# KP hierarchy for Hurwitz-type cohomological field theories

Reinier Kramer  University of Alberta, Edmonton, AB, T6G 2G1, Canada
<EMAIL_ADDRESS>
###### Abstract.

We generalise a result of Kazarian regarding Kadomtsev-Petviashvili
integrability for single Hodge integrals to general cohomological field
theories related to Hurwitz-type counting problems or hypergeometric tau-
functions. The proof uses recent results on the relations between
hypergeometric tau-functions and topological recursion, as well as the DOSS
correspondence between topological recursion and cohomological field theories.
As a particular case, we recover the result of Alexandrov of KP integrability
for triple Hodge integrals with a Calabi-Yau condition.

###### Contents

  1. 1 Introduction
  2. 2 Prerequisites on the KP hierarchy and topological recursion
    1. 2.1 The KP hierarchy
    2. 2.2 Space of tau-functions and Lie action
    3. 2.3 Single Hodge integrals
    4. 2.4 Hypergeometric KP tau-functions and topological recursion
    5. 2.5 Topological recursion and cohomological field theories
    6. 2.6 The Mariño-Vafa formula and KP for topological vertex amplitudes
  3. 3 KP hierarchy for intersection numbers
    1. 3.1 The change of variables
    2. 3.2 KP for intersection numbers
    3. 3.3 Finiteness of the transformation
    4. 3.4 The case of triple Hodge integrals

##  1\. Introduction

The moduli spaces of curves are a central object in modern algebraic geometry,
and have been studied intensively. In particular, their intersection theory is
a subject of ongoing research. The space $\overline{\mathcal{M}}_{g,n}$ has
$n$ line bundles $\mathbb{L}_{i}$ whose fibres at a point are the cotangent
lines at the $i$th point of the represented curve, and a rank-$g$ Hodge bundle
$\mathbb{E}$ whose fibres are the space of one-forms at the curve. Their Chern
classes are defined to be $\psi_{i}\coloneqq c_{1}(\mathbb{L}_{i})$ and
$\lambda_{j}\coloneqq c_{j}(\mathbb{E})$, respectively. Moreover, the spaces
$\overline{\mathcal{M}}_{g,n}$ for different $g$ and $n$ have many structure
maps between them, and many classes behave well under these maps. A collection
of classes on all $\overline{\mathcal{M}}_{g,n}$ satisfying certain coherence
axioms with respect to the structure maps are called cohomological field
theories (CohFTs), and these play an important role in enumerative geometry of
curves. One well-known example is the total Hodge class
$\Lambda(t)=\sum\lambda_{i}t^{i}$.

By the Witten-Kontsevich theorem [Wit91, Kon92], moduli spaces of curves have
many relations to areas of mathematical physics and integrable hierarchies. In
particular, this theorem proves that a generating function of the intersection
numbers of $\psi$-classes is a tau-function of the Korteweg-de Vries
hierarchy.

Furthermore, the Ekedahl-Lando-Shapiro-Vainshtein formula [ELSV01] relates
single Hodge integrals, i.e. intersection numbers of $\Lambda(-1)$ with
$\psi$-classes, to simple single Hurwitz numbers, counting ramified coverings
of $\mathbb{P}^{1}_{\mathbb{C}}$ with only simple ramifications (with profile
$(2,1,1,1,\dotsc)$) except for one point. Hurwitz numbers themselves also give
a large class of tau-functions of Toda or Kadomtsev-Petviashvili hierarchies
(of which the KdV hierarchy is a reduction), as noted by Okounkov [Oko00].

Kazarian [Kaz09] interpreted the ELSV formula as a change of variables from
the generating function of single Hodge integrals to a tau-function of the
Kadomtsev-Petviashvili hierarchy, using the result of Okounkov on simple
single Hurwitz numbers.

All of these results have strong relations to Chekhov-Eynard-Orantin
topological recursion [CEO06, EO07], a succesful way of encoding many counting
problem with a natural genus expansion into a spectral curve with a
recursively defined collection of multidifferentials, which should be
generating functions of the counts. The Witten-Kontsevich $\psi$-intersection
numbers can be encoded this way, and this is somehow the base case of the
theory. Many types of Hurwitz numbers obey topological recursion as well,
starting with [BM08, BEMS11] for the first case of simple Hurwitz numbers, and
culminating in the works of Bychkov-Dunin-Barkowski-Kazarian-Shadrin [BDKS20,
BDKS20a], which prove topological recursion for two large families of
hypergeometric KP tau-functions, encompassing nearly all previously-studied
cases of Hurwitz numbers.

In another direction, there is a general correspondence between topological
recursion and intersection numbers of CohFTs [Eyn14, DOSS14], which vastly
generalises the ELSV formula when combined with the results on topological
recursion for Hurwitz numbers.

A particularly interesting case is the conjecture of Mariño and Vafa [MV02] on
a further generalisation of the ELSV formula, proved independently in [LLZ03,
OP04]. This Mariño-Vafa formula relates triple Hodge integrals with a Calabi-
Yau condition to topological vertex amplitudes, i.e. Gromov-Witten invariants
of $\mathbb{C}^{3}$. Topological recursion was conjectured for toric Calabi-
Yau threefolds by Bouchard-Klemm-Mariño-Pasquetti [BKMP09]. It was first
proved in [Che18, Zho09] for $\mathbb{C}^{3}$, as well as in [Eyn11] as an
example of the general correspondence of theorem 2.12, while the general BKMP
conjecture was proved in [EO15].

Both the space of CohFTs and the space of KP tau-functions have an action of
an infinite-dimensional group, respectively the Givental group and the
Heisenberg-Virasoro group. As certain elements of these spaces have been
identified by Witten-Kontsevich and Kazarian, and different integrable
hierarchies have been constructed for general CohFTs by Dubrovin-Zhang [DZ01]
and Buryak [Bur15], one may ask how general the relation is with KP
specifically, and the group actions are a natural tool to study this question.

Alexandrov [Ale21] showed that in the case of a rank-one CohFT, the orbits of
the Witten-Kontsevich CohFT/tau-function under these two different group
actions have an intersection which is only two-dimensional, and contains
exactly the triple Hodge integrals that appear in the Mariño-Vafa formula. As
a consequence, Alexandrov generalises Kazarian’s result to show that the
generating function of Calabi-Yau triple Hodge integrals satisfies the KP
hierarchy after a linear change of variables.

### Results of this paper

We give a new viewpoint on the relation found by Alexandrov, by generalising
Kazarian’s proof in [Kaz09] to all hypergeometric KP tau-functions satisfying
topological recursion, using the above results. This yields a change of
variables coming from the function $X$ for any hypergeometric tau-function
preserving the KP hierarchy after removing the unstable terms of the tau-
function. When topological recursion holds, this resulting tau-function can be
interpreted as the generating function of the cohomological field theory.

In general, the change of variables contains infinite linear combinations.
However, we identify when the linear combinations are actually finite, and
find a finite-dimensional family for each CohFT rank. In the rank one case,
this recovers exactly the triple Hodge integrals, in a particular
parametrisation. For higher rank, this family seems to fit within Alexandrov’s
deformed generalised Kontsevich model [Ale21a].

Interestingly, the function $X$ may also be a Möbius transformation. In this
case, there is no unstable correction term, and this can be interpreted as
certain independence of the parametrisation of the spectral curve. This also
resolves the meaning behind Kazarian’s change of coordinates, as voiced in
[Kaz09, Remark 2.6]: “The definition for the change (6) looks unmotivated. […]
The only motivation that we can provide here is that ‘it works’.” There is
quite a freedom of choice, but the particular choice Kazarian made reduces to
the finite-dimensional family indicated above.

### Open questions

Single and triple Hodge integrals have been studied intensively in relation to
Dubrovin-Zhang hierarchies, yielding relations to the intermediate long wave
(ILW) hierarchy and the fractional Volterra hierarchy, cf. [Bur15a, Bur16,
LYZZ21]. The relation between those results and the current work are still
unclear, and will be discussed elsewhere. Between the first and second
versions of this preprint, Liu-Wang-Zhang [LWZ21] related the ILW hierarchy to
a limit of fractional Volterra hierarchy viewed as a reduction of the 2D Toda
hierarchy, possibly giving a new avenue to relating to the current paper.

The family where the linear change of variables is finite seems like an
interesting and natural deformation of Witten’s $r$-spin class, keeping a
single ramification point, but splitting the pole of $dx$. However, this
family seems mostly unknown, with the exception of Alexandrov’s work mentioned
above. It may be interesting to investigate it more closely, in order to
better understand the deformation of higher-order zeroes of $dx$.

Currently, there is a gap in the literature on limits of spectral curves,
which in particular limits the validity of the proof theorem 2.6, and hence
the applicability of the main theorem of this paper, to $dx$ with simple
zeroes. Future work with Borot, Bouchard, Chidambaram, and Shadrin will fix
this, and will investigate more generally the applicability of limit arguments
for topological recursion.

For the BKP hierarchy, similar results should hold. In particular, Alexandrov
and Shadrin [AS21] proved an adapted topological recursion for a large class
of hypergeometric BKP tau-functions, analogous to theorem 2.6. The analogous
ELSV-Eynard-DOSS correspondence between this kind of topological recursion and
cohomological field theories has not appeared in the literature, but the
special case of completed cycles spin Hurwitz numbers is treated in work of
the author with Giacchetto and Lewański [GKL21].

### Outline of the paper

Section 2 contains prerequisites. In sections 2.1 and 2.2, we give a short
introduction to the Kadomtsev-Petviashvili hierarchy and its space of
solutions. In section 2.3, we recall the main ideas from [Kaz09], which we
will generalise. In sections 2.4 and 2.5, we recall recent results on
hypergeometric tau-functions and their relations to topological recursion and
cohomological field theories, and state our main theorem, which is theorem
2.16. We also introduce, in section 2.6, the generating function of triple
Hodge numbers, which is the main motivating example of this paper.

In section 3, we prove the main result. Firstly, in section 3.1, we find a
change of variables, for any hypergeometric tau-function, that preserves the
property of being a tau-function after removal of unstable terms, corollary
3.5. In section 3.2, we restrict to the case where topological recursion
holds, and use this machinery to obtain tau-functions of intersection numbers,
proving our main result. We also determine, in section 3.3, the exact
conditions for the change of variables to be finite, in a specific sense.
Finally, in section 3.4, we return to the triple Hodge integrals, and prove an
explicit version of the main theorem for this case.

### Notation

We work over the field of complex numbers $\mathbb{C}$. We will use the
function $\varsigma(z)\coloneqq e^{\frac{z}{2}}-e^{-\frac{z}{2}}$ and
$\mathcal{S}(z)=\frac{\varsigma(z)}{z}$, $\mu$ and $\nu$ will denote
partitions, and
$z_{\mu}\coloneqq\prod_{i=1}^{\mu_{1}}i^{m_{i}(\mu)}m_{i}(\mu)!$, where
$m_{i}(\mu)$ is the number of parts of $\mu$ of size $i$. We will also
consistently write $n\coloneqq\ell(\mu)$ and $\llbracket
n\rrbracket\coloneqq\\{1,\dotsc,n\\}$.

### On the origin of this paper

An earlier version of this text, only concerning triple Hodge integrals, was
written in 2018, shortly after A. Alexandrov informed me of his result. That
version appeared in my PhD dissertation [Kra19, Chapter 10]. This paper is an
updated and extended version of that chapter.

### Acknowledgments

I would like to thank A. Alexandrov for informing me of his theorem on triple
Hodge integrals, S. Shadrin for introducing me to the subject and suggesting
generalising Kazarian’s method to this case, and both of them, G. Carlet, N.
Chidambaram, and A. Giacchetto for many interesting discussions. I would also
like to thank A. Alexandov and S. Shadrin for suggesting improvements to a
previous version of this paper, and P. Norbury for finding errors by
calculating example 2.18, spurring me to double-check and do this example
myself.

Research conducted for this paper is supported by the Netherlands Organization
for Scientific Research, the Max-Planck-Gesellschaft, the Natural Sciences and
Engineering Research Council of Canada, and the Pacific Institute for the
Mathematical Sciences (PIMS). The research and findings may not reflect those
of these institutions.

The University of Alberta respectfully acknowledges that we are situated on
Treaty 6 territory, traditional lands of First Nations and Métis people.

##  2\. Prerequisites on the KP hierarchy and topological recursion

In this section, we review some standard notions on the KP hierarchy and its
relations to the infinite Grassmannian. We give the main outline of Kazarian’s
proof of KP for single Hodge integrals, which we will use as a blueprint for
our results. We also recall the class of hypergeometric tau-functions, which
fulfills a central role in this paper, as well as its relation to topological
recursion and cohomological field theories. Finally, we recall the Mariño-Vafa
formula for triple Hodge integrals and show it fits in the setup.

###  2.1. The KP hierarchy

The Kadomtsev-Petviashvili hierarchy is an infinite set of evolutionary
differential equations in infinitely many variables. It is a very well-studied
system, and some introductions into the subject can be found in [Dic03, Kha99,
MJD00].

Let $\underline{t}=\\{t_{i}\\}_{i\geq 1}$ be a set of independent variables
and $\partial\coloneqq\frac{\partial}{\partial t_{1}}$. Define the pseudo-
differential operator (i.e. a Laurent series in $\partial^{-1}$ with
coefficients functions in $\underline{t}$ with composition defined formally)

(1) $L=\partial+u_{1}\partial^{-1}+u_{2}\partial^{-2}+\dotsc\,.$

where the $u_{j}$ are dependent variables in the $t_{i}$. For a pseudo-
differential operator $O$, define $O_{+}$ to be its purely differential part,
the part without powers of $\partial^{-1}$. The _Lax formulation of the KP
hierarchy_ is given by the system of equations

(2) $\frac{\partial L}{\partial t_{i}}=\big{[}(L^{i})_{+},L\big{]}\,.$

This is a system of partial differential equations for the $u_{j}$, and they
can be interpreted as the compatibility equations for the system

(3) $\displaystyle L\Psi$ $\displaystyle=z\Psi$
$\displaystyle\frac{\partial\Psi}{\partial t_{i}}=(L^{i})_{+}\Psi\,.$

The function $\Psi$ is called the _Baker-Akhiezer_ function. The first non-
trivial equation, the KP equation, is

(4) $3\frac{\partial^{2}u_{1}}{\partial t_{2}^{2}}-\frac{\partial}{\partial
t_{1}}\Big{(}4\frac{\partial u_{1}}{\partial t_{3}}-12u_{1}\frac{\partial
u_{1}}{\partial t_{1}}-\frac{\partial^{3}u_{1}}{\partial
t_{1}^{3}}\Big{)}=0\,.$

The Baker-Akhiezer function can be written in the form

(5)
$\Psi=\frac{\tau\big{(}\big{\\{}t_{k}-\frac{z^{-k}}{k}\big{\\}}\big{)}}{\tau(\\{t_{k}\\})}e^{\xi(\underline{t},z)}\,,\qquad\xi(\underline{t},z)=\sum_{k=1}^{\infty}t_{k}z^{k}\,.$

Here $\tau$ is a single function, called a tau-function, dependent on the
$t_{k}$, and all dependent variables can be expressed in terms of this one
function. This way, the entire hierarchy can be rewritten as bilinear
equations for $\tau$ called _Hirota equations_ :

(6)
$\operatorname*{Res}_{z=\infty}dz\,e^{\xi(\underline{t},z)-\xi(\underline{t}^{\prime},z)}\tau\big{(}\big{\\{}t_{k}-\frac{1}{kz^{k}}\big{\\}}\big{)}\tau\big{(}\big{\\{}t_{k}^{\prime}+\frac{1}{kz^{k}}\big{\\}}\big{)}\,.$

Writing $F=\log\tau$, we find $u_{1}=\partial^{2}\log\tau$, and the first two
equations are

(7) $\displaystyle 0$ $\displaystyle=3\frac{\partial^{2}F}{\partial
t_{2}^{2}}-4\frac{\partial^{2}F}{\partial t_{3}\partial
t_{1}}+\frac{\partial^{4}F}{\partial
t_{1}^{4}}+6\Big{(}\frac{\partial^{2}F}{\partial t_{1}^{2}}\Big{)}^{2}\,;$ (8)
$\displaystyle 0$ $\displaystyle=2\frac{\partial^{2}F}{\partial t_{3}\partial
t_{2}}-3\frac{\partial^{2}F}{\partial t_{4}\partial
t_{1}}+\frac{\partial^{4}F}{\partial t_{2}\partial
t_{1}^{3}}+6\frac{\partial^{2}F}{\partial t_{2}\partial
t_{1}}\frac{\partial^{2}F}{\partial t_{1}^{2}}\,.$

###  2.2. Space of tau-functions and Lie action

The space of solutions of the KP hierarchy is an infinite-dimensional
Grassmannian [SS83], which is usually Plücker embedded in a Fock space, i.e. a
highest weight module of a certain Clifford algebra. The Hirota equations are
then the Plücker relations defining the Grassmannian inside the Fock space. By
the boson-fermion correspondence, this can also be expressed in terms of
symmetric functions, which is the viewpoint we will adopt here.

######  Definition 2.1.

We write $\Lambda\coloneqq\mathbb{C}\llbracket p_{1},p_{2},\dotsc\rrbracket$
for the _space of symmetric functions_ , also called the _bosonic Fock space_
(of type A). Here the $p_{k}$ are power-sum functions
$p_{k}=\sum_{i}X_{i}^{k}$ in some countably infinite variable set
$\underline{X}=\\{X_{i}\\}$.

For other symmetric functions in $\underline{X}$, e.g. the Schur functions
$s_{\lambda}$, we write $s_{\lambda}(\underline{p})\coloneqq
s_{\lambda}(\underline{X})$.

The space of symmetric functions has a projective action of the Lie algebra
$\mathfrak{gl}(\infty)$, the algebra of infinite square matrices
$(a_{ij})_{i,j\in\mathbb{Z}+\frac{1}{2}}$.111In order to make the Lie bracket
well-defined, some decay condition is needed. A common choice is restriction
to finitely many diagonals, but there are other options, see e.g. [SS83]. We
will remain agnostic on this choice, as in this paper, the required
convergence in guaranteed by our constructions. This space has a standard
basis given by $E_{kl}=(\delta_{ik}\delta_{jkl})_{ij}$. Define the _vertex
operator_

(9)
$Z(z,w)=\frac{1}{z-w}\bigg{(}\exp\Big{(}\sum_{j=1}^{\infty}(z^{j}-w^{j})p_{j}\Big{)}\exp\Big{(}-\sum_{k=1}^{\infty}(z^{-k}-w^{-k})\frac{1}{k}\frac{\partial}{\partial
p_{k}}\Big{)}-1\bigg{)}\,.$

Then expanding this vertex operator as

(10) $Z(z,w)=\sum_{i,j\in\mathbb{Z}+\frac{1}{2}}Z_{ij}z^{i+1/2}w^{-j-1/2}\,,$

the assignment $E_{ij}\mapsto Z_{ij}$ is a projective representation of
$\mathfrak{gl}(\infty)$, i.e. a representation of a central extension
$\widehat{\mathfrak{gl}(\infty)}$.

The matrices $\alpha_{k}=\sum_{l\in\mathbb{Z}+\frac{1}{2}}E_{l-k,l}$ give rise
to the following operators on $\Lambda$:

(11) $a_{k}\coloneqq\begin{cases}p_{k}&k>0\\\ -k\frac{\partial}{\partial
p_{-k}}&k<0\\\ 0&k=0\end{cases}\,.$

We also define the following operators:

(12) $\displaystyle L_{m}$
$\displaystyle\coloneqq\frac{1}{2}\sum_{i=-\infty}^{\infty}\vcentcolon\mathrel{a_{i}a_{m-i}}\vcentcolon\,,$
$\displaystyle M_{l}$
$\displaystyle\coloneqq\frac{1}{6}\sum_{i,j=-\infty}^{\infty}\vcentcolon\mathrel{a_{i}a_{j}a_{l-i-j}}\vcentcolon\,,$

where the $\vcentcolon\mathrel{\;}\vcentcolon$, the _normal ordering_ , means
one should order the operators inside in order of decreasing index. All of
these operators are in $\widehat{\mathfrak{gl}(\infty)}$.

######  Theorem 2.2 ([SS83]).

Under the identification $t_{k}=\frac{p_{k}}{k}$, the space of KP tau-
functions is the orbit of $1\in\Lambda$ under the action of
$\widehat{\mathfrak{gl}(\infty)}$.

###  2.3. Single Hodge integrals

In [Kaz09], Kazarian considered the generating function for single Hodge
integrals,

(13)
$F_{\textup{H}}(u;T_{0},T_{1},T_{2},\dotsc)\coloneqq\sum_{g,n}\frac{1}{n!}\sum_{d_{1},\dotsc,d_{n}\geq
0}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda(-u^{2})\prod_{i=1}^{n}\psi_{i}^{d_{i}}T_{d_{i}}\,,$

and showed that its exponent, $Z_{\textup{H}}\coloneqq\exp(F_{\textup{H}})$,
is a tau-function for the KP hierarchy, after a certain change of coordinates.
Explicitly, this change of coordinates is given as follows: define

(14) $D=(u+z)^{2}z\frac{\partial}{\partial z}\,.$

Then we define the $T_{d}$ in terms of other coordinates $q_{k}$ by the linear
correspondence

(15) $\displaystyle q_{k}$ $\displaystyle\leftrightarrow z^{k}\,,$
$\displaystyle T_{d}\leftrightarrow D^{d}z\,.$

The proof consists of three steps, and makes essential use of the ELSV formula
[ELSV01] to transform this generating function into a generating function of
Hurwitz numbers.

The first step, [Kaz09, Theorem 2.2], is the observation that the generating
function for single simple Hurwitz numbers is a tau-tunction for the KP
hierarchy. This is a well-known result, see [Oko00]. In fact, the single
simple Hurwitz generating function can be obtained from the trivial
$\tau$-function $1$ by the action of two very explicit elements of the Lie
group associated to $\widehat{\mathfrak{gl}(\infty)}$. The second step,
[Kaz09, Theorem 2.3], uses the ELSV formula to rewrite the Hurwitz generating
function (after subtracting the unstable geometries) as a generating function
for single Hodge integrals. This introduces certain combinatorial factors,
that suggest a certain change of coordinates, which is encoded by the equation
$X(z)=\frac{z}{1+\beta z}e^{-\frac{\beta z}{1+\beta z}}$. After this change of
coordinates, we obtain $Z_{\textup{H}}$, viewed as a function in $q$’s.

The third step, [Kaz09, Theorem 2.5] shows that a certain class of coordinate
changes preserves solutions of the KP hierarchy, after they are modified with
a quadratic function. In essence, this coordinate change is given
infinitesimally by the flow along the differential part of an
$A\in\widehat{\mathfrak{gl}(\infty)}$, whose polynomial part is exactly the
added quadratic function. In this specific case, this quadratic function is
exactly the $(0,2)$ part of the Hurwitz generating function.

In this paper, we will generalise this proof scheme to a more general setting.
We will start from a general hypergeometric tau-function in the sense of
theorem 2.6 below, corresponding to the first point of the proof.

We obtain a change of coordinates coming from this formalism that can always
be completed to an automorphism of KP when correcting with the $H_{0,2}$ of
equation 26, without any further assumption, corresponding to the third point
of the proof.

If we restrict to the class of hypergeometric tau-functions satisfying
topological recursion, we can use the correspondence between topological
recursion and cohomological field theories of Eynard and Dunin-Barkowski-
Orantin-Shadrin-Spitz [Eyn14, DOSS14], which generalises the ELSV formula and
hence gives the second step.

In the particular case of triple Hodge integrals, the role of the ELSV formula
is taken by the Mariño-Vafa formula. For explanations on all the required
notions and notation, see the following sections.

###  2.4. Hypergeometric KP tau-functions and topological recursion

An important class of KP tau-functions is given by the hypergeometric tau-
functions [KMMM95, OS01, OS01a], for which we will use the results and
notation of [BDKS20]. In two large families of examples, these satisfy Eynard-
Orantin topological recursion [EO07] (or its generalisation to non-simple
ramification given by Bouchard-Eynard [BE13]), which we define first. We will
confine ourselves to the case of rational spectral curves, as this is the
appropriate setting for the Hurwitz-type problems covered.

######  Definition 2.3 ([EO07, BE13]).

A _rational spectral curve_ is a quadruple
$\mathcal{C}=(\Sigma=\mathbb{P}^{1},dx,dy,B)$, where $dx$ and $dy$ are
meromorphic one-forms on $\Sigma$ with no common zeroes, only simple poles of
$dx$, and without poles of $dy$ at zeroes of $dx$, and
$B=B(z_{1},z_{2})=\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}}$ is a symmetric
$(1,1)$-form on $\Sigma\times\Sigma$. Write $R\subset\Sigma$ for the set of
zeroes of $dx$, and $r_{a}$ for the order of vanishing of $dx$ at $a\in R$.

On a rational spectral curve, define a set of symmetric multidifferentials
$\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$ on $\Sigma^{n}$ via _topological
recursion_ as follows: first, define the unstable cases by
$\omega_{0,1}\coloneqq ydx$ (this need only be defined locally near the
$a_{i}$ using any primitive of $y$) and $\omega_{0,2}\coloneqq B$. Then, for
$2g-2+(n+1)>0$, the stable range, define

(16) $\omega_{g,n+1}(z_{[n]},z_{n+1})\coloneqq\sum_{a\in
R}\sum_{\\{0\\}\subsetneq
I\subset\\{0,\dotsc,r_{a}-1\\}}\operatorname*{Res}_{z=a}\frac{\int_{a}^{z}\omega_{0,2}(\mathord{\cdot},z_{n+1})}{\prod_{i=2}^{m}\big{(}\omega_{0,1}(z)-\omega_{0,1}(\sigma_{a}^{i}(z))\big{)}}\mathcal{W}_{g,|I|+1,n}(\sigma_{a}^{I}(z);z_{[n]})\,,$

where $\sigma_{a}$ is a generator of the local deck transformations of a
primitive of $dx$ at $a$, and

(17)
$\mathcal{W}_{g,m,n}(\zeta_{[m]};z_{[n]})\coloneqq\sum^{\prime}_{\begin{subarray}{c}\mu\vdash[m]\\\
\bigsqcup_{k=1}^{l(\mu)}N_{k}=[n]\\\ \sum
g_{k}=g+l(\mu)-n\end{subarray}}\prod_{k=1}^{l(\mu)}\omega_{g_{k},|\mu_{k}|+|N_{k}|}(\zeta_{\mu_{k}},z_{N_{k}})$

where the prime on the summation means exclusion of any
$(g_{k},|\mu_{k}|+|N_{k}|)=(0,1)$.

######  Remark 2.4.

Often, the definition of spectral curves involves functions $x$ and $y$, in
stead of their derivatives. However, these functions may not be defined
globally on $x$, e.g. they may – and in this paper will – contain logarithmic
terms. As most of the theory of topological recursion (with the notable
exception of the global topological recursion of Bouchard-Eynard [BE13]) only
depends on the derivatives, I have chosen to use this as a definition.

######  Theorem 2.5 ([BEO15, BS17]).

Let $\mathcal{C}$ be a rational spectral curve with simple zeroes of $dx$. A
collection $\\{\omega_{g,n}\\}_{g\geq 0,n\geq 1}$ with $\omega_{0,1}=ydx$ and
$\omega_{0,2}=B$ satisfies topological recursion if and only if the following
hold:

  * •

Meromorphicity: For $2g-2+n>0$, $\omega_{g,n}$ extends to a meromorphic form
on $\Sigma^{n}$;

  * •

Linear loop equation: For any $g$, $n$, and $a\in R$,

(18) $\omega_{g,n+1}(z,z_{\llbracket
n\rrbracket})+\omega_{g,n+1}(\sigma_{a}(z),z_{\llbracket n\rrbracket})$

is holomorphic near $z=a$ and has a simple zero at $z=a$;

  * •

Quadratic loop equation: For any $g$, $n$, and $a\in R$,

(19) $\omega_{g-1,n+2}(z,\sigma_{a}(z),z_{\llbracket
n\rrbracket})+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\ I\sqcup J=\llbracket
n\rrbracket\end{subarray}}\omega_{g_{1},|I|_{1}+1}(z,z_{I})\omega_{g_{2},|J|+1}(\sigma_{a}(z),z_{J})$

is holomorphic near $z=a$ and has a double zero at $z=a$;

  * •

Projection property: For $2g-2+n>0$,

(20) $\omega_{g,n}(z_{\llbracket n\rrbracket})=\sum_{a_{1},\dotsc,a_{n}\in
R}\Big{(}\prod_{j=1}^{n}\operatorname*{Res}_{\zeta_{j}=a_{j}}\int_{a_{j}}^{z_{j}}\omega_{0,2}(z_{j},\mathord{\cdot})\Big{)}\omega_{g,n}(\zeta_{\llbracket
n\rrbracket})\,.$

If only the meromorphicity and linear and quadratic loop equations hold, the
problem is said to satisfy blobbed topological recursion, cf. [BS17]. In this
case, the $\omega_{g,n}$ are determined by the spectral curve along with their
holomorphic parts at ramification points.

One important reason to consider topological recursion is that the
$\omega_{g,n}$ will often encode enumerative invariants in their Taylor series
expansion around a given point of the spectral curve in a given coordinate.
For us, this is also the case, as we consider the class given by the following
theorem:

######  Theorem 2.6 ([BDKS20, BDKS20a]).

Consider two formal power series

(21)
$\hat{\psi}(\hbar^{2},y)\coloneqq\sum_{k=1}^{\infty}\sum_{m=0}^{\infty}c_{k,m}y^{k}\hbar^{2m}\,,\qquad\hat{y}(\hbar^{2},z)\coloneqq\sum_{k=1}^{\infty}\hat{y}_{k}(\hbar^{2})z^{k}\coloneqq\sum_{k=1}^{\infty}\sum_{m=0}^{\infty}s_{k,m}z^{k}\hbar^{2m}\,,$

and their associated _hypergeometric KP tau-function_ or _Orlov-Scherbin
partition function_

(22)
$Z(\underline{p})=e^{F(\underline{p})}=\sum_{\nu\in\mathcal{P}}\exp\Big{(}\sum_{\square\in\nu}\hat{\psi}(\hbar^{2},-\hbar
c_{\square})\Big{)}s_{\nu}(\underline{p})s_{\nu}\big{(}\big{\\{}\frac{\hat{y}_{k}(\hbar^{2})}{\hbar}\big{\\}}\big{)}\,.$

Define

(23) $\displaystyle\psi(y)$ $\displaystyle\coloneqq\hat{\psi}(0,y)\,,\qquad$
$\displaystyle y(z)$ $\displaystyle\coloneqq\hat{y}(0,z)\,,\qquad$
$\displaystyle x(z)$ $\displaystyle\coloneqq\log z-\psi(y(z))\,,$
$\displaystyle X(z)$ $\displaystyle\coloneqq e^{x(z)}\,,$ $\displaystyle D$
$\displaystyle\coloneqq\frac{\partial}{\partial x}\,,$ $\displaystyle Q$
$\displaystyle\coloneqq z\frac{dx}{dz}$

and write

(24)
$H_{n}\coloneqq\sum_{k_{1},\dotsc,k_{n}=1}^{\infty}\frac{\partial^{n}F}{\partial
p_{k_{1}}\dotsb\partial p_{k_{n}}}\bigg{|}_{p=0}X_{1}^{k_{1}}\dotsb
X_{n}^{k_{n}}\,.$

Then these can be decomposed as

(25) $H_{n}=\sum_{g=0}^{\infty}\hbar^{2g-2+n}H_{g,n}\,,$

with $H_{g,n}$ independent of $\hbar$, and

(26) $DH_{0,1}(X(z))=y(z)\,,\qquad
H_{0,2}(X(z_{1}),X(z_{2}))=\log\Big{(}\frac{z_{1}^{-1}-z_{2}^{-1}}{X_{1}^{-1}-X_{2}^{-1}}\Big{)}\,.$

If moreover $\frac{d\psi(y)}{dy}\big{|}_{y=y(z)}$ and $\frac{dy(z)}{dz}$ have
analytic continuations to meromorphic functions in $z$ and all coefficients of
positive powers of $\hbar^{2}$ in $\hat{\psi}(\hbar^{2},y(z))$ and
$\hat{y}(\hbar^{2},z)$ are rational functions of $z$ whose singular points are
disjoint from the zeroes of $dx$, then the $n$-point differentials

(27) $\omega_{g,n}\coloneqq d_{1}\dotsb
d_{n}H_{g,n}+\delta_{g,0}\delta_{n,2}\frac{dX_{1}\,dX_{2}}{(X_{1}-X_{2})^{2}}$

can be extended analytically to $(\mathbb{P}^{1})^{n}$ as global rational
forms, and the collection of $n$-point differentials satisfies meromorphicity
and the linear and quadratic loop equations, i.e. blobbed topological
recursion, for the curve
$(\mathbb{P}^{1},dx(z),dy(z),B=\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$.

Finally, if $\hat{\psi}$ and $\hat{y}$ belong to one of the two families

Family I $\displaystyle\hat{\psi}(\hbar^{2},y)$
$\displaystyle=\mathcal{S}(\hbar\partial_{y})P_{1}(y)+\log\Big{(}\frac{P_{2}(y)}{P_{3}(y)}\Big{)}\,;$
$\displaystyle\hat{y}(\hbar^{2},z)$
$\displaystyle=\frac{R_{1}(z)}{R_{2}(z)}\,,$ Family II
$\displaystyle\hat{\psi}(\hbar^{2},y)$ $\displaystyle=\alpha y\,;$
$\displaystyle\hat{y}(\hbar^{2},z)$
$\displaystyle=\frac{R_{1}(z)}{R_{2}(z)}+\mathcal{S}(\hbar
z\partial_{z})^{-1}\log\Big{(}\frac{R_{3}(z)}{R_{4}(z)}\Big{)}\,,$

where $\alpha\in\mathbb{C}^{\times}$ and the $P_{i}$ and $R_{j}$ are arbitrary
polynomials such that $\psi(y)$ and $y(z)$ are non-zero, but vanishing at
zero, and no singular points of $y$ are mapped to branch points by $x$, then
the $n$-point differentials also satisfy the projection property, and hence
topological recursion, for the curve above.

######  Remark 2.7.

It is possible to allow for constant terms in $\hat{\psi}$ in equation 21, but
using quasihomogeneity of the $s_{\nu}$ in equation 22, this can be absorbed
in a recaling of the argument of $\hat{y}$. From the spectral curve point of
view, this follows from the fact that the two curves

(28) $\begin{cases}X_{1}(z)&=ze^{-\psi\circ y(z)+\log a}=aze^{-\psi\circ
y(z)}\\\
y_{1}(z)&=y(z)\end{cases}\qquad\text{and}\qquad\begin{cases}X_{2}(z^{\prime})&=z^{\prime}e^{-\psi\circ
y(\frac{z^{\prime}}{a})}\\\
y_{2}(z^{\prime})&=y(\frac{z^{\prime}}{a})\end{cases}$

can be identified via $z^{\prime}=az$.

######  Remark 2.8.

We will consistently use the symbol $x$ for the function which is part of the
spectral curve data and $X$ for its exponential, which is the expansion
parameter for this class of Hurwitz problems.

We will need different parts of this theorem for the different parts of the
proof. In particular, topological recursion is needed to obtain intersection
numbers.

There is one small generalisation that can be made in theorem 2.6 by using a
homogeneity property, using that equation 22 is a KP tau-function identically
in $\hbar$.

######  Lemma 2.9.

There is a $\mathbb{C}^{\times}$ action on pairs $(\hat{\psi},\hat{y})$
induced by rescaling of $\hbar$ in equation 22 as follows:

(29)
$\lambda\cdot\big{(}\hat{\psi}(\hbar^{2},y),\hat{y}(\hbar^{2},z)\big{)}=\big{(}\hat{\psi}(\lambda^{-2}\hbar^{2},\lambda^{-1}y),\lambda\hat{y}(\lambda^{-2}\hbar^{2},z)\big{)}\,.$

This acts on $H_{g,n}$ as $\lambda\cdot H_{g,n}=\lambda^{2-2g-n}H_{g,n}$ and
on the spectral curve by

(30) $\lambda\cdot(\mathbb{P}^{1},dx(z),dy(z),B)=(\mathbb{P}^{1},dx(z),\lambda
dy(z),B)\,.$

Hence it is compatible with the homogeneity of topological recursion of e.g.
[EO09, Section 4.1].

######  Corollary 2.10.

Family II of theorem 2.6 can be extended to

(31) $\hat{\psi}(\hbar^{2},y)=\alpha
y\,;\qquad\hat{y}(\hbar^{2},z)=\frac{R_{1}(z)}{R_{2}(z)}+\lambda\mathcal{S}(\lambda^{-1}\hbar
z\partial_{z})^{-1}\log\Big{(}\frac{R_{3}(z)}{R_{4}(z)}\Big{)}\,,$

with the same conditions on $\alpha$, the $P_{i}$, and the $R_{j}$ as before,
and $\lambda\in\mathbb{C}^{\times}$.

###### Proof.

The constant $\lambda$ has been absorbed in $\alpha$ and the $R_{i}$ where
possible. ∎

A similar argument for Family I does not lead to an extended class, as this
family is already invariant under the torus action.

###  2.5. Topological recursion and cohomological field theories

Topological recursion is strongly related to intersection theory of the moduli
spaces of curves: there is a quite general correspondence between spectral
curves and certain coherent collections of intersection classes in the moduli
spaces. These coherent collections are cohomological field theories, which
were orignally defined by Kontsevich and Manin [KM94] to axiomatise Gromov-
Witten theory.

######  Definition 2.11 ([KM94]).

Let $V$ be a vector space with a non-degenerate bilinear form $\eta$ and a
distinguished vector $\mathds{1}$. A _cohomological field theory with flat
unit (CohFT)_ on $(V,\eta,\mathds{1})$ is a collection of maps

(32) $\Omega_{g,n}\colon V^{\otimes n}\to
H^{*}(\overline{\mathcal{M}}_{g,n})\,,$

for all $g\geq 0$, $n\geq 1$ such that $2g-2+n>0$, such that

  * •

$\Omega_{g,n}$ is $\mathfrak{S}_{n}$-equivariant with respect to simultaneous
permutation of the factors and the marked points;

  * •

with respect to the glueing maps

(33)
$\rho\colon\overline{M}_{g-1,n+2}\to\overline{\mathcal{M}}_{g,n}\,,\qquad\sigma\colon\overline{M}_{g,|I|+1}\times\overline{M}_{h,|J|+1}\to\overline{M}_{g+h,|I\sqcup
J|}\,,$

we get

$\displaystyle\rho^{*}\Omega_{g,n}(v_{\llbracket n\rrbracket})$
$\displaystyle=\Omega_{g-1,n+2}(v_{\llbracket
n\rrbracket}\otimes\eta^{\dagger})\,,$
$\displaystyle\sigma^{*}\Omega_{g+h,|I|+|J|}(v_{I}\otimes v_{J})$
$\displaystyle=\Omega_{g,|I|+1}\otimes\Omega_{h,|J|+1}(v_{I}\otimes\eta^{\dagger}\otimes
v_{J})\,;$

  * •

With respect to the forgetful maps

(34) $\pi\colon\overline{M}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}\,,$

we have

(35) $\pi^{*}\Omega_{g,n}(v_{\llbracket
n\rrbracket})=\Omega_{g,n+1}(v_{\llbracket n\rrbracket}\otimes\mathds{1})\,.$

There is a large group acting on the space of CohFTs, called the Givental
group [Giv01, Sha09, Tel12]. It consists of
$R(u)\in\mathord{\mathrm{Id}}+u\operatorname{End}(V)\llbracket u\rrbracket$
such that $R(u)R^{\dagger}(-u)=\mathord{\mathrm{Id}}$. It is called the unit-
preserving action in case one also considers CohFTs without unit.

######  Theorem 2.12 ([Eyn14, DOSS14, BKS20]).

Consider a compact rational spectral curve $(\mathbb{P}^{1},dx,dy,B)$, and
define $V^{*}$ to be the space of residueless meromorphic one-forms on
$\mathbb{P}^{1}$ with poles only at $a\in R$ of order at most $r_{a}+1$.
Choose a basis $\\{d\xi^{j}\\}_{j\in J}$ of $V^{*}$ with dual basis $e_{j}$
and define $d\xi^{j}_{k}=(d\circ\frac{1}{dx})^{k}d\xi^{j}$. Then

(36) $\omega_{g,n}(z_{1},\dotsc,z_{n})=\sum_{j_{1},\dotsc,j_{n}\in
J}\int_{\overline{\mathcal{M}}_{g,n}}\Omega_{g,n}(e_{j_{1}}\otimes\dotsb\otimes
e_{j_{n}})\prod_{i=1}^{n}\sum_{k_{i}=0}^{\infty}\psi_{i}^{k_{i}}d\xi_{k_{i}}^{j_{i}}(z_{i})\,,$

where $\Omega$ is a cohomological field theory on $V$, given explicitly by
acting on a direct sum of Witten $r_{a}$-spin classes for all ramification
points of order $r_{a}$ by an $R$ determined by the spectral curve.

######  Remark 2.13.

The results of [Eyn14, BKS20] do not mention CohFTs, but rather give a
relation between local spectral curves and intersection numbers. In order to
obtain a CohFT, a condition is required, cf. [LPSZ17, Equation (17)]. As noted
in [DNOPS19, Section 2.6], in case the spectral curve is compact and $dx$ is
meromorphic with simple zeroes, this condition is satisfied by [Eyn14,
Appendix B]. In case of higher order zeroes, the same holds, using [BKS20,
Section 7.2.3].

######  Remark 2.14.

The space $V^{*}$ is naturally related to the projection property of theorem
2.5: the $d\xi_{k}^{j}$ span the image of the projection operator. Its
dimension, the rank of the CohFT, equals the degree of the divisor of zeroes
of $dx$.

There are two common choices for the basis $d\xi^{j}$, depending on a local
coordinate $\zeta_{a}$ around a ramification point $a$ such that
$x(z)=\zeta_{a}(z)^{r_{a}}+x(a)$. One is
$d\xi^{a,k}(z)=\operatorname*{Res}_{z^{\prime}=a}\Big{(}\int_{a}^{z^{\prime}}B(z,\cdot)\Big{)}\frac{d\zeta(z^{\prime})}{\zeta(z^{\prime})^{k}}$,
with $1\leq k\leq r_{a}-1$, cf. [BKS20, Equation (80)], while the other is
$\xi^{a}(z)=\int^{z}\frac{B(\zeta_{a},\cdot)}{d\zeta_{a}}\big{|}_{\zeta_{a}=0}$,
in case $r_{a}=2$, cf. [GKL21, Equation (2.23)]. Both have merit, depending on
the situation, but they are not compatible.

Furthermore, several normalisation conventions exist for the recursion
operator linking $d\xi^{j}_{k}$ to $d\xi^{j}_{k+1}$. These different
conventions can be related by rescaling $\Omega$ and the correlators, using
that the integrand must be of degree $3g-3+n$.

So the $\omega_{g,n}$ we are concerned with can be expanded in different ways:
as a formal series around $X=0$ by theorem 2.6, and on a basis of meromorphic
differentials with poles at the zeroes of $dx$ by theorem 2.12. The change of
variables we require is found by relating these different expansions.

In order to apply the Eynard-DOSS correspondence to get a good change of
variables, we will want to take a different basis of $V^{*}$. It turns out to
be useful to relate to powers of our preferred coordinate $z$, so the basis we
take is
$\xi^{j}\coloneqq\big{(}\frac{dx}{dz}\big{)}^{-1}z^{j}=\frac{d}{dx}\frac{z^{j+1}}{j+1}$.

######  Definition 2.15.

Let $\Omega$ be a cohomological field theory on a space $(V,\eta)$ with a
basis $\\{e_{j}\\}_{j\in J}$. Its _generating function_ $G_{\Omega}$ is
defined as

(37) $G_{\Omega}(\\{T^{j}_{k}\mid k\geq 0,j\in
J\\})\coloneqq\sum_{\begin{subarray}{c}g,n\\\
2g-2+n>0\end{subarray}}\frac{\hbar^{2g-2+n}}{n!}\sum_{\begin{subarray}{c}j_{i},\dotsc,j_{n}\in
J\end{subarray}}\int_{\overline{\mathcal{M}}_{g,n}}\Omega(e_{j_{1}}\otimes\dotsb\otimes
e_{j_{n}})\prod_{i=1}^{n}\sum_{k_{i}=1}^{\infty}\psi_{i}^{k_{i}}T^{j_{i}}_{k_{i}}\,,$

where we write $\\{T^{j}\\}$ for the basis of $V^{*}$ dual to $\\{e_{j}\\}$
and $T^{j}_{k}=T^{j}\otimes p_{k}$.

The main theorem of this paper is the following:

######  Theorem 2.16.

If a cohomological field theory $\Omega$ is obtained from theorem 2.12 applied
to either family in theorem 2.6, then the exponential of
$G_{\Omega}(\underline{T}(\underline{q}))$ is a KP tau-function in
$\\{t_{d}=\frac{q_{d}}{d}\\}$, where the $T^{j}_{k}(\underline{q})$ are
defined by

(38) $T_{-1}^{j}=\frac{1}{j+1}q_{j+1}\,,\qquad
T_{k+1}^{j}=\sum_{m=1}^{\infty}\sum_{l=0}^{\infty}m\mathcal{T}_{l}q_{m+l}\frac{\partial}{\partial
q_{m}}T_{k}^{j}\,,\quad\text{with $\mathcal{T}_{l}$ given by}\quad
Q(z)^{-1}=\sum_{l=0}^{\infty}\mathcal{T}_{l}z^{l}\,.$

The proof of this theorem is given in proposition 3.8.

######  Remark 2.17.

The proof of this theorem does not use anything specific to the families
mentioned, it just requires topological recursion to obtain a cohomological
field theory. As soon as topological recursion is proved for another
hypergeometric tau-function and the spectral curve fits in the scope of
theorem 2.12, this theorem generalises.

The first two KP equations in $q$ variables are

(39) $\displaystyle 0$ $\displaystyle=\frac{\partial^{2}F}{\partial
q_{2}^{2}}-\frac{\partial^{2}F}{\partial q_{3}\partial
q_{1}}+\frac{1}{12}\frac{\partial^{4}F}{\partial
q_{1}^{4}}+\frac{1}{2}\Big{(}\frac{\partial^{2}F}{\partial
q_{1}^{2}}\Big{)}^{2}\,;$ (40) $\displaystyle 0$
$\displaystyle=\frac{\partial^{2}F}{\partial q_{3}\partial
q_{2}}-\frac{\partial^{2}F}{\partial q_{4}\partial
q_{1}}+\frac{1}{6}\frac{\partial^{4}F}{\partial q_{2}\partial
q_{1}^{3}}+\frac{\partial^{2}F}{\partial q_{2}\partial
q_{1}}\frac{\partial^{2}F}{\partial q_{1}^{2}}\,.$

######  Example 2.18 (Naive single Hodge).

Let us consider the functions

(41) $\hat{\psi}(\hbar^{2},y)=y\,,\qquad\hat{y}(\hbar^{2},z)=z\,.$

Then we find

(42) $x(z)=\log z-z\,,\qquad X(z)=z\,e^{-z}\,,\qquad Q=1-z\,.$

This is the usual shape of spectral curve for simple Hurwitz numbers [BM08,
BEMS11] , so the CohFT associated to it by theorem 2.12 is the single Hodge
class $\Lambda(-1)$ via the ELSV formula [ELSV01]. In this case, writing
$T_{k}=\sum_{m=1}^{\infty}c_{k,m}q_{m}$, theorem 2.16 yields

(43) $c_{k+1,m}=\sum_{j=0}^{\infty}jc_{k,j}\,.$

Along with the initial condition $c_{-1,m}=\delta_{m,1}$, this shows that
$c_{k,m}=\genfrac{\\{}{\\}}{0.0pt}{}{k+m}{m}$ for $m>-1$, the Stirling numbers
of the second kind. In particular,

(44) $\displaystyle T_{0}$
$\displaystyle=q_{1}+q_{2}+q_{3}+q_{4}+q_{5}+...\,,$ $\displaystyle T_{1}$
$\displaystyle=q_{1}+3q_{2}+6q_{3}+10q_{4}+15q_{5}+...\,,$ $\displaystyle
T_{2}$ $\displaystyle=q_{1}+7q_{2}+25q_{3}+65q_{4}+140q_{5}+...$

Note that these are infinite sums, in contrast to the ones Kazarian found in
[Kaz09], cf. equations 14 and 15, even though both are related to single Hodge
integrals. This phenomenon is explained by the arbitrary choice of a rational
parametrisation of the spectral curve, formalised in corollary 3.6.

Using the intersection numbers

(45) $\displaystyle\int_{\bar{\mathcal{M}}_{0,3}}1$
$\displaystyle=\int_{\bar{\mathcal{M}}_{0,4}}\psi_{i}=1\,,$ (46)
$\displaystyle\int_{\bar{\mathcal{M}}_{1,1}}\lambda_{1}$
$\displaystyle=\int_{\bar{\mathcal{M}}_{1,1}}\psi_{1}=\int_{\bar{\mathcal{M}}_{1,2}}\psi_{i}^{2}=\int_{\bar{\mathcal{M}}_{1,2}}\psi_{1}\psi_{2}=\int_{\bar{\mathcal{M}}_{1,2}}\lambda_{1}\psi_{1}=\frac{1}{24}\,,$

we see that

(47)
$G_{\Lambda(-1)}(\underline{T})=\hbar\big{(}\frac{1}{6}T_{0}^{3}+\frac{1}{24}T_{1}-\frac{1}{24}T_{0}\big{)}+\hbar^{2}\big{(}\frac{1}{6}T_{0}^{3}T_{1}+\frac{1}{48}T_{1}^{2}+\frac{1}{24}T_{0}T_{2}-\frac{1}{24}T_{0}T_{1}\big{)}+\mathcal{O}(\hbar^{3})\,.$

From this, we obtain

(48)
$\displaystyle\frac{\partial^{2}G_{\Lambda(-1)}(\underline{T}(\underline{q}))}{\partial
q_{2}^{2}}$
$\displaystyle=\hbar\sum_{k>0}q_{k}+\hbar^{2}\bigg{(}\sum_{k,l>0}\Big{(}\frac{l(l+1)}{2}+3\Big{)}q_{k}q_{l}+\frac{2\cdot
3^{2}}{48}+\frac{2\cdot 7}{24}-\frac{2\cdot
3}{24}\bigg{)}+\mathcal{O}(\hbar^{3})\,,$
$\displaystyle\frac{\partial^{2}G_{\Lambda(-1)}(\underline{T}(\underline{q}))}{\partial
q_{1}q_{3}}$
$\displaystyle=\hbar\sum_{k>0}q_{k}+\hbar^{2}\bigg{(}\sum_{k,l>0}\Big{(}\frac{l(l+1)}{2}+\frac{7}{2}\Big{)}q_{k}q_{l}+\frac{2\cdot
6}{48}+\frac{25+1}{24}-\frac{6+1}{24}\bigg{)}+\mathcal{O}(\hbar^{3})\,,$
$\displaystyle\frac{\partial^{4}G_{\Lambda(-1)}(\underline{T}(\underline{q}))}{\partial
q_{1}^{4}}$ $\displaystyle=\hbar^{2}\frac{24}{6}+\mathcal{O}(\hbar^{3})\,,$
$\displaystyle\frac{\partial^{2}G_{\Lambda(-1)}(\underline{T}(\underline{q}))}{\partial
q_{1}^{2}}$ $\displaystyle=\hbar\sum_{k>0}q_{k}+\mathcal{O}(\hbar^{2})\,,$

which does show that $G_{\Lambda(-1)}(\underline{T}(\underline{q}))$ solves
equation 39 up to second order in $\hbar$.

I would like to thank P. Norbury for using this example to check the results
of this paper.

###  2.6. The Mariño-Vafa formula and KP for topological vertex amplitudes

A particularly interesting family of hypergeometric tau-functions is given by
the theory of the topological vertex, or triple Hodge integrals. For the
triple Hodge integrals, the ELSV-type formula required is the Mariño-Vafa
formula [MV02]. This theory is the particular case for $\mathbb{C}^{3}$ of the
Gromov-Witten theory of toric Calabi-Yau threefolds, conjectured by Bouchard-
Klemm-Mariño-Pasquetti [BKMP09] to satisfy topological recursion. The case we
are interested in was proved in [Che18, Zho09], as well as in [Eyn11] as an
example of the general correspondence of theorem 2.12, while the general BKMP
conjecture was proved in [EO15].

In this section, we use the triple Hodge integrals as an example of our
general theory, using methods slightly adapted to this special case. We will
see in section 3.3 why this case is particularly nice.

######  Definition 2.19.

The _triple Hodge cohomological field theory with Calabi-Yau condition_ is the
one-dimensional CohFT
$\mathop{\textup{TH}}_{g,n}(a,b,c)=\Lambda(a)\Lambda(b)\Lambda(c)$, where the
parameters $a,b,c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$.

We write

(49) $\ G_{\textup{TH}}(a,b,c;\underline{T})\coloneqq
G_{\mathop{\textup{TH}}(a,b,c)}(\underline{T})\,.$

An adapted application of theorem 2.16 is given in the following theorem. This
theorem has already been proved by Alexandrov [Ale21], here we give a new
proof.

######  Theorem 2.20 ([Ale21, Theorem 2]).

Define $T_{0}(\underline{q})\coloneqq q_{1}$,
$T_{k+1}(\underline{q})\coloneqq\sum_{m=1}^{\infty}m(u^{2}q_{m}+u\frac{w+2}{\sqrt{w+1}}q_{m+1}+q_{m+2})\frac{\partial}{\partial
q_{m}}T_{k}$. Then

(50)
$G_{\textup{TH}}\big{(}-u^{2},-u^{2}w,\frac{u^{2}w}{w+1};\\{T_{k}(\underline{q})\\}\big{)}$

is a solution of the KP hierarchy with respect to the variables
$\\{t_{d}=\frac{q_{d}}{d}\\}$, identically in $u$ and $w$.

In this particular case, we will make slightly different choices to end up
with the formulation above.

######  Remark 2.21.

Note that the triple $a=-u^{2}$, $b=-u^{2}w$, $c=\frac{u^{2}w}{w+1}$ does
indeed satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$, and moreover any
triple satisfying this condition can be written this way.

######  Remark 2.22.

In the limit $w\to 0$, this theorem reduces to the main theorem, 2.1, of
[Kaz09]. In the limit $u\to 0$, it reduces to the Witten-Kontsevich theorem
[Wit91, Kon92]: in that limit $T_{d}\to(2d-1)!!q_{2d+1}$ and independence of
even parameters reduces the KP hierarchy to the KdV hierarchy.

Before giving the Mariño-Vafa formula, note that in genus zero

(51)
$\int_{\overline{\mathcal{M}}_{0,n}}\frac{\Lambda(a)\Lambda(b)\Lambda(c)}{\prod_{i=1}^{n}1-\mu_{i}\psi_{i}^{d_{i}}}=|\mu|^{n-3}$

for $n\geq 3$, and this serves as a definition for $n=1,2$. These terms are
not included in $G_{\textup{TH}}$.

######  Theorem 2.23 (Mariño-Vafa formula, [MV02, LLZ03, OP04]).

There is a relation between triple Hodge integrals and characters of symmetric
groups, as follows:

(52)
$\begin{split}\exp\bigg{(}\sum_{\mu}\sum_{g=0}^{\infty}\frac{(w+1)^{g+n-1}}{|\operatorname{Aut}\mu|}\prod_{i=1}^{n}\frac{\prod_{j=1}^{\mu_{i}-1}(\mu_{i}+jw)}{(\mu_{i}-1)!}&\int_{\overline{\mathcal{M}}_{g,n}}\\!\\!\frac{\Lambda(-1)\Lambda(-w)\Lambda\big{(}\frac{w}{w+1}\big{)}}{\prod_{i=1}^{n}(1-\mu_{i}\psi_{i})}\,\beta^{2g-2+n+|\mu|}\,p_{\mu}\bigg{)}\\\
&=\sum_{m=0}^{\infty}\sum_{\mu,\nu\vdash
m}\frac{\chi^{\nu}_{\mu}}{z_{\mu}}e^{(1+\frac{w}{2})\beta
f_{2}(\nu)}\prod_{\square\in\nu}\frac{\beta w}{\varsigma(\beta
wh_{\square})}p_{\mu}\,.\end{split}$

On the right-hand side the sum is over all partitions $\nu$ of size equal to
$|\mu|$, the product is over all boxes in the Young diagram of $\nu$, and
$h_{\square}$ is the hook length of the box $\square$. Furthermore,
$f_{2}(\nu)=\frac{1}{2}\sum_{j}(\nu_{j}-j+\tfrac{1}{2})^{2}-(-j+\tfrac{1}{2})^{2}$
is the shifted symmetric sum of squares.

######  Remark 2.24.

Even though it seems the triple Hodge class in this formula only depends on
one parameter, $w$, the parameter $\beta$ can be interpreted in this way as
well, entering as a cohomological grading parameter. Hence, the formula does
govern the entire generating function of triple Hodge integrals.

In the limit $w\to 0$, the Mariño-Vafa formula reduces to the ELSV formula, as
the product over boxes simplifies to the hook length formula for the dimension
of the $\mathfrak{S}_{|\mu|}$-representation associated to $\nu$.

######  Remark 2.25.

This formula is perfectly well-behaved for $w=-1$, but theorem 2.20 does not
make sense in this case. From the general theorem 2.16, we will see that in
this case $X$ is a Möbius transformation, and hence conforms to corollary 3.6.

By symmetry in the arguments of the $\Lambda$-classes, the point $w=-1$ is
equivalent to the limit $w\to\infty$, which in the conventional formulation of
the Mariño-Vafa formula is the initial condition for the cut-and-join equation
used to prove the formula, see [Zho03, Theorem 3.3]. In this case, the
integral reduces to
$\int_{\overline{\mathcal{M}}_{g,1}}\lambda_{g}\psi^{2g-2}$ by Mumford’s
relation. These integrals were calculated by Faber and Pandharipande [FP00].

The right-hand side of the Mariño-Vafa formula is a hypergeometric KP tau-
function, which can be seen explicitly by the following lemma. In essence this
lemma was used by both [LLZ03, OP04] to prove the Mariño-Vafa formula.

######  Lemma 2.26.

Relabel parameters in equation 52 by $\beta\to\frac{\hbar}{w}$ and
$p_{k}\to\big{(}\frac{\beta w}{\hbar}\big{)}^{k}p_{k}$ to obtain

(53)
$\begin{split}\exp\bigg{(}\sum_{\mu}\sum_{g=0}^{\infty}\frac{(w+1)^{g+n-1}}{|\operatorname{Aut}\mu|}\prod_{i=1}^{n}\frac{\prod_{j=1}^{\mu_{i}-1}(\mu_{i}+jw)}{(\mu_{i}-1)!}\int_{\overline{\mathcal{M}}_{g,n}}\\!\\!&\frac{\Lambda(-1)\Lambda(-w)\Lambda\big{(}\frac{w}{w+1}\big{)}}{\prod_{i=1}^{n}(1-\mu_{i}\psi_{i})}\,\hbar^{2g-2+n}\beta^{|\mu|}w^{|\mu|+2-2g-n}\,p_{\mu}\bigg{)}\\\
&=\sum_{m=0}^{\infty}\sum_{\mu,\nu\vdash
m}\frac{\chi^{\nu}_{\mu}}{z_{\mu}}e^{(\frac{1}{w}+\frac{1}{2})\hbar
f_{2}(\nu)}\prod_{\square\in\nu}\frac{\beta w}{\varsigma(\hbar
h_{\square})}p_{\mu}\,.\end{split}$

This right-hand side may alternatively be written as a hypergeometric KP tau-
function in the shape of theorem 2.6, with

$\hat{\psi}(\hbar^{2},y)=-\frac{y}{w}\,,\qquad\hat{y}(\hbar^{2},z)=\sum_{k=1}^{\infty}\frac{1}{k\mathcal{S}(\hbar
k)}(\beta wz)^{k}\,,\qquad X(z)=z(1-\beta wz)^{1/w}\,.$

This fits in Family II.

###### Proof.

By basic theory of symmetric functions, $\sum_{\mu\vdash
m}\frac{\chi^{\nu}_{\mu}}{z_{\mu}}p_{\mu}=s_{\nu}(\underline{p})$. Also, by
[OP04, Equations (0.6), (0.7)],

$\frac{1}{\prod_{\square\in\nu}q^{h_{\square}/2}-q^{-h_{\square}/2}}=q^{-|\nu|/2-f_{2}(\nu)/2}s_{\nu}(1,q^{-1},q^{-2},\dotsc)$

where here the $q^{-k}$ are the ‘usual’ variables, i.e. the ones in which
$s_{\nu}$ is symmetric, not the power sum variables.

Writing $q=e^{\hbar}$ and using that
$f_{2}(\nu)=\sum_{\square\in\nu}c_{\square}$ gives

$\displaystyle\sum_{m=0}^{\infty}\sum_{\mu,\nu\vdash
m}\frac{\chi^{\nu}_{\mu}}{z_{\mu}}e^{(\frac{1}{w}+\frac{1}{2})\hbar
f_{2}(\nu)}\prod_{\square\in\nu}\frac{\beta w}{\varsigma(\hbar
h_{\square})}p_{\mu}$ $\displaystyle=\sum_{m=0}^{\infty}\sum_{\nu\vdash
m}s_{\nu}(\underline{p})e^{\sum_{\square\in\nu}\frac{\hbar}{w}c_{\square}}s_{\nu}\Big{(}\Big{\\{}\frac{\beta
w}{e^{\hbar(l+\frac{1}{2})}}\Big{\\}}_{l=0}^{\infty}\Big{)}\,.$

To revert to power-sum variables, we use that
$p_{l}\Big{(}\Big{\\{}\frac{\beta
w}{e^{\hbar(l+\frac{1}{2})}}\Big{\\}}_{l=0}^{\infty}\Big{)}=\frac{(\beta
w)^{k}}{\varsigma(\hbar k)}$, and inserting this in the definition of
$\hat{y}$ yields the result. ∎

If one were to relabel in stead by $\beta\to\beta\hbar$ and
$p_{k}\to\hbar^{-k}p_{k}$, i.e. just to naively introduce a parameter $\hbar$,
one would not end up in Family II, but in the extension of corollary 2.10.
These two choices are related by lemma 2.9, for $\lambda=\beta w$.

Zhou [Zho10] also explored this relation between triple Hodge integrals and
integrable hierarchies, extending it to the 2-Toda hierarchy and to certain
relative Gromov-Witten theories.

##  3\. KP hierarchy for intersection numbers

In this section, we will formulate and prove the main theorem, generalising
Kazarian’s method to the generating functions of intersection numbers coming
from hypergeometric tau-functions.

###  3.1. The change of variables

We will interpret any $X(z)$ defined by equations 21 and 23 as giving a change
of coordinates. For this, define a linear correspondence $\Theta$ between
power series in $X$ or $z$ on the one hand and linear series in $p$ or $q$ on
the other by

(54) $\displaystyle p_{k}$ $\displaystyle\leftrightarrow X^{k}\,,$
$\displaystyle q_{m}$ $\displaystyle\leftrightarrow z^{m}\,.$

This defines a change of coordinates as follows:

######  Definition 3.1.

We define a linear morphism between power series in $\\{p_{m}\\}_{m\geq 1}$
and $\\{q_{d}\\}_{d\geq 1}$ by

(55) $p_{k}(\underline{q})=\sum_{m=k}^{\infty}c_{k}^{m}q_{m}\qquad\text{with
$c_{k}^{m}$ given by}\qquad X^{k}=\sum_{m=k}^{\infty}c_{k}^{m}z^{m}\,.$

In order to make this change of coordinates and remain within the realm of
solutions of the KP hierarchy, we should flow along the action of the infinite
general linear algebra. Hence, we should find the infinitesimal flow
associated to this change. For this, we introduce a flow parameter $\beta$ by

(56) $X_{\beta}(z)\coloneqq\frac{1}{\beta}X(\beta z)=ze^{-\psi(y(\beta
z))}\,,$

such that $X_{0}(z)=z$ and $X_{1}(z)=X(z)$.

######  Lemma 3.2.

For $X_{\beta}(z)\coloneqq\frac{1}{\beta}X(\beta z)$, where
$X(z)=z+\mathcal{O}(z^{2})$, and with $Q(z)\coloneqq\frac{z}{X}\frac{dX}{dz}$,
the flow along $\beta$ of the function $X_{\beta}$ is given by

(57) $\frac{\partial X_{\beta}}{\partial\beta}=\Big{(}1-\frac{1}{Q(\beta
z)}\Big{)}\frac{z}{\beta}\frac{\partial X_{\beta}}{\partial
z}=\frac{1}{\beta}\big{(}Q(\beta z)-1\big{)}X_{\beta}\,.$

###### Proof.

By definition of $Q$, $X=\frac{z}{Q(z)}\frac{dX}{dz}$. Therefore,

$\displaystyle\frac{\partial X_{\beta}}{\partial\beta}$
$\displaystyle=\frac{\partial}{\partial\beta}\Big{(}\frac{1}{\beta}X(\beta
z)\Big{)}=\frac{z}{\beta}\frac{dX}{dz}\Big{|}_{z\to\beta
z}-\frac{1}{\beta^{2}}X(\beta z)$
$\displaystyle=\frac{z}{\beta}\frac{dX}{dz}\Big{|}_{z\to\beta
z}-\frac{1}{\beta^{2}}\Big{(}\frac{z}{Q(z)}\frac{dX}{dz}\Big{)}\Big{|}_{z\to\beta
z}=\Big{(}1-\frac{1}{Q(\beta z)}\Big{)}\frac{z}{\beta}\frac{\partial
X_{\beta}}{\partial z}\,.\qed$

We will use this with [Kaz09, Theorem 2.5], which uses the
$\widehat{\mathfrak{gl}(\infty)}$ action on $\tau$-functions:

######  Theorem 3.3 ([Kaz09]).

In the situation of a correspondence like equation 55, there is a quadratic
function $Q(\underline{p})$ such that the transformation sending an arbitrary
series $\Phi(\underline{p})$ to the series
$\Psi(\underline{q})=(\Phi+Q)\big{|}_{p\to p(\underline{q})}$ is an
automorphism of the KP hierarchy: it sends solutions to solutions.

The function $Q(\underline{p})$ is not unique.

######  Proposition 3.4.

In the general situation of theorem 2.6, without analytic assumptions, the
quadratic function for the change of variables of definition 3.1 can be taken
to be $-\frac{1}{2}\Theta(H_{0,2})$.

###### Proof.

Consider the more general linear correspondence $\Theta_{\beta}$ between power
series in $X_{\beta}$ or $z$ on the one hand and linear series in $p$ or $q$
on the other by

$\displaystyle p_{k}$ $\displaystyle\leftrightarrow X_{\beta}(z)^{k}\,,$
$\displaystyle q_{m}$ $\displaystyle\leftrightarrow z^{m}\,.$

This gives a linear morphism between power series in $\\{p_{m}\\}_{m\geq 1}$
and $\\{q_{d}\\}_{d\geq 1}$ by

$\displaystyle p_{k}(\beta;\underline{q})$
$\displaystyle=\sum_{m=k}^{\infty}c_{k}^{m}q_{m}$ with $c_{k}^{m}$ given by
$\displaystyle X_{\beta}^{k}$
$\displaystyle=\sum_{m=k}^{\infty}c_{k}^{m}z^{m}\,,$

such that $p_{k}(0;\underline{q})=q_{k}$.

Under $\Theta_{\beta}$, the operator $z^{m+1}\frac{\partial}{\partial z}$
transforms into $\sum_{k=1}^{\infty}kq_{m+k}\frac{\partial}{\partial q_{k}}$,
which is the differential part of $L_{m}(\underline{q})$. The polynomial part
of this operator is

$\frac{1}{2}\sum_{k=1}^{m-1}q_{k}q_{m-k}\,,$

which under the correspondence transforms into

$\frac{1}{2}\sum_{k=1}^{m-1}z_{1}^{k}z_{2}^{m-k}=\frac{1}{2}z_{1}z_{2}\frac{z_{1}^{m-1}-z_{2}^{m-1}}{z_{1}-z_{2}}=-\frac{1}{2}\frac{z_{1}^{m-1}-z_{2}^{m-1}}{z_{1}^{-1}-z_{2}^{-1}}\,.$

Therefore, the correction to be made to lemma 3.2 to obtain a KP-preserving
flow is found by the substitution $f(z)z\frac{\partial}{\partial
z}\to-\frac{1}{2}\frac{1}{z_{1}^{-1}-z_{2}^{-1}}\big{(}z_{1}^{-1}f(z_{1})-z_{2}^{-1}f(z_{2})\big{)}$
for a series $f(z)\in z\mathbb{C}\llbracket z\rrbracket$. Note that
$\frac{1}{\beta}\Big{(}1-\frac{1}{Q(\beta z)}\Big{)}$ satisfies these
requirements, and we find that the differential operator of lemma 3.2 needs to
be completed by

$-\frac{1}{2\beta(z_{1}^{-1}-z_{2}^{-1})}\bigg{(}z_{1}^{-1}\big{(}1-\frac{1}{Q(\beta
z_{1})}\big{)}-z_{2}^{-1}\big{(}1-\frac{1}{Q(\beta
z_{2})}\big{)}\bigg{)}=\frac{1}{2\beta(z_{1}^{-1}-z_{2}^{-1})}\bigg{(}\frac{1}{z_{1}Q(\beta
z_{1})}-\frac{1}{z_{2}Q(\beta z_{2})}\bigg{)}-\frac{1}{2\beta}\,.$

By a similar calculation as for lemma 3.2,

$\frac{\partial z}{\partial\beta}\Big{|}_{X\text{
const.}}=\frac{1}{\beta}\Big{(}\frac{1}{Q(\beta z)}-1\Big{)}z\,,$

from which it follows that

$\displaystyle-\frac{\partial H_{0,2}}{\partial\beta}\Big{|}_{X\text{
const.}}$
$\displaystyle=-\frac{\partial}{\partial\beta}\log\Big{(}\frac{z_{1}^{-1}-z_{2}^{-1}}{X_{1}^{-1}-X_{2}^{-1}}\Big{)}\Big{|}_{X\text{
const.}}$
$\displaystyle=\frac{1}{z_{1}^{-1}-z_{2}^{-1}}\Big{(}z_{1}^{-2}\frac{\partial
z_{1}}{\partial\beta}\Big{|}_{X\text{ const.}}-z_{2}^{-2}\frac{\partial
z_{2}}{\partial\beta}\Big{|}_{X\text{ const.}}\Big{)}$
$\displaystyle=\frac{1}{\beta(z_{1}^{-1}-z_{2}^{-1})}\Big{(}\frac{1}{z_{1}Q(\beta
z_{1})}-\frac{1}{z_{2}Q(\beta z_{2})}\Big{)}-\frac{1}{\beta}\,,$

which, up to a factor $2$, is exactly the polynomial correction needed.

From these calculations, we find that

$A\coloneqq\Big{(}1-\frac{1}{Q(\beta
z)}\Big{)}\frac{z}{\beta}\frac{\partial}{\partial z}-\frac{1}{2}\frac{\partial
H_{0,2}}{\partial\beta}\Big{|}_{X\text{ const.}}$

corresponds to a linear combination of $L_{m}$ under $\Theta_{\beta}$, and
hence preserves KP. Now consider a KP tau-function $\Phi(\underline{p})$ and
define the function
$Z(\beta,\underline{q})=\exp\big{(}\Phi(\underline{p}(\beta,\underline{q})-\frac{1}{2}\Theta(H_{0,2}))\big{)}$.
Then

(58) $\displaystyle\frac{\partial}{\partial\beta}Z$
$\displaystyle=\Big{(}\sum_{k=1}^{\infty}\frac{\partial
p_{k}(\beta,\underline{q})}{\partial\beta}\frac{\partial}{\partial
p_{k}}-\frac{1}{2}\Theta\Big{(}\frac{\partial
H_{0,2}}{\partial\beta}\Big{|}_{X\text{ const.}}\Big{)}\Big{)}Z$ (59)
$\displaystyle=\Theta\Big{(}\Big{(}1-\frac{1}{Q(\beta
z)}\Big{)}\frac{z}{\beta}\frac{\partial}{\partial z}-\frac{1}{2}\frac{\partial
H_{0,2}}{\partial\beta}\Big{|}_{X\text{ const.}}\Big{)}Z(\beta)$ (60)
$\displaystyle=\Theta(A)\,Z(\beta)$

As $Z(0)=Z$, and $\Theta(A)$ preserves $\tau$-functions of KP, this
automorphism does indeed preserve solutions. ∎

######  Corollary 3.5.

For $Z(\underline{p})$ defined by equation 22,
$Z(\underline{p})\exp\big{(}-\Theta(\hbar^{-1}H_{0,1}+\frac{1}{2}H_{0,2})\big{)}\big{|}_{p\to
p(\underline{q})}$ is also a KP tau-function, whose logarithm does not contain
unstable terms.

###### Proof.

As all equations in the KP hierarchy only contain at least second derivatives
of $F$, addition of a linear term $-\Theta(\hbar^{-1}H_{0,1})$ preserves
solutions. By proposition 3.4, subtracting the $(0,2)$ term and changing
$p\mapsto p(\underline{q})$ is an automorphism as well. ∎

######  Corollary 3.6.

In case $X(z)$ is a Möbius transformation with the shape of equation 23, i.e.
$X(z)=\frac{az}{1+bz}$ (taking into account remark 2.7), this quadratic
function can be taken to be $0$.

###### Proof.

By direct calculation,

$H_{0,2}(z_{1},z_{2})=\log\Big{(}\frac{z_{1}^{-1}-z_{2}^{-1}}{X(z_{1})^{-1}-X(z_{2})^{-1}}\Big{)}=\log
a\,.$

Comparing this with the proof of proposition 3.4, the quadratic correction is
needed to complete the operator $A$, which only depends on $\frac{\partial
H_{0,2}}{\partial\beta}$. As this vanishes in the present case, we may as well
omit the entire correction. ∎

######  Remark 3.7.

The usual $B$-function of topological recursion,

(61)
$B(z_{1},z_{2})=\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}}=d_{1}d_{2}\log(z_{1}^{-1}-z_{2}^{-1})\,,$

is invariant under all Möbius transformations, so $d_{1}d_{2}H_{0,2}$ vanishes
if $X$ is any Möbius transformation. However, this is not the case for
$H_{0,2}$ itself: it is invariant under a one-dimensional subgroup, changes by
a constant under the two-dimensional subgroup above, but under other Möbius
transformations also changes by addition of terms like $\log z_{i}$.

Viewed another way, these more general Möbius transformations would take us
out of the realm of formal power series in $z$. However, in a space of
functions, a shift $z\mapsto z+c$ does preserve the KP hierarchy, so if the
formal power series converges to a function on a large enough domain, this
shift does preserve KP. This argument is essentially taken from [Kaz09,
Section 8]. In particular, under the ‘natural analytic assumptions’ of
[BDKS20a, section 1.3], i.e. the assumptions in the second part of theorem
2.6, the $H_{g,n}$ do extend to rational functions on all of $\mathbb{P}^{1}$,
so this shift is well-defined.

###  3.2. KP for intersection numbers

Now we will restrict to the cases where topological recursion, and hence
theorem 2.12, can be used, in order to relate to intersection numbers. In this
case, the following holds from equation 36.

(62)
$F(\underline{p})=\bigg{(}\hbar^{-1}H_{0,1}+\frac{1}{2}H_{0,2}+\\!\\!\\!\sum_{2g-2+n>0}\\!\\!\frac{\hbar^{2g-2+n}}{n!}\\!\\!\\!\\!\sum_{j_{1},\dotsc,j_{n}\in
J}\int_{\overline{\mathcal{M}}_{g,n}}\\!\\!\\!\Omega_{g,n}(e_{j_{1}}\otimes\dotsb\otimes
e_{j_{n}})\prod_{i=1}^{n}\sum_{k_{i}=0}^{\infty}\psi_{i}^{k_{i}}\xi_{k_{i}}^{j_{i}}(z_{i})\bigg{)}\bigg{|}_{X_{i}^{k_{i}}\to
p_{k_{i}}},$

if we define
$\xi_{k}^{j}(z)\coloneqq\int_{z^{\prime}=r_{j}}^{z}d\xi^{j}_{k}(z^{\prime})$,
noting that due to the shape of the $H_{g,n}$ in equations 24 and 25 and
$X(z)$ in equation 23, the $H_{g,n}$ have no constant terms in $z_{i}$.

Under the correspondence $p_{k}\leftrightarrow X^{k}$, $q_{m}\leftrightarrow
z^{m}$ of definition 3.1, we define $T_{k}^{j}$ by

(63)
$T^{j}_{k}(\underline{p})\leftrightarrow\frac{1}{dx}d\xi_{k}^{j}(z)=D^{k+1}\frac{z^{j+1}}{j+1}\,,$

with $D$ as in equation 23. Explicitly,

(64) $T_{-1}^{j}=\frac{1}{j+1}q_{j+1}\,,\qquad
T_{k+1}^{j}=\sum_{m=1}^{\infty}\sum_{l=0}^{\infty}m\mathcal{T}_{l}q_{m+l}\frac{\partial}{\partial
q_{m}}T_{k}^{j}\,,\quad\text{with $\mathcal{T}_{l}$ given by}\quad
Q(z)^{-1}=\sum_{l=0}^{\infty}\mathcal{T}_{l}z^{l}\,.$

Note that, even though the recursion operator for the $T_{k}^{j}$ may have
infinitely many terms, its alternate description via equation 63 ensures they
are well-defined in $\Lambda$.

Therefore, by definition,

(65)
$\displaystyle\Theta\big{(}\hbar^{-1}H_{0,1}+\frac{1}{2}H_{0,2}\big{)}+G_{\Omega}(T(\underline{p}))$
$\displaystyle\quad=\Theta\big{(}\hbar^{-1}H_{0,1}+\frac{1}{2}H_{0,2}\big{)}+\\!\\!\sum_{2g-2+n>0}\\!\\!\frac{\hbar^{2g-2+n}}{n!}\\!\\!\\!\\!\sum_{j_{1},\dotsc,j_{n}\in
J}\int_{\overline{\mathcal{M}}_{g,n}}\\!\\!\\!\Omega_{g,n}(e_{j_{1}}\otimes\dotsb\otimes
e_{j_{n}})\prod_{i=1}^{n}\sum_{k_{i}=0}^{\infty}\psi_{i}^{k_{i}}T_{k_{i}}^{j_{i}}(\underline{p})$

is the logarithm of a tau-function, where $\\{e_{j}\\}$ is the dual basis to
the basis $\\{d\xi_{0}^{j}\\}$ of $V^{*}$.

######  Proposition 3.8.

Suppose the pair of functions $(\hat{\psi},\hat{y})$ lies in family I or II of
theorem 2.6, and let $\Omega$ be the cohomological field theory associated to
the related topological recursion via theorem 2.12. Then

(66) $Z_{\Omega}(\underline{q})=\exp(G_{\Omega}(T(p(\underline{q}))))$

is a KP tau-function.

###### Proof.

Apply corollary 3.5 to the exponent of equation 65. ∎

###  3.3. Finiteness of the transformation

The operator $A$ in the proof of proposition 3.4 corresponds to a finite sum
of $L_{m}$ if and only if $Q(z)^{-1}$ is a polynomial in $z$. As this case
seems particularly nice, we will investigate it here. Note that this condition
dependent on the parameter $z$ on the spectral curve, cf. the difference
between example 2.18 and section 2.3.

Write $P(z)=Q(z)^{-1}$ for this polynomial, and write $r+1$ for its degree.
From equation 23, it follows that $P(0)=1$, so we may write

(67) $P(z)=\prod_{j=1}^{r+1}(1-c_{j}z)\,.$

We immediately see that $dx(z)=\frac{dz}{zP(z)}$, and hence the spectral curve
has a unique ramification point, $\infty$, of ramification index $r$. This is
also the rank of the associated Frobenius algebra. But we can do better. By
calculating the residues in $v$ of

(68) $\frac{v^{r+1}dv}{(1-vz)\prod_{k=1}^{r+1}(v-c_{k})}\,$

and using that they sum to zero, one can check that (if all $c_{j}$ are
distinct)222If some $c_{j}$ coincide, the residue argument still holds, but
the result changes.

(69)
$\frac{dx}{dz}=\frac{1}{z}+\sum_{j=1}^{r+1}\frac{c_{j}^{r+1}}{\prod_{k\neq
j}(c_{j}-c_{k})}\frac{1}{1-c_{j}z}\,,$

from which we see that

(70) $x(z)=\log z-\sum_{j=1}^{r+1}\prod_{k\neq
j}\big{(}1-\frac{c_{k}}{c_{j}}\big{)}^{-1}\log(1-c_{j}z)\,.$

If $r=0$, $dx$ has two (simple) poles, and hence no zeroes. In fact, in this
case, $X$ is a Möbius transformation.

If $r=1$, this recovers the triple Hodge curve, studied in section 3.4 below.

If $r>1$, the Frobenius algebra is not semi-simple: it seems to be a
deformation of the algebra corresponding to Witten’s $r+1$-spin cohomological
field theory, which is given by $x=y^{r+1}$, cf. [Wit93, DNOPS18, BCEG21].
This class fits in Alexandrov’s theory of the deformed generalised Kontsevich
model [Ale21a]: it seems like it is a complementary subspace of the polynomial
deformations of the Witten $r+1$-spin theory.

Interestingly, except for special choices of $c_{j}$, these cases seem not to
be covered in the two families in theorem 2.6 for which topological recursion
is proved (for any choice of $y$). Even the $r=1$ case does not fall in that
scope, unless $\frac{c_{1}}{c_{2}}\in\mathbb{Q}$.

###  3.4. The case of triple Hodge integrals

Let us now consider the special case of triple Hodge integrals. The approach
taken in this section overlaps with the previous results, but is also slightly
different in details, adapted to this specific problem. For example, we do not
use the formal parameter $\hbar$, but make another convenient choice. In this
case, the ELSV-type formula is completely explicit, and there is no need to
take the detour via topological recursion.

The coordinate change we want to perform is inspired by the Mariño-Vafa
formula.

(71)
$\begin{split}F\big{(}w,\beta;\underline{p}\big{)}&=\log\bigg{(}\sum_{m=0}^{\infty}\sum_{\mu,\nu\vdash
m}\frac{\chi^{\nu}_{\mu}}{z_{\mu}}e^{(1+\frac{w}{2})\beta
f_{2}(\nu)}\prod_{\square\in\nu}\frac{\beta w}{\varsigma(\beta
wh_{\square})}p_{\mu}\bigg{)}\\\
&=\sum_{\mu}\sum_{g=0}^{\infty}\frac{(w+1)^{g+n-1}}{|\operatorname{Aut}\mu|}\prod_{i=1}^{n}\frac{\prod_{j=1}^{\mu_{i}-1}(\mu_{i}+jw)}{(\mu_{i}-1)!}\int_{\overline{\mathcal{M}}_{g,n}}\\!\\!\frac{\Lambda(-1)\Lambda(-w)\Lambda\big{(}\frac{w}{w+1}\big{)}}{\prod_{i=1}^{n}(1-\mu_{i}\psi_{i})}\beta^{2g-2+n+|\mu|}p_{\mu}\,.\end{split}$

As
$2g-2+n+|\mu|=\frac{2}{3}\dim\overline{\mathcal{M}}_{g,n}+\sum_{i=1}^{n}(\mu_{i}+\frac{1}{3})$
and
$g+n-1=\frac{1}{3}\dim\overline{\mathcal{M}}_{g,n}+\sum_{i=1}^{n}\frac{2}{3}$,
we get after rewriting $u\coloneqq\beta^{\frac{1}{3}}(w+1)^{\frac{1}{6}}$

(72)
$\begin{split}F\big{(}w,\beta;\underline{p}\big{)}&=\sum_{\mu}\frac{1}{|\operatorname{Aut}\mu|}\sum_{g=0}^{\infty}\prod_{i=1}^{n}u^{4}\frac{\prod_{j=1}^{\mu_{i}-1}(\mu_{i}+jw)\beta}{(\mu_{i}-1)!}\int_{\overline{\mathcal{M}}_{g,n}}\frac{\Lambda(-u^{2})\Lambda(-u^{2}w)\Lambda\big{(}\frac{u^{2}w}{w+1}\big{)}}{\prod_{i=1}^{n}(1-\mu_{i}u^{2}\psi_{i})}p_{\mu}\\\
&=\sum_{g=0}^{\infty}\sum_{n=1}^{\infty}\frac{1}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda(-u^{2})\Lambda(-u^{2}w)\Lambda\Big{(}\frac{u^{2}w}{w+1}\Big{)}\prod_{i=1}^{n}\sum_{d=0}^{\infty}T_{d}(\underline{p})\psi_{i}^{d}\\\
&=G_{\textup{TH}}\Big{(}-u^{2},-u^{2}w,\frac{u^{2}w}{w+1};T(\underline{p})\Big{)}+H_{0,1}+\frac{1}{2}H_{0,2}\,,\end{split}$

where

(73)
$T_{d}(\underline{p})\coloneqq\sum_{m=1}^{\infty}\frac{\prod_{j=1}^{m-1}(m+jw)}{(m-1)!}m^{d}u^{2d+4}\beta^{m-1}p_{m}\,.$

Hence, our goal is to show that this change of variables and addition of the
unstable terms preserves solutions of the KP hierarchy.

######  Lemma 3.9.

The following two expressions are inverse to each other:

(74) $X(z)=\frac{z}{1+(w+1)\beta z}\bigg{(}\frac{1+\beta z}{1+(w+1)\beta
z}\bigg{)}^{\frac{1}{w}}\,;\qquad
z(X)=\sum_{m=1}^{\infty}\frac{\prod_{j=1}^{m-1}(m+jw)}{(m-1)!}\beta^{m-1}X^{m}\,.$

###### Proof.

This can be proved by a residue calculation. Start from the formula for $X(z)$
with $\beta=1$ and write $z(X)=\sum_{m=1}^{\infty}C_{m}X^{m}$. Then
$C_{m}=\operatorname*{Res}_{X=0}z\,X^{-m}\frac{dX}{X}$, and

$\displaystyle\frac{dX}{X}$
$\displaystyle=\frac{dz}{z}+\frac{d(1+z)^{\frac{1}{w}}}{(1+z)^{\frac{1}{w}}}+\frac{d(1+(w+1)z)^{-\frac{w+1}{w}})}{(1+(w+1)z)^{-\frac{w+1}{w}}}$
$\displaystyle=\frac{dz}{z}+\frac{1}{w}\frac{dz}{1+z}-\frac{(w+1)^{2}}{w}\frac{dz}{1+(w+1)z}$
$\displaystyle=\frac{dz}{z(1+z)(1+(w+1)z)}\,.$

Therefore,

$\displaystyle C_{m}$
$\displaystyle=\operatorname*{Res}_{X=0}z\,X^{-m}\frac{dz}{z(1+z)(1+(w+1)z)}$
$\displaystyle=\operatorname*{Res}_{z=0}z^{-m}(1+z)^{-\frac{m}{w}-1}(1+(w+1)z)^{m\frac{w+1}{w}-1}dz$
$\displaystyle=\operatorname*{Res}_{z=0}z^{-m}\sum_{k=0}^{\infty}\frac{\prod_{i=0}^{k-1}(-\frac{m}{w}-1-i)}{k!}z^{k}\sum_{l=0}^{\infty}\frac{\prod_{j=0}^{l-1}(m\frac{w+1}{w}-1-j)}{l!}(w+1)^{l}z^{l}dz$
$\displaystyle=\operatorname*{Res}_{z=0}z^{-m}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{k}(\frac{m}{w}+i)}{k!}(-1)^{k}z^{k}\sum_{l=0}^{\infty}\frac{\prod_{j=m-l}^{m-1}(\frac{m}{w}+j)}{l!}(w+1)^{l}z^{l}dz$
$\displaystyle=\sum_{k=0}^{m-1}\frac{\prod_{i=1}^{k}(\frac{m}{w}+i)}{k!}\frac{\prod_{j=k+1}^{m-1}(\frac{m}{w}+j)}{(m-k-1)!}(-1)^{k}(w+1)^{m-k-1}$
$\displaystyle=\frac{\prod_{j=1}^{m-1}(\frac{m}{w}+j)}{(m-1)!}\sum_{k=0}^{m-1}\binom{m-1}{k}(-1)^{k}(w+1)^{m-k-1}$
$\displaystyle=\frac{\prod_{j=1}^{m-1}(\frac{m}{w}+j)}{(m-1)!}w^{m-1}=\frac{\prod_{j=1}^{m-1}(m+jw)}{(m-1)!}\,.$

Finally, $\beta$ can be introduced in this formula by scaling $z\to\beta z$,
$X\to\beta X$. ∎

######  Corollary 3.10.

The expressions for $X(z)$ in lemma 3.9 and lemma 2.26 are related by a Möbius
transformation

(75) $z\mapsto\frac{z}{1+(w+1)\beta z}\,.$

Hence, by corollary 3.6, they require the same correction term for their
induced linear change of variables.

We see that in this particular case we may obtain the function $X$ in two
different ways: from the general theory of theorem 2.6, or from the specific
shape of the Mariño-Vafa formula, theorem 2.23. In fact, the second choice is
nothing but choosing the spectral curve coordinate $z$ to equal $\xi$ (which
is unique in this case), or in other words $T_{0}=q_{1}$.

######  Remark 3.11.

Under the correspondence of theorem 2.12, the rank of the cohomological field
theory corresponds to the number of zeroes of $dx$, counted with
multiplicities. So for rank one, $dx$ can only have one zero, and hence must
have three poles. By Möbius transformation, we may place the zero at infinity,
and two of the poles at $0$ and $-1$, from which we find that $dx$ must
correspond to the $\frac{dX}{X}$ found in the proof of lemma 3.9. This may
explain in part why Alexandrov [Ale21] finds only the triple Hodge CohFT in
the intersection of the orbits of the Givental and Heisenberg-Virasoro groups.
However, $dx$ is not the only datum of a spectral curve, and while
$\mathbb{P}^{1}$ is rigid and has a unique $B$, it is not clear why there is
no freedom in the choice of $dy$.

######  Lemma 3.12.

The series $X(z)$ from lemma 3.9 satisfies the differential equation

(76) $\frac{\partial X}{\partial\beta}(z)=-\big{(}(w+2)z+(w+1)\beta
z^{2})z\frac{\partial X}{\partial z}(z)\,.$

###### Proof.

For $X(z)=\frac{z}{1+(w+1)z}\Big{(}\frac{1+z}{1+(w+1)z}\Big{)}^{\frac{1}{w}}$,
we get $Q(z)^{-1}=(1+z)(1+(w+1)z)$, which using lemma 3.2 immediately yields
the result. ∎

We use this lemma in combination with the linear correspondence of definition
3.1, slightly adapted as follows: define a linear correspondence $\Theta$
between power series in $X$ or $z$ on the one hand and linear series in $p$ or
$\tilde{q}$ on the other by

(77) $\displaystyle p_{k}$ $\displaystyle\leftrightarrow X^{k}\,,$
$\displaystyle\tilde{q}_{m}$ $\displaystyle\leftrightarrow z^{m}\,.$

######  Definition 3.13.

We define a linear morphism between power series in $\\{p_{m}\\}_{m\geq 1}$
and $\\{\tilde{q}_{d}\\}_{d\geq 1}$ by

(78) $\displaystyle p_{k}(\tilde{\underline{q}})$
$\displaystyle=\sum_{m=k}^{\infty}c_{k}^{m}\tilde{q}_{m}$ with $c_{k}^{m}$
given by $\displaystyle X^{k}$
$\displaystyle=\sum_{m=k}^{\infty}c_{k}^{m}z^{m}\,.$

Under the correspondence $p_{k}\leftrightarrow X^{k}$,
$\tilde{q}_{m}\leftrightarrow z^{m}$, we have

(79) $\displaystyle T_{d}(\underline{p})$
$\displaystyle\leftrightarrow(u^{2}D)^{d}u^{4}z\,;$ $\displaystyle D$
$\displaystyle\coloneqq X\frac{\partial}{\partial X}=\big{(}1+\beta
z\big{)}\big{(}1+(w+1)\beta z\big{)}z\frac{\partial}{\partial z}\,.$

In terms of $\tilde{q}$-variables, this gives

(80) $\displaystyle T_{d}$
$\displaystyle=u^{2}\sum_{m=1}^{\infty}m\big{(}\tilde{q}_{m}+(w+2)\beta\tilde{q}_{m+1}+(w+1)\beta^{2}\tilde{q}_{m+2}\big{)}\frac{\partial}{\partial\tilde{q}_{m}}T_{d-1}\,;$
$\displaystyle T_{0}$ $\displaystyle=u^{4}\tilde{q}_{1}\,.$

If we write $q_{m}\coloneqq u^{4m}\tilde{q}_{m}$, and using
$\beta=\frac{u^{3}}{\sqrt{w+1}}$, this becomes

(81) $\displaystyle T_{d}$
$\displaystyle=\sum_{m=1}^{\infty}m\Big{(}u^{4m+2}\tilde{q}_{m}+(w+2)\frac{u^{3}}{\sqrt{w+1}}u^{4m+2}\tilde{q}_{m+1}+u^{6}u^{4m+2}\tilde{q}_{m+2}\Big{)}\frac{1}{u^{4m}}\frac{\partial}{\partial\tilde{q}_{m}}T_{d-1}$
(82)
$\displaystyle=\sum_{m=1}^{\infty}m\Big{(}u^{2}q_{m}+\frac{u(w+2)}{\sqrt{w+1}}q_{m+1}+q_{m+2}\Big{)}\frac{\partial}{\partial
q_{m}}T_{d-1}\,;$ (83) $\displaystyle T_{0}$ $\displaystyle=q_{1}\,.$

This is exactly the definition given in theorem 2.20.

######  Corollary 3.14.

For $X(z)=\frac{z}{1+(w+1)\beta z}\big{(}\frac{1+\beta z}{1+(w+1)\beta
z}\big{)}^{\frac{1}{w}}$, the quadratic correction of theorem 3.3 is
$Q=-\frac{1}{2}\Theta(H_{0,2})$.

###### Proof.

The function $X(z)$ satisfies the conditions of theorem 2.6, so we may apply
proposition 3.4. ∎

Now we are ready to prove the main result on KP integrability of triple Hodge
integrals.

###### Proof of theorem 2.20.

By lemma 2.26, $F$ is a solution of the KP hierarchy in the variables
$t_{k}\coloneqq\frac{p_{k}}{k}$. In the same way as for proposition 3.8, now
using corollaries 3.14 and LABEL:THHurwitztoTHgen,

(84)
$G_{\textup{TH}}\Big{(}-u^{2},-wu^{2},\frac{wu^{2}}{w+1};\\{T_{d}(p(\tilde{q}))\\}\Big{)}$

is a solution of the KP hierarchy in the variables $\frac{\tilde{q}_{m}}{m}$.
As the KP hierarchy is quasi-homogeneous, rescaling $\tilde{q}_{m}\to q_{m}$
preserves solutions. This completes the proof. ∎

######  Remark 3.15.

The result in this subsection do hold for $w=-1$ (ignoring powers of $u$), but
in this specific case $X(z)$ is a Möbius transformation, so it reduces to the
setting of corollary 3.6. From another point of view, in this case the change
of coordinates equation 55 is an isomorphism, whereas it gives a half-
dimensional subspace in all other cases. Equations for this half-dimensional
space, in the linear Hodge case, were found in [Ale15], cf. also [GW17] for a
reformulation. These can be viewed as a deformation of the reduction from KP
to KdV. Similar equations should exist for triple Hodge integrals as well, but
clearly none of this works for $w=-1$.

In light of section 3.3, one may expect a deformation of the reduction from KP
to $r$-KdV or $r$-Gelfand-Dickey for the families found there.

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