#  $b$-Hurwitz numbers from Whittaker vectors for $\mathcal{W}$-algebras

Nitin K. Chidambaram  School of Mathematics, University of Edinburgh, James
Clerk Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, U.K.
<EMAIL_ADDRESS>,  Maciej Dołęga  Institute of Mathematics, Polish
Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland.
<EMAIL_ADDRESS>and  Kento Osuga  Graduate School of Mathematical Sciences,
University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-0041, Japan
<EMAIL_ADDRESS>
###### Abstract.

We show that $b$-Hurwitz numbers with a rational weight are obtained by taking
an explicit limit of a Whittaker vector for the $\mathcal{W}$-algebra of type
$A$. Our result is a vast generalization of several previous results that
treated the monotone case, and the cases of quadratic and cubic polynomial
weights. It also provides an interpretation of the associated Whittaker vector
in terms of generalized branched coverings that might be of independent
interest. Our result is new even in the special case $b=0$ that corresponds to
classical hypergeometric Hurwitz numbers, and implies that they are governed
by the topological recursion of Eynard–Orantin. This gives an independent
proof of the recent result of Bychkov–Dunin-Barkowski–Kazarian–Shadrin.

###### Contents

  1. 1 Introduction
  2. 2 Cut-and-join equation for weighted $b$-Hurwitz numbers
  3. 3 Airy structures from $\mathcal{W}$-algebra representations
  4. 4 $b$-Hurwitz numbers from the Airy structure
  5. 5 Topological recursion for weighted Hurwitz numbers

##  1\. Introduction

Hurwitz theory, in its simplest form, counts branched coverings of algebraic
curves. First considered in 1891 by Hurwitz [Hur91] who showed that Hurwitz
numbers can be expressed in terms of counting permutation factorizations,
Hurwitz theory has recently been found to have fascinating connections to
integrable hierarchies, Gromov–-Witten theory, and topological recursion
([Pan00, ELSV01, OP06, GJ08, Eyn14] to mention a few). In the last two
decades, various generalizations of simple Hurwitz numbers have been studied
in many different contexts.

The most notable cases include Grothendieck dessins d’enfants and more
generally constellations [BMS00], monotone Hurwitz numbers [GGPN14], orbifold
Hurwitz numbers [JPT11], and monotone orbifold Hurwitz numbers [KLS19] among
others. An approach unifying all these different classes was introduced in
[GPH17] which defined $G$-weighted Hurwitz numbers
$H^{(0)}_{G;g}(\mu_{1},\dots,\mu_{n})$, where the weight $G(z):=\sum_{i\geq
0}g_{i}z^{i}$ is a formal power series. The generating function of weighted
Hurwitz numbers is a tau-function of hypergeometric type of the KP, or more
generally the 2-Toda integrable hierarchies [OS00], and is given by the
following explicit formula:

(1) $\tau_{G}^{(0)}:=\sum_{\begin{subarray}{c}n\geq 0,\\\ \lambda\vdash
n\end{subarray}}\check{s}_{\lambda}(\mathbf{p})\prod_{\square\in\lambda}\frac{G(\hbar\cdot
c(\square))}{\hbar}=\exp\left(\sum_{\begin{subarray}{c}g\in\mathbb{Z}_{\geq
0},\\\ n\in\mathbb{Z}_{\geq
1}\end{subarray}}\hbar^{2g-2+n}\sum_{\mu\colon\ell(\mu)=n}H^{(0)}_{G;g}(\mu_{1},\dots,\mu_{n})\cdot
p_{\mu}\right),$

where $\check{s}_{\lambda}(\mathbf{p})$ is an appropriately normalized Schur
function expressed in the basis of power-sum symmetric functions
$\mathbf{p}=(p_{k})_{k\geq 1}$, and $c(\square):=x-y$ is the content of the
box $\square=(x,y)$.

###  1.1. $b$-deformed weighted Hurwitz numbers

For many classical examples of Hurwitz numbers (i.e., for specific choices of
the weight $G$), the generating function $\tau^{(0)}_{G}$ can be realized as
the partition function of a matrix model. Such matrix models admit a natural
one-parameter deformation, which first arose in the context of log-gases with
the inverse temperature $\beta$, and it is well-understood that its partition
function admits an expansion in terms of Jack polynomials
$J_{\lambda}^{(\alpha)}(\mathbf{p})$ with the parameter
$\alpha=\frac{2}{\beta}$ [For10, BH03, BEMPF12]. Motivated by these results
and the universality of weighted Hurwitz numbers, Chapuy and the second author
defined in [CD22] $G$-weighted $b$-Hurwitz numbers
$H^{(b)}_{G;g}(\mu_{1},\dots,\mu_{\ell})$, a one-parameter deformation of
classical weighted Hurwitz numbers, satisfying an analogous relation to (1):

$\tau_{G}^{(b)}:=\sum_{\begin{subarray}{c}n\geq 0,\\\ \lambda\vdash
n\end{subarray}}\check{J}_{\lambda}^{(1+b)}(\mathbf{p})\prod_{\square\in\lambda}\frac{G(\hbar\cdot
c_{1+b}(\square))}{\hbar}=\exp\left(\sum_{\begin{subarray}{c}g\in\frac{1}{2}\mathbb{Z}_{\geq
0},\\\ n\in\mathbb{Z}_{\geq
1}\end{subarray}}\frac{\hbar^{2g-2+n}}{1+b}\sum_{\mu\colon\ell(\mu)=n}H^{(b)}_{G;g}(\mu_{1},\dots,\mu_{n})\cdot
p_{\mu}\right),$

where $c_{1+b}(x,y):=(b+1)\cdot x-y-b$ is the $b$-deformation of the content,
and $\check{J}^{(1+b)}_{\lambda}(\mathbf{p})$ is an appropriately normalized
Jack polynomial with the condition
$\check{J}_{\lambda}^{(1)}(\mathbf{p})=\check{s}_{\lambda}(\mathbf{p})$ so
that the $H^{(b)}_{G;g}$ agree with the $H^{(0)}_{G;g}$ of (1) when $b=0$ (see
LABEL:{sub:Jack} for more details).

The reparametrization $\alpha=1+b$ comes from a fascinating enumerative
property which was conjectured in several special cases covered by weighted
$b$-Hurwitz numbers – for any $\mu_{1},\dots,\mu_{n}$ the quantity $|\mu|\cdot
H^{(b)}_{G;g}(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}_{\geq
0}[b][g_{0},g_{1},\dots]$ is a polynomial in $b$ with non-negative integer
coefficients. This conjecture was proved in full generality by Chapuy and the
second author in [CD22]. They interpreted weighted $b$-Hurwitz numbers as an
explicit one-parameter interpolation between classical (complex) Hurwitz
theory and its non-orientable (real) version. Additionally, in the case of a
rational weight $G(z)$, which is the topic of our article, Ruzza recently
found in [Ruz23] an elegant combinatorial interpretation of
$H^{(b)}_{G;g}(\mu_{1},\dots,\mu_{n})$ in terms of colored $b$-monotone
Hurwitz maps introduced in [BCD23].

Weighted $b$-Hurwitz numbers have already been proved to encompass many models
from mathematical physics, enumerative geometry, and combinatorics: specific
polynomial weights cover the case of Gaussian $\beta$-ensembles [BCD23], the
monotone case was shown in [BCD23] to cover the $\beta$-deformation of the
Harish-Chandra/Itzykson–Zuber (HCIZ) integral introduced by Herz in [Her55] as
a multivariate Bessel function and reintroduced in the physics literature in
[BH03] (see also [BEMPF12]) and the $\beta$-deformation of the
Brézin–Gross–Witten integral introduced in [MMS11]. The case of Jacobi
$\beta$-ensembles and Laguerre $\beta$-ensembles are also covered by
rationally-weighted $b$-Hurwitz numbers as recently shown in [Ruz23]. The very
special case $b=1$ of simple Hurwitz numbers was rediscovered in [BF21, HM22]
in the context of surgery theory, tropical geometry and combinatorics (see
[BD22] for different and more general combinatorial models).

###  1.2. Cut-and-join equation and $\mathcal{W}$-constraints

Some of the main tools in understanding the structure of classical Hurwitz
numbers are cut-and-join equations that determine $H^{(0)}_{G;g}$ uniquely.
These equations can be obtained by interpreting Hurwitz numbers as
factorizations of a permutation into transpositions, and by analyzing the
effect of multiplying a permutation by a transposition, which either cuts a
cycle into two new cycles, or joins two cycles into a new cycle. In classical
examples where the weight $G(z)$ is relatively simple, one can understand
these equations in terms of the Jucys–Murphy elements that further lead to
methods employing the representation theory of the symmetric group and so-
called semi-infinite wedge representations (see for instance [BDK+23] and
references therein). Cut-and-join equations turned out to be crucial in
proving various structural properties of Hurwitz numbers. They were used to
prove the Eynard–Orantin topological recursion (discussed in more detail in
Section 1.3) for the classical [BMn08, EMS11], the monotone [GGPN14, DDM17a]
and other models of Hurwitz numbers, polynomiality and ELSV-type formulas
[GJV05, BDK+23].

On the other hand, finding cut-and-join equations for the arbitrary $b$
requires new ideas due to the lack of an analogous
combinatorial/representation theoretic interpretation. An approach based on
the theory of Jack polynomials was proposed by Chapuy and the second author in
[CD22], where they found a Lax structure that allowed them to obtain certain
cut-and-join-type equations for $\tau_{G}^{(b)}$. These equations are of
infinite degree unless $G(z)$ is a polynomial, which makes it difficult to
extract some interesting information from them (extracing constraints that
refine the cut-and-join-type equation is highly non-trivial even in the
polynomial case, see [BN23] for the case of cubic polynomial weight).

However, in the special case of monotone $b$-Hurwitz numbers, [BCD23] found a
different cut-and-join equation and used it to produce Virasoro constraints
(also obtained by [Yan11] prior to discovering the enumerative interpretation
of $\tau_{G}^{(b)}$). One of the key contributions of our paper is the proof
of a cut-and-join-type equation of finite degree for arbitrary rational
weights, extending the monotone case. More importantly, we find that the
underlying structure of this cut-and-join equation is closely related to the
structure of a vertex algebra known as the $\mathcal{W}$-algebra (a
generalization of the Virasoro algebra). An important insight is to use
lattice paths as a combinatorial tool that simultaneously encodes certain
operators acting on $\tau^{(b)}_{G}$ as well as the $\mathcal{W}$-algebra
structure. As a consequence we show that $\tau^{(b)}_{G}$ for any rational
weight $G(z)$ can be obtained from a Whittaker vector for a
$\mathcal{W}$-algebra of type $A$ via some simple substitutions and limits.

More concretely, we work with the algebra
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ (where the level $\mathsf{k}$ is
related to $b$), and a specific representation of it, defined using (20).
Then, we use Airy structure techniques (introduced in [KS18, BBC+18, BBCC21])
to construct a Whittaker vector $\mathcal{Z}$ for
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ satisfying the following
constraints

$\mathsf{W}^{i}_{k}\mathcal{Z}=\Omega_{i}\delta_{k,0}\mathcal{Z}\,\qquad\forall
i\in[r],k\in\mathbb{Z}_{\geq 0},$

where $\big{(}\mathsf{W}^{i}_{k}\big{)}_{k\in\mathbb{Z},i\in[r]}$ are the
modes of a set of strong generators for
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$, and $\Omega_{i}$ is a certain
function of the parameters appearing in the rational weight $G(z)$. Note that
the vector $\mathcal{Z}$ is not the highest weight vector of the
representation we are working in, but rather a Whittaker vector (see Remark
4.1 for more details). An abbreviated version of one of our main results
Theorem 4.8 is as follows.

######  Theorem 1.1.

Consider the generating function $\tau^{(b)}_{G}$ of weighted $b$-Hurwitz
numbers for an arbitrary rational weight, say
$G(z)=\frac{\prod_{i=1}^{p}(P_{i}+z)}{\prod_{i=1}^{q}(Q_{i}-z)}$ for some
$p,q\geq 0$. The function $\tau^{(b)}_{G}$ is an (explicit) limit of the
Whittaker vector $\mathcal{Z}$ for
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ where $r=\max(p,q+2)$.

A slight variant of Theorem 1.1 states that the Gaiotto vector from the
celebrated AGT conjecture [AGT10, SV13, MO19, BBCC21] is closely related to
the generating function of certain weighted $b$-Hurwitz numbers, which might
shed new light on the relation between Toda conformal blocks and the Nekrasov
instanton partition function (see Remark 4.9 for more details).

###  1.3. Topological recursion

In various contexts [BBC+18, BBCC21], it has been proved using the formalism
of Airy structures that $\mathcal{W}$-constraints are closely related to the
topological recursion (TR) formalism, originally invented by Eynard–Orantin
[EO07]. TR is a universal procedure which produces certain enumerative
invariants $\omega_{g,n}$, called correlators for a fixed topology of genus
$g$ with $n$ boundaries. These correlators are built recursively from the
initial data of a spectral curve $(\Sigma,x,y,\omega_{0,2})$, consisting of a
Riemann surface $\Sigma$, two non-constant meromorphic functions $x,y$ on
$\Sigma$, and a choice of a canonical symmetric bi-differential $\omega_{0,2}$
on $\Sigma\times\Sigma$.

[BCU] proves that the Whittaker vector $\mathcal{Z}$ at the self-dual level
$\mathsf{k}=1-r$, which corresponds to choosing $b=0$, can be computed using
topological recursion on a certain spectral curve. Combining this with Theorem
1.1, and taking appropriate limits allow us to deduce that the classical
rationally weighted Hurwitz numbers can be computed by TR, which was first
proved in [BDBKS20]. This is one of the most important structural results in
Hurwitz theory, and we provide a new and independent proof of it in Section 5
from the $\mathcal{W}$-algebra perspective:

######  Theorem 1.2.

Consider the rational weight
$G(z)=\frac{\prod_{i=1}^{p}(P_{i}+z)}{\prod_{i=1}^{q}(Q_{i}-z)}$ for some
$p,q\geq 0$. Let $\omega_{g,n}$ denote the correlators obtained from TR on the
spectral curve
$\left({\mathbb{C}\mathbb{P}}^{1},x,\omega_{0,1}=ydx,\omega_{0,2}\right)$,
with

$x(z)=\frac{z}{G(z)},\quad
y(z)=G(z),\quad\omega_{0,2}(z_{1},z_{2})=\frac{\mathsf{d}z_{1}\mathsf{d}z_{2}}{(z_{1}-z_{2})^{2}}.$

Then, expanding the correlators $\omega_{g,n}(z_{1},\ldots,z_{n})$ near
$z_{1},\dots,z_{n}=0$ gives $G$-weighted Hurwitz numbers:

$\omega_{g,n}(z_{1},\dots,z_{n})=\sum_{\mu_{1},\dots,\mu_{n}\in\mathbb{Z}_{\geq
1}}H^{(0)}_{G;g}(\mu_{1},\dots,\mu_{n})|\operatorname{Aut}(\mu)|\prod_{i=1}^{n}\mu_{i}x_{i}(z)^{\mu_{i}-1}\mathsf{d}x(z_{i})\,.$

Let us comment briefly on the history of this theorem. Before the breakthrough
of [ACEH20] where Hurwitz numbers weighted by arbitrary polynomials $G$ was
proved to be governed by TR, each model of interest was treated case by case
over a decade: [BMn08, EMS11] (simple Hurwitz numbers), [DDM17b] (monotone),
[KPS22] (monotone orbifold) to name just a few. Finally, [BDBKS20] gave a
uniform proof of a more general result, that in particular, covers the case of
arbitrary rational weights $G$ Theorem 1.2 (see also [BCCGF22, BDBKS22] for
the most recent developments concerning double Hurwitz numbers that we do not
consider in this paper).

The existing proofs of Theorem 1.2 rely heavily on the KP integrability of the
generating function $\tau^{(0)}_{G}$ that is specific to the $b=0$ case and
does not have even a conjectural extension to arbitrary $b$. In our proof, we
use the underlying $\mathcal{W}$-algebra structure controlling Hurwitz theory
for arbitrary $b$ as we prove in Theorem 1.1, and the close relationship
between $\mathcal{W}$-algebras and the Eynard–Orantin topological recursion.
In fact, this leads to the following natural question:

What is the correct generalization (if any) of the Eynard–Orantin topological
recursion that will govern enumerative properties of weighted $b$-Hurwitz
numbers?

While the $\mathcal{W}$-constraints of (1.2) hold for any $b$, the analytic
side that mimics the original construction of Eynard–Orantin is still under
investigation. There are two differing approaches to extend the Eynard–Orantin
topological recursion to arbitrary $b$. One approach, which has been developed
in [CEM11, BE19, BBCC21] is to replace the initial data of a spectral curve by
a non-commutative spectral curve (essentially, a $D$-module on the spectral
curve) and goes by the name of non-commutative topological recursion. The
underlying $\mathcal{W}$-algebra structure of this non-commutative topological
recursion was explored thoroughly in [BBCC21] for a specific example, which,
in Hurwitz theory corresponds to the case where $1/G(z)$ is a polynomial (see
Remark 4.9). While it seems difficult to extract analytic properties of
$b$-Hurwitz numbers directly from non-commutative TR, this connection is an
interesting problem that deserves further study.

Another promising approach is the so-called refined topological recursion. A
refinement was first attempted by Chekhov and Eynard [CE06] in the case of
$\beta$-deformed matrix models, and its mathematical formulation for
degree-$2$ curves was recently carried out by Kidwai and the third author in
[KO23, Osu23]. The initial data is upgraded to a refined spectral curve which
comes with a new differential $\omega_{\frac{1}{2},1}$. In subsequent work we
will show that our main result Theorem 1.1 implies that the refined TR
computes the tau function of some classes of weighted $b$-Hurwitz numbers.
This will explain a mysterious structural difference between rational
parameterizations of generating series of orientable and non-orientable
combinatorial maps (see [DL22] and references therein).

All in all, our main theorems give the first answer to the natural question of
relating weighted $b$-Hurwitz numbers with TR, and form the foundation for
further investigation into this topic.

###  1.4. Notation

Given an integer $n\geq 1$, we will denote the set $\\{1,\dots,n\\}$ by $[n]$,
and given integers $2\leq m\leq n$, we define the set
$[m..n]:=[n]\setminus[m-1]$ ($[0]:=\emptyset$ by convention). We denote by
$\mathcal{F}_{<}(A,B)$ a set of strictly increasing functions $f\colon
A\rightarrow B$, i.e. $f(a)<f(a^{\prime})$ for all $a<a^{\prime}$. We use a
bold face to indicate that we are working with a family of indeterminates
indexed by positive integers, e.g.
$\mathbf{p}=(p_{1},p_{2},\dots),\mathbf{g}=(g_{1},g_{2},\dots),\mathbf{x}=(x_{1},x_{2},\dots)$,
etc. For a ring $R$ we denote by $R[\mathbf{x}]$, $R(\mathbf{x})$,
$R\llbracket\mathbf{x}\rrbracket$, and $R(\\!(\mathbf{x})\\!)$ the polynomial,
rational, formal power series, and formal Laurent series, respectively, rings
over $R$. For a function $F$ of the variables $\mathbf{p}$, we use the
notation $[p_{i_{1}}\cdots p_{i_{n}}]F$ to denote the coefficient of the
monomial $p_{i_{1}}\cdots p_{i_{n}}$ in $F$.

In this paper we work with a non-trivial reparametrization of
$\tau_{G}^{(b)}$, which reveals more remarkable properties of $b$-Hurwitz
numbers. We will denote the reparametrized function by
$\tau_{G}^{(\mathfrak{b})}$, where the main parameter is
$\mathfrak{b}=\sqrt{\alpha}^{-1}-\sqrt{\alpha}$111A similar parametrization
was discovered in [DF16], where $\mathfrak{b}$ is the main parameter of
interest, in the context of some open problems concerning Jack polynomials and
free probability (see (10)). We summarize the relation between the four
equivalent parameters $\alpha,b,\mathsf{k},\mathfrak{b}$ that will appear in
our paper:

(2)
$\mathfrak{b}=\sqrt{\alpha}^{-1}-\sqrt{\alpha}=-\frac{b}{\sqrt{1+b}}=\mathsf{k}+r-1.$

### Acknowledgements

NKC acknowledges the support of the ERC Starting Grant 948885, and the Royal
Society University Research Fellowship. MD is supported by Narodowe Centrum
Nauki, grant 2021/42/E/ST1/00162. KO is supported by JSPS KAKENHI Grant number
22KJ0715 and 23K12968. We would like to thank the organizers of the program
"TR SALENTO 2022", where our collaboration on this project begun. We would
also like to thank Gaëtan Borot and Thomas Creutzig for helpful discussions.

##  2\. Cut-and-join equation for weighted $b$-Hurwitz numbers

In this section, we will derive a specific cut-and-join equation for weighted
$b$-Hurwitz numbers, which will turn out to be related to
$\mathcal{W}$-algebra representations in the later sections.

###  2.1. Jack polynomials

We call $\lambda$ an integer partition of $n$ if $\lambda$ is a finite
sequence $\lambda_{1}\geq\cdots\geq\lambda_{\ell}>0$ of nonnegative integers
that sums up to $n=:|\lambda|$. We denote it by $\lambda\vdash n$, and its
number of parts $\ell$ is denoted $\ell(\lambda)$. Sometimes we use the
notation $\lambda=(1^{m_{1}(\lambda)},2^{m_{2}(\lambda)},\dots)$, where
$m_{i}(\lambda)$ denotes the number of parts of $\lambda$ that are equal to
$i$. The set of partitions of the same size is linearly ordered:

$\lambda\leq\mu\iff\sum_{i=1}^{j}\lambda_{i}\leq\sum_{i=1}^{j}\mu_{i}\qquad\forall
j.$

We will interchangeably use the terms _an integer partition $\lambda$_ and _a
Young diagram $\lambda$_ and we associate the latter with the collection of
boxes $\\{\lambda\ni\square:=(x,y)\colon 1\leq y\leq\ell(\lambda),1\leq
x\leq\lambda_{y}\\}$. For such a box $\square=(x,y)\in\lambda$ we define its
$\alpha$-deformed content as the quantity
$c_{\alpha}(x,y):=\alpha(x-1)-(y-1)$. This quantity plays an important role in
theory of Jack polynomials, that we will briefly describe following [Sta89,
Mac95].

Let $R$ be a ring and let $R(\alpha)[\mathbf{p}]$ denote the polynomial ring
in infinitely many variables $\mathbf{p}=(p_{k})_{k\geq 1}$ over the ring
$R(\alpha)$. The following operator, called the _Laplace–Beltrami operator_ ,
acts on $R(\alpha)[\mathbf{p}]$:

$D_{\alpha}:=\frac{1}{2}\left(\sum_{k,\ell\geq 1}\alpha k\ell\cdot
p_{k+\ell}\partial_{p_{k}}\partial_{p_{\ell}}+\sum_{k,\ell\geq
1}\alpha(k+\ell)\cdot p_{k}p_{\ell}\partial_{p_{k+\ell}}+(\alpha-1)\sum_{k\geq
1}k(k-1)p_{k}\partial_{p_{k}}\right).$

_Jack polynomials_ $J^{(\alpha)}_{\lambda}$ are elements of
$\mathbb{Q}(\alpha)[\mathbf{p}]$ that are indexed by integer partitions
$\lambda$. Consider the following grading on the ring
$\mathbb{Q}(\alpha)[\mathbf{p}]$ defined on the basis:
$\deg(\alpha):=0,\deg(p_{k}):=k$. Then, the Jack polynomial
$J^{(\alpha)}_{\lambda}$ can be characterized (up to a normalization factor)
as the unique homogeneous polynomial of degree $|\lambda|$ such that:

  1. (1)

it is an eigenfunction of the Laplace–Beltrami operator:
$D_{\alpha}J^{(\alpha)}_{\lambda}=\sum_{\square\in\lambda}c_{\alpha}(\square)J^{(\alpha)}_{\lambda},$

  2. (2)

the transition matrix from $\\{J^{{(\alpha)}}_{\lambda}\\}_{|\lambda|=n}$ to
the monomial symmetric functions $\\{m_{\lambda}(\mathbf{x})\\}_{|\lambda|=n}$
is lower triangular after identifying $p_{k}$ with the power-sum symmetric
functions $p_{k}=p_{k}(\mathbf{x}):=\sum_{i}x_{i}^{k}$.

We will work with the _integral Jack polynomials_ which are normalized such
that the coefficient $[p_{1}^{|\lambda|}]J^{(\alpha)}_{\lambda}=1$. The
quantity

$j^{(\alpha)}_{\lambda}:=\prod_{\square\in\lambda}\operatorname{hook}_{\lambda}(\square)\cdot\operatorname{hook}_{\lambda}(\square)^{\prime},$

with

$\operatorname{hook}_{\lambda}(\square):=\alpha(\lambda_{y}-x)+\lambda_{x}^{\prime}-y+1,\quad\operatorname{hook}_{\lambda}(\square)^{\prime}=\alpha(\lambda_{y}-x+1)+(\lambda_{x}^{\prime}-y)$

is the normalization factor. One can define a scalar product on
$\mathbb{Q}(\alpha)[\mathbf{p}]$ for which the $J_{\lambda}^{(\alpha)}$ give
an orthogonal basis with the norm
$\|J_{\lambda}^{(\alpha)}\|^{2}=j_{\lambda}^{(\alpha)}$. One can show that
this is precisely the $\alpha$-deformed Hall scalar product for which
$p_{\lambda}:=\prod_{i=1}^{\ell(\lambda)}p_{\lambda_{i}}$ is orthogonal with
the classical norm:

$\langle
p_{\lambda},p_{\lambda}\rangle=\alpha^{\ell(\lambda)}\frac{\prod_{i\geq
1}m_{i}(\lambda)!i^{m_{i}(\lambda)}}{|\lambda|!}.$

Let us define the operators $(\mathsf{J}^{1}_{k})_{k\in\mathbb{Z}}$ acting on
$R[\tilde{\mathbf{p}}]\llbracket\hbar\rrbracket$ by

(3) $\mathsf{J}^{1}_{k}=\begin{cases}\hbar
k\cdot\partial_{\tilde{p}_{k}}&\text{ if }k>0,\\\ 0&\text{ if }k=0,\\\
\hbar\tilde{p}_{-k}&\text{ if }k<0.\end{cases}$

Since
$[\mathsf{J}^{1}_{k},\mathsf{J}^{1}_{\ell}]=k\hbar^{2}\delta_{k+\ell,0}$, the
action of $(\mathsf{J}^{1}_{k})_{k\in\mathbb{Z}}$ on
$R[\tilde{\mathbf{p}}]\llbracket\hbar\rrbracket$ is a representation of the
Heisenberg algebra. Furthermore, with the change of variables
$p_{k}=\sqrt{\alpha}^{-1}\tilde{p}_{k}$ one has
$\partial_{p_{k}}=\sqrt{\alpha}\partial_{\tilde{p}_{k}}$, hence
$J^{(\alpha)}_{\lambda}$ can be considered as an element of
$\mathbb{Q}(\sqrt{\alpha})[\tilde{\mathbf{p}}]$ and the rescaled
Laplace–Beltrami operator
$\tilde{D}_{\alpha}:=\frac{\hbar}{\sqrt{\alpha}}D_{\alpha}$ can be written as

(4) $\tilde{D}_{\alpha}:=\frac{\hbar^{-2}}{2}\sum_{k\geq
1}\mathsf{J}^{1}_{-k}\left(\sum_{\ell\geq
1}\mathsf{J}^{1}_{\ell}\mathsf{J}^{1}_{k-\ell}-(k-1)\hbar\cdot\mathfrak{b}\cdot\mathsf{J}^{1}_{k}\right),$

and

(5)
$\tilde{D}_{\alpha}J^{(\alpha)}_{\lambda}=\hbar\sum_{\square\in\lambda}\tilde{c}_{\alpha}(\square)J^{(\alpha)}_{\lambda},$

where $\tilde{c}_{\alpha}(x,y):=\sqrt{\alpha}(x-1)-\sqrt{\alpha}^{-1}(y-1)$,
and $\mathfrak{b}$ is given by (2).

###  2.2. Nazarov–Sklyanin operators and co-transition measure

Consider the (infinite) row vector
$\mathsf{J}^{1}_{+}=(\mathsf{J}^{1}_{k})_{{k}\in\mathbb{Z}_{\geq 1}}$ and
dually the column vector
$\mathsf{J}^{1}_{-}=(\mathsf{J}^{1}_{-k})_{{k}\in\mathbb{Z}_{\geq 1}}$. Let
$L=(L_{k,\ell})_{k,\ell\in\mathbb{Z}_{\geq 1}}$ be the infinite matrix defined
by $L_{k,\ell}:=\mathsf{J}^{1}_{k-\ell}-\delta_{k,\ell}\,k\hbar\mathfrak{b}$,
for all $k,\ell\in\mathbb{Z}_{\geq 1}$:

$L=\begin{bmatrix}-\hbar\mathfrak{b}&\mathsf{J}^{1}_{-1}&\mathsf{J}^{1}_{-2}&\cdots\\\
\mathsf{J}^{1}_{1}&-2\hbar\mathfrak{b}&\mathsf{J}^{1}_{-1}&\ddots\\\
\mathsf{J}^{1}_{2}&\mathsf{J}^{1}_{1}&-3\hbar\mathfrak{b}&\ddots\\\
\vdots&\ddots&\ddots&\ddots\end{bmatrix}.$

Define the following generating series:

(6) $\sum_{\ell\geq
0}B^{(\alpha)}_{\ell+2}(\lambda)z^{-\ell-1}=z-(z+\sqrt{\alpha}^{-1}\ell(\lambda))\prod_{i=1}^{\ell(\lambda)}\frac{z+\sqrt{\alpha}^{-1}(i-1)-\sqrt{\alpha}\lambda_{i}}{z+\sqrt{\alpha}^{-1}i-\sqrt{\alpha}\lambda_{i}}.$

The main result of Nazarov–Sklyanin [NS13, Theorem 2] is a construction of a
family of commuting operators that includes the Euler and Laplace–Beltrami
operators, whose eigenfunctions are Jack polynomials with explicit
eigenvalues:

######  Theorem 2.1.

The following equality holds true for all $\ell\geq 0$:

$\displaystyle\left(\mathsf{J}_{-}L^{\ell}\mathsf{J}_{+}\right)J^{(\alpha)}_{\lambda}$
$\displaystyle=\hbar^{\ell+2}B^{(\alpha)}_{\ell+2}(\lambda)\cdot
J^{{(\alpha)}}_{\lambda}.$

Note that the operators $\hbar^{-2}\mathsf{J}_{-}\mathsf{J}_{+}$ and
$\frac{\hbar^{-2}}{2}\mathsf{J}_{-}(L+\hbar\mathfrak{b}\operatorname{Id})\mathsf{J}_{+}$
are precisely the Euler and rescaled Laplace–Beltrami operators, and (5) is a
special case of Theorem 2.1.

The coefficients $B^{(\alpha)}_{\ell+2}(\lambda)$ can be interpreted as the
Boolean cumulants of a transition measure of Kerov, or equivalently as the
moments of a co-transition measure of Kerov, both introduced in [Ker93] to
study random Young diagrams. It appears that this fact was not known to
Nazarov and Sklyanin, and a connection to probability was first mentioned in
[Mol15]. Later this connection was described in greater generality and
Nazarov–Sklyanin operators were used to prove the LLN and CLT for Jack
measures [Mol23, CDM23], and Lassalle’s positivity conjecture [BDD23]. We will
use this connection here to find an explicit formula for the action of
$\mathsf{J}_{-}(L+\hbar\mathfrak{b}\operatorname{Id})^{\ell}\mathsf{J}_{+}$ on
Jack polynomials that will be crucial in the next section.

Define a probability measure $\nu^{(\alpha)}_{\lambda}$, called a _co-
transition measure_ , as the measure uniquely characterized by its Cauchy
transform expanded around infinity:

(7)
$\sqrt{\alpha}|\lambda|\int_{\mathbb{R}}\frac{d\nu^{(\alpha)}_{\lambda}(x)}{z-x}=z-(z+\sqrt{\alpha}^{-1}\ell(\lambda))\prod_{i=1}^{\ell(\lambda)}\frac{z+\sqrt{\alpha}^{-1}(i-1)-\sqrt{\alpha}\lambda_{i}}{z+\sqrt{\alpha}^{-1}i-\sqrt{\alpha}\lambda_{i}}.$

The fact that $\nu^{(\alpha)}_{\lambda}$ is a probability measure is due to
Kerov [Ker00, Lemma (2.6)]. Note that $\nu^{(\alpha)}_{\lambda}$ is supported
on a discrete set located at the $\alpha$-contents of the outer corners of
$\lambda$ that can be removed to obtain a smaller diagram $\mu$:

$y\in\operatorname{supp}(\nu^{(\alpha)}_{\lambda})\iff\mu\subset\lambda,|\mu|=|\lambda|-1,y=\big{(}\tilde{c}_{\alpha}(\lambda\setminus\mu)-\mathfrak{b}\big{)}.$

Therefore

(8)
$\nu^{(\alpha)}_{\lambda}=\sum_{\mu\nearrow\lambda}f_{\mu}^{\lambda}\cdot\delta_{\big{(}\tilde{c}_{\alpha}(\lambda\setminus\mu)-\mathfrak{b}\big{)}},$

where $\mu\nearrow\lambda$ means that we sum over all Young diagrams $\mu$
that are obtained from $\lambda$ by removing a box. It was proven by Kerov
[Ker00, Theorem (6.7)] that the masses $f_{\mu}^{\lambda}$ can be computed as:

(9)
$f_{\mu}^{\lambda}=\frac{\langle\mathsf{J}^{1}_{-1}J^{(\alpha)}_{\mu},J^{(\alpha)}_{\lambda}\rangle}{\hbar\sqrt{\alpha}|\lambda|j_{\mu}^{(\alpha)}}.$

We will use the relation between Nazarov–Sklyanin operators and the co-
transition measure to prove the following lemma that will be crucial in the
next subsection.

######  Lemma 2.2.

Let $\lambda\vdash n+1$. The following identity holds true:

$\big{(}\mathsf{J}_{-}(L+\hbar\mathfrak{b}\operatorname{Id})^{\ell}\mathsf{J}_{+}\big{)}J^{(\alpha)}_{\lambda}=\frac{\hbar}{\sqrt{\alpha}}\sum_{\mu\vdash
n}\frac{\langle\mathsf{J}^{1}_{-1}J_{\mu}^{(\alpha)},J_{\lambda}^{(\alpha)}\rangle}{j^{(\alpha)}_{\mu}}\left(\hbar\tilde{c}_{\alpha}(\lambda\setminus\mu)\right)^{\ell}J^{(\alpha)}_{\lambda}.$

###### Proof.

Fix $\lambda\vdash n+1$. It is well-known (see [Sta89]) that
$\langle\mathsf{J}^{1}_{-1}J_{\mu}^{(\alpha)},J_{\lambda}^{(\alpha)}\rangle$
is nonzero only if $\mu\nearrow\lambda$. Therefore (8) and (9) imply that

$\hbar\sqrt{\alpha}\cdot|\lambda|\cdot\nu_{\lambda}^{(\alpha)}=\sum_{\mu\vdash
n}\langle\mathsf{J}^{1}_{-1}J_{\mu}^{(\alpha)},J_{\lambda}^{(\alpha)}\rangle\frac{\delta_{\big{(}\tilde{c}_{\alpha}(\lambda\setminus\mu)-\mathfrak{b}\big{)}}}{j_{\mu}^{(\alpha)}}.$

This formula, together with (6) and (7), immediately gives that for all
$\ell\geq 0$

$B^{(\alpha)}_{\ell+2}(\lambda)=\sum_{\mu\vdash
n}\frac{\langle\mathsf{J}^{1}_{-1}J_{\mu}^{(\alpha)},J_{\lambda}^{(\alpha)}\rangle}{\hbar\sqrt{\alpha}j_{\mu}^{(\alpha)}}\left(\tilde{c}_{\alpha}(\lambda\setminus\mu)-\mathfrak{b}\right)^{\ell}.$

We conclude by applying Theorem 2.1. ∎

###  2.3. Weighted $b$-Hurwitz numbers and cut-and-join equation

Following [CD22] we define the generating function of $G$-weighted $b$-Hurwitz
numbers $H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{\ell})$ by the following
identity:

(10) $\tau_{G}^{(\mathfrak{b})}:=\sum_{n\geq
0}\left(\frac{\sqrt{\alpha}}{\hbar}\right)^{n}\sum_{\lambda\vdash
n}\frac{J_{\lambda}^{(\alpha)}(\sqrt{\alpha}\tilde{\mathbf{p}})}{j_{\lambda}^{(\alpha)}}\prod_{\square\in\lambda}G(\hbar\cdot\tilde{c}_{\alpha}(\square))=\\\
=\exp\left(\sum_{g\in\frac{1}{2}\cdot\mathbb{Z}_{\geq
0}}\sum_{n\in\mathbb{Z}_{\geq
1}}\hbar^{2g-2+n}\sum_{\mu\colon\ell(\mu)=n}H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})\cdot\tilde{p}_{\mu}\right),$

where $G(z)=\sum_{i=0}^{\infty}g_{i}z^{i}$. These are so-called _single_
$G$-weighted $b$-Hurwitz numbers, and the $\tau_{G}^{(\mathfrak{b})}$ given by
(10) corresponds to the specialization
$\tau_{G}^{(b)}(\mathbf{p},\mathbf{q})\big{|}_{q_{k}=\delta_{k,1}}$ of the
generating function of double $G$-weighted $b$-Hurwitz numbers introduced in
[CD22] after the reparametrization

$\tilde{p}_{k}=\sqrt{\alpha}^{-1}p_{k},\ \ \hbar=\sqrt{\alpha},\ \
\mathfrak{b}=-\sqrt{\alpha}^{-1}b.$

It is convenient to add a redundant parameter $t$ by the substitution
$\tau_{G}^{(\mathfrak{b})}(t,\hbar,\tilde{\mathbf{p}}):=\tau_{G}^{(\mathfrak{b})}\big{|}_{\tilde{p}_{k}=t^{k}\tilde{p}_{k}}$.
It is not hard to show that

$\hbar
t\partial_{t}\log\tau_{G}^{(\mathfrak{b})}(t,\hbar,\tilde{\mathbf{p}})\in\mathbb{Q}(\alpha)[\sqrt{\alpha}\tilde{\mathbf{p}},\mathbf{g},\hbar\sqrt{\alpha}^{-1}]\llbracket
t\rrbracket.$

However, a much stronger result due to Chapuy and the second author holds,
which explains the meaning of the parameter $\sqrt{\alpha}\cdot\mathfrak{b}$:

######  Theorem 2.3 ([CD22]).

For any partition $\mu$ one has

$\alpha^{g}\cdot|\mu|H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}_{\geq
0}[-\sqrt{\alpha}\cdot\mathfrak{b}][\mathbf{g}].$

Moreover this polynomial is a generating series of certain labeled real
meromorphic functions $f$ of degree $\deg(f)=|\mu|$ with ramification profile
$(\mu_{1},\dots,\mu_{n})$ over $\infty$ such that each such function $f$ is
counted with a weight
$(-\sqrt{\alpha}\cdot\mathfrak{b})^{\nu(f)}\frac{\deg(f)}{|\operatorname{Aut}(f)|}\cdot
P_{f}(\mathbf{g})$, where $P_{f}(\mathbf{g})\in\mathbb{Z}_{\geq
0}[g_{1},g_{2},\dots]$ is an explicit polynomial that depends only on the
number of branched points of $f$ over other critical points, and $\nu(f)$ is a
non-negative integer that equals to $0$ if and only if $f$ is a double copy of
a complex meromorphic function.

Theorem 2.3 is the abbreviated version of the main result of [CD22], and it is
stated here to highlight topological/combinatorial relevance of the
$G$-weighted $b$-Hurwitz numbers; we will not use it here and therefore refer
to [CD22] for the relevant definitions and details.

The reparametrization introduced here is motivated by our main result that
gives an explicit relation between the reparametrized function
$\tau^{(\mathfrak{b})}_{G}$ and the Airy structure coming from representations
of $\mathcal{W}$-algebras. In particular, one can notice by comparing (10)
with the parametrization used in Section 1.1 that
$H^{(b)}_{G;g}(\mu_{1},\dots,\mu_{n})=\alpha^{g}\cdot
H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})$, and we will show that
$|\mu|\tilde{H}_{G;g}^{(\mathfrak{b})}(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}_{\geq
0}[-\mathfrak{b},\mathbf{g}]$, which eliminates the dependence on $\alpha$ and
is stronger than Theorem 2.3. In order to do this, we intend to find an
explicit differential operator of finite degree that annihilates
$\tau_{G}^{(\mathfrak{b})}$ and uniquely determines it by the initial
condition $\tau_{G}^{(\mathfrak{b})}\big{|}_{t=0}=1$ — this is why we
introduced the parameter $t$.

######  Remark 2.4.

In the following we will mostly work with $G$ being a rational function. A
generic rational function can be parametrized either by
$G_{u}(z)=u\frac{\prod_{i=1}^{n}(P_{i}+z)}{\prod_{j=1}^{m}(Q_{j}-z)}$ for some
$u,\mathbf{P},\mathbf{Q}$, or by
$\check{G}_{u}(z)=u\frac{\prod_{i=1}^{n}(1+P_{i}z)}{\prod_{i=1}^{m}(1-Q_{j}z)}$.
Note that the change of variables $u\mapsto\frac{P_{1}\cdots
P_{n}}{Q_{1}\cdots Q_{m}}u,P_{i}\mapsto P_{i}^{-1},Q_{j}^{-1}\mapsto Q_{j}$
for all $i\in[n]$, $j\in[m]$ transform $G_{u}$ into $\check{G}_{u}$. Moreover,
$u$ can be absorbed into the parameter $t$ of $\tau^{(\mathfrak{b})}$ by
substituting $t\mapsto ut$, so that we can equivalently work with
$G(z)=\frac{\prod_{i=1}^{n}(P_{i}+z)}{\prod_{j=1}^{m}(Q_{j}-z)}$ or
$\check{G}(z)=\frac{\prod_{i=1}^{n}(1+P_{i}z)}{\prod_{j=1}^{m}(1-Q_{j}z)}$, as
their generating functions are related by

(11)
$\tau^{(\mathfrak{b})}_{G}(t,\tilde{\mathbf{p}},\mathbf{P},\mathbf{Q},\hbar)=\tau^{(\mathfrak{b})}_{\check{G}}\left(t\cdot\frac{P_{1}\cdots
P_{n}}{Q_{1}\cdots
Q_{m}},\tilde{\mathbf{p}},\mathbf{P}^{-1},\mathbf{Q}^{-1},\hbar\right).$

Note that $\check{G}(z)$ considered as a formal power series has positive
integer coefficients $\check{G}(z)\in\mathbb{Z}_{\geq
0}[\mathbf{P},\mathbf{Q}]$ therefore it is a more natural parametrization from
an enumerative point of view as Theorem 2.3 implies that
$\alpha^{g}\cdot|\mu|H_{\check{G};g}^{(b)}(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}_{\geq
0}[-\sqrt{\alpha}\cdot\mathfrak{b},\mathbf{P},\mathbf{Q}]$ and each monomial
in $-\sqrt{\alpha}\cdot\mathfrak{b},\mathbf{P},\mathbf{Q}$ is counting certain
generalized branched coverings. However, the parametrization $G(z)$ is suited
much better for our purpose of connecting $\tau^{(\mathfrak{b})}_{G}$ with
$\mathcal{W}$-algebras, so we will mostly assume the weight function to be of
the form $G=\frac{\prod_{i=1}^{n}(P_{i}+z)}{\prod_{j=1}^{m}(Q_{j}-z)}$.

####  2.3.1. Weighted lattice paths

Sometimes, operators annihilating $\tau^{(\mathfrak{b})}_{G}$ can be expressed
in a particularly elegant form involving weighted lattice paths. This point of
view was applied in [Mol23, CDM23] to Nazarov–Sklyanin operators and it turned
out to be very useful in studying the structure of Jack polynomials [BDD23].
In the following we further develop this approach to connect $G$-weighted
$b$-Hurwitz numbers with $\mathcal{W}$-algebras.

All the paths we consider will be directed paths on the lattice
$\mathbb{Z}^{2}$ where integral points $(x,y)\in\mathbb{Z}^{2}$ are connected
by steps of the form $(1,k)$ with $k\in\mathbb{Z}$. We will refer to the first
coordinate of the integral points as the $X$-coordinate and the second
coordinate as the $Y$-coordinate. We only consider such paths and hence we
will simply refer to them as paths.

######  Definition 2.5.

Suppose that $x<x^{\prime}$ and $y,y^{\prime}$ are integers. We define
$\Gamma_{(x,y)\to(x^{\prime},y^{\prime})}$ to be the set of _paths_ starting
from $(x,y)$ and finishing at $(x^{\prime},y^{\prime})$, and we say that a
path $\gamma\in\Gamma_{(x,y)\to(x^{\prime},y^{\prime})}$ has length
$x^{\prime}-x$, so that it contains $x^{\prime}-x$ steps.

Consider three integers $x<x^{\prime}<x^{\prime\prime}$. Given two paths, say
$\gamma\in\Gamma_{(x,y)\to(x^{\prime},y^{\prime})}$ and
$\gamma^{\prime}\in\Gamma_{(x^{\prime},y^{\prime})\to(x^{\prime\prime},y^{\prime\prime})}$,
we can construct a new path
$\gamma\cup\gamma^{\prime}\in\Gamma_{(x,y)\to(x^{\prime\prime},y^{\prime\prime})}$
by concatenation. We also consider special types of paths, called bridges,
that stay above the $X$-axis.

######  Definition 2.6.

Suppose that $x<x^{\prime}$ and $y,y^{\prime}$ are integers. We define a
_bridge_ to be a path $\gamma\in\Gamma_{(x,y)\to(x^{\prime},y^{\prime})}$ such
that all the integral points of $\gamma$ (except the points $(x,y)$ and
$(x^{\prime},y^{\prime})$) have non-negative $Y$-coordinate. We denote the set
of bridges between $(x,y)$ and $(x^{\prime},y^{\prime})$ by $\Gamma^{\geq
0}_{(x,y)\to(x^{\prime},y^{\prime})}$.

In the following, we will assign various operator-valued weights to a path
$\gamma$. For a family of indeterminates $\mathbf{u}:=(u_{1},\dots,u_{r})$ and
for a path $\gamma$ of length $r$ we assign a weight
$\widetilde{\operatorname{wt}}(\gamma\,|\mathbf{u}):=\widetilde{\operatorname{wt}}(s_{1}|\mathbf{u})\cdots\widetilde{\operatorname{wt}}(s_{r}|\mathbf{u})$,
where $s_{i}$ is the $i$-th step of $\gamma$ and

(12)
$\widetilde{\operatorname{wt}}(s_{i}|\mathbf{u})=\begin{cases}\mathsf{J}^{1}_{-k}&\text{
if }s_{i}=(1,k),k\neq 0,\\\ u_{i}-\hbar\mathfrak{b}\ell&\text{ if
}s_{i}=(1,0),\text{ and connects }(i-1,\ell)\text{ with }(i,\ell).\end{cases}$

######  Theorem 2.7.

Let
$G(z)=\frac{\prod_{i=1}^{n}\left(P_{i}+z\right)}{\prod_{i=1}^{m}\left(Q_{i}-z\right)}$.
The generating function $\tau^{(\mathfrak{b})}_{G}$ defined by (10) is the
unique formal power series
$\tau\in\mathbb{Q}(\mathbf{Q})[\mathbf{P},\tilde{\mathbf{p}},\mathfrak{b}](\\!(\hbar)\\!)\llbracket
t\rrbracket$ which satisfies the following PDE

(13) $t\cdot\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(n+1,0)}}\widetilde{\operatorname{wt}}(\gamma\,|(0,\mathbf{P}))\tau=(-1)^{m}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(m+2,-1)}}\widetilde{\operatorname{wt}}(\gamma\,|(0,-\mathbf{Q},0))\tau$

with the initial condition $\tau=1+O(t)$, where
$(0,\mathbf{P})=(0,P_{1},\dots,P_{n})$ and
$(0,-\mathbf{Q},0)=(0,-Q_{1},\dots,-Q_{m},0)$.

We will deduce Theorem 2.7 from a more general result.

######  Theorem 2.8.

Suppose that there exist invertible formal power series $F(z):=\sum_{i\geq
0}f_{i}z^{i}$, and $H(z):=\sum_{i\geq 0}h_{i}z^{i}$ such that

(14) $G(z):=\sum_{i\geq 0}g_{i}z^{i}=H(z)\cdot F(z)^{-1}.$

The generating function $\tau^{(\mathfrak{b})}_{G}$ defined by (10) is the
unique formal power series
$\tau\in\mathbb{Q}[\tilde{\mathbf{p}},f_{0}^{-1},f_{1},f_{2},\dots,\mathbf{h},\mathfrak{b}](\\!(\hbar)\\!)\llbracket
t\rrbracket$ which satisfies the following PDE

(15) $t\sum_{i\geq 0}h_{i}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(i+1,0)}}\widetilde{\operatorname{wt}}(\gamma\,|0)\tau=\sum_{i\geq
0}f_{i}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(i+2,-1)}}\widetilde{\operatorname{wt}}(\gamma\,|0)\tau.$

with the initial condition $\tau=1+O(t)$. Moreover
$|\mu|H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}_{\geq
0}[-\mathfrak{b}][\mathbf{g}]$ is a polynomial in $-\mathfrak{b},\mathbf{g}$
with non-negative integer coefficients. In addition, when $2g$ is odd (resp.
even), the non-zero coefficients of
$|\mu|H^{(\mathfrak{b})}_{G;g}(\mu_{1},\ldots,\mu_{n})$ have odd (resp. even)
powers of $-\mathfrak{b}$.

######  Remark 2.9.

Note that one can always pick $H(z)=G(z)$, and $F(z)=1$. For this choice, (15)
recovers the cut-and-join equation [CD22, Eq. (62)].

Recall that $\operatorname{Ad}_{A}(B):=[A,B]$ is the adjoint operator and that
$\tilde{D}_{\alpha}$ is the Laplace–Beltrami operator given by (4). Before we
prove Theorem 2.8 we need the following lemma.

######  Lemma 2.10.

For any $\ell\geq 0$ one has:

$\operatorname{Ad}^{i}_{\tilde{D}_{\alpha}}(\mathsf{J}^{1}_{-1})=\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(i+1,0)}}\widetilde{\operatorname{wt}}(\gamma\,|0).$

###### Proof.

Chapuy and the second author proved in [CD22, Theorem 4.7] that

$\operatorname{Ad}^{i}_{\tilde{D}_{\alpha}}(\mathsf{J}^{1}_{-1})=\sum_{k\geq
0}\mathsf{J}^{1}_{-k-1}\partial_{y_{k}}\Lambda_{Y}^{i}y_{0},$

where

$\Lambda_{Y}:=\sum_{k,\ell\geq
1}y_{k+\ell-1}\partial_{y_{\ell-1}}\mathsf{J}^{1}_{k}+\sum_{k,\ell\geq
1}y_{\ell-1}\partial_{y_{k+\ell-1}}\mathsf{J}^{1}_{-k}-\hbar\mathfrak{b}\cdot\sum_{k\geq
0}k\cdot y_{k}\partial_{y_{k}}.$

Therefore it is enough to prove that

$\Lambda_{Y}^{i}y_{0}=\sum_{k\geq 0}y_{k}\sum_{\gamma\in\Gamma^{\geq
0}_{(1,k)\to(i+1,0)}}\widetilde{\operatorname{wt}}(\gamma\,|0)$

with the convention that $\Gamma_{(x,y)\to(x,y)}=\Gamma^{\geq
0}_{(x,y)\to(x,y)}=\\{(x,y)\\}$, and
$\widetilde{\operatorname{wt}}((x,y)\,|0)=1$. This identity follows easily by
induction on $i$ by noticing that

$\Lambda_{Y}y_{\ell}=\sum_{k\geq 0}y_{k}\sum_{\gamma\in\Gamma^{\geq
0}_{(1,k)\to(2,\ell)}}\widetilde{\operatorname{wt}}(\gamma\,|0).\qed$

###### Proof of Theorem 2.8.

In order to make the notation lighter we write
$\tau=\tau^{(\mathfrak{b})}_{G}$,
$J_{\lambda}=J_{\lambda}^{{(\alpha)}}(\sqrt{\alpha}\tilde{\mathbf{p}})$,
$j_{\lambda}=j_{\lambda}^{{(\alpha)}}$,
$c(\square)=\tilde{c}_{\alpha}(\square)$, and we define

$\tilde{J}_{\lambda}:=\frac{J_{\lambda}}{j_{\lambda}}\left(\frac{\sqrt{\alpha}}{\hbar}\right)^{|\lambda|}\prod_{\square\in\lambda}G(\hbar\tilde{c}(\square)).$

Notice that

$h_{0}\cdot\mathsf{J}^{1}_{-1}\tilde{J}_{\lambda}=h_{0}\cdot\frac{\hbar}{\sqrt{\alpha}}\sum_{\lambda\nearrow\mu}\frac{\langle\mathsf{J}^{1}_{-1}J_{\lambda},J_{\mu}\rangle}{j_{\lambda}}G(\hbar\tilde{c}(\mu\setminus\lambda))^{-1}\tilde{J}_{\mu}\\\
=\frac{\hbar}{\sqrt{\alpha}}\sum_{\lambda\nearrow\mu}\frac{\langle\mathsf{J}^{1}_{-1}J_{\lambda},J_{\mu}\rangle}{j_{\lambda}}\left(\sum_{i\geq
0}f_{i}(\hbar\tilde{c}(\mu\setminus\lambda))^{i}-G(\hbar\tilde{c}(\mu\setminus\lambda))^{-1}\sum_{i\geq
1}h_{i}(\hbar\tilde{c}(\mu\setminus\lambda))^{i}\right)\tilde{J}_{\mu}$

by our assumption (14). Therefore

$[t^{n+1}]t\cdot h_{0}\cdot\mathsf{J}^{1}_{-1}\cdot\tau$

can be decomposed as a difference of two terms:

$A_{1}=\frac{\hbar}{\sqrt{\alpha}}\sum_{\lambda\vdash n}\sum_{\mu\vdash
n+1}\bigg{(}\sum_{i\geq
0}f_{i}(\hbar\tilde{c}(\mu\setminus\lambda))^{i}\bigg{)}\frac{\langle\mathsf{J}^{1}_{-1}J_{\lambda},J_{\mu}\rangle}{j_{\lambda}}\tilde{J}_{\mu}=\bigg{(}\sum_{i\geq
0}f_{i}\mathsf{J}_{-}(L+\hbar\mathfrak{b}\operatorname{Id})^{i}\mathsf{J}_{+}\bigg{)}[t^{n+1}]\tau$

by Lemma 2.2, and

$A_{2}=\frac{\hbar}{\sqrt{\alpha}}\sum_{\lambda\vdash n}\sum_{\mu\vdash
n+1}\sum_{i\geq
1}h_{i}(\hbar\tilde{c}(\mu\setminus\lambda))^{i}G(\hbar\tilde{c}(\mu\setminus\lambda))^{-1}\frac{\langle\mathsf{J}^{1}_{-1}J_{\lambda},J_{\mu}\rangle}{j_{\lambda}}\tilde{J}_{\mu}.$

We claim that

$A_{2}=\bigg{(}\sum_{i\geq
1}h_{i}\operatorname{Ad}^{i}_{\tilde{D}_{\alpha}}(\mathsf{J}^{1}_{-1})\bigg{)}[t^{n}]\tau,$

which is equivalent to proving that for each $i\geq 1$ one has

$\sum_{\lambda\vdash
n}\operatorname{Ad}^{i}_{\tilde{D}_{\alpha}}(\mathsf{J}^{1}_{-1})\tilde{J}_{\lambda}=\frac{\hbar}{\sqrt{\alpha}}\sum_{\lambda\vdash
n}\sum_{\mu\vdash
n+1}(\hbar\tilde{c}(\mu\setminus\lambda))^{i}G(\hbar\tilde{c}(\mu\setminus\lambda))^{-1}\frac{\langle\mathsf{J}^{1}_{-1}J_{\lambda},J_{\mu}\rangle}{j_{\lambda}}\tilde{J}_{\mu}.$

For $i=0$ the above equality is immediate from the definition, and for $i\geq
1$ we will proceed by induction:

$\sum_{\lambda\vdash
n}\operatorname{Ad}^{i}_{\tilde{D}_{\alpha}}(\mathsf{J}^{1}_{-1})\tilde{J}_{\lambda}=\frac{\hbar}{\sqrt{\alpha}}\sum_{\lambda\vdash
n}\sum_{\mu\vdash
n+1}(\hbar\tilde{c}(\mu\setminus\lambda))^{i-1}G(\hbar\tilde{c}(\mu\setminus\lambda))^{-1}\frac{\langle\mathsf{J}^{1}_{-1}J_{\lambda},J_{\mu}\rangle}{j_{\lambda}}\cdot\\\
\cdot\left(\tilde{D}_{\alpha}\tilde{J}_{\mu}-\hbar\sum_{\square\in\lambda}\tilde{c}(\square)\tilde{J}_{\mu}\right)=\frac{\hbar}{\sqrt{\alpha}}\sum_{\lambda\vdash
n}\sum_{\mu\vdash
n+1}(\hbar\tilde{c}(\mu\setminus\lambda))^{i}G(\hbar\tilde{c}(\mu\setminus\lambda))^{-1}\frac{\langle\mathsf{J}^{1}_{-1}J_{\lambda},J_{\mu}\rangle}{j_{\lambda}}\tilde{J}_{\mu},$

where we have used (5). Finally, Lemma 2.10 implies that

$A_{2}=\sum_{i=1}^{\infty}h_{i}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(i+1,0)}}\widetilde{\operatorname{wt}}(\gamma\,|0)[t^{n}]\tau$

and the fact that

$A_{1}=\sum_{i=0}^{\infty}f_{i}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(i+2,-1)}}\widetilde{\operatorname{wt}}(\gamma\,|0)[t^{n+1}]\tau$

follows easily from the interpretation of the operator
$\mathsf{J}_{-}(L+\hbar\mathfrak{b}\operatorname{Id})^{i}\mathsf{J}_{+}$ as a
sum over paths – see for instance [CDM23, Theorem (3.10)].

To prove uniqueness, suppose that there are two different solutions
$\tau_{1}$, and $\tau_{2}$ of (15). Then,
$\tau^{\prime}:=\tau_{1}-\tau_{2}=O(t)$ also satisfies (15). Let $k\geq 1$ be
the smallest $k$ such that $[t^{k}]\tau^{\prime}\neq 0$. Our assumption on $k$
guarantees that the LHS of (15) acting on $\tau^{\prime}_{G}$ gives a formal
power series which is $O(t^{k+1})$. On the other hand, extracting the
coefficient $[t^{k}]$ of the action of the RHS of (15) on $\tau^{\prime}$ is
the same as the action of the RHS of (15) on $[t^{k}]\tau^{\prime}$. One can
introduce a grading on
$\mathbb{Q}[\tilde{\mathbf{p}},f_{0}^{-1},f_{1},\dots,\mathbf{h},\mathfrak{b}](\\!(\hbar)\\!)\llbracket
t\rrbracket$ by giving the only non-zero grading to
$\deg(f_{i})=\deg(h_{i})=1$ for $i\in\mathbb{Z}_{\geq 1}$. Then the RHS of
(15) can be written as

$f_{0}\sum_{k\geq 1}\mathsf{J}^{1}_{-k}\mathsf{J}^{1}_{k}+\text{ terms of
degree greater or equal to }1.$

Note that $\sum_{k\geq
1}\mathsf{J}^{1}_{-k}\mathsf{J}^{1}_{k}\tilde{p}_{\mu}=\hbar^{2}|\mu|\tilde{p}_{\mu}$
for any partition $\mu$. In particular let $A_{k}\neq 0$ denote the smallest
degree term of $[t^{k}]\tau^{\prime}$. Then the action of the RHS on
$[t^{k}]\tau^{\prime}$ is equal to

$kf_{0}\hbar^{2}\cdot A_{k}+\text{higher degree terms}\neq 0,$

which contradicts that this power series is $O(t^{k+1})$. This finishes the
proof of the uniqueness.

Finally, (15) implies (by taking $H(z)=G(z),F(z)=1$) that
$|\mu|H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})\in\mathbb{Q}[\mathfrak{b}][\mathbf{g}]$.
But Theorem 2.3 says that
$\sqrt{\alpha}^{2g}\cdot|\mu|H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}_{\geq
0}[-\mathfrak{b}\cdot\sqrt{\alpha}][\mathbf{g}]=\mathbb{Z}_{\geq
0}[\alpha-1][\mathbf{g}]$, therefore
$|\mu|H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})\in\mathbb{Q}[\mathfrak{b}][\mathbf{g}]$
is a polynomial of the same parity as $2g$. The fact that
$\mathbb{Q}[\mathfrak{b}]$ could be replaced by $\mathbb{Z}_{\geq
0}[-\mathfrak{b}]$ follows easily by comparing the coefficients of
$|\mu|H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})$ with the coefficients
of $\alpha^{g}|\mu|H^{(\mathfrak{b})}_{G;g}(\mu_{1},\dots,\mu_{n})$. ∎

######  Remark 2.11.

In the case of a rational weight of the form
$\check{G}(z):=\frac{\prod_{i=1}^{n}\left(1+P_{i}z\right)}{\prod_{i=1}^{m}\left(1-Q_{i}z\right)}$
we can prove directly that
$|\mu|H_{\check{G}}^{(\mathfrak{b})}(\mu_{1},\dots,\mu_{n})\in\mathbb{Z}_{\geq
0}[-\mathfrak{b}][\mathbf{P},\mathbf{Q}]$ without referring to [CD22]. Indeed,
note that the latter statement is equivalent to showing that $\hbar
t\partial_{t}\log(\tau^{(\mathfrak{b})}_{\check{G}})\in\mathbb{Z}_{\geq
0}[-\mathfrak{b}][\mathbf{P},\mathbf{Q},\tilde{\mathbf{p}},\hbar]\llbracket
t\rrbracket$. Note however that $\hbar
t\partial_{t}\log(\tau^{(\mathfrak{b})}_{\check{G}})=\sum_{k=1}^{\infty}\tilde{p}_{k}\mathsf{J}^{1}_{k}\log(\tau^{(\mathfrak{b})}_{\check{G}})$,
and we can prove a stronger statement that
$\mathsf{J}^{1}_{k}\log(\tau^{(\mathfrak{b})}_{\check{G}})\in\mathbb{Z}_{\geq
0}[-\mathfrak{b}][\mathbf{P},\mathbf{Q},\tilde{\mathbf{p}},\hbar]\llbracket
t\rrbracket$. This statement can be proven using (40), which refines (13), and
then making an appropriate change of variables given by (11). Equation (40)
produces a system of partial differential equations involving
$\\{\mathsf{J}^{1}_{k}\log(\tau^{(\mathfrak{b})}_{\check{G}})\\}_{k\geq 1}$
that allows one to compute
$\mathsf{J}^{1}_{k}\log(\tau^{(\mathfrak{b})}_{\check{G}})$ inductively order
by order w.r.t $t$, and since it involves sums of terms that were inductively
assumed to lie in $\mathbb{Z}_{\geq
0}[-\mathfrak{b}][\mathbf{P},\mathbf{Q},\tilde{\mathbf{p}},\hbar]\llbracket
t\rrbracket$, the statement follows. It is not our intention to focus on this
alternative proof, so we leave the details to the interested reader.

###### Proof of Theorem 2.7.

By taking $H(z)=\prod_{i=1}^{n}(P_{i}+z)$, $F(z)=\prod_{i=1}^{m}(Q_{i}-z)$ and
applying Theorem 2.8 we have that $\tau^{(\mathfrak{b})}_{G}$ satisfies the
following equation

$t\sum_{i=0}^{n}e_{n-i}(\mathbf{P})\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(i+1,0)}}\widetilde{\operatorname{wt}}(\gamma\,|0)\tau^{(\mathfrak{b})}_{G}=(-1)^{m}\sum_{i=0}^{m}e_{m-i}(-\mathbf{Q})\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(i+2,-1)}}\widetilde{\operatorname{wt}}(\gamma\,|0)\tau^{(\mathfrak{b})}_{G},$

where $e_{i}(\mathbf{x})$ is the $i$-th elementary symmetric function. In
order to prove (13), it is enough to notice that

$\sum_{i=0}^{n}e_{n-i}(\mathbf{P})\sum_{\gamma^{\prime}\in\Gamma^{\geq
0}_{(0,-1)\to(i+1,0)}}\widetilde{\operatorname{wt}}(\gamma^{\prime}\,|0)=\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(n+1,0)}}\widetilde{\operatorname{wt}}(\gamma\,|(0,\mathbf{P})),$

and similarly

$\sum_{i=0}^{m}e_{m-i}(-\mathbf{Q})\sum_{\gamma^{\prime}\in\Gamma^{\geq
0}_{(0,-1)\to(i+2,-1)}}\widetilde{\operatorname{wt}}(\gamma^{\prime}\,|0)=\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(n+2,-1)}}\widetilde{\operatorname{wt}}(\gamma\,|(0,-\mathbf{Q},0).$

We will explain only the first equality, as the second one follows by the same
argument. This identity follows directly by interpreting the product of the
weights $\widetilde{\operatorname{wt}}(s_{i}\,|(0,\mathbf{P}))$ associated
with the horizontal steps as the sum over all subsets of the horizontal steps,
and then removing the selected steps at places $j_{1}<\dots<j_{n-i}$ (thus,
obtaining a path $\gamma^{\prime}\in\Gamma^{\geq 0}_{(0,-1)\to(i+1,0)}$) and
associating the weight $P_{j_{1}-1}\cdots
P_{j_{n-i}-1}\cdot\widetilde{\operatorname{wt}}(\gamma^{\prime}\,|0)$.
Finally, the fact that there is a unique solution $\tau$ in the ring
$\mathbb{Q}(\mathbf{Q})[\mathbf{P},\tilde{\mathbf{p}},\mathfrak{b}](\\!(\hbar)\\!)\llbracket
t\rrbracket$ which is slightly bigger than the ring
$\mathbb{Q}[\tilde{\mathbf{p}},f_{0}^{-1},f_{1},f_{2},\dots,\mathbf{h},\mathfrak{b}](\\!(\hbar)\\!)\llbracket
t\rrbracket$ given in Theorem 2.8 follows the argument given there verbatim.
Indeed, using the same notation as in the proof of Theorem 2.8 it is enough to
multiply $[t^{k}]\tau^{\prime}$ by a polynomial
$c(\mathbf{Q})\in\mathbb{Q}[\mathbf{Q}]$ so that
$[t^{k}]c(\mathbf{Q})\cdot\tau^{\prime}\in\mathbb{Q}[\mathbf{Q},\mathbf{P},\tilde{\mathbf{p}},\mathfrak{b}](\\!(\hbar)\\!)$.
Then $c(\mathbf{Q})\tau^{\prime}$ also satisfies (13). Then the argument from
the proof of Theorem 2.8 can be applied with the grading
$\deg(P_{i})=\deg(Q_{i})=1$. ∎

##  3\. Airy structures from $\mathcal{W}$-algebra representations

The formalism of Airy structures provides a set of conditions under which a
sequence of differential operators has a unique solution of a certain form.
Typically, the differential operators arise from representations of vertex
algebras, and the solution is a partition function that encodes interesting
geometric information. In this section, we construct an Airy structure using
the principal $\mathcal{W}$ algebra of type $A$ whose partition function will
be related to weighted $b$-Hurwitz numbers.

###  3.1. Airy structures

We introduce the notion of Airy structures as studied in [KS18, BBC+18,
BCJ22]. We shall mostly adopt the notation and presentation of Airy structures
in [BCJ22], and refer the reader there for the proofs.

Before defining Airy structures in general, we need to set up some notation
and terminology. Consider a countable index set $A$. We will denote the set of
variables $\\{x_{a}\\}_{a\in A}$ by $x_{A}$, the differential operator
$\frac{\partial}{\partial x_{a}}$ by $\partial_{a}$ and the set of
differential operators $\\{\frac{\partial}{\partial x_{a}}\\}_{a\in A}$ by
$\partial_{A}$. We introduce a formal parameter $\hbar$ and the notion of
$\hbar$-degree given by $\deg_{\hbar}\hbar=1$, $\deg_{\hbar}x_{a}=0$ and
$\deg_{\hbar}\partial_{a}=0$ for all $a\in A$, in order to distinguish from
degree of polynomials in $x_{A}$ where we view $x_{a}$ as degree 1.

Using this $\hbar$-grading, we can define the completed Rees Weyl algebra
$\widehat{\mathcal{D}}_{A}^{\hbar}$ (see [BCJ22, Definition 2.12]), whose
elements $S\in\widehat{\mathcal{D}}_{A}^{\hbar}$ are differential operators of
the form

$S=\sum_{n\in\mathbb{Z}_{\geq
0}}\hbar^{n}\sum_{\begin{subarray}{c}m,k\in\mathbb{Z}_{\geq 0}\\\
m+k=n\end{subarray}}\sum_{a_{1},\ldots,a_{m}\in
A}s^{(n,k)}_{a_{1},\ldots,a_{m}}(x_{A})\partial_{a_{1}}\ldots\partial_{a_{m}},$

where the $s^{(n,k)}_{a_{1},\ldots,a_{m}}(x_{A})$ are polynomials of degree
$\leq k$.

We are ready to define an Airy structure in
$\widehat{\mathcal{D}}_{A}^{\hbar}$ following [BCJ22] now.

######  Definition 3.1.

Let $\mathcal{I}\subset\widehat{\mathcal{D}}_{A}^{\hbar}$ be a left ideal. We
say that the ideal $\mathcal{I}$ is an _Airy structure_ (or _Airy ideal_) if
there exist operators $\mathsf{H}_{a}\in\widehat{\mathcal{D}}_{A}^{\hbar}$,
for all $a\in A$, such that:

  1. a)

The collection of operators $\\{\mathsf{H}_{a}\\}_{a\in A}$ is bounded (see
[BCJ22, Definition 2.15] for a definition of boundedness222Boundedness allows
us to take infinite linear combinations of the operators $H_{a}$ without
divergent sums appearing, see [BCJ22, Section 2.2].).

  2. b)

The left ideal $\mathcal{I}$ can be written as

$\mathcal{I}=\left\\{\sum_{a\in A}c_{a}\mathsf{H}_{a}\ |\
c_{a}\in\widehat{\mathcal{D}}_{A}^{\hbar}\right\\},$

which consists of finite and infinite (if $A$ is countably infinite)
$\widehat{\mathcal{D}}_{A}^{\hbar}$-linear combinations of the
$\mathsf{H}_{a}$.

  3. c)

The operators $\mathsf{H}_{a}$ take the form

$\mathsf{H}_{a}=\hbar\partial_{a}+O(\hbar^{2}).$

  4. d)

The left ideal $\mathcal{I}$ satisfies the property:

$[\mathcal{I},\mathcal{I}]\subseteq\hbar^{2}\mathcal{I}.$

We need a slight generalization of the above notion of Airy structures as we
would like to incorporate so-called Whittaker shifts into our Airy structures
(see Section 3.3.1). More precisely, consider a set of formal commuting
variables $\\{\Lambda_{a}\\}_{a\in A}$ with $\deg_{\hbar}\Lambda_{a}=0$. Then,
we define a shifted Airy structure as follows.

######  Definition 3.2.

Let the collection of operators $\\{\mathsf{H}_{a}\\}_{a\in A}$ generate an
Airy structure. Consider the ideal
$\mathcal{I}^{\Lambda}\subset\widehat{\mathcal{D}}_{A}^{\hbar}\llbracket\Lambda_{A}\rrbracket$
generated333In the context of Airy structures, the word generated will always
mean generated by infinite linear combinations. by the operators
$\mathsf{H}_{a}-\Lambda_{a}$ for every $a\in A$, viewed as elements of
$\widehat{\mathcal{D}}_{A}^{\hbar}\llbracket\Lambda_{A}\rrbracket$. Then the
ideal $\mathcal{I}^{\Lambda}$ is said to be a _shifted Airy structure_ if it
satisfies condition $(d)$ from Definition 3.1, i.e.,

$[\mathcal{I}^{\Lambda},\mathcal{I}^{\Lambda}]\subseteq\hbar^{2}\mathcal{I}^{\Lambda}.$

The reason that (shifted) Airy structures are interesting is the following
theorem that guarantees a unique solution $Z$ (of a certain form) to the
equations $\mathsf{H}_{a}Z=\Lambda_{a}Z$ for all $a\in A$.

######  Theorem 3.3.

Let
$\mathcal{I}^{\Lambda}\subset\widehat{\mathcal{D}}_{A}^{\hbar}\llbracket\Lambda_{A}\rrbracket$
be a shifted Airy structure. Then there exists a unique solution $Z$ of the
form

$Z=\exp\left(\sum_{\begin{subarray}{c}g\in\frac{1}{2}\mathbb{Z}_{\geq
0},n\in\mathbb{Z}_{\geq
1}\end{subarray}}\hbar^{2g-2+n}F_{g,n}(x_{A})\right)\,,$

to the differential equations $\mathcal{I}^{\Lambda}\cdot Z=0$ (i.e.,
$\mathsf{H}_{a}Z=\Lambda_{a}Z$, for all $a\in A$), given the initial condition
$Z\big{|}_{x_{A}=0}=1$. The $F_{g,n}(x_{A})$ are formal power series in the
$\Lambda_{A}$, whose coefficients are homogeneous polynomials of degree $n$
with $F_{g,n}(0)=0$, i.e.
$F_{g,n}(x_{A})\in\mathbb{C}[x_{A}]\llbracket\Lambda_{A}\rrbracket$. We call
$Z$ the _partition function_ of the $\Lambda$-shifted Airy structure
$\mathcal{I}^{\Lambda}$.

For Airy structures, this theorem was first proved in [KS18]. In the case of
shifted Airy structures, the generalized version presented above can be proved
easily following the arguments of [BCJ22, Theorem 2.28], but a thorough
treatment of shifted Airy structures will appear in [BBC+]. An alternative to
working with shifted Airy structures is the approach of [BBCC21] – first,
rescale all the parameters $\Lambda_{a}$ by $\hbar^{2}$ to get an honest Airy
structure and its associated partition function, and then undo the rescaling
to recover the partition function of the shifted Airy structure.

Airy structures were invented as an algebraic framework in which to understand
the Eynard–Orantin topological recursion and its variants. In general, it’s
not an easy task to construct examples of Airy structures. Almost all
interesting examples of Airy structures, in which the partition function has a
geometric interpretation, come from representations of vertex algebras such as
$\mathcal{W}$-algebras [BBC+18, BBCC21]. As we will see in this paper, the
Airy structures of interest for $b$-Hurwitz theory also come from
$\mathcal{W}$-algebras.

###  3.2. The algebra $\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$

In this section, we give a brief introduction to the principal
$\mathcal{W}$-algebra of $\mathfrak{gl}_{r}$ at level $\mathsf{k}$, which we
will denote by $\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$. A standard
reference for $\mathcal{W}$-algebras and their representation theory is
[Ara17]. In this paper, we will only study the algebra
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ via its free field embedding
into a Heisenberg vertex operator algebra (VOA). Throughout this section, we
work over the ring $\mathbb{Q}[\hbar]$, where $\hbar$ is a formal parameter
that keeps track of the conformal dimension of the $\mathcal{W}$-algebra.

A Heisenberg VOA of rank $r$, say $\mathcal{H}_{r}$, is strongly and freely
generated as a vertex algebra by $r$ fields
$\mathsf{J}^{1}(z),\mathsf{J}^{2}(z),\ldots,\mathsf{J}^{r}(z)$, whose mode
decomposition is

$\mathsf{J}^{a}(z)=\sum_{k\in\mathbb{Z}}\mathsf{J}^{a}_{k}z^{-k-1},\qquad
a=1,\ldots,r\,,$

and the modes satisfy the following commutation relations:

$[\mathsf{J}^{a_{1}}_{k_{1}},\mathsf{J}^{a_{2}}_{k_{2}}]=\hbar^{2}k_{1}\delta_{a_{1},a_{2}}\delta_{k_{1}+k_{2},0}.$

At any level $\mathsf{k}\in\mathbb{C}$, the vertex algebra
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ is also strongly and freely
generated by $r$ fields $\mathsf{W}^{1}(z),\ldots,\mathsf{W}^{r}(z)$ of
conformal dimensions $1,\ldots,r$ respectively. A convenient choice of these
generators is provided by the quantum Miura transform, which embeds the
algebra $\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ into the Heisenberg VOA
$\mathcal{H}_{r}$. An explicit expression for the fields $\mathsf{W}^{i}(z)$
is given in [AM17, Corollary 3.12]:

(16)
$\displaystyle\sum_{i=0}^{r}\mathsf{W}^{i}(z)(\hbar\mathfrak{b})^{r-i}\partial_{z}^{r-i}=\left(\hbar\mathfrak{b}\partial_{z}+\mathsf{J}^{1}(z)\right)\cdots\left(\hbar\mathfrak{b}\partial_{z}+\mathsf{J}^{r}(z)\right),$

where we set $\mathsf{W}^{0}(z)=1$ by convention. The parameter $\mathfrak{b}$
is related to the level $\mathsf{k}$ as $\mathfrak{b}=\mathsf{k}+r-1$. We use
the following mode convention for the generating fields $\mathsf{W}^{i}(z)$,
for $i=1,\ldots,r$.

$\mathsf{W}^{i}(z)=\sum_{k\in\mathbb{Z}}\mathsf{W}^{i}_{k}z^{-k-i}\,.$

For $i=1,2,3$, (16) yields the following formulae for the modes
$\mathsf{W}^{i}_{k}$:

$\displaystyle\mathsf{W}^{1}_{k}$
$\displaystyle=\sum_{a_{1}=1}^{r}\mathsf{J}^{a_{1}}_{k},$
$\displaystyle\mathsf{W}^{2}_{k}$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq a_{1}<a_{2}\leq r,\\\
k_{1}+k_{2}=k\end{subarray}}\mathsf{J}^{a_{1}}_{k_{1}}\mathsf{J}^{a_{2}}_{k_{2}}-\mathfrak{b}\hbar(k+1)\sum_{a=1}^{r}(a-1)\mathsf{J}^{a}_{k}$
$\displaystyle\mathsf{W}^{3}_{k}$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq a_{1}<a_{2}<a_{3}\leq r,\\\
k_{1}+k_{2}+k_{3}=k\end{subarray}}\mathsf{J}^{a_{1}}_{k_{1}}\mathsf{J}^{a_{2}}_{k_{2}}\mathsf{J}^{a_{3}}_{k_{3}}-\mathfrak{b}\hbar\sum_{\begin{subarray}{c}1\leq
a_{1}<a_{2}\leq r,\\\
k_{1}+k_{2}=k\end{subarray}}\big{(}k_{1}(a_{1}-1)+k_{2}(2a_{1}-a_{2})+a_{1}+a_{2}-3\big{)}\mathsf{J}^{a_{1}}_{k_{1}}\mathsf{J}^{a_{2}}_{k_{2}}+$
$\displaystyle+\mathfrak{b}^{2}\hbar^{2}(k+2)(k+1)\sum_{a=1}^{r}\frac{(a-1)(a-2)}{2}\mathsf{J}^{a}_{k}.$

For higher values of $i$ and for generic $\mathfrak{b}$, explicit expressions
for the modes $\mathsf{W}^{i}_{k}$ that one obtains directly from (16) are
rather complicated.

####  3.2.1. A combinatorial description of the modes $\mathsf{W}^{i}_{k}$

For our purposes, it’s essential to find a tractable formula for these modes
$\mathsf{W}^{i}_{k}$, and this is the main result of this subsection. More
precisely, we will provide a combinatorial formula for the modes
$\mathsf{W}^{i}_{k}$ in terms of weighted paths as introduced in Section
2.3.1.

In addition to the notion of paths considered already, we define colored
paths, which are paths whose steps carry an associated integral weight.

######  Definition 3.4.

Fix an integer $r\geq 1$. A _colored path_ is a tuple $(\gamma,f)$, where
$\gamma$ is a path in $\Gamma_{(x,y)\to(x^{\prime},y^{\prime})}$, and
$f:[x^{\prime}-x]\to[r]$ is a function that associates an integer to each step
of the path $\gamma$.

For every colored path $(\gamma,f)$ with $\gamma$ in
$\Gamma_{(x,y)\to(x^{\prime},y^{\prime})}$, we associate a weight
$\operatorname{wt}(\gamma,f):=\operatorname{wt}_{1}(\gamma,f)\cdots\operatorname{wt}_{x^{\prime}-x}(\gamma,f)$
as

$\operatorname{wt}_{j}(\gamma,f)=\begin{cases}\mathsf{J}^{f(j)}_{\gamma_{j}}&\text{
if }\gamma_{j}\neq 0,\\\
\mathsf{J}^{f(j)}_{0}-\mathfrak{b}\hbar\big{(}\sum_{l=j+1}^{x^{\prime}-x}\gamma_{l}+x^{\prime}-x-j\big{)}&\text{
if }\gamma_{j}=0,\end{cases}$

where we denote the increment of the $j$-th step of the path by $-\gamma_{j}$,
i.e., if the $j$-th segment is from $(x_{j},y_{j})$ to
$(x_{j}+1,y^{\prime}_{j})$, then $\gamma_{j}=-(y^{\prime}_{j}-y_{j})$. See
Fig. 1 for examples of paths and the associated weights, including a
combinatorial interpretation of the sum appearing in the weight of a
horizontal step.

Figure 1. Generic colored path associated with the operators
$\mathsf{W}^{i}_{k}$ by (17). Here $k=2,i=7$, and the associated weight is
equal to
$\mathsf{J}^{f(1)}_{-1}\mathsf{J}^{f(2)}_{4}(\mathsf{J}^{f(3)}_{0}-3\mathfrak{b}\hbar)\mathsf{J}^{f(4)}_{-2}(\mathsf{J}^{f(5)}_{0}-3\mathfrak{b}\hbar)\mathsf{J}^{f(6)}_{3}\mathsf{J}^{f(7)}_{-2}$.

With this set up, we can prove the following combinatorial formula for the
modes $\mathsf{W}^{i}_{k}$.

######  Lemma 3.5.

We have the following combinatorial interpretation of the modes
$\mathsf{W}^{i}_{k}$ for all $1\leq i\leq r$ and for all $k\in\mathbb{Z}$:

(17)
$\mathsf{W}^{i}_{k}=\sum_{\begin{subarray}{c}\gamma\in\Gamma_{(0,k)\to(i,0)},\\\
f\in\mathcal{F}_{<}([i],[r])\end{subarray}}\operatorname{wt}(\gamma,f),$

where we recall that $f\in\mathcal{F}_{<}([i],[r])$ means that we are summing
over strictly increasing colorings.

###### Proof.

In this proof, we will need to work with the algebra
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r-1})$ realized as an embedding into
the Heisenberg VOA generated by the fields
$\mathsf{J}^{2}(z),\ldots,\mathsf{J}^{r}(z)$. Then the algebra
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r-1})$ is generated by $(r-1)$
generators, say $\mathsf{W}^{i,r-1}(z)$, defined by the following Miura
transform:

$\sum_{i=0}^{r-1}\mathsf{W}^{i,r-1}(z)(\hbar\mathfrak{b}\partial_{z})^{r-1-i}:=(\hbar\mathfrak{b}\partial_{z}+\mathsf{J}^{2}(z))\cdots(\hbar\mathfrak{b}\partial_{z}+\mathsf{J}^{r}(z))\,.$

In this proof, we denote the generators of
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ given by the Miura transform
(16) as $\mathsf{W}^{i,r}(z)$, to emphasize the $r$-dependence and avoid
confusion. Then, the quantum Miura transform for
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ can be written as

$\sum_{i=0}^{r}{\mathsf{W}}^{i,r}(z)(\hbar\mathfrak{b}\partial_{z})^{r-i}=\left(\hbar\mathfrak{b}\partial_{z}+\mathsf{J}^{1}(z)\right)\cdot\sum_{i=0}^{r-1}\mathsf{W}^{i,r-1}(z)(\hbar\mathfrak{b}\partial_{z})^{r-1-i},$

which, in terms of the modes, gives

(18)
${\mathsf{W}}^{i,r}_{k}=\mathsf{W}^{i,r-1}_{k}+\sum_{k_{1}+k_{2}=k}\left(\mathsf{J}^{1}_{k_{1}}-\delta_{k_{1},0}\hbar\mathfrak{b}(k_{2}+i-1)\right)\cdot\mathsf{W}^{i-1,r-1}_{k_{2}}.$

In addition to our existing convention that
$\mathsf{W}^{0,r}_{k}=\delta_{k,0}$, we set $\mathsf{W}^{r,r-1}_{k}=0$ and
$\mathsf{W}^{-1,r-1}_{k}=0$ for any $k\in\mathbb{Z}$.

With this setup, we can prove (17) by induction with respect to the
lexicographic order on $(i,r)$. It is clear that the formula holds for $(1,r)$
for all $r$. Assume that it is true for all $(i^{\prime},r^{\prime})<(i,r)$.
Then, the RHS of (17) can be decomposed into two terms – either the first step
is colored by $f(1)=1$, or $f(1)>1$. When $f(1)=1$, the weight
$\operatorname{wt}_{1}(\gamma,f)=\mathsf{J}^{1}_{k_{1}}-\delta_{k_{1},0}\mathfrak{b}\hbar\big{(}k_{2}+i-1\big{)}$,
where the first step of $\gamma$ ends at $(1,k_{2})$ and $k_{1}=k-k_{2}$. This
gives the following formula for the RHS of (17):

(19) $\sum_{\begin{subarray}{c}\gamma\in\Gamma_{(0,k)\to(i,0)},\\\
f\in\mathcal{F}_{<}([i],[2..r])\end{subarray}}\operatorname{wt}(\gamma,f)+\sum_{k_{1}+k_{2}=k}\left(\mathsf{J}^{1}_{k_{1}}-\delta_{k_{2},0}\hbar\mathfrak{b}(k_{2}+i-1)\right)\cdot\sum_{\begin{subarray}{c}\tilde{\gamma}\in\Gamma_{(0,k_{2})\to(i-1,0)},\\\
\tilde{f}\in\mathcal{F}_{<}([i-1],[2..r])\end{subarray}}\operatorname{wt}(\tilde{\gamma},\tilde{f}),$

where $\tilde{\gamma}$ is the path obtained from $\gamma$ by removing the
first step and shifting it one unit to the left, so that $\tilde{\gamma}$
starts at $(0,k_{2})$. Note that the weight of a path is invariant under a
horizontal shift. For $j>1$, we define $\tilde{f}(j-1):=f(j)$ such that
$\operatorname{wt}_{j}(\gamma,f)=\operatorname{wt}_{j-1}(\tilde{\gamma},\tilde{f})$.
The inductive hypothesis gives us the following formula for $j\leq i$:

$\mathsf{W}^{j,r-1}_{k}=\sum_{\begin{subarray}{c}\gamma\in\Gamma_{(0,k)\to(j,0)},\\\
f\in\mathcal{F}_{<}([j],[2..r])\end{subarray}}\operatorname{wt}(\gamma,f),$

and hence (19) is equivalent to the recursion (18), which finishes the proof.
∎

####  3.2.2. Algebra of modes

In order to construct an Airy structure, we will need to work with the algebra
of modes of the vertex algebra $\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$.
We consider the algebra $\mathcal{A}^{\hbar}$, which is a suitable completion
of the algebra of modes, with the parameter $\hbar$ keeping track of the
conformal dimension of the generating fields (see [BBCC21, Section 3.1] for a
definition).

The strategy to find an Airy structure is to pick a set of modes of
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$, such that the ideal generated
by these modes will be a subalgebra of $\mathcal{A}^{\hbar}$. In addition, we
may want to pick a subset of these modes, and shift them to obtain Whittaker
vectors. Various such subsets of modes were studied in [BBC+18], and we
present one of these subsets that is relevant for us. In fact, we prove an
upgraded version that allows us to shift the zero modes $\mathsf{W}^{i}_{0}$
later on.

######  Lemma 3.6.

Consider the set of modes $S$ defined as

$S:=\left\\{\mathsf{W}^{i}_{k}:i\in[r]\,,k\geq 0\right\\},$

and the subset of modes
$S^{\prime}:=\left\\{\mathsf{W}^{i}_{k}:i\in[r]\,,k\geq 1\right\\}\subset S$.
Then, for any two modes
$\mathsf{W}^{i_{1}}_{k_{1}},\mathsf{W}^{i_{2}}_{k_{2}}$ in $S$, we have

$[\mathsf{W}^{i_{1}}_{k_{1}},\mathsf{W}^{i_{2}}_{k_{2}}]\in\hbar^{2}\mathcal{A}^{\hbar}\cdot
S^{\prime}.$

###### Proof.

The weaker statement that
$[\mathsf{W}^{i_{1}}_{k_{1}},\mathsf{W}^{i_{2}}_{k_{2}}]\in\hbar^{2}\mathcal{A}^{\hbar}\cdot
S$ is a direct application of [BBC+18, Proposition 3.14] to the strong
generators of $\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ (up to the
rescaling by $\hbar$).

To prove the lemma, we need to show that terms appearing in the commutator
$[\mathsf{W}^{i_{1}}_{k_{1}},\mathsf{W}^{i_{2}}_{k_{2}}]$ can be ordered such
that the rightmost mode of each monomial is not $\mathsf{W}^{i}_{0}$. Using
Li’s filtration [Li05], any monomial can be normally ordered to take the form
(up to a constant prefactor)

$\mathsf{W}^{j_{1}}_{\ell_{1}}\cdots\mathsf{W}^{j_{n}}_{\ell_{n}},$

for some $n\geq 1$, where $\ell_{n}=\max(\ell_{1},\dots,\ell_{n})$ and
$\ell_{1}+\cdots+\ell_{n}=k_{1}+k_{2}$. If $k_{1}+k_{2}\geq 1$, we must have
$\ell_{n}\geq 1$, and thus the zero modes do not appear as the rightmost mode
in the monomial.

For the case $k_{1}=k_{2}=0$, we need to use the fact that the Zhu algebra of
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ is commutative [Ara17, Theorem
4.16.3]. Recall that the Zhu algebra in general is defined as a certain
quotient of the algebra of zero modes [Ara17, Section 3.12]. In the case of
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$, the Zhu algebra is generated by
monomials of the form
$\mathsf{W}^{i_{1}}_{k_{1}}\cdots\mathsf{W}^{i_{n}}_{k_{n}}$ such that
$\sum_{i=1}^{n}k_{i}=0$, modulo relations
$\mathsf{W}^{j_{1}}_{\ell_{1}}\cdots\mathsf{W}^{j_{n}}_{\ell_{n}}\cdot\mathsf{W}^{j_{n+1}}_{\ell_{n+1}}\cdots\mathsf{W}^{j_{n+m}}_{\ell_{n+m}}=0$
whenever $\sum_{i=1}^{n}\ell_{i}=-\sum_{j=n+1}^{n+m}\ell_{j}<0$.

Then, we know that the commutator
$[\mathsf{W}^{i_{1}}_{0},\mathsf{W}^{i_{2}}_{0}]$ is a linear combination of
terms of the form
$\mathsf{W}^{j_{1}}_{\ell_{1}}\cdots\mathsf{W}^{j_{n}}_{\ell_{n}}\cdot\mathsf{W}^{j_{n+m}}_{\ell_{n+1}}\cdots\mathsf{W}^{j_{n+m}}_{\ell_{n+m}}$,
where $\sum_{i=1}^{n}\ell_{i}=-\sum_{j=n+1}^{n+m}\ell_{j}<0$. Again using Li’s
filtration, we can order
$\mathsf{W}^{j_{n+1}}_{\ell_{n+1}}\cdots\mathsf{W}^{j_{n+1}}_{\ell_{n+m}}$
such that $\ell_{n+m}>0$, and hence we are done. ∎

###  3.3. An Airy structure for weighted $b$-Hurwitz numbers

Now that we have introduced the notion of Airy structures and the algebra
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$, we can go ahead and construct
the Airy structures we are after. We will consider a specific representation
of the vertex algebra $\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$, look at a
certain subset of modes and prove that they form an Airy structure. This Airy
structure will turn out to have a trivial partition function i.e. $Z=1$, but
by doing appropriate shifts, we will obtain a shifted Airy structure with an
interesting partition function.

######  Remark 3.7.

It may help the reader to keep in mind that the partition function of the
shifted Airy structure that we will eventually construct will be related to
the generating function of weighted $b$-Hurwitz numbers with the following
weight:

$G(z)=\frac{\prod_{i=1}^{r}(P_{i}+z)}{\prod_{i=1}^{r-2}(Q_{i}-z)}\,$

where $r\geq 2$ is an integer. While this is not the most general weight we
are interested in, one can take appropriate limits to reduce to an arbitrary
rational weight (see Section 4.3.2 for a more detailed explanation).

We will first construct a representation of the Heisenberg VOA
$\mathcal{H}_{r}$. Consider the ring
$R=\mathbb{C}(\mathbf{P},\mathbf{Q},\mathfrak{b})[\mathbf{x}]$, which is the
polynomial ring in infinitely many variables
$\mathbf{x}=\\{x^{a}_{k}\\}_{k\geq 1,a\in[r]}$ over the field of rational
functions in $\mathfrak{b}$, $\mathbf{P}=\\{P_{1},\dots,P_{r}\\}$ and
$\mathbf{Q}=\\{Q_{0},Q_{1},\dots,Q_{r-2}\\}$ — we will set $Q_{0}=0$ in
Section 4. Consider the vector space

$\widehat{\mathcal{M}}:=\prod_{n\in\mathbb{Z}_{\geq 0}}\hbar^{n}\left(R_{\leq
n}\right),$

where $R_{\leq n}$ denotes polynomials in the $x^{a}_{k}$ of degree $\leq n$.
A typical element of $\widehat{\mathcal{M}}$ is of the form
$\sum_{n=0}^{\infty}\hbar^{n}P_{n}$, where $P_{n}$ is a polynomial in the
$x^{a}_{k}$ of degree at most $n$. This vector space $\widehat{\mathcal{M}}$
is a module for the Heisenberg VOA via the following mode assignment:

(20)
$\begin{split}\mathsf{J}^{a}_{k}=\left\\{\begin{array}[]{lr}\hbar\partial_{x^{a}_{k}}&k>0,\\\
\hbar(-k)x^{a}_{-k}+(-1)^{r}\frac{1}{\Lambda}\delta_{k,-1}\delta_{a,r}&k<0,\\\
Q_{a-1}-\hbar\mathfrak{b}(a-1)&a\in[r-1],k=0\\\
-\sum_{a^{\prime}=1}^{r}P_{a^{\prime}}-\sum_{a^{\prime}=0}^{r-2}Q_{a^{\prime}}-\hbar\mathfrak{b}(a-1)&a=r,k=0,\\\
\end{array}\right.\end{split}$

Note that while the $\mathsf{J}^{1}_{k}$ have been defined previously in (3),
it is not a conflict of notation as we will eventually substitute
$\tilde{p}_{k}=kx^{1}_{k}$ (see Theorem 4.7). As the quantum Miura
transformation gives an embedding
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})\hookrightarrow\mathcal{H}_{r}$,
$\widehat{\mathcal{M}}$ is also a module for
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ by restriction. By abuse of
notation, we will henceforth use both $\mathsf{W}^{i}_{k}$ and
$\mathsf{J}^{a}_{k}$ to denote the modes of the generators of
$\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ and $\mathcal{H}_{r}$,
respectively, in the representation $\widehat{\mathcal{M}}$.

We would like to claim that the set of operators
$\left\\{\mathsf{W}^{i}_{k}:i\in[r],k\geq\delta_{i,1}\right\\}$ in the
representation $\widehat{\mathcal{M}}$ generates an Airy structure. However,
this set of operators does not satisfy condition $(c)$ of Definition 3.1 of
Airy structures. Indeed, for an operator $P$ in
$\widehat{\mathcal{D}}_{A}^{\hbar}$, let
$\pi_{i}(P):=\hbar^{i}\cdot[\hbar^{i}]P$ denote the $\hbar$-degree $i$ term of
$P$. Then we have the following two problems: first
$\pi_{0}(\mathsf{W}^{i}_{0})\neq 0$, and second
$\pi_{1}(\mathsf{W}^{i}_{k})\neq\hbar\partial_{x^{\bullet}_{\bullet}}$.

To address the first issue, let us define new operators
$\widetilde{\mathsf{W}}^{i}_{k}$, where we remove some terms by hand,
including all the $\hbar$-degree zero terms. Define $\mathsf{V}^{i}$ for any
$i\in[r]$ as the following combinations of the Heisenberg zero modes:

$\mathsf{V}^{i}:=\sum_{f\in\mathcal{F}_{<}([i],[r])}\big{(}\mathsf{J}^{f(1)}_{0}-\hbar\mathfrak{b}(i-1)\big{)}\cdots\big{(}\mathsf{J}^{f(i-1)}_{0}-\hbar\mathfrak{b}\big{)}\mathsf{J}^{f(i)}_{0}.$

Define, for any $i\in[r]$, and any $k\geq 0$,

(21)
$\widetilde{\mathsf{W}}^{i}_{k}:=\mathsf{W}^{i}_{k}-\mathsf{V}^{i}\delta_{k,0}.$

To address the second issue, we need to diagonalize the set of operators
$\left\\{\widetilde{\mathsf{W}}^{i}_{k}:i\in[r],k\geq\delta_{i,1}\right\\}$.
Assuming that $Q_{0},Q_{1},\dots,Q_{r-2}$ are pairwise distinct, define
operators $\widetilde{\mathsf{H}}^{a}_{k}$, for any $a\in[r]$ and any $k\geq
1$:

(22)
$\displaystyle\widetilde{\mathsf{H}}^{r}_{k}:=\widetilde{\mathsf{W}}^{1}_{k}+\sum_{a\neq
r}\sum_{\ell=0}^{k-1}\frac{((-1)^{r}\Lambda)^{\ell+1}\left(-e_{1}(\mathbf{P})-e_{1}(\mathbf{Q})-Q_{a-1}\right)^{\ell}}{\prod_{a^{\prime}\neq
a,r}\left(Q_{a^{\prime}-1}-Q_{a-1}\right)}\sum_{i=1}^{r}(-Q_{a-1})^{r-i}\widetilde{\mathsf{W}}^{i}_{k-1-\ell}\,$
.
$\displaystyle\widetilde{\mathsf{H}}^{a}_{k}:=-\sum_{\ell=0}^{k-1}\frac{((-1)^{r}\Lambda)^{\ell+1}\left(-e_{1}(\mathbf{P})-e_{1}(\mathbf{Q})-Q_{a-1}\right)^{\ell}}{\prod_{a^{\prime}\neq
a,r}\left(Q_{a^{\prime}-1}-Q_{a-1}\right)}\sum_{i=1}^{r}(-Q_{a-1})^{r-i}\widetilde{\mathsf{W}}^{i}_{k-1-\ell},\
\text{ for }a\in[r-1]$ .

These operators $\widetilde{\mathsf{H}}^{a}_{k}$ are diagonal in the following
sense.

######  Lemma 3.8.

For any $1\leq a\leq r$ and $k\geq 1$, we have

$\widetilde{\mathsf{H}}^{a}_{k}=\hbar\partial_{x^{a}_{k}}+O(\hbar^{2})\,.$

###### Proof.

First notice that $\pi_{0}(\mathsf{J}^{a}_{k})=0$ for all $k>0$. Lemma 3.5
implies that $\mathsf{W}^{i}_{k}$ can be written as a linear combination of
operators associated with colored paths and the only paths in (17) that
contribute in $\hbar$-degree zero are paths that do not have any steps of
negative increment. Since $k\geq 0$, the only such path appears when $k=0$ and
it is the path that runs horizontally from $(0,0)$ to $(i,0)$. Therefore (17)
implies that

$\mathsf{W}^{i}_{k}=\delta_{k,0}\sum_{f\in\mathcal{F}_{<}([i],[r])}\big{(}\mathsf{J}^{f(1)}_{0}-\hbar\mathfrak{b}(i-1)\big{)}\cdots\big{(}\mathsf{J}^{f(i-1)}_{0}-\hbar\mathfrak{b}\big{)}\mathsf{J}^{f(i)}_{0}+O(\hbar)=\delta_{k,0}\mathsf{V}^{i}+O(\hbar),$

which further implies that $\pi_{0}(\widetilde{\mathsf{W}}^{i}_{k})=0$.
Similarly, the only paths in (17) that contribute in $\hbar$-degree one are
paths that have precisely one step of negative increment. Moreover,
$\pi_{0}(\mathsf{J}^{a}_{k})=0$ for all $k<0$ except for $a=r$ and $k=-1$.
Therefore the only paths in (17) that contribute in $\hbar$-degree one are
paths that have precisely one step of negative increment – either they finish
with a step from $(i-1,-1)$ to $(i,0)$ colored by $r$, or they have precisely
one step of negative increment and all the other steps are flat. Thus, (17)
implies that the corresponding $\hbar$-degree one part is given by

$\pi_{1}(\widetilde{\mathsf{W}}^{i}_{k})=\frac{(-1)^{r}}{\Lambda}\sum_{a=1}^{r-1}\hbar\partial_{x^{a}_{k+1}}\pi_{0}(e_{i-2}(\mathsf{J}^{1}_{0},\dots,\hat{\mathsf{J}^{a}_{0}},\dots,\mathsf{J}^{r-1}_{0}))+\sum_{a=1}^{r}\hbar\partial_{x^{a}_{k}}\pi_{0}(e_{i-1}(\mathsf{J}^{1}_{0},\dots,\hat{\mathsf{J}^{a}_{0}},\dots,\mathsf{J}^{r}_{0}))\delta_{k>0},$

for any $i\in[r]$ and $k\geq 0$. In generating series form, we can express
this formula as

$\sum_{i=1}^{r}u^{r-i}\pi_{1}(\widetilde{\mathsf{W}}^{i}_{k})=\frac{(-1)^{r}}{\Lambda}\sum_{a=1}^{r-1}\hbar\partial_{x^{a}_{k+1}}\prod_{a^{\prime}\neq
a,r}(u+\pi_{0}(\mathsf{J}^{a^{\prime}}_{0}))+\sum_{a=1}^{r}\hbar\partial_{x^{a}_{k}}\prod_{a^{\prime}\neq
a}(u+\pi_{0}(\mathsf{J}^{a^{\prime}}_{0}))\delta_{k>0}\,.$

For any $a\neq r$ we can invert it by dividing by
$\prod_{a^{\prime}=1}^{r}(u+\pi_{0}(\mathsf{J}^{a^{\prime}}_{0}))$ and taking
the residue at $u=-\pi_{0}(\mathsf{J}^{a}_{0})$, which for $k\geq 0$ gives

(23)
$\frac{-(-1)^{r}\Lambda\pi_{0}(\mathsf{J}^{r}_{0}-\mathsf{J}^{a}_{0})}{\prod_{a^{\prime}\neq
a}\pi_{0}(\mathsf{J}^{a^{\prime}}_{0}-\mathsf{J}^{a}_{0})}\sum_{i=1}^{r}(-\pi_{0}(\mathsf{J}^{a}_{0}))^{r-i}\pi_{1}(\widetilde{\mathsf{W}}^{i}_{k})=\hbar\partial_{x^{a}_{k+1}}+(-1)^{r}\Lambda\pi_{0}(\mathsf{J}^{r}_{0}-\mathsf{J}^{a}_{0})\hbar\partial_{x^{a}_{k}}\delta_{k>0}.$

We complete diagonalization by removing the term
$(-1)^{r}\Lambda\hbar\partial_{x^{a}_{k}}\pi_{0}(\mathsf{J}^{r}_{0}-\mathsf{J}^{a}_{0})\delta_{k>0}$
on the RHS of (23), which leads to the definition of
$\widetilde{\mathsf{H}}^{a}_{k}$ for $a\neq r$ by comparing with (20). When
$a=r$, and $k\geq 1$ we have
$\mathsf{J}^{r}_{k}=\widetilde{\mathsf{W}}^{1}_{k}-\sum_{a=1}^{r-1}\mathsf{J}^{a^{\prime}}_{k}$
for any $k\geq 1$, hence we define
$\widetilde{\mathsf{H}}^{r}_{k}=\widetilde{\mathsf{W}}^{1}_{k}-\sum_{a^{\prime}=1}^{r-1}\widetilde{\mathsf{H}}^{a^{\prime}}_{k}$.
∎

Finally, we are ready to construct an Airy structure.

######  Proposition 3.9.

Assume that $Q_{0},Q_{1},\dots,Q_{r-2}$ are pairwise distinct. Then, the ideal
$\mathcal{I}$ generated by the set of operators

$\widetilde{\mathsf{W}}^{i}_{k}\qquad\text{ for
}i\in[r]\,,\,k\geq\delta_{i,1}$

is an Airy structure.

###### Proof.

We will work with the diagonalized set of operators
$\widetilde{\mathsf{H}}^{a}_{k}$ as defined in (22). The set of operators

(24) $\left\\{\widetilde{\mathsf{H}}^{a}_{k}:a\in[r]\,,k\geq 1\right\\}$

also generate the ideal $\mathcal{I}$ as the change of basis (22) is
invertible.

We need to check the conditions $(a)-(d)$ to be an Airy structure from
Definition 3.1.

  1. (a)

The operators $\widetilde{\mathsf{W}}^{i}_{k}$ are bounded by [BCJ22, Lemma
2.43], which proves that any set of non-negative modes of a vertex algebra
obtained from a representation of a Heisenberg VOA is bounded. The operators
$\widetilde{\mathsf{H}}^{a}_{k}$ are also bounded as they are finite linear
combinations of the $\widetilde{\mathsf{W}}^{i}_{k}$.

  2. (b)

As we mentioned at the start of the proof, the ideal $\mathcal{I}$ is
generated by the set of operators (24).

  3. (c)

$\widetilde{\mathsf{H}}^{a}_{k}=\hbar\partial_{x^{a}_{k}}+O(\hbar^{2})$ by
Lemma 3.8.

  4. (d)

Lemma 3.6 shows that the ideal generated by the set of operators
$\left\\{\mathsf{W}^{i}_{k}:i\in[r],k\geq 0\right\\}$ satisfies condition
$(d)$ to be an Airy structure. In fact, it also allows us to shift the subset
of operators $\\{\mathsf{W}^{i}_{0}:i\in[r]\\}$ to get the subset
$\\{\widetilde{\mathsf{W}}^{i}_{0}:i\in[r]\\}$, and the ideal $\mathcal{I}$
generated by the resulting operators will still satisfy condition $(d)$. (Note
that $\widetilde{\mathsf{W}}^{1}_{0}=0$, and hence we may disregard it.) ∎

While we have constructed an Airy structure, the partition function associated
to it can easily seen to be trivial. As we know that there is a unique
solution by Theorem 3.3 all that needs to be done is to check that $Z=1$ is a
solution to the set of equations

$\mathsf{W}^{i}_{k}Z=\mathsf{V}^{i}\delta_{k,0}Z\qquad\text{ for
}i\in[r]\,,\,k\geq\delta_{i,1}\,.$

As this is a straightforward check, and we do not need this result in this
paper, we leave it as an exercise for the interested reader.

####  3.3.1. Shifted Airy structure

In order to get the partition function that we are interested in, we need to
shift the zero modes $\mathsf{W}^{i}_{0}$. More precisely, we have the
following theorem.

######  Theorem 3.10.

Assume that $Q_{0},Q_{1},\dots,Q_{r-2}$ are pairwise distinct. Then, the ideal
$\mathcal{I}^{L}$ generated by the set of operators

(25)
$\mathsf{W}^{i}_{k}-\mathsf{V}^{i}\delta_{k,0}-L_{i}\delta_{k,0}\qquad\text{
for }i\in[r]\,,\,k\geq\delta_{i,1}\,,$

where $\left(L_{i}\right)_{i\in[2..r]}$ are formal commuting variables of
$\deg_{\hbar}L_{i}=0$, is a shifted Airy structure.

###### Proof.

We know from Proposition 3.9, that the ideal generated by the operators
$\mathsf{W}^{i}_{k}-\mathsf{V}^{i}\delta_{k,0}$ is an Airy structure. So all
we need to check to get a shifted Airy structure is that the condition
$[\mathcal{I}^{L},\mathcal{I}^{L}]\subseteq\hbar^{2}\mathcal{I}^{L}$ is
satisfied. Lemma 3.6 does the job, as it says that the modes
$\mathsf{W}^{i}_{0}$ can be shifted freely without changing the commutation
relations. ∎

Recall that by Theorem 3.3 of Airy structures, the shifted Airy structure
$\mathcal{I}^{L}$ has a unique partition function, say $\mathcal{Z}^{L}$ of
the following form:

$\mathcal{Z}^{L}=\exp\left(\sum_{\begin{subarray}{c}g\in\frac{1}{2}\mathbb{Z}_{\geq
0},n\in\mathbb{Z}_{>0}\end{subarray}}\hbar^{2g-2+n}F_{g,n}\right)\,$

where the $F_{g,n}$, _a priori_ , are elements of
$\mathbb{C}(\\!(\mathbf{P},\mathbf{Q},\Lambda)\\!)(\mathfrak{b})[\mathbf{x}]\llbracket
L_{2},\ldots,L_{r}\rrbracket$, and in this case the partition function
$\mathcal{Z}^{L}$ will indeed be non-trivial.

We need to study the properties of the partition function $\mathcal{Z}^{L}$,
especially the dependence of the $F_{g,n}$ on the $L_{i}$, as we will want to
specialize the $L_{i}$ to functions of $\mathbf{P}$ and $\mathfrak{b}$ later
on (see (29)). Let us define the expansion coefficients of $F_{g,n}$ as
follows:

$F_{g,n}=\frac{1}{n!}\sum_{\begin{subarray}{c}a_{1},\ldots,a_{n}\in[r]\\\
k_{1},\ldots,k_{n}\in\mathbb{Z}_{>0}\end{subarray}}F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}\prod_{i=1}^{n}x^{a_{i}}_{k_{i}}.$

such that $F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}$ is an element of
$\mathbb{C}(\\!(\mathbf{P},\mathbf{Q},\Lambda)\\!)(\mathfrak{b})\llbracket
L_{2},\ldots,L_{r}\rrbracket$. We can show that the
$F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}$ actually live in
$\mathbb{Q}(\mathbf{Q})[\mathbf{P},\mathfrak{b},\Lambda][L_{2},\ldots,L_{r}]$:

######  Proposition 3.11.

The $F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}$ of the partition function depend
polynomially on $\Lambda,\mathfrak{b},\mathbf{P}$ and $L_{2},\cdots,L_{r}$ and
rationally on $\mathbf{Q}$. More precisely, we have

$\frac{F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}}{\Lambda^{k_{1}+\cdots+k_{n}}}\in\mathbb{Q}(\mathbf{Q})[\mathbf{P},\mathfrak{b}][L_{2},\ldots,L_{r}].$

###### Proof.

In this proof we will denote by $(I|A)$ a tuple of two ordered sets
$I=\\{i_{1},\dots,i_{d}\\}\subset\mathbb{Z}^{d}_{>0}$, and
$A=\\{a_{1},\dots,a_{d}\\}\subset[r]^{d}$ of the same cardinality
$|(I|A)|:=d$. For such a tuple we denote
$\mathbf{x}^{A}_{I}:=\prod_{j=1}^{|A|}x_{i_{j}}^{a_{j}}$, and
$\partial_{\mathbf{x}^{A}_{I}}:=\prod_{j=1}^{|A|}\partial_{x_{i_{j}}^{a_{j}}}$.
Equations (17), (20), and (21) imply that

$\Lambda\widetilde{\mathsf{W}}_{k}^{i}=\sum_{d+h=1}^{i}\sum_{|(I_{1}|A_{1})\sqcup(I_{2}|A_{2})|=d}\hbar^{h}K_{k,h}^{i}\big{[}\begin{smallmatrix}{A}_{1}&{A}_{2}\\\
{I}_{1}&{I}_{2}\end{smallmatrix}\big{]}\cdot\hbar^{d}\cdot\mathbf{x}^{A_{1}}_{I_{1}}\cdot\partial_{\mathbf{x}^{A_{2}}_{I_{2}}},$

where $K_{k,h}^{i}\big{[}\begin{smallmatrix}{A}_{1}&{A}_{2}\\\
{I}_{1}&{I}_{2}\end{smallmatrix}\big{]}\in\mathbb{Q}[\mathbf{P},\mathbf{Q},\mathfrak{b},\Lambda]$
vanish unless
$\sum_{i^{\prime\prime}\in{I}_{2}}i^{\prime\prime}-\sum_{i^{\prime}\in{I}_{1}}i^{\prime}\leq
k+1$. Therefore (22) and Lemma 3.8 further imply that

(26)
$\hbar\partial_{x_{k}^{a}}=\widetilde{\mathsf{H}}_{k}^{a}+\sum_{d+h=2}^{r}\sum_{|(I_{1}|A_{1})\sqcup(I_{2}|A_{2})|=d}\hbar^{h}U_{k,h}^{a}\big{[}\begin{smallmatrix}{A}_{1}&{A}_{2}\\\
{I}_{1}&{I}_{2}\end{smallmatrix}\big{]}\cdot\hbar^{d}\cdot\mathbf{x}^{A_{1}}_{I_{1}}\cdot\partial_{\mathbf{x}^{A_{2}}_{I_{2}}},$

where $U_{k,h}^{a}\big{[}\begin{smallmatrix}{A}_{1}&{A}_{2}\\\
{I}_{1}&{I}_{2}\end{smallmatrix}\big{]}\in\mathbb{Q}(\mathbf{Q})[\mathbf{P},\mathfrak{b},\Lambda]$
vanish unless
$\sum_{i^{\prime\prime}\in{I}_{2}}i^{\prime\prime}-\sum_{i^{\prime}\in{I}_{1}}i^{\prime}\leq
k$. Moreover (22) together with the shifted Airy structure (25) gives that

(27)
$\frac{1}{\mathcal{Z}^{L}}\widetilde{\mathsf{H}}_{k}^{a}\mathcal{Z}^{L}=\widetilde{U}_{k}^{a}\in\Lambda^{k}\cdot\mathbb{Q}(\mathbf{Q})[\mathbf{P},L_{2},\dots,L_{r}]$

for any $a\in[r],k\geq 1$, where the coefficients $\widetilde{U}_{k}^{a}$ can
be written explicitly from (22).

Note that we can extract the coefficients of $\mathcal{Z}^{L}$ as follows

$F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}=[\hbar^{2g-1+n}]\left(\prod_{l=2}^{n}\partial_{x_{k_{l}}^{a_{l}}}\frac{1}{\mathcal{Z}^{L}}(\hbar\partial_{x_{k_{1}}^{a_{1}}})\mathcal{Z}^{L}\right)\Bigg{|}_{\mathbf{x}=0}.$

Applying (26) with $k=k_{1},a=a_{1}$, and comparing it with (27), we obtain
the following identity

$\displaystyle F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}=\delta_{n,1}\delta_{g,0}\widetilde{U}_{k_{1}}^{a_{1}}+$
$\displaystyle\sum_{d+h=2}^{r}\sum_{l=1}^{d}\sum_{\begin{subarray}{c}|(I|A)\sqcup(I_{1}|A_{1})\sqcup\cdots\sqcup(I_{l}|A_{l})|=d,\\\
|(I_{1}|A_{1})|,\cdots,|(I_{l}|A_{l})|\geq 1\end{subarray}}\cdot$ (28)
$\displaystyle\sum_{\begin{subarray}{c}(I^{\prime}_{1}|A^{\prime}_{1})\sqcup\cdots\sqcup(I^{\prime}_{l}|A^{\prime}_{l})\sqcup({I}|{A})=\\\
=(\\{k_{2},\dots,k_{n}\\}|\\{a_{2},\dots,a_{n}\\})\end{subarray}}$
$\displaystyle\sum_{\begin{subarray}{c}g_{1},\cdots,g_{l}\in\frac{1}{2}\mathbb{Z}_{>0},\\\
2\sum_{m}(g_{m}-1+|{I}_{m}|)=2g-h\end{subarray}}U_{k_{1},h}^{a_{1}}\big{[}\begin{smallmatrix}{A}&{A}_{1}\cdots{A}_{l}\\\
{I}&{I}_{1}\cdots{I}_{l}\end{smallmatrix}\big{]}\prod_{m=1}^{l}F_{g_{m},|I^{\prime}_{m}|+|{I}_{m}|}\big{[}\begin{smallmatrix}A^{\prime}_{m}&{A}_{m}\\\
I^{\prime}_{m}&{I}_{m}\end{smallmatrix}\big{]}.$

While the formula above looks complicated (it follows by applying the Leibnitz
rule repeatedly and we leave its verification to the interested reader), we
only need the following property: the RHS is a finite combination over
$\mathbb{Q}(\mathbf{Q})[\mathbf{P},\mathfrak{b},\Lambda][L_{2},\ldots,L_{r}]$
of products of terms of the form
$F_{g^{\prime},n^{\prime}}\big{[}\begin{smallmatrix}a^{\prime}_{1}&\cdots&a^{\prime}_{n^{\prime}}\\\
k^{\prime}_{1}&\cdots&k^{\prime}_{n^{\prime}}\end{smallmatrix}\big{]}$ with
$2g^{\prime}-2+n^{\prime}\leq 2g-2+n$ and
$k^{\prime}_{1}+\cdots+k^{\prime}_{n^{\prime}}\leq k_{1}+\cdots+k_{n}$.
Indeed, the condition
$\sum_{i\in{I}_{m}}i+\sum_{i^{\prime}\in{I}^{\prime}_{m}}i^{\prime}\leq
k_{1}+\cdots+k_{n}$ follows from the vanishing condition for
$U_{k_{1},h}^{a_{1}}\big{[}\begin{smallmatrix}{A}&A_{1}\cdots A_{l}\\\
{I}&I_{1}\cdots I_{l}\end{smallmatrix}\big{]}$. Similarly,

$2g_{m}-2+|I^{\prime}_{m}|+|I_{m}|\leq
2\sum_{m}(g_{m}-1+|I_{m}|)+|I^{\prime}_{m}|-|I_{m}|=2g-h+|I^{\prime}_{m}|-|I_{m}|\leq
2g-2+n,$

and the equality holds if and only if $h=0$,
$I^{\prime}_{m}=\\{k_{2},\dots,k_{n}\\}$ (thus, $I=\emptyset$), $|I_{m}|=1$
and $g_{m}=g$. Since $d\geq 2$ these conditions imply that $l\geq 2$,
therefore the vanishing condition for
$U_{k_{1},h}^{a_{1}}\big{[}\begin{smallmatrix}{A}&A_{1}\cdots A_{l}\\\
{I}&I_{1}\cdots I_{l}\end{smallmatrix}\big{]}$ implies that $I_{m}=\\{y\\}$
with $y<k_{1}$. Thus, we conclude that for
$F_{g^{\prime},n^{\prime}}\big{[}\begin{smallmatrix}a^{\prime}_{1}&\cdots&a^{\prime}_{n^{\prime}}\\\
k^{\prime}_{1}&\cdots&k^{\prime}_{n^{\prime}}\end{smallmatrix}\big{]}$ with
$2g^{\prime}-2+n^{\prime}=2g-2+n$ we necessarily have a strict inequality
$k^{\prime}_{1}+\cdots+k^{\prime}_{n^{\prime}}<k_{1}+\cdots+k_{n}$.
Consequently, (28) can be used to compute the
$F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}$ recursively on
$K=k_{1}+\cdots+k_{n}$ and then on $\chi=2g-2+n$ in the lexicographic order
and the claim $F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}\in\mathbb{Q}(\mathbf{Q})[\mathbf{P},\mathfrak{b},\Lambda][L_{2},\ldots,L_{r}]$
follows inductively.

To prove the specific form of the dependence on $\Lambda$ let us
simultaneously rescale $x^{a}_{k}\mapsto c^{-k}\,x^{a}_{k}$ and
$\Lambda\mapsto c\,\Lambda$ for some $c\in\mathbb{C}^{*}$, which corresponds
to $\mathsf{J}_{k}^{a}\mapsto c^{k}\mathsf{J}_{k}^{a}$ and
$\widetilde{\mathsf{W}}^{i}_{k}\mapsto c^{k}\,\widetilde{\mathsf{W}}^{i}_{k}$.
Then, as (25) remains unchanged under this rescaling, so does the solution
$\mathcal{Z}^{L}$. Thus, every
$F_{g,n}\big{[}\begin{smallmatrix}a_{1}&a_{2}&\cdots&a_{n}\\\
k_{1}&k_{2}&\cdots&k_{n}\end{smallmatrix}\big{]}$ is a monomial in $\Lambda$
of degree $k_{1}+\cdots+k_{n}$, which finishes the proof. ∎

As there are only a finite number of positive integers $k_{1},\dots,k_{n}$
with the fixed sum $k_{1}+\cdots+k_{n}=k$, we have the following corollary of
Proposition 3.11.

######  Corollary 3.12.

The expansion coefficients $F_{g,n}$ of the partition function
$\mathcal{Z}^{L}$ are elements of the ring
$\mathbb{Q}(\mathbf{Q})[\mathbf{P},\mathfrak{b},L_{2},\ldots,L_{r},\mathbf{x}]\llbracket\Lambda\rrbracket$.

##  4\. $b$-Hurwitz numbers from the Airy structure

In this section, we continue the study of the partition function
$\mathcal{Z}^{L}$ of the shifted Airy structure. After performing various
substitutions, we will prove that $\mathcal{Z}^{L}$ reduces to the generating
function $\tau^{(\mathfrak{b})}_{G}$ of weighted $b$-Hurwitz numbers for an
arbitrary rational weight $G(z)$. As a consequence, we will prove a stronger
set of constraints for $\tau^{(\mathfrak{b})}_{G}$, from which the cut-and-
join equation derived in Theorem 2.7 follows.

###  4.1. Reduction of the Airy structure

Recall that the partition function $\mathcal{Z}^{L}$ depends on the parameters
$\mathbf{P},\mathbf{Q},\Lambda,\mathfrak{b},\mathbf{x}$ and the
$\left(L_{i}\right)_{i\in[2..r]}$. We will prove that $\mathcal{Z}^{L}$
reduces to $\tau^{(\mathfrak{b})}_{G}$ for the weight
$G(z)=\frac{\prod_{i=1}^{r}(P_{i}+z)}{\prod_{i=1}^{r-2}(Q_{i}-z)}$ after the
following substitutions:

S1:

set $Q_{0}=0$,

S2:

set all the variables $x^{a}_{k}=0$ unless $a=1$, i.e.,

$x^{a}_{k}=0,\qquad a\in[2..r],\quad k\in\mathbb{Z}_{\geq 1},$

S3:

for any $i\in[2..r]$, we set the $L_{i}$ to be the following:

(29) $L_{i}=\sum_{1\leq n_{1}<\dots<n_{i}\leq
r}\prod_{m=1}^{i}\big{(}-P_{r+1-n_{m}}-\hbar\mathfrak{b}(r-n_{m}+m-1)\big{)}-\mathsf{V}^{i}.$

When $\mathfrak{b}=0$, it is known [BDBKS20] that these $G$-weighted Hurwitz
numbers can be obtained by topological recursion on a certain spectral curve
of degree $r$ (see Section 5.1 for the explicit form of the curve). This
perspective of topological recursion provides the following motivation for the
$\mathfrak{b}=0$ part of the substitutions S1–S3.

  * S1:

The mode $\mathsf{J}_{0}^{a}$ is related to the residue of $\omega_{0,1}$ at
the $a$-th preimage of $x^{-1}(\infty)=(0,Q_{1},..,Q_{r-2},\infty)$ from which
we expect that $Q_{0}=0$. As the only condition on the Airy structure is that
the $Q_{0},\ldots,Q_{r-2}$ are pairwise disjoint, we may impose $Q_{0}=0$, as
long as we assume that $Q_{i}\in\mathbb{C}^{*}$ for $i\in[r-2]$.

  * S2:

For the spectral curve given in Section 5.1, the expansion of the correlators
$\omega_{g,n}(z_{1},..,z_{n})$ at $z_{i}=0$ produces the corresponding
weighted Hurwitz numbers. On the other hand, [BCU] (see Theorem 5.1) proves
that the expansion of $\omega_{g,n}(z_{1},..,z_{n})$ at $z_{i}=0$ is related
to the coefficients $F_{g,n}\big{[}\begin{smallmatrix}1&\cdots&1\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}$ of $\mathcal{Z}^{L}$. In other
words, the part of $\mathcal{Z}^{L}$ that survives after setting $x_{k}^{a}=0$
for all $k\geq 1,a\in[2..r]$ gives Hurwitz numbers.

  * S3:

The choice of $L_{i}$ when $\mathfrak{b}=0$ is again inspired by [BDBKS20] and
[BCU], in particular the form of the (global) loop equations.

The above observations do not help determine the $\mathfrak{b}$-dependence of
$L_{i}$, because refinements of the results of [BDBKS20, BCU] for arbitrary
$\mathfrak{b}$ are still under investigation. The $\mathfrak{b}$-corrections
are dictated by the form of the cut-and-join equation from Theorem 2.7 (see
Section 4.3.2 for more details).

It is worth emphasizing that the substitution S3 (29) for the $L_{i}$ is
allowed as the $F_{g,n}$ is proved to be a polynomial in $L_{i}$ in Corollary
3.12. Thus, after substitution, we will not have arbitrarily negative powers
in the $\mathbf{P}$ and $\mathbf{Q}$, and the resulting $F_{g,n}$ will belong
to
$\mathbb{Q}(\mathbf{Q})[\mathbf{P},\mathfrak{b},\mathbf{x}]\llbracket\Lambda\rrbracket$.
We will denote the function obtained after the substitutions S1 and S3 simply
by $\mathcal{Z}$, without a superscript. Then, $\mathcal{Z}$ has the form

$\mathcal{Z}=\exp\left(\sum_{\begin{subarray}{c}g\in\frac{1}{2}\mathbb{Z}_{\geq
0},n\in\mathbb{Z}_{>0}\end{subarray}}\hbar^{2g-2+n}F_{g,n}\right)\,$

where $F_{g,n}$ is an element of
$\mathbb{Q}(\mathbf{Q})[\mathbf{P},\mathfrak{b},\mathbf{x}]\llbracket\Lambda\rrbracket$,
and satisfies the constraints

(30)
$\mathsf{W}^{i}_{k}\mathcal{Z}=\Omega_{i}\delta_{k,0}\mathcal{Z}\qquad\text{
for }i\in[r]\,,\,k\geq 0\,,$

where we define the $\Omega_{i}$ for $i\in[r]$ as

(31) $\Omega_{i}:=\sum_{1\leq n_{1}<\dots<n_{i}\leq
r}\prod_{m=1}^{i}\big{(}-P_{r+1-n_{m}}-\mathfrak{b}\hbar(r-n_{m}+m-1)\big{)}\,.$

Note that $\mathsf{W}^{1}_{0}=\sum_{a=1}^{r}\mathsf{J}_{0}^{a}=\Omega_{1}$,
and hence we have included the (trivial) constraint for $\mathsf{W}^{1}_{0}$
in (30). It will also be convenient in the following to define
$\Omega_{0}:=1$. We will refer to the set of equations (30) as
$\mathcal{W}$-constraints. We will denote the function obtained after all the
substitutions S1–S3 by $\mathcal{Z}^{\operatorname{red}}$, and refer to it as
the reduced partition function.

######  Remark 4.1.

It is important to note that the state $\mathcal{Z}$ satisfying (30) is not
the highest weight vector for the representation of
${\mathcal{W}}^{\mathsf{k}}(\mathfrak{gl}_{r})$ that we are considering in
this paper. Indeed, the highest weight vector in this representation is the
partition function $1$ of the Airy structure $\mathcal{I}$ of Proposition 3.9.
This vector $\mathcal{Z}$ is obtained by a shift of the zero modes
$\mathsf{W}^{i}_{0}$ and we will refer to it as a Whittaker vector (extending
the terminology used in [SV13, BBCC21]).

###  4.2. Constraints for $\mathcal{Z}^{\operatorname{red}}$

The goal of this section is to find a set of constraints for the reduced
partition function $\mathcal{Z}^{\operatorname{red}}$ which will determine it
uniquely. Naively, one might want to apply the substitutions S1–S3 directly to
the operators $\mathsf{W}^{i}_{k}$ appearing in the $\mathcal{W}$-constraints
(30). However, the resulting operators will not annihilate the reduced
partition function $\mathcal{Z}^{\operatorname{red}}$. Indeed, the recursive
process of solving the $\mathcal{W}$-constraints (30) to obtain $\mathcal{Z}$
mixes together the various $a$, and hence, cannot directly be reduced to a
recursive formula purely in terms of the
$F_{g,n}\big{[}\begin{smallmatrix}1&\cdots&1\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}$.

Instead, we find a sequence of different operators $\widehat{D}^{r}_{k}$,
defined as certain combinations of the $\mathsf{W}^{i}_{k}$, that annihilate
$\mathcal{Z}$. These operators $\widehat{D}^{r}_{k}$ are constructed such that
the substitution S2 commutes with the action of $\widehat{D}^{r}_{k}$ on
$\mathcal{Z}$. Then the operators $D_{k}$, defined as the reduction of
$\widehat{D}^{r}_{k}$ under substitution S2, annihilate
$\mathcal{Z}^{\operatorname{red}}$, and are closely related to the cut-and-
join equation of Theorem 2.7.

####  4.2.1. The $\widehat{D}$-operators

As before, we fix an integer $r\geq 2$. Let us begin by defining certain
differential operators $\widehat{D}^{i}_{k}$. For $i=0$, we set
$\widehat{D}^{0}_{k}:=0$ for any $k\in\mathbb{Z}$. Then we recursively define,
for any $i\in[r]$ and $k\in\mathbb{Z}$,

(32)
$\widehat{D}^{i}_{k}:=(-1)^{i+1}(\mathsf{W}^{i}_{k}-\delta_{k,0}\Omega_{i})+\sum_{\begin{subarray}{c}k_{2}\geq
0\\\
k_{1}+k_{2}=k\end{subarray}}\left(\mathsf{J}^{1}_{k_{1}}-\delta_{k_{1},0}\hbar\mathfrak{b}(k_{2}+i-1)\right)\widehat{D}^{i-1}_{k_{2}}\,.$

From the definition of $\Omega_{1}$, it is easy to see that
$\widehat{D}^{1}_{0}=0$.

######  Lemma 4.2.

We have the following constraints for the function $\mathcal{Z}$:

$\widehat{D}^{i}_{k}\mathcal{Z}=0\qquad\text{ for }i\in[r]\,,\,k\geq 0\,.$

###### Proof.

The statement of the lemma follows directly from the the
$\mathcal{W}$-constraints of equation (30), and the definition (32) of the
operators $\widehat{D}^{i}_{k}$. ∎

In order to prove certain properties of the operators $\widehat{D}^{i}_{k}$,
we will find a combinatorial interpretation for them in terms of weighted
lattice paths. We will use the combinatorial formula for the operators
$\mathsf{W}^{i}_{k}$ in terms of weighted colored paths proved in Lemma 3.5.
Recall the notion of bridges from Definition 2.6. Here is a combinatorial
formula for the $\widehat{D}$-operators, which is illustrated in Fig. 2.

Figure 2. Left: A typical path contributing to the operator
$\widehat{D}^{i}_{k}$ from the last sum in (33). Right: A path contributing to
the operator $\widehat{D}^{r}_{k}$ from the last sum in (33) that doesn’t
vanish after substituting $x^{a}_{k}=0$ for all $a\in[r],k\in\mathbb{Z}_{\geq
1}$. Here the asssociated weight after the substitution is equal to
$\mathsf{J}^{1}_{2}\mathsf{J}^{1}_{-1}\mathsf{J}^{1}_{2}(Q_{f(1)-1}-\mathfrak{b}\hbar(f(1)+j-3)\cdots(Q_{f(j)-1}-\mathfrak{b}\hbar(f(j-1)-1)(-1)^{r+1}\Lambda^{-1}$.

######  Lemma 4.3.

For any $i\in[r]$ and $k\geq 0$, we have the following combinatorial
interpretation of the $\widehat{D}$-operators:

$\displaystyle\widehat{D}^{i}_{k}$
$\displaystyle=(-1)^{i+1}\sum_{\begin{subarray}{c}\gamma\in\Gamma_{(0,k)\to(i,0)},\\\
f\in\mathcal{F}_{<}([i],[2..r])\end{subarray}}\operatorname{wt}(\gamma,f)+\sum_{j=0}^{i}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(i-j,0)}}(-1)^{j}\Omega_{j}\prod_{u=0}^{i-j}\operatorname{wt}_{u}(\gamma\cup\gamma_{0},\mathbf{1})$
(33)
$\displaystyle+\sum_{j=1}^{i-1}\sum_{n=1}^{\infty}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(i-j,-n)}}\sum_{\begin{subarray}{c}\gamma^{\prime}\in\Gamma_{(i-j,-n)\to(i,0)},\\\
f\in\mathcal{F}_{<}([j],[2..r])\end{subarray}}\operatorname{wt}(\gamma\cup\gamma^{\prime},\tilde{f})\,,$

where $\tilde{f}$ is the coloring of $\gamma\cup\gamma^{\prime}$ that colors
each step of $\gamma$ by $1$, and the $\ell$-th step of $\gamma^{\prime}$ by
$f(\ell)$, and the coloring $\mathbf{1}$ colors each step of the associated
path by $1$. The path $\gamma_{0}$ is a path that starts at $(i-j,0)$ and runs
along the $X$-axis till $(i,0)$ for any $0\leq j\leq i$, and
$\operatorname{wt}_{0}(\gamma,f):=1$ by convention.

###### Proof.

We will prove the lemma by induction on $i$. When $i=1$, and $k\geq 1$ we see
that (33) reduces to the combinatorial formula (17) for the operators
$\mathsf{W}^{1}_{k}$. As for the case $i=1$ and $k=0$, recall that
$\Omega_{0}=1$, and we directly obtain
$\widehat{D}^{1}_{0}=\sum_{a=1}^{r}\mathsf{J}^{a}_{0}-\Omega_{1}=0$.

Now assume that (33) holds for all $\widehat{D}^{\ell}_{k}$, such that
$\ell\leq i-1$ and all $k\geq 0$. Then, by plugging in the formula (19) for
the $\mathsf{W}^{i}_{k}$ into the definition (32) of the
$\widehat{D}^{i}_{k}$, and using the inductive hypothesis for
$\widehat{D}^{i-1}_{k_{2}}$, we get the following formula:

$\widehat{D}^{i}_{k}=(-1)^{i+1}\sum_{\begin{subarray}{c}\gamma\in\Gamma_{(0,k)\to(i,0)},\\\
f\in\mathcal{F}_{<}([i],[2..r])\end{subarray}}\operatorname{wt}(\gamma,f)+(-1)^{i}\delta_{k,0}\Omega_{i}+(-1)^{i+1}\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\
k_{1}<0\end{subarray}}\left(\mathsf{J}^{1}_{k_{1}}-\delta_{k_{1},0}\hbar\mathfrak{b}(k_{2}+i-1)\right)\cdot\\\
\cdot\sum_{\begin{subarray}{c}\gamma\in\Gamma_{(0,k_{2})\to(i-1,0)},\\\
f\in\mathcal{F}_{<}([i-1],[2..r])\end{subarray}}\operatorname{wt}(\gamma,f)+\sum_{\begin{subarray}{c}k_{2}\geq
0\\\
k_{1}+k_{2}=k\end{subarray}}\left(\mathsf{J}^{1}_{k_{1}}-\delta_{k_{1},0}\hbar\mathfrak{b}(k_{2}+i-1)\right)\cdot\left(\sum_{j=1}^{i-1}(-1)^{j}\Omega_{j}\cdot\right.\\\
\cdot\left.\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k_{2})\to(i-1-j,0)}}\prod_{u=0}^{i-1-j}\operatorname{wt}_{u}(\gamma\cup\gamma_{0},\mathbf{1})+\sum_{j=1}^{i-2}\sum_{n=1}^{\infty}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k_{2})\to(i\\!-\\!1\\!-\\!j,\\!-\\!n)}}\sum_{\begin{subarray}{c}\gamma^{\prime}\in\Gamma_{(i\\!-\\!1\\!-\\!j,\\!-\\!n)\to(i,0)},\\\
f\in\mathcal{F}_{<}([j],[2..r])\end{subarray}}(\\!-\\!1)^{j}\operatorname{wt}(\gamma\\!\cup\\!\gamma^{\prime},\tilde{f})\right),$

where $\tilde{f}$ is the coloring that colors all the steps of $\gamma$ by $1$
and the $l$-th step of $\gamma^{\prime}$ by $f(l)$ and the path $\gamma_{0}$
is the path that starts from $(i-1-j,0)$ and runs along the $X$-axis until
$(i-1,0)$. The factor
$\left(\mathsf{J}^{1}_{k_{1}}-\delta_{k_{1},0}\hbar\alpha_{0}(k_{2}+i-1)\right)$
is the weight associated to the first step of a path that starts from $(0,k)$
and ends at $(i,0)$, such that its first step has increment $-k_{1}$, and
hence we will concatenate this step with the paths that appear to the right
(after shifting those paths by one unit in the positive $X$-direction) to
obtain the following expression:

$\widehat{D}^{i}_{k}=(-1)^{i+1}\sum_{\begin{subarray}{c}\gamma\in\Gamma_{(0,k)\to(i,0)},\\\
f\in\mathcal{F}_{<}([i],[2..r])\end{subarray}}\operatorname{wt}(\gamma,f)+\sum_{j=1}^{i}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(i-j,0)}}(-1)^{j}\Omega_{j}\prod_{u=0}^{i-j}\operatorname{wt}_{u}(\gamma\cup\gamma_{0},\mathbf{1})\\\
+\sum_{n=1}^{\infty}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(1,\\!-\\!n)}}\sum_{\begin{subarray}{c}\gamma^{\prime}\in\Gamma_{(1,\\!-\\!n)\to(i,0)},\\\
f\in\mathcal{F}_{<}([i\\!-\\!1],[2..r])\end{subarray}}(-1)^{i-1}\operatorname{wt}(\gamma\cup\gamma^{\prime},\tilde{f})+\\\
+\sum_{j=1}^{i-2}\sum_{n=1}^{\infty}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(i-j,-n)}}\sum_{\begin{subarray}{c}\gamma^{\prime}\in\Gamma_{(i\\!-\\!j,\\!-\\!n)\to(i,0)},\\\
f\in\mathcal{F}_{<}([j],[2..r])\end{subarray}}(-1)^{j}\operatorname{wt}(\gamma\cup\gamma^{\prime},\tilde{f}),$

where $\gamma_{0}$ is now the path along the $X$-axis from $(i-j,0)$ to
$(i,0)$. We have combined the term $(-1)^{i}\delta_{k,0}\Omega_{i}$ with the
sum over the rest of the $\Omega_{j}$ to get the above expression. The sum
over $j$ in the last line, can be expanded to include $j=i-1$, which absorbs
the second to last line into the last line, and the resulting expression
matches (33), which concludes the proof. ∎

####  4.2.2. The operators $D_{k}$

Recall that one of the motivations for defining the new
$\widehat{D}$-operators is precisely to be able to do the substitutions
$(x^{a}_{\ell})_{a\in[2..r],\ell\geq 1}=0$ and obtain operators that
annihilate the reduced partition function $\mathcal{Z}^{\operatorname{red}}$.
For this purpose, we define new operators $D_{k}$ as follows. For any $k\geq
0$, define

(34) $D_{k}:=\Lambda\,\left(\widehat{D}^{r}_{k}\big{|}_{\,x^{a}_{\ell}=0\text{
for }a\in[2..r]\,,\,\ell\geq 1}\right).$

The operators $D_{k}$ depend on the integer $r\geq 2$, but we omit this for
notational simplicity.

We show that the operators $D_{k}$ annihilate the reduced partition function
$\mathcal{Z}^{\operatorname{red}}$. During the course of the proof, we also
prove an explicit combinatorial formula for $D_{k}$.

######  Proposition 4.4.

Consider the operators $D_{k}$ defined in (34).

  1. (a)

The reduced partition function $\mathcal{Z}^{\operatorname{red}}$ satisfies
the following constraints,

$D_{k}\mathcal{Z}^{\operatorname{red}}=0\qquad\text{ for }k\geq 0\,.$

  2. (b)

The operators $D_{k}$ take the following form for any $k\geq 0$:

(35) $D_{k}=\Lambda\sum_{j=0}^{r}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r\\!-\\!j,0)}}\prod_{u=0}^{r-j}\operatorname{wt}_{u}(\gamma\\!\cup\\!\gamma_{0},\mathbf{1})\sum_{1\leq
n_{1}<\cdots<n_{j}\\!\leq
r}\prod_{m=1}^{j}\big{(}P_{r\\!-\\!n_{m}\\!+\\!1}\\!+\\!\mathfrak{b}\hbar(r\\!-\\!n_{m}\\!+\\!m\\!-\\!1)\big{)}+\\\
+(-1)^{r+1}\sum_{j=0}^{r-1}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r\\!-\\!j,\\!-\\!1)}}\prod_{u=0}^{r-j}\operatorname{wt}_{u}(\gamma\\!\cup\\!\gamma_{0},\mathbf{1})\sum_{1\leq
n_{1}<\cdots<n_{j\\!-\\!1}\leq
r\\!-\\!2}\prod_{m=1}^{j}\big{(}\\!-\\!Q_{r\\!-\\!n_{m}\\!-\\!1}+\mathfrak{b}\hbar(r\\!-\\!n_{m}\\!+\\!m\\!-\\!2)\big{)},$

where $\gamma_{0}$ is the path that runs along the $X$-axis from
$(r-j,0)\to(r,0)$, and $\gamma_{-1}$ is the path that is horizontal from
$(r-j,-1)$ to $(r-1,-1)$ and then goes up to $(r,0)$.

###### Proof.

We will prove that for all $k\geq 0$,

(36)
$D_{k}\mathcal{Z}^{\operatorname{red}}=\left(\widehat{D}^{r}_{k}\mathcal{Z}\right)\bigg{|}_{x^{a}_{\ell}=0\text{
for }a\in[2..r]\,,\,\ell\geq 1},$

which implies part (a) of the proposition. Indeed, Lemma 4.2 specialized to
the case $i=r$ states that $\widehat{D}^{r}_{k}\mathcal{Z}=0$ for all $k\geq
0$.

Notice first that the reduction of the modes for $a\neq 1$ does not affect the
action of the $\mathsf{J}^{1}_{k}$, i.e., we have

$\left(\mathsf{J}^{1}_{k_{1}}\cdots\mathsf{J}^{1}_{k_{n}}\mathcal{Z}\right)\bigg{|}_{x^{a}_{\ell}=0\text{
for }a\in[2..r]\,,\,\ell\geq
1}=\mathsf{J}^{1}_{k_{1}}\cdots\mathsf{J}^{1}_{k_{n}}\mathcal{Z}^{\operatorname{red}}\,.$

With this in mind, let us look at the three terms appearing in formula (33)
for $\widehat{D}^{r}_{k}$, which could contribute to the RHS of (36). The
first term in (33) vanishes identically when $i=r$. Only paths where the
coloring is always $1$ appear in the second term, and hence survives in its
entirety.

Let us analyze the third term now, which involves paths
$\gamma^{\prime}\in\Gamma_{(j,-n)\to(r,0)}$ colored by $[2..r]$. As $n\geq 1$,
all these paths $\gamma^{\prime}$ start strictly below the $X$-axis, and as
they end on the $X$-axis, they must touch the $X$-axis at some point. The
segment that goes up and touches the $X$-axis will then contribute the weight
$\mathsf{J}^{a}_{-\ell}$ for some $a\in[2..r]$ and $\ell\geq 1$. Unless $a=r$
and $\ell=1$, in which case
$\mathsf{J}^{r}_{-1}=x^{r}_{1}+\frac{(-1)^{r}}{\Lambda}$ (see (20)), all these
terms vanish under the substitution $x^{a}_{\ell}=0$ for $a\in[2..r],\ell\geq
1$. This means that the only paths $\gamma^{\prime}$ that survive are such
that $n=1$, are horizontal from $(r-j,-1)$ to $(r-1,-1)$, and the last step
$\gamma^{\prime}_{r}$ is colored by $r$ and connects $(r-1,-1)$ to $(r,0)$
(such a path is illustrated on the right side in Fig. 2). The weights of these
surviving paths are polynomials in the $\mathsf{J}^{1}_{\ell}$ and hence are
not affected by the substitution, as noted before. This proves (36), and hence
part (a) of the lemma.

As for part (b) of the proposition, the preceding analysis gives all the terms
in $\widehat{D}^{r}_{k}$ that survive after the reduction $x^{a}_{\ell}=0$.
Collecting them together, we get

$\frac{D_{k}}{\Lambda}=\sum_{j=0}^{r}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r\\!-\\!j,0)}}(\\!-\\!1)^{j}\Omega_{j}\prod_{u=0}^{r-j}\operatorname{wt}_{u}(\gamma\cup\gamma_{0},\mathbf{1})+\sum_{j=1}^{r-1}\sum_{\begin{subarray}{c}\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r\\!-\\!j,\\!-\\!1)},\\\
f\in\mathcal{F}_{<}([j\\!-\\!1],[2..r\\!-\\!1]\end{subarray}}(-1)^{j}\operatorname{wt}(\gamma\cup\gamma_{-1},\tilde{f}_{-1})\bigg{|}_{x^{r}_{1}=0}\,,$

where $\gamma_{-1}$ is the path that is horizontal from $(r-j,-1)$ to
$(r-1,-1)$ and then goes up to $(r,0)$, and $\tilde{f}_{-1}$ is the coloring
that colors all the steps of $\gamma$ by $1$, the $\ell$-th step of
$\gamma_{-1}$ by $f(\ell)$ for any $\ell\in[j-1]$, and the last step of
$\gamma_{-1}$ by $r$. Now, we can extract the contributions coming from
$\gamma_{-1}$ explicitly, which gives

$D_{k}=\Lambda\sum_{j=0}^{r}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r-j,0)}}(-1)^{j}\Omega_{j}\prod_{u=0}^{r-j}\operatorname{wt}_{u}(\gamma\cup\gamma_{0},\mathbf{1})+(-1)^{r+1}\sum_{j=1}^{r-1}\cdot\\\
\cdot\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r-j,-1)}}\left(\prod_{u=0}^{r-j}\operatorname{wt}_{u}(\gamma\cup\gamma_{-1},\mathbf{1})\right)\sum_{1\leq
n_{1}<\cdots<n_{j-1}\leq
r-2}\prod_{m=1}^{j-1}\left(-Q_{r-n_{m}-1}+\hbar\mathfrak{b}(r+m-n_{m}-2)\right)\,,$

where we have set $n_{m}=r-f(j-m)$. Finally we plug in the expression (31) for
the $\Omega_{j}$, and $\Omega_{0}=1$ to get the expression (35) stated in part
(b) of the proposition. ∎

Having proved that the operators $D_{k}$ annihilate the reduced partition
function $\mathcal{Z}^{\operatorname{red}}$, we would like to understand them
better. We will find a simpler expression for the $D_{k}$ which will help us
compare these operators with the cut-and-join operators. Before that, we need
to state a technical lemma which will be useful in the proof of the
simplification.

######  Lemma 4.5.

We have the following identity between polynomials in
$u_{1},\dots,u_{\ell},w_{1},\dots,w_{\ell}$:

(37) $\prod_{k=1}^{\ell}(u_{k}+w_{k})=\sum_{I\sqcup J=[\ell]}\prod_{i\in
I}(u_{i}+|J\cap[i]|)\prod_{j\in J}(w_{j}-(|J\cap[j-1]|))\,.$

###### Proof.

We will prove the lemma by induction on $\ell$. For $\ell=1$, the statement is
obvious. Let $F_{\ell}(\mathbf{u},\mathbf{w})$ denote the polynomial that
appears on the RHS of (37). For any $\ell>1$,
$F_{\ell}(\mathbf{u},\mathbf{w})$ can be split into two parts, depending on
whether $1$ appears in $I$ or $J$:

$u_{1}\sum_{I\sqcup J=[2..\ell]}\prod_{i\in I}(u_{i}+|J\cap[i]|)\prod_{j\in
J}(w_{j}-(|J\cap[j-1]|))+\\\ +w_{1}\sum_{I\sqcup J=[2..\ell]}\prod_{i\in
I}(u_{i}+1+|J\cap[i]|)\prod_{j\in J}(w_{j}-(1+|J\cap[j-1]|))=\\\
=u_{1}F_{\ell-1}(\mathbf{u}^{\prime},\mathbf{w}^{\prime})+w_{1}F_{\ell-1}(\mathbf{u}^{\prime\prime},\mathbf{w}^{\prime\prime}),$

where we define $u_{i}^{\prime}=u_{i+1}$, $u_{i}^{\prime\prime}=u_{i+1}+1$,
$w_{i}^{\prime}=w_{i+1}$, $w_{i}^{\prime\prime}=w_{i+1}-1$ for $i\in[\ell-1]$.
Then, using the inductive hypothesis for $F_{\ell-1}$, we get

$F_{\ell}(\mathbf{u},\mathbf{w})=u_{1}\prod_{k=2}^{\ell}(u_{k}+w_{k})+w_{1}\prod_{k=2}^{\ell}(u_{k}+w_{k})=\prod_{k=1}^{\ell}(u_{k}+w_{k}),$

which completes the proof. ∎

Now, we prove a simplified formula for the operators $D_{k}$ in terms of
weighted paths with the weight $\widetilde{\operatorname{wt}}$ (defined in
(12)) instead of the weight $\operatorname{wt}$.

######  Lemma 4.6.

For any $k\geq 0$, we have the following combinatorial formula for the
operators $D_{k}$:

(38) $D_{k}=\Lambda\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r,0)}}\widetilde{\operatorname{wt}}\left(\gamma|\left(\mathbf{P}\right)\right)+(-1)^{r-1}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r-1,-1)}}\widetilde{\operatorname{wt}}\left(\gamma|\left(-\mathbf{Q},0\right)\right),$

where we define $\left(\mathbf{P}\right):=\left(P_{1},\ldots,P_{r}\right)$ and
$\left(-\mathbf{Q},0\right):=\left(-Q_{1},\ldots,-Q_{r-2},0\right)$.

###### Proof.

Recall equation (35) for the operators $D_{k}$. Let us consider the first term
that depends on $\mathbf{P}$. Consider a path $\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r,0)}$, and let $1\leq n_{1}<\cdots<n_{m}\leq r$ denote the
positions of the horizontal steps in $\gamma$, i.e., $\gamma$ has a segment
$s_{n}=(1,0)$ if and only if $n\in\\{n_{1},\ldots,n_{m}\\}$. Let $s_{n_{i}}$
connect $(n_{i}-1,y_{i})$ to $(n_{i},y_{i})$. For any subset $J\subset[m]$, we
define the path $\gamma_{J}\in\Gamma^{\geq 0}_{(0,k)\to(r-|J|,0)}$ as the path
obtained from $\gamma$ by removing all the horizontal steps $s_{n}$ for any
$n=n_{j}$ with $j\in J$. Then, we claim that the associated weights coincide:

(39)
$\widetilde{\operatorname{wt}}\left(\gamma|\left(\mathbf{P}\right)^{\operatorname{op}}\right)=\sum_{J\subset[m]}\prod_{\ell=1}^{r-|J|}\operatorname{wt}_{\ell}(\gamma_{J}\cup\gamma_{0},\mathbf{1})\prod_{j\in
J}\left(P_{r+1-n_{j}}+\hbar\mathfrak{b}\left(r-n_{j}+\left|J\cap[j-1]\right|\right)\right),$

where $\gamma_{0}$ is the path running along the $X$-axis from $(r-j,0)$ to
$(r,0)$, and we use the notation
$\left(\mathbf{P}\right)^{\operatorname{op}}=\left(P_{r},P_{r-1},\ldots,P_{1}\right)$.
Notice that the contributions from the weights $\operatorname{wt}$ and
$\widetilde{\operatorname{wt}}$ coincide for non-horizontal steps. Thus, in
order to prove the claim, it suffices to prove the contributions to (39) for
horizontal steps. Hence, (39) is equivalent to the following equation:

$\frac{\prod_{i=1}^{m}\left(P_{r+1-n_{i}}\\!-\\!\hbar\mathfrak{b}y_{i})\right)}{(-\hbar\mathfrak{b})^{m}}=\sum_{I\sqcup
J=[m]}\prod_{i\in I}\big{(}y_{i}+r-n_{i}+|J\cap[i]|\big{)}\prod_{j\in
J}\left(\frac{-P_{r+1-n_{j}}}{\hbar\mathfrak{b}}\\!-\\!r\\!+\\!n_{j}\\!-\\!\left|J\cap[j-1]\right|\right),$

which is precisely the content of Lemma 4.5 with the choice
$w_{j}=\frac{-P_{r+1-n_{j}}}{\hbar\mathfrak{b}}-r+n_{j}$ and
$u_{i}=y_{i}+r-n_{i}$.

Finally, by considering all possible choices of $m$, we get a bijection
between the set of paths $\gamma\in\Gamma^{\geq 0}_{(0,k)\to(r,0)}$ that
appears in (38), and the set of paths $\gamma_{J}\in\Gamma^{\geq
0}_{(0,k)\to(r-j,0)}$ along with the choice of an integer $0\leq j\leq r$ and
integers $1\leq n_{1}<\cdots<n_{j}\leq r$ appearing in the first term of (35).
Thus, summing (39) over all paths $\gamma\in\Gamma^{\geq 0}_{(0,k)\to(r,0)}$
and shifting $n_{j}\to r+1-n_{j}$ on the LHS, gives that the sum
$\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r,0)}}\widetilde{\operatorname{wt}}\left(\gamma|\left(\mathbf{P}\right)\right)$
is equal to

$\sum_{j=0}^{r}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(r-j,0)}}\prod_{u=1}^{r-j}\operatorname{wt}_{u}(\gamma\cup\gamma_{0},\mathbf{1})\sum_{1\leq
n_{1}<\dots<n_{j}\leq
r}\prod_{m=1}^{j}\big{(}P_{r+1-n_{m}}+\hbar\mathfrak{b}(r-n_{m}+m-1)\big{)}.$

A completely analogous argument works for the second term in (38) involving
the $\mathbf{Q}$, which proves the lemma. ∎

###  4.3. Relation to $b$-Hurwitz numbers

By comparing the explicit form of the operators $D_{k}$ with the cut-and-join
equation of Theorem 2.7, we can prove one of our main results – the function
$\mathcal{Z}^{\operatorname{red}}$ matches the tau function
$\tau^{(\mathfrak{b})}_{G}$ of weighted $b$-Hurwitz numbers with the weight
$G(z)=\frac{\prod_{i=1}^{r}(P_{i}+z)}{\prod_{i=1}^{r-2}(Q_{i}-z)}$, after an
identification between the variables $x^{1}_{k}$ and $\tilde{p}_{k}$. By
taking appropriate limits, we can obtain the weighted $b$-Hurwitz generating
function for arbitrary rational weights.

####  4.3.1. $\mathcal{Z}^{\operatorname{red}}$ as a $b$-Hurwitz generating
function

We have the following theorem.

######  Theorem 4.7.

We have the following equality between the reduced partition function
$\mathcal{Z}^{\operatorname{red}}$ and the weighted b-Hurwitz tau function
$\tau_{G}^{(\mathfrak{b})}$, for weight
$G(z)=\frac{\prod_{i=1}^{r}(P_{i}+z)}{\prod_{i=1}^{r-2}(Q_{i}-z)}$:

$\tau^{(\mathfrak{b})}_{G}=\mathcal{Z}^{\operatorname{red}}\big{|}_{x^{1}_{k}=\frac{\tilde{p}_{k}}{k},\,\Lambda=t}\,.$

###### Proof.

Let us prove that the function
$\mathcal{Z}^{\operatorname{red}}\big{|}_{x^{1}_{k}=\frac{\tilde{p}_{k}}{k}}$
is a solution to the cut-and-join equation derived in Theorem 2.7. Indeed,
consider the operator

$C=\sum_{k\geq 0}\mathsf{J}^{1}_{-k-1}D_{k}.$

Note that $\mathsf{J}^{1}_{-k-1}$ is the weight of a segment that starts at
$(-1,-1)$ and ends at $(0,k)$. We can concatenate this segment with the paths
that appear in (38) for the operators $D_{k}$, and shift this combined path
one step to the right to obtain the following formula for the operator $C$:

$C=\Lambda\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(r+1,0)}}\widetilde{\operatorname{wt}}\left(\gamma|\left(0,\mathbf{P}\right)\right)+(-1)^{r-1}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,-1)\to(r,-1)}}\widetilde{\operatorname{wt}}\left(\gamma|\left(0,-\mathbf{Q},0\right)\right),$

where we denote $\left(0,\mathbf{P}\right):=\left(0,P_{1},\ldots,P_{r}\right)$
and $\left(0,-\mathbf{Q},0\right):=\left(0,-Q_{1},\ldots,-Q_{r-2},0\right)$.
Part (a) of Proposition 4.4 implies that
$C\mathcal{Z}^{\operatorname{red}}=0$. In this form, it is easy to recognize
$C$ as the cut-and-join operator of Theorem 2.7 after the substitutions
$x^{1}_{k}=\frac{\tilde{p}_{k}}{k}$ for $k>0$ and $\Lambda=t$.

Moreover, Corollary 3.12 gives that
$\mathcal{Z}^{\operatorname{red}}\big{|}_{x^{1}_{k}=\frac{\tilde{p}_{k}}{k},\Lambda=t}$
is an element of
$\mathbb{Q}(\mathbf{Q})[\mathbf{P},\tilde{\mathbf{p}},\mathfrak{b}](\\!(\hbar)\\!)\llbracket
t\rrbracket$ of the form $1+O(t)$. Such a solution of the cut-and-join
equation is proved to be unique in Theorem 2.7, and hence we get the statement
of the theorem.∎

####  4.3.2. Arbitrary rational weights

In this section, we explain how one can get constraints for
$\tau^{(\mathfrak{b})}_{G}$ for an arbitrary rational weight, which we fix to
be
$G(z)=\frac{\prod_{i=1}^{n}\left(P_{i}+z\right)}{\prod_{i=1}^{m}\left(Q_{i}-z\right)}$
(where $m$ is not necessarily equal to $n-2$) by taking appropriate limits of
the parameters $\mathbf{Q},\mathbf{P}$.

######  Theorem 4.8.

The generating function $\tau^{(\mathfrak{b})}_{G}$ of $b$-Hurwitz numbers
weighted by
$G(z)=\frac{\prod_{i=1}^{n}\left(P_{i}+z\right)}{\prod_{i=1}^{m}\left(Q_{i}-z\right)}$,
for any integers $n,m\geq 0$, can be obtained from the reduced partition
function $\mathcal{Z}^{\operatorname{red}}$ for $r=\max(m+2,n)$ as

$\lim_{\begin{subarray}{c}Q_{m+1},\ldots,Q_{r-2}\to\infty,\\\
P_{n+1},\ldots,P_{r}\to\infty\end{subarray}}\mathcal{Z}^{\operatorname{red}}\big{|}_{\Lambda=t\frac{Q_{m+1}\cdots
Q_{r-2}}{P_{n+1}\cdots P_{r}}\,,\,x^{1}_{k}=\frac{\tilde{p}_{k}}{k}}.$

In addition, $\tau^{(\mathfrak{b})}_{G}$ satisfies the following constraints

(40) $t\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(n,0)}}\widetilde{\operatorname{wt}}\left(\gamma|\left(\mathbf{P}\right)\right)\tau^{(\mathfrak{b})}_{G}+(-1)^{m+1}\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(m+1,-1)}}\widetilde{\operatorname{wt}}\left(\gamma|\left(-\mathbf{Q},0\right)\right)\tau^{(\mathfrak{b})}_{G}=0$

for any $k\geq 0$, where we denote
$\left(\mathbf{P}\right)=\left(P_{1},\dots,P_{n}\right)$ and
$\left(-\mathbf{Q},0\right)=\left(-Q_{1},\dots,-Q_{m},0\right)$.

###### Proof.

This theorem follows from (11). Indeed, let $\check{G}(z)$ be the rational
function associated with $G$ as in Remark 2.4, and let
$G^{\prime}(z):=\frac{\prod_{i=1}^{r}\left(P_{i}+z\right)}{\prod_{i=1}^{r-2}\left(Q_{i}-z\right)}$,
and $\check{G}^{\prime}(z)$ be the associated function. Let
$\mathbf{P}_{r}:=(\mathbf{P},P_{n+1},\dots,P_{r})$ and
$\mathbf{Q}_{r-2}:=(\mathbf{Q},Q_{m+1},\dots,Q_{r-2})$. Then

$\tau^{(\mathfrak{b})}_{G}(t,\mathbf{P},\mathbf{Q})=\tau^{(\mathfrak{b})}_{\check{G}}(t\cdot\frac{P_{1}\cdots
P_{n}}{Q_{1}\cdots Q_{m}},\mathbf{P}^{-1},\mathbf{Q}^{-1})=\\\
=\tau^{(\mathfrak{b})}_{\check{G}^{\prime}}\left(t\cdot\frac{P_{1}\cdots
P_{n}}{Q_{1}\cdots
Q_{m}},\mathbf{P}_{r}^{-1},\mathbf{Q}_{r-2}^{-1}\right)\bigg{|}_{\begin{subarray}{c}Q^{-1}_{m+1},\ldots,Q^{-1}_{r-2}=0,\\\
P^{-1}_{n+1},\ldots,P^{-1}_{r}=0\end{subarray}}=\lim_{\begin{subarray}{c}Q_{m+1},\ldots,Q_{r-2}\to\infty,\\\
P_{n+1},\ldots,P_{r}\to\infty\end{subarray}}\tau^{(\mathfrak{b})}_{G^{\prime}}\left(t\frac{Q_{m+1}\cdots
Q_{r-2}}{P_{n+1}\cdots P_{r}},\mathbf{P}_{r},\mathbf{Q}_{r-2}\right)$

which is equal to
$\lim_{\begin{subarray}{c}Q_{m+1},\ldots,Q_{r-2}\to\infty,\\\
P_{n+1},\ldots,P_{r}\to\infty\end{subarray}}\mathcal{Z}^{\operatorname{red}}\big{|}_{\Lambda=t\frac{Q_{m+1}\cdots
Q_{r-2}}{P_{n+1}\cdots P_{r}}\,,\,x^{1}_{k}=\frac{\tilde{p}_{k}}{k}}$ by
Theorem 4.7. Similarly, $\tau^{(\mathfrak{b})}_{G}$ satisfies (40) if and only
if $\tau^{(\mathfrak{b})}_{\check{G}}$ satisfies (40) after the change of
variables as in Remark 2.4. It was shown in the proof of Theorem 2.7 that the
latter is equal to

$\bigg{[}t\sum_{i=0}^{n}e_{i}(\mathbf{P})\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(i,0)}}\widetilde{\operatorname{wt}}(\gamma\,|0)-\sum_{i=0}^{m}e_{i}(-\mathbf{Q})\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(i+1,-1)}}\widetilde{\operatorname{wt}}(\gamma\,|0)\bigg{]}\tau^{(\mathfrak{b})}_{\check{G}}=\\\
=\bigg{[}t\sum_{i=0}^{r}e_{i}(\mathbf{P}_{r})\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(i,0)}}\widetilde{\operatorname{wt}}(\gamma\,|0)-\sum_{i=0}^{r-2}e_{i}(-\mathbf{Q}_{r-2})\sum_{\gamma\in\Gamma^{\geq
0}_{(0,k)\to(i+1,-1)}}\widetilde{\operatorname{wt}}(\gamma\,|0)\tau^{(\mathfrak{b})}_{\check{G}^{\prime}}\bigg{]}\bigg{|}_{\begin{subarray}{c}Q_{m+1},\ldots,Q_{r-2}=0,\\\
P_{n+1},\ldots,P_{r}=0\end{subarray}}$

and we conclude as before by applying Theorem 4.7, and the expression (38) for
the $D_{k}$. ∎

This is a vast generalization of recent results that were limited to the cases
$(n,m)=(0,1),(1,0),(2,0)$ [BCD23] and $(n,m)=(3,0)$ [BN23]. Notice that
Theorem 4.8 proves a stronger set of constraint on the generating function
$\tau^{(\mathfrak{b})}_{G}$ as compared to Theorem 2.7. Indeed multiplying
equation (40) on the left by $\mathsf{J}^{1}_{-k-1}$ and summing over
$k\in\mathbb{Z}_{\geq 0}$ gives the cut-and-join equation proved in Theorem
2.7.

######  Remark 4.9.

The case of $G(z)=\frac{1}{\prod_{i=1}^{r-1}(Q_{i}-z)}$ can be treated without
taking limits as in Theorem 4.8. Instead, we could consider the representation
of $\mathcal{W}^{\mathsf{k}}(\mathfrak{gl}_{r})$ given by

$\begin{split}\mathsf{J}^{a}_{k}=\left\\{\begin{array}[]{lr}\hbar\partial_{x^{a}_{k}}&k>0,\\\
\hbar(-k)x^{a}_{-k}&k<0,\\\
Q_{a-1}-\hbar\mathfrak{b}(a-1)&a\in[r],k=0\end{array}\right.\end{split}$

with $Q_{0}=0$ as before. Then the Whittaker vector $\widehat{\mathcal{Z}}$
(whose existence and uniqueness follows from a slight modification of the Airy
structure of [BBCC21]) satisfying

$\mathsf{W}^{i}_{k}\widehat{\mathcal{Z}}=\delta_{i,r}\delta_{k,1}t\widehat{\mathcal{Z}}\quad
i\in[r],k\geq 1\,,$

is such that its reduction $\widehat{\mathcal{Z}}^{\operatorname{red}}$ (under
the substitutions S1,S2) matches the generating function
$\tau^{(\mathfrak{b})}_{G}$ on the nose. We do not provide a proof, but the
interested reader can follow the combinatorial methods of the previous
sections to derive this result. We mention this result as this Whittaker
vector is a slight variant of the Gaiotto state that appears in the AGT
correspondence [AGT10, SV13, MO19, BBCC21].

##  5\. Topological recursion for weighted Hurwitz numbers

In this section we prove that, when $\mathfrak{b}=0$, rationally weighted
Hurwitz numbers can be computed using the Eynard–Orantin topological recursion
[EO07] on an associated spectral curve. This provides a totally different
proof of one of the main theorems of [BDBKS20] using $\mathcal{W}$-algebra
representations. Our proof relies on the weighted $b$-Hurwitz interpretation
of the Airy structure partition function $\mathcal{Z}$ proved in Section 4.
Throughout this section, we set $\mathfrak{b}=0$.

###  5.1. A spectral curve for $\mathcal{Z}$

Let us give a very brief introduction to the topological recursion (TR)
formalism. TR takes as input data a spectral curve
$(\Sigma,x,\omega_{0,1},\omega_{0,2})$, where $\Sigma$ is a Riemann surface,
$x$ is a meromorphic function on $\Sigma$, $\omega_{0,1}$ is a meromorphic
differential on $\Sigma$ and $\omega_{0,2}$ is a fundamental bidifferential on
$\Sigma\times\Sigma$ (i.e., a symmetric meromorphic bidifferential, whose only
poles consist of a double pole on the diagonal with biresidue 1). From a
spectral curve, TR produces symmetric meromorphic differentials
$\left(\omega_{g,n}\right)_{g\in\mathbb{Z}_{\geq 0}\,,\,n\in\mathbb{Z}_{\geq
1}}$ called correlators on $\Sigma^{\times n}$, in the range $2g-2+n>0$. For
the explicit topological recursion formula, various nice properties satisfied
by the correlators and their appearance in many different contexts in
geometry, see for instance [Eyn14].

We are interested in the curve
$\Sigma\subset\mathbb{P}^{1}_{x}\times\mathbb{P}^{1}_{y}$, cut out by the
equation

(41)
$\sum_{i=0}^{r}(-y)^{r-i}\left(\frac{e_{i}(P_{1},\dots,P_{r})}{(-x)^{i}}+(-1)^{r}\frac{e_{i-1}(Q_{0},Q_{1},\dots,Q_{r-2})}{\Lambda\,x^{i-1}}\right)=0,$