# On the strong DR/DZ equivalence conjecture

Xavier Blot  X. B.: Korteweg-de Vriesinstituut voor Wiskunde, Universiteit van
Amsterdam, Postbus 94248, 1090GE Amsterdam, Nederland<EMAIL_ADDRESS>,
Danilo Lewański  D. L.: Dipartimento di Matematica, Informatica e Geoscienze,
Università degli studi di Trieste, Via Weiss 2, 34128 Trieste, Italia
<EMAIL_ADDRESS>and  Sergei Shadrin  S. S.: Korteweg-de
Vriesinstituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248,
1090GE Amsterdam, Nederland<EMAIL_ADDRESS>
###### Abstract.

We establish the Miura equivalence of two integrable systems associated to a
semi-simple cohomological field theory: the double ramification hierarchy of
Buryak and the Dubrovin-Zhang hierarchy. This equivalence was conjectured by
Buryak and further refined by Buryak, Dubrovin, Guéré, and Rossi.

###### Contents

  1. 1 Introduction
    1. 1.1 Cohomological field theories
    2. 1.2 Dubrovin-Zhang hierarchy
    3. 1.3 Buryak’s double ramification hierarchy
    4. 1.4 Normal Miura equivalence
    5. 1.5 Generalizations and approach to the proof
    6. 1.6 Structure of the paper
    7. 1.7 Acknowledgments
  2. 2 The $A=B$ relation in the Gorenstein quotient
    1. 2.1 Basic notation for trees
    2. 2.2 Leveled stable rooted trees
    3. 2.3 Definition of the $B$ class
    4. 2.4 The conjectural $A=B$ relation revisited
    5. 2.5 The $A=B$ identity in the Gorenstein quotient
    6. 2.6 The trivial CohFT
  3. 3 The master relation in the Gorenstein quotient
    1. 3.1 Main definitions and statements
      1. 3.1.1 Pre-stable star rooted trees
      2. 3.1.2 Classes assigned to vertices
      3. 3.1.3 Classes assigned to trees
      4. 3.1.4 Classes assigned to sets of trees
    2. 3.2 Intersection with a divisor
    3. 3.3 Intersection with a kappa class
    4. 3.4 Intersection with a $\psi$ class at a regular leg
  4. 4 Proof of the main theorem
    1. 4.1 Strategy of proof of Theorem 2.3
      1. 4.1.1 Step 1
      2. 4.1.2 Step 2
      3. 4.1.3 Step 3
    2. 4.2 Step $1$: the key combinatorial lemma
    3. 4.3 Step $2$: equivalence between the master relation and DR/DZ
    4. 4.4 Step 3: proof of the master relation in the Gorenstein quotient

##  1\. Introduction

###  1.1. Cohomological field theories

Let $V=\langle e_{1},\dots,e_{N}\rangle$ be a finite dimensional vector space
equipped with a symmetric nondegenerate bilinear form $\eta$. A cohomological
field theory (CohFT) [23] is a system of linear maps

(1.1) $\displaystyle c_{g,n}\colon V^{\otimes n}\to
H^{*}({\overline{\mathcal{M}}}_{g,n}),\quad 2g-2+n>0,$

such that

  1. (1)

The maps $c_{g,n}$ are equivariant with respect to the $S_{n}$-action
permuting the factors of $V$ in $V^{\otimes n}$ and the labels of the marked
points in ${\overline{\mathcal{M}}}_{g,n}$.

  2. (2)

$\pi^{*}c_{g,n}(\otimes_{i=1}^{n}e_{\alpha_{i}})=c_{g,n+1}(\otimes_{i=1}^{n}e_{\alpha_{i}}\otimes
e_{1})$, where
$\pi\colon{\overline{\mathcal{M}}}_{g,n+1}\to{\overline{\mathcal{M}}}_{g,n}$
is the map that forgets the last marked point. Moreover,
$c_{0,3}(e_{\alpha}\otimes e_{\beta}\otimes e_{1})=\eta(e_{\alpha}\otimes
e_{\beta})=:\eta_{\alpha\beta}$.

  3. (3)

$\mathrm{gl}^{*}c_{g,n}(\otimes_{i=1}^{n}e_{\alpha_{i}})=\eta^{\mu\nu}c_{g_{1},n_{1}+1}(\otimes_{i\in
I_{1}}e_{\alpha_{i}}\otimes e_{\mu})\otimes c_{g_{2},n_{2}+1}(\otimes_{j\in
I_{2}}e_{\alpha_{j}}\otimes e_{\nu})$, where $I_{1}\sqcup
I_{2}=\\{1,\dots,n_{1}+n_{2}\\}$, $|I_{1}|=n_{1}$, $|I_{2}|=n_{2}$,
$g_{1}+g_{2}=g$, and
$\mathrm{gl}\colon{\overline{\mathcal{M}}}_{g_{1},n_{1}+1}\times{\overline{\mathcal{M}}}_{g_{2},n_{2}+1}\to{\overline{\mathcal{M}}}_{g,n}$
is the corresponding gluing map.

  4. (4)

$\mathrm{gl}^{*}c_{g,n}(\otimes_{i=1}^{n}e_{\alpha_{i}})=\eta^{\mu\nu}c_{g-1,n+2}(\otimes_{i=1}^{n}e_{\alpha_{i}}\otimes
e_{\mu}\otimes e_{\nu})$, where
$\mathrm{gl}\colon{\overline{\mathcal{M}}}_{g-1,n+2}\to{\overline{\mathcal{M}}}_{g,n}$
is the corresponding gluing map.

CohFTs capture the algebraic properties of enumerative problems in different
contexts, among which Gromov-Witten theory. They are fully classified in the
case $c_{0,3}$ defines a structure of a semi-simple Frobenius algebra on $V$
in [26]. In particular, under this assumption they consist of tautological
classes.

One associates to a CohFT its potential

(1.2) $\displaystyle
F\coloneqq\sum\frac{\varepsilon^{2g}}{n!}\left(\int_{{\overline{\mathcal{M}}}_{g,n}}c_{g,n}(\otimes_{i=1}^{n}e_{\alpha_{i}})\prod_{i=1}^{n}\psi_{i}^{d_{i}}\right)\prod_{i=1}^{n}t^{\alpha_{i},d_{i}}$

and a partition function $Z\coloneqq\exp(\epsilon^{-2}F)$.

###  1.2. Dubrovin-Zhang hierarchy

One can associate to a semi-simple CohFT a Hamiltonian integrable hierarchy of
hydrodynamic type. This construction was initially developed by Dubrovin and
Zhang [18], and subsequently by Liu, Wang, and Zhang [24], in the framework of
calibrated Frobenius manifolds. In the context of CohFTs, calibration refers
to an extra homogeneity property.

A more general approach can be applied to any semi-simple CohFT and it is
developed in [10, 11]. In a nutshell, one can define a new system of
coordinates

(1.3) $\displaystyle
w^{\alpha,d}\coloneqq\partial_{x}^{d}\eta^{\alpha\beta}\frac{\partial^{2}F}{\partial
t^{\beta,0}t^{1,0}}\big{|}_{t^{1,0}\to t^{1,0}+x}.$

The double derivatives of $F$ are proved to be differential polynomials in
these new coordinates,

(1.4) $\displaystyle\frac{\partial^{2}F}{\partial t^{\alpha,p}\partial
t^{\beta,q}}=\Omega_{\alpha,p;\beta,q}(\\{w^{\gamma,d}\\}),$

and the following evolutionary integrable system possesses a polynomial
Hamiltonian structure of hydrodynamic type with a Poisson bracket
$K^{\mathrm{DZ}}=\eta^{\alpha\beta}{\partial}_{x}+O(\epsilon^{2})$ and a tau
structure:

(1.5) $\displaystyle\frac{\partial w^{\alpha}}{\partial
t^{\beta,q}}=\partial_{x}\eta^{\alpha\gamma}\Omega_{\gamma,0;\beta,q}.$

This system is described explicitly in a number of examples: for the trivial
CohFT it is the KdV hierarchy [27, 22], see also [2], and for the Witten
$r$-spin CohFT it is the Gelfand-Dickey hierarchy [28, 19].

###  1.3. Buryak’s double ramification hierarchy

Buryak constructed a different integrable system associated to a CohFT in [5].
Let $u^{1},\ldots,u^{N}$ be formal dependent variables, and let
$u^{\alpha,d}\coloneqq{\partial}_{x}^{d}u^{\alpha}$. Define differential
polynomials $P^{\alpha}_{\beta,d}$ by

(1.6) $\displaystyle P^{\alpha}_{\beta,d}$
$\displaystyle\coloneqq\eta^{\alpha\gamma}\sum_{\begin{subarray}{c}g,n\geq
0,\,2g+n>0\\\ k_{1},\ldots,k_{n}\geq 0\\\
\sum_{j=1}^{n}k_{j}=2g\end{subarray}}\frac{\varepsilon^{2g}}{n!}\prod_{j=1}^{n}u^{\alpha_{j},k_{j}}\mathrm{Coef}_{a_{1}^{k_{1}}\cdots
a_{n}^{k_{n}}}I^{g,d}_{\gamma,\beta,\alpha_{1},\dots,\alpha_{n}},$

where $I^{g,d}_{\gamma,\beta,\alpha_{1},\dots,\alpha_{n}}$ is a polynomial of
degree $2g$ in the variables $a_{1},\dots,a_{n}$ given by

(1.7) $\displaystyle
I^{g,d}_{\gamma,\beta,\alpha_{1},\dots,\alpha_{n}}\coloneqq\left(\int_{{\overline{\mathcal{M}}}_{g,n+2}}\lambda_{g}{\mathrm{DR}_{g}\bigg{(}0,-\sum_{j=1}^{n}a_{j},a_{1},\ldots,a_{n}\bigg{)}}\psi_{1}^{d}c_{g,n+2}\Big{(}e_{\beta}\otimes
e_{\gamma}\otimes\bigotimes_{j=1}^{n}e_{\alpha_{j}}\Big{)}\right).$

Buryak’s DR hierarchy [5] is the following integrable system of evolutionary
PDEs:

(1.8)
$\displaystyle\frac{{\partial}u^{\alpha}}{{\partial}t^{\beta,d}}={\partial}_{x}P^{\alpha}_{\beta,d}.$

It possesses the Hamiltonian structure given by
$\eta^{\alpha\beta}{\partial}_{x}$ and a tau structure. Note, however, that
$u^{\alpha}$ are not in general the normal coordinates for the tau structure.

The further properties of this integrable system and its generalizations are
studied in a number of papers, see [13, 6, 7, 15]. The importance of the
Buryak’s approach to integrable systems associated to semi-simple CohFTs is in
particular justified by that fact that it admits a quantization with respect
to its Poisson structure, see [12, 7].

###  1.4. Normal Miura equivalence

In general the Miura group acts on (Hamiltonian) integrable systems by the
diffeomorphic changes of dependent variables, see e.g. [18]. More refined are
the so-called normal Miura transformations that transform a tau-symmetric
Hamiltonian hierarchy written in normal coordinates to a tau-symmetric
Hamiltonian hierarchy written in normal coordinates.

Let $\tilde{u}^{\alpha}$ be the normal coordinates for Buryak’s double
ramification hierarchy. The normal coordinates for the Dubrovin-Zhang
hierarchy are $w^{\alpha}$. The following conjecture is proposed in [6,
Conjecture 7.3]:

######  Conjecture 1.1 (Strong DR/DZ equivalence conjecture).

For any semi-simple cohomological field theory there exists a differential
polynomial $\mathcal{P}$ in coordinates $w^{\alpha}$ (and it has a very
explicit description that we omit here) such that the normal Miura
transformation defined as

(1.9) $\displaystyle
w^{\alpha}\mapsto\tilde{u}^{\alpha}(w)=w^{\alpha}+\eta^{\alpha\gamma}{\partial}_{x}\\{\mathcal{P},\bar{h}^{\mathrm{DZ}}_{\mu,0}\\}_{K^{DZ}}$

maps the Dubrovin-Zhang hierarchy to Buryak’s double ramification hierarchy
written in the normal coordinates.

In the statement of the conjecture $\bar{h}^{\mathrm{DZ}}_{\mu,0}$ is the
Hamiltonian of the time flow $\partial_{t^{\mu,0}}$ for the Dubrovin-Zhang
hierarchy; $\\{\bullet,\bullet\\}_{K^{DZ}}$ denotes the Poisson bracket of the
Dubrovin-Zhang hierarchy. This conjecture refines an earlier conjecture of
Buryak [5] on Miura equivalence of the Dubrovin-Zhang and double ramification
hierarchies.

The main application of our work is the following theorem:

######  Theorem 1.2.

Conjecture 1.1 holds.

###  1.5. Generalizations and approach to the proof

There is a number of generalizations of the strong DR/DZ equivalence
conjecture for semi-simple CohFTs above. In particular, the condition of semi-
simplicity can be dropped (but then one has to prove that the Dubrovin-Zhang
integrable system is given by differential polynomials in the dependent
variables), and similar constructions can be applied to the so-called partial
CohFTs and F-CohFTs (in the latter case we get just a system of conservation
laws rather than a Hamiltonian system). These constructions are available in
[1, 14, 16], and the corresponding conjectures there state that

  1. (1)

On the Dubrovin-Zhang side, the corresponding integrable system is represented
by differential polynomials.

  2. (2)

The Dubrovin-Zhang integrable system is normal Miura equivalent to the double
ramification system.

We refer to [16, Section 4] for the precise statement and further details. In
[16], following and generalizing earlier results in [6, 8], these conjectures
are reduced to a system of conjectural tautological relations in the moduli
space of curves known as a generalization of the $A=B$ relation. The initial
$A=B$ relation itself was proposed in [8] and designed to imply Conjecture
1.1. It is convenient to let the term “$A=B$ relation” refer to the whole
system of (generalized) $A=B$ relations.

In this paper we prove the $A=B$ relation as well as all its generalization
from [16] in the Gorenstein quotient. In other words we prove the relations
after intersection with tautological classes of the complementary dimension
(Theorem 2.3). As a consequence, our result implies the entire system of
conjectures for not-necessary semi-simple CohFTs, as well as partial CohFTs
and F-CohFTs under the assumption that the classes involved in these
structures are tautological. Since this is always the case for semi-simple
CohFTs, as a corollary Theorem 2.3 implies Conjecture 1.1 (Theorem 1.2). We
remark that Theorem 2.3 is the main result of this paper, and we moreover
refer further to [16, Section 4] for a list of corollaries descending from it.

###  1.6. Structure of the paper

This paper is by no means self-closed as it is based on an enormous amount of
prior work; we expect the reader to be familiar with the theory of (infinite-
dimensional) integrable systems of evolutionary type along the lines of [18]
and [6] as well as the intersection theory of the moduli spaces of curves and
the structure of its tautological ring, along the lines of [20] and [17].
Moreover, it might be instructive to follow the big steps in the literature
where the conjecture of Buryak and its generalizations were further developed,
which we summarise in the following.

  * •

Buryak’s double ramification hierachy and the original conjecture are
formulated in [5]; the conjecture is proved there for the trivial CohFT;

  * •

The conjecture is refined to be an explicitly presented normal Miura
transformation in [6]; the strong DR/DZ equivalence is proved there for the
trivial CohFT;

  * •

The refined conjecture is reduced to a system of relations in the tautological
ring of the moduli space of curves in [8];

  * •

The tautological relations is further generalized to a form that we study here
and is connected to a number of more general settings explained in [16];

  * •

The relations expressions are slightly simplified in [3], in line with a twin
system of relations that involve the so-called $\Omega$-classes;

  * •

Some new combinatorial insights, that in particular allowed to prove the twin
system of tautological relations, are obtained in [4];

  * •

Finally, here we prove the desired relations in the Gorenstein quotient.

Starting from this point we only talk about classes in the tautological ring
of the moduli space of curves. In Section 2 we introduce the conjectural
tautological relations, the generalized $A=B$ relations, in a bit more compact
form than they were presented before, and recall some basic facts about their
intersections with $\psi$-classes that follow from the strong DR/DZ
equivalence for the trivial CohFT.

In Section 3 we introduce a new system of conjectural tautological relations,
the so-called master relations, which are strongly inspired by our work on the
twin system of tautological relations and localization techniques. We also
make some steps towards establishing their vanishing in the Gorenstein
quotient.

Finally, in Section 4 we use a combinatorial relation between the generalized
$A=B$ relations and the master relations in two different directions. In one
direction, we show that the generalized $A=B$ relations are equivalent to the
master relations, and in particular this equivalence holds in the Gorenstein
quotient. And in the other direction, we transfer the properties of $A=B$
cycles implied by the DR/DZ correspondence for the trivial CohFT to complete
the proof that the master relations vanish in the Gorenstein quotient. This
proves that the generalized $A=B$ relations hold in the Gorenstein quotient.

###  1.7. Acknowledgments

X. B. and S. S. were supported by the Netherlands Organization for Scientific
Research. D. L. is supported by the University of Trieste, by the INFN under
the national project MMNLP (APINE) section of Trieste, by the INdAM group
GNSAGA, and by the PRIN project 2022 “Geometry of algebraic structures:
moduli, invariants, deformations”. The authors thank A. Buryak and P. Rossi
for useful discussions and collaboration on closely related topics.

##  2\. The $A=B$ relation in the Gorenstein quotient

###  2.1. Basic notation for trees

Let $\mathrm{SRT}_{g,n,m}$ be the set of stable rooted trees of total genus
$g$, with $n$ regular legs $\sigma_{1},\dots,\sigma_{n}$ and $m$ extra legs
$\sigma_{n+1},\dots,\sigma_{n+m}$, which we refer to as “frozen” legs and must
always be attached to the root vertex. For a $T\in\mathrm{SRT}_{g,n,m}$ we use
the following notation:

  * •

$H(T)$ is the set of half-edges of $T$.

  * •

$L(T),L_{r}(T),L_{f}(T)\subset H(T)$ are the sets of all, regular, and frozen
legs of $T$, respectively. $L(T)=L_{r}(T)\sqcup L_{f}(T)$.

  * •

$H_{e}(T)\coloneqq H(T)\setminus L(T)$.

  * •

$\iota\colon H_{e}(T)\to H_{e}(T)$ is the involution that interchanges the
half-edges that form an edge.

  * •

$E(T)$ is the set of edges of $T$, $E\cong H_{e}(T)/\iota$.

  * •

$H_{+}(T)\subset H(T)$ is the set of the so-called “positive” half-edges that
consists of all regular legs of $T$ and of half-edges in $H(T)\setminus L(T)$
directed away from the root at the vertices where they are attached,
$H_{+}(T)\cong E(T)\cup L_{r}(T)$;

  * •

$H_{-}(T)\subset H(T)$ is the set of the so-called “negative” half-edges that
consists of all frozen legs of $T$ and of half-edges in $H(T)\setminus L(T)$
directed towards the root at the vertices where they are attached,
$H_{-}(T)\cong E(T)\cup L_{f}(T)$;

  * •

$V(T),V_{nr}(T)$ are the sets of vertices and non-root vertices of $T$.

  * •

$v_{r}\in V(T)$ is the root vertex of $T$; $V(T)=\\{v_{r}(T)\\}\sqcup
V_{nr}(T)$.

  * •

For a $v\in V(T)$, $H(v),H_{+}(v),H_{-}(v)$ are all, positive, and negative
half-edges attached to $v$, respectively. Obviously, $|H_{-}(v_{r})|=m$ and
for any $v\in V_{nr}(T)$ we have $|H_{-}(v)|=1$.

  * •

For a $v\in V(T)$ let $g(v)\in\mathbb{Z}_{\geq 0}$ be the genus assigned to
$v$. The stability condition means that

$\chi(v)\coloneqq 2g(v)-2+|H(v)|>0.$

The genus condition reads

$\sum_{v\in V(T)}g(v)=g.$

  * •

We say that a vertex or a (half-)edge $x$ is a descendant of a vertex or a
(half-)edge $y$ if $y$ is on the unique path connecting $x$ to $v_{r}$.

  * •

For an $h\in H_{+}(T)$ let $DL(h)$ be the set of all legs that are descendants
to $h$, including $h$ itself. Note that $DL(h)\subseteq L_{r}(T)$ for any
$h\in H_{+}(T)$ and $DL(l)=\\{l\\}$ for $l\in L_{r}(T)$.

  * •

For an $h\in H_{+}(T)$ let $DH(h)$ be the set of all positive half-edges that
are descendants to $h$, _excluding_ $h$. For instance, for $l\in L_{r}(T)$ we
have $DH(l)=\emptyset$, and for $h\in H_{+}(T)\setminus L_{r}(T)$ we have
$DH(h)\supseteq DL(h)$.

  * •

For an $e\in E(T)$ let $DL(e)$ be the set of all legs that are descendants to
$e$. Note that $DL(e)\subseteq L_{r}(T)$ for any $e\in E(T)$.

  * •

For an $v\in V(T)$ let $DL(v)$ be the set of all regular legs that are
descendants to $v$. In particular, $DL(v_{r})=L_{r}(T)$.

  * •

For a $v\in V(T)$ let $DV(v)\subset V(T)$ be the subset of all vertices that
are descendants of $v$, including $v$ itself. For instance, $DV(v_{r})=V(T)$.
Let

$D\chi(v)\coloneqq\sum_{v^{\prime}\in DV(v)}\chi(v^{\prime}).$

Consider the polynomial ring $Q\coloneqq\mathbb{Q}[a_{1},\dots,a_{n}]$ and
define $a\colon H_{+}(T)\to Q$, $a\colon E(T)\to Q$, and $a\colon V(T)\to Q$
(abusing notation we use the same symbol $a$ for all these maps) by

$\displaystyle a(\sigma_{i})$ $\displaystyle\coloneqq a_{i},$ $\displaystyle
i=1,\dots,n;$ $\displaystyle a(h)$ $\displaystyle\coloneqq\textstyle\sum_{l\in
DL(h)}a(l),$ $\displaystyle h\in H_{+}(T);$ $\displaystyle a(e)$
$\displaystyle\coloneqq\textstyle\sum_{l\in DL(e)}a(l),$ $\displaystyle e\in
E(T);$ $\displaystyle a(v)$ $\displaystyle\coloneqq\textstyle\sum_{l\in
DL(v)}a(l),$ $\displaystyle v\in V(T).$

###  2.2. Leveled stable rooted trees

We enhance the structure of a stable rooted tree to what we call a degree-
labeled stable rooted tree (of genus $g$, with $n$ regular and $m$ frozen
legs). To this end we take a stable rooted tree $T\in\mathrm{SRT}_{g,n,m}$ and
assign to each $v\in V(T)$ an extra degree label $p(v)\in\mathbb{Z}_{\geq 0}$
such that $p(v)\leq 3g(v)-3+|H(v)|$. Let $\mathcal{P}(T)$ denote the set of
degree label functions on $T$. A degree-labeled stable rooted tree is a pair
$(T,p)$, where $T\in\mathrm{SRT}_{g,n,m}$ and $p\in\mathcal{P}$.

Our next goal is to assign to a degree-labeled stable rooted tree $(T,p)$ a
so-called admissible level function. A function $\ell\colon
V(T)\to\mathbb{Z}_{\geq 0}$ is called an admissible level function if the
following conditions are satisfied:

  * •

The value of $\ell$ on the root vertex is zero ($\ell(v_{r})=0$).

  * •

If $v^{\prime}\in DV(v)$ and $v^{\prime}\not=v$, then
$\ell(v^{\prime})>\ell(v)$.

  * •

There are no empty levels, that is, for any $0\leq i\leq\max\ell(V(T))$ the
set $\ell^{-1}(i)$ is non-empty.

  * •

For every $0\leq i\leq\max\ell(V(T))-1$ we have inequality

$|\\{v\in V(T)\,|\,\ell(v)\leq i\\}|-1+\sum_{\begin{subarray}{c}v\in V(T)\\\
\ell(v)\leq i\end{subarray}}p(v)\leq\sum_{\begin{subarray}{c}v\in V(T)\\\
\ell(v)\leq i\end{subarray}}2g(v)-2+m.$

For instance, if $T$ has more than one vertex, then $p(v_{r})\leq
2g(v_{r})-2+m$.

Let $\mathcal{L}(T,p)$ denote the set of admissible level functions on
$(T,p\in\mathcal{P}(T))$.

The set $\mathrm{LDLSRT}_{g,n,m}$ of leveled degree-labeled stable rooted
trees consists of triples $(T,p,\ell)$, where $T\in\mathrm{SRT}_{g,n,m}$,
$p\in\mathcal{P}(T)$, and $\ell\in\mathcal{L}(T,p)$.

###  2.3. Definition of the $B$ class

The goal of this section is to define the so-called $B$ class in the
tautological ring of ${\overline{\mathcal{M}}}_{g,n+m}$.

Let $(T,p,\ell)\in\mathrm{LDLSRT}_{g,n,m}$. Assign to each $v\in V(T)$ the
moduli space of curves ${\overline{\mathcal{M}}}_{g(v),|H(v)|}$, where the
first $|H_{+}(v)|$ marked points correspond to the positive half-edges
attached to $v$ and ordered in an arbitrary but fixed way and the the last
$|H_{-}(v)|$ marked points correspond to the negative half-edges attached to
$v$, also ordered in some arbitrary but fixed way. Consider the class

$\displaystyle\Psi(v)\coloneqq\left(\prod_{i=1}^{|H_{+}(v)|}\frac{1}{1-a(h_{i})\psi_{i}}\right)_{p(v)}\in
R^{p}({\overline{\mathcal{M}}}_{g(v),|H(v)|})\otimes_{\mathbb{Q}}Q$

For each $(g,n.m)$ such that $2g-2+n+m>0$ define the class

$B^{m}_{g,n}\in R^{*}({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q$

as

(2.1)
$B^{m}_{g,n}\coloneqq\sum_{(T,p,\ell)\in\mathrm{SRT}(g,n,m)}(-1)^{\max\ell(V(T))}\biggl{(}\prod_{e\in
E(T)}a(e)\biggr{)}(b_{T})_{*}\bigotimes_{v\in V(T)}\Psi(v)$

Here $(b_{T})_{*}$ is the boundary pushforward map that acts from
$\bigotimes_{v\in
V(T)}R^{*}({\overline{\mathcal{M}}}_{g(v),|H(v)|})\otimes_{\mathbb{Q}}Q$ to
$R^{*}({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q$. The class
$B^{m}_{g,n}$ has the feature that its component in $R^{d}$ is a homogeneous
polynomial of degree $d$ in $a_{1},\dots,a_{n}$, $d=0,\dots,3g-3+n+m$.

###  2.4. The conjectural $A=B$ relation revisited

Consider the moduli space

(2.2)
$\displaystyle\overline{\mathcal{M}}_{g}^{\sim}(\mathbb{P}^{1},a_{1},\dots,a_{n},-\sum_{i=1}^{n}a_{i})$

of rubber stable maps to $(\mathbb{P}^{1},0,\infty)$. Let
$s\colon{\overline{\mathcal{M}}}_{g}^{\sim}(\mathbb{P}^{1},a_{1},\dots,a_{n},-\sum_{i=1}^{n}a_{i})\to{\overline{\mathcal{M}}}_{g,n+1}$
be the projection to the source curve, and $\lambda_{g}$ the lift of the
lambda class with respect to this projection. Let
$t\colon{\overline{\mathcal{M}}}_{g}^{\sim}(\mathbb{P}^{1},a_{1},\dots,a_{n},-\sum_{i=1}^{n}a_{i})\to
LM_{2g-1+n}$ be the projection to the target curve, where $LM_{2g-1+n}$
denotes the Losev-Manin space with $2g-1+n$ marked points. Let
$\tilde{\psi}_{0}$ be the pull-back by $t$ of the $\psi$-class at the point
$0$ in Losev-Manin space. Define

(2.3) $\displaystyle A^{1}_{g,n}\coloneqq
s_{*}\left(\frac{\lambda_{g}}{1-\tilde{\psi}_{0}}\left[{\overline{\mathcal{M}}}_{g}^{\sim}\left(\mathbb{P}^{1},a_{1},\dots,a_{n},-\sum\nolimits_{i=1}^{n}a_{i}\right)\right]^{\mathrm{vir}}\right)\in
R^{*}({\overline{\mathcal{M}}}_{g,n+1})\otimes_{\mathbb{Q}}Q.$

Note that $(A^{1}_{g,n})_{<2g}=0$ and
$(A^{1}_{g,n})_{2g}=\lambda_{g}\mathrm{DR}_{g}(a_{1},\dots,a_{n},-\sum_{i=1}^{n}a_{i})$.
Note also that for any $d\geq 2g$ the class $(A^{1}_{g,n})_{d}$ is a
homogeneous polynomial of degree $d$ in $a_{1},\dots,a_{n}$.

######  Conjecture 2.1.

The following three statements hold:

  1. (1)

For any $(g,n,m)$ such that $g\geq 0$, $n\geq 1$, $m\geq 2$, we have

(2.4) $\displaystyle\deg B^{m}_{g,n}\leq 2g-2+m.$

  2. (2)

For any $(g,n)$ such that $g\geq 0$, $n\geq 1$, $2g-1+n>0$, we have

(2.5) $\displaystyle\deg(B^{1}_{g,n}-A^{1}_{g,n})\leq 2g-1.$

  3. (3)

For any $(g,n)$ such that $g\geq 0$, $n\geq 1$, $2g-2+n>0$, we have

(2.6)
$\displaystyle\deg(B^{0}_{g,n}-\frac{1}{\sum_{i=1}^{n}a_{i}}\pi_{*}A^{1}_{g,n})\leq
2g-2,$

where
$\pi\colon{\overline{\mathcal{M}}}_{g,n+1}\to{\overline{\mathcal{M}}}_{g,n}$
is the map that forgets the last marked point.

In all three statement $\deg$ refers either to cohomological degree or to the
homogenenous degree in $a_{1},\dots,a_{n}$. We use the convention that a
polynomial of negative degree is the zero polynomial. Note that the third
statement follows from the second statement. We have the following lemma:

######  Lemma 2.2.

Conjecture 2.1 is equivalent to [16, Conjectures 1, 2, and 3].

###### Proof.

We have to show that our definition of the class $A^{1}_{g,n}$ coincides with
the definition given in [16] (see also [3, Section 2.4]). To this end, we
perform the intersection of the class $\tilde{\psi}_{0}$ with the moduli space
of rubber maps using the technics of [17], this yields exactly the same trees
in $\mathrm{SRT}_{g,n,1}$ decorated by double ramification cycles as in the
original definition of $A^{1}_{g,n}$. The only difference is the combinatorial
coefficient that we have to compare.

In the case of the original definition, the combinatorial coefficient of a
tree $T\in\mathrm{SRT}_{g,n,1}$ is equal to

(2.7) $\displaystyle C_{1}(T)=\prod_{v\in V(T)}\frac{\chi(v)}{D\chi(v)}.$

For the computation of (2.3) we consider a set rigorous level structures
$\mathcal{R}(T)$, that is, the set of one-to-one maps $f\colon
V(T)\to\\{1,\dots,|V(T)|\\}$ such that $f(v_{1})>f(v_{2})$ if $v_{1}\in
DV(v_{2})$. In particular, $f(v_{r})=1$. Then the coefficient implied by (2.3)
is equal to

(2.8) $\displaystyle
C_{2}(T)=\sum_{f\in\mathcal{R}(T)}\prod_{i=1}^{|V(T)|}\frac{\chi(f^{-1}(i))}{\chi(f^{-1}(i))+\cdots+\chi(f^{-1}(|V(T)|))}.$

Let us show that $C_{1}(T)=C_{2}(T)$. To this end, observe that both the
definition of $C_{1}$ and $C_{2}$ can be applied to forests, moreover,

(2.9) $\displaystyle C_{1}(T_{1}\sqcup\cdots
T_{k})=\prod_{i=1}^{k}C_{1}(T_{i})\quad\text{and}\quad C_{2}(T_{1}\sqcup\cdots
T_{k})=\prod_{i=1}^{k}C_{2}(T_{i}).$

Moreover, the factor in $C_{1}(T)$ corresponding to $v_{r}$ coincides with the
factor for $v_{r}$ in each summand in $C_{2}(T)$. Thus we can remove the root
vertex and use the factorization for the forests above to prove
$C_{1}(T)=C_{2}(T)$ by induction on the size of the tree. ∎

Various simplifications and partial results towards Conjecture 2.1 are
available in [3, 9, 16, 21, 25].

###  2.5. The $A=B$ identity in the Gorenstein quotient

The main statement that we prove in this paper is the following theorem:

######  Theorem 2.3 (main theorem).

Conjecture 2.1 holds in the Gorenstein quotient. Namely, the following three
statements hold:

  1. (1)

For any $(g,n,m)$ such that $g\geq 0$, $n\geq 1$, $m\geq 2$, and for any
$\alpha\in R^{*}({\overline{\mathcal{M}}}_{g,n+m})$ we have

(2.10) $\displaystyle\deg\left(\int_{{\overline{\mathcal{M}}}_{g,n+m}}\alpha
B^{m}_{g,n}\right)\leq 2g-2+m.$

  2. (2)

For any $(g,n)$ such that $g\geq 0$, $n\geq 1$, $2g-1+n>0$, and for any
$\alpha\in R^{*}({\overline{\mathcal{M}}}_{g,n+1})$ we have

(2.11)
$\displaystyle\deg\left(\int_{{\overline{\mathcal{M}}}_{g,n+1}}\alpha(B^{1}_{g,n}-A^{1}_{g,n})\right)\leq
2g-1.$

  3. (3)

For any $(g,n)$ such that $g\geq 0$, $n\geq 1$, $2g-2+n>0$, and for any
$\alpha\in R^{*}({\overline{\mathcal{M}}}_{g,n})$ we have

(2.12)
$\displaystyle\deg\left(\int_{{\overline{\mathcal{M}}}_{g,n}}\alpha(B^{0}_{g,n}-\frac{1}{\sum_{i=1}^{n}a_{i}}\pi_{*}A^{1}_{g,n})\right)\leq
2g-2.$

######  Remark 2.4.

The third statement is a corollary of the second one that one can obtain by
the projection formula.

As a corollary, the first statement of this theorem implies the polynomiality
of the system of Dubrovin-Zhang conservation laws associated to an arbitrary
F-CohFT whose classes belong to the tautological ring. The second statement
implies that the latter hierarchy is Miura equivalent, in a very explicit way,
to the DR hierarchy of conservation laws associated to the same F-CohFT, also
under assumption that its classes belong to the tautological ring. Finally,
the third statement implies the strong DR/DZ correspondence for (partial)
CohFTs, also under assumption that the classes of (partial) CohFT belong to
the tautological ring. See [16, Section 4] and [6, 8].

One important lemma that allows to establish the strong DR/DZ correspondence
for any semi-simple cohomological field theory is a direct corollary of the
classification proved in [26]:

######  Lemma 2.5 (Corollary of [26]).

The classes of any semi-simple cohomological field theory belong to the
tautological ring.

###  2.6. The trivial CohFT

An important case when the Dubrovin-Zhang hierarchy and the DR/DZ equivalence
are fully understood is the case of the trivial CohFT [5, 6]. In particular,
this implies the following special cases of Theorem 2.3:

######  Lemma 2.6.

The following three statements hold:

  1. (1)

For any $(g,n,m)$ such that $g\geq 0$, $n\geq 1$, $m\geq 2$, and for any
monomial $\prod_{i=1}^{m}\psi_{n+i}^{b_{i}}\in
R^{*}({\overline{\mathcal{M}}}_{g,n+m})$ we have

(2.13)
$\displaystyle\deg\left(\int_{{\overline{\mathcal{M}}}_{g,n+m}}\left(\prod_{i=1}^{m}\psi_{n+i}^{b_{i}}\right)B^{m}_{g,n}\right)\leq
2g-2+m.$

  2. (2)

For any $(g,n)$ such that $g\geq 0$, $n\geq 1$, $2g-1+n>0$, we have

(2.14)
$\displaystyle\deg\left(\int_{{\overline{\mathcal{M}}}_{g,n+1}}(B^{1}_{g,n}-A^{1}_{g,n})\right)\leq
2g-1.$

  3. (3)

For any $(g,n)$ such that $g\geq 0$, $n\geq 1$, $2g-2+n>0$ we have

(2.15)
$\displaystyle\deg\left(\int_{{\overline{\mathcal{M}}}_{g,n}}(B^{0}_{g,n}-\frac{1}{\sum_{i=1}^{n}a_{i}}\pi_{*}A^{1}_{g,n})\right)\leq
2g-2.$

###### Proof.

The second and the third statement are literally equivalent to the statements
of the strong DR/DZ correspondence in this case, see [5, 6, 16]. The first
statement is equivalent by [16, Section 4] to the fact that the second and
higher derivatives of the potential $F(t_{0},t_{1},\dots)$ corresponding to
the string solution are known to be differential polynomials in the dependent
variables in this case, see e. g. [18]. ∎

######  Remark 2.7.

Note that in the case $m=1$ we don’t have a ready statement on the degree of
$\int_{{\overline{\mathcal{M}}}_{g,n+1}}\psi_{n+1}^{b}(B^{1}_{g,n}-A^{1}_{g,n})$
for $b>0$. The reason for this is that the first derivatives of the potential
$F$ with respect to the variables $t_{b}$, $b>0$, play no direct role in the
construction of integrable systems in this case.

##  3\. The master relation in the Gorenstein quotient

###  3.1. Main definitions and statements

The definition and overall idea to use what we call _master relation_ comes
from a parallel paper [4], where we use a close relative of this relation to
prove the conjectures on the so-called $\Omega$-classes posed in [3].

####  3.1.1. Pre-stable star rooted trees

Fix $m\geq 1$, $n\geq 1$, and $g\geq 0$. Let $\mathrm{PSSRT}_{g,n,m}$ be the
set of pre-stable star rooted trees, that is, the rooted trees with one root
vertex, where the frozen legs $\sigma_{n+1},\dots,\sigma_{n+m}$ are attached,
and all other vertices are connected by an edge to the root (hence the term
“star”). We also demand that no regular legs are attached to the root vertex.

The graphs that we obtain are quite similar to the graphs that form a subset
of $\mathrm{SRT}_{g,n,m}$, and we extend the definitions applying the same
notation for various concepts related to these graphs. However, since we allow
non-stable vertices, $\mathrm{PSSRT}_{g,n,m}$ is not quite a subset of
$\mathrm{SRT}_{g,n,m}$.

####  3.1.2. Classes assigned to vertices

Let $T\in\mathrm{PSSRT}_{g,n,m}$. As in the case of trees in
$\mathrm{SRT}_{g,n,m}$, we assign to each $v\in V(T)$ the moduli space of
curves ${\overline{\mathcal{M}}}_{g(v),|H(v)|}$, where the first $|H_{+}(v)|$
marked points correspond to the positive half-edges attached to $v$ and
ordered in an arbitrary but fixed way and the the last $|H_{-}(v)|$ marked
points correspond to the negative half-edges attached to $v$, also ordered in
some arbitrary but fixed way. The cases of non-stable pairs $(g(v),|H(v)|)$
will be treated separately, but informally one can think of a natural
extension of the classes we use for stable vertices to the unstable moduli
spaces.

For the root vertex, we consider the class

(3.1) $\displaystyle\Psi(v_{r})$
$\displaystyle\coloneqq\prod_{i=1}^{|E(T)|}\frac{1}{1-a(h_{i})\psi_{i}}\in
R^{*}({\overline{\mathcal{M}}}_{g(v_{r}),|E(T)|+m})\otimes_{\mathbb{Q}}Q$

Note that for pre-stable star rooted trees, $H_{+}(v_{r})$ can be identified
with $E(T)$. In the exceptional unstable case $g(v_{r})=0,m=1,|E(T)|=1$ we
formally assign to the root vertex the following class:

(3.2) $\displaystyle a(h_{1})^{-1}\in
R^{-1}({\overline{\mathcal{M}}}_{0,2})\otimes_{\mathbb{Q}}Q[a(h_{1})^{-1}],$

where the negative cohomological degree is just formally assigned to allow to
treat this case non-exceptionally in what follows.

For a non-root root vertex $v$, we consider the class

(3.3) $\displaystyle{\mathbb{D}}(v)$
$\displaystyle\coloneqq\frac{\lambda_{g(v)}\mathrm{DR}_{g(v)}\big{(}a(h_{1}),\dots,a(h_{|H_{+}(v)|}),-a(v)\big{)}}{1-a(v)\psi_{|H(v)|}}\in
R^{*}({\overline{\mathcal{M}}}_{g(v),|H(v)|})\otimes_{\mathbb{Q}}Q$

Note that in this case $a(v)=-a(h_{H(v)})=a(h_{1})+\cdots+a(h_{|H_{+}(v)|})$.
In the exceptional unstable case $g(v)=0,H_{+}(v)=1$ we formally assign to
this vertex the following class:

(3.4) $\displaystyle a(h_{1})^{-1}\in
R^{-1}({\overline{\mathcal{M}}}_{0,2})\otimes_{\mathbb{Q}}Q[a(h_{1})^{-1}],$

where, as it was for the root vertex, the negative cohomological degree is
just formally assigned to allow to treat this case non-exceptionally in what
follows.

####  3.1.3. Classes assigned to trees

Now, let us assume that $2g-2+n+m>0$ and assign to a tree
$T\in\mathrm{PSSRT}_{g,n,m}$ a class $\Xi(T)\in
R^{*}({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q[u,u^{-1}]$. Here
$u$ is a new formal variable to control the dimension. We have:

(3.5) $\displaystyle\Xi(T)\coloneqq u^{2g-2+m}\left(\prod_{e\in
E(T)}\frac{a(e)}{u}\right)(b_{T})_{*}\left(\left(\sum_{d=-1}^{\infty}\frac{\Psi(v_{r})_{d}}{(-u)^{d}}\right)\otimes\bigotimes_{v\in
V\setminus\\{v_{r}\\}}\left(\sum_{d=-1}^{\infty}\frac{{\mathbb{D}}(v)_{d}}{u^{d}}\right)\right),$

where $(b_{T})_{*}$ is the boundary pushforward map from $\bigotimes_{v\in
V(T)}R^{*}({\overline{\mathcal{M}}}_{g(v),|H(v)|})$ to
$R^{*}({\overline{\mathcal{M}}}_{g,n+m})$ extended by linearity to the
rational functions in $a_{i}$ and Laurent polynomials in $u$. In the case of
pre-stable trees $b_{T}$ is assumed to contract the unstable components. Note
that under the assumption $2g-2+n+m>0$ the dependence of the resulting formula
on $a_{i}$’s is purely polynomial, so $\Xi(T)\in
R^{*}({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q[u,u^{-1}]$.

######  Remark 3.1.

Note that $[u^{2g-2+m-d}]\Xi(T)\in
R^{d}({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q$ and $\Xi(T)_{d}$
is a homogeneous polynomial of degree $d$ in $a_{1},\dots,a_{n}$.

######  Remark 3.2.

As a side remark, note that in the case $2g-2+n+m=0$, that is, $g=0,n=m=1$, we
have just one tree with two vertices, both unstable, and we can extend the
definition of the class $\Xi(T)$ given above to produce the class
$a_{1}^{-1}u^{0}\in
R^{-1}({\overline{\mathcal{M}}}_{0,2})\otimes_{\mathbb{Q}}Q[a_{1}^{-1}][u,u^{-1}]$.

####  3.1.4. Classes assigned to sets of trees

Let $2g-2+n+m>0$. Consider the following Laurent polynomial in a formal
variable $u$ and a polynomial in the variables $a_{1},\dots,a_{n}$:

(3.6)
$\displaystyle\Xi^{m}_{g,n}\coloneqq\sum_{T\in\mathrm{PSSRT}_{g,n,m}}\Xi(T).$

######  Remark 3.3.

Note that $(\Xi^{m}_{g,n})_{d}\coloneqq[u^{2g-2+m-d}]\Xi^{m}_{g,n}\in
R^{d}({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q$ and
$(\Xi^{m}_{g,n})_{d}$ is a homogeneous polynomial of degree $d$ in
$a_{1},\dots,a_{n}$.

######  Conjecture 3.4 (master relation).

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

(3.7) $\displaystyle\Xi^{m}_{g,n}\in
R^{*}({\overline{\mathcal{M}}}_{g,n+m})\otimes_{\mathbb{Q}}Q[u].$

In other words, for any $d\geq 2g-1+m$ we have

(3.8) $\displaystyle(\Xi^{m}_{g,n})_{d}=0.$

The master relation is equivalent to Conjecture 2.1, the precise statement of
this equivalence and its proof are delayed to the next section, see Theorem
4.5. For our applications, that is, for the case of tautological CohFTs and
F-CohFTs, in particular for all semi-simple CohFTs, a weaker statement is
sufficient:

######  Theorem 3.5 (master relation in the Gorenstein quotient).

For any $g\geq 0$, $m,n\geq 1$, $2g-2+n+m>0$, and for any $\alpha\in
R^{*}({\overline{\mathcal{M}}}_{g,n+m})$ we have

(3.9)
$\displaystyle\int_{{\overline{\mathcal{M}}}_{g,n+m}}\alpha\Xi^{m}_{g,n}\in
Q[u].$

In other words, for any $\alpha\in R^{*}({\overline{\mathcal{M}}}_{g,n+m})$ we
have

(3.10)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g,n+m}}\alpha\Xi^{m}_{g,n}|_{u=1}\leq
2g-2+m.$

The latter statement can also be rephrased as the vanishing of
$\int_{{\overline{\mathcal{M}}}_{g,n+m}}\alpha(\Xi^{m}_{g,n})_{d}$ for $d\geq
2g-1+m$ and any $\alpha\in R^{*}({\overline{\mathcal{M}}}_{g,n+m})$, and we
call this vanishing the master relation in the Gorenstein quotient.

The rest of this section is devoted to a reduction of a proof of this theorem
to the following statement:

######  Lemma 3.6.

Theorem 3.5 holds for any $\alpha$ if it holds for the classes
$\alpha=\prod_{i=1}^{m}\psi_{n+i}^{b_{i}}$ for any
$b_{1},\dots,b_{m}\in\mathbb{Z}_{\geq 0}$.

###  3.2. Intersection with a divisor

For practical computations it is convenient to extend the definition above to
$n=0$, where $\Xi^{m}_{g,0}$ is defined for $2g-2+m>0$ and is set to zero.

Let
$\rho_{1}\colon{\overline{\mathcal{M}}}_{g-1,n+m+2}\to{\overline{\mathcal{M}}}_{g,n+m}$
be the boundary map.

######  Lemma 3.7.

We have

(3.11) $\displaystyle\rho_{1}^{*}(\Xi^{m}_{g,n})_{d}=(\Xi^{m+2}_{g-1,n})_{d}.$

######  Remark 3.8.

Note that if $d\geq 2g-1+m$, then $d\geq 2(g-1)-1+(m+2)$, so the degree
remains in the range that should vanish in the Gorenstein quotient once it was
there before the pull-back.

Let
$\rho_{2}\colon{\overline{\mathcal{M}}}_{g_{1},n_{1}+m_{1}+1}\times{\overline{\mathcal{M}}}_{g_{2},n_{2}+m_{2}+1}\to{\overline{\mathcal{M}}}_{g,n+m}$
be the boundary map and assume that $m_{1}$ and $m_{2}$ are non-zero.

######  Lemma 3.9.

We have

(3.12)
$\displaystyle\rho_{2}^{*}(\Xi^{m}_{g,n})_{d}=\sum_{d_{1}+d_{2}=d}(\Xi^{m_{1}+1}_{g_{1},n_{1}})_{d_{1}}\otimes(\Xi^{m_{2}+1}_{g_{2},n_{2}})_{d_{2}}.$

In this formula we assume that the arguments of $\Xi$-classes correspond to
the marked points on the corresponding components.

######  Remark 3.10.

Note that if $d\geq 2g-1+m$, then either $d_{1}\geq 2g_{1}-1+(m_{1}+1)$ or
$d_{2}\geq 2g_{2}-1+(m_{2}+1)$, so the degree on at least one of the two
components is in the range that should vanish in the Gorenstein quotient once
the degree of the original class was in the vanishing range before the pull-
back.

Let
$\rho_{3}\colon{\overline{\mathcal{M}}}_{g_{1},n_{1}+m+1}\times{\overline{\mathcal{M}}}_{g_{2},n_{2}+1}\to{\overline{\mathcal{M}}}_{g,n+m}$
be the boundary map and assume that all points with the labels
${n+1},\dots,{n+m}$ are on the first component, and $n_{2}$ points on the
second component have labels in $I\subset\\{1,\dots,n\\}$, $|I|=n_{2}$.

######  Lemma 3.11.

We have

(3.13) $\displaystyle\rho_{3}^{*}(\Xi^{m}_{g,n})_{d}$
$\displaystyle=\sum_{d_{1}+d_{2}=d}(\Xi^{m+1}_{g_{1},n_{1}})_{d_{1}}\otimes(\Xi^{1}_{g_{2},n_{2}})_{d_{2}}$
$\displaystyle\quad+(\Xi^{m}_{g_{1},n_{1}+1})_{d-2g_{2}}(\\{a_{j}\\}_{j\not\in
I},a_{I})\otimes\lambda_{g_{2}}\mathrm{DR}_{g_{2}}\Big{(}\\{a_{i}\\}_{i\in
I},-a_{I}\Big{)}.$

In this formula we assume that the arguments of $\Xi$-classes correspond to
the marked points on the corresponding components. In the last summand we use
$a_{i}\coloneqq\sum\nolimits_{i\in I}a_{i}$.

######  Remark 3.12.

Note that if $d\geq 2g-1+m$, then in the first summand either $d_{1}\geq
2g_{1}-1+(m+1)$ or $d_{2}\geq 2g_{2}-1+1$, so the degree on at least one of
the two components is in the range that should vanish in the Gorenstein
quotient once the degree of the original class was in the vanishing range
before the pull-back. In the second summand, $d-2g_{2}\geq 2g_{1}-1+m$ once
$d\geq 2g-1+m$.

The proof of all three lemmas is a straightforward application of the
instructions on intersection of tautological classes given in [20] and a
formula for the intersection of a double ramification cycle with a divisor in
[17].

###  3.3. Intersection with a kappa class

Lemmata in Section 3.2 reduce the computation of
$\int_{{\overline{\mathcal{M}}}_{g,n+m}}\alpha\Xi^{m}_{g,n}$ for an arbitrary
class $\alpha$ to the computation of the same intersection numbers for the
classes $\alpha=\kappa_{c_{1},\dots,c_{l}}\prod_{i=1}^{n+m}\psi_{i}^{b_{i}}$.
To this end, we have the following statement:

######  Lemma 3.13.

We have

(3.14)
$\displaystyle\int_{{\overline{\mathcal{M}}}_{g,n+m}}\kappa_{c_{1},\dots,c_{l}}\prod_{i=1}^{n+m}\psi_{i}^{b_{i}}\Xi^{m}_{g,n}=\int_{{\overline{\mathcal{M}}}_{g,n+l+m}}\prod_{i=1}^{n}\psi_{i}^{b_{i}}\prod_{i=1}^{l}\psi_{n+i}^{c_{i}+1}\prod_{i=1}^{m}\psi_{n+l+i}^{b_{n+i}}\Xi^{m}_{g,n+l}|_{a_{n+1}=\cdots=a_{n+l}=0}.$

###### Proof.

Let
$\pi_{n+1}\colon{\overline{\mathcal{M}}}_{g,n+1+m}\to{\overline{\mathcal{M}}}_{g,n+m}$
be the map that forgets the $(n+1)$-st marked point and shifts the labels of
the last $m$ marked points by $-1$. Then
$\pi_{n+1}^{*}\Xi^{m}_{g,n}=\Xi^{m}_{g,n+1}|_{a_{n+1}=0}$. In order to obtain
(3.14) we iterate this formula $l$ times and use the projection formula for
$\kappa_{c_{1},\dots,c_{l}}=(\pi_{n+1}\cdots\pi_{n+l})_{*}\prod_{i=1}^{l}\psi_{n+i}^{c_{i}+1}$.
∎

######  Remark 3.14.

Equation (3.14) preserves the degree, that is, it is bounded by $2g-2+m$ on
the left hand side if and only if it is bounded by $2g-2+m$ on the right hand
side. In particular, together with Lemmata of Section 3.2, this reduces
Theorem 3.5 to a weaker statement, where the class $\alpha$ is a product of
$\psi$-classes.

###  3.4. Intersection with a $\psi$ class at a regular leg

######  Lemma 3.15.

For any $i=1,\dots,n$, we have:

(3.15)
$\displaystyle{a_{i}\psi_{i}}(\Xi^{m}_{g,n})_{d}=(\sigma_{i})^{*}(\Xi^{m+1}_{g,n-1}(\\{a_{j}\\}_{j\not=i}))_{d+1}-(\Xi^{m}_{g,n})_{d+1}$
$\displaystyle+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\ n_{1}+n_{2}=n\\\ i\in
I\subseteq\\{1,\dots,n\\}\\\
|I|=n_{2}\end{subarray}}a_{I}(\rho_{3})_{*}\left((\Xi^{m}_{g_{1},n_{1}+1})_{d-2g_{2}}(\\{a_{j}\\}_{j\not\in
I},a_{I})\otimes\lambda_{g_{2}}\mathrm{DR}_{g_{2}}\Big{(}\\{a_{i}\\}_{i\in
I},-a_{I}\Big{)}\right).$

Here in the first term by $\sigma_{i}$ we mean the relabeling of the points

(3.16) $\displaystyle(1,\dots,n+m)\to(1,\dots,i-1,i+1,\dots,n+m,i).$

In the last sum we use $a_{I}\coloneqq\sum\nolimits_{j\in I}a_{j}$ and assume
that $2g_{1}-1+n_{1}+m>0$ and $2g_{2}-1+n_{2}>0$, $\rho_{3}$ is the boundary
map described above.

###### Proof.

We just apply the following formula adapted from [17] to each term on the left
hand side where the $i$-th marked point lies on a double ramification cycle,
and then regroup the terms:

(3.17)
$\displaystyle(a_{i}\psi_{i}+a\psi_{n+1})\lambda_{g}\mathrm{DR}_{g}(a_{1},\dots,a_{n},-a)=$
$\displaystyle\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\ n_{1}+n_{2}=n\\\ i\in
I\subseteq\\{1,\dots,n\\}\\\
|I|=n_{2}\end{subarray}}a_{I}\rho_{*}\left(\lambda_{g_{1}}\mathrm{DR}_{g_{1}}(\\{a_{j}\\}_{j\not\in
I},-a,a_{I})\otimes\lambda_{g_{2}}\mathrm{DR}_{g_{2}}(\\{a_{i}\\}_{i\in
I},-a_{I})\right),$

where $a=\sum_{i=1}^{n}a_{i}$, $a_{I}=\sum_{i\in I}a_{i}$, and
$\rho\colon{\overline{\mathcal{M}}}_{g_{1},n_{1}+2}\times{\overline{\mathcal{M}}}_{g_{2},n_{2}+1}$
is the boundary map that glues the last marked points on both components into
a node and maps the first $(n_{1}+1)$ points on the first component to the
points with the indices $j\not\in I$ and $(n+1)$ and the first $n_{2}$ points
on the second component to $i\in I$. ∎

######  Remark 3.16.

Note that once on the left hand side of (3.15) $d\geq 2g-1+m$, then in all
three summands on the right hand side of Equation (3.15) we have $d+1\geq
2g-1+(m+1)$, $d+1\geq 2g-1+m$, and $d-2g_{2}\geq 2g_{1}-1+m$ in the first,
second, and third summand respectively.

Lemma 3.15 together with Remark 3.16 imply that the desired vanishing in the
intersection numbers on the right hand side of Equation 3.14 follows from the
vanishing of

(3.18)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g,n+m}}\prod_{i=1}^{m}\psi_{n+i}^{b_{1}}\Xi^{m}_{g,n}|_{u=1}$

in degrees $\geq 2g-1+m$ for all $g\geq 0$, $n,m\geq 1$ and all
$b_{1},\dots,b_{m}\in\mathbb{Z}_{\geq 0}$. Together with Lemma 3.13 and
Lemmata of Section 3.2, this completes the proof of Lemma 3.6.

##  4\. Proof of the main theorem

###  4.1. Strategy of proof of Theorem 2.3

We first give a short outline of the proof and then we expand it to make it
more precise.

####  4.1.1. Step 1

Using a combinatorial argument of [4] adapted to our case, we state a lemma
which controls the difference between either: the $B$-class and the
$\Xi$-class when $m\geq 2$, or the $A-B$-class and the $\Xi$-class when $m=1$.

####  4.1.2. Step 2

A first consequence of this lemma is that the main theorem (Theorem 1.2) is
equivalent to the master relation in the Gorenstein quotient (Theorem 3.5).

####  4.1.3. Step 3

We prove the master relation in the Gorenstein quotient. By a sequence of
explicit intersections with tautological classes, we reduced Theorem 3.5 for
any $\alpha$ to $\alpha=\prod_{i=1}^{m}\psi_{n+i}^{b_{i}}$, see Lemma 3.6. To
prove this special case, we use that the main theorem is already established
in some special cases, see Lemma 2.6, together with a second use of the lemma
of step $1$.

###  4.2. Step $1$: the key combinatorial lemma

We describe a result in [4] in quite general terms adapted to the needs of our
argument.

######  Lemma 4.1 (Corollary of [4]).

The following two statements hold:

  1. (1)

Let $m\geq 2$, $g\geq 0$, $n\geq 1$. Then, for any $d\geq 2g-1+m$,

(4.1) $\displaystyle(B^{m}_{g,n})_{d}-(-1)^{d}(\Xi^{m}_{g,n})_{d}$

is represented as a linear combination of tautological classes supported on
graphs in $\mathrm{SRT}_{g,n,m}$ with at least one edge, where in each summand
either one non-root vertex $v$ is decorated by
$(\Xi^{1}_{g(v),|H_{+}(v)|})_{d(v)}$ with $d(v)\geq 2g(v)$ or the root vertex
is decorated by $(B^{m}_{g(v_{r}),|H_{+}(v_{r})|})_{d(v_{r})}$ with
$d(v_{r})\geq 2g(v_{r})-1+m$.

  2. (2)

Let $m=1$, $g\geq 0$, $n\geq 1$, $2g-1+n>0$. Then, for any $d\geq 2g$,

(4.2) $\displaystyle(B^{1}_{g,n}-A^{1}_{g,n})_{d}-(-1)^{d}(\Xi^{1}_{g,n})_{d}$

is represented as a linear combination of tautological classes supported on
graphs in $\mathrm{SRT}_{g,n,1}$ with at least one edge, where in each summand
either one non-root vertex $v$ is decorated by
$(\Xi^{1}_{g(v),|H_{+}(v)|})_{d(v)}$ with $d(v)\geq 2g(v)$ or the root vertex
is decorated by
$(B^{1}_{g(v_{r}),|H_{+}(v_{r})|}-A^{1}_{g(v_{r}),|H_{+}(v_{r})|})_{d(v_{r})}$
with $d(v_{r})\geq 2g(v_{r})$.

######  Remark 4.2.

In all cases, the structure of the tautological classes supported on the trees
$T$ in $\mathrm{SRT}_{g,n,m}$ used in Lemma 4.1 is similar to the one used in
the definition of $B^{m}_{g,n}$: each vertex is decorated by a class in
$R^{*}({\overline{\mathcal{M}}}_{g(v),|H(v)|})\otimes_{\mathbb{Q}}Q$, whose
component of cohomological degree $d$ is a homogeneous polynomial in $a_{i}$’s
of degree $d$ (this can be $\Xi^{1}_{g(v),|H_{+}(v)|}$, $\Psi(v)$, or
${\mathbb{D}}(v)$, or eventually
$(B^{m}_{g(v_{r}),|H_{+}(v_{r})|}-\delta_{m,1}A^{1}_{g(v_{r}),|H_{+}(v_{r})|})$
for the root vertex), and we multiply the push-forward $(b_{T})_{*}$ of the
tensor product of such classes by $\prod_{e\in E(T)}a(e)$.

######  Remark 4.3.

As we see from the statement of Lemma 4.1, is it more convenient to use the
class $\tilde{\Xi}^{m}_{g,n}\coloneqq\sum_{d\geq
0}(-1^{d})(\Xi^{m}_{g,n})_{d}$.

######  Remark 4.4.

Since the vertices of the graphs in $\mathrm{SRT}_{g,n,m}$ with at least one
edge have strictly less negative Euler characteristic than $2g-2+m+n$, Lemma
4.1 allows to perform various inductive arguments. In particular, this lemma
has two immediate corollaries, both proved by induction on $(g,n)$ with fixed
$m$.

###  4.3. Step $2$: equivalence between the master relation and DR/DZ

It follows by induction using the key lemma that the master relation is
equivalent to Conjecture 2.1, more precisely:

######  Theorem 4.5.

The following equivalences hold:

  1. (1)

For any $(g,n)$ such that $g\geq 0$, $n\geq 1$, $2g-1+n>0$

$\begin{cases}\deg\Xi_{g,n}^{1}\leq 2g-1,\\\
\deg\Xi_{g^{\prime},n^{\prime}}^{1}\leq 2g^{\prime}-1,\ \text{for all}\
g^{\prime}\leq g,\,n^{\prime}\leq
n,\,2g^{\prime}+n^{\prime}<2g+n\end{cases}\qquad\Longleftrightarrow\qquad\\\
\begin{cases}\deg(B^{1}_{g,n}-A^{1}_{g,n})\leq 2g-1,\\\
\deg(B^{1}_{g^{\prime},n^{\prime}}-A^{1}_{g^{\prime},n^{\prime}})\leq
2g^{\prime}-1,\ \text{for all}\ g^{\prime}\leq g,\,n^{\prime}\leq
n,\,2g^{\prime}+n^{\prime}<2g+n.\end{cases}$

  2. (2)

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

$\begin{cases}\deg\Xi_{g,n}^{m}\leq 2g-2+m,\\\
\deg\Xi_{g^{\prime},n^{\prime}}^{m}\leq 2g^{\prime}-2+m,\ \text{for all}\
g^{\prime}\leq g,\,n^{\prime}\leq n,\,2g^{\prime}+n^{\prime}<2g+n\\\
\deg\Xi_{g^{\prime},n^{\prime}}^{1}\leq 2g^{\prime}-1,\ \text{for all}\
g^{\prime}\leq g,\,n^{\prime}\leq
n,\,2g^{\prime}+n^{\prime}<2g+n\end{cases}\quad\Longleftrightarrow\quad\\\
\begin{cases}\deg B^{m}_{g,n}\leq 2g-2+m,\\\ \deg
B^{m}_{g^{\prime},n^{\prime}}\leq 2g^{\prime}-2+m,\ \text{for all}\
g^{\prime}\leq g,\,n^{\prime}\leq n,\,2g^{\prime}+n^{\prime}<2g+n\\\
\deg(B^{1}_{g^{\prime},n^{\prime}}-A^{1}_{g^{\prime},n^{\prime}})\leq
2g^{\prime}-1,\ \text{for all}\ g^{\prime}\leq g,\,n^{\prime}\leq
n,\,2g^{\prime}+n^{\prime}<2g+n.\end{cases}$

In particular, this equivalence holds in the Gorenstein quotient. Thus it
suffices to prove the master relation in the Gorenstein quotient, that is
Theorem 3.5, to prove the main theorem.

###  4.4. Step 3: proof of the master relation in the Gorenstein quotient

We prove Theorem 3.5 for $\alpha=\prod_{i=1}^{m}\psi_{n+i}^{b_{i}}$ which is
implies Theorem 3.5 for any $\alpha$ by Lemma 3.6.

First, using the key lemma we deduce by induction the following statement:

######  Corollary 4.6.

The following statements hold

  1. (1)

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

(4.3)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g,n+1}}\psi_{n+1}^{b}\left(B^{1}_{g,n}-A^{1}_{g,n}-\tilde{\Xi}^{1}_{g,n}\right)\leq
2g-1$

if

(4.4)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g^{\prime},n^{\prime}+1}}\psi_{n+1}^{b}\left(B^{1}_{g^{\prime},n^{\prime}}-A^{1}_{g^{\prime},n^{\prime}}\right)\leq
2g-1$

holds for any $g^{\prime}\geq 0$, $n^{\prime}\geq 1$, such that
$g^{\prime}\leq g$, and $2g^{\prime}+n^{\prime}<2g+n$, and also

(4.5)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g^{\prime},n^{\prime}+1}}\tilde{\Xi}^{1}_{g^{\prime},n^{\prime}}\leq
2g^{\prime}-1$

holds for any $g^{\prime}\geq 0$, $n^{\prime}\geq 1$, such that
$g^{\prime}\leq g$, and $2g^{\prime}+n^{\prime}<2g+n$.

  2. (2)

Fix $m\geq 2$. For any $g\geq 0$, $n\geq 1$, and
$b_{1},\dots,b_{m}\in\mathbb{Z}_{\geq 0}$,

(4.6)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g,n+m}}\left(\prod_{i=1}^{m}\psi_{n+i}^{b_{i}}\right)\left(B^{m}_{g,n}-\tilde{\Xi}^{m}_{g,n}\right)\leq
2g-2+m$

if

(4.7)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g^{\prime},n^{\prime}+m}}\left(\prod_{i=1}^{m}\psi_{n^{\prime}+i}^{b_{i}}\right)B^{m}_{g^{\prime},n^{\prime}}\leq
2g^{\prime}-2+m$

holds for any $g^{\prime}\geq 0$, $n^{\prime}\geq 1$, such that
$g^{\prime}\leq g$, and $2g^{\prime}+n^{\prime}<2g+n$, and also

(4.8)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g^{\prime},n^{\prime}+1}}\tilde{\Xi}^{1}_{g^{\prime},n^{\prime}}\leq
2g^{\prime}-1$

holds for any $g^{\prime}\geq 0$, $n^{\prime}\geq 1$, such that
$g^{\prime}\leq g$, and $2g^{\prime}+n^{\prime}<2g+n$.

We recall our convention that a polynomial with negative degree is the zero
polynomial.

Then, we combine the statements of Lemma 2.6 and Corollary 4.6. Inductively,
we obtain the following statement:

######  Lemma 4.7.

We have

(4.9)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g,n+m}}\left(\prod_{i=1}^{m}\psi_{n+i}^{b_{i}}\right)\tilde{\Xi}^{m}_{g,n}\leq
2g-2+m$

for $m\geq 2$ and any $b_{1},\dots,b_{m}\in\mathbb{Z}_{\geq 0}$ and

(4.10)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g,n+1}}\tilde{\Xi}^{1}_{g,n}\leq
2g-1.$

Note that we still don’t have the statement for the degree of
$\int_{{\overline{\mathcal{M}}}_{g,n+1}}\psi_{n+1}^{b}\tilde{\Xi}^{1}_{g,n}$
for any $b\in\mathbb{Z}_{\geq 0}$. However, by the pushforward formula which
implies the string equation, we have

(4.11)
$\displaystyle\int_{{\overline{\mathcal{M}}}_{g,n+2}}\psi_{n+1}^{b}\psi_{n+2}^{0}\tilde{\Xi}^{2}_{g,n}=\int_{{\overline{\mathcal{M}}}_{g,n+1}}\psi_{n+1}^{b-1}\tilde{\Xi}^{1}_{g,n}+\left(\sum_{i=1}^{n}a_{i}\right)\int_{{\overline{\mathcal{M}}}_{g,n+1}}\psi_{n+1}^{b}\tilde{\Xi}^{1}_{g,n}.$

Hence, by induction on $b$, we obtain the following corollary of Lemma 4.7:

######  Corollary 4.8.

We have

(4.12)
$\displaystyle\deg\int_{{\overline{\mathcal{M}}}_{g,n+1}}\psi_{n+1}^{b}\tilde{\Xi}^{1}_{g,n}\leq
2g-1$

for any $b\in\mathbb{Z}_{\geq 0}$.

This completes the proof of Theorem 3.5 by Lemma 3.6, and hence of Theorem
2.3.

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# Unmasking noise transients masquerading as intermediate-mass black hole
binaries

Sayantan Ghosh<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Technology Bombay, Powai, Mumbai 400 076, India  Koustav Chandra
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400 076, India  Archana Pai<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India

###### Abstract

In the first three observation runs, ground-based gravitational wave (GW)
detectors have observed close to $\sim 100$ compact binary coalescence (CBC)
events. The GW detection rates for CBCs are expected to increase with
improvements in the sensitivity of the International Gravitational-Wave
Observatory Network (IGWN). However, with improved sensitivity, non-Gaussian
instrumental transients or “glitches” are expected to adversely affect GW
searches and characterisation algorithms. The most detrimental effect is due
to short-duration glitches, which mimic the morphology of short-duration GW
transients, in particular intermediate-mass black hole (IMBH) binaries. They
can be easily misidentified as astrophysical signals by current searches, and
if included in astrophysical analyses, glitches mislabelled as IMBH binaries
can affect IMBH population studies. In this work, we introduce a new
similarity metric that quantifies the consistency of astrophysical parameters
across the detector network and helps to distinguish between IMBH binaries and
short-duration, loud glitches which mimic IMBH binaries. We develop this
method using a simulated set of IMBH binary signals and a collection of noise
transients identified during the third observing run of the Advanced LIGO and
Advanced Virgo detectors.

intermediate-mass black hole — glitches — gravitational waves

††preprint: APS/123-QED

##  I  Introduction

In recent years, interest has surged in the detection and characterisation of
intermediate-mass black holes (IMBHs), which are black holes having masses
between $10^{2}\mathrm{M_{\odot}}$ and $10^{5}\mathrm{M_{\odot}}$ Greene  _et
al._ [2020]. They are believed to be seeds of supermassive black holes (SMBHs)
($M>10^{5}\mathrm{M_{\odot}}$). In general, the observation and
characterisation of IMBHs are expected to lead to a better understanding of
how supermassive black holes form, which will eventually enable us to study
the formation and growth of galaxies. Greene  _et al._ [2020]

The ground-based GW detectors Advanced LIGO Aasi  _et al._ [2015] and Advanced
Virgo Acernese  _et al._ [2015] are sensitive to IMBH binary mergers producing
remnants in the lower ($10^{2}-10^{3}\mathrm{M_{\odot}}$) IMBH mass range
Abbott  _et al._ [2019a], thus opening up new prospects for detecting and
characterising IMBH binaries. These detectors have completed three observing
runs - O1 , O2 and O3 Nitz  _et al._ [2019, 2020, 2021, 2023]; Abbott  _et
al._ [2019b, 2021a, 2021b] and the fourth observation run O4 has begun
recently. To date, $\sim$ 100 compact binary coalescence (CBC) events have
been detected, most of which are binary black hole (BBH) mergers. The hallmark
of O3 was the first confident IMBH binary event - GW190521 Abbott  _et al._
[2020a, b]. This signal, under the quasi-spherical black hole binary
hypothesis, is consistent with the merger between two very massive black holes
(masses $85M_{\odot}$ and $66M_{\odot}$), leaving behind an IMBH remnant of
mass $\sim 142M_{\odot}$. There was a dedicated IMBH binary search for the O3
run Abbott  _et al._ [2022]; Chandra  _et al._ [2021]; Drago  _et al._ [2021],
which declared GW190521 as the only significant IMBH binary event along with a
couple of marginal candidates.

IMBH binary signals detectable in the LIGO-Virgo bandwidth have short duration
($<0.1s$), and are dominated by the merger and ringdown phases, with a small
contribution to the signal-to-noise ratio (SNR) coming from the inspiral part.
This makes detecting and characterising IMBH binaries challenging. Non-
Gaussian, non-stationary noise transients or “glitches” appearing in the
strain data adversely affect search sensitivity and astrophysical studies
Abbott  _et al._ [2022]. Many glitch classes are detrimental to IMBH searches.
Due to their durations ($<0.1s$) and frequency bandwidths (10 to a few hundred
Hz) being similar to those of IMBH binary signals Davis [2019], some glitch
classes, such as Tomtes and Blips, are prone to be flagged as IMBH binary
signals by the current detection pipelines. 200214_224526, one of the marginal
candidates detected in the O3 IMBH search, is a classic case of glitches
masquerading as an IMBH binary event. Although this trigger was detected by
the coherent waveburst (cWB) pipeline, subsequent investigations revealed that
this trigger was of instrumental origin Abbott  _et al._ [2022].

It may not always be possible to ascertain whether a particular transient is
of instrumental or astrophysical origin. For example, the candidate
200114_020818 was detected in Hanford (LHO), Livingston (LLO) and Virgo
detectors with a network SNR of 14.5 and false alarm rate (FAR) of
$0.06\mathrm{yr}^{-1}$ by the O3 IMBH search Abbott  _et al._ [2022]. When
this candidate was analysed with the three waveform models — IMRPhenomXPHM
Pratten  _et al._ [2021], SEOBNRv4PHM Ossokine  _et al._ [2020] and NRSur7dq4
Varma  _et al._ [2019], the posteriors obtained were inconsistent with one
another. It is quite likely that the inconsistency between the posteriors is
due to waveform systematics. Still, there is also another possibility that
this candidate is a culmination of chance coincident glitches in multiple
detectors. So, this candidate could neither be confidently ruled out as a
glitch nor could it be flagged as a confident IMBH binary event. Also, the
coherence test did not provide sufficient evidence in support of the coherent
signal hypothesis Abbott  _et al._ [2022]. Thus, it is crucial to develop
additional independent tests that can increase confidence in deciding the
nature of transients of doubtful origin. From an astrophysical standpoint,
distinguishing between IMBH binaries and IMBH binary mimickers is paramount.
This is because the incorrectly classified noisy transients can severely
affect the astrophysical population studies of IMBH binaries, which are rare
events compared to stellar BBH mergers.

In order to minimise the effect of glitches, GW search algorithms use a
variety of distinct classes of vetoes Abbott  _et al._ [2018], gating Usman
_et al._ [2016] and test statistics that can discriminate between signals and
glitches Allen [2005]. In the parameter estimation (PE) study, a test was
developed in Veitch and Vecchio [2010] that assesses if a given trigger is of
noise origin in the multi-detector Bayesian framework. This coherence test
computes the Bayes Factor between the hypothesis that the data contains a
coherent CBC signal across the detectors ($H_{\mathrm{coh}}$) against the
hypothesis that the data contains incoherent instrumental features
($H_{\mathrm{inc}}$). The evidence for $H_{\mathrm{coh}}$ is computed by
assuming that the incoming astrophysical signal is characterised by the same
signal parameter vector $\vec{\theta}$ in all detectors. The evidence for
$H_{\mathrm{inc}}$ is the product of the evidence in the individual detectors,
allowing for different, and thus inconsistent parameters across the detector
network.

In addition, there have been several attempts to address the problem of
distinguishing signals from glitches in the detector characterisation stage.
BayesWave approaches this problem in a morphology-independent way, by
projecting both signals and glitches onto the wavelet basis Cornish and
Littenberg [2015]; Cornish  _et al._ [2021]. Parameterised models of
frequently occurring glitch types have been developed using probabilistic
principal component analysis Merritt  _et al._ [2021]. Recently, efforts have
been made to develop population models of glitches by projecting the glitches
into the parameter space for astrophysical models Ashton  _et al._ [2022].

In this work, we develop a new, simplistic approach that helps distinguish
between IMBH binary signals and short-duration glitches. This method assumes
that regardless of the waveform model, the posteriors of any astrophysical
transient should be consistent between the detectors. Glitches, being of local
origin, may produce inconsistent posteriors. We use Jensen Shannon divergence
(JSD) Lin [1991] as a similarity metric to quantify the consistency between
parameter estimates. We develop this method using a simulation campaign in the
IMBH binary parameter space and glitches in the PyCBC-IMBH search Chandra  _et
al._ [2021]. Finally, we apply this method to several high-mass
($M_{T}^{\mathrm{src}}>65\mathrm{M_{\odot}}$) GW events and marginal
candidates in the GW catalogues.

The paper is structured as follows: Section II.1 reviews the basics of
Bayesian parameter inference of GW signals. Section II.2 introduces a JSD-
based similarity metric that quantifies the parameter consistency between two
detectors. Section II.3 describes the astrophysical simulations and glitches
analysed in this work. Section II.4 defines the new test statistic used in
this work and also summarises the main results. Finally, section III discusses
the application of the method developed in this work on a curated set of GW
triggers and section IV summarises the main conclusions of this work.

##  II  Methodology

###  II.1  Parameter Estimation in the Bayesian Framework

The calibrated strain data $d(t)$ in the interferometric GW detectors is
modelled as:

$d(t)=n(t)+s(t;\vec{\theta})\,,$ (1)

where, $n(t)$ is the detector noise and $s(t;\vec{\theta})$ is the GW signal.
The noise $n(t)$ can be modelled as:

$n(t)=n_{G}(t)+n_{NG}(t)\,,$ (2)

where $n_{G}(t)$ is Gaussian and wide-sense stationary. $n_{NG}(t)$ denotes
glitches, which are non-Gaussian and non-stationary. The posterior of the
parameters $\vec{\theta}$ describing the signal is given by Veitch and Vecchio
[2010]; Thrane and Talbot [2019]:

$p(\vec{\theta}|d,W)=\frac{\mathcal{L}(d|W,\vec{\theta})\pi(\vec{\theta}|W)}{\mathcal{Z}}\,,$
(3)

where $d$ is the strain data, $W$ is the waveform approximant which models the
signal, $\mathcal{L}(d|W,\vec{\theta})$ is the likelihood function, which
gives the probability of obtaining the data $d$ given the waveform model $W$
and the parameters $\vec{\theta}$. Under the assumption that the noise is
Gaussian, the likelihood takes a Gaussian form. $\pi(\vec{\theta}|W)$ is the
prior distribution on the parameters $\vec{\theta}$. $\mathcal{Z}$ is called
the model evidence because it gives the total probability of the model $W$
producing the data $d$, marginalised over all parameters.

$\mathcal{Z}=\int\mathcal{L}(d|W,\vec{\theta})\pi(\vec{\theta}|W)d\vec{\theta}.$
(4)

The posterior distribution of a single parameter is then obtained by
marginalising the multi-dimensional posterior over all the other parameters,
also known as nuisance parameters.

###  II.2 Jensen Shannon Divergence: A measure of two-detector parameter
consistency

An astrophysical signal $s(t;\vec{\theta})$ (in our case, an IMBH binary
signal) arrives at two detectors with a time difference consistent with the
light travel time between the two detectors. The signal amplitude in each
detector is a linear combination of the plus and cross polarisations, weighted
by the antenna pattern functions, which give the directional sensitivity of a
detector to a particular polarisation. The phase evolution of the signal in
each detector depends on the intrinsic parameters, i.e, the masses and spins
of the merging black holes. We expect the signal parameters estimated
separately from the data in two detectors to be consistent with each other,
barring the noise power spectral density (PSD) effects pertaining to the noise
in each detector. This feature should not depend on the waveform model used.
Of course, different waveform models might introduce biases, but these biases
are also expected to be consistent across the detectors. On the other hand, if
there are two non-astrophysical but temporally chance-coincident noise
transients($n_{NG}(t)$) in the two detectors which happened to trigger the
same template, then the parameters estimated from the strain data in the two
detectors (assuming that they are of astrophysical origin) may not be
consistent with each other, as local disturbances cause the noisy transients.
A matched filter’s output is simply the glitch’s projection on a particular
template and does not need to describe the entire glitch morphology. However,
the PE samples provide a more complete description of the morphology of the
transient. Thus, we expect that the posterior distributions of the
astrophysical parameters corresponding to two detectors will be consistent for
an incoming astrophysical signal and will be either non-overlapping or
minimally overlapping for chance coincident glitches.

In our approach, we evaluate the likelihoods by assuming that a quasi-circular
quadrupole waveform model well describes the transient. Subsequently, we
assess the compatibility of the inferred posteriors of astrophysical
parameters across the detector network. Unlike the coherence test, which
compares the evidence for the coherent and incoherent hypotheses, we use
Jensen Shannon divergence (JSD) Lin [1991] to quantify how distinguishable the
posteriors in different detectors are. This provides us with a bounded,
symmetric measure of the difference between the information content in two
probability distributions. It can be computed for both one-dimensional and
multi-dimensional distributions. Mathematically, it is defined as follows. Let
$\theta$ be one of the parameters describing an IMBH binary signal. Let
$p_{H}(\theta)$ and $p_{L}(\theta)$ be the posteriors of $\theta$ obtained
from PE runs with LHO and LLO data respectively.

The JSD between $p_{H}(\theta)$ and $p_{L}(\theta)$ is defined as Lin [1991]:

$\begin{split}\mathrm{JSD}_{\theta}(p_{H}||p_{L})=\frac{1}{2}\int
p_{H}(\theta)\log_{2}\frac{p_{H}(\theta)}{p_{\mathrm{avg}}(\theta)}d\theta+\\\
\frac{1}{2}\int
p_{L}(\theta)\log_{2}\frac{p_{L}(\theta)}{p_{\mathrm{avg}}(\theta)}d\theta\,,\end{split}$
(5)

where,

$p_{\mathrm{avg}}(\theta)=\frac{1}{2}(p_{H}(\theta)+p_{L}(\theta))$ (6)

We can further extend this definition to two dimensions as follows:

${}\begin{split}\mathrm{JSD}_{\theta_{1},\theta_{2}}^{\mathrm{2D}}(p_{H}||p_{L})=\frac{1}{2}\int
p_{H}(\theta_{1},\theta_{2})\log_{2}\frac{p_{H}(\theta_{1},\theta_{2})}{p_{\mathrm{avg}}(\theta_{1},\theta_{2})}d\theta_{1}d\theta_{2}\\\
+\frac{1}{2}\int
p_{L}(\theta_{1},\theta_{2})\log_{2}\frac{p_{L}(\theta_{1},\theta_{2})}{p_{\mathrm{avg}}(\theta_{1},\theta_{2})}d\theta_{1}d\theta_{2}\,,\end{split}$
(7)

where $\theta_{1}$ and $\theta_{2}$ are two parameters.

For two identical distributions, the JSD is zero. The greater the value of the
JSD, the greater the dissimilarity between the two distributions, with the
maximum possible value of unity. For signals, we expect the posteriors
corresponding to the two detectors to be very similar to each other, and hence
the JSD is expected to be close to zero. For chance-coincident glitches, we
expect the posteriors corresponding to the two detectors to be dissimilar;
hence, the JSD is expected to be close to 1.

###  II.3 Simulations

We simulate 1000 IMBH binary signals with the model NRSur7dq4 Varma  _et al._
[2019], a quasi-spherical, multipolar waveform model calibrated to numerical
relativity simulations. The parameters of these simulated signals are chosen
randomly from the following
ranges:$M_{T}^{\mathrm{det}}\sim\mathcal{U}(150,500)M_{\odot}$, mass ratio
$\sim\mathcal{U}(1,6)$, the spin magnitudes
$\chi_{1},\chi_{2}\sim\mathcal{U}(0,0.9)$, and the directions of spins are
chosen from an isotropic distribution. These signals are added to different
realisations of Gaussian noise coloured by the PSD of the two Advanced LIGO
detectors in the O3 run, with the LHO-LLO network SNR
$\sim\mathcal{U}(10,20)$. We chose this SNR window because most of the
observed GW events lie in this range.

Often, obtaining glitches for a playground study is a challenging task. In GW
astronomy, the noise background study is carried out by artificially sliding
the data of one detector with respect to the other detector by unphysical time
delays. The search algorithm is processed on this data, and the chance-
coincident noisy events provide the statistic for the noise background. As a
playground, here we consider $\sim 500$ pairs of LHO-LLO glitches in the
background data of the O3 PyCBC-IMBH search - an optimised matched-filter-
based search for IMBH binaries Chandra  _et al._ [2021]. The pairs of glitches
that we consider were template-coincident as well as temporally coincident in
the time-shifted background constructed for estimating the noise statistic for
the search.

We perform parameter estimation (PE) runs with the LHO and LLO data separately
for both signals and glitches with IMRPhenomXAS Pratten  _et al._ [2020]. We
use this waveform model for our recovery as it is a frequency-domain quasi-
circular quadrupole waveform model. This allowed us to use likelihoods that
are explicitly marginalised over the effective phase, distance and time,
thereby reducing the number of dimensions and speeding up likelihood
evaluations Thrane and Talbot [2019]. We understand that for signals, the
difference between the injection and recovery models can introduce a bias in
the estimates of the parameters. However, we expect this bias to be similar in
the two detectors, and it should not affect the JSD study between the
posteriors. We note that here our goal is not to recover the injected
parameters but to assess the consistency between the parameter estimates in
the two detectors. By no means do we propose to use these parameter estimates
to infer the properties of the underlying signal.

All PE runs are performed with PyCBC Inference Biwer  _et al._ [2019]. We use
the dynesty sampler Speagle [2020] and employ an aligned-spin prior, which is
uniform in the detector-frame component masses, uniform in the aligned-spin
magnitudes, isotropic in sky location and uniform in Euclidean volume. We take
a lower frequency cutoff of 15 Hz for likelihood evaluations.

Figure 1: The top figure in the left panel shows the whitened data in LHO for
GW190521. The bottom figure in the left panel shows the Q-transform of the LHO
data. The right panel shows the LHO and the LLO posteriors, which are largely
overlapping.

Figure 2: The top figure in the left panel shows the whitened data around the
time of occurrence of a Tomte glitch in LHO. The bottom figure in the left
panel shows the Q-transform. The right panel shows the LHO and LLO posteriors
of the Tomte glitch shown in the left panel and its counterpart in LLO, which
triggered the same template. These posteriors are minimally overlapping.

As described in section II.1, the intrinsic parameters determine the phase
evolution of the signal. The temporal morphology of the signal depends on the
phase evolution. For a quasi-circular waveform model (such as IMRPhenomXAS),
the relevant intrinsic parameters are the detector-frame component masses
$m_{1}^{\mathrm{det}}$ and $m_{2}^{\mathrm{det}}$ (which are related to the
source-frame masses $m_{1}^{\mathrm{src}}$ and $m_{2}^{\mathrm{src}}$ by
$m_{1,2}^{\mathrm{det}}=m_{1,2}^{\mathrm{src}}(1+z)$, $z$ being the redshift)
and the effective spin parameter $\chi_{\mathrm{eff}}$ Santamaria  _et al._
[2010]; Damour [2001]; Racine [2008] which quantifies the degree of alignment
of the component spins along the direction of the orbital angular momentum.
$\chi_{\mathrm{eff}}$ is defined as follows:

$\chi_{\mathrm{eff}}=\frac{m_{1}^{\mathrm{det}}\chi_{1z}+m_{2}^{\mathrm{det}}\chi_{2z}}{m_{1}^{\mathrm{det}}+m_{2}^{\mathrm{det}}}\,,$
(8)

where $\chi_{1z}$ and $\chi_{2z}$ are the components of the spins along the
direction of the orbital angular momentum.

Fig. 1 shows the whitened data, the Q-transform, and the corner plot for the
posteriors of $m_{1}^{\mathrm{det}},m_{2}^{\mathrm{det}}$ and
$\chi_{\mathrm{eff}}$ using the LHO and LLO data for GW190521. Fig. 2 shows
the same for a typical Tomte glitch. The duration, frequency range and overall
morphologies of the signal and the glitches look similar. However, the
posteriors corresponding to LHO and LLO are consistent for the signal
(GW190521) and minimally overlap for the glitches.

###  II.4 Test Statistic $\mathcal{R}(\mathcal{J})$

In this subsection, we construct a test statistic that ranks candidates based
on the consistency of the posteriors across the detector network. For each
simulated signal and each pair of background glitches, we calculate
$\mathrm{JSD}_{\theta_{1},\theta_{2}}^{\mathrm{2D}}(p_{H}||p_{L})$, where
$p_{H}$ and $p_{L}$ are the LHO and LLO posteriors respectively. We take all
possible combinations of the parameters -
$(m_{1}^{\mathrm{det}},m_{2}^{\mathrm{det}})$,
$(m_{1}^{\mathrm{det}},\chi_{\mathrm{eff}})$ and
$(m_{2}^{\mathrm{det}},\chi_{\mathrm{eff}})$. Then, we construct their mean
$\mathcal{J}$ as:

${}\begin{split}\mathcal{J}=\frac{1}{3}[\mathrm{JSD}^{\mathrm{2D}}(m_{1}^{\mathrm{det}},m_{2}^{\mathrm{det}})+\mathrm{JSD}^{\mathrm{2D}}(m_{1}^{\mathrm{det}},\chi_{\mathrm{eff}})+\\\
\mathrm{JSD}^{\mathrm{2D}}(m_{2}^{\mathrm{det}},\chi_{\mathrm{eff}})]\end{split}$
(9)

Due to redundancy, we drop the symbol $p_{H}||p_{L}$ and shift the subscript
of Eq. (7) (which consists of the parameters for which the JSD is calculated)
inside the parentheses.

We compute $\mathcal{J}$ as defined in Eq. (9) for all simulated signals and
background glitches. Panel (a) of Fig. 3 shows the $\mathcal{J}$ distributions
for signals and glitches. We observe that a large number of signals have lower
$\mathcal{J}$ values than glitches, although there is some overlap between the
two distributions.

Figure 3: Panel (a) shows the normalised histograms of $\mathcal{J}$ for
signals (blue stepped) and glitches (black stepped), along with their KDEs
(blue solid for signals and black solid for glitches). Panel (b) shows 1 - CDF
(blue solid) for the signal distribution and CDF (black solid) for the glitch
distribution. Panel (c) shows the distributions of the $\mathcal{R}$ statistic
for signals and glitches constructed from the $\mathcal{J}$ distributions
shown in panel (a). In panels (a) and (b), the vertical blue dashed line shows
the $\mathcal{J}$ value of GW190521 and the vertical black dashed line shows
the $\mathcal{J}$ value of the pair of coincident glitches shown in Fig. 2.

Here is the scenario in which we are working: We assume that a trigger with
$\mathcal{J}=\mathcal{J}_{\mathrm{test}}$ is already flagged as a GW event
with some significance in terms of the FAR. The event may not have a very high
significance value and may not have been detected in all the searches, like
the 200114_020818 event. In such a scenario, the null hypothesis is that the
trigger is of astrophysical origin ($\mathcal{H}_{\mathrm{S}}$). The
hypothesis that the trigger is of instrumental origin
($\mathcal{H}_{\mathrm{G}}$) is the alternative hypothesis. We note that
$1-\mathrm{CDF}^{S}$ 111$\mathrm{CDF}$ denotes cumulative density function is
the right-tailed p-value for the signal distribution and will signify the
probability of the event being consistent with the null hypothesis. Likewise,
$\mathrm{CDF}^{G}$ is the left-tailed p-value for the glitch distribution.
Then, high $1-\mathrm{CDF}^{S}(\mathcal{J}=\mathcal{J}_{\mathrm{test}})$ and
low $\mathrm{CDF}^{G}(\mathcal{J}=\mathcal{J}_{\mathrm{test}})$ imply strong
evidence in favour of $\mathcal{H}_{\mathrm{S}}$ and against
($\mathcal{H}_{\mathrm{G}}$), and the reverse is true for low
$1-\mathrm{CDF}^{S}(\mathcal{J}=\mathcal{J}_{\mathrm{test}})$ and high
$\mathrm{CDF}^{G}(\mathcal{J}=\mathcal{J}_{\mathrm{test}})$. Panel (b) in Fig.
3 shows $1-\mathrm{CDF}^{S}$ for the signal distribution and
$\mathrm{CDF}^{G}$ for the glitch distribution. Based on the above rationale,
we define a test statistic that quantifies the evidence in favour of rejecting
$\mathcal{H}_{\mathrm{S}}$ as follows:

${}\mathcal{R}(\mathcal{J})=\frac{\mathrm{CDF}^{G}(\mathcal{J})}{1-\mathrm{CDF}^{S}(\mathcal{J})}\\\
,$ (10)

By construction, $\mathcal{R}\gg 1$ implies that the candidate is more likely
to be of instrumental origin, whereas $\mathcal{R}\ll 1$ implies that the
candidate is more likely to be of astrophysical origin.

Name | Network SNR | FAR ($\mathrm{yr}^{-1}$) | $\mathcal{J}$ | $\mathcal{R}$ | p-value
---|---|---|---|---|---
GW150914 | 26.0 |  $\leq$ $10^{-7}$ | 0.009 | 0.002 | 0.921
GW190519_153544 | 15.9 |  $\leq$ $10^{-5}$ | 0.011 | 0.005 | 0.882
GW200224_222234 | 20.0 |  $\leq$ $10^{-5}$ | 0.016 | 0.022 | 0.793
GW190727_060333 | 11.7 |  $\leq$ $10^{-5}$ | 0.018 | 0.032 | 0.755
GW190503_185404 | 12.2 |  $\leq$ $10^{-5}$ | 0.018 | 0.032 | 0.758
GW190521_074359 | 25.9 |  $\leq$ $10^{-5}$ | 0.029 | 0.110 | 0.588
GW190602_175927 | 12.1 |  $\leq$ $10^{-5}$ | 0.038 | 0.185 | 0.495
GW191222_033537 | 12.5 |  $\leq$ $10^{-5}$ | 0.045 | 0.268 | 0.430
GW190706_222641 | 13.4 | $5.0\times 10^{-5}$ | 0.013 | 0.012 | 0.839
GW190521 | 14.3 | $2.0\times 10^{-4}$ | 0.013 | 0.010 | 0.849
GW190421_213856 | 10.7 | $7.7\times 10^{-4}$ | 0.023 | 0.065 | 0.670
GW200219_094415 | 10.7 | $9.9\times 10^{-4}$ | 0.057 | 0.469 | 0.327
GW190701_203306 | 11.2 | $5.7\times 10^{-3}$ | 0.019 | 0.036 | 0.741
200114_020818 | 14.5 | $6.0\times 10^{-2}$ | 0.241 | 31.873 | 0.017
200214_224526 | 13.1 | $1.3\times 10^{-1}$ | 0.355 | 175.549 | 0.005
Table 1: Shows the results of applying the method developed in this work to
real events having source -frame total mass $\geq 65M_{\odot}$ and network SNR
$\geq 10$. Abbott  _et al._ [2019b, 2021a, 2021b]

We fit KDEs to the distributions of $\mathcal{J}$ for signals and glitches as
shown in panel (a) of Fig. 3. With the assumption that real signals and
glitches will follow these KDEs (also shown in panel (a) of Fig. 3), we
generate points from the signal KDE and the glitch KDE and treat them like our
“test signals” and “test glitches”. Panel (c) of Fig. 3 shows the
distributions of $\mathcal{R}$ for test signals and test glitches. Appendix A
explains the relation between $\mathcal{R}$ and the commonly used likelihood
ratio (LR) statistic, derived from the KDEs in panel (a) of Fig. 3. It shows
that there is a one-to-one correspondence between $\mathcal{R}$ and LR, and
the Receiver Operating Characteristic (ROC) plots of these two statistics,
which show that they have comparable distinguishing powers. Here, we use the
$\mathcal{R}$ statistic.

##  III Application to candidate events

Table 1 summarises the results of applying our method to events from
Gravitational Wave Open Science Center (GWOSC) with LHO-LLO. We apply our
method to 13 GW-tagged events having source-frame total mass more than
$65M_{\odot}$ (median values) and 2 marginal candidates observed in the LHO
and LLO detectors with network SNR above 10. Although our primary interest is
in the IMBH mass range ($10^{2}\mathrm{M_{\odot}}$ to
$10^{5}\mathrm{M_{\odot}}$), we extend the lower mass limit to
$65\mathrm{M_{\odot}}$ to test the robustness of our method.

We calculate the p-values computed from the distribution of $\mathcal{R}$.
Significant GW events are expected to have $\mathcal{R}<1$, and high p-values.
Coincident glitches mimicking signals will have $\mathcal{R}>1$ and low
p-values.

GW190521, the only significant IMBH binary, has an $\mathcal{R}$ value of 0.01
and a p-value of 0.85, which clearly supports $\mathcal{H}_{\mathrm{S}}$. The
marginal candidate 200214_224526 has an $\mathcal{R}$ value of 176, with a
p-value of 0.005. The low p-value is consistent with the instrumental origin
of this event.

The second marginal candidate 200114_020818 has $\mathcal{R}$ value of 32 and
a p-value of 0.017, which indicates that this candidate is less likely to be
of astrophysical origin.

.

##  IV  Discussion and Conclusion

Certain classes of noisy glitches mimic the massive BBH events, especially in
the IMBH binary parameter space. Several approaches are employed in the
searches such as glitch-specific veto methods, machine learning approaches in
the post-processing stage Abbott  _et al._ [2018]; Usman  _et al._ [2016];
Allen [2005]; Zevin  _et al._ [2017] In this work, we took a new parameter
consistency approach to investigate the possibility of loud, short-duration
IMBH-like GW triggers being of instrumental origin. Here, we have demonstrated
the potential of the 2D Jensen Shannon divergence (JSD) to distinguish between
IMBH binaries and short-duration glitches with comparable SNRs in at least two
advanced interferometric detectors. We defined a new test statistic
$\mathcal{R}$, synonymous with likelihood ratio (LR), based on 2D JSD to
quantify the consistency of posteriors for the intrinsic parameters across the
detector network. A pair of coincident triggers giving a low value of
$\mathcal{R}$ has a high degree of consistency between the detectors and hence
is more likely to be of astrophysical origin. On the other hand, a high value
of $\mathcal{R}$ implies a lack of consistency between the detectors, and this
indicates that the triggers in question may be of instrumental/environmental
origin. We also emphasize that the method is quite independent of the choice
of waveform model. Different waveform models can introduce bias in the
estimated parameters. However, if the trigger is of astrophysical origin, the
bias in the parameters will be consistent across the detectors, provided the
signal SNR is not too different across the detectors.

We computed the p-values of events to quantify the statistical significance of
our classification. We demonstrated this method with the massive BBH merger
events and marginal candidates in the observational runs. We observed that all
events reported with high significance give low $\mathcal{R}$ values and high
p-values as expected. 200214_224526, whose origin is confirmed to be
instrumental, gives a p-value of 0.005. 200114_020818, which is currently of
unknown origin, has a p-value of 0.017, which indicates that this candidate is
less likely to be of astrophysical origin.

Our method can be used in a variety of ways. It can be used in conjunction
with detector characterisation to add further confidence in the
classification. Further, it can also be folded in the main search pipelines,
similar to the $\chi^{2}$ statistic Allen [2005] used in PyCBC-based searches
Usman  _et al._ [2016]; Chandra  _et al._ [2021] to weigh the candidate events
appropriately. Some of these approaches will be taken up in the future.

Similar to JSD, several other ways exist to assess the
similarity/dissimilarity between probability distributions. Our work is a
proof of concept demonstrating one such metric’s use. Other common measures of
the dissimilarity between two probability distributions are - Jeffreys
distance Jeffreys [1946], Kolmogorov Smirnov statistic KOLMOGOROV [1933],
Bhattacharyya distance Bhattacharyya [1943] and many more. The exercise that
we have performed in this work can be performed with other measures as well
which we plan to explore for optimal choice.

We restricted our study to a 2-detector network consisting of LHO and LLO with
comparable sensitivities. The extension of this method to networks with three
or more detectors may be straightforward. However, the difference in SNRs due
to the difference in detector sensitivity as well as antenna pattern of the
detectors on the JSD distributions needs to be investigated.

##  Appendix A  $\mathcal{R}$ and the likelihood ratio

Figure 4: Panel (a) shows the variation of $\mathcal{R}$ with $\mathcal{J}$.
The blue and grey shaded regions represent the $68\%$ confidence intervals of
$\mathcal{R}$ for signals and glitches, respectively. Panel (b) shows the
relation between the LR and the newly defined $\mathcal{R}$ statistic.
$\mathcal{R}$ increases monotonically with LR. The blue circles are the test
signals, and the black circles are the test glitches. Panel (c) shows the ROC
curves of the LR and $\mathcal{R}$. The y-axis is the probability of detecting
a glitch correctly, and the x-axis is the probability of a signal being
misclassified as a glitch.

In this appendix, we show the variation of the $\mathcal{R}$ statistic with
$\mathcal{J}$ and demonstrate that the LR statistic and the $\mathcal{R}$
statistic have a one-to-one correspondence and that they have comparable
distinguishing powers.

Panel (a) of Fig. 4 shows the variation of $\mathcal{R}$ with $\mathcal{J}$.
As expected, $\mathcal{R}$ increases monotonically with $\mathcal{J}$. In
panel (a) of Fig. 3, we plot the KDEs for signals and glitches.

We generate $10^{4}$ samples from both the signal KDE and the glitch KDE and
treat them like our “test signals” and “test glitches”. We calculate
$\mathcal{R}$ and LR for each of these test cases. Panel (b) of Fig. 4, shows
that the new test statistic $\mathcal{R}$ is a monotonically increasing
function of LR. A value of LR corresponds to a unique value of $\mathcal{R}$
and vice-versa. Panel (c) of Fig. 4 shows the Receiver Operating
Characteristic (ROC) curves for $\mathcal{R}$ and LR. We see that they are
identical, i.e., the distinguishing power of $\mathcal{R}$ is equal to that of
LR. This indicates that we can use either $\mathcal{R}$ or LR as our test
statistic.

###### Acknowledgements.

We thank Kipp Cannon, Edoardo Milotti, Florent Robinet, Thomas Dent and Marco
Drago for their valuable comments. This research has made use of data,
software, and/or web tools obtained from the Gravitational Wave Open Science
Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the
LIGO Scientific Collaboration and the Virgo Collaboration. Virgo is funded by
the French Centre National de Recherche Scientifique (CNRS), the Italian
Istituto Nazionale della Fisica Nucleare (INFN), and the Dutch Nikhef, with
contributions by Polish and Hungarian institutes. This material is based upon
work supported by NSF’s LIGO Laboratory, which is a major facility fully
funded by the National Science Foundation. The authors are grateful for
computational resources provided by the LIGO Laboratory and supported by
National Science Foundation Grants PHY0757058 and PHY-0823459. KC acknowledges
the MHRD, the Government of India, for the fellowship support. AP acknowledges
the support from SERB-Power fellowship grant SPF/2021/000036, DST, India. This
document has LIGO DCC No LIGO-P2200227.

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  *[GW]: gravitational wave
  *[CBC]: compact binary coalescence
  *[CBCs]: compact binary coalescence
  *[IGWN]: International Gravitational-Wave Observatory Network
  *[IMBH]: intermediate-mass black hole
  *[IMBHs]: intermediate-mass black hole
  *[SMBHs]: supermassive black hole
  *[BBH]: binary black hole
  *[SNR]: signal-to-noise ratio
  *[cWB]: coherent waveburst
  *[LHO]: Hanford
  *[LLO]: Livingston
  *[FAR]: false alarm rate
  *[PE]: parameter estimation
  *[JSD]: Jensen Shannon divergence
  *[PSD]: power spectral density
  *[KDEs]: kernel density estimate
  *[KDE]: kernel density estimate
  *[LR]: likelihood ratio
  *[ROC]: Receiver Operating Characteristic
  *[GWOSC]: Gravitational Wave Open Science Center