#  KP integrability of triple Hodge integrals. III.
Cut-and-join description, KdV reduction, and topological recursions

Alexander Alexandrov  IBS Center for Geometry and Physics, Pohang University
of Science and Technology (POSTECH), 77 Cheongam-ro, Nam-gu, Pohang,
Gyeongbuk, 37673, Korea  alexandrovsash at gmail.com

###### Abstract.

In this paper, we continue our investigation of the triple Hodge integrals
satisfying the Calabi–Yau condition. For the tau-functions, which generate
these integrals, we derive the complete families of the Heisenberg–Virasoro
constraints. We also construct several equivalent versions of the cut-and-join
operators. These operators describe the algebraic version of topological
recursion. For the specific values of parameters associated with the KdV
reduction, we prove that these tau-functions are equal to the generating
functions of intersection numbers of $\psi$ and $\kappa$ classes. We interpret
this relation as a symplectic invariance of the Chekhov–Eynard–Orantin
topological recursion and prove this recursion for the general $\Theta$-case.

###### 2020 Mathematics Subject Classification:

37K10, 14N35, 81R10, 14H70, 14N10

Keywords: enumerative geometry, Hodge integrals, tau-functions, KP hierarchy,
Virasoro constraints, cut-and-join operator, topological recursion

###### Contents

  1. 0.1 Algebraic topological recursion
  2. 0.2 Intersection theory and KP integrability
  3. 0.3 KdV reduction
  4. 1 Symmetries of KP hierarchy and Heisenberg–Virasoro algebra
  5. 2 Algebraic topological recursion and cut-and-join operators
    1. 2.1 General cut-and-join description
    2. 2.2 Algebraic topological recursion
  6. 3 Cut-and-join operators for KW and BGW tau-functions
    1. 3.1 Intersection numbers
    2. 3.2 Heisenberg–Virasoro constraints
    3. 3.3 Heisenberg–Virasoro group action and cut-and-join operators
  7. 4 Triple Hodge integrals and KP hierarchy
    1. 4.1 KP integrability
    2. 4.2 Heisenberg–Virasoro and dimension constraints
    3. 4.3 Factorizations of the Virasoro group elements
    4. 4.4 Cut-and-join description of the triple Hodge integrals
  8. 5 KdV reduction
    1. 5.1 Translations and $\kappa$-classes
    2. 5.2 KdV reduction for triple Hodge integrals
  9. 6 Tau-functions identification and topological recursion
    1. 6.1 Chekhov–Eynard–Orantin topological recursion
    2. 6.2 Topological recursion for triple Hodge integrals
    3. 6.3 Topological recursion for triple $\Theta$-Hodge integrals
  10. A Topological expansion of $\log\tau_{q,p}^{(\alpha)}$

## Introduction

###  0.1. Algebraic topological recursion

The Chekhov–Eynard–Orantin topological recursion is an effective and universal
tool in mathematical physics, enumerative geometry, and combinatorics [EO07,
EO09]. For the basic models, this recursion provides the solution to the
Virasoro constraints. However, there are other ways to solve them, and
sometimes they suit better for the particular needs. For instance, the
Kontsevich–Witten and Brézin–Gross–Witten tau-functions, which are the basic
elements of the Chekhov–Eynard–Orantin topological recursion, can be described
by explicit formulas in terms of the second order differential operators
[Ale11, Ale18].

This description results from a solution of an equation given by the
combination of the Virasoro constraints. It was recently noted [MMMR21] that
the partition functions of some interesting matrix models are completely
specified by a unique equation of this type. In this paper, we prove a vast
generalization of this observation. Namely, we prove that the solution of a
rather general equation, which includes the Euler operator as the leading
term, exists and is unique. This solution can be constructed recursively.

For the generating functions with the natural topological expansion this
recursive procedure may have a transparent topological meaning. In this case
the recursion can be interpreted as the algebraic topological recursion. As
this recursion is given by the differential operators (also known as cut-and-
join operators), it is rather simple computationally. It is not known in
general how to construct the equation leading to the cut-and-join solution. In
some cases this equation can be constructed from the Virasoro and
W-constrains, satisfied by the generating functions. One of the main goals of
this paper is to provide the cut-and-join description for an interesting
family of the enumerative geometry invariants, namely for the triple Hodge
integrals satisfying the Calabi–Yau condition.

###  0.2. Intersection theory and KP integrability

Let $\overline{\mathcal{M}}_{g,n}$ be the Deligne–Mumford compactification of
the moduli space of all compact Riemann surfaces of genus $g$ with $n$
distinct marked points. It is empty unless the stability condition

(0.1) $\displaystyle 2g-2+n>0$

is obeyed.

On $\overline{\mathcal{M}}_{g,n}$ consider four families of cohomological
classes: $\psi_{k}$, $\kappa_{k}$, $\lambda_{k}$, and $\Theta_{g,n}$. For each
marking index $i$ we consider the cotangent line bundle
${\mathbb{L}}_{i}\rightarrow\overline{\mathcal{M}}_{g,n}$, whose fiber over a
point $[\Sigma,z_{1},\ldots,z_{n}]\in\overline{\mathcal{M}}_{g,n}$ is the
complex cotangent space $T_{z_{i}}^{*}\Sigma$ of $\Sigma$ at $z_{i}$. Then
$\psi_{i}\in H^{2}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ is the first
Chern class of ${\mathbb{L}}_{i}$. With the forgetful map for the marked point
$n+1$,
$\pi:\overline{\mathcal{M}}_{g,n+1}\rightarrow\overline{\mathcal{M}}_{g,n}$,
we define the Miller–Morita–Mumford tautological classes [Mum83],
$\kappa_{k}:=\pi_{*}\psi_{n+1}^{k+1}\in
H^{2k}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$. For the Hodge bundle
$\mathbb{E}$, a rank $g$ vector bundle over $\overline{\mathcal{M}}_{g,n}$, we
consider the $i$th Chern classes $\lambda_{i}=c_{i}({\mathbb{E}})$ and their
generating function $\Lambda_{g}(u)=\sum_{i=0}^{g}u^{i}\lambda_{i}$. Finally,
following Norbury [Nor17], we introduce $\Theta$-classes, $\Theta_{g,n}\in
H^{4g-4+2n}(\overline{\mathcal{M}}_{g,n})$, which are also related to the
super Riemann surfaces [Nor20]. We refer the reader to [Nor17, Nor18, Nor20]
for a detailed description of $\Theta$-classes.

By the Kontsevich–Witten theorem [Kon92, Wit91] the generating function of the
intersection numbers

(0.2)
$\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}$

is a tau-function of the KdV hierarchy. For their $\Theta$-version,

(0.3)
$\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}},$

KdV integrability of the generating function was established by Norbury
[Nor17]. Relation between the intersection theory over the moduli spaces and
solitonic integrable hierarchies is rather mysterious, and only a few
instances of this relation are known so far.

The Kontsevich–Witten (KW) and Brézin–Gross–Witten (BGW) tau-functions,
corresponding to the intersection numbers of $\psi$ and $\Theta$ classes, play
very important role in the modern mathematical physics and enumerative
geometry. In particular, they are the main ingredients of such universal
constructions as the Chekhov–Eynard–Orantin topological recursion [EO07, EO09]
and the Givental decomposition [Giv01, Tel12]. These tau-functions share a lot
of common properties, in particular they satisfy the similar looking Virasoro
constraints [FKN91, DVV91, GN92], and it is convenient to consider them
simultaneously. The Virasoro constraints allow us [Ale11, Ale18] to derive the
cut-and-join description of these tau-functions and to prove the algebraic
version of topological recursion, associated with the expansion with respect
to the negative Euler characteristics of the punctured Riemann surface. We
claim that it is natural to consider all types of intersection numbers and
their $\Theta$-versions simultaneously. In many respects $\Theta$-versions are
a bit simpler, perhaps because of the supersymmetry behind them.

In this paper, we investigate triple Hodge integrals satisfying the Calabi–Yau
condition,

(0.4)
$\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}^{1-\alpha}\Lambda_{g}(-q)\Lambda_{g}(-p)\Lambda_{g}\left(\frac{pq}{p+q}\right)\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}$

for $\alpha=0$ and $\alpha=1$. For $\alpha=1$ these integrals appear at the
localization techniques for the Gromov–Witten invariants on the three-
dimensional target manifolds [GP99]. They constitute the topological vertex,
which provides a diagrammatic method to compute topological string amplitudes
on non-compact toric Calabi–Yau three-folds [AKMnV05, LLLZ09]. String theory
interpretation of these integrals for $\alpha=0$ is not known yet.

It appears that the generating functions of these intersection numbers have a
surprisingly simple description in terms of integrable hierarchies. Namely,
these generating functions are tau-functions of the KP hierarchy [Ale20].
These tau-functions are certain deformations of the KW and BGW tau-functions.
This generalizes the result of Kazarian for the linear Hodge integrals
[Kaz09].

In this paper, we continue our investigation of the tau-functions for the
triple Hodge integrals satisfying the Calabi–Yau condition. In particular, for
these tau-functions using their relation to the KW and BGW tau-functions, we
derive a complete set of the Heisenberg–Virasoro constraints, see Theorem 1.
These linear constrains completely specify a formal series solution. However,
they are not sufficient to construct the cut-and-join description. In Theorem
2 we construct this description using the KP hierarchy group of symmetries.
Resulting algebraic topological recursion, given by the third order
differential operators, is very convenient computationally. We use it to
compute the first few terms of the topological expansion of the tau-functions
for the triple Hodge integrals satisfying the Calabi–Yau condition, see
Appendix A.

###  0.3. KdV reduction

Let us also consider the intersection numbers of $\psi$, $\kappa$ and $\Theta$
classes

(0.5)
$\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}^{1-\alpha}e^{\sum_{j=1}^{\infty}s_{j}\kappa_{j}}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}.$

According to [MZ00, Nor20] the generating functions of these intersection
numbers can be identified with the KW (for $\alpha=1$) and BGW (for
$\alpha=0$) tau-functions with shifted arguments.

For $p=-2q$ the generating functions of the triple Hodge integrals satisfying
the Calabi–Yau condition (0.4) are the tau-functions of the KdV hierarchy
[Ale21]. In Theorem 4 we prove that in this case the tau-functions coincide
with the generating functions of intersection numbers (0.5) for the specific
values of $s_{k}$ variables. The proof is based on the Virasoro constraints
and the explicit form of the group element of the KP hierarchy symmetry. A
geometric interpretation of this relation is not clear yet.

This theorem allows us to identify the intersection numbers (0.4) with (0.5)
for arbitrary genus $g$ and number of marked points $n$ and specific values of
parameters. We interpret this identification in terms of the symplectic
invariance of the Chekhov–Eynard–Orantin topological recursion. For
$\alpha=1$, where the topological recursion for (0.4) is known [BMn08, Che18,
Zho09], we use the symplectic invariance to find an independent proof of the
relations between the intersection numbers. For $\alpha=0$ we prove the
topological recursion for the general triple $\Theta$-Hodge integrals
satisfying the Calabi–Yau condition.

### Organization of the paper

In Section 1 we remind the reader some basic properties of the
Heisenberg–Virasoro symmetry algebra of the KP hierarchy. Section 2 is devoted
to the origin of the cut-and-join description. In particular, we derive it as
a solution of the differential equation given by a rather general deformation
of the Euler operator and its relation to the topological expansion. In
Section 3 we remind the main properties of the KW and BGW tau-functions, and
investigate how these properties transform under the action of the Virasoro
group. In Section 4 we construct the Heisenberg–Virasoro constraints and the
cut-and-join operators for the tau-functions, which describe the triple Hodge
integrals satisfying the Calabi–Yau condition. Section 5 is devoted to the
reduction to the KdV hierarchy for the specific values of the parameters. We
prove that in this case the tau-functions coincide with the tau-functions for
the intersection numbers containing the Miller–Morita–Mumford classes. In
Section 6 we describe this relation using symplectic invariance of the
Chekhov–Eynard–Orantin topological recursion and prove this recursion for the
general triple $\Theta$-Hodge integrals satisfying the Calabi–Yau condition.

### Acknowledgments

This work was supported by the Institute for Basic Science (IBS-R003-D1).

##  1\. Symmetries of KP hierarchy and Heisenberg–Virasoro algebra

It is well known that a certain central extension of the algebra
$\operatorname{\mathfrak{gl}}(\infty)$ and corresponding group
$\operatorname{GL}(\infty)$ act on the space of KP tau-functions [FKN92,
MJD00]. From the boson-fermion correspondence we also know how to identify the
operators, acting on the tau-functions with the operators, acting on the Sato
Grassmannian. Often the Sato Grassmannian parametrization is more convenient,
and we will use it for computations. In this section, we remind the reader the
properties of the important Heisenberg–Virasoro subalgebra of
$\operatorname{\mathfrak{gl}}(\infty)$; for more details see, e.g., [Ale15].

Consider the space ${\mathbb{C}}[\\![{\bf t}]\\!]$ of formal power series of
infinitely many variables ${\bf t}=\\{t_{1},t_{2},\dots\\}$. We will denote by
$\widehat{\dots}$ differential operators acting on this space. The
Heisenberg–Virasoro subalgebra ${\mathcal{L}}$ of
$\operatorname{\mathfrak{gl}}(\infty)$ is generated by the operators

(1.1) $\widehat{J}_{k}=\begin{cases}\displaystyle{\frac{\partial}{\partial
t_{k}}\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\quad k>0},\\\\[2.0pt]
\displaystyle{0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\quad
k=0,\\\\[2.0pt] \displaystyle{-kt_{-k}\,\,\,\,\,\mathrm{for}\quad
k<0,}\end{cases}$

the unit, and the Virasoro operators

(1.2)
$\widehat{L}_{m}=\frac{1}{2}\sum_{a+b=-m}abt_{a}t_{b}+\sum_{k=1}^{\infty}kt_{k}\frac{\partial}{\partial
t_{k+m}}+\frac{1}{2}\sum_{a+b=m}\frac{\partial^{2}}{\partial t_{a}\partial
t_{b}}.$

These operators satisfy the commutation relations

$\displaystyle\left[\widehat{J}_{k},\widehat{J}_{m}\right]$
$\displaystyle=k\delta_{k,-m},$ (1.3)
$\displaystyle\left[\widehat{L}_{k},\widehat{J}_{m}\right]$
$\displaystyle=-m\widehat{J}_{k+m},$
$\displaystyle\left[\widehat{L}_{k},\widehat{L}_{m}\right]$
$\displaystyle=(k-m)\widehat{L}_{k+m}+\frac{1}{12}\delta_{k,-m}(k^{3}-k)$

and generate a Lie algebra. We decompose ${\mathcal{L}}$ as
${\mathcal{L}}={\mathcal{L}}_{+}\oplus{\mathcal{L}}_{0}\oplus{\mathcal{L}}_{-}$,
where ${\mathcal{L}}_{+}$ (${\mathcal{L}}_{-}$) are generated by the operators
$\widehat{L}_{k}$ and $\widehat{J}_{k}$ with positive (negative) $k$, and
${\mathcal{L}}_{0}$ is generated by $\widehat{L}_{0}$ and a unit.

The Heisenberg–Virasoro group ${\mathcal{V}}$ is generated by the operators
$\widehat{J}_{k}$, $\widehat{L}_{k}$ and a unit,

(1.4) ${\mathcal{V}}:=\left.\\{C\,e^{\sum
a_{k}\widehat{J}_{k}+b_{k}\widehat{L}_{k}}\right|a_{k},b_{k},C\in{\mathbb{C}}\\}.$

For this group we also have a decomposition,
${\mathcal{V}}={\mathcal{V}}_{+}\oplus{\mathcal{V}}_{0}\oplus{\mathcal{V}}_{-}$.
This group acts on the space of the tau-functions of the KP hierarchy.

The Heisenberg–Virasoro algebra (1) is a central extension of the algebra of
operators, acting on the Sato Grassmannian and generated by

(1.5)
${\mathtt{j}}_{m}=z^{m},\,\,\,\,\,\,\,\,\,{\mathtt{l}}_{m}=-z^{m}\left(z\frac{\partial}{\partial
z}+\frac{m+1}{2}\right).$

Operators ${\mathtt{l}}_{m}$ generate the Witt algebra. These operators obey
the commutation relations (1) with omitted central term. Thanks to this
relation between the commutation relations it is convenient to use the
operators ${\mathtt{j}}_{m}$ and ${\mathtt{l}}_{m}$ for computations in the
group ${\mathcal{V}}$.

The action of the ${\mathcal{V}}_{0}$ and ${\mathcal{V}}_{-}$ groups on the
space of the functions of ${\bf t}$ is rather simple – it is equivalent to the
linear change of the variables supplemented by the multiplication by a
function. Below we will mostly focus on the group ${\mathcal{V}}_{+}$.

##  2\. Algebraic topological recursion and cut-and-join operators

In this section, we describe how a single equation of a certain type satisfied
by the generating functions can lead to the recursion relations and the cut-
and-join description which specify a unique solution. Moreover, for the
generating functions of the enumerative geometry and mathematical physics this
recursion can have a natural topological interpretation.

###  2.1. General cut-and-join description

Let us consider (in general, infinitely many) formal variables ${\bf
q}=\\{q_{1},q_{2},q_{3},\dots\\}$. We associate with them positive integer
degrees $a_{k}\in{\mathbb{Z}}_{>0}$. Consider corresponding Euler operator

(2.1) $E=\sum_{k}a_{k}q_{k}\frac{\partial}{\partial q_{k}}.$

We assume that there are only finitely many variables with any finite degree,
that is, $\\#q_{k}(a_{k}<n)<\infty$ for any $0<n<\infty$. We associate the
degree $-a_{k}$ to the operators $\frac{\partial}{\partial q_{k}}$ and
consider a differential equation

(2.2) $\left(E-\sum_{k=1}^{\infty}kV_{k}\right)\cdot Z({\bf q})=0,$

where ${V}_{k}$ are some differential operators of degree $k$ with
coefficients given by formal series in variables ${\bf q}$. It appears that
such an equation always has a unique solution.

######  Proposition 2.1.

Equation (2.2) has a unique solution in ${\mathbb{C}}[\\![{\bf q}]\\!]$ up to
a normalization.

###### Proof.

Let us find this solution explicitly. Consider the expansion

(2.3) $Z({\bf q})=\sum_{k=0}^{\infty}Z_{k}({\bf q}),$

where $Z_{0}\in{\mathbb{C}}$ is a constant, and the degree of $Z_{k}$ is $k$,
that is, $E\cdot Z_{k}=kZ_{k}$. Then the term of degree $k$ on the left hand
side of (2.2) should vanish,

(2.4) $E\cdot Z_{k}-\sum_{j=0}^{k-1}(k-j){V}_{k-j}\cdot Z_{j}=0.$

Since $E\cdot Z_{k}=kZ_{k}$, the unique solution of (2.2) is given by the
recursion

(2.5) $Z_{k}=\frac{1}{k}\sum_{j=0}^{k-1}(k-j){V}_{k-j}Z_{j}.$

Because of the finiteness of the variables with a given finite degree, all
$Z_{j}$ are polynomials, and the action on them of all finite degree operators
is well defined. Arbitrary constant $Z_{0}$ fixes the total normalization of
the solution, which completes the proof. ∎

Equation (2.5) provides a recursion which completely specifies the solution
for a given initial condition $Z_{0}=Z({\bf 0})$ which can be fixed by $Z({\bf
0})=1$. This solution is described by the operators $V_{k}$. There are several
possibilities depending on the relations between the operators $V_{k}$.

  * •

If ${V}_{k}=\delta_{k,r}{V}_{r}$ for some $r>0$, we have an explicit solution

(2.6) $Z({\bf q})=\exp{\left({V}_{r}\right)}\cdot 1.$

  * •

In general, if the operators ${V}_{k}$ constitute a commutative family,
$\left[{V}_{k},{V}_{m}\right]=0$ for all $k,m>0$, then

(2.7) $Z({\bf q})=\exp\left({\sum_{k=1}^{\infty}{V}_{k}}\right)\cdot 1.$

If $V_{k}=0$ for all $k>r$ for some $r$, then the sum in the right hand side
of (2.5) contains at most $r$ terms.

  * •

If the operators $V_{k}$ do not commute, then the solution is given by the
ordered exponential, which we denote by $``\exp\left(\dots\right)"$,

(2.8) $Z({\bf q})=``\exp\left({\sum_{k=1}^{\infty}{V}_{k}}\right)"\cdot 1.$

######  Remark 2.1.

Equation of the form (2.2) is never unique. Therefore, in general it is not
clear how to choose the best representative of equations of this type and the
corresponding solution (2.8). At least for some interesting models arising in
enumerative geometry and matrix models theory the generating functions can be
represented by (2.6).

In particular, one can consider the functions of the variables $t_{k}$. We
introduce a ${\mathbb{Z}}$ grading on functions and differential operators on
this space by assigning

(2.9) $\deg t_{k}=k,\quad\quad\deg\frac{\partial}{\partial t_{k}}=-k.$

Then $\deg\widehat{L}_{k}=\deg\widehat{J}_{k}=-k$ and degree of the central
element of the Heisenberg–Virasoro algebra is zero. The Euler operator
$E=\sum_{k=1}^{\infty}kt_{k}\frac{\partial}{\partial t_{k}}$ coincides with
$\widehat{L}_{0}$. Let us consider a differential equation

(2.10) $\left(\widehat{L}_{0}-\sum_{k=1}^{\infty}k\widehat{V}_{k}\right)\cdot
Z({\bf t})=0,$

where $\widehat{V}_{k}$ are some differential operators (not necessarily from
the $\operatorname{\mathfrak{gl}}(\infty)$ algebra associated with the KP
hierarchy) with $\deg\widehat{V}_{k}=k$. From Proposition 2.1 it follows that
such an equation always have a unique solution. The following simple statement
is a generalization of the observation of [MMMR21]:

######  Corollary 2.2.

Equation (2.10) has a unique solution in ${\mathbb{C}}[\\![{\bf t}]\\!]$ up to
a normalization.

######  Remark 2.2.

For the KP hierarchy one can construct the equation of the form (2.10) using
the product of the canonical Kac–Schwarz operators [Ale21]. In this case the
constructed operators $\widehat{V}_{k}$ belong to the
$\operatorname{\mathfrak{gl}}(\infty)$ algebra of KP symmetries. However,
already for the Kontsevich–Witten and Brézin–Gross–Witten tau-functions the
answer, provided by this construction, does not coincide with the simplest
description obtained from the solution of the Virasoro constraints, see
Section 3.2.

###  2.2. Algebraic topological recursion

Generating functions in enumerative geometry and mathematical physics often
depend on a parameter $\hbar$ associated to some topological expansion. Often
it is related to the Euler characteristics $\chi$ of certain Riemann surface
with punctures, in this case the topological expansion is given by
$Z=\exp\sum_{\chi}\hbar^{-\chi}F_{\chi}$, or

(2.11) $Z=\sum_{k=0}^{\infty}\hbar^{k}Z^{(k)}.$

Assume that the generating function $Z$ is described by the equation (2.10),
the more general equation (2.2) can be considered similarly. Let the
generating function $Z$ also satisfy the dimension constraint

(2.12) $a\hbar\frac{\partial}{\partial\hbar}Z=\widehat{L}_{0}\cdot Z$

for some $a\in{\mathbb{Z}}_{>0}$.

######  Remark 2.3.

The constraint (2.12) makes the parameter $\hbar$ redundant, however, it is
convenient to use it.

Assume that the operators in (2.10) are of the form
$\widehat{V}_{k}=\hbar^{\frac{k}{a}}\widehat{W}_{k}$, where $\widehat{W}_{k}$
do not depend on $\hbar$. Then they are nontrivial only if
$k/a\in{\mathbb{Z}}$, and the equation (2.10) leads to

(2.13)
$a\hbar\frac{\partial}{\partial\hbar}Z=\sum_{k=1}^{\infty}ka\,\hbar^{k}\widehat{W}_{ka}\cdot
Z.$

######  Proposition 2.3.

For any initial condition $Z^{(0)}$ if a solution to the equation (2.13) of
the form (2.11) exists, it is unique.

###### Proof.

Indeed, if we substitute (2.11) into (2.13), we get a recursion

(2.14)
${Z}^{(k)}=\frac{1}{k}\sum_{j=0}^{k-1}(k-j)\widehat{W}_{a(k-j)}\cdot{Z}^{(j)},$

which allows us to find all ${Z}^{(k)}$ recursively starting from the initial
condition $Z^{(0)}$. ∎

Note that we do not use Proposition 2.1, therefore, Proposition 2.3 is valid
even if the generating function does not satisfy any equation (2.2). However,
if this equation is satisfied, then $Z^{(0)}$ is a constant and the solution
always exists.

Equation of the form (2.14) describes the algebraic topological recursion. The
solution is given by ordered exponential

(2.15)
$Z=``\exp\left({\sum_{k=1}^{\infty}\hbar^{k}\widehat{W}_{ka}}\right)"\cdot
Z^{(0)},$

which simplifies as in (2.6) or (2.7) for the particular cases. Operators
$\widehat{W}_{k}$ describe the change of topology, that’s why below slightly
abusing notation we will call them the cut-and-join operators.

In the simplest case $\widehat{W}_{ka}=\delta_{k,1}\widehat{{W}}_{a}$, and the
generating function is given by

(2.16) $Z=\exp\left({\hbar\widehat{W}_{a}}\right)\cdot Z^{(0)}.$

In this case the algebraic topological recursion have a particularly simple
form

(2.17) $Z^{(k)}=\frac{1}{k}\widehat{W}_{a}\cdot Z^{(k-1)}.$

Let us list some enumerative geometry problems (and associated matrix models),
which can be described by the cut-and-join operators

  * •

Fat graphs or maps (Hermitian matrix model), [MS09];

  * •

Intersection theory of $\psi$ classes on the moduli spaces (Kontsevich model),
[Ale11];

  * •

Intersection theory of $\psi$ and Norbury’s classes on the moduli spaces
(Brézin–Gross–Witten model), [Ale18];

  * •

Weighted Hurwitz numbers (certain multi-matrix models), [GJ97] for ordinary
Hurwitz numbers, [ACEH18] and references therein for the weighted Hurwitz
numbers;

  * •

Open intersection theory (Kontsevich–Penner model), [Ale17];

  * •

Intersection theory of $\psi$ and Witten’s classes (Generalized Kontsevich
model), [Zho13, MMM21].

The last two examples are described by the non-commuting families of the cut-
and-join operators.

The Chekhov–Eynard–Orantin topological recursion [EO07, EO09] also describes a
recursive procedure in the negative Euler characteristics. However, many
properties of this recursion, briefly outlined in Section 6, are essentially
different from those of the algebraic topological recursion (2.14).

The algebraic recursion (2.14) is given by differential operators with
polynomial coefficients, while the Chekhov–Eynard–Orantin recursion is given
in terms of the residues on the spectral curve. The topological recursion
(2.14) is formulated in terms of the coefficients of the generating function
$Z$, or non-connected correlation functions, while the Chekhov–Eynard–Orantin
topological recursion is naturally formulated in terms of connected
correlation functions, or the coefficients of expansion of $\log Z$. The
recursion (2.14) is linear in the non-connected contributions, while the
Chekhov–Eynard–Orantin recursion in quadratic in connected correlation
functions.

The topological recursion (2.14) is often very convenient for computations. It
would be interesting to find a relation between the algebraic topological
recursion, described in this paper, and geometrical topological recursion of
Chekhov–Eynard–Orantin. In particular, there should be a way to construct the
cut-and-join operators from the topological recursion data. We believe that it
is given by some version of the inverse Laplace transform [EMS11].

######  Conjecture 2.1.

The Chekhov–Eynard–Orantin topological recursion with simple ramification
points can always be described by the cut-and-join formula (2.16) with
operator $\widehat{W}_{a}$ cubic in $\widehat{J}_{k}$. Higher ramification
points correspond to the higher cut-and-join operators.

It is also unclear how to directely relate the cut-and-join description to the
matrix integrals.

If two generating functions $Z$ and ${Z}^{*}$ are related to each other by

(2.18) ${Z}^{*}=\widehat{U}\cdot Z,$

where the operator $\widehat{U}$ does not depend on $\hbar$, then their cut-
and-join operators are also related by conjugation. Namely, if $Z$ satisfies
(2.13), then

(2.19)
${Z}^{*}=``\exp\left({\sum_{k=1}^{\infty}\hbar^{k}\widehat{W}_{ka}^{*}}\right)"\cdot
Z^{(0)*},$

where $\widehat{W}_{k}^{*}=\widehat{U}\,\widehat{W}_{k}\,\widehat{U}^{-1}$ and
$Z^{(0)*}=\widehat{U}\cdot Z^{(0)}$ are independent of $\hbar$. For the case
(2.16) we have

(2.20) ${Z}^{*}=\exp\left({\hbar\widehat{W}_{a}^{*}}\right)\cdot Z^{(0)*}.$

For a more general case with $\widehat{U}$ dependent on $\hbar$ sometimes it
is also possible to relate the cut-and-join descriptions.

In Sections 3 and 4 we consider the generating functions of the intersection
numbers on the moduli spaces with a natural topological expansion. Namely, the
KW and BGW tau-functions satisfy (2.12) for $a=3$ and $a=1$ respectively. From
the Virasoro constraints it follows that they are given by a cut-and-join
description of the form (2.16), see (3.21) below. The generating functions of
the triple Hodge integrals satisfying the Calabi–Yau condition satisfy the
dimension constraints given by Proposition 4.1. For the corresponding Euler
operator we do not have a description of the form (2.2). However, these tau-
functions are related to the KW and BGW tau-functions by (2.18), where
$\widehat{U}$ are the elements of the Virasoro group, see (4.50). Using the
conjugation, we derive the cut-and-join description (2.20) for these
generating functions.

##  3\. Cut-and-join operators for KW and BGW tau-functions

###  3.1. Intersection numbers

Denote by $\overline{\mathcal{M}}_{g,n}$ the Deligne–Mumford compactification
of the moduli space of all compact Riemann surfaces of genus $g$ with $n$
distinct marked points. It is a non-singular complex orbifold of dimension
$3g-3+n$. It is empty unless the stability condition

(3.1) $\displaystyle 2g-2+n>0$

is satisfied.

For each marking index $i$ consider the cotangent line bundle
${\mathbb{L}}_{i}\rightarrow\overline{\mathcal{M}}_{g,n}$, whose fiber over a
point $[\Sigma,z_{1},\ldots,z_{n}]\in\overline{\mathcal{M}}_{g,n}$ is the
complex cotangent space $T_{z_{i}}^{*}\Sigma$ of $\Sigma$ at $z_{i}$. Let
$\psi_{i}\in H^{2}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ denote the first
Chern class of ${\mathbb{L}}_{i}$. We consider the intersection numbers

(3.2)
$\displaystyle\left<\tau_{a_{1}}\tau_{a_{2}}\cdots\tau_{a_{n}}\right>_{g}=\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}.$

The integral on the right hand side of (3.2) vanishes unless the stability
condition (3.1) is obeyed, all $a_{i}$ are non-negative integers, and the
dimension constraint

(3.3) $3g-3+n=\sum_{i=1}^{n}a_{i}$

holds. Let $T_{i}$, $i\geq 0$, be formal variables and consider

(3.4)
$\tau_{KW}=\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}F_{g,n}\right),$

where

(3.5) $F_{g,n}=\sum_{a_{1},\ldots,a_{n}\geq
0}\left<\tau_{a_{1}}\tau_{a_{2}}\cdots\tau_{a_{n}}\right>_{g}\frac{\prod
T_{a_{i}}}{n!}.$

Witten’s conjecture [Wit91], proved by Kontsevich [Kon92], states that the
partition function $\tau_{KW}$ becomes a tau-function of the KdV hierarchy
after the change of variables $T_{k}=(2k+1)!!t_{2k+1}$. Integrability of the
Kontsevich–Witten tau-function immediately follows from Kontsevich’s matrix
integral representation, for more details see e.g. [Ale21] and references
therein.

The Brézin–Gross–Witten tau-function is another solution of the KdV hierarchy,
which is closely related to the moduli spaces. Namely, this tau-function
governs the intersection theory with the insertions of the fascinating
Norbury’s $\Theta$-classes. Norbury’s $\Theta$-classes are the cohomology
classes, $\Theta_{g,n}\in H^{4g-4+2n}(\overline{\mathcal{M}}_{g,n})$, which
are also related to the super Riemann surfaces [Nor20]. We refer the reader to
[Nor17, Nor18] for a detailed description. Let us consider the intersection
numbers

(3.6)
$\left<\tau_{a_{1}}\tau_{a_{2}}\cdots\tau_{a_{n}}\right>_{g}^{\Theta}=\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}.$

######  Remark 3.1.

For the $\Theta$-classes, higher degrees vanish because of the dimension
constrains. Indeed, $\dim\overline{\mathcal{M}}_{g,n}=3g-3+n$ and
$\Theta_{g,n}\in H^{4g-4+2n}(\overline{\mathcal{M}}_{g,n})$, therefore
$\Theta_{g,n}^{2}=0$.

Again, the integral on the right hand side vanishes unless the stability
condition (3.1) is satisfied, all $a_{i}$ are non-negative integers, and the
dimension constraint

(3.7) $g-1=\sum_{i=1}^{n}a_{i}$

holds. Consider the generating function of the intersection numbers of
$\Theta$ and $\psi$ classes

(3.8) $F^{\Theta}_{g,n}=\sum_{a_{1},\ldots,a_{n}\geq
0}\left<\tau_{a_{1}}\tau_{a_{2}}\cdots\tau_{a_{n}}\right>_{g}^{\Theta}\frac{\prod
T_{a_{i}}}{n!}$

then, we have a direct analog of the Kontsevich–Witten tau-function:

######  Theorem ([Nor17]).

The generating function

(3.9)
$\tau_{\Theta}=\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}F_{g,n}^{\Theta}\right)$

becomes a tau-function of the KdV hierarchy after the change of variables
$T_{k}=(2k+1)!!t_{2k+1}$.

Norbury also proved, that $\tau_{\Theta}$ is nothing but a tau-function of the
BGW model,

(3.10) $\tau_{\Theta}=\tau_{BGW}.$

KdV integrability of the BGW model follows from the relation to the
generalized Kontsevich model and was established by Mironov, Morozov and
Semenoff [MMS96].

###  3.2. Heisenberg–Virasoro constraints

It appears that the KW and BGW tau-functions share a lot of common properties
and it is natural to consider them simultaneously. Let us denote

(3.11) $\tau_{1}={\tau}_{KW},\quad\quad\tau_{0}={\tau}_{BGW}.$

The dimension constrains (3.3), (3.7) can be represented as

(3.12)
$\begin{split}(1+2\alpha)\hbar\frac{\partial}{\partial\hbar}{\tau}_{\alpha}=\widehat{L}_{0}\cdot{\tau}_{\alpha},\,\,\,\,\,\,\,\,\,\alpha=0,1.\end{split}$

Below we always assume that the index $\alpha$ runs from $0$ to $1$,

(3.13) $\alpha\in\\{0,1\\}.$

######  Remark 3.2.

Tau-functions ${\tau}_{\alpha}$ are KdV tau-functions; however, it is
convenient to consider them in the context of the KP hierarchy. So, below we
use the Heisenberg–Virasoro algebra of KP symmetries, introduced in Section 1.

Both $\tau_{\alpha}$ are KdV tau-functions, hence satisfy the Heisenberg
constraints (or the KdV reduction constraints)

(3.14) $\frac{\partial}{\partial
t_{2k}}{\tau}_{\alpha}=0,\,\,\,\,\,\,\,\,\,\,\,\,k>0.$

Moreover, they satisfy the Virasoro constraints [FKN91, DVV91, GN92]

(3.15)
$\begin{split}\widehat{L}_{k}^{\alpha}\cdot{\tau}_{\alpha}&=0,\,\,\,\,k\geq-\alpha,\\\
\end{split}$

where the Virasoro operators are given by

(3.16)
$\begin{split}\widehat{L}_{k}^{\alpha}&=\frac{1}{2}\widehat{L}_{2k}-\frac{1}{2\hbar}\frac{\partial}{\partial
t_{2k+1+2\alpha}}+\frac{\delta_{k,0}}{16}\in{\mathcal{L}}.\end{split}$

These operators satisfy the commutation relations

(3.17)
$\left[\widehat{L}_{k}^{\alpha},\widehat{L}_{k}^{\alpha}\right]=(k-m)\widehat{L}_{k+m}^{\alpha},\quad\quad
k,m\geq-\alpha.$

Combining (3.14) with (3.15) one gets

(3.18)
$\widehat{L}_{0}\cdot\tau_{\alpha}=(1+2\alpha)\hbar\widehat{W}_{\alpha}\cdot\tau_{\alpha},$

where

(3.19)
$\widehat{W}_{\alpha}=\frac{1}{2\alpha+1}\sum_{k=0}^{\infty}(2k+1)t_{2k+1}\left(\widehat{L}_{2k-2\alpha}+\frac{\delta_{k,\alpha}}{8}\right).$

Therefore, from (3.12) we have

(3.20)
$\hbar\frac{\partial}{\partial\hbar}\tau_{\alpha}=\hbar\widehat{W}_{\alpha}\cdot\tau_{\alpha}.$

This is the relation of the form (2.13), it leads to the cut-and-join
description for the KW and BGW tau-functions [Ale11, Ale18]

(3.21) $\tau_{\alpha}=\exp\left({\hbar\widehat{W}_{\alpha}}\right)\cdot 1.$

######  Remark 3.3.

There is a certain arbitrariness in the construction of the cut-and-join
operators. Since the tau-functions $\tau_{\alpha}$ depend only on odd times,
it is possible to consider operators $\widehat{W}_{\alpha}$ with omitted even
times $t_{2k}$ and corresponding derivatives,

(3.22)
$\begin{split}\widehat{W}_{0}^{*}&=\sum_{k,m\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}\left(kmt_{k}t_{m}\frac{\partial}{\partial
t_{k+m-1}}+\frac{1}{2}(k+m+1)t_{k+m+1}\frac{\partial^{2}}{\partial
t_{k}\partial t_{m}}\right)+\frac{t_{1}}{8},\\\
\widehat{W}_{1}^{*}&=\frac{1}{3}\sum_{k,m\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}\left(kmt_{k}t_{m}\frac{\partial}{\partial
t_{k+m-3}}+\frac{1}{2}(k+m+3)t_{k+m+3}\frac{\partial^{2}}{\partial
t_{k}\partial t_{m}}\right)+\frac{t_{1}^{3}}{3!}+\frac{t_{3}}{8}.\end{split}$

Then an equivalent cut-and-join description is given by the these operators,

(3.23) $\tau_{\alpha}=\exp\left({\hbar\widehat{W}_{\alpha}^{*}}\right)\cdot
1.$

If we consider the topological expansion of the tau-functions
$\tau_{\alpha}=\sum_{k=0}^{\infty}\tau_{\alpha}^{(k)}\hbar^{k}$, then we have
the algebraic topological recursion

(3.24)
$\begin{split}\tau_{\alpha}^{(k)}&=\frac{1}{k}\widehat{W}_{\alpha}\cdot\tau_{\alpha}^{(k-1)}\\\
&=\frac{1}{k}\widehat{W}_{\alpha}^{*}\cdot\tau_{\alpha}^{(k-1)}\end{split}$

with the initial condition $\tau_{\alpha}^{(0)}=1$. Moreover, we can separate
the contributions with different numbers of marked points. Namely, let
$\tau_{\alpha}^{(k)}=\sum_{n=1}^{(2\alpha+1)(k-1)}\tau_{\alpha,n}^{(k)}$,
where $\tau_{\alpha,n}^{(k)}$ is a homogeneous polynomial in ${\bf t}$ of
degree $n$. Let also

(3.25)
$\begin{split}\widehat{W}_{0,1}^{*}&=\sum_{k,m\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}kmt_{k}t_{m}\frac{\partial}{\partial
t_{k+m-1}}+\frac{t_{1}}{8},\\\
\widehat{W}_{0,-1}^{*}&=\frac{1}{2}\sum_{k,m\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}(k+m+1)t_{k+m+1}\frac{\partial^{2}}{\partial
t_{k}\partial t_{m}},\\\
\widehat{W}_{1,1}^{*}&=\frac{1}{3}\sum_{k,m\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}kmt_{k}t_{m}\frac{\partial}{\partial
t_{k+m-3}}+\frac{t_{3}}{8},\\\
\widehat{W}_{1,-1}^{*}&=\frac{1}{6}\sum_{k,m\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}(k+m+3)t_{k+m+3}\frac{\partial^{2}}{\partial
t_{k}\partial t_{m}},\end{split}$

and $\widehat{W}_{0,3}^{*}=0$, $\widehat{W}_{1,3}^{*}=\frac{t_{1}^{3}}{3!}$.
Then we have the recursion

(3.26)
$\tau_{\alpha,n}^{(k)}=\frac{1}{k}\left(\widehat{W}^{*}_{\alpha,3}\cdot\tau_{\alpha,n-3}^{(k-1)}+\widehat{W}^{*}_{\alpha,1}\cdot\tau_{\alpha,n-1}^{(k-1)}+\widehat{W}^{*}_{\alpha,-1}\cdot\tau_{\alpha,n+1}^{(k-1)}\right)$

with the initial condition $\tau_{\alpha,n}^{(0)}=\delta_{n,0}$.

The cut-and-join description allows us to find the topological expansion of
the tau-functions

(3.27)
$\log\tau_{0}={\frac{t_{{1}}}{8}}\hbar+{\frac{{t_{{1}}}^{2}}{16}}{\hbar}^{2}+\left({\frac{{t_{{1}}}^{3}}{24}}+{\frac{9\,t_{{3}}}{128}}\right){\hbar}^{3}+\left({\frac{{t_{{1}}}^{4}}{32}}+{\frac{27\,t_{{3}}t_{{1}}}{128}}\right){\hbar}^{4}+\left({\frac{{t_{{1}}}^{5}}{40}}+{\frac{27\,t_{{3}}{t_{{1}}}^{2}}{64}}+{\frac{225\,t_{{5}}}{1024}}\right){\hbar}^{5}\\\
+\left({\frac{{t_{{1}}}^{6}}{48}}+{\frac{45\,t_{{3}}{t_{{1}}}^{3}}{64}}+{\frac{1125\,t_{{5}}t_{{1}}}{1024}}+{\frac{567\,{t_{{3}}}^{2}}{1024}}\right){\hbar}^{6}+O(\hbar^{7}),$
(3.28)
$\log\tau_{1}=\left({\frac{{t_{{1}}}^{3}}{6}}+{\frac{t_{{3}}}{8}}\right)\hbar+\left({\frac{t_{{3}}{t_{{1}}}^{3}}{2}}+{\frac{5\,t_{{5}}t_{{1}}}{8}}+{\frac{3\,{t_{{3}}}^{2}}{16}}\right){\hbar}^{2}\\\
+\left({\frac{15\,t_{{1}}t_{{3}}t_{{5}}}{4}}+{\frac{3\,{t_{{3}}}^{2}{t_{{1}}}^{3}}{2}}+{\frac{5\,t_{{5}}{t_{{1}}}^{4}}{8}}+{\frac{35\,t_{{7}}{t_{{1}}}^{2}}{16}}+{\frac{3\,{t_{{3}}}^{3}}{8}}+{\frac{105\,t_{{9}}}{128}}\right){\hbar}^{3}+O(\hbar^{4}).$

###  3.3. Heisenberg–Virasoro group action and cut-and-join operators

In this section, we investigate how the subgroup ${\mathcal{V}}_{+}$ of the
Heisenberg–Virasoro group, see Section 1, acts on the KdV tau-functions
$\tau_{\alpha}$, $\alpha\in\\{0,1\\}$. In general, the action of this subgroup
is not well-defined and can lead to divergences. In this section, we do not
address this potential issue, assuming that all considered functions are well
defined.

Let us consider the subgroups of the Heisenberg–Virasoro group, generated by
the operators of the form

(3.29)
${\mathcal{V}}_{+}\ni\widehat{G}_{\alpha}:=\exp\left(\sum_{k=1}^{\infty}\tilde{a}_{k}\widehat{L}_{k}\right)\exp\left(\hbar^{-1}\sum_{k=2+2\alpha}^{\infty}{v}_{k}\widehat{J}_{k}\right).$

######  Remark 3.4.

One can easily relax the constraint on the range of summation and consider
more general operators. However, this makes the formulas less compact, so we
prefer to deal with a less general case, which is sufficient for our needs.

We consider the functions $\widehat{G}_{\alpha}\cdot\tau_{\alpha}$. The tau-
functions $\tau_{\alpha}$ satisfy the KdV reduction constraints,
$\widehat{J}_{2k}\cdot\tau_{\alpha}=0$ for $k>0$, therefore we may assume that
${v}_{2k}=0$. Using the Virasoro constrains (3.15) we can rewrite action of
the operator (3.29) on the tau-function $\tau_{\alpha}$ in terms of the
Virasoro group only. For this purpose we introduce a bijection between the
space of the Virasoro operators

(3.30) $\widehat{V}_{\bf a}=\exp\left({\sum_{k>0}a_{k}\widehat{L}_{k}}\right)$

and the space of all formal series of the form $f(z)\in
z+z{\mathbb{C}}[\\![z]\\!]$. Recall that the operators ${\mathtt{l}}_{k}$ are
given by (1.5).

######  Definition 1.

For the Virasoro group element $\widehat{V}$ consider

(3.31)
$\Xi(\widehat{V})=e^{\sum_{k>0}a_{k}{\mathtt{l}}_{k}}\,z\,e^{-\sum_{k>0}a_{k}{\mathtt{l}}_{k}}\in
z+z{\mathbb{C}}[\\![z]\\!].$

For a given set of parameters $a_{k}\in{\mathbb{C}}$, $k\in{\mathbb{Z}}_{>0}$
we denote such a series associated to the operator ${\widehat{V}}_{\bf a}$ by
$f(z;{\bf a})=\Xi(\widehat{V}_{\bf a})$. We also introduce the inverse formal
series $h(z;{\bf a})$,

(3.32) $f(h(z;{\bf a});{\bf a})=z.$

For any series $f(z)\in z+z{\mathbb{C}}[\\![z]\\!]$ we construct a
corresponding element of the Virasoro group

(3.33) $\widehat{V}=\Xi^{-1}(f(z)),$

where $a_{k}$ are defined implicitly by

(3.34)
$f(z)=e^{\sum_{k>0}a_{k}{\mathtt{l}}_{k}}\,z\,e^{-\sum_{k>0}a_{k}{\mathtt{l}}_{k}}.$

For the inverse group element we have

(3.35) $\widehat{V}^{-1}=\Xi^{-1}(h(z)).$

Multiplication of the Virasoro group elements corresponds to the composition
of the associated functions

(3.36) $\Xi(\widehat{V}_{1}\widehat{V}_{2})=f_{2}(f_{1}(z)).$

To any sequence of coefficients $a_{k}\in{\mathbb{C}}$,
$k\in{\mathbb{Z}}_{>0}$, we also associate the following formal series

(3.37)
${v}^{\alpha}(z)=\frac{1}{(2\alpha+1)}(z^{2\alpha+1}-f(z;{\bf{a}})^{2\alpha+1})\in
z^{2+2\alpha}{\mathbb{C}}[\\![z]\\!]$

with the coefficients

(3.38) $v_{k}^{\alpha}({\bf{a}})=[z^{k}]{v}^{\alpha}(z),$

where for any formal Laurent series $F(z)=\sum b_{k}z^{k}$ we introduce
$[z^{k}]F(z):=b_{k}$.

######  Lemma 3.1.

We have

(3.39)
$\exp\left({\hbar^{-1}\sum_{k=\alpha+1}v_{k}^{\alpha}({\bf{a}})\frac{\partial}{\partial
t_{2k+1}}}\right)\cdot\tau_{\alpha}=\exp\left({\sum_{k>0}a_{2k}\widehat{L}_{2k}}\right)\cdot\tau_{\alpha}.$

###### Proof.

The statement of the lemma follows from Lemma 1.1 of [Ale15]. ∎

From this lemma it follows that

(3.40)
$\begin{split}\widehat{G}_{\alpha}\cdot\tau_{\alpha}&=\exp\left(\sum_{k=1}^{\infty}\tilde{a}_{k}\widehat{L}_{k}\right)\exp\left(\hbar^{-1}\sum_{k=\alpha+1}^{\infty}{v}_{2k+1}\widehat{J}_{2k+1}\right)\cdot\tau_{\alpha}\\\
&=\exp\left(\sum_{k=1}^{\infty}\tilde{a}_{k}\widehat{L}_{k}\right)\exp\left(\sum_{k=1}^{\infty}{a}_{2k}\widehat{L}_{2k}\right)\cdot\tau_{\alpha},\end{split}$

where the coefficients ${a}_{k}$ are given implicitly by the relation (3.37)
with ${v}_{2k+1}=v_{2k+1}^{\alpha}({\bf{a}})$. Therefore, we always can reduce
the action of the operator (3.29) to the action of equivalent operator (3.30)
from the Virasoro group.

Let us consider arbitrary operator $\widehat{V}_{\bf a}$ of the form (3.30)
and its action on the tau-function

(3.41) $\tau_{\alpha}^{\bf a}=\widehat{V}_{\bf a}\cdot\tau_{\alpha}.$

for some coefficients $a_{k}$. $\tau_{\alpha}^{\bf a}$ are tau-functions of
the KP hierarchy. Let us derive the Heisenberg–Virasoro constraints and the
cut-and-join description for the tau-functions $\tau_{\alpha}^{\bf a}$.

We introduce

(3.42) $\begin{split}{\widehat{J}}_{k}^{{\bf a}}=\widehat{V}_{\bf
a}\widehat{J}_{k}\widehat{V}_{\bf a}^{-1},\quad\quad{\widehat{L}}_{k}^{{\bf
a}}=\widehat{V}_{\bf a}\widehat{L}_{k}\widehat{V}_{\bf a}^{-1}.\end{split}$

From the commutation relations of the Heisenberg–Virasoro algebra
${\mathcal{L}}$ it follows that these operators are given by the linear
combinations of $\widehat{J}_{m}$ and $\widehat{L}_{m}$ operators. Let us also
introduce the coefficients

(3.43)
$\rho[k,m]=[z^{m}]f(z;{\bf{a}})^{k},\quad\quad\quad\sigma[k,m]=[z^{m+1}]\frac{f(z;{\bf{a}})^{k+1}}{f^{\prime}(z;{\bf{a}})},$

where $f(z,{\bf a})=\Xi(\widehat{V}_{\bf a})$. They are polynomials in ${\bf
a}$.

From the relation between the operators $\widehat{J}_{k}$, $\widehat{L}_{k}$,
${\mathtt{l}}_{k}$, ${\mathtt{j}}_{k}$ and Definition 1 we have

######  Lemma 3.2.

(3.44) $\displaystyle\widehat{J}_{k}^{{\bf a}}$
$\displaystyle=\sum_{m=k}^{\infty}\rho[k,m]\widehat{J}_{m},$ $\displaystyle
k\in{\mathbb{Z}},$ $\displaystyle\widehat{L}_{k}^{{\bf a}}$
$\displaystyle=\sum_{m=k}^{\infty}\sigma[k,m]\widehat{L}_{m}+\frac{a_{2}}{2}\delta_{k,-2},\quad\quad$
$\displaystyle k\geq-2.$

Consider the operators

(3.45) $\begin{split}\widehat{J}_{k}^{{\bf
a}}&=\frac{1}{2}\widehat{J}_{2k}^{{\bf a}},\\\ \widehat{L}_{k}^{\alpha,{\bf
a}}&=\frac{1}{2}\widehat{L}_{2k}^{\bf
a}-\frac{1}{2\hbar}\widehat{J}_{2k+1+2\alpha}^{{\bf
a}}+\frac{\delta_{k,0}}{16}.\end{split}$

Then

######  Lemma 3.3.

Tau-functions $\tau_{\alpha}^{\bf a}$ satisfy the Heisenberg constraints

(3.46) $\widehat{J}_{2k}^{{\bf a}}\cdot\tau_{\alpha}^{\bf a}=0,\quad k>0,$

and the Virasoro constraints

(3.47) $\begin{split}\widehat{L}_{k}^{\alpha,{\bf a}}\cdot\tau_{\alpha}^{\bf
a}&=0,\quad k\geq-\alpha.\end{split}$

These operators satisfy the commutation relations of the Heisenberg–Virasoro
algebra

(3.48) $\displaystyle\left[\widehat{J}_{k}^{{\bf a}},\widehat{J}_{m}^{{\bf
a}}\right]$ $\displaystyle=0,$ $\displaystyle k,m>0,$
$\displaystyle\left[\widehat{L}_{k}^{\alpha,{\bf a}},\widehat{J}_{m}^{{\bf
a}}\right]$ $\displaystyle=k\widehat{J}_{k+m}^{{\bf a}},$ $\displaystyle
k\geq-\alpha,\,m>0,$ $\displaystyle\left[\widehat{L}_{k}^{\alpha,{\bf
a}},\widehat{L}_{m}^{\alpha,{\bf a}}\right]$
$\displaystyle=(k-m)\widehat{L}_{k+m}^{\alpha,{\bf a}},\quad\quad$
$\displaystyle k,m\geq-\alpha.$

###### Proof.

Operators $\widehat{J}_{k}^{{\bf a}}$ and $\widehat{L}_{k}^{\alpha,{\bf a}}$
are given by the conjugation of the constraints (3.14) and (3.15), in
particular

(3.49) $\widehat{L}_{k}^{\alpha,{\bf a}}=\widehat{V}_{\bf
a}\widehat{L}_{k}^{\alpha}\widehat{V}_{\bf a}^{-1}.$

∎

If ${\bf a}$ do not depend on $\hbar$, then the relation (3.41) is of the form
(2.18), therefore we can derive the cut-and-join description of
$\tau_{\alpha}^{\bf a}$ from the expression (3.19). Let

(3.50) $\widehat{{W}}_{\alpha}^{{\bf
a}}=\frac{1}{2\alpha+1}\sum_{k=0}^{\infty}\widehat{J}_{-2k-1}^{{\bf
a}}\left(\widehat{L}_{2k-2\alpha}^{{\bf
a}}+\frac{\delta_{k,\alpha}}{8}\right).$

This is a cut-and-join operator for $\tau_{\alpha}^{\bf a}$.

######  Lemma 3.4.

(3.51) $\tau_{\alpha}^{\bf a}=\exp\left({\hbar\widehat{{W}}_{\alpha}^{{\bf
a}}}\right)\cdot 1.$

###### Proof.

We use the cut-and-join representation (3.21). Since $\widehat{L}_{k}\cdot
1=0$ for $k\geq 0$, we have

(3.52) $\begin{split}\widehat{V}_{\bf a}\cdot\tau_{\alpha}&=\widehat{V}_{\bf
a}\exp\left({\hbar\widehat{W}_{\alpha}}\right)\widehat{V}_{\bf a}^{-1}\cdot
1\\\ &=\exp\left({\hbar\widehat{{W}}_{\alpha}^{{\bf a}}}\right)\cdot
1,\end{split}$

where $\widehat{{W}}_{\alpha}^{{\bf a}}:=\widehat{V}_{\bf
a}\widehat{W}_{\alpha}\widehat{V}_{\bf a}^{-1}$ coincides with (3.50). ∎

Because of the certain arbitrariness of the cut-and-join description of the
tau-functions $\tau_{\alpha}$, see Remark 3.3, we also construct another
version of the cut-and-join operators for the tau-functions
$\tau_{\alpha}^{\bf a}$. Let

(3.53) $\begin{split}{\widehat{W}_{0}^{*{\bf
a}}}&=\sum_{k,m\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}\left({\widehat{J}}_{-k}^{\bf
a}{\widehat{J}}_{-m}^{\bf a}{\widehat{J}}_{k+m-1}^{\bf
a}+\frac{1}{2}{\widehat{J}}^{\bf a}_{-k-m-1}{\widehat{J}}_{k}^{\bf
a}{\widehat{J}}_{m}^{\bf a}\right)+\frac{{\widehat{J}}_{-1}^{\bf a}}{8},\\\
{\widehat{W}_{1}^{*{\bf
a}}}&=\frac{1}{3}\sum_{k,m\in{\mathbb{Z}}_{\operatorname{odd}}^{+}}^{\infty}\left({\widehat{J}}_{-k}^{\bf
a}{\widehat{J}}_{-m}^{\bf a}{\widehat{J}}_{k+m-3}^{\bf
a}+\frac{1}{2}{\widehat{J}}_{-k-m-3}^{\bf a}{\widehat{J}}_{k}^{\bf
a}{\widehat{J}}_{m}^{\bf a}\right)+\frac{({\widehat{J}}_{-1}^{\bf
a})^{3}}{3!}+\frac{{\widehat{J}}_{-3}^{\bf a}}{24}.\end{split}$

######  Lemma 3.5.

(3.54) $\tau_{\alpha}^{\bf a}=\exp\left(\hbar\,{\widehat{W}_{\alpha}^{*{\bf
a}}}\right)\cdot 1.$

Note that the operators ${\widehat{W}_{\alpha}^{\bf a}}$ and
${\widehat{W}_{\alpha}^{*{\bf a}}}$ only contain the cubic and linear terms in
$\widehat{J}_{k}$.

##  4\. Triple Hodge integrals and KP hierarchy

In [Ale20, Ale21] the author has proven that the generating functions of the
triple Hodge integrals satisfying the Calabi–Yau condition are tau-functions
of the KP hierarchy. These tau-functions can be related to the tau-functions
$\tau_{\alpha}$ considered in Section 3 by certain elements of the subgroup
${\mathcal{V}}_{+}$ of the Heisenberg–Virasoro group. The Virasoro constraints
allow us to reduce these group elements to the elements of the Virasoro group.
In this section, using the arguments of Section 3.3 we will derive the cut-
and-join description and the Heisenberg–Virasoro constraints for the
generating functions of the triple Hodge integrals satisfying the Calabi–Yau
condition.

###  4.1. KP integrability

Let us remind the description of the generating functions of the triple Hodge
integrals satisfying the Calabi–Yau condition by KP tau-functions [Ale20,
Ale21]. Consider the linear change of variables

(4.1) $\begin{split}T_{0}^{q,p}({\bf t})&=t_{1},\\\ T_{k}^{q,p}({\bf
t})&=\sum_{m=1}^{\infty}mt_{m}\left(q\frac{\partial}{\partial
t_{m}}+\frac{2q+p}{\sqrt{p+q}}\frac{\partial}{\partial
t_{m-1}}+\frac{\partial}{\partial t_{m-2}}\right)\cdot T_{k-1}^{q,p}({\bf
t}),\quad k>1,\end{split}$

where $q$ and $p$ are two parameters with $q+p\neq 0$ if $q\neq 0$.

We also introduce the coefficients $a_{k}^{q,p}$. They are given implicitly
through the Definition 1,

(4.2) $f(z;{\bf a}^{q,p})=f(z),$

where the series on the right hand side is given by

(4.3) $f(z)\hbox{d}f(z)=\frac{z\hbox{d}z}{(1+\sqrt{p+q}z)(1+qz/\sqrt{p+q})}.$

In particular,

(4.4) $a_{1}^{q,p}=\frac{1}{3}{\frac{p+2\,q}{\sqrt{p+q}}},\quad
a_{2}^{q,p}=-\frac{1}{12}{\frac{{p}^{2}+pq+{q}^{2}}{p+q}},\quad
a_{3}^{q,p}={\frac{31\,{p}^{3}+69\,{p}^{2}q+21\,p{q}^{2}+14\,{q}^{3}}{1080\,\left(p+q\right)^{3/2}}}.$

Let ${\mathtt{x}}(z)=\frac{f^{2}(z)}{2}=z^{2}/2+\dots$. If $p\neq 0$ and
$q\neq 0$ we have

(4.5)
${\mathtt{x}}(z)=\frac{p+q}{pq}\log\left(1+\frac{qz}{\sqrt{p+q}}\right)-\frac{1}{p}\log(1+\sqrt{p+q}z)$

and

(4.6)
$f(z)=\sqrt{2\frac{p+q}{pq}\log\left(1+\frac{qz}{\sqrt{p+q}}\right)-\frac{2}{p}\log(1+\sqrt{p+q}z)}.$

For $q=0$ it degenerates to

(4.7) ${\mathtt{x}}(z)=\frac{z}{\sqrt{p}}-\frac{1}{p}\log(1+\sqrt{p}z),$

and for $p=0$ it degenerates to

(4.8)
${\mathtt{x}}(z)=\frac{1}{q}\log(1+\sqrt{q}z)-\frac{1}{\sqrt{q}}\frac{z}{1+\sqrt{q}z}.$

Let us note that the change of variables (4.1) is generated by the operator

(4.9)
$\frac{\partial}{\partial{\mathtt{x}}}=\frac{(1+\sqrt{p+q}z)(1+qz/\sqrt{p+q})}{z}\frac{\partial}{\partial
z}.$

Namely, the functions

(4.10)
$\Phi_{k}(z):=\left(-\frac{\partial}{\partial{\mathtt{x}}}\right)^{k}\cdot\frac{1}{z}$

define this change of variables if we associate $kt_{k}$ with $z^{-k}$ and
$T_{k}^{q,p}$ with $\Phi_{k}(z)$.

Let

(4.11) ${\mathtt{y}}_{0}(z)=\frac{1}{z}$

and

(4.12)
${\mathtt{y}}_{1}(z)=\int_{0}^{z}{\mathtt{y}}_{0}(\eta)\hbox{d}{\mathtt{x}}(\eta)=\int_{0}^{z}\frac{\hbox{d}{\mathtt{x}}(\eta)}{\eta},$

where $\hbox{d}{\mathtt{x}}(\eta)=f(\eta)\hbox{d}f(\eta)$. For $p\neq 0$ and
$q\neq 0$ we have

(4.13)
${\mathtt{y}}_{1}(z)=\frac{\sqrt{p+q}}{p}\left(\log\left(1+\sqrt{p+q}z\right)-\log\left(1+\frac{qz}{\sqrt{p+q}}\right)\right).$

For $q=0$ it reduces to

(4.14) ${\mathtt{y}}_{1}(z)=\frac{1}{\sqrt{p}}\log\left(1+\sqrt{p}z\right)$

and for $p=0$ it degenerates to

(4.15) ${\mathtt{y}}_{1}(z)=\frac{z}{1+\sqrt{q}z}.$

The series ${\mathtt{x}}$ and ${\mathtt{y}}_{1}$ satisfy the equation

(4.16)
$p\exp\left(-q{\mathtt{x}}(z)\right)=(p+q)\exp\left(\frac{q}{\sqrt{p+q}}{\mathtt{y}}_{1}(z)\right)-q\exp\left(\sqrt{p+q}{\mathtt{y}}_{1}(z)\right).$

Let us consider the Hodge bundle $\mathbb{E}$, a rank $g$ vector bundle over
$\overline{\mathcal{M}}_{g,n}$. For $\lambda_{i}=c_{i}({\mathbb{E}})$ we set

(4.17) $\Lambda_{g}(u)=\sum_{i=0}^{g}u^{i}\lambda_{i}.$

Below we focus on the case of cubic Hodge integrals

(4.18)
$\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}^{1-\alpha}\Lambda_{g}(u_{1})\Lambda_{g}(u_{2})\Lambda_{g}(u_{3})\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}$

satisfying an additional Calabi–Yau condition

(4.19) $\frac{1}{u_{1}}+\frac{1}{u_{2}}+\frac{1}{u_{3}}=0.$

For this case it is convenient to use the parametrization

(4.20) $u_{1}=-p,u_{2}=-q,u_{3}=\frac{pq}{p+q}.$

We introduce the generating functions of the triple Hodge integrals satisfying
the Calabi–Yau condition, with (for $\alpha=0$) and without (for $\alpha=1$)
Norbury’s $\Theta$-classes, namely

(4.21) $Z_{q,p}^{(\alpha)}({\bf
T})=\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}{\mathcal{F}}_{g,n}^{\alpha}\right),$

where

(4.22) ${\mathcal{F}}_{g,n}^{\alpha}=\sum_{a_{1},\dots,a_{n}\geq 0}\frac{\prod
T_{a_{i}}}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}^{1-\alpha}\Lambda_{g}(-q)\Lambda_{g}(-p)\Lambda_{g}\left(\frac{pq}{p+q}\right)\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}.$

Define the coefficients ${v}_{j}^{(\alpha)}$ by

(4.23)
$\begin{split}\sum_{j=2+2\alpha}^{\infty}{v}_{j}^{(\alpha)}z^{j}&=\int_{0}^{z}\left(f(\eta)^{2\alpha-1}-{\mathtt{y}}_{\alpha}(\eta)\right)\hbox{d}{\mathtt{x}}(\eta).\end{split}$

In particular, for $\alpha=0$ one has

(4.24)
$\begin{split}v^{(0)}_{k}&=[z^{k}]\left(f(z)-{\mathtt{y}}_{1}(z)\right).\end{split}$

Let us introduce an element of the Virasoro group

(4.25)
$\widehat{V}_{q,p}=\Xi^{-1}(f(z))=\exp\left({\sum_{k\in{\mathbb{Z}}_{>0}}a_{k}^{q,p}\widehat{L}_{k}}\right).$

Then we have

######  Theorem ([Ale20]).

The generating functions of triple Hodge integrals and $\Theta$-Hodge
integrals satisfying the Calabi–Yau condition in the variables ${{\bf
T}^{q,p}(\bf t)}$ are tau-functions of the KP hierarchy,

(4.26) $\tau_{q,p}^{(\alpha)}({\bf t})=Z_{q,p}^{(\alpha)}({{\bf T}^{q,p}(\bf
t)}).$

They are related to the KW and BGW tau-functions by the elements of the
Heisenberg–Virasoro group

(4.27) $\tau_{q,p}^{(\alpha)}({\bf
t})=\exp\left({\hbar^{-1}\sum_{j=2+2\alpha}^{\infty}v_{j}^{(\alpha)}\frac{\partial}{\partial
t_{j}}}\right)\widehat{V}_{q,p}\cdot\tau_{\alpha}({\bf t}).$

The proof of the theorem is based on the comparison between the Givental
operators and the elements of the Heisenberg–Virasoro group ${\mathcal{V}}$ of
KP symmetries. For the $p=0$ and $\alpha=1$ case, corresponding to the linear
Hodge integrals, the KP integrability was proved by Kazarian [Kaz09]. Relation
between tau-functions given by the elements of the Heisenberg–Virasoro group
elements for this case was derived in [Ale15]. Identification of this group
element and the Givental group element has been found in [LW16]. Alternative
proof of the integrability part of this theorem for $\alpha=1$, based on the
Mariño–Vafa formula, is given by Kramer [Kra21].

######  Remark 4.1.

In this paper we focus on the tau-functions $\tau_{q,p}^{(\alpha)}$ and do not
investigate the properties of the generating functions $Z_{q,p}^{(\alpha)}$ in
the original variables. Virasoro constraints and cut-and-join description for
them can be found with the help of the Givental formalism [Giv01, Tel12] and
will be considered elsewhere.

###  4.2. Heisenberg–Virasoro and dimension constraints

We start with the permutation of the translation operator with the element of
the Virasoro group in (4.27). Namely, Lemma 3.2 leads to the relation

(4.28) $\tau_{q,p}^{(\alpha)}({\bf
t})=\widehat{V}_{q,p}\exp\left({\hbar^{-1}\sum_{j=2+2\alpha}\tilde{v}_{j}^{(\alpha)}\frac{\partial}{\partial
t_{j}}}\right)\cdot\tau_{\alpha}({\bf t}),$

where from (4.23) for the coefficients $\tilde{v}_{j}^{(\alpha)}$ we have

(4.29)
$\begin{split}\sum_{j=2+2\alpha}^{\infty}\tilde{v}_{j}^{(\alpha)}z^{j}=\int_{0}^{h(z)}\left(f(\eta)^{2\alpha-1}-{\mathtt{y}}_{\alpha}(\eta)\right)\hbox{d}{\mathtt{x}}(\eta).\end{split}$

Here $h(z)$ is a series, inverse to $f(z)$ of (4.5), that is
$h(f(z))=f(h(z))=z$, hence

(4.30)
$\sum_{j=2+2\alpha}^{\infty}\tilde{v}_{j}^{(\alpha)}z^{j}=\int_{0}^{z}\left(\eta^{2\alpha}-\eta{\mathtt{y}}_{\alpha}(h(\eta))\right)\hbox{d}\eta.$

Since $\tau_{\alpha}({\bf t})$ are tau-function of the KdV hierarchy, we can
neglect the translations of the even times $t_{2k}$ and

(4.31) $\tau_{q,p}^{(\alpha)}({\bf
t})=\widehat{V}_{q,p}\exp\left({\hbar^{-1}\sum_{j=1+\alpha}\tilde{v}_{2j+1}^{(\alpha)}\frac{\partial}{\partial
t_{2j+1}}}\right)\cdot\tau_{\alpha}({\bf t}).$

To describe the Heisenberg–Virasoro constraints for the tau-functions
$\tau_{q,p}^{(\alpha)}$ let us consider

(4.32) $\displaystyle\widehat{J}_{k}^{(q,p)}$
$\displaystyle=\frac{1}{2}\sum_{m=2k}^{\infty}\rho[2k,m]\widehat{J}_{m},$
$\displaystyle\widehat{L}_{k}^{\alpha,(q,p)}$
$\displaystyle=\frac{1}{2}\sum_{m=2k}^{\infty}\sigma[2k,m]\widehat{L}_{m}-\frac{1}{2\hbar}\sum_{m=2k+1+2\alpha}^{\infty}\chi_{\alpha}[2k,m]\widehat{J}_{m}+\frac{\delta_{k,0}}{16}-\frac{\delta_{k,-1}}{48}{\frac{{p}^{2}+pq+{q}^{2}}{p+q}}.$

Here the coefficients $\sigma[k,m]$, $\rho[k,m]$, and $\chi_{\alpha}[k,m]$ are
given by

(4.33)
$\rho[k,m]=[z^{m}]f(z)^{k},\quad\quad\sigma[k,m]=[z^{m+1}]\frac{f(z)^{k+1}}{f^{\prime}(z)},\quad\quad\chi_{\alpha}[k,m]=[z^{m}]f(z)^{k+2}{\mathtt{y}}_{\alpha}(z),$

where the function $f(z)$ is given by (4.6).

######  Theorem 1.

The tau-functions of triple Hodge integrals satisfy the Heisenberg–Virasoro
constraints

(4.34) $\displaystyle\widehat{J}_{k}^{(q,p)}\cdot{\tau}_{q,p}^{(\alpha)}$
$\displaystyle=0,\quad$ $\displaystyle k>0,$
$\displaystyle\widehat{L}_{k}^{\alpha,(q,p)}\cdot{\tau}_{q,p}^{(\alpha)}$
$\displaystyle=0,\quad\quad$ $\displaystyle k\geq-\alpha.$

These operators satisfy the commutation relations of the Heisenberg–Virasoro
algebra

(4.35)
$\displaystyle\left[\widehat{J}_{k}^{(q,p)},\widehat{J}_{m}^{(q,p)}\right]$
$\displaystyle=0,$ $\displaystyle k,m>0,$
$\displaystyle\left[\widehat{L}_{k}^{\alpha,(q,p)},\widehat{J}_{m}^{(q,p)}\right]$
$\displaystyle=k\widehat{J}_{k+m}^{(q,p)},$ $\displaystyle
k\geq-\alpha,\,m>0,$
$\displaystyle\left[\widehat{L}_{k}^{\alpha,(q,p)},\widehat{L}_{m}^{\alpha,(q,p)}\right]$
$\displaystyle=(k-m)\widehat{L}_{k+m}^{\alpha,(q,p)},\quad\quad$
$\displaystyle k,m\geq-\alpha.$

###### Proof.

Consider the conjugation of the constraints (3.14) and (3.15) by the
translation operator

(4.36)
$\exp\left({\hbar^{-1}\sum_{j=1+\alpha}\tilde{v}_{2j+1}^{(\alpha)}\frac{\partial}{\partial
t_{2j+1}}}\right).$

The Heisenberg constraints do not change, and the conjugated Virasoro
constraints are given by the operators

(4.37)
$\frac{1}{2}\widehat{L}_{2k}-\frac{1}{2\hbar}\sum_{j=2k+1+2\alpha}^{\infty}[z^{j}]z^{2k+2}{\mathtt{y}}_{\alpha}(h(z))\frac{\partial}{\partial
t_{j}}+\frac{\delta_{k,0}}{16}.$

Then the conjugation by the Virasoro group element $\widehat{V}_{q,p}$ is
described by Lemma 3.2. Note that the value of $a_{2}$ in (3.44) is provided
by (4.4). ∎

For the case of linear Hodge integrals ($p=0$, $\alpha=1$) the
Heisenberg–Virasoro constraints were derived in [Ale15]. A more convenient
basis of constraints was obtained in [GW17]. Constraints in Theorem 1 provide
a generalization of this basis for arbitrary $p$ and $\alpha\in\\{0,1\\}$.

######  Remark 4.2.

The Heisenberg–Virasoro constraints can also be derived from the Kac–Schwarz
description, constructed in [Ale21].

The sting equation for $\alpha=1$ has a particularly simple form. We have

(4.38) $\displaystyle\sigma[-2,m]$
$\displaystyle=[z^{m+1}]\frac{1}{f^{\prime}(z)f(z)}$
$\displaystyle=[z^{m+2}](1+\sqrt{p+q}z)(1+qz/\sqrt{p+q})$
$\displaystyle=\delta_{m,-2}+\frac{2q+p}{\sqrt{p+q}}\delta_{m,-1}+q\delta_{m,0}$

and

(4.39)
$\chi_{1}[-2,m]=[z^{m}]{\mathtt{y}}_{1}=[z^{m}]\frac{\sqrt{p+q}}{p}\left(\log\left(1+\sqrt{p+q}z\right)-\log\left(1+\frac{qz}{\sqrt{p+q}}\right)\right).$

Then

(4.40)
$\widehat{L}_{-1}^{1,(q,p)}=\frac{1}{2}\widehat{L}_{-2}+\frac{2q+p}{2\sqrt{p+q}}\widehat{L}_{-1}+\frac{q}{2}\widehat{L}_{0}-\frac{1}{2\hbar}\sum_{m=1}^{\infty}\chi_{1}[-2,m]\widehat{J}_{m}-\frac{1}{48}{\frac{{p}^{2}+pq+{q}^{2}}{p+q}}$

annihilates the tau-function

(4.41) $\widehat{L}_{-1}^{1,(q,p)}\cdot{\tau}_{q,p}^{(1)}=0.$

From the dimension constraints we have

(4.42) $\begin{split}\left(q\frac{\partial}{\partial
q}+p\frac{\partial}{\partial
p}+\sum_{a=0}^{\infty}\left(a+\frac{1}{2}\right)T_{a}\frac{\partial}{\partial
T_{a}}-\frac{2\alpha+1}{2}\hbar\frac{\partial}{\partial\hbar}\right)\cdot
Z_{q,p}^{(\alpha)}=0.\end{split}$

Let

(4.43) $E=2q\frac{\partial}{\partial q}+2p\frac{\partial}{\partial
p}+\widehat{L}_{0}$

be the Euler operator. Then from relations between $Z^{(\alpha)}$’s and
$\tau_{\alpha}$’s we have

######  Proposition 4.1.

(4.44)
$\begin{split}E\cdot\tau_{q,p}^{(\alpha)}=(2\alpha+1)\hbar\frac{\partial}{\partial\hbar}\tau_{q,p}^{(\alpha)}.\end{split}$

######  Remark 4.3.

From this proposition it immediately follows that one of the parameters $p$,
$q$ and $\hbar$ in the tau-functions $\tau_{q,p}^{(\alpha)}$ is redundant.

For the Euler operator (4.43) we do not know the relation of the form (2.2).
However, tau-functions ${\tau}_{q,p}^{(\alpha)}$ have a simple cut-and-join
description, which leads to the algebraic topological recursion. To derive it
we will use (2.18), where $\widehat{U}$ is a Virasoro group element described
in the next section.

###  4.3. Factorizations of the Virasoro group elements

In this section, we will describe the relation between the tau-functions
$\tau_{\alpha}$ and ${\tau}_{q,p}^{(\alpha)}$ given by the Virasoro group
element. Let us use Lemma 3.1 to substitute the translation operators in
(4.31) with the equivalent elements of the Virasoro group,

(4.45)
$\exp\left({\hbar^{-1}\sum_{j=1+\alpha}^{\infty}\tilde{v}_{2j+1}^{(\alpha)}\frac{\partial}{\partial
t_{2j+1}}}\right)\cdot\tau_{\alpha}({\bf
t})=\widehat{V}^{(\alpha)}_{\Delta}\cdot\tau_{\alpha}({\bf t}).$

Here the Virasoro group operators $\widehat{V}^{(\alpha)}_{\Delta}$ contain
only even generators $\widehat{L}_{2k}$. By Definition 1 these operators can
be described by the formal series

(4.46) $f_{\Delta}^{(\alpha)}(z)=\Xi[\widehat{V}^{(\alpha)}_{\Delta}].$

Here from (4.30) we have

(4.47)
$\begin{split}f_{\Delta}^{(0)}(z)=Y(z),\quad\quad\frac{f_{\Delta}^{(1)}(z)^{3}}{3}=\int_{0}^{z}Y(\eta)\eta\hbox{d}\eta,\end{split}$

where

(4.48)
$Y(z)=\frac{1}{2}\left({\mathtt{y}}_{1}(h(z))-{\mathtt{y}}_{1}(h(-z))\right)$

is the odd part of ${\mathtt{y}}_{1}(h(z))$. To derive the first equation we
use

(4.49)
${\mathtt{y}}_{1}(h(z))=\int_{0}^{h(z)}\frac{f(z)\hbox{d}f(z)}{\eta}=\int_{0}^{z}\frac{\eta\hbox{d}\eta}{h(\eta)}.$

Combining (4.45) with (4.28) one gets

(4.50) $\tau_{q,p}^{(\alpha)}({\bf
t})=\widehat{V}_{q,p}^{(\alpha)}\cdot\tau_{\alpha}({\bf t}),$

where

(4.51)
$\widehat{V}_{q,p}^{(\alpha)}=\widehat{V}_{q,p}\,\widehat{V}^{(\alpha)}_{\Delta}\in{\mathcal{V}}_{+}$

is an element of the Virasoro group.

Let us find a different, more convenient factorization of this Virasoro group
element $\widehat{V}_{q,p}^{(\alpha)}$. For $p\neq 0$ and $q\neq 0$ we
introduce

(4.52)
$\tilde{h}(z)=\frac{p}{q\sqrt{p+q}}\frac{\exp\left(2\frac{q}{\sqrt{p+q}}z\right)-1}{1-\exp\left(-2\frac{p}{\sqrt{p+q}}z\right)}-\frac{1}{\sqrt{p+q}}.$

For $p=0$ it degenerates to

(4.53)
$\tilde{h}(z)=\frac{\exp\left(2\sqrt{q}z\right)-1}{2qz}-\frac{1}{\sqrt{q}}$

and for $q=0$ it degenerates to

(4.54)
$\tilde{h}(z)=\frac{2z}{1-\exp\left(-2\sqrt{p}z\right)}-\frac{1}{\sqrt{p}}.$

Inverse function is given by the series

(4.55)
$\begin{split}\tilde{f}(z)&=z-\frac{1}{3}\,{\frac{p+2\,q}{\sqrt{p+q}}}z^{2}+{\frac{2\,{p}^{2}+5\,pq+5\,{q}^{2}}{9(p+q)}}z^{3}-\frac{2}{135}{\frac{11\,{p}^{3}+39\,{p}^{2}q+51\,p{q}^{2}+34\,{q}^{3}}{(p+q)^{3/2}}}z^{4}\\\
&+{\frac{52\,{p}^{4}+236\,q{p}^{3}+429\,{p}^{2}{q}^{2}+386\,p{q}^{3}+193\,{q}^{4}}{405\,\left(p+q\right)^{2}}}z^{5}+O(z^{6}).\end{split}$

It is easy to see that both $\tilde{h}(z)$ and $\tilde{f}(z)$ are formal
series in $z$ of the form
$z+\sum_{k=2}^{\infty}\frac{c_{k}}{(p+q)^{\frac{k-1}{2}}}z^{k}$, where
$c_{k}\in{\mathbb{Q}}[q,p]$ is a polynomial of degree $k-1$. Set

(4.56) $\widehat{\tilde{V}}=\Xi^{-1}(\tilde{f}(z)).$

Consider an element of the Virasoro group
$\widehat{\tilde{V}}^{-1}\widehat{V}_{q,p}$. Then

(4.57) $\Xi[\widehat{\tilde{V}}^{-1}\widehat{V}_{q,p}]=f(\tilde{h}(z))$

is a composition of (4.52) and (4.6). Let the coefficients ${\bf
b}^{(\alpha)}$ be given by

(4.58)
$\exp\left({\sum_{k\in{\mathbb{Z}}_{>0}}{b}_{k}^{(\alpha)}\widehat{L}_{k}}\right)=\widehat{\tilde{V}}^{-1}\,\widehat{V}_{q,p}\,\widehat{V}^{(\alpha)}_{\Delta},$

then the Virasoro group element $\widehat{V}_{q,p}^{(\alpha)}$ can be
factorized as

(4.59)
$\widehat{V}_{q,p}^{(\alpha)}=\widehat{\tilde{V}}\exp\left({\sum_{k\in{\mathbb{Z}}_{>0}}{b}_{k}^{(\alpha)}\widehat{L}_{k}}\right).$

We have

(4.60)
$f(z,{\bf{b}^{(\alpha)}})=\Xi(\widehat{\tilde{V}}^{-1}\,\widehat{V}_{q,p}\,\widehat{V}^{(\alpha)}_{\Delta})=f_{\Delta}^{(\alpha)}(f(\tilde{h}(z))).$

The reason to introduce the Virasoro group element $\widehat{\tilde{V}}$ is
the following

######  Lemma 4.2.

The function $\Xi[\widehat{\tilde{V}}^{-1}\widehat{V}_{q,p}]$ given by (4.57)
is inverse to $Y(z)$, given by (4.48), or

(4.61) $f(\tilde{h}(Y(z)))=z.$

###### Proof.

Assume that $p\neq 0$ and $q\neq 0$. Recall that ${\mathtt{x}}$ and
${\mathtt{y}}_{1}$ satisfy the equation (4.16),

(4.62)
$p\exp\left(-q{\mathtt{x}}(z)\right)=(p+q)\exp\left(\frac{q}{\sqrt{p+q}}{\mathtt{y}}_{1}(z)\right)-q\exp\left(\sqrt{p+q}{\mathtt{y}}_{1}(z)\right).$

Let us change the variable $z\mapsto h(z)$. We get

(4.63)
$p\exp\left(-qz^{2}/2\right)=(p+q)\exp\left(\frac{q}{\sqrt{p+q}}{\mathtt{y}}_{1}(h(z))\right)-q\exp\left(\sqrt{p+q}{\mathtt{y}}_{1}(h(z))\right),$

or, if we change the sign of $z$,

(4.64)
$p\exp\left(-qz^{2}/2\right)=(p+q)\exp\left(\frac{q}{\sqrt{p+q}}{\mathtt{y}}_{1}(h(-z))\right)-q\exp\left(\sqrt{p+q}{\mathtt{y}}_{1}(h(-z))\right).$

Making the inverse transformation $z\mapsto f(z)$, we have

(4.65)
$p\exp\left(-q{\mathtt{x}}(z)\right)=(p+q)\exp\left(\frac{q}{\sqrt{p+q}}\chi\right)-q\exp\left(\sqrt{p+q}\chi\right),$

where we denote

(4.66) $\chi={\mathtt{y}}_{1}(h(-f(z))).$

If we substitute the expression for ${\mathtt{x}}(z)$ here, we get

(4.67)
$p\frac{(1+\sqrt{p+q}z)^{\frac{q}{p}}}{\left(1+qz/\sqrt{p+q}\right)^{\frac{p+q}{p}}}=(p+q)\exp\left(\frac{q}{\sqrt{p+q}}\chi\right)-q\exp\left(\sqrt{p+q}\chi\right),$

which, after a straightforward computation leads to the identity

(4.68)
$\frac{p}{q\sqrt{p+q}}\frac{\left(\frac{1+\sqrt{p+q}z}{1+qz/\sqrt{p+q}}\right)^{\frac{q}{p}}\exp\left(-\frac{q}{\sqrt{p+q}}\chi\right)-1}{1-\left(\frac{1+qz/\sqrt{p+q}}{1+\sqrt{p+q}z}\right)\exp\left(\frac{p}{\sqrt{p+q}}\chi\right)}-\frac{1}{\sqrt{p+q}}=z.$

The left hand side of this equation is equal to
$\tilde{h}\left(\frac{1}{2}\left({\mathtt{y}}_{1}(z)-\chi\right)\right)$;
therefore, we have

(4.69)
$\tilde{h}\left(\frac{1}{2}\left({\mathtt{y}}_{1}(z)-{\mathtt{y}}_{1}(h(-f(z)))\right)\right)=z.$

If we apply $f(z)$ to the both sides of this equation, we have

(4.70)
$f\left(\tilde{h}\left(\frac{1}{2}\left({\mathtt{y}}_{1}(z)-{\mathtt{y}}_{1}(h(-f(z)))\right)\right)\right)=f(z).$

After the change of variables $z\mapsto h(z)$ we have

(4.71)
$f\left(\tilde{h}\left(\frac{1}{2}\left({\mathtt{y}}_{1}(h(z))-{\mathtt{y}}_{1}(h(-z))\right)\right)\right)=z.$

Then the statement of the lemma for $q\neq 0$ and $p\neq 0$ follows from the
definition of $Y(z)$, (4.48). Special cases with $p=0$ or $q=0$ can be
considered similarly as the certain limits. ∎

From this lemma and (4.47) it follows that the Virasoro group elements (4.58)
are described by the functions

(4.72) $\displaystyle f(z;{\bf{b}}^{(0)})$
$\displaystyle=Y(f(\tilde{h}(z)))=z,$
$\displaystyle\frac{f(z;{\bf{b}}^{(1)})^{3}}{3}$
$\displaystyle=\int_{0}^{z}\eta\,\hbox{d}{\mathtt{x}}(\tilde{h}(\eta)).$

In particular, ${\bf{b}}^{(0)}={\bf 0}$ and
$\widehat{\tilde{V}}^{-1}\,\widehat{V}_{q,p}\,\widehat{V}^{(0)}_{\Delta}=1$,
therefore, $\widehat{V}_{q,p}^{(0)}=\widehat{\tilde{V}}$.

The last integral in (4.72) can be computed explicitly as a series,

(4.73)
$\frac{f(z;{\bf{b}}^{(1)})^{3}}{3}=\sum_{k=1}^{\infty}\frac{4^{k}B_{2k}}{(2k+1)!}\frac{(p+q)^{2k+1}-q^{2k+1}-p^{2k+1}}{pq(p+q)^{k}}z^{2k+1}.$

Here $B_{2k}$ are the Bernoulli numbers defined by

(4.74)
$\frac{ze^{z}}{e^{z}-1}=1+\frac{z}{2}+\sum_{k=1}^{\infty}\frac{B_{2k}z^{2k}}{(2k)!}.$

######  Remark 4.4.

For $p=0$ we have

(4.75)
$\left.\frac{f(z;{\bf{b}}^{(1)})^{3}}{3}\right|_{p=0}=\frac{z^{2}}{\sqrt{q}}\coth(\sqrt{q}z)-\frac{z}{q^{\frac{5}{2}}}$

which, up to inversion and rescaling of the variable, coincides with the
function describing one of the components of the Virasoro group element in
[Ale15, Section 2.2]. This observation indicates that the factorization (4.59)
is a generalization for $p\neq 0$ of the factorization of the group element
considered in [Ale15].

######  Proposition 4.3.

The Virasoro group element $\widehat{V}_{q,p}^{(\alpha)}$ by Definition 1
corresponds to the series

(4.76)
$f_{\alpha}(z)=\Xi(\widehat{V}_{q,p}^{(\alpha)})=f(\tilde{f}(z);{\bf{b}^{(\alpha)}}),$

where $f(z,{\bf{b}}^{(\alpha)})$ and $\tilde{f}$ are given by (4.72) and the
function inverse to (4.52). We have

(4.77)
$f_{0}(z)=\tilde{f}(z),\quad\quad\frac{f_{1}(z)^{3}}{3}=\int_{0}^{z}\tilde{f}(\eta){\hbox{d}}{\mathtt{x}}(\eta).$

###  4.4. Cut-and-join description of the triple Hodge integrals

The cut-and-join description of the tau-function for the triple Hodge
integrals satisfying the Calabi–Yau condition follows from the cut-and-join
description for the KW and BGW tau-functions, see Section 3.2. Let

(4.78)
$\widehat{{W}}_{\alpha}^{q,p}=\frac{1}{2\alpha+1}\sum_{k=0}^{\infty}\widehat{J}_{-2k-1}^{\alpha}\left(\widehat{L}_{2k-2\alpha}^{\alpha}+\frac{\delta_{k,\alpha}}{8}\right).$

Here

(4.79) $\displaystyle\widehat{J}_{k}^{\alpha}$
$\displaystyle=\widehat{V}_{q,p}^{(\alpha)}\widehat{J}_{k}\left(\widehat{V}_{q,p}^{(\alpha)}\right)^{-1},$
$\displaystyle\widehat{L}_{k}^{\alpha}$
$\displaystyle=\widehat{V}_{q,p}^{(\alpha)}\widehat{L}_{k}\left(\widehat{V}_{q,p}^{(\alpha)}\right)^{-1},$

are the conjugated operators of the Heisenberg–Virasoro algebra. By Lemma 3.2
they are given by the linear combinations

(4.80) $\displaystyle\widehat{J}_{k}^{\alpha}$
$\displaystyle=\sum_{m=k}^{\infty}\rho_{\alpha}[k,m]\widehat{J}_{m},$
$\displaystyle k\in{\mathbb{Z}},$ $\displaystyle\widehat{L}_{k}^{\alpha}$
$\displaystyle=\sum_{m=k}^{\infty}\sigma_{\alpha}[k,m]\widehat{L}_{m}+c_{\alpha}\delta_{k,-2},\quad\quad$
$\displaystyle k\geq-2,$

where

(4.81)
$\rho_{\alpha}[k,m]=[z^{m}]f_{\alpha}(z)^{k},\quad\quad\sigma_{\alpha}[k,m]=[z^{m+1}]\frac{f_{\alpha}(z)^{k+1}}{f^{\prime}_{\alpha}(z)}$

with $f_{\alpha}$ given by Proposition 4.3 and

(4.82)
$c_{\alpha}=\left(\frac{\delta_{\alpha,1}}{90}-\frac{1}{18}\right)\frac{p^{2}+pq+q^{2}}{p+q}.$

Because of a certain arbitrariness of the cut-and-join description of the tau-
functions $\tau_{\alpha}$, see Remark 3.3, we can also construct different
versions of the cut-and-join operators for the tau-functions
$\tau_{q,p}^{(\alpha)}$. For example, consider the operators
$\widehat{W}_{\alpha}^{*q,p}$ given by (3.53) with $\widehat{J}_{k}^{a}$
substituted by $\widehat{J}_{k}^{\alpha}$.

######  Theorem 2.

(4.83)
$\begin{split}{\tau}_{q,p}^{(\alpha)}&=\exp\left(\hbar\,\widehat{W}_{\alpha}^{q,p}\right)\cdot
1\\\ &=\exp\left(\hbar\,{\widehat{W}_{\alpha}^{*q,p}}\right)\cdot
1.\end{split}$

###### Proof.

The statement immediately follows from Lemma 3.4, Lemma 3.5, and Proposition
4.3. ∎

Operators $\widehat{W}_{\alpha}^{q,p}$ and ${\widehat{W}_{\alpha}^{*q,p}}$ do
not belong to the $\operatorname{\mathfrak{gl}}(\infty)$ algebra of KP
hierarchy symmetries.

Expressions for the coefficients $\rho_{\alpha}$ and $\sigma_{\alpha}$ can be
simplified. Namely, for $\alpha=0$, since $f_{0}(z)=\tilde{f}(z)$, we have

(4.84) $\begin{split}\rho_{0}[k,m]&=[z^{m}]\tilde{f}(z)^{k}=\frac{1}{2\pi
i}\oint\frac{\hbox{d}\tilde{h}(z)}{\tilde{h}(z)^{m+1}}z^{k},\\\
\sigma_{0}[k,m]&=[z^{m+1}]\frac{\tilde{f}(z)^{k+1}}{\tilde{f}^{\prime}(z)}=\frac{1}{2\pi
i}\oint\frac{\tilde{h}^{\prime}(z)\hbox{d}\tilde{h}(z)}{\tilde{h}(z)^{m+2}}z^{k+1}.\end{split}$

Both $(p+q)^{(m-k)/2}\rho_{0}[k,m]\in{\mathbb{Q}}[q,p]$ and
$(p+q)^{(m-k)/2}\sigma_{0}[k,m]\in{\mathbb{Q}}[q,p]$ are polynomials of the
total degree $m-k$.

Let us expand the cut-and-join operator

(4.85)
$\widehat{{W}}_{0}^{q,p}=\sum_{k=0}^{\infty}\widehat{{W}}_{0,1-k}^{q,p},$

where $\deg\widehat{{W}}_{0,k}^{q,p}=k$. The leading term coincides with the
cut-and-join operator for the BGW model (3.19),
$\widehat{{W}}_{0,1}^{q,p}=\widehat{W}_{0}$. For the sub-leading term

(4.86)
$\widehat{{W}}_{0,0}^{q,p}=\sum_{k=0}^{\infty}\left(\rho_{0}[-2k-1,-2k]\widehat{J}_{-2k}\widehat{L}_{2k}+\sigma_{0}[2k,2k+1]\widehat{J}_{-2k-1}\widehat{L}_{2k+1}\right)$

it is possible to find corresponding coefficients $\rho_{0}$ and $\sigma_{0}$
explicitly. From (4.84) and (4.52) for the non-negative $k$ we immediately get

(4.87)
$\rho_{0}[-2k-1,-2k]=\frac{2k+1}{3}\frac{p+2q}{\sqrt{p+q}},\quad\quad\sigma_{0}[2k,2k+1]=-\frac{2k-1}{3}\frac{p+2q}{\sqrt{p+q}}.$

Therefore, we have

(4.88)
$\begin{split}\widehat{{W}}_{0,0}^{q,p}&=\frac{1}{3}\frac{p+2q}{\sqrt{p+q}}\sum_{k=0}^{\infty}\left((2k+1)\widehat{J}_{-2k}\widehat{L}_{2k}-(2k-1)\widehat{J}_{-2k-1}\widehat{L}_{2k+1}\right).\end{split}$

In the same way for the next term we have

(4.89)
$\begin{split}\widehat{{W}}_{0,-1}^{q,p}&=\sum_{k=0}^{\infty}\left(\rho_{0}[-2k-1,-2k+1]\widehat{J}_{-2k+1}\left(\widehat{L}_{2k}+\frac{\delta_{k,0}}{8}\right)\right.\\\
&\left.+\rho_{0}[-2k-1,-2k]\sigma_{0}[2k,2k+1]\widehat{J}_{-2k}\widehat{L}_{2k+1}+\sigma_{0}[2k,2k+2]\widehat{J}_{-2k-1}\widehat{L}_{2k+2}\right),\end{split}$

where

(4.90)
$\begin{split}\rho_{0}[-2k-1,-2k+1]&=-\frac{2k+1}{9}\left((k-1)\frac{p^{2}}{p+q}+(4k-1)q\right),\\\
\sigma_{0}[2k,2k+2]&=\frac{1}{9}\left(\left(2\,{k}^{2}+k-2\right){\frac{{p}^{2}}{p+q}}+\left(8{k}^{2}-2k-2\right)q\right).\end{split}$

For $\alpha=1$ the coefficients $\rho_{1}$ and $\sigma_{1}$ can be expressed
in terms of the residues of the combinations of functions $\tilde{h}(z)$ and
$f(z;{\bf{b}}^{(1)})$, given by (4.52) and (4.73), namely

(4.91) $\begin{split}\rho_{1}[k,m]&=[z^{m}]f_{1}(z)^{k}=\frac{1}{2\pi
i}\oint\frac{\hbox{d}\tilde{h}(z)}{\tilde{h}(z)^{m+1}}f(z;{\bf{b}}^{(1)})^{k},\\\
\sigma_{1}[k,m]&=[z^{m+1}]\frac{f_{1}(z)^{k+1}}{f^{\prime}_{1}(z)}=\frac{1}{2\pi
i}\oint\frac{\tilde{h}^{\prime}(z)\hbox{d}\tilde{h}(z)}{\tilde{h}(z)^{m+2}}\frac{f(z;{\bf{b}}^{(1)})^{k+1}}{f^{\prime}(z;{\bf{b}}^{(1)})}.\end{split}$

By construction, the cut-and-join operators $\widehat{W}_{\alpha}^{q,p}$ and
$\widehat{W}_{\alpha}^{*q,p}$ are cubic in $\widehat{J}_{k}$. Moreover, they
do not depend on $\hbar$. Hence, the cut-and-join formulas (4.83) describe the
algebraic topological recursion. Namely, if we consider the topological
expansion of the tau-functions

(4.92)
${\tau}_{q,p}^{(\alpha)}=\sum_{k=0}^{\infty}\hbar^{k}{\tau}_{q,p}^{(\alpha,k)},$

then the coefficients of this expansion satisfy

(4.93)
$\begin{split}{\tau}_{q,p}^{(\alpha,k)}&=\frac{1}{k}\widehat{W}_{\alpha}^{q,p}\cdot{\tau}_{q,p}^{(\alpha,k-1)}\\\
&=\frac{1}{k}\widehat{W}_{\alpha}^{*q,p}\cdot{\tau}_{q,p}^{(\alpha,k-1)}\end{split}$

with the initial condition ${\tau}_{q,p}^{(\alpha,0)}=1$. The operators
$\widehat{W}_{\alpha}^{q,p}$ and $\widehat{W}_{\alpha}^{*q,p}$ contain
infinitely many terms, however, only a finite number of them contribute to
each step of recursion. Moreover, for a contributions with the fixed numbers
of marked points we have a recursion similar to (3.26). In Appendix A we
provide the first few terms $F_{k}^{\alpha}({\bf t})$ of the expansion of
$\log{\tau}_{q,p}^{(\alpha)}$,

(4.94) ${\tau}_{q,p}^{(\alpha)}({\bf
t})=\exp\left(\sum_{k=1}^{\infty}\hbar^{k}F_{k}^{\alpha}({\bf t})\right)$

obtained by this recursion with the cut-and-join operators
$\widehat{W}_{\alpha}^{q,p}$ using Maple.

For $q=0$ or $p=0$ we get the cut-and-join description for the tau-function of
the linear Hodge integrals.

######  Remark 4.5.

Using this approach one can also restore other types of cut-and-join-like
representations of the generating functions of the triple Hodge integrals. For
example, it is possible to find a description similar to the one constructed
in [GHW19] for the KW tau-function.

######  Remark 4.6.

We believe that there is a universal distinguished choice of the cut-and-join
operators associated to some natural representation theory interpretation. It
may be related to the quantum Airy structures [BBC+18, BKS20].

##  5\. KdV reduction

For $p=-2q$ the tau-functions $\tau^{(\alpha)}_{q,p}$ do not depend on even
times, hence, they are solutions of the KdV hierarchy, a 2-reduction of the KP
hierarchy [Ale20]. These tau-functions have particularly nice properties, and
can be identified with the KW and BGW tau-functions with shifted arguments. We
will investigate them in this and the next sections.

######  Remark 5.1.

In [DLYZ20] it was shown that in this case the tau-function
$\tau^{(1)}_{q,-2q}$ is related to the discrete KdV hierarchy (aka the
Volterra lattice hierarchy). We expect that $\tau^{(0)}_{q,-2q}$ is also
related to the Volterra hierarchy in the similar way.

###  5.1. Translations and $\kappa$-classes

With the forgetful map
$\pi:\overline{\mathcal{M}}_{g,n+1}\rightarrow\overline{\mathcal{M}}_{g,n}$ we
define the Miller–Morita–Mumford tautological classes [Mum83],
$\kappa_{k}:=\pi_{*}\psi_{n+1}^{k+1}\in
H^{2k}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$. According to Manin and
Zograf [MZ00], insertion of these classes into the intersection numbers (3.2)
can be described by the translation of the variables $t_{k}$ responsible for
the insertion of the $\psi$-classes. Let us consider the generating functions
of the intersection numbers of $\psi$ and $\kappa$ classes with (for
$\alpha=0$) and without (for $\alpha=1$) Norbury’s $\Theta$-classes

(5.1) $F_{g,n}^{\alpha}({\bf T},{\bf s})=\sum_{a_{1},\ldots,a_{n}\geq
0}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}^{1-\alpha}e^{\sum_{j=1}^{\infty}s_{j}\kappa_{j}}\psi_{1}^{a_{1}}\psi_{2}^{a_{2}}\cdots\psi_{n}^{a_{n}}\frac{\prod
T_{a_{i}}}{n!}$

and

(5.2) $\tau_{\alpha}({\bf t},{\bf
s})=\left.\exp\left(\sum_{g=0}^{\infty}\sum_{n=0}^{\infty}\hbar^{2g-2+n}F_{g,n}^{\alpha}({\bf
T},{\bf s})\right)\right|_{T_{k}=(2k+1)!!t_{2k+1}}.$

We introduce the polynomials $q_{j}({\bf s})$, defined by the generating
function

(5.3)
$1-\exp\left(-\sum_{j=1}^{\infty}s_{j}z^{j}\right)=\sum_{j=1}^{\infty}q_{j}({\bf
s})z^{j},$

and we put $q_{k}({\bf s})=0$ for $k<1$. For $k\geq 1$ they are nothing but
the negative of the elementary Schur functions

(5.4) $q_{j}({\bf s})=-p_{j}(-{\bf s}),$

where

(5.5)
$\exp\left(\sum_{j=1}^{\infty}s_{j}z^{j}\right)=\sum_{j=0}^{\infty}p_{j}({\bf
s})z^{j}.$

Then the generating functions $\tau_{\alpha}({\bf t},{\bf s})$ can be obtained
from the generating functions $\tau_{\alpha}({\bf t})$ without
$\kappa$-classes, see Section 3.1, by translation

######  Theorem 3 ([MZ00, Nor20]).

For $\alpha\in\\{0,1\\}$

(5.6) $\tau_{\alpha}({\bf t},{\bf
s})=\tau_{\alpha}\left(\left\\{t_{2k+1}+\frac{1}{\hbar}\frac{q_{k-\alpha}({\bf
s})}{(2k+1)!!}\right\\}\right).$

###  5.2. KdV reduction for triple Hodge integrals

From the change of the variables (4.1) it is clear that for $p=-2q$ the tau-
functions $\tau_{q,p}^{(\alpha)}({\bf t})$ do not depend on even variables
$t_{2k}$, therefore they are tau-function of the KdV hierarchy. Let us put
$q=-u^{2}$ with new parameter $u$, so that $p=2u^{2}$. This parametrization
corresponds to the insertion of the triple Hodge class of the form
$\Lambda(-2u^{2})\Lambda(-2u^{2})\Lambda(u^{2})$.

######  Remark 5.2.

From Proposition 4.1 it follows that the parameter $u$ is redundant. However,
we will leave it as a free parameter to make the limit $u=0$ more transparent.

In this case the series $\tilde{f}(z)$, see (4.55), have a simple explicit
form

(5.7)
$\begin{split}\tilde{f}(z)&=\frac{1}{2u}\log\left(\frac{1+uz}{1-uz}\right)\\\
&=\frac{1}{u}\operatorname{arctanh}(uz)\\\
&=\sum_{k=0}^{\infty}\frac{u^{2k}z^{2k+1}}{2k+1}.\end{split}$

It is an odd function. The Virasoro group operators
$\widehat{V}^{(\alpha)}_{q,p}$ in (4.50), relating $\tau_{\alpha}$ to the tau-
functions for the triple Hodge integrals, contain only even positive
components of the Virasoro operators. Therefore, using Lemma 3.1 one
equivalently describes them by translation operators

(5.8) $\tau_{-u^{2},2u^{2}}^{(\alpha)}({\bf
t})=\exp\left({\hbar^{-1}\sum_{j=1+\alpha}v_{2j+1}^{(\alpha)}(u)\frac{\partial}{\partial
t_{2j+1}}}\right)\cdot\tau_{\alpha}({\bf t}).$

Let us find the coefficients $v_{2j+1}^{(\alpha)}(u)$ using Lemma 3.1 and
Proposition 4.3.

For $\alpha=0$ these coefficients are given by the generating function

(5.9) $\sum_{k=1}^{\infty}v_{2k+1}^{(0)}(u)z^{2k+1}=z-\tilde{f}(z),$

therefore,

(5.10) $v_{2k+1}^{(0)}(u)=-\frac{u^{2k}}{2k+1}.$

By Theorem 3 this translation of the variables corresponds to the insertion of
the $\kappa$-classes with parameters $s^{0}_{k}$ given by $q_{k}({\bf
s}^{0})=-(2k-1)!!u^{2k}$. From (5.3) we have

(5.11)
$\sum_{k=1}^{\infty}s_{k}^{0}z^{k}=-\log\left(1+\sum_{k=1}^{\infty}(2k-1)!!\left(zu^{2}\right)^{k}\right).$

For instance, $s_{1}^{0}=-u^{2}$, $s_{2}^{0}=-\frac{5}{2}u^{4}$,
$s_{3}^{0}=-\frac{37}{3}u^{6}$. In general,
$s_{k}^{0}=-\frac{a_{k}}{k}u^{2k}$, where the sequence $a_{k}$ is described in
[SI20, A004208].

To find the coefficients for $\alpha=1$ we need the specialization of (4.3),

(5.12) $\hbox{d}{\mathtt{x}}=\frac{zdz}{1-u^{2}z^{2}}.$

From Proposition 4.3 we have

(5.13)
$\begin{split}\frac{1}{3}f_{1}(z)^{3}&=\int_{0}^{z}\tilde{f}(\eta)\hbox{d}{\mathtt{x}}(\eta)\\\
&=\frac{1}{u}\int_{0}^{z}\operatorname{arctanh}(u\eta){\frac{\eta\hbox{d}\eta}{1-u^{2}{\eta}^{2}}}\\\
&=\frac{1}{2u^{2}}\int_{0}^{z}\eta\left(\frac{\partial}{\partial\eta}\operatorname{arctanh}(u\eta)^{2}\right)\hbox{d}\eta.\end{split}$

From Lemma 3.1 we know that the coefficients satisfy

(5.14)
$\sum_{k=1}^{\infty}v_{2k+1}^{(1)}(u)z^{2k+1}=\frac{1}{3}\left(z^{3}-f_{1}(z)^{3}\right),$

therefore,

(5.15) $v_{2k+3}^{(1)}(u)=-\frac{u^{2k}}{2k+3}\sum_{j=0}^{k}\frac{1}{2j+1}.$

Here we use the series expansion of $\operatorname{arctanh}(z)^{2}$, see e.g.
[SI20, A004041].

Using Theorem 3 one can find the values of the parameters $s_{k}$, associated
to these translations,

(5.16)
$\begin{split}\exp\left(-\sum_{j=1}^{\infty}s_{j}^{1}z^{j}\right)&=1-\sum_{k=1}^{\infty}(2k+3)!!v_{2j+3}^{(1)}(u)z^{k}\\\
&=\sum_{k=0}^{\infty}(2k+1)!!\left(\sum_{j=0}^{k}\frac{1}{2j+1}\right)\left(zu^{2}\right)^{k}.\end{split}$

In particular, $s_{1}^{1}=-4u^{2}$, $s_{2}^{2}=-15u^{4}$,
$s_{3}^{1}=-\frac{316}{3}u^{6}$.

Therefore, for ${\bf s}^{\alpha}$ given by (5.11) and (5.16) we have the
following relation between the generating functions of different types of
intersection numbers:

######  Theorem 4.

The generating functions of the triple Hodge integrals satisfying Calabi–Yau
condition for $p=-2q=2u^{2}$ coincide with the generation functions of the
intersection numbers containing $\kappa$ classes,

(5.17) $\tau^{(\alpha)}_{-u^{2},2u^{2}}({\bf t})=\tau_{\alpha}({\bf t},{\bf
s^{\alpha}}).$

######  Remark 5.3.

Geometric interpretation of this identity is not clear yet. It would be
interesting to find if it follows from the relations between the cohomology
classes on ${\overline{\mathcal{M}}_{g,n}}$.

This theorem implies that for investigation of the tau-functions
$\tau^{(\alpha)}_{-u^{2},2u^{2}}({\bf t})$ we can use known properties of the
KW and BGW tau-functions. In particular, the tau-functions
$\tau^{(\alpha)}_{-u^{2},2u^{2}}({\bf t})$ obey the Virasoro constraints,
obtained from the constraints (3.15) by a translation of variables
$t_{k}\mapsto t_{k}+\hbar^{-1}v_{2j+1}^{(\alpha)}(u)$. This basis in the space
of constraints can be more convenient when one, constructed in Section 4.2.

Let us consider the identification of the intersection numbers for the given
genus $g$ and number of the marked points $n$. Using the change of variables
(4.1) we can reformulate Theorem 4 in terms of these intersection numbers.
Namely, for $p=-2q=2u^{2}$ the change of the variables (4.1) is given by

(5.18) $T^{-u^{2},2u^{2}}_{k}({\bf
t})=\sum_{m=1}^{\infty}mt_{m}\left(\frac{\partial}{\partial
t_{m-2}}-u^{2}\frac{\partial}{\partial
t_{m}}\right)T^{-u^{2},2u^{2}}_{k-1}({\bf t})$

with $T^{-u^{2},2u^{2}}_{0}=t_{1}$. It is generated by

(5.19)
$\frac{\partial}{\partial{\mathtt{x}}}=\frac{1-u^{2}z^{2}}{z}\frac{\partial}{\partial
z}.$

Namely, the functions (4.10),

(5.20)
$\Phi_{k}(z):=\left(-\frac{\partial}{\partial{\mathtt{x}}}\right)^{k}\cdot\frac{1}{z},$

define a change of variables, if we associate $T_{k}=(2k+1)!!t_{2k+1}$ with
$(2k-1)!!z^{-2k-1}$ and $T^{-u^{2},2u^{2}}_{k}$ with $\Phi_{k}(z)$.

For $g,n\geq 0$ let us introduce the correlation functions

(5.21) $W_{g,n}^{\alpha}(z_{1},\dots,z_{n})=\sum_{a_{1},\dots,a_{n}\geq
0}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}^{1-\alpha}\Lambda(-2u^{2})\Lambda(-2u^{2})\Lambda(u^{2})\prod_{j=1}^{n}\psi_{j}^{a_{j}}\Phi_{a_{j}}(z_{j})$

and

(5.22)
$\tilde{W}_{g,n}^{\alpha}(z_{1},\dots,z_{n})=\sum_{a_{1},\dots,a_{n}\geq
0}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}^{1-\alpha}e^{\sum_{j=1}^{\infty}s_{j}^{\alpha}\kappa_{j}}\prod_{j=1}^{n}\psi_{j}^{a_{j}}\left(-\frac{1}{z_{j}}\frac{\partial}{\partial
z_{j}}\right)^{a_{j}}z_{j}^{-1}.$

Then the following statement is equivalent to Theorem 4:

######  Corollary 5.1.

For $g,n\geq 0$

(5.23)
$W_{g,n}^{\alpha}(z_{1},\dots,z_{n})=\tilde{W}_{g,n}^{\alpha}(z_{1},\dots,z_{n}).$

For $u=0$ this relation is trivial. Note that for the non-stable cases with
$2g-2+n\leq 0$ both ${W}_{g,n}^{\alpha}$ and $\tilde{W}_{g,n}^{\alpha}$ vanish
identically. For $n=0$ we have

(5.24)
$\int_{\overline{\mathcal{M}}_{g}}\Theta_{g,0}^{1-\alpha}\Lambda_{g}(-2)\Lambda_{g}(-2)\Lambda_{g}(1)=\int_{\overline{\mathcal{M}}_{g}}\Theta_{g,0}^{1-\alpha}e^{\sum_{j=1}^{\infty}s_{j}^{\alpha}\kappa_{j}}.$

For $\alpha=1$ according to Faber and Pandharipande [FP00] one has

(5.25)
$\int_{\overline{\mathcal{M}}_{g}}\Lambda_{g}(-q)\Lambda_{g}(-p)\Lambda_{g}(\frac{pq}{p+q})=\frac{1}{2}\left(\frac{q^{2}p^{2}}{q+p}\right)^{g-1}\frac{B_{2g}}{2g}\frac{B_{2g-2}}{2g-2}\frac{1}{(2g-2)!}$

for $g\geq 2$. For $p=-2q=2u^{2}$ it reduces to

(5.26)
$\int_{\overline{\mathcal{M}}_{g}}\Lambda_{g}(-2u^{2})\Lambda_{g}(-2u^{2})\Lambda_{g}(u^{2})=2^{2g-3}u^{6g-6}\frac{B_{2g}}{2g}\frac{B_{2g-2}}{2g-2}\frac{1}{(2g-2)!}.$

It would be interesting to derive this expression using the right hand side of
(5.24).

##  6\. Tau-functions identification and topological recursion

In this section, we will describe the Chekhov–Eynard–Orantin topological
recursion for the triple Hodge integrals satisfying the Calabi–Yau condition
and consider Theorem 4 from the point of view of this recursion.

###  6.1. Chekhov–Eynard–Orantin topological recursion

The Chekhov–Eynard–Orantin topological recursion is a universal construction,
which allows to recover an infinite tower of correlation function starting
from a (non necessarily globally defined) spectral curve $\Sigma$, two
meromorphic differentials on it, $\hbox{d}x$ and $\hbox{d}y$, and a canonical
bilinear differential $B(z_{1},z_{2})$ for $z_{1},z_{2}\in\Sigma$. We call the
set

(6.1) $S=\left(\Sigma,B,x,y\right)$

the topological recursion data. Let us briefly review the formalism of
topological recursion, see [EO07, EO09, CN19] for more details.

In this paper we restrict ourselves to the spectral curves of genus zero,
$\Sigma=\mathbb{C}\mathrm{P}^{1}$. Consider $\mathbb{C}\mathrm{P}^{1}$ with a
fixed global coordinate $z$, then we can specify the spectral curve by the
functions $x(z)$ and $y(z)$. Consider the differential $\hbox{d}x$ with an
assumption that $\hbox{d}x$ is a rational differential with simple critical
points $p_{1},\dots,p_{N}$. On $\mathbb{C}\mathrm{P}^{1}$ the canonical
bilinear differential is the Cauchy differentiation kernel

(6.2)
$B_{C}(z_{1},z_{2})=\frac{\hbox{d}z_{1}\hbox{d}z_{2}}{(z_{1}-z_{2})^{2}}.$

We assume that the function $y$ is meromorphic in the neighbourhoods of the
zeros of $\hbox{d}x$. We allow the spectral curve to be irregular [CN19], that
is, the poles of $y$ may coincide with the zeroes of $\hbox{d}x$.

With this input, on the Cartesian product $\Sigma^{n}$ we construct a system
of symmetric differentials (or correlators) $\omega_{g,n}$, $g\geq 0$, $n\geq
1$, given by

(6.3) $\displaystyle\omega_{0,1}(z_{1})=y(z_{1})\hbox{d}x(z_{1});$
$\displaystyle\omega_{0,2}(z_{1},z_{2})=B_{C}(z_{1},z_{2});$

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

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

where $\sigma_{i}$ is the deck transformation of $x$ near $p_{i}$,
$i=1,\dots,N$. For the stable cases $2g-2+n>0$ these differentials are
meromorphic. We wouldn’t go into the discussion of this version of this
topological recursion, for more detail e.g. [EO07]. As the spectral curve
$\Sigma=\mathbb{C}\mathrm{P}^{1}$ and the canonical bilinear differential
$B=B_{C}$ are the same for all considered cases, we will denote the
topological recursion data by $S=\left(x(z),y(z)\right)$, and also call it the
spectral curve.

The $g$th symplectic invariant for $g\geq 2$ associated to the spectral curve
$S=\left(x,y\right)$ is defined by

(6.5) $F^{(g)}=\frac{1}{2-2g}\sum_{i=1}^{N}\operatorname{Res}_{z\to
p_{i}}\Phi(z)\omega_{g,1}(z),$

where $\Phi(z)=\int^{z}y\hbox{d}x$ is a primitive of $y(z)\hbox{d}x(z)$.
Definitions of $F^{(0)}$ and $F^{(1)}$ can be found in [EO07]. We will
identify $\omega_{g,0}=F^{(g)}$. All correlators $\omega_{g,n}$ with
$2-2g-n<0$ are called stable, and the others are called unstable.

The symplectic invariants are expected to be invariant under the symplectic
transformations which preserve the symplectic form $\hbox{d}x\wedge\hbox{d}y$.
However, the invariance under the exchange transformation $x\leftrightarrow y$
is still challengeable. It is known that in general the correlators
$\omega_{g,n}$ are not invariant under the symplectic transformations.
Nevertheless, there are some symplectic transformations which leave them
invariant. Let us consider an interesting class of such transformations.

Namely, consider the transformation [EO09] of the form

(6.6) $\tilde{x}=g(x),\quad\quad\quad\tilde{y}=\frac{1}{g^{\prime}(x)}y,$

where $g(x)$ is an analytic function near the zeros of $\hbox{d}x$. This
transformation preserves the symplectic form
$\hbox{d}x\wedge\hbox{d}y=\hbox{d}\tilde{x}\wedge\hbox{d}\tilde{y}$,
therefore, it is symplectic. We assume that both spectral curves $S=(x,y)$ and
$\tilde{S}=(\tilde{x},{\tilde{y}})$ are of genus zero.

######  Proposition 6.1.

If the zeroes of $\hbox{d}\tilde{x}$ are simple and the set of these zeroes
coincides with the set of the zeroes of $\hbox{d}x$, then for the two spectral
curves $S=(x,y)$ and $\tilde{S}=(\tilde{x},\tilde{y})$ we have

(6.7)
$\omega_{g,n}(z_{1},\dots,z_{n})=\tilde{\omega}_{g,n}(z_{1},\dots,z_{n})$

for all stable cases.

###### Proof.

For $n=0$ the statement was established in [EO09]. The canonical bilinear
differentials for two spectral curves are the same and coincide with the
Cauchy differentiation kernel (6.2). Since $x(\sigma_{i}(z))=x(z)$ in the
neighbourhood of $p_{i}$, we have
$\tilde{x}(\sigma_{i}(z))=g({x}(\sigma_{i}(z)))=\tilde{x}(z)$. Therefore,
$\tilde{\sigma}_{i}(z)=\sigma_{i}(z)$ and

(6.8)
$\omega_{0,1}(z_{1})-\omega_{0,1}(\sigma_{i}(z_{1}))=\tilde{\omega}_{0,1}(z_{1})-\tilde{\omega}_{0,1}(\tilde{\sigma}_{i}(z_{1})).$

We see that the initial conditions and the kernel of the topological recursion
for $S$ and $\tilde{S}$ are the same, which completes the proof. ∎

With a topological recursion data we associate a generating function made of
only stable correlators with neglected unstable terms,

(6.9)
$\exp\left(\sum_{2g-2+n>0}\frac{\hbar^{2g-2+n}}{n!}\left.\omega_{g,n}(z_{1},\dots,z_{n})\right|_{\Psi_{k}(z_{i})=T_{k}}\right),$

where $\Psi_{k}(z)=\hbox{d}\left(-\frac{\partial}{\partial
x}\right)^{k}z^{-1}$.

From now on we consider only the case $N=1$, that is, the differential
$\hbox{d}x$ has a unique zero. The KW tau-function $\tau_{1}({\bf t})$ is
described by the topological recursion on the Airy curve $x=\frac{1}{2}y^{2}$
[EO09]

(6.10) $S=\left(\frac{1}{2}z^{2},z\right).$

It was conjectured in [Ale18] and proven in [DN18] that the BGW tau-function
$\tau_{0}({\bf t})$ is described by the topological recursion on the Bessel
curve $xy^{2}=\frac{1}{2}$ with the topological recursion data

(6.11) $S=\left(\frac{1}{2}z^{2},\frac{1}{z}\right).$

For these cases the stable correlators of topological recursion are related to
the intersection numbers

(6.12)
$\omega_{g,n}^{\alpha,0}(z_{1},\dots,z_{n})=\hbox{d}_{1}\dots\hbox{d}_{n}\sum_{a_{1},\dots,a_{n}\geq
0}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}^{1-\alpha}\prod_{j=1}^{n}\psi_{j}^{a_{j}}\left(-\frac{1}{z_{j}}\frac{\partial}{\partial
z_{j}}\right)^{a_{j}}z_{j}^{-1}.$

for $\alpha=1$ and $\alpha=0$ respectively.

For these two tau-functions translation of variables ${\bf t}$ leads to the
change of $y$ described by Proposition 4.8 of Chekhov–Norbury:

######  Proposition 6.2 ([CN19]).

The generating function of the topological recursion data

(6.13)
$S=\left(\frac{1}{2}z^{2},z^{2\alpha-1}+\sum_{k=2\alpha}^{\infty}y_{k}z^{k}\right)$

is obtained via translation of the appropriate tau-function

(6.14)
$\tau_{\alpha}\left(\left\\{t_{2k+1}-\frac{1}{\hbar}\frac{y_{2k-1}}{2k+1}\right\\}\right).$

Note that only $t_{2k+1}$ with $k\geq\alpha+1$ are translated.

######  Remark 6.1.

Actually, the proposition of Chekhov–Norbury is more general and describes
$y=\sum_{k=2\alpha-1}^{\infty}y_{k}z^{k}$ with $y_{2\alpha-1}\neq 1$. We do
not need this more general version here.

######  Corollary 6.3.

The functions $\tau_{\alpha}({\bf t},{\bf s^{\alpha}})$ in Theorem 4 are
described by the topological recursion on the genus zero spectral curves
$S_{\alpha}$ with

(6.15)
$S_{0}=\left(\frac{1}{2}z^{2},z^{-1}+\sum_{k=1}^{\infty}u^{2k}z^{2k-1}\right)=\left(\frac{1}{2}z^{2},\frac{1}{z(1-u^{2}z^{2})}\right)$

and

(6.16)
$S_{1}=\left(\frac{1}{2}z^{2},z+\sum_{k=1}^{\infty}u^{2k}z^{2k+1}\sum_{j=0}^{k}\frac{1}{2j+1}\right)=\left(\frac{1}{2}z^{2},\frac{1}{2u(1-u^{2}z^{2})}\log\frac{1+uz}{1-uz}\right).$

The stable correlators of topological recursion for these curves are related
to the functions (5.22) by

(6.17)
$\omega_{g,n}^{\alpha,u}=\hbox{d}_{1}\dots\hbox{d}_{n}\tilde{W}_{g,n}^{\alpha}(z_{1},\dots,z_{n}).$

Spectral curves in Corollary 6.3 are described by the equations

(6.18) $x(1-2u^{2}x)^{2}y^{2}=\frac{1}{2}$

and

(6.19)
$u^{-2}\left(1-e^{2u(1-2u^{2}x)y}\right)^{2}=2x\left(1+e^{2u(1-2u^{2}x)y}\right)^{2}$

respectively. Locally at $x=0$ their behavior is described by the Bessel and
Airy curves correspondingly. For both curves the only zero of
$\hbox{d}x=z\hbox{d}z$ is simple and is located at $z=0$. The curve for
$\alpha=1$ is not algebraic, similar to the case of the Weil–Petersson volumes
[EO09].

###  6.2. Topological recursion for triple Hodge integrals

In this section, we will focus on the case $\alpha=1$, the case $\alpha=0$ is
considered in Section 6.3. Topological recursion for the triple Hodge
integrals of the form (0.4) for $\alpha=1$ is described by Bouchard–Mariño
conjecture [BMn08], which was proven by Chen and Zhou [Che18, Zho09]. In this
section we remind the reader these results basically following [Zho09].

To formulate the results of Chen and Zhou let us describe the relation between
the combination of the Hodge classes used in these papers and
$\Lambda_{g}(p)$,

(6.20) $\Lambda_{g}^{\vee}(a)=a^{g}\Lambda_{g}\left(-\frac{1}{a}\right).$

Let

(6.21) $\phi_{b}(z)=\left(X\frac{\partial}{\partial
X}\right)^{b}\frac{\frac{1}{z}-(a+1)}{(a+1)^{2}},$

where

(6.22)
$X=\frac{a^{a}}{(a+1)^{a+1}}\left(1-(a+1)z\right)\left(1+\frac{(a+1)z}{a}\right)^{a}.$

Then $X$ and

(6.23) $Y=\frac{a}{a+1}+z$

satisfy the spectral curve equation

(6.24) $X=Y^{a}-Y^{a+1}.$

Consider the topological recursion for the genus zero spectral curve

(6.25) $S=\left(\log X,\log Y\right)$

and

(6.26) $\displaystyle\omega_{0,1}(z_{1})=\log Y(z_{1})\,\hbox{d}\log
X(z_{1});$ $\displaystyle\omega_{0,2}(z_{1},z_{2})=B_{C}(z_{1},z_{2}).$

Then the stable correlators are related to the triple Hodge integrals
satisfying the Calabi–Yau condition by [Che18, Zho09]

(6.27) $\omega_{g,n}(z_{1},\dots,z_{n})\\\
=(-1)^{n}(a(a+1))^{g+n-1}\sum_{a_{1},\dots,a_{n}\geq
0}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_{g}(-1)\Lambda_{g}\left(-\frac{1}{a}\right)\Lambda_{g}\left(\frac{1}{a+1}\right)\prod_{j=1}^{n}\psi_{j}^{a_{j}}\hbox{d}\phi_{a_{i}}(z_{i}).$

Under the rescaling $y\mapsto\epsilon y$ the correlators of the topological
recursion change as

(6.28) $\omega_{g,n}\mapsto\epsilon^{2-2g-n}\omega_{g,n}.$

Let us consider this transformation with $\epsilon=-\sqrt{a(a+1)}$ for the
spectral curves (6.25), then for the new spectral curve $S=\left(\log
X,\log\tilde{Y}\right)$ the stable correlators (6.27) acquire the form

(6.29) $\tilde{\omega}_{g,n}(z_{1},\dots,z_{n})=\sum_{a_{1},\dots,a_{n}\geq
0}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_{g}(-1)\Lambda_{g}\left(-\frac{1}{a}\right)\Lambda_{g}\left(\frac{1}{a+1}\right)\prod_{j=1}^{n}\psi_{j}^{a_{j}}\hbox{d}\tilde{\phi}_{a_{i}}(z_{i}).$

Here $\tilde{Y}=(Y)^{-\sqrt{a(a+1)}}$ and

(6.30) $\begin{split}\tilde{\phi}_{b}(z)&=\sqrt{a(a+1)}{\phi}_{b}(z)\\\
&=\left(X\frac{\partial}{\partial
X}\right)^{b}\frac{\sqrt{a}}{(a+1)^{\frac{3}{2}}}\left(\frac{1}{z}-(a+1)\right).\end{split}$

To identify the functions $\tilde{\phi}_{k}(z)$ with the coefficients
$\Phi_{k}(z)$ in (4.10) let us make the change of the variable

(6.31)
$\frac{\sqrt{a}}{(a+1)^{\frac{3}{2}}}\left(\frac{1}{z}-(a+1)\right)\mapsto\frac{1}{z}.$

Then $X(z)$ and $\tilde{Y}(z)$ get the form

(6.32)
$X(z)=\frac{a^{a}}{(a+1)^{a+1}}\frac{\left(1+\sqrt{\frac{a+1}{a}}z\right)^{a}}{\left(1+\sqrt{\frac{a}{a+1}}z\right)^{a+1}},\quad\quad\tilde{Y}=\left(\frac{a}{a+1}\frac{1+\sqrt{\frac{a+1}{a}}z}{1+\sqrt{\frac{a}{a+1}}z}\right)^{-\sqrt{a(a+1)}}.$

To identify the triple Hodge class of (6.29) with the one in (4.22) one should
consider $q,p\in\left\\{1,\frac{1}{a},-\frac{1}{a+1}\right\\}$. Put $q=1$,
$p=-\frac{1}{a+1}$ or $p=\frac{1}{a}$. Then (4.5) and (4.13) are given by

(6.33)
$\begin{split}{\mathtt{x}}(z)&=(a+1)\log\left(1+\sqrt{\frac{a}{a+1}}z\right)-a\log\left(1+\sqrt{\frac{a+1}{a}}z\right),\\\
{\mathtt{y}}_{1}(z)&=\sqrt{a(a+1)}\left(\log\left(1+\sqrt{\frac{a+1}{a}z}\right)-\log\left(1+\sqrt{\frac{a}{a+1}z}\right)\right).\end{split}$

This pair is symplectically equivalent to the pair $\log X(z)$ and
$\log\tilde{Y}(z)$,

(6.34) $-\log
X(z)={\mathtt{x}}(z)-\log\left(\frac{a^{a}}{(a+1)^{a+1}}\right),\quad\quad-\log\tilde{Y}(z)={\mathtt{y}}_{1}(z)-\sqrt{a(a+1)}\log\frac{a+1}{a},$

that is $\hbox{d}\log
X\wedge\hbox{d}\log\tilde{Y}=\hbox{d}{\mathtt{x}}\wedge\hbox{d}{\mathtt{y}}_{1}$.
It is easy to see that for $\left(\log X,\log\tilde{Y}\right)$ and
$({\mathtt{x}},{\mathtt{y}}_{1})$ all stable correlators of topological
recursion coincide. In particular,

(6.35) $\left(X\frac{\partial}{\partial
X}\right)^{b}\frac{1}{z}=\left(-\frac{\partial}{\partial{\mathtt{x}}}\right)^{k}\cdot\frac{1}{z}.$

and the functions $\Phi_{k}(z)$ and $\phi_{k}(z)$, given (4.10) and (6.21)
coincide with each other.

For arbitrary $p$ and $q$ let us consider

(6.36)
$\omega_{g,n}^{(\alpha),p,q}(z_{1},\dots,z_{n})=\hbox{d}_{1}\dots\hbox{d}_{n}\sum_{a_{1},\dots,a_{n}\geq
0}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_{g}(-p)\Lambda_{g}\left(-q\right)\Lambda_{g}\left(\frac{pq}{p+q}\right)\prod_{j=1}^{n}\psi_{j}^{a_{j}}\Phi_{a_{i}}(z_{i}),$

where $\Phi_{a}$’s are given by (4.10). If we transform
$p\mapsto\epsilon^{2}p$, $q\mapsto\epsilon^{2}q$ and $z\mapsto\epsilon^{-1}z$,
then ${\mathtt{x}}\mapsto\epsilon^{-2}{\mathtt{x}}$ and
${\mathtt{y}}_{\alpha}\mapsto\epsilon^{1-2\alpha}{\mathtt{y}}_{\alpha}$,
therefore
$\omega_{g,n}^{(\alpha),p,q}(z_{1},\dots,z_{n})\mapsto\epsilon^{(2\alpha+1)(2g-2+n)}\omega_{g,n}^{(\alpha),p,q}(z_{1},\dots,z_{n})$.
Using this transformation for $\alpha=1$ we can substitute the values
$q_{0}=1$ and $p_{0}=-\frac{1}{a+1}$ or $p_{0}=\frac{1}{a}$ in (6.29) by
arbitrary $q$ and $p$. Therefore, the tau-function $\tau^{(1)}_{q,p}$ and
correlators $\omega_{g,n}^{(1),p,q}$ are governed by topological recursion
with $S=({\mathtt{x}},{\mathtt{y}}_{1})$. This spectral curve is given by the
equation (4.16).

Let us put $p=-2q=2u^{2}$. Then for $\alpha=1$ the triple Hodge classes in
(6.36) coincide with the triple Hodge classes in (5.21), and for the stable
cases with $n>0$ we have

(6.37) $\omega_{g,n}^{(1),2u^{2},-u^{2}}(z_{1},\dots,z_{n})=d_{1}\dots
d_{n}W_{g,n}^{1}(z_{1},\dots,z_{n}).$

We have

(6.38)
$\begin{split}{\mathtt{x}}&=-\frac{1}{2u^{2}}\log\left(1-u^{2}z^{2}\right),\\\
{\mathtt{y}}_{1}&=\frac{1}{2u}\log\frac{1+uz}{1-uz}.\end{split}$

If we compare the curve $S=({\mathtt{x}},{\mathtt{y}}_{1})$ to the curve
$S_{1}$ in (6.16) with $x=\frac{1}{2}z^{2}$ and
$y=\frac{1}{2u(1-u^{2}z^{2})}\log\frac{1+uz}{1-uz}$, we have

(6.39)
${\mathtt{x}}=g({x}),\quad\quad{\mathtt{y}}_{1}=\frac{1}{g^{\prime}({x})}{y},$

where $x=\frac{z^{2}}{2}$ and

(6.40) $g(x)=-\frac{1}{2u^{2}}\log(1-2u^{2}x).$

We see that the curve $({\mathtt{x}},{\mathtt{y}}_{1})$ is related to
$({x},{y})$ by a symplectic transformation of the form (6.6).

Now from Proposition 6.1 it follows that stable correlators for the
topological recursion data $S=({\mathtt{x}},{\mathtt{y}}_{1})$ and (6.16)
coincide. This provides an independent proof of Corollary 5.1 for $\alpha=1$.

###  6.3. Topological recursion for triple $\Theta$-Hodge integrals

In this section, we will consider the Chekhov–Eynard–Orantin topological
recursion for the triple $\Theta$-Hodge integrals satisfying the Calabi–Yau
condition ($\alpha=0$). Namely, for arbitrary $q$ and $p$ we prove an analog
of the Bouchard–Mariño topological recursion for this case.

Let us consider the triple $\Theta$-Hodge integrals satisfying the Calabi–Yau
condition for arbitrary $q$ and $p$. For
$\omega_{g,n}^{(0),p,q}(z_{1},\dots,z_{n})$, given by (6.36), we have

######  Proposition 6.4.

Stable correlators $\omega_{g,n}^{(0),p,q}$ are given by topological recursion
for

(6.41) $\begin{split}S&=({\mathtt{x}},{\mathtt{y}}_{0})\\\
&=\left(\frac{p+q}{pq}\log\left(1+\frac{qz}{\sqrt{p+q}}\right)-\frac{1}{p}\log(1+\sqrt{p+q}z),\frac{1}{z}\right).\end{split}$

###### Proof.

The statement follows from the results of Chekhov and Norbury [CN19] and
almost literally repeats the proof of Theorem 12 in [Nor18]. Let us give only
a brief outline of the proof. The generating function of the triple Hodge
integrals $Z^{(1)}_{p,q}$ given by (4.21) can be described by Givental’s
formalism. For $Z^{(0)}_{p,q}$, according to [Nor18, Proposition 3.9], we have
a similar looking Givental description with $\tau_{1}$ substituted by
$\tau_{0}$ and a different translation operator. The details of both
Givental’s descriptions are given in [Ale20, Section 3].

By [DBOSS14, CN19] Givental’s description is related to the topological
recursion. The topological recursion for $\tau_{p,q}^{(1)}$ is described by
$S=({\mathtt{x}},{\mathtt{y}}_{1})$, see Section 6.2. Then, according to
[Nor18], the $\Theta$-version of the same tau-function is described by
substitution of ${\mathtt{y}}_{1}$ with
$\frac{\partial{\mathtt{y}}_{1}}{\partial{\mathtt{x}}}$. From (4.12) we have

(6.42)
$\frac{\partial{\mathtt{y}}_{1}}{\partial{\mathtt{x}}}={\mathtt{y}}_{0}$

which completes the proof.

∎

For for $p=-2q=2u^{2}$ the stable correlators are related to (5.21) with
$\alpha=0$,

(6.43)
$\begin{split}\omega_{g,n}^{(0),2u^{2},-u^{2}}(z_{1},\dots,z_{n})&=d_{1}\dots
d_{n}W_{g,n}^{0}(z_{1},\dots,z_{n})\\\ \end{split}$

Then these correlators are given by topological recursion:

######  Corollary 6.5.

The stable correlators $\omega_{g,n}^{(0),2u^{2},-u^{2}}$ are given by the
topological recursion with

(6.44) $S=\left(-\frac{1}{2u^{2}}\log(1-u^{2}z^{2}),\frac{1}{z}\right).$

###### Proof.

This corollary has an independent proof, based on the symplectic invariance.
From Corollary 5.1 it follows that the stable correlators
$\omega_{g,n}^{(0),2u^{2},-u^{2}}$ are equal to $d_{1}\dots
d_{n}\tilde{W}^{0}_{g,n}$. From Corollary 6.3 these correlators are given by
the topological recursion with the data

(6.45) $S=\left(\frac{1}{2}z^{2},\frac{1}{z(1-u^{2}z^{2})}\right).$

These $x$ and $y$ are symplectically equivalent to (6.44) for the symplectic
transformation (6.6) with $g(x)$ given by (6.40), therefore, the statement
follows from Proposition 6.1. ∎

The curve (6.44) is described by the equation

(6.46) $u^{-2}y^{2}(1-e^{-2u^{2}x})=1,$

which for $u=0$ reduces to the Bessel curve $2xy^{2}=1$.

##  Appendix A Topological expansion of $\log\tau_{q,p}^{(\alpha)}$

(A.1) $\displaystyle\begin{aligned} F_{1}^{0}&=\frac{1}{8}t_{1},\end{aligned}$
(A.2) $\displaystyle\begin{aligned}
F_{2}^{0}&=\frac{1}{16}\,{t_{{1}}}^{2}-{\frac{1}{128}}\,{\frac{{p}^{2}+pq+{q}^{2}}{p+q}},\end{aligned}$
(A.3) $\displaystyle\begin{aligned}
F_{3}^{0}&={\frac{9}{128}}\,t_{{3}}+\frac{1}{24}\,{t_{{1}}}^{3}+{\frac{3}{64}}\,{\frac{\left(p+2\,q\right)t_{{2}}}{\sqrt{p+q}}}-{\frac{1}{128}}\,{\frac{\left(2\,{p}^{2}-pq-{q}^{2}\right)t_{{1}}}{p+q}}\end{aligned}$
(A.4) $\displaystyle\begin{aligned}
F_{4}^{0}&={\frac{27}{128}}\,t_{{3}}t_{{1}}+\frac{1}{32}\,{t_{{1}}}^{4}+{\frac{9}{64}}\,{\frac{\left(p+2\,q\right)t_{{1}}t_{{2}}}{\sqrt{p+q}}}-{\frac{3}{128}}\,{\frac{\left({p}^{2}-2\,pq-2\,{q}^{2}\right){t_{{1}}}^{2}}{p+q}}\\\
&+{\frac{1}{512}}\,{\frac{{p}^{4}+2\,{p}^{3}q+3\,{p}^{2}{q}^{2}+2\,p{q}^{3}+{q}^{4}}{\left(p+q\right)^{2}}}\end{aligned}$
(A.5) $\displaystyle\begin{aligned}
F_{5}^{0}&={\frac{225}{1024}}\,t_{{5}}+{\frac{27}{64}}\,t_{{3}}{t_{{1}}}^{2}+\frac{1}{40}\,{t_{{1}}}^{5}+{\frac{75}{256}}\,{\frac{\left(p+2\,q\right)t_{{4}}}{\sqrt{p+q}}}+{\frac{9}{32}}\,{\frac{\left(p+2\,q\right)t_{{2}}{t_{{1}}}^{2}}{\sqrt{p+q}}}\\\
&+{\frac{3}{512}}\,{\frac{t_{{3}}\left(4\,{p}^{2}+79\,pq+79\,{q}^{2}\right)}{p+q}}-{\frac{1}{64}}\,{\frac{\left(2\,{p}^{2}-7\,pq-7\,{q}^{2}\right){t_{{1}}}^{3}}{p+q}}\\\
&-{\frac{1}{512}}\,{\frac{\left(22\,{p}^{3}+21\,{p}^{2}q-69\,p{q}^{2}-46\,{q}^{3}\right)t_{{2}}}{\left(p+q\right)^{3/2}}}\\\
&+{\frac{1}{1024}}\,{\frac{\left(8\,{p}^{4}-6\,{p}^{3}q-5\,{p}^{2}{q}^{2}+2\,p{q}^{3}+{q}^{4}\right)t_{{1}}}{\left(p+q\right)^{2}}}\end{aligned}$
(A.6) $\displaystyle\begin{aligned}
F_{6}^{0}&={\frac{1125}{1024}}\,t_{{5}}t_{{1}}+{\frac{567}{1024}}\,{t_{{3}}}^{2}+{\frac{45}{64}}\,t_{{3}}{t_{{1}}}^{3}+\frac{1}{48}\,{t_{{1}}}^{6}\\\
&+{\frac{375}{256}}\,{\frac{\left(p+2\,q\right)t_{{1}}t_{{4}}}{\sqrt{p+q}}}+{\frac{189}{256}}\,{\frac{\left(p+2\,q\right)t_{{2}}t_{{3}}}{\sqrt{p+q}}}+{\frac{15}{32}}\,{\frac{\left(p+2\,q\right)t_{{2}}{t_{{1}}}^{3}}{\sqrt{p+q}}}\\\
&+{\frac{3}{256}}\,{\frac{\left(10\,{p}^{2}+229\,pq+229\,{q}^{2}\right)t_{{3}}t_{{1}}}{p+q}}+{\frac{63}{256}}\,{\frac{\left({p}^{2}+4\,pq+4\,{q}^{2}\right){t_{{2}}}^{2}}{p+q}}\\\
&-{\frac{5}{128}}\,{\frac{\left({p}^{2}-5\,pq-5\,{q}^{2}\right){t_{{1}}}^{4}}{p+q}}\\\
&-{\frac{1}{512}}\,{\frac{\left(110\,{p}^{3}-21\,{p}^{2}q-723\,p{q}^{2}-482\,{q}^{3}\right)t_{{2}}t_{{1}}}{\left(p+q\right)^{3/2}}}\\\
&+{\frac{1}{512}}\,{\frac{\left(10\,{p}^{4}-35\,{p}^{3}q-11\,{p}^{2}{q}^{2}+48\,p{q}^{3}+24\,{q}^{4}\right){t_{{1}}}^{2}}{\left(p+q\right)^{2}}}\\\
&-{\frac{1}{12288}}\,{\frac{17\,{p}^{6}+51\,{p}^{5}q+105\,{p}^{4}{q}^{2}+125\,{p}^{3}{q}^{3}+105\,{p}^{2}{q}^{4}+51\,p{q}^{5}+17\,{q}^{6}}{\left(p+q\right)^{3}}}\end{aligned}$
(A.7) $\displaystyle\begin{aligned}
F_{7}^{0}&={\frac{55125\,t_{{7}}}{32768}}+{\frac{3375}{1024}}\,t_{{5}}{t_{{1}}}^{2}+{\frac{1701}{512}}\,{t_{{3}}}^{2}t_{{1}}+{\frac{135}{128}}\,t_{{3}}{t_{{1}}}^{4}+{\frac{1}{56}}\,{t_{{1}}}^{7}+{\frac{55125}{16384}}\,{\frac{\left(p+2\,q\right)t_{{6}}}{\sqrt{p+q}}}\\\
&+{\frac{1125}{256}}\,{\frac{\left(p+2\,q\right)t_{{4}}{t_{{1}}}^{2}}{\sqrt{p+q}}}+{\frac{567}{128}}\,{\frac{\left(p+2\,q\right)t_{{3}}t_{{2}}t_{{1}}}{\sqrt{p+q}}}+{\frac{45}{64}}\,{\frac{\left(p+2\,q\right)t_{{2}}{t_{{1}}}^{4}}{\sqrt{p+q}}}\\\
&+{\frac{1875}{32768}}\,{\frac{\left(26\,{p}^{2}+173\,pq+173\,{q}^{2}\right)t_{{5}}}{p+q}}+{\frac{9}{512}}\,{\frac{\left(20\,{p}^{2}+521\,pq+521\,{q}^{2}\right)t_{{3}}{t_{{1}}}^{2}}{p+q}}\\\
&+{\frac{189}{128}}\,{\frac{\left({p}^{2}+4\,pq+4\,{q}^{2}\right){t_{{2}}}^{2}t_{{1}}}{p+q}}-{\frac{3}{128}}\,{\frac{\left(2\,{p}^{2}-13\,pq-13\,{q}^{2}\right){t_{{1}}}^{5}}{p+q}}\\\
&-{\frac{25}{4096}}\,{\frac{\left(67\,{p}^{3}-387\,{p}^{2}q-1563\,p{q}^{2}-1042\,{q}^{3}\right)t_{{4}}}{\left(p+q\right)^{3/2}}}\\\
&-{\frac{3}{512}}\,{\frac{\left(110\,{p}^{3}-147\,{p}^{2}q-1101\,p{q}^{2}-734\,{q}^{3}\right)t_{{2}}{t_{{1}}}^{2}}{\left(p+q\right)^{3/2}}}\\\
&-{\frac{3}{32768}}\,{\frac{\left(1548\,{p}^{4}+9796\,{p}^{3}q-7681\,{p}^{2}{q}^{2}-34954\,p{q}^{3}-17477\,{q}^{4}\right)t_{{3}}}{\left(p+q\right)^{2}}}\\\
&+{\frac{1}{1024}}\,{\frac{\left(40\,{p}^{4}-250\,{p}^{3}q+63\,{p}^{2}{q}^{2}+626\,p{q}^{3}+313\,{q}^{4}\right){t_{{1}}}^{3}}{\left(p+q\right)^{2}}}\\\
&+{\frac{1}{16384}}\,{\frac{\left(1052\,{p}^{5}+308\,q{p}^{4}-4561\,{q}^{2}{p}^{3}-284\,{q}^{3}{p}^{2}+4135\,{q}^{4}p+1654\,{q}^{5}\right)t_{{2}}}{\left(p+q\right)^{5/2}}}\\\
&-{\frac{1}{32768}}\,{\frac{\left(272\,{p}^{6}-236\,{p}^{5}q-176\,{p}^{4}{q}^{2}+115\,{p}^{3}{q}^{3}+45\,{p}^{2}{q}^{4}-15\,p{q}^{5}-5\,{q}^{6}\right)t_{{1}}}{\left(p+q\right)^{3}}}\end{aligned}$

(A.8) $\displaystyle\begin{aligned}
F_{1}^{1}&=\frac{1}{8}t_{{3}}+\frac{1}{6}{t_{{1}}}^{3}+\frac{1}{12}{\frac{\left(p+2\,q\right)t_{{2}}}{\sqrt{p+q}}}-\frac{1}{24}{\frac{{p}^{2}t_{{1}}}{p+q}},\end{aligned}$
(A.9) $\displaystyle\begin{aligned}
F_{2}^{1}&=\frac{1}{2}\,t_{{3}}{t_{{1}}}^{3}+\frac{5}{8}\,t_{{5}}t_{{1}}+\frac{3}{16}\,{t_{{3}}}^{2}\\\
&+\frac{5}{6}\,{\frac{\left(p+2\,q\right)t_{{4}}t_{{1}}}{\sqrt{p+q}}}+\frac{1}{4}\,{\frac{\left(p+2\,q\right)t_{{3}}t_{{2}}}{\sqrt{p+q}}}+\frac{1}{3}\,{\frac{\left(p+2\,q\right)t_{{2}}{t_{{1}}}^{3}}{\sqrt{p+q}}}\\\
&+\frac{1}{8}\,{\frac{\left({p}^{2}+12\,pq+12\,{q}^{2}\right)t_{{3}}t_{{1}}}{p+q}}+\frac{1}{12}\,{\frac{\left({p}^{2}+4\,pq+4\,{q}^{2}\right){t_{{2}}}^{2}}{p+q}}+\frac{1}{6}\,q{t_{{1}}}^{4}\\\
&-\frac{1}{12}\,{\frac{\left({p}^{3}-{p}^{2}q-9\,p{q}^{2}-6\,{q}^{3}\right)t_{{1}}t_{{2}}}{\left(p+q\right)^{3/2}}}\\\
&-\frac{1}{48}\,{\frac{q\left(2\,{p}^{2}-pq-{q}^{2}\right){t_{{1}}}^{2}}{p+q}}\\\
&-{\frac{1}{5760}}\,{\frac{{p}^{2}{q}^{2}}{p+q}},\end{aligned}$

(A.10) $\displaystyle\begin{aligned}
F_{3}^{1}&=\frac{5}{8}\,t_{{5}}{t_{{1}}}^{4}+\frac{3}{2}\,{t_{{3}}}^{2}{t_{{1}}}^{3}+{\frac{35}{16}}\,t_{{7}}{t_{{1}}}^{2}+{\frac{15}{4}}\,t_{{5}}t_{{3}}t_{{1}}+\frac{3}{8}\,{t_{{3}}}^{3}+{\frac{105}{128}}\,t_{{9}}\\\
&+{\frac{35}{8}}\,{\frac{\left(p+2\,q\right)t_{{6}}{t_{{1}}}^{2}}{\sqrt{p+q}}}+\frac{3}{4}\,{\frac{\left(p+2\,q\right){t_{{3}}}^{2}t_{{2}}}{\sqrt{p+q}}}+{\frac{35}{16}}\,{\frac{\left(p+2\,q\right)t_{{8}}}{\sqrt{p+q}}}+\frac{5}{6}\,{\frac{\left(p+2\,q\right)t_{{4}}{t_{{1}}}^{4}}{\sqrt{p+q}}}\\\
&+2\,{\frac{\left(p+2\,q\right)t_{{3}}t_{{2}}{t_{{1}}}^{3}}{\sqrt{p+q}}}+\frac{5}{2}\,{\frac{\left(p+2\,q\right)t_{{5}}t_{{2}}t_{{1}}}{\sqrt{p+q}}}+5\,{\frac{\left(p+2\,q\right)t_{{4}}t_{{3}}t_{{1}}}{\sqrt{p+q}}}\\\
&+\frac{10}{3}\,{\frac{\left({p}^{2}+4\,pq+4\,{q}^{2}\right)t_{{4}}t_{{2}}t_{{1}}}{p+q}}+{\frac{9}{8}}\,{\frac{\left({p}^{2}+8\,pq+8\,{q}^{2}\right){t_{{3}}}^{2}t_{{1}}}{p+q}}\\\
&+{\frac{5}{48}}\,{\frac{\left(23\,{p}^{2}+140\,pq+140\,{q}^{2}\right)t_{{5}}{t_{{1}}}^{2}}{p+q}}+\frac{1}{4}\,{\frac{\left({p}^{2}+10\,pq+10\,{q}^{2}\right)t_{{3}}{t_{{1}}}^{4}}{p+q}}\\\
&+{\frac{7}{288}}\,{\frac{\left(76\,{p}^{2}+391\,pq+391\,{q}^{2}\right)t_{{7}}}{p+q}}+\frac{2}{3}\,{\frac{\left({p}^{2}+4\,pq+4\,{q}^{2}\right){t_{{2}}}^{2}{t_{{1}}}^{3}}{p+q}}\\\
&+\frac{1}{2}\,{\frac{\left({p}^{2}+4\,pq+4\,{q}^{2}\right)t_{{3}}{t_{{2}}}^{2}}{p+q}}+\frac{1}{2}\,{\frac{\left({p}^{3}+17\,{p}^{2}q+45\,p{q}^{2}+30\,{q}^{3}\right)t_{{3}}t_{{2}}t_{{1}}}{\left(p+q\right)^{3/2}}}\\\
&+\frac{1}{12}\,{\frac{\left({p}^{3}+79\,{p}^{2}q+231\,p{q}^{2}+154\,{q}^{3}\right)t_{{4}}{t_{{1}}}^{2}}{\left(p+q\right)^{3/2}}}\\\
&+{\frac{11}{12}}\,{\frac{\left(p+2\,q\right)qt_{{2}}{t_{{1}}}^{4}}{\sqrt{p+q}}}+\frac{1}{9}\,{\frac{\left({p}^{3}+6\,{p}^{2}q+12\,p{q}^{2}+8\,{q}^{3}\right){t_{{2}}}^{3}}{\left(p+q\right)^{3/2}}}\\\
&+{\frac{7}{192}}\,{\frac{\left(10\,{p}^{3}+167\,{p}^{2}q+441\,p{q}^{2}+294\,{q}^{3}\right)t_{{6}}}{\left(p+q\right)^{3/2}}}\\\
&-\frac{1}{16}\,{\frac{\left(2\,{p}^{4}-2\,{p}^{3}q-101\,{p}^{2}{q}^{2}-198\,p{q}^{3}-99\,{q}^{4}\right)t_{{3}}{t_{{1}}}^{2}}{\left(p+q\right)^{2}}}+{\frac{5}{24}}\,{q}^{2}{t_{{1}}}^{5}\\\
&-{\frac{1}{384}}\,{\frac{\left(57\,{p}^{4}-236\,{p}^{3}q-2769\,{p}^{2}{q}^{2}-5066\,p{q}^{3}-2533\,{q}^{4}\right)t_{{5}}}{\left(p+q\right)^{2}}}\\\
&-\frac{1}{6}\,{\frac{\left({p}^{4}-2\,{p}^{3}q-26\,{p}^{2}{q}^{2}-48\,p{q}^{3}-24\,{q}^{4}\right){t_{{2}}}^{2}t_{{1}}}{\left(p+q\right)^{2}}}\\\
&-{\frac{1}{1440}}\,{\frac{\left(37\,{p}^{5}+520\,{p}^{4}q-178\,{p}^{3}{q}^{2}-5172\,{p}^{2}{q}^{3}-7580\,p{q}^{4}-3032\,{q}^{5}\right)t_{{4}}}{\left(p+q\right)^{5/2}}}\\\
&-\frac{1}{24}\,{\frac{\left(7\,{p}^{3}-4\,{p}^{2}q-54\,p{q}^{2}-36\,{q}^{3}\right)qt_{{2}}{t_{{1}}}^{2}}{\left(p+q\right)^{3/2}}}\\\
&-{\frac{1}{144}}\,{\frac{\left(9\,{p}^{2}-8\,pq-8\,{q}^{2}\right){q}^{2}{t_{{1}}}^{3}}{p+q}}\\\
&+{\frac{1}{960}}\,{\frac{\left(7\,{p}^{6}-16\,q{p}^{5}-239\,{q}^{2}{p}^{4}-170\,{p}^{3}{q}^{3}+605\,{q}^{4}{p}^{2}+828\,p{q}^{5}+276\,{q}^{6}\right)t_{{3}}}{\left(p+q\right)^{3}}}\\\
&+{\frac{1}{2880}}\,{\frac{\left(21\,{p}^{5}-2\,{p}^{4}q-120\,{p}^{3}{q}^{2}-40\,{p}^{2}{q}^{3}+60\,p{q}^{4}+24\,{q}^{5}\right)qt_{{2}}}{\left(p+q\right)^{5/2}}}\\\
&+{\frac{7}{5760}}\,{\frac{{p}^{4}{q}^{2}t_{{1}}}{\left(p+q\right)^{2}}}.\end{aligned}$

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