# Elements of spin Hurwitz theory: closed algebraic formulas, blobbed
topological recursion, and a proof of the Giacchetto–Kramer–Lewański
conjecture

Alexander Alexandrov  Center for Geometry and Physics, Institute for Basic
Science (IBS), Pohang 37673, Korea<EMAIL_ADDRESS>and  Sergey Shadrin
Korteweg-de Vriesinstituut voor Wiskunde, Universiteit van Amsterdam, Postbus
94248, 1090GE Amsterdam, Nederland<EMAIL_ADDRESS>
###### Abstract.

In this paper, we discuss the properties of the generating functions of spin
Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary
ramification profiles, we construct the weighed sums which are given by
Orlov’s hypergeometric solutions of the 2-component BKP hierarchy. We derive
the closed algebraic formulas for the correlation functions associated with
these tau-functions, and under reasonable analytical assumptions we prove the
loop equations (the blobbed topological recursion). Finally, we prove a
version of topological recursion for the spin Hurwitz numbers with the spin
completed cycles (a generalized version of the Giacchetto–Kramer–Lewański
conjecture).

###### 2020 Mathematics Subject Classification:

37K10, 14N10, 81R10, 33C80

###### Contents

  1. 1 Introduction
    1. 1.1 Topological recursion and integrability
    2. 1.2 BKP and spin Hurwitz theory
    3. 1.3 Giacchetto–Kramer–Lewański conjecture and its generalization
    4. 1.4 Notation
    5. 1.5 Organization of the paper
    6. 1.6 Acknowledgments
  2. 2 Neutral fermions and boson-fermion correspondence
    1. 2.1 Neutral fermions
    2. 2.2 Vertex operators
    3. 2.3 Boson-fermion correspondence
  3. 3 Hypergeometric tau-functions and weighted spin Hurwitz numbers
    1. 3.1 Hypergeometric tau-functions of 2-BKP hierarchy
    2. 3.2 Spin Hurwitz numbers
    3. 3.3 From spin Hurwitz numbers to 2-BKP hierarchy
    4. 3.4 Weighted spin Hurwitz numbers
    5. 3.5 From BKP to KP
  4. 4 Diagonal group element and $n$-point functions
    1. 4.1 Operators $\mathbb{J}_{k}$
    2. 4.2 Topological expansion
    3. 4.3 Preliminary formulas for $H_{n}^{\bullet}$ and $W_{n}^{\bullet}$
    4. 4.4 Closed algebraic formula for $W_{g,n}$
    5. 4.5 Special cases
      1. 4.5.1 Stable terms for $n=1$
  5. 5 Loop equations
    1. 5.1 Assumptions
    2. 5.2 Blobbed topological recursion
    3. 5.3 Proof of Theorem 5.1
  6. 6 Formulas for $H_{g,n}$
  7. 7 Topological recursion for spin Hurwitz number with completed cycles
    1. 7.1 Topological recursion in the odd situation
    2. 7.2 Quasi-polynomiality
    3. 7.3 Giacchetto–Kramer–Lewański conjecture and its generalization

##  1\. Introduction

###  1.1. Topological recursion and integrability

It is well known that the Chekhov–Eynard–Orantin topological recursion [EO07]
is closely related to integrability. However, the details of a general
relationship between the two phenomena remain unclear. Although topological
recursion is believed to be a universal property for a huge class of
enumerative geometry and physics problems, the proofs of its validity are
often model-dependent and technically involved, and at this point, despite the
lack of understanding of the general relationship, various universal
properties of integrability often help to prove topological recursion.

One of the most general applications of integrability to topological recursion
is given by the weighted Hurwitz numbers. The generating functions of the
weighted Hurwitz numbers are hypergeometric tau-functions of the $2$-component
KP ($2$-KP) hierarchy. The study of topological recursion for the general
hypergeometric solutions of the $2$-KP hierarchy was initiated in [ACEH18,
ACEH20] (subsuming a huge list of particular examples known before). Many
elements of the general construction including quantum and classical spectral
curves were properly identified there. However, the topological recursion was
proved only for an infinite-dimensional family of solutions with polynomial
weight functions and finite sets of the second times of the $2$-KP hierarchy.
Topological recursion for the much more general families of the hypergeometric
solutions of the $2$-KP hierarchy was proved in [BDBKS22, BDBKS20]. The proof
there is based on the free field description of the KP hierarchy, more
specifically, on the free fermion construction and the boson-fermion
correspondence.

These results indicate that the same line of reasoning can be applied to any
integrable hierarchy with free fermion description. In this paper, we describe
topological recursion for the hypergeometric solutions of the $2$-component
BKP ($2$-BKP) hierarchy. A well-known neutral fermion description of the BKP
hierarchy allows us to follow the general approach for the $2$-KP hierarchy,
derived in [BDBKS22, BDBKS20]. Many steps can be repeated without essential
changes, but some specifics of the $2$-BKP case (mostly important, the built-
in oddness of the parametrizations) require extra analysis and lead to new
phenomena. We derive the general closed algebraic formulas for the correlation
functions in the $2$-BKP case and prove the blobbed topological recursion
[BS17], that is, the linear and quadratic loop equations.

###  1.2. BKP and spin Hurwitz theory

The BKP hierarchy is believed to govern the spin Hurwitz numbers in
essentially the same way as the KP hierarchy governs the ordinary Hurwitz
numbers [MMN20]. However, the important construction of the weighted spin
Hurwitz numbers (in the sense of [GPH17]) is still unavailable in the
literature. In this paper we show how to construct integrable generating
functions of spin Hurwitz numbers for arbitrary ramification profiles and
number of the branch points. These generating functions are Orlov’s
hypergeometric tau-functions of the $2$-component BKP hierarchy [Orl03], and
the weights associated with the ramifications serve as parameters. It is not
clear at the moment how to reduce naturally the number of parameters and to
define the direct analogs of weighted Hurwitz numbers in the spin case. To
this end, we suggest two possible candidates for the elementary weight
functions.

It is well known that the tau-functions of the KP and BKP hierarchies are
related to each other by a simple quadratic relation [DJKM82]. Following Orlov
[Orl03], we describe this relation for the hypergeometric tau-functions.
Namely, for any hypergeometric tau-function of the $2$-BKP hierarchy we find
the corresponding tau-function of the $2$-KP hierarchy. It is easy to see that
such KP tau-function is not unique. This relation between tau-functions should
lead to the non-trivial relations between the spin and ordinary Hurwitz
numbers.

###  1.3. Giacchetto–Kramer–Lewański conjecture and its generalization

Additional input and motivation to study the correlation functions of the
corresponding hypergeometric $2$-BKP tau-functions comes from a recent work of
Giacchetto, Kramer, and Lewański [GKL21]. They study in detail the theory of
so-called spin Hurwitz numbers with completed cycles, both single and double,
whose elements occur naturally in a number of other works in relation to
computation of the volumes of strata in the moduli spaces of holomorphic
differentials [EOP08] and Gromov–Witten theory of Kähler surfaces [Lee20,
Introduction].

Remarkably, Giacchetto, Kramer, and Lewański propose a conjectural statement
on $\mathbb{Z}_{2}$-equivariant version of topological recursion for the
correlation differentials of these numbers, and they prove that the statement
on topological recursion is equivalent to an ELSV-type formula for spin
Hurwitz numbers with completed cycles that expresses these numbers in terms of
the Chiodo classes twisted by the Witten $2$-spin class.

Using the formulas for the correlation functions and loop equations we prove a
natural generalization of the Giacchetto–Kramer–Lewański conjecture, that is,
a $\mathbb{Z}_{2}$-equivariant version of topological recursion for the double
spin Hurwitz numbers with arbitrary finite linear combinations of the spin
completed cycles.

Let us remark that while with the motivation coming from [GKL21] we focus on
this particular family of spin Hurwitz numbers, we expect that our
modification of the methods of [BDBKS22, BDBKS20] should immediately work for
other families of the generating functions of the spin Hurwitz numbers,
analogous to the families investigated in [BDBKS20]. We also expect that the
integrable approach to the topological recursion in the BKP case should be as
universal as for the KP case. Moreover, without significant modifications, it
should also work for other integrable hierarchies described by free fermions.

###  1.4. Notation

A partition $\lambda$ is strict, if
$\lambda_{1}>\lambda_{2}>\lambda_{3}>\dots>\lambda_{\ell(\lambda)}>\lambda_{\ell(\lambda)+1}=0$,
where $\ell(\lambda)$ is the length of the partition. We denote the set of
strict partitions, including the empty one, by $\operatorname{SP}$. A
partition $\lambda$ is odd if all parts in $\lambda$ are odd. We denote the
set of odd partitions, including the empty one, by $\operatorname{OP}$. For a
partition $\lambda$ by $\lambda(k)$ we denote the number of parts equal to
$k$.

###  1.5. Organization of the paper

In Section 2 we recall the neutral fermion description of the BKP hierarchy.
In Section 3 we explain how to construct the generating functions of the spin
Hurwitz numbers that solve the 2-BKP hierarchy and how these tau-functions can
be related to the generating functions of the ordinary Hurwitz numbers.
Section 4 is devoted to the correlation functions for the general
hypergeometric tau-functions of the 2-BKP hierarchy. In Section 5 we prove
that these correlation functions, under mild analytic assumptions, satisfy
linear and quadratic loop equations. In Section 6 we derive explicit
expressions for the $n$-point correlation functions. In Section 7 we use these
expressions to prove the topological recursion for the spin Hurwitz numbers
with the spin completed cycles.

###  1.6. Acknowledgments

The work of A. A. was supported by the Institute for Basic Science
(IBS-R003-D1). The work of S. S. was supported by the Netherlands Organization
for Scientific Research.

The authors thank A. Giacchetto, R. Kramer, and D. Lewański for useful
discussions and an anonymous referee for the suggested improvements.

##  2\. Neutral fermions and boson-fermion correspondence

In this section we remind the reader the neutral fermion formalism and boson-
fermion correspondence in the framework of the BKP hierarchy. More detailed
descriptions can be found in [DJKM82, You89, vdL95, Orl03].

###  2.1. Neutral fermions

Let $\phi_{k}$, $k\in{\mathbb{Z}}$, be the neutral free fermions satisfying
the canonical anticommutation relations

(2.1) $\left\\{\phi_{k},\phi_{m}\right\\}=(-1)^{k}\delta_{k+m,0}.$

Note that $\phi_{0}^{2}=1/2$. These relations define the Clifford algebra as
an associative algebra.

For the vacuum vector $\left|0\right>$ and the co-vacuum $\left<0\right|$,
satisfying

(2.2)
$\phi_{m}\left|0\right>=0,\,\,\,\,\,\left<0\right|\phi_{-m}=0,\,\,\,\,\,m<0$

the elements $\phi_{k_{1}}\phi_{k_{2}}\dots\phi_{k_{m}}\left|0\right>$ with
$k_{1}>k_{2}>\dots>k_{m}\geq 0$ form a basis of the neutral fermion Fock space
${\mathcal{F}}_{B}$,

(2.3)
${\mathcal{F}}_{B}=\operatorname{span}\left\\{\phi_{k_{1}}\phi_{k_{2}}\dots\phi_{k_{m}}\left|0\right>\
|\ k_{1}>k_{2}>\dots>k_{m}\geq 0\right\\},$

and its dual

(2.4)
${\mathcal{F}}_{B}^{*}=\operatorname{span}\left\\{\left<0\right|\phi_{k_{m}}\dots\phi_{k_{2}}\phi_{k_{1}}\
|\ k_{1}<k_{2}<\dots<k_{m}\leq 0\right\\}.$

The space ${\mathcal{F}}_{B}$ splits into two subspaces

(2.5) ${\mathcal{F}}_{B}={\mathcal{F}}_{B}^{0}\oplus{\mathcal{F}}_{B}^{1},$

where ${\mathcal{F}}_{B}^{0}$ and ${\mathcal{F}}_{B}^{1}$ denote the subspaces
with even and odd numbers of generators $\phi_{k}$, respectively. The same
decomposition exists for ${\mathcal{F}}_{B}^{*}$.

There is a nondegenerate bilinear pairing
${\mathcal{F}}_{B}\times{\mathcal{F}}_{B}^{*}\rightarrow{\mathbb{C}}$, and the
pairing of $\left<U\right|\in{\mathcal{F}}_{B}^{*}$ and
$\left|V\right>\in{\mathcal{F}}_{B}$ is denoted by $\left<U|V\right>$ with

(2.6) $\left<0|0\right>=1.$

The vacuum expectation values of an element $a$ of the Clifford algebra is a
pairing of $\left<0\right|$ and $a\left|0\right>$, which is denoted by
$\left<0\right|a\left|0\right>$. It is uniquely defined by the anticommutation
relations (2.1), property (2.2), and the following relation:

(2.7) $\left<0\right|\phi_{0}\left|0\right>=0.$

In particular, if $a$ is an odd element of the Clifford algebra, then
$\left<0\right|a\left|0\right>=0$. It is easy to see that the bases in (2.3)
and (2.4) are orthogonal. Let us focus on the space ${\mathcal{F}}_{B}^{0}$
and its dual. The basis can be labelled by strict partitions
$\lambda\in\operatorname{SP}$ in the following way:

(2.8)
$\left|\lambda\right>=\begin{cases}\phi_{\lambda_{1}}\phi_{\lambda_{2}}\dots\phi_{\lambda_{\ell(\lambda)}}\left|0\right>&\mathrm{for}\,\,\,\ell(\lambda)=0\mod
2,\\\
\sqrt{2}\phi_{\lambda_{1}}\phi_{\lambda_{2}}\dots\phi_{\lambda_{\ell(\lambda)}}\phi_{0}\left|0\right>&\mathrm{for}\,\,\,\ell(\lambda)=1\mod
2,\end{cases}$

and similarly for ${\mathcal{F}}_{B}^{0*}$. From the anticommutation relations
we have

(2.9) $\left<\lambda|\mu\right>=(-1)^{|\lambda|}\delta_{\lambda,\mu}.$

It is easy to see that

(2.10) $\left<0\right|\phi_{k}\phi_{m}\left|0\right>=\delta_{k+m,0}H[m],$

where

(2.11) $H[m]=\begin{cases}\displaystyle{0}&\mathrm{for}\quad m<0,\\\
\displaystyle{\frac{1}{2}}&\mathrm{for}\quad m=0,\\\
\displaystyle{(-1)^{m}}\,\,\,\,\,\,&\mathrm{for}\quad m>0.\end{cases}$

Bilinear combinations of neutral fermions $\phi_{k}\phi_{m}$ satisfy the
commutation relations of the Lie algebra $B_{\infty}$. Let
$\left(E_{i,j}\right)_{k,l}=\delta_{i,k}\delta_{j,l}$ be the standard basis of
the matrix units $\left.\left\\{E_{{i,j}}\right|i,j\in{\mathbb{Z}}\right\\}$.
Then $\phi_{k}\phi_{-m}$ corresponds [DJKM82] to

(2.12) $F_{k,m}=(-1)^{m}E_{k,m}-(-1)^{k}E_{-m,-k}$

with the commutation relations

(2.13)
$\left[F_{a,b},F_{c,d}\right]=(-1)^{b}\delta_{b,c}F_{a,d}-(-1)^{a}\delta_{a+c,0}F_{-b,d}+(-1)^{b}\delta_{b+d,0}F_{c,-a}-(-1)^{a}\delta_{a,d}F_{c,b}.$

For the bilinear combinations of neutral fermions we introduce the normal
ordering by

(2.14)
$:\mathrel{\phi_{k}\phi_{m}}:=\phi_{k}\phi_{m}-\left<0\right|\phi_{k}\phi_{m}\left|0\right>.$

It is skewsymmetric

(2.15) $:\mathrel{\phi_{k}\phi_{m}}:=-:\mathrel{\phi_{m}\phi_{k}}:,$

in particular, $:\mathrel{\phi_{k}\phi_{k}}:=0$. The normal ordered quadratic
combinations of neutral fermions satisfy the commutation relations of a
central extension of the algebra $B_{\infty}$:

(2.16)
$\displaystyle[:\mathrel{\phi_{a}\phi_{b}}:,:\mathrel{\phi_{c}\phi_{d}}:]$
$\displaystyle=(-1)^{b}\delta_{b+c,0}:\mathrel{\phi_{a}\phi_{d}}:-(-1)^{a}\delta_{a+c,0}:\mathrel{\phi_{b}\phi_{d}}:$
$\displaystyle\quad+(-1)^{b}\delta_{b+d,0}:\mathrel{\phi_{c}\phi_{a}}:-(-)^{a}\delta_{a+d,0}:\mathrel{\phi_{c}\phi_{b}}:$
$\displaystyle\quad+(\delta_{c,b}\delta_{a,d}-\delta_{a-c,0}\delta_{b-d,0})((-1)^{a}H[b]-(-1)^{b}H[a]),$

where $H[a]$ is given by (2.11). The operator $:\mathrel{\phi_{k}\phi_{-m}}:$
corresponds to the projective representation of the Lie algebra $B_{\infty}$,
and will also be denoted by $\hat{F}_{k,m}$.

Let us consider the generating function

(2.17) $\phi(z)=\sum_{k\in{\mathbb{Z}}}\phi_{k}z^{k}.$

It satisfies the anti-commutation relation

(2.18) $\left\\{\phi(z),\phi(w)\right\\}=\delta(z+w).$

Here we introduce the delta-function

(2.19) $\delta(z-w)=\sum_{k\in{\mathbb{Z}}}\left(\frac{z}{w}\right)^{k}.$

It satisfies

(2.20) $\delta(z-w)f(z)=\delta(z-w)f(w)$

for any formal series $f(z)\in{\mathbb{C}}[\\![z,z^{-1}]\\!]$ and can be
represented as

(2.21)
$2\delta(z+w)=\iota_{|z|>|w|}\frac{z-w}{z+w}-\iota_{|w|>|z|}\frac{z-w}{w+z},$

where $\iota_{|z|>|w|}$ is the operation of Laurent series expansion in the
region $|z|>|w|$.

Quadratic combinations of the generating functions $\phi(z)$ generate a Lie
algebra with the following commutation relations

(2.22)
$\left[\phi_{1}(z_{1})\phi(w_{1}),\phi(z_{2})\phi(w_{2})\right]=\delta(w_{1}+z_{2})\phi(z_{1})\phi(w_{2})-\delta(z_{1}+z_{2})\phi(w_{1})\phi(w_{2})\\\
+\delta(w_{1}+w_{2})\phi(z_{2})\phi(z_{1})-\delta(z_{1}+w_{2})\phi(z_{2})\phi(w_{1}).$

For the normal ordered operator we have

(2.23)
$\phi(z)\phi(w)=:\mathrel{\phi(z)\phi(w)}:+\frac{1}{2}\iota_{|z|>|w|}\frac{z-w}{z+w}.$

###  2.2. Vertex operators

For $k\in{\mathbb{Z}}_{\operatorname{odd}}$ we introduce bosonic operators

(2.24)
$J_{k}=\frac{1}{2}\sum_{m\in{\mathbb{Z}}}(-1)^{m+1}:\mathrel{\phi_{m}\phi_{-m-k}}:$

satisfying a commutation relation of the Heisenberg algebra

(2.25) $\left[J_{k},J_{m}\right]=\frac{k}{2}\delta_{k+m,0}.$

From (2.2) we have:

(2.26) $J_{m}\left|0\right>=0,\,\,\,\,\,\left<0\right|J_{-m}=0,\,\,\,\,\,m>0.$

Let us consider the vertex operator for the BKP hierarchy introduced in
[DJKM82],

(2.27)
$\widehat{V}^{(1)}_{B}(z)=\exp\left(\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}z^{k}t_{k}\right)\exp\left(-2\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}\frac{1}{kz^{k}}\frac{\partial}{\partial
t_{k}}\right).$

These operators satisfy the anticommutation relation

(2.28)
$\left\\{\widehat{V}^{(1)}_{B}(z),\widehat{V}^{(1)}_{B}(w)\right\\}=2\delta(z+w)$

similar to the relation (2.18).

It is convenient to introduce generating functions of the bosonic operators:

(2.29) $J_{+}({\bf
t})=\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}t_{k}J_{k},\,\,\,\,\,\,\,J_{-}({\bf
s})=\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}s_{k}J_{-k}.$

Then one has

(2.30) $\begin{split}\widehat{V}^{(1)}_{B}(z)\left<0\right|e^{J_{+}({\bf
t})}&=2\left<0\right|\phi_{0}e^{J_{+}({\bf t})}\phi(z),\\\
\widehat{V}^{(1)}_{B}(z)\left<0\right|\phi_{0}e^{J_{+}({\bf
t})}&=\left<0\right|e^{J_{+}({\bf t})}\phi(z).\end{split}$

Let us consider a bilinear combination of the vertex operators

(2.31)
$\widehat{Y}_{B}(z,w)=\frac{1}{2}\widehat{V}^{(1)}_{B}(z)\widehat{V}^{(1)}_{B}(w).$

Using the anti-commutation relation (2.28) it is easy to show that the vertex
operators $\widehat{Y}_{B}(z,w)$ satisfy a commutation relation equivalent to
the relation (2.22) for the bilinear combinations $\phi(z)\phi(w)$:

(2.32)
$\displaystyle\left[\widehat{Y}_{B}(z_{1},w_{1}),\widehat{Y}_{B}(z_{2},w_{2})\right]$
$\displaystyle=\delta(w_{1}+z_{2})\widehat{Y}_{B}(z_{1},w_{2})-\delta(z_{1}+z_{2})\widehat{Y}_{B}(w_{1},w_{2})$
$\displaystyle\quad+\delta(w_{1}+w_{2})\widehat{Y}_{B}(z_{2},z_{1})-\delta(z_{1}+w_{2})\widehat{Y}_{B}(z_{2},w_{1}).$

It is also convenient to consider its regularized version, corresponding to
$:\mathrel{\phi(z)\phi(-w)}:$

(2.33)
${\widehat{V}}^{(2)}_{B}(z,w)=\widehat{Y}_{B}(z,w)-\frac{1}{2}\iota_{|z|>|w|}\frac{z-w}{{z}+w},$

where for the second term we assume the series expansion in $|z|>|w|$. This
expression has no pole at $z=-w$, moreover, it is antisymmetric with respect
to the permutation of $z$ and $w$. These vertex operators can be represented
as

(2.34)
$\widehat{V}^{(2)}_{B}(z,w)=\frac{1}{2}\frac{z-w}{z+w}\left(e^{\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}t_{k}(z^{k}+w^{k})}e^{-2\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}\left(\frac{1}{kz^{k}}+\frac{1}{kw^{k}}\right)\frac{\partial}{\partial
t_{k}}}-1\right).$

From (2.30) it follows that

(2.35) ${\widehat{V}}^{(2)}_{B}(z,w)\left<0\right|e^{J_{+}({\bf
t})}=\left<0\right|e^{J_{+}({\bf t})}:\mathrel{\phi(z)\phi(w)}:.$

###  2.3. Boson-fermion correspondence

For the neutral fermions the boson-fermion correspondence describes an
isomorphism [You89]

(2.36) $\sigma_{B}^{i}:\,\,\,\,\,{\mathcal{F}}_{B}^{i}\simeq
B^{(i)}={\mathbb{C}}[\\![t_{1},t_{3},t_{5}\dots]\\!]$

for $i=0,1$. Here

(2.37) $\sigma_{B}^{i}(\left|i\right>)=1,$

where we introduce $\left|1\right>=\sqrt{2}\phi_{0}\left|0\right>$, and for
both $i=0,1$ we have

(2.38)
$\sigma_{B}^{i}J_{-k}(\sigma_{B}^{i})^{-1}=\frac{k}{2}t_{k},\,\,\,\,\,\sigma_{B}^{i}J_{k}(\sigma_{B}^{i})^{-1}=\frac{\partial}{\partial
t_{k}}$

for $k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}$. The boson-fermion
correspondence is given by

(2.39)
$\sigma_{B}^{i}(\left|a\right>)=\begin{cases}\displaystyle{\left<1\right|e^{J_{+}({\bf
t})}\left|a\right>}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\quad\left|a\right>\in{\mathcal{F}}_{B}^{1},\\\\[4.0pt]
\displaystyle{\left<0\right|e^{J_{+}({\bf
t})}\left|a\right>}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\quad\left|a\right>\in{\mathcal{F}}_{B}^{0},\end{cases}$

where $\left<1\right|=\sqrt{2}\left<0\right|\phi_{0}$. The boson-fermion
correspondence between two different representations of the central extension
of the $B_{\infty}$ algebra is given by

(2.40)
$\sigma_{B}^{i}:\mathrel{\phi(z)\phi(w)}:(\sigma_{B}^{i})^{-1}={\widehat{V}}^{(2)}_{B}(z,w).$

Below we will work only with ${\mathcal{F}}_{B}^{0}$ component of the
fermionic Fock space and its bosonic counterpart.

Relation between the Schur Q-functions and the BKP hierarchy, in particular,
is described by the following result of You:

######  Theorem 2.1 ([You89]).

For the states (2.8) the boson-fermion correspondence yields

(2.41)
$\sigma_{B}^{0}(\left|\lambda\right>)=2^{-\ell(\lambda)/2}Q_{\lambda}({\bf
t}/2).$

Here $Q_{\lambda}$ are the Schur Q-functions (see Section III.8 of [Mac95] for
definition and details).

It was shown by Date, Jimbo, Kashiwara, and Miwa [DJKM82] that for any group
element of the central extension of the algebra $B_{\infty}$,

(2.42)
$G=\exp\left(\sum_{k,m\in{\mathbb{Z}}}a_{km}:\mathrel{\phi_{k}\phi_{m}}:\right),$

the bosonic image of the fermionic state $Ge^{J_{-}({\bf s})}\left|0\right>$
solves the 2-BKP hierarchy. Namely,

(2.43) $\tau({\bf t},{\bf s})=\left<0\right|e^{J_{+}({\bf t})}Ge^{J_{-}({\bf
s})}\left|0\right>$

is a tau-function of 2-BKP hierarchy.

##  3\. Hypergeometric tau-functions and weighted spin Hurwitz numbers

In this section we suggest a way to construct the weighed sums of the spin
Hurwitz numbers which solve the $2$-BKP hierarchy. There is a certain
ambiguity associated to the choice of the weights, and we discuss two natural
candidates for the role of the elementary weight functions.

###  3.1. Hypergeometric tau-functions of 2-BKP hierarchy

Following Orlov [Orl03] we consider a set of parameters $T_{n}$,
$n\in{\mathbb{Z}}$, such that $T_{n}=-T_{-n}$. In particular, $T_{0}=0$. Then

(3.1)
$\sum_{k\in{\mathbb{Z}}}(-1)^{k}T_{k}:\mathrel{\phi_{k}\phi_{-k}}:=2\sum_{k\in{\mathbb{Z}}_{+}}(-1)^{k}T_{k}:\mathrel{\phi_{k}\phi_{-k}}:=2\sum_{k\in{\mathbb{Z}}_{+}}(-1)^{k}T_{k}\hat{F}_{k,k}.$

Consider the group element

(3.2)
${\mathcal{D}}=\exp\left(\sum_{k\in{\mathbb{Z}}}(-1)^{k}T_{k}:\mathrel{\phi_{k}\phi_{-k}}:\right),$

then

(3.3) $\tau({\bf t},{\bf s})=\left<0\right|e^{J_{+}({\bf
t})}{\mathcal{D}}e^{J_{-}({\bf s})}\left|0\right>$

is a tau-function of the $2$-component BKP hierarchy symmetric in the
variables $t_{k}$ and $s_{k}$. From Theorem 2.1 it follows that this tau-
function has an equivalent description [Orl03]

(3.4) $\tau({\bf t},{\bf
s})=\sum_{\lambda\in\operatorname{SP}}2^{-\ell(\lambda)}e^{2T_{\lambda_{1}}+\dots+2T_{\lambda_{\ell(\lambda)}}}Q_{\lambda}({\bf
t}/2)Q_{\lambda}({\bf s}/2).$

Here $\operatorname{SP}$ is the set of all strict partitions including the
empty one. These are the hypergeometric tau-functions of the 2-BKP hierarchy.

Consider $T(x)$, an odd function such that $T(k)=T_{k}$ for
$k\in{\mathbb{Z}}$. For future applications it is natural to introduce the
topological expansion parameter $\hbar$. Consider a new _even_ function
$\overline{\psi}(z)$ such that

(3.5) $\overline{\psi}(\hbar(z+1/2))=T(z+1)-T(z).$

We assume that $\overline{\psi}(z)$ is itself a series in $\hbar^{2}$,
$\overline{\psi}(\hbar^{2},z)=\sum_{d=0}^{\infty}\hbar^{2g}\psi_{2d}(z)$,
where $\psi_{2d}(z)$ is an even formal power series in $z$, therefore $T_{k}$
also depend on $\hbar$. The constant term of this series,
$\psi_{0}=\overline{\psi}(0,z)$, is also denoted by $\psi=\psi(z)$.

######  Remark 3.1.

Our definition of the parameters $T_{k}$ corresponds to the doubled parameters
of [Orl03] with the inverse sign.

###  3.2. Spin Hurwitz numbers

Spin Hurwitz numbers, which count the ramified coverings with sign coming from
spin structure, were introduces by Eskin, Okounkov, and Pandharipande [EOP08].
Using TQFT, Gunningham [Gun16] found a combinatorial expression for all genera
spin Hurwitz numbers, which uses the representation theory of Sergeev’s group.
In this section we recall this combinatorial expression. We address the reader
to [EOP08, Gun16, Lee18, Lee20, MMN20, GKL21] for the basic definitions and
properties. Different authors use different conventions, our notation is
consistent with that of [GKL21].

For any set of variables or parameters $r_{k}$ and any partition $\mu$ let us
denote

(3.6) $r_{\mu}=\prod_{j=1}^{\ell(\mu)}r_{\mu_{j}}.$

Let $\operatorname{OP}(d)$ and $\operatorname{SP}(d)$ be the sets of odd
partitions and strict partitions of the size $d$ respectively. Then the Schur
Q-functions can be expanded as

(3.7)
$Q_{\lambda}=2^{\frac{\ell(\lambda)-\delta(\lambda)}{2}}\sum_{\mu\in\operatorname{OP}(|\lambda|)}\frac{\zeta^{\lambda}_{\mu}}{z_{\mu}}p_{\mu}$

with the inverse relation

(3.8)
$p_{\mu}=2^{-\ell(\mu)}\sum_{\lambda\in\operatorname{SP}(|\mu|)}2^{-\frac{\ell(\lambda)+\delta(\lambda)}{2}}\zeta^{\lambda}_{\mu}Q_{\lambda}.$

Here $p_{k}=kt_{k}$ are the independent variables,

(3.9) $\delta(\mu)=\begin{cases}0,\,\,\,\,\,\,{\rm for}\,\,{\rm
even}\,\,\ell(\mu)\\\ 1,\,\,\,\,\,\,{\rm for}\,\,{\rm
odd}\,\,\ell(\mu)\end{cases}$

and $z_{\mu}=\prod_{k}\mu(k)!k^{\mu(k)}$.

The characters of the Sergeev group $\zeta_{\mu}^{\rho}$ satisfy the
orthogonality relations

(3.10)
$\sum_{\mu\in\operatorname{OP}(d)}2^{-\ell(\mu)-\delta(\sigma)}\frac{\zeta_{\mu}^{\rho}\zeta_{\mu}^{\sigma}}{z_{\mu}}=\delta_{\rho,\sigma}$

and

(3.11)
$\sum_{\lambda\in\operatorname{SP}(d)}2^{-\ell(\sigma)-\delta(\lambda)}\frac{\zeta_{\sigma}^{\lambda}\zeta_{\rho}^{\lambda}}{z_{\sigma}}=\delta_{\rho,\sigma}.$

Let us also introduce the central characters

(3.12) $f^{\lambda}_{\mu}=\frac{2^{d}d!}{2^{\ell(\mu)}z_{\mu}\dim
V^{\lambda}}\zeta_{\mu}^{\lambda}.$

Here

(3.13) $\dim
V^{\lambda}=\zeta^{\lambda}_{1^{d}}=2^{\frac{\delta(\lambda)-\ell(\lambda)}{2}}d!Q_{\lambda}\Big{|}_{p_{k}=\delta_{k,1}}$

is the dimension of the irreducible supermodule associated with the strict
partition $\lambda$.

Let us consider the disconnected spin Hurwitz numbers for the
${\mathbb{C}}{\mathrm{P}}^{1}$ with the ramifications at $k$ branch points
given by odd partitions $\mu_{1},\dots,\mu_{k}$ with $|\mu_{j}|=d$. The
Gunningham formula [Gun16, Lee18] describes them in terms of the central
characters of the Sergeev group:

(3.14)
$H_{d}^{\theta}(\mu_{1},\dots,\mu_{k})=2^{-d-\sum_{i=1}^{k}\ell^{*}(\mu_{i})/2}\sum_{\lambda\in\operatorname{SP}(d)}2^{-\delta(\lambda)}\left(\frac{\dim
V^{\lambda}}{d!}\right)^{2}\prod_{j=1}^{k}f_{\mu_{j}}^{\lambda},$

where $\ell^{*}(\mu)=|\mu|-\ell(\mu)$ is the colength of the partition $\mu$.

###  3.3. From spin Hurwitz numbers to 2-BKP hierarchy

Let us single out two of the $k$ partitions and denote them by $\mu$ and
$\nu$. Using Equation (3.12) we can rewrite the spin Hurwitz numbers (3.14) as
follows

(3.15)
$H_{d}^{\theta}(\mu_{1},\dots,\mu_{k-2},\mu,\nu)=2^{-\frac{1}{2}(\ell(\mu)+\ell(\nu)+\sum_{i=1}^{k-2}\ell^{*}(\mu_{i}))}\sum_{\lambda\in\operatorname{SP}(d)}2^{-\delta(\lambda)}\frac{\zeta_{\mu}^{\lambda}}{z_{\mu}}\frac{\zeta_{\nu}^{\lambda}}{z_{\nu}}\prod_{j=1}^{k-2}f_{\mu_{j}}^{\lambda}.$

Let us introduce $k-2$ families of weights $r^{(j)}_{m}$ for $1\leq j\leq
k-2$, $m\in{\mathbb{Z}}_{+}$, associated with $k-2$ branch points. Then for
the spin Hurwitz numbers (3.15) we introduce their weighted combinations

(3.16) $H_{d,{\bf
r}}^{\theta}(\nu,\mu)=\sum_{\mu_{1},\dots,\mu_{k-2}\in\operatorname{OP}(d)}H_{d}^{\theta}(\mu_{1},\dots,\mu_{k-2},\mu,\nu)R_{\mu_{1}}(r^{(1)})\dots
R_{\mu_{k-2}}(r^{(k-2)}),$

where

(3.17)
$R_{\mu}(r)=\hbar^{-\ell^{*}(\mu)}2^{-\ell(\mu)/2}\sum_{\sigma\in\operatorname{SP}(d)}\frac{\dim
V^{\sigma}}{d!2^{d/2}}2^{-\delta(\sigma)}\zeta_{\mu}^{\sigma}r_{\sigma}.$

Let us stress that the trivial ramifications $\mu_{j}=1^{d}$ are allowed in
the summation. For the empty partition we put $R_{\emptyset}=1$. If we compare
this expression with the decomposition of the functions $p_{\mu}(Q_{\sigma})$
in the basis of Schur Q-functions (3.8), then using Equation (3.13) we get

(3.18)
$R_{\mu}(r)=\hbar^{-\ell^{*}(\mu)}2^{-\ell^{*}(\mu)/2}p_{\mu}(Q_{\sigma}(\delta_{k,1})r_{\sigma}).$

In particular,

(3.19)
$\begin{split}R_{[1]}(r)&=\frac{1}{2}Q_{[1]}(\delta_{k,1})r_{1}=r_{1},\\\
R_{[1,1]}(r)&=\frac{1}{2}Q_{[2]}(\delta_{k,1})r_{2}=r_{2},\\\
R_{[3]}(r)&={(2\hbar^{2})}^{-1}\left(\frac{1}{2}Q_{[3]}(\delta_{k,1})r_{3}-\frac{1}{2}Q_{[2,1]}(\delta_{k,1})r_{2}r_{1}\right)=\frac{1}{3\hbar^{2}}(r_{3}-r_{2}r_{1}),\\\
R_{[1,1,1]}(r)&=\frac{1}{2}Q_{[3]}(\delta_{k,1})r_{3}+\frac{1}{4}Q_{[2,1]}(\delta_{k,1})r_{2}r_{1}=\frac{1}{3}(2r_{3}+r_{2}r_{1}).\end{split}$

Definition (3.17) is justified by the following observation: from the
orthogonality relation (3.10) it follows that

(3.20)
$\sum_{\mu\in\operatorname{OP}(d)}2^{-\ell^{*}(\mu)/2}f_{\mu}^{\lambda}R_{\mu}(r)=r_{\lambda}.$

Therefore

(3.21) $H_{d,{\bf
r}}^{\theta}(\nu,\mu)=2^{-\frac{\ell(\mu)+\ell(\nu)}{2}}\sum_{\lambda\in\operatorname{SP}(d)}2^{-\delta(\lambda)}\frac{\zeta_{\mu}^{\lambda}}{z_{\mu}}\frac{\zeta_{\nu}^{\lambda}}{z_{\nu}}\prod_{j=1}^{k-2}r^{(j)}_{\lambda}.$

By the Riemann–Hurwitz formula

(3.22) $2-2g=\ell(\mu)+\ell(\nu)-\sum_{i=1}^{k-2}\ell^{*}(\mu_{i}),$

where $g$ is the genus of the covering curve. Consider the following
generating function

(3.23) $\tau({\bf t},{\bf
s})=\sum_{d=0}^{\infty}\sum_{\mu,\nu\in\operatorname{OP}(d)}\hbar^{2g-2+\ell(\mu)+\ell(\nu)}2^{-\frac{\ell(\mu)+\ell(\nu)}{2}}H_{d,{\bf
r}}^{\theta}(\nu,\mu)\prod_{j=1}^{\ell(\mu)}\mu_{j}\prod_{j=1}^{\ell(\nu)}\nu_{j}t_{\mu}s_{\nu}.$

Then for any choice of parameters $r_{k}^{(j)}$ from (3.21) we have

######  Theorem 3.1.

The generating function $\tau({\bf t},{\bf s})$ is a hypergeometric tau-
function of the 2-BKP hierarchy

(3.24) $\tau({\bf t},{\bf
s})=\sum_{\lambda\in\operatorname{SP}}\frac{Q_{\lambda}({\bf
t}/2)Q_{\lambda}({\bf
s}/2)}{2^{\ell(\mu)}}\prod_{j=1}^{k-2}r^{(j)}_{\lambda}.$

This tau-function can be identified with (3.4) if one puts

(3.25) $T_{m}=\frac{1}{2}\sum_{j=1}^{k-2}\log r_{m}^{(j)}.$

Similarly to the case of ordinary Hurwitz numbers, we can consider the limit
when the maximal number of the branch points $k$ tends to infinity.

###  3.4. Weighted spin Hurwitz numbers

In the previous section we have constructed the weighted sums of the spin
Hurwitz numbers that lead to the tau-functions of the 2-BKP hierarchy. While
working with arbitrary parameters $r_{k}^{(j)}$ allow us to trace more
information about the spin Hurwitz numbers from the properties of the tau-
function, similarly to the case of the ordinary Hurwitz numbers [Zog15, GPH17,
ACEH20] we would like to introduce the distinguished weights, parametrized by
one parameter $c_{j}$, $j=1,\dots,k-2$ for each of $k-2$ points.

By analogy with the ordinary weighted Hurwitz numbers, see [GPH17] and, more
specifically, in [ACEH20, Equation (3.1)], one would tend to put
$R_{\mu}(r^{(j)})=c_{j}^{\ell^{*}(\mu)}$. For this choice the weighted spin
Hurwitz numbers (3.16) would be independent of $\hbar$. However, it is easy to
see that $R_{\mu}$ should depend on $\hbar$ – this is clear from (3.19).
Therefore, for the combinations of the spin Hurwitz numbers defined by (3.16)
we need some “completion” of the partitions for the rational weight functions,

(3.26) $R_{\mu}(r^{(j)})=\sum_{k=0}^{\infty}R_{\mu}^{(k)}\hbar^{2k},$

a new effect of spin Hurwitz numbers which is absent in the theory of ordinary
weighted Hurwitz numbers.

Let us consider the generating function (3.24) for the case with the maximal
number of the branch points $k=3$. We claim that this tau-function can be
considered as a generating function of a spin version of dessins d’enfants. We
also put $s_{k}=\delta_{k,1}\hbar^{-1}$, therefore the non-trivial branching
is allowed only at two points. Then the tau-function (3.24) reduces to

(3.27) $\tau({\bf t})=\sum_{\lambda\in\operatorname{SP}}\frac{Q_{\lambda}({\bf
t}/2)Q_{\lambda}(\delta_{k,1})}{2^{\ell(\mu)+|\lambda|}\,\hbar^{|\lambda|}}r_{\lambda},$

where $r_{\lambda}=r^{(1)}_{\lambda}$. From the orthogonality relation (3.11)
and Equation (3.23) it follows that

(3.28) $\tau({\bf
t})=\sum_{\mu\in\operatorname{OP}}\hbar^{-\ell(\mu)}2^{-\frac{|\mu|+\ell(\mu)}{2}}\prod_{j=1}^{\ell(\mu)}\mu_{j}t_{\mu}\frac{R_{\mu}(r)}{z_{\mu}}.$

To relate $R_{\mu}$ to $\overline{\psi}(z)$ we use the results of Section 4
below. From Equation (3.28) it follows that the coefficients $R_{\mu}(r)$’s
are proportional to the coefficients of the correlation functions
$W_{n}^{\bullet}$, namely

(3.29)
$W_{n}^{\bullet}=\hbar^{-n}\sum_{\mu\in\operatorname{OP},\ell(\mu)=n}2^{-\frac{|\mu|+n}{2}}R_{\mu}(r)\sum_{\sigma\in
S_{n}}X_{\sigma(1)}^{\mu_{1}}\dots X_{\sigma(n)}^{\mu_{n}}.$

If we require $R_{\mu}^{(0)}=c^{\ell^{*}(\mu)}$, then for $n=1$ the leading
term of equation (3.29) reduces to

(3.30)
$W_{0,1}(x)=\frac{1}{2}\sum_{k\in{{\mathbb{Z}}_{\operatorname{odd}}^{+}}}\left(\frac{c}{\sqrt{2}}\right)^{k-1}X^{k}.$

After identification of this expression with the general expression for the
correlation function given by Proposition 4.4 we find $\psi(z)$,

(3.31) $\psi(z)=\frac{1}{2}\log\frac{1+\sqrt{1+2c^{2}z^{2}}}{2}.$

One can identify it with $\overline{\psi}$, however, it is also possible to
consider the $\hbar$-deformations.

Another possibility is to consider

(3.32) $\overline{\psi}(z)=\frac{1}{2}\log\left(1+\frac{c^{2}z^{2}}{2}\right)$

The associated functions $R_{\mu}$, (3.19) are rational functions of $c$,

(3.33) $\begin{split}R_{[1]}(r)&=1+\frac{1}{8}c^{2}\hbar^{2},\\\
R_{[1,1]}(r)&=\left(1+\frac{1}{8}c^{2}\hbar^{2}\right)\left(1+\frac{9}{8}c^{2}\hbar^{2}\right),\\\
R_{[3]}(r)&=c^{2}\left(1+\frac{1}{8}c^{2}\hbar^{2}\right)\left(1+\frac{9}{8}c^{2}\hbar^{2}\right),\\\
R_{[1,1,1]}(r)&=\left(1+\frac{1}{8}c^{2}\hbar^{2}\right)\left(1+\frac{9}{8}c^{2}\hbar^{2}\right)\left(1+\frac{17}{8}c^{2}\hbar^{2}\right).\end{split}$

For $c=1$ the generating function $\tau({\bf t})$ for this choice of
parametrization can be identified with the generalized BGW tau-function
[Ale23],

(3.34) $\left.\tau_{BGW}({\bf
t}/2)\right|_{N^{2}\mapsto-\frac{2}{\hbar^{2}}}=\tau({\bf
t},{\delta_{k,1}\hbar^{-1}}).$

###  3.5. From BKP to KP

It is well known that the solutions of the BKP hierarchy are related to the
solutions of the KP hierarchy for the particular choice of the variables
[DJKM82]. Following [Orl03], in this section we consider this relation for the
hypergeometric tau-functions of both hierarchies. Namely, we relate any
hypergeometric 2-BKP tau-function (3.4) to a hypergeometric tau-function of
the 2-KP hierarchy.

Let us consider a 2-component system of neutral fermions $\phi_{k}^{(a)}$,
$a=1,2$, satisfying the anti-commutation relations

(3.35)
$\left\\{\phi_{j}^{(k)},\phi_{l}^{(m)}\right\\}=(-1)^{j}\delta_{k,m}\delta_{j+l,0}.$

Following [JM83] we can relate them to the charged free fermions

(3.36)
$\phi_{j}^{(1)}=\frac{\psi_{j}+(-1)^{j}\psi_{-j}^{*}}{\sqrt{2}},\,\,\,\,\,\,\,\,\,\phi_{j}^{(2)}=\sqrt{-1}\frac{\psi_{j}-(-1)^{j}\psi_{-j}^{*}}{\sqrt{2}}.$

for $j\in{\mathbb{Z}}$. We immediately have

(3.37)
$\psi_{j}\psi_{j}^{*}+\psi_{-j}^{*}\psi_{-j}=(-1)^{j}\left(\phi_{j}^{(1)}\phi_{-j}^{(1)}+\phi_{j}^{(1)}\phi_{-j}^{(1)}\right).$

Consider the bosonic operators for the charged fermions

(3.38)
$J_{k}^{\operatorname{KP}}=\sum_{j\in{\mathbb{Z}}}:\mathrel{\psi_{j}\psi_{j+k}^{*}}:.$

Then for odd $k$ we have

(3.39) $J_{k}^{\operatorname{KP}}=J_{k}^{(1)}+J_{k}^{(2)},$

where the bosonic operators $J_{k}^{(j)}$ are given in terms of the
corresponding neutral fermions by Equation (2.24).

Let us consider the hypergeometric 2-KP tau-function

(3.40) $\tau_{\operatorname{KP}}({\bf t},{\bf
s})=\left<0\right|e^{J^{\operatorname{KP}}_{+}({\bf
t})}e^{2\sum_{j=1}^{\infty}T_{j}(:\mathrel{\psi_{j}\psi_{j}^{*}}:-:\mathrel{\psi_{-j}^{*}\psi_{-j}}:)}e^{J^{\operatorname{KP}}_{-}({\bf
s})}\left|0\right>,$

where $T_{k}$ are some parameters and $J_{\pm}^{\operatorname{KP}}({\bf
t})=\sum_{k=1}^{\infty}t_{k}J_{\pm k}^{\operatorname{KP}}$. Note that the
group element in (3.40) is not the most general diagonal group element.
However, for this choice of the the group element we have a simple relation
between this tau-function and a tau-function of the 2-BKP hierarchy. If all
even time variables vanish, $t_{2k}=s_{2k}=0$ for $k\in{\mathbb{Z}}_{+}$, then
in terms of the neutral fermions we have

(3.41) $\left.\tau_{\operatorname{KP}}({\bf t},{\bf
s})\right|_{t_{2k}=s_{2k}=0}=\left<0\right|e^{J^{(1)}_{+}({\bf
t})+J^{(2)}_{+}({\bf
t})}e^{2\sum_{j=1}^{\infty}(-1)^{j}T_{j}(:\mathrel{\phi^{(1)}_{j}\phi^{(1)}_{-j}}:+:\mathrel{\phi^{(2)}_{j}\phi^{(2)}_{-j}}:)}e^{J^{(1)}_{-}({\bf
s})+J^{(2)}_{-}({\bf s})}\left|0\right>$

Therefore [Orl03]

(3.42) $\left.\tau_{\operatorname{KP}}({\bf t},{\bf
s})\right|_{t_{2k}=s_{2k}=0}=\tau({\bf t},{\bf s})^{2},$

where

(3.43) $\tau({\bf t},{\bf s})=\left<0\right|e^{J_{+}({\bf
t})}e^{2\sum_{j=1}^{\infty}(-1)^{j}T_{j}:\mathrel{\phi_{j}\phi_{-j}}:}e^{J_{-}({\bf
s})}\left|0\right>$

is a hypergeometric tau-function of 2-BKP hierarchy (3.4). By definition, it
depends only on odd times $t_{2k+1}$ and $s_{2k+1}$.

Let us compare the expansions of the tau-functions $\tau_{\operatorname{KP}}$
and $\tau$ in terms of the corresponding sets of the Schur functions. For the
hypergeometric 2-KP tau-function (3.40) one has

(3.44) $\tau_{KP}({\bf t},{\bf
s})=\sum_{\lambda}e^{2\sum_{(i,j)\in\lambda}\overline{\psi}(\hbar(j-i-1/2))}s_{\lambda}({\bf
t})s_{\lambda}({\bf s}),$

where $s_{\lambda}$ are the ordinary Schur functions and the sum runs over all
partitions. These tau-functions are generating functions of the ordinary
weighted Hurwitz numbers. For the 2-component BKP tau-function (3.43) we have

(3.45) $\tau({\bf t},{\bf
s})=\sum_{\lambda\in\operatorname{SP}}e^{2\sum_{(i,j)\in\lambda}\overline{\psi}(\hbar(j-1/2))}\frac{Q_{\lambda}({\bf
t}/2)Q_{\lambda}({\bf s}/2)}{2^{\ell(\lambda)}}.$

We see that for any hypergeometric tau-function of 2-BKP there exist a
hypergeometric tau-function of 2-KP satisfying (3.42). It is easy to see that
such 2-KP tau-function is not unique. Let $\lambda^{\prime}$ denotes the
transpose partition of $\lambda$. Then, as it follows i.e. from the Giambelli
formula,

(3.46) $\left.s_{\lambda}({\bf
t})\right|_{t_{2k}=0}=\left.s_{\lambda^{\prime}}({\bf t})\right|_{t_{2k}=0}$

and for the Equation (3.44) we have

(3.47) $\begin{split}\left.\tau_{KP}({\bf t},{\bf
s})\right|_{t_{2k}=s_{2k}=0}&=\left.\sum_{\lambda}e^{2\sum_{(i,j)\in\lambda}\overline{\psi}(\hbar(j-i-1/2))}s_{\lambda^{\prime}}({\bf
t})s_{\lambda^{\prime}}({\bf s})\right|_{t_{2k}=s_{2k}=0}\\\
&=\left.\sum_{\lambda}e^{2\sum_{(i,j)\in{\lambda^{\prime}}}\overline{\psi}(\hbar(j-i-1/2))}s_{\lambda}({\bf
t})s_{\lambda}({\bf s})\right|_{t_{2k}=s_{2k}=0}\\\
&=\left.\sum_{\lambda}e^{2\sum_{(i,j)\in{\lambda}}\overline{\psi}(\hbar(i-j-1/2))}s_{\lambda}({\bf
t})s_{\lambda}({\bf s})\right|_{t_{2k}=s_{2k}=0}.\end{split}$

Therefore, (3.42) is also satisfied for the tau-function (3.44) with
$\overline{\psi}(z)$ substituted by $\overline{\psi}(-z-\hbar)$.

Hence, we can relate any generating function of the spin Hurwitz numbers
(3.45) to the generating function of the ordinary Hurwitz numbers (3.44).
Moreover, we have at least two different hypergeometric tau-functions of the
2-KP hierarchy, corresponding to a given hypergeometric tau-function of the
2-BKP hierarchy. We expect that this identification should lead to a non-
trivial relation between spin and ordinary Hurwitz numbers.

Let us consider a few examples. If $T(x)=ax$, then $\overline{\psi}(z)=a$ is a
constant, and the tau-function of the 2-BKP hierarchy is very simple

(3.48) $\begin{split}\tau({\bf t},{\bf
s})&=\sum_{\lambda\in\operatorname{SP}}e^{2a|\lambda|}\frac{Q_{\lambda}({\bf
t}/2)Q_{\lambda}({\bf s}/2)}{2^{\ell(\lambda)}}\\\
&=\exp\left(a\sum_{k\in{\mathbb{Z}}^{+}_{\operatorname{odd}}}kt_{k}s_{k}\right).\end{split}$

More complicated example corresponds to $T(x)=\frac{bx^{3}}{3}+ax$ for some
$a$ and $b$. It is associated with to
$\overline{\psi}(z)=\frac{b}{2\hbar^{2}}z^{2}+\frac{b}{12}+a$. The identity
(3.42) for this case with $a=\frac{2}{3}b$ and $\hbar=1$ was proven by Lee
[Lee20]. On the KP side the generating function, considered by Lee, is given
by the last line of (3.47).

##  4\. Diagonal group element and $n$-point functions

In this section we prove explicit closed algebraic formulas for the
correlation functions $W_{g,n}$.

###  4.1. Operators $\mathbb{J}_{k}$

For the diagonal group element (3.2) introduce the operators

(4.1) ${\mathbb{J}}_{k}={\mathcal{D}}^{-1}J_{k}{\mathcal{D}},$

Our first goal is to provide a few explicit formulas for these operators. To
this end, we introduce a fermionic operator

(4.2)
$\begin{split}{\mathcal{E}}(u,a)&=:\mathrel{\phi(a^{-1}e^{u/2})\phi(-a^{-1}e^{-u/2})}:\\\
&=\sum_{k,m\in{\mathbb{Z}}}(-1)^{m}:\mathrel{\phi_{m-k}\phi_{-m}}:a^{k}e^{(m-k/2)u}\\\
&=\sum_{k,m\in{\mathbb{Z}}}(-1)^{m}\hat{F}_{m-k,m}a^{k}e^{(m-k/2)u}.\end{split}$

Let

(4.3) ${\mathcal{S}}(z)=\frac{e^{z/2}-e^{-z/2}}{z}.$

Then in terms of the bosonic operators (2.24) the operator
${\mathcal{E}}(u,a)$ is a reparametrization of the operator
$\widehat{V}^{(2)}_{B}$ given by (2.34), and can be represented as

(4.4)
${\mathcal{E}}(u,a)=\frac{1}{2}\frac{1+e^{-u}}{1-e^{-u}}\left(\exp\left(2u\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}a^{-k}{\mathcal{S}}(ku)J_{-k}\right)\exp\left(2u\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}a^{k}{\mathcal{S}}(ku)J_{k}\right)-1\right).$

######  Proposition 4.1.

The operators ${\mathbb{J}}_{k}$ belong to the image of the projective
representation of $B_{\infty}$ for all $k\in{\mathbb{Z}}_{\operatorname{odd}}$

(4.5)
${\mathbb{J}}_{k}=\frac{1}{2}\sum_{m\in{\mathbb{Z}}}(-1)^{m+1}e^{T_{k+m}-T_{-k-m}-T_{m}+T_{-m}}\hat{F}_{m,m+k}$

and

(4.6)
${\mathbb{J}}_{k}=\left.\frac{1}{2}[a^{k}]e^{2T(\partial_{u}+1/2a\partial_{a})-2T(\partial_{u}-1/2a\partial_{a})}{\mathcal{E}}(u,a)\right|_{u=0}.$

###### Proof.

From (2.13) we have

(4.7)
$\left[\sum_{a\in{\mathbb{Z}}}(-1)^{a}T_{a}F_{a,a},F_{k,m}\right]=(T_{k}-T_{-k}-T_{m}+T_{-m})F_{k,m}.$

Hence

(4.8)
${\mathcal{D}}^{-1}\hat{F}_{k,m}{\mathcal{D}}=e^{-T_{k}+T_{-k}+T_{m}-T_{-m}}\hat{F}_{k,m}$

and

(4.9)
${\mathbb{J}}_{k}=\frac{1}{2}\sum_{m\in{\mathbb{Z}}}(-1)^{m+1}e^{T_{k+m}-T_{-k-m}-T_{m}+T_{-m}}\hat{F}_{m,m+k},$

or, equivalently

(4.10)
${\mathbb{J}}_{k}=\frac{1}{2}\sum_{m\in{\mathbb{Z}}}(-1)^{m+k+1}e^{T_{m}-T_{-m}-T_{m-k}+T_{k-m}}\hat{F}_{m-k,m}.$

Comparing it to (4.2) we get

(4.11)
${\mathbb{J}}_{k}=\left.\frac{1}{2}[a^{k}]e^{T(\partial_{u}+1/2a\partial_{a})-T(-\partial_{u}-1/2a\partial_{a})-T(\partial_{u}-1/2a\partial_{a})+T(-\partial_{u}+1/2a\partial_{a})}{\mathcal{E}}(u,a)\right|_{u=0}.$

∎

###  4.2. Topological expansion

In terms of $\overline{\psi}$ we can rewrite the formula for
${\mathbb{J}}_{k}$, $k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}$, as

(4.12) $\displaystyle{\mathbb{J}}_{k}$
$\displaystyle=\frac{1}{2}\sum_{m\in{\mathbb{Z}}}(-1)^{m+1}e^{T_{k+m}-T_{-k-m}-T_{m}+T_{-m}}\hat{F}_{m,m+k}$
$\displaystyle=\frac{1}{2}\sum_{m\in{\mathbb{Z}}}(-1)^{m+1}\exp\left(2\sum_{i=1}^{k}\overline{\psi}(\hbar^{2},\hbar(m-\frac{1}{2}+i))\right)\hat{F}_{m,m+k}$

Let
$\phi_{k}(y)\coloneqq\exp\left(2\sum_{i=1}^{k}\overline{\psi}(\hbar^{2},y+\hbar(-\frac{k}{2}-\frac{1}{2}+i))\right)=\exp\left(2k\frac{{\mathcal{S}}(k\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}(\hbar^{2},y)\right)$.
Then

(4.13) $\displaystyle{\mathbb{J}}_{k}$
$\displaystyle=\frac{1}{2}\sum_{m\in{\mathbb{Z}}}(-1)^{m+1}\phi_{k}(\hbar(m+\frac{k}{2}))\hat{F}_{m,m+k}$
$\displaystyle=-\frac{1}{2}\sum_{r=0}^{\infty}\partial_{y}^{r}\phi_{k}(y)\Big{|}_{y=0}\cdot[u^{r}a^{k}]{\mathcal{E}}(\hbar
u,-a)$
$\displaystyle=-\frac{1}{4}\sum_{r=0}^{\infty}\partial_{y}^{r}\exp\left(2k\frac{{\mathcal{S}}(k\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}(\hbar^{2},y)\right)\Big{|}_{y=0}$
$\displaystyle\quad[u^{r}a^{k}]\frac{1+e^{-\hbar u}}{1-e^{-\hbar
u}}\exp\left(2\hbar
u\sum_{l\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}(-a)^{-l}{\mathcal{S}}(l\hbar
u)J_{-l}\right)\exp\left(2\hbar
u\sum_{l\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}(-a)^{l}{\mathcal{S}}(l\hbar
u)J_{l}\right)$
$\displaystyle=\frac{1}{4}\sum_{r=0}^{\infty}\partial_{y}^{r}\exp\left(2k\frac{{\mathcal{S}}(k\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}(\hbar^{2},y)\right)\Big{|}_{y=0}$
$\displaystyle\quad[u^{r}a^{k}]\frac{e^{\hbar u/2}+e^{-\hbar
u/2}}{u\hbar\mathcal{S}(u\hbar)}\exp\left(2\hbar
u\sum_{l\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}a^{-l}{\mathcal{S}}(l\hbar
u)J_{-l}\right)\exp\left(2\hbar
u\sum_{l\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}a^{l}{\mathcal{S}}(l\hbar
u)J_{l}\right).$

We also consider an arbitrary series
$\overline{y}(\hbar^{2},z)=\sum_{d=0}^{\infty}\hbar^{2d}y_{d}(z)$, where each
$y_{d}(z)$ is an odd formal power series in $z$. The constant term of this
series, $y_{0}=\overline{y}(0,z)$, is also denoted by $y=y(z)$. The prime
object of our interest in this and the subsequent sections are the
$\hbar$-expansions of the (disconnected) $n$-point functions

(4.14)
$H_{n}^{\bullet}=\left.\sum_{k_{1},\dots,k_{n}\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}\frac{\partial^{n}\tau(\bf
t,\hbar^{-1}{\bf s})}{\partial t_{k_{1}}\dots\partial t_{k_{1}}}\right|_{{\bf
t}={\bf 0}}\frac{X_{1}^{k_{1}}}{k_{1}}\dots\frac{X_{1}^{k_{n}}}{k_{n}}$

and

(4.15) $W_{n}^{\bullet}=D_{1}\cdots D_{n}H_{n}^{\bullet},$

where $D_{i}\coloneqq X_{i}\partial_{X_{i}}$. Then from the definition of the
operators ${\mathbb{J}}_{m}$ we have

(4.16)
$H_{n}^{\bullet}=\sum_{m_{1},\dots,m_{n}\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}\frac{X_{1}^{m_{1}}\dots
X_{n}^{m_{n}}}{m_{1}\dots
m_{n}}\left<0\right|{\mathbb{J}}_{m_{1}}\dots\mathbb{J}_{m_{n}}e^{\sum_{d=0}^{\infty}\hbar^{2d}\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}\frac{J_{-k}}{\hbar
k}[z^{k}]y_{d}(z)}\left|0\right>$

and

(4.17) $\displaystyle W_{n}^{\bullet}$
$\displaystyle=\sum_{m_{1},\dots,m_{n}\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}X_{1}^{m_{1}}\dots
X_{n}^{m_{n}}\left<0\right|\mathbb{J}_{m_{1}}\dots\mathbb{J}_{m_{n}}e^{\sum_{d=0}^{\infty}\hbar^{2d}\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}\frac{J_{-k}}{\hbar
k}[z^{k}]y_{d}(z)}\left|0\right>.$

Using the inclusion-exclusion formulas, we define the _connected_ $n$-point
functions $H_{n}$ and $W_{n}$, $n\geq 1$, and they expand in $\hbar$ as

(4.18) $\displaystyle H_{n}$
$\displaystyle=\sum_{g=0}^{\infty}H_{g,n}\hbar^{2g-2+n};$ $\displaystyle
W_{n}$ $\displaystyle=\sum_{g=0}^{\infty}W_{g,n}\hbar^{2g-2+n}.$

###  4.3. Preliminary formulas for $H_{n}^{\bullet}$ and $W_{n}^{\bullet}$

Denote

(4.19) $B(z,w)\coloneqq\frac{zw}{(z-w)^{2}}+\frac{zw}{(z+w)^{2}}.$

######  Proposition 4.2.

We have the following formula for $H_{n}^{\bullet}$:

(4.20) $\displaystyle
H_{n}^{\bullet}=\sum_{m_{1},\dots,m_{n}\in\mathbb{Z}_{\operatorname{odd}}^{+}}\prod_{i=1}^{n}\frac{1}{m_{i}}X_{i}^{m_{i}}\sum_{r_{1},\dots,r_{n}=0}^{\infty}\prod_{i=1}^{n}\partial_{y}^{r_{i}}\exp\left(2m_{i}\frac{{\mathcal{S}}(m_{i}\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}\right)\Big{|}_{y=0}$
$\displaystyle[\prod_{i=1}^{n}u_{i}^{r_{i}}z_{i}^{m_{i}}]\prod_{i=1}^{n}\frac{e^{\frac{\hbar
u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}\prod_{1\leq i<j\leq
n}\left(e^{\hbar^{2}u_{i}u_{j}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\mathcal{S}(\hbar
u_{j}z_{j}\partial_{z_{j}})B(z_{i},z_{j})}-1\right),$

where $\overline{\psi}=\overline{\psi}(\hbar^{2},y)$ and
$\overline{y}_{i}=\overline{y}(\hbar^{2},z_{i})$. An analogous formula for
$W_{n}^{\bullet}$ is obtained by replacing
$\prod_{i=1}^{n}\frac{1}{m_{i}}X_{i}^{m_{i}}$ in Equation (4.20) by
$\prod_{i=1}^{n}X_{i}^{m_{i}}$.

###### Proof.

This formula should be understood as an expansion in the sector
$|z_{1}|\ll|z_{2}|\ll\cdots\ll|z_{n}|\ll 1$, and it comes directly from the
commutation of the operators $\mathbb{J}_{k}$ given in Equation (4.13), once
one observes that

(4.21) $\left[J_{+}(z),J_{-}(w)\right]=\frac{1}{4}B(z,w).$

We refer also to an argument in [BDBKS22, Section 3.2], which does exactly the
same in a bit different situation. ∎

###  4.4. Closed algebraic formula for $W_{g,n}$

We use a change of variables of exactly the type as in [ACEH18], namely,

(4.22) $X=ze^{-2\psi(y(z))}.$

Define $D\coloneqq X\partial_{X}$ and define $Q$ by $D=Q^{-1}z\partial_{z}$.
In the case we have variables $X_{1},\dots,X_{n}$, we define $z_{i}$ by
$X_{i}\coloneqq X(z_{i})$, and furthermore we use the notation $D_{i}\coloneqq
X_{i}\partial_{X_{i}}$, $Q_{i}\coloneqq z_{i}/X_{i}\cdot dX_{i}/dz_{i}$,
$\overline{y}_{i}=\overline{y}(\hbar^{2},z_{i})$, $y_{i}=y(z_{i})$,
$\overline{\psi}_{i}=\psi(\hbar^{2},y_{i})$, and $\psi_{i}=\psi(y_{i})$.

######  Theorem 4.1.

For $g\geq 0$, $n\geq 2$, $2g-2+n>0$, we have:

(4.23) $\displaystyle
W_{g,n}=[\hbar^{2g-2+n}]\sum_{\begin{subarray}{c}j_{1},\dots,j_{n},\\\
r_{1},\dots,r_{n}=0\end{subarray}}^{\infty}\left[\prod_{i=1}^{n}D_{i}^{j_{i}}[t_{i}^{j_{i}}]\frac{1}{Q_{i}}e^{-2t_{i}\psi_{i}}\partial_{y_{i}}^{r_{i}}e^{2t_{i}\frac{{\mathcal{S}}(t_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}}[u_{i}^{r_{i}}]\right]$
$\displaystyle\prod_{i=1}^{n}\frac{e^{\frac{\hbar u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\left(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}-y_{i}\right)}\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right)$

Here $\Gamma_{n}$ is the set of all connected simple graphs on $n$ vertices
$v_{1},\dots,v_{n}$, and $E_{\gamma}$ is the set of edges of $\gamma$.

######  Remark 4.1.

It is an explicit closed algebraic formula of the same type as in [BDBKS22,
BDBKS20]. In particular the sum over ${j_{1},\dots,j_{n},r_{1},\dots,r_{n}}$
is finite for every $(g,n)$.

######  Remark 4.2.

Note that $y_{i}$ and hence $\partial_{y_{i}}$ are odd in $z_{i}$. Note also
that $\overline{\psi}_{i}$ and hence
$\frac{{\mathcal{S}}(t_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}$
are even in $z_{i}$. Note also that $D_{i}$ and $Q_{i}$ are even in $z_{i}$.
Note also that in the second line the coefficient of $u_{i}^{r_{i}}$ for odd
$r_{i}$ is even in $z_{i}$ and the coefficient of $u_{i}^{r_{i}}$ for even
$r_{i}$ is odd in $z_{i}$. These observations imply that the right hand side
of (4.23) is necessarily odd in $z_{1},\dots,z_{n}$.

######  Remark 4.3.

Note that the structure of the formula suggests that there might be non-
trivial poles along the diagonals $z_{i}=z_{j}$ and antidiagonals
$z_{i}=-z_{j}$, but in fact the statement of the theorem in particular implies
that these polar parts cancel and the resulting formula is non-singular at the
diagonals and antidiagonals. Cf. a discussion in [BDBKS22, Remark 1.3 and
Corollary 4.10].

###### Proof of Theorem 4.1.

Recall the formula for $W_{n}^{\bullet}$ in Proposition 4.2. Passing to the
connected $n$-point functions $W_{n}$ via inclusion-exclusion formula we
replace

(4.24) $\displaystyle\prod_{1\leq i<j\leq
n}\left(e^{\hbar^{2}u_{i}u_{j}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\mathcal{S}(\hbar
u_{j}z_{j}\partial_{z_{j}})B(z_{i},z_{j})}-1\right)$

with

(4.25) $\displaystyle\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right).$

Using the series expansion in $u_{1},\dots,u_{n}$ (cf. [BDBKS22, Lemma 4.5]),
which is applicable only if $n\geq 2$ (hence the restriction on $n$ in the
statement of the theorem) we can rewrite the formula as

(4.26) $\displaystyle W_{n}$
$\displaystyle=\sum_{m_{1},\dots,m_{n}\in\mathbb{Z}_{\operatorname{odd}}^{+}}\prod_{i=1}^{n}X_{i}^{m_{i}}\cdot\sum_{r_{1},\dots,r_{n}=0}^{\infty}\prod_{i=1}^{n}\partial_{y}^{r_{i}}e^{2m_{i}\frac{{\mathcal{S}}(m_{i}\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}_{i}}$
$\displaystyle\quad[\prod_{i=1}^{n}u_{i}^{r_{i}}z_{i}^{m_{i}}]\prod_{i=1}^{n}\frac{e^{\frac{\hbar
u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\left(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}-y_{i}\right)}$
$\displaystyle\quad\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right).$

The next observation that we use is the following. Replace the summation in
Equation (4.17) from
$\sum_{m_{1},\dots,m_{n}\in\mathbb{Z}_{\operatorname{odd}}^{+}}$ to
$\sum_{m_{1},\dots,m_{n}\in\mathbb{Z}_{\operatorname{odd}}}$. This replacement
changes the disconnected $W_{n}^{\bullet}$, but when we pass to the connected
ones, this replacement just changes the $W_{0,2}$ by adding to it the singular
term $\frac{1}{4}B(X_{1},X_{2})$ (hence the condition $2g-2+n>0$ in the
statement of the theorem), cf. [BDBKS22, Proposition 4.1]. With this
adjustment, we have for $g\geq 0$, $n\geq 2$, $2g-2+n>0$:

(4.27) $\displaystyle W_{g,n}$
$\displaystyle=[\hbar^{2g-2+n}]\sum_{m_{1},\dots,m_{n}\in\mathbb{Z}_{\operatorname{odd}}}\prod_{i=1}^{n}X_{i}^{m_{i}}[z_{i}^{m_{i}}]\cdot\sum_{r_{1},\dots,r_{n}=0}^{\infty}\prod_{i=1}^{n}\partial_{y}^{r_{i}}e^{2m_{i}\frac{{\mathcal{S}}(m_{i}\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}_{i}}$
$\displaystyle\quad[\prod_{i=1}^{n}u_{i}^{r_{i}}]\prod_{i=1}^{n}\frac{e^{\frac{\hbar
u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\left(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}-y_{i}\right)}$
$\displaystyle\quad\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right).$

Now note two things. First,
$\partial_{y}^{r_{i}}\exp\left(2m_{i}\frac{{\mathcal{S}}(m_{i}\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}_{i}\right)$
is even in $z_{i}$ for even $r_{i}$ and odd in $z_{i}$ for odd $r_{i}$. On the
other hand, in the expression

(4.28)
$\displaystyle[\prod_{i=1}^{n}u_{i}^{r_{i}}z_{i}^{m_{i}}]\prod_{i=1}^{n}\frac{e^{\frac{\hbar
u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\left(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}-y_{i}\right)}$
$\displaystyle\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right)$

we only have terms with $r_{i}+m_{i}$ odd. This means that the whole
expression in which we take the coefficients of
$[\prod_{i=1}^{n}z_{i}^{m_{i}}]$ is odd in $z_{1},\dots,z_{n}$, and thus we
can extend the summation to ${m_{1},\dots,m_{n}\in\mathbb{Z}}$. Second, we can
use the trick that for a polynomial $g$ in $m$ we can replace $\sum X^{m}g(m)$
by $\sum_{j=o}^{\infty}D^{j}X^{m}[t^{j}]g(t)$. These two ideas allow to
rewrite (4.27) as

(4.29) $\displaystyle W_{g,n}$
$\displaystyle=[\hbar^{2g-2+n}]\sum_{j_{1},\dots,j_{n}=0}^{\infty}D_{i}^{j_{i}}[t_{i}^{j_{i}}]\sum_{m_{1},\dots,m_{n}\in\mathbb{Z}}\prod_{i=1}^{n}X_{i}^{m_{i}}[z_{i}^{m_{i}}]e^{2m_{i}\psi_{i}}$
$\displaystyle\quad\sum_{r_{1},\dots,r_{n}=0}^{\infty}\prod_{i=1}^{n}e^{-2t_{i}\psi_{i}}\partial_{y}^{r_{i}}e^{2m_{i}\frac{{\mathcal{S}}(m_{i}\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}_{i}}$
$\displaystyle\quad[\prod_{i=1}^{n}u_{i}^{r_{i}}]\prod_{i=1}^{n}\frac{e^{\frac{\hbar
u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\left(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}-y_{i}\right)}$
$\displaystyle\quad\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right)$

(here for each $r_{1},\dots,r_{n}$ the second line expands in $\hbar$ with the
coefficients that are manifestly polynomial in $t_{1},\dots,t_{n}$). Finally,
we apply the Lagrange–Bührmann formula for $X_{i}=z_{i}e^{-2\psi_{i}}$ to
Equation (4.29) (cf. [BDBKS22, Lemma 4.7]) and obtain the statement of the
theorem. ∎

###  4.5. Special cases

In this section we discuss the formulas for $W_{g,n}$ for $(g,n)=(0,2)$ and
$n=1$, in the variable $z$ related to $X$ by $X=ze^{-2\psi(y(z))}$. We begin
with unstable terms $(g,n)=(0,2)$ and $(0,1)$.

######  Proposition 4.3.

For $(g,n)=(0,2)$ we have:

(4.30)
$W_{0,2}=\frac{1}{4Q_{1}Q_{2}}B(z_{1},z_{2})-\frac{1}{4}B(X_{1},X_{2}).$

###### Proof.

Indeed, as we discussed in the proof of Theorem 4.1, the change of summation
from $m\in Z^{+}_{\operatorname{odd}}$ to $m\in Z_{\operatorname{odd}}$ and
the commutation rules for $J_{+}(X_{1}),J_{-}(X_{2})$ imply that the $(0,2)$
term gets a correction. We have:

(4.31) $\displaystyle W_{0,2}+\frac{1}{4}B(X_{1},X_{2})$
$\displaystyle=\sum_{m_{1},m_{2}\in\mathbb{Z}_{\operatorname{odd}}}X_{1}^{m_{1}}X_{2}^{m_{2}}[z_{1}^{m_{1}}z_{2}^{m_{2}}]e^{2m_{1}\psi_{1}+2m_{2}\psi_{2}}\frac{1}{4}B(z_{1},z_{2})$
$\displaystyle=\sum_{m_{1},m_{2}\in\mathbb{Z}}X_{1}^{m_{1}}X_{2}^{m_{2}}[z_{1}^{m_{1}}z_{2}^{m_{2}}]e^{2m_{1}\psi_{1}+2m_{2}\psi_{2}}\frac{1}{4}B(z_{1},z_{2})$
$\displaystyle=\frac{1}{4Q_{1}Q_{2}}B(z_{1},z_{2}).$

In the second line, in order to change the summation from
$m_{1},m_{2}\in\mathbb{Z}_{\operatorname{odd}}$ to $m_{1},m_{2}\in\mathbb{Z}$,
we use that $B(z_{1},z_{2})$ is odd in $z_{1}$ and $z_{2}$ and
$\psi_{1}=\psi(y_{1})$ (respectively, $\psi_{2}=\psi(y_{2})$) is even in
$z_{1}$ (respectively, $z_{2}$). ∎

######  Proposition 4.4.

For $(g,n)=(0,1)$ we have: $W_{0,1}(X)=y(z)/2$.

###### Proof.

It is a straightforward computation. First, recall that

(4.32) $\displaystyle H_{0,1}(X)$
$\displaystyle=[\hbar^{-1}]\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}\frac{X^{m}}{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}(\hbar^{2},y)}\Big{|}_{y=0}[u^{r}z^{m}]\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar
u}{2}}}{4u\hbar\mathcal{S}(u\hbar)}e^{u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}(\hbar^{2},z)}$
$\displaystyle=\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}\frac{X^{m}}{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\psi(y)}\Big{|}_{y=0}[u^{r}z^{m}]\frac{e^{uy(z)}}{2u}.$

With this equation, in order to compute $W_{0,1}$, we consider its
differential. We have:

(4.33) $\displaystyle DW_{0,1}(X)$
$\displaystyle=\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\psi(y)}\Big{|}_{y=0}[u^{r}z^{m}]z\partial_{z}\frac{e^{uy(z)}}{2u}$
$\displaystyle=\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\psi(y)}\Big{|}_{y=0}[u^{r}z^{m}]\frac{e^{uy(z)}QDy(z)}{2}$
$\displaystyle=\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}[z^{m}]e^{2m\psi(y(z)))}\frac{QDy(z)}{2}$
$\displaystyle=\sum_{m\in\mathbb{Z}}X^{m}[z^{m}]e^{2m\psi(y(z)))}\frac{QDy(z)}{2}$
$\displaystyle=\frac{1}{2}Dy(z).$

Hence $W_{0,1}(X)=y(z)/2$. ∎

####  4.5.1. Stable terms for $n=1$

Consider $g\geq 1$, $n=1$, that is, we consider
$W_{1}=\sum_{g=0}^{\infty}\hbar^{2g-1}W_{g,1}(X)$.

######  Proposition 4.5.

Under the change of variables $X=ze^{-2\psi(y(z))}$ we have:

(4.34) $\displaystyle
W_{1}(X)=\frac{y}{2\hbar}+\sum_{j=1}^{\infty}D^{j-1}[t^{j}]e^{-2t\psi+2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\frac{Dy}{2\hbar}$
$\displaystyle+\sum_{j,r=0}^{\infty}D^{j}[t^{j}]\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar
u}{2}}}{4u\hbar\mathcal{S}(u\hbar)}e^{u\left(\mathcal{S}(\hbar
uz\partial_{z})\overline{y}-y\right)}\right).$

Here $y=y(z)$, $\overline{y}=\overline{y}(\hbar^{2},z)$, $\psi=\psi(y)$,
$\overline{\psi}=\overline{\psi}(\hbar^{2},y)$, and $D=X\partial_{X}$.

###### Proof.

By direct commutation of the operators, we have:

(4.35) $\displaystyle W_{1}(X)$
$\displaystyle=\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\Big{|}_{y=0}[u^{r}z^{m}]\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar
u}{2}}}{4u\hbar\mathcal{S}(u\hbar)}e^{u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}$
$\displaystyle=\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\Big{|}_{y=0}[u^{r}z^{m}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar
u}{2}}}{4u\hbar\mathcal{S}(u\hbar)}e^{u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}-\frac{e^{uy}}{2u\hbar}\right)$
$\displaystyle\quad+\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\Big{|}_{y=0}[u^{r}z^{m}]\frac{e^{uy}}{2u\hbar}$

The first summand is regular in $u$, so it can be computed as in the proof of
Theorem 4.1:

(4.36)
$\displaystyle\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\Big{|}_{y=0}[u^{r}z^{m}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar
u}{2}}}{4u\hbar\mathcal{S}(u\hbar)}e^{u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}-\frac{e^{uy}}{2u\hbar}\right)$
$\displaystyle=\sum_{m\in\mathbb{Z}}X^{m}[z^{m}]\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar
u}{2}}}{4u\hbar\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}-\frac{1}{2u\hbar}\right)$
$\displaystyle=\sum_{j,r=0}^{\infty}D^{j}[t^{j}]\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar
u}{2}}}{4u\hbar\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}\right).$

The second summand of (4.35) can be computed by differentiation.

(4.37) $\displaystyle
D\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\Big{|}_{y=0}[u^{r}z^{m}]\frac{e^{uy}}{2u\hbar}$
$\displaystyle=\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\Big{|}_{y=0}[u^{r}z^{m}]\frac{e^{uy(z)}QDy}{2\hbar}$
$\displaystyle=\sum_{m\in\mathbb{Z}_{\operatorname{odd}}^{+}}X^{m}\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}z^{m}]\frac{QDy}{2\hbar}=\sum_{m\in\mathbb{Z}}X^{m}[z^{m}]\sum_{r=0}^{\infty}\partial_{y}^{r}e^{2m\frac{{\mathcal{S}}(m\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\frac{QDy}{2\hbar}$
$\displaystyle=\sum_{j=0}^{\infty}D^{j}[t^{j}]e^{-2t\psi+2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\frac{Dy}{2\hbar}=\frac{Dy}{2\hbar}+D\sum_{j=1}^{\infty}D^{j-1}[t^{j}]e^{-2t\psi+2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\frac{Dy}{2\hbar}.$

Combining these two computations, we obtain the statement of the proposition.
∎

##  5\. Loop equations

Consider the change of variables $X=ze^{-2\psi(y(z))}$. In this section we
make a number of extra assumptions of analytical nature on the coefficients
$\psi_{2d}$ and $y_{2d}$ of the $\hbar^{2}$-expansions of $\overline{\psi}$
and $\overline{y}$, and with these assumptions we prove the linear and
quadratic loop equations for the $W_{g,n}$’s that we computed in the closed
form in the previous section (or, more precisely, for the symmetric
differentials that we construct from $W_{g,n}$’s).

###  5.1. Assumptions

Let $z$ be a global affine coordinate on $\mathbb{C}\mathrm{P}^{1}$. We assume
that $\psi^{\prime}(y(z))$ and $y^{\prime}(z)$ can be analytically extended to
rational functions on $\mathbb{C}\mathrm{P}^{1}$. These assumptions imply that
$X=ze^{-2\psi(y(z))}$ extends to a global function on
$\mathbb{C}\mathrm{P}^{1}$ and $d\log X$ is a rational $1$-form in the global
coordinate $z$.

The rational $1$-form $d\log X$ has a finite number of zeros, and we assume
that all zeros of $d\log X$ are simple. We also assume that the coefficients
of the positive degrees of $\hbar$ in the series
$\overline{\psi}(\hbar^{2},y(z))$ and $\overline{y}(\hbar^{2},z)$ are rational
functions in $z$ as well, and their singular points are disjoint from the
zeros of $d\log X$.

######  Proposition 5.1.

Under these assumptions the symmetric $n$-differentials

(5.1) $\omega_{g,n}\coloneqq
2^{1-g}W_{g,n}(X_{1},\dots,X_{n})\prod_{i=1}^{n}d\log
X_{i}+\delta_{g,0}\delta_{n,2}\frac{1}{2}B(X_{1},X_{2})d\log X_{1}d\log
X_{2},\quad g\geq 0,n\geq 1,$

analytically extend to global rational differentials on
$(\mathbb{C}\mathrm{P}^{1})^{n}$ for $2g-2+n>0$.

###### Proof.

This statement follows directly from the structure of formulas given in
Equations (4.23) and (4.34). ∎

Note the factor $2^{1-g}$. It is a compensation for the fact that the natural
$W_{0,1}$ and $W_{0,2}$ that we obtained in the previous section are twice
less than the formulas one might expect from the point of view of the spectral
curve topological recursion, see Section 7.1.

###  5.2. Blobbed topological recursion

Let $p$ be a simple zero point of $d\log X$. Let $\sigma$ denote the deck
transformation of $X$ near $p$.

######  Definition 5.1.

We say that the system of symmetric $n$-differentials
$\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$ satisfies the linear loop equations at
$p$ if for any $g\geq 0$, $n\geq 0$,

(5.2) $\omega_{g,n+1}(w,z_{\llbracket
n\rrbracket})+\omega_{g,n+1}(\sigma(w),z_{\llbracket n\rrbracket})$

is holomorphic at $w\to p$ and vanishes at $w=p$.

We say that the system of symmetric $n$-differentials
$\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$ satisfies the quadratic loop equations
at $p$ if for any $g\geq 0$, $n\geq 0$, $(g,n)\not=(1,0)$,

(5.3) $\omega_{g-1,n+2}(w,\sigma(w),z_{\llbracket
n\rrbracket})+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\ I_{1}\sqcup
I_{2}=\llbracket
n\rrbracket\end{subarray}}\omega_{g_{1},n_{1}+1}(w,z_{I_{2}})\omega_{g_{2},n_{2}+1}(\sigma(w),z_{I_{2}})$

is holomorphic at $w\to p$ and has a zero of order at least two at $w=p$. In
the case $(g,n)=(1,0)$ we require the same property, but we remove the
singularity from the first summand, that is, we replace
$\omega_{0,2}(w,\sigma(w))$ with

(5.4) $\displaystyle\left(\omega_{0,2}-\frac{1}{2}B(X_{1},X_{2})d\log
X_{1}d\log X_{2}\right)|_{X_{1}=X(w),X_{2}=X(\sigma(w))}$
$\displaystyle=2W_{0,2}(X_{1},X_{2})d\log X_{1}d\log
X_{2}|_{X_{1}=X(w),X_{2}=X(\sigma(w))}.$

If all zero points of $d\log X$ are simple and $\omega_{g,n}$’s satisfy the
linear and quadratic loop equations at each of them, then we say that the
system of symmetric $n$-differentials $\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$
satisfies the blobbed topological recursion [BS17].

######  Theorem 5.1.

Under the analytic assumptions listed in Section 5.1 the system of symmetric
differentials (5.1) satisfies the blobbed topological recursion.

###  5.3. Proof of Theorem 5.1

Consider the connected correlation function defined as

(5.5) $\displaystyle\mathcal{W}_{g,n}=\ $
$\displaystyle[\hbar^{2g-2+n}]\sum_{m_{1},\dots,m_{n}\in{\mathbb{Z}}_{\operatorname{odd}}}X_{1}^{m_{1}}\dots
X_{n}^{m_{n}}$ $\displaystyle\left<0\right|\frac{J_{m_{1}}}{\hbar
m_{1}}[a^{0}]\mathcal{E}(\hbar v,a)J_{m_{2}}\dots
J_{m_{n}}{\mathcal{D}}e^{\sum_{d=0}^{\infty}\hbar^{2d}\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}\frac{J_{-k}}{\hbar
k}[z^{k}]y_{d}(z)}\left|0\right>^{\circ}.$

(here by $\left<0\right|-\left|0\right>^{\circ}$ we mean the connected vacuum
expectation obtained by inclusion-exclusion formula from the disconnected
one).

######  Lemma 5.2.

For $\sum_{m\in{\mathbb{Z}}_{\operatorname{odd}}}J_{m}X^{m}/(\hbar
m)=\sum_{m\in{\mathbb{Z}}}J_{m}X^{m}/(\hbar m)$ we have:

(5.6) $\displaystyle\left[\sum_{m\in{\mathbb{Z}}}\frac{J_{m}X^{m}}{\hbar
m},[a^{0}]\mathcal{E}(\hbar v,a)\right]=v{\mathcal{S}}(\hbar
vX\partial_{X})\left(\frac{\mathcal{E}(\hbar v,X)+\mathcal{E}(-\hbar
v,X)}{2}\right).$

###### Proof.

A straightforward computation using Equations (4.4) and (2.25). ∎

######  Remark 5.1.

Note that $\mathcal{E}(u,X)=-\mathcal{E}(-u,-X)$. Hence the right hand side of
Equation (5.6) is odd in $X$. Note also that the right hand side of Equation
(5.6) is manifestly odd in $v$.

Our next goal is to compute the coefficients of $v^{1}$ and $v^{3}$ in (5.6)
applied to the covacuum.

######  Lemma 5.3.

We have:

(5.7) $\displaystyle\left<0\right|[v^{1}]v{\mathcal{S}}(\hbar
vX\partial_{X})\left(\frac{\mathcal{E}(\hbar v,X)+\mathcal{E}(-\hbar
v,X)}{2}\right)$
$\displaystyle=2\left<0\right|\sum_{m\in\mathbb{Z}^{+}_{\operatorname{odd}}}X^{m}J_{m};$
(5.8) $\displaystyle\left<0\right|[v^{3}]v{\mathcal{S}}(\hbar
vX\partial_{X})\left(\frac{\mathcal{E}(\hbar v,X)+\mathcal{E}(-\hbar
v,X)}{2}\right)$
$\displaystyle=\hbar^{2}\left(\frac{1}{6}(X\partial_{X})^{2}+\frac{1}{6}\right)\left<0\right|\sum_{m\in\mathbb{Z}^{+}_{\operatorname{odd}}}X^{m}J_{m}$
$\displaystyle\quad+\hbar^{2}\frac{4}{3}\left<0\right|\Big{(}\sum_{m\in\mathbb{Z}^{+}_{\operatorname{odd}}}X^{m}J_{m}\Big{)}^{3}$

###### Proof.

A straightforward computation using Equation (4.4). ∎

######  Corollary 5.4.

We have:

(5.9) $\displaystyle\mathcal{W}_{g,n}(X_{\llbracket n\rrbracket})={}$
$\displaystyle v\Bigg{(}2W_{g,n}(X_{\llbracket
n\rrbracket})\Bigg{)}+v^{3}\Bigg{(}\frac{4}{3}\Big{(}W_{g-2,n+2}(X_{1},X_{1},X_{1},X_{\llbracket
n\rrbracket\setminus 1})$
$\displaystyle+3\sum_{\begin{subarray}{c}g_{1}+g_{2}=g-1\\\ I_{1}\sqcup
I_{2}=\llbracket n\rrbracket\setminus
1\end{subarray}}W_{g_{1},n_{1}+1}(X_{1},X_{I_{1}})W_{g_{2},n_{2}+2}(X_{1},X_{1},X_{I_{2}})$
$\displaystyle+\sum_{\begin{subarray}{c}g_{1}+g_{2}+g_{3}=g\\\ I_{1}\sqcup
I_{2}\sqcup I_{3}=\llbracket n\rrbracket\setminus
1\end{subarray}}W_{g_{1},n_{1}+1}(X_{1},X_{I_{1}})W_{g_{2},n_{2}+1}(X_{1},X_{I_{2}})W_{g_{3},n_{3}+1}(X_{1},X_{I_{3}})\Big{)}$
$\displaystyle+\left(\frac{1}{6}(X_{1}\partial_{X_{1}})^{2}+\frac{1}{6}\right)W_{g-1,n}(X_{\llbracket
n\rrbracket})\Bigg{)}+O(v^{5}),$

where we have to substitue $W_{0,2}(X_{i},X_{j})+\frac{1}{4}B(X_{i},X_{j})$
instead of $W_{0,2}(X_{i},X_{j})$ in all instances when the arguments are not
the same, that is, $i\not=j$.

Now, repeating _mutatis mutandis_ the arguments of the proofs of Theorem 4.1
and Propositions 4.3, 4.4, and 4.5, we obtain closed algebraic formulas for
$\mathcal{W}_{g,n}$.

######  Lemma 5.5.

Under the change of variables $X=ze^{-2\psi(y(z))}$ we have:

(5.10) $\displaystyle\mathcal{W}_{g,n}(X_{\llbracket
n\rrbracket})=[\hbar^{2g-2+n}]\sum_{\begin{subarray}{c}j_{1},\dots,j_{n},\\\
r_{1},\dots,r_{n}=0\end{subarray}}^{\infty}\left[\prod_{i=2}^{n}D_{i}^{j_{i}}[t_{i}^{j_{i}}]\frac{1}{Q_{i}}e^{-2t_{i}\psi_{i}}\partial_{y_{i}}^{r_{i}}e^{2t_{i}\frac{{\mathcal{S}}(t_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}}[u_{i}^{r_{i}}]\right]$
$\displaystyle\left[D_{1}^{j_{1}}[t_{1}^{j_{1}}]\frac{1}{Q_{1}}e^{-2t_{1}\psi_{1}}\partial_{y_{1}}^{r_{1}}e^{2t_{1}\frac{{\mathcal{S}}(t_{1}\hbar\partial_{y_{1}})}{\mathcal{S}(\hbar\partial_{y_{1}})}\overline{\psi}_{1}}v\mathcal{S}(t_{1}\hbar\partial_{y_{1}})(e^{vy_{1}}+e^{-vy_{1}})[u_{1}^{r_{1}}]\right]$
$\displaystyle\prod_{i=1}^{n}\frac{e^{\frac{\hbar u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\left(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}-y_{i}\right)}\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right)$

for $n\geq 2$, $(g,n)\not=(0,2)$. Here $\Gamma_{n}$ is the set of all
connected simple graphs on $n$ vertices $v_{1},\dots,v_{n}$, and $E_{\gamma}$
is the set of edges of $\gamma$.

In the case $(g,n)=(0,2)$ we have

(5.11)
$\mathcal{W}_{0,2}=\frac{v(e^{vy_{1}}+e^{-vy_{1}})}{4Q_{1}Q_{2}}B(z_{1},z_{2})$

In the case $(g,n)=(0,1)$ we have

(5.12) $\mathcal{W}_{0,1}=\frac{1}{2}(e^{vy_{1}}-e^{-vy_{1}}).$

In the case $g\geq 1$, $n=1$ we have

(5.13) $\displaystyle\mathcal{W}_{g,1}={}$
$\displaystyle[\hbar^{2g-1}]\sum_{\begin{subarray}{c}j_{1},r_{1}=0\end{subarray}}^{\infty}D_{1}^{j_{1}}[t_{1}^{j_{1}}]\frac{1}{Q_{1}}e^{-2t_{1}\psi_{1}}\partial_{y_{1}}^{r_{1}}e^{2t_{1}\frac{{\mathcal{S}}(t_{1}\hbar\partial_{y_{1}})}{\mathcal{S}(\hbar\partial_{y_{1}})}\overline{\psi}_{1}}v\mathcal{S}(t_{1}\hbar\partial_{y_{1}})(e^{vy_{1}}+e^{-vy_{1}})$
$\displaystyle\qquad\qquad\qquad[u_{1}^{r_{1}}]\frac{e^{\frac{\hbar
u_{1}}{2}}+e^{-\frac{\hbar
u_{1}}{2}}}{4u_{1}\hbar\mathcal{S}(u_{1}\hbar)}e^{-u_{1}y_{1}+u_{1}\mathcal{S}(\hbar
u_{1}z_{1}\partial_{z_{1}})\overline{y}_{1}}$
$\displaystyle+[\hbar^{2g-1}]\sum_{j_{1}=1}^{\infty}D^{j_{1}-1}[t^{j_{1}}]e^{-2t_{1}\psi_{1}+2t_{1}\frac{{\mathcal{S}}(t_{1}\hbar\partial_{y_{1}})}{\mathcal{S}(\hbar\partial_{y_{1}})}\overline{\psi}_{1}}v\mathcal{S}(t_{1}\hbar\partial_{y_{1}})(e^{vy_{1}}+e^{-vy_{1}})\frac{Dy}{2}.$

In all these formulas we use $y_{i}=y(z_{i})$,
$\overline{y}_{i}=\overline{y}(\hbar^{2},z_{i})$, $\psi_{i}=\psi(y_{i})$,
$\overline{\psi}_{i}=\overline{\psi}(\hbar^{2},y_{i})$, $Q_{i}=Q(z_{i})$,
$X_{i}=X(z_{i})$,
$D_{i}=X_{i}\partial_{X_{i}}=Q_{i}^{-1}z_{i}\partial_{z_{i}}$.

###### Proof.

First, observe that $[a^{0}]\mathcal{E}(\hbar v,a)$ can be represented as the
coefficient of $[\epsilon^{1}]$ in the expression

(5.14) $\displaystyle e^{\epsilon[a^{0}]\mathcal{E}(\hbar
v,a)}=e^{\epsilon\sum_{k\in\mathbb{Z}}(-1)^{k}e^{\hbar
vk}:\phi_{k}\phi_{-k}:}=e^{\epsilon\sum_{k\in\mathbb{Z}_{+}}(-1)^{k}(e^{\hbar
vk}-e^{-\hbar vk}):\phi_{k}\phi_{-k}:}$

For $T(k)=\frac{1}{2\hbar}(e^{\hbar vk}-e^{-\hbar vk})$ we have
$T(k+1)-T(k)=\Delta\overline{\psi}(\hbar^{2},\hbar(k+\frac{1}{2}))$, where

(5.15)
$\displaystyle\Delta\overline{\psi}(\hbar^{2},y)=\frac{1}{2\hbar}(e^{\frac{\hbar
v}{2}}-e^{-\frac{hv}{2}})(e^{vy}+e^{-vy})=v+v^{3}\left(\frac{\hbar^{2}}{24}+\frac{y^{2}}{2}\right)+O(v^{5}).$

Define $\widetilde{\mathbb{J}}_{k}$ as the conjugation of ${\mathbb{J}}_{k}$
with the operator given in (5.14). It is operator of exactly the same type as
${\mathbb{J}}_{k}$, we just replace $\overline{\psi}(\hbar^{2},y)$ by
$\overline{\psi}(\hbar^{2},y)+\epsilon\Delta\overline{\psi}(\hbar^{2},y)$ in
its definition. By (4.13) we have:

(5.16) $\displaystyle\widetilde{\mathbb{J}}_{k}$
$\displaystyle=\frac{1}{4}\sum_{r=0}^{\infty}\partial_{y}^{r}\exp\left(2k\frac{{\mathcal{S}}(k\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}(\overline{\psi}(\hbar^{2},y)+\epsilon\Delta\overline{\psi}(\hbar^{2},y))\right)\Big{|}_{y=0}$
$\displaystyle\quad[u^{r}a^{k}]\frac{e^{\hbar u/2}+e^{-\hbar
u/2}}{u\hbar\mathcal{S}(u\hbar)}\exp\left(2\hbar
u\sum_{l\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}a^{-l}{\mathcal{S}}(l\hbar
u)J_{-l}\right)\exp\left(2\hbar
u\sum_{l\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}a^{l}{\mathcal{S}}(l\hbar
u)J_{l}\right).$

Let $\mathcal{W}_{n}=\sum_{g=0}^{\infty}\hbar^{2g-2+n}\mathcal{W}_{g,n}$. Then

(5.17) $\displaystyle\mathcal{W}_{n}(X_{\llbracket n\rrbracket})$
$\displaystyle=[\epsilon^{1}]\sum_{m_{1},\dots,m_{n}\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}\frac{1}{m_{1}}X_{1}^{m_{1}}\dots
X_{n}^{m_{n}}$
$\displaystyle\quad\left<0\right|\widetilde{\mathbb{J}}_{m_{1}}\dots\mathbb{J}_{m_{n}}e^{\sum_{d=0}^{\infty}\hbar^{2d}\sum_{k\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}\frac{J_{-k}}{\hbar
k}[z^{k}]y_{d}(z)}\left|0\right>^{\circ}.$

By commutation of the operators, we have:

(5.18) $\displaystyle\mathcal{W}_{n}={}$
$\displaystyle\sum_{m_{1},\dots,m_{n}\in\mathbb{Z}_{\operatorname{odd}}^{+}}\prod_{i=1}^{n}X_{i}^{m_{i}}\cdot\sum_{r_{1},\dots,r_{n}=0}^{\infty}\frac{1}{m_{1}}[\epsilon^{1}]\partial_{y}^{r_{1}}\exp\left(2m_{1}\frac{{\mathcal{S}}(m_{1}\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}(\overline{\psi}+\epsilon\Delta\overline{\psi}(\hbar^{2},y))\right)\Big{|}_{y=0}$
$\displaystyle\prod_{i=2}^{n}\partial_{y}^{r_{i}}\exp\left(2m_{i}\frac{{\mathcal{S}}(m_{i}\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}\right)\Big{|}_{y=0}[\prod_{i=1}^{n}u_{i}^{r_{i}}z_{i}^{m_{i}}]\prod_{i=1}^{n}\frac{e^{\frac{\hbar
u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}$
$\displaystyle\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right),$

where $\Gamma_{n}$ is the set of all connected simple graphs on $n$ vertices
$v_{1},\dots,v_{n}$, and $E_{\gamma}$ is the set of edges of $\gamma$.

As in the proof of Theorem 4.1 and Propositions 4.4 and 4.5, we use that for
any formal power series $G(y)$ in $y$ and $F(u)$ in $u$, we have

(5.19)
$\displaystyle\sum_{r=0}^{\infty}\partial_{y}^{r}G(y)|_{y=0}[u^{r}]F(u)=\sum_{r=0}^{\infty}\partial_{y}^{r}G(y)[u^{r}]e^{-uy}F(u)$

([BDBKS22, Lemma 4.5]). Applying it to Equation (5.18) in the cases $n\geq 2$,
$(g,n)\not=(0,2)$ (these cases have to be treated separately, it is the same
situation as in the proof of Theorem 4.1), we obtain:

(5.20) $\displaystyle\mathcal{W}_{n}={}$
$\displaystyle\sum_{m_{1},\dots,m_{n}\in\mathbb{Z}_{\operatorname{odd}}^{+}}\prod_{i=1}^{n}X_{i}^{m_{i}}[z_{i}^{m_{i}}]e^{2m_{i}\psi_{i}}\cdot\sum_{r_{1},\dots,r_{n}=0}^{\infty}$
$\displaystyle
e^{-2m_{1}\psi_{1}}\partial_{y_{1}}^{r_{1}}\left(\exp\left(2m_{1}\frac{{\mathcal{S}}(m_{1}\hbar\partial_{y_{1}})}{\mathcal{S}(\hbar\partial_{y_{1}})}\overline{\psi}_{1}\right)v\mathcal{S}(m_{1}\hbar\partial_{y_{1}})(e^{vy_{1}}+e^{-vy_{1}})\right)$
$\displaystyle\prod_{i=2}^{n}e^{-2m_{i}\psi_{i}}\partial_{y_{i}}^{r_{i}}\exp\left(2m_{i}\frac{{\mathcal{S}}(m_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}\right)[\prod_{i=1}^{n}u_{i}^{r_{i}}]\prod_{i=1}^{n}\frac{e^{\frac{\hbar
u_{i}}{2}}+e^{-\frac{\hbar
u_{i}}{2}}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{u_{i}\left(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}-y\right)}$
$\displaystyle\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right),$

Starting from this point all further steps just repeat the computations made
in the proof of Theorem 4.1. We use three ideas:

  * •

extend the summation to $m_{1},\dots,m_{n}\in\mathbb{Z}$;

  * •

capture the polynomial dependence on $m_{1},\dots,m_{n}$ replacing their
entrances by $t_{1},\dots,t_{n}$ and applying
$\prod_{i=1}^{n}\sum_{j_{i}=1}^{\infty}D_{i}^{j_{i}}[t_{i}^{j_{i}}]$;

  * •

apply Lagrange–Bührmann formula for the change of variables.

This completes the proof of Equation (5.10). All other equations stated in the
lemma are obtained by small variations of this argument, which repeat the
corresponding special cases in the proofs of Propositions 4.3, 4.4, and 4.5. ∎

######  Corollary 5.6.

The functions $\mathcal{W}_{g,n}$, $g\geq 0$, $n\geq 1$, are formal power
series in $v$, whose coefficients are rational functions in the variables
$z_{\llbracket n\rrbracket}$, that near each simple zero point $p$ of $d\log
X$ satisfy the property that

(5.21) $\mathcal{W}_{g,n}(z_{1},z_{\llbracket n\rrbracket\setminus
1})+\mathcal{W}_{g,n}(\sigma(z_{1}),z_{\llbracket n\rrbracket\setminus 1})$

is holomorphic at $z_{1}\to p$. Here $\sigma$ is the deck transformation of
$X$ at $p$.

###### Proof.

This follows directly from the structure of the formulas in Lemma 5.5. We
apply $D_{1}^{j_{1}}$ to a rational function that has a simple pole at $w\to
p$ (coming from the factor $1/Q_{1}$). A function with at most simple pole
automatically satisfies (5.21), and the operator $D_{1}$ preserves this
property. ∎

###### Proof of Theorem 5.1.

Fix a zero point $p$ of $d\log X$ (which by assumption is simple) and let
$\sigma$ be the deck transformation of $X$ near this point. For any function
$f(z)$ defined in the neighborhood of $p$ we define

(5.22) $\mathsf{S}_{z}f(z)=f(z)+f(\sigma(z)).$

Then the linear loop equations at the point $p$ for the symmetric
differentials expressed as in Equation (5.1) can be equivalently rewritten as

(5.23) $\displaystyle\mathsf{S}_{z_{1}}W_{g,n}(z_{\llbracket n\rrbracket})$

is holomorphic at $z\to p$ for any $(g,n)$. Corollary 5.6 applied to the
coefficients of $[v^{1}]$ in $\mathcal{W}_{g,n}$ implies that it is indeed the
case.

Note also that Corollary 5.6 applied to the coefficients of $[v^{3}]$ in
$\mathcal{W}_{g,n}$ implies that $\mathsf{S}_{z_{1}}[v^{3}]\mathcal{W}_{g,n}$
is holomorphic at $z\to p$ for any $(g,n)$. Using explicit formula for
$[v^{3}]2^{3-g}\mathcal{W}_{g,n}$ given in Equation (5.9) and the linear loop
equations, we conclude that

(5.24) $\displaystyle\mathsf{S}_{z_{1}}\Big{(}$ $\displaystyle
2^{1-(g-2)}W_{g-2,n+2}(z_{1},z_{1},z_{1},z_{\llbracket n\rrbracket\setminus
1})$ $\displaystyle+3\sum_{\begin{subarray}{c}g_{1}+g_{2}=g-1\\\ I_{1}\sqcup
I_{2}=\llbracket n\rrbracket\setminus
1\end{subarray}}2^{1-g_{1}}W_{g_{1},n_{1}+1}(z_{1},z_{I_{1}})2^{1-g_{2}}W_{g_{2},n_{2}+2}(z_{1},z_{1},z_{I_{2}})$
$\displaystyle+\sum_{\begin{subarray}{c}g_{1}+g_{2}+g_{3}=g\\\ I_{1}\sqcup
I_{2}\sqcup I_{3}=\llbracket n\rrbracket\setminus
1\end{subarray}}2^{1-g_{1}}W_{g_{1},n_{1}+1}(z_{1},z_{I_{1}})2^{1-g_{3}}W_{g_{2},n_{2}+1}(z_{1},z_{I_{2}})2^{1-g_{3}}W_{g_{3},n_{3}+1}(z_{1},z_{I_{3}})\Big{)}$

is holomorphic at $z\to p$ for any $(g,n)$. Here we abuse the notation a
little bit since each time we use $2W_{0,2}(z_{i},z_{j})$ with $i\not=j$, we
actually mean $\frac{1}{2}B(z_{i},z_{j})$.

This particular system of equations is studied in a bit different situation in
[BKL+21, Lemma 20]. The main difference between our situation and the one
studied in [BKL+21, Lemma 20] is the choice of $B$, which is the standard
Bergman kernel in [BKL+21], but it does not affect the proof in any step.
Another difference is the rescaling of $W_{g,n}$ by $2^{1-g}$ in the
definition of $\omega_{g,n}$’s, but both (5.24) and the quadratic loop
equations are homogeneous with respect to this rescaling.

So, adjusted in our situation [BKL+21, Lemma 20] proves that the holomorphy of
the expression given in (5.24) implies the quadratic loop equations for the
symmetric differentials $\omega_{g,n}$ given by Equation (5.1), under the
condition that $y$ does not vanish at $z=p$. The latter condition is obviously
satisfied in our situation. Indeed, the point $p$ satisfies the equation
$1-2p\psi^{\prime}(y(p))y^{\prime}(p)=0$. On the other hand, $\psi^{\prime}$
is an odd function in $y$, so at any point $z$ where $y(z)=0$, we have
$1-2z\psi^{\prime}(y(z))y^{\prime}(z)=1$. Therefore, $y$ does not vanish at
$z=p$. Hence the symmetric differentials $\omega_{g,n}$ satisfy the quadratic
loop equations. ∎

##  6\. Formulas for $H_{g,n}$

In this section we derive expressions for $H_{g,n}$ by integration of the
earlier derived expressions for $W_{g,n}$. Since the case of $W_{g,1}$ was a
bit special, we firstly perform a separate computation for $H_{g,1}$.

######  Proposition 6.1.

For $g\geq 1$ we have:

(6.1) $\displaystyle
H_{g,1}=[\hbar^{2g}]\sum_{j=2}^{\infty}D^{j-2}[t^{j}]e^{-2t\psi+2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\frac{Dy}{2}$
$\displaystyle+[\hbar^{2g}]\sum_{j=1}^{\infty}D^{j-1}[t^{j}]\sum_{r=0}^{\infty}\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}\right)$
$\displaystyle+[\hbar^{2g}]\int_{0}^{z}\left(\frac{1}{S(\hbar\partial_{y})}\overline{\psi}-\psi\right)y^{\prime}dz+[\hbar^{2g}]\int_{0}^{z}\frac{dz}{2z}\left(\overline{y}-y\right).$

Here, as usual, we use $y=y(z)$, $\overline{y}=\overline{y}(\hbar^{2},z)$,
$\psi=\psi(y)=\psi(y(z))$,
$\overline{\psi}=\overline{\psi}(\hbar^{2},y)=\overline{\psi}(\hbar^{2},y(z))$,
$Q=Q(z)$, $X=X(z)$, $D=X\partial_{X}=Q^{-1}z\partial_{z}$.

###### Proof.

Recall that for $g\geq 1$

(6.2) $\displaystyle W_{g,1}=\ $
$\displaystyle[\hbar^{2g}]\sum_{j=1}^{\infty}D^{j-1}[t^{j}]e^{-2t\psi+2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\frac{Dy}{2}$
$\displaystyle+[\hbar^{2g}]\sum_{j,r=0}^{\infty}D^{j}[t^{j}]\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}\right).$

Hence, for $g\geq 1$

(6.3) $\displaystyle
H_{g,1}=[\hbar^{2g}]\sum_{j=2}^{\infty}D^{j-2}[t^{j}]e^{-2t\psi+2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\frac{Dy}{2}$
$\displaystyle+[\hbar^{2g}]\sum_{j=1}^{\infty}D^{j-1}[t^{j}]\sum_{r=0}^{\infty}\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}\right)$
$\displaystyle+[\hbar^{2g}]\int_{0}^{z}dz\frac{Q}{z}[t^{1}]e^{-2t\psi+2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\frac{Dy}{2}$
$\displaystyle+[\hbar^{2g}]\int_{0}^{z}dz\frac{Q}{z}[t^{0}]\sum_{r=0}^{\infty}\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}\right)$

(note that the constant term in $z$ of this expression vanishes). The third
term here can be computed as

(6.4)
$\displaystyle[\hbar^{2g}]\int_{0}^{z}dz\frac{Q}{z}[t^{1}]e^{-2t\psi+2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}\frac{Dy}{2}$
$\displaystyle=[\hbar^{2g}]\int_{0}^{z}dz\left(\frac{1}{S(\hbar\partial_{y})}\overline{\psi}-\psi\right)\frac{QDy}{z}$
$\displaystyle=[\hbar^{2g}]\int_{0}^{z}\left(\frac{1}{S(\hbar\partial_{y})}\overline{\psi}-\psi\right)y^{\prime}dz.$

The fourth term can be computed as

(6.5)
$\displaystyle[\hbar^{2g}]\int_{0}^{z}dz\frac{Q}{z}[t^{0}]\sum_{r=0}^{\infty}\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}\right)$
$\displaystyle=[\hbar^{2g}]\int_{0}^{z}\frac{dz}{z}[u^{0}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}\right)$
$\displaystyle=[\hbar^{2g}]\int_{0}^{z}\frac{dz}{2z}\left(\overline{y}-y\right)=[\hbar^{2g}]\int_{0}^{z}\frac{dz}{2z}\left(\overline{y}-y\right).$

Combining these formulas, we obtain the statement of the proposition. ∎

In the case $n=2$ we have the following formula for $H_{g,2}$.

######  Proposition 6.2.

In the case $g=0$ we have

(6.6) $\displaystyle
H_{0,2}=\frac{1}{4}\log\frac{(z_{1}-z_{2})(X_{1}+X_{2})}{(z_{1}+z_{2})(X_{1}-X_{2})}.$

For $g\geq 0$ we have:

(6.7) $\displaystyle H_{g,2}=\ $
$\displaystyle[\hbar^{2g}]\sum_{\begin{subarray}{c}j_{1},j_{2}=1\\\
r_{1},r_{2}=0\end{subarray}}^{\infty}\Bigg{[}\prod_{i=1}^{2}D_{i}^{j_{i}-1}[t_{i}^{j_{i}}]\frac{1}{Q_{i}}e^{-2t_{i}\psi_{i}}\partial_{y_{i}}^{r_{i}}e^{2t_{i}\frac{{\mathcal{S}}(t_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}}[u_{i}^{r_{i}}]$
$\displaystyle\frac{e^{\hbar u_{i}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}\Bigg{]}\left(e^{\hbar^{2}u_{1}u_{2}\mathcal{S}(\hbar
u_{1}z_{1}\partial_{z_{1}})\mathcal{S}(\hbar
u_{2}z_{2}\partial_{z_{2}})B(z_{1},z_{2})}-1\right)$
$\displaystyle+[\hbar^{2g}]\sum_{\begin{subarray}{c}j=1\\\
r=0\end{subarray}}^{\infty}\Bigg{[}D_{1}^{j-1}[t^{j}]\frac{1}{Q_{1}}e^{-2t\psi_{1}}\partial_{y_{1}}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y_{1}})}{\mathcal{S}(\hbar\partial_{y_{1}})}\overline{\psi}_{1}}[u^{r}]$
$\displaystyle\frac{e^{\hbar u/2}+e^{-\hbar
u/2}}{4u\hbar\mathcal{S}(u\hbar)}e^{-uy_{1}+u\mathcal{S}(\hbar
uz_{1}\partial_{z_{1}})\overline{y}_{1}}\Bigg{]}\frac{1}{2}\hbar
u\mathcal{S}(\hbar
uz_{1}\partial_{z_{1}})\Big{(}\frac{z_{1}}{z_{1}-z_{2}}-\frac{z_{1}}{z_{1}+z_{2}}\Big{)}$
$\displaystyle+[\hbar^{2g}]\sum_{\begin{subarray}{c}j=1\\\
r=0\end{subarray}}^{\infty}\Bigg{[}D_{2}^{j-1}[t^{j}]\frac{1}{Q_{2}}e^{-2t\psi_{2}}\partial_{y_{2}}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y_{2}})}{\mathcal{S}(\hbar\partial_{y_{2}})}\overline{\psi}_{2}}[u^{r}]$
$\displaystyle\frac{e^{\hbar u/2}+e^{-\hbar
u/2}}{4u\hbar\mathcal{S}(u\hbar)}e^{-uy_{2}+u\mathcal{S}(\hbar
uz_{2}\partial_{z_{2}})\overline{y}_{2}}\Bigg{]}\frac{1}{2}\hbar
u\mathcal{S}(\hbar
uz_{2}\partial_{z_{2}})\Big{(}\frac{z_{2}}{z_{2}-z_{1}}-\frac{z_{2}}{z_{2}+z_{1}}\Big{)}.$

Here we use the notation $y_{i}=y(z_{i})$,
$\overline{y}_{i}=\overline{y}(\hbar^{2},z_{i})$, $\psi_{i}=\psi(y_{i})$,
$\overline{\psi}_{i}=\overline{\psi}(\hbar^{2},y_{i})$, $Q_{i}=Q(z_{i})$,
$X_{i}=X(z_{i})$,
$D_{i}=X_{i}\partial_{X_{i}}=Q_{i}^{-1}z_{i}\partial_{z_{i}}$ for $i=1,2$.

###### Proof.

Note that all formulas above are odd in both arguments, hence they vanish if
any of their arguments vanishes. Hence it is enough to check that
$D_{1}D_{2}H_{g,2}=W_{g,2}$. In the case $g=0$ it is a straightforward
computation. For $g\geq 1$ we recall the relevant special case of Equation
(4.23):

(6.8) $\displaystyle W_{g,2}=\ $
$\displaystyle[\hbar^{2g}]\sum_{\begin{subarray}{c}j_{1},j_{2}\\\
r_{1},r_{2}=0\end{subarray}}^{\infty}\left[\prod_{i=1}^{2}D_{i}^{j_{i}}[t_{i}^{j_{i}}]\frac{1}{Q_{i}}e^{-2t_{i}\psi_{i}}\partial_{y_{i}}^{r_{i}}e^{2t_{i}\frac{{\mathcal{S}}(t_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}}[u_{i}^{r_{i}}]\right]$
$\displaystyle\prod_{i=1}^{2}\frac{e^{\hbar u_{i}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}\left(e^{\hbar^{2}u_{1}u_{2}\mathcal{S}(\hbar
u_{1}z_{1}\partial_{z_{1}})\mathcal{S}(\hbar
u_{2}z_{2}\partial_{z_{2}})B(z_{1},z_{2})}-1\right).$

Note that

(6.9)
$\displaystyle\sum_{j,r=0}^{\infty}D^{j}[t^{j}]\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]=\frac{1}{Q}[u^{0}]+\sum_{r=0}^{\infty}\sum_{j=1}^{\infty}D^{j}[t^{j}]\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]$

The second summand here can be trivially integrated by applying $D^{-1}$. In
order to integrate the cases of application of $\frac{1}{Q}[u^{0}]$ we observe
that for any $\gamma\in\Gamma_{n}$ the coefficient of $u_{i}^{0}$ in

(6.10) $\displaystyle\frac{1}{Q_{i}}[u_{i}^{0}]\frac{e^{\hbar
u_{1}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}\left(e^{\hbar^{2}u_{i}u_{k}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})B(z_{i},z_{k})}-1\right)$
$\displaystyle=D_{i}\frac{1}{2}\hbar u_{k}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\Big{(}\frac{z_{k}}{z_{k}-z_{i}}-\frac{z_{k}}{z_{k}+z_{i}}\Big{)}.$

(here $k=2$ if $i=1$ and $k=1$ if $i=2$), which also admits application of
$D_{i}^{-1}$. In particular, if we apply this term for both variables, we
have:

(6.11)
$\displaystyle\prod_{i=1}^{2}\left[\frac{1}{Q_{i}}[u_{i}^{0}]\frac{e^{\hbar
u_{i}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}\right]$
$\displaystyle\left(e^{\hbar^{2}u_{1}u_{2}\mathcal{S}(\hbar
u_{1}z_{1}\partial_{z_{1}})\mathcal{S}(\hbar
u_{2}z_{2}\partial_{z_{2}})B(z_{1},z_{2})}-1\right)$
$\displaystyle=\frac{1}{4Q_{1}Q_{2}}B(z_{1},z_{2}),$

so this case doesn’t contribute to $[\hbar^{2g}]$, $g\geq 1$. Combining these
computations with the application of $D_{1}^{-1}D_{2}^{-1}$, we obtain the
statement of the proposition. ∎

Finally, in the general case of $n\geq 3$ we have the following expression for
$H_{g,n}$.

######  Proposition 6.3.

For a $\gamma\in\Gamma_{n}$ let $I_{\gamma}$ denote the subset of vertices of
$\gamma$ of index $\geq 2$. Let $K_{\gamma}\subset E_{\gamma}$ be the subset
of the set of edges that connect a vertex of index $1$ to another vertex. When
we write $(v_{i},v_{k})\in K_{\gamma}$, we assume that $v_{i}$ is the vertex
of index $1$ (and, therefore, $v_{k}\in I_{\gamma}$). We have:

(6.12) $\displaystyle H_{g,n}=\ $
$\displaystyle[\hbar^{2g-2+n}]\sum_{\gamma\in\Gamma_{n}}\prod_{i\in
I_{\gamma}}\left[\sum_{r_{i}=0}^{\infty}\sum_{j_{i}=1}^{\infty}D_{i}^{j_{i}-1}[t_{i}^{j_{i}}]\frac{1}{Q_{i}}e^{-2t_{i}\psi_{i}}\partial_{y_{i}}^{r_{i}}e^{2t_{i}\frac{{\mathcal{S}}(t_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}}[u_{i}^{r_{i}}]\right]$
$\displaystyle\prod_{i\in I_{\gamma}}\frac{e^{\hbar u_{i}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}$
$\displaystyle\prod_{(v_{k},v_{\ell})\in E_{\gamma}\setminus
K_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right)$
$\displaystyle\prod_{(v_{i},v_{k})\in K_{\gamma}}\Bigg{(}\frac{1}{2}\hbar
u_{k}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\Big{(}\frac{z_{k}}{z_{k}-z_{i}}-\frac{z_{k}}{z_{k}+z_{i}}\Big{)}+$
$\displaystyle\left[\sum_{r_{i}=0}^{\infty}\sum_{j_{i}=1}^{\infty}D_{i}^{j_{i}-1}[t_{i}^{j_{i}}]\frac{1}{Q_{i}}e^{-2t_{i}\psi_{i}}\partial_{y_{i}}^{r_{i}}e^{2t_{i}\frac{{\mathcal{S}}(t_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}}[u_{i}^{r_{i}}]\right]$
$\displaystyle\frac{e^{\hbar u_{i}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}(\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}\left(e^{\hbar^{2}u_{i}u_{k}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})B(z_{i},z_{k})}-1\right)\Bigg{)}.$

Here, as usual, we use $y_{i}=y(z_{i})$,
$\overline{y}_{i}=\overline{y}(\hbar^{2},z_{i})$, $\psi_{i}=\psi(y_{i})$,
$\overline{\psi}_{i}=\overline{\psi}(\hbar^{2},y_{i})$, $Q_{i}=Q(z_{i})$,
$X_{i}=X(z_{i})$,
$D_{i}=X_{i}\partial_{X_{i}}=Q_{i}^{-1}z_{i}\partial_{z_{i}}$.

###### Proof.

Note that $H_{g,n}$ as given in Equation (6.12) vanishes if we set any of its
variables to zero (since it is odd in each of its variables). So, the only
thing that we have to check is that indeed $D_{1}\cdots D_{n}H_{g,n}=W_{g,n}$
as given by Equation (4.23). Recall Equation (4.23):

(6.13) $\displaystyle W_{g,n}=$
$\displaystyle[\hbar^{2g-2+n}]\sum_{\begin{subarray}{c}j_{1},\dots,j_{n},\\\
r_{1},\dots,r_{n}=0\end{subarray}}^{\infty}\left[\prod_{i=1}^{n}D_{i}^{j_{i}}[t_{i}^{j_{i}}]\frac{1}{Q_{i}}e^{-2t_{i}\psi_{i}}\partial_{y_{i}}^{r_{i}}e^{2t_{i}\frac{{\mathcal{S}}(t_{i}\hbar\partial_{y_{i}})}{\mathcal{S}(\hbar\partial_{y_{i}})}\overline{\psi}_{i}}[u_{i}^{r_{i}}]\right]$
$\displaystyle\prod_{i=1}^{n}\frac{e^{\hbar u_{i}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}$
$\displaystyle\sum_{\gamma\in\Gamma_{n}}\prod_{(v_{k},v_{\ell})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{k}u_{\ell}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\mathcal{S}(\hbar
u_{\ell}z_{\ell}\partial_{z_{\ell}})B(z_{k},z_{\ell})}-1\right).$

Note that

(6.14)
$\displaystyle\sum_{j,r=0}^{\infty}D^{j}[t^{j}]\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]=\frac{1}{Q}[u^{0}]+\sum_{r=0}^{\infty}\sum_{j=1}^{\infty}D^{j}[t^{j}]\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}].$

The second summand here can be trivially integrated by applying $D^{-1}$. In
order to integrate the cases of application of $\frac{1}{Q}[u^{0}]$ we observe
that for any $\gamma\in\Gamma_{n}$ the coefficient of $u_{i}^{0}$ in

(6.15) $\displaystyle\frac{e^{\hbar u_{i}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}\prod_{(v_{i},v_{k})\in
E_{\gamma}}\left(e^{\hbar^{2}u_{i}u_{k}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})B(z_{i},z_{k})}-1\right)$

is non-trivial if and only if $i$ has index $1$ in $\gamma$. Then there is
only one edge $(e_{i},e_{k})\in E_{\gamma}$ that is attached to the vertex
$i$. In this case,

(6.16) $\displaystyle\frac{1}{Q_{i}}[u_{i}^{0}]\frac{e^{\hbar
u_{i}/2}+e^{-\hbar
u_{i}/2}}{4u_{i}\hbar\mathcal{S}(u_{i}\hbar)}e^{-u_{i}y_{i}+u_{i}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\overline{y}_{i}}\left(e^{\hbar^{2}u_{i}u_{k}\mathcal{S}(\hbar
u_{i}z_{i}\partial_{z_{i}})\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})B(z_{i},z_{k})}-1\right)$
$\displaystyle=\frac{1}{Q_{i}}\frac{1}{2}\hbar u_{k}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})B(z_{i},z_{k})=D_{i}\frac{1}{2}\hbar
u_{k}\mathcal{S}(\hbar
u_{k}z_{k}\partial_{z_{k}})\Big{(}\frac{z_{k}}{z_{k}-z_{i}}-\frac{z_{k}}{z_{k}+z_{i}}\Big{)},$

and we can apply $D_{i}^{-1}$ to the latter expression. This explains the
special summands for $(v_{i},v_{k})\in K_{\gamma}$ in Equation (6.12) and
completes the proof of the proposition. ∎

##  7\. Topological recursion for spin Hurwitz number with completed cycles

The goal of this Section is to prove a conjecture proposed by Giacchetto,
Kramer, and Lewański. In our terms, it concerns the symmetric
$n$-differentials constructed from Orlov’s hypergeometric 2-BKP tau-functions
for $\overline{\psi}=\frac{1}{2}{\mathcal{S}}(\hbar\partial_{y})y^{2s}$ and
$\overline{y}=z$. But in fact we consider a more general situation, with
$\overline{\psi}=\frac{1}{2}{\mathcal{S}}(\hbar\partial_{y})P(y)$ and
$\overline{y}=y=R(z)$, where $P$ is an arbitrary even polynomial in $y$ and
$R$ is an arbitrary odd polynomial in $z$, since the arguments in this more
general situation do not differ from the ones for the
Giacchetto–Kramer–Lewański situation.

######  Remark 7.1.

Note that if we put
$\overline{\psi}(y)=\frac{1}{2}{\mathcal{S}}(\hbar\partial_{y})P(y)$, then the
weight for the KP hypergeometric tau-function (3.44) does not coincide with
the deformation, considered in [BDBKS20]. Therefore, if in the relation (3.42)
one of the tau-functions, $\tau_{KP}$ or $\tau$, is described by a suitable
version of topological recursion, the other one is not described by it.

###  7.1. Topological recursion in the odd situation

Consider $\mathbb{C}\mathrm{P}^{1}$ with a fixed global coordinate $z$, and
with two functions, $X$ and $y$ such that $X(-z)=-X(z)$ and $y(-z)=-y(-z)$,
with an extra assumption that $dX/X$ is a rational differential with the
simple critical points $p_{1},\dots,p_{N}$ (it is clear that $N$ must be even
and the set of critical points is invariant under $z\leftrightarrow-z$) and
$y$ is holomorphic near the critical points with $dy|_{p_{i}}\not=0$. It is
not necessary but both convenient and sufficient for our goals to assume that
$y$ is meromorphic. Let

(7.1)
$\displaystyle\mathcal{B}(z_{1},z_{2})\coloneqq\frac{1}{2}\left(\frac{1}{(z_{1}-z_{2})^{2}}+\frac{1}{(z_{1}+z_{2})^{2}}\right)dz_{1}dz_{2}.$

With this input we construct a system of symmetric differentials
$\omega_{g,n}$, $g\geq 0$, $n\geq 1$, given by

(7.2) $\displaystyle\omega_{0,1}(z_{1})=y(z_{1})d\log X(z_{1});$
$\displaystyle\omega_{0,2}(z_{1},z_{2})=\mathcal{B}(z_{1},z_{2});$

and for $2g-2+n>0$ we use the recursion

(7.3) $\displaystyle\omega_{g,n}(z_{1},\dots,z_{n})\coloneqq\ $
$\displaystyle\frac{1}{2}\sum_{i=1}^{N}\operatorname{Res}_{z\to
p_{i}}\frac{\int_{z}^{\sigma_{i}(z)}\mathcal{B}(z_{1},\cdot)}{\omega_{0,1}(\sigma_{i}(z_{1}))-\omega_{0,1}(z_{1})}\Bigg{(}\omega_{g-1,n+1}(z,\sigma_{i}(z),z_{\llbracket
n\rrbracket\setminus\\{1\\}})$
$\displaystyle+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g,I_{1}\sqcup
I_{2}={\llbracket n\rrbracket\setminus\\{1\\}}\\\
(g_{1},|I_{1}|),(g_{2},|I_{2}|)\not=(0,0)\end{subarray}}\omega_{g_{1},1+|I_{1}|}(z,z_{I_{1}})\omega_{g_{2},1+|I_{2}|}(\sigma_{i}(z),z_{I_{2}})\Bigg{)},$

where $\sigma_{i}$ is the deck transformation of $X$ near $p_{i}$,
$i=1,\dots,N$. We wouldn’t go into the discussion of this peculiar version of
this topological recursion, as it should be done in a more general equivariant
setup.

For our goals it is sufficient to state the following equivalent reformulation
of this version of topological recursion, which is completely parallel to
[BS17, Theorem 2.2] and [BEO15, Section 1].

######  Lemma 7.1.

A system of meromorphic symmetric differentials $\omega_{g,n}$, $2g-2+n>0$ is
obtained from the given starting data (that includes the formulas for
$\omega_{0,1}$ and $\omega_{0,2}$) by topological recursion (7.3) if and only
if

  1. (1)

This system of differentials satisfies the blobbed topological recursion (see
Definition 5.1).

  2. (2)

For any $g\geq 0,n\geq 1,2g-2+n>0$

(7.4) $\displaystyle\omega_{g,n}(z_{\llbracket
n\rrbracket})=\sum_{i_{1},\dots,i_{n}=1}^{N}\Bigg{(}\prod_{j=1}^{n}\operatorname{Res}_{w_{j}\to
p_{i_{j}}}\int_{p_{i_{j}}}^{w_{i}}\mathcal{B}(\cdot,z_{j})\Bigg{)}\omega_{g,n}(w_{\llbracket
n\rrbracket})$

(this is the so-called projection property).

###### Proof.

The same argument as in [BS17, Section 2.4]. ∎

If we represent the symmetric differential $\omega_{g,n}$ as
$\omega_{g,n}=2^{1-g}W_{g,n}\prod_{i=1}^{n}d\log X_{i}$,
$\omega_{0,2}=2W_{0,2}d\log X_{1}d\log X_{2}+\mathcal{B}(X_{1},X_{2})$, where
$W_{g,n}=D_{1}\cdots D_{n}H_{g,n}$, $D_{i}=X_{i}\partial_{X_{i}}$, then the
linear loop equations in combination with the projection property can be
equivalently reformulated in terms of $H_{g,n}$. This reformulation can be
directly applied in the odd case that we consider here, and we recall it and
prove for the particular $n$-point functions of spin Hurwitz numbers with
completed cycles in the next section, Section 7.2.

###  7.2. Quasi-polynomiality

The goal of this section is to prove some special property of the functions
$H_{g,n}$, and, as a corollary, $W_{g,n}$’s that is sometimes called quasi-
polynomiality in the literature and in the context of topological recursion is
equivalent to a combination of the so-called projection property and the liner
loop equations. We refer to [BDBKS20, Section 3] for a full discussion.

Recall that with
$\overline{\psi}=\frac{1}{2}{\mathcal{S}}(\hbar\partial_{y})P(y)$,
$P(-y)=P(y)$ is a polynomial, and $\overline{y}=y=R(z)$, $R(-z)=-R(z)$ is a
polynomial we have $X=z\exp(-P(R(z)))$. Let
$p_{1},\dots,p_{N}\in\mathbb{C}\mathrm{P}^{1}$ be the critical points of $X$.
Here $N=\deg P\cdot\deg R\in 2\mathbb{Z}$, we assume that all critical points
are simple, and the set of critical points is obviously invariant under the
involution $z\leftrightarrow-z$.

Define the space $\Theta_{n}$ as the linear span of functions
$\prod_{i=1}^{N}f_{i}(z_{i})$, where each $f_{i}(z_{i})$ is a rational
function on $\mathbb{C}\mathrm{P}^{1}$, $f_{i}(-z_{i})=-f_{i}(z_{i})$, $f_{i}$
has poles only at the points $p_{1},\dots,p_{N}$, and the principal part of
$f_{i}$ at $p_{k}$, $k=1,\dots,N$, is odd with respect to the corresponding
deck transformation $\sigma_{k}$ of function $X$ near $p_{k}$. The last
condition can be reformulated as a requirement that for any $k=1,\dots,N$ the
locally defined function $f_{i}(z_{i})+f_{i}(\sigma_{k}z_{i})$ is holomorphic
at $z_{i}\to p_{k}$.

######  Proposition 7.2.

In the case $\overline{\psi}=\frac{1}{2}{\mathcal{S}}(\hbar\partial_{y})P(y)$,
$P(-y)=P(y)$ is a polynomial, and $\overline{y}=y=R(z)$, $R(-z)=-R(z)$ is a
polynomial, the functions $H_{g,n}$ belong to the space $\Theta_{n}$, for any
$n\geq 1$, $g\geq 0$ such that $2g-2+n>0$.

###### Proof.

In the proof we analyze the formulas obtained in Propositions 6.1, 6.2, and
6.3. It is clear from the structure of the formulas (6.1), (6.7), and (6.12)
that with our assumptions $H_{g,n}$ are rational functions in
$z_{1},\dots,z_{n}$.

Consider $H_{g,n}$ as a function of $z_{1}$, treating the rest of the
variables as parameters. From the shape of the formula we see that it might
have poles at $z_{1}\to\pm z_{i}$, $i=2,\dots,n$, $z_{1}\to\infty$, and at the
zeros of $Q$. In this case $Q=z\partial_{z}\log X=1+z\partial_{z}P(R(z))$, and
its zeros are exactly $p_{1},\dots,p_{N}$.

From Remark 4.3 it follows that there are no singularities at $z_{1}=\pm
z_{i}$, $i=2,\dots,n$.

In all terms of the formulas (6.1), (6.7), and (6.12) the principal part at
$z_{1}\to p_{k}$ is generated by the iterative application of the operator
$D_{1}=X_{1}\partial_{X_{1}}=Q(z_{1})^{-1}z_{1}\partial_{z_{1}}$ to a function
that is either holomorphic at $z_{1}\to p_{k}$ (as in the first summand of
(6.1)), or has a simple pole at $z_{1}\to p_{k}$ (as in the second summand of
(6.1), where we divide a function holomorphic at $z_{1}\to p_{k}$ by
$Q(z_{1})$). Holomorphic functions and functions with a simple pole
automatically have principal parts at $z_{1}\to p_{k}$ that are odd with
respect to the deck transformation at $p_{k}$, and the operator $D_{1}$
preserves this property (while increasing the order of the pole at $p_{k}$).

Let us now check that there is no pole at $z_{1}\to\infty$. Note that the
terms that really look special, the last two summands in Equation (6.1),
vanish with our assumptions (and that is crucially important since for any
other choice of $\overline{\psi}$ and $\overline{y}$ with given $\psi=P$ and
$y=R$ it wouldn’t be the case). To all other terms in the formulas (6.1),
(6.7), and (6.12) the same rough estimation of the order of pole is
applicable, cf. [BDBKS20, Lemma 4.6]. We perform it here only for the second
summand in Equation (6.1), since in all other cases the analysis is exactly
the same. To this end, consider

(7.5)
$\displaystyle[\hbar^{2g}]\sum_{j=1}^{\infty}D^{j-1}[t^{j}]\sum_{r=0}^{\infty}\frac{1}{Q}e^{-2t\psi}\partial_{y}^{r}e^{2t\frac{{\mathcal{S}}(t\hbar\partial_{y})}{\mathcal{S}(\hbar\partial_{y})}\overline{\psi}}[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{-uy+u\mathcal{S}(\hbar
uz\partial_{z})\overline{y}}\right)$
$\displaystyle=[\hbar^{2g}]\sum_{j=1}^{\infty}D^{j-1}[t^{j}]\sum_{r=0}^{\infty}\frac{1}{Q}(\partial_{y}+2tP^{\prime}(y))^{r}e^{2t({{\mathcal{S}}(t\hbar\partial_{y})}-1)P(y)}$
$\displaystyle\qquad\qquad\qquad\quad\quad\quad\quad[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{u(\mathcal{S}(\hbar
uz\partial_{z})-1)R(z)}\right)$

Note that the operator $D=Q^{-1}z\partial z$ decreases the order of pole at
$z\to\infty$ by $\deg Q=\deg P\deg R$. The same holds for the factor $Q^{-1}$
alone. This means that the order of pole in (7.5) at $z\to\infty$ is equal to
the order of pole at $z\to\infty$ of

(7.6)
$\displaystyle[\hbar^{2g}]\sum_{r=0}^{\infty}(\partial_{y}+2tP^{\prime}(y))^{r}e^{2t({{\mathcal{S}}(t\hbar\partial_{y})}-1)P(y)}\Big{|}^{\prime}_{t=z^{-\deg
P\deg R}}$ $\displaystyle\qquad\qquad[u^{r}]\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{u(\mathcal{S}(\hbar
uz\partial_{z})-1)R(z)}\right),$

where by $|^{\prime}$ we mean that we only select the terms with $\deg t\geq
1$. With this substitution observe that each application of the operator
$\partial_{y}+2tP^{\prime}(y)$ decreases the order at $z\to\infty$ by $\deg
R$. Thus the order of pole of (7.6) at $z\to\infty$ is equal to the order of
pole at $z\to\infty$ of

(7.7)
$\displaystyle[\hbar^{2g}]e^{2t({{\mathcal{S}}(t\hbar\partial_{y})}-1)P(y)}\left(\frac{e^{\frac{\hbar
u}{2}}+e^{-\frac{\hbar u}{2}}}{4u\mathcal{S}(u\hbar)}e^{u(\mathcal{S}(\hbar
uz\partial_{z})-1)R(z)}\right)\Big{|}^{\prime}_{t=z^{-\deg P\deg
R}}\Big{|}^{\prime\prime}_{u=z^{-\deg R}},$

where by $|^{\prime\prime}$ we mean that we only select the terms with $\deg
u\geq 0$. The latter expression is manifestly regular at $z\to\infty$.

Finally, extending our arguments to all variables $z_{1},\dots,z_{n}$, we
obtain that $H_{g,n}(z_{1},\dots,z_{n})$, $2g-2+n>0$, is a rational function
that in each of its variables has poles only at the points $p_{1},\dots,p_{N}$
with the odd principal parts with respect to the corresponding deck
transformations. This immediately implies that $H_{g,n}\in\Theta_{n}$. ∎

###  7.3. Giacchetto–Kramer–Lewański conjecture and its generalization

Consider the $n$-functions $H_{g,n}$ constructed from Orlov’s hypergeometric
BKP tau-functions for
$\overline{\psi}=\frac{1}{2}{\mathcal{S}}(\hbar\partial_{y})P(y)$ and
$\overline{y}=y=R(z)$, where $P$ is an arbitrary even polynomial in $y$ and
$R$ is an arbitrary odd polynomial in $z$. Recall $X=X(z)=z\exp(-P(R(z)))$.
Recall that we defined $W_{g,n}=D_{1}\cdots D_{n}H_{g,n}$, and we set

(7.8) $\displaystyle\omega_{g,n}(z_{\llbracket n\rrbracket})\coloneqq
2^{1-g}W_{g,n}(X_{\llbracket
n\rrbracket})\prod_{i=1}^{n}\frac{dX_{i}}{X_{i}}+\delta_{g,0}\delta_{n,2}\frac{1}{2}B(X_{1},X_{2})d\log
X_{1}d\log X_{2},$

With this assignment, it follows from Propositions 4.3 and 4.4 that

(7.9) $\displaystyle\omega_{0,1}(z)=y\,d\log
X\qquad\text{and}\qquad\omega_{0,2}(z_{1},z_{2})=\mathcal{B}(z_{1},z_{2}).$

For all other $\omega_{g,n}$, $g\geq 0$, $n\geq 1$, $2g-2+n>0$, we have the
following theorem

######  Theorem 7.1 (Generalized Giacchetto–Kramer–Lewański conjecture).

The symmetric $n$-differentials $\omega_{g,n}$ are obtained by the odd
topological recursion (7.3) for the initial data $X=z\exp(-P(R(z)))$ and
$y=R(z)$.

###### Proof.

According to Lemma 7.1 we have to check the blobbed topological recursion and
the projection property. The blobbed topological recursion follows from
Theorem 5.1, which is proved in a much more general situation (it is obvious
that the analytic assumptions listed in Section 5.1 are satisfied). On the
other hand, the linear loop equations and the projection property are
equivalent to the statement of Proposition 7.2. ∎

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