# Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz
numbers

Vincent Bouchard<EMAIL_ADDRESS>,  Reinier Kramer
<EMAIL_ADDRESS>and  Quinten Weller<EMAIL_ADDRESS>University of Alberta,
Edmonton, AB, T6G 2G1, Canada  New York University, 726 Broadway, New York, NY
10003, U.S.A.

###### Abstract.

Given a spectral curve with exponential singularities (which we call a
“transalgebraic spectral curve”), we extend the definition of topological
recursion to include contributions from the exponential singularities in a way
that is compatible with limits of sequences of spectral curves. This allows us
to prove the topological recursion/quantum curve correspondence for a large
class of transalgebraic spectral curves. As an application, we find that
Atlantes Hurwitz numbers, which were previously thought to fall outside the
scope of topological recursion, satisfy (our extended version of) topological
recursion, and we construct the corresponding quantum curve directly from
topological recursion.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
      1. 1.1.1 An observation
      2. 1.1.2 A resolution
    2. 1.2 Main results
    3. 1.3 Outline of the paper
    4. 1.4 Notation
    5. 1.5 Acknowledgments
  2. 2 Topological recursion on meromorphic spectral curves
    1. 2.1 Spectral curves
    2. 2.2 Topological recursion
  3. 3 Transalgebraic spectral curves
    1. 3.1 Transalgebraic functions
    2. 3.2 Transalgebraic spectral curves
    3. 3.3 Transalgebraic spectral curves as limits
  4. 4 Topological recursion on transalgebraic spectral curves
    1. 4.1 Rewriting topological recursion
    2. 4.2 Topological recursion on transalgebraic spectral curves
    3. 4.3 Essential singularities only contribute for $n=1$
  5. 5 Quantum curves
    1. 5.1 The topological recursion/quantum curve correspondence
    2. 5.2 Quantum curves for transalgebraic spectral curves
    3. 5.3 A particular example
  6. 6 Topological recursion for Atlantes Hurwitz numbers
    1. 6.1 Hurwitz numbers
      1. 6.1.1 $r$-completed cycles Hurwitz numbers
      2. 6.1.2 Atlantes Hurwitz numbers
    2. 6.2 The projection property for Atlantes Hurwitz numbers
    3. 6.3 Topological recursion for Atlantes Hurwitz numbers
    4. 6.4 Relation between the Atlantes and $r$-completed cycles Hurwitz quantum curves
  7. 7 Conclusion and open questions
  8. A Extensions of [BE17] to the transalgebraic case
  9. B Proof of proposition 6.13

##  1\. Introduction

###  1.1. Motivation

Topological recursion [EO07] is a method to recursively define a collection of
multidifferentials on a given object, called a spectral curve. It was
originally developed to solve loop equations coming from matrix models
[CEO06], but has applications to many other areas of mathematics as well:
among others intersection theory on moduli spaces of curves [EO07, Eyn11,
DOSS14], volumes of moduli spaces [EO07a, And+19], Hurwitz theory [BM08,
EMS11, BHLM14, DKPS19], Gromov–Witten theory [BKMP08, EO15, DOSS14, NS14,
FLZ17, FLZ20, GKLS22], maps [BCDG19], free probability [BCGLS21],
supersymmetric gauge theories [BBCC21], and integrable systems of various
kinds [BEM17, EGMO21, BDKS20a].

The spectral curve of topological recursion is usually taken to be a Riemann
surface $\Sigma$, together with two meromorphic functions $x$ and $y$ on it,
and a symmetric bidifferential $B$. The recursion only depends on the local
behaviour of the spectral curve near the ramification points of $x$, which
were originally required to be simple and regular points of $y$. These
conditions on the ramification points have been lifted to a far higher
generality [BHLMR13, BE13, CN19, DN18, BBCCN18, BKS20], allowing for higher
ramification orders as well as certain poles of $y$.

Although topological recursion itself is local in nature, it behaves better if
the spectral curve is global, i.e. if $x$ and $y$ are meromorphic functions on
a compact Riemann surface. In this case, the functions $x$ and $y$ satisfy a
polynomial relation $P(x,y)=0$, and, according to the topological
recursion/quantum curve correspondence conjecture [BE09, GS11] (see also
[Nor15]), one should be able to quantise such an equation. Explicitly, there
should exist an operator $\hat{P}(\hat{x},\hat{y},\hslash)$, where
$\hat{x}=x\cdot$ and $\hat{y}=\hslash\frac{d}{dx}$, such that
$P=\hat{P}(x,y,0)$, and such that it annihilates the wave function

$\psi(x(z))=\exp\left[\sum_{n=1}^{\infty}\sum_{g=0}^{\infty}\frac{\hslash^{2g+n-2}}{n!}\int^{z}\cdots\int^{z}\left(\omega_{g,n}-\delta_{g,0}\delta_{n,2}\frac{dx(z_{1})dx(z_{2})}{(x(z_{1})-x(z_{2}))^{2}}\right)\right]\,,$
(1)

that is, $\hat{P}\psi=0$. This is a subtle issue, in part due to non-unique
quantisation and integration paths. Nevertheless, the correspondence was
proven to hold for a large class of genus zero algebraic spectral curves with
arbitrary ramification in [BE17], and for all algebraic spectral curves with
simple ramification in [EG19, MO22, EGMO21].

Compact spectral curves exhibit more nice features: topological recursion is
related to intersection theory of the moduli spaces of curves in a general
setup [Eyn11, DOSS14], and in case the spectral curve is compact, the
intersection theory is well-behaved and largely independent of the choice of
bidifferential [DNOPS17].

But what if the spectral curve is of a form where $x$ and $y$ do not satisfy
an algebraic relation? This occurs in a large class of examples, mostly
related to hypergeometric tau-functions and Hurwitz theory, see e.g. [BEMS10,
EMS11, DKPS19, BDKS20, BDKS20a], where the spectral curve is usually of the
form $x(z)=ze^{-\psi(y(z))}$, for some series $\psi(y)$ and $y(z)$. In
important examples, both $\psi$ and $y$ are polynomial, so the function $x$
has an essential singularity. Does the topological recursion/quantum curve
correspondence conjecture hold for these cases?

In certain cases, for instance in [MSS13], and also the more general setup of
[ALS16], it is shown that the differentials $\omega_{g,n}$ produced by
topological recursion are generating series for certain types of Hurwitz
numbers. Using this interpretation, it is then proven that a quantum curve
exists, albeit a fairly complicated one. However, the quantum curves were
constructed from the enumerative interpretation in terms of Hurwitz numbers,
rather than from the spectral curve itself, which is somewhat unsatisfactory.
Can we construct the quantum curves directly from topological recursion, in
the spirit of [BE17, EG19, MO22, EGMO21]?

####  1.1.1. An observation

In fact, this project started with the following observation. Consider
topological recursion on the spectral curve $\mathcal{S}$ given by the
equation

$y-e^{x^{r}y^{r}}=0,$ (2)

for $r\in\mathbb{Z}_{\geq 1}$.111Our spectral curve looks a bit different from
[MSS13, DKPS19], but this is because we are taking the one-form $\omega_{0,1}$
to be $ydx$ instead of $y\frac{dx}{x}$. Ultimately, it is the same spectral
curve. As shown in [DKPS19], the differentials $\omega_{g,n}$ produced by
topological recursion are generating series for $r$-completed cycles (also
called $r$-spin) Hurwitz numbers. Moreover, using the semi-infinite wedge
space interpretation of Hurwitz numbers, it was proven in [MSS13] that these
Hurwitz numbers satisfy a quantum curve in the sense above. The resulting
quantum curve is a quantisation of the spectral curve, but a rather
complicated one:

$\hat{P}=\hat{y}-\hat{x}^{1/2}{\rm
e}^{\frac{1}{r+1}\sum_{i=0}^{r}\hat{x}^{-1}(\hat{x}\hat{y})^{i}\hat{x}(\hat{x}\hat{y})^{r-i}}\hat{x}^{-1/2},$
(3)

with $\hat{x}=x\cdot$ and $\hat{y}=\hslash\frac{d}{dx}$.

To obtain this quantum curve directly from topological recursion, in the
spirit of [BE17], one approach is to consider a sequence of genus zero compact
spectral curves $\mathcal{S}_{N}$, specified by rational functions $x_{N}$ and
$y_{N}$, such that $x_{N}$ and $y_{N}$ go to the above $x$ and $y$ in the
limit $N\to\infty$. (Schematically,
$\lim_{N\to\infty}\mathcal{S}_{N}=\mathcal{S}$.) For all positive integers
$N$, topological recursion produces differentials $\omega_{g,n}^{N}$. From
these differentials, one can construct a wave function $\psi_{N}$ as in (1).
If the genus zero spectral curves for finite $N$ fall within the class studied
in [BE17], then we know right away that there exist quantum curves
$\hat{P}_{N}$ such that $\hat{P}_{N}\psi_{N}=0$ for all positive integers $N$,
and we can construct $\hat{P}_{N}$ explicitly. Finally, we can take the limit
$N\to\infty$ to get a quantum curve $\hat{P}_{\infty}$.

Assuming that the $N\to\infty$ limit of the differentials $\omega_{g,n}^{N}$
recovers the differentials $\omega_{g,n}$ of the $N\to\infty$ spectral curve,
which we could rewrite schematically as the condition

$\lim_{N\to\infty}\left(\omega_{g,n}^{N}[\mathcal{S}_{N}]\right)=\omega_{g,n}\left[\lim_{N\to\infty}\mathcal{S}_{N}\right],$
(4)

then the limiting quantum curve $\hat{P}_{\infty}$ should annihilate the wave
function $\psi$ constructed from the differentials $\omega_{g,n}$ by (1).
However, the quantum curve $\hat{P}_{\infty}$ that we obtain in this way reads

$\hat{P}_{\infty}=\hat{y}-e^{(\hat{x}\hat{y})^{r}},$ (5)

which is not the same as the quantum curve (3) that was obtained for
$r$-completed cycles Hurwitz numbers! Both are quantisations of the same
spectral curve, but they are certainly different, and annihilate different
wave functions. What is going on?

Meanwhile, the quantum curve $\hat{P}_{\infty}$ from (5) already appeared in
the work of [ALS16], where it was proved to annihilate the wave function
constructed from differentials that are generating series for another type of
Hurwitz numbers, known as Atlantes Hurwitz numbers. In fact, it was already
noticed in that paper that the quantum curve $\hat{P}_{\infty}$ from (5) and
the quantum curve $\hat{P}$ from (3) are both quantisations of the same
spectral curve. Since it was known that topological recursion on this spectral
curve produces generating series for $r$-completed cycle Hurwitz numbers, this
observation was taken as an indication that Atlantes Hurwitz numbers fall
outside the scope of topological recursion. To quote [ALS16]: “We have an
example where the dequantization of the quantum curve doesn’t give a spectral
curve suitable for the corresponding topological recursion.” They also state:
“We can conclude that the dequantization of
$\hat{y}-e^{\hat{x}^{r}\hat{y}^{r}}$ cannot be the spectral curve for the
atlantes Hurwitz numbers, suitable for the construction of the topological
recursion.”

But… is this really the end of the story?

####  1.1.2. A resolution

In this paper we resolve this conundrum and propose an explanation for this
observation. The key is that for topological recursion to commute with limits
of sequences of spectral curves as in (4), exponential singularities of the
limiting spectral curve must be taken into account. More precisely, given a
spectral curve with exponential singularities, one can construct differentials
$\omega_{g,n}$ by using topological recursion ignoring the exponential
singularities, as it has been done so far in the literature. But, as we
propose in this paper, one can also construct another set of differentials,
call them $\omega_{g,n}^{\infty}$, using an extension of topological recursion
that includes contributions from exponential singularities (informally
considered as “ramification points of infinite order”). In general, for a
given spectral curve with exponential singularities, the differentials
$\omega_{g,n}^{\infty}$ and $\omega_{g,n}$ will be distinct. It turns out
that, as we prove in this paper, topological recursion commutes with limits of
sequences of spectral curves as in (4) only if the differentials on the right-
hand-side are the differentials $\omega_{g,n}^{\infty}$ that include
contributions from exponential singularities.

This explains the observation above. For the spectral curve (2), the
differentials $\omega_{g,n}$ that ignore the exponential singularities are
generating series for $r$-completed cycles Hurwitz numbers, as shown in
[DKPS19]. However, as we show in the current paper, the differentials
$\omega_{g,n}^{\infty}$ that include contributions from the exponential
singularity are generating series for Atlantes Hurwitz numbers. This shows
that Atlantes Hurwitz numbers do fall within the scope of topological
recursion, once the formalism is properly extended to include contributions
from exponential singularities.

Furthermore, since we show that topological recursion (properly extended to
include contribution from exponential singularities) commutes with limits of
sequences of spectral curves, we obtain directly that the wave function
$\psi_{\infty}$ constructed from the differentials $\omega_{g,n}^{\infty}$ is
annihilated by the quantum curve $\hat{P}_{\infty}$ from (5). This provides a
construction of the quantum curve for Atlantes Hurwitz numbers directly from
topological recursion, and explains why it differs from the quantum curve for
$r$-completed cycles Hurwitz numbers that was obtained in [MSS13].

###  1.2. Main results

We propose an extension of topological recursion that includes contributions
from exponential singularities of the spectral curves, i.e. points
$p\in\Sigma$ where $x(z)\sim M_{0}(z)e^{M_{1}(z)}$ as $z\to p$ and $M_{0}$,
$M_{1}$ are meromorphic functions with $p$ a pole of $M_{1}$. We will consider
these singularities as ramification points of infinite order. Such functions
are called “transalgebraic”, and hence we will call these spectral curves
“transalgebraic spectral curves”.

As the topological recursion formula involves sums over local deck
transformations, infinite order ramification points require infinite sums, and
this leads to multiple issues; chief amongst these is the definition of the
residue at what may not be an isolated singularity. Instead of dealing with
these issues directly, we construct topological recursion on transalgebraic
spectral curves as a limit of topological recursion on sequences of finite
degree, meromorphic, spectral curves, as in (4). For this definition to make
sense, we need to make sure that the $N\to\infty$ limit of the differentials
exist and satisfies desired properties, which we do.

Furthermore, we show that while the definition of topological recursion on
transalgebraic spectral curves is fairly complicated (having to do with limits
of sequences of spectral curves, although we do provide a formula applicable
in some specific cases), in the end it is not too bad: for any given
transalgebraic spectral curve, the formal definition has to be used only
finitely many times, after which the exponential singularity does not
contribute extra terms anymore.

With this construction in hand, we study the topological recursion/quantum
curve correspondence, with the aim of constructing quantum curves directly
from topological recursion for transalgebraic spectral curves. For a subclass
of transalgebraic curves, which we call regular, we adapt the argument of
[BE17] to construct the quantum curves associated to these transalgebraic
spectral curves.

As an application, we show that Atlantes Hurwitz numbers, which were
introduced in [ALS16] as an example of Hurwitz numbers not satisfying
topological recursion, do fit in our transalgebraic framework. We show that
the differentials constructed from topological recursion (suitably extended to
include contributions from exponential singularities) on the spectral curve
(2) are generating series for Atlantes Hurwitz numbers (while the
differentials constructed from the usual topological recursion that ignores
exponential singularities on the “same” spectral curve222In fact, as we will
explain, it is better to think of these two spectral curves as distinct
spectral curves, in the following sense. While the functions $x$ and $y$, as
well as the symmetric bidifferential $B$, are formally the same “rules”, the
two spectral curves have two different Riemann surfaces. For the spectral
curve for $r$-completed cycles Hurwitz numbers, we take the Riemann surface to
be $\Sigma=\mathbb{C}$, with the exponential singularity of $x$ and $y$ at
infinity removed, while for the spectral curve for Atlantes Hurwitz numbers we
take the Riemann surface to be $\Sigma=\mathbb{P}^{1}$, which includes the
exponential singularity (see Examples 2.6 and 3.8). are generating series for
$r$-completed cycles Hurwitz numbers). Finally, we prove that the
corresponding wave function is annihilated by the quantum curve (5) directly
from topological recursion.

###  1.3. Outline of the paper

In Section 2 we define spectral curves and review the topological recursion
framework for meromorphic spectral curves with arbitrary ramification. In
Section 3 we define transalgebraic spectral curves and explain how they can be
realized as limits of sequences of meromorphic spectral curves. We then
proceed in Section 4 with the definition of topological recursion on
transalgebraic spectral curves and prove various properties of this extension
of topological recursion, including the fact that essential singularities
contribute to topological recursion only in a finite number of steps.

In Section 5 we prove the topological recursion/quantum curve correspondence
for a large class of transalgebraic spectral curves, which we call regular.
This class includes the spectral curve for $r$-completed cycles and Atlantes
Hurwitz numbers. We focus on this particular curve in Section 6, where we show
that topological recursion on this transalgebraic spectral curve (including
the exponential singularity) produces generating functions for Atlantes
Hurwitz numbers. We conclude with open questions in Section 7.

Appendix A provides the extension of the results of [BE17] needed to study the
topological recursion/quantum curve correspondence for transalgebraic spectral
curves, while Appendix B contains the proof of proposition 6.13 about Atlantes
Hurwitz numbers.

###  1.4. Notation

We set $\mathcal{S}(z)=\frac{\sinh(z/2)}{z}$.

For a natural number $n$, we define $\llbracket
n\rrbracket\coloneqq\\{1,\dotsc,n\\}$. Given a set $S$ indexed by another set
$I$, i.e $S=\\{s_{i}\,|\,i\in I\\}$, and given a subset $J\subseteq I$, we
denote $s_{J}\coloneqq\\{s_{i}\,|\,i\in J\\}$. In particular, $s_{I}=S$. For a
set $Z$, we use the notation $\mu\vdash Z$ to indicate that $\mu$ is a set
partition of $Z$; the length of this partition is then denoted by $l(\mu)$.

For a curve $\Sigma$ with coordinate $z$, we denote the induced coordinates on
$\Sigma^{n}$ by $\\{z_{1},\dotsc,z_{n}\\}$. With the previously given
notation, this can be denoted $z_{\llbracket n\rrbracket}$.

Given $C=\\{z\\}\cup z_{\llbracket i\rrbracket}\subset\Sigma$ and
$C^{\prime}=C\setminus\\{z\\}$ sets of cardinality $n+1$ and $n$,
respectively, take a symmetric $n$-differential $\eta$ and define

$\operatorname*{Res}_{C=z}\eta(z_{\llbracket
n\rrbracket})\coloneqq\operatorname*{Res}_{C^{\prime}=z}\eta(z_{\llbracket
n\rrbracket})\coloneqq\operatorname*{Res}_{z_{1}=z}\cdots\operatorname*{Res}_{z_{n}=t}\eta(z_{\llbracket
n\rrbracket})\,.$ (6)

Since $\eta$ is symmetric, this notation makes sense as the order in which we
take the residues does not matter. In a similar spirit, we also define

$\operatorname*{Res}_{z=C^{\prime}}\coloneqq\sum_{z_{0}\in
C^{\prime}}\operatorname*{Res}_{z=z_{0}}\,.$ (7)

For any set of points $C\subset\Sigma$ we denote by $z^{C}$ one arbitrarily
chosen point in this set. Lastly, as we will have to take many residues at
once, we define, for fixed points $a_{1},\dots,a_{n}\in\Sigma$

$\operatorname*{Res}_{\begin{subarray}{c}z_{l}=a_{l}\\\
l=1,\dots,n\end{subarray}}\coloneqq\operatorname*{Res}_{z_{1}=a_{1}}\cdots\operatorname*{Res}_{z_{n}=a_{n}}\,,$
(8)

along with the obvious generalisations to our residues over sets notations.

###  1.5. Acknowledgments

The authors acknowledge support from the National Science and Engineering
Research Council of Canada. R.K. acknowledges support from the Pacific
Institute for the Mathematical Sciences. The research and findings may not
reflect those of these institutions. The University of Alberta respectfully
acknowledges that we are situated on Treaty 6 territory, traditional lands of
First Nations and Métis people.

##  2\. Topological recursion on meromorphic spectral curves

###  2.1. Spectral curves

One of the main goals of this paper is to extend the definition of topological
recursion to spectral curves with exponential singularities, which we will
call transalgebraic spectral curves. Let us start by recalling the usual
formulation of topological recursion, in the Bouchard-Eynard formalism [BE13],
which extends the original Eynard-Orantin formalism [EO07] to higher order
ramification. We start with the definition of a spectral curve.

######  Definition 2.1.

A _spectral curve_ is a quadruple $\mathcal{S}=(\Sigma,x,y,B)$, where:

  1. (1)

$\Sigma$ is a Riemann surface;

  2. (2)

$x$ and $y$ are functions on $\Sigma$ that are holomorphic except potentially
at a finite number of points and that separate points, i.e. $(x,y)$ is
injective;

  3. (3)

$B$ is a symmetric bi-differential on $\Sigma\times\Sigma$ with a double pole
on the diagonal with biresidue $1$.

We say that the spectral curve is _meromorphic_ if $x$ and $y$ are meromorphic
on $\Sigma$ and the ramification locus $R$ of $x$, which is the set of zeros
of $dx$ and poles of $x$ of order $\geq 2$, is finite.

The usual Bouchard-Eynard formalism will correspond to the case of meromorphic
spectral curves, whereas our extension will be to a certain class of non-
meromorphic spectral curves. Let us now review some of the basic features of
meromorphic spectral curves.

We write $r_{a}$ for the ramification order of $x$ at $a\in R$. For a point
$z\in\Sigma$, we write $\mathfrak{f}(z)=x^{-1}(x(z))$ for the fibre, and
$\mathfrak{f}^{\prime}(z)=\mathfrak{f}(z)\setminus\\{z\\}$. Also, if $a\in R$
and $z$ is close to $a$, we write $\mathfrak{f}_{a}(z)$ for the local Galois
conjugates of $z$, and again
$\mathfrak{f}^{\prime}_{a}(z)=\mathfrak{f}(z)\setminus\\{z\\}$. We note that
while $\mathfrak{f}_{a}(z)$ is always finite of cardinality $r_{a}$,
$\mathfrak{f}(z)$ may be countably infinite, as $x:\Sigma\to\mathbb{P}^{1}$ is
not necessarily a finite branched covering if $\Sigma$ is non-compact.

For $a\in R$, define a local coordinate $\zeta_{a}$ on a neighbourhood of $a$
by $x=x(a)+\zeta_{a}^{r_{a}}$ if $x(a)\neq\infty$ and $x=\zeta_{a}^{-r_{a}}$
if $x(a)=\infty$. Then the one-form $\omega_{0,1}\coloneqq ydx$ has an
expansion:

$\omega_{0,1}\coloneqq ydx=\sum_{k=-l}^{\infty}t^{a}_{k}\zeta_{a}^{k-1}dt_{a}$
(9)

for some $l$ and $t^{a}_{k}$. Let $s_{a}\coloneqq\min\\{k\,|\,t^{a}_{k}\neq
0\text{ and }r_{a}\nmid k\\}$. The following admissibility condition on
spectral curves is required to make sense of topological recursion [BBCCN18]:

######  Definition 2.2.

A meromorphic spectral curve is _admissible_ if for every point $a\in R$,
$s_{a}$ and $r_{a}$ are coprime, and either $s_{a}\leq-1$, or $1\leq s_{a}\leq
r_{a}+1$ with $r_{a}=\pm 1\pmod{s_{a}}$.

######  Remark 2.3.

The ramification points $a\in R$ with $s_{a}\leq-1$ never contribute to the
Eynard-Bouchard topological recursion [BBCCN18]. This means that any
ramification point $a$ that is a pole of $x$ may be dropped from the
topological recursion unless $dy$ has a zero at $a$. As a result it is common
practice in the topological recursion literature to sloppily refer to the set
of zeros of $dx$ as all the ramification points of $x$ [EO07, BE13]. We will
see that, for our purposes, it is critical to include the poles of $x$ of
order greater than two as ramification points.

A particularly interesting class of spectral curves is when the Riemann
surface $\Sigma$ is compact.

######  Definition 2.4.

A spectral curve is _compact_ if $\Sigma$ is a compact connected Riemann
surface and $B(z_{1},z_{2})$ has no poles except on the diagonal
$z_{1}=z_{2}$.333This condition on $B$ is equivalent to requiring that $B$ be
normalised on a chosen Torelli marking of the compact Riemann surface
$\Sigma$.

Compact meromorphic spectral curves have nice geometric properties. If
$\Sigma$ is compact, $x$ and $y$ are two meromorphic functions on a compact
Riemann surface, and hence they identically satisfy an algebraic equation

$P(x,y)=0,$ (10)

where $P$ is a polynomial. (Note that if $\Sigma$ is non-compact, $x$ and $y$
may still satisfy a relation as above, but $P$ could no longer be a polynomial
– see example 2.6.)

We also note note that for compact meromorphic spectral curves, since $x$ is a
meromorphic function on a compact Riemann surface $\Sigma$,
$x:\Sigma\to\mathbb{P}^{1}$ is a finite degree branched covering. This means
that $\mathfrak{f}(z)$ is finite and of cardinality given by the degree of $x$
if $z\notin R$.

######  Example 2.5.

Consider the spectral curve $\mathcal{S}$ with $\Sigma=\mathbb{P}^{1}$,
$x=z^{r}$, $y=z^{s-r}$, and $B=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$, with
$r,s$ integers such that $r\geq 2$, $1\leq s\leq r+1$, and $r=\pm 1\pmod{s}$.
This is a compact meromorphic spectral curve, and $x$ and $y$ satisfy the
algebraic equation $x^{r-s}y^{r}-1=0$ if $s<r$, or $y^{r}-x=0$ if $s=r+1$. One
can also check that the spectral curve is admissible. In particular, this is
the fundamental $(r,s)$ spectral curve studied in [BBCCN18].

######  Example 2.6.

Consider the spectral curve $\mathcal{S}$ with $\Sigma=\mathbb{C}$,
$x=ze^{-z^{r}}$, $y=e^{z^{r}}$, and
$B=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$, with $r\in\mathbb{Z}_{\geq 1}$. As
the function $x$ is meromorphic on $\mathbb{C}$ (it is in fact holomorphic),
this spectral curve is meromorphic, but it is not compact. One can check that
it is admissible. The functions $x$ and $y$ satisfy the relation

$y-e^{x^{r}y^{r}}=0,$ (11)

which is not algebraic. This spectral curve will play an important role in the
following. As proven in [DKPS19], the differentials $\omega_{g,n}$ produced by
topological recursion on this spectral curve are generating functions for
$r$-completed cycles Hurwitz numbers. We will come back to this enumerative
geometric interpretation in section 6.444To avoid confusion, we remark that
for this spectral curve $y$ is often define in the literature via
$\omega_{0,1}=yd\log x$ instead of $\omega_{0,1}=ydx$, which gives $y=z$
instead of $y=e^{z^{r}}$. The two definitions are of course equivalent, as it
simply amounts to redefining $y\mapsto xy$. In this paper, for all spectral
curves, the one-form will be defined as $\omega_{0,1}=ydx$.

######  Remark 2.7.

We emphasize here that the choice of Riemann surface $\Sigma$ in the
definition of a spectral curve is very important. For instance, if we replace
the Riemann surface $\mathbb{P}^{1}$ by $\mathbb{C}$ in example 2.5, it should
be considered as a different spectral curve, since the pole of $x$ at infinity
is not included in the Riemann surface. In this case, as $x$ is meromorphic on
$\mathbb{P}^{1}$ and holomorphic on $\mathbb{C}$, the usual topological
recursion (which applies to meromorphic spectral curves) can be used to
calculate correlators $\omega_{g,n}$ for both spectral curves, and it happens
that the correlators coincide, since the pole at infinity does not contribute
to the topological recursion. But this will not always be the case.

For instance, one may want to consider the spectral curve of example 2.6, but
with the Riemann surface $\Sigma=\mathbb{P}^{1}$, which includes the
exponential singularity of $x$ at infinity. We claim that the usual
topological recursion does not apply in this case, as it only applies to
meromorphic spectral curves; instead, one should use the extended version that
we propose, in which the exponential singularity generally contributes. As we
will see, with this extended definition topological recursion produces
different correlators (for $r\geq 2$) depending on whether the essential
singularity at infinity is included in the Riemann surface or not, i.e.
whether $\Sigma$ is taken to be $\mathbb{P}^{1}$ or $\mathbb{C}$ in the
spectral curve of example 2.6.

###  2.2. Topological recursion

The standard definition of topological recursion applies to admissible
meromorphic spectral curves. It does not however require the spectral curve to
be compact; for instance, it can be applied to both spectral curves in
examples 2.5 and 2.6. Out of the data of the spectral curve, a collection of
symmetric differentials $\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$ are recursively
computed. As mentioned in the introduction, topological recursion is
interesting because for many spectral curves, the correlators $\omega_{g,n}$
that it produces are generating functions for interesting enumerative
invariants, such as Hurwitz numbers, Gromov-Witten invariants, etc.

For the definition of (Bouchard–Eynard) topological recursion, recall the
notation given in section 1.4.

######  Definition 2.8.

Given an admissible meromorphic spectral curve $\mathcal{S}=(\Sigma,x,y,B)$,
_topological recursion_ gives a procedure to define multi-differentials
$\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$, recursive on $2g-2+n$, as follows: the
base cases are $\omega_{0,1}=ydx$ and $\omega_{0,2}=B$, and the recursive step
is

$\omega_{g,n+1}(z_{0},z_{\llbracket n\rrbracket})\coloneqq\sum_{a\in
R}\operatorname*{Res}_{z=a}\sum_{\emptyset\neq
Z\subseteq\mathfrak{f}^{\prime}_{a}(z)}K_{|Z|+1}(z_{0},z,Z)\mathcal{W}_{g,n,|Z|+1}(z,Z\mid
z_{\llbracket n\rrbracket})\,,$ (12)

where

$K_{|Z|+1}(z_{0},z,Z)\coloneqq\frac{\int_{*}^{z}B(z_{0},\cdot)}{\prod_{z^{\prime}\in
Z}(y(z^{\prime})-y(z))dx(z)}$ (13)

is the _recursion kernel_ and

$\mathcal{W}_{g,n,|Z|}(Z\mid z_{\llbracket
n\rrbracket})\coloneqq\sum_{\begin{subarray}{c}\mu\vdash Z\\\
\bigsqcup_{k=1}^{\ell(\mu)}N_{k}=\llbracket n\rrbracket\\\ \sum
g_{k}=g+\ell(\mu)-n\end{subarray}}^{\prime}\prod_{k=1}^{\ell(\mu)}\omega_{g_{k},|\mu_{k}|+|N_{k}|}(\mu_{k},z_{N_{k}})\,,$
(14)

where the prime on the summation means we omit any term containing
$\omega_{0,1}$. The differentials $\omega_{g,n}$ are often called
_correlators_ due to their origin in matrix models.

For future reference, we also define

$\mathcal{E}_{g,n,|Z|}(Z\mid z_{\llbracket
n\rrbracket})\coloneqq\sum_{\begin{subarray}{c}\mu\vdash Z\\\
\bigsqcup_{k=1}^{\ell(\mu)}N_{k}=\llbracket n\rrbracket\\\ \sum
g_{k}=g+\ell(\mu)-n\end{subarray}}\prod_{k=1}^{\ell(\mu)}\omega_{g_{k},|\mu_{k}|+|N_{k}|}(\mu_{k},z_{N_{k}})\,,$
(15)

where terms containing $\omega_{0,1}$ are now included.

Topological recursion was originally obtained as the unique solution to the
loop equations of matrix models that satisfies a particular normalisation
condition. This can be defined more mathematically, in terms of the so-called
abstract loop equations, with the normalisation condition often called the
“projection property”.

######  Proposition 2.9 ([BBCCN18, Appendix C]).

A collection of multidifferentials $\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$
satisfies topological recursion if and only if it satisfies the _higher
abstract loop equations_ :

$\sum_{\begin{subarray}{c}Z\subset\mathfrak{f}_{a}(z)\\\
|Z|=i\end{subarray}}\mathcal{E}_{g,n,i}(Z\,|\,z_{\llbracket
n\rrbracket})=\mathcal{O}\Big{(}z^{-r_{a}\mathfrak{d}_{a}^{i}}\big{(}\frac{dz}{z}\big{)}^{i}\Big{)}\quad\text{as
}z\to a\,,$ (16)

where

$\mathfrak{d}_{a}^{i}\coloneqq-1-\Big{\lfloor}\frac{s_{a}(i-1)}{r_{a}}\Big{\rfloor}\,,$
(17)

and the _projection property_ : if $2g-2+n\geq 0$,

$\omega_{g,n+1}(z_{0},z_{\llbracket n\rrbracket})=\sum_{a\in
R}\operatorname*{Res}_{z=a}\Big{(}\int_{a}^{z}\omega_{0,2}(z_{0},\mathord{\cdot})\Big{)}\omega_{g,n+1}(z,z_{\llbracket
n\rrbracket})$ (18)

Collections $(\omega_{g,n})_{g,n}$ satisfying the projection property may also
be called _polarised_ , and the choice of $\omega_{0,2}=B$ is sometimes
referred to as a choice of polarisation.

Another property of the multi-differentials, which seems not to have appeared
in the literature in this generality, is a bound on their pole order. We give
that property in the next lemma, which generalises [BE13, Proposition 9] and
[DN18, Section 7].

######  Lemma 2.10.

Let the multi-differentials $\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$ be obtained
by topological recursion on the admissible meromorphic spectral curve
$\mathcal{S}=(\Sigma,x,y,B)$. Then, the pole order of $\omega_{g,n}$ in each
variable at a point $a\in\Sigma$ is bounded by $(s_{a}-1)(2g-2+n)+2g$.

###### Proof.

The proof is exactly the same as that of [BE13, Proposition 9], which is the
$s_{a}=r_{a}+1$ case, noting that the pole order of the $k$-th recursion
kernel in general is $(k-1)(s_{a}-1)$, by the definition of $s_{a}$.

Note that the original proof does not handle the possibility of
$\omega_{0,2}$s in $\mathcal{W}^{g}_{k,n}$ with both arguments coupled to the
kernel in the recursive step. Each such occurrence adds a double pole, so if
we call the number of these occurrences $b$, then the pole order of each term
in $\mathcal{W}^{g}_{k,n}$ is at most

$(s_{a}-1)(2(g+\ell(\mu)-k)-2k+n+k)+2(g+\ell(\mu)+b-k)\,.$

Then we can use that $\ell(\mu)+b\leq k$, by the definition of $b$, so this
can be bounded by $(s_{a}-1)(2g-k+n)+2g$, which is the same as found without
the diagonal poles, so this omission does not change the pole order. ∎

##  3\. Transalgebraic spectral curves

Our goal is to extend the definition of topological recursion to spectral
curves where $x$ is not meromorphic on the Riemann surface $\Sigma$. More
precisely, we wish to study spectral curves where $x$ has exponential
singularities at some isolated points on $\Sigma$. The key example to keep in
mind is example 2.6: we want to consider this spectral curve, but with the
Riemann surface $\Sigma=\mathbb{P}^{1}$, which includes the exponential
singularity of $x$ at infinity.

###  3.1. Transalgebraic functions

Let us define more precisely the class of functions that we are interested in.
We first define exponential singularities.

######  Definition 3.1 ([BP15b, Pér19]).

Let $\Sigma$ be a Riemann surface and $z\in\Sigma$. A function $f$ is said to
have an _exponential singularity_ at $z$ if it is holomorphic and non-zero on
some punctured open neighbourhood $U\setminus\\{z\\}$ of $z$, but cannot be
extended to a meromorphic function on $U$.

The _exponential order_ of $f$ at $z$ is defined to be

$\operatorname{Erd}_{f}(z)=\inf\big{\\{}d\in\mathbb{R}_{\geq
0}\,\big{|}\,\limsup_{w\to z}|w-z|^{d}\log|f(w)|<\infty\big{\\}}\,.$ (19)

A transalgebraic function on $\Sigma$ is a function that is holomorphic on
$\Sigma$ except potentially at a finite number of points, where it can have
either poles or exponential singularities. As such, it is a natural
generalization of meromorphic functions, where we allow not only poles but
also exponential singularities. More precisely:

######  Definition 3.2 ([Pér19]).

Let $\Sigma$ be a Riemann surface. Let $\mathcal{T}_{n}(\Sigma)$ be the space
of _transalgebraic functions_ on $\Sigma$ with at most $n\in\mathbb{Z}_{\geq
0}$ zeros, poles, and exponential singularities, which consists of all non-
zero holomorphic functions on $\Sigma\setminus S$ for some $S\subset\Sigma$
such that $|S|\leq n$ and such that for any $z\in S$,
$\operatorname{Erd}_{f}(z)<\infty$.

We define the _class of transalgebraic functions_ on $\Sigma$ as

$\mathcal{T}(\Sigma)=\bigcup_{n\in\mathbb{Z}_{\geq 0}}\mathcal{T}_{n}(\Sigma)$
(20)

It is a natural question to ask why we insist that there are finitely many
zeros and poles in the previous definition, but it is rather straightforward
to see that this condition is required for ramification points to be isolated.
Of course, by the great Picard theorem, all points in $\mathbb{P}^{1}$ are
obtained infinitely often as one approaches an essential singularity save for
possibly two points and, fixing an affine coordinate on $\mathbb{P}^{1}$, we
may put those two points (if they exist) at zero and infinity via a change of
coordinates. The following proposition tells us that if we didn’t have exactly
two such points, occasionally called _Picard points_ in the literature, then
the exponential singularity would be a cluster point of ramification points;
it seems natural to exclude such things from the perspective of the
topological recursion.

######  Proposition 3.3.

Let $\pi:B_{\varepsilon}(0)\setminus\\{0\\}\to\mathbb{P}^{1}$ be a branched
covering from the punctured disk of radius $\varepsilon>0$ to $\mathbb{P}^{1}$
with an exponential singularity at zero. Assume the only Picard point of $\pi$
is infinity, and if infinity is a Picard point $\varepsilon$ is small enough
that $\pi$ never takes the value infinity. Then $\pi$ has a ramification point
in $B_{\varepsilon}(0)\setminus\\{0\\}$.

###### Proof.

Proceed by contradiction and assume $\pi:B_{\varepsilon}(0)\setminus\\{0\\}\to
C$ is an honest covering map where $C=\pi(B_{\varepsilon}(0)\setminus\\{0\\})$
is either $\mathbb{C}$ or $\mathbb{C}_{\infty}\cong\mathbb{P}^{1}$. As $C$ is
simply connected the monodromy group of $\pi$ is trivial. Ergo, there exists a
right inverse map (non-unique) $\pi^{-1}:C\to
B_{\varepsilon}(0)\setminus\\{0\\}$ so that $\pi\circ\pi^{-1}={\rm
id}_{B_{\varepsilon}(0)\setminus\\{0\\}}$. If we define the image of
$\pi^{-1}$ to be $B$ then the map $\pi^{\prime}:C\to B$ with
$\pi^{\prime}\equiv\pi^{-1}$ is biholomorphic and, in particular, a
homeomorphism so $B$ is simply connected; therefore, ${\pi^{\prime}}^{-1}:B\to
C$ is the universal cover of $C$ so there exists a covering map $\varphi:B\to
B_{\varepsilon}(0)\setminus\\{0\\}$ such that
$\pi\circ\varphi={\pi^{\prime}}^{-1}$. As ${\pi^{\prime}}^{-1}$ has the
inverse $\pi^{\prime}$, it is bijective so $\varphi$ must be injective. As
$\varphi$ is a covering map, it must be surjective so it is in fact a
homeomorphism. Thus, $B_{\varepsilon}(0)\setminus\\{0\\}$ is simply connected,
an obvious contradiction. ∎

Thus, if we only have one (or zero) Picard points then, for any
$\varepsilon>0$ sufficiently small, we get a ramification point in the disk of
radius $\varepsilon$ about the exponential singularity so that the exponential
singularity must be an accumulation point of ramification points.

It turns out that transalgebraic functions on compact Riemann surfaces have a
very simple form.

######  Theorem 3.4 ([Pér19, Theorem 2.17]).

The space of transalgebraic functions on a compact Riemann surface $\Sigma$ is
equal to the space of functions of the form

$f(z)=M_{0}(z)e^{M_{1}(z)}\,,$ (21)

where $M_{0}$ and $M_{1}$ are meromorphic functions on $\Sigma$ and $M_{0}\neq
0$.

The choice of $M_{0}$ and $M_{1}$ in (21) is not quite unique: we can add to
$M_{1}$ a (local) constant $c$, and multiply $M_{0}$ by $e^{-c}$ without
changing $f$.

Given a transalgebraic function $f$ on $\Sigma$, its differential $df$ is not
usually a meromorphic one-form. However, it turns out that $d\log f$ always
is, which is the content of the next lemma.

######  Proposition 3.5 ([Pér19, Lemma 2.15]).

Let $f\in\mathcal{T}(\Sigma)$. The logarithmic derivative $d\log f$ is a
meromorphic differential on $\Sigma$ with integer residues. Conversely, if $f$
is function on $\Sigma$ which is non-zero holomorphic outside a finite set and
is such that $d\log f$ is meromorphic with integer residues at poles, then
$f\in\mathcal{T}(\Sigma)$.

For more on transalgebraic functions and their underlying geometry (including
the so-called log-Riemann surfaces), see [BP15, BP15a, BP15b, Pér19].

###  3.2. Transalgebraic spectral curves

Looking back at the definition of spectral curves definition 2.1, $x$ and $y$
were only required to be holomorphic at all but finitely many of points. In
particular, they may be transalgebraic functions on $\Sigma$.

######  Definition 3.6.

Let $\mathcal{S}=(\Sigma,x,y,B)$ be a spectral curve. We say that it is
_transalgebraic_ if $x$ and $y$ are transalgebraic functions on $\Sigma$ such
that $xy$ is a meromorphic function on $\Sigma$.

Because of the requirement that $xy$ is meromorphic, the one-form
$\omega_{0,1}=ydx$ is meromorphic on $\Sigma$, since by proposition 3.5 we
know that $\frac{dx}{x}$ is always meromorphic and by assumption $xy$ is
meromorphic. Interestingly, all correlators $\omega_{g,n}$ produced by the
topological recursion will still be meromorphic.

For topological recursion, we would like to consider $x$ as a branched
covering $x:\Sigma\to\mathbb{P}^{1}$. Even though $x$ is transalgebraic on
$\Sigma$, it can still be thought of as a branched covering [BP13]. However,
if $x$ has exponential singularities, the covering will not be of finite
degree. Nevertheless, it makes sense, and we can define its ramification locus
as follows.

######  Definition 3.7.

Let $x$ be a transalgebraic function on $\Sigma$. If $x$ is not meromorphic,
we define its _degree_ to equal $\infty$. We consider all the exponential
singularities $a$ of $x$ as ramification points, with ramification order
$r_{a}=\infty$. We write $R_{\infty}$ for the collection of exponential
singularities, $R_{0}$ for the collection of finite ($r_{a}<\infty$)
ramification points, and note $R=R_{0}\cup R_{\infty}$. Following [BP15b], we
will sometimes refer to the elements of $R_{\infty}$ as infinite ramification
points.

We now focus on compact transalgebraic spectral curves, for which $x$ is a
transalgebraic function on a compact Riemann surface $\Sigma$. In this case,
by Theorem 3.4, we can write

$x(z)=M_{0}(z)e^{M_{1}(z)}$ (22)

for some meromorphic functions $M_{0}$ and $M_{1}$ on $\Sigma$ with $M_{0}\neq
0$. We also define $M_{2}(z)\coloneqq x(z)y(z)$, which is another meromorphic
function on $\Sigma$. We will use the $M_{0},M_{1}$ and $M_{2}$ notation
throughout the paper.

######  Example 3.8.

The typical example of a compact transalgebraic spectral curve is example 2.6
but with $\Sigma=\mathbb{P}^{1}$. In other words, we consider the spectral
curve $\mathcal{S}_{\infty}=(\Sigma,x,y,B)$ with $\Sigma=\mathbb{P}^{1}$,
$x=ze^{-z^{r}}$, $y=e^{z^{r}}$, and
$B=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$, where $r\in\mathbb{Z}_{\geq 1}$.
In this case, $M_{0}(z)=z$, $M_{1}(z)=-z^{r}$, and $M_{2}(z)=x(z)y(z)=z$.
$R_{0}$ contains the $r$ finite ramification points at the solutions of
$z^{r}=\frac{1}{r}$, while $R_{\infty}$ has a single point at $\infty$.

In topological recursion we are interested in the behaviour of $x$ near its
ramification points. Around a finite ramification point in $R_{0}$, the
behaviour of $x$ is, locally at least, identical to meromorphic curves. So let
us focus on the exponential singularities $a\in R_{\infty}$, which correspond
to the poles of $M_{1}(z)$.

######  Lemma 3.9.

Let $a\in R_{\infty}$. Suppose that $M_{1}$ has a pole of order $m_{1}$ at
$a$, and that $M_{0}$ has either a zero of order $m_{0}$ (for $m_{0}\geq 0$)
or a pole of order $|m_{0}|$ (for $m_{0}<0$) at $a$. Then there exists a local
coordinate $\zeta$ near $a$ such that

$x(\zeta)=e^{\zeta^{-m_{1}}},m_{0}=0,\quad
x(\zeta)=z^{m_{0}}e^{-\frac{m_{0}}{m_{1}}z^{-m_{1}}},m_{0}\neq 0.$ (23)

###### Proof.

If $m_{0}=0$, this is obvious as near $a$ $\log(x)$ is a well defined
meromorphic function with a pole of order $m_{1}$ at $a$, so there exists a
local coordinate such that $\zeta^{-m_{1}}=\log(x(\zeta))$. If $m_{0}\neq 0$
we may take $z$ as the coordinate such that, near $a$, $M_{0}(z)=z^{m_{0}}$.
We can then solve the relation

$M_{1}(z)+\frac{m_{0}}{m_{1}}\zeta^{-m_{1}}-m_{0}\log\left(\zeta/z\right)=0,$
(24)

recursively for the coefficients $\zeta=\sum_{n\geq 1}a_{n}z^{n}$ where
$a_{1}\neq 0$. This constructs $\zeta$ in terms of $z$. ∎

However, unlike in the case of finite ramification points we see that there
are infinitely many different choices of $\zeta$ corresponding to the branch
choices for the $m_{1}$th root and the logarithm. In other words,
$\mathfrak{f}_{a}(z)$ is countably infinite. Moreover, unlike the finite case,
even in the local coordinate $\zeta$, the local deck transformations have no
simple expression in terms of elementary functions (except when $m_{0}=0$).
One can derive series expansions around a ramification point of the form (in
terms of the local coordinate $\zeta$)

$\sum_{n\geq 0}s_{n}\zeta^{nm_{1}+1},$ (25)

where $s_{0}$ is an $m_{1}$th root of unity and $s_{1}$ is
$s_{0}\log(s_{0}^{m_{0}})/m_{0}$ for some branch choice of the logarithm and
$m_{0}\neq 0$ (the explicit expansion is given below for $m_{0}=0$). Here,
different choices of the $m_{1}$th root of unity and different branches of the
logarithm will yield different local deck transformations. The radius of
convergence of these series will depend on the choice of logarithm; as we will
see shortly, there is no open set on which all such expansions converge.

When $m_{0}=0$ we may explicitly solve for the deck transformations. Indexing
by $k\in\mathbb{Z}$ and $m=0,1,\dots,m_{1}-1$ and then denoting the deck
transformations as $\sigma_{a}^{k,m}$ we find

$\sigma_{a}^{k,m}(z)=\frac{\vartheta^{m}\zeta}{\left(1+2\pi
ik\zeta^{m_{1}}\right)^{1/m_{1}}}\stackrel{{\scriptstyle\zeta\to
0}}{{=}}\vartheta^{m}\zeta\sum_{n=0}^{\infty}\binom{-1/m_{1}}{n}(2\pi
ik)^{n}\zeta^{m_{1}n},$ (26)

where the radius of convergence is $|\zeta|<|2\pi k|^{-1}$ and
$\vartheta=\exp(2\pi i/m_{1})$ is a primitive $m_{1}$th root of unity.

To proceed with an examination of these deck transformations when $m_{0}\neq
0$ we fix some notation. By equation 25, each local deck transformation is
uniquely defined by the first and second coefficient in its expansion about
$a$ (more abstractly, this is due to the unique lifting property; see
[BP15b]). The first coefficient is an $m_{1}$th root of unity which we fix as
$s_{0}=\vartheta^{m}$. The second coefficient is
$s_{0}\log(s_{0}^{m_{0}})/m_{0}$. If we fix a choice of $\log$ with a branch
cut chosen along an irrational angle in the complex plane (in particular, it
must not exclude any power of $\vartheta$), then the choices of $s_{1}$ are in
one-to-one correspondence with the integers. We denote the local deck
transformation with first coefficient $\vartheta^{m}$ and second coefficient
$\vartheta^{m}(2\pi im/m_{1}-2\pi ik/m_{0})$ as $\sigma_{a}^{k,m}$.

To proceed, we first find solutions for the partial inverses of $x$ in terms
of the Lambert $W$ function

$x=\zeta^{m_{0}}e^{-\frac{m_{0}}{m_{1}}\zeta^{-m_{1}}}\Rightarrow
x^{-m_{1}/m_{0}}=\zeta^{-m_{1}}e^{\zeta^{-m_{1}}}\Rightarrow\zeta=\left[W_{k}\left(x^{-m_{1}/m_{0}}\right)\right]^{-1/m_{1}},$
(27)

where $W_{k}$ is the $k$th branch of the Lambert $W$ function defined by the
relation

$z=we^{w}\Leftrightarrow\exists k\in\mathbb{Z}\ni w=W_{k}(z).$ (28)

Normally, $W_{0}$ denotes the principal branch that is real-valued on the non-
negative half of the real line and $W_{-1}$ is the branch that is real-valued
on the interval $[-1/e,0]$. Otherwise, there is no standard convention in the
literature regarding the choices of branches of the Lambert $W$ function; one
such choice is given in [CGHJK96], which we will use in the following.

Our deck transformations then take the form

$\sigma_{a}^{k,m}(\zeta)=\vartheta^{m^{\prime}}\left[W_{k^{\prime}}\left(\varphi^{l}\zeta^{-m_{1}}e^{\zeta^{-m_{1}}}\right)\right]^{-1/m_{1}},$
(29)

where $\varphi=\exp(-2\pi im_{1}/m_{0})$ and $m^{\prime},k^{\prime},l$ are
integers. We then note the following expansion of the Lambert $W$ function
from [CGHJK96], which is valid for all non-zero $k$ when $\log(|z|)$ is
sufficiently large

$W_{k}(z)=\log(z)+2\pi ik-\log(\log(z)+2\pi ik)\\\
+\sum_{a=0}^{\infty}(-1)^{a}\sum_{b=1}^{\infty}\frac{1}{b!}\genfrac{[}{]}{0.0pt}{}{a+b}{a+1}(\log(z)+2\pi
ik)^{-a-b}\log^{b}(\log(z)+2\pi ik),$ (30)

where $\genfrac{[}{]}{0.0pt}{}{n_{1}}{n_{2}}$ denotes an unsigned Stirling
number of the first kind and $\log^{b}$ denotes the logarithm composed with
itself $b$ times; all the logarithms in the above expansion are the principal
branch of the logarithm. The above expansion is also valid when $k=0$ provided
$|z|$ is sufficiently large; for small $z$ the principal branch satisfies
$W_{0}(z)\sim z$. Using the above expansion we find, for $\zeta$ a small
positive number

$\sigma_{a}^{k,m}(\zeta)=\vartheta^{m^{\prime}}\zeta-\frac{\vartheta^{m^{\prime}}}{m_{1}}\left(2\pi
ik^{\prime}-2\pi
i\frac{m_{1}}{m_{0}}l\right)\zeta^{m_{1}+1}+\mathcal{O}(\zeta^{2m_{1}+1}).$
(31)

This gives us that $m^{\prime}=m$ and that $k^{\prime}$ and $l$ should satisfy

$\frac{m_{1}}{m_{0}}l-k^{\prime}=m-\frac{m_{1}}{m_{0}}k,$ (32)

where $l$ must be chosen so that $-\frac{2\pi im_{1}l}{m_{0}}\in(-\pi,\pi)$.
Recalling that $m=0,\dots,m_{1}-1$ we see that $k=\mathcal{O}(k^{\prime})$ as
$k\to\infty$. This leads us to the following lemma, which considers the
asymptotic behaviour as the chosen branch of the logarithm becomes ‘large’ in
some sense.

######  Lemma 3.10.

For $m_{0}\neq 0$ (the corresponding $m_{0}=0$ formula is obvious from (26))

$\begin{split}\sigma_{a}^{m,k}(\zeta)&=\vartheta^{m}(2\pi
im_{1}k/m_{0})^{-1/m_{1}}\left(1+\mathcal{O}\left(\frac{\log(|k|)}{k}\right)\right),\quad|k|\to\infty,\\\
\frac{d\sigma_{a}^{m,k}}{d\zeta}(\zeta)&=\vartheta^{m}\zeta^{-m_{1}-1}(2\pi
im_{1}k/m_{0})^{-1/m_{1}-1}\left(1+\mathcal{O}\left(\frac{\log(|k|)}{k}\right)\right),\quad|k|\to\infty.\end{split}$
(33)

###### Proof.

This is a direct consequence of the fact that
$k\sim\frac{m_{0}}{m_{1}}k^{\prime},\hskip 2.84526pt|k|\to\infty$, and the
expansion (30). ∎

###  3.3. Transalgebraic spectral curves as limits

Compact transalgebraic spectral curves naturally arise as limits of sequences
of compact meromorphic spectral curves. The guiding light behind our
definition of topological recursion for transalgebraic spectral curves is that
it should commute with taking such limits. Schematically, if $\mathcal{S}_{N}$
is a sequence of compact meromorphic spectral curves such that
$\lim_{N\to\infty}\mathcal{S}_{N}$ is a compact transalgebraic spectral curve,
and the $\omega^{N}_{g,n}[\mathcal{S}_{N}]$ are the correlators constructed
from usual topological recursion on $\mathcal{S}_{N}$, we want the correlators
$\omega_{g,n}[\lim_{N\to\infty}\mathcal{S}_{N}]$ associated to the
transalgebraic spectral curve $\lim_{N\to\infty}\mathcal{S}_{N}$ to satisfy

$\lim_{N\to\infty}\left(\omega_{g,n}^{N}[\mathcal{S}_{N}]\right)=\omega_{g,n}\left[\lim_{N\to\infty}\mathcal{S}_{N}\right].$
(34)

Therefore, we should study such sequences of spectral curves. Considering such
limits will also allow us to extend the notion of admissibility from
definition 2.2 to exponential singularities.

Therefore, consider such a sequence of compact meromorphic spectral curves
$\mathcal{S}_{N}=(\Sigma,x_{N},y_{N},B)$, such that $x_{N}\to x$ and $y_{N}\to
y$ as $N\to\infty$, where $x$ and $y$ are transalgebraic functions on $\Sigma$
with $xy$ meromorphic. Explicitly, we will consider the sequence

$x_{N}=M_{0}\left(1+(\tau-1)\frac{M_{1}}{N}\right)^{-N}\left(1+\tau\frac{M_{1}}{N}\right)^{N}\,,\qquad
y_{N}=\frac{M_{2}}{x_{N}}\,,$ (35)

which converges compactly to

$x=M_{0}e^{M_{1}},\qquad y=\frac{M_{2}}{x},$ (36)

away from the poles of $M_{1}$.

######  Remark 3.11.

We introduce the parameter $\tau$ for two reasons. First, we will see in
Theorem 4.3 that the limiting correlators do not depend on the choice of
$\tau$, which is evidence that our definition of topological recursion on
transalgebraic spectral curves is the correct one for the limiting curve and
not an artefact of the particular sequence chosen. Secondly, when constructing
quantum curves, we will see that we get a priori different quantum curves for
each choice of $\tau$. However, at least in the cases of interest in this
paper, we will see that this $\tau$ dependence can be naturally transformed
away.

For the spectral curves $\mathcal{S}_{N}$, we divide the ramification points
of $x_{N}$, denoted collectively as $R^{N}$, into two sets of ramification
points:

  1. (1)

$R^{N}_{\infty}=\\{M_{1}=-\frac{N}{\tau}\\}\cup\\{M_{1}=\frac{N}{1-\tau}\\}\cup\\{M_{1}=\infty\\}$
consists of the ramification points colliding at essential singularities of
$x$;

  2. (2)

$R^{N}_{0}=R^{N}\setminus R^{N}_{\infty}$ consists of those ramification
points not colliding at essential singularities of $x$.

Let us consider what admissibility means for ramification points of
transalgebraic spectral curves. The notion of admissibility at the
ramification points in $R_{0}$ is clear: it should be the same as for
algebraic spectral curves, that of definition 2.2. At the points in
$R_{\infty}$ we need a new definition based on the notion of admissibility for
points in $R_{\infty}^{N}$.

We distinguish two cases for an exponential singularity $a\in R_{\infty}$
depending on whether $M_{2}=xy$ has a pole at $a$ or not. Let $a\in
R_{\infty}$, which means that it is a pole of $M_{1}$, and suppose that $xy$
has a pole at $a$. Then, for finite $N$, we have $s_{a}\leq-1$ and therefore
$\mathcal{S}_{N}$ is admissible at $a$ by definition 2.2 and the correlators
will be regular at $a$. Moreover, for sufficiently large $N$, $M_{2}$ will be
regular and non-zero at the other points $a^{\prime}$ in $R_{\infty}^{N}$ that
collide at the essential singularity $a$, and hence $s_{a^{\prime}}=1$ in
definition 2.2 at all such points so $\mathcal{S}_{N}$ is admissible at all
such points. Thus, if $xy$ has a pole at $a$, each spectral curve in the
sequence is admissible, and so it makes intuitive sense that the limit of
these curves should be admissible.

There appears to be significant challenges in defining the topological
recursion in the case where $xy$ does not have a pole at an exponential
singularity of $x$. As it appears in no cases of interest, it is not done
here555Curiously, via limiting arguments, it seems that the modularity
condition for admissibility should be $m_{0}\pmod{m_{1}}=\pm 1\pmod{s_{a}}$
where $m_{0}$ and $m_{1}$ are the order of $M_{0}$ and $M_{1}$, respectively,
at $a$, which very naturally generalises definition 2.2.. Therefore, we will
define admissibility at infinite ramification points as follows.

######  Definition 3.12.

Let $\mathcal{S}$ be a compact transalgebraic spectral curve. We say that it
is _admissible_ if both of the following conditions are satisfied:

  1. (1)

it is admissible in the sense of definition 2.2 at the finite ramification
points in $R_{0}$;

  2. (2)

$xy=M_{2}$ has poles at the infinite ramification points in $R_{\infty}$.

##  4\. Topological recursion on transalgebraic spectral curves

We are now ready to define topological recursion on transalgebraic spectral
curves. Recall that we want topological recursion to commute with limits of
sequences of meromorphic spectral curves, as stated schematically in (34). If
$\mathcal{S}_{N}$ is a sequence of compact meromorphic spectral curves such
that $\lim_{N\to\infty}\mathcal{S}_{N}$ is a compact transalgebraic spectral
curve, and the $\omega_{g,n}^{N}[\mathcal{S}_{N}]$ are the correlators
constructed from usual topological recursion on $\mathcal{S}_{N}$, we want the
correlators $\omega_{g,n}[\lim_{N\to\infty}\mathcal{S}_{N}]$ associated to the
transalgebraic spectral curve $\lim_{N\to\infty}\mathcal{S}_{N}$ to satisfy

$\lim_{N\to\infty}\left(\omega_{g,n}^{N}[\mathcal{S}_{N}]\right)=\omega_{g,n}\left[\lim_{N\to\infty}\mathcal{S}_{N}\right].$
(37)

However, it is not straightforward to define topological recursion in this
limit. The usual formulation of topological recursion considers residues at
ramification points, and to define the integrand one needs to sum over local
deck transformations at those ramification points. In other words, one needs
to take the pullback of the pushforward of a one-form under the map $x$. To
include infinite ramification points in the topological recursion, one would
need to take the pullback of the pushforward of a one-form under the local map
$x=\zeta^{m_{0}}e^{-\frac{m_{0}}{m_{1}}\zeta^{-m_{1}}}$. Using lemma 3.10, one
can see that for a $1$-form $\eta$ that is holomorphic at all essential
singularities of $x$, $x^{*}x_{*}\eta$ is well-defined in the sense that the
sum over deck transformations is convergent. However, for the topological
recursion, we want to look at forms with poles at the essential singularities.
In this case the sum in $x^{*}x_{*}\eta$ is not absolutely convergent, but
there is a natural way to define the principal value.666In particular, we sum
as follows: we first sum over the index $m$, then sum over the sign of $k$ if
$k\neq 0$, and then finally sum from $k=1,\dots,\infty$ (at some point adding
in the $k=0$ term). Then, one can see the sum from $k=1$ to $k=\infty$ _is
absolutely convergent_ if the $1$-form $\eta$ has, at $a\in R_{\infty}$, a
pole of order no more than $\operatorname{Erd}_{x}(a)$ for each $a\in
R_{\infty}$. However, even after summing, the resulting differential may not
have an isolated singularity at the essential singularities of $x$. Thus,
defining the residue at these points becomes highly changing. One possible
approach is to pushforward the entire TR integrand to the $x$ plane, where it
should be meromorphic as a function of $x$, but then it is not clear what
residues in the $x$ plane one should be taking as essential singularities do
not have well-defined branchpoints, although $x=0,\infty$ (the Picard points)
are the most compelling candidates. Even presuming one does all this in a
reasonable manner, proving the $N\to\infty$ limit commutes in the sense of
(34) remains a daunting challenge.

Instead, our approach consists in first rewriting the topological recursion in
a different way, which trades out the sum over the deck transformations of $x$
for a sum over ramification points, coinciding points, and deck
transformations of $y$. We present this rewriting for compact meromorphic
spectral curves in the next section, and then generalise it to transalgebraic
spectral curves.777This rewriting of topological recursion for compact
meromorphic spectral curves is inspired by private notes of Nitin K.
Chidambaram.

###  4.1. Rewriting topological recursion

Let us review some of the notation required for this rewriting that was
introduced in section 1.4. Let $C=\\{t,t_{1},\dots,t_{i}\\}\subset\Sigma$ and
$C^{\prime}=C\setminus\\{t\\}$ be sets of cardinality $i+1$ and $i$,
respectively. For a symmetric $i$-differential $\eta$ we set, by definition

$\operatorname*{Res}_{C=t}\eta(t_{1},\dots,t_{i})=\operatorname*{Res}_{C^{\prime}=t}\eta(t_{1},\dots,t_{i})=\operatorname*{Res}_{t_{1}=t}\cdots\operatorname*{Res}_{t_{i}=t}\eta(t_{1},\dots,t_{i})\,.$
(38)

Then, for a set $C\subset\Sigma$, we denoted by $t^{C}$ one arbitrarily chosen
element in this set. Finally, for the purposes of taking many residues in a
compact notation, we defined

$\operatorname*{Res}_{\begin{subarray}{c}t_{l}=a_{l}\\\
l=1,\dots,n\end{subarray}}=\operatorname*{Res}_{t_{1}=a_{1}}\cdots\operatorname*{Res}_{t_{n}=a_{n}}\,.$
(39)

With this notation (see section 1.4 for more details), we may proceed to the
theorem of this section.

######  Theorem 4.1.

Let $\mathcal{S}$ be a compact meromorphic admissible spectral curve888To be
precise, we also need to assume here that $\mathcal{S}$ can be “fully
globalised”, in the language of [BBCKS]. What this means is that we can
replace in topological recursion the sums over local deck transformations
$\mathfrak{f}^{\prime}_{a}(t)$ at the ramification points with a sum over the
whole fibre $\mathfrak{f}^{\prime}(t)$, and perform manipulations along the
lines of [BE13, Theorem 5]. All spectral curves considered in the present
paper are fully globalisable, according to the conditions determined in
[BBCKS].  and write $Y(t)=y^{-1}(y(t))$. Then the correlators of topological
recursion satisfy the alternative recursive formula

$\begin{split}\omega_{g,n+1}(z_{0},z_{\llbracket
n\rrbracket})&=\operatorname*{Res}_{t=R}\sum_{m=2}^{{\deg}(x)}\int_{*}^{t}B(z_{0},\cdot)\sum_{C_{1},\dots,C_{j}\vdash
t_{\llbracket
m-1\rrbracket}}\\!\\!\\!\frac{(-1)^{1-\delta_{j,m-1}}}{j!}\\!\\!\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=R,z_{\llbracket
n\rrbracket},Y(t)\\\
l=1,\dots,j\end{subarray}}\operatorname*{Res}_{\begin{subarray}{c}C_{l}=t^{C_{l}}\\\
l=1,\dots,j\end{subarray}}\\\
&\quad\left(\prod_{l=1}^{j}\frac{1}{x(t)-x(t^{C_{l}})}\prod_{t_{0}\in
C_{l}\setminus\\{t^{C_{l}}\\}}\frac{1}{x(t_{0})-x(t^{C_{l}})}\right)\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(y(t)-y(t_{l}))}\,,\end{split}$ (40)

where we slightly abuse notation by writing that the $C_{l}$ partition the
dummy integration variables outside the integral.

###### Proof.

Starting with topological recursion from definition 2.8, we first replace the
sums over local deck transformations $\mathfrak{f}^{\prime}_{a}(t)$ by a sum
over the whole fibre $\mathfrak{f}^{\prime}(t)$, as in [BE13, Theorem 5] (see
footnote 8 and [BBCKS]). Then, we perform the rewriting:

$\sum_{\emptyset\neq
Z\subseteq\mathfrak{f}^{\prime}(t)}K_{|Z|+1}(z_{0},t,Z)\mathcal{W}_{g,n,|Z|+1}(t,Z\,|\,z_{\llbracket
n\rrbracket})=\\\ \left(\int_{*}^{t}B(z_{0},\cdot)\right)\\!\\!\sum_{m\geq
2}\sum_{\emptyset\neq\\{\zeta_{1},\dots,\zeta_{m-1}\\}\subseteq\mathfrak{f}^{\prime}(t)}\operatorname*{Res}_{\begin{subarray}{c}t_{l}=\zeta_{l}\\\
l=1,\dots,m-1\end{subarray}}\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(x(t_{l})-x(t))(y(t)-y(t_{l}))}\,.$ (41)

In our new writing, the summand is well-defined when two or more of the
elements of $Z=\\{\zeta_{1},\dots,\zeta_{m-1}\\}$ coincide, it will just
result in higher order poles at such a $t_{l}=\zeta_{l}$. Therefore, instead
of summing over subsets of $\mathfrak{f}^{\prime}(t)$, we will sum over tuples
of size at most $\deg x-1$, but then we need to subtract tuples with repeating
terms. This gives us two main terms: the original sum plus the added terms
where two or more $t_{l}$ coincide and the subtracted terms where two or more
$t_{l}$ coincide. We first examine the first term, for fixed size of the tuple
$m-1$,

$\begin{split}\sum_{\zeta_{1},\dots,\zeta_{m-1}\in\mathfrak{f}^{\prime}(t)}&\frac{1}{(m-1)!}\operatorname*{Res}_{\begin{subarray}{c}t_{l}=\zeta_{l}\\\
l=1,\dots,m-1\end{subarray}}\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(x(t_{l})-x(t))(y(t)-y(t_{l}))}\\\
=&\frac{1}{(m-1)!}\operatorname*{Res}_{\begin{subarray}{c}t_{l}=R,z_{\llbracket
n\rrbracket},Y(t)\\\
l=1,\dots,m-1\end{subarray}}\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(x(t)-x(t_{l}))(y(t)-y(t_{l}))}.\end{split}$
(42)

All we have done here is used the fact that $\Sigma$ is a compact Riemann
surface so the sum of all the residues of any differential must be zero. That
we only pick up residues at the listed points is because the $\omega_{g,n}$
only have poles at coinciding points and ramification points.

We now wish to apply the same logic to the terms with the coinciding points.
To this end, we want to know what happens when $j\leq m-1$ of the same $t_{l}$
are specialised to the same sheet. So we consider

$\begin{split}&\sum_{\zeta\in\mathfrak{f}^{\prime}(t)}\operatorname*{Res}_{\begin{subarray}{c}t_{l}=\zeta\\\
l=1,\dots,j\end{subarray}}\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(x(t_{l})-x(t))(y(t)-y(t_{l}))}\\\
=&\sum_{\zeta\in\mathfrak{f}^{\prime}(t)}\operatorname*{Res}_{\begin{subarray}{c}t_{l}=\zeta\\\
l=1,\dots,j-1\end{subarray}}\frac{\mathcal{W}_{g,n,m}(t,t_{1},\dots,t_{j-1},\zeta,t_{j+1},\dots,t_{m-1}\,|\,z_{\llbracket
n\rrbracket})}{dx(t)(y(t)-y(\zeta))\prod_{\begin{subarray}{c}l=1\\\ l\neq
j\end{subarray}}^{m-1}(x(t_{l})-x(t))(y(t)-y(t_{l}))}\\\
=&\sum_{\zeta\in\mathfrak{f}^{\prime}(t)}\operatorname*{Res}_{t_{j}=\zeta}\operatorname*{Res}_{\begin{subarray}{c}t_{l}=t_{j}\\\
l=1,\dots,j-1\end{subarray}}\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{(x(t_{j})-x(t))(y(t)-y(t_{j}))\prod_{\begin{subarray}{c}l=1\\\
l\neq j\end{subarray}}^{m-1}(x(t_{l})-x(t))(y(t)-y(t_{l}))}\\\
=&\operatorname*{Res}_{t_{j}=R,z_{\llbracket
n\rrbracket},Y(t)}\operatorname*{Res}_{\begin{subarray}{c}t_{l}=t_{j}\\\
l=1,\dots,j-1\end{subarray}}\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}|z_{\llbracket
n\rrbracket})}{(x(t)-x(t_{j}))(y(t)-y(t_{j}))\prod_{\begin{subarray}{c}l=1\\\
l\neq j\end{subarray}}^{m-1}(x(t_{l})-x(t))(y(t)-y(t_{l}))}.\end{split}$ (43)

Hence, the subtracted terms with the coinciding deck transformations may be
written as

$\begin{split}&-\sum_{j=1}^{m-2}\frac{1}{j!}\sum_{\zeta_{1},\dots,\zeta_{j}\in\mathfrak{f}^{\prime}(t)}\sum_{C_{1},\dots,C_{j}\vdash
t_{\llbracket
m-1\rrbracket}}\operatorname*{Res}_{\begin{subarray}{c}C_{l}=\zeta_{l}\\\
l=1,\dots,j\end{subarray}}\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}|z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(x(t_{l})-x(t))(y(t)-y(t_{l}))}\\\
=&-\sum_{j=1}^{m-2}\frac{1}{j!}\sum_{C_{1},\dots,C_{j}\vdash t_{\llbracket
m-1\rrbracket}}\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=R,z_{\llbracket
n\rrbracket},Y(t)\\\
l=1,\dots,j\end{subarray}}\operatorname*{Res}_{\begin{subarray}{c}C_{l}=t^{C_{l}}\\\
l=1,\dots,j\end{subarray}}\\\ &\hskip
28.45274pt\left(\prod_{l=1}^{j}\frac{1}{x(t)-x(t^{C_{l}})}\prod_{t_{0}\in
C_{l}\setminus\\{t^{C_{l}}\\}}\frac{1}{x(t_{0})-x(t^{C_{l}})}\right)\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}|z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(y(t)-y(t_{l}))}\,,\end{split}$ (44)

and the desired result then follows. ∎

###  4.2. Topological recursion on transalgebraic spectral curves

With the topological recursion for compact meromorphic spectral curves
rewritten as in theorem 4.1, we are in a position of topological recursion for
compact transalgebraic spectral curves. We will present the definition of the
topological recursion for transalgebraic spectral curves straight away, and
then spend the rest of the section arguing and demonstrating why it works.

######  Definition 4.2.

Let $\mathcal{S}=\left(\Sigma,x,y,B\right)$ be a compact transalgebraic
admissible spectral curve, with $x=M_{0}\exp(M_{1})$, $y=M_{2}/x$ and
$M_{0},M_{1},M_{2}$ meromorphic functions on $\Sigma$. Fix $\tau\in\mathbb{C}$
and define the sequence of spectral curves
$\mathcal{S}_{N}=\left(\Sigma,x_{N},y_{N},B\right)$, where

$x_{N}=M_{0}\left(1+(\tau-1)\frac{M_{1}}{N}\right)^{-N}\left(1+\tau\frac{M_{1}}{N}\right)^{N}\,,\quad
y_{N}=M_{2}/x_{N}\,.$ (45)

Then, if $\omega_{g,n}^{N}$ are the correlators constructed by topological
recursion for the spectral curve $\mathcal{S}_{N}$, we _define_ the
correlators of the spectral curve $\mathcal{S}$ as the $N\to\infty$ limit of
the $\omega_{g,n}^{N}$.

This defines nothing if the limit depends on $\tau$ or does not yield well-
defined meromorphic correlators. The main result of this section is the
following theorem, which shows that these issues do not occur, and, therefore,
that the above definition makes sense.

######  Theorem 4.3.

Let $\mathcal{S}=\left(\Sigma,x,y,B\right)$ be a compact transalgebraic
admissible spectral curve. Then the $\omega_{g,n}$ constructed from definition
4.2 are well-defined meromorphic differentials on $\Sigma^{n}$ and do not
depend on the choice of $\tau$.

###### Proof.

Our strategy will be to first prove the that $\omega_{g,n}$ are well-defined
for $\tau=0$, and then show that the limit is independent of $\tau$. The proof
is divided into eight steps.

_First step: start the induction and flip contours._

We proceed inductively in the $\tau=0$ case on $-\chi_{g,n}=2g+n-2$. For
$-\chi_{g,n}=-1,0$ (corresponding to $\omega_{0,1}$ and $\omega_{0,2}$) the
result holds trivially, so we may proceed to the induction step. For finite
$N$ we may use theorem 4.1 to write

$\begin{split}\omega^{N}_{g,n+1}(z_{0},z_{\llbracket
n\rrbracket})&=\operatorname*{Res}_{t=R^{N}}\sum_{m=2}^{\deg(x_{N})}\left(\int_{*}^{t}B(z_{0},\cdot)\right)\sum_{C_{1},\dots,C_{j}\vdash
t_{\llbracket
m-1\rrbracket}}\frac{(-1)^{1-\delta_{j,m-1}}}{j!}\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=R^{N},z_{\llbracket
n\rrbracket},\mathcal{Y}(t)\\\
l=1,\dots,j\end{subarray}}\operatorname*{Res}_{\begin{subarray}{c}C_{l}=t^{C_{l}}\\\
l=1,\dots,j\end{subarray}}\\\
&\quad\left(\prod_{l=1}^{j}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}\prod_{t_{0}\in
C_{l}\setminus\\{t^{C_{l}}\\}}\frac{x_{N}(t^{C_{l}})}{x_{N}(t_{0})-x_{N}(t^{C_{l}})}\right)\frac{\mathcal{W}_{g,n,m}^{N}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(M_{2}(t)-M_{2}(t_{l}))}\,,\end{split}$ (46)

where $\mathcal{Y}(t)=(xy)^{-1}\big{(}(xy)(\\{t\\})\big{)}$. This is slightly
different from theorem 4.1, so a couple of remarks are in order so it is clear
how we get here:

  * •

in the denominator of the integrand in the original topological recursion we
rewrote $y_{N}(t)-y_{N}(\sigma(t))=(M_{2}(t)-M_{2}(\sigma(t)))/x_{N}(t)$ so we
ended up with $M_{2}=xy=x_{N}y_{N}$ in the denominator and $x_{N}(t)$ in the
numerator, where $\sigma(t)\in\mathfrak{f}^{\prime}(t)$ is a deck
transformation;

  * •

as $x_{N}(t)=x_{N}(\sigma(t))$ for every deck transformation, we can choose
which deck transformation we take the argument of $x_{N}$ to be at; in
particular, we take $j$ of them to just be $t$ and the other $m-1-j$ to be
precisely those deck transformations that gives us $t^{C_{l}}$;

  * •

when we flipped the contour, we then had to pick out residues at
$\mathcal{Y}(t)$ rather than $Y(t)$.

_Second step: $R_{\infty}^{N}$ does not contribute._

We now observe that the residues at $t^{C_{l}}=R_{\infty}^{N}$ vanish for
sufficiently large $N$. Namely, for the points in $R_{\infty}^{N}$ that
satisfy $M_{1}=N$, $x_{N}(t^{C_{l}})$ has a pole of order $N$. As
$x_{N}(t^{C_{l}})$ appears in the denominator one time with no corresponding
$x_{N}(t^{C_{l}})$ in the numerator, the overall integrand in the variable
$t^{C_{l}}$ gains a zero of order $N$. We claim the rest of the integrand has
a pole of at worst uniformly bounded order in $N$. At these ramification
points $\omega_{0,1}^{N}$ has simple poles, and, for sufficiently large $N$,
$M_{2}$ will be regular and non-zero, which means that $s_{a}=1$ at these
points. From Lemma 2.10 we then know that the $\omega_{g,n}^{N}$ have poles of
order no more than $2g$ and so the $\mathcal{W}_{g,n,m}^{N}$ have poles of
bounded order in $N$. Similarly, $M_{2}$ is meromorphic and constant in $N$.
For $x_{N}(t^{C_{l}})/(x_{N}(t_{0})-x_{N}(t^{C_{l}}))$, the $x_{N}$ appears in
both the denominator and the numerator. Finally, we need to examine taking the
residues at $C_{l}=t^{C_{l}}$. This will be a residue of a pole of order no
more than three (two from a potential $\omega_{0,2}$ contribution plus one for
the difference of the $x_{N}$ in the denominator). Thus, this residue may be
replaced by multiplication by $(t_{0}-t^{C_{l}})^{3}$ and twice
differentiating by $t_{0}$, for each $t_{0}\in C_{l}\setminus\\{t^{C_{l}}\\}$,
before taking the limit as $t^{C_{l}}\to t_{0}$. By the quotient rule for
differentiation, we will have the same total power of derivatives of $x_{N}$
in the numerator and denominator, just in different combinations and orders of
differentiation. Thus, at the residues at points where $M_{1}=N$ we may drop
the residue in $t^{C_{l}}$.

Now we examine the residues in $t^{C_{l}}$ where the point at which the
residue is taken satisfies $M_{1}=\infty$. Here, when we take the residue at
$C_{l}=t^{C_{l}}$, as discussed previously, this corresponds to derivatives.
Here though, the pole counting is a little more subtle so we do it explicitly.
In particular, observe

$\begin{split}\frac{x_{N}(t^{C_{l}})\omega_{0,2}(t_{0},t^{C_{l}})(t_{0}-t^{C_{l}})^{3}}{(x_{N}(t_{0})-x_{N}(t^{C_{l}}))dt_{0}dt^{C_{l}}}&=\frac{x_{N}(t^{C_{l}})}{x_{N}^{\prime}(t^{C_{l}})}-(t_{0}-t^{C_{l}})\frac{x_{N}(t^{C_{l}})x_{N}^{\prime\prime}(t^{C_{l}})}{x^{\prime}_{N}(t^{C_{l}})^{2}}\\\
&+(t_{0}-t^{C_{l}})^{2}\left(\frac{x_{N}(t^{C_{l}})x_{N}^{\prime\prime}(t^{C_{l}})^{2}}{4x_{N}^{\prime}(t^{C_{l}})^{3}}-\frac{x_{N}(t^{C_{l}})x_{N}^{\prime\prime\prime}(t^{C_{l}})}{x_{N}^{\prime}(t^{C_{l}})^{2}}\right)+\mathcal{O}\left((t_{0}-t^{C_{l}})^{3}\right)\,.\end{split}$
(47)

In the constant term and the $t_{0}-t^{C_{l}}$ term, there is no pole at
$t^{C_{l}}$ equalling a pole of $M_{1}$. However, the $(t_{0}-t^{C_{l}})^{2}$
term has a simple pole here. On the other hand, the $\omega_{g,n}^{N}$ are
regular at these points (this is because we are in the $s_{a}\leq-1$ case at
these points; see remark 2.3) and $M_{2}(t_{0})$, which has at least a simple
pole by admissibility, appears in the denominator. Thus, in terms with an
$\omega_{0,2}(t_{0},t^{C_{l}})$ we do not have contributions from these
points. For terms without this factor, the pole at $t_{0}=t^{C_{l}}$ is
simple. Thus, observing

$\frac{x_{N}(t^{C_{l}})(t_{0}-t^{C_{l}})}{x_{N}(t_{0})-x_{N}(t^{C_{l}})}=\frac{x_{N}(t^{C_{l}})}{x_{N}^{\prime}(t^{C_{l}})}+\mathcal{O}\left(t_{0}-t^{C_{l}}\right)\,,$
(48)

we see the same argument still holds. In summary, we may replace the residues
in each $t^{C_{l}}$ at all the points in $R_{N}$ with just those at
$R_{N}^{0}$.

_Third step: integrand well-defined._

This shows that the integrand in $t$ is well defined in the limit: indeed, we
may commute the limit in $N\to\infty$ with the residues (integrals) in the
$t_{0}\in C_{l}$ and $t^{C_{l}}$ using dominated convergence and use the
induction assumption that $\omega_{g,n}^{N}\to\omega_{g,n}$. Note that,
although it may appear that the sum over $m$ becomes infinite in the limit,
for any fixed $g$ and $n$ only finitely many terms are non-zero so commuting
the limit with this sum is entirely trivial.

_Fourth step: integral well-defined at $R_{0}$._

However, we want the integral, not just the integrand, to be well-defined in
the limit. To this end, we note that the contributions from the residues at
$t=R_{0}^{N}$ go to the contributions at $t=R_{0}$ in the limit by pulling the
limit in $N$ inside each integral using dominated convergence, as before, and
applying the induction assumption. However, this simple argument will not work
for the residues at $t=R_{\infty}$ as these points can collide in the limit.

_Fifth step: work in coordinate $w$ for integrals at $R_{\infty}$._

To deduce that the integral must be well-defined in the limit, we will
pushforward to work in the $M_{1}$ plane where all elements of
$R_{\infty}^{N}$ fall at $M_{1}=N,\infty$. To move to the origin, let
$w=1/M_{1}$. For a deck transformation $\sigma$ of $M_{2}$, i.e.,
$\sigma(t)\in\mathcal{Y}(t)$, we define a corresponding transformation $\nu$
through $\nu(w)=\nu(1/M_{1}(t))=1/M_{1}(\sigma(t))$. Although $\nu$ may depend
on $t$, and so is multi-valued, we sum over $\nu$ at every step; this is well-
defined. In completing this sum, as from now on we will suppress this detail,
note that the sum over $\nu$ must include both the sum over $\sigma$ (from all
elements of $\mathcal{Y}(t)$) and partial inverse of $M_{1}$ (from the fact we
pushforward in $M_{1}$). Let $\nu_{1},\dotsc,\nu_{r}$ be all such non-trivial
$\nu$ and note that, for general $N$, $\nu_{p}(w=1/N)\neq 1/N$ for all
$p=1,\dotsc,r$. We claim that the integrand in $t$ of topological recursion,
pushed forward under $M_{1}$ and then written in the coordinate $w$, takes the
following form (where $\exp_{N}(z)=(1-z/N)^{-N}$):

$\frac{\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(w^{-1})^{d}+\mathcal{N}_{N}^{d-1}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(w^{-1})^{d-1}+\dotsb+\mathcal{N}_{N}^{0}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(w^{-1})^{d}+\mathcal{D}_{N}^{d-1}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(w^{-1})^{d-1}+\dotsb+\mathcal{D}^{N}_{0}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}\,,$ (49)

where each of the $\mathcal{N}_{N}^{k}$ and $\mathcal{D}_{N}^{k}$ are
meromorphic functions999Due to the presence of $\int_{*}^{t}B(z_{0},\cdot)$ in
the integrand, this is not strictly true for non-zero genus. However, all we
want to do is integrate, so we may slightly misuse terminology. with the order
of their zeros and poles at $w=1/N$ bounded uniformly in $N$ and it is assumed
$\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$ is not
identically zero.

To get this expansion, first note we can write every derivative of $x_{N}$ as
$\exp_{N}(M_{1})=\exp_{N}(1/w)$ times a sequence of meromorphic functions with
the order of their poles and zeros bounded uniformly in $N$. Then, when we
take the residues at $t_{0}=t^{C_{l}}$, we get expansion like (47) and (48),
which we may put over a common denominator.

Then, we will claim that when we take the residues at
$t^{C_{l}}=R_{N}^{0},z_{\llbracket n\rrbracket},\mathcal{Y}(t)$ the total
number of factors of derivatives of $x_{N}(t)$ in the denominator is greater
than or equal to those in the numerator in each term; putting everything over
a common denominator and pushing forward results in an expression of the form
(49). The fact that we get this same power behaviour will be demonstrated in
the proceeding (sixth) step and for now can be taken as a claim. Like any
fraction, such an expression is non-unique as we may multiply the numerator
and the denominator by the same factor without changing the total value;
however, the ratio $\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$ is
unique and all we will eventually care about.

_Sixth step: properties of $\mathcal{N}/\mathcal{D}$._

We now need to verify a couple of important properties of
$\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$.
First, we claim it has no pole at $w=0$. Second, we claim that it does not
have essential singularity in the limit and its only potential pole near $w=0$
is the one at $w=1/N$. Note that the only place an essential singularity could
come from are the $\exp_{N}(1/\nu_{p}(w))$. Finally, along the way, it will
become clear that, as we have stated, the degree of the numerator and
denominator in $exp_{N}(1/w)$ must be the same and that the ratio
$\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$ is
well-defined.

The proof for these claims is a bit more involved. We will examine the
individual terms that we put over a common denominator before pushing forward
to the $w$-plane; by examining the ratio of the coefficient of the highest
power of $x_{N}$ in the numerator to the one in the denominator, we can deduce
the behaviour of the coefficients in the fraction put over a common
denominator (before pushing forward, $x_{N}$ is $\exp_{N}(M_{1})$ times a
meromorphic function $M_{0}$, so looking at leading powers of $x_{N}$ is the
same as looking at leading powers of $\exp_{N}(M_{1})$). As there are no poles
at $t\in R_{\infty}$, there are none at $w=0$, and the lack of
$\exp_{N}(M_{1}(\sigma(t)))$ factors in the leading order will give us the no
essential singularities result. To this end we first perform the residues at
$C_{l}=t^{C_{l}}$ in the integrand in $t$ and will be left with an expression
of the form

$\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=R_{N}^{0},z_{\llbracket
n\rrbracket},\mathcal{Y}(t)\\\
l=1,\dotsc,j\end{subarray}}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}\frac{f_{N}(t,t^{C_{l}},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{(M_{2}(t)-M_{2}(t^{C_{l}}))^{|C_{l}|}}\,,$ (50)

where $f_{N}$ is a differential in all its arguments except $t$. Furthermore,
$f_{N}$ is meromorphic in $t$ and $t^{C_{l}}$ and remains so in the limit
(note that in (47) and (48) the derivatives in $x_{N}$ appear in the same
power in the numerator and denominator so we may cancel out the factor of
$\exp_{N}(M_{1})$), and there is no pole in $t$ or $t^{C_{l}}$ at the poles of
$M_{1}$ (we established this before in the second step to show that the
residues at $t^{C_{l}}=R_{\infty}$ do not contribute).

First we examine the residues

$\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=R_{N}^{0},z_{\llbracket
n\rrbracket}\\\
l=1,\dotsc,j\end{subarray}}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}\frac{f_{N}(t,t^{C_{l}},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{(M_{2}(t)-M_{2}(t^{C_{l}}))^{|C_{l}|}}\,.$ (51)

Here, the residues in the $t^{C_{l}}$ are taken at points that do not depend
on $t$. Thus, due to the pole of $M_{2}$ at the poles of $M_{1}$ guaranteed by
admissibility, these residues will all contribute some sort of zero to the
ratio of leading order coefficients in powers of $x_{N}$. Finally, it is
trivial that these residues will never contribute $\exp_{N}(M_{1}(\sigma(t)))$
as nothing depends on $\sigma(t)$.

However, a pathology can occur here. In the limit we can have two or more
different elements of $R_{0}^{N}$ collide; if this happens, the independent
contribution of each to the final integrand will not be well-defined and only
the sum over the residues at these colliding points is well-defined in the
limit. Let us examine this case and assure ourselves that this presents no
issues for the well-definedness of the ratio
$\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$ in
the limit. First note that these colliding points cannot be poles of $x_{N}$
as the location of the poles of $x_{N}$ in $R_{0}^{N}$ do not depend on $N$ as
they are just the poles of $M_{0}$ and the poles of $M_{0}$ will never collide
with the zeros of $(1-M_{1}/N)^{N+1}dx_{N}=(1-M_{1}/N)dM_{0}+M_{0}dM_{1}$.
Thus, the general scenario we must examine is when we have
$a^{1}_{N},\dots,a^{k}_{N}\in R_{0}^{N}$ which are all zeros of $dx_{N}$ and
all collide in the limit. We then examine the following expression in which we
computed the residue in terms of derivatives

$\sum_{i=1}^{k}\lim\limits_{t^{C_{l}}\to
a^{i}_{N}}\frac{1}{M!}\frac{d^{M}}{d(t^{C_{l}})^{M}}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}\frac{f_{N}(t,t^{C_{l}},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{(M_{2}(t)-M_{2}(t^{C_{l}}))^{|C_{l}|}}\,,$ (52)

where $M\in\mathbb{Z}_{\geq 0}$ is chosen large enough so all the limits in
$t^{C_{l}}$ are finite. By using the product rule we may write this expression
as

$\frac{1}{M!}\sum_{h=1}^{M}\sum_{i=1}^{k}\frac{x_{N}(t)}{(x_{N}(t)-x_{N}(a^{i}_{N}))^{h}}F_{N}^{h}(t,a^{i}_{N},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket n\rrbracket})\,,$ (53)

where the $F_{N}^{h}$ are meromorphic functions in $t$ such that the order of
all its zeros and poles (in $t$) are uniformly bounded in $N$. Denoting this
uniform bound by $N_{0}$, we see for sufficiently large $N$ the prefactors of
the $F_{N}^{h}$ in the above expression must have zeros and poles of order
larger than $N_{0}$ (except possibly for $h=1$ if $a_{N}^{i}$ is a zero of
$x_{N}$ for some $i$ and all $N$ sufficiently large). As the sum over $h$ must
be well defined in the limit, we can conclude that each individual term in the
sum over $n$ is well-defined in the limit. Therefore, the $h=1$ expression

$\frac{1}{M!}\sum_{i=1}^{k}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(a^{i}_{N})}F_{N}^{1}(t,a^{i}_{N},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket n\rrbracket})\\\
=\frac{x_{N}(t)}{M!}\prod_{i=1}^{k}\left[x_{N}(t)-x_{N}(a_{N}^{i})\right]^{-1}\sum_{i=1}^{k}F_{N}^{1}(t,a^{i}_{N},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})\prod_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{k}\left[x_{N}(t)-x_{N}(a_{N}^{j})\right]\,,$ (54)

is well-defined in the limit. Observing that this is degree $k$ in $x_{N}(t)$
in both the numerator and the denominator we see that this is precisely the
desired leading order in $x_{N}$ expression that will contribute to the ratio
$\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$,
whereas the terms other than $j=1$ will contribute only to terms lower order
in $\exp_{N}(1/w)^{d}$.

More involved are the contributions from the residues at $\mathcal{Y}(t)$.
Here, we divide these into three sub-cases:

  * (i)

we may take the residues at $\sigma(t)\in\mathcal{Y}^{\prime}(t)$ that do not
preserve $M_{1}$, these correspond to the $\nu_{1},\dotsc,\nu_{r}$ and are
where the limit of $\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$ may
have potential essential singularities;

  * (ii)

we may take the residues at $\sigma(t)\in\mathcal{Y}^{\prime}(t)$ that
preserve $M_{1}$, i.e., $M_{1}\circ\sigma=M_{1}$;

  * (iii)

we may take the residue at $t$ itself.

Starting with sub-case (i), take an element
$\sigma(t)\in\mathcal{Y}^{\prime}(t)$, fix an $l$, and inspect the following
residue

$\operatorname*{Res}_{t^{C_{l}}=\sigma(t)}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}\frac{f_{N}(t,t^{C_{l}},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{(M_{2}(t)-M_{2}(t^{C_{l}}))^{|C_{l}|}}\,,$ (55)

where $f_{N}$ is as before. Here we have a pole of order $|C_{l}|$ at
$\sigma(t)$ in the variable $t^{C_{l}}$. We can therefore calculate the
residue with the formula

$\lim_{t^{C_{l}}\to\sigma(t)}\frac{(-1)^{|C_{l}|}}{|C_{l}|!}\frac{d^{|C_{l}|-1}}{d(t^{C_{l}})^{|C_{l}|-1}}\frac{x_{N}(t)f_{N}(t,t^{C_{l}},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{x_{N}(t)-x_{N}(t^{C_{l}})}\left(\frac{t^{C_{l}}\sigma(t)}{M_{2}(t^{C_{l}})-M_{2}(t)}\right)^{|C_{l}|}\,.$
(56)

If we take the derivatives of the $1/(x_{N}(t)-x_{N}(t^{C_{l}}))$, we will end
up with subleading terms in powers of $x_{N}$; these therefore do not concern
our analysis. All we care about when we take derivatives of $f$, is that
derivatives cannot create poles. Finally, we have the expansion

$\left(\frac{t^{C_{l}}\sigma(t)}{M_{2}(t^{C_{l}})-M_{2}(\sigma(t))}\right)^{|C_{l}|}=\frac{1}{M_{2}^{\prime}(\sigma(t))^{|C_{l}|}}\sum_{k=0}^{\infty}S_{k}(\sigma(t))(t^{C_{l}}-\sigma(t))^{k}\,,$
(57)

where $S_{k}$ has a pole of order at most $k$ at elements of $R_{\infty}$
(poles of $M_{1}$). The pre-factor has a zero of at least order $|C_{l}|$ (as
$M_{2}$ has a pole at all elements of $R_{\infty}$), and the only $S_{k}$ that
can contribute are those with $k\leq|C_{l}|$. Thus, for the leading terms in
$x_{N}$, we will never get poles. Furthermore, from the above discussion, we
see the $\exp_{N}(1/\nu_{p}(w))$ will never enter the leading order power in
$\exp_{N}(1/w)$.

Now, we move on to sub-case (ii): the deck transformations that preserve
$M_{1}$; let $\sigma(t)$ be such a deck transformation and examine the
expression

$\lim_{t^{C_{l}}\to\sigma(t)}\frac{(-1)^{|C_{l}|}}{|C_{l}|!}\frac{d^{|C_{l}|-1}}{dt_{C_{l}}^{|C_{l}|-1}}\frac{x_{N}(t)f_{N}(t,t^{C_{l}},t_{\llbracket
i-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{x_{N}(t)-x_{N}(t^{C_{l}})}\left(\frac{t^{C_{l}}\sigma(t)}{M_{2}(t^{C_{l}})-M_{2}(t)}\right)^{|C_{l}|}\,,$
(58)

which is the same as the prior case as nothing in the steps changes up to this
point. The only thing that changes in analysing this expression is that when
we take derivatives of the $x_{N}(t)/(x_{N}(t)-x_{N}(t^{C_{l}}))$ factor we do
not end up with only subleading terms as $x_{N}(\sigma(t))$ now has a factor
of $\exp(M_{1}(t))$. If we take $k$ derivatives of this factor, we get a pole
of order at most $k$ in the ratio of the coefficients of the leading powers of
$x_{N}$. Thus, we still cannot get a pole as we have the factor of
$M_{2}^{\prime}(\sigma(t))^{-|C_{l}|}$, as before.

Finally, we examine sub-case (iii), where we take the residue at
$t^{C_{l}}=t$:

$\operatorname*{Res}_{t^{C_{l}}=t}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}\frac{f_{N}(t,t^{C_{l}},t_{\llbracket
i-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{(M_{2}(t)-M_{2}(t^{C_{l}}))^{|C_{l}|}}.$ (59)

Here, we may have a pole of order at most $|C_{l}|+3$. In particular, we get a
pole of order one from the $x_{N}(t)/(x_{N}(t)-x_{N}(t^{C_{l}}))$ factor, a
pole of order $|C_{l}|$ from the difference of the $M_{2}$ in the denominator,
and a potential double pole in $f_{N}$ at $t^{C_{l}}=t$ due to possible
presence of an $\omega_{0,2}(t,t^{C_{l}})$. Here, however, the
$M_{2}^{\prime}(t)^{-|C_{l}|}$ has at least a zero of order $2|C_{l}|$. Using
identical arguments with the expansions of the individual factors, this case
will not create an undesired pole in
$\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$.

_Seventh step: integral well-defined._

With these properties established, we can prove the $\omega_{g,n}^{N}$ are
well-defined in the limit. First note

$\begin{split}\operatorname*{Res}_{w=1/N}\frac{\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(1/w)^{d}+\dotsb}{\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(1/w)^{d}+\dotsb}&=\operatorname*{Res}_{w=1/N}\frac{\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}\left(1+\mathcal{O}(w-1/N)^{N}\right)\\\
&=\operatorname*{Res}_{w=1/N}\frac{\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}\,.\end{split}$ (60)

As we have established $\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$ has
no pole at $w=0$, we may change the residue at $w=1/N$ to a contour integral
about a small circle around $w=0$. Then, using dominated convergence to bring
the limit as $N\to\infty$ inside the contour, we conclude

$\lim_{N\to\infty}\operatorname*{Res}_{w=1/N}\frac{\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(1/w)^{d}+\dotsb}{\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(1/w)^{d}+\dotsb}=\operatorname*{Res}_{w=0}\frac{\mathcal{N}_{\infty}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d}_{\infty}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}.$ (61)

_Eighth step: independence of $\tau$._

Finally, we claim that all choices of $\tau$ yield the same result in the
$N\to\infty$. To establish this, we first prove
$\partial_{\tau}^{m}\omega_{g,n+1}^{N}(z_{0},z_{\llbracket
n\rrbracket})|_{\tau=0}$ exists and goes to zero for every value of
$m\in\mathbb{Z}_{\geq 1}$ as $N\to\infty$ proceeding inductively on
$-\chi_{g,n}=2g+n-2$.

First note that the result is straightforward for $\omega_{0,1}^{N}$ and
trivial for $\omega_{0,2}$. So we may proceed directly to the induction step
and assume the result holds for all prior correlators. We first argue that we
may commute derivatives in $\tau$ with all residues in (46). To do this, we
transform all residues into contour integrals; even if the point at which the
residue is being taken depends on $\tau$, the contour may be taken to be
locally constant in $\tau$. Then we may commute the derivatives in $\tau$ with
the $\tau$-independent contour integrals and, as the derivatives in $\tau$
cannot create new poles, we may switch all contour integrals back to the same
residues.

In an identical manner to the second step of the proof, we wish to argue the
residues at $t^{C_{l}}=R^{N}_{\infty}$ do not contribute. For the points in
$R^{N}_{\infty}$ that satisfy $M_{1}=N/(1-\tau)$, $x_{N}(t^{C_{l}})$ has a
pole of order $N$ and the argument proceeds identically to the argument in the
second step. Similarly, for the points where $M_{1}=\infty$ the expansions
(47) and (48) are the same as before and the identical argument works for
general choices of $\tau$.

The only new thing to check is that the points in $R^{N}_{\infty}$ that
satisfy $M_{1}=-N/\tau$ do not contribute. Here $x_{N}$ has a zero of order
$N$ rather than a pole of order $N$. Denoting the collection of these points
as $V_{N}$, by (50) we look at the expression (with the $\tau$ dependence
suitably inserted)

$\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=V_{N}\end{subarray}}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}\frac{f^{\tau}_{N}(t,t^{C_{l}},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{(M_{2}(t)-M_{2}(t^{C_{l}}))^{|C_{l}|}}\,,$ (62)

where, as before, $f^{\tau}_{N}$ will only have poles of order bounded
uniformly in $N$. It will be important to observe, that as the correlators are
regular at the poles of $M_{1}$ and $M_{2}$ has poles at the poles of $M_{1}$
that $f_{N}^{\tau}$ has no pole at the poles of $M_{1}$ for all $\tau$. Then
we note, for any $a_{N}\in V_{N}$,

$\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}=1+\mathcal{O}((t-t^{C_{l}})^{N}),$
(63)

so the expression in (62) is in fact equal to

$\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=V_{N}\end{subarray}}\frac{f^{\tau}_{N}(t,t^{C_{l}},t_{\llbracket
m-1\rrbracket}\setminus C_{l}\,|\,z_{\llbracket
n\rrbracket})}{(M_{2}(t)-M_{2}(t^{C_{l}}))^{|C_{l}|}}\eqcolon
F^{\tau}_{N}(t,t^{C_{l}},t_{\llbracket m-1\rrbracket}\setminus
C_{l}\,|\,z_{\llbracket n\rrbracket}).$ (64)

We now claim that $[\partial^{m}F^{\tau}_{N}]_{\tau=0}\equiv 0$ for all
$m\in\mathbb{Z}_{\geq 0}$, which would mean, at least locally near $\tau=0$,
that these points do not contribute. To prove this we wish to commute the
$\tau$ derivatives with the residue in the above expression. There is a slight
subtlety, as all points in $V_{N}$ collide at poles of $M_{1}$ in the limit
$\tau\to 0$. However, as the $f_{N}^{\tau}$ are regular at the poles of
$M_{1}$, near $\tau=0$ we may draw a contour around each of the poles of
$M_{1}$ that includes all points in $V_{N}$ (but no other poles of the
integrand) and commute the $\tau$ derivatives with this contour.

Next, we note that $f_{N}^{0}=f_{N}$ has no pole at the poles of $M_{1}$ so
$[\partial^{m}f^{\tau}_{N}]_{\tau=0}$ is in fact regular at the poles of
$M_{1}$ for all $m\in\mathbb{Z}_{\geq 0}$. Then,
$[\partial^{m}F^{\tau}_{N}]_{\tau=0}$ just involves taking residues of
$[\partial^{m}f^{\tau}_{N}]_{\tau=0}$ at the poles of $M_{1}$ and therefore
vanishes.

As argued previously, we can commute the $\tau$ derivatives with the residues
and obtain

$\begin{split}\partial_{\tau}^{k}\omega^{N}_{g,n+1}(z_{0},z_{\llbracket
n\rrbracket})&=\operatorname*{Res}_{t=R^{N}}\sum_{m=2}^{\deg(x_{N})}\\!\\!\\!\left(\int_{*}^{t}B(z_{0},\cdot)\right)\\!\\!\\!\sum_{C_{1},\dots,C_{j}\vdash
t_{\llbracket
m-1\rrbracket}}\\!\\!\\!\\!\\!\\!\frac{(-1)^{1-\delta_{j,m-1}}}{j!}\\!\\!\\!\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=R^{N}_{0},z_{\llbracket
n\rrbracket},\mathcal{Y}(t)\\\
l=1,\dots,j\end{subarray}}\operatorname*{Res}_{\begin{subarray}{c}C_{l}=t^{C_{l}}\\\
l=1,\dots,j\end{subarray}}\\\
&\quad\partial_{\tau}^{k}\left(\prod_{l=1}^{j}\frac{x_{N}(t)}{x_{N}(t)-x_{N}(t^{C_{l}})}\\!\prod_{t_{0}\in
C_{l}\setminus\\{t^{C_{l}}\\}}\\!\\!\frac{x_{N}(t^{C_{l}})}{x_{N}(t_{0})-x_{N}(t^{C_{l}})}\right)\frac{\mathcal{W}_{g,n,m}^{N}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{m-1}(M_{2}(t)-M_{2}(t_{l}))}\,,\end{split}$ (65)

The expression that the derivative in $\tau$ hits in the above equation
depends on $\tau$ through the $x_{N}$ and the
$\omega_{g^{\prime},n^{\prime}}^{N}$. $\partial_{\tau}^{i}x_{N}$ goes to zero
in the limit $N\to\infty$ and
$\partial_{\tau}^{i}\omega_{g^{\prime},n^{\prime}}^{N}$ goes to zero by the
induction assumption. Thus, the entire second line (i.e., the part inside all
the residues) is converging to zero as $N\to\infty$. As all the residues in
the variables $C_{l}$ may just be converted into integrals101010If two or more
ramification points in $R^{N}_{0}$ collide in the limit, then we will have to
write one contour integral around the point in $R_{0}$ they collide at. we get
that the entire integrand in $t$ is converging to zero. Analogously to proving
the existence of the limit when $\tau=0$, we must now argue the integral
itself goes to zero.

For the residues at $t=R^{N}_{0}$ the fact that the integrand is going to zero
is clear as we may take the $N\to\infty$ limit inside the contour integral; so
we concentrate on the residues at $t=R^{N}_{\infty}$. Here, for generic
$\tau$, we will have residues at solutions of $M_{1}(z)=N/(1-\tau),-N/\tau$.
When we set $\tau=0$, we end up with no poles at $M_{1}(z)=\infty$, even after
taking derivatives, as we cannot create poles by taking derivatives. For our
purposes, we may therefore neglect the residues at $M_{1}(z)=-N/\tau$, as they
will drop out in the end.

Thus, at these points, we need to take $\tau$ derivatives of the analogous
expression to (49) where the $\mathcal{N}$ and $\mathcal{D}$ coefficients
acquire the suitable $\tau$ dependence and
$\exp_{N}(w^{-1})=(1+(\tau-1)/(Nw))^{-N}(1+\tau/(Nw))^{N}$ is appropriately
modified111111$\tau$ derivatives commute with the pushforward in $M_{1}$ as
$M_{1}$ does not depend on $\tau$.. After taking $m$ $\tau$ derivatives, the
ratio of the new leading order coefficients will have no pole at $w=\tau=0$ by
the quotient rule and the fact that the $\tau$ derivative can only decrease
the order of poles at $w=0$. We may then conclude that the same argument with
the $N\to\infty$ limit holds

$\begin{split}\operatorname*{Res}_{w=(1-\tau)/N}&\partial_{\tau}^{m}\frac{\mathcal{N}_{N}^{d,\tau}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(1/w)^{d}+\dotsb}{\mathcal{D}^{d,\tau}_{N}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp_{N}(1/w)^{d}+\dotsb}\\\
&=\operatorname*{Res}_{w=(1-\tau)/N}\partial_{\tau}^{m}\frac{\mathcal{N}_{N}^{d,\tau}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d,\tau}_{N}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}\left(1+\mathcal{O}(w-(1-\tau)/N)^{N}\right)\\\
&=\operatorname*{Res}_{w=(1-\tau)/N}\partial_{\tau}^{m}\frac{\mathcal{N}_{N}^{d,\tau}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d,\tau}_{N}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}\,.\end{split}$ (66)

but this time, after taking the derivatives in $\tau$, the ratio of the
leading order coefficients is converging to zero.

Finally, we argue that this result, namely that
$\partial_{\tau}^{m}\omega_{g,n+1}^{N}(z_{0},z_{\llbracket
n\rrbracket})|_{\tau=0}$ exists and goes to zero in the limit, perhaps
unsurprisingly, actually establishes the theorem. Note that we have the
following expansion for sufficiently small $\tau$ and generic choices of
$z_{1},\dotsc,z_{n}\in\Sigma$

$\omega_{g,n+1}^{N}(z_{0},z_{\llbracket
n\rrbracket})=\sum_{m=0}^{\infty}\frac{\tau^{m}}{m!}\partial_{\tau}^{m}\omega_{g,n+1}^{N}(z_{0},z_{\llbracket
n\rrbracket})|_{\tau=0}\,.$ (67)

Denote the radius of convergence of this sum as $\rho_{N}(z_{0},z_{\llbracket
n\rrbracket})$. We claim that $\rho_{N}\to\infty$ provided $z_{i}\notin
R\quad\forall i=0,\dots,n$. To prove this claim we examine the singularity
structure of $\omega_{g,n+1}^{N}(z_{0},z_{\llbracket
n\rrbracket})=\omega_{g,n+1}^{N}(z_{0},z_{\llbracket n\rrbracket};\tau)$. This
is straightforward as these $\omega_{g,n+1}^{N}$ only have poles at
ramification points.

The ramification points of $x_{N}$ can be put into three categories: solutions
of $M_{1}(z)=(1-\tau)/N$; solutions of $M_{1}(z)=-\tau/N$; poles of $M_{1}$;
poles of $M_{0}$; zeros of $dx_{N}$ that converge to elements of $R_{0}$. For
the first two cases we clearly see that, for fixed $z$, these singularities go
to infinity in the $\tau$ plane. The next two cases can never create
singularities in the $\tau$ plane as, by assumption, $z_{i}\notin R$. The
fifth and final case is dealt with by computing

$\left(1+(\tau-1)\frac{M_{1}}{N}\right)^{N+1}\left(1+\tau\frac{M_{1}}{N}\right)^{-N+1}dx_{N}=\left(1+(\tau-1)\frac{M_{1}}{N}\right)\left(1+\tau\frac{M_{1}}{N}\right)dM_{0}+M_{0}dM_{1}\,,$
(68)

and noting that, for a fixed point on $\Sigma$ that is not a zero of this
differential in the limit, the only zero this has in the $\tau$ plane shoots
off to infinity in the limit $N\to\infty$.

As $\rho_{N}$ is the distance between zero and the nearest singularity in the
$\tau$ plane we indeed have that $\rho_{N}\to\infty$. Thus, given any fixed
$\tau$ we are inside the radius of convergence, i.e., $|\tau|<\rho_{N}$, for
all $N$ sufficiently large so we may commute the limit in $N$ with the
infinite sum. This proves the theorem. ∎

Now that we know that correlators for transalgebraic spectral curves are well-
defined, we can easily prove a number of corollaries.

######  Corollary 4.4.

Let $\mathcal{S}$ be a compact transalgebraic admissible spectral curve. For
$2g+n-2\geq 1$, the correlators $\omega_{g,n}$ constructed by topological
recursion on $\mathcal{S}$ satisfy the following properties:

  * •

Symmetry: the $\omega_{g,n}$ are symmetric in all of their $n$ variables.

  * •

Pole structure: the $\omega_{g,n}$ have poles only at the ramification points
of $x$.

  * •

Residueless: the $\omega_{g,n}$ have vanishing residue at all points.

  * •

Homogeneity: rescaling $\omega_{0,1}$ by a constant $c\in\mathbb{C}^{*}$ to
$c\omega_{0,1}$ results in a rescaling $\omega_{g,n}\to
c^{2-2g-n}\omega_{g,n}$.

###### Proof.

These properties are well-known for ordinary topological recursion, and were
proved in [EO07, BE13, BBCCN18]. They carry over as they hold for each curve
in our sequence of spectral curves. ∎

We also give a direct formula for topological recursion on transalgebraic
spectral curves, in a wide variety of cases, without using a sequence of
correlators and taking limits.

######  Lemma 4.5.

Let $\mathcal{S}$ be a compact transalgebraic admissible spectral curve. If
$M_{1}$ is a well-defined function of $M_{2}$ we may use the following formula
to recursively compute the correlators of topological recursion.

$\begin{split}\omega_{g,n+1}(z_{0},z_{\llbracket
n\rrbracket})&=\operatorname*{Res}_{t=R}\sum_{m=2}^{\deg(x)}\left(\int_{*}^{t}B(z_{0},\cdot)\right)\sum_{C_{1},\dotsc,C_{j}\vdash
t_{\llbracket
m-1\rrbracket}}\frac{(-1)^{1-\delta_{j,m-1}}}{j!}\operatorname*{Res}_{\begin{subarray}{c}t^{C_{l}}=R_{0},Z,\mathcal{Y}(t)\\\
l=1,\dotsc,j\end{subarray}}\operatorname*{Res}_{\begin{subarray}{c}C_{l}=t^{C_{l}}\\\
l=1,\dotsc,j\end{subarray}}\\\
&\left(\prod_{l=1}^{j}\frac{x(t)}{x(t)-x(t^{C_{l}})}\prod_{t_{0}\in
C_{l}\setminus\\{t^{C_{l}}\\}}\frac{x(t)}{x(t_{0})-x(t^{C_{l}})}\right)\frac{\mathcal{W}_{g,n,m}(t,t_{\llbracket
m-1\rrbracket}\,|\,z_{\llbracket
n\rrbracket})}{\prod_{l=1}^{i-1}((xy)(t)-(xy)(t_{l}))}\,,\end{split}$ (69)

where $\mathcal{Y}(t)=M_{2}^{-1}\big{(}M_{2}(t)\big{)}$ and the residues at
the infinite ramification points $R_{\infty}$ are defined as

$\operatorname*{Res}_{t=R_{\infty}}\leftrightarrow\frac{1}{(2g-1)!}\lim_{w\to
0^{+}}\frac{d^{2g-1}}{dw^{2g-1}}{M_{1}}_{*}\,,$ (70)

where the expression on the right is to be interpreted as follows: we take the
pushforward under the map $M_{1}$ and define $w=1/M_{1}$ so the infinite
ramification points are all located at $w=0$; the formula is then the standard
one for a pole of order $2g$ at $w=0$ except we take the limit as $w\to 0$
along the positive real axis.

###### Proof.

Adopting the notation of the proof of theorem 4.3, as
$\mathcal{N}_{N}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{N}(w\,|\,z_{0},z_{\llbracket n\rrbracket})$ has
a pole of order at most $2g$ at $w=1/N$, we will have that
$\mathcal{N}_{\infty}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})/\mathcal{D}^{d}_{\infty}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})$ will have a pole of order no more than $2g$ at $w=0$. Ergo, we
can compute the topological recursion in the limit (note here, by assumption,
there are no $\nu$). By definition, this is

$\begin{split}&\frac{1}{(2g-1)!}\lim_{w\to
0^{+}}\frac{d^{2g-1}}{dw^{2g-1}}\frac{\mathcal{N}_{\infty}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp(d/w)+\dotsb}{\mathcal{D}^{d}_{\infty}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})\exp(d/w)+\dotsb}\\\ =&\frac{1}{(2g-1)!}\lim_{w\to
0^{+}}\frac{d^{2g-1}}{dw^{2g-1}}\frac{\mathcal{N}_{\infty}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d}_{\infty}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}\left(1+\mathcal{O}(\exp(-w^{-1}))\right)\\\
=&\frac{1}{(2g-1)!}\lim_{w\to
0^{+}}\frac{d^{2g-1}}{dw^{2g-1}}\frac{\mathcal{N}_{\infty}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d}_{\infty}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}=\operatorname*{Res}_{w=0}\frac{\mathcal{N}_{\infty}^{d}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}{\mathcal{D}^{d}_{\infty}(w\,|\,z_{0},z_{\llbracket
n\rrbracket})}.\qed\end{split}$ (71)

######  Remark 4.6.

We believe that this formula works even when $M_{1}$ is not a well-defined
function of $M_{2}$, but it is tricky to establish due to the possibility of
the $\exp(1/\nu(w))$ contributing to the leading order coefficients in the
limit. If one could prove that the $\exp(1/\nu(w))$ do not contribute to the
leading order in the limit, the lemma would hold even when $M_{1}$ is not a
well-defined function of $M_{2}$.

###  4.3. Essential singularities only contribute for $n=1$

As highlighted in remark 2.7, given transalgebraic functions $x$ and $y$ with
exponential singularities on a compact Riemann surface $\Sigma$, with $xy$
meromorphic, one can define two distinct spectral curves:

  1. (1)

A compact transalgebraic spectral curve, where the Riemann surface is taken to
be $\Sigma$ itself;

  2. (2)

A non-compact meromorphic spectral curve, where the Riemann surface is taken
to be $\Sigma\setminus R_{\infty}$ (which ignores the essential singularities
of $x$).

In this section, we show that topological recursion on the compact
transalgebraic spectral curve differs from topological recursion on the non-
compact meromorphic spectral curve at only finitely many steps. More
explicitly, what this means is that for all but finitely many $(g,n)$, the
$\omega_{g,n}$ defined by topological recursion on transalgebraic spectral
curves (definition 4.2) may also be calculated using the recursive step of
standard topological recursion (definition 2.8) ignoring essential
singularities.

We will prove this by analysing the $N$-dependence of the $\omega_{g,n}^{N}$;
we show that $\lim_{N\to\infty}\omega_{g,n}^{N}$ has no principal part at
$R_{\infty}$ for all but finitely many $(g,n)$.

We begin by examining the loop equations of proposition 2.9. We wish to prove
that the $i$th loop equation (corresponding to $\mathcal{E}_{g,n,i}$) can be
written with all the same sheets rather than all different ones. First we
define the non-regularised $(0,2)$ correlator

$\bar{\omega}_{0,2}(z_{1},z_{2})\coloneqq\omega_{0,2}(z_{1},z_{2})-\frac{dx(z_{1})dx(z_{2})}{(x(z_{1})-x(z_{2}))^{2}}=-\sum_{\zeta\in\mathfrak{f}^{\prime}(z_{1})}\omega_{0,2}(\zeta,z_{2})+\textup{regular}\,.$
(72)

Then we have the following result.

######  Proposition 4.7.

On a meromorphic spectral curve,

$\sum_{\begin{subarray}{c}Z\subset\mathfrak{f}_{a}(z)\\\
|Z|=i\end{subarray}}\mathcal{E}_{g,n,i}(Z\,|\,z_{\llbracket
n\rrbracket})=(-1)^{i-1}(i-1)!\sum_{z^{\prime}\in\mathfrak{f}_{a}(z)}\bar{\mathcal{E}}_{g,n,i}(z^{\prime},\dotsc,z^{\prime}\,|\,z_{\llbracket
n\rrbracket})+\mathcal{O}(z^{r_{a}-1-2g}dz^{i})$ (73)

for a local coordinate $z$ near a ramification point $a$ with $s_{a}=1$, where
the bar indicates we replace any occurrence of $\omega_{0,2}$ with
$\bar{\omega}_{0,2}$.

For an admissible transalgebraic spectral curve $\mathcal{S}$, for fixed $g$,
either side of the expression is holomorphic of arbitrarily high vanishing
order on $\mathcal{S}_{N}$ near points $a\in R_{\infty}^{N}$ as $N\to\infty$.

###### Proof.

We will repeatedly apply the linear loop equations on the meromorphic spectral
curves converging to our transalgebraic one. The linear loop equation, the
$i=1$ case of proposition 2.9, always holds up to
$\mathcal{O}(x^{0}(z)dx(z))=\mathcal{O}(z^{r_{a}-1}dz)$. By assumption,
$s_{a}=1$, so by lemma 2.10, the pole order of $\omega_{g,n}$ at $a$ is
bounded by $2g$.

Writing $\zeta_{k}$ for the local Galois conjugates of a $\zeta_{1}$,

$\sum_{k_{1}}\sum_{k_{2}\neq k_{1}}\cdots\sum_{k_{i}\neq
k_{1},\dots,k_{i-1}}\mathcal{E}_{g,n,i}(\zeta_{k_{1}},\dotsc,\zeta_{k_{i}}(z)\,|\,z_{\llbracket
n\rrbracket})\\\ \sim-\sum_{k_{1}}\sum_{k_{2}\neq
k_{1}}\cdots\sum_{k_{i-1}\neq
k_{1},\dots,k_{i-2}}\sum_{j=1}^{i-1}\mathcal{E}_{g,n,i}(\zeta_{k_{1}},\dotsc,\zeta_{k_{i-1}},\overline{\zeta_{k_{j}}}\,|\,z_{\llbracket
n\rrbracket})\,,$ (74)

where the bar over the entry indicates we should replace every instance of
$\omega_{0,2}$ evaluated at this entry with $\bar{\omega}_{0,2}$ and $\sim$
means up to $\mathcal{O}(z^{r_{a}-1-2g}dz^{i})$. We then note the summation is
entirely symmetric in $k_{1},\dots,k_{i-1}$ (we are summing over all
permutations of $i-1$ indices where no two indices are the same) so all terms
in the sum over $j$ are equal to the term with $j=1$ and therefore the above
is equal to

$-(i-1)\sum_{k_{1}}\sum_{k_{2}\neq k_{1}}\cdots\sum_{k_{i-1}\neq
k_{1},\dots,k_{i-2}}\mathcal{E}_{g,n,i}(\zeta_{k_{1}},\dotsc,\zeta_{k_{i-1}},\overline{\zeta_{k_{1}}}\,|\,z_{\llbracket
n\rrbracket})\\\ \sim(i-1)\sum_{k_{1}}\sum_{k_{2}\neq
k_{1}}\cdots\sum_{k_{i-2}\neq
k_{1},\dots,k_{i-3}}\sum_{j=1}^{i-2}\mathcal{E}_{g,n,i}(\zeta_{k_{1}},\dotsc,\zeta_{k_{i-2}},\overline{\zeta_{k_{j}}},\overline{\zeta_{k_{1}}}\,|\,z_{\llbracket
n\rrbracket}).$ (75)

By the same argument we may remove the sum over $j$ by picking up a factor of
$i-2$. Repeating this argument a further $i-3$ times yields the desired first
result.

For the second, we use that for any $a\in R_{\infty}^{N}$, by admissibility,
$s_{a}=1$, so we may apply the first result, and moreover the order of the
loop equation from proposition 2.9 is $\mathcal{O}(z^{r_{a}-i}dz^{i})$, as
$s_{a}=1$ and $i\leq r_{a}$. Then $r_{a}=\mathcal{O}(N)$, so indeed the
vanishing order of either side grows arbitrarily large as $N\to\infty$. ∎

Now we use our rewritten loop equations to derive the desired $N$-dependence
of the $\omega_{g,n}^{N}$. Fixing an essential singularity $a$ of $x$, we
define a local coordinate near this essential singularity through
$\zeta^{-m_{1}}=M_{1}$; using this notation, the essential singularity
corresponds to $\zeta(a)=0$. Setting $\tau=0$ in $x_{N}$, let $\vartheta$ be a
primitive $m_{1}$th root of unity and define the coordinate $t$ such that
$t^{-N}=x_{N}$ so that
$t(\zeta)=M_{0}(\zeta)^{-1/N}(1-\frac{1}{\zeta^{m_{1}}N})$ where the branch of
the $N$th root is chosen such that $M_{0}(\zeta=\vartheta^{m}N^{-1/m_{1}})$
does not lie on the cut for any values of $m$ and $N$ and the limit value of
$t$ ($\lim_{N\to\infty}t$ is a constant function) is not a pole of any of the
correlators. Our claim is then

######  Lemma 4.8.

With the above conditions, the principal part of
$\omega_{g,n+1}^{N}(t,z_{\llbracket n\rrbracket})$ at
$\zeta(t)=\vartheta^{m}N^{-1/m_{1}}$ is given by

$\operatorname*{Res}_{\zeta=\vartheta^{m}N^{-1/m_{1}}}\Big{(}\int_{\vartheta^{m}N^{-1/m_{1}}}^{\zeta}\\!\\!\\!\omega_{0,2}(t,\mathord{\cdot})\Big{)}\omega_{g,n+1}^{N}(\zeta,z_{\llbracket
n\rrbracket})=N^{1-n+\frac{m_{2}}{m_{1}}\chi_{g,n+1}}\sum_{l=1-2g}^{-1}w^{l,m}_{g,n}(z_{\llbracket
n\rrbracket})d\xi_{l}^{m}(t)\,,$ (76)

where

$d\xi_{l}^{m}(t^{\prime})=\operatorname*{Res}_{\zeta=\vartheta^{m}N^{-1/m_{1}}}\Big{(}\int_{\vartheta^{m}N^{-1/m_{1}}}^{\zeta}\\!\\!\\!\omega_{0,2}(t^{\prime},\mathord{\cdot})\Big{)}t(\zeta)^{l}dt(\zeta),\qquad
l<0\,,$ (77)

$\chi_{g,n}=2-2g-n$ is the Euler characteristic, and
$w^{l,m}_{g,n}=\mathcal{O}(N^{0})$.

We require a lemma for the proof.

######  Lemma 4.9.

With the notation as above,

$\bar{\omega}_{0,2}^{N}(t,t)=\frac{dt\,du}{(t-u)^{2}}-\frac{dt^{N}du^{N}}{(t^{N}-u^{N})^{2}}\bigg{|}_{u=t}+\mathcal{O}(t^{0}dt^{2})=\frac{(N-1)(N+1)(5N-6)}{24N}\frac{dt^{2}}{t^{2}}+\mathcal{O}(t^{0}dt^{2})\,.$
(78)

###### Proof.

The first equality holds by general invariance of the principal part of
$\frac{dzdw}{(z-w)^{2}}$ under change of local coordinates.

The second equality is a direct calculation, using geometric series.

$\begin{split}\frac{dt\,du}{(t-u)^{2}}-\frac{dt^{N}du^{N}}{(t^{N}-u^{N})^{2}}&=\bigg{(}\Big{(}\sum_{m=0}^{N-1}t^{N-m-1}u^{m}\Big{)}^{2}-N^{2}t^{N-1}u^{N-1}\bigg{)}\frac{dt\,du}{(t^{N}-u^{N})^{2}}\\\
&=\bigg{(}N(N-1)t^{N-1}u^{N-1}-\\!\sum_{k=0}^{N-2}\\!(k+1)\Big{(}t^{2N-2-k}u^{k}+t^{k}u^{2N-2-k}\Big{)}\bigg{)}\frac{dt\,du}{(t^{N}-u^{N})^{2}}\\\
&=\sum_{k=0}^{N-2}(k+1)\Big{(}t^{2N-2-k}u^{k}+t^{k}u^{2N-2-k}-2t^{N-1}u^{N-1}\Big{)}\frac{dt\,du}{(t^{N}-u^{N})^{2}}\\\
&=\sum_{k=0}^{N-2}(k+1)\sum_{l=0}^{N-2-k}\frac{(t^{2N-3-k-l}u^{k+l}-t^{k+l}u^{2N-3-k-l})dt\,du}{(t-u)\big{(}\sum_{m=0}^{N-1}t^{N-1-m}u^{m}\big{)}^{2}}\\\
&=\sum_{j=0}^{N-2}\sum_{k=0}^{j}(k+1)\frac{(t^{2N-3-j}u^{j}-t^{j}u^{2N-3-j})dt\,du}{(t-u)\big{(}\sum_{m=0}^{N-1}t^{N-1-m}u^{m}\big{)}^{2}}\\\
&=\sum_{j=0}^{N-2}\frac{(j+1)(j+2)}{2}\frac{(t^{2N-3-j}u^{j}-t^{j}u^{2N-3-j})dt\,du}{(t-u)\big{(}\sum_{m=0}^{N-1}t^{N-1-m}u^{m}\big{)}^{2}}\\\
&=\sum_{j=0}^{N-2}\frac{(j+1)(j+2)}{2}t^{j}u^{j}\sum_{i=0}^{2N-4-2j}\frac{t^{2N-4-2j-i}u^{i}dt\,du}{\big{(}\sum_{m=0}^{N-1}t^{N-1-m}u^{m}\big{)}^{2}}\,.\end{split}$

At this point, we may set $u=t$, and obtain

$\begin{split}\bar{\omega}_{0,2}^{N}(t,t)&=\sum_{j=0}^{N-2}\frac{(j+1)(j+2)}{2}(2N-3-2j)\frac{t^{2N-4}dt^{2}}{\big{(}Nt^{N-1}\big{)}^{2}}+\mathcal{O}(t^{0}dt^{2})\\\
&=\frac{(N-1)(N+1)(5N-6)}{24N}\frac{dt^{2}}{t^{2}}+\mathcal{O}(t^{0}dt^{2})\,.\qed\end{split}$

###### Proof of lemma 4.8.

The principal part is given by the left-hand side because of the projection
property of proposition 2.9.

To prove the equality, we use induction on the negative of the Euler
characteristic $-\chi_{g,n}=2g-2+n$. We start with $\omega_{0,1}$. However, we
modify $\omega_{0,1}$ locally near each ramification point
$z=\vartheta^{m}N^{1/m_{1}}$ by subtracting
$-N^{-1}y(\vartheta^{m}N^{1/m_{1}})d\log(x_{N})$; as this is a pure function
of $x$, this modification does not affect topological recursion in any way,
and it ensures $\omega_{0,1}$ satisfies the local linear loop equations at the
ramification points colliding at $a$. Examining the expansion coefficients in
$t$ of $M_{2}$ around $t(z=\vartheta^{m}N^{1/m_{1}})=0$,

$M_{2}(t)=N^{\frac{m_{2}}{m_{1}}}\vartheta^{mm_{2}}\sum_{l=0}^{\infty}y_{l}\xi_{l}(t)\,,$

where the $y_{l}$, to leading order in $N$, do not depend on $N$ or $m$. As
$\omega_{0,1}(t)=-N(M_{2}(t)-M_{2}(t=0))dt/t$, the claimed result holds in
this case. The claim for $\omega_{0,2}$ also holds:

$\omega_{0,2}(t,z_{1})=\sum_{l=1}^{\infty}w^{l,m}_{0,2}(z_{1})\xi_{l}^{m}(t),\qquad
z\to\vartheta^{m}N^{1/m_{1}}\,.$

However, we will be particularly concerned with not $\omega_{0,2}$ but
$\bar{\omega}_{0,2}$. There are two cases we need to examine when there is
such a term in
$\mathcal{E}_{g,n,i}(\zeta_{k_{1}},\dotsc,\zeta_{k_{i}}\,|\,z_{\llbracket
n\rrbracket})$. The first is when a term has a factor of

$\bar{\omega}_{0,2}(\zeta_{k_{a}},z_{b})=\omega_{0,2}(\zeta_{k_{a}},z_{b})-\frac{dx_{N}(z)dx_{N}(z_{b})}{(x_{N}(z)-x_{N}(z_{b}))}.$

Here, at the ramification points of interest, $t=0$, we note that the second
term has a zero of order $N-1$ and so will not need to be taken into account
in examining the loop equations. The first term is just $\omega_{0,2}$ and so
follows the claim.

The second case is when there is a factor of
$\bar{\omega}_{0,2}(\zeta_{k_{a}},\zeta_{k_{a}})$. In this case, by lemma 4.9,
$\bar{\omega}_{0,2}(\zeta_{k_{a}},\zeta_{k_{a}})=\mathcal{O}(N^{2})\frac{dt^{2}}{t^{2}}$.

Now we are equipped to perform the induction step. Let $\alpha$ be a primitive
$N$th root of unity and examine the loop equation from proposition 4.7

$\sum_{r=1}^{N}\mathcal{E}_{g,n,i}(\alpha^{r}t,\dotsc,\alpha^{r}t\,|\,z_{\llbracket
n\rrbracket})=\sum_{\mu\vdash\llbracket
i\rrbracket}\sum_{\begin{subarray}{c}\sqcup_{k=1}^{l(\mu)}N_{k}=\llbracket
n\rrbracket\\\
\sum_{k=1}^{l(\mu)}g_{k}=g+l(\mu)-n\end{subarray}}\prod_{k=1}^{l(\mu)}\omega_{g_{k},|\mu_{k}|+|N_{k}|}(\alpha^{r}t,\dotsc,\alpha^{r}t,z_{N_{k}}).$

Let us first examine the terms that contain a factor of $\omega_{g,n+1}$.
These are

$i\sum_{r=1}^{N}\omega_{g,n+1}(\alpha^{r}t,z_{\llbracket
n\rrbracket})\omega_{0,1}(\alpha^{r}t)^{i-1}.$

Each factor of $\omega_{0,1}$ gives an $N$ dependence of
$N^{1+\frac{m_{2}}{m_{1}}}$ and the sum over $r$ gives an additional
(possible) factor of $N$. Thus the highest order $N$ dependence the
coefficient of $\omega_{g,n+1}(\alpha^{r}t,z_{\llbracket n\rrbracket})$ can
have is $N^{1+(i-1)(1+\frac{m_{2}}{m_{1}})}$.

Now let us examine the highest order $N$ dependence another term may have
using the induction assumption. Examining the expansion of
$\bar{\omega}_{0,2}(\alpha^{r}t,\alpha^{r}t)$ and the induction assumption, we
see that the highest order occurs when the partition $\mu$ consists only of
singletons and pairs where each pair has genus zero and none of the
$z_{\llbracket n\rrbracket}$ to yield a
$\bar{\omega}_{0,2}(\alpha^{r}t,\alpha^{r}t)$ factor. We will calculate the
largest possible $N$ dependence of such a term. Let $s$ be the number of
singletons and $d$ the number of pairs. The pairs give us $d$ factors
$\bar{\omega}_{0,2}(\alpha^{r}t,\alpha^{r}t)$, which give an $N$ dependence of
$N^{2d}$. The singletons give a total $N$ dependence, by the induction
assumption, of $N^{2s-s-n+\frac{m_{2}}{m_{1}}(i-1+\chi_{g,n+1})}$. The sum
then gives us an additional factor of $N$, for a total dependence of (using
that $2d+s=i$)

$N^{2d+s-n+\frac{m_{2}}{m_{1}}(i-1+\chi_{g,n+1})+1}=N^{i+1-n+\frac{m_{2}}{m_{1}}(i-1+\chi_{g,n+1})}.$

Thus, $\omega_{g,n+1}$ indeed has the claimed $N$ dependence and we are done.
∎

As a direct consequence of lemma 4.8, we find the following corollary.

######  Corollary 4.10.

Let $a$ be a pole of $M_{1}$. Then all correlators $\omega_{g,n}^{N}$ with
$2gm_{2}\geq(2-n)(m_{1}+m_{2})$ have vanishing principal part at $a$ in the
limit as $N\to\infty$; in particular, this includes all correlators with
$n\geq 2$.

###### Proof.

Given that $t=M_{0}(z)^{-1/N}(1-z^{m_{1}}/N)$, we have the following
expansion, where the $a_{l}=\mathcal{O}(N^{0})$ are order one coefficients
that, to leading order in $N$, do not depend on $m$:

$t=\sum_{l=1}^{\infty}\frac{a_{l}}{\vartheta^{ml}N^{l/m_{1}}}(z-\vartheta^{m}N^{1/m_{1}})^{l}.$

Ergo, lemma 4.8 implies that we have the expansion

$\omega_{g,n+1}^{N}(z,z_{\llbracket
n\rrbracket})=N^{1-n+\frac{m_{2}}{m_{1}}\chi_{g,n+1}}\sum_{l=2}^{2g}\frac{A_{l,m}(z_{\llbracket
n\rrbracket})}{(z/(\vartheta^{m}N^{1/m_{1}})-1)^{l}}d\frac{z}{\vartheta^{m}N^{1/m_{1}}}+\mathcal{O}\left((z-\vartheta^{m}N^{1/m_{1}})^{0}\right)\,,$
(79)

where the $A_{l,m}=\mathcal{O}(N^{0})$. So if
$1-n+\frac{m_{2}}{m_{1}}\chi_{g,n+1}-\frac{1}{m_{1}}<0$, the limit
$N\to\infty$ vanishes. Changing $n+1\to n$, this is equivalent to the
condition in the corollary. ∎

######  Corollary 4.11.

The correlators $\omega_{g,n}$ with $2g-2+n>0$ are regular at essential
singularities where $m_{2}\geq m_{1}$; in particular, this includes all
essential singularities where $M_{1}$ has only a simple pole ($m_{1}=1$).

###### Proof.

For $n>1$ we have established the correlators may never have poles. For $n=1$
we know the correlators do not have poles if $(2g-1)m_{2}\geq m_{1}$ by the
previous corollary; this is trivially true if $m_{2}\geq m_{1}$ as $g\geq 1$.
∎

######  Proposition 4.12.

Let $\mathcal{S}$ be a compact transalgebraic admissible spectral curve. For
$2gm_{2}\geq(2-n)(m_{1}+m_{2})$, the correlators $\omega_{g,n}$ defined via
topological recursion on $\mathcal{S}$ (definition 4.2) are regular at all
essential singularities $a\in R_{\infty}$. In particular, this includes all
correlators with $n\geq 2$.

The correlators $\omega_{g,n}$ satisfying the condition above may be
calculated via the topological recursion of definition 2.8 with residues only
at the finite ramification points, but where the $\omega_{g_{k},n_{k}}$ on the
right-hand side of equation 12 are obtained by the topological recursion of
definition 4.2.

###### Proof.

The first statement follows immediately from corollary 4.10 and the definition
of transalgebraic topological recursion as a limit.

For the second statement, note that the individual contributions of
ramification points in equation 12 are continuous as the spectral curve varies
without the type of ramification changing.121212To be precise, this is proven
for ramification points of arbitrary order in [BBCKS]. This is the case for
the $a\in R_{0}^{N}$, which converge to $R_{0}$. By the projection property,
equation 18, $\omega_{g,n}^{N}$ are the sums of their principal parts, and by
corollary 4.10, the limit of the contributions at elements of $R_{\infty}^{N}$
vanishes. As $\omega_{g,n}$ is defined as the limit, this proves the second
statement. ∎

######  Remark 4.13.

Proposition 4.12 does _not_ mean that for $2gm_{2}\geq(2-n)(m_{1}+m_{2})$, the
correlators $\omega_{g,n}$ calculated via topological recursion on the
transalgebraic spectral curve (definition 4.2) are equal to the ones
calculated from topological recursion on the non-compact meromorphic spectral
curve that ignores essential singularities. Rather, the contributions from the
essential singularities at higher Euler characteristic propagate to all
$(g,n)$ through the recursion at finite ramification points.

However, as the inequality only fails for $n=1$ and
$2g-1\leq\frac{m_{1}}{m_{2}}$, it does mean that for any given transalgebraic
spectral curve, the limit definition of topological recursion only has to be
used a finite number of times, and can then be disregarded for the remaining
correlator calculations.

Now we provide a bound on the order of the poles of the correlators at the
infinite ramification points.

######  Proposition 4.14.

Let $\mathcal{S}$ be a compact transalgebraic admissible spectral curve. Let
$a\in R_{\infty}$ be an infinite ramification point. Suppose that $M_{1}$ has
a pole of order $m_{1}$ at $a$, and let $m_{2}$ be the order of the pole of
$xy$ at $a$. Then $\omega_{g,1}$ has a pole of order no greater than
$m_{2}(1-2g)+m_{1}+1$ at $a$.

###### Proof.

Begin with the expansion we found before in equation 79, which for $n=0$ reads

$\omega_{g,1}^{N}(z)=N^{1+\frac{m_{2}}{m_{1}}(1-2g)}\sum_{l=2}^{2g}\frac{A_{l,m}}{(z/(\vartheta^{m}N^{1/m_{1}})-1)^{l}}d\frac{z}{\vartheta^{m}N^{1/m_{1}}}+\mathcal{O}\left((z-\vartheta^{m}N^{1/m_{1}})^{0}\right)\,,$
(80)

where $A_{l,m}=\mathcal{O}(N^{0})$. We then change coordinate $z=\zeta^{-1}$,
such that $\zeta(a)=0$ and $\zeta^{m_{1}}=M_{1}$, so we obtain

$\omega_{g,1}^{N}(\zeta)\sim
N^{1+\frac{m_{2}}{m_{1}}(1-2g)-\frac{1}{m_{1}}}\sum_{l=2}^{2g}\frac{A_{l,m}\vartheta^{-m}}{(\vartheta^{-m}N^{-1/m_{1}}\zeta^{-1}-1)^{l}}d\zeta^{-1}\,,$
(81)

where we ignore the non-polar part, as it does not contribute to the poles for
an admissible spectral curve.

For a fixed small $\zeta$, take $N$ large enough that $N^{-1/m_{1}}\zeta^{-1}$
is small, and use this to expand denominators as geometric series. We see that
the coefficient of $\zeta^{-k}d\zeta^{-1}=-\zeta^{-k-2}d\zeta$ is then
$\mathcal{O}(N^{1+\frac{m_{2}}{m_{1}}(1-2g)-\frac{k+1}{m_{1}}})$, so for this
to persist in the limit, we require that

$\begin{split}0&\leq 1+\frac{m_{2}}{m_{1}}(1-2g)-\frac{k+1}{m_{1}}\\\ 0&\leq
m_{1}+m_{2}(1-2g)-k-1\\\ k&\leq m_{1}+m_{2}(1-2g)-1\,,\end{split}$

which after a shift of $2$ for $k$ exactly gives the order in the proposition.
∎

We then conjecture that the principal parts of the correlators at essential
singularities actually take a nice form.

######  Conjecture 4.15.

Let $\mathcal{S}$ be a compact transalgebraic admissible spectral curve. The
contribution to the correlator $\omega_{g,n}$ from the infinite ramification
points $R_{\infty}$ is given explicitly by the following formula

$\begin{split}&\sum_{a\in
R_{\infty}}\operatorname*{Res}_{t=a}\left(\int_{a}^{t}\omega_{0,2}(z_{0},\cdot)\right)\omega_{g,n}(t,z_{\llbracket
n-1\rrbracket})\\\ =\delta_{n,1}\frac{(2^{1-2g}-1)B_{2g}}{(2g)!}&\sum_{a\in
R_{\infty}}\operatorname*{Res}_{t=a}\left(\int_{a}^{t}\omega_{0,2}(z_{0},\cdot)\right)d\left(\frac{d}{dM_{2}(t)}\right)^{2g-1}\log(x(t))\\\
=\delta_{n,1}\frac{(2^{1-2g}-1)B_{2g}}{(2g)!}&\sum_{a\in
R_{\infty}}\operatorname*{Res}_{t=a}\left(\int_{a}^{t}\omega_{0,2}(z_{0},\cdot)\right)dM_{2}(t)\left(\frac{d}{dM_{2}(t)}\right)^{2g}M_{1}(t)\,,\end{split}$
(82)

where $B_{2g}$ denotes the $2g$th Bernoulli number. This formula implies that
the only correlators with poles at essential singularities are those with
$n=1$. Furthermore, for admissible spectral curves, poles of $M_{1}$ are poles
of $M_{2}$, so only finitely many $\omega_{g,1}$ have poles.

The evidence we have for this conjecture is as follows:

  * •

the conjecture agrees with the bounds on the order of the poles in proposition
4.14;

  * •

if we replace $x$ by $x_{N}$ (with $\tau=0$) and $R_{\infty}$ by the solutions
of $M_{1}=N$ in the above formula we reproduce the correct pole structure for
finite $N$ and $n=1$, i.e., a pole of order $2g$ (see lemma 2.10 with
$s_{a}=1$);

  * •

the $\omega_{g,n}$ will have vanishing principal parts at essential
singularities for $n\geq 2$ by corollary 4.10;

  * •

we prove the conjecture for the $r$-Atlantes Hurwitz curves in corollary 6.17;

  * •

the conjecture holds for $g=1$ by the following proposition.

######  Proposition 4.16.

For $g=1$, 4.15 holds.

###### Proof.

For $n\geq 2$ the result holds by corollary 4.10, so we concentrate on the
$n=1$ case. Take a large positive integer $N$ with a corresponding primitive
$N$th root of unity $\alpha$ and fix a ramification point of $x_{N}$ (we use
$\tau=0$), which we denote $a$, such that $M_{1}(a)=N$. Fixing a local
coordinate $t^{-N}=x_{N}$ we write the quadratic local loop equation for
$\omega^{N}_{1,1}$ at $a$131313When we consider the local loop equations about
$a$ we need $\omega^{N}_{0,1}$ to be regular at $a$. This is accomplished by
taking the local $\omega^{N}_{0,1}$ to be
$\left[M_{2}(\alpha^{i}t)-M_{2}(a)\right]d\log(x_{N}(t))$, which differs from
the original by a pure function of $x_{N}$ and therefore does not change the
other correlators. See the proof of lemma 4.8 for further explanation.

$\frac{1}{2!}\frac{dx_{N}(t)}{x_{N}(t)}\sum_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{N}2\left[M_{2}(\alpha^{i}t)-M_{2}(a)\right]\omega_{1,1}(\alpha^{j}t)+\frac{1}{2!}\sum_{\begin{subarray}{c}i,j=1\\\
i\neq
j\end{subarray}}^{N}\omega_{0,2}(\alpha^{i}t,\alpha^{j}t)=\mathcal{O}\left(\frac{dx_{N}(t)^{2}}{x_{N}(t)}\right).$
(83)

By lemma 2.10 (with $s_{a}=1$) $\omega^{N}_{1,1}$ has only a double pole at
$a$; denote the coefficient of this double pole as $A$. Therefore,

$\left[M_{2}(\alpha^{i}t)-M_{2}(a)\right]\omega_{1,1}(\alpha^{j}t)=-\alpha^{i-j}M_{2}^{\prime}(a)\frac{A}{N}\frac{dx_{N}(t)}{x_{N}(t)}+\mathcal{O}(dt),$
(84)

where we assume that $N$ is chosen so large that $M_{2}^{\prime}(a)\neq 0$.
Furthermore, from [BBCCN18, Lemma A.5] we have

$\frac{1}{2!}\sum_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{N}\omega_{0,2}(\alpha^{i}t,\alpha^{j}t)=-\frac{N^{2}-1}{24N}\left(\frac{dx_{N}(t)}{x_{N}(t)}\right)^{2}.$
(85)

Putting these two results together we obtain

$A=\frac{1}{M_{2}^{\prime}(a)}\frac{N^{2}-1}{24N},$ (86)

and we can then repackage this result in a more suggestive form (the base
point $*$ is arbitrary)

$\operatorname*{Res}_{t=a}\left(\int_{*}^{t}\omega_{0,2}(z_{0},\cdot)\right)\omega^{N}_{1,1}(t)=-\frac{N^{2}-1}{24N^{2}}\operatorname*{Res}_{t=a}\left(\int_{*}^{t}\omega_{0,2}(z_{0},\cdot)\right)d\frac{d}{dM_{2}(t)}\log(x_{N}(t)).$
(87)

This holds for every $a$ with $M_{1}(a)=N$. Ergo, we can sum both sides over
all such $a$. Moreover, near the elements of $R_{\infty}$ (poles of $M_{1}$)
the integrands will only have poles at such $a$s; we can therefore replace
this sum over residues by integration over a contour $\Gamma$ that is the
disjoint union of small circles around each element of $R_{\infty}$