# Bi-Hamiltonian recursion, Liu–Pandharipande relations, and vanishing terms
of the second Dubrovin–Zhang bracket

Francisco Hernández Iglesias  Korteweg-de Vriesinstituut voor Wiskunde,
Universiteit van Amsterdam, Postbus 94248, 1090GE Amsterdam, Nederland
<EMAIL_ADDRESS>and  Sergey Shadrin  Korteweg-de Vriesinstituut
voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090GE Amsterdam,
Nederland<EMAIL_ADDRESS>
###### Abstract.

The Dubrovin–Zhang hierarchy is a Hamiltonian infinite-dimensional integrable
system associated to a semi-simple cohomological field theory or,
alternatively, to a semi-simple Dubrovin–Frobenius manifold. Under an extra
assumption of homogeneity, Dubrovin and Zhang conjectured that there exists a
second Poisson bracket that endows their hierarchy with a bi-Hamiltonian
structure. More precisely, they gave a construction for the second bracket,
but the polynomiality of its coefficients in the dispersion parameter
expansion is yet to be proved.

In this paper we use the bi-Hamiltonian recursion and a set of relations in
the tautological rings of the moduli spaces of curves derived by Liu and
Pandharipande in order to analyze the second Poisson bracket of Dubrovin and
Zhang. We give a new proof of a theorem of Dubrovin and Zhang that the
coefficients of the dispersion parameter expansion of the second bracket are
rational functions with prescribed singularities. We also prove that all terms
in the expansion of the second bracket in the dispersion parameter that cannot
be realized by polynomials because they have negative degree do vanish, thus
partly confirming the conjecture of Dubrovin and Zhang.

###### Contents

  1. 1 Introduction
    1. 1.1 Organization of the paper
    2. 1.2 Acknowledgments
  2. 2 Hamiltonian structures
    1. 2.1 Changes of coordinates
    2. 2.2 Differential rational functions
  3. 3 Dubrovin–Zhang hierarchy
    1. 3.1 Cohomological field theories
    2. 3.2 The principal hierarchy
    3. 3.3 The full hierarchy
  4. 4 The second bracket
    1. 4.1 Conformality and bi-Hamiltonian recursion
    2. 4.2 Uniqueness theorem
    3. 4.3 Dubrovin–Zhang structural theorem
  5. 5 Liu–Pandharipande relations
    1. 5.1 Relation among the tautological classes
    2. 5.2 Implications for $\partial_{x}$-derivatives of two-point functions
    3. 5.3 Variation for $\tilde{\mathsf{E}}$
  6. 6 Vanishing terms of the second bracket
    1. 6.1 Equivalent form without the variational derivative
    2. 6.2 Vanishing terms

##  1\. Introduction

Dubrovin–Frobenius manifolds were first introduced in [Dub96] as a way to
study homogeneous solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV)
associativity equations [Wit90, DVV91] in a coordinate-free way. In this work
the relation between Dubrovin–Frobenius manifolds, topological field theories
(TFTs) and integrable systems is first explored: namely, one can construct two
flat metrics related to a Dubrovin–Frobenius manifold, which give a pair of
compatible Poisson brackets $(P,K)$ of hydrodynamic type. Then, starting from
the Casimirs of $P$, $\bar{h}_{\alpha,-1}=\int\eta_{\alpha\beta}v^{\beta}dx$,
and under the assumption that there are no common Casimirs, one can apply a
bi-Hamiltonian recursion algorithm [Mag78] to obtain a dispersionless
hierarchy of the form

(1) $\displaystyle\frac{\partial v^{\alpha}}{\partial
t^{\beta}_{q}}=P^{\alpha\gamma}\frac{\delta\bar{h}_{\beta,q}}{\delta
v^{\gamma}}.$

In [DZ01], Dubrovin and Zhang further explore the relationship between
Dubrovin–Frobenius manifolds and integrable systems and deform this hierarchy
via a quasi-Miura transformation

(2) $\displaystyle
w^{\alpha}=v^{\alpha}+\sum_{g=1}^{\infty}\epsilon^{2g}Q^{\alpha}_{g}(v,v_{1},\dots,v_{3g})$

given by weighted homogeneous differential rational functions $Q^{\alpha}_{g}$
to obtain the full dispersive hierarchy, which is known as the Dubrovin–Zhang
(DZ) hierarchy. They conjecture in [DZ01] that the transformed equations,
Hamiltonians and brackets are differential polynomials in the coordinates
$w^{\alpha}$.

In [BPS12a, BPS12b] this conjecture is partially proved in a more general
setting: the DZ hierarchy is constructed from a semi-simple cohomological
field theory (CohFT), without an assumption of homogeneity. It is proved in
_op. cit._ that the equations, Hamiltonians, tau structure, and first bracket
of the DZ hierarchy are polynomial.

The main goal of this paper is to analyze the second Poisson bracket of the
Dubrovin–Zhang hierarchy. We start with a conformal semi-simple cohomological
field theory, thus the construction of Dubrovin applied to the underlying
Dubrovin–Frobenius manifold gives the second Poisson bracket in the
dispersionless limit, and the quasi-Miura transformation (2) produces a
possibly singular Poisson structure. We have two tools to analyze it: the bi-
Hamiltonian recursion and the tautological relations in cohomology of the
moduli spaces of curves or, more precisely, the differential equations that
they imply on various structures of the Dubrovin–Zhang hierarchies.

The bi-Hamiltonian recursion appears to be sufficient to uniquely determine
the second Dubrovin–Zhang bracket, and we also use it to give a new proof of a
structural result of Dubrovin and Zhang on its possible singularities. Bi-
Hamiltonian recursion also implies that the constant term of the second
Dubrovin–Zhang bracket is a differential polynomial.

As a source of suitable tautological relations we use the work of Liu and
Pandharipande [LP11]. The relations that they derive there appear to be
exactly enough to prove the vanishing of all terms in the second
Dubrovin–Zhang bracket whose standard degree is negative. Remarkably, the
dimensional inequalities of the Liu–Pandharipande relations match exactly the
standard degree count for the terms of the second Dubrovin–Zhang bracket in
the equations that we derive from the bi-Hamiltonian recursion, so the
Liu–Pandharipande relations say nothing about the non-negative standard degree
terms of the second bracket.

###  1.1. Organization of the paper

In Section 2 we set the appropriate formalism to work with Hamiltonian and bi-
Hamiltonian structures. In Section 3 we recall the construction of the
principal Dubrovin-Zhang hierarchy from a CohFT and endow it with a
Hamiltonian structure of hydrodynamic type. We then deform it to the full
hierarchy and show that it inherits a Hamiltonian structure. In Section 4 we
prove that in the case the underlying CohFT is conformal, the DZ hierarchy is
also bi-Hamiltonian, with the second Hamiltonian structure having
singularities of a very particular type (we reprove a result of Dubrovin and
Zhang on that) and being uniquely determined by the bi-Hamiltonian recursion
relation. In particular, we prove there that the constant term of the second
bracket it polynomial. In Section 5 we recall the Liu–Pandharipande relations
in the tautological ring, and summarize the most important corollaries. In
Section 6 we prove that all terms that must vanish for degree reasons once the
conjecture of Dubrovin and Zhang on polynomiality of the second bracket holds
actually do vanish.

###  1.2. Acknowledgments

We thank A. Buryak and G. Carlet for useful discussions. We also thank the
anonymous referees for useful remarks. The authors were supported by the
Netherlands Organization for Scientific Research. S. S. was also partially
supported by International Laboratory of Cluster Geometry NRU HSE, RF
Government grant, ag. № 75-15-2021-608 dated 08.06.2021.

##  2\. Hamiltonian structures

In this section, we explain the $\theta$-formalism first introduced in [Get02]
and further developed in [LZ13] and [DLZ18] to work with Hamiltonian and bi-
Hamiltonian structures. The main addition to their theory is the completion of
the differential polynomial algebra to allow certain singularities.

Let $M$ be a formal germ of an $N$-dimensional smooth manifold. We define a
formal supermanifold $\hat{M}$ by describing its ring of functions. A system
of local coordinates on $\hat{M}$ is given by
$\\{u^{1},\dots,u^{N},\theta_{1},\dots,\theta_{N}\\}$, where
$\\{u^{1},\dots,u^{N}\\}$ is a system of local coordinates of $M$ and
$\\{\theta_{1},\dots,\theta_{N}\\}$ are the corresponding dual coordinates.
Note the latter are Grassmann variables, i. e.,
$\theta_{\alpha}\theta_{\beta}+\theta_{\beta}\theta_{\alpha}=0$ for all
$\alpha,\beta$.

Consider now the infinite jet space of $\hat{M}$, $J^{\infty}(\hat{M})$. A
system of local coordinates in $J^{\infty}(\hat{M})$ is given by
$\\{u^{\alpha}_{p},\theta_{\beta}^{q}\\}_{1\leq\alpha,\beta\leq N}^{p,q\geq
0}$ and we identify $u^{\alpha}_{0}=u^{\alpha}$ and
$\theta_{\alpha}^{0}=\theta_{\alpha}$, $\alpha=1,\dots,N$.

######  Definition 2.1.

The _differential polynomial algebra_ $\hat{\mathcal{A}}$ is defined as

(3)
$\displaystyle\hat{\mathcal{A}}=\mathbb{C}[[u^{\alpha}_{0}=u^{\alpha},u^{\alpha}_{1},u^{\alpha}_{2},\dots,\theta_{\alpha}^{0}=\theta_{\alpha},\theta_{\alpha}^{1},\theta_{\alpha}^{2},\dots\
|\ 1\leq\alpha\leq N]].$

The term differential polynomial in $u$ (respectively, in $u,\theta$) means
for us a polynomial in $u^{\alpha}_{s}$, $s\geq 1$, (respectively, in
$u^{\alpha}_{s}$, $s\geq 1$, and $\theta_{\alpha}^{s}$, $s\geq 0$) with formal
power series in $u^{\alpha}$ as coefficients.

The differential polynomial algebra admits two gradations: the standard
gradation

(4) $\displaystyle\deg u^{\alpha}_{s}=\deg\theta_{\alpha}^{s}=s,\qquad s\geq
0.$

and the super gradation

(5) $\displaystyle\deg\theta_{\alpha}^{s}=1,\qquad\deg u^{\alpha}_{s}=0,\qquad
s\geq 0.$

Let $\hat{\mathcal{A}}^{p}$ and $\hat{\mathcal{A}}_{p}$ be the degree $p$
components of $\hat{\mathcal{A}}$ with respect to the super and standard
gradations, respectively. Let
$\hat{\mathcal{A}}^{p}_{d}=\hat{\mathcal{A}}^{p}\cap\hat{\mathcal{A}}_{d}$.
Note that $\mathcal{A}=\hat{\mathcal{A}}^{0}$ is the differential polynomial
algebra on $M$. In the following, Einstein’s summation convention applies to
Greek indices, but not to Latin ones. The space $\hat{\mathcal{A}}$ is endowed
with a total derivative

(6) $\displaystyle\partial_{x}=\sum_{p\geq
0}\left(u^{\alpha}_{p+1}\frac{\partial}{\partial
u^{\alpha}_{p}}+\theta_{\alpha}^{p+1}\frac{\partial}{\partial\theta_{\alpha}^{p}}\right),$

which preserves the super gradation on $\hat{\mathcal{A}}$ and increases the
standard gradation by $1$.

######  Definition 2.2.

The space of polyvector fields $\hat{\mathcal{F}}$ is defined as the quotient
space of $\hat{\mathcal{A}}$ by $\partial_{x}\hat{\mathcal{A}}$ and the
constant functions.

The projection map is denoted by
$\int:\hat{\mathcal{A}}\rightarrow\hat{\mathcal{F}}$. Since $\partial_{x}$ is
homogeneous with respect to the gradations, the subspaces
$\hat{\mathcal{F}}^{p}$, $\hat{\mathcal{F}}_{d}$ and
$\hat{\mathcal{F}}^{p}_{d}$ are well defined. The elements of
$\hat{\mathcal{F}}^{p}$ are called $p$-vectors. It is possible to write a
$p$-vector as a sum of its homogeneous components in the standard degree: for
$f\in\hat{\mathcal{F}}^{p}$, we write

(7) $\displaystyle f=f_{0}+f_{1}+f_{2}+\dots,\qquad
f_{k}\in\hat{\mathcal{F}}^{p}_{k}.$

It will be useful later to introduce a formal parameter $\epsilon$ (the
dispersion parameter) to keep track of the degree by rescaling
$x\mapsto\epsilon x$. That is, if $s=\min\\{k|f_{k}\not=0\\}$, then we write

(8) $\displaystyle f=f_{s}+\epsilon f_{s+1}+\epsilon^{2}f_{s+2}+\dots,\qquad
f_{k}\in\hat{\mathcal{F}}^{p}_{k}$

Having defined the functionals, the goal is to define a graded Lie bracket on
$J^{\infty}(\hat{M})$ that extends the usual Schouten bracket on $\hat{M}$,
given by

(9) $\displaystyle[P,Q]=\frac{\partial
P}{\partial\theta_{\alpha}}\frac{\partial Q}{\partial
u^{\alpha}}+(-1)^{p}\frac{\partial P}{\partial u^{\alpha}}\frac{\partial
Q}{\partial\theta_{\alpha}}$

for $P\in\hat{\mathcal{A}}^{p}_{0}$ and $Q\in\hat{\mathcal{A}}^{q}_{0}$. For
this purpose, define the variational derivatives of $f\in\hat{\mathcal{A}}$:

(10) $\displaystyle\frac{\delta f}{\delta\theta_{\gamma}}=\sum_{p\geq
0}(-\partial_{x})^{p}\frac{\partial
f}{\partial\theta_{\gamma}^{p}},\qquad\frac{\delta f}{\delta
u^{\gamma}}=\sum_{p\geq 0}(-\partial_{x})^{p}\frac{\partial f}{\partial
u^{\gamma}_{p}}.$

Since
$\frac{\delta}{\delta\theta_{\gamma}}\circ\partial_{x}=\frac{\delta}{\delta
u^{\gamma}}\circ\partial_{x}=0$, the operators above can be defined on the
space of functionals $\hat{\mathcal{F}}$. Now we can define the bracket

(11)
$\displaystyle[\cdot,\cdot]:\hat{\mathcal{F}}^{p}\times\hat{\mathcal{F}}^{q}$
$\displaystyle\longrightarrow\hat{\mathcal{F}}^{p+q-1}$ (12)
$\displaystyle(P,Q)$ $\displaystyle\longmapsto[P,Q]=\int\left(\frac{\delta
P}{\delta\theta_{\alpha}}\frac{\delta Q}{\delta
u^{\alpha}}+(-1)^{p}\frac{\delta P}{\delta u^{\alpha}}\frac{\delta
Q}{\delta\theta_{\alpha}}\right)dx,$

which we call the Schouten bracket on $J^{\infty}(\hat{M})$. The next theorem
shows that this bracket gives $\hat{\mathcal{F}}$ a graded Lie algebra
structure:

######  Theorem 2.3 ([LZ11]).

Let
$P\in\hat{\mathcal{F}}^{p},Q\in\hat{\mathcal{F}}^{q},R\in\hat{\mathcal{F}}^{r}$.
Then

  * •

$[P,Q]=(-1)^{pq}[Q,P]$.

  * •

$(-1)^{pr}[[P,Q],R]+(-1)^{qp}[[Q,R],P]+(-1)^{rq}[[R,P],Q]=0$.

Finally, we are ready to define Hamiltonian and bi-Hamiltonian structures.

######  Definition 2.4.

A Poisson bivector or Hamiltonian structure is a bivector
$P\in\hat{\mathcal{F}}^{2}$ such that $[P,P]=0$. A bi-Hamiltonian structure is
a pair $(P_{1},P_{2})$ of Poisson bivectors satisfying $[P_{1},P_{2}]=0$.

######  Remark 2.5 (Poisson bivectors, Poisson brackets, Poisson operators).

A Poisson bivector $P$ defines a Poisson bracket on $\hat{\mathcal{F}}^{0}$

(13)
$\displaystyle\\{\cdot,\cdot\\}_{P}\colon\hat{\mathcal{F}}^{0}\times\hat{\mathcal{F}}^{0}$
$\displaystyle\longrightarrow\hat{\mathcal{F}}^{0}$ (14) $\displaystyle(F,G)$
$\displaystyle\longmapsto\\{F,G\\}_{P}=[[P,F],G].$

On the other hand, given a Poisson bracket $\\{\cdot,\cdot\\}_{P}$, there
exist unique $P^{\alpha\beta}_{s}\in\mathcal{A}$ satisfying $\sum_{s\geq
0}P^{\alpha\beta}_{s}\partial_{x}^{s}=\sum_{s\geq
0}(-1)^{s+1}\partial_{x}^{s}P^{\beta\alpha}_{s}$ such that the Poisson
operator $P^{\alpha\beta}=\sum_{s\geq 0}P^{\alpha\beta}_{s}\partial_{x}^{s}$
gives the bracket

(15) $\displaystyle\\{F,G\\}_{P}=\int\frac{\delta F}{\delta
u^{\alpha}}P^{\alpha\beta}\frac{\delta G}{\delta u^{\beta}}dx.$

It is clear that for

(16) $\displaystyle
P=\frac{1}{2}\int\theta_{\alpha}P^{\alpha\beta}(\theta_{\beta})dx=\frac{1}{2}\sum_{s\geq
0}\int P^{\alpha\beta}_{s}\theta_{\alpha}\theta_{\beta}^{s}dx$

both definitions of $\\{\cdot,\cdot\\}_{P}$ coincide. Thus, we will use the
terms Poisson bivector, Poisson operator and Poisson bracket interchangeably
from this point.

######  Example 2.6.

Particularly interesting examples of Poisson bivectors are those of
_hydrodynamic type_ , i.e., those of the standard degree $1$.
$P\in\hat{\mathcal{F}}^{2}_{1}$ is a Poisson bivector of hydrodynamic type if
and only if it takes the form (see [DN83])

(17) $\displaystyle
P=\frac{1}{2}\int(g^{\alpha\beta}(u)\theta_{\alpha}\theta_{\beta}^{1}+\Gamma^{\alpha\beta}_{\gamma}(u)u^{\gamma}_{1}\theta_{\alpha}\theta_{\beta})dx,$

where $g$ is a flat metric on $M$ and $\Gamma^{\alpha\beta}_{\gamma}$ are the
contravariant Christoffel symbols of its Levi-Civita connection.

The Poisson bivectors considered in this paper will be deformations in even
degrees of Poisson bivectors of hydrodynamic type, i.e.,
$P\in\hat{\mathcal{F}}^{2}$ such that $P|_{\epsilon=0}$ is a Poisson bracket
of hydrodynamic type. For any such $P$, we can find unique
$P^{\alpha\beta}_{g,s}\in\mathcal{A}_{2g+1-s}$ satisfying
$\sum_{s=0}^{2g+1}P^{\alpha\beta}_{g,s}\partial_{x}^{s}=\sum_{s=0}^{2g+1}(-1)^{s+1}\partial_{x}^{s}P^{\beta\alpha}_{g,s}$
such that $P$ can be written as

(18) $\displaystyle
P=\frac{1}{2}\int\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{2g+1}P^{\alpha\beta}_{g,s}\theta_{\alpha}\theta_{\beta}^{s}dx.$

###  2.1. Changes of coordinates

We want the Schouten bracket to remain invariant under changes of coordinates.
First, we introduce the groups of transformations that will be considered.

######  Definition 2.7.

A _Miura transformation_ is a formal change of coordinates
$u^{\alpha}\rightarrow\tilde{u}^{\alpha}$ of the form

(19)
$\displaystyle\tilde{u}^{\alpha}=u^{\alpha}+\sum_{k=1}^{\infty}\epsilon^{k}F^{\alpha}_{k}(u;u_{1},\dots,u_{k})$

where $F^{\alpha}_{k}\in\mathcal{A}_{k}$.

In the literature, Miura transformations of the form (19) with an arbitrary
diffeomorphism $F^{\alpha}_{0}(u)$ as the leading term are often considered.
However, all (quasi-)Miura transformations studied in this paper have
$F^{\alpha}_{0}(u)=u^{\alpha}$, thus justifying our more restricted
definition, usually known as _Miura transformations close to the identity_. It
is important to study the behavior of the Schouten bracket under Miura
transformations:

######  Proposition 2.8 ([LZ11]).

A Miura transformation (19) $u^{\alpha}\rightarrow\tilde{u}^{\alpha}$ induces
a change of variables

(20) $\displaystyle\theta_{\beta}=\sum_{s\geq
0}(-\partial_{x})^{s}\left(\frac{\partial\tilde{u}^{\alpha}}{\partial
u_{s}^{\beta}}\tilde{\theta}_{\alpha}\right)$

such that the Schouten bracket remains invariant.

###### Proof.

Let us sketch the proof with the help of the formulas given in [LZ11]. The
change for the variational derivatives is given by

(21) $\displaystyle\frac{\delta}{\delta\tilde{u}^{\alpha}}=\sum_{p\geq
0}(-\partial_{x})^{p}\circ\frac{\partial
u^{\beta}}{\partial\tilde{u}^{\alpha}_{p}}\circ\frac{\delta}{\delta
u^{\beta}},$

Analogously,

(22) $\displaystyle\frac{\delta}{\delta\tilde{\theta}_{\alpha}}=\sum_{t\geq
0}(-\partial_{x})^{t}\circ\frac{\partial\theta_{\beta}}{\partial\tilde{\theta}_{\alpha}^{t}}\circ\frac{\delta}{\delta\theta_{\beta}}.$

From (20), we get

(23)
$\displaystyle\frac{\partial\theta_{\beta}}{\partial\tilde{\theta}_{\alpha}^{t}}=(-1)^{t}\sum_{s\geq
0}\begin{pmatrix}s+t\\\
t\end{pmatrix}(-\partial_{x})^{s}\left(\frac{\partial\tilde{u}^{\alpha}}{\partial
u^{\beta}_{s+t}}\right),$

so

(24) $\displaystyle\frac{\delta}{\delta\tilde{\theta}_{\alpha}}=\sum_{t\geq
0}\partial_{x}^{t}\circ\sum_{s\geq 0}\begin{pmatrix}s+t\\\
t\end{pmatrix}(-\partial_{x})^{s}\left(\frac{\partial\tilde{u}^{\alpha}}{\partial
u^{\beta}_{s+t}}\right)\circ\frac{\delta}{\delta\theta_{\beta}}.$

Let $P,Q\in\hat{\mathcal{F}}$. The result follows after replacing (21) and
(24) in the expression

(25) $\displaystyle[P,Q]=\int\frac{\delta
P}{\delta\tilde{u}^{\alpha}}\frac{\delta Q}{\delta\tilde{\theta}_{\alpha}}dx$

∎

######  Example 2.9 (Transformation rule for Poisson brackets).

Let $P=\frac{1}{2}\int\theta_{\alpha}P^{\alpha\beta}(\theta_{\beta})$ be a
Poisson bivector, and consider the change of variables (19). In the new
coordinates, $P$ takes the form

(26) $\displaystyle P$ $\displaystyle=\frac{1}{2}\int\sum_{s\geq
0}(-\partial_{x})^{s}\left(\frac{\partial\tilde{u}^{\gamma}}{\partial
u_{s}^{\alpha}}\tilde{\theta}_{\gamma}\right)P^{\alpha\beta}\left(\sum_{t\geq
0}(-\partial_{x})^{t}\left(\frac{\partial\tilde{u}^{\sigma}}{\partial
u_{s}^{\beta}}\tilde{\theta}_{\sigma}\right)\right)dx$
$\displaystyle=\frac{1}{2}\int\sum_{s,t\geq
0}\tilde{\theta}_{\gamma}\frac{\partial\tilde{u}^{\gamma}}{\partial
u_{s}^{\alpha}}\partial_{x}^{s}\circ
P^{\alpha\beta}\circ(-\partial_{x})^{t}\circ\frac{\partial\tilde{u}^{\sigma}}{\partial
u_{s}^{\beta}}(\tilde{\theta}_{\sigma})dx.$

Therefore, the transformed Poisson operator $\tilde{P}^{\gamma\sigma}$ is
given by

(27) $\displaystyle\tilde{P}^{\gamma\sigma}=\sum_{s,t\geq
0}\frac{\partial\tilde{u}^{\gamma}}{\partial
u_{s}^{\alpha}}\partial_{x}^{s}\circ
P^{\alpha\beta}\circ(-\partial_{x})^{t}\circ\frac{\partial\tilde{u}^{\sigma}}{\partial
u_{s}^{\beta}}.$

###  2.2. Differential rational functions

For applications in enumerative geometry the spaces $\hat{\mathcal{A}}$ and
$\hat{\mathcal{F}}$ are too restrictive. We will complete them by allowing
certain singularities.

######  Definition 2.10.

A differential rational function of type $(1,1)$ is a function $f$ of the form

(28) $\displaystyle f=f_{s}+\epsilon f_{s+1}+\epsilon^{2}f_{s+2}+\dots,$

where the functions $f_{k}$ only depend on finitely many derivatives
$u,u_{1},\dots,u_{r}$ and take the form

(29) $\displaystyle
f_{k}=(u^{1}_{1})^{k}\sum_{l=0}^{\infty}\frac{P_{k,l}}{(u^{1}_{1})^{l}}$

with $P_{k,l}=P_{k,l}(u_{\bullet}^{\bullet},\theta_{\bullet}^{\bullet})$ is a
homogeneous differential polynomial of standard degree $l$ that does not
depend on $u^{1}_{1}$. The space of differential rational functions of type
$(1,1)$ is denoted by $\hat{\mathcal{B}}$. The space of rational polyvector
fields of type $(1,1)$ is defined as the quotient of $\hat{\mathcal{B}}$ by
$\partial_{x}\hat{\mathcal{B}}$ and the constant functions and denoted by
$\hat{\mathcal{Q}}$.

The next step is to enlarge the Miura group to allow transformations given by
differential rational functions.

######  Definition 2.11.

A _quasi-Miura transformation_ is a change of variables of the form

(30)
$\displaystyle\tilde{u}^{\alpha}=u^{\alpha}+\sum_{k=1}^{\infty}\epsilon^{k}F^{\alpha}_{k}(u;u_{1},\dots,u_{n_{k}}).$

where the functions $F^{\alpha}_{k}$ are homogeneous rational functions in the
derivatives $u_{1},\dots,u_{n_{k}}$ of standard degree $k$. A quasi-Miura
transformation is of $(1,1)$-type if it takes the form

(31) $\displaystyle\tilde{u}^{\alpha}=u^{\alpha}+\sum_{k=1}^{\infty}(\epsilon
u^{1}_{1})^{k}\sum_{l=0}^{\infty}\frac{F^{\alpha}_{k,l}(u;u_{1},\dots,u_{n_{k}})}{(u^{1}_{1})^{l}},$

where $F^{\alpha}_{k,l}$ is a differential polynomial of degree $l$ with
$\frac{\partial F^{\alpha}_{k,l}}{\partial u^{1}_{1}}=0$.

######  Remark 2.12.

  * •

The key aspect of the space $\hat{\mathcal{Q}}$ is that all notions introduced
before for $\hat{\mathcal{F}}$ are still well-defined for $\hat{\mathcal{Q}}$:
variational derivatives, the Schouten bracket and both gradations. It is also
possible to define Hamiltonian and bi-Hamiltonian structures of type $(1,1)$
in the same way it was done for their polynomial analogues.

  * •

Note the proof of Proposition 2.8 does not use polynomiality at any point. The
only requirement is that the variational derivatives still make sense for the
transformed polyvector fields, so it still holds if we consider quasi-Miura
transformations of type $(1,1)$ instead of Miura transformations.

  * •

Applying (31) to an element of $\hat{\mathcal{F}}^{r}$ yields an element of
$\hat{\mathcal{Q}}^{r}$. The key question is whether this new element is
polynomial in the variables $\tilde{u}$.

##  3\. Dubrovin–Zhang hierarchy

In this section we recall the construction of the DZ hierarchy as done in
[DZ01, BPS12a, BPS12b] as well as some of its most important properties. For
the full details, we refer the reader to those articles.

###  3.1. Cohomological field theories

In this section, we briefly recall the definition and the most important
properties of a cohomological field theory, firstly introduced by Kontsevich
and Manin in [KM94]. Let $V$ be an $N$-dimensional vector space over
$\mathbb{C}$ equipped with a scalar product $(\cdot,\cdot)$. Choose a basis
$\\{e_{1},e_{2},\dots,e_{N}\\}$ of $V$ and let
$\eta_{\alpha\beta}=(e_{\alpha},e_{\beta})$. In the rest of the paper we use
$\eta_{\alpha\beta}$ to lower indices and its inverse $\eta^{\alpha\beta}$ to
raise them.

######  Definition 3.1 (CohFT).

A _cohomological field theory_ with unit $e_{1}$ is a collection of linear
homomorphisms

(32) $\displaystyle c_{g,n}:V^{\otimes n}\rightarrow
H^{2*}\left(\overline{\mathcal{M}}_{g,n};\mathbb{C}\right),\qquad 2g-2+n>0$

such that

  * •

$c_{g,n}$ is $S_{n}$-equivariant, where $S_{n}$ acts on $V^{\otimes n}$ by
permutation of the factors and on
$H^{2*}\left(\overline{\mathcal{M}}_{g,n};\mathbb{C}\right)$ by permutation of
the marked points.

  * •

For any gluing map
$\rho:\overline{\mathcal{M}}_{g_{1},n_{1}+1}\times\overline{\mathcal{M}}_{g_{2},n_{2}+1}\rightarrow\overline{\mathcal{M}}_{g_{1}+g_{2},n_{1}+n_{2}}$,

(33)
$\displaystyle\rho^{*}c_{g_{1}+g_{2},n_{1}+n_{2}}(v_{1}\otimes\dots\otimes
v_{n_{1}+n_{2}})=$ $\displaystyle c_{g_{1},n_{1}+1}(v_{1}\otimes\dots\otimes
v_{n_{1}}\otimes
e_{\alpha})\eta^{\alpha\beta}c_{g_{2},n_{2}+1}(v_{n_{1}+1}\otimes\dots\otimes
v_{n_{2}}\otimes e_{\beta}).$

  * •

For any gluing map
$\sigma:\overline{\mathcal{M}}_{g-1,n+2}\rightarrow\overline{\mathcal{M}}_{g,n}$,

(34) $\displaystyle\sigma^{*}c_{g,n}(v_{1}\otimes\dots\otimes
v_{n})=c_{g-1,n+2}(v_{1}\otimes\dots\otimes v_{n}\otimes e_{\alpha}\otimes
e_{\beta})\eta^{\alpha\beta}.$

  * •

For any forgetful map
$\pi:\overline{\mathcal{M}}_{g,n+1}\rightarrow\overline{\mathcal{M}}_{g,n}$,

(35) $\displaystyle\pi^{*}c_{g,n}(v_{1}\otimes\dots\otimes
v_{n})=c_{g,n+1}(v_{1}\otimes\dots\otimes v_{n}\otimes e_{1})$

  * •

$c_{0,3}(v_{1}\otimes v_{2}\otimes e_{1})=(v_{1},v_{2})$

We associate to a CohFT $\\{c_{g,n}\\}$ a formal power series in the variables
$\\{t^{\alpha}_{d}\\}^{1\leq\alpha\leq N}_{d\geq 0}$. Firstly, define the
correlation functions as

(36)
$\displaystyle\langle\prod_{i=1}^{n}\tau_{d_{i}}(v_{i})\rangle_{g}\coloneqq\int_{\overline{\mathcal{M}}_{g,n}}c_{g,n}(\otimes_{i=1}^{n}v_{i})\prod_{i=1}^{n}\psi_{i}^{d_{i}},$

where $\psi_{i}$ is the psi-class at the $i$-th marked point. Define the
partition function and the potential as

(37) $\displaystyle\tau\coloneqq\exp(\epsilon^{-2}F),\qquad
F\coloneqq\sum_{g\geq 0}\epsilon^{2g}F_{g},\qquad
F_{g}\coloneqq\sum_{\begin{subarray}{c}n\geq 0\\\
2g-2+n>0\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},\dots,d_{n}\geq
0\end{subarray}}\langle\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{g}\prod_{i=1}^{n}t_{d_{i}}^{\alpha_{i}}.$

The following identities satisfied by the potential will be used in the text
(see e. g. [Wit91]):

  * •

String equation

(38) $\displaystyle\frac{\partial F}{\partial t_{0}^{1}}=\sum_{p\geq
0}t_{p+1}^{\alpha}\frac{\partial F}{\partial
t^{\alpha}_{p}}+\frac{1}{2}\eta_{\alpha\beta}t_{0}^{\alpha}t_{0}^{\beta}+\epsilon^{2}\langle\tau_{0}(e_{1})\rangle_{1}.$

  * •

Dilaton equation

(39) $\displaystyle\frac{\partial F}{\partial
t_{1}^{1}}=\epsilon\frac{\partial F}{\partial\epsilon}+\sum_{p\geq
0}t^{\alpha}_{p}\frac{\partial F}{\partial
t^{\alpha}_{p}}-2F+\epsilon^{2}\frac{N}{24}.$

  * •

WDVV (associativity) equations

(40) $\displaystyle\frac{\partial^{3}F_{0}}{\partial
t^{\alpha_{1}}_{d_{1}}\partial t^{\alpha_{2}}_{d_{2}}\partial
t_{0}^{\alpha}}\eta^{\alpha\beta}\frac{\partial^{3}F_{0}}{\partial
t^{\alpha_{3}}_{d_{3}}\partial t^{\alpha_{4}}_{d_{4}}\partial
t_{0}^{\beta}}=\frac{\partial^{3}F_{0}}{\partial
t^{\alpha_{2}}_{d_{2}}\partial t^{\alpha_{3}}_{d_{3}}\partial
t_{0}^{\alpha}}\eta^{\alpha\beta}\frac{\partial^{3}F_{0}}{\partial
t^{\alpha_{1}}_{d_{1}}\partial t^{\alpha_{4}}_{d_{4}}\partial t_{0}^{\beta}}.$

  * •

Topological recursion relation in genus 0 (TRR-0):

(41) $\displaystyle\frac{\partial^{3}F_{0}}{\partial
t^{\alpha_{1}}_{d_{1}+1}\partial t^{\alpha_{2}}_{d_{2}}\partial
t^{\alpha_{3}}_{d_{3}}}=\frac{\partial^{2}F_{0}}{\partial
t^{\alpha_{1}}_{d_{1}}\partial
t^{\alpha}_{0}}\eta^{\alpha\beta}\frac{\partial^{3}F_{0}}{\partial
t^{\beta}_{0}\partial t^{\alpha_{2}}_{d_{2}}\partial t^{\alpha_{3}}_{d_{3}}}$

  * •

The Liu–Pandharipande relations. We recall them in Section 5.

There are many other universal differential equations for $F_{g}$, $g\geq 0$.
Basically, any relation in the tautological ring of the moduli space of curves
implies such an equation.

###  3.2. The principal hierarchy

From now on, all CohFTs considered in this article are semi-simple, i. e., the
functions

(42) $\displaystyle
c^{\lambda}_{\alpha\beta}(t)=\eta^{\lambda\gamma}\frac{\partial^{3}F^{\textrm{Frob}}}{\partial
t^{\gamma}\partial t^{\alpha}\partial t^{\beta}},$

where $F^{\textrm{Frob}}=F_{0}|_{t^{\bullet}_{\geq 1}=0}$ give the structure
constants of a semi-simple associative algebra at $t=0$.

Let $t^{1}_{0}\mapsto t^{1}_{0}+x$. Consider the _two-point correlators in
genus $g$_

(43)
$\displaystyle\Omega^{[g]}_{\alpha,p;\beta,q}=\frac{\partial^{2}F_{g}}{\partial
t^{\alpha}_{p}\partial t^{\beta}_{q}}$

and the variables

(44) $\displaystyle v_{\beta}=\frac{\partial^{2}F_{0}}{\partial
t^{\beta}_{0}\partial t^{1}_{0}},\qquad
v^{\alpha}=\eta^{\alpha\beta}v_{\beta},\qquad
v^{\alpha}_{k}=\partial_{x}^{k}v^{\alpha}.$

######  Proposition 3.2 ([Dub96, BPS12b]).

The two point correlators in genus 0 are given by

(45)
$\displaystyle\Omega^{[0]}_{\alpha,p;\beta,q}(t_{0},t_{1},t_{2},\dots)=\Omega^{[0]}_{\alpha,p;\beta,q}(v,0,0,\dots)$

As a consequence of the tau-symmetry of $\Omega^{[g]}_{\alpha,p;\beta,q}$,
i.e., the expression

(46) $\displaystyle\frac{\partial}{\partial
t^{\gamma}_{r}}\Omega^{[g]}_{\alpha,p;\beta,q}$

being invariant under any permutation of
$(\alpha,p)\leftrightarrow(\beta,q)\leftrightarrow(\gamma,r)$, the variables
$v^{\alpha}$ satisfy the system of equations

(47) $\displaystyle\frac{\partial v^{\alpha}}{\partial
t^{\beta}_{q}}=\eta^{\alpha\gamma}\partial_{x}(\Omega^{[0]}_{\gamma,0;\beta,q}).$

The goal is to rewrite the equations (47) in Hamiltonian form. First, define
the Hamiltonian densities

(48) $\displaystyle h_{\alpha,p}(v)\coloneqq\Omega^{[0]}_{\alpha,p+1;1,0}.$

Consider the Poisson operator of hydrodynamic type
$P^{\alpha\beta}=\eta^{\alpha\beta}\partial_{x}$ or, equivalently by Remark
2.5, the Poisson bivector

(49) $\displaystyle
P=\frac{1}{2}\int\theta_{\alpha}\eta^{\alpha\beta}\theta^{1}_{\beta}dx\in\hat{\mathcal{F}}^{2}_{1}.$

######  Proposition 3.3 ([Dub96, BPS12b]).

The Hamiltonians $\bar{h}_{\alpha,p}=\int h_{\alpha,p}dx$ satisfy:

(50) $\displaystyle\frac{\delta\bar{h}_{\alpha,p}}{\delta
v^{\gamma}}\eta^{\gamma\sigma}\partial_{x}\frac{\delta\bar{h}_{\beta,q}}{\delta
v^{\sigma}}=\partial_{x}\Omega^{[0]}_{\alpha,p+1;\beta,q}.$

In particular, they Poisson-commute
$\\{\bar{h}_{\alpha,p},\bar{h}_{\beta,q}\\}_{P}=0$ and the system (47) can be
rewritten as a Hamiltonian system, called the principal or dispersionless
Dubrovin–Zhang hierarchy:

(51) $\displaystyle\frac{\partial v^{\alpha}}{\partial
t^{\beta}_{q}}=\eta^{\alpha\gamma}\partial_{x}\frac{\delta\bar{h}_{\beta,q}}{\delta
v^{\gamma}}$

###  3.3. The full hierarchy

The principal hierarchy constructed above “forgets” the information of the
CohFT carried by $F_{g}$ for $g\geq 1$, in other words, no information is lost
if we set $\epsilon=0$ at the beginning. Here we construct the full hierarchy.
Consider the variables $w$ and the two point correlators:

(52) $\displaystyle w^{\alpha}=\eta^{\alpha\beta}\frac{\partial^{2}F}{\partial
t^{\beta}_{0}\partial
t^{1}_{0}},\qquad\Omega_{\alpha,p;\beta,q}=\frac{\partial^{2}F}{\partial
t^{\alpha}_{p}\partial
t^{\beta}_{q}}=\sum_{g=0}^{\infty}\epsilon^{2g}\Omega^{[g]}_{\alpha,p;\beta,q}$

As a consequence of the tau-symmetry of $\Omega_{\alpha,p;\beta,q}$, i.e., the
expression

(53) $\displaystyle\frac{\partial}{\partial
t^{\gamma}_{r}}\Omega_{\alpha,p;\beta,q}$

being invariant under any permutation of
$(\alpha,p)\leftrightarrow(\beta,q)\leftrightarrow(\gamma,r)$, the variables
$w^{\alpha}$ satisfy the system of equations

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

The goal is to endow the system (54) with a Hamiltonian structure as it was
done for the principal hierarchy in Section 3.2. For this, we recall an
important result, known as the $(3g-2)$-property:

######  Proposition 3.4 (see e. g. [BPS12b]).

For $g\geq 1$, there exist functions $P_{0}^{[g]},\dots,P_{3g-2}^{[g]}$ such
that

(55) $\displaystyle
F_{g}(t_{0},t_{1},\dots)=F_{g}(P_{0}^{[g]}(v,v_{1},\dots,v_{3g-2}),\dots,P_{3g-2}^{[g]}(v,v_{1},\dots,v_{3g-2}),0,0,\dots)$

As a consequence of the $(3g-2)$-property, $\Omega^{[g]}$ only depends on
$v,v_{1},\dots,v_{3g}$, so the expansion of $w$ in terms of $v$ takes the form

(56) $\displaystyle
w^{\alpha}=v^{\alpha}+\eta^{\alpha\beta}\sum_{g=1}^{\infty}\epsilon^{2g}\Omega^{[g]}_{\beta,0;1,0}(v,v_{1},\dots,v_{3g}).$

The next proposition shows that the string and dilaton equations give (56) the
required regularity to use the framework developed in Section 2.

######  Proposition 3.5 (see [BDGR20]).

The transformation $v^{\alpha}\rightarrow w^{\alpha}$ is a quasi-Miura
transformation of $(1,1)$-type.

###### Proof.

Dilaton equation (39) implies the following relation between the coefficients
of $F_{g}$

(57)
$\displaystyle[\prod_{i=1}^{n}t_{d_{i}}^{\alpha_{i}}]F_{g}=\frac{(2g-3+n)!d!}{(2g-3+n+d)!}[\prod_{i=1}^{n}t_{d_{i}}^{\alpha_{i}}(t_{1}^{1})^{d}]F_{g}.$

This allows us to rewrite $F_{g}$ ($g\geq 2$) in the following form:

(58) $\displaystyle F_{g}$ $\displaystyle=\sum_{\begin{subarray}{c}n\geq
0\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},d_{2},\dots,d_{n}\geq 0\\\
(\alpha_{k},d_{k})\not=(1,1)\end{subarray}}\langle\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{g}\prod_{i=1}^{n}t_{d_{i}}^{\alpha_{i}}\sum_{d\geq
0}\frac{(2g-3+n+d)!}{(2g-3+n)!d!}(t^{1}_{1})^{d}$
$\displaystyle=\sum_{\begin{subarray}{c}n\geq
0\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},d_{2},\dots,d_{n}\geq 0\\\
(\alpha_{k},d_{k})\not=(1,1)\end{subarray}}\langle\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{g}\prod_{i=1}^{n}t_{d_{i}}^{\alpha_{i}}\left(\frac{1}{1-t^{1}_{1}}\right)^{2g-2+n}$
$\displaystyle=\left(\frac{1}{1-t^{1}_{1}}\right)^{2g-2}\sum_{\begin{subarray}{c}n\geq
0\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},d_{2},\dots,d_{n}\geq 0\\\
(\alpha_{k},d_{k})\not=(1,1)\end{subarray}}\langle\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{g}\prod_{i=1}^{n}\left(\frac{t_{d_{i}}^{\alpha_{i}}}{1-t^{1}_{1}}\right).$

For $F_{0}$ and $F_{1}$ we have to mind the unstable correlation functions:

(59) $\displaystyle F_{1}$ $\displaystyle=\sum_{\begin{subarray}{c}n\geq
1\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},d_{2},\dots,d_{n}\geq 0\\\
(\alpha_{k},d_{k})\not=(1,1)\end{subarray}}\langle\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{1}\prod_{i=1}^{n}\left(\frac{t_{d_{i}}^{\alpha_{i}}}{1-t^{1}_{1}}\right)+\sum_{d\geq
0}\frac{1}{(d+1)!}\langle(\tau_{1}(e_{1}))^{d+1}\rangle_{1}(t^{1}_{1})^{d+1}$
$\displaystyle=\sum_{\begin{subarray}{c}n\geq
1\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},d_{2},\dots,d_{n}\geq 0\\\
(\alpha_{k},d_{k})\not=(1,1)\end{subarray}}\langle\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{1}\prod_{i=1}^{n}\left(\frac{t_{d_{i}}^{\alpha_{i}}}{1-t^{1}_{1}}\right)+\langle\tau_{1}(e_{1})\rangle_{1}\sum_{d\geq
0}\frac{1}{(d+1)}(t^{1}_{1})^{d+1}$
$\displaystyle=\sum_{\begin{subarray}{c}n\geq
1\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},d_{2},\dots,d_{n}\geq 0\\\
(\alpha_{k},d_{k})\not=(1,1)\end{subarray}}\langle\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{1}\prod_{i=1}^{n}\left(\frac{t_{d_{i}}^{\alpha_{i}}}{1-t^{1}_{1}}\right)+\frac{N}{24}\log\left(\frac{1}{1-t^{1}_{1}}\right);$
(60) $\displaystyle F_{0}$
$\displaystyle=\left(1-t^{1}_{1}\right)^{2}\sum_{\begin{subarray}{c}n\geq
3\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},d_{2},\dots,d_{n}\geq 0\\\
(\alpha_{k},d_{k})\not=(1,1)\end{subarray}}\langle\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{0}\prod_{i=1}^{n}\left(\frac{t_{d_{i}}^{\alpha_{i}}}{1-t^{1}_{1}}\right),$

where the last formula follows from the fact that
$\langle\tau_{d_{1}}(e_{\alpha_{1}})\tau_{d_{2}}(e_{\alpha_{2}})\tau_{1}(e_{1})\rangle_{0}=0$
for all $\alpha_{1},\alpha_{2},d_{1},d_{2}$. By the string equation (38) we
have

(61) $\displaystyle v^{\alpha}$
$\displaystyle=\eta^{\alpha\beta}\frac{\partial^{2}F_{0}}{\partial
t^{\beta}_{0}\partial t_{0}^{1}}=t_{0}^{\alpha}+\sum_{p\geq
0}\eta^{\alpha\beta}t_{p+1}^{\gamma}\frac{\partial^{2}F_{0}}{\partial
t^{\gamma}_{p}\partial t_{0}^{\beta}}=t_{0}^{\alpha}+\mathcal{O}(t^{2})$ (62)
$\displaystyle v^{1}_{1}$ $\displaystyle=1+t_{1}^{1}+\sum_{p\geq
0}t_{1}^{\gamma}t_{p+1}^{\mu}\eta^{1\beta}\frac{\partial^{3}F_{0}}{\partial
t_{0}^{\gamma}\partial t_{0}^{\beta}\partial t_{p}^{\mu}}+\sum_{p\geq
1}t_{p+1}^{\gamma}\eta^{1\beta}\frac{\partial^{3}F_{0}}{\partial
t_{0}^{1}\partial t_{0}^{\beta}\partial
t_{p}^{\gamma}}=1+t^{1}_{1}+\mathcal{O}(t^{2})$

To get the dependence of $v^{1}_{1}$ on $t_{1}^{1}$ we compute

(63) $\displaystyle v_{1}^{1}=\eta^{1\mu}\frac{\partial^{3}F_{0}}{\partial
t_{0}^{\mu}\partial t_{0}^{1}\partial
t_{0}^{1}}=\frac{1}{1-t_{1}^{1}}\sum_{n\geq
0}\frac{1}{n!}\sum_{\begin{subarray}{c}1\leq\alpha_{1},\dots,\alpha_{n}\leq
N\\\ d_{1},d_{2},\dots,d_{n}\geq 0\\\
(\alpha_{k},d_{k})\not=(1,1)\end{subarray}}\eta^{1\mu}\langle\tau_{0}(e_{\mu})\tau_{0}(e_{1})^{2}\prod_{i=1}^{n}\tau_{d_{i}}(e_{\alpha_{i}})\rangle_{0}\prod_{i=1}^{n}\left(\frac{t_{d_{i}}^{\alpha_{i}}}{1-t_{1}^{1}}\right);$
(68) $\displaystyle
v_{1}^{1}\bigg{|}_{\vbox{\Let@\restore@math@cr\default@tag\halign{\hfil$\m@th\scriptstyle#$&$\m@th\scriptstyle{}#$\cr&t^{\alpha}_{p}=0\\\
&(\alpha,p)\not=(1,1)\crcr}}}=\eta^{1\mu}\frac{\partial^{3}F_{0}}{\partial
t_{0}^{\mu}\partial t_{0}^{1}\partial
t_{0}^{1}}\bigg{|}_{\vbox{\Let@\restore@math@cr\default@tag\halign{\hfil$\m@th\scriptstyle#$&$\m@th\scriptstyle{}#$\cr&t^{\alpha}_{p}=0\\\
&(\alpha,p)\not=(1,1)\crcr}}}=\eta^{1\mu}\frac{1}{1-t^{1}_{1}}\langle\tau_{0}(e_{\mu})\tau_{0}(e_{1})^{2}\rangle_{0}=\frac{1}{1-t_{1}^{1}}.$

Therefore, from (62) and (68), we get:

(69) $\displaystyle
v_{1}^{1}=\frac{1}{1-t^{1}_{1}}\left(1+\mathcal{O}\left(\frac{t^{\cdot}_{\cdot}}{1-t_{1}^{1}}\right)\right).$

Taking $k$ derivatives of (61), we have:

(70) $\displaystyle v_{k}^{\alpha}=\sum_{p\geq
0}\eta^{\alpha\beta}t_{p+1}^{\gamma}\frac{\partial^{k+2}F_{0}}{\partial
t_{p}^{\gamma}\partial
t_{0}^{\beta}\partial(t_{0}^{1})^{k}}=\left(\frac{1}{1-t^{1}_{1}}\right)^{k}\left(\frac{t_{k}^{\alpha}}{1-t_{1}^{1}}+\mathcal{O}\left(\frac{t^{\cdot}_{\cdot}}{1-t_{1}^{1}}\right)^{2}\right),\qquad(\alpha,k)\not=(1,1).$

Thus,

(71)
$\displaystyle\frac{v_{k}^{\alpha}}{(v_{1}^{1})^{k}}=\frac{t_{k}^{\alpha}}{1-t_{1}^{1}}+\mathcal{O}\left(\frac{t^{\cdot}_{\cdot}}{1-t_{1}^{1}}\right)^{2}.$

In conclusion, the functions

(72)
$\displaystyle\Omega^{[g]}_{\alpha,p;\beta,q}(v):=\frac{\partial^{2}F_{g}}{\partial
t_{p}^{\alpha}\partial t_{q}^{\beta}}$

are of the form $(v_{1}^{1})^{2g}S$, where $S$ is a formal power series in
$v^{\alpha}_{p}/(v_{1}^{1})^{p}$. Since we know they only depend on
$v,v_{1},\dots,v_{3g}$ as a consequence of Proposition 3.4, we can conclude

(73) $\displaystyle\Omega^{[g]}_{\alpha,p;\beta,q}=(v_{1}^{1})^{2g}\sum_{k\geq
0}\frac{R_{k}}{(v_{1}^{1})^{k}}$

for $R_{k}$ a differential polynomial depending on $v,v_{1},\dots,v_{3g}$ but
not on $v_{1}^{1}$ of standard degree $k$. Thus, we can write the
transformation (56) as

(74) $\displaystyle
w^{\alpha}=v^{\alpha}+\eta^{\alpha\beta}\sum_{g=1}^{\infty}\epsilon^{2g}\Omega^{[g]}_{\beta,0;1,0}=v^{\alpha}+\sum_{g\geq
1}(\epsilon v_{1}^{1})^{2g}\sum_{k\geq
0}\frac{R^{\alpha}_{g,k}(v,v_{1},\dots,v_{3g})}{(v_{1}^{1})^{k}},$

where $R^{\alpha}_{g,k}$ is a differential polynomial in $v$ not depending on
$v_{1}^{1}$ of standard degree $k$. ∎

######  Remark 3.6.

Another consequence of Proposition 3.4 is that we can equivalently define the
Hamiltonian densities (48) in terms of the full two-point correlators
$\Omega$, since they differ by a total derivative:

(75) $\displaystyle
h_{\alpha,p}(w,\dots,w_{3g})=\Omega_{\alpha,p+1;1,0}=\Omega^{[0]}_{\alpha,p+1;1,0}+\partial_{x}\left(\sum_{g=1}^{\infty}\epsilon^{2g}\frac{\partial
F_{g}}{\partial t^{\alpha}_{p+1}}\right).$

For the particular case $p=-1$, this means

(76) $\displaystyle
w^{\alpha}=v^{\alpha}+\partial_{x}\left(\eta^{\alpha\beta}\sum_{g=1}^{\infty}\epsilon^{2g}\frac{\partial
F_{g}}{\partial t^{\beta}_{0}}\right)$

so $\int v^{\alpha}dx=\int w^{\alpha}dx$.

We can now write the full hierarchy in Hamiltonian form. The transformation
(56) induces a transformation on the Poisson bracket $P$ (49) as explained in
Example 2.9. Explicitly, the deformed Poisson bracket is given by

(77) $\displaystyle A^{\alpha\beta}=\sum_{s\geq
0}A^{\alpha\beta}_{s}\partial_{x}^{s}:=\sum_{e,f\geq 0}\frac{\partial
w^{\alpha}}{\partial v^{\mu}_{e}}\partial_{x}^{e}\circ
P^{\mu\nu}\circ(-\partial_{x})^{f}\circ\frac{\partial w^{\beta}}{\partial
v^{\nu}_{f}}$

The full hierarchy (54) is thus given by

(78) $\displaystyle\frac{\partial w^{\alpha}}{\partial
t_{q}^{\beta}}=A^{\alpha\gamma}\frac{\delta\bar{h}_{\beta,q}}{\delta
w^{\gamma}}.$

Finally, we are ready to state the main result of [BPS12b]:

######  Theorem 3.7 ([BPS12b]).

  * •

The functions $\Omega_{\alpha,p;\beta,q}$ are differential polynomials in $w$,
that is,
$\Omega_{\alpha,p;\beta,q}=\sum_{g=0}^{\infty}\epsilon^{2g}\Omega^{g}_{\alpha,p;\beta,q}(w,\dots,w_{3g})$,
where $\Omega^{g}_{\alpha,p;\beta,q}$ is a differential polynomial in $w$ of
standard degree $2g$ .

  * •

The Poisson bracket of the full hierarchy
$A^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{2g+1}A_{g,s}^{\alpha\beta}\partial_{x}^{s}$
is polynomial in $w$, i.e., the functions $A_{g,s}^{\alpha\beta}$ are
differential polynomials in $w$ of standard degree $2g+1-s$.

As an immediate consequence of this theorem together with Remark 3.6, the
Hamiltonian densities are polynomial in $w$, and so are the equations of the
full hierarchy (78).

######  Remark 3.8.

Note that
$\Omega^{0}_{\alpha,p;\beta,q}(w)=\Omega^{[0]}_{\alpha,p;\beta,q}(v)|_{v^{\alpha}=w^{\alpha}}$,
but the relation between $\Omega^{g}_{\alpha,p;\beta,q}(w,\dots,w_{3g})$ and
$\Omega^{[g]}_{\alpha,p;\beta,q}(v,\dots,v_{3g})$ is much more involved.

##  4\. The second bracket

###  4.1. Conformality and bi-Hamiltonian recursion

In Sections 3.1 and 3.2 we have constructed an integrable hierarchy whose
solutions are generated from the partition function of a CohFT. CohFTs are
designed to capture the universal basic properties of the Gromov–Witten
theories, and it is possible to introduce an extra homogeneity property, which
is designed to reflect the computation of the degrees of the Gromov–Witten
classes.

For the purposes of this article, we rewrite the homogeneity condition
explicitly as the conformality of the CohFT (as it is done in e. g. [BRS21]),
and this extra condition implies the existence of another Hamiltonian
structure $K$ of hydrodynamic type compatible with $P$, i.e., $[K,P]=0$, for
which the Hamiltonians $\bar{h}_{\alpha,p}$ are bi-Hamiltonian conserved
quantities, i.e., they satisfy $[K,[P,\bar{h}_{\alpha,p}]]=0$.

######  Definition 4.1.

A CohFT $\\{c_{g,n}\\}$ is called conformal if there exist constants
$\mathsf{q}_{\beta}^{\alpha}$, ${\mathsf{b}}^{\alpha}$ and $\mathsf{d}$ such
that $\mathsf{q}_{1}^{\alpha}=\delta_{1}^{\alpha}$,
$\mathsf{q}^{\alpha}_{\beta}+\eta^{\alpha\mu}\mathsf{q}^{\nu}_{\mu}\eta_{\nu\beta}=(2-\mathsf{d})\delta^{\alpha}_{\beta}$,
and

(79)
$\displaystyle\left(\frac{1}{2}\deg-(g-1)\mathsf{d}-m\right)c_{g,m}(\otimes^{m}_{i=1}e_{\beta_{i}})+\sum_{i=1}^{m}\mathsf{q}^{\mu}_{\beta_{i}}c_{g,m}(\otimes_{j=1}^{i-1}e_{\beta_{j}}\otimes
e_{\mu}\otimes\otimes_{j=i+1}^{m}e_{\beta_{j}})$
$\displaystyle+\pi_{*}c_{g,m+1}(\otimes_{i=1}^{m}e_{\beta_{i}}\otimes{\mathsf{b}}^{\gamma}e_{\gamma})=0,$

where $\deg$ acts on the $k$-th cohomology by multiplication by $k$ and
$\pi\colon\overline{\mathcal{M}}_{g,m+1}\to\overline{\mathcal{M}}_{g,m}$ is
the standard map forgetting the last marked point.

In terms of the logarithm of the partition function
$F=\epsilon^{2}\log\tau=\sum_{g=0}^{\infty}\epsilon^{2g}F_{g}$, this means

(80) $\displaystyle\left(\sum_{d\geq
0}(\mathsf{q}^{\gamma}_{\mu}-d\delta^{\gamma}_{\mu})t^{\mu}_{d}\frac{\partial}{\partial
t_{d}^{\gamma}}+{\mathsf{b}}^{\gamma}\frac{\partial}{\partial
t^{\gamma}_{0}}-\sum_{d\geq
0}\eta^{\alpha\mu}{\mathsf{b}}^{\beta}\langle\tau_{0}(e_{\alpha})\tau_{0}(e_{\beta})\tau_{0}(e_{\gamma})\rangle_{0}t^{\gamma}_{d+1}\frac{\partial}{\partial
t^{\mu}_{d}}+\frac{(3-\mathsf{d})}{2}\epsilon\frac{\partial}{\partial\epsilon}\right)F$
$\displaystyle=(3-\mathsf{d})F+\frac{1}{2}{\mathsf{b}}^{\gamma}\langle\tau_{0}(e_{\alpha})\tau_{0}(e_{\beta})\tau_{0}(e_{\gamma})\rangle_{0}t_{0}^{\alpha}t_{0}^{\beta}+\epsilon^{2}{\mathsf{b}}^{\gamma}\langle\tau_{0}(e_{\gamma})\rangle_{1}.$

It is convenient to introduce notation for a part of this equation which is a
vector field on the big phase space:

(81) $\displaystyle\tilde{\mathsf{E}}\coloneqq\sum_{d\geq
0}(\mathsf{q}^{\gamma}_{\mu}-d\delta^{\gamma}_{\mu})t^{\mu}_{d}\frac{\partial}{\partial
t_{d}^{\gamma}}+{\mathsf{b}}^{\gamma}\frac{\partial}{\partial
t^{\gamma}_{0}}-\sum_{d\geq
0}\eta^{\alpha\mu}{\mathsf{b}}^{\beta}\langle\tau_{0}(e_{\alpha})\tau_{0}(e_{\beta})\tau_{0}(e_{\gamma})\rangle_{0}t^{\gamma}_{d+1}\frac{\partial}{\partial
t^{\mu}_{d}}.$

Also, denote by $\tilde{R}^{\alpha}_{\beta}$ the matrix

(82) $\displaystyle\tilde{R}^{\alpha}_{\beta}$
$\displaystyle\coloneqq\frac{\mathsf{d}-1}{2}\delta^{\alpha}_{\beta}+\mathsf{q}^{\alpha}_{\beta};$
$\displaystyle M^{\alpha}_{\beta}$
$\displaystyle\coloneqq\eta^{\alpha\mu}{\mathsf{b}}^{\gamma}\langle\tau_{0}(e_{\mu})\tau_{0}(e_{\beta})\tau_{0}(e_{\gamma})\rangle_{0}.$

By direct computation we obtain the following useful lemma that explains the
action of $\tilde{\mathsf{E}}$ on the double derivatives of $F$,

(83) $\displaystyle\Omega^{[g]}_{\alpha,0;\beta,p}$
$\displaystyle\coloneqq\frac{\partial^{2}F_{g}}{\partial
t^{\alpha}_{0}\partial t^{\beta}_{p}};$
$\displaystyle\Omega^{[g]}_{\alpha,0;\beta,-1}$
$\displaystyle\coloneqq\eta_{\alpha\beta}\delta_{g,0}.$

######  Lemma 4.2.

We have:

(84)
$\displaystyle\tilde{\mathsf{E}}\Omega^{[g]}_{\alpha,0;\beta,p}+(g(3-\mathsf{d})-1)\Omega^{[g]}_{\alpha,0;\beta,p}+\tilde{R}^{\gamma}_{\alpha}\Omega^{[g]}_{\gamma,0;\beta,p}$
$\displaystyle=(p+1-\tilde{R})^{\gamma}_{\beta}\Omega^{[g]}_{\alpha,0;\gamma,p}+M^{\gamma}_{\beta}\Omega^{[g]}_{\alpha,0;\gamma,p-1};$
(85)
$\displaystyle\tilde{\mathsf{E}}\partial_{x}\Omega^{[g]}_{\alpha,0;\beta,p}+g(3-\mathsf{d})\partial_{x}\Omega^{[g]}_{\alpha,0;\beta,p}+\tilde{R}^{\gamma}_{\alpha}\partial_{x}\Omega^{[g]}_{\gamma,0;\beta,p}$
$\displaystyle=(p+1-\tilde{R})^{\gamma}_{\beta}\partial_{x}\Omega^{[g]}_{\alpha,0;\gamma,p}+M^{\gamma}_{\beta}\partial_{x}\Omega^{[g]}_{\alpha,0;\gamma,p-1}.$

Let $F^{\textrm{Frob}}=F_{0}|_{t^{\bullet}_{\geq 1}=0}$. As a consequence of
the string equation (38), $v^{\alpha}|_{t^{\bullet}_{\geq
1}=0}=t_{0}^{\alpha}$. We have the following system of equations for
$F^{\textrm{Frob}}$, derived from the homogeneity (80) and WDVV (40)
equations:

(86) $\displaystyle\frac{\partial^{3}F^{\textrm{Frob}}}{\partial
v^{\alpha_{1}}\partial v^{\alpha_{2}}\partial
v^{\alpha}}\eta^{\alpha\beta}\frac{\partial^{3}F^{\textrm{Frob}}}{\partial
v^{\alpha_{3}}\partial v^{\alpha_{4}}\partial
v_{0}^{\beta}}=\frac{\partial^{3}F^{\textrm{Frob}}}{\partial
v^{\alpha_{2}}\partial v^{\alpha_{3}}\partial
v^{\alpha}}\eta^{\alpha\beta}\frac{\partial^{3}F^{\textrm{Frob}}}{\partial
v^{\alpha_{1}}\partial v^{\alpha_{4}}\partial v^{\beta}},$ (87)
$\displaystyle(\mathsf{q}^{\gamma}_{\mu}v^{\mu}+{\mathsf{b}}^{\gamma})\frac{\partial}{\partial
v^{\gamma}}F^{\textrm{Frob}}=(3-\mathsf{d})F^{\textrm{Frob}}+\frac{1}{2}{\mathsf{b}}^{\gamma}\langle\tau_{0}(e_{\alpha})\tau_{0}(e_{\beta})\tau_{0}(e_{\gamma})\rangle_{0}v^{\alpha}v^{\beta}.$

The system above realizes the function $F^{\textrm{Frob}}$ as the potential of
a Dubrovin–Frobenius manifold, identifying

(88) $\displaystyle
E^{\alpha}=\mathsf{q}^{\alpha}_{\mu}v^{\mu}+{\mathsf{b}}^{\alpha}$

with the coefficients of the Euler vector field. For the theory of
Dubrovin–Frobenius manifolds, we refer the reader to [Dub96, Dub99]. For the
purposes of this article, we only need the following results

######  Proposition 4.3 ([Dub96]).

If $F^{\textrm{Frob}}(v)$ satisfies the system (86)-(87), there exists a non-
degenerate flat metric $g^{\alpha\beta}$ with Christoffel symbols
$b^{\alpha\beta}_{\gamma}$ such that the Poisson operator of hydrodynamic type

(89) $\displaystyle
K^{\alpha\beta}=g^{\alpha\beta}\partial_{x}+b^{\alpha\beta}_{\gamma}v^{\gamma}_{1}$

is compatible with $P^{\alpha\beta}=\eta^{\alpha\beta}\partial_{x}$. Moreover,
the explicit expressions of $g$ and $b$ are given by:

(90) $\displaystyle
g^{\alpha\beta}=\eta^{\alpha\gamma}\eta^{\beta\nu}E^{\mu}c_{\mu\gamma\nu},\qquad
b^{\alpha\beta}_{\gamma}=c^{\alpha\delta}_{\gamma}\tilde{R}^{\beta}_{\delta},\qquad$

where
$c^{\alpha\beta}_{\gamma}=\eta^{\alpha\mu}\eta^{\beta\nu}\frac{\partial^{3}F^{\textrm{Frob}}}{\partial
v^{\mu}\partial v^{\nu}\partial v^{\gamma}}$ are the structure constants of
the Frobenius algebra.

######  Remark 4.4.

Under the conformality assumption, the principal hierarchy shown in section
3.2 as constructed in [BPS12b] coincides with the Dubrovin–Zhang construction
of the principal hierarchy starting from a Dubrovin–Frobenius manifold in
[DZ01], for a particular choice of calibration.

For the two compatible Poisson brackets, $P$ and $K$, we have a bi-Hamiltonian
recursion relation:

######  Proposition 4.5 (see e. g. [DZ98]).

The following equations hold:

(91)
$\displaystyle\\{\cdot,\bar{h}_{\beta,-1}\\}_{K}=\\{\cdot,\bar{h}_{\mu,0}\\}_{P}(1-\tilde{R})^{\mu}_{\beta}$
(92)
$\displaystyle\\{\cdot,\bar{h}_{\beta,d}\\}_{K}=\\{\cdot,\bar{h}_{\mu,d+1}\\}_{P}(d+2-\tilde{R})^{\mu}_{\beta}+\\{\cdot,\bar{h}_{\mu,d}\\}_{P}M^{\mu}_{\beta},\quad
d\geq 0.$

###### Proof.

All the arguments in this article are based on the bi-Hamiltonian recursion
relation above, which is the central piece to prove the main results of the
text. That is why, despite its proof being well-known, we reproduce it here.
We compute the Hamiltonian vector fields of (91) term by term: the LHS (after
multiplication by $\eta^{\alpha\beta}$) becomes

(93) $\displaystyle g^{\lambda\nu}\partial_{x}\left(\frac{\delta
v^{\alpha}}{\delta
v^{\nu}}\right)+b^{\lambda\nu}_{\gamma}v_{1}^{\gamma}\frac{\delta
v^{\alpha}}{\delta
v^{\nu}}=b^{\lambda\alpha}_{\gamma}v_{1}^{\gamma}=\tilde{R}^{\alpha}_{\delta}c^{\lambda\delta}_{\gamma}v_{1}^{\gamma}$

On the other hand, the RHS equals:

(94)
$\displaystyle\eta^{\alpha\beta}\eta^{\lambda\nu}\partial_{x}\left(\frac{\delta\bar{h}_{\mu,0}}{\delta
v^{\nu}}\right)(1-\tilde{R})^{\mu}_{\beta}=\eta^{\alpha\beta}\eta^{\lambda\nu}\partial_{x}\left(\Omega^{[0]}_{\mu,0;\nu,0}\right)(1-\tilde{R})^{\mu}_{\beta}$
(95)
$\displaystyle=\eta^{\alpha\beta}\eta^{\lambda\nu}c_{\mu\nu\gamma}v_{1}^{\gamma}(1-\tilde{R})^{\mu}_{\beta}=c^{\lambda}_{\mu\gamma}v_{1}^{\gamma}\eta^{\mu\beta}\tilde{R}^{\alpha}_{\beta}=\tilde{R}^{\alpha}_{\beta}c^{\lambda\beta}_{\gamma}v_{1}^{\gamma},$

where we have used that
$\tilde{R}^{\alpha}_{\beta}\eta^{\beta\gamma}+\tilde{R}^{\gamma}_{\beta}\eta^{\alpha\beta}=\eta^{\alpha\gamma}$.
Thus, both sides are equal and the relation (91) holds. To prove (92), first
note that

(96) $\displaystyle
g^{\alpha\mu}\partial_{x}+b^{\alpha\mu}_{\gamma}v_{1}^{\gamma}=\partial_{x}\circ
g^{\alpha\mu}-b^{\mu\alpha}_{\gamma}v_{1}^{\gamma}$

Thus, the LHS equals

(97)
$\displaystyle\partial_{x}(g^{\alpha\mu}\Omega^{[0]}_{\beta,d;\mu,0})-\partial_{x}(\tilde{R}^{\alpha}_{\delta}\eta^{\delta\theta}\Omega^{[0]}_{\beta,d+1;\theta,0})$

where the second summand comes from the computation

(98)
$\displaystyle\tilde{R}^{\alpha}_{\delta}\eta^{\delta\theta}\partial_{x}(\Omega^{[0]}_{\beta,d+1;\theta,0})$
$\displaystyle=\tilde{R}^{\alpha}_{\delta}\eta^{\delta\theta}\Omega^{[0]}_{\beta,d;\lambda,0}\eta^{\lambda\sigma}\partial_{x}\Omega^{[0]}_{\sigma,0;\theta,0}=\tilde{R}^{\alpha}_{\delta}\eta^{\delta\theta}\Omega^{[0]}_{\beta,d;\lambda,0}\eta^{\lambda\sigma}c_{\sigma\theta\gamma}v_{1}^{\gamma}$
$\displaystyle=\tilde{R}^{\alpha}_{\delta}c^{\lambda\delta}_{\gamma}v_{1}^{\gamma}\Omega^{[0]}_{\beta,d;\lambda,0}=b^{\lambda\alpha}_{\gamma}v_{1}^{\gamma}\Omega^{[0]}_{\beta,d;\lambda,0}.$

Here we have used TRR-0 (41) in the first equality. Thus, equation (92) is
equivalent to

(99) $\displaystyle
g^{\alpha\mu}\Omega^{[0]}_{\beta,d;\mu,0}-\tilde{R}^{\alpha}_{\delta}\eta^{\delta\theta}\Omega^{[0]}_{\beta,d+1;\theta,0}=\eta^{\alpha\theta}\Omega^{[0]}_{\mu,d+1;\theta,0}(d+2-\tilde{R})^{\mu}_{\beta}+\eta^{\alpha\theta}\Omega^{[0]}_{\mu,d;\theta,0}M^{\mu}_{\beta}.$

Firstly, using TRR-0 (41), we compute the first summand:

(100) $\displaystyle
g^{\alpha\mu}\Omega^{[0]}_{\beta,d;\mu,0}=E^{\nu}\eta^{\alpha\theta}\frac{\partial}{\partial
v^{\nu}}\Omega^{[0]}_{\theta,0;\lambda,0}\eta^{\mu\lambda}\Omega^{[0]}_{\beta,d;\mu,0}=\eta^{\alpha\theta}E^{\mu}\frac{\partial}{\partial
v^{\mu}}\Omega^{[0]}_{\beta,d+1;\theta,0}$

Secondly, we set $\epsilon=t^{\bullet}_{\geq 1}=0$ in (84):

(101) $\displaystyle E^{\mu}\frac{\partial}{\partial
v^{\mu}}\Omega^{[0]}_{\beta,d+1;\theta,0}-\Omega^{[0]}_{\beta,d+1;\theta,0}+\tilde{R}^{\gamma}_{\theta}\Omega^{[0]}_{\beta,d+1;\gamma,0}=(d+2-\tilde{R})^{\gamma}_{\beta}\Omega^{[0]}_{\gamma,d+1;\theta,0}+M^{\gamma}_{\beta}\Omega^{[0]}_{\gamma,d;\theta,0}$

Combining this last equation with the identity
$\tilde{R}^{\alpha}_{\beta}\eta^{\beta\gamma}+\tilde{R}^{\gamma}_{\beta}\eta^{\alpha\beta}=\eta^{\alpha\gamma}$
we see that (99) holds. ∎

As before, we can apply the transformation (56) to obtain a deformed bracket:

(102) $\displaystyle B^{\alpha\beta}=\sum_{s\geq
0}B^{\alpha\beta}_{s}\partial_{x}^{s}:=\sum_{e,f\geq 0}\frac{\partial
w^{\alpha}}{\partial v^{\mu}_{e}}\partial_{x}^{e}\circ
K^{\mu\nu}\circ(-\partial_{x})^{f}\circ\frac{\partial w^{\beta}}{\partial
v^{\nu}_{f}}.$

Since (56) is a quasi-Miura transformation of $(1,1)$ type by Proposition 3.5,
and it only has terms of even degree in $\epsilon$, the second bracket admits
an $\epsilon$-expansion of the form

(103) $\displaystyle
B^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}B_{g,s}^{\alpha\beta}\partial_{x}^{s},$

where $B^{\alpha\beta}_{g,s}$ is a homogeneous differential rational function
of type $(1,1)$ and degree $2g+1-s$. Note that the max $s$ for a given $g$ is
$3g+1$ as a consequence of Proposition 3.4.

Equations (91) and (92) can be reformulated in terms of the Hamiltonian vector
fields:

(104)
$\displaystyle[K,\bar{h}_{\beta,d}]=[P,\bar{h}_{\mu,d+1}](d+2-\tilde{R})^{\mu}_{\beta}+[P,\bar{h}_{\mu,d}]M^{\mu}_{\beta},\
d\geq-1.$

Moreover, as a consequence of Proposition 3.5, the Schouten bracket invariance
proved in Proposition 2.8 still holds and the system (104) can be rewritten as

(105)
$\displaystyle[B,\bar{h}_{\beta,d}]=[A,\bar{h}_{\mu,d+1}](d+2-\tilde{R})^{\mu}_{\beta}+[A,\bar{h}_{\mu,d}]M^{\mu}_{\beta},\
d\geq-1.$

It is illustrative to see how the bi-Hamiltonian recursion relation is
preserved in terms of operators instead of the Schouten bracket. Let
$L^{\alpha}_{\mu}=\sum_{e}\frac{\partial w^{\alpha}}{\partial
v^{\mu}_{e}}\partial_{x}^{e}$ and
$(L^{*})^{\beta}_{\nu}=\sum_{f}(-\partial_{x})^{f}\circ\frac{\partial
w^{\beta}}{\partial v^{\nu}_{f}}$, then:

(106) $\displaystyle B^{\alpha\beta}\frac{\delta}{\delta
w^{\beta}}(\bar{h}_{\gamma,d})$ $\displaystyle=L^{\alpha}_{\mu}\circ
K^{\mu\nu}\circ(L^{*})^{\beta}_{\nu}\circ\frac{\delta}{\delta
w^{\beta}}(\bar{h}_{\gamma,d})$ $\displaystyle=L^{\alpha}_{\mu}\circ
K^{\mu\nu}\circ\frac{\delta}{\delta v^{\nu}}(\bar{h}_{\gamma,d})$
$\displaystyle=L^{\alpha}_{\mu}\circ P^{\mu\nu}\circ\frac{\delta}{\delta
v^{\nu}}\left((d+2-\tilde{R})^{\lambda}_{\gamma}\bar{h}_{\lambda,d+1}+M^{\lambda}_{\gamma}\bar{h}_{\lambda,d}\right)$
$\displaystyle=L^{\alpha}_{\mu}\circ
P^{\mu\nu}\circ(L^{*})^{\beta}_{\nu}\circ\frac{\delta}{\delta
w^{\beta}}\left((d+2-\tilde{R})^{\lambda}_{\gamma}\bar{h}_{\lambda,d+1}+M^{\lambda}_{\gamma}\bar{h}_{\lambda,d}\right)$
$\displaystyle=A^{\alpha\beta}\frac{\delta}{\delta
w^{\beta}}\left((d+2-\tilde{R})^{\lambda}_{\gamma}\bar{h}_{\lambda,d+1}+M^{\lambda}_{\gamma}\bar{h}_{\lambda,d}\right).$

In other words, $B^{\alpha\beta}$ is an operator of the form (103) satisfying
(106). These are the main equations that we are going to use in the rest of
the paper, recycling the same idea thrice. Firstly, we will show that these
two conditions determine $B^{\alpha\beta}$ uniquely. Secondly, we will show
that the functions $B^{\alpha\beta}_{g,s}$ are of the form
$B^{\alpha\beta}_{g,s}=C^{\alpha\beta}_{g,s}/\det(\eta^{-1}\partial_{x}\Omega^{0})^{n_{g,s}}$,
where $n_{g,s}\in\mathbb{Z}_{\geq 0}$ and $C^{\alpha\beta}_{g,s}$ is a
differential polynomial in $w$, thus giving an alternative proof to [DZ01,
Theorem 4.2.14] for the second bracket without resorting to the loop
equations. Finally, we will prove the vanishing $B^{\alpha\beta}_{g,s}=0$ for
$2g+2\leq s\leq 3g+1$, which is a necessary condition for $B^{\alpha\beta}$ to
be polynomial.

###  4.2. Uniqueness theorem

######  Theorem 4.6.

Let
$C^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}C^{\alpha\beta}_{g,s}\partial_{x}^{s}$
be a Poisson operator of type $(1,1)$ in $w$-coordinates satisfying

(107)
$\displaystyle[C,\bar{h}_{\beta,d}]=[A,\bar{h}_{\mu,d+1}](d+2-\tilde{R})^{\mu}_{\beta}+[A,\bar{h}_{\mu,d}]M^{\mu}_{\beta},\
d\geq-1.$

Then $C=B$.

###### Proof.

Let $D=C-B$. Then we have

(108) $\displaystyle[D,\bar{h}_{\beta,d}]=0,$

or, equivalently,

(109)
$\displaystyle\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}D_{g,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\delta\bar{h}_{\beta,d}}{\delta
w^{\gamma}}\right)=0,$

for all $1\leq\beta\leq N$, $d\geq-1$.

#### Genus $0$

Expanding (109) in $\epsilon$ and taking the $g=0$ term, we have that

(110) $\displaystyle D^{\alpha\gamma}_{0,0}\left(\frac{\partial
h^{[0]}_{\beta,d}}{\partial
w^{\gamma}}\right)+D^{\alpha\gamma}_{0,1}\partial_{x}\left(\frac{\partial
h^{[0]}_{\beta,d}}{\partial w^{\gamma}}\right)=0,$

where
${h}_{\beta,d}={h}_{\beta,d}^{[0]}+\epsilon^{2}{h}_{\beta,d}^{[1]}+\dots$ is
the genus expansion of ${h}_{\beta,d}$ in the $w$ coordinates. Note that we
have replaced the variational derivatives
$\delta\bar{h}^{[0]}_{\beta,d}/\delta w^{\gamma}$ by the partial derivatives
$\partial h^{[0]}_{\beta,d}/\partial w^{\gamma}$ because the Hamiltonian
density $h^{[0]}_{\beta,d}$ does not depend on $w^{\bullet}_{\geq 1}$. For
$d=-1$

(111) $\displaystyle D^{\alpha\gamma}_{0,0}\left(\frac{\partial
h^{[0]}_{\beta,-1}}{\partial
w^{\gamma}}\right)+D^{\alpha\gamma}_{0,1}\partial_{x}\left(\frac{\partial
h^{[0]}_{\beta,-1}}{\partial
w^{\gamma}}\right)=D^{\alpha\gamma}_{0,0}\eta_{\beta\gamma}=0,$

so we can conclude that $D^{\alpha\gamma}_{0,0}=0$. We are left with

(112) $\displaystyle D^{\alpha\gamma}_{0,1}\partial_{x}\left(\frac{\partial
h^{[0]}_{\beta,d}}{\partial w^{\gamma}}\right)=0.$

Recall that the dispersionless Hamiltonians
$h^{[0]}_{\alpha,d}(w)=h_{\alpha,d}(v)|_{v^{\alpha}\to w^{\alpha}}$ satisfy
TRR-0 (41), which implies that

(113) $\displaystyle\frac{\partial^{2}h^{[0]}_{\beta,d}}{\partial
w^{\alpha}\partial w^{\gamma}}=\tilde{c}_{\alpha\gamma}^{\sigma}\frac{\partial
h^{[0]}_{\beta,d-1}}{\partial w^{\sigma}},$

where
$\tilde{c}_{\alpha\gamma}^{\sigma}=\eta^{\sigma\beta}\frac{\partial^{3}F^{\textrm{Frob}}}{\partial
v^{\mu}\partial v^{\nu}\partial v^{\gamma}}|_{v^{\alpha}\to w^{\alpha}}$. Now

(114) $\displaystyle D_{0,1}^{\alpha\gamma}\partial_{x}\left(\frac{\partial
h^{[0]}_{\beta,0}}{\partial
w^{\gamma}}\right)=D_{0,1}^{\alpha\gamma}w_{1}^{\mu}\tilde{c}^{\sigma}_{\gamma\mu}\frac{\partial
h^{[0]}_{\beta,-1}}{\partial
w^{\sigma}}=D_{0,1}^{\alpha\gamma}w_{1}^{\mu}\tilde{c}^{\sigma}_{\gamma\mu}\eta_{\beta\sigma}=0,$

so

(115) $\displaystyle
D_{0,1}^{\alpha\gamma}w^{\mu}_{1}\tilde{c}^{\lambda}_{\gamma\mu}=0,\quad\lambda=1,\dots,N.$

To show that the last equation implies $D_{0,1}^{\alpha\gamma}=0$, we need the
following lemma:

######  Lemma 4.7.

The matrix $\eta^{-1}\partial_{x}\Omega^{0}$, written with indices as
$\tilde{c}^{\alpha}_{\gamma\mu}w^{\mu}_{1}=\eta^{\alpha\beta}\partial_{x}\Omega^{0}_{\beta,0;\gamma,0}$,
is invertible in the $(1,1)$ class.

###### Proof.

The string equation implies that
$\tilde{c}^{\alpha}_{\gamma\mu}w^{\mu}_{1}-\delta^{\alpha}_{\gamma}w^{1}_{1}$
is a differential poynomial that does not depend on $w^{1}_{1}$. ∎

So, we can conclude that $D^{\alpha\gamma}_{0,1}=0$.

#### Induction on $g$

We proceed by induction on $g$. The case $g=0$ has already been proven. Assume
$D^{\alpha\beta}_{r,s}=0$ for all $r\leq g-1$ and for all $s=0,\dots,3r+1$.
The coefficient of $\epsilon^{2g}$ in (109) is

(116)
$\displaystyle\sum_{r=0}^{g-1}\sum_{s=0}^{3r+1}D_{r,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\delta\bar{h}^{[g-r]}_{\beta,d}}{\delta
w^{\gamma}}\right)+\sum_{s=0}^{3g+1}D_{g,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\delta\bar{h}^{[0]}_{\beta,d}}{\delta
w^{\gamma}}\right)=0.$

By induction hypothesis, the first summand vanishes, so

(117)
$\displaystyle\sum_{s=0}^{3g+1}D_{g,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\partial
h^{[0]}_{\beta,d}}{\partial w^{\gamma}}\right)=0.$

Firstly, choosing $d=-1$ implies that $D_{g,0}^{\alpha\gamma}=0$, so

(118)
$\displaystyle\sum_{s=1}^{3g+1}D_{g,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\partial
h^{[0]}_{\beta,d}}{\partial w^{\gamma}}\right)=0.$

Secondly, by the chain rule

(119) $\displaystyle\partial_{x}^{s}\left(\frac{\partial
h^{[0]}_{\beta,d}}{\partial
w^{\gamma}}\right)=\sum_{m=1}^{s}\frac{\partial^{m+1}h^{[0]}_{\beta,d}}{\partial
w^{\gamma}\partial w^{\mu_{1}}\dots\partial
w^{\mu_{m}}}BP^{\mu_{1}\dots\mu_{m}}_{s,m},$

where $BP^{\mu_{1}\dots\mu_{m}}_{s,m}$ is a homogeneous differential
polynomial of degree $s$ in the variables $w^{\mu}_{p}$, where
$\mu=\mu_{1}\dots\mu_{m}$ and $p=1,\dots,s-m+1$. For this proof, we only need
the explicit form of

(120) $\displaystyle
BP^{\mu_{1}\dots\mu_{s}}_{s,s}=w_{1}^{\mu_{1}}w_{1}^{\mu_{2}}\dots
w_{1}^{\mu_{s}}.$

Iterating (113) $m$ times yields

(121) $\displaystyle\frac{\partial^{m+1}h^{[0]}_{\beta,d}}{\partial
w^{\gamma}\partial w^{\mu_{1}}\dots\partial
w^{\mu_{m}}}=\sum_{k=1}^{m}BQ^{(m,k),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}\frac{\partial
h^{[0]}_{\beta,d-k}}{\partial w^{\lambda}},$

where $BQ^{(m,k),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}$ is a function in $w$
that can be written in terms of the functions
$\tilde{c}^{\alpha}_{\beta\gamma}$ and their partial derivatives. For this
proof, we only need the explicit form of

(122) $\displaystyle
BQ^{(m,m),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}=\tilde{c}^{\lambda_{1}}_{\gamma\mu_{1}}\tilde{c}^{\lambda_{2}}_{\lambda_{1}\mu_{2}}\dots\tilde{c}^{\lambda}_{\lambda_{m-1}\mu_{m}}.$

Inserting (119) and (121) in (118), and changing the order of summation yields

(123)
$\displaystyle\sum_{k=1}^{3g+1}\left(\sum_{m=k}^{3g+1}\left(\sum_{s=m}^{3g+1}D_{g,s}^{\alpha\gamma}BP^{\mu_{1}\dots\mu_{m}}_{s,m}\right)BQ^{(m,k),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}\right)\frac{\partial
h^{[0]}_{\beta,d-k}}{\partial w^{\lambda}}=0.$

Choosing $d=0$ kills all terms except the one with $k=1$, so

(124)
$\displaystyle\sum_{m=1}^{3g+1}\left(\sum_{s=m}^{3g+1}D_{g,s}^{\alpha\gamma}BP^{\mu_{1}\dots\mu_{m}}_{s,m}\right)BQ^{(m,1),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}=0,$

vanishes and so does

(125)
$\displaystyle\sum_{k=2}^{3g+1}\left(\sum_{m=k}^{3g+1}\left(\sum_{s=m}^{3g+1}D_{g,s}^{\alpha\gamma}BP^{\mu_{1}\dots\mu_{m}}_{s,m}\right)BQ^{(m,k),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}\right)\frac{\partial
h^{[0]}_{\beta,d-k}}{\partial w^{\lambda}}=0.$

Choosing $d=1,2,\dots 3g$ in the same way shows that

(126)
$\displaystyle\sum_{m=k}^{3g+1}\left(\sum_{s=m}^{3g+1}D_{g,s}^{\alpha\gamma}BP^{\mu_{1}\dots\mu_{m}}_{s,m}\right)BQ^{(m,k),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}=0,\qquad
k=1,\dots,3g+1.$

Let $k=3g+1$. By (120) and (122), we have

(127) $\displaystyle D_{g,3g+1}^{\alpha\gamma}w_{1}^{\mu_{1}}\dots
w_{1}^{\mu_{3g+1}}\tilde{c}_{\gamma\mu_{1}}^{\lambda_{1}}\tilde{c}^{\lambda_{2}}_{\lambda_{1}\mu_{2}}\dots\tilde{c}^{\lambda}_{\lambda_{3g,\mu_{3g+1}}}=0.$

Regrouping the terms

(128) $\displaystyle
D^{\alpha\gamma}_{g,3g+1}w_{1}^{\mu_{1}}\tilde{c}^{\lambda_{1}}_{\gamma\mu_{1}}w_{1}^{\mu_{2}}\tilde{c}^{\lambda_{2}}_{\lambda_{1}\mu_{2}}\dots
w_{1}^{\mu_{3g}}\tilde{c}^{\lambda_{3g}}_{\lambda_{3g-1},\mu_{3g}}w_{1}^{\mu_{3g+1}}\tilde{c}^{\lambda}_{\lambda_{3g},\mu_{3g+1}}=0.$

By Lemma 4.7, the factor
$w_{1}^{\mu_{3g+1}}\tilde{c}^{\lambda}_{\lambda_{3g},\mu_{3g+1}}$ can be
canceled out, meaning the remaining terms must be zero
$D^{\alpha\gamma}_{g,3g+1}w_{1}^{\mu_{1}}\tilde{c}^{\lambda_{1}}_{\gamma\mu_{1}}w_{1}^{\mu_{2}}\tilde{c}^{\lambda_{2}}_{\lambda_{1}\mu_{2}}\dots
w_{1}^{\mu_{3g}}\tilde{c}^{\lambda_{3g}}_{\lambda_{3g-1},\mu_{3g}}=0$.
Iterating these cancellations shows that

(129) $\displaystyle D^{\alpha\gamma}_{g,3g+1}=0.$

Replacing this in (126) yields

(130)
$\displaystyle\sum_{m=k}^{3g}\left(\sum_{s=m}^{3g}D_{g,s}^{\alpha\gamma}BP^{\mu_{1}\dots\mu_{m}}_{s,m}\right)BQ^{(m,k),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}=0,\qquad
k=1,\dots,3g.$

Taking the $k=3g$ term implies, by the same argument as before, that
$D^{\alpha\gamma}_{g,3g}=0$. Repeating for $k=3g-1,3g-2,\dots,1$ shows

(131) $\displaystyle D^{\alpha\gamma}_{g,s}=0,\qquad s=0,1,\dots,3g+1,$

which completes the proof. ∎

###  4.3. Dubrovin–Zhang structural theorem

An argument based on bi-Hamiltonian recursion as in the proof of Theorem 4.6
is insufficient to show that the functions $B_{g,s}^{\alpha\beta}$ are
polynomial. However, it is enough to derive a new proof of the following
weaker structural result, which has been already proved in [DZ01] using the
loop equation.

######  Theorem 4.8.

The second Poisson operator of the Dubrovin–Zhang hierarchy $B^{\alpha\beta}$
can be expanded as

(132) $\displaystyle
B^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}B^{\alpha\beta}_{g,s}\partial_{x}^{s},$

where $B^{\alpha\beta}_{g,s}$ is a homogeneous differential rational function
in the variables $w$ of degree $2g+1-s$ of the form

(133) $\displaystyle
B^{\alpha\beta}_{g,s}=\frac{C^{\alpha\beta}_{g,s}}{D^{n_{g,s}}},$

where
$D=D(w;w_{1})=\det(\tilde{c}^{\lambda}_{\gamma\mu}w_{1}^{\mu})=\det(\eta^{-1}\partial_{x}\Omega^{0})$,
$C^{\alpha\beta}_{g,s}$ is a differential polynomial not divisible by $D$ and
$n_{g,s}\in\mathbb{Z}_{\geq 0}$.

###### Proof.

Recall $B$ satisfies equation (106)

(134) $\displaystyle B^{\alpha\beta}\frac{\delta}{\delta
w^{\beta}}(\bar{h}_{\gamma,d})=A^{\alpha\beta}\frac{\delta}{\delta
w^{\beta}}\left((d+2-\tilde{R})^{\lambda}_{\gamma}\bar{h}_{\lambda,d+1}+M^{\lambda}_{\gamma}\bar{h}_{\lambda,d}\right),$

whose right hand side is polynomial as a consequence of Theorem 3.7. We know
that the $g=0$ term of the expansion of $B$ equals
$K^{\alpha\beta}|_{v^{\alpha}\rightarrow w^{\alpha}}$, which is polynomial,
but we will proceed analogously to Theorem 4.6 even from $g=0$ to illustrate
the methods of the proof. Expanding the expression

(135)
$\displaystyle\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}B_{g,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\delta\bar{h}_{\beta,d}}{\delta
w^{\gamma}}\right),$

which is polynomial, and taking the $g=0$ term implies that

(136) $\displaystyle B_{0,0}^{\alpha\gamma}\left(\frac{\partial
h_{\beta,d}^{[0]}}{\partial
w^{\gamma}}\right)+B_{0,1}^{\alpha\gamma}\partial_{x}\left(\frac{\partial
h_{\beta,d}^{[0]}}{\partial w^{\gamma}}\right)$

is polynomial. Choosing $d=-1$ shows immediately that $B_{0,0}^{\alpha\gamma}$
is polynomial, so we know

(137) $\displaystyle B_{0,1}^{\alpha\gamma}\partial_{x}\left(\frac{\partial
h_{\beta,d}^{[0]}}{\partial w^{\gamma}}\right)$

is polynomial. As in the proof of Theorem 4.6, we apply a corollary of TRR-0
(113) and choose $d=0$ to show that

(138) $\displaystyle
B_{0,1}^{\alpha\gamma}w^{\mu}_{1}\tilde{c}^{\lambda}_{\gamma\mu}$

is polynomial. By Lemma 4.7, the matrix
$(\eta^{-1}\partial_{x}\Omega^{0})^{\lambda}_{\gamma}=w^{\mu}_{1}\tilde{c}^{\lambda}_{\gamma\mu}$
is invertible. We can write its inverse as
$((\eta^{-1}\partial_{x}\Omega^{0})^{-1})^{\gamma}_{\lambda}=\frac{1}{D}T^{\gamma}_{\lambda}$,
where $T^{\gamma}_{\lambda}$ is the transpose of the adjugate of
$\eta^{-1}\partial_{x}\Omega^{0}$, hence a differential polynomial, and
$D=\det(w^{\mu}_{1}\tilde{c}^{\lambda}_{\gamma\mu})$ is its determinant.
Therefore, multiplying the polynomial expression (138) by
$(\eta^{-1}\partial_{x}\Omega^{0})^{-1}$ implies $B_{0,1}^{\alpha\gamma}$ can
be written as the quotient of a differential polynomial by $D$. In other
words, we can write

(139) $\displaystyle
B_{0,1}^{\alpha\gamma}=\frac{C^{\alpha\gamma}_{0,1}}{D^{n_{0,1}}}$

for $C^{\alpha\gamma}_{0,1}$ a differential polynomial not divisible by $D$
and $n_{0,1}\leq 1$. Assume

(140) $\displaystyle
B^{\alpha\beta}_{r,s}=\frac{C^{\alpha\beta}_{r,s}}{D^{n_{r,s}}}$

with $C^{\alpha\beta}_{r,s}$ a differential polynomial not divisible by $D$
for all $r\leq g-1$ and for all $s\leq 3r+1$. Let
$n=\max_{\begin{subarray}{c}0\leq r\leq g-1\\\ 0\leq s\leq
3r+1\end{subarray}}(n_{r,s})$. The coefficient of $\epsilon^{2g}$ in (135) is

(141)
$\displaystyle\sum_{r=0}^{g-1}\sum_{s=0}^{3r+1}B_{r,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\delta\bar{h}_{\beta,d}^{[g-r]}}{\delta
w^{\gamma}}\right)+\sum_{s=0}^{3g+1}B_{g,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\delta\bar{h}_{\beta,d}^{[0]}}{\delta
w^{\gamma}}\right),$

which is polynomial. By induction hypothesis, the first summand is a
differential polynomial divided by $D^{n}$, and so is

(142)
$\displaystyle\sum_{s=0}^{3g+1}B_{g,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\partial{h}_{\beta,d}^{[0]}}{\partial
w^{\gamma}}\right).$

Choosing $d=-1$ implies

(143) $\displaystyle
B_{g,0}^{\alpha\gamma}=\frac{C_{g,0}^{\alpha\gamma}}{n_{g,0}}$

with $n_{g,0}\leq n$, so

(144)
$\displaystyle\sum_{s=1}^{3g+1}B_{g,s}^{\alpha\gamma}\partial_{x}^{s}\left(\frac{\partial{h}_{\beta,d}^{[0]}}{\partial
w^{\gamma}}\right)$

can be written as a differential polynomial divided by $D^{n}$ as well. As in
the proof of Theorem 4.6, we apply iteratively the chain rule (119), TRR-0
(121) and choose $d=0,1,\dots,3g$ to conclude that

(145)
$\displaystyle\sum_{m=k}^{3g+1}\left(\sum_{s=m}^{3g+1}B_{g,s}^{\alpha\gamma}BP^{\mu_{1}\dots\mu_{m}}_{s,m}\right)BQ^{(m,k),\lambda}_{\gamma\mu_{1}\dots\mu_{m}}$

is a differential polynomial divided by $D^{n}$ for all $k=1,\dots,3g+1$. Let
$k=3g+1$, then

(146) $\displaystyle
B^{\alpha\gamma}_{g,3g+1}w_{1}^{\mu_{1}}\tilde{c}^{\lambda_{1}}_{\gamma\mu_{1}}w_{1}^{\mu_{2}}\tilde{c}^{\lambda_{2}}_{\lambda_{1}\mu_{2}}\dots
w_{1}^{\mu_{3g+1}}\tilde{c}^{\lambda}_{\lambda_{3g}\mu_{3g+1}}=\frac{R^{\alpha\gamma}_{3g,3g+1}}{D^{n}},$

where $R^{\alpha\gamma}_{3g,3g+1}$ is a differential polynomial. Multiplying
this identity by the matrix
$((\eta^{-1}\partial_{x}\Omega^{0})^{-1})^{\gamma}_{\lambda}=\frac{1}{D}T^{\gamma}_{\lambda}$
from the right $3g+1$ times, we obtain that

(147) $\displaystyle
B^{\alpha\gamma}_{g,3g+1}=\frac{C^{\alpha\gamma}_{3g,3g+1}}{D^{n_{g,3g+1}}},$

where $C_{3g,3g+1}^{\alpha\gamma}$ is a differential polynomial and
$n_{g,3g+1}\leq n+3g+1$.

Taking the $k=3g$ term in (145) shows that

(148) $\displaystyle
B^{\alpha\gamma}_{g,3g}w_{1}^{\mu_{1}}\tilde{c}^{\lambda_{1}}_{\gamma\mu_{1}}w_{1}^{\mu_{2}}\tilde{c}^{\lambda_{2}}_{\lambda_{1}\mu_{2}}\dots
w_{1}^{\mu_{3g}}\tilde{c}^{\lambda_{3g}}_{\lambda_{3g-1}\mu_{3g}}$
$\displaystyle+B^{\alpha\gamma}_{g,3g+1}\left(BP^{\mu_{1},\dots,\mu_{3g}}_{3g+1,3g}BQ^{(3g,3g)\lambda}_{\gamma\mu_{1}\dots\mu_{3g}}+BP^{\mu_{1}\dots\mu_{3g+1}}_{3g+1,3g+1}BQ^{(3g+1,3g)\lambda}_{\gamma,\mu_{1},\dots,\mu_{3g}}\right)$

is a differential polynomial divided by $D^{n}$. Therefore,

(149) $\displaystyle
B^{\alpha\gamma}_{g,3g}w_{1}^{\mu_{1}}\tilde{c}^{\lambda_{1}}_{\gamma\mu_{1}}w_{1}^{\mu_{2}}\tilde{c}^{\lambda_{2}}_{\lambda_{1}\mu_{2}}\dots
w_{1}^{\mu_{3g}}\tilde{c}^{\lambda_{3g}}_{\lambda_{3g-1}\mu_{3g}}=\frac{R_{3g,3g}^{\alpha\gamma}}{D^{n+3g+1}},$

where $R_{3g,3g}^{\alpha\gamma}$ is a differential polynomial. Multiplying
this identity by the matrix
$((\eta^{-1}\partial_{x}\Omega^{0})^{-1})^{\gamma}_{\lambda}=\frac{1}{D}T^{\gamma}_{\lambda}$
from the right $3g$ times, we obtain that

(150) $\displaystyle
B^{\alpha\gamma}_{g,3g+1}=\frac{C^{\alpha\gamma}_{3g,3g}}{D^{n_{g,3g}}},$

where $C_{3g,3g}$ is a differential polynomial and $n_{g,3g}\leq n+6g+1$.

Repeating this argument for $k=3g-1,3g-2,\dots,1$ shows

(151) $\displaystyle
B^{\alpha\gamma}_{g,s}=\frac{C^{\alpha\gamma}_{g,s}}{D^{n_{g,s}}}$

with $C^{\alpha\gamma}_{g,s}$ a differential polynomial not divisible by $D$.
∎

######  Remark 4.9.

It is easy to track through the proof of Theorem 4.8 an estimate for the
degrees of the denominators $n_{g,s}$. To make these estimates sharper, one
can use the polynomiality in genus $0$ and $1$ [DZ98], and the result of
Theorem 6.8 below, which states that $B_{g,s}^{\alpha\beta}=0$ for $s\geq
2g+2$. But, of course, the conjecture of Dubrovin and Zhang suggests that
$n_{g,s}=0$.

######  Remark 4.10.

The combinatorics of the argument in the proofs of Theorem 4.6 and Theorem 4.8
basically reflects what happens when one replaces the $\psi$-classes by their
pull-backs from the moduli spaces with less number of marked points (cf.
[BPS12b, Equation 3] or [LP11, Proof of Theorem 4]). We make this point
precise in Section 6.

Let us also formulate one extra bit of polynomiality of the second
Dubrovin–Zhang bracket that follows directly from the proof of Theorem 4.8:

######  Theorem 4.11.

The constant term of the second Poisson operator of the Dubrovin–Zhang
hierarchy, $\sum_{g=0}^{\infty}\epsilon^{2g}B^{\alpha\beta}_{g,0}$, is a
differential polynomial.

##  5\. Liu–Pandharipande relations

###  5.1. Relation among the tautological classes

Fix sets of indices $I_{1}$ and $I_{2}$ such that $I_{1}\sqcup
I_{2}=\\{1,\dots,n\\}$. Let
$\Delta_{g_{1},g_{2}}\subset\overline{\mathcal{M}}_{g,n}$ denote a divisor in
$\overline{\mathcal{M}}_{g,n}$ whose generic points are represented by two-
component curves intersecting at a node, where the two components have genera
$g_{1},g_{2}$ and contain the points with the indices $I_{1},I_{2}$,
respectively. Note that if $g_{i}=0$, then $|I_{i}|$ must be at least $2$, for
the stability condition.

Let $n_{1}=|I_{1}|$, $n_{2}=|I_{2}|$. For each $\Delta_{g_{1},g_{2}}$ we
consider the map
$\iota_{g_{1},g_{2}}\colon\overline{\mathcal{M}}_{g_{1},n_{1}+1}\times\overline{\mathcal{M}}_{g_{2},n_{2}+1}\to\overline{\mathcal{M}}_{g,n}$
that glues the last marked points into a node and whose image is
$\Delta_{g_{1},g_{2}}$. Let $\psi_{\circ 1}$ (respectively, $\psi_{\circ 2}$)
denote the psi classes at the marked points on the first (respectively,
second) component that are glued into the node.

######  Proposition 5.1 ([LP11, Proposition 1]).

For any $g\geq 0$, $n\geq 4$, $I_{1}$ and $I_{2}$ such that $I_{1}\sqcup
I_{2}=\\{1,\dots,n\\}$ and $|I_{1}|,|I_{2}|\geq 2$, and an arbitrary $r\geq 0$
we have:

(152) $\sum_{\begin{subarray}{c}g_{1},g_{2}\geq 0\\\
g_{1}+g_{2}=g\end{subarray}}\sum_{\begin{subarray}{c}a_{1},a_{2}\geq 0\\\
a_{1}+a_{2}=\\\
2g-3+n+r\end{subarray}}(-1)^{a_{1}}(\iota_{g_{1},g_{2}})_{*}\psi_{\circ
1}^{a_{1}}\psi_{\circ 2}^{a_{2}}=0.$

Let $n=k+1$, $I_{1}=\\{1,\dots,k-1\\}$, $I_{2}=\\{k,k+1\\}$, and consider the
map $\pi\colon\overline{\mathcal{M}}_{g,k+1}\to\overline{\mathcal{M}}_{g,k}$
that forgets the last marked point. We apply the push-forward $\pi_{*}$ to the
left hand side of (152) and to the left hand side of (152) multiplied by
$\psi_{k+1}$ in order to obtain the following corollaries.

######  Corollary 5.2.

For any $g\geq 0$, $k\geq 3$, $I_{1}=\\{1,\dots,k-1\\}$ and $I_{2}=\\{k\\}$,
and an arbitrary $r\geq 0$ we have:

(153) $\sum_{\begin{subarray}{c}g_{1}\geq 0,g_{2}>0\\\
g_{1}+g_{2}=g\end{subarray}}\sum_{\begin{subarray}{c}a_{1},a_{2}\geq 0\\\
a_{1}+a_{2}=\\\
2g-2+k+r\end{subarray}}g_{2}(-1)^{a_{1}}(\iota_{g_{1},g_{2}})_{*}\psi_{\circ
1}^{a_{1}}\psi_{\circ 2}^{a_{2}}=0.$

######  Corollary 5.3.

For any $g\geq 0$, $k\geq 3$, $I_{1}=\\{1,\dots,k-1\\}$ and $I_{2}=\\{k\\}$,
and an arbitrary $r\geq 0$ we have:

(154) $(-1)^{k+r}\psi_{k}^{2g-2+k+r}+\sum_{\begin{subarray}{c}g_{1}\geq
0,g_{2}>0\\\
g_{1}+g_{2}=g\end{subarray}}\sum_{\begin{subarray}{c}a_{1},a_{2}\geq 0\\\
a_{1}+a_{2}=\\\
2g-3+k+r\end{subarray}}(-1)^{a_{1}}(\iota_{g_{1},g_{2}})_{*}\psi_{\circ
1}^{a_{1}}\psi_{\circ 2}^{a_{2}}=0.$

Taking yet another pushforward, we have the following corollary:

######  Corollary 5.4 ([LP11, Proposition 2]).

For any $g\geq 1$, $I_{1}=\\{1\\}$, $I_{2}=\\{2\\}$, and an arbitrary $r\geq
0$ we have:

(155)
$-\psi_{1}^{2g+r}+(-1)^{r}\psi_{2}^{2g+r}+\sum_{\begin{subarray}{c}g_{1}\geq
0,g_{2}>0\\\
g_{1}+g_{2}=g\end{subarray}}\sum_{\begin{subarray}{c}a_{1},a_{2}\geq 0\\\
a_{1}+a_{2}=\\\
2g-1+r\end{subarray}}(-1)^{a_{1}}(\iota_{g_{1},g_{2}})_{*}\psi_{\circ
1}^{a_{1}}\psi_{\circ 2}^{a_{2}}=0.$

###  5.2. Implications for $\partial_{x}$-derivatives of two-point functions

Equations (153) and (154) imply a number of identities for the functions
$\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,p}$. In order to formulate these
identities in a useful way for the computational scheme presented in Section
4, we introduce a new notation.

Let
$\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\overline{0}}\coloneqq\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,0}$,
$s\geq 0$, and for $p\geq 1$ we set

(156)
$\displaystyle\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\overline{p}}\coloneqq\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,p}-\sum_{q=0}^{p-1}\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\mu,\overline{q}}\eta^{\mu\nu}\Omega^{[0]}_{\nu,0;\beta,p-q-1}.$

In other words, in the expansion of
$\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\overline{p}}$ we use the pull-
back of $\psi^{p}$ from $\overline{\mathcal{M}}_{g,s+2}$ at the point with the
primary field $\beta$.

######  Lemma 5.5.

For $s\geq 1$, $p\geq 2g+s$ we have:

(157) $\displaystyle\sum_{\begin{subarray}{c}g_{1}\geq 0,g_{2}>0\\\
g_{1}+g_{2}=g\end{subarray}}\sum_{q=0}^{p}g_{2}(-1)^{q}\partial_{x}^{s}\Omega^{[g_{1}]}_{\alpha,0;\mu,\overline{q}}\eta^{\mu\nu}\Omega^{[g_{2}]}_{\beta,0;\nu,\overline{p-q}}=0.$

######  Lemma 5.6.

For $s\geq 1$, $p\geq 2g+s$ we have:

(158)
$\displaystyle\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\overline{p}}=\sum_{\begin{subarray}{c}g_{1}\geq
0,g_{2}>0\\\
g_{1}+g_{2}=g\end{subarray}}\sum_{q=0}^{p-1}(-1)^{p-q-1}\partial_{x}^{s}\Omega^{[g_{1}]}_{\alpha,0;\mu,\overline{q}}\eta^{\mu\nu}\Omega^{[g_{2}]}_{\beta,0;\nu,\overline{p-q-1}}.$

Recall also the notation
$\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,-1}\coloneqq\delta_{s,0}\delta_{g,0}\eta_{\alpha\beta}$.
We set
$\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\widetilde{-1}}\coloneqq\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,{-1}}$
and for $p\geq 0$

(159)
$\displaystyle\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\widetilde{p}}\coloneqq\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,p}-\sum_{q=-1}^{p-1}\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\mu,\widetilde{q}}\eta^{\mu\nu}\Omega^{[0]}_{\nu,0;\beta,p-q-1}.$

Of course, if $g>0$ or $s>0$, then
$\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\widetilde{p}}=\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\overline{p}}$,
but for $g=s=0$ and $p\geq 0$,
$\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\widetilde{p}}=0$. With this
extra piece of notation we can include the case $s=0$ in Lemma 5.6 in the
following way:

######  Lemma 5.7.

For $s\geq 0$, $p\geq 2g+s$ we have:

(160)
$\displaystyle\partial_{x}^{s}\Omega^{[g]}_{\alpha,0;\beta,\widetilde{p}}=\sum_{\begin{subarray}{c}g_{1}\geq
0,g_{2}>0\\\
g_{1}+g_{2}=g\end{subarray}}\sum_{q=-1}^{p-1}(-1)^{p-q-1}\partial_{x}^{s}\Omega^{[g_{1}]}_{\alpha,0;\mu,\widetilde{q}}\eta^{\mu\nu}\Omega^{[g_{2}]}_{\beta,0;\nu,\overline{p-q-1}}.$

Lemmata 5.5, 5.6, and 5.7 are direct corollaries of Corollaries 5.2, 5.3, and
5.4, respectively. It is a rather standard translation of tautological
relations into differential equations for the coefficients of the genus
expansion of the logarithm of the partition function, see e. g. [LP11, Proof
of Theorem 4]. Another exposition of a detailed step-by-step instruction how
one can translate a tautological relation into a PDE is presented in [FSZ10,
Section 2.1.3].

###  5.3. Variation for $\tilde{\mathsf{E}}$

In fact, as it is explained in [LP11, Proof of Theorem 4], all lemmata in the
previous section work without any change once we replace the operator
$\partial_{x}^{s}$ with an arbitrary $s$-vector field on the big phase space.
The actual result that we use below is a variation of Lemma 5.6 that is
related to the vector field $\tilde{\mathsf{E}}$.

Let
$\tilde{\mathsf{E}}\partial_{x}\Omega^{[g]}_{\alpha,0;\beta,\overline{0}}\coloneqq\tilde{\mathsf{E}}\partial_{x}\Omega^{[g]}_{\alpha,0;\beta,0}$,
and for $p\geq 1$ we set

(161)
$\displaystyle\tilde{\mathsf{E}}\partial_{x}\Omega^{[g]}_{\alpha,0;\beta,\overline{p}}\coloneqq\tilde{\mathsf{E}}\partial_{x}\Omega^{[g]}_{\alpha,0;\beta,p}-\sum_{q=0}^{p-1}\tilde{\mathsf{E}}\partial_{x}\Omega^{[g]}_{\alpha,0;\mu,\overline{q}}\eta^{\mu\nu}\Omega^{[0]}_{\nu,0;\beta,p-q-1}.$

In other words, in the expansion of
$\tilde{\mathsf{E}}\partial_{x}\Omega^{[g]}_{\alpha,0;\beta,\overline{p}}$ we
use the pull-back of $\psi^{p}$ from $\overline{\mathcal{M}}_{g,4}$ at the
point with the primary field $\beta$.

######  Lemma 5.8.

For $p\geq 2g+2$ we have:

(162)
$\displaystyle\tilde{\mathsf{E}}\partial_{x}\Omega^{[g]}_{\alpha,0;\beta,\overline{p}}=\sum_{\begin{subarray}{c}g_{1}\geq
0,g_{2}>0\\\
g_{1}+g_{2}=g\end{subarray}}\sum_{q=0}^{p-1}(-1)^{p-q-1}\tilde{\mathsf{E}}\partial_{x}\Omega^{[g_{1}]}_{\alpha,0;\mu,\overline{q}}\eta^{\mu\nu}\Omega^{[g_{2}]}_{\beta,0;\nu,\overline{p-q-1}}.$

##  6\. Vanishing terms of the second bracket

The goal of this section is to prove that all terms of the second
Dubrovin–Zhang bracket that have negative degree, and therefore cannot be
polynomial, do vanish. The argument goes as follows.

We start with two essential steps to simplify the problem. First, we replace
the operator $B$ with a different operator $\tilde{B}$ that has equivalent
vanishing properties but satisfies a simplified version of the bi-Hamiltonian
recursion. Second, we employ a triangular structure with respect to the
$\epsilon$-degree of the change of variables from $v$ coordinates to $w$
coordinates in order to reduce the problem to the vanishing of the negative
degree terms of the operator $\tilde{B}$ in the $v$ coordinates.

The latter observation allows us to consider the simplified version of the bi-
Hamiltonian recursion in the $v$ coordinates, for which the
$\epsilon$-expansion of the $\Omega$-functions has geometric meaning, as it
coincides with the expansion in the $t$ variables. This lets us apply various
geometric observations from Section 5 and homogeneity properties from Section
4 to derive the desired vanishing statement about $\tilde{B}$.

###  6.1. Equivalent form without the variational derivative

Recall the bi-Hamiltonian recursion:

(163) $B^{\alpha\beta}\frac{\delta}{\delta
w^{\beta}}\int\Omega_{1,0;\gamma,p+1}dx=A^{\alpha\beta}\frac{\delta}{\delta
w^{\beta}}\left(\int\Omega_{1,0;\mu,p+2}dx\,(p+2-\tilde{R})^{\mu}_{\gamma}+\int\Omega_{1,0;\mu,p+1}dx\,M^{\mu}_{\gamma}\right)$

for $p\geq-1$. Note that
$A^{\alpha\beta}=\partial_{x}\circ\tilde{A}^{\alpha\beta}$, where
$\tilde{A}^{\alpha\beta}=\eta^{\alpha\beta}+O(\epsilon^{2})=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{2g}\tilde{A}^{\alpha\beta}_{g,s}\partial_{x}^{s}$,
where $\tilde{A}^{\alpha\beta}_{g,s}$ are differential polynomials in the
coordinates $w$, and the standard gradation of $\tilde{A}^{\alpha\beta}_{g,s}$
is $\deg\tilde{A}^{\alpha\beta}_{g,s}=2g-s$. Consider the inverse operator,
$\tilde{A}^{-1}_{\alpha\beta}$. It has exactly the same properties as
$\tilde{A}^{\alpha\beta}$, namely, it expands as
$\tilde{A}^{-1}_{\alpha\beta}=\eta_{\alpha\beta}+O(\epsilon^{2})=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{2g}(\tilde{A}^{-1}_{g,s})_{\alpha\beta}\partial_{x}^{s}$,
where $(\tilde{A}^{-1}_{g,s})_{\alpha\beta}$ are differential polynomials, and
the standard gradation of $(\tilde{A}^{-1}_{g,s})_{\alpha\beta}$ is
$\deg(\tilde{A}^{-1}_{g,s})_{\alpha\beta}=2g-s$. Define
$\tilde{B}^{\alpha\beta}\coloneqq
B^{\alpha\mu}\tilde{A}^{-1}_{\mu\nu}\eta^{\nu\beta}$. This operator satisfies
a simplified version of bi-Hamiltonian recursion:

######  Lemma 6.1.

We have:

(164)
$\displaystyle\tilde{B}^{\alpha\beta}\Omega_{\beta,0;\gamma,p}=\eta^{\alpha\beta}\partial_{x}\left(\Omega_{\beta,0;\mu,p+1}\,(p+2-\tilde{R})^{\mu}_{\gamma}+\Omega_{\beta,0;\mu,p}\,M^{\mu}_{\gamma}\right).$

###### Proof.

The way the operator $A^{\alpha\beta}$ acts on the variational derivatives of
the Hamiltonians implies that

(165) $\displaystyle\tilde{A}^{\alpha\beta}\frac{\delta}{\delta
w^{\beta}}\int\Omega_{1,0;\gamma,p+1}dx=\eta^{\alpha\beta}\Omega_{\beta,0;\gamma,p},$

hence the statement of the lemma. ∎

Recall that
$B^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}B^{\alpha\beta}_{g,s}\partial_{x}^{s}$,
with $\deg B_{g,s}^{\alpha\beta}=2g+1-s$. It is easy to see that
$\tilde{B}^{\alpha\beta}$ has expansion with exactly the same properties,
namely,
$\tilde{B}^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}\tilde{B}^{\alpha\beta}_{g,s}\partial_{x}^{s}$,
with $\deg\tilde{B}_{g,s}^{\alpha\beta}=2g+1-s$. Moreover,

######  Lemma 6.2.

(1) The coefficients of the operator $B^{\alpha\beta}$ are differential
polynomials if and only if the coefficients of the operator
$\tilde{B}^{\alpha\beta}$ are differential polynomials.

(2) The coefficients $B^{\alpha\beta}_{g,s}$, $g\geq 0$, $2g+2\leq s\leq
3g+1$, vanish if and only if the coefficients $\tilde{B}^{\alpha\beta}_{g,s}$,
$g\geq 0$, $2g+2\leq s\leq 3g+1$, vanish.

###### Proof.

Both statements follow from the polynomiality of
$\tilde{A}^{-1}_{\alpha\beta}$. ∎

Finally, it is a bit easier to work in the $v$ coordinates instead of the $w$
coordinates, but then, of course, all polynomiality properties are destroyed.
The vanishing properties are, however, preserved. Namely, consider the
expansion of the operator $\tilde{B}^{\alpha\beta}$ in the $v$ coordinates:
$\tilde{B}^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}\tilde{B}^{\alpha\beta}_{[g],s}\partial_{x}^{s}$.

######  Lemma 6.3.

The coefficients $\tilde{B}^{\alpha\beta}_{g,s}$, $g\geq 0$, $2g+2\leq s\leq
3g+1$, vanish if and only if the coefficients
$\tilde{B}^{\alpha\beta}_{[g],s}$, $g\geq 0$, $2g+2\leq s\leq 3g+1$, vanish.

###### Proof.

Indeed, the change of variables from $w$ to $v$ in
$\tilde{B}^{\alpha\beta}(w(v,\epsilon),\epsilon)$ does not affect the terms
$\tilde{B}^{\alpha\beta}_{g,s}$ such that
$\tilde{B}^{\alpha\beta}_{g^{\prime},s}=0$ for all $g^{\prime}<g$. More
precisely, under this condition
$\tilde{B}^{\alpha\beta}_{[g],s}(v)=\tilde{B}^{\alpha\beta}_{g,s}(w)|_{w=v}$.
The same argument applies also to the change of variables from $v$ to $w$.

Now we proof the lemma by induction on $g$. The base of induction is obvious,
and if we prove the equivalence of the vanishings for any $g^{\prime}<g$, then
for any $g^{\prime}<g$ the top non-vanishing terms in $w$ (respectively, $v$)
coordinates are $\tilde{B}^{\alpha\beta}_{g^{\prime},s}$ (respectively,
$\tilde{B}^{\alpha\beta}_{[g^{\prime}],s}$) with $s=2g^{\prime}+1$. Since
$2g^{\prime}+1<2g+2$ for any $g^{\prime}<g$, the vanishing of
$\tilde{B}^{\alpha\beta}_{g,s}$, $s\geq 2g+2$ is equivalent to the vanishing
of $\tilde{B}^{\alpha\beta}_{[g],s}$, $s\geq 2g+2$. ∎

######  Remark 6.4.

Assume that the vanishing of $\tilde{B}^{\alpha\beta}_{g,s}$ (or,
equivalently, $\tilde{B}^{\alpha\beta}_{[g],s}$) is proved for $g\geq 0$,
$s\geq 2g+2$. Then the same argument as in the proof of Lemma 6.3 implies that
the polynomiality of $\tilde{B}^{\alpha\beta}_{g,2g}$ and
$\tilde{B}^{\alpha\beta}_{g,2g+1}$ is equivalent to the polynomiality of
$\tilde{B}^{\alpha\beta}_{[g],2g}$ and $\tilde{B}^{\alpha\beta}_{[g],2g+1}$,
respectively, for any $g\geq 0$.

###  6.2. Vanishing terms

Consider the expansion of the operator $\tilde{B}^{\alpha\beta}$ in the $v$
coordinates:

(166)
$\displaystyle\tilde{B}^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}\tilde{B}^{\alpha\beta}_{[g],s}\partial_{x}^{s}.$

######  Proposition 6.5.

We have $\tilde{B}^{\alpha\beta}_{[g],s}=0$ for $s\geq 2g+2$, $g\geq 0$.

Before we proceed with the proof of Proposition 6.5, let us note that the
equation that determines $\tilde{B}^{\alpha\beta}_{[g],s}$ (once
$\tilde{B}^{\alpha\beta}_{[h],t}$ are known for $h<g$ and for $h=g$, $t>s$),
i.e., what corresponds to the $s$-th summand of (126) in the proof of Theorem
4.6 or, analogously, the $s$-th summand of (145) in the proof of Theorem 4.8,
can be compactly written as follows:

######  Lemma 6.6.

We have:

(167) $\displaystyle\sum_{\begin{subarray}{c}g_{1},g_{2},t\geq 0\\\
g_{1}+g_{2}=g\end{subarray}}\tilde{B}^{\alpha\beta}_{[g_{1}],t}\partial_{x}^{t}\Omega^{[g_{2}]}_{\beta,0;\gamma,\widetilde{s-1}}=\eta^{\alpha\beta}\tilde{R}^{\mu}_{\beta}\partial_{x}\Omega^{[g]}_{\mu,0;\gamma,\overline{s}}+\tilde{\mathsf{E}}\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,\overline{s}}+g(3-\mathsf{d})\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,\overline{s}}.$

###### Proof.

Consider the genus $g$ component of Equation (164) with $p=s-1$ (recall that
we use the $v$ coordinates for all ingredients of the formula):

(168) $\displaystyle\sum_{\begin{subarray}{c}g_{1},g_{2},t\geq 0\\\
g_{1}+g_{2}=g\end{subarray}}\tilde{B}^{\alpha\beta}_{[g_{1}],t}\partial_{x}^{t}\Omega^{[g_{2}]}_{\beta,0;\gamma,s-1}=\eta^{\alpha\beta}\partial_{x}\left(\Omega^{[g]}_{\beta,0;\mu,s}\,(s+1-\tilde{R})^{\mu}_{\gamma}+\Omega^{[g]}_{\beta,0;\mu,s-1}\,M^{\mu}_{\gamma}\right).$

Then we apply (85) with $p=s$. We obtain:

(169) $\displaystyle\sum_{\begin{subarray}{c}g_{1},g_{2},t\geq 0\\\
g_{1}+g_{2}=g\end{subarray}}\tilde{B}^{\alpha\beta}_{[g_{1}],t}\partial_{x}^{t}\Omega^{[g_{2}]}_{\beta,0;\gamma,s-1}=\eta^{\alpha\beta}\tilde{R}^{\mu}_{\beta}\partial_{x}\Omega^{[g]}_{\mu,0;\gamma,{s}}+\tilde{\mathsf{E}}\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,{s}}+g(3-\mathsf{d})\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,{s}}.$

Let us prove that Equation (169) implies the statement of the lemma by
induction on $s$. The base is the case $s=0$; in this case Equations (167) and
(169) are equivalent. Assume the lemma is proved for all $s<S$. Then, for
$s=S$ we have:

(170) $\displaystyle\sum_{\begin{subarray}{c}g_{1},g_{2},t\geq 0\\\
g_{1}+g_{2}=g\end{subarray}}\tilde{B}^{\alpha\beta}_{[g_{1}],t}\partial_{x}^{t}\Omega^{[g_{2}]}_{\beta,0;\gamma,\widetilde{S-1}}-\eta^{\alpha\beta}\tilde{R}^{\mu}_{\beta}\partial_{x}\Omega^{[g]}_{\mu,0;\gamma,\overline{S}}-\tilde{\mathsf{E}}\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,\overline{S}}-g(3-\mathsf{d})\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,\overline{S}}$
$\displaystyle=\sum_{\begin{subarray}{c}g_{1},g_{2},t\geq 0\\\
g_{1}+g_{2}=g\end{subarray}}\tilde{B}^{\alpha\beta}_{[g_{1}],t}\partial_{x}^{t}\Omega^{[g_{2}]}_{\beta,0;\gamma,{S-1}}-\eta^{\alpha\beta}\tilde{R}^{\mu}_{\beta}\partial_{x}\Omega^{[g]}_{\mu,0;\gamma,{S}}-\tilde{\mathsf{E}}\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,{S}}-g(3-\mathsf{d})\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,{S}}$
$\displaystyle\quad-\sum_{q=0}^{S-1}\Bigg{(}\sum_{\begin{subarray}{c}g_{1},g_{2},t\geq
0\\\
g_{1}+g_{2}=g\end{subarray}}\tilde{B}^{\alpha\beta}_{[g_{1}],t}\partial_{x}^{t}\Omega^{[g_{2}]}_{\beta,0;\mu,\widetilde{q-1}}-\eta^{\alpha\beta}\tilde{R}^{\mu}_{\beta}\partial_{x}\Omega^{[g]}_{\mu,0;\mu,\overline{q}}$
$\displaystyle\qquad\qquad\quad-\tilde{\mathsf{E}}\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\mu,\overline{q}}-g(3-\mathsf{d})\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\mu,\overline{q}}\Bigg{)}\eta^{\mu\nu}\Omega^{[0]}_{\nu,0;\gamma,S-q-1}.$

This equality follows from directly from the definitions (156), (159), and
(161). Now, the right hand side of this equality is equal to zero: the first
line by Equation (169), and the sum over $q$ by the induction assumption. This
implies that (167) holds for $s=S$ and completes the inductive proof of the
lemma. ∎

###### Proof of Proposition 6.5.

We assume that using the computational scheme in Theorems 4.6 and 4.8 we
already proved by induction the vanishing of $\tilde{B}^{\alpha\beta}_{[h],t}$
for $h<g$, $t\geq 2h+2$ and for $h=g$, $t>s$. Also, recall that
$\tilde{B}^{\alpha\beta}_{[g_{1}],t}\partial_{x}^{t}\Omega^{[g_{2}]}_{\beta,0;\gamma,\widetilde{s-1}}=0$
for $0\leq t<s-3g_{2}$ for dimensional reasons ($\psi_{t+2}^{>3g_{2}-1+t}$
vanishes on $\overline{\mathcal{M}}_{g_{2},t+2}$), which we use below for
$g_{1}=g$ and $g_{2}=0$. Then we have:

(171) $\displaystyle\sum_{\begin{subarray}{c}g_{1},g_{2},t\geq 0\\\
g_{1}+g_{2}=g\end{subarray}}\tilde{B}^{\alpha\beta}_{[g_{1}],t}\partial_{x}^{t}\Omega^{[g_{2}]}_{\beta,0;\gamma,\widetilde{s-1}}=\tilde{B}^{\alpha\beta}_{[g],s}\partial_{x}^{s}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{s-1}}+\sum_{h=0}^{g-1}\sum_{t=0}^{2h+1}\tilde{B}^{\alpha\beta}_{[h],t}\partial_{x}^{t}\Omega^{[g-h]}_{\beta,0;\gamma,\widetilde{s-1}}$
$\displaystyle=\tilde{B}^{\alpha\beta}_{[g],s}\partial_{x}^{s}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{s-1}}+\sum_{h=0}^{g-1}\sum_{r=-1}^{s-2}(-1)^{s-r}\sum_{\begin{subarray}{c}h_{1},h_{2}\geq
0\\\
h_{1}+h_{2}=h\end{subarray}}\sum_{t=0}^{2h_{1}+1}\tilde{B}^{\alpha\beta}_{[h_{1}],t}\partial_{x}^{t}\Omega^{[h_{2}]}_{\beta,0;\mu,\widetilde{r}}\eta^{\mu\nu}\Omega^{[g-h]}_{\gamma,0;\nu,\overline{s-2-r}}$
$\displaystyle=\tilde{B}^{\alpha\beta}_{[g],s}\partial_{x}^{s}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{s-1}}+\sum_{h=0}^{g-1}\sum_{r=-1}^{s-2}(-1)^{s-r}\Bigg{(}\eta^{\alpha\beta}\tilde{R}^{\xi}_{\beta}\partial_{x}\Omega^{[h]}_{\xi,0;\mu,\overline{r+1}}+\tilde{\mathsf{E}}\eta^{\alpha\beta}\partial_{x}\Omega^{[h]}_{\beta,0;\mu,\overline{r+1}}$
$\displaystyle\phantom{=\
}+h(3-\mathsf{d})\eta^{\alpha\beta}\partial_{x}\Omega^{[h]}_{\beta,0;\mu,\overline{r+1}}\Bigg{)}\eta^{\mu\nu}\Omega^{[g-h]}_{\gamma,0;\nu,\overline{s-2-r}}.$

Here for the second equality we use Lemma 5.7, and for the third equality we
use Equation (167) for $g=h$ and $s=r+1$. Note that the condition of Lemma 5.7
is indeed satisfied: since $t\leq 2h+1$ and $2g+2\leq s$, we have
$2(g-h)+t\leq s-1$.

On the other hand, for $s\geq 2g+2$ (this inequality is crucially important
for the second summand, for the first and the third ones $s\geq 2g+1$ would be
sufficient), we have from Lemmata 5.6 and 5.8 the following:

(172)
$\displaystyle\eta^{\alpha\beta}\tilde{R}^{\mu}_{\beta}\partial_{x}\Omega^{[g]}_{\mu,0;\gamma,\overline{s}}+\tilde{\mathsf{E}}\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,\overline{s}}+g(3-\mathsf{d})\eta^{\alpha\beta}\partial_{x}\Omega^{[g]}_{\beta,0;\gamma,\overline{s}}$
$\displaystyle=\sum_{h=0}^{g-1}\sum_{r=0}^{s-1}(-1)^{s-r-1}\Bigg{(}\eta^{\alpha\beta}\tilde{R}^{\xi}_{\beta}\partial_{x}\Omega^{[h]}_{\xi,0;\mu,\overline{r}}+\tilde{\mathsf{E}}\eta^{\alpha\beta}\partial_{x}\Omega^{[h]}_{\beta,0;\mu,\overline{r}}+g(3-\mathsf{d})\eta^{\alpha\beta}\partial_{x}\Omega^{[h]}_{\beta,0;\mu,\overline{r}}\Bigg{)}\eta^{\mu\nu}\Omega^{[g-h]}_{\gamma,0;\nu,\overline{s-1-r}}.$

Substituting this expression and Equation (171) in Equation (167), we obtain:

(173)
$\displaystyle\tilde{B}^{\alpha\beta}_{[g],s}\partial_{x}^{s}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{s-1}}=(3-\mathsf{d})\eta^{\alpha\beta}\sum_{h=0}^{g-1}\sum_{r=0}^{s-1}(-1)^{s-r-1}(g-h)\partial_{x}\Omega^{[h]}_{\beta,0;\mu,\overline{r}}\eta^{\mu\nu}\Omega^{[g-h]}_{\gamma,0;\nu,\overline{s-1-r}}.$

For $s-1\geq 2g+1$ the right hand side of this equation is equal to zero by
Lemma 5.5. Note that

(174)
$\displaystyle\partial_{x}^{s}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{s-1}}=\delta_{\beta}^{\mu_{1}}\delta_{\gamma}^{\nu_{s}}\prod_{i=1}^{s}\partial_{x}\Omega^{[0]}_{\mu_{i},0;\nu_{i},0}\prod_{i=1}^{s-1}\eta^{\nu_{i}\mu_{i+1}}.$

(we prove it below in Lemma 6.7).

Now, since $\partial_{x}^{s}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{s-1}}$ as
given by (174) is invertible, we can use the vanishing of the right hand side
of Equation (173) to conclude that $\tilde{B}^{\alpha\beta}_{[g],s}=0$. ∎

######  Lemma 6.7.

Equation (174) holds for $s\geq 0$.

###### Proof.

We prove it by induction on $s$. The full induction statement is the
following. For any $s\geq 0$

(175)
$\displaystyle\partial_{x}^{t}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{s-1}}=\begin{cases}\delta_{\beta}^{\mu_{1}}\delta_{\gamma}^{\nu_{s}}\prod_{i=1}^{s}\partial_{x}\Omega^{[0]}_{\mu_{i},0;\nu_{i},0}\prod_{i=1}^{s-1}\eta^{\nu_{i}\mu_{i+1}}&t=s;\\\
0&0\leq t<s.\end{cases}$

For $s=0$ it is the definition of
$\Omega^{[0]}_{\beta,0;\gamma,\widetilde{-1}}$. For the induction step we have
to recall the topological recursion relation in genus $0$ (41), which implies
that for $p\geq 0$

(176)
$\displaystyle\partial_{x}\Omega^{[0]}_{\alpha,0;\beta,p}=\partial_{x}\Omega^{[0]}_{\alpha,0;\mu,0}\eta^{\mu\nu}\Omega^{[0]}_{\nu,0;\beta,p-1},$

and, therefore,

(177)
$\displaystyle\partial_{x}\Omega^{[0]}_{\alpha,0;\beta,\widetilde{p}}=\partial_{x}\Omega^{[0]}_{\alpha,0;\mu,0}\eta^{\mu\nu}\Omega^{[0]}_{\nu,0;\beta,\widetilde{p-1}};$

Assume (175) is proved for $s\leq S$. Then for $t=0$ and $s=S+1$ the required
vanishing follows directly from the induction assumption applied to the right
hand side of Equation (159). If $t\geq 1$, then for $s=S+1$ we use (177) to
obtain

(178)
$\displaystyle\partial_{x}^{t}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{S}}=\partial_{x}^{t-1}\left(\partial_{x}\Omega^{[0]}_{\beta,0;\xi,0}\eta^{\xi\zeta}\Omega^{[0]}_{\zeta,0;\gamma,\widetilde{S-1}}\right)=\sum_{u=0}^{t-1}\binom{t-1}{u}\partial_{x}^{u+1}\Omega^{[0]}_{\beta,0;\xi,0}\eta^{\xi\zeta}\partial_{x}^{t-1-u}\Omega^{[0]}_{\zeta,0;\gamma,\widetilde{S-1}}.$

If $1\leq t\leq S$, then $t-1-u<S$ for any $u=0,\dots,t-1$, and then this
expression is equal to zero by the induction assumption. Let $t=S+1$. Then
$t-1-u=S-u<S$ for $u=1,\dots,t-1$, and therefore the corresponding summands
are equal to zero by the induction assumption. Thus for $t=S+1$ we have

(179)
$\partial_{x}^{S+1}\Omega^{[0]}_{\beta,0;\gamma,\widetilde{S}}=\partial_{x}\Omega^{[0]}_{\beta,0;\xi,0}\eta^{\xi\zeta}\partial_{x}^{S}\Omega^{[0]}_{\zeta,0;\gamma,\widetilde{S-1}}$

Substitution of the non-vanishing case of the Equation (175) for $s=S$ into
this formula proves the non-vanishing case of the Equation (175) for $s=S+1$,
which completes the step of induction and proves the lemma. ∎

Now we are ready to state and prove our main theorem, which appears to be a
direct corollary of Proposition 6.5.

######  Theorem 6.8.

Consider the $\epsilon$-expansion of the second Dubrovin–Zhang bracket in the
coordinates $w$:
$B^{\alpha\beta}=\sum_{g=0}^{\infty}\epsilon^{2g}\sum_{s=0}^{3g+1}B^{\alpha\beta}_{g,s}\partial_{x}^{s}$.
We have $B^{\alpha\beta}_{g,s}=0$ for $g\geq 0$, $s\geq 2g+2$.

###### Proof.

Lemmata 6.2 and 6.3 imply that the statement of the theorem is equivalent to
the statement of Proposition 6.5. ∎

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