# Polynomial relations among kappa classes on the moduli space of curves

Maxim Kazarian  National Research University Higher School of Economics,
Skolkovo Institute of Science and Technology<EMAIL_ADDRESS>and  Paul
Norbury  School of Mathematics and Statistics, University of Melbourne, VIC
3010, Australia<EMAIL_ADDRESS>
###### Abstract.

We construct an infinite collection of universal—independent of
$(g,n)$—polynomials in the Miller-Morita-Mumford classes $\kappa_{m}\in
H^{2m}(\overline{\cal M}_{g,n},\mathbb{Q})$, defined over the moduli space of
genus $g$ stable curves with $n$ labeled points. We conjecture vanishing of
these polynomials in a range depending on $g$ and $n$.

###### 2020 Mathematics Subject Classification:

14H10; 14D23; 32G15

###### Contents

  1. 1 Introduction
    1. 1.1 The construction
  2. 2 Vanishing of intersection numbers
    1. 2.1 Topological recursion for the $\kappa$-potential
    2. 2.2 Regularity of potentials
      1. 2.2.1 Intersections with the class $\mathbb{J}$.
      2. 2.2.2 Intersections with $\kappa$ classes
  3. 3 Relation to the Brézin-Gross-Witten tau function
    1. 3.1 KdV tau functions
    2. 3.2 BGW tau function from the limit of intersection numbers
  4. 4 Cohomological field theories
    1. 4.1 Push-forward and pull-back properties
    2. 4.2 CohFTs without unit
  5. 5 Geometric constructions
    1. 5.1 Hodge CohFT
    2. 5.2 Spin CohFT
    3. 5.3 Monotone Hurwitz numbers

##  1\. Introduction

Let $\overline{\cal M}_{g,n}$ be the moduli space of genus $g$ stable
curves—curves with only nodal singularities and finite automorphism group—with
$n$ labeled points disjoint from nodes, for any
$g,n\in\mathbb{N}=\\{0,1,2,...\\}$ satisfying $2g-2+n>0$. Define the forgetful
map $\pi:\overline{\cal M}_{g,n+1}\to\overline{\cal M}_{g,n}$ that forgets the
last point, and define $\omega_{\pi}$ to be the relative dualising sheaf of
the map $\pi$. There are natural sections $p_{i}:\overline{\cal
M}_{g,n}\to\overline{\cal M}_{g,n+1}$ of the map $\pi$ for each $i=1,...,n$.
Define $L_{i}=p_{i}^{*}\omega_{\pi}$ to be the line bundle
$L_{i}\to\overline{\cal M}_{g,n}$ with fibre above $[(C,p_{1},\ldots,p_{n})]$
given by $T_{p_{i}}^{*}C$ and $\psi_{i}=c_{1}(L_{i})\in H^{2}(\overline{\cal
M}_{g,n},\mathbb{Q})$ to be its first Chern class. Define the Hodge bundle
$\mathbb{E}_{g}=\pi_{*}\omega_{\pi}$ which is a rank $g$ bundle over
$\overline{\cal M}_{g,n}$ and $\lambda_{i}=c_{i}(\mathbb{E}_{g})\in
H^{2i}(\overline{\cal M}_{g,n},\mathbb{Q})$ to be its $i$th Chern class for
$i=1,...,g$. Define the Miller-Morita-Mumford classes
$\kappa_{m}=\pi_{*}\psi^{m+1}_{n+1}\in H^{2m}(\overline{\cal
M}_{g,n},\mathbb{Q})$ where $\pi$ is the forgetful map $\overline{\cal
M}_{g,n+1}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\overline{\cal
M}_{g,n}$. We will refer to $\psi_{i}$ and $\kappa_{m}$ as $\psi$ classes,
respectively $\kappa$ classes.

###  1.1. The construction

Consider $\\{s_{i}\in\mathbb{Q}\mid i>0\\}$ defined by

$\exp\left(-\sum_{i>0}s_{i}t^{i}\right)=\sum_{k=0}^{\infty}(-1)^{k}(2k+1)!!t^{k}$

and $\\{\sigma_{i}\in\mathbb{Q}\mid i>0\\}$ defined by

$\exp\left(-\sum_{i>0}\sigma_{i}t^{i}\right)=\sum_{k=0}^{\infty}(-1)^{k}k!t^{k}.$

Define

$\mathbb{K}=K_{0}+K_{1}+K_{2}+...=\exp(\sum s_{i}\kappa_{i})\in
H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$

and

$\mathbb{J}=J_{0}+J_{1}+J_{2}+...=\exp(\sum\sigma_{i}\kappa_{i})\in
H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$

where $K_{m},J_{m}\in H^{2m}(\overline{\cal M}_{g,n},\mathbb{Q})$. This
defines two sequences of homogeneous polynomials in $\kappa_{m}$. The first
few polynomials are given by:

$K_{1}=3\kappa_{1},\quad K_{2}=\frac{3}{2}(3\kappa_{1}^{2}-7\kappa_{2}),\quad
K_{3}=\frac{3}{2}(3\kappa_{1}^{3}-21\kappa_{1}\kappa_{2}+46\kappa_{3})$
$K_{4}=\frac{9}{8}(3\kappa_{1}^{4}-42\kappa_{1}^{2}\kappa_{2}+49\kappa_{2}^{2}+184\kappa_{1}\kappa_{3}-562\kappa_{4})$

and

$J_{1}=\kappa_{1},\quad J_{2}=\frac{1}{2}(\kappa_{1}^{2}-3\kappa_{2}),\quad
J_{3}=\frac{1}{6}(\kappa_{1}^{3}-9\kappa_{1}\kappa_{2}+26\kappa_{3})$
$J_{4}=\frac{1}{24}(\kappa_{1}^{4}-18\kappa_{1}^{2}\kappa_{2}+27\kappa_{2}^{2}+104\kappa_{1}\kappa_{3}-426\kappa_{4}).$

######  Conjecture 1.

The following polynomial relations among the $\kappa_{j}$ hold:

(1) $\displaystyle K_{m}(\kappa_{1},...,\kappa_{m})$
$\displaystyle=0\quad\text{for}\quad
m>2g-2+n\quad\text{except}\quad(m,n)=(3g-3,0);$ (4) $\displaystyle
J_{m}(\kappa_{1},...,\kappa_{m})$
$\displaystyle=0\quad\text{for}\quad\left\\{\begin{array}[]{ll}m>2g-2+n&\\\
m=2g-2+n,&n>1.\end{array}\right.$

The conjecture is trivially true when $m>3g-3+n$, and hence says nothing about
$g=0$. Evidence for the conjecture is given by the following theorem.

######  Theorem 1.

For any polynomial $P(\psi_{1},...,\psi_{n})\in
H^{2m}(\overline{\mathcal{M}}_{g,n})$ with $n>0$ and $m<g-1$,

$\int_{\overline{\cal M}_{g,n}}\mathbb{K}\cdot P(\psi_{1},...\psi_{n})=0.$

Theorem 1 is the first step towards a proof of a weak version of Conjecture 1
involving the tautological ring $RH^{m}(\overline{\cal M}_{g,n})\subset
H^{2m}(\overline{\cal M}_{g,n},\mathbb{Q})$, [16]—see Corollary 2.8 in Section
2.1. An analogous statement for $\int_{\overline{\cal M}_{g,n}}\mathbb{J}\cdot
P(\psi_{1},...\psi_{n})$ is reduced to a purely combinatorial statement,
without reference to the moduli space of stable curves, in Section 2.2.1.

The construction of the polynomials in $\kappa$ classes $J_{m}$ and $K_{m}$
resembles the following construction of polynomials in $\kappa$ classes used
to produce polynomial relations among $\kappa$ classes over the moduli space
of curves of compact type $\cal M_{g,n}^{\text{ct}}$. A stable curve is of
compact type if its dual graph is a tree, and we have:

$\cal M_{g,n}\subset\cal M_{g,n}^{\text{ct}}\subset\overline{\cal M}_{g,n}.$

In [32], Pandharipande found relations among $\kappa$ classes over the moduli
space of curves of compact type $\cal M_{g,n}^{c}$. Pixton [34] found explicit
formulae for these relations in a form strikingly similar to the construction
of $\mathbb{J}$ and $\mathbb{K}$. Define $\\{p_{i}\in\mathbb{Q}\mid i>0\\}$ by

$\exp\left(-\sum_{i>0}p_{i}t^{i}\right)=\sum_{k=0}^{\infty}(-1)^{k}(2k-1)!!t^{k}$

where $(-1)!!=1$. Define

$\mathbb{P}=P_{0}+P_{1}+P_{2}+...=\exp(\sum p_{i}\kappa_{i})\in H^{*}(\cal
M_{g,n}^{\text{ct}},\mathbb{Q})$

where $P_{m}\in H^{2m}(\overline{\cal M}_{g,n},\mathbb{Q})$.

######  Theorem 2 (Pixton 2013).

$P_{m}=0\in H^{2m}(\cal M_{g,n}^{\text{ct}},\mathbb{Q}),\quad 2m>2g-2+n.$

Theorem 2 appears in [34, Section 3.4], see also [31, Section 3.5], as part of
a more general collection of relations.

Due to the natural restriction properties of the classes $\mathbb{K}$ and
$\mathbb{J}$ to lower dimensional strata in the moduli space of stable curves,
the conjectural vanishing of the polynomials $K_{m}$ and $J_{m}$ can be
reduced to properties of the (non-vanishing) polynomials $K_{2g-2+n}$ and
$J_{2g-2+n}$. This is proven in Propositions 5.2 and 5.3 and motivates
Conjectures 2, 3 and 4 below.

The following two conjectures seek to characterise the polynomials
$J_{2g-2+n}$ and $K_{2g-2+n}$ in such a way to give a possible mechanism for
the proof of Conjecture 1. As usual $\kappa_{i}$, hence also $J_{m}$ and
$K_{m}$, represents a class in
$H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ for any $(g,n)$. If we write
the subscript $m=m(g,n)$ as a function of $g$ and $n$, we have fixed $g$ and
$n$. In particular, $J_{2g-2+n},K_{2g-2+n}\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$.

######  Conjecture 2.

$J_{2g-2+n}=\left\\{\begin{array}[]{cc}(-1)^{g}\lambda_{g-2}\lambda_{g}\in
H^{*}(\overline{\mathcal{M}}_{g},\mathbb{Q})&n=0\\\
(-1)^{g-1}\lambda_{g-1}\lambda_{g}\in
H^{*}(\overline{\mathcal{M}}_{g,1},\mathbb{Q})&n=1\end{array}\right.$

Conjecture 2 has been checked up to $g=4$ using the Sage package admcycles [8]
which is based on Pixton’s relations [33, 34].

######  Conjecture 3.

For $g>1$

$K_{2g-2+n}=\left(\prod_{i=1}^{n}\psi_{i}\right)\pi^{*}K_{2g-2}$

and $K_{2g-2}$ is the unique degree $2g-2$ polynomial up to scale with the
property that all of its pullbacks times the product of $\psi$ classes are
polynomials in $\kappa$ classes.

Conjecture 3 is compatible with the push-forward property

$\pi_{*}K_{2g-2+n+1}=(2g-2+n)K_{2g-2+n}$

proven in Section 4.1. Moreover, an easy consequence of the push-forward
property is the following proven $g=1$ version of Conjecture 3. For $K_{1}\in
H^{*}(\overline{\cal M}_{1,1},\mathbb{Q})$,

$K_{n}=\left(\prod_{i=2}^{n}\psi_{i}\right)\pi^{*}K_{1}\in
H^{*}(\overline{\cal M}_{1,n},\mathbb{Q}).$

The uniqueness part of Conjecture 3 can be restated as follows. For $g>1$, a
homogeneous degree $2g-2$ polynomial $P(\kappa_{1},...,\kappa_{2g-2})$
satisfies the property that $\left(\prod_{i=1}^{n}\psi_{i}\right)\pi^{*}P\in
H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$ is a polynomial in $\kappa$ classes
for each $n>0$ iff $P(\kappa_{1},...,\kappa_{2g-2})$ is a scalar multiple of
$K_{2g-2}$.

Conjectures 2 and 3 imply Conjecture 1 via vanishing of well chosen
restriction maps—see Section 5, Propositions 5.2 and 5.3. Thus, expressions
for $\lambda_{g-2}\lambda_{g}$ and $\lambda_{g-1}\lambda_{g}$ in terms of
known polynomials in the $\kappa$ classes, such as

$\lambda_{2}\lambda_{3}=J_{5}=\tfrac{1}{120}\kappa_{1}^{5}-\tfrac{1}{4}\kappa_{1}^{3}\kappa_{2}+\tfrac{9}{8}\kappa_{1}\kappa_{2}^{2}+\tfrac{13}{6}\kappa_{1}^{2}\kappa_{3}-\tfrac{13}{2}\kappa_{2}\kappa_{3}-\tfrac{71}{4}\kappa_{1}\kappa_{4}+\tfrac{461}{5}\kappa_{5}\in
H^{*}(\overline{\cal M}_{3},\mathbb{Q})$

are enough to prove the vanishing $J$-polynomial relations among $\kappa$
classes. Similarly, properties of the action of the operator
$\psi_{n+1}\pi^{*}$ on the ring of $\kappa$ polynomials would be enough to
prove the vanishing $K$-polynomial relations among $\kappa$ classes.

In [30] a collection of cohomology classes $\Theta_{g,n}\in
H^{2(2g-2+n)}(\overline{\cal M}_{g,n},\mathbb{Q})$, conjecturally related to
the supermoduli space of curves $\widehat{\cal M}_{g,n}$, [29, 35], are
defined. We recall the construction in Section 5.2.

######  Conjecture 4.

For $2g-2+n>0$,

$K_{2g-2+n}=\Theta_{g,n}.$

It is proven in [30] that for $n\geq 0$,
$\Theta_{g,n+1}=\psi_{n+1}\pi^{*}\Theta_{g,n}$. Hence Conjecture 4 implies
Conjecture 3 and thus also the vanishing $K_{m}(\kappa_{1},...,\kappa_{m})=0$
for $m>2g-2+n$ except $(m,n)=(3g-3,0)$. A summary of the implications is as
follows:

Conjecture 2 \+ Conjecture 3 $\ \Rightarrow\ $ Conjecture 1

Conjecture 4 $\Rightarrow$ Conjecture 3 $\Rightarrow$ (1) of Conjecture 1.

We prove Theorem 1 in Section 2 via regularity of Virasoro equations, or
equivalently topological recursion, depending on a parameter. We also show
that a weak version of Conjecture 1, which considers vanishing intersection
numbers, can be reduced to verifying the regularity of generating functions
depending on a parameter and produced via topological recursion, following the
proof of Theorem 1. We relate the classes $\mathbb{K}$ to the Brézin-Gross-
Witten tau function and conjecturally to the classes $\Theta_{g,n}$ in
Sections 3 and 5.

Acknowledgements. PN would like to thank Rahul Pandharipande, Aaron Pixton,
Johannes Schmitt and Mehdi Tavakol for useful conversations. MK is supported
by the National Research University Higher School of Economics. PN was
supported under the Australian Research Council Discovery Projects funding
scheme project number DP180103891.

##  2\. Vanishing of intersection numbers

In this section we consider intersection numbers $\int_{\overline{\cal
M}_{g,n}}\mathbb{K}\cdot\omega$ and $\int_{\overline{\cal
M}_{g,n}}\mathbb{J}\cdot\omega$ for any tautological class $\omega\in
RH^{*}(\overline{\cal M}_{g,n})$. We prove Theorem 1 regarding vanishing of
intersection numbers of $\mathbb{K}$ with polynomials in $\psi$ classes via
regularity of a related potential. Using this same idea, we prove that the
more general vanishing of intersection numbers with $\omega\in
RH^{m}(\overline{\mathcal{M}}_{g,n})$ for $m<g-1$, i.e. a weak version of
Conjecture 1, is equivalent to a purely combinatorial problem stated in terms
of regularity of correlators, without reference to the moduli space of curves.

###  2.1. Topological recursion for the $\kappa$-potential

Consider two sets of independent variables $s=(s_{1},s_{2},\dots)$ and
$t=(t_{0},t_{1},\dots)$. Set

$\mathcal{K}=\exp\Bigl{(}\sum_{i=1}^{\infty}s_{i}\kappa_{i}\Bigr{)}$

and introduce the function

${\cal
F}(\hbar,t,s)=\sum_{g,n}\frac{\hbar^{g}}{n!}\sum_{k_{1},\dots,k_{n}}\int_{\overline{\cal
M}_{g,n}}\mathcal{K}\;\psi_{1}^{k_{1}}\dots\psi_{n}^{k_{n}}\;t_{k_{1}}\dots
t_{k_{n}}$

where the sum is over all $g,n\in\mathbb{N}$ satisfying $2g-2+n>0$ and
$(k_{1},\dots,k_{n})\in\mathbb{N}^{n}$. This is the generating function for
intersection numbers of all possible monomials in $\kappa$ classes and $\psi$
classes. If we set $s_{i}=0$ for all $i$, this specializes to the Kontsevich-
Witten potential enumerating intersection numbers of $\psi$-monomials,

${\cal F}(\hbar,t,0)=F^{KW}(\hbar,t).$

Define polynomials $h_{i}=h_{i}(s)$ by the formal expansion

(5) $1+h_{1}t+h_{2}t^{2}+\dots=\exp(-\sum s_{i}t^{i}).$

######  Theorem 3 ([26]).

The full potential ${\cal F}(\hbar,t,s)$ is obtained from the Kontsevich-
Witten potential $F^{KW}(\hbar,t)$ by a suitable shift of variables

${\cal
F}(\hbar,t,s)=F^{KW}(\hbar,t_{0},t_{1},t_{2}-h_{1}(s),t_{3}-h_{2}(s),\dots).$

######  Remark 2.1.

The Kontsevich-Witten potential is a power series in the variables
$t_{0},t_{1},\hbar$ whose coefficients are polynomials in the remaining
variables. Therefore, the Taylor expansion of $F^{KW}$ at the point
$t=(0,0,-h_{1},-h_{2},\dots)$ is well defined.

###### Proof.

There is a standard inductive procedure reducing computation of intersection
numbers involving $\kappa$ classes to those involving just $\psi$ classes.
This procedure is based on the following relations describing the behaviour of
$\kappa$ classes under the forgetful map $\pi:\overline{\cal
M}_{g,n+1}\to\overline{\cal M}_{g,n}$:

$\kappa_{m}=\pi_{*}(\psi_{n+1}^{m+1}),\qquad\psi_{n+1}^{a}\pi^{*}(\kappa_{m})=\psi_{n+1}^{a}(\kappa_{m}-\psi_{n+1}^{m}),~{}a>0.$

These relations imply relations between intersection numbers which are
equivalent to the following linear partial differential equations:

(6) $V_{m}{\cal F}=0,\quad m=1,2,\dots,\qquad V_{m}=\frac{\partial}{\partial
s_{m}}-\sum_{i=0}^{\infty}h_{i-m-1}(s)\frac{\partial}{\partial t_{i}}.$

These equations mean that the function ${\cal F}$ is constant along the phase
curves of the pairwise commuting vector fields $V_{m}$ and thus along the
whole distribution generated by these fields. The integral subvariety of this
distribution passing through the point $(t,s)$ intersects the subspace $s=0$
at the point $(q,0)$ where
$q=(t_{0},t_{1},t_{2}-h_{1}(s),t_{3}-h_{2}(s),\dots)$. This implies the
equality of the theorem. ∎

######  Remark 2.2.

Equation (5) provides an invertible change of $h$ and $s$ coordinates. From
now on, it will be more convenient for us to apply this change and to use the
coordinates $h_{1},h_{2},\dots$ instead of $s_{1},s_{2},\dots$ in all
considerations. So, for example, by $\mathcal{K}(h)$ we mean the class

$\mathcal{K}(h)=\exp\bigl{(}(-h_{1})\,\kappa_{1}+(\frac{1}{2}h_{1}^{2}-h_{2})\,\kappa_{2}+(-\frac{1}{3}h_{1}^{3}+h_{1}h_{2}-h_{3})\,\kappa_{3}+\dots\bigr{)}$

which is the exponential $e^{\sum s_{i}\kappa_{i}}$ expressed in terms of $h$
coordinates, and similarly for the $\kappa$-potential ${\cal F}$. In
$h$-coordinates, the equalities (6) take a simpler form

$\frac{\partial{\cal F}}{\partial h_{m}}+\frac{\partial{\cal F}}{\partial
t_{m+1}}=0,\quad m=1,2,\dots.$

######  Corollary 2.3.

The potential ${\cal F}$ obeys the following Virasoro constraints. For any
$m\geq-1$ we have

$\frac{1}{2}\sum_{i+j=m-1}(2i+1)!!(2j+1)!!\left(\hbar\frac{\partial^{2}{\cal
F}}{\partial t_{i}\partial t_{j}}+\frac{\partial{\cal F}}{\partial
t_{i}}\frac{\partial{\cal F}}{\partial t_{j}}\right)+\\\
\sum_{k-i=m}(t_{i}-h_{i-1})\frac{(2k+1)!!}{(2i-1)!!}\frac{\partial{\cal
F}}{\partial
t_{k}}+\delta_{m,-1}\frac{t_{0}^{2}}{2}+\delta_{m,0}\frac{\hbar}{8}=0.$

where we set $h_{0}=1$, $h_{-1}=0$.

###### Proof.

This follows directly from the Virasoro constraints for the Kontsevich-Witten
potential. ∎

Here is an equivalent form of the Virasoro constraints. Define the following
partial differential operators:

$L_{m}=\frac{\hbar}{2}\sum_{i+j=m-1}(2i+1)!!(2j+1)!!\frac{\partial^{2}}{\partial
t_{i}\partial
t_{j}}+\sum_{k-i=m}t_{i}\frac{(2k+1)!!}{(2i-1)!!}\frac{\partial}{\partial
t_{k}}+\delta_{m,-1}\frac{t_{0}^{2}}{2}+\delta_{m,0}\frac{\hbar}{8}.$

Then we have

$\left(L_{m}-\sum_{k-i=m}h_{i-1}\frac{(2k+1)!!}{(2i-1)!!}\frac{\partial}{\partial
t_{k}}\right)e^{\hbar^{-1}{\cal F}}=0,\qquad m=-1,0,1,2,\dots$

There is another equivalent way to represent the Virasoro constraints, namely,
in the form of _topological recursion_ [14]. Let us expand ${\cal
F}=\sum_{g,n}\hbar^{g}F_{g,n}$ where $F_{g,n}$ is the contribution of
intersection numbers on a single moduli space,

$F_{g,n}=\frac{1}{n!}\sum_{k_{1},\dots,k_{n}}\int_{\overline{\cal
M}_{g,n}}\mathcal{K}(h)\;\psi_{1}^{k_{1}}\dots\psi_{n}^{k_{n}}\;t_{k_{1}}\dots
t_{k_{n}}.$

There is a different way to collect these intersection numbers into a
generating polynomial, namely, using the so called _correlator differentials_

$w_{g,n}(z_{1},\dots,z_{n})=\sum_{k_{1},\dots,k_{n}}\int_{\overline{\cal
M}_{g,n}}\mathcal{K}(h)\;\psi_{1}^{k_{1}}\dots\psi_{n}^{k_{n}}~{}\prod_{i=1}^{n}\frac{(2k_{i}+1)!!dz_{i}}{z^{2k_{i}+2}_{i}}.$

Equivalently, introduce the _loop insertion operators_

$\delta_{i}=\sum_{k=0}\frac{(2k+1)!!dz_{i}}{z^{2k+2}_{i}}\frac{\partial}{\partial
t_{k}}.$

Then we have

$w_{g,n}=\delta_{1}\dots\delta_{n}F_{g,n}.$

######  Theorem 4.

The differentials $w_{g,n}$ satisfy the relations of topological recursion for
the spectral curve

$x=\frac{1}{2}z^{2},\qquad
y=z+\sum_{k=1}^{\infty}h_{k}\frac{z^{2k+1}}{(2k+1)!!}.$

In this statement, we treat $x(z)$ as a global function on the complex line
$\mathbb{C}$ with coordinate $z$ having a ramification of order $2$ at the
origin, while $y(z)$ is just a formal power expansion near the point $z=0$.

######  Corollary 2.4.

Any potential of topological recursion with the spectral curve whose
$x$-function is given by $x(z)=\frac{1}{2}z^{2}$ is a specialization of the
$\kappa$-potential ${\cal F}$ for suitable choices of the parameters $h_{i}$
(or $s_{i}$), and thus is equal to the Kontsevich-Witten potential $F^{KW}$
with a suitable shift of times $t_{i}$ for $i\geq 2$.

###### Proof of Theorem 4.

Sum up all Virasoro equations of Corollary 2.3 multiplying the $m$th equation
by $\frac{2dz^{2}}{z^{2m+2}}$ where $z$ is a new variable. Then, denoting

$\mathbf{t}(z)=\sum_{k=0}^{\infty}t_{k}\frac{z^{2k}dz}{(2k-1)!!},\quad\eta(z)=y\,dx=\sum_{k=0}^{\infty}h_{k}\frac{z^{2k+2}dz}{(2k+1)!!}$

and observing $\delta{\cal F}=\sum_{k=0}^{\infty}\frac{\partial{\cal
F}}{\partial t_{k}}\frac{(2k+1)!!dz}{z^{2k+2}}$, we collect all Virasoro
equations into a single one that we call the _master loop equation_

$\hbar\Bigl{(}\delta^{2}{\cal F}+\frac{dz^{2}}{4z^{2}}\Bigr{)}+(\delta{\cal
F}+\mathbf{t}(z)-\eta(z))^{2}=O(z^{2}),$

in the Laurent expansion in $z$. The original Virasoro equations can be
extracted from the above relation by selecting the coefficient of
$dz^{2}z^{2i}$ for non-positive $i$. Equivalently, this equation can be
written as

(7) $\delta{\cal F}=\frac{1}{2\eta(z)}\left(\hbar\Bigl{(}\delta^{2}{\cal
F}+\frac{dz^{2}}{4z^{2}}\Bigr{)}+(\delta{\cal F})^{2}+2\,\mathbf{t}\,{\cal
F}+\mathbf{t}^{2}\right)+O(1)$

Now, select homogeneous terms of fixed degrees $(g,n)$ and apply
$\delta_{2}\dots\delta_{n}$ to the obtained equation. Then the loop equation
takes the following form. _Redefine_ in the unstable cases

$w_{0,1}(z_{1})=0,\qquad w_{0,2}(z_{1},z_{2})=\delta_{2}\mathbf{t}(z_{1})$

and denote

$\mathcal{W}_{g,n}(z,z_{N})=w_{g-1,n+1}(z,z,z_{N})+\sum_{\begin{smallmatrix}g_{1}+g_{2}=g\\\
I\sqcup
J=N\end{smallmatrix}}w_{g_{1},|I|+1}(z,z_{I})w_{g_{2},|J|+1}(z,z_{J}).$

Here, we use the notation $N=\\{2,3,\ldots,n\\}$ and
$z_{I}=\\{z_{i_{1}},z_{i_{2}},\ldots,z_{i_{k}}\\}$ for
$I=\\{i_{1},i_{2},\ldots,i_{k}\\}$. Then we obtain

(8) $w_{g,n}(z,z_{N})=\frac{\mathcal{W}_{g,n}(z,z_{N})}{2\eta(z)}+O(1)$

in the expansion in $z$ at $z=0$. Notice that $w_{g,n}(z,z_{N})$ contains only
negative powers of the coordinate $z$, therefore, the last equation determines
it uniquely by induction: $w_{g,n}(z,z_{N})$ as a meromorphic $1$-form in $z$
is the principal part of the pole at $z=0$ of the form on the right hand side
of the equation. The obtained relation is nothing but an equivalent form of
the conventional topological recursion relation. In order to observe it, we
remark that the projection to the principal part of the pole can be written as
the residue

$w_{g,n}(z,z_{N})=\mathop{\text{res}}_{\zeta=0}\frac{\mathcal{W}_{g,n}(\zeta,z_{N})}{2\eta(\zeta)}\frac{\zeta\,dz}{z^{2}-\zeta^{2}}$

Remark also that $w_{0,2}(z_{1},z_{2})$ is the odd part of the _Bergman
Kernel_ $B(z_{1},z_{2})=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$:

$w_{0,2}(z_{1},z_{2})=\delta_{2}\mathbf{t}(z_{1})=\sum_{k=0}^{\infty}\frac{dz_{2}(2k+1)!!}{z_{2}^{2k+2}}\frac{dz_{1}z_{1}^{2k}}{(2k-1)!!}=\\\
\frac{(z_{1}^{2}+z_{2}^{2})dz_{1}dz_{2}}{(z_{1}^{2}-z_{2}^{2})^{2}}=\frac{1}{2}(B(z_{1},z_{2})-B(-z_{1},z_{2})),$

and the form
$\frac{\zeta\,dz}{z^{2}-\zeta^{2}}=\frac{1}{2}\int_{-\zeta}^{\zeta}B(z,\cdot)$
is the odd part of its primitive. ∎

###  2.2. Regularity of potentials

We now introduce a parameter $\epsilon$ into various potentials, and prove
that regularity in $\epsilon$ is equivalent to vanishing of some coefficients.
This is enough to prove Theorem 1 and reduces a weak version of Conjecture 1
to a regularity property which is verified numerically in small cases.

###### Proof of Theorem 1.

Let us make an additional rescaling in the series ${\cal F}$ and in the
corresponding correlator differentials $w_{g,n}$ by replacing the class
$\mathcal{K}(h)$ in their definition with
$\sum_{m}\epsilon^{2g-2+n-m}\mathcal{K}_{m}(h)$ where $\mathcal{K}_{m}(h)$ is
the degree $m$ homogeneous component of $\mathcal{K}(h)$. With this rescaling,
the classes $\mathcal{K}_{m}(h)$ with $m>2g-2+n$ are multiplied with negative
powers of the parameter $\epsilon$. Therefore, the fact that intersection
numbers with these classes vanish is equivalent to the fact that the resulting
rescaled polynomials $F_{g,n}$ and differentials $w_{g,n}$ are regular at
$\epsilon=0$, that is, there exists limits $\lim\limits_{\epsilon\to
0}F_{g,n}$, or equivalently, there exist limits $\lim\limits_{\epsilon\to
0}w_{g,n}$. The rescaled correlator differentials also obey topological
recursion with the following rescaled spectral curve

$x=\frac{1}{2}z^{2},\qquad
y=\epsilon^{-1}\sum_{k=0}^{\infty}h_{k}\epsilon^{-k}\frac{z^{2k+1}}{(2k+1)!!}.$

The rescaling $h_{k}\to h_{k}\epsilon^{-k}$ corresponds to the factor
$\epsilon^{-m}$ of $\mathcal{K}_{m}(h)$. And introducing an additional global
factor $\epsilon^{-1}$ of $y$ results in the factor $\epsilon^{2g-2+n}$ for
the whole class $\mathcal{K}(h)$.

The class $\mathbb{K}$ of Theorem 1 corresponds to the specialization
$h_{k}=(-1)^{k}(2k+1)!!$ in $\mathcal{K}(h)$. The $y$-function of the
corresponding (rescaled) spectral curve is

(9)
$y=\epsilon^{-1}\sum_{k=0}^{\infty}(-1)^{k}(2k+1)!!\epsilon^{-k}\frac{z^{2k+1}}{(2k+1)!!}=\frac{z}{z^{2}+\epsilon}.$

Then we observe that this function possesses the property that

$y^{-1}=z+\epsilon z^{-1}$

is regular in $\epsilon$. This observation implies immediately the statement
of Theorem 1 for the class $\mathbb{K}$ by induction since negative powers of
$\epsilon$ do not appear in the recursion (8). ∎

######  Remark 2.5.

As was mentioned above, the topological recursion relation is equivalent to
the loop equation which is, in turn, an equivalent form of Virasoro
constraints. Therefore, the above arguments that use topological recursion for
the correlator differentials can be translated to the arguments involving
Virasoro constraints for the potential ${\cal F}$. For the case of the
$y$-function (9) the loop equation (7) specializes to

$\delta{\cal F}=(1+\epsilon
z^{-2})\frac{1}{2\,dz}\left(\hbar\Bigl{(}\delta^{2}{\cal
F}-\frac{dz^{2}}{4z^{2}}\Bigr{)}+(\delta{\cal F})^{2}+2\,\mathbf{t}\,{\cal
F}+\mathbf{t}^{2}\right)+O(1)$

Taking the coefficients of the fixed (negative) powers of $z$, we obtain the
Virasoro constraints in the following form

(10) $\bigl{(}(2m+1)!!\frac{\partial}{\partial
t_{m}}-\epsilon\;L_{m-1}-L_{m}\bigr{)}e^{\hbar^{-1}{\cal F}}=0,\quad
m=0,1,2,\dots.$

These equations determine the function uniquely. They are regular in
$\epsilon$, therefore, the solution is also regular in $\epsilon$.

######  Remark 2.6.

Topological recursion is defined in [14] for the spectral curve
$x=\frac{1}{2}z^{2}$, $y=y(z)$ in the following form which is equivalent to
the approach described in Section 2.1. Define

$\omega_{0,2}(z_{1},z_{2})=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}.$

and define $\omega_{g,n}$ for $2g-2+n>0$ recursively via the following
equation.

$\begin{split}\omega_{g,n}&(z_{1},z_{S})\\\
&=\mathop{\text{Res}}_{z=0}K(z_{1},z)\bigg{[}\omega_{g-1,n+1}(z,-z,z_{S})+\sum_{\begin{subarray}{c}h+h^{\prime}=g\\\
I\sqcup
J=S\end{subarray}}^{\circ}\omega_{h,1+|I|}(z,z_{I})\,\omega_{h^{\prime},1+|J|}(-z,z_{J})\bigg{]}.\end{split}$

Here, $S=\\{2,3,\ldots,n\\}$ and
$z_{I}=\\{z_{i_{1}},z_{i_{2}},\ldots,z_{i_{k}}\\}$ for
$I=\\{i_{1},i_{2},\ldots,i_{k}\\}\subseteq S$. The symbol $\circ$ over the
inner summation means that we exclude any term that involves $\omega_{0,1}$.
The recursion kernel is defined by:

$K(z_{1},z)=\frac{1}{2}\frac{\int_{-z}^{z}\omega_{0,2}(z_{1},\cdot)}{[y(z)-y(-z))]dx(z)}.$

The proof of Theorem 1 restated in these terms uses the fact that $K(z_{1},z)$
depends on $y^{-1}$ hence it is regular in $\epsilon$ which implies that each
$\omega_{g,n}$ is also regular in $\epsilon$.

####  2.2.1. Intersections with the class $\mathbb{J}$.

The class $\mathbb{J}$ corresponds to the specialization $h_{k}=(-1)^{k}k!$ in
$\mathcal{K}(h)$. The $y$-function of the corresponding (rescaled) spectral
curve is (it was first found in [13])

$y=\epsilon^{-1}\sum_{k=0}^{\infty}(-1)^{k}k!\epsilon^{-k}\frac{z^{2k+1}}{(2k+1)!!}=\frac{2\text{\
arcsinh}(z/\sqrt{2\epsilon})}{\sqrt{z^{2}+2\epsilon}}.$

In this case, we have that

$y^{-1}=\frac{\sqrt{z^{2}+2\epsilon}}{2\,\text{\
arcsinh}(z/\sqrt{2\epsilon})}=\frac{\epsilon}{z}+\frac{z}{3}-\frac{z^{3}}{45\epsilon}+\frac{z^{5}}{189\epsilon^{2}}-\frac{23z^{7}}{14175\epsilon^{3}}+\dots$

is a Laurent series in $z$ whose coefficients are not regular in $\epsilon$
any more. It turns out, however, that due to some cancellation all negative
powers of $\epsilon$ disappear in the course of the recursion and the
differentials $w_{g,n}$ prove to be regular in $\epsilon$.

$\displaystyle\omega_{0,3}$
$\displaystyle=\epsilon\prod_{i=1}^{3}\frac{dz_{i}}{z_{i}^{2}}$
$\displaystyle\omega_{0,4}$
$\displaystyle=\left(\epsilon+\epsilon^{2}\sum_{i=1}^{4}\frac{3}{z_{i}^{2}}\right)\prod_{i=1}^{4}\frac{dz_{i}}{z_{i}^{2}}$
$\displaystyle\omega_{1,1}$
$\displaystyle=\frac{1}{24}\left(1+\epsilon\frac{3}{z_{1}^{2}}\right)\frac{dz_{1}}{z_{1}^{2}}$
$\displaystyle\omega_{1,2}$
$\displaystyle=\frac{1}{8}\left(\epsilon\sum_{i=1}^{2}\frac{2}{z_{i}^{2}}+\epsilon^{2}\left(\sum_{i=1}^{2}\frac{5}{z_{i}^{4}}+\frac{3}{z_{1}^{2}z_{2}^{2}}\right)\right)\prod_{i=1}^{2}\frac{dz_{i}}{z_{i}^{2}}$
$\displaystyle\omega_{2,1}$
$\displaystyle=\frac{1}{1920}\left(-\frac{2}{z_{1}^{2}}+\epsilon\frac{130}{z_{1}^{4}}+\epsilon^{2}\frac{1015}{z_{1}^{6}}+\epsilon^{3}\frac{1575}{z_{1}^{8}}\right)\frac{dz_{1}}{z_{1}^{2}}$

We expect that the regularity in $\epsilon$ holds true for all $(g,n)$ with
$2g-2+n>0$. Thus, the proof is reduced to a purely combinatorial problem which
starts with an explicit spectral curve and asks for vanishing behaviour of
some coefficients of its $\omega_{g,n}$.

Let us turn to the (presumingly) top non-vanishing term. Consider the function

$F^{J}(\hbar,t_{0},t_{1},...)=\sum_{g,n,\vec{k}}\frac{\hbar^{g}}{n!}\int_{\overline{\cal
M}_{g,n}}J_{2g-2+n}\cdot\prod_{j=1}^{n}\psi_{j}^{k_{j}}t_{k_{j}}.$

Corresponding to the (conjectural) limit $\epsilon\to 0$ for the above
specialization of the potential ${\cal F}$. The numerical experiments show
that the $\epsilon=0$ limit of $w_{g,n}$ can be nonvanishing for $n=1$ only.
This means that all terms of $F^{J}$ of homogeneous degree $>1$ in
$t$-variables are vanishing,

$F^{J}(\hbar,t_{0},t_{1},...)=\frac{t_{0}}{24}\hbar-\frac{t_{1}}{2880}\hbar^{2}+\frac{t_{2}}{120960}\hbar^{3}-\frac{t_{3}}{3225600}\hbar^{4}+\\\
\frac{t_{4}}{63866880}\hbar^{5}-\frac{691}{697426329600}t_{5}\hbar^{6}+\frac{1}{13284311040}t_{6}\hbar^{7}+\dots{}$

Hence it seems that $F^{J}$ is linear in $t_{i}$ and given by the unproven
formula

(11)
$F^{J}(\hbar,t_{0},t_{1},...)=\sum_{g>0}\frac{(g-1)!B_{2g}}{(2g)!2^{g}}\cdot\hbar^{g}t_{g-1}$

where $B_{2g}$ is a Bernoulli number. It is proven in [17, Theorem 1] that

$\int_{\overline{\cal
M}_{g,1}}\lambda_{g-1}\lambda_{g}\psi^{g-1}=\int_{\overline{\cal
M}_{g}}\lambda_{g-1}\lambda_{g}\kappa_{g-2}=\frac{(-1)^{g-1}}{2^{2g-1}(2g-1)!!}\frac{B_{2g}}{2g}$

and from (11), which is only calculated numerically for small $g$,

$\int_{\overline{\cal
M}_{g,1}}J_{g,1}\psi^{g-1}=\frac{1}{2^{2g-1}(2g-1)!!}\frac{B_{2g}}{2g}$

so they agree up to $(-1)^{g-1}$. These properties give evidence towards
Conjecture 2, that $J_{g,1}=(-1)^{g-1}\lambda_{g-1}\lambda_{g}$. This is
discussed further in Section 5.1.

####  2.2.2. Intersections with $\kappa$ classes

We now discuss verification of trivial intersection pairings of the classes
$K_{m}$ and $J_{m}$ for $m>2g-2+n$ with arbitrary monomials in $\psi$ classes
and $\kappa$ classes. Let us denote by $h^{*}$ the specialization of Theorem 1
of $h$ parameters, that is, either $h_{k}^{*}=(-1)^{k}(2k-1)!!$ or
$h_{k}^{*}=(-1)^{k}k!$. Consider the potential enumerating intersection
numbers of the class $\mathcal{K}(h^{*})$ with both $\psi$ classes and
$\kappa$ monomials. It involves additional parameters $h_{1},h_{2},\dots$ and
is obtained by replacement of the class $\mathcal{K}$ in the definitions of
${\cal F}$ and $w_{g,n}$ with the product $\mathcal{K}(h^{*})\mathcal{K}(h)$.
The product $\mathcal{K}(h^{*})\mathcal{K}(h)$ of two exponents corresponds to
the shift $s_{k}\to s_{k}+s_{k}^{*}$ of $s$-coordinates in the class
$\mathcal{K}$, or, in terms of $h$-coordinates, it corresponds to the change
$h_{k}\to\sum_{i+j=k}h_{i}^{*}h_{j}$. Taking also into account the
$\epsilon$-rescaling separating intersection numbers with the classes
$\mathcal{K}_{m}(h^{*})$ with $m\leq 2g-2+n$ and those with $m>2g-2+n$, we
arrive at the following assertion.

######  Proposition 2.7.

The correlator differentials $w_{g,n}$ enumerating intersection numbers with
the classes $\mathbb{K}$ and $\mathbb{J}$ of Theorem 1 with arbitrary
monomials in $\psi$ classes and $\kappa$ classes satisfy relations of
topological recursion with the spectral curve

(12) $x=\frac{1}{2}z^{2},\qquad
y=\sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)!!}\sum_{i+j=k}\epsilon^{-i-1}h_{i}^{*}h_{j}$

where the sum is over $i,j\in\mathbb{N}=\\{0,1,2,...\\}$ and we set
$h_{0}^{*}=1=h_{0}$. Choose $h_{k}^{*}=(-1)^{k}(2k-1)!!$ for the case
$\mathbb{K}$ and $h_{k}^{*}=(-1)^{k}k!$ for the case $\mathbb{J}$, while
$h_{1},h_{2},\dots$ are formal parameters.

Starting the recursion we find the differentials $w_{g,n}$ and thus the
components $F_{g,n}$ of the partition function for small $g$ and $n$. For the
class $\mathbb{K}$ they are as follows. (We prefer to demonstrate $F_{g,n}$’s
rather than $w_{g,n}$’s since they are more compact)

$\displaystyle F_{0,3}$ $\displaystyle=\frac{t_{0}^{3}\epsilon}{6},$
$\displaystyle F_{1,1}$
$\displaystyle=\epsilon\left(\frac{t_{1}}{24}-\frac{h_{1}t_{0}}{24}\right)+\frac{t_{0}}{8},$
$\displaystyle F_{0,4}$
$\displaystyle=\epsilon^{2}\left(\frac{1}{6}t_{0}^{3}t_{1}-\frac{1}{24}h_{1}t_{0}^{4}\right)+\frac{t_{0}^{4}\epsilon}{8},$
$\displaystyle F_{1,2}$
$\displaystyle=\epsilon^{2}\left(\frac{1}{48}\left(2h_{1}^{2}-h_{2}\right)t_{0}^{2}+t_{0}\left(\frac{t_{2}}{24}-\frac{h_{1}t_{1}}{12}\right)+\frac{t_{1}^{2}}{48}\right)+\epsilon\left(\frac{t_{0}t_{1}}{4}-\frac{3}{16}h_{1}t_{0}^{2}\right)+\frac{t_{0}^{2}}{16},$
$\displaystyle F_{2,1}$
$\displaystyle=\epsilon^{3}\left(..\right)+\epsilon^{2}\left(..\right)+\epsilon\left(\frac{\left(627h_{1}^{2}-203h_{2}\right)t_{0}}{1920}-\frac{407h_{1}t_{1}}{1920}+\frac{107t_{2}}{1920}\right)-\frac{9h_{1}t_{0}}{128}+\frac{3t_{1}}{128}.$

For the class $\mathbb{J}$ these components are

$\displaystyle F_{0,3}$ $\displaystyle=\frac{t_{0}^{3}\epsilon}{6},$
$\displaystyle F_{1,1}$
$\displaystyle=\epsilon\left(\frac{t_{1}}{24}-\frac{h_{1}t_{0}}{24}\right)+\frac{t_{0}}{24},$
$\displaystyle F_{0,4}$
$\displaystyle=\epsilon^{2}\left(\frac{1}{6}t_{0}^{3}t_{1}-\frac{1}{24}h_{1}t_{0}^{4}\right)+\frac{t_{0}^{4}\epsilon}{24},$
$\displaystyle F_{1,2}$
$\displaystyle=\epsilon^{2}\left(\frac{1}{48}\left(2h_{1}^{2}-h_{2}\right)t_{0}^{2}+t_{0}\left(\frac{t_{2}}{24}-\frac{h_{1}t_{1}}{12}\right)+\frac{t_{1}^{2}}{48}\right)+\epsilon\left(\frac{t_{0}t_{1}}{12}-\frac{1}{16}h_{1}t_{0}^{2}\right),$
$\displaystyle F_{2,1}$
$\displaystyle=\epsilon^{3}\left(..\right)+\epsilon^{2}\left(..\right)+\epsilon\left(\frac{\left(157h_{1}^{2}-50h_{2}\right)t_{0}}{5760}-\frac{17h_{1}t_{1}}{960}+\frac{13t_{2}}{2880}\right)+\frac{h_{1}t_{0}}{2880}-\frac{t_{1}}{2880}.$

In both cases these polynomials contain only non-negative powers of
$\epsilon$. This does not follow automatically from the recursion since in
both cases the Laurent expansion of $y^{-1}$ in $z$ contains negative powers
of $\epsilon$. However, we expect that these polynomials are regular in
$\epsilon$ for all $g$ and $n$ which is equivalent to the fact that the
intersection with classes $K_{m}$ and $J_{m}$ for $m>2g-2+n$ is trivial for
arbitrary tautological classes. Moreover, we observe that in the $J$ case the
limit for $\epsilon\to 0$ is nonvanishing in the case $n=1$ only, as expected.

We say that a class in the tautological ring
$RH^{*}(\overline{\mathcal{M}}_{g,n})\subset
H^{2*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ vanishes weakly if it
vanishes under the intersection pairing with all other classes in
$RH^{*}(\overline{\mathcal{M}}_{g,n})$. In particular, in those cases where
the tautological ring $RH^{*}(\overline{\mathcal{M}}_{g,n})$ is perfect
Conjecture 1 follows from its weak version.

######  Corollary 2.8.

If the correlators of the spectral curve (12) are regular in $\epsilon$ then
Conjecture 1 holds weakly, i.e. the intersection pairing of $K_{m}$ with all
classes in $RH^{*}(\overline{\mathcal{M}}_{g,n})$ vanishes for $m>2g-2+n$.

###### Proof.

Let $G_{g,n}$ be the finite set of stable graphs of type $(g,n)$. Each
$\Gamma\in G_{g,n}$ defines a stratum $\phi_{\Gamma}:\overline{\cal
M}_{\Gamma}\to\overline{\cal M}_{g,n}$ and any tautological class $\omega\in
RH^{m}(\overline{\mathcal{M}}_{g,n})$ is given by a sum over stable graphs

$\omega=\sum_{\Gamma\in G_{g,n}}(\phi_{\Gamma})_{*}p_{\Gamma}$

where $p_{\Gamma}=\bigotimes_{v\in\Gamma}p_{v}$ is a product of polynomials
$p_{v}$ in $\psi$ classes and $\kappa$ classes defined at each vertex $v$ of
$\Gamma$. Now

$\int_{\overline{\cal M}_{g,n}}\mathbb{K}\cdot\omega=\sum_{\Gamma\in
G_{g,n}}\int_{\overline{\cal
M}_{g,n}}\mathbb{K}\cdot(\phi_{\Gamma})_{*}p_{\Gamma}=\sum_{\Gamma\in
G_{g,n}}\int_{\overline{\cal M}_{\Gamma}}\mathbb{K}_{\Gamma}\cdot p_{\Gamma}$

where the second equality uses
$\phi_{\Gamma}^{*}\mathbb{K}=\mathbb{K}_{\Gamma}$ which consists of a factor
of $\mathbb{K}$ at each vertex of $\Gamma$. Let $E(\Gamma)$ and $V(\Gamma)$ be
the number of interior edges, respectively vertices, of $\Gamma$. When
$V(\Gamma)=1$ and $E(\Gamma)=0$, $\omega$ is a polynomial in $\psi$ classes
and $\kappa$ classes, and the vanishing of $\mathbb{K}\cdot\omega$ follows
from regularity in $\epsilon$ of the correlators of the spectral curve (12).

More generally, if $V(\Gamma)=1$ and $E(\Gamma)>0$, then

$g-1>\deg\omega=E(\Gamma)+\deg p_{v}=g-g_{v}+\deg p_{v}$

hence $\deg p_{v}<g_{v}-1$, and

$\int_{\overline{\cal M}_{g,n}}\mathbb{K}\cdot\omega=\int_{\overline{\cal
M}_{g_{v},n_{v}}}\mathbb{K}\cdot p_{v}=0$

where $(g_{v},n_{v})=(g-E(\Gamma),n+2E(\Gamma))$ and the second equality
follows from the initial case of $V(\Gamma)=1$ and $E(\Gamma)=0$.

In full generality, we have

$g-1>\deg\omega=E(\Gamma)+\sum_{v}\deg p_{v}$

hence there exists a vertex $v$ for which $p_{v}<g_{v}-1$ since otherwise

$g-1>\deg\omega\geq
E(\Gamma)+\sum_{v}(g_{v}-1)=E(\Gamma)-V(\Gamma)+\sum_{v}g_{v}=b_{1}(\Gamma)+\sum_{v}g_{v}=g$

which is a contradiction. On that vertex, $\int_{\overline{\cal
M}_{g(v),n(v)}}\mathbb{K}\cdot p_{v}=0$ by the $V(\Gamma)=1$ cases proven
above. Hence $\mathbb{K}\cdot\omega=0$ as required. ∎

##  3\. Relation to the Brézin-Gross-Witten tau function

###  3.1. KdV tau functions

A tau function $Z(t_{0},t_{1},...)$ of the KdV hierarchy (equivalently the KP
hierarchy in odd times $p_{2k+1}=t_{k}/(2k+1)!!$) gives rise to a solution
$U=\hbar\frac{\partial^{2}}{\partial t_{0}^{2}}\log Z$ of the KdV hierarchy

(13) $U_{t_{1}}=UU_{t_{0}}+\frac{\hbar}{12}U_{t_{0}t_{0}t_{0}},\quad
U(t_{0},0,0,...)=f(t_{0}).$

The first equation in the hierarchy is the KdV equation (13), and later
equations $U_{t_{k}}=P_{k}(U,U_{t_{0}},U_{t_{0}t_{0}},...)$ for $k>1$
determine $U$ uniquely from $U(t_{0},0,0,...)$. See [27] for the full
definition.

The Brézin-Gross-Witten solution
$U^{\text{BGW}}=\hbar\frac{\partial^{2}}{\partial t_{0}^{2}}\log
Z^{\text{BGW}}$ of the KdV hierarchy arises out of a unitary matrix model
studied in [4, 20]. It is defined by the initial condition

$\hbar\frac{\partial^{2}}{\partial t_{0}^{2}}\log
Z^{\text{BGW}}(t_{0},0,0,...)=U^{\text{BGW}}(t_{0},0,0,...)=\frac{\hbar}{8(1-t_{0})^{2}}.$

It arises out of topological recursion applied to the spectral curve
$(x,y)=(\frac{1}{2}z^{2},1/z)$ [9] and more generally to any spectral curve
with points locally of this form near a zero of $dx$, [7].

The low genus $g$ terms (= coefficient of $\hbar^{g-1}$) of $\log
Z^{\text{BGW}}$ are

$\displaystyle\log Z^{\text{BGW}}$
$\displaystyle=-\frac{1}{8}\log(1-t_{0})+\hbar\frac{3}{128}\frac{t_{1}}{(1-t_{0})^{3}}+\hbar^{2}\frac{15}{1024}\frac{t_{2}}{(1-t_{0})^{5}}$
$\displaystyle\hskip
128.0374pt+\hbar^{2}\frac{63}{1024}\frac{t_{1}^{2}}{(1-t_{0})^{6}}+O(\hbar^{3})$
$\displaystyle=(\frac{1}{8}t_{0}+\frac{1}{16}t_{0}^{2}+\frac{1}{24}t_{0}^{3}+...)+\hbar(\frac{3}{128}t_{1}+\frac{9}{128}t_{0}t_{1}+...)$
$\displaystyle\hskip
113.81102pt+\hbar^{2}(\frac{15}{1024}t_{2}+\frac{63}{1024}t_{1}^{2}+...)+...$

The tau function $Z^{\text{BGW}}(\hbar,t_{0},t_{1},...)$ shares many
properties of the famous Kontsevich-Witten tau function
$Z^{\text{KW}}(\hbar,t_{0},t_{1},...)$ introduced in [37]. The Kontsevich-
Witten tau function $Z^{\text{KW}}$ is defined by the initial condition
$U^{\text{KW}}(t_{0},0,0,...)=t_{0}$ for
$U^{\text{KW}}=\hbar\frac{\partial^{2}}{\partial t_{0}^{2}}\log
Z^{\text{KW}}$. It arises out of topological recursion applied to the curve
$(x,y)=(\frac{1}{2}z^{2},z)$ [15].The low genus terms of $\log Z^{\text{KW}}$
are

$\log
Z^{\text{KW}}(\hbar,t_{0},t_{1},...)=\hbar^{-1}(\frac{t_{0}^{3}}{3!}+\frac{t_{0}^{3}t_{1}}{3!}+\frac{t_{0}^{4}t_{2}}{4!}+...)+\frac{t_{1}}{24}+...$

######  Theorem 5 (Witten-Kontsevich 1992 [24, 37]).

$Z^{\text{KW}}(\hbar,t_{0},t_{1},...)=\exp\sum_{g,n}\hbar^{g-1}\frac{1}{n!}\sum_{\vec{k}\in\mathbb{N}^{n}}\int_{\overline{\cal
M}_{g,n}}\prod_{i=1}^{n}\psi_{i}^{k_{i}}t_{k_{i}}$

is a tau function of the KdV hierarchy.

###  3.2. BGW tau function from the limit of intersection numbers

By the proof of Theorem 1 the $\epsilon$-rescaled potential for intersection
numbers of the class $\mathbb{K}$ with $\psi$ monomials, with the constant
term removed, admits a limit for $\epsilon\to 0$. The limit is a similar
potential enumerating intersection numbers of $\psi$ monomials with the
component $K_{2g-2+n}$ of $\mathbb{K}$. Denote this limit by $F^{K}$,

$F^{K}(\hbar,t_{0},t_{1},...)=\sum_{g,n,\vec{k}}\frac{\hbar^{g}}{n!}\int_{\overline{\cal
M}_{g,n}}K_{2g-2+n}\cdot\prod_{j=1}^{n}\psi_{j}^{k_{j}}t_{k_{j}}.$

######  Theorem 6.

$e^{\hbar^{-1}F^{K}}$ is the BGW tau function.

###### Proof.

We provide two proofs of this fact. First, by (the proof of) Theorem 1, the
potential for the intersection numbers with the whole class $\mathbb{K}$ is
govern by the topological recursion with the spectral curve
$(x,y)=(\frac{1}{2}z^{2},\frac{z}{\epsilon+z^{2}})$ which is equivalent to the
Virasoro constraints (10). This implies the existence of the limit
$\epsilon\to 0$ for the potential, and also the fact that the limit satisfies
Virasoro equations of the form

$\bigl{(}(2m+1)!!\frac{\partial}{\partial
t_{m}}-L_{m}\bigr{)}e^{\hbar^{-1}F^{K}}=0,\quad m=0,1,2,\dots.$

These are exactly Virasoro constraints for the BGW potential, and these
equations determine the function uniquely, see e.g. [19]. As a byproduct of
this proof, we recover the spectral curve
$(x,y)=(\frac{1}{2}z^{2},\frac{1}{z})$ for the BGW potential $F^{K}$ found in
[15].

For the second proof, we observe that the potential ${\cal F}$ of Sect. 2.1
being the Kontsevich-Witten potential with shifted times, is a solution to the
KdV hierarchy for any choice of parameters $h_{i}$. It follows that the
potential $F^{K}$ as a limit of a family of solutions is also a solution of
the KdV hierarchy.

It remains to check the initial conditions for $F^{K}|_{t_{1}=t_{2}=\dots=0}$.
This part of the potential enumerates intersection numbers of $K_{2g-2+n}$
with no insertion of $\psi$ classes, that is, those of the form
$\int_{\overline{\cal M}_{g,n}}K_{2g-2+n}$. By dimensional reason, these
numbers are nonzero in the case $g=1$ only, and by Theorem 3 we have

$F^{K}(t_{0},0,0,\dots)=\hbar\sum_{n=0}^{\infty}\int_{\overline{\cal
M}_{1,n}}K_{n}\frac{t_{0}^{n}}{n!}=\hbar
F_{1}^{\text{KW}}(t_{0},0,3!!,-5!!,7!!,\dots).$

Fortunately, the $g=1$ component $F_{1}^{\text{KW}}$ of the Kontsevich-Witten
potential is known. By the Itzykson-Zuber anzatz [22] (a sequence of the
string and dilaton equations), it is given by

$F_{1}^{\text{KW}}=-\frac{1}{24}\log\Bigl{(}1-\sum_{k=0}^{\infty}t_{k+1}\frac{Y^{k}}{k!}\Bigr{)}$

where $Y=Y(t_{0},t_{1},\dots)$ is the solution of the implicit equation

$Y=\sum_{k=0}^{\infty}t_{k}\frac{Y^{k}}{k!}.$

Denote $y(t_{0})=Y(t_{0},0,3!!,-5!!,7!!,\dots)$. Then the implicit equation
specializes to

$y=1+\sum_{k=2}^{\infty}(-1)^{k}(2k-1)!!\frac{y^{k}}{k!}=t_{0}+y+\frac{1}{\sqrt{1+2y}}-1$

which gives $\frac{1}{\sqrt{2y+1}}=1-t_{0}$. Then,
$1-\sum_{k=0}^{\infty}t_{k+1}\frac{Y^{k}}{k!}$ specializes to

$1-\sum_{k=2}^{\infty}(-1)^{k}(2k-1)!!\frac{y^{k-1}}{(k-1)!}=\frac{1}{(1+2y)^{3/2}}=(1-t_{0})^{3},$

and we get finally

$F^{K}(t_{0},0,0,\dots)=-\frac{\hbar}{24}\log(1-t_{0})^{3}=-\frac{\hbar}{8}\log(1-t_{0}),$
$\frac{\partial^{2}F^{K}(t_{0},0,0,\dots)}{\partial
t_{0}^{2}}=\frac{\hbar}{8(1-t_{0})^{2}}.$

This coincides with the initial condition for the BGW potential. ∎

In Section 5.2 we relate Theorem 6 to a conjectural relationship between a
family of classes $\Theta_{g,n}\in H^{2(2g-2+n)}(\overline{\cal
M}_{g,n},\mathbb{Q})$ and the BGW potential.

##  4\. Cohomological field theories

A cohomological field theory is a pair $(V,\eta)$ composed of a finite-
dimensional complex vector space $V$ equipped with a metric $\eta$ and a
sequence of $S_{n}$-equivariant maps.

$\Omega_{g,n}:V^{\otimes n}\to H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$

that satisfy compatibility conditions from inclusion of strata:

$\phi_{\text{irr}}:\overline{\cal M}_{g-1,n+2}\to\overline{\cal
M}_{g,n},\quad\phi_{h,I}:\overline{\cal M}_{h,|I|+1}\times\overline{\cal
M}_{g-h,|J|+1}\to\overline{\cal M}_{g,n},\ \ I\sqcup J=\\{1,...,n\\}$

given by

(14) $\displaystyle\phi_{\text{irr}}^{*}\Omega_{g,n}(v_{1}\otimes...\otimes
v_{n})$ $\displaystyle=\Omega_{g-1,n+2}(v_{1}\otimes...\otimes
v_{n}\otimes\Delta)$ (15)
$\displaystyle\phi_{h,I}^{*}\Omega_{g,n}(v_{1}\otimes...\otimes v_{n})$
$\displaystyle=\pi_{1}^{*}\Omega_{h,|I|+1}\otimes\pi_{2}^{*}\Omega_{g-h,|J|+1}\big{(}\bigotimes_{i\in
I}v_{i}\otimes\Delta\otimes\bigotimes_{j\in J}v_{j}\big{)}$

where $\Delta\in V\otimes V$ is dual to the metric $\eta\in V^{*}\otimes
V^{*}$, and $\pi_{i}$, $i=1,2$ is projection onto the $i$th factor of
$\overline{\cal M}_{h,|I|+1}\times\overline{\cal M}_{g-h,|J|+1}$. When $n=0$,
$\Omega_{g}:=\Omega_{g,0}\in H^{*}(\overline{\cal M}_{g},\mathbb{Q})$. The
CohFT has a unit if there exists a vector $1\\!\\!1\in V$ which satisfies

(16) $\Omega_{0,3}(1\\!\\!1\otimes v_{1}\otimes v_{2})=\eta(v_{1},v_{2}).$

We also consider the case when $\Omega_{g,n}$ satisfies (14) and (15) but not
(16) and refer to this as a CohFT without unit.

The CohFT has flat unit if $1\\!\\!1\in H$ is compatible with the forgetful
map $\pi:\overline{\cal M}_{g,n+1}\to\overline{\cal M}_{g,n}$ by

(17) $\Omega_{g,n+1}(1\\!\\!1\otimes v_{1}\otimes...\otimes
v_{n})=\pi^{*}\Omega_{g,n}(v_{1}\otimes...\otimes v_{n})$

for $2g-2+n>0$.

Given a CohFT $\Omega=\\{\Omega_{g,n}\\}$ and a basis $\\{e_{1},...,e_{N}\\}$
of $V$, the partition function of $\Omega$ is defined by:

$Z_{\Omega}(\hbar,\\{t^{\alpha}_{k}\\})=\exp\sum_{g,n,\vec{k}}\frac{\hbar^{g-1}}{n!}\int_{\overline{\cal
M}_{g,n}}\Omega_{g,n}(e_{\alpha_{1}}\otimes...\otimes
e_{\alpha_{n}})\cdot\prod_{j=1}^{n}\psi_{j}^{k_{j}}\prod
t^{\alpha_{j}}_{k_{j}}$

for $\alpha_{i}\in\\{1,...,N\\}$ and $k_{j}\in\mathbb{N}$.

In the case of one-dimensional CohFTs, i.e. $\dim V=1$, with unit $1\\!\\!1\in
V$ that satisfies $\eta(1\\!\\!1,1\\!\\!1)=1$ (which is related to the more
general $\eta(1\\!\\!1,1\\!\\!1)\in\mathbb{C}$ by a simple rescaling), we
identify $\Omega_{g,n}$ with its image $\Omega_{g,n}(1\\!\\!1^{\otimes n})$,
and write $\Omega_{g,n}\in H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$.

Given two CohFTs $\Omega^{i}$ defined on $(V_{i},\eta_{i})$, $i=1,2$, they
define a CohFT $\Omega^{1}\otimes\Omega^{2}$ on $(V_{1}\otimes
V_{2},\eta_{1}\otimes\eta_{2})$ via the cup products
$\Omega^{1}_{g,n}(u_{1}\otimes...\otimes
u_{n})\cup\Omega^{2}_{g,n}(v_{1}\otimes...\otimes v_{n})$, and known as the
tensor product. The trivial CohFT $1\\!\\!1$ defined by $1\\!\\!1_{g,n}=1\in
H^{0}(\overline{\cal M}_{g,n},\mathbb{Q})$ acts as a unit with respect to
tensor product. An invertible CohFT is any one-dimensional CohFT $\Omega^{1}$
such that there exists $\Omega^{2}$, its inverse, with
$\Omega^{1}\otimes\Omega^{2}=1\\!\\!1$. Invertible CohFTs were classified by
Manin-Zograf [26] and Teleman [36]. From the properties
$\phi_{\text{irr}}^{*}\kappa_{m}=\kappa_{m}$ and
$\phi_{h,I}^{*}\kappa_{m}=\pi_{1}^{*}\kappa_{m}+\pi_{2}^{*}\kappa_{m}$, proven
in [3], and $\phi_{\text{irr}}^{*}\mu_{m}=\mu_{m}$ and
$\phi_{h,I}^{*}\mu_{m}=\pi_{1}^{*}\mu_{m}+\pi_{2}^{*}\mu_{m}$, for
$\mu_{k}=ch_{k}(\mathbb{E})$, which uses the decomposition of the restriction
of the Hodge bundle to the boundary, Manin and Zograf [26] proved that
$\exp{\sum_{i\geq 0}(s_{i}\kappa_{i}+t_{i}\mu_{2i+1})}$ defines an invertible
CohFT. The converse was proven by Teleman in [36] giving the following
theorem.

######  Theorem 7 ([26, 36]).

For any $\\{s_{i},t_{i}\\}_{i\in\mathbb{N}}\subset\mathbb{C}$

$\Omega_{g,n}=\exp{\sum_{i\geq 0}(s_{i}\kappa_{i}+t_{i}\mu_{2i+1})}$

defines an invertible CohFT and any invertible CohFT is of this form.

In particular, $\Omega_{g,n}^{\mathbb{K}}=\mathbb{K}$ and
$\Omega_{g,n}^{\mathbb{J}}=\mathbb{J}$ define invertible CohFTs.

###  4.1. Push-forward and pull-back properties

The polynomials $K_{m}(\kappa_{1},...,\kappa_{m})$ and
$J_{m}(\kappa_{1},...,\kappa_{m})$ satisfy push-forward and pull-back
relations with respect to the forgetful map $\pi:\overline{\cal
M}_{g,n+1}\to\overline{\cal M}_{g,n}$.

(18) $\pi_{*}J_{m+1}=(2g-2+n-m)J_{m},\quad\pi_{*}K_{m+1}=(6g-6+3n-2m)K_{m}.$
(19)
$\pi^{*}J_{m}=\sum_{i=0}^{m}(-1)^{i}i!\psi_{n+1}^{i}J_{m-i},\qquad\pi^{*}K_{m}=\sum_{i=0}^{m}(-1)^{i}(2i+1)!!\psi_{n+1}^{i}K_{m-i}.$

The push-forward and pull-back relations are a consequence of a more general
result. For any $v=(a,b)\in\mathbb{C}^{2}$, define

$\mathbb{L}^{v}=L^{v}_{0}+L^{v}_{1}t+L^{v}_{2}t^{2}+...=\exp(\sum\ell^{v}_{j}\kappa_{j}t^{j})$

for $\ell^{v}_{j}$ defined by (20) and (21).

######  Definition/Lemma 1.

For any $v=(a,b)\in\mathbb{C}^{2}$, define the sequence $\\{\ell^{v}_{j}\mid
j>0\\}$ in either of the following two equivalent ways:

(20) $\exp\left(\sum\ell_{j}^{v}t^{j}\right)=1+at-
bt^{2}\frac{d}{dt}\sum_{j>0}\ell_{j}^{v}t^{j}$ (21)
$\exp\left(-\sum\ell_{j}^{v}t^{j}\right)=\sum_{k=0}^{\infty}(-1)^{k}t^{k}\prod_{m=0}^{k-1}(a+bm).$

###### Proof.

Define $f(t)$ by the right hand side of (21). Then, shifting the summation
index by $1$, we get

$f(t)=1+\sum_{k=0}^{\infty}(-1)^{k+1}t^{k+1}\prod_{m=0}^{k}(a+mb)=\\\
1-t\sum_{k=0}^{\infty}(a+bk)(-1)^{k}t^{k}\prod_{m=0}^{k-1}(a+mb)=1-t(a+b\,t\frac{d}{dt})f(t).$

The obtained differential equation is equivalent to (20). ∎

The classes $L^{v}_{k}$ for any $v=(a,b)\in\mathbb{C}^{2}$ are proven below to
satisfy push-forward relations using (20) and pull-back relations using (21).
These relations specialise to (18) and (19) since
$\mathbb{L}^{(1,1)}=\mathbb{J}$ and $\mathbb{L}^{(3,2)}=\mathbb{K}$, i.e. if
we substitute $(a,b)=(1,1)$ or $(3,2)$ into (21), the right hand side becomes

$\sum_{k=0}^{\infty}(-1)^{k}t^{k}\prod_{m=0}^{k-1}(1+m)=\sum_{k=0}^{\infty}(-1)^{k}k!t^{k}$

and

$\sum_{k=0}^{\infty}(-1)^{k}t^{k}\prod_{m=0}^{k-1}(3+2m)=\sum_{k=0}^{\infty}(-1)^{k}(2k+1)!!t^{k}$

and the relations (18) and (19) are immediate corollaries of Lemmas 4.1 and
4.2. We have abused notation and identified
$\mathbb{K}(t)=K_{0}+K_{1}t+K_{2}t^{2}+...$ with $\mathbb{K}=\mathbb{K}(1)$
since $t$ simply keeps track of degree, and similarly with $\mathbb{J}$.

######  Lemma 4.1.

For any $(a,b)\in\mathbb{C}$

$\pi_{*}L^{v}_{m+1}=a(2g-2+n)L^{v}_{m}-bmL^{v}_{m}.$

###### Proof.

Each $L^{v}_{m}$ is a degree $m$ polynomial in $\\{\kappa_{j}\\}$. We can
calculate its push-forward as follows:

$\displaystyle\pi_{*}\mathbb{L}^{v}=$
$\displaystyle\pi_{*}\exp\sum\ell^{v}_{i}\kappa_{i}t^{i}=\pi_{*}\exp\sum\ell^{v}_{i}(\pi^{*}\kappa_{i}+\psi^{i}_{n+1})t^{i}$
$\displaystyle=$
$\displaystyle\pi_{*}\left(\exp(\sum\ell^{v}_{i}\pi^{*}\kappa_{i}t^{i})\exp(\sum\ell^{v}_{i}\psi^{i}_{n+1}t^{i})\right)$
$\displaystyle=$
$\displaystyle\pi_{*}\left(\pi^{*}(\exp(\sum\ell^{v}_{i}\kappa_{i}t^{i}))\exp(\sum\ell^{v}_{i}\psi^{i}_{n+1}t^{i})\right)$
$\displaystyle=$
$\displaystyle\mathbb{L}^{v}\cdot\pi_{*}\exp\sum\ell^{v}_{i}\psi^{i}_{n+1}t^{i}$
$\displaystyle=$
$\displaystyle\mathbb{L}^{v}\cdot\pi_{*}\left(1+a\psi_{n+1}t-b\sum_{j>0}j\ell^{v}_{j}\psi^{j+1}_{n+1}t^{j+1}\right)$
$\displaystyle=$
$\displaystyle\mathbb{L}^{v}\cdot\left(a(2g-2+n)t-b\sum_{j>0}j\ell^{v}_{j}\kappa_{j}t^{j+1}\right)$
$\displaystyle=$ $\displaystyle\mathbb{L}^{v}\cdot a(2g-2+n)t-b\sum
kL^{v}_{k}t^{k+1}.$

The third last equality uses the differential equation (20) with
$t\mapsto\psi_{n+1}t$. The last equality is obtained via differentiation with
respect to $t$ of $\mathbb{L}^{v}=\sum
L^{v}_{k}t^{k}=\exp(\sum\ell^{v}_{j}\kappa_{j}t^{j})$ to get

$\sum
kL^{v}_{k}t^{k-1}=\mathbb{L}^{v}\cdot\sum_{j>0}j\ell^{v}_{j}\kappa_{j}t^{j-1}.$

The coefficient of $t^{m+1}$ in the expression for $\pi_{*}\mathbb{L}^{v}$
yields $\pi_{*}L^{v}_{m+1}=a(2g-2+n)L^{v}_{m}-bmL^{v}_{m}$ as required. ∎

######  Lemma 4.2.

For any $(a,b)\in\mathbb{C}$

$\pi^{*}L^{v}_{k}=\sum_{i=1}^{k}(-1)^{i}\psi_{n+1}^{i}L^{v}_{k-i}\prod_{m=0}^{i-1}(a+mb).$

###### Proof.

$\displaystyle\pi^{*}\mathbb{L}^{v}$
$\displaystyle=\pi^{*}\exp(\sum\ell^{v}_{i}\kappa_{i}t^{i})=\exp(\sum\ell^{v}_{i}\pi^{*}\kappa_{i}t^{i})=\exp(\sum\ell^{v}_{i}(\kappa_{i}-\psi_{n+1}^{i})t^{i}$
$\displaystyle=\mathbb{L}^{v}\exp\left(-\sum\ell^{v}_{i}\psi_{n+1}^{i}t^{i})\right)$
$\displaystyle=\mathbb{L}^{v}\sum_{i=1}^{\infty}(-1)^{i}\psi_{n+1}^{i}t^{i}\prod_{m=0}^{i-1}(a+mb)$

where the last equality uses (21). The coefficient of $t^{k}$ yields the
lemma. ∎

######  Remark 4.3.

There is a combinatorial proof of Lemma 1 in the case of $v=(1,1)$ which
considers connected and disconnected permutations of $(1,...,n)$.

###  4.2. CohFTs without unit

A one-dimensional CohFT without unit is a collection of classes
$\Omega_{g,n}\in H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$ that satisfies
(14) and (15). It may be that $\Omega_{0,3}=0$ as in the following examples.

######  Proposition 4.4.

The sequences $\Omega_{g,n}=K_{2g-2+n}$ and $\Omega^{\prime}_{g,n}=J_{2g-2+n}$
are CohFTs without unit and $\Omega^{\prime}_{g,n}=0$ for $n>1$ iff Conjecture
1 holds.

###### Proof.

We will treat the case of $\Omega_{g,n}=K_{2g-2+n}$ first. One direction is
straightforward. If

(22) $K_{m}(\kappa_{1},...,\kappa_{m})=0\quad\text{for}\quad m>2g-2+n,\ n>0$

then for $\phi:\overline{\cal M}_{g_{1},n_{1}+1}\times\overline{\cal
M}_{g_{2},n_{2}+1}\to\overline{\cal M}_{g,n}$ where $g_{1}+g_{2}=g$,
$n_{1}+n_{2}=n$ and $p_{i}$, $i=1,2$ is projection onto the $i$th factor

(23) $\phi^{*}K_{2g-2+n}=\sum_{i+j=2g-2+n}p_{1}^{*}K_{i}\cdot
p_{2}^{*}K_{j}=p_{1}^{*}K_{2g_{1}-2+n_{1}+1}\cdot
p_{2}^{*}K_{2g_{2}-2+n_{2}+1}.$

The first equality uses $\phi^{*}\exp(\sum s_{i}\kappa_{i})=p_{1}^{*}\exp(\sum
s_{i}\kappa_{i})\cdot p_{2}^{*}\exp(\sum s_{i}\kappa_{i})$. For the second
equality, $i+j=2g-2+n$ implies that for all but one summand in (23) either
$i>2g_{1}-2+n_{1}+1$, hence $K_{i}=0$ by (22), or $j>2g_{2}-2+n_{2}+1$ in
which case $K_{j}=0$ again by (22). Also $\phi_{\text{irr}}^{*}\exp(\sum
s_{i}\kappa_{i})=\exp(\sum s_{i}\kappa_{i})$ implies that
$\phi_{\text{irr}}^{*}K_{2g-2+n}=K_{2(g-1)-2+n+2}$ and together with (23) we
see that $\Omega_{g,n}=K_{2g-2+n}$ is a CohFT without unit.

For the other direction, suppose that $\Omega_{g,n}=K_{2g-2+n}$ is a CohFT
without unit. For $k>0$ and $n>0$, consider the following stable graph
$\Gamma\in G_{g,n+k}$:

$\Gamma=$$g$000$\cdots$$\vdots$n-1k

which defines the stratum:

$\overline{\cal M}_{\Gamma}=\overline{\cal M}_{g,n}\times\overbrace{\cal
M_{0,3}\times...\times\cal M_{0,3}}^{k\text{
factors}}\stackrel{{\scriptstyle\phi_{\Gamma}}}{{\longrightarrow}}\overline{\cal
M}_{g,n+k}.$

For $m=2g-2+n+k>2g-2+n$,

$0=\Omega_{g,n}\otimes\Omega_{0,3}^{\otimes
k}=\phi_{\Gamma}^{*}\Omega_{g,n+k}=\phi_{\Gamma}^{*}K_{m}=K_{m}\otimes
1^{\otimes k}$

where the first equality uses $\Omega_{0,3}=0$ which vanishes for degree
reasons, the second equality uses the assumption that
$\Omega_{g,n}=K_{2g-2+n}$ is a CohFT without unit, and the final equality uses
the degree $m$ part of the restriction

$\phi_{\Gamma}^{*}\mathbb{K}=\phi_{\Gamma}^{*}\exp(\sum
s_{i}\kappa_{i})=\exp(\sum s_{i}\kappa_{i})\otimes\exp(\sum
s_{i}\kappa_{i})^{\otimes k}=\exp(\sum s_{i}\kappa_{i})\otimes 1^{\otimes k}.$

Hence $K_{m}=0$ for $m>2g-2+n$ and $n>0$. When $n=0$, for $2g-2<m<3g-3$ from
the push-forward formula (18) we have

$K_{m}=\frac{1}{6g-6-2m}\pi_{*}K_{m+1}=0$

since we have shown that $K_{m+1}=0$ in $H^{*}(\overline{\cal
M}_{g,1},\mathbb{Q})$. Although $K_{3g-2}=0$ in $H^{*}(\overline{\cal
M}_{g,1},\mathbb{Q})$, the push-forward formula (18) $\pi_{*}K_{3g-2}=0\times
K_{3g-3}$ gives no information about $K_{3g-3}\in H^{*}(\overline{\cal
M}_{g},\mathbb{Q})$, and in particular $K_{3g-3}$ can be non-vanishing.

For the case $\Omega_{g,n}=J_{2g-2+n}$ the argument is exactly the same as for
$K_{2g-2+n}$ except for the push-forward formula (18) in the case $n=0$,

$J_{m}=\frac{1}{2g-2-m}\pi_{*}J_{m+1}=0$

which proves that $J_{m}=0$ in $H^{*}(\overline{\cal M}_{g},\mathbb{Q})$ for
$m>2g-2$ (including $m=3g-3$). Hence

$J_{m}(\kappa_{1},...,\kappa_{m})=0\quad\text{for}\quad\left\\{\begin{array}[]{ll}m>2g-2+n&\\\
m=2g-2+n,&n>1.\end{array}\right.$

The extra vanishing $J_{2g-2+n}=0$ for $n>1$ follows from the assumption
$\Omega^{\prime}_{g,n}=0$ for $n>1$. ∎

The push-forward relations (18) have an interpretation in terms of a shift of
1-dimensional CohFTs. Given a 1-dimensional CohFT $\Omega_{g,n}\in
H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$ define its shift depending on a
parameter $\epsilon$ by:

$\Omega_{g,n}(\epsilon)=\sum_{m\geq
0}\pi_{*}^{m}\Omega_{g,n+m}\cdot\frac{\epsilon^{m}}{m!}.$

This defines a CohFT for values $\epsilon$ where the sum converges.

######  Lemma 4.5.

$\mathbb{J}(\epsilon):=\sum_{m\geq
0}\pi_{*}^{m}\mathbb{J}\cdot\frac{\epsilon^{m}}{m!}=\sum
d_{k}(\epsilon)J_{k}t^{k},\quad d_{k}(\epsilon)=e^{-(k+\chi)\epsilon}$
$\mathbb{K}(\epsilon):=(1-\epsilon)^{2g-2+n}\sum_{m\geq
0}\pi_{*}^{m}\mathbb{K}\cdot\frac{\epsilon^{m}}{m!}=\sum
c_{k}(\epsilon)K_{k}t^{k},\quad c_{k}(\epsilon)=(1-\epsilon)^{2(k+\chi)}.$

The factor $(1-t)^{2g-2+n}$ is chosen in the definition of
$\mathbb{K}(\epsilon)$ for convenience. Note that any factor of
$\lambda^{2g-2+n}$ sends a CohFT to a CohFT.

###### Proof.

Iterating (18), we have

$\pi_{*}^{m}J_{k}=M^{m}J_{k-m},\quad M=2g-2+n-k.$

Hence the coefficient of $J_{k}t^{k}$ in $\sum_{m\geq
0}\pi_{*}^{m}\mathbb{J}\cdot\frac{\epsilon^{m}}{m!}$ is
$e^{M\epsilon}=e^{-(k+\chi)\epsilon}$ as required.

For $K_{k}$, we iterate (18) to get

$\pi_{*}^{m}K_{k}=M(M-1)...(M-m+1)K_{k-m},\quad M=3(2g-2+n)-2k-1.$

Hence the coefficient of $K_{k}t^{k}$ in $\sum_{m\geq
0}\pi_{*}^{m}\mathbb{K}\cdot\frac{\epsilon^{m}}{m!}$ is
$\displaystyle\sum_{k\geq 0}\binom{M+k}{k}\epsilon^{k}=(1-\epsilon)^{-M-1}$,
for $|\epsilon|<1$. When we multiply by $(1-t)^{2g-2+n}$, the coefficient of
$K_{k}t^{k}$ in $\mathbb{K}(\epsilon)$ is $(1-\epsilon)^{2(k+\chi)}$ as
required. ∎

The shift is equivalent to rescaling $(a,b)$ in Definition 1. Then
$\mathbb{J}(\epsilon)$ corresponds to $(a,b)=(e^{-\epsilon},e^{-\epsilon})$
and $\mathbb{K}(\epsilon)$ corresponds to
$(a,b)=(3(1-\epsilon)^{2},2(1-\epsilon)^{2})$. We see that
$\mathbb{J}(\epsilon)$ is defined for all $\epsilon\in\mathbb{C}$ and
$\mathbb{K}(\epsilon)$ is defined for all $|\epsilon|<1$ and can be
analytically continued to $\epsilon\in\mathbb{C}-\\{1\\}$. We can restate
Conjecture 1 using $\mathbb{J}(\epsilon)$ and $\mathbb{K}(\epsilon)$:

$\text{Conjecture
}\ref{conj}\Leftrightarrow\lim_{\epsilon\to-\infty}\mathbb{J}(\epsilon)\text{
exists\quad and}\quad\lim_{\epsilon\to 1}\mathbb{K}(\epsilon)\text{ exists}.$

##  5\. Geometric constructions

In this section we give three geometric constructions which may give a way to
prove the vanishing results involving the classes $\mathbb{J}$ and
$\mathbb{K}$.

###  5.1. Hodge CohFT

The Hodge bundle $\mathbb{E}_{g}\to\overline{\cal M}_{g,n}$ satisfies natural
restriction properties on the boundary:

(24)
$\begin{array}[]{l}0\to\mathbb{E}_{g-1}\to\phi_{\text{irr}}^{*}\mathbb{E}_{g}\to\mathcal{O}_{\overline{\cal
M}_{g-1,n+2}}\to 0\\\ \\\
0\to\pi_{1}^{*}\mathbb{E}_{h}\oplus\pi_{2}^{*}\mathbb{E}_{g-h}\to\phi_{h,I}^{*}\mathbb{E}_{g}\to
0\end{array}$

where $\pi_{i}$ is projection onto the $i$th factor of $\overline{\cal
M}_{h,|I|+1}\times\overline{\cal M}_{g-h,|J|+1}$. A consequence of the
restriction properties is that the total Chern class $c(\mathbb{E}_{g})$ of
the Hodge bundle defines a CohFT. The relation
$\pi^{*}\mathbb{E}_{g}=\mathbb{E}_{g}$ implies that the CohFT has flat unit.

Similarly, the bundle $\mathbb{E}_{g}\oplus\mathbb{E}_{g}$ satisfies natural
restriction properties and its total Chern class
$c(\mathbb{E}_{g})^{2}=c(\mathbb{E}_{g}\oplus\mathbb{E}_{g})$ defines a CohFT.
Alternatively, the square of any CohFT satisfies the properties of a CohFT.
Define:

$\Omega^{\text{Hod}2}_{g,n}:=c(\mathbb{E})^{2}=(1+\lambda_{1}+...+\lambda_{g})^{2}.$

Flatness of $\mathbb{E}\oplus\mathbb{E}^{\vee}$ implies
$c(\mathbb{E}\oplus\mathbb{E}^{\vee})=c(\mathbb{E})c(\mathbb{E}^{\vee})=1$
hence $\lambda_{g}^{2}=0$ and $\lambda_{g-1}^{2}=2\lambda_{g-2}\lambda_{g}$ so

$\displaystyle\Omega^{\text{Hod}2}_{g,n}$
$\displaystyle=\lambda_{g}^{2}+2\lambda_{g-1}\lambda_{g}+\lambda_{g-1}^{2}+2\lambda_{g-2}\lambda_{g}+...$
$\displaystyle=(-1)^{g+n}2^{2-n}\Lambda_{g,n}+\deg_{<2g-2+n}\text{ terms}$

where

(25)
$\Lambda_{g,n}=\left\\{\begin{array}[]{cc}(-1)^{g}\lambda_{g-2}\lambda_{g}&n=0\\\
(-1)^{g-1}\lambda_{g-1}\lambda_{g}&n=1\\\ 0&n>1\end{array}\right.$

######  Proposition 5.1.

The family of classes $\\{\Lambda_{g,n}\\}$ defines a CohFT without unit, i.e.
they satisfy the pull-back properties (14) and (15).

###### Proof.

We can prove this via the relation with the degree $2g-2+n$ terms of
$\Omega^{\text{Hod}2}_{g,n}$, or directly as follows. The Hodge classes
satisfy the restriction property:

$\phi_{\text{irr}}^{*}\lambda_{k}=\lambda_{k},\quad\phi_{h,I}^{*}\lambda_{k}=\sum_{a+b=k}\lambda_{a}\otimes\lambda_{b}.$

Thus $\phi_{\text{irr}}^{*}\lambda_{g}=0$ since $\lambda_{g}=0$ in
$H^{*}(\overline{\cal M}_{g-1,n+2},\mathbb{Q})$ and
$\phi_{h,I}^{*}\lambda_{g}=\lambda_{h}\otimes\lambda_{g-h}$ again since higher
degree $\lambda_{k}=0$ for $k>h$ in the first factor and $k>g-h$ in the second
factor. The same argument gives
$\phi_{h,I}^{*}\lambda_{g-1}=\lambda_{h}\otimes\lambda_{g-h-1}+\lambda_{h-1}\otimes\lambda_{g-h}$.
Thus

$\phi_{\text{irr}}^{*}\lambda_{g}\lambda_{g-1}=0,\quad\phi_{h,I}^{*}\lambda_{g}\lambda_{g-1}=\lambda_{h}^{2}\otimes\lambda_{g-h}\lambda_{g-h-1}+\lambda_{h}\lambda_{h-1}\otimes\lambda_{g-h}^{2}.$

We have $c(\mathbb{E}\oplus\mathbb{E}^{\vee})=1$ for $\mathbb{E}$ the Hodge
bundle and $\mathbb{E}^{\vee}$ its dual, so in particular $\lambda_{g}^{2}=0$
for $g>0$. Thus $\phi_{h,I}^{*}\lambda_{g}\lambda_{g-1}=0$ for $h(g-h)>0$
leaving only
$\phi_{0,I}^{*}\lambda_{g}\lambda_{g-1}=1\otimes\lambda_{g}\lambda_{g-1}$. On
$\overline{\cal M}_{g,1}$, all boundary divisors have positive genus
irreducible components, so we conclude that the pullback of
$\lambda_{g}\lambda_{g-1}\in H^{*}(\overline{\cal M}_{g,1},\mathbb{Q})$ to the
boundary is 0. Hence we have proven that

$\phi^{*}_{\Gamma}\Lambda_{g,1}=0$

as required since any boundary divisor $\overline{\cal
M}_{\Gamma}\subset\overline{\cal M}_{g,1}$ involves a component with $n>1$.

For the pull-back $\phi^{*}_{\Gamma}\Lambda_{g,0}$ we have
$\phi_{\text{irr}}^{*}\lambda_{g}=0$ hence
$\phi_{\text{irr}}^{*}\Lambda_{g}=0$ as required on $H^{*}(\overline{\cal
M}_{g-1,2},\mathbb{Q})$. Finally,

$\displaystyle\phi_{h,I}^{*}\lambda_{g-2}\lambda_{g}$
$\displaystyle=\lambda_{h-2}\lambda_{h}\otimes\lambda_{g-h}^{2}+\lambda_{h-1}\lambda_{h}\otimes\lambda_{g-1}\lambda_{g}+\lambda_{h}^{2}\otimes\lambda_{g-2}\lambda_{g}$
$\displaystyle=\lambda_{h-1}\lambda_{h}\otimes\lambda_{g-h-1}\lambda_{g-h}$

and $(-1)^{h-1}(-1)^{g-h-1}=(-1)^{g}$ shows that the signs agree yielding

$\phi_{h,I}^{*}\Lambda_{g}=\Lambda_{h,1}\otimes\Lambda_{g-h,1}.$

∎

The interest in $\Lambda_{g,n}$ is due to the equality checked via admcycles

$J_{2g-2+n}=\Lambda_{g,n},\quad 2g-2+n\leq 6.$

We expect this equality to hold for all $g$ and $n$ in which case Proposition
4.4 would apply to conclude (4) in Conjecture 1.

######  Proposition 5.2.

If

(26)
$J_{2g-2+n}=\left\\{\begin{array}[]{cc}(-1)^{g}\lambda_{g-2}\lambda_{g}&n=0\\\
(-1)^{g-1}\lambda_{g-1}\lambda_{g}&n=1\end{array}\right.$

then

$J_{m}(\kappa_{1},...,\kappa_{m})=0\quad\text{for}\quad\left\\{\begin{array}[]{ll}m>2g-2+n&\\\
m=2g-2+n,&n>1.\end{array}\right.$

###### Proof.

For $\Lambda_{g,n}$ defined in (25), the more general equality

$J_{2g-2+n}=\Lambda_{g,n},\quad 2g-2+n>0$

follows from the initial cases of $n=0,1$ given by (26). Equivalently, (26)
implies that $J_{2g-2+n}=0$ for $n>1$, which we now show. For even $n=2m>0$
consider the stable graph $\Gamma\in G_{g+m}$ with a single vertex and $m$
loops which defines the map

$\phi_{\Gamma}:\overline{\cal M}_{g,n}\to\overline{\cal M}_{g+m}$

by identifying the labeled points $p_{i}$, say $p_{1}\sim p_{2}$, $p_{3}\sim
p_{4}$, …, $p_{n-1}\sim p_{n}$ to produce a stable curve of genus $g+m$. Then
in $H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$

$J_{2g-2+n}=\phi_{\Gamma}^{*}J_{2(g+m)-2}=\phi_{\Gamma}^{*}(-1)^{g+m}\lambda_{g+m-2}\lambda_{g+m}=0$

where the first equality uses the degree $2g-2+n$ part of the restriction

$\phi^{*}\mathbb{J}=\phi_{\Gamma}^{*}\exp(\sum\sigma_{i}\kappa_{i})=\exp(\sum\sigma_{i}\kappa_{i})=\mathbb{J},$

the second equality uses $J_{2(g+m)-2}=(-1)^{g+m}\lambda_{g+m-2}\lambda_{g+m}$
which is the assumption (26), and the third equality uses
$0=\phi_{\Gamma}^{*}\lambda_{g+m}\in H^{*}(\overline{\cal
M}_{g,n},\mathbb{Q})$ since $g+m>g$.

For odd $n=2m+1>1$ the argument is similar. The stable graph $\Gamma\in
G_{g+m,1}$ with a single vertex, $m$ loops and a leaf defines

$\phi_{\Gamma}:\overline{\cal M}_{g,n}\to\overline{\cal M}_{g+m,1}$

by identifying the labeled points $p_{1}\sim p_{2}$, $p_{3}\sim p_{4}$, …,
$p_{n-2}\sim p_{n-1}$. Then in $H^{*}(\overline{\cal M}_{g,n},\mathbb{Q})$

$J_{2g-2+n}=\phi_{\Gamma}^{*}J_{2(g+m)-1}=\phi_{\Gamma}^{*}(-1)^{g+m-1}\lambda_{g+m-1}\lambda_{g+m}=0$

as above.

Thus, the assumption (26) implies that $J_{2g-2+n}=\Lambda_{g,n}$ is a CohFT
without unit and $J_{2g-2+n}=0$ for $n>1$. Hence Proposition 4.4 applies to
deduce the vanishing of $J$ polynomials in the range given by Conjecture 1. ∎

######  Proposition 5.3.

For $g>1$ and $n>0$, if

(27) $K_{2g-2+n}=\left(\prod_{i=1}^{n}\psi_{i}\right)\pi^{*}K_{2g-2}$

then $K_{m}(\kappa_{1},...,\kappa_{m})=0$ for $m>2g-2+n$ except
$(m,n)=(3g-3,0)$.

###### Proof.

The proof is similar to the proof of Proposition 4.4. For $k>0$ and $n>0$,
consider the stable graph $\Gamma\in G_{g,n+k}$ given by

$\Gamma=$$g$000$\cdots$$\vdots$n-1k

and its associated stratum:

$\overline{\cal M}_{\Gamma}=\overline{\cal M}_{g,n}\times\overbrace{\cal
M_{0,3}\times...\times\cal M_{0,3}}^{k\text{
factors}}\stackrel{{\scriptstyle\phi_{\Gamma}}}{{\longrightarrow}}\overline{\cal
M}_{g,n+k}.$

Then

(28) $0=\phi_{\Gamma}^{*}K_{2g-2+n+k}=K_{2g-2+n+k}\otimes 1^{\otimes k}$

where the first equality uses the assumption
$K_{2g-2+n+k}=\left(\prod_{i=1}^{n+k}\psi_{i}\right)\pi^{*}K_{2g-2}$, so in
particular there is a $\psi_{i}$ which vanishes on the corresponding factor of
$\cal M_{0,3}$. The second equality uses the degree $2g-2+n+k$ part of the
restriction

$\phi_{\Gamma}^{*}\mathbb{K}=\phi_{\Gamma}^{*}\exp(\sum
s_{i}\kappa_{i})=\exp(\sum s_{i}\kappa_{i})\otimes\exp(\sum
s_{i}\kappa_{i})^{\otimes k}=\exp(\sum s_{i}\kappa_{i})\otimes 1^{\otimes k}.$

Hence $K_{m}=0$ for $m>2g-2+n$ and $n>0$. The $n=0$ cases uses the $n=1$ case
and the push-forward formula when $2g-2<m<3g-3$

$K_{m}=\frac{1}{6g-6-2m}\pi_{*}K_{m+1}=0.$

∎

######  Remark 5.4.

By essentially the same proof, a weak version of Proposition 5.3 also holds.
If

(29) $\int_{\overline{\cal
M}_{g,n}}\Big{(}K_{2g-2+n}-\Big{(}\prod_{i=1}^{n}\psi_{i}\Big{)}\pi^{*}K_{2g-2}\Big{)}\cdot\omega=0$

for all tautological classes $\omega\in RH^{*}(\overline{\cal M}_{g,n})$ then
$\int_{\overline{\cal M}_{g,n}}K_{m}\cdot\omega=0$ for all $\omega\in
RH^{*}(\overline{\cal M}_{g,n})$ and $m>2g-2+n$ except $(m,n)=(3g-3,0)$. In
the proof, simply take the intersection of (28) with
$(\phi_{\Gamma})_{*}(\omega\otimes 1^{\otimes k})$ for any tautological class
$\omega\in RH^{*}(\overline{\cal M}_{g,n})$ .

###  5.2. Spin CohFT

We define here a two dimensional CohFT arising out of the total Chern class of
a natural bundle defined over the moduli space of spin curves, analogous to
the Hodge bundle and its related CohFT. Define the moduli space of stable
twisted spin curves [1] by

$\overline{\cal M}_{g,n}^{\rm
spin}=\\{(\mathcal{C},\theta,p_{1},...,p_{n},\phi)\mid\phi:\theta^{2}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\omega_{\mathcal{C}}^{\text{log}}\\}.$

The pair $(\theta,\phi)$ is a spin structure on $\mathcal{C}$ where
$\omega_{\mathcal{C}}^{\text{log}}$ and $\theta$ are line bundles over the
stable twisted curve $\mathcal{C}$ with labeled orbifold points $p_{j}$ and
$\deg\theta=g-1+\frac{1}{2}n$. A line bundle $L$ over a twisted curve
$\mathcal{C}$ is a locally equivariant bundle over the local charts, such that
at each nodal point there is an equivariant isomorphism of fibres. Hence each
orbifold point $p$ associates a representation of $\mathbb{Z}_{2}$ on $L|_{p}$
acting by multiplication by $\exp(\pi i\sigma_{p})$ for
$\sigma_{p}\in\\{0,1\\}$. The equivariant isomorphism of fibres over nodal
points forces a balanced condition at each node $p$ given by
$\sigma_{p_{+}}=\sigma_{p_{-}}$ for $p_{\pm}$ corresponding to the node on
each irreducible component.

Define a vector bundle over $\overline{\cal M}_{g,n}^{\rm spin}$ using the
dual bundle $\theta^{\vee}$ on each stable twisted curve as follows. Denote by
$\mathcal{E}$ the universal spin structure on the universal stable twisted
spin curve over $\overline{\cal M}_{g,n}^{\rm spin}$. Given a map
$S\to\overline{\cal M}_{g,n}^{\rm spin}$, $\mathcal{E}$ pulls back to $\theta$
giving a family $(\mathcal{C},\theta,p_{1},...,p_{n},\phi)$ where
$\pi:\mathcal{C}\to S$ has stable twisted curve fibres,
$p_{i}:S\to\mathcal{C}$ are sections with orbifold isotropy $\mathbb{Z}_{2}$
and
$\phi:\theta^{2}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\omega_{\mathcal{C}/S}^{\text{log}}=\omega_{\mathcal{C}/S}(p_{1},..,p_{n})$.
The derived push-forward sheaf of $\mathcal{E}^{\vee}$ over $\overline{\cal
M}_{g,n}^{\rm spin}$ gives rise to a bundle as follows. The space of global
sections of $\pi_{*}\mathcal{E}^{\vee}$ vanishes, i.e.
$R^{0}\pi_{*}\mathcal{E}^{\vee}=0$, since

$\deg\theta^{\vee}=1-g-\frac{1}{2}n<0$

and furthermore, for any irreducible component
$\mathcal{C}^{\prime}\stackrel{{\scriptstyle i}}{{\to}}\mathcal{C}$,

$\deg\theta^{\vee}|_{\mathcal{C}^{\prime}}<0$

since
$i^{*}\omega_{\mathcal{C}/S}^{\text{log}}=\omega_{\mathcal{C}^{\prime}/S}^{\text{log}}$
and $\mathcal{C}^{\prime}$ is stable guarantees its log canonical bundle has
negative degree. Thus $R^{1}\pi_{*}\mathcal{E}^{\vee}$ defines the following
bundle.

######  Definition 5.5.

Define a bundle $E_{g,n}=-R\pi_{*}\mathcal{E}^{\vee}$ over $\overline{\cal
M}_{g,n}^{\rm spin}$ with fibre $H^{1}(\theta^{\vee})$.

The moduli space of stable twisted spin curves decomposes into components
labeled by $\vec{\sigma}\in\\{0,1\\}^{n}$ satisfying
$\displaystyle|\vec{\sigma}|+n\in 2\mathbb{Z}$:

$\overline{\cal M}_{g,n}^{\rm spin}=\bigsqcup_{\vec{\sigma}}\overline{\cal
M}_{g,n,\vec{\sigma}}^{\rm spin}$

where $\overline{\cal M}_{g,n,\vec{\sigma}}^{\rm spin}$ consists of spin
curves with induced representation at labeled points determined by
$\vec{\sigma}$ as follows. The representations induced by $\theta$ at the
labeled points produce a vector
$\vec{\sigma}=(\sigma_{1},...,\sigma_{n})\in\\{0,1\\}^{n}$ which determine the
connected component. The number of $p_{i}$ with $\lambda_{p_{i}}=0$ is even
due to evenness of the degree of the push-forward sheaf
$|\theta|:=\rho_{*}\mathcal{O}_{\mathcal{C}}(\theta)$ on the coarse curve $C$,
[23]. In the smooth case, the boundary type of a spin structure is determined
by an associated quadratic form applied to each of the $n$ boundary classes
which vanishes since it is a homological invariant, again implying that the
number of $p_{i}$ with $\lambda_{p_{i}}=0$ is even. Each component
$\overline{\cal M}_{g,n,\vec{\sigma}}^{\rm spin}$ is connected except when
$|\vec{\sigma}|=n$, in which case there are two connected components
determined by their Arf invariant, and known as even and odd spin structures.
This follows from the case of smooth spin curves proven in [28].

Restricted to $\overline{\cal M}_{g,n,\vec{\sigma}}^{\rm spin}$, the bundle
$E_{g,n}$ has rank

(30) $\text{rank\ }E_{g,n}=2g-2+\tfrac{1}{2}(n+|\vec{\sigma}|)$

by the following Riemann-Roch calculation. Orbifold Riemann-Roch takes into
account the representation information

$\displaystyle h^{0}(\theta^{\vee})-h^{1}(\theta^{\vee})$
$\displaystyle=1-g+\deg\theta^{\vee}-\sum_{i=1}^{n}\lambda_{p_{i}}=1-g+1-g-\tfrac{1}{2}n-\tfrac{1}{2}|\vec{\sigma}|$
$\displaystyle=2-2g-\tfrac{1}{2}(n+|\vec{\sigma}|).$

We have $h^{0}(\theta^{\vee})=0$ since $\deg\theta^{\vee}=1-g-\frac{1}{2}n<0$,
and the restriction of $\theta^{\vee}$ to any irreducible component
$C^{\prime}$, say of type $(g^{\prime},n^{\prime})$, also has negative degree,
$\deg\theta^{\vee}|_{C^{\prime}}=1-g^{\prime}-\frac{1}{2}n^{\prime}<0$. Hence
$h^{1}(\theta^{\vee})=2g-2+\frac{1}{2}(n+|\vec{\sigma}|)$. Thus
$H^{1}(\theta^{\vee})$ gives fibres of a rank
$2g-2+\frac{1}{2}(n+|\vec{\sigma}|)$ vector bundle.

The forgetful map $p:\overline{\cal M}_{g,n}^{\rm spin}\to\overline{\cal
M}_{g,n}$ and its restriction $p^{\bar{\sigma}}:\overline{\cal M}_{g,n}^{\rm
spin}\to\overline{\cal M}_{g,n,\bar{\sigma}}$ forgets the spin structure and
maps the twisted curve to its underlying coarse curve.

######  Definition 5.6.

On $\mathbb{C}^{2}=\langle e_{0},e_{1}\rangle$ with metric
$\eta(e_{i},e_{j})=\frac{1}{2}\delta_{ij}$ define the CohFT

$\Omega^{\text{spin}}_{g,n}(e_{\sigma_{1}}\otimes...\otimes
e_{\sigma_{n}}):=p^{\bar{\sigma}}_{*}c(E_{g,n})\in H^{*}(\overline{\cal
M}_{g,n},\mathbb{Q}).$

The different components of the moduli space behave like the different
components of the moduli space of stable maps used in the CohFT associated to
Gromov-Witten invariants. The proof that $\\{\Omega^{\text{spin}}_{g,n}\\}$
defines a CohFT is a special case of a more general result in [25] which
proves that push-forwards of classes defined over moduli spaces of $r$-spin
curves known as Chiodo classes produce dimension $r$ CohFTs. The proof in [25]
uses the Givental action on CohFTs, or instead one can use natural restriction
properties of the bundles $E_{g,n}$ analogous to (24) proven in [30].

The CohFT vanishes in degrees greater than $2g-2+n$ = the largest rank of
$E_{g,n}$ on components of $\overline{\cal M}_{g,n}^{\rm spin}$, hence its
degree $2g-2+n$ terms define a CohFT without unit:

$\Theta_{g,n}:=(-1)^{n}2^{g-1+n}p_{*}c_{2g-2+n}(E_{g,n})\in H^{4g-4+2n}.$

For example, $\Theta_{1,1}=3\psi_{1}$ and $\Theta_{0,n}=0$ for all $n$. The
factor of 2 in the bivector $\Delta=2(e_{0}\otimes e_{0}+e_{1}\otimes e_{1})$
leads to the rescaling by a power of 2, which is quite naturally given by
$2^{1-g}$, and we choose the further rescaling by $(-2)^{2g-2+n}$ to produce
the elegant formula involving the Brézin-Gross-Witten tau function of the KdV
hierarchy in Proposition 5.7 below. Define the partition function

$Z^{\Theta}(\hbar,t_{0},t_{1},...)=\exp\sum_{g,n,\vec{k}}\frac{\hbar^{g-1}}{n!}\int_{\overline{\cal
M}_{g,n}}\Theta_{g,n}\cdot\prod_{j=1}^{n}\psi_{j}^{k_{j}}\prod t_{k_{j}}.$

######  Proposition 5.7.

$Z^{\text{BGW}}(\hbar,t_{0},t_{1},...)=Z^{\Theta}(\hbar,t_{0},t_{1},...)+O(\hbar^{7}).$

######  Remark 5.8.

The $O(\hbar^{7})$ term is conjectured in [30] to vanish which is a
consequence of Conjecture 4, that $K_{2g-2+n}=\Theta_{g,n}$.

###### Proof.

The strategy of proof is to prove equality of intersection numbers given by
$Z^{\Theta}(\hbar,t_{0},t_{1},...)=e^{\hbar^{-1}F^{K}}$ and then to use
Theorem 6. Such intersection

1\. Reduce the equality to finitely many terms for each genus $g$ as follows.
It is proven in [30] that the classes $\Theta_{g,n}$ satisfy the pull-back
property

$\Theta_{g,n+1}=\psi_{n+1}\cdot\pi^{*}\Theta_{g,n}$

which implies that $Z^{\Theta}(\hbar,t_{0},t_{1},...)$ satisfies the following
dilaton equation.

(31) $\frac{\partial}{\partial
t_{0}}Z^{\Theta}(\hbar,t_{0},t_{1},...)=\sum_{i=0}^{\infty}(2i+1)t_{i}\frac{\partial}{\partial
t_{i}}Z^{\Theta}(\hbar,t_{0},t_{1},...)+\frac{1}{8}Z^{\Theta}(\hbar,t_{0},t_{1},...)$

A consequence is that the intersection numbers $\int_{\overline{\cal
M}_{g,n}}\Theta_{g,n}\cdot\prod_{j=1}^{n}\psi_{j}^{k_{j}}$ for $n\geq g$ are
uniquely determined by those for $n<g$. Moreover,
$Z^{\text{BGW}}(\hbar,t_{0},t_{1},...)$ also satisfies (31) hence equality of
$\log Z^{\Theta}$ and $\log Z^{\text{BGW}}$ for a given genus $g$ follows from
the finitely many intersection numbers for $(g,n)$ where $n<g$.

2\. Assumption: (29) holds for all $(g,n)$ with $g\leq 7$ and $n\leq g-1$.
Using the software admcycles we can only directly calculate terms on the RHS
up to genus 4 so we instead use a weak argument as described in Remark 5.4.
Given the assumption above, let $\omega\in RH^{g-1}(\overline{\cal M}_{g,n})$
be a polynomial in $\kappa$ classes and $\psi$ classes for $g\leq 7$ and
$n\leq g-1$. Then for $g>1$

$\displaystyle\int_{\overline{\cal M}_{g,n}}K_{2g-2+n}\cdot\omega$
$\displaystyle=\int_{\overline{\cal
M}_{g,n}}\psi_{n}\pi^{*}K_{2g-2+n-1}\cdot\omega=\int_{\overline{\cal
M}_{g,n-1}}K_{2g-2+n-1}\pi_{*}(\psi_{n}\omega_{n})$
$\displaystyle=\int_{\overline{\cal
M}_{g,n-1}}K_{2g-2+n-1}\omega_{n-1}=\int_{\overline{\cal
M}_{g}}\pi^{*}K_{2g-2}\cdot\omega_{0}(\kappa_{1},\kappa_{2},...,\kappa_{g-1})$

where the first equality used the assumption (29). Also, $\omega_{n}:=\omega$
and inductively $\pi_{*}(\psi_{k}\omega_{k})=\omega_{k-1}\in
RH^{g-1}(\overline{\cal M}_{g,k-1})$ is a polynomial in $\kappa$ classes and
$\psi$ classes. This uses the inductive hypothesis that $\omega_{k}$ is a
polynomial in $\kappa$ classes and $\psi$ classes, together with the property
for $j<k$, $\psi_{k}\psi_{j}=\psi_{k}\pi^{*}\psi_{j}\in RH^{*}(\overline{\cal
M}_{g,k-1})$ and push-forward properties of $\kappa$ classes obtained by
replacing $\kappa_{j}$ by $\kappa_{\ell_{j}}=\pi^{*}\kappa_{j}+\psi_{n}^{j}$
and push-forward all summands. When $g=1$, the right hand side is instead
$\int_{\overline{\cal M}_{1,1}}K_{1}\cdot p$ for $p\in\mathbb{Q}$ a constant.
The same inductive argument proves for $g>1$

$\int_{\overline{\cal M}_{g,n}}\Theta_{g,n}\cdot\omega=\int_{\overline{\cal
M}_{g}}\Theta_{g}\cdot\omega_{0}(\kappa_{1},\kappa_{2},...,\kappa_{g-1})$

and again when $g=1$, the right hand side is $\int_{\overline{\cal
M}_{1,1}}\Theta_{1,1}\cdot p$ for $p\in\mathbb{Q}$ a constant.

By a result of Faber and Pandharipande [18, Proposition 2], a homogeneous
degree $g-1$ monomial in the $\kappa$ classes is equal in the tautological
ring to the sum of boundary terms, i.e. the sum of push-forwards of
polynomials in $\psi$ classes and $\kappa$ classes by the the maps
$(\phi_{\Gamma})_{*}$. But the pull-back of $K_{2g-2}$ and $\Theta_{g}$ to any
boundary divisor is a tensor product of $K_{2g^{\prime}-2+n^{\prime}}$ terms,
respectively $\Theta_{g^{\prime},n^{\prime}}$ terms, for $g^{\prime}<g$. The
pull-back of $K_{2g-2}$ uses the pull-back of the exponential $\mathbb{K}$
together with the weak vanishing of $K_{m}$ for $m>2g^{\prime}-2+n^{\prime}$.
By induction, $K_{2g-2+n}$ and $\Theta_{g,n}$ agree weakly, for lower genus,
hence

$\int_{\overline{\cal M}_{g,n}}(K_{2g-2+n}-\Theta_{g,n})\cdot\omega=0$

for any polynomial $\omega$ in $\psi$ classes and $\kappa$ classes (and in
fact any tautological class) so in particular any polynomial in $\psi$
classes. By Theorem 6 this proves the required result under assumption 1
above, for $g\leq 7$ and $n\leq g-1$ and hence by the reduction in 1 above,
for $g\leq 7$ and $n\geq 0$.

It remains to prove the assumption that (29) holds for all $(g,n)$ with $g\leq
7$ and $n\leq g-1$. This is a relation between intersection numbers involving
only polynomials in $\psi$ classes and $\kappa$ classes which have been well-
studied. It is achieved using different software, in particular by admcycles
for $g\leq 7$ and $n\leq g-1$, hence the proposition follows. ∎

###  5.3. Monotone Hurwitz numbers

The class $\mathbb{K}=1+K_{1}+K_{2}+\dots$ of Theorem 1 naturally arises also
in the study of monotone Hurwitz numbers. The _monotone Hurwitz numbers_
$\vec{h}_{g,(k_{1},\dots,k_{n})}$ enumerate factorizations in the symmetric
group $S(\sum k_{i})$ into transpositions of the form

$(a_{1},b_{1})\circ(a_{2},b_{2})\circ\dots\circ(a_{m},b_{m}),\quad
a_{i}<b_{i},$

such that the product has cyclic type $(k_{1},\dots,k_{n})$, the integers $m$
and $g$ are related by the Riemann-Hurwitz relation $m=2g-2+n+\sum k_{i}$, the
subgroup generated by transpositions acts transitively on the set of permuted
elements, and also the following additional monotonicity condition is
satisfied: the bigger numbers $b_{i}$ of the pairs of transposed elements
satisfy the inequality $b_{1}\leq b_{2}\leq\dots\leq b_{m}$.

For relations of the present section it would be more convenient to consider a
slightly rescaled version of the class $\mathbb{K}$, namely, define

$\mathbb{K}^{*}=1-K_{1}+K_{2}-K_{3}+\dots=\exp\big{(}\sum(-1)^{i}s_{i}\kappa_{i}\big{)}$

where $s_{i}$ are as in Section 1.1. We have proved in Section 2 that the
differentials $\omega_{g,n}$ of topological recursion with the rational
spectral curve

$x=\frac{z^{2}}{2},\qquad y=\frac{z}{1-z^{2}}.$

are given explicitly for $2g-2+n>0$ by

(32) $w_{g,n}(z_{1},\dots,z_{n})=\sum_{k_{1},\dots,k_{n}}\int_{\overline{\cal
M}_{g,n}}\mathbb{K}^{*}\;\psi_{1}^{k_{1}}\dots\psi_{n}^{k_{n}}~{}\prod_{i=1}^{n}\frac{(2k_{i}+1)!!dz_{i}}{z^{2k_{i}+2}_{i}}$

(it corresponds to the specialization $\epsilon=-1$ in (9)).

It turns out that the monotone Hurwitz numbers are governed by the very same
topological recursion with a special choice of the expansion point $z=1$ and
the local coordinate $X$ at this point defined by the change $z=\sqrt{1-4X}$.

######  Proposition 5.9 ([10]).

In the stable range, that is, for each $(g,n)$ with $2g-2+n>0$, the power
expansions of the above differential $\omega_{g,n}(z_{1},\dots,z_{n})$ in the
local coordinates $X_{1},\dots,X_{n}$ obtained by the substitution
$z_{i}=\sqrt{1-4X_{i}}$ are given by

$\omega_{g,n}=\sum_{k_{1},\dots,k_{n}}\vec{h}_{g,(k_{1},\dots,k_{n})}\prod_{i=1}^{n}k_{i}X_{i}^{k_{i}-1}dX_{i}.$

###### Proof.

In fact, a slightly different but equivalent form of topological recursion for
the monotone Hurwitz numbers is derived in [10]. Its spectral curve data
corresponds to the differential $\omega_{0,1}=-Y\,dX$ with

$X(Z)=\frac{Z-1}{Z^{2}},\quad Y(Z)=-Z,\qquad X_{i}=X(Z).$

Here $Z$ is the global affine coordinate on the spectral curve
$\Sigma=\mathbb{C}P^{1}$ and $X$ is a local coordinate at $Z=\infty$ that
extends as a degree $2$ global rational function on the spectral curve.
Applying an affine change $Z=\frac{2}{1-z}$ of the coordinate on the spectral
curve we can rewrite its equation in an equivalent form

$x(z)=\frac{1}{2}-2X=\frac{z^{2}}{2},\quad y(z)=-\frac{1}{2}Y=\frac{1}{1-z}.$

Both spectral curves produce the same topological recursion since
$-Y\,dX=-y\,dx$. The expansion point $Z=\infty$ corresponds to $z=1$ and the
function $X$ takes in the $z$ coordinate the form
$X=\frac{1}{4}-\frac{1}{2}x=\frac{1-z^{2}}{4}$ with the inverse change
$z=\sqrt{1-4X}$. Besides, adding an even summand to the $y$-function does not
affect the differentials $\omega_{g,n}$ of topological recursion for
$(g,n)\neq(0,1)$. Therefore, we can replace the $y$-function in the above
equation of the spectral curve by its odd part with respect to the involution
$z\mapsto-z$ which is

$\frac{1}{2}\Bigl{(}\frac{1}{1-z}-\frac{1}{1+z}\Bigr{)}=\frac{z}{1-z^{2}}.$

This proves an equivalence of the two forms of topological recursion for the
monotone Hurwitz numbers. ∎

In recent work [5], the structure of partition functions for a large class of
Hurwitz numbers, which includes simple and monotone Hurwitz numbers, has been
studied. One property of this structure is the relationship of these Hurwitz
numbers to topological recursion.

The proposition implies that the monotone Hurwitz numbers can be expressed as
linear combinations of the intersection numbers involving $\psi$ and $\kappa$
classes. More explicitly, it is easy to compute that the form
$\frac{(2k+1)!!dz}{z^{2k+2}}$ is given in the local coordinate $X$ at $z=1$ by

$\frac{(2k+1)!!dz}{z^{2k+2}}=\sum_{i=1}^{\infty}(2i+1)(2i+3)\dots(2(i+k)-1)\binom{2i}{i}i\,X^{i-1}dX.$

Substituting into (32) and selecting the coefficient of a particular monomial
in $X$-variables we arrive at the following known analog of the ELSV formula,
see [2, 11, 12]. Denote

$\Phi_{k}(\psi)=1+(2k+1)\psi+(2k+1)(2k+3)\psi^{2}+\dots=\sum_{i=0}^{\infty}\left(\prod_{j=1}^{i}(2(j+k)-1)\right)\psi^{i}$

where we use the convention that the empty product is 1.

######  Corollary 5.10.

The monotone Hurwitz numbers in the case $2g-2+n>0$ can be computed as the
following intersection numbers

$\displaystyle\vec{h}_{g,(k_{1},\dots,k_{n})}$
$\displaystyle=\prod_{i=1}^{n}\\!\\!\binom{2k_{i}}{k_{i}}~{}\vec{P}_{g,n}(k_{1},\dots,k_{n}),$
$\displaystyle\vec{P}_{g,n}(k_{1},\dots,k_{n})$
$\displaystyle=\int_{\overline{\cal
M}_{g,n}}\mathbb{K}^{*}\cdot\Phi_{k_{1}}(\psi_{1})\dots\Phi_{k_{n}}(\psi_{n}).$

Note that, by definition, $\vec{P}_{g,n}$ is a symmetric polynomial in
$k_{1},\dots,k_{n}$. A consequence of Theorem 1 is that for $g>1$,

$\vec{P}_{g,n}(-\tfrac{1}{2},-\tfrac{1}{2},\dots,-\tfrac{1}{2})=0$

since $\Phi_{-\frac{1}{2}}(\psi)=1$ and $\deg\mathbb{K}^{*}<3g-3+n$ weakly.
More generally,

$\deg\Phi_{-\frac{1}{2}(2m+1)}(\psi)=m$

hence by Theorem 1 the following vanishing properties of the polynomials
$\vec{P}_{g,n}$ evaluated on negative half-integers holds.

######  Corollary 5.11.

For $m_{i}\in\mathbb{N}$,

$\sum_{i=1}^{n}m_{i}<g-1\quad\Rightarrow\quad\vec{P}_{g,n}\big{(}-\tfrac{1}{2}(2m_{1}+1),\dots,-\tfrac{1}{2}(2m_{n}+1)\big{)}=0.$

By direct calculation, via the Witten-Kontsevich Theorem 5 or with admcycles,
one can verify:

$\int_{\overline{\cal
M}_{g}}K_{3g-3}=(-1)^{g}\frac{B(2g)}{2g(2g-2)}=|\chi(\cal M_{g})|,\quad 2\leq
g\leq 9$

where $\chi(\cal M_{g})$ is the orbifold Euler characteristic of the moduli
space of curves. Using the spectral curve and recursions between the 1-point
correlators proven in [6] it is likely that we can prove this formula for all
$g\geq 2$, using a method analogous to the calculation of the orbifold Euler
characteristic of the moduli space of curves by Harer and Zagier in [21] from
their recursions between the 1-point correlators.

## References

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