_{a}$ of $\Theta(\mathrm{z})$, but appears to be new.
In contrast to the potential in (14), the function $U$ also depends on a point
$b$ on the Riemann surface $\mathcal{R}$ which is distinct from $a$, but
otherwise arbitrary.

We also prove
Main theorem Part II
The function
$\phi(x,y,t):=\mathrm{D}_{b}\ln{\Theta^{*}_{ab}\Theta(x\mathbf{v}_{0}(a)+y\mathbf{v}_{1}(a)+t\mathbf{v}_{2}(a)+\mathbf{d}})$,
$(x,y,t)\in\mathbb{R}^{3}$, solves the Schwarzian KP equation:

(3)
$\Big{(}\frac{\phi_{t}}{\phi_{x}}-\frac{1}{2}\\{\phi;x\\}\Big{)}_{x}-\frac{3}{2}\Big{(}\frac{\phi_{y}}{\phi_{x}}\Big{)}_{y}-\frac{3}{4}\Big{(}\frac{\phi_{y}^{2}}{\phi_{x}^{2}}\Big{)}_{x}=0$

where $\\{\phi;x\\}$ denotes the Schwarzian derivative along $x$:
$\\{\phi;x\\}:=\frac{\phi_{xxx}}{\phi_{x}}-\frac{3}{2}\Big{(}\frac{\phi_{xx}}{\phi_{x}}\Big{)}^{2}$,
where the indices denote partial derivatives with respect to the respective
variable, and where $\mathbf{v}_{j}$, $j=0,1,2$ has the components $v_{ij}$,
$i=1,\ldots,g$ defined in (4).

The solution $\phi$ in terms of multi-dimensional theta functions for the
Schwarzian KP equation seems to be new. The Schwarzian KP equation (3)
appeared first in the Painlevé analysis of the KP equation in [23] as a
singularity manifold equation. Its integrability was established in [3]. As in
the case of the KP equation, a reduction to a Schwarzian KdV and Boussinesq
equation is possible.

The paper is organised as follows: in section 2 we collect some basic
definitions of quantities defined on a compact Riemann surface and known facts
on Fay’s identities. In section 3 we rederive identities (12) and (16) from
identity (10) and prove the first part of the main theorem. In section 4 this
is applied to integrable PDEs. We add some concluding remarks in section 5.

##  2\. Preliminaries

In this section, we will collect some basic definitions and known facts on
Fay’s identities and applications.

###  2.1. Basic definitions

In this paper we always consider a Riemann surface $\mathcal{R}$ of genus
$g\in\mathbb{N}$ equipped with a canonical basis of cycles
$a_{1},\ldots,a_{g},b_{1},\ldots,b_{g}$ satisfying the intersection conditions

$a_{i}\circ b_{j}=\delta_{ij},\quad a_{i}\circ a_{j}=0,\quad b_{i}\circ
b_{j}=0,\quad i,j=1,\ldots,g.$

The $g$-dimensional vector of holomorphic 1-forms is denoted by
$\mathrm{d}\omega$ and normalized by
$\int_{a_{i}}\mathrm{d}\omega_{j}=\delta_{ij}$, $i,j=1,\ldots,g$. The matrix
of $b$-periods $\mathbb{B}_{ij}=\int_{b_{i}}\mathrm{d}\omega_{j}$,
$i,j=1,\ldots,g$, is a Riemann matrix, i.e., it is symmetric and has a
positive definite imaginary part. The Abel map
$\omega:P\mapsto\int_{P_{0}}^{P}\mathrm{d}\omega$ is a bijective map from the
Riemann surface $\mathcal{R}$ into the _Jacobian_
$Jac(\mathcal{R}):=\mathbb{C}^{g}/\Lambda$ where $\Lambda$ is the lattice
formed by the periods of the holomorphic 1-forms,

$\Lambda=\left\\{\mathrm{m}+\mathbb{B}\mathrm{n}:m,n\in\mathbb{Z}^{g}\right\\}.$

The expansion of the Abel map at a point $P\in\mathcal{R}$ near a point
$a\in\mathcal{R}$ is written in the form,

(4) $\omega_{i}(P)=\sum_{j=0}^{\infty}v_{ij}\frac{\tau^{j}}{j!},\quad
i=1,\ldots,g,$

where $\tau$ is a local parameter in the vicinity of $a$ containing also $P$.
We define the derivatives acting on a function $f(z)$, $z\in\mathbb{C}^{g}$ as

(5) $\begin{split}D_{a}&:=\sum_{i=1}^{g}v_{i0}\partial_{z_{i}},\quad
D_{a}^{\prime}:=\sum_{i=1}^{g}v_{i1}\partial_{z_{i}},\quad
D_{a}^{\prime\prime}:=\sum_{i=1}^{g}\frac{v_{i2}}{2}\partial_{z_{i}},\\\
D_{a}^{(n)}&:=\sum_{i=1}^{g}\frac{v_{in}}{n!}\partial_{z_{i}},\quad
n\in\mathbb{N}.\end{split}$

Multi-dimensional theta functions are the building blocks of meromorphic
functions on Riemann surfaces. The theta function with characteristic
$\left[\mathrm{p},\mathrm{q}\right]$ is defined as an infinite series,

(6)
$\Theta_{\mathrm{p}\mathrm{q}}(\mathrm{z},\mathbb{B})=\sum\limits_{\mathrm{N}\in\mathbb{Z}^{g}}\exp\left\\{\mathrm{i}\pi\left\langle\mathbb{B}\left(\mathrm{N}+\mathrm{p}\right),\mathrm{N}+\mathrm{p}\right\rangle+2\pi\mathrm{i}\left\langle\mathrm{z}+\mathrm{q},\mathrm{N}+\mathrm{p}\right\rangle\right\\}\;,$

with $\mathrm{z}\in\mathbb{C}^{g}$ and $\mathrm{p}$,
$\mathrm{q}\in{\mathbb{R}}^{g}$, where $\left\langle\cdot,\cdot\right\rangle$
denotes the Euclidean scalar product
$\left\langle\mathrm{N},\mathrm{z}\right\rangle=\sum_{i=1}^{g}N_{i}z_{i}$. The
properties of the Riemann matrix ensure that the series converges absolutely
and that the theta function is an entire function on $\mathbb{C}^{g}$. A
characteristic is called _singular_ if the corresponding theta function
vanishes identically. Half-integer characteristics with
$2\mathrm{p},2\mathrm{q}\in\mathbb{Z}^{g}$ are called _even_ if
$4\langle\mathrm{p},\mathrm{q}\rangle=0\mbox{ mod }2$ and _odd_ otherwise.
Theta functions with odd (even) characteristic are odd (even) functions of the
argument $\mathrm{z}$. The theta function with characteristic is related to
the Riemann theta function $\Theta$, the theta function with zero
characteristic $\Theta:=\Theta_{\mathrm{00}}$, via

(7)
$\Theta_{\mathrm{pq}}(\mathrm{z},\mathbb{B})=\Theta(\mathrm{z}+\mathbb{B}\mathrm{p}+\mathrm{q})\exp\left\\{\mathrm{i}\pi\left\langle\mathbb{B}\mathrm{p},\mathrm{p}\right\rangle+2\pi\mathrm{i}\left\langle\mathrm{p},\mathrm{z}+\mathrm{q}\right\rangle\right\\}\;.$

A theta function with a nonsingular half-integer characteristic is denoted by
$\Theta^{*}$.

###  2.2. Fay’s identities

Theta functions on Jacobians satisfy Fay’s celebrated trisecant identity [7].
It can be seen as a generalization of the classical relation between cross
ration functions for four arbitrary points $a$, $b$, $c$, $d$ in the euclidean
plane

(8)
$\lambda_{abcd}=\frac{\Theta^{*}(\smallint_{a}^{b})\Theta^{*}(\smallint_{c}^{d})}{\Theta^{*}(\smallint_{a}^{d})\Theta^{*}(\smallint_{c}^{b})}\;,$

which is a function on $\mathcal{R}$ that vanishes for $a=b$ and $c=d$ and has
poles for $a=d$ and $b=c$.

######  Theorem 2.1 (Fay [7]).

Let $a$, $b$, $c$, $d$ be four points on the Riemann surface $\mathbb{R}$.
Then with the above definitions the following identity holds

(9) $\displaystyle\lambda_{cabd}\,\Theta(\mathrm{z}+\smallint_{b}^{c})$
$\displaystyle\Theta(\mathrm{z}+\smallint_{a}^{d})+\lambda_{cbad}\,\Theta(\mathrm{z}+\smallint_{a}^{c})\,\Theta(\mathrm{z}+\smallint_{b}^{d})$
$\displaystyle=\Theta(\mathrm{z})\;\Theta(\mathrm{z}+\smallint_{b}^{c}+\smallint_{a}^{d})\;,$

$\forall\mathrm{z}\in\mathbb{C}^{g}$. The integration paths in (9) have to be
chosen in a way not to intersect the canonical cycles.

Degenerated versions of Fay’s identity lead to identities for derivatives of
theta functions. In the limit $d\to b$, one finds for (9)

######  Corollary 2.2 (Fay [7]).

Let $a$, $b$, $c$ be points on the Riemann surface $\mathcal{R}$. Then the
following identity holds,

(10)
$D_{b}\ln\frac{\Theta(\mathrm{z}+\smallint_{a}^{c})}{\Theta(\mathrm{z})}=p_{1}(a,b,c)+p_{2}(a,b,c)\frac{\Theta(\mathrm{z}+\smallint_{b}^{c})\Theta(\mathrm{z}+\smallint_{a}^{b})}{\Theta(\mathrm{z}+\smallint_{a}^{c})\Theta(\mathrm{z})},$

where

$\displaystyle p_{1}(a,b,c)$
$\displaystyle=D_{b}\ln\frac{\Theta^{*}(\smallint_{a}^{b})}{\Theta^{*}(\smallint_{c}^{b})},$
(11) $\displaystyle p_{2}(a,b,c)$
$\displaystyle=\frac{\Theta^{*}(\smallint_{a}^{c})D_{b}\Theta^{*}}{\Theta^{*}(\smallint_{b}^{c})\Theta^{*}(\smallint_{b}^{a})}.$

In the limit $c\to a$, equation (10) yields

######  Corollary 2.3 (Fay [7]).

Let $a$, $b$ be points on the Riemann surface $\mathcal{R}$. Then the
following identity holds,

(12)
$D_{a}D_{b}\ln\Theta(\mathrm{z})=q_{1}(a,b)+q_{2}(a,b)\frac{\Theta(\mathrm{z}+\smallint_{b}^{a})\Theta(\mathrm{z}+\smallint_{a}^{b})}{\Theta(\mathrm{z})^{2}},$

where

$\displaystyle q_{1}(a,b)$
$\displaystyle=D_{a}D_{b}\ln\Theta^{*}(\smallint_{a}^{b}),$ (13)
$\displaystyle q_{2}(a,b)$
$\displaystyle=\frac{D_{a}\Theta^{*}D_{b}\Theta^{*}}{\Theta^{*}(\smallint_{a}^{b})^{2}}.$

In the limit $b\to a$, equation (12) can be cast into the form (we suppress
the index $a$ at the derivatives)

######  Corollary 2.4 (Fay [7]).

The following identity holds on the Riemann surface $\mathcal{R}$,

(14)
$D^{4}\ln\Theta(\mathrm{z})+6(D^{2}\ln\Theta(\mathrm{z}))^{2}+3D^{\prime}D^{\prime}\ln\Theta(\mathrm{z})-2DD^{\prime\prime}\ln\Theta(\mathrm{z})+c_{1}D^{2}\ln\Theta(\mathrm{z})+c_{2}=0,$

where

(15)
$c_{1}=2\frac{D^{\prime\prime}\Theta^{*}}{\Theta^{*}}-4\frac{D^{3}\Theta^{*}}{D\Theta^{*}}-3\left(\frac{D^{\prime}\Theta^{*}}{D\Theta^{*}}\right)^{2};$

the constant $c_{2}$ can be obtained by expanding $q_{1}$ and $q_{2}$ (13) in
the considered limit to fourth order in the local parameter near $a$.

Kalla generalized the identity (12) to

######  Theorem 2.5 (Kalla [11]).

Let $a$, $b$ be points on $\mathcal{R}$, then the following identity holds,

(16)
$\begin{split}0=&D_{a}^{\prime}\ln\frac{\Theta(\mathrm{z}+\smallint_{a}^{b})}{\Theta(\mathrm{z})}+D_{a}^{2}\ln\frac{\Theta(\mathrm{z}+\smallint_{a}^{b})}{\Theta(\mathrm{z})}+\left(D_{a}\ln\frac{\Theta(\mathrm{z}+\smallint_{a}^{b})}{\Theta(\mathrm{z})}-K_{1}(a,b)\right)^{2}\\\
&+2D_{a}^{2}\ln\Theta(\mathrm{z})+K_{2}(a,b)\end{split}$

where $K_{1}(a,b)$, $K_{2}(a,b)$ depend on the points $a$, $b$, but not on
$\mathrm{z}$.

##  3\. Proof of Main theorem Part I

In this section we will consider various identities following from Fay’s
identity (10) in the limit $c\to a$. First we identify the known relations
(12) and (16), then we prove the first part of the main theorem.

To this end we write (10) in the form

(17)
$V^{2}D_{b}\frac{V(c)}{V}=D_{b}\Theta^{*}\Theta^{*}_{ac}\Theta_{ab}\Theta_{bc},$

where we have put

(18) $V(c)=\Theta_{ac}\Theta^{*}_{cb},\quad V=V(a)=\Theta\Theta^{*}_{ab}.$

###  3.1. Known identities

For identity (17)) we consider a Taylor expansion in the limit $c\to a$ in the
local parameter $\tau$ in (4). In lowest order we get (12) in the form

(19) $D_{a}D_{b}\ln
V=\frac{1}{V^{2}}D_{b}\Theta^{*}D_{a}\Theta^{*}\Theta_{ab}\Theta_{ba}.$

In order $\tau^{2}$, we get for (17)

$\frac{V^{2}}{2}D_{b}(D_{a}^{2}\ln V+D_{a}^{\prime}\ln V+(D_{a}\ln
V)^{2})=D_{b}\Theta^{*}D_{a}\Theta^{*}\Theta_{ab}\Theta_{ba}\left(\frac{D_{a}^{\prime}\Theta^{*}}{2D_{a}\Theta^{*}}+D_{a}\ln\Theta_{ba}\right),$

which can be written with (19) in the form

(20) $\frac{1}{2}D_{b}(D_{a}^{2}\ln V+D_{a}^{\prime}\ln V+(D_{a}\ln
V)^{2})=D_{a}D_{b}\ln
V\left(\frac{D_{a}^{\prime}\Theta^{*}}{2D_{a}\Theta^{*}}+D_{a}\ln\Theta_{ba}\right).$

Since this identity holds for all $\mathrm{z}\in\mathbb{C}$, it also holds for
$\mathrm{z}$ replaced by $\mathrm{z}+\int_{a}^{b}$. This means
$\Theta\mapsto\Theta_{ab}$, $\Theta_{ba}\mapsto\Theta$. As shown by Kalla
[11], the difference between identity (20) and (20) after this shift of
$\mathrm{z}$ reads

(21)
$\begin{split}0=&\frac{1}{2}D_{b}\left(D_{a}^{\prime}\ln\frac{\Theta_{ab}}{\Theta}+\frac{D_{a}^{2}\Theta_{ab}}{\Theta_{ab}}-\frac{D_{a}^{2}\Theta}{\Theta}+2D_{a}\ln\Theta^{*}_{ba}D_{a}\ln\frac{\Theta_{ab}}{\Theta}\right)\\\
&+D_{a}D_{b}\ln\Theta\Theta^{*}_{ab}D_{a}\ln\Theta_{ba}-D_{a}D_{b}\ln\Theta_{ab}\Theta^{*}_{ab}D_{a}\ln\Theta-\frac{D_{a}^{\prime}\Theta^{*}}{2D_{a}\Theta^{*}}D_{a}D_{b}\ln\frac{\Theta_{ab}}{\Theta}\end{split}.$

With (12), we get

(22)
$\begin{split}0=&\frac{1}{2}D_{b}\left(D_{a}^{\prime}\ln\frac{\Theta_{ab}}{\Theta}+D_{a}^{2}\ln\frac{\Theta_{ab}}{\Theta}+\left(D_{a}\ln\frac{\Theta_{ab}}{\Theta}\right)^{2}+2D_{a}^{2}\ln\Theta\Theta^{*}_{ba}\right)\\\
&+\left(D_{a}\ln\Theta^{*}_{ba}-\frac{D_{a}^{\prime}\Theta^{*}}{2D_{a}\Theta^{*}}\right)D_{a}D_{b}\ln\frac{\Theta_{ab}}{\Theta}\end{split}.$

Introducing the derivative
$\nabla_{b}:=\sum_{i=1}^{g}v_{i0}(b)\partial_{\mathrm{z}_{i}}$ acting only on
$\mathrm{z}$, we can write (22) in the form

(23)
$\begin{split}0=&\nabla_{b}\left\\{\frac{1}{2}\left(D_{a}^{\prime}\ln\frac{\Theta_{ab}}{\Theta}+D_{a}^{2}\ln\frac{\Theta_{ab}}{\Theta}\right)+\left(D_{a}\ln\Theta^{*}_{ba}-\frac{D_{a}^{\prime}\Theta^{*}}{2D_{a}\Theta^{*}}\right)D_{a}\ln\frac{\Theta_{ab}}{\Theta}\right.\\\
&\left.+D_{a}^{2}\ln\Theta\Theta^{*}_{ab}+\frac{1}{2}\left(D_{a}\ln\frac{\Theta_{ab}}{\Theta}\right)^{2}\right\\}\end{split}.$

This implies that relation (16) holds with

(24)
$\begin{split}K(a,b)=&D_{a}^{\prime}\ln\frac{\Theta_{ab}}{\Theta}+D_{a}^{2}\ln\frac{\Theta_{ab}}{\Theta}+2\left(D_{a}\ln\Theta^{*}_{ba}-\frac{D_{a}^{\prime}\Theta^{*}}{2D_{a}\Theta^{*}}\right)D_{a}\ln\frac{\Theta_{ab}}{\Theta}\\\
&+2D_{a}^{2}\ln\Theta\Theta^{*}_{ab}+\left(D_{a}\ln\frac{\Theta_{ab}}{\Theta}\right)^{2}\end{split},$

where $K(a,b)$ just depends on $a$, $b$, but not on $\mathrm{z}$. It can be
computed for instance by putting $\mathrm{z}=0$ on the right hand side of
(24). This reproduces the proof of Theorem 2.5 from [11].

###  3.2. Third order in $\tau$

In third order of the local parameter $\tau$ we get for (10)

(25) $\begin{split}0=&D_{b}\left(\frac{1}{6}D_{a}^{\prime\prime}\ln
V+\frac{1}{2}D_{a}D_{a}^{\prime}\ln V+\frac{1}{2}D_{a}^{\prime}\ln VD_{a}\ln
V+\frac{1}{6}D_{a}^{3}\ln V\right.\\\ &\left.+\frac{1}{2}D_{a}^{2}\ln
VD_{a}\ln V+\frac{1}{6}(D_{a}\ln V)^{3}\right)\\\ &-D_{a}D_{b}\ln
V\left(\frac{D_{a}^{\prime\prime}\Theta^{*}}{6D_{a}\Theta^{*}}+\frac{D_{a}^{3}\Theta^{*}}{6D_{a}\Theta^{*}}+\frac{D_{a}^{\prime}\Theta^{*}}{2D_{a}\Theta^{*}}D_{a}\ln\Theta_{ba}+\frac{1}{2}D_{a}^{\prime}\ln\Theta_{ba}+\frac{D_{a}^{2}\Theta_{ba}}{2\Theta_{ba}}\right),\end{split}$

where we have used (19). With (20), we can replace $\Theta_{ba}$ in (25) to
obtain a relation only involving $V$. To this end we put

(26) $F:=\frac{1}{2}D_{b}\left(D_{a}^{\prime}\ln V+D_{a}^{2}\ln V+(D_{a}\ln
V)^{2}\right)$

as well as

(27) $C_{1}:=\frac{D_{a}^{\prime}\Theta^{*}}{2D_{a}\Theta^{*}}$

which leads to (20) in the form

(28) $D_{a}\ln\Theta_{ba}=\frac{F}{D_{a}D_{b}\ln V}-C_{1}.$

In addition we put

(29) $\begin{split}G:=&D_{b}\left(\frac{1}{6}D_{a}^{\prime\prime}\ln
V+\frac{1}{2}D_{a}D_{a}^{\prime}\ln V+\frac{1}{2}D_{a}^{\prime}\ln VD_{a}\ln
V\right.\\\ &\left.+\frac{1}{6}D_{a}^{3}\ln V+\frac{1}{2}D_{a}^{2}\ln
VD_{a}\ln V+\frac{1}{6}(D_{a}\ln V)^{3}\right)\end{split}$

and

(30)
$C_{2}=\frac{D_{a}^{\prime\prime}\Theta^{*}}{6D_{a}\Theta^{*}}+\frac{D_{a}^{3}\Theta^{*}}{6D_{a}\Theta^{*}}.$

with which (25) takes the form

(31) $G=D_{a}D_{b}\ln
V\left(C_{2}+C_{1}D_{a}\ln\Theta_{ba}+\frac{1}{2}D_{a}^{\prime}\ln\Theta_{ba}+\frac{1}{2}D_{a}^{2}\ln\Theta_{ba}+\frac{1}{2}(D_{a}\ln\Theta_{ba})^{2}\right).$

Eliminating $\Theta_{ba}$ with (28) from (31) leads in a first step to

(32) $\frac{G}{D_{a}D_{b}\ln V}-\frac{1}{2}D_{a}\left(\frac{F}{D_{a}D_{b}\ln
V}\right)-\frac{F^{2}}{2(D_{a}D_{b}\ln
V)^{2}}=C_{2}-\frac{C_{1}^{2}}{2}+\frac{1}{2}D_{a}^{\prime}\ln\Theta_{ba}.$

Differentiating with respect to $D_{a}$ (note that the derivatives of the odd
theta functions in $c_{1}$, $c_{2}$ vanish), we obtain with (28)

(33) $D_{a}\left(\frac{G}{D_{a}D_{b}\ln
V}-\frac{1}{2}D_{a}\left(\frac{F}{D_{a}D_{b}\ln
V}\right)-\frac{F^{2}}{2(D_{a}D_{b}\ln
V)^{2}}\right)=\frac{1}{2}D_{a}^{\prime}\left(\frac{F}{D_{a}D_{b}\ln
V}\right).$

We have with $U:=D_{b}\ln V$

(34) $\begin{split}&\frac{G}{D_{a}D_{b}\ln
V}-\frac{1}{2}D_{a}\left(\frac{F}{D_{a}D_{b}\ln
V}\right)-\frac{F^{2}}{2(D_{a}D_{b}\ln V)^{2}}\\\
&=\frac{D_{a}^{\prime\prime}U}{6D_{a}U}+\frac{D_{a}^{\prime}D_{a}U}{4D_{a}U}+\frac{1}{2}D_{a}^{\prime}\ln
V\\\
&-\frac{D_{a}^{3}U}{12D_{a}U}+\frac{(D_{a}^{2}U)^{2}}{8(D_{a}U)^{2}}-\frac{(D_{a}^{\prime}U)^{2}}{8(D_{a}U)^{2}},\end{split}.$

Thus we get for (33)

(35)
$\begin{split}0&=\frac{D_{a}^{\prime\prime}D_{a}U}{6D_{a}U}-\frac{D_{a}^{\prime\prime}UD_{a}^{2}U}{(6D_{a}U)^{2}}-\frac{D_{a}^{4}U}{12D_{a}U}+\frac{D_{a}^{3}UD_{a}^{2}U}{3(D_{a}U)^{2}}\\\
&-\frac{(D_{a}^{2}U)^{3}}{4(D_{a}U)^{3}}+\frac{(D_{a}^{\prime}U)^{2}D_{a}^{2}U}{4(D_{a}U)^{3}}-\frac{(D_{a}^{\prime})^{2}DU}{4D_{a}U}.\end{split}$

identical to equation (2) which concludes the proof. Note that there are terms
in (33) without a derivative $D_{a}$, but remarkably these terms all cancel
leaving (34) a relation for terms all involving this derivative.

##  4\. Applications to integrable PDEs

In this section we will apply relation (2) to integrable equations, prove the
second part of the main theorem and consider various reductions on special
Riemann surfaces as known for the KP case.

###  4.1. Main theorem Part II

To prove the second part of the main theorem, we define the function
$\phi(x,y,t):=\mathrm{D}_{b}\ln{\Theta^{*}_{ab}\Theta(x\mathbf{v}_{0}(a)+y\mathbf{v}_{1}(a)+t\mathbf{v}_{2}(a)+\mathbf{d}})$
and show that it solves the Schwarzian KP equation (3).

With our previous notations $\phi=D_{b}\ln V=U$, moreover we can identify:
$D_{a}=\partial_{x}$, $D_{a}^{\prime}=\partial_{y}$ and
$D_{a}^{\prime\prime}=\partial_{t}$. Inserting the function $\phi$ into (3) we
get:

$\displaystyle\frac{D_{a}^{\prime\prime}D_{a}U}{D_{a}U}-\frac{D_{a}^{2}U}{(D_{a}U)^{2}}D_{a}^{\prime\prime}U-\frac{1}{2}\frac{D_{a}^{4}U}{D_{a}U}+\frac{2}{(D_{a}U)^{2}}D_{a}^{2}UD_{a}^{3}U$
$\displaystyle-\frac{3}{2(D_{a}U)^{3}}(D_{a}U)^{3}+\frac{3}{2}\frac{D_{a}^{2}U}{(D_{a}U)^{3}}(D^{\prime}_{a}U)^{2}-\frac{3}{2}\frac{(D^{\prime}_{a})^{2}U}{D_{a}U}=0$

which is equivalent to (2) thus proving this part of the theorem.

######  Remark 4.1.

It is well known, see for instance [19, 1], that one way to represent
meromorphic functions on a Riemann surface is in terms of second logarithmic
derivatives of theta functions. The function $D_{b}\ln\Theta^{*}_{ab}\Theta$
is a priori not independent of the path between $a$ and $b$ in
$\int_{a}^{b}\mathrm{d}\omega$. Both points have to be in the same fundamental
polygon. A possibility to avoid this condition would be to consider
$U=\partial_{x}^{-1}u$, where $u:=D_{a}D_{b}\ln V$ is path independent; the
anti-derivative is defined as $\partial_{x}^{-1}=\int_{a}^{x}dx^{\prime}$.

The Schwarzian KP equation can also be written in the form

(36)
$\left(\frac{U_{t}}{U_{x}}-\frac{1}{2}\frac{U_{xxx}}{U_{x}}+\frac{3}{4}\frac{U_{xx}^{2}}{U_{x}^{2}}\right)_{x}+\frac{3}{2}\frac{U_{xx}}{U_{x}^{3}}U_{y}^{2}-\frac{3}{2}\frac{U_{yy}}{U_{x}}=0.$

Integrating with respect to $x$, we get

(37)
$U_{t}-\frac{1}{2}U_{xxx}+\frac{3}{4}\frac{U_{xx}^{2}-U_{y}^{2}}{U_{x}}+\frac{3}{2}U_{x}\partial_{x}^{-1}\left(\frac{U_{xy}U_{y}}{U_{x}^{2}}-\frac{U_{yy}}{U_{x}}\right),$

which can be written in the form

(38)
$U_{t}-\frac{1}{2}U_{xxx}+\frac{3}{4}\frac{U_{xx}^{2}-U_{y}^{2}}{U_{x}}-\frac{3}{2}U_{x}W_{y},$

where $W_{x}:=U_{y}/U_{x}$. This is equation (13) in [3] after the change of
time $t\mapsto 2t$.

###  4.2. Reductions on special Riemann surfaces

Let us restrict our attention to the special case of the Riemann surface
$\mathcal{R}$ being hyperelliptic, i.e., given by the zero locus of the
polynomial $P(\lambda,\mu)=\mu^{2}-\prod_{j=1}^{N}(\lambda-\lambda_{j})$ where
$\lambda_{j}\in\mathbb{C}$, $j=1,\ldots,N$, and $N=2g+1$ or $N=2g+2$, and
denote by $\pi:\mathcal{R}\longrightarrow\mathbb{CP}^{1}$ the projection onto
the Riemann sphere.
If $a$ is a branch point of $\pi$ then the hyperelliptic involution locally
reads $\sigma:k_{a}\mapsto-k_{a}$ and its pullback on $\omega_{j}$ is given
by: $\sigma^{*}\omega_{j}=-\omega_{j}$, hence the Taylor expansion of
$\omega_{j}$ around $a$ must be even in $k_{a}$, i.e $\mathbf{v}_{1}=0$ and
thus $\mathrm{D_{a}^{\prime}}=0$. Eliminating $\Theta_{ba}$ from (31) via
(26), we get

######  Corollary 4.2.

Identity (2) reduces on hyperelliptic surfaces with $a$ being a branch point
to

(39) $0=\frac{1}{6}D_{a}^{\prime\prime}D_{b}\ln
V-\frac{1}{12}D_{a}^{3}D_{b}\ln V-C_{2}D_{a}D_{b}\ln
V+\frac{1}{8}\frac{(D_{a}^{2}D_{b}\ln V)^{2}}{D_{a}D_{b}\ln V}.$

If we put again $U=D_{b}\ln V$, $D_{a}^{\prime\prime}=\partial_{t}$ and
$D_{a}=\partial_{x}$, we get for (39)

(40)
$\frac{1}{6}U_{t}-\frac{1}{12}U_{xxx}-C_{2}U_{x}+\frac{1}{8}\frac{U_{xx}^{2}}{U_{x}}=0.$

This implies

######  Corollary 4.3.

The function
$\phi(x,t):=\mathrm{D}_{b}\ln{\Theta^{*}_{ab}\Theta(x\mathbf{v}_{0}(a)+t\mathbf{v}_{2}(a)+\mathbf{d}})$
solves the Schwarzian KdV equation:

(41) $\frac{1}{6}\frac{\phi_{t}}{\phi_{x}}-\frac{1}{12}\\{\phi;x\\}=C_{2}.$

As in Chapter 3.4 of [1] for KP, there is also a reduction to a Schwarzian
Boussinesq equation. If the surface $\mathcal{R}$ is given by a trigonal
curve, i.e., a curve on which a meromorphic function with a third order pole
at a point $a\in\mathcal{R}$ and no other singularities exists. A simple
example of such a curve is

(42) $\mu^{4}=\prod_{i=1}^{4}(\lambda-E_{i}).$

In this case $D_{a}^{\prime\prime}=0$ which leads for (2) to

(43)
$0=-\frac{D_{a}^{4}U}{12D_{a}U}+\frac{D_{a}^{3}UD_{a}^{2}U}{3(D_{a}U)^{2}}\\\
-\frac{(D_{a}^{2}U)^{3}}{4(D_{a}U)^{3}}+\frac{(D_{a}^{\prime}U)^{2}D_{a}^{2}U}{4(D_{a}U)^{3}}-\frac{(D_{a}^{\prime})^{2}DU}{4D_{a}U}.$

The function
$\phi(x,t):=\mathrm{D}_{b}\ln{\Theta^{*}_{ab}\Theta(x\mathbf{v}_{0}(a)+t\mathbf{v}_{1}(a)+\mathbf{d}})$
then gives a solution to the Schwarzian Boussinesq equation [23]

(44)
$\Big{(}-\frac{1}{2}\\{\phi;x\\}\Big{)}_{x}-\frac{3}{2}\Big{(}\frac{\phi_{t}}{\phi_{x}}\Big{)}_{t}-\frac{3}{4}\Big{(}\frac{\phi_{t}^{2}}{\phi_{x}^{2}}\Big{)}_{x}=0$

##  5\. Conclusion

In this paper we have studied degenerations of Fay’s identities in higher
order of the local parameter $\tau$ near one of the points. The starting point
was Fay’s identity for 3 points on a Riemann surface in the form (17). The
case of order $\tau^{3}$ was studied in detail as well as its application to
the Schwarzian KP equation. It appears straight forward to generalize this
approach to higher orders of the parameter $\tau$. A standard Taylor expansion
of the quantity $V(c)$ yields for the left hand side of equation (17)

(45) $D_{b}\sum_{m=1}^{\infty}\frac{1}{m!V}\left(\tau
D_{a}+\frac{\tau^{2}}{2}D_{a}^{\prime}+\frac{\tau^{3}}{6}D_{a}^{\prime\prime}+\ldots\frac{\tau^{k}}{k!}D_{a}^{(k)}+\ldots\right)^{m}V.$

Thus one gets in order $\tau^{n}$

(46)
$D_{b}\left(\frac{D_{a}^{n}V}{n!V}+\frac{nD_{a}^{n-1}D_{a}^{\prime}V}{2(n-1)!V}+\ldots+\frac{D_{a}^{(n)}V}{n!V}\right).$

The same expansion can be obtained on the right hand side of (17) for
$\Theta^{*}_{ac}$, where all even order derivatives vanish for symmetry
reasons, and for $\Theta_{bc}$ leading to derivatives of $\Theta_{ba}$. As in
section 3, the latter terms can be replaced via (28) by derivatives of
$\Theta$. As in (33), it will be necessary to differentiate with respect to
$D_{a}$ in general in order to eliminate all terms with $\Theta_{ba}$. It is
beyond the scope of the current paper to detail the resulting relations and to
establish a potential relation to integrable PDEs and whether these are from a
hierarchy of Schwarzian KP equations. An interesting question is also whether
a similar approach can be applied to the degeneration of identity (12) in the
limit $b\to a$ in higher orders of the local parameter near $a$, which would
lead to a generalisation of relation (14). This will be the subject of future
research.

Another interesting aspect would be to relate the present work to the bilinear
approach studied in [4]. In this article the authors describe the tau and
Baker-Akhiezer functions associated to some generalized KP hierarchy (the
Schwarzian KP hierarchy being one of them). Their derivation is based on a
generalized Hirota’s bilinear identity based on the equation:

$\int_{\partial
G}\chi(\nu,\mu;g_{1})g_{1}(\nu)g_{2}^{-1}(\nu)\chi(\lambda,\nu;g_{2})d\nu=0$

Where (following the notations of [4]) $G$ is the unit disk,
$(\lambda,\mu)\in\mathbb{C}^{2}$ are spectral parameters,
$g_{1}(\nu)=g(\nu,\mathbf{x})=\exp{\big{(}\sum_{i=1}^{\infty}x_{i}\nu^{-i}\big{)}}$,
$g_{2}(\nu)=g(\nu,\mathbf{x^{\prime}})$ and $\chi(\lambda,\mu)$ is an unknown
meromorphic function in both variables.
The associated tau function is given by:

$\chi(\lambda,\mu,\mathbf{x})=\frac{1}{(\lambda-\mu)}\frac{\tau(\mathbf{x}-[\lambda]+[\mu])}{\tau(\mathbf{x})}$

where $\mathbf{x}+[\mu]=x_{i}+[\mu]_{i}$, with $[\mu]_{i}=\frac{1}{i}\mu^{i}$,
$0\leq i<\infty$. They proved that this tau function satisfies the following
addition formula for $a$, $b$, $c$ and $d$ some arbitrary complex numbers :

$\begin{split}&(a-c)(d-b)\tau(\mathbf{x}+[a]+[c])\tau(\mathbf{x}+[d]+[b])+\\\
&+(d-a)(b-c)\tau(\mathbf{x}+[d]+[a])\tau(\mathbf{x}+[b]+[c])+\\\
&+(b-a)(d-c)\tau(\mathbf{x}+[b]+[a])\tau(\mathbf{x}+[d]+[c])=0\end{split}$

As established in the seminal paper [21], this addition formula is nothing
more than Fay’s trisecant identity when a tau function can be written in terms
of Riemann theta functions. All these considerations lead to the following
conjecture:

######  Conjecture 5.1.

There exists a quadratic form
$\mathbf{Q}(\mathbf{t}):=\sum_{i,j=1}^{4}Q_{ij}t_{i}t_{j}$,
$Q_{ij}\in\mathbb{C}$, such that the function

$\tau(\mathbf{t}):=\exp{(\mathbf{Q}(\mathbf{t}))}\Theta(t_{1}\mathbf{v}_{0}(a)+t_{2}\mathbf{v}_{1}(a)+t_{3}\mathbf{v}_{2}(a)+t_{4}\mathbf{v_{0}}(b)+\mathbf{d})$

is a tau function for the Schwarzian KP equation 3 in the sense of [4], the
variables $t_{1}$, $t_{2}$, $t_{3}$ being respectively identified with $x$,
$y$, $t$ and $t_{4}$ is an auxiliary parameter.

This conjecture relies on the fact that the Schwarzian KP hierarchy is
generated in a very similar fashion as the classical KP hierarchy [4]. Hence a
tau function for the full Schwarzian KP hierarchy could be of the form
$\tilde{\tau}:=\exp{(\mathbf{Q}(\mathbf{t}))}\Theta(\mathbb{V}\mathbf{t}+\mathbf{d})$,
with $\mathbb{V}$ a $g\times\infty$ matrix whose columns are the vectors
associated to the different ”time” variables, and the higher order
degenerations discussed above could have an explicit and compact form when
expressed with Hirota’s symbols.

##  Appendix A Interesting links with other integrable systems

To allow for an independent reading, we collect in this appendix some facts on
the Schwarzian PDEs appearing naturally in the context of higher order
degenerations of Fay’s identity. The Schwarzian KP, KdV and Boussinesq
equations originally appear in a series of papers by Weiss [23], [24], in an
extensive study of the Painlevé property of many integrable PDEs. These three
equations were shown to be in some sense prototypical PDEs satisfying this
property. They are also linked to some integrable systems appearing in physics
through Bäcklund and Miura transforms. We would like to explain these
relationships in this section.

First we can observe that these equations are invariant under a Möbius
transformation, in the sense that if the function $\phi$ solves the Schwarzian
KP, Boussinesq or KdV equations then $\psi:=\frac{A\phi+B}{C\phi+D}$ with $AD-
BC\neq 0$ is also a solution of these equations. Moreover this transformation
plays the role of the Bäcklund transform for these three equations.
If $\phi(x,y,t)$ solves the Schwarzian KP equation
$\Big{(}\frac{\phi_{t}}{\phi_{x}}+\\{\phi;x\\}\Big{)}_{x}+\Big{(}\frac{\phi_{y}}{\phi_{x}}\Big{)}_{y}+\frac{1}{2}\Big{(}\frac{\phi_{y}^{2}}{\phi_{x}^{2}}\Big{)}_{x}=0$
then the Bäcklund transformed function
$u(x,y,t):=12\frac{\partial^{2}}{\partial x^{2}}\ln{\phi}+v$ with
$v=3\frac{\phi_{xx}^{2}}{\phi_{x}^{2}}-4\frac{\phi_{xxx}}{\phi_{x}}-\frac{\phi_{t}}{\phi_{x}}-\frac{\phi_{y}^{2}}{\phi_{x}^{2}}$
is a solution of the KP equation $u_{tx}+u_{x}^{2}+uu_{xx}+u_{xxxx}+u_{yy}=0$.

Similarly the Schwarzian KdV equation is related to the classical KdV equation
via the same type of Bäcklund transformation: let $\phi(x,t)$ be a solution of
$\frac{\phi_{t}}{\phi_{x}}-\\{\phi;x\\}=\lambda$ then
$u=12\frac{\partial^{2}}{\partial x^{2}}\ln{\phi}+v$ is a solution of
$u_{t}+u_{xxx}+uu_{x}=0$ where $v$ is given by
$v=3\frac{\phi_{xx}^{2}}{\phi_{x}^{2}}-4\frac{\phi_{xxx}}{\phi_{x}}-\frac{\phi_{t}}{\phi_{x}}$
(the very same Bäcklund transformation relates solutions of the Schwarzian
Boussinesq equation to solutions of the classical Boussinesq equation).
Moreover if we set $\lambda=0$ then
$w:=2\frac{\phi_{x}}{\phi}+\frac{\phi_{xx}}{\phi_{x}}$ solves the modified KdV
equation $w_{t}+\frac{\partial}{\partial
x}\big{(}w_{xx}-\frac{w^{3}}{2}\big{)}=0$.
Finally, under the change of variables (Miura transform):

$x\to\phi,\quad t\to t,\quad\phi\to x$

which implies

$\\{\phi;x\\}=-\phi_{x}^{2}\\{x;\phi\\},\quad\phi_{x}=\frac{1}{x_{\phi}},\quad
x_{t}=-\frac{\phi_{t}}{\phi_{x}}.$

Therefore, under this change of variables, the Schwarzian KdV equation becomes

$x_{\phi}^{2}x_{t}=\lambda x_{\phi}^{2}+\\{x;\phi\\},$

or replacing the Schwarzian derivative $\\{x;\phi\\}$ by its explicit
expression,

$x_{t}=\lambda-\frac{1}{2}\Big{(}\frac{1}{x_{\phi}}\Big{)}_{\phi\phi}+\frac{3}{2}\Big{(}\frac{1}{x_{\phi}}\Big{)}_{\phi}^{2}.$

With $v:=x_{\phi}^{-1}$, the previous equation is equivalent to

$v_{t}=v^{3}v_{\phi\phi\phi}$

i.e. the Dym equation.

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inequalities of Step 4 must be sharp. It follows that $\Delta h$ approximately
takes the prescribed values and that $\Delta h\approx u_{2}(x)T_{\Delta
h}u_{2}(y)+u_{2}(y)T_{\Delta h}u_{2}(x)$; but as in Section 7 these two
properties contradict the definitions of $C_{1}$ and $C_{2}$ in Proposition 6.
The rest of the proof of Theorem 4 for $\tau_{k}<e^{k}$ continues as in the
case $k=3$.

We now turn to $\tau_{k}>e^{k}$. A graph is said to be $2$-star-like if all of
its vertices have degree 1 or 2. In particular, all $k$-cycles are 2-star-
like. Theorem 1.1 of [14] then states that the optimizing graphon for
$\tau_{k}$ slightly above $e^{3}$ is bipodal with the parameters $(a,b,c,d)$
taking the approximate forms indicated in Theorem 4, only with errors that are
$o(1)$ or $o(\Delta\tau)$ instead of the $O(\Delta\tau)$ or
$O(\Delta\tau^{2})$ errors claimed in Theorem 4.

All that remains is to sharpen the estimates and compute the entropy. We
follow the same procedure as in Section 9, only with the expressions (76) and
(80) for $\Delta\tau$ replaced with

$\displaystyle\Delta\tau_{k}$ $\displaystyle=$
$\displaystyle\left(\frac{c}{1-c}\right)ke^{k-2}\mu^{2}+ke^{k-3}c^{2}\mu^{2}(\Delta
a+\Delta b-2\Delta d)$ $\displaystyle+ke^{k-4}c^{2}\mu^{2}\left(c(\Delta
a-\Delta d)^{2}+(1-c)(\Delta d-\Delta b)^{2}\right)$
$\displaystyle+\frac{k(k-5)e^{k-4}c^{2}\mu^{4}}{2(1-c)^{2}}+O(c^{3}).$

The first three terms are the multiples of $\langle 1,T_{\delta
g^{2}}1\rangle$, $\langle 1,T_{\delta g^{3}}1\rangle$ and $\langle 1,T_{\delta
g^{4}}1\rangle$, respectively, discussed in Step 1 of the proof for
$\tau<e^{k}$. The last term involves the product of two factors of $\langle
1,T_{\delta g^{2}}1\rangle$ and only occurs when $k\geq 7$. All other terms in
the expansion of $\Delta\tau_{k}$, including $\operatorname{Tr}(\delta
g^{k})$, are of higher order.

The upshot is that the calculation is the same as for triangles, only with a
coefficient of $ke^{k-2}$ instead of $3e$ in the leading term, and with
different $O(c^{2})$ terms. Adjusting the $O(c^{2})$ terms does not affect the
computation of $(a,b,c,d)$ to the order specified in Theorem 4. The change
from $3e$ to $ke^{k-2}$ in the leading term does change the $O(\Delta\tau)$
terms in the expansions of $c$ and $b$, but does not affect the leading
expressions for $d$ or $a$, or the fact that the errors are indeed
$O(\Delta\tau)^{2})$ for $b$ and $c$ and are $O(\Delta\tau)$ for $a$ and $d$.
Plugging these values of $(a,b,c,d)$ into the formula for the entropy then
yields the estimate (30).

### Data availability statement:

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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