# Linearization of Virasoro symmetries associated with semisimple Frobenius
manifolds

Si-Qi Liu, Zhe Wang, Youjin Zhang

###### Abstract.

For any semisimple Frobenius manifold, we prove that a tau-symmetric
bihamiltonian deformation of its Principal Hierarchy admits an infinite family
of linearizable Virasoro symmetries if and only if all the central invariants
of the corresponding deformation of the bihamiltonian structure are equal to
$\frac{1}{24}$. As an important application of this result, we prove that the
Dubrovin-Zhang hierarchy associated with the semisimple Frobenius manifold
possesses a bihamiltonian structure which can be represented in terms of
differential polynomials.

###### Key words and phrases:

Frobenius manifold, Principal Hierarchy, Bihamiltonian structure, Virasoro
symmetry, Loop equation

###### Contents

  1. 1 Introduction
  2. 2 Frobenius manifold and its loop equation
  3. 3 Linearization of Virasoro symmetries
    1. 3.1 Bihamiltonian structure and quasi-triviality
    2. 3.2 Proof of the main theorem
  4. 4 Conclusion

##  1\. Introduction

The deep relationship between 2D topological field theories (2DTFT) and
integrable hierarchies has become one of the central research topics in
mathematical physics since the proof of the Witten conjecture[24] by
Kontsevich [17], which relates the intersection numbers on the moduli space of
stable curves to the well-known Korteweg-de Vries (KdV) hierarchy. In [8],
Dubrovin introduced the notion of Frobenius manifolds as a geometric
interpretation of the genus zero part of a 2DTFT, and showed that the genus
zero partition function of a 2DTFT is the tau-function of a particular
solution of a bihamiltonian integrable hierarchy, which is called the
Principal Hierarchy of the Frobenius manifold corresponding to the 2DTFT.

In [13] Dubrovin and the third-named author constructed a certain deformation
of the Principal Hierarchy associated with a semisimple Frobenius manifold and
conjectured that the full genera partition function of the 2DTFT corresponding
to the Frobenius manifold is the tau-function of a solution of the deformed
integrable hierarchy. This particular deformed integrable hierarchy is called
the topological deformation of the Principal Hierarchy and it is also called
the Dubrovin-Zhang (DZ) hierarchy in the literature. It is constructed by
performing a quasi-Miura transformation to the Principal Hierarchy and
consists of bihamiltonian evolutionary PDEs. In [4, 5], Buryak, Posthuma and
Shadrin proved that the DZ hierarchy and its first Hamiltonian structure can
be represented by differential polynomials. However, the polynomiality of the
second Hamiltonian structure of the DZ hierarchy remains unproved. In [16],
Hernández Iglesias and Shadrin provided some evidences which support the
validity of the polynomiality of the second Hamiltonian structure by using the
geometry of the moduli space of the stable curves.

In this paper, we prove the polynomiality of the second Hamiltonian structure
of the DZ hierarchy.The approach of our proof consists of the following two
steps:

Step 1. To prove that any polynomial tau-symmetric bihamiltonian deformation
of the Principal Hierarchy possesses an infinite family of Virasoro
symmetries.

Step 2. To prove that the Virasoro symmetries constructed in Step 1. can be
linearized (in a sense that will be explained later) if the central invariants
of the bihamiltonian structure of the deformed integrable hierarchy are all
equal to $\frac{1}{24}$.

In [13], it is proved that the integrable hierarchy obtained in these two
steps is unique and it is exactly the DZ hierarchy. This implies that the
bihamiltonian structure of the DZ hierarchy is polynomial. We have finished
the first step in [19, 20, 21] by developing the theory of super tau-covers of
bihamiltonian integrable hierarchies and variational bihamiltonian
cohomologies. In particular we have the following theorem.

######  Theorem 1.1 ([19]).

For the Principal Hierarchy associated with a semisimple Frobenius manifold
and any of its tau-symmetric bihamiltonian deformations, there exists a unique
deformation of its Virasoro symmetries such that they are symmetries of the
deformed integrable hierarchy. Moreover, the action of the Virasoro symmetries
on the tau-function $Z$ of the deformed integrable hierarchy can be
represented in the form

$\frac{\partial Z}{\partial s_{m}}=L_{m}Z+O_{m}Z,\quad m\geq-1,$

where $L_{m}$ are the Virasoro operators constructed in [12] and $O_{m}$ are
some differential polynomials, and the flows $\frac{\partial}{\partial s_{m}}$
satisfy the Virasoro commutation relations

$\left[\frac{\partial}{\partial s_{k}},\frac{\partial}{\partial
s_{l}}\right]=(l-k)\frac{\partial}{\partial s_{k+l}},\quad k,l\geq-1.$

In this paper, we proceed to finish the second step. Let us first explain this
step in a more explicit way. It is proved in [10] that the tau-structure of
the deformed Principal Hierarchy changes if we choose different
representatives for the deformation of the bihamiltonian structure under Miura
type transformations. In particular, the differential polynomials $O_{m}$
described in the above theorem change accordingly after performing a Miura
type transformation to the deformed integrable hierarchy. So by the term to
linearize the Virasoro symmetries we mean to find a suitable Miura type
transformation such that after performing this transformation, the
differential polynomials $O_{m}$ become zero (see Sect. 3 for a detailed
description). We prove the following theorem.

######  Main Theorem.

For a given tau-symmetric bihamiltonian deformation for the Principal
Hierarchy associated with a semisimple Frobenius manifold, the Virasoro
symmetries of the deformed integrable hierarchy can be linearized if and only
if the central invariants of the deformed bihamiltonian structure are all
equal to $\frac{1}{24}$, i.e., in this case, by choosing a suitable
representative of the bihamiltonian structure, the action of the Virasoro
symmetries on the tau-function $Z$ of the deformed integrable hierarchy can be
represented in the form

$\frac{\partial Z}{\partial s_{m}}=L_{m}Z,\quad m\geq-1.$

As an immediate corollary, we have the following polynomiality theorem for the
DZ hierarchy.

######  Theorem 1.2.

The bihamiltonian structure of the DZ hierarchy associated with a semisimple
Frobenius manifold can be represented by differential polynomials.

This paper is organized as follows. In Sect. 2, we recall some basic facts
about the Frobenius manifolds and their loop equations. In Sect. 3 we give the
prove of the main theorem. In Sect. 4 we make some concluding remarks.

##  2\. Frobenius manifold and its loop equation

The notion of Frobenius manifolds [8, 13] gives a coordinate-free description
of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) associativity equation [7,
23]. An $n$-dimensional Frobenius manifold $M$ is an analytic manifold
equipped with a flat (pseudo-)Riemannian metric $\eta$ and a $(1,2)$-tensor
$c$ which define Frobenius algebra structures on the tangent spaces $T_{p}M$
for any $p\in M$ with flat unit vector field. Furthermore these data satisfy
certain homogeneous conditions. Let $v^{1},\cdots,v^{n}$ be flat coordinates
of $M$ with $\frac{\partial}{\partial v^{1}}$ being the unit vector field,
then the local structure of $M$ can be described by a solution
$F(v^{1},\cdots,v^{n})$ of the following WDVV equation:

$\partial_{\alpha}\partial_{\beta}\partial_{\lambda}F\eta^{\lambda\mu}\partial_{\mu}\partial_{\gamma}\partial_{\delta}F=\partial_{\delta}\partial_{\beta}\partial_{\lambda}F\eta^{\lambda\mu}\partial_{\mu}\partial_{\gamma}\partial_{\alpha}F,$

where $(\eta^{\lambda\mu})=(\eta_{\lambda\mu})^{-1}$ and
$\partial_{\alpha}=\frac{\partial}{\partial v^{\alpha}}$. Here and henceforth
the summation over repeated upper and lower Greek indices is assumed. This
function $F$ is called the potential of the Frobenius manifold $M$, and it is
related to the tensors $\eta$ and $c$ by

$\eta_{\alpha\beta}=\partial_{1}\partial_{\alpha}\partial_{\beta}F,\quad
c_{\alpha\beta}^{\gamma}=\eta^{\gamma\lambda}\partial_{\lambda}\partial_{\alpha}\partial_{\beta}F.$

The potential is required to be quasi-homogeneous with respect to an Euler
vector field of the form

(2.1)
$E=\sum_{\alpha=1}^{n}\left(\left(1-\frac{d}{2}-\mu_{\alpha}\right)v^{\alpha}+r_{\alpha}\right)\partial_{\alpha},$

i.e.,

$E(F)=(3-d)F+\frac{1}{2}A_{\alpha\beta}v^{\alpha}v^{\beta}+B_{\alpha}v^{\alpha}+C.$

Here $\mu_{\alpha}$, $r_{\alpha}$, $A_{\alpha\beta}$, $B_{\alpha}$ and $C$ are
some constants and the constant $d$ is called the charge of $M$.

The Frobenius algebra structure on $TM$ yields a deformed flat connection

(2.2) $\tilde{\nabla}_{X}Y=\nabla_{X}Y+zX\cdot
Y,\quad\forall\,X,Y\in\Gamma(TM),\quad z\in\mathbb{C},$

here $\nabla$ is the Levi-Civita connection of $\eta$ and $X\cdot Y=c(X,Y)$.
It can be extended to be an affine connection on $M\times\mathbb{C}^{*}$ by
regarding $z$ as the coordinate of $\mathbb{C}^{*}$ and by requiring

$\tilde{\nabla}_{\partial_{z}}X=\partial_{z}X+E\cdot
X-\frac{1}{z}\mu(X),\quad\tilde{\nabla}_{\partial_{z}}\partial_{z}=\tilde{\nabla}_{X}\partial_{z}=0,$

where $X$ is a vector field on $M\times\mathbb{C}^{*}$ whose $\partial_{z}$
component is zero and $\mu=\mathrm{diag}(\mu_{1},\cdots,\mu_{n})$ is a
diagonal matrix regarding as a section of the vector bundle
$\mathrm{End}(TM)$.

A system of flat coordinates $\tilde{v}_{1},\cdots,\tilde{v}_{n}$ of the
deformed flat connection can be chosen to have the following form:

$(\tilde{v}^{1}(v,z),\cdots,\tilde{v}^{n}(v,z))=(\theta_{1}(v,z),\cdots,\theta_{n}(v,z))z^{\mu}z^{R},$

here $R$ is a certain constant matrix. The functions $\theta_{\alpha}(v,z)$
are analytic at $z=0$ and has the expansion $\theta_{\alpha}(v,z)=\sum_{p\geq
0}\theta_{\alpha,p}(v)z^{p}$. The coefficients $\theta_{\alpha,p}$ satisfy the
following recursion relations:

$\theta_{\alpha,0}=\eta_{\alpha\beta}v^{\beta},\quad\partial_{\beta}\partial_{\gamma}\theta_{\alpha,p+1}=c^{\lambda}_{\beta\gamma}\partial_{\lambda}\theta_{\alpha,p},\quad
p\geq 0.$

They also satisfy the quasi-homogeneous condition

$E(\partial_{\beta}\theta_{\alpha,p})=(p+\mu_{\alpha}+\mu_{\beta})\partial_{\beta}\theta_{\alpha,p}+\sum_{k=1}^{p}(R_{k})^{\gamma}_{\alpha}\partial_{\beta}\theta_{\gamma,p-k}.$

The Principal Hierarchy associated with $M$ is defined to be the following
mutually commuting flows:

(2.3) $\frac{\partial v^{\lambda}}{\partial
t^{\alpha,p}}=\eta^{\lambda\gamma}\partial_{x}(\partial_{\gamma}\theta_{\alpha,p+1}),\quad\alpha=1,\cdots,n,\quad
p\geq 0.$

Since $\frac{\partial v^{\lambda}}{\partial t^{1,0}}=\partial_{x}v^{\lambda}$,
we will identify the time variable $t^{1,0}$ with the spacial variable $x$. We
will also use the notation $v^{\lambda,k}=\partial_{x}^{k}v^{\lambda}$ in what
follows. The Principal Hierarchy (2.3) is bihamiltonian with respect to the
following Poisson pencil:

(2.4) $\displaystyle\\{v^{\alpha}(x),v^{\beta}(y)\\}_{1}^{[0]}$
$\displaystyle=\eta^{\alpha\beta}\delta^{\prime}(x-y),$ (2.5)
$\displaystyle\\{v^{\alpha}(x),v^{\beta}(y)\\}_{2}^{[0]}$
$\displaystyle=g^{\alpha\beta}(v(x))\delta^{\prime}(x-y)+\Gamma^{\alpha\beta}_{\gamma}v^{\gamma}_{x}(x)\delta(x-y).$

Here the functions $g^{\alpha\beta}$ and $\Gamma^{\alpha\beta}_{\gamma}$ are
given by

$g^{\alpha\beta}=E^{\varepsilon}c^{\alpha\beta}_{\varepsilon},\quad\Gamma^{\alpha\beta}_{\gamma}=\left(\frac{1}{2}-\mu_{\beta}\right)c^{\alpha\beta}_{\gamma}$

with $c^{\alpha\beta}_{\gamma}=\eta^{\alpha\lambda}c^{\beta}_{\lambda\gamma}$.
In terms of the first Hamiltonian structure, the flows of the Principal
Hierarchy can be represented as the following Hamiltonian systems:

$\frac{\partial v^{\lambda}}{\partial
t^{\alpha,p}}=\\{v^{\lambda},\int\theta_{\alpha,p+1}\\}_{1}^{[0]}.$

It is also tau-symmetric, i.e., the densities of the Hamiltonians satisfy the
symmetry property

$\frac{\partial\theta_{\alpha,p}}{\partial
t^{\beta,q}}=\frac{\partial\theta_{\beta,q}}{\partial t^{\alpha,p}}.$

The tau-symmetric property of the Principal Hierarchy enables us to define a
set of functions $\Omega_{\alpha,p;\beta,q}^{[0]}(v)$ for
$\alpha,\beta=1,\cdots,n$ and $p,q\geq 0$ such that the following identities
hold true:

$\displaystyle\Omega^{[0]}_{\alpha,p;1,0}=\theta_{\alpha,p},\quad\Omega^{[0]}_{\alpha,p;\beta,0}=\partial_{\beta}\theta_{\alpha,p+1};$
$\displaystyle\Omega_{\alpha,p;\beta,q}^{[0]}=\Omega_{\beta,q;\alpha,p}^{[0]},\quad\frac{\partial\Omega_{\alpha,p;\beta,q}^{[0]}}{\partial
t^{\lambda,k}}=\frac{\partial\Omega_{\lambda,k;\beta,q}^{[0]}}{\partial
t^{\alpha,p}}.$

Let us introduce the tau-cover of the Principal Hierarchy as follows:

$\frac{\partial\mathcal{F}_{0}}{\partial
t^{\alpha,p}}=f_{\alpha,p}^{[0]},\quad\frac{\partial
f_{\beta,q}^{[0]}}{\partial
t^{\alpha,p}}=\Omega_{\alpha,p;\beta,q}^{[0]},\quad\frac{\partial
v^{\beta}}{\partial
t^{\alpha,p}}=\eta^{\beta\lambda}\partial_{x}\Omega_{\lambda,0;\alpha,p}^{[0]}.$

The unknown function $\mathcal{F}_{0}$ is called the genus zero free energy
and the functions $f_{\alpha,p}^{[0]}$ are called the genus zero one-point
functions.

In [12], an infinite set of Virasoro symmetries of the tau-cover of the
Principal Hierarchy was constructed. These symmetries are closely related to
the differential operators

$L_{m}=a_{m}^{\alpha,p;\beta,q}\varepsilon^{2}\frac{\partial^{2}}{\partial
t^{\alpha,p}\partial
t^{\beta,q}}+{b}_{m;\alpha,p}^{\beta,q}t^{\alpha,p}\frac{\partial}{\partial
t^{\beta,q}}+c_{m;\alpha,p;\beta,q}\frac{1}{\varepsilon^{2}}t^{\alpha,p}t^{\beta,q}+\frac{1}{4}\delta_{m,0}\mathrm{tr}\left(\frac{1}{4}-\mu^{2}\right).$

where $m\geq-1$ and $a_{m}^{\alpha,p;\beta,q}$, ${b}_{m;\alpha,p}^{\beta,q}$,
$c_{m;\alpha,p;\beta,q}$ are some constants determined from the monodromy data
of the Frobenius manifold $M$. They can be represented in the form

(2.6) $\displaystyle\frac{\partial\mathcal{F}_{0}}{\partial s_{m}}$
$\displaystyle=a_{m}^{\alpha,p;\beta,q}f_{\alpha,p}^{[0]}f_{\beta,q}^{[0]}+{b}_{m;\alpha,p}^{\beta,q}t^{\alpha,p}f_{\beta,q}^{[0]}+c_{m;\alpha,p;\beta,q}t^{\alpha,p}t^{\beta,q},$
(2.7) $\displaystyle\frac{\partial f_{\alpha,p}^{[0]}}{\partial s_{m}}$
$\displaystyle=\frac{\partial}{\partial
t_{\alpha,p}}\frac{\partial\mathcal{F}_{0}}{\partial
s_{m}},\quad\frac{\partial v^{\alpha}}{\partial
s_{m}}=\eta^{\alpha\beta}\frac{\partial}{\partial t^{1,0}}\frac{\partial
f_{\beta,0}^{[0]}}{\partial s_{m}}.$

These flows satisfy the Virasoro commutation relations

$\left[\frac{\partial}{\partial s_{k}},\frac{\partial}{\partial
s_{l}}\right]=(l-k)\frac{\partial}{\partial s_{k+l}},\quad k,l\geq-1.$

In [13], a deformation of the Principal Hierarchy associated with a semisimple
Frobenius manifold $M$ is constructed by requiring that the actions of the
Virasoro symmetries on the tau-function

$Z=\exp\biggl{(}\frac{1}{\varepsilon^{2}}\mathcal{F}_{0}+\sum_{g\geq
1}\varepsilon^{2g-2}F_{g}\bigl{(}v,v_{x},v_{xx},\cdots,v^{(m_{g})}\bigr{)}\biggr{)}$

of the deformed integrable hierarchy are represented by

(2.8) $\frac{\partial Z}{\partial s_{m}}=L_{m}Z,\quad m\geq-1.$

This procedure is called the linearization of the Virasoro symmetries, and the
functions $F_{g}$ are called the genus $g$ free energy of the deformed
integrable hierarchy, which are determined by the equations

(2.9) $\displaystyle\frac{\partial\Delta F}{\partial s_{m}}=$
$\displaystyle\,a_{m}^{\alpha,p;\beta,q}\left(2\frac{\partial\mathcal{F}_{0}}{\partial
t^{\alpha,p}}\frac{\partial\Delta F}{\partial
t^{\beta,q}}+\varepsilon^{2}\frac{\partial\Delta F}{\partial
t^{\alpha,p}}\frac{\partial\Delta F}{\partial
t^{\beta,q}}+\frac{\partial^{2}(\mathcal{F}_{0}+\varepsilon^{2}\Delta
F)}{\partial t^{\alpha,p}\partial t^{\beta,q}}\right)$
$\displaystyle+{b}_{m;\alpha,p}^{\beta,q}t^{\alpha,p}\frac{\partial\Delta
F}{\partial
t^{\beta,q}}+\frac{1}{4}\delta_{m,0}\mathrm{tr}\left(\frac{1}{4}-\mu^{2}\right),\quad
m\geq-1,$

here we denote

$\Delta F=\sum_{g\geq
1}\varepsilon^{2g-2}F_{g}\bigl{(}v,v_{x},v_{xx},\cdots,v^{(m_{g})}\bigr{)}.$

Denote by $\mathcal{D}_{m}$ the following derivation:

(2.10)
$\mathcal{D}_{m}=2a_{m}^{\alpha,p;\beta,q}\frac{\partial\mathcal{F}_{0}}{\partial
t^{\alpha,p}}\frac{\partial}{\partial
t^{\beta,q}}+{b}_{m;\alpha,p}^{\beta,q}t^{\alpha,p}\frac{\partial}{\partial
t^{\beta,q}}-\frac{\partial}{\partial s_{m}},\quad m\geq-1,$

then the equations in (2.9) can be rewritten as

(2.11) $\displaystyle\mathcal{D}_{m}\Delta F=$
$\displaystyle\,-a_{m}^{\alpha,p;\beta,q}\frac{\partial^{2}\mathcal{F}_{0}}{\partial
t^{\alpha,p}\partial
t^{\beta,q}}-\varepsilon^{2}a_{m}^{\alpha,p;\beta,q}\left(\frac{\partial\Delta
F}{\partial t^{\alpha,p}}\frac{\partial\Delta F}{\partial
t^{\beta,q}}+\frac{\partial^{2}\Delta F}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)$
$\displaystyle-\frac{1}{4}\delta_{m,0}\mathrm{tr}\left(\frac{1}{4}-\mu^{2}\right),\quad
m\geq-1.$

By comparing the coefficients of $\varepsilon^{2g-2}$ for $g\geq 1$, we see
that we can represent the above equations as

$\displaystyle\mathcal{D}_{m}F_{1}=$
$\displaystyle\,-a_{m}^{\alpha,p;\beta,q}\frac{\partial^{2}\mathcal{F}_{0}}{\partial
t^{\alpha,p}t^{\beta,q}}-\frac{1}{4}\delta_{m,0}\mathrm{tr}\left(\frac{1}{4}-\mu^{2}\right),$
$\displaystyle\mathcal{D}_{m}F_{g}=$
$\displaystyle\,-a_{m}^{\alpha,p;\beta,q}\left(\sum_{k=1}^{g-1}\frac{\partial
F_{k}}{\partial t^{\alpha,p}}\frac{\partial F_{g-k}}{\partial
t^{\beta,q}}+\frac{\partial^{2}F_{g-1}}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right),\quad g\geq 2.$

To solve equations (2.11) for all $m\geq-1$, the authors of [13] reorganized
them into the form

(2.12) $\displaystyle\mathcal{D}(\lambda)\Delta F=$
$\displaystyle\,-\sum_{m\geq-1}\frac{1}{\lambda^{m+2}}a_{m}^{\alpha,p;\beta,q}\left(\frac{\partial^{2}\mathcal{F}_{0}}{\partial
t^{\alpha,p}\partial t^{\beta,q}}+\varepsilon^{2}\frac{\partial\Delta
F}{\partial t^{\alpha,p}}\frac{\partial\Delta F}{\partial
t^{\beta,q}}+\varepsilon^{2}\frac{\partial^{2}\Delta F}{\partial
t^{\alpha,p}\partial t^{\beta,q}}\right)$
$\displaystyle-\frac{1}{4\lambda^{2}}\mathrm{tr}\left(\frac{1}{4}-\mu^{2}\right),$

where $\mathcal{D}(\lambda)=\sum_{m\geq-1}\mathcal{D}_{m}/\lambda^{m+2}$. An
important observation made in [13] is that the equation (2.12) can be
rewritten as an equation for $\Delta F$ on the jet space, which has the
following explicit form (Lemma 4.2.6 of [13]):

(2.13) $\displaystyle\sum_{r\geq 0}\frac{\partial\Delta F}{\partial
v^{\gamma,r}}\partial_{x}^{r}\left(\frac{1}{E-\lambda}\right)^{\gamma}+\sum_{r\geq
1}\frac{\partial\Delta F}{\partial
v^{\gamma,r}}\sum_{k=1}^{r}\binom{r}{k}\partial_{x}^{k-1}\partial_{1}p_{\alpha}G^{\alpha\beta}\partial_{x}^{r-k+1}\partial^{\gamma}p_{\beta}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}(\partial_{\lambda}p_{\alpha})*(\partial_{\lambda}p_{\beta})G^{\alpha\beta}-\frac{1}{4\lambda^{2}}\mathrm{tr}\left(\frac{1}{4}-\mu^{2}\right)$
$\displaystyle+\frac{\varepsilon^{2}}{2}\sum\left(\frac{\partial\Delta
F}{\partial v^{\gamma,k}}\frac{\partial\Delta F}{\partial
v^{\rho,l}}+\frac{\partial^{2}\Delta F}{\partial v^{\gamma,k}\partial
v^{\rho,l}}\right)\partial_{x}^{k+1}\partial^{\gamma}p_{\alpha}G^{\alpha\beta}\partial_{x}^{l+1}\partial^{\rho}p_{\beta}$
$\displaystyle+\frac{\varepsilon^{2}}{2}\sum\frac{\partial\Delta F}{\partial
v^{\gamma,k}}\partial_{x}^{k+1}\left[\nabla\frac{\partial
p_{\alpha}}{\partial\lambda}\cdot\nabla\frac{\partial
p_{\beta}}{\partial\lambda}\cdot v_{x}\right]^{\gamma}G^{\alpha\beta},$

here $E$ is the Euler vector field (2.1), $p_{\alpha}=p_{\alpha}(v;\lambda)$
are the periods of the Frobenius manifold $M$, $G^{\alpha\beta}$ is a constant
matrix and $*$ is a multiplication defined on the space of conserved
quantities of the Principal Hierarchy. One may refer to Sect. 4 of [13] for
details. The equation (2.13) is called the loop equation of the Frobenius
manifold $M$ and it plays a central role in the construction of the DZ
hierarchy.

It is proved that the loop equation (2.13) has a unique solution up to
addition of constants if the Frobenius manifold $M$ is semisimple. Recall that
a Frobenius manifold $M$ is called semisimple if the Frobenius algebra
structure on $T_{p}M$ is semisimple for generic $p\in M$. For a semisimple
Frobenius manifold, there exists local coordinates $u^{1},\cdots u^{n}$ such
that $\frac{\partial}{\partial u^{1}},\cdots,\frac{\partial}{\partial u^{n}}$
form a basis of idempotents of the Frobenius algebra on $TM$, i.e.,

$c\left(\frac{\partial}{\partial u^{i}},\frac{\partial}{\partial
u^{j}}\right)=\delta_{i,j}\frac{\partial}{\partial u^{i}},\quad
i,j=1,\cdots,n.$

Such coordinates are unique up to permutations and are called the canonical
coordinates of $M$. The following theorem is proved in [13].

######  Theorem 2.1 ([13]).

If the Frobenius manifold $M$ is semisimple, then the loop equation (2.13) has
a unique, up to the addition of constants, solution

$\Delta F=\sum_{g\geq
1}\varepsilon^{2g-2}F_{g}\bigl{(}v,v_{x},v_{xx},\cdots,v^{(3g-2)}\bigr{)}.$

In particular we have

$\displaystyle F_{1}$
$\displaystyle=\log\frac{\tau_{I}(u)}{J^{1/24}(u)}+\frac{1}{24}\sum_{i=1}^{n}\log
u^{i}_{x},$ $\displaystyle F_{g}$ $\displaystyle\in
C^{\infty}(v)\left[v_{x},v_{xx},\cdots,v^{(3g-2)}\right]\left[\frac{1}{u^{1}_{x}\cdots
u^{n}_{x}}\right],\quad g\geq 2,$

where $u^{1},\cdots,u^{n}$ are canonical coordinates of $M$, $\tau_{I}(u)$ is
the isomonodromic tau-function of $M$ and $J(u)$ is the Jacobian
$\det\left(\frac{\partial v^{\alpha}}{\partial u^{i}}\right)$.

By using the above theorem, the DZ hierarchy associated with a semisimple
Frobenius manifold $M$ is obtained by performing the quasi-Miura
transformation

$v^{\alpha}\mapsto
w^{\alpha}=v^{\alpha}+\eta^{\alpha\beta}\varepsilon^{2}\frac{\partial^{2}\Delta
F}{\partial t^{\beta,0}\partial t^{1,0}},\quad\alpha=1,\cdots,n$

to the Principal Hierarchy (2.3) and $w^{\alpha}$ are called the normal
coordinates of the DZ hierarchy. The tau-cover of the DZ hierarchy can be
obtained by the same quasi-Miura transformation from that of the Principal
Hierarchy, it can be represented as

$\varepsilon\frac{\partial\mathcal{F}}{\partial
t^{\alpha,p}}=f_{\alpha,p},\quad\varepsilon\frac{\partial
f_{\beta,q}}{\partial
t^{\alpha,p}}=\Omega_{\alpha,p;\beta,q},\quad\frac{\partial
w^{\beta}}{\partial
t^{\alpha,p}}=\eta^{\beta\lambda}\partial_{x}\Omega_{\lambda,0;\alpha,p},$

where the functions $\Omega_{\alpha,p;\beta,q}$ is given by

$\Omega_{\alpha,p;\beta,q}=\Omega_{\alpha,p;\beta,q}^{[0]}+\varepsilon^{2}\frac{\partial^{2}\Delta
F}{\partial t^{\alpha,p}\partial t^{\beta,q}}.$

We call $\mathcal{F}$, $f_{\alpha,p}$ and $\Omega_{\alpha,p;\beta,q}$ the free
energy, the one-point functions and the two-point functions of the DZ
hierarchy respectively. Moreover the tau-cover of the DZ hierarchy admits
linearized Virasoro symmetries

$\frac{\partial\mathcal{F}}{\partial
s_{m}}=\exp(-\mathcal{F})L_{m}(\exp(\mathcal{F})),\quad\frac{\partial
f_{\alpha,p}}{\partial s_{m}}=\varepsilon\frac{\partial}{\partial
t^{\alpha,p}}\frac{\partial\mathcal{F}}{\partial s_{m}},\quad\frac{\partial
w^{\alpha}}{\partial
s_{m}}=\varepsilon\eta^{\alpha\beta}\frac{\partial}{\partial
t^{1,0}}\frac{\partial f_{\beta,0}}{\partial s_{m}},$

where $m\geq-1$.

##  3\. Linearization of Virasoro symmetries

###  3.1. Bihamiltonian structure and quasi-triviality

Throughout this section, we fix a semisimple Frobenius manifold $M$ of
dimension $n$. Let us consider a deformation of the bihamiltonian structure
(2.4),(2.5) of the Principal Hierarchy of $M$ with constant central
invariants. It follows from [10] that the deformed bihamiltonian structure
uniquely determines a deformation of the Principal Hierarchy which is tau-
symmetric. In another word, the deformed bihamiltonian structure determines a
deformation of the tau-cover of the Principal Hierarchy, which can be
represented by

$\varepsilon\frac{\partial\mathcal{F}}{\partial
t^{\alpha,p}}=f_{\alpha,p},\quad\varepsilon\frac{\partial
f_{\beta,q}}{\partial
t^{\alpha,p}}=\Omega_{\alpha,p;\beta,q},\quad\frac{\partial
w^{\beta}}{\partial
t^{\alpha,p}}=\eta^{\beta\lambda}\partial_{x}\Omega_{\lambda,0;\alpha,p}.$

Here the two-point functions $\Omega_{\alpha,p;\beta,q}$ are differential
polynomials in $w^{1},\cdots,w^{n}$ whose leading terms are
$\Omega_{\alpha,p;\beta,q}^{[0]}$ after we replace each $w^{\lambda}$ by
$v^{\lambda}$. The unknown functions $w^{\alpha}$ are called the normal
coordinates of the deformed integrable hierarchy. In terms of the normal
coordinates, we can represent the deformed bihamiltonian structure in the form

$\\{w^{\alpha}(x),w^{\beta}(y)\\}_{a}=\sum_{g\geq
0}\sum_{k=0}^{2g+1}\varepsilon^{2g}P_{a;g,k}^{\alpha\beta}(x)\delta^{(2g+1-k)}(x-y),\quad
a=1,2.$

Here $P_{a;g,k}^{\alpha\beta}$ are differential polynomials in
$w^{1},\cdots,w^{n}$ of differential degree $k$. Recall that the ring of
differential polynomials is graded with respect to the differential degree
$\deg_{x}$ defined by $\deg_{x}w^{\alpha,r}=r$. It was proved in [9] that such
a deformation is quasi-trivial, i.e., there exists a unique quasi-Miura
transformation

(3.1) $v^{\alpha}\mapsto w^{\alpha}=v^{\alpha}+\sum_{g\geq
1}\varepsilon^{2g}Q^{\alpha}_{g}(v_{x},v_{xx},\cdots,v^{(3g)}),\quad
Q_{g}^{\alpha}\in\mathcal{S}_{2g}$

such that the Poisson bracket $\\{-,-\\}_{a}$ is obtained from
$\\{-,-\\}_{a}^{[0]}$ by the above transformation. Here we denote

$\mathcal{S}=C^{\infty}(v)\left[v_{x},v_{xx},\cdots\right]\left[\frac{1}{u^{1}_{x}\cdots
u^{n}_{x}}\right].$

Note that $\mathcal{S}$ is also graded by the differential degree with
$\deg_{x}u^{i}_{x}=1$ and we denote by $\mathcal{S}_{d}$ the subspace consists
of homogeneous elements with differential degree $d$.

Due to the fact that the deformed bihamiltonian structure is tau-symmetric, it
follows from Theorem 3.8.24 of [10] that the functions $Q_{g}^{\alpha}$ in the
quasi-Miura transformation in (3.1) can be represented in the form

(3.2) $Q_{g}^{\alpha}=\eta^{\alpha\beta}\frac{\partial^{2}}{\partial
t^{\beta,0}\partial t^{1,0}}T_{g}(v,v_{x},\cdots,v^{(m_{g})}),\quad g\geq 2.$

To prove the main theorem of the present paper, we first need to show that
$T_{g}\in\mathcal{S}_{2g-2}$ and $m_{g}\leq 3g-2$.

Let us denote by $\mathcal{C}$ the space of smooth functions depending on the
jet variables $u,u_{x},u_{xx},\cdots$. It is clear that $\mathcal{S}$ is a
subspace of $\mathcal{C}$. The derivation $\partial_{x}$ can be viewed as a
linear operator on $\mathcal{C}$ by

$\partial_{x}=\sum_{i=1}^{n}\sum_{s\geq 0}u^{i,s+1}\frac{\partial}{\partial
u^{i,s}}.$

Consider the following operators $\delta_{i}$ of variational derivatives on
$\mathcal{C}$ defined by

$\delta_{i}=\sum_{s\geq 0}(-\partial_{x})^{s}\frac{\partial}{\partial
u^{i,s}},\quad i=1,\cdots,n.$

We have the following lemma.

######  Lemma 3.1.

The following properties for the operators $\delta_{i}$ hold true:

  1. (1)

For any function $f\in\mathcal{C}$, $\delta_{i}\left(\partial_{x}f\right)=0$.

  2. (2)

If an element $f\in\mathcal{S}_{\geq 2}$ satisfies $\delta_{i}f=0$, then there
exists $g\in\mathcal{S}$ such that $f=\partial_{x}g$.

###### Proof.

The first property is trivial, so we only need to prove the second one. Let us
denote by $\mathcal{S}^{(N)}$ the subspace of $\mathcal{S}$ consists of
elements that do not depend on $u^{i,k}$ for any $i=1,\cdots,n$ and $k>N$.
Without loss of generality, we can assume that $f\in\mathcal{S}_{d}$ for
$d\geq 2$ and that $f\in\mathcal{S}^{(N)}$ for a certain integer $N$.

It follows from the definition of the operator $\delta_{i}$ that

$\frac{\partial}{\partial
u^{j,2N}}(\delta_{i}f)=(-1)^{N}\frac{\partial}{\partial u^{j,N}}\frac{\partial
f}{\partial u^{i,N}}.$

So by using $\delta_{i}f=0$ we see that $f$ must have the form

(3.3) $f=\sum_{i=1}^{n}f_{i}u^{i,N}+g,\quad
g,f_{1},\cdots,f_{n}\in\mathcal{S}^{(N-1)}.$

From the above expression of the function $f$, we arrive at the following
identity

$0=\frac{\partial}{\partial
u^{j,2N-1}}(\delta_{i}f)=(-1)^{N}\left(\frac{\partial f_{i}}{\partial
u^{j,N-1}}-\frac{\partial f_{j}}{\partial u^{i,N-1}}\right).$

Therefore there exists $h\in\mathcal{C}$ such that

$f_{i}=\frac{\partial h}{\partial u^{i,N-1}},\quad i=1,\cdots,n.$

If $N\neq 2$, it is clear that $h$ can be chosen such that
$h\in\mathcal{S}^{(N-1)}$. If $N=2$, we can show that $h$ can be chosen to
have the form

$h=\sum_{i=1}^{n}a_{i}\log u^{i}_{x}+b,\quad a_{i}\in C^{\infty}(u),\
b\in\mathcal{S}^{(1)}.$

However by using the assumption $f\in\mathcal{S}_{\geq 2}$, it follows from
(3.3) that $f_{i}\in\mathcal{S}_{\geq 0}$ and therefore all smooth functions
$a_{i}$ must vanish. We conclude that we can always assume
$h\in\mathcal{S}^{(N-1)}$ and consequently
$f-\partial_{x}h\in\mathcal{S}^{(N-1)}$. Now the lemma can be proved by
induction on $N$. ∎

######  Proposition 3.2.

The functions $T_{g}$ given in (3.2) lie in the space
$\mathcal{S}_{2g-2}^{(3g-2)}$ for $g\geq 2$.

###### Proof.

It follows from (3.2) that
$\eta_{1\alpha}Q^{\alpha}_{g}=\partial_{x}^{2}T_{g}$ for $g\geq 2$. Therefore
by using Lemma 3.1 we know that
$\partial_{x}T_{g}\in\mathcal{S}_{2g-1}^{(3g-1)}$. By using this lemma again,
we arrive at the fact that $T_{g}\in\mathcal{S}_{2g-2}^{(3g-2)}$. The
proposition is proved. ∎

###  3.2. Proof of the main theorem

Let us fix a tau-symmetric deformation of the bihamiltonian structure
(2.4),(2.5) of the Principal Hierarchy associated with $M$. We denote by $Z$
the tau-function of the deformed integrable hierarchy determined by the
deformation of the bihamiltonian structure and by $w^{\alpha}$ the normal
coordinates of the deformed integrable hierarchy. Then the tau-cover of the
deformed integrable hierarchy can be written in the form

(3.4) $\varepsilon\frac{\partial\mathcal{F}}{\partial
t^{\alpha,p}}=f_{\alpha,p},\quad\varepsilon\frac{\partial
f_{\beta,q}}{\partial
t^{\alpha,p}}=\Omega_{\alpha,p;\beta,q},\quad\frac{\partial
w^{\beta}}{\partial
t^{\alpha,p}}=\eta^{\beta\lambda}\partial_{x}\Omega_{\lambda,0;\alpha,p}.$

According to Theorem 1.1, the deformed integrable hierarchy admits Virasoro
symmetries $\frac{\partial}{\partial s_{m}}$ whose actions on $Z$ are given by

(3.5) $\frac{\partial Z}{\partial s_{m}}=L_{m}Z+O_{m}Z,\quad m\geq-1,$

where $O_{m}$ are differential polynomials. It is proved in [10] that after
applying a Miura type transformation

$w^{\alpha}\mapsto\tilde{w}^{\alpha}=w^{\alpha}+\sum_{g\geq
1}\varepsilon^{2g}A^{\alpha}_{g},$

where $A^{\alpha}_{g}$ are differential polynomials in $w^{1},\cdots,w^{n}$ of
differential degree $2g$, the tau-cover of the deformed integrable hierarchy
changes accordingly:

(3.6)
$\mathcal{\tilde{F}}=\mathcal{F}+G,\quad\tilde{f}_{\alpha,p}=f_{\alpha,p}+\varepsilon\frac{\partial
G}{\partial
t^{\alpha,p}},\quad\tilde{w}^{\alpha}=w^{\alpha}+\varepsilon^{2}\eta^{\alpha\beta}\frac{\partial^{2}G}{\partial
t^{\beta,0}\partial t^{1,0}},$

where $G=\sum_{g\geq 1}\varepsilon^{2g-2}G_{g}$, and $G_{g}$ are homogeneous
differential polynomials of differential degree $2g-2$. Here
$\mathcal{\tilde{F}}$, $\tilde{f}_{\alpha,p}$ and $\tilde{w}^{\alpha}$ are the
free energy, one-point functions and normal coordinates of the tau-cover of
the deformed integrable hierarchy after applying the Miura type
transformation. Conversely, any $G=\sum_{g\geq 1}\varepsilon^{2g-2}G_{g}$
defines a Miura type transformation and changes the tau-cover of the deformed
integrable hierarchy according to (3.6). Let us first study how the Miura type
transformations affect the Virasoro symmetries (3.5).

After applying a Miura type transformation given by (3.6), the tau-function
$Z$ is changed to $\tilde{Z}=Z\exp(G)$. Then the actions of the Virasoro
symmetries (3.5) on $\tilde{Z}$ are represented in the form

$\frac{\partial\tilde{Z}}{\partial
s_{m}}=L_{m}\tilde{Z}+\tilde{O}_{m}\tilde{Z},\quad m\geq-1,$

where $\tilde{O}_{m}$ is given by

$\displaystyle\tilde{O}_{m}=$ $\displaystyle\,O_{m}-\sum
a_{m}^{\alpha,p;\beta,q}\left(2\varepsilon f_{\alpha,p}\frac{\partial
G}{\partial t^{\beta,q}}+\varepsilon^{2}\frac{\partial G}{\partial
t^{\alpha,p}}\frac{\partial G}{\partial
t^{\beta,q}}+\varepsilon^{2}\frac{\partial^{2}G}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)$ $\displaystyle+\frac{\partial G}{\partial
s_{m}}-\sum{b}_{m;\alpha,p}^{\beta,q}t^{\alpha,p}\frac{\partial G}{\partial
t^{\beta,q}},\quad m\geq-1.$

It follows from (3.4) and (3.5) that for any differential polynomial $Q$ in
$w^{1},\cdots,w^{n}$, the actions of the Virasoro symmetries on $Q$ are given
by

$\frac{\partial Q}{\partial s_{m}}=2\varepsilon\sum
a_{m}^{\alpha,p;\beta,q}f_{\alpha,p}\frac{\partial Q}{\partial
t^{\beta,q}}+X_{m}+\sum{b}_{m;\alpha,p}^{\beta,q}t^{\alpha,p}\frac{\partial
Q}{\partial t^{\beta,q}},\quad m\geq-1,$

where $X_{m}$ is a certain differential polynomial determined by $L_{m}$ and
$Q$. For this reason it is easy to see that $\tilde{O}_{m}$ is also a
differential polynomial. Hence to linearize the Virasoro symmetries is
equivalent to solve the following equations for a differential polynomial $G$:

(3.7) $\displaystyle\frac{\partial G}{\partial s_{m}}=$ $\displaystyle\,\sum
a_{m}^{\alpha,p;\beta,q}\left(2\varepsilon f_{\alpha,p}\frac{\partial
G}{\partial t^{\beta,q}}+\varepsilon^{2}\frac{\partial G}{\partial
t^{\alpha,p}}\frac{\partial G}{\partial
t^{\beta,q}}+\varepsilon^{2}\frac{\partial^{2}G}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)$
$\displaystyle+\sum{b}_{m;\alpha,p}^{\beta,q}t^{\alpha,p}\frac{\partial
G}{\partial t^{\beta,q}}-O_{m},\quad m\geq-1.$

Let us make the expansion

$G=\sum_{g\geq 1}\varepsilon^{2g-2}G_{g},\quad O_{m}=\sum_{g\geq
1}\varepsilon^{2g-2}O_{m;g},$

where $G_{g}$ and $O_{m;g}$ are differential polynomials of differential
degree $2g-2$. Then by comparing the leading terms of the equations (3.7), we
obtain the following equations for $G_{1}$:

$\mathcal{D}_{m}(G_{1})=O_{m,1},\quad m\geq-1,$

where $\mathcal{D}_{m}$ is the operator defined in (2.10). Once such a
differential polynomial $G_{1}$ is found, we can apply the Miura type
transformation

$\mathcal{F}\mapsto\mathcal{F}+G_{1},\quad{f}_{\alpha,p}\mapsto
f_{\alpha,p}+\varepsilon\frac{\partial G_{1}}{\partial
t^{\alpha,p}},\quad{w}^{\alpha}\mapsto
w^{\alpha}+\varepsilon^{2}\eta^{\alpha\beta}\frac{\partial^{2}G_{1}}{\partial
t^{\beta,0}\partial t^{1,0}}$

to the tau-cover of the deformation of the Principal Hierarchy. If we still
denote by (3.5) the Virasoro symmetries of the deformed integrable hierarchy
after the above Miura type transformation, then we see that $O_{m}=\sum_{g\geq
2}\varepsilon^{2g-2}O_{m;g}$. In order to eliminate $O_{m;2}$ from the
representations of the Virasoro symmetries, we need to find a differential
polynomial $G_{2}$ such that

$\mathcal{D}_{m}(G_{2})=O_{m,2},\quad m\geq-1$

and perform a Miura type transformation

$\mathcal{F}\mapsto\mathcal{F}+\varepsilon^{2}G_{2},\quad{f}_{\alpha,p}\mapsto
f_{\alpha,p}+\varepsilon^{3}\frac{\partial G_{2}}{\partial
t^{\alpha,p}},\quad{w}^{\alpha}\mapsto
w^{\alpha}+\varepsilon^{4}\eta^{\alpha\beta}\frac{\partial^{2}G_{2}}{\partial
t^{\beta,0}\partial t^{1,0}}.$

By continuing this procedure recursively, we are able to prove the main
theorem. Before turning to the actual proof, we first make some preparations.

In what follows, we will use $u^{i}=u^{i}(w)$ to denote the canonical
coordinates of the semisimple Frobenius manifold $M$. Let us recall some basic
facts related to the canonical coordinates following [13, 8]. In terms of
$u^{1},\cdots,u^{n}$, the flat metric $\eta$ of $M$ has the expression

$\eta=\sum_{i=1}^{n}f_{i}(u)(du^{i})^{2},\quad f_{i}(u)\neq 0.$

We denote by $\psi_{i1}=\sqrt{f_{i}}$ the Lamé coefficients, and by
$\gamma_{ij}=(\psi_{i1})^{-1}\partial_{i}\psi_{j1}$ the rotation coefficients
for $i\neq j$. Following [13] we say that $M$ is reducible if there exists a
partition of the set $\\{1,2,\cdots,n\\}$ into two nonempty disjoint subsets
$I,J$ such that

$\gamma_{ij}=0,\quad\forall\,i\in I,\forall\,j\in J,$

and say that $M$ is irreducible if it is not reducible. Note that if the
dimension $n$ of $M$ satisfies $n\geq 2$, the irreducibility of $M$ implies
that for each $i$ there exists $j\neq i$ such that $\gamma_{ij}\neq 0$.

We define a strict partial order $\prec$ on the space of homogeneous
differential polynomials $\mathcal{A}_{d}$ with differential degree $d\geq 1$.
For any $W\in\mathcal{A}_{d}$, we can represent it in the form

$W=\sum W_{\mu_{1},\cdots,\mu_{n}}(u)u^{1,(\mu_{1})}\cdots u^{n,(\mu_{n})},$

where each $\mu_{i}=(\mu_{i,1},\mu_{i,2},\cdots)$ is a partition of a non-
negative integer $|\mu_{i}|$ and

$u^{i,(\mu_{i})}=u^{i,\mu_{i,1}}u^{i,\mu_{i,2}}\cdots.$

Note that $|\mu_{1}|+\cdots+|\mu_{n}|=d$, hence after reordering
$(\mu_{1},\cdots,\mu_{n})$ we obtain a partition of $d$. Then the relation
$\prec$ on $\mathcal{A}_{d}$ is induced by the total order on the space of all
partitions of $d$. To illustrate this definition, we consider the partial
order on $\mathcal{A}_{4}$ for a $2$-dimensional Frobenius manifold. For
example, we have the relations

$(u^{2,1})^{3}u^{1,1}\prec(u^{1,1})^{2}u^{2,2}\prec u^{1,3}u^{2,1}\prec
u^{2,4},$

which correspond to the following relations of the partitions of $4$:

$(1,1,1,1)\prec(2,1,1)\prec(3,1)\prec(4).$

There are also some monomials that are not comparable with respect to the
above partial order, for example $u^{1,3}u^{2,1}$ and $u^{1,1}u^{2,3}$ are not
comparable.

######  Lemma 3.3.

For $k\geq 1$, we have

$\mathcal{D}(\lambda)u^{i,k}=-\left(1+\frac{k}{2}\right)\frac{u^{i,k}}{(u^{i}-\lambda)^{2}}-ku^{i,k}\sum_{j\neq
i}\frac{\psi_{j1}}{\psi_{i1}}\gamma_{ij}\left(\frac{1}{u^{j}-\lambda}-\frac{1}{u^{i}-\lambda}\right)+R_{i,k},$

here $R_{i,k}$ consists of terms smaller than $u^{i,k}$ with respect to the
order $\prec$. In particular, $R_{i,1}=0$.

###### Proof.

In terms of the canonical coordinates, we can write $\mathcal{D}(\lambda)$ in
the form

$\mathcal{D}(\lambda)=\sum_{i=1}^{n}\sum_{r\geq
0}\partial_{x}^{r}\left(\frac{1}{u^{i}-\lambda}\right)\frac{\partial}{\partial
u^{i,r}}+\sum_{i=1}^{n}\sum_{r\geq 1}B_{i,r}\frac{\partial}{\partial
u^{i,r}}.$

It is proved in [13] that the functions $B_{i,r}$ satisfy the relations

$\displaystyle B_{i,1}$
$\displaystyle=-\frac{1}{2}\frac{u^{i,1}}{(u^{i}-\lambda)^{2}}-u^{i,1}\sum_{j\neq
i}\frac{\psi_{j1}}{\psi_{i1}}\gamma_{ij}\left(\frac{1}{u^{j}-\lambda}-\frac{1}{u^{i}-\lambda}\right)$
$\displaystyle=\frac{\partial u^{i}}{\partial
v^{\zeta}}\eta^{\zeta\lambda}\partial_{\rho}\partial_{\lambda}p_{\alpha}G^{\alpha\beta}\frac{\partial
p_{\beta}}{\partial v^{1}}v^{\rho,1},$ $\displaystyle B_{i,r}$
$\displaystyle=\partial_{x}B_{i,r-1}+\partial_{x}^{r-1}\left(\frac{\partial
u^{i}}{\partial
v^{\zeta}}\eta^{\zeta\lambda}\partial_{\rho}\partial_{\lambda}p_{\alpha}v^{\rho,1}\right)G^{\alpha\beta}\frac{\partial
p_{\beta}}{\partial v^{1}},\quad r\geq 2.$

The lemma is proved by a straightforward computation. ∎

######  Lemma 3.4 ( Lemma 4.2.2 of [13]).

Let $F\in\mathcal{S}^{(N)}$, then $\mathcal{D}(\lambda)F$ is a rational
function in $\lambda$ with poles at most of order $N+1$ at each
$\lambda=u^{i}$ and it is regular at $\lambda=\infty$. The coefficients of
$(\lambda-u^{i})^{-N-1}$ are given by $-C_{N}(u^{i,1})^{N}\frac{\partial
F}{\partial u^{i,N}}$ where $C_{N}$ are positive numbers given by

$C_{N}=N!+2^{-N}\sum_{l=1}^{N}\binom{N}{l}(2l-3)!!(2N-2l+1)!!.$

######  Proposition 3.5.

Let $M$ be a semisimple irreducible Frobenius manifold of dimension $n\geq 2$,
and $F\in\mathcal{S}_{d}$ with $d\geq 1$ satisfy the identity

$\mathcal{D}(\lambda)F=H$

for a certain $H\in\mathcal{A}_{d}$, then we have $F\in\mathcal{A}_{d}$.

###### Proof.

Let us assume that $F\in\mathcal{S}^{(N)}$, $H\in\mathcal{A}^{(N)}$ for a
certain positive integer $N$, here
$\mathcal{A}^{(N)}=\mathcal{A}\cap\mathcal{S}^{(N)}$. Then we can represent
$F$ in the following form:

$F=\sum_{(k_{1},\cdots,k_{n})\neq
0}W_{k_{1},\cdots,k_{n}}(u^{1,N})^{k_{1}}\cdots(u^{n,N})^{k_{n}}+R,\quad
W_{k_{1},\cdots,k_{n}},R\in\mathcal{S}^{(N-1)}.$

By using Lemma 3.4 we know that the coefficients of $(\lambda-u^{i})^{-N-1}$
in $\mathcal{D}(\lambda)F$ are given by

(3.8) $\sum_{(k_{1},\cdots,k_{n})\neq
0}-k_{i}C_{N}(u^{i,1})^{N}W_{k_{1},\cdots,k_{n}}(u^{1,N})^{k_{1}}\cdots(u^{i,N})^{k_{i}-1}\cdots(u^{n,N})^{k_{n}}.$

If $N=1$, then for each term
$W_{k_{1},\cdots,k_{n}}(u)(u^{1,1})^{k_{1}}\cdots(u^{n,1})^{k_{n}}$ with
$k_{i}\neq 0$ for some $i$, we must have $k_{i}>0$ due to the assumption that
$\mathcal{D}(\lambda)F$ is a differential polynomial. Therefore we conclude
that $F\in\mathcal{A}$, hence in what follows we assume $N\geq 2$.

It follows from (3.8) that

$k_{i}C_{N}(u^{i,1})^{N}W_{k_{1},\cdots,k_{n}}\in\mathcal{A},\quad
i=1,\cdots,n.$

So it is easy to see that for a term
$W_{k_{1},\cdots,k_{n}}(u^{1,N})^{k_{1}}\cdots(u^{n,N})^{k_{n}}$, if there
exists $i\neq j$ such that $k_{i}\neq 0$ and $k_{j}\neq 0$, then we must have
$W_{k_{1},\cdots,k_{n}}\in\mathcal{A}$. In another word, we can uniquely
represent $F$ in the following form:

$F=R+P+\sum_{i=1}^{n}\sum_{r}V^{i}_{r}\frac{(u^{i,N})^{k_{i,r}}}{(u^{i,1})^{l_{i,r}}},\quad
R\in\mathcal{S}^{(N-1)},P\in\mathcal{A}^{(N)},V^{i}_{r}\in\mathcal{A}^{(N-1)},$

here $V^{i}_{r}$ do not depend on $u^{i,1}$ and $l_{i,r},k_{i,r}>0$. Let us
proceed to show that

$\frac{\partial V^{i}_{r}}{\partial u^{j,s}}=0,\quad j\neq i,\ s\geq 0.$

For each $j\neq i$, we denote by $s$ the maximal integer such that
$\frac{\partial V^{i}_{r}}{\partial u^{j,s}}\neq 0$. Let us consider the
coefficients of

(3.9)
$\frac{1}{(\lambda-u^{j})^{s+1}}\frac{(u^{i,N})^{k_{i,r}}}{(u^{i,1})^{l_{i,r}}}$

in the expression of $\mathcal{D}(\lambda)F$. It is clear that both $R$ and
$P$ give no contributions to this term and we see, by using Lemma 3.4 again,
that the coefficient of (3.9) is given by

$-\frac{\partial V^{i}_{r}}{\partial u^{j,s}}C_{s}(u^{j,1})^{s}.$

This contradicts the fact that $\mathcal{D}(\lambda)F$ is a differential
polynomial and therefore $\frac{\partial V^{i}_{r}}{\partial u^{j,s}}=0$ for
any $j\neq i$ and $s\geq 0$. So we can uniquely represent $F$ in the form

$F=R+P+\sum_{i=1}^{n}\sum_{l_{1};k_{2},\cdots,k_{N}}G^{i}_{k_{1},\cdots,k_{N}}(u^{i})\frac{(u^{i,2})^{k_{2}}\cdots(u^{i,N})^{k_{N}}}{(u^{i,1})^{l_{1}}},\quad
l_{1},k_{N}>0.$

For each index $i$, let us pick up a nonzero term

$G^{i}_{k_{1},\cdots,k_{N}}(u^{i})\frac{(u^{i,2})^{k_{2}}\cdots(u^{i,N})^{k_{N}}}{(u^{i,1})^{l_{1}}}$

such that $l_{1}$ is maximal and its numerator is maximal with respect to
$\prec$. Then we can apply Lemma 3.3 and compute the coefficients of

$\frac{1}{\lambda-u^{j}}\frac{(u^{i,2})^{k_{2}}\cdots(u^{i,N})^{k_{N}}}{(u^{i,1})^{l_{1}}},\quad
j\neq i$

in the expression of $\mathcal{D}(\lambda)F$, which are given by

$G^{i}_{k_{1},\cdots,k_{N}}(2k_{2}+3k_{3}\cdots+Nk_{N}-l_{1})\frac{\psi_{j1}}{\psi_{i1}}\gamma_{ij},\quad
j\neq i.$

Due to the assumption that $F\in\mathcal{S}_{d}$, we see that
$2k_{2}+\cdots+Nk_{N}-l_{1}=d\geq 1$ and it follows from the irreducibility of
$M$ that we can choose $j\neq i$ such that $\gamma_{ij}\neq 0$. Therefore by
using $\mathcal{D}(\lambda)F\in\mathcal{A}$, we conclude that
$G^{i}_{k_{1},\cdots,k_{N}}=0$, which contradicts to our choice. Hence we
conclude that $F$ must have the form

$F=R+P,\quad R\in\mathcal{S}^{(N-1)},P\in\mathcal{A}.$

We then arrive at

$\mathcal{D}(\lambda)(R)=H-\mathcal{D}(\lambda)(P)\in\mathcal{A},\quad
R\in\mathcal{S}^{(N-1)}.$

Then we can finish the proof of the proposition by doing induction on $N$. ∎

Now we are ready to prove the main theorem.

###### Proof of the Main Theorem.

The ‘only if’ part is proved by combining the results of [11, 13, 25]. In
fact, due to the linearization of the Virasoro symmetries, it follows from
Theorem 2.1 that the quasi-Miura transformation (3.1) must be given by

$Q_{g}^{\alpha}=\eta^{\alpha\beta}\frac{\partial^{2}F_{g}}{\partial
t^{\beta,0}\partial t^{1,0}},\quad g\geq 1,$

where $F_{g}$ are described as in Theorem 2.1. Thus the explicit formula of
the genus one component of the deformed bihamiltonian structure is given by
[11] and its central invariants are all equal to $\frac{1}{24}$ [25].

Let us turn to prove the ‘if’ part. If the dimension $n$ of the semisimple
Frobenius manifold $M$ is equal to $1$, then it is well known that the
corresponding Principal Hierarchy is the dispersionless KdV hierarchy. It
admits a deformation given by the deformed bihamiltonian structure

$\displaystyle\\{w(x),w(y)\\}_{1}=\delta^{\prime}(x-y),$
$\displaystyle\\{w(x),w(y)\\}_{2}=w(x)\delta^{\prime}(x-y)+\frac{1}{2}w^{\prime}(x)\delta(x-y)+\frac{\varepsilon^{2}}{8}\delta^{\prime\prime\prime}(x-y).$

This particular deformation determines an integrable hierarchy with linearized
Virasoro symmetries [13, 17, 24]. Hence in what follows we assume the
dimension $n$ of the semisimple Frobenius manifold $M$ satisfies $n\geq 2$ and
without loss of generality, we assume $M$ is irreducible.

Recall that the central invariants completely determine the equivalent class
of deformations of a semisimple bihamiltonian structure of hydrodynamic type
under Miura type transformation. Therefore if the central invariants of the
deformed bihamiltonian structure are all equal to $\frac{1}{24}$, then there
exists a Miura type transformation such that, after applying this
transformation, the genus one component of the deformed bihamiltonian
structure coincides with the one given in [11]. This Miura type transformation
cancels $O_{m,1}$ due to the result of [13, 12]. Now let us do indunction and
assume that we have canceled $O_{m,k}$ for all $m\geq-1$ and $k\leq g$ for a
certain $g\geq 1$ in the representations of the Virasoro symmetries (3.5).
Then we are going to solve $G_{g+1}$ such that

(3.10) $\mathcal{D}(\lambda)G_{g+1}=O_{g+1}(\lambda),\quad
O_{g+1}(\lambda)=\sum_{\lambda\geq-1}\frac{1}{\lambda^{m+2}}O_{m;g+1}.$

It follows from Proposition 3.2 that the quasi-Miura transformation of the
deformed bihamiltonian structure can be represented by

$v^{\alpha}\mapsto
w^{\alpha}=v^{\alpha}+\varepsilon^{2}Q^{\alpha}_{1}+\sum_{k\geq
2}\varepsilon^{2k}\eta^{\alpha\beta}\frac{\partial^{2}}{\partial
t^{\beta,0}\partial
t^{1,0}}T_{k}\bigl{(}v,v_{x},\cdots,v^{(3g-2)}\bigr{)},\quad
T_{k}\in\mathcal{S}_{2k-2}^{(3k-2)}.$

Due to the induction hypothesis and the uniqueness of the quasi-Miura
transformation, we see that

$Q_{1}^{\alpha}=\eta^{\alpha\beta}\frac{\partial^{2}F_{1}^{DZ}}{\partial
t^{\beta,0}\partial t^{1,0}},\quad T_{k}=F_{k}^{DZ},\quad k\leq g,$

here we denote by $F^{DZ}_{k}$ the solution of the loop equation (2.13) given
by Theorem 2.1.

On the other hand, from the tau-structure (3.4) of the deformed integrable
hierarchy it follows that the genus expansion of the free energy
$\mathcal{F}=\varepsilon^{-2}\mathcal{F}_{0}+\sum_{k\geq
1}\varepsilon^{2k-2}F_{k}$ satisfies the identities

$F_{1}=F_{1}^{DZ},\quad F_{k}=T_{k},\quad k\geq 2.$

Similar to the the derivation of the loop equation given in [13], we obtain
from the expressions of the Virasoro symmetries

$\frac{\partial Z}{\partial s_{m}}=L_{m}Z+O_{m}Z$

and the genus zero Virasoro symmetries (2.6) that the function $F_{g+1}$
satisfies the equation

$\displaystyle\mathcal{D}(\lambda)F_{g+1}=$
$\displaystyle\,\frac{1}{2}\left(\sum_{k=1}^{g-1}\frac{\partial
F_{k}}{\partial v^{\gamma,k}}\frac{\partial F_{g-k}}{\partial
v^{\rho,l}}+\frac{\partial^{2}F_{g}}{\partial v^{\gamma,k}\partial
v^{\rho,l}}\right)\partial_{x}^{k+1}\partial^{\gamma}p_{\alpha}G^{\alpha\beta}\partial_{x}^{l+1}\partial^{\rho}p_{\beta}$
$\displaystyle+\frac{1}{2}\sum\frac{\partial F_{g}}{\partial
v^{\gamma,k}}\partial_{x}^{k+1}\left[\nabla\frac{\partial
p_{\alpha}}{\partial\lambda}\cdot\nabla\frac{\partial
p_{\beta}}{\partial\lambda}\cdot
v_{x}\right]^{\gamma}G^{\alpha\beta}-O_{g+1}(\lambda).$

Recall that the loop equation (2.13) satisfied by $F_{g+1}^{DZ}$ is given by

$\displaystyle\mathcal{D}(\lambda)F_{g+1}^{DZ}=$
$\displaystyle\,\frac{1}{2}\left(\sum_{k=1}^{g-1}\frac{\partial
F_{k}^{DZ}}{\partial v^{\gamma,k}}\frac{\partial F_{g-k}^{DZ}}{\partial
v^{\rho,l}}+\frac{\partial^{2}F_{g}^{DZ}}{\partial v^{\gamma,k}\partial
v^{\rho,l}}\right)\partial_{x}^{k+1}\partial^{\gamma}p_{\alpha}G^{\alpha\beta}\partial_{x}^{l+1}\partial^{\rho}p_{\beta}$
$\displaystyle+\frac{1}{2}\sum\frac{\partial F_{g}^{DZ}}{\partial
v^{\gamma,k}}\partial_{x}^{k+1}\left[\nabla\frac{\partial
p_{\alpha}}{\partial\lambda}\cdot\nabla\frac{\partial
p_{\beta}}{\partial\lambda}\cdot v_{x}\right]^{\gamma}G^{\alpha\beta}.$

Therefore it follows from the induction hypothesis that equation (3.10) has a
solution

$G_{g+1}=F_{g+1}^{DZ}-F_{g+1}\in\mathcal{S}_{2g}$

such that

$\mathcal{D}(\lambda)G_{g+1}=O_{g+1}.$

Note that $O_{g+1}(\lambda)\in\mathcal{A}_{2g}$ and $g\geq 1$, then by
applying Proposition 3.5 we see that $G_{g+1}$ is also a differential
polynomial. The theorem is proved. ∎

##  4\. Conclusion

In [13], a program for classifying the integrable hierarchies of topological
type was proposed, the main goal of which is to describe a 2DTFT completely
via an integrable hierarchy. These integrable hierarchies should satisfy the
following four axioms:

Axiom BH:

The integrable hierarchy possesses a bihamiltonian structure with
dispsersionless limit of hydrodynamic type.

Axiom QT:

The integrable hierarchy is quasi-trivial, meaning that there is a quasi-Miura
transformation that changes the integrable hierarchy to its dispersionless
limit.

Axiom TS:

The integrable hierarchy possesses a tau structure.

Axiom LVS:

The integrable hierarchy possesses Virasoro symmetries which are given by
linear actions of Virasoro operators on the tau function.

Such an integrable hierarchy is called of topological type. The central
problem regarding this classification program is the existence and uniqueness
of such kinds of integrable hierarchies. In [13], it is proved that starting
from the axiom BH and axiom TS, the dispersionless limit of such an integrable
hierarchy corresponds to a Frobenius manifold or a degenerate one. On the
other hand, for a given caliberated semisimple Frobenius manifold, there
exists a unique deformation of its Principal Hierarchy satisfies all four
axioms, which is just the DZ hierarchy as it is constructed in [13]. The
remaining work to be finished is to prove that the bihamiltonian structure of
DZ hierarchy can be represented in terms of differential polynomials. In [4,
5], Buryak, Posthuma and Shadrin proved that the first Hamiltonian structure
of the DZ hierarchy can be represented by differential polynomials by
considering the infinitesimal transformation of Givental’s group action [15,
18]. In the present paper we prove the polynomiality property of the
bihamiltonian structure of the DZ hierarchy by using the super tau-covers of
integrable hierarchies[20] and the associated variational bihamiltonian
cohomologies[21, 19].

Our strategy is to study the relations among these four axioms. The relations
among the first three were studied in the papers [9, 10, 14, 22]. Under the
semisimplicity condition, axiom QT can be derived from axiom BH, and axiom TS
can also be derived form axiom BH by requiring that the bihamiltonian
structure is flat exact[10]. The present paper together with the work [19, 20,
21] gives a clear description of the relation between axiom BH and axiom LVS,
that is, the axiom LVS can also be derived from axiom BH by requiring that the
central invariants of the flat exact bihamiltonian structure are all equal to
$\frac{1}{24}$. To conclude, we have the following theorem.

######  Theorem 4.1.

For a given semisimple Frobenius manifold, a bihamiltonian deformation of its
Principal Hierarchy is equivalent to the DZ hierarchy under Miura type
transformations if and only if the central invariants of the deformed
bihamiltonian structure are all equal to $\frac{1}{24}$.

A possible application of this theorem is to study the so-called DR/DZ
equivalence conjecture in the semisimple setting. In [2], Buryak constructed
an integrable hierarchy, called the double ramification (DR) hierarchy,
starting from any cohomological field theory (CohFT). He showed that the genus
zero part of the DR hierarchy coincides with the Principal Hierarchy of the
Frobenius manifold given by the genus zero part of the CohFT. It was
conjectured that the DR hierarchy is equivalent to the DZ hierarchy for any
semisimple CohFT under Miura type transformations, and this is called the
DR/DZ equivalence conjecture. Although the conjecture is still open, there
have been many evidences and verifications for this conjecture. In the paper
[6], a conjectural formula was given for the bihamiltonian structure of the DR
hierarchy and under the assumption of the correctness of their conjectural
formula, the central invariants were computed and they are all equal to
$\frac{1}{24}$. In the papers [1, 3], the DR/DZ equivalence conjecture was
proved up to genus one approximation and in particular the conjectural formula
for the bihamiltonian structure of the DR hierarchy in [6] is correct up to
genus one. Therefore by applying the above theorem, the DR/DZ equivalence
conjecture can be proved in the semisimple setting by confirming the
correctness of the conjectural formula for the bihamiltonian structure of the
DR hierarchy.

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Si-Qi Liu,

Department of Mathematical Sciences, Tsinghua University
Beijing 100084, P.R. China
<EMAIL_ADDRESS>
Zhe Wang,

Department of Mathematical Sciences, Tsinghua University
Beijing 100084, P.R. China
<EMAIL_ADDRESS>
Youjin Zhang,

Department of Mathematical Sciences, Tsinghua University
Beijing 100084, P.R. China
<EMAIL_ADDRESS>