Variational Bihamiltonian Cohomologies and Integrable Hierarchies II: Virasoro Symmetries

Abstract

We prove that for any tau-symmetric bihamiltonian deformation of the tau-cover of the Principal Hierarchy associated with a semisimple Frobenius manifold, the deformed tau-cover admits an infinite set of Virasoro symmetries.

1 Introduction

This is the second one of the series of papers devoted to the study of deformations of the Virasoro symmetries of bihamiltonian integrable hierarchies. In the first one [23], we developed a cohomology theory on the space of differential forms of the infinite jet space of a super manifold for a given bihamiltonian structure of hydrodynamic type, and we call such a cohomology theory the variational bihamiltonian cohomology. It can be viewed as a generalization of the bihamiltonian cohomology introduced in [12], and it provides a suitable tool for us to study deformations of the Virasoro symmetries of bihamiltonian integrable hierarchies.

The purpose of the present paper is to prove the following theorem.

Theorem 1

(Main Theorem). For a given tau-symmetric bihamiltonian deformation of the Principal Hierarchy associated with a semisimple Frobenius manifold, there exists a unique deformation of the Virasoro symmetries of the tau-cover of the Principal Hierarchy such that they are symmetries of the tau-cover of the deformed integrable hierarchy. Moreover, the action of the Virasoro symmetries on the tau-function $\mathcal {Z}$ of the deformed integrable hierarchy can be represented in the form

$$\begin{aligned} \frac{\partial \mathcal Z}{\partial s_m} = L_m\mathcal Z+O_m\mathcal Z,\quad m\ge -1, \end{aligned}$$

(1.1)

where $L_m$ are the Virasoro operators constructed in [11] and $O_m$ are some differential polynomials, and the flows $\frac{\partial }{\partial s_m}$ satisfy the Virasoro commutation relations

$$\begin{aligned} \left[ \frac{\partial }{\partial s_k}, \frac{\partial }{\partial s_l}\right] = (l-k)\frac{\partial }{\partial s_{k+l}},\quad k,l\ge -1. \end{aligned}$$

Let us briefly explain the basic idea for proving this theorem. Consider the following system of evolutionary PDEs with time variable t and spacial variable x:

$$\begin{aligned} \frac{\partial u^i}{\partial t} = A^i_j(u)u^j_x+\varepsilon \left( B^i_j(u)u^j_{xx}+C^i_{jk}(u)u^j_xu^k_x\right) +\dots ,\quad i = 1,\dots , n. \end{aligned}$$

(1.2)

We assume that this system is bihamiltonian with respect to the bihamiltonian structure $(P_0,P_1)$ whose leading term is semisimple. We can associate with it a super extension by introducing odd unknown functions $\theta _i$ for $i = 1,\dots ,n$, and by adding odd flows $\frac{\partial }{\partial \tau _0}$ and $\frac{\partial }{\partial \tau _1}$ which correspond respectively to the Hamiltonian structure $P_0$ and $P_1$ (see [22] and Sect. 2.3 given below for details). Thus the super extension of (1.2) consists of the flows $\frac{\partial }{\partial t}$, $\frac{\partial }{\partial \tau _0}$ and $\frac{\partial }{\partial \tau _1}$ for the unknown functions $u^i$ and $\theta _i$, and the fact that the system (1.2) is bihamiltonian with respect to $(P_0,P_1)$ is equivalent to the following commutation relations:

$$\begin{aligned} \left[ \frac{\partial }{\partial t}, \frac{\partial }{\partial \tau _i}\right] = 0,\quad \left[ \frac{\partial }{\partial \tau _i}, \frac{\partial }{\partial \tau _j}\right] = 0,\quad i,j = 0,1. \end{aligned}$$

Example 1

Consider the following Korteweg-de Vries (KdV) equation:

$$\begin{aligned} \frac{\partial u}{\partial t} = uu_x+\frac{\varepsilon ^2}{12}u_{xxx}. \end{aligned}$$

It admits a bihamiltonian structure given by the following Poisson brackets:

$$\begin{aligned} \{u(x),u(y)\}_0&= \delta '(x-y),\\\{u(x),u(y)\}_1&=u(x) \delta '(x-y)+\frac{u_x}{2}\delta (x-y)+\frac{\varepsilon ^2}{8}\delta '''(x-y). \end{aligned}$$

We introduce an odd unknown function $\theta$, and construct the following super extension of the KdV equation:

$$\begin{aligned} \frac{\partial u}{\partial t}&= uu_x+\frac{\varepsilon ^2}{12}u_{xxx},\quad \frac{\partial \theta }{\partial t} = u\theta _x+\frac{\varepsilon ^2}{12}\theta _{xxx}, \end{aligned}$$

(1.3)

$$\begin{aligned} \frac{\partial u}{\partial \tau _0}&= \theta _x,\quad \frac{\partial u}{\partial \tau _1} = u\theta _x+\frac{1}{2}u_x\theta +\frac{\varepsilon ^2}{8}\theta _{xxx},\end{aligned}$$

(1.4)

$$\begin{aligned} \frac{\partial \theta }{\partial \tau _0}&=0,\quad \frac{\partial \theta }{\partial \tau _1}=\frac{1}{2} \theta \theta _x. \end{aligned}$$

(1.5)

It is easy to check directly that the flows in the extended system mutually commute.

Remark 1

The flow (1.3) also appeared in [1].

According to the theory of bihamiltonian cohomology developed in [7], we know that if there is another system of evolutionary PDEs given by the flow $\frac{\partial }{\partial {\hat{t}}}$ satisfying the commutation relation

$$\begin{aligned} \left[ \frac{\partial }{\partial {\hat{t}}}, \frac{\partial }{\partial \tau _0}\right] = \left[ \frac{\partial }{\partial {\hat{t}}}, \frac{\partial }{\partial \tau _1}\right] =0, \end{aligned}$$

(1.6)

then it gives a symmetry of the system (1.2), i.e.,

$$\begin{aligned} \left[ \frac{\partial }{\partial t}, \frac{\partial }{\partial {\hat{t}}}\right] = 0. \end{aligned}$$

We note that the above results is for a flow $\frac{\partial }{\partial {\hat{t}}}$ given by differential polynomials. In order to use the above results to consider symmetries of more general forms such as Virasoro symmetries, we introduce the notion of super tau-cover of a bihamiltonian integrable hierarchy and develop the theory of variational bihamiltonian cohomology [23]. Then we are able to consider the commutation relation between the Virasoro symmetries $\frac{\partial }{\partial s_m}$ and the odd flows $\frac{\partial }{\partial \tau _0}$ and $\frac{\partial }{\partial \tau _1}$. By applying the results of variational bihamiltonian cohomology proved in [23], we manage to prove the main theorem. A more detailed description of the idea for proving this theorem is given at the end of Sect. 3.2 and in Sect. 3.3.

We organize this paper as follows. In Sect. 2, we construct the super tau-cover of a given tau-symmetric bihamiltonian deformation of the Principal Hierarchy associated with a semisimple Frobenius manifold. This construction builds a bridge which relates Virasoro symmetries to bihamiltonian structures. In Sect. 3, we explain how the problem of deformations of the Virasoro symmetries can be solved via the theory of the variational bihamiltonian cohomology. In Sect. 4 we give the proof of the main theorem. Finally in Sect. 5, we make some concluding remarks.

2 Super Tau-Covers of Bihamiltonian Integrable Hierarchies

2.1 Bihamiltonian structures on infinite jet spaces

Let us start by recalling the basic construction of bihamiltonian structures as local functionals on infinite jet spaces. One may refer to [20] for a detailed introduction to this topic.

Let M be a smooth manifold of dimension n and ${\hat{M}}$ be the super manifold of dimension (n|n) obtained by reversing the parity of the fibers of the cotangent bundle of M. In another word, if we choose a local canonical coordinate system $(u^1,\dots ,u^n;\theta _1,\dots ,\theta _n)$ on $T^*M$, then ${\hat{M}}$ can be described locally by the same chart while regarding the fiber coordinates as odd variables:

$$\begin{aligned} \theta _i\theta _j+\theta _j\theta _i = 0,\quad i,j=1,\dots ,n. \end{aligned}$$

We say that the odd coordinates $\theta _i$ are dual to $u^i$. The transition functions between two local trivializations $(u^1,\dots ,u^n;\theta _1,\dots ,\theta _n)$ and $(w^1,\dots ,w^n;\phi _1,\dots ,\phi _n)$ are given by the same formula as those of the cotangent bundle:

$$\begin{aligned} \phi _\alpha = \frac{\partial u^\beta }{\partial w^\alpha }\theta _\beta ,\quad \alpha = 1,\dots ,n. \end{aligned}$$

Here and henceforth, summation over repeated upper and lower Greek indices is assumed.

Denote by $J^\infty ({\hat{M}})$ the infinite jet bundle of ${\hat{M}}$. It is a fiber bundle over ${\hat{M}}$ with fiber ${\mathbb {R}}^\infty$. If we choose a local chart $(u^1,\dots ,u^n;\theta _1,\dots ,\theta _n)$ of ${\hat{M}}$, a trivialization can be realized by choosing the fiber coordinates being $(u^{\alpha ,s};\theta _\alpha ^s)$ for $\alpha = 1,\dots ,n$ and $s\ge 1$. The transition functions between different charts are given by the chain rule

$$\begin{aligned} w^{\alpha ,1}&= \frac{\partial w^\alpha }{\partial u^\beta }u^{\beta ,1},\quad w^{\alpha ,2} = \frac{\partial w^\alpha }{\partial u^\beta }u^{\beta ,2}+\frac{\partial ^2w^\alpha }{\partial u^\beta \partial u^\gamma }u^{\beta ,1} u^{\gamma ,1},\dots ,\\\phi _\alpha ^1&= \frac{\partial u^\beta }{\partial w^\alpha }\theta _\beta ^1+\frac{\partial ^2 u^\beta }{\partial w^\gamma \partial w^\alpha }\frac{\partial w^\gamma }{\partial u^\lambda }u^{\lambda ,1}\theta _\beta ,\dots . \end{aligned}$$

Denote by $\hat{\mathcal A}$ the ring of differential polynomials, locally it is given by

$$\begin{aligned}\hat{{\mathcal {A}}} = C^\infty (u)[[u^{\alpha ,s+1}, \theta ^s_\alpha \mid \alpha = 1,\dots ,n;s\ge 0]].\end{aligned}$$

It is graded with respect to the super degree $\deg _\theta$ defined by

$$\begin{aligned} \deg _{\theta }u^{\alpha ,s} = 0,\quad \deg _{\theta }\theta _\alpha ^s = 1,\quad \alpha = 1,\dots ,n,\ s\ge 0. \end{aligned}$$

Here and henceforth we use the notation $u^{\alpha ,0} = u^{\alpha }$, $\theta _\alpha ^0 = \theta _\alpha$. The set of homogeneous elements with super degree p is denoted by $\hat{\mathcal A}^p$.

Introduce a global vector field $\partial _x$ on $J^\infty ({\hat{M}})$ which is locally described by

$$\begin{aligned} \partial _x = \sum _{s\ge 0}u^{\alpha ,s+1}\frac{\partial }{\partial u^{\alpha ,s}}+\theta _\alpha ^{s+1}\frac{\partial }{\partial \theta _\alpha ^s}. \end{aligned}$$

Then we have $u^{\alpha ,s} = \partial _x^su^\alpha$ and $\theta _\alpha ^s = \partial _x^s\theta _\alpha$. Hence we can grade the ring $\hat{\mathcal A}$ with respect to the differential degree $\deg _{\partial _x}$ defined by

$$\begin{aligned} \deg _{\partial _x}u^{\alpha ,s} = s,\quad \deg _{\partial _x}\theta _\alpha ^s = s,\quad \alpha = 1,\dots ,n,\ s\ge 0. \end{aligned}$$

We use the notation $\hat{\mathcal A}_d$ to denote the set of homogeneous elements with differential degree d, and $\hat{\mathcal A}^p_d = \hat{\mathcal A}^p\cap \hat{\mathcal A}_d$.

Using the vector field $\partial _x$, one can construct the space $\hat{\mathcal F}$ of local functionals via the quotient $\hat{\mathcal F}:=\hat{\mathcal A}/\partial _x\hat{\mathcal A}$. Since the vector field $\partial _x$ is homogeneous with respect to both the super degree and the differential degree, the quotient space $\hat{\mathcal F}$ admits natural gradations induced from $\hat{\mathcal A}$ and we will use the notation $\hat{\mathcal F}^p$, $\hat{\mathcal F}_d$ and $\hat{\mathcal F}^p_d$ to denote the corresponding subspaces of homogeneous elements. For any element $f\in \hat{\mathcal A}$, we will use $\int f\in \hat{\mathcal F}$ to denote its image of the natural projection $\pi :\hat{\mathcal A}\rightarrow \hat{\mathcal F}$.

For a differential polynomial $f\in \hat{\mathcal A}$, one may define the variational derivatives by

$$\begin{aligned} \frac{\delta f}{\delta u^\alpha } = \sum _{s\ge 0}(-\partial _x)^s\frac{\partial f}{\partial u^{\alpha ,s}},\quad \frac{\delta f}{\delta \theta _\alpha } = \sum _{s\ge 0}(-\partial _x)^s\frac{\partial f}{\partial \theta _\alpha ^s}. \end{aligned}$$

It is easy to verify that the variational derivatives annihilate the elements in $\partial _x\hat{\mathcal A}$, hence they are also well-defined on the quotient space $\hat{\mathcal F}$. For any $F\in \hat{\mathcal F}$, we have

$$\begin{aligned} \frac{\delta F}{\delta u^\alpha } = \sum _{s\ge 0}(-\partial _x)^s\frac{\partial f}{\partial u^{\alpha ,s}},\quad \frac{\delta F}{\delta \theta _\alpha } = \sum _{s\ge 0}(-\partial _x)^s\frac{\partial f}{\partial \theta _\alpha ^s}, \end{aligned}$$

with $f\in \hat{\mathcal A}$ being an arbitrary lift of F such that $F = \int f$. With the help of the notion of the variational derivatives, one can define the so-called Schouten-Nijenhuis bracket, which is a bilinear map $[-,-]:\hat{\mathcal F}\times \hat{\mathcal F}\rightarrow \hat{\mathcal F}$ defined by

$$\begin{aligned} \left[ P, Q\right] = \int \left( \frac{\delta P}{\delta \theta _\alpha }\frac{\delta Q}{\delta u^\alpha }+(-1)^p\frac{\delta P}{\delta u^\alpha }\frac{\delta Q}{\delta \theta _\alpha }\right) ,\quad P\in \mathcal {{\hat{F}}}^p,\, Q\in \mathcal {{\hat{F}}}^q. \end{aligned}$$

This bracket satisfies the graded commutation relation

$$\begin{aligned}\left[ P, Q\right] = (-1)^{pq}\left[ Q, P\right] ,\quad P\in \mathcal {{\hat{F}}}^p,\, Q\in \mathcal {{\hat{F}}}^q,\end{aligned}$$

and the graded Jacobi identity

$$\begin{aligned}&(-1)^{rp}[[P,Q],R]+(-1)^{pq}[[Q,R],P]+(-1)^{qr}[[R,P],Q]\\&\quad =0,\quad P\in \mathcal {{\hat{F}}}^p,\, Q\in \mathcal {{\hat{F}}}^q,\, R\in \mathcal {{\hat{F}}}^r. \end{aligned}$$

Any local functional $P\in \hat{\mathcal F}^p$ gives rise to a graded derivation

$$\begin{aligned} D_P = \sum _{s\ge 0}\partial _x^s\left( \frac{\delta P}{\delta \theta _\alpha }\right) \frac{\partial }{\partial u^{\alpha ,s}}+(-1)^p\partial _x^s\left( \frac{\delta P}{\delta u^\alpha }\right) \frac{\partial }{\partial \theta ^{s}_\alpha }\in \mathrm {Der}(\hat{\mathcal A})^{p-1}. \end{aligned}$$

(2.1)

Here the space $\mathrm {Der}(\hat{\mathcal A})^p\ (p\in {\mathbb {Z}})$ is the space of linear maps

$$\begin{aligned} D:\hat{\mathcal A}^q\rightarrow \hat{\mathcal A}^{q+p} \end{aligned}$$

satisfying the graded Leibniz rule

$$\begin{aligned} D(fg) = D(f)\,g+(-1)^{kp}f\,D(g),\quad f\in \hat{\mathcal A}^k,\ g\in \hat{\mathcal A}. \end{aligned}$$

Denote $\mathrm {Der}(\hat{\mathcal A})= \oplus _{p\in {\mathbb {Z}}}\mathrm {Der}(\hat{\mathcal A})^p$, then it is a graded Lie algebra with the graded commutator

$$\begin{aligned} {[}D_1,D_2] = D_1\circ D_2-(-1)^{kl}D_2\circ D_1,\quad D_1\in \mathrm {Der}(\hat{\mathcal A})^k,\ D_2\in \mathrm {Der}(\hat{\mathcal A})^l, \end{aligned}$$

and it is also graded by the differential degree

$$\begin{aligned} \mathrm {Der}(\hat{\mathcal A})_d = \{D\in \mathrm {Der}(\hat{\mathcal A})\mid D(\hat{\mathcal A}_k)\subseteq \hat{\mathcal A}_{k+d}\}, \end{aligned}$$

and we denote

$$\begin{aligned} \mathrm {Der}(\hat{\mathcal A})^p_d = \mathrm {Der}(\hat{\mathcal A})^p\cap \mathrm {Der}(\hat{\mathcal A})_d.\end{aligned}$$

(2.2)

For $P\in \hat{\mathcal F}^p$ and $Q\in \hat{\mathcal F}^q$, we have the following useful identities:

$$\begin{aligned} D_{[P,Q]}&= (-1)^{p-1}\left[ D_P, D_Q\right] , \end{aligned}$$

(2.3)

$$\begin{aligned} \frac{\delta }{\delta u^\alpha }[P,Q]&= D_P\left( \frac{\delta Q}{\delta u^\alpha }\right) +(-1)^{pq}D_Q\left( \frac{\delta P}{\delta u^\alpha }\right) , \end{aligned}$$

(2.4)

$$\begin{aligned} \frac{\delta }{\delta \theta _\alpha }[P,Q]&= (-1)^{p-1}D_P\left( \frac{\delta Q}{\delta \theta _\alpha }\right) -(-1)^{(p-1)q}D_Q\left( \frac{\delta P}{\delta \theta _\alpha }\right) . \end{aligned}$$

(2.5)

A Hamiltonian structure is defined as a local functional $P\in \hat{\mathcal F}^2$ such that $[P,P] = 0$. We can associate a matrix valued differential operator ${\mathcal {P}} = ({\mathcal {P}}^{\alpha \beta })$ with ${\mathcal {P}}^{\alpha \beta } = \sum _{s\ge 0}{\mathcal {P}}^{\alpha \beta }_s\partial _x^s$ to any bivector $P\in \hat{\mathcal F}^2$, where ${\mathcal {P}}^{\alpha \beta }_s\in \hat{\mathcal A}$ are defined by

$$\begin{aligned} \frac{\delta P}{\delta \theta _\alpha } = \sum _{s\ge 0}\mathcal P^{\alpha \beta }_s\theta _\beta ^s,\quad \alpha = 1,\dots ,n. \end{aligned}$$

If P is a Hamiltonian structure, then we call ${\mathcal {P}}$ the Hamiltonian operator of P.

Theorem 2

([10]). Let $P\in \hat{\mathcal F}^2_1$. Denote the differential operator associated with P by

$$\begin{aligned} {\mathcal {P}}^{\alpha \beta } = g^{\alpha \beta }(u)\partial _x+\Gamma ^{\alpha \beta }_\gamma (u) u^{\gamma ,1},\quad \det (g^{\alpha \beta })\ne 0, \end{aligned}$$

then P is a Hamiltonian structure if and only if $g = (g_{\alpha \beta }) = (g^{\alpha \beta })^{-1}$ defines a flat metric on M and the Christoffel symbols of the Levi–Civita connection of g are given by $\Gamma _{\alpha \beta }^\gamma = -g_{\alpha \lambda }\Gamma ^{\lambda \gamma }_\beta$.

A Hamiltonian structure P satisfying the conditions of the above theorem is called of hydrodynamic type. It follows from the above theorem that there exists a local coordinate system $(v^\alpha ;\sigma _\alpha )$ on ${\hat{M}}$ such that

$$\begin{aligned} P = \frac{1}{2}\int \eta ^{\alpha \beta }\sigma _\alpha \sigma _\beta ^1, \end{aligned}$$

where $\eta ^{\alpha \beta }$ is a constant non-degenerate matrix. The coordinates $v^\alpha$ and $\sigma _\alpha$ are called flat coordinates of P.

A bihamiltonian structure $(P_0,P_1)$ is a pair of Hamiltonian structures which satisfies an additional compatibility condition $[P_0,P_1] = 0$. Assume that the bihamiltonian structure is of hydrodynamic type, then according to Theorem 2, we have two flat contravariant metrics $g^{\alpha \beta }_0$ and $g^{\alpha \beta }_1$. We say that this bihamiltonian structure is semisimple if the roots of the characteristic equation

$$\begin{aligned} \det \left( g_1^{\alpha \beta }-\lambda g_0^{\alpha \beta }\right) = 0 \end{aligned}$$

are distinct and not constant. In this case, the roots $\lambda ^1(u),\dots ,\lambda ^n(u)$ can serve as local coordinates of M and they are called the canonical coordinates of the semisimple bihamiltonian structure. It is proved in [14], in terms of the canonical coordinates, that

$$\begin{aligned} P_0= & {} \frac{1}{2}\int \sum _{i,j=1}^n \left( \delta _{i,j}f^i(\lambda )\theta _i\theta _i^1+ A^{ij}\theta _i\theta _j\right) ,\quad \\ P_1= & {} \frac{1}{2}\int \sum _{i,j=1}^n\left( \delta _{i,j} g^i(\lambda )\theta _i\theta _i^1+ B^{ij}\theta _i\theta _j\right) , \end{aligned}$$

where $f^i$ are non-vanishing functions, $g^i = \lambda ^if^i$ and the functions $A^{ij}$ and $B^{ij}$ are given by

$$\begin{aligned} A^{ij} = \frac{1}{2}\left( \frac{f^i}{f^j}\frac{\partial f^j}{\partial \lambda ^i}\lambda ^{j,1}-\frac{f^j}{f^i}\frac{\partial f^i}{\partial \lambda ^j}\lambda ^{i,1}\right) ,\quad B^{ij} = \frac{1}{2}\left( \frac{g^i}{f^j}\frac{\partial f^j}{\partial \lambda ^i}\lambda ^{j,1}-\frac{g^j}{f^i}\frac{\partial f^i}{\partial \lambda ^j}\lambda ^{i,1}\right) .\nonumber \\ \end{aligned}$$

(2.6)

Here by abusing the notation, we still use $\theta _i$ to denote the fiber coordinates of ${\hat{M}}$ dual to $\lambda ^i$. We also call $\lambda ^i$ and $\theta _i$ the canonical coordinates of $(P_0,P_1)$.

From now on, for a semisimple bihamiltonian structure $(P_0,P_1)$ of hydrodynamic type, we will use $(v^\alpha ;\sigma _\alpha )$ to denote flat coordinates of $P_0$ such that

$$\begin{aligned} P_0= & {} \frac{1}{2}\int \eta ^{\alpha \beta }\sigma _\alpha \sigma _\beta ^1,\quad \\ P_1= & {} \frac{1}{2}\int g^{\alpha \beta }(v)\sigma _\alpha \sigma _\beta ^1+\Gamma ^{\alpha \beta }_\gamma (v)v^{\gamma ,1}\sigma _\alpha \sigma _\beta , \end{aligned}$$

and we will use $(u^i;\theta _i)$ to denote the canonical coordinates for $(P_0,P_1)$ such that

$$\begin{aligned} P_0= & {} \frac{1}{2}\int \sum _{i,j=1}^n\left( \delta _{i,j} f^i(u)\theta _i\theta _i^1+ A^{ij}\theta _i\theta _j\right) ,\quad \nonumber \\ P_1= & {} \frac{1}{2}\int \sum _{i,j=1}^n\left( \delta _{i,j} u^if^i(u)\theta _i\theta _i^1+ B^{ij}\theta _i\theta _j\right) . \end{aligned}$$

(2.7)

Here and henceforth we do not assume summations over repeated upper and lower Latin indices.

In terms of the notations introduced above, a system of evolutionary PDEs

$$\begin{aligned} \frac{\partial u^\alpha }{\partial t} = X^\alpha ,\quad X^\alpha \in \hat{\mathcal A}^0 \end{aligned}$$

can be represented by a local functional $X = \int X^\alpha \theta _{\alpha }$, and it is called a bihamiltonian system if there exists a bihamiltonian structure $(P_0,P_1)$ and two Hamiltonians $G,H\in \hat{\mathcal F}^0$ such that

$$\begin{aligned} X = -[G,P_0] =-[H,P_1]. \end{aligned}$$

Example 2

The KdV equation

$$\begin{aligned} \frac{\partial u}{\partial t} = uu_x+\frac{\varepsilon ^2}{12}u_{xxx} \end{aligned}$$

(2.8)

can be represented by $X = \int (uu_x+\frac{\varepsilon ^2}{12}u_{xxx})\theta$. Its bihamiltonian structure is given by

$$\begin{aligned} P_0 = \frac{1}{2}\int \theta \theta _x,\quad P_1 = \frac{1}{2}\int u\theta \theta ^1+\frac{\varepsilon ^2}{8}\theta \theta ^3. \end{aligned}$$

The two Hamiltonians with respect to the bihamiltonian structure are given by

$$\begin{aligned} X = -\left[ \int \frac{u^3}{6}-\frac{\varepsilon ^2}{24}u_x^2, P_0\right] = -\left[ \int \frac{u^2}{3}, P_1\right] . \end{aligned}$$

2.2 Frobenius manifolds and super tau-covers of the Principal Hierarchies

In this subsection, we first recall some basic facts of Frobenius manifolds and the construction of the associated Principal Hierarchies following the work of [4,5,6, 12]. Then we recall the construction of the super tau-covers of the Principal Hierarchies given in [22].

The notion of Frobenius manifolds is a geometric description of genus zero 2D topological field theories. An n-dimensional Frobenius manifold M can be locally described by a solution $F(v^1,\dots ,v^n)$ of the following Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) associativity equations [2, 29]:

$$\begin{aligned} \partial _\alpha \partial _\beta \partial _\lambda F\eta ^{\lambda \mu }\partial _\mu \partial _\gamma \partial _\delta F = \partial _\delta \partial _\beta \partial _\lambda F\eta ^{\lambda \mu }\partial _\mu \partial _\gamma \partial _\alpha F. \end{aligned}$$

(2.9)

Here $\partial _\alpha = \frac{\partial }{\partial v^\alpha }$ and we require that $(\eta _{\alpha \beta }) := (\partial _1\partial _\alpha \partial _\beta F)$ is a constant non-degenerate matrix with inverse $(\eta ^{\alpha \beta })$. The function $F(v^1,\dots ,v^n)$ is called the potential of M and it defines a Frobenius algebra structure on TM:

$$\begin{aligned} \langle \partial _\alpha ,\partial _\beta \rangle = \eta _{\alpha \beta },\quad \partial _\alpha \cdot \partial _\beta = c^{\gamma }_{\alpha \beta }\partial _\gamma , \end{aligned}$$

where the functions $c^{\gamma }_{\alpha \beta }$ are defined by

$$\begin{aligned} \quad c^{\gamma }_{\alpha \beta } = \eta ^{\gamma \lambda }c_{\lambda \alpha \beta },\quad c_{\lambda \alpha \beta } = \partial _\lambda \partial _\alpha \partial _\beta F. \end{aligned}$$

The potential F is required to be quasi-homogeneous in the sense that there exists a vector field

$$\begin{aligned} E = \sum _{\alpha =1}^n \left( \left( 1-\frac{d}{2}-\mu _\alpha \right) v^\alpha +r_\alpha \right) \partial _\alpha , \end{aligned}$$

called the Euler vector field, such that

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

Here the diagonal matrix $\mu = \mathrm {diag}(\mu _1,\dots ,\mu _n)$ is part of the monodromy data of M which satisfies the identity

$$\begin{aligned} (\mu _\alpha +\mu _\beta )\eta _{\alpha \beta } = 0,\quad \forall \, \alpha ,\beta . \end{aligned}$$

(2.10)

It is also assumed that $\mu _1 = -d/2$ and $r_1 = 0$.

An important property of Frobenius manifolds is that the affine connection

$$\begin{aligned} {{\tilde{\nabla }}}_X Y = \nabla _X Y+zX\cdot Y,\quad \forall \, X,Y\in \Gamma (TM),\quad z\in {\mathbb {C}} \end{aligned}$$

is flat for arbitrary z, here $\nabla$ is the Levi–Civita connection of the flat metric $(\eta _{\alpha \beta })$. It can be extended to a flat connection on $M\times {\mathbb {C}}^*$ by viewing z as the coordinate on ${\mathbb {C}}^*$ and defining

$$\begin{aligned} {{\tilde{\nabla }}}_{\partial _z}X = \partial _z X+E\cdot X-\frac{1}{z} \mu X,\quad {{\tilde{\nabla }}}_{\partial _z}\partial _z = {{\tilde{\nabla }}}_{X}\partial _z = 0. \end{aligned}$$

The connection ${{\tilde{\nabla }}}$ is called the deformed flat connection or the Dubrovin connection. For such a flat connection, one can find a system of flat coordinates of the form

$$\begin{aligned} ({\tilde{v}}^1(v,z),\dots ,{\tilde{v}}^n(v,z)) = (h_1(v,z),\dots ,h_n(v,z))z^\mu z^R. \end{aligned}$$

Here R is a constant matrix. The constant matrices $\eta$, $\mu$ and R form the monodromy data of M at $z=0$. The matrix R can be decomposed into a finite sum $R = R_1+\dots +R_m$, and they satisfy the conditions

$$\begin{aligned} \left[ \mu , R_k\right] = kR_k,\quad \eta _{\alpha \gamma }(R_k)^\gamma _\beta =(-1)^{k+1}\eta _{\beta \gamma }(R_k)^\gamma _\alpha . \end{aligned}$$

(2.11)

The functions $h_\alpha (v,z)$ are analytic at $z=0$ and has the expansion $h_\alpha (v,z) =\sum _{p\ge 0}h_{\alpha ,p}(v)z^p$. The coefficients $h_{\alpha ,p}$ satisfy the recursion relations

$$\begin{aligned} h_{\alpha ,0} = \eta _{\alpha \beta }v^\beta ,\quad \partial _\beta \partial _\gamma h_{\alpha ,p+1} = c^\lambda _{\beta \gamma }\partial _\lambda h_{\alpha ,p},\quad p\ge 0, \end{aligned}$$

the quasi-homogeneous and normalization conditions

$$\begin{aligned}&E(\partial _\beta h_{\alpha ,p}) = (p+\mu _\alpha +\mu _\beta )\partial _\beta h_{\alpha ,p}+\sum _{k=1}^p(R_k)^\gamma _\alpha \partial _\beta h_{\gamma ,p-k},\\&\langle \nabla h_\alpha (v,z),\nabla h_\beta (v,-z)\rangle = \eta _{\alpha \beta }. \end{aligned}$$

A choice of the functions $h_{\alpha ,p}$ satisfying all the above-mentioned conditions is called a calibration of M, and a Frobenius manifold M is called calibrated if such a choice is fixed. In what follows, we assume that the Frobenius manifolds we consider are calibrated.

The Principal Hierarchy associated with a Frobenius manifold M is a bihamiltonian integrable hierarchy of hydrodynamic type. Denote by $\sigma _\alpha$ the odd variables dual to the flat coordinates $v^\alpha$, then the Principal Hierarchy can be described by the local functionals $X_{\alpha ,p}\in \hat{\mathcal F}^1$ of the form

$$\begin{aligned} X_{\alpha ,p} = \int \eta ^{\lambda \gamma }\partial _x(\partial _\gamma h_{\alpha ,p+1})\sigma _\lambda ,\quad \alpha = 1,\dots ,n,\quad p\ge 0, \end{aligned}$$

(2.12)

or equivalently, we can represent the integrable hierarchy as follows:

$$\begin{aligned} \frac{\partial v^\lambda }{\partial t^{\alpha ,p}} = \eta ^{\lambda \gamma }\partial _x(\partial _\gamma h_{\alpha ,p+1}). \end{aligned}$$

Define two local functionals

$$\begin{aligned} P_0 = \frac{1}{2}\int \eta ^{\alpha \beta }\sigma _\alpha \sigma _\beta ^1,\quad P_1 = \frac{1}{2}\int g^{\alpha \beta }\sigma _\alpha \sigma _\beta ^1+\Gamma ^{\alpha \beta }_\gamma v^{\gamma ,1}\sigma _\alpha \sigma _\beta , \end{aligned}$$

(2.13)

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

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

with $c^{\alpha \beta }_\gamma = \eta ^{\alpha \lambda }c^\beta _{\lambda \gamma }$, then we have the following theorem.

Theorem 3

([5]). Let M be a Frobenius manifold, then

• 1. The local functionals $P_0$, $P_1$ defined in (2.13) form a bihamiltonian structure which is exact in the sense that

  $$\begin{aligned} P_0 = [Z,P_1],\quad Z = \int \sigma _{1}. \end{aligned}$$

• 2. The Principal Hierarchy $X_{\alpha ,p}$ associated with M is bihamiltonian with respect to the bihamiltonian structure $(P_0,P_1)$ and

  $$\begin{aligned} X_{\alpha ,p} = -\left[ H_{\alpha ,p}, P_0\right] ,\quad H_{\alpha ,p} = \int h_{\alpha ,p+1}. \end{aligned}$$

• 3. The following bihamiltonian recursion relation holds true:

  $$\begin{aligned}{}[H_{\alpha ,p-1},P_1]=\left( p+\frac{1}{2}+\mu _\alpha \right) [H_{\alpha ,p},P_0]+\sum _{k=1}^p \left( R_k\right) ^\gamma _\alpha [H_{\gamma , p-k},P_0],\quad p\ge 0. \end{aligned}$$

Another important property satisfied by the Principal Hierarchy is that it is tau-symmetric. Let us define the functions $\Omega _{\alpha ,p;\beta ,q}$ for $\alpha ,\beta = 1,\dots ,n$ and $p,q\ge 0$ by the generating function

$$\begin{aligned} \sum _{p\ge 0,q\ge 0}\Omega _{\alpha ,p;\beta ,q}(v)z_1^pz_2^q = \frac{\langle \nabla h_\alpha (v,z_1),\nabla h_\beta (v,z_2)\rangle - \eta _{\alpha \beta }}{z_1+z_2}. \end{aligned}$$

They have the following properties [5]:

$$\begin{aligned} \Omega _{\alpha ,p;1,0}= & {} h_{\alpha ,p},\quad \displaystyle \Omega _{\alpha ,p;\beta ,0} = \partial _\beta h_{\alpha ,p+1}, \\ \Omega _{\alpha ,p;\beta ,q}= & {} \Omega _{\beta ,q;\alpha ,p},\quad \displaystyle \frac{\partial \Omega _{\alpha ,p;\beta ,q}}{\partial t^{\lambda ,k}}= \displaystyle \frac{\partial \Omega _{\lambda ,k;\beta ,q}}{\partial t^{\alpha ,p}}.\end{aligned}$$

It follows from these identities that one can extend the Principal Hierarchy by introducing another family of unknown functions $f_{\alpha ,p}$ satisfying the following equations:

$$\begin{aligned} \frac{\partial f_{\alpha ,p}}{\partial t^{\beta ,q}} = \Omega _{\alpha ,p;\beta ,q}, \quad \frac{\partial v^\alpha }{\partial t^{\beta ,q}} = \eta ^{\alpha \lambda }\partial _x\Omega _{\lambda ,0;\beta ,q}. \end{aligned}$$

(2.14)

The system (2.14) is called the tau-cover of the Principal Hierarchy.

In order to study the relation between bihamiltonian structures and Virasoro symmetries, we introduced the notion of the super tau-cover of the Principal Hierarchy of a Frobenius manifold in [22]. Let us briefly recall its construction which provides the main motivation of our work presented in the next subsection. We first introduce a family of odd unknown functions $\sigma _{\alpha ,k}^s$ for $s,k\ge 0$ with $\sigma _{\alpha ,0}^s = \sigma ^s_\alpha$. In what follows we will also use $\sigma _{\alpha ,k}$ to denote $\sigma _{\alpha ,k}^0$. We extend the action of $\partial _x$ to include these new odd variables as follows:

$$\begin{aligned} \partial _x = \sum _{s\ge 0}v^{\alpha ,s+1}\frac{\partial }{\partial v^{\alpha ,s}}+\sum _{k,s\ge 0}\sigma _{\alpha ,k}^{s+1}\frac{\partial }{\partial \sigma _{\alpha ,k}^s}. \end{aligned}$$

These odd variables are required to satisfy the following bihamiltonian recursion relation:

$$\begin{aligned} \eta ^{\alpha \beta }\sigma _{\beta ,k+1}^1 = g^{\alpha \beta }\sigma _{\beta ,k}^1+\Gamma _\gamma ^{\alpha \beta }v^{\gamma ,1}\sigma _{\beta ,k},\quad \alpha =1,\dots , n;\, k\ge 0. \end{aligned}$$

(2.15)

We also introduce a family of odd flows $\frac{\partial }{\partial \tau _m}$ for $m\ge 0$. The first two odd flows are determined by the bihamiltonian structure $(P_0,P_1)$ as follows:

$$\begin{aligned} \frac{\partial v^\alpha }{\partial \tau _i} = \frac{\delta P_i}{\delta \sigma _\alpha },\quad \frac{\partial \sigma _\alpha }{\partial \tau _i} = \frac{\delta P_i}{\delta v^\alpha },\quad i = 0,1. \end{aligned}$$

Note that $\frac{\partial }{\partial \tau _i} = D_{P_i}$ for $i =0,1$, where $D_{P_i}$ is defined by (2.1). The actions of $\frac{\partial }{\partial \tau _i}$ can be extended to all the other odd variables $\sigma _{\alpha ,k}$ such that the flows $\frac{\partial }{\partial \tau _i}$ are compatible with the recursion relation (2.15). Furthermore we can define infinitely many odd flows $\frac{\partial }{\partial \tau _m}$ for $m\ge 2$ which can be viewed as flows corresponding to certain non-local Hamiltonian structures.

We have the following theorem.

Theorem 4

([22]). We have the following mutually commuting flows associated with any given Frobenius manifold M:

$$\begin{aligned} \frac{\partial v^\alpha }{\partial t^{\beta ,p}}&= \eta ^{\alpha \gamma }{(\partial _\lambda \partial _\gamma h_{\beta ,p+1})v^{\lambda ,1}},\quad \frac{\partial \sigma _{\alpha ,k}}{\partial t^{\beta ,p}} = \eta ^{\gamma \varepsilon }(\partial _\alpha \partial _\varepsilon h_{\beta ,p+1})\sigma _{\gamma ,k}^1,\\ \frac{\partial v^\alpha }{\partial \tau _m}&= \eta ^{\alpha \beta }\sigma _{\beta ,m}^1,\quad \frac{\partial \sigma _{\alpha ,k}}{\partial \tau _m} = -\frac{\partial \sigma _{\alpha ,m}}{\partial \tau _k} = \Gamma ^{\gamma \beta }_\alpha \sum _{i = 0}^{m-k-1}\sigma _{\beta ,k+i}\sigma _{\gamma ,m-i-1}^1,\quad 0\le k\le m, \end{aligned}$$

where $\alpha ,\beta =1,\dots ,n$, and $m, p\ge 0$. These flows are well-defined in the sense that they are compatible with the recursion relation (2.15).

The system described in Theorem 4 is a super extension of the Principal Hierarchy, since the reduction obtained by setting all the odd variables to be zero yields the original Principal Hierarchy. The super extension of the tau-cover (2.14) can be constructed by introducing another family of odd variables $\Phi ^{m}_{\alpha ,p}$ for $p,m\ge 0$ and we call it the super tau-cover of the Principal Hierarchy. It is given in the following theorem.

Theorem 5

([22]). The mutually commuting flows

$$\begin{aligned} \frac{\partial f_{\alpha ,p}}{\partial t^{\beta ,q}}&= \Omega _{\alpha ,p;\beta ,q},\\ \frac{\partial f_{\alpha ,p}}{\partial \tau _m}&= \Phi _{\alpha ,p}^m,\\ \frac{\partial \Phi _{\alpha ,p}^m}{\partial t^{\beta ,q}}&= \frac{\partial \Omega _{\alpha ,p;\beta ,q}}{\partial \tau _m},\\ \frac{\partial \Phi _{\alpha ,p}^m}{\partial \tau _k}&= \Delta _{\alpha ,p}^{k,m}, \end{aligned}$$

together with the ones presented in Theorem 4, give the super tau-cover of the Principal Hierarchy associated with M, where $\Delta _{\alpha ,p}^{k,m}$ are defined by the formula

$$\begin{aligned} \Delta _{\alpha ,p}^{k,m}=-\Delta _{\alpha ,p}^{m,k}=\eta ^{\gamma \lambda }\partial _\lambda h_{\alpha ,p}\Gamma _\gamma ^{\delta \mu }\left( \sum _{i=0}^{k-m-1}\sigma _{\mu ,m+i}\sigma _{\delta ,k-i-1}^1\right) ,\quad k\ge m. \end{aligned}$$

It was shown in [22] that the odd variables $\Phi _{\alpha ,p}^m$ satisfy the recursion relation

$$\begin{aligned} -\left( \frac{2p-1}{2}+\mu _\alpha \right) \Phi _{\alpha ,p}^m = \left( \frac{1}{2}+\mu _\lambda \right) \eta ^{\lambda \varepsilon }(\partial _\lambda h_{\alpha ,p})\sigma _{\varepsilon ,m}+\sum _{k=1}^p(R_k)^\xi _\alpha \Phi _{\xi ,p-k}^m-\Phi _{\alpha ,p-1}^{m+1} \end{aligned}$$

with the initial condition $\Phi ^m_{\alpha ,0} = \sigma _{\alpha ,m}$. So when the diagonal matrix $\mu$ of the Frobenius manifold M satisfies the condition $\frac{1-2k}{2}\notin \mathrm {Spec}(\mu )$ for any $k = 1,2,\dots$, all the variables $\Phi ^m_{\alpha ,p}$ are linear combinations of $\sigma _{\varepsilon ,k}$ with coefficients being smooth functions of $v^1,\dots ,v^n$.

For an arbitrary tau-symmetric bihamiltonian deformation of the Principal Hierarchy associated with a semisimple Frobenius manifold, we are to construct in the remaining part of this section its super extension and super tau-cover by generalizing the constructions given in Theorems 4 and 5.

2.3 Super extensions of bihamiltonian integrable hierarchies

In this subsection, we construct a super extension for a given bihamiltonian integrable hierarchy with hydrodynamic limit.

We fix an n-dimensional smooth manifold M and a semisimple bihamiltonian structure $(P_0^{[0]},P_1^{[0]})$ of hydrodynamic type defined on $J^\infty ({\hat{M}})$. Let us choose $(w^\alpha ;\phi _{\alpha })$ as local coordinates on ${\hat{M}}$. Recall that a Miura type transformation is a choice of n differential polynomials $\tilde{w}^1,\dots , {\tilde{w}}^n\in \hat{\mathcal A}^0_{\ge 0}$ such that

$$\begin{aligned} \det \left( \frac{\partial {\tilde{w}}^\alpha _0}{\partial w^\beta }\right) \ne 0, \end{aligned}$$

where ${\tilde{w}}^\alpha _0$ is the differential degree zero component of ${\tilde{w}}^\alpha$. By defining ${\tilde{w}}^{\alpha ,s} = \partial _x^s\tilde{w}^\alpha$, it is easy to see that we can represent any differential polynomial in $w^{\alpha ,s}$ by a differential polynomial in $\tilde{w}^{\alpha ,s}$. Therefore a Miura type transformation can be viewed as a special type of change of coordinates on $J^\infty (M)$. The extension of the Miura type transformations on $J^\infty ({\hat{M}})$ is given by the following theorem [25].

Theorem 6

([25]). A Miura type transformation induces a change of coordinates from $(w^{\alpha ,s};\phi _\alpha ^s)$ to $(\tilde{w}^{\alpha ,s};{{\tilde{\phi }}}_\alpha ^s)$ given by

$$\begin{aligned} \phi _\alpha ^s = \partial _x^s\sum _{t\ge 0}(-\partial _x)^t\left( \frac{\partial \tilde{w}^{\beta }}{\partial w^{\alpha ,t}}{{\tilde{\phi }}}_\beta \right) . \end{aligned}$$

Now let $(P_0,P_1)$ be any given deformation of $(P_0^{[0]},P_1^{[0]})$. Denote by ${\mathcal {P}}_0$ and ${\mathcal {P}}_1$ the Hamiltonian operators of $P_0$ and $P_1$ in the coordinates $(w^{\alpha ,s};\phi _\alpha ^s)$. We introduce another family of odd variables $\phi ^s_{\alpha ,m}$ for $m\ge 0$ and extend the vector field $\partial _x$ to the following one:

$$\begin{aligned} \partial _x = \sum _{s\ge 0}w^{\alpha ,s+1}\frac{\partial }{\partial w^{\alpha ,s}}+\sum _{k,s\ge 0}\phi _{\alpha ,k}^{s+1}\frac{\partial }{\partial \phi _{\alpha ,k}^s}. \end{aligned}$$

In what follows we also use the notations $\phi ^s_{\alpha ,0} = \phi ^s_\alpha$ and $\phi ^0_{\alpha ,m} = \phi _{\alpha ,m}$. Inspired by (2.15), we require that these new odd variables satisfy the recursion relations

$$\begin{aligned} {\mathcal {P}}_0^{\alpha \beta }\phi _{\beta ,m+1} = \mathcal P_1^{\alpha \beta }\phi _{\beta ,m},\quad m\ge 0. \end{aligned}$$

(2.16)

We first show that (2.16) is well defined in the sense that it is invariant under Miura type transformations.

Proposition 1

The Miura type transformation from $(w^{\alpha ,s};\phi _\alpha ^s)$ to $({\tilde{w}}^{\alpha ,s};{{\tilde{\phi }}}_\alpha ^s)$ induces a transformation for the new odd variables $\phi ^s_{\alpha ,m}$ given by

$$\begin{aligned} \phi _{\alpha ,m}^s = \partial _x^s\sum _{t\ge 0}(-\partial _x)^t\left( \frac{\partial \tilde{w}^{\beta }}{\partial w^{\alpha ,t}}{{\tilde{\phi }}}_{\beta ,m}\right) ,\quad m\ge 1, \end{aligned}$$

(2.17)

such that the recursion relation (2.16) is invariant.

Proof

Denote by ${\mathcal {P}}_i$ and $\tilde{{\mathcal {P}}_i}$ the Hamiltonian operator of $P_i$ in the coordinates $(w^{\alpha ,s};\phi _\alpha ^s)$ and $({\tilde{w}}^{\alpha ,s};{{\tilde{\phi }}}_\alpha ^s)$ respectively for $i = 0,1$. Then it is well known that

$$\begin{aligned} \tilde{{\mathcal {P}}_i}^{\alpha \beta } = \sum _{s\ge 0}\frac{\partial \tilde{w}^\alpha }{\partial w^{\lambda ,s}}\partial _x^s\circ {\mathcal {P}}_i^{\lambda \varepsilon }\circ \sum _{t\ge 0}(-\partial _x)^t\circ \frac{\partial {\tilde{w}}^\beta }{\partial w^{\varepsilon ,t}}. \end{aligned}$$

Therefore by using the relation (2.16), it is easy to see that:

$$\begin{aligned} \tilde{{\mathcal {P}}_0}^{\alpha \beta }{{\tilde{\phi }}}_{\beta ,m+1}&=\sum _{s\ge 0}\frac{\partial {\tilde{w}}^\alpha }{\partial w^{\lambda ,s}}\partial _x^s\circ {\mathcal {P}}_0^{\lambda \varepsilon }\circ \sum _{t\ge 0}(-\partial _x)^t\circ \frac{\partial {\tilde{w}}^\beta }{\partial w^{\varepsilon ,t}}({{\tilde{\phi }}}_{\beta ,m+1})\\&=\sum _{s\ge 0}\frac{\partial {\tilde{w}}^\alpha }{\partial w^{\lambda ,s}}\partial _x^s\circ {\mathcal {P}}_0^{\lambda \varepsilon }\phi _{\varepsilon ,m+1}\\&=\sum _{s\ge 0}\frac{\partial {\tilde{w}}^\alpha }{\partial w^{\lambda ,s}}\partial _x^s\circ {\mathcal {P}}_1^{\lambda \varepsilon }\phi _{\varepsilon ,m}\\&=\sum _{s\ge 0}\frac{\partial {\tilde{w}}^\alpha }{\partial w^{\lambda ,s}}\partial _x^s\circ {\mathcal {P}}_1^{\lambda \varepsilon }\circ \sum _{t\ge 0}(-\partial _x)^t\circ \frac{\partial {\tilde{w}}^\beta }{\partial w^{\varepsilon ,t}}({{\tilde{\phi }}}_{\beta ,m})\\&=\tilde{{\mathcal {P}}_1}^{\alpha \beta }{{\tilde{\phi }}}_{\beta ,m}. \end{aligned}$$

Thus we see that the recursion relations (2.16) are preserved under the change of coordinates (2.17). The proposition is proved.$\square$

By using the theory of bihamiltonian cohomology [7] for $(P_0^{[0]},P_1^{[0]})$, we can choose a coordinate system $(v^{\alpha ,s};\sigma _\alpha ^s)$ such that

$$\begin{aligned} P_0 = P_0^{[0]} = \frac{1}{2}\int \eta ^{\alpha \beta }\sigma _\alpha \sigma _\beta ^1, \end{aligned}$$

and the $\hat{\mathcal F}^2_2$ component of $P_1$ vanishes. From now on, we will always use $(v^{\alpha ,s};\sigma _\alpha ^s)$ to denote the coordinate system described above. We use the notation $\hat{\mathcal A}^+$ to denote the extension of $\hat{\mathcal A}$ by including the odd variables $\sigma _{\alpha ,m}^s$ for $m\ge 1$ satisfying the recursion relations

$$\begin{aligned} \eta ^{\alpha \beta }\sigma _{\beta ,m+1}^1 = \mathcal P_1^{\alpha \beta }\sigma _{\beta ,m},\quad m\ge 0. \end{aligned}$$

(2.18)

As before we use the notation $\sigma ^s_{\alpha ,0} = \sigma ^s_\alpha$ and $\sigma ^0_{\alpha ,m} = \sigma _{\alpha ,m}$. We will still use $\partial _x$ to denote the vector field on $\hat{\mathcal A}^+$ defined by

$$\begin{aligned} \partial _x = \sum _{s\ge 0}v^{\alpha ,s+1}\frac{\partial }{\partial v^{\alpha ,s}}+\sum _{s,m\ge 0}\sigma _{\alpha ,m}^{s+1}\frac{\partial }{\partial \sigma _{\alpha ,m}^s}. \end{aligned}$$

For any element $f\in \hat{\mathcal A}^+$, we say that f is local if it can be represented by an element of $\hat{\mathcal A}$ and we say that f is non-local if it is not local. Note that on the space $\hat{\mathcal A}^+$, the super degree is still well defined by setting the super degree of $\sigma _{\alpha ,m}^s$ being 1. We will use $\hat{\mathcal A}^{+,p}$ to denote the set of homogeneous elements with super degree p.

Example 3

Consider the following bihamiltonian structure of the KdV equation (2.8):

$$\begin{aligned} {\mathcal {P}}_0 = \partial _x,\quad {\mathcal {P}}_1 = v\partial _x+\frac{1}{2} v_x+\frac{\varepsilon ^2}{8}\partial _x^3. \end{aligned}$$

We introduce odd variables $\sigma _m^s$ for $s, m\ge 0$ such that they satisfy the recursion relations

$$\begin{aligned} \sigma _{m+1}^1 = v\sigma _{m}^1+\frac{1}{2} v_x\sigma _{m}+\frac{\varepsilon ^2}{8}\sigma _{m}^3,\quad m\ge 0. \end{aligned}$$

Then the ring $\hat{\mathcal A}^+$ is given by the quotient

$$\begin{aligned} \hat{\mathcal A}^+ = C^\infty (v)[[v^{(s+1)},\sigma _m^s\mid m,s\ge 0]]/J, \end{aligned}$$

where J is the differential ideal generated by

$$\begin{aligned} v\sigma _{m}^1+\frac{1}{2} v_x\sigma _{m}+\frac{\varepsilon ^2}{8}\sigma _{m}^3-\sigma _{m+1}^1,\quad m\ge 0. \end{aligned}$$

Then we see that $\sigma _1^1$ is local but $\sigma _2^1$ is non-local.

Definition 1

For $k,l\ge 0$, we define the shift operators

$$\begin{aligned} T_k:\hat{\mathcal A}^1\rightarrow \hat{\mathcal A}^{+,1},\quad T_{k,l}:\hat{\mathcal A}^2\rightarrow \hat{\mathcal A}^{+,2} \end{aligned}$$

to be the linear operators given by

$$\begin{aligned}&T_k(f\sigma _{\alpha ,0}^s)= {} f\sigma _{\alpha ,k}^s,\quad f\in \hat{\mathcal A}^0, \\&T_{k,l}= {} -T_{l,k},\quad T_{k,l}(f\sigma _{\alpha ,0}^t\sigma _{\beta ,0}^s) = f\sum _{i=0}^{l-k-1}\sigma _{\alpha ,k+i}^t\sigma _{\beta ,l-i-1}^s,\quad k\le l,\ f\in \hat{\mathcal A}^0. \end{aligned}$$

In particular, $T_{k,k} = 0$.

The following lemmas are obvious from the above definition.

Lemma 1

The shift operators $T_k$ and $T_{k,l}$ commute with $\partial _x$.

Lemma 2

The shift operators $T_k$ and $T_{k,l}$ are globally defined, i.e. they are invariant under Miura type transformations.

Example 4

Using the shift operators, the recursion relation (2.18) can be represented by the following formula

$$\begin{aligned} T_{m+1}\frac{\delta P_0}{\delta \sigma _{\alpha ,0}} = T_m \frac{\delta P_1}{\delta \sigma _{\alpha ,0}},\quad m\ge 0. \end{aligned}$$

(2.19)

Example 5

The recursion relation (2.18) can also be represented by the following formula:

$$\begin{aligned} T_{m+1}\left( D_{P_0}f\right) = T_m\left( D_{P_1}f\right) ,\quad m\ge 0,\ f\in \hat{\mathcal A}^0, \end{aligned}$$

(2.20)

here $D_{P_i}$ are the derivations defined in (2.1). Indeed, when $f = v^\alpha$ we recover the relation (2.19); for general $f\in \hat{\mathcal A}^0$, by definition (2.1), we have

$$\begin{aligned} T_{m+1}\left( D_{P_0}f\right) = T_{m+1}\sum _{s\ge 0}\frac{\partial f}{\partial v^{\alpha ,s}}\partial _x^s\frac{\delta P_0}{\delta \sigma _{\alpha ,0}} = T_{m}\sum _{s\ge 0}\frac{\partial f}{\partial v^{\alpha ,s}}\partial _x^s\frac{\delta P_1}{\delta \sigma _{\alpha ,0}} = T_m\left( D_{P_1}f\right) . \end{aligned}$$

With the help of the shift operators, we can generalize the construction given in the previous subsection. We first introduce the following notation.

Definition 2

We define a family of odd derivations $\frac{\partial }{\partial \tau _m}$ on $\hat{\mathcal A}^+$ by

$$\begin{aligned} \frac{\partial v^\alpha }{\partial \tau _m} = T_m\frac{\delta P_0}{\delta \sigma _{\alpha ,0}},\quad \frac{\partial \sigma _{\alpha ,k}}{\partial \tau _m} = T_{k,m}\frac{\delta P_1}{\delta v^\alpha },\quad \left[ \frac{\partial }{\partial \tau _m}, \partial _x\right] = 0. \end{aligned}$$

(2.21)

In particular, for $f\in \hat{\mathcal A}$ we have

$$\begin{aligned} \frac{\partial f}{\partial \tau _0} = D_{P_0}f,\quad \frac{\partial f}{\partial \tau _1} = D_{P_1}f, \end{aligned}$$

(2.22)

and for $f\in \hat{\mathcal A}^0$ we have

$$\begin{aligned} \frac{\partial f}{\partial \tau _m} = T_m\frac{\partial f}{\partial \tau _0},\quad m\ge 0. \end{aligned}$$

(2.23)

We need to check that that this definition is well-defined, i.e., it is compatible with the recursion relations (2.19).

Lemma 3

The following identity holds true for any $X\in \hat{\mathcal A}^1$ and $m,k\ge 0$:

$$\begin{aligned} \frac{\partial }{\partial \tau _k}T_m(X) = T_{m,k}\left( D_{P_1}(X)\right) -T_{m+1,k}\left( D_{P_0}(X)\right) . \end{aligned}$$

Proof

Since all the operators are linear, we may assume $X = f\sigma _{\beta ,0}^l$ for some $f\in \hat{\mathcal A}^0$ and $l\ge 0$. We first assume that $k\ge m$. The case $k=m$ can be easily verified as follows:

$$\begin{aligned} \frac{\partial }{\partial \tau _m}(f\sigma _{\beta ,m}^l) = T_m\left( D_{P_0}f\right) \sigma _{\beta ,m}^l = T_{m,m+1}\left( D_{P_0}(f\sigma _{\beta ,0}^l)\right) . \end{aligned}$$

Here we use the fact that $P_0 = P_0^{[0]}$ and $D_{P_0}\sigma _{\beta ,0} = 0$. Now we assume $k\ge m+1$, then by using the definition of the shift operators and the recursion relation (2.19) we obtain the following identities:

$$\begin{aligned} \frac{\partial }{\partial \tau _k}(f\sigma _{\beta ,m}^l)&= T_k\left( D_{P_0}f\right) \sigma _{\beta ,m}^l+T_{m,k}\left( fD_{P_1}\sigma _{\beta ,0}^l\right) \\&= T_{k-1}\left( D_{P_1}f\right) \sigma _{\beta ,m}^l+T_{m,k}\left( fD_{P_1}\sigma _{\beta ,0}^l\right) \\&=T_{m,k}\left( \left( D_{P_1}f\right) \sigma _{\beta ,0}^l\right) \\ {}&\quad -\sum _{i=0}^{k-m-2}T_{m+i}\left( D_{P_1}f\right) \sigma _{\beta ,k-i-1}^l+T_{m,k}\left( fD_{P_1}\sigma _{\beta ,0}^l\right) \\&= T_{m,k}\left( D_{P_1}(f\sigma _{\beta ,0}^l)\right) -\sum _{i=0}^{k-m-2}T_{m+i+1}\left( D_{P_0}f\right) \sigma _{\beta ,k-i-1}^l\\&=T_{m,k}\left( D_{P_1}(f\sigma _{\beta ,0}^l)\right) -T_{m+1,k}\left( D_{P_0}(f\sigma _{\beta ,0}^l)\right) . \end{aligned}$$

The case $k<m$ is proved in exactly the same way. The lemma is proved.$\square$

Proposition 2

The flows $\frac{\partial }{\partial \tau _k}$ are compatible with the recursion relation (2.19), i.e.,

$$\begin{aligned} \frac{\partial }{\partial \tau _k}T_{m+1}\frac{\delta P_0}{\delta \sigma _{\alpha ,0}} = \frac{\partial }{\partial \tau _k}T_{m}\frac{\delta P_1}{\delta \sigma _{\alpha ,0}},\quad m,k\ge 0. \end{aligned}$$

Proof

Using the fact that

$$\begin{aligned} P_0 = P_0^{[0]} = \frac{1}{2}\int \eta ^{\alpha \beta }\sigma _\alpha \sigma _\beta ^1, \end{aligned}$$

it is easy to obtain the following identities:

$$\begin{aligned} \frac{\partial }{\partial \tau _k}T_{m+1}\frac{\delta P_0}{\delta \sigma _{\alpha ,0}} = \frac{\partial }{\partial \tau _k}\eta ^{\alpha \beta }\sigma _{\beta ,m+1}^1 = T_{m+1,k}\left( \eta ^{\alpha \beta }\partial _x\frac{\delta P_1}{\delta v^\beta }\right) = T_{m+1,k}\left( D_{P_1}\frac{\delta P_0}{\delta \sigma _{\alpha ,0}}\right) . \end{aligned}$$

Since $[P_0,P_1] = 0$, it follows from the identity (2.5) that

$$\begin{aligned} \frac{\partial }{\partial \tau _k}T_{m+1}\frac{\delta P_0}{\delta \sigma _{\alpha ,0}} = -T_{m+1,k}\left( D_{P_0}\frac{\delta P_1}{\delta \sigma _{\alpha ,0}}\right) . \end{aligned}$$

Thus by using $[P_1,P_1] = 0$ and Lemma 3 we finish the proof of the proposition.$\square$

Proposition 3

The odd flows $\frac{\partial }{\partial \tau _m}$ mutually commute, i.e.,

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _m}, \frac{\partial }{\partial \tau _k}\right] = 0,\quad m,\,k\ge 0. \end{aligned}$$

Proof

By the definition of the flows $\frac{\partial }{\partial \tau _m}$, it is easy to see that

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _m}, \frac{\partial }{\partial \tau _k}\right] v^\alpha = \eta ^{\alpha \beta }\partial _xT_{k,m}\frac{\delta P_1}{\delta v^\beta }+\eta ^{\alpha \beta }\partial _xT_{m,k}\frac{\delta P_1}{\delta v^\beta } = 0. \end{aligned}$$

To show the commutation relation

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _l}, \frac{\partial }{\partial \tau _k}\right] \sigma _{\alpha ,m} = 0, \end{aligned}$$

it suffices to verify the case $m = 0$ due to the recursion relations (2.18). By using the trivial relation

$$\begin{aligned}\left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial \tau _0}\right] \sigma _{\alpha ,0} = 0,\end{aligned}$$

the recursion relations (2.18), and by induction on k, we arrive at

$$\begin{aligned}\left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial \tau _0}\right] \sigma _{\alpha ,k} = 0,\quad k\ge 0.\end{aligned}$$

This commutation relation is equivalent to

$$\begin{aligned}\left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial \tau _k}\right] \sigma _{\alpha ,0} = 0\end{aligned}$$

due to the definition of the odd flows. By using induction again we arrive at the identity

$$\begin{aligned}\left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial \tau _k}\right] \sigma _{\alpha ,l} = 0\end{aligned}$$

for any $l\ge 0$. Therefore we have

$$\begin{aligned} \frac{\partial }{\partial \tau _k}\frac{\partial \sigma _{\alpha ,0}}{\partial \tau _l} = -\frac{\partial }{\partial \tau _k}\frac{\partial \sigma _{\alpha ,l}}{\partial \tau _0} = \frac{\partial }{\partial \tau _0}\frac{\partial \sigma _{\alpha ,l}}{\partial \tau _k}. \end{aligned}$$

It follows from the definition of the odd flows that the right hand side is anti-symmetric with respect to the indices k, l, hence we prove that

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _l}, \frac{\partial }{\partial \tau _k}\right] \sigma _{\alpha ,0} = 0. \end{aligned}$$

The proposition is proved.$\square$

Now let $X_i\in \hat{\mathcal F}^1$, $i\in I$ be a family of bihamiltonian vector fields with respect to an index set I, i.e., each $X_i$ satisfies the equations $[X_i,P_0] = [X_i,P_1] = 0$. Recall that the family $\{X_i\}$ corresponds to a bihamiltonian integrable hierarchy given by

$$\begin{aligned} \frac{\partial v^\alpha }{\partial t_i} = \frac{\delta X_i}{\delta \sigma _{\alpha ,0}},\quad i\in I. \end{aligned}$$

(2.24)

In what follows we will extend this integrable hierarchy such that it becomes a system of mutually commuting vector fields on $\hat{\mathcal A}^+$.

Definition 3

For a bihamiltonian vector field $X\in \hat{\mathcal F}^1$ we associate it with the following system of PDEs on $\hat{\mathcal A}^+$:

$$\begin{aligned} \frac{\partial v^\alpha }{\partial t_X} = D_Xv^\alpha ,\quad \frac{\partial \sigma _{\alpha ,m}}{\partial t_X} = T_m D_X\sigma _{\alpha ,0},\quad \left[ \frac{\partial }{\partial t_X}, \partial _x\right] = 0. \end{aligned}$$

It is called the super extended flow associated with X.

Proposition 4

The super extended flow $\frac{\partial }{\partial t_X}$ associated with a bihamiltonian vector field X is compatible with the recursion relation (2.19), i.e.,

$$\begin{aligned} \frac{\partial }{\partial t_X}T_{m+1}\frac{\delta P_0}{\delta \sigma _{\alpha ,0}} = \frac{\partial }{\partial t_X}T_{m}\frac{\delta P_1}{\delta \sigma _{\alpha ,0}}. \end{aligned}$$

(2.25)

Proof

From Definition 3 of the flow $\frac{\partial }{\partial t_X}$, it is easy to see that

$$\begin{aligned} \frac{\partial }{\partial t_X}T_{m+1}\frac{\delta P_0}{\delta \sigma _{\alpha ,0}} = T_{m+1}D_X\frac{\delta P_0}{\delta \sigma _{\alpha ,0}},\quad \frac{\partial }{\partial t_X}T_{m}\frac{\delta P_1}{\delta \sigma _{\alpha ,0}}= T_{m}D_X\frac{\delta P_1}{\delta \sigma _{\alpha ,0}}. \end{aligned}$$

On the other hand from the fact that $[X,P_0] = [X,P_1] = 0$ and the identity (2.5), we see that (2.25) is equivalent to the following identity:

$$\begin{aligned} T_{m+1}D_{P_0}\frac{\delta X}{\delta \sigma _{\alpha ,0}} = T_{m}D_{P_1}\frac{\delta X}{\delta \sigma _{\alpha ,0}}, \end{aligned}$$

which holds true due to (2.20). The proposition is proved.$\square$

Proposition 5

Let X and Y be two bihamiltonian vector fields, then their associated super extended flows commute.

Proof

From the theory of the bihamiltonian cohomology [7] we know that $[X,Y] = 0$, hence it follows from (2.3) that

$$\begin{aligned} \left[ \frac{\partial }{\partial t_X}, \frac{\partial }{\partial t_Y}\right] v^\alpha = 0,\quad \left[ \frac{\partial }{\partial t_X}, \frac{\partial }{\partial t_Y}\right] \sigma _{\alpha ,0} = 0. \end{aligned}$$

By using Definition 3 we also have

$$\begin{aligned} \left[ \frac{\partial }{\partial t_X}, \frac{\partial }{\partial t_Y}\right] \sigma _{\alpha ,m} = T_m\left[ \frac{\partial }{\partial t_X}, \frac{\partial }{\partial t_Y}\right] \sigma _{\alpha ,0} = 0. \end{aligned}$$

The proposition is proved.$\square$

Now let us prove that the super extended flow associated with a bihamiltonian vector field commutes with the odd flows $\frac{\partial }{\partial \tau _m}$.

Lemma 4

For any ${\mathcal {D}}\in \mathrm {Der}(\hat{\mathcal A})^0$ satisfying the condition $[\mathcal D,\partial _x] = 0$, we extend its action to $\hat{\mathcal A}^+$ by setting

$$\begin{aligned} {\mathcal {D}}\sigma _{\alpha ,m} = T_m{\mathcal {D}}\sigma _{\alpha ,0},\quad m\ge 0. \end{aligned}$$

Then the following identities hold true:

$$\begin{aligned} T_k\circ \left( {\mathcal {D}}|_{\hat{\mathcal A}^1}\right) = \left( {\mathcal {D}}|_{\hat{\mathcal A}^{+,1}}\right) \circ T_k;\quad T_{k,l}\circ \left( {\mathcal {D}}|_{\hat{\mathcal A}^2}\right) = \left( {\mathcal {D}}|_{\hat{\mathcal A}^{+,2}}\right) \circ T_{k,l},\quad k,l\ge 0. \end{aligned}$$

Proof

The first identity is obvious from the definition $\mathcal D\sigma _{\alpha ,k} = T_k{\mathcal {D}}\sigma _{\alpha ,0}$. The second one can also be verified by using the definition of the shift operator $T_{k,l}$. The lemma is proved.$\square$

Proposition 6

The odd flows $\frac{\partial }{\partial \tau _m}$ commute with the super extended flow associated with a bihamiltonian vector field X.

Proof

Using Lemma 4 and the definition of the odd flows, it is easy to see that

$$\begin{aligned} \frac{\partial }{\partial t_X}\frac{\partial v^\alpha }{\partial \tau _m} = \frac{\partial }{\partial t_X}T_m\frac{\delta P_0}{\delta \sigma _{\alpha ,0}} = T_mD_X\left( \frac{\delta P_0}{\delta \sigma _{\alpha ,0}}\right) ,\quad \frac{\partial }{\partial \tau _m}\frac{\partial v^\alpha }{\partial t_X} = T_mD_{P_0}\left( \frac{\partial v^\alpha }{\partial t_X}\right) . \end{aligned}$$

Therefore it follows from $[X,P_0] = 0$ and the identity (2.5) that

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _m}, \frac{\partial }{\partial t_X}\right] v^\alpha = 0. \end{aligned}$$

Similarly, by using Lemma 4 again we have

$$\begin{aligned} \frac{\partial }{\partial t_X}\frac{\partial \sigma _{\alpha ,k}}{\partial \tau _m} = \frac{\partial }{\partial t_X}T_{k,m}\frac{\delta P_1}{\delta v^\alpha } = T_{k,m}D_X\left( \frac{\delta P_1}{\delta v^\alpha }\right) . \end{aligned}$$

On the other hand, by using Lemma 3 and the fact that $[P_0,X] = 0$ and $\frac{\delta P_0}{\delta v^\alpha } = 0$, we obtain

$$\begin{aligned} \frac{\partial }{\partial \tau _m}\frac{\partial \sigma _{\alpha ,k}}{\partial t_X} = -\frac{\partial }{\partial \tau _m}T_k\left( D_X\left( \frac{\delta X}{\delta v^\alpha }\right) \right) = -T_{k,m}\left( D_{P_1}\left( \frac{\delta X}{\delta v^\alpha }\right) \right) . \end{aligned}$$

Thus by using (2.4) and $[X,P_1] = 0$, we can conclude that

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _m}, \frac{\partial }{\partial t_X}\right] \sigma _{\alpha ,k} = 0. \end{aligned}$$

The proposition is proved.$\square$

Let us summarize the constructions given in this subsection in the following theorem.

Theorem 7

Let $(P_0,P_1)$ be a bihamiltonian structure with semisimple hydrodynamic leading terms and $\{X_i\}$ be a family of bihamiltonian vector fields, then we have the following super extended integrable hierarchy:

$$\begin{aligned}\frac{\partial v^\alpha }{\partial t_i}= & {} D_{X_i}v^\alpha ,\quad \frac{\partial \sigma _{\alpha ,m}}{\partial t_i} = T_m D_{X_i}\sigma _{\alpha ,0},\\ \frac{\partial v^\alpha }{\partial \tau _m}= & {} T_m\frac{\delta P_0}{\delta \sigma _{\alpha ,0}},\quad \frac{\partial \sigma _{\alpha ,k}}{\partial \tau _m} = T_{k,m}\frac{\delta P_1}{\delta v^\alpha }. \end{aligned}$$

The flows in this hierarchy mutually commute.

2.4 Deformations of the super tau-covers

In this subsection, we construct super tau-covers for tau-symmetric bihamiltonian deformations of the Principal Hierarchy associated with a semisimple Frobenius manifold. Let us first recall how to construct the deformations of the tau-cover (2.14) of the Principal Hierarchy following [9].

We fix a semisimple Frobenius manifold M and use $(P_0^{[0]},P_1^{[0]})$ to denote the bihamiltonian structure (2.13). We denote the two-point functions in the tau-cover (2.14) by $\Omega ^{[0]}_{\alpha ,p;\beta ,q}$ and denote the Hamiltonian densities of $P_0^{[0]}$ by $h_{\alpha ,p}^{[0]}$, which are equal to $\Omega ^{[0]}_{\alpha ,p;1,0}$. Let $(P_0,P_1)$ be a deformation of $(P_0^{[0]},P_1^{[0]})$, then it determines a unique deformation of the Principal Hierarchy according to [7, 26]. By using the results proved in [13], we know that a bihamiltonian deformation of the Principal Hierarchy is tau-symmetric if and only if the central invariants of the deformation of the bihamiltonian structure are constants. In such a case, after an appropriate Miura type transformation the Hamiltonian structure $P_0$ can be represented in the form

$$\begin{aligned} P_0 = P_0^{[0]} = \frac{1}{2}\int \eta ^{\alpha \beta }\sigma _\alpha \sigma _\beta ^1, \end{aligned}$$

and $P_1$ has no $\hat{\mathcal F}^2_2$ components. Moreover, we can also require that the condition of exactness of the bihamiltonian structure is preserved [9, 13]:

$$\begin{aligned} P_0 = [Z,P_1],\quad Z = \int \sigma _1. \end{aligned}$$

In what follows, we will always assume that $P_0$ and Z take the above forms. We still use the same notation $X_{\alpha ,p}\in \hat{\mathcal F}^1$, as we have already used in (2.12) for the flows of the Principal Hierarchy, to denote the unique deformed flows of the Principal Hierarchy, and we will also use $\frac{\partial }{\partial t^{\alpha ,p}}$ to denote the vector field $D_{X_{\alpha ,p}}$. Let $H_{\alpha ,p}\in \hat{\mathcal F}^0$ be the unique deformations of the Hamiltonians of the Principal Hierarchy such that

$$\begin{aligned} X_{\alpha ,p} = -\left[ H_{\alpha ,p}, P_0\right] ,\quad \alpha = 1,\dots ,n,\quad p\ge 0, \end{aligned}$$

(2.26)

and

$$\begin{aligned} H_{\alpha ,-1}:=\int \eta _{\alpha \beta }v^\beta . \end{aligned}$$

(2.27)

Let us define

$$\begin{aligned} h_{\alpha ,p} = D_ZH_{\alpha , p},\quad \alpha = 1,\dots ,n,\quad p\ge 0. \end{aligned}$$

(2.28)

Note that we use an index convention that is different from the one used in [9].

Proposition 7

([9]). We have the following results:

• 1. ${D}_{{X}_{1,0}} = \partial _x$.

• 2. The functionals $H_{\alpha ,p}$ defined in (2.26), (2.27) and the differential polynomials defined in (2.28) satisfy the relations

  $$\begin{aligned} H_{\alpha ,p} = \int h_{\alpha ,p+1},\quad p\ge -1. \end{aligned}$$

We also have the following proposition and theorem on properties of the Hamiltonians and two-point functions of the deformed Principal Hierarchy.

Proposition 8

The following bihamiltonian recursion relation holds true:

$$\begin{aligned}{}[H_{\alpha ,p-1},P_1]=\left( p+\frac{1}{2}+\mu _\alpha \right) [H_{\alpha ,p},P_0] +\sum _{k=1}^p \left( R_k\right) ^\gamma _\alpha [H_{\gamma , p-k},P_0],\quad p\ge 0.\nonumber \\ \end{aligned}$$

(2.29)

Proof

Denote

$$\begin{aligned} Y_{\alpha ,p} = [H_{\alpha ,p-1},P_1]-\left( p+\frac{1}{2}+\mu _\alpha \right) [H_{\alpha ,p},P_0]-\sum _{k=1}^p \left( R_k\right) ^\gamma _\alpha [H_{\gamma , p-k},P_0],\quad p\ge 0, \end{aligned}$$

then from Theorem 3 we know that $Y_{\alpha ,p}\in \hat{\mathcal F}^1_{\ge 2}$. Since the flows $X_{\alpha ,p} = -[H_{\alpha ,p},P_0]$ are bihamiltonian, we conclude that $Y_{\alpha ,p}$ is also a bihamiltonian vector field. Therefore by using the theory of bihamiltonian cohomology developed in [7] we arrive at $Y_{\alpha ,p} = 0$.

$\square$

Theorem 8

([9]). There exist differential polynomials $\Omega _{\alpha ,p;\beta ,q}$ such that they are deformations of $\Omega _{\alpha ,p;\beta ,q}^{[0]}$, and satisfy the following properties:

• 1. $\partial _x\Omega _{\alpha ,p;\beta ,q} =\frac{\partial h_{\beta ,q}}{\partial t^{\alpha ,p}}$.

• 2. $\Omega _{\alpha ,p;\beta ,q} = \Omega _{\beta ,q;\alpha ,p}$ and $\Omega _{\alpha ,p;1,0} = h_{\alpha ,p}$.

• 3. $\frac{\partial \Omega _{\alpha ,p;\beta ,q}}{\partial t^{\lambda ,k}}= \frac{\partial \Omega _{\lambda ,k;\beta ,q}}{\partial t^{\alpha ,p}}$.

To construct the tau-cover of the deformed Principal Hierarchy, we introduce the following normal coordinates as in [12]:

$$\begin{aligned} w_\alpha = h_{\alpha ,0},\quad w^\alpha = \eta ^{\alpha \beta }w_\beta , \end{aligned}$$

(2.30)

then we see that the differential polynomials $w^\alpha$ and $v^\alpha$ are related by a Miura type transformation. In particular, it follows from Proposition 7 that

$$\begin{aligned} H_{\alpha ,-1} =\int w_\alpha = \int \eta _{\alpha \beta }v^\beta . \end{aligned}$$

(2.31)

In terms of the normal coordinates, the tau-cover of the deformed Principal Hierarchy can be represented in the form (cf. (2.14))

$$\begin{aligned} \frac{\partial f_{\alpha ,p}}{\partial t^{\beta ,q}} = \Omega _{\alpha ,p;\beta ,q}, \quad \frac{\partial w^\alpha }{\partial t^{\beta ,q}} = \eta ^{\alpha \lambda }\partial _x\Omega _{\lambda ,0;\beta ,q}. \end{aligned}$$

(2.32)

Let us proceed to construct its super tau-cover. To this end we introduce odd variables $\Phi _{\alpha ,p}^m$, as we do for the super tau-cover of the Principal Hierarchy, such that

$$\begin{aligned} \partial _x\Phi _{\alpha ,p}^m = \frac{\partial h_{\alpha ,p}}{\partial \tau _m},\quad m\ge 0. \end{aligned}$$

(2.33)

Here the odd flows $\frac{\partial }{\partial \tau _m}$ are defined by (2.21). By using Theorem 8 we obtain the following identity:

$$\begin{aligned} \frac{\partial }{\partial t^{\beta ,q}}\frac{\partial h_{\alpha ,p}}{\partial \tau _m} = \partial _x\frac{\partial \Omega _{\alpha ,p;\beta ,q}}{\partial \tau _m}. \end{aligned}$$

Therefore we conclude that the following definitions of the evolutions of the odd variables $\Phi _{\alpha ,p}^m$ along the flows $\frac{\partial }{\partial t^{\beta ,q}}$ are compatible with (2.33):

$$\begin{aligned} \frac{\partial \Phi _{\alpha ,p}^m}{\partial t^{\beta ,q}} = \frac{\partial \Omega _{\alpha ,p;\beta ,q}}{\partial \tau _m}. \end{aligned}$$

To define the evolutions of $\Phi _{\alpha ,p}^m$ along the odd flows $\frac{\partial }{\partial \tau _k}$, we need the following lemma.

Lemma 5

There exist differential polynomials $F_{\alpha ,p}\in \hat{\mathcal A}^2$ such that

$$\begin{aligned} \frac{\partial }{\partial \tau _k}\frac{\partial h_{\alpha ,p}}{\partial \tau _m} = T_{m,k}\partial _xF_{\alpha ,p},\quad \alpha = 1,\dots ,n;\ p,m,k\ge 0. \end{aligned}$$

Proof

By using the definition (2.1) of $D_{P_i}$ and the fact that the vector fields $X_{\alpha ,p}$ are bihamiltonian, it is easy to see that

$$\begin{aligned} \int D_{P_1}D_{P_0}h_{\alpha ,p} = [P_1,[P_0,H_{\alpha ,p-1}]] = 0. \end{aligned}$$

Therefore from (2.22) it follows that there exists $F_{\alpha ,p}\in \hat{\mathcal A}^2$ such that

$$\begin{aligned} \frac{\partial }{\partial \tau _1}\frac{\partial h_{\alpha ,p}}{\partial \tau _0} = \partial _xF_{\alpha ,p}. \end{aligned}$$

Now from Lemma 3 and (2.23) it follows that

$$\begin{aligned} \frac{\partial }{\partial \tau _k}\frac{\partial h_{\alpha ,p}}{\partial \tau _m} = \frac{\partial }{\partial \tau _k}T_m\frac{\partial h_{\alpha ,p}}{\partial \tau _0} = T_{m,k}\partial _xF_{\alpha ,p}. \end{aligned}$$

The lemma is proved.$\square$

Now we are ready to construct the super tau-cover of the deformed Principal Hierarchy.

Theorem 9

Let M be a semisimple Frobenius manifold and $(P_0,P_1)$ be a deformation of the bihamiltonian structure (2.13) with constant central invariants, then the following flows together with the super extended flows associated with $\frac{\partial }{\partial t^{\alpha ,p}}$ form the super tau-cover of the deformed Principal Hierarchy:

$$\begin{aligned} \frac{\partial f_{\alpha ,p}}{\partial t^{\beta ,q}}&= \Omega _{\alpha ,p;\beta ,q},\quad \frac{\partial f_{\alpha ,p}}{\partial \tau _m}= \Phi _{\alpha ,p}^m,\\ \frac{\partial \Phi _{\alpha ,p}^m}{\partial t^{\beta ,q}}&= \frac{\partial \Omega _{\alpha ,p;\beta ,q}}{\partial \tau _m},\quad \frac{\partial \Phi _{\alpha ,p}^m}{\partial \tau _k}= T_{m,k}F_{\alpha ,p}. \end{aligned}$$

Recall that when the diagonal matrix $\mu$ of a Frobenius manifold M satisfies the condition $\frac{1-2k}{2}\notin \mathrm {Spec}(\mu )$ for any $k = 1,2,\dots$, the odd variables $\Phi _{\alpha ,p}^m$ for the super tau-cover of the Principal Hierarchy are redundant since they can be represented by elements in $\hat{\mathcal A}^+$. Proposition 9 that we are to give below shows that the deformed super tau-cover has the same property, which will play an important role in our consideration of the deformation of the Virasoro symmetries.

We start with the definition of generalized shift operators.

Definition 4

We define the shift operators ${\hat{T}}_k$ for $k\ge 0$ to be the linear operators from $\hat{\mathcal A}^{+,1}$ to $\hat{\mathcal A}^{+,1}$ such that

$$\begin{aligned} {\hat{T}}_k(f\sigma _{\alpha ,m}^l) = f\sigma _{\alpha ,m+k}^l,\quad f\in \hat{\mathcal A}^0,\ m\ge 0. \end{aligned}$$

The following lemma is easy to verify.

Lemma 6

The operators ${\hat{T}}_k$ commute with $\partial _x$ and are compatible with the recursion relation (2.18).

Lemma 7

If $\Phi _{\alpha ,p}^0$ can be represented by an element in $\hat{\mathcal A}^{+,1}$, then so does $\Phi _{\alpha ,p}^m$. In this case $\Phi _{\alpha ,p}^m = {\hat{T}}_m\Phi _{\alpha ,p}^0$ for $m\ge 1$.

Proof

By using the relations (2.33) and (2.23), it is easy to see that

$$\begin{aligned} \partial _x{\hat{T}}_m\Phi _{\alpha ,p}^0 = {\hat{T}}_m \frac{\partial h_{\alpha ,p}}{\partial \tau _0} = \frac{\partial h_{\alpha ,p}}{\partial \tau _m} = \partial _x \Phi _{\alpha ,p}^m. \end{aligned}$$

The lemma is proved.$\square$

We will use the notation $\Phi _{\alpha ,p}^m\in \hat{\mathcal A}^+$ to mean that $\Phi _{\alpha ,p}^m$ can be represented by an element in $\hat{\mathcal A}^+$.

Proposition 9

We have $\Phi _{\alpha ,0}^m\in \hat{\mathcal A}^{+}$ and

$$\begin{aligned} \left( \prod _{k=1}^p\left( k-\frac{1}{2}+\mu _\alpha \right) \right) \Phi _{\alpha ,p}^m\in \hat{\mathcal A}^+,\quad p\ge 1. \end{aligned}$$

(2.34)

Proof

It follows from Lemma 7 that we only need to prove this lemma for $m = 0$. For $\Phi _{\alpha ,0}^0$, it is easy to see from (2.31) that there exist differential polynomials $g_{\alpha }\in \hat{\mathcal A}^0_{ \ge 1}$ such that

$$\begin{aligned} h_{\alpha ,0} = \eta _{\alpha \beta }v^\beta +\partial _xg_{\alpha }. \end{aligned}$$

(2.35)

Therefore we arrive at

$$\begin{aligned} \Phi _{\alpha ,0}^0 = \sigma _{\alpha ,0}+\frac{\partial g_\alpha }{\partial \tau _0}\in \hat{\mathcal A}. \end{aligned}$$

(2.36)

We proceed to consider $\Phi _{\alpha ,p}^0$ for $p\ge 1$. By using Proposition 7 we can rewrite (2.29) as follows:

$$\begin{aligned} \left[ \int h_{\alpha ,p}, P_1\right] =\left( p+\frac{1}{2}+\mu _\alpha \right) \left[ \int h_{\alpha ,p+1}, P_0\right] +\sum _{k=1}^p \left( R_k\right) ^\gamma _\alpha \left[ \int h_{\gamma ,p-k+1}, P_0\right] .\nonumber \\ \end{aligned}$$

(2.37)

By taking $p=0$ in(2.37) we get

$$\begin{aligned} \left( \frac{1}{2}+\mu _\alpha \right) \int \frac{\partial h_{\alpha ,1}}{\partial \tau _0} = \int \frac{\partial h_{\alpha ,0}}{\partial \tau _1}, \end{aligned}$$

so there exists a differential polynomial $p_{\alpha ,1}\in \hat{\mathcal A}^1$ such that

$$\begin{aligned} \left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial h_{\alpha ,1}}{\partial \tau _0} = \frac{\partial h_{\alpha ,0}}{\partial \tau _1}+\partial _x p_{\alpha ,1}. \end{aligned}$$

Therefore we have

$$\begin{aligned} \left( \frac{1}{2}+\mu _\alpha \right) \Phi _{\alpha ,1}^0 = \Phi _{\alpha ,0}^1+p_{\alpha ,1}\in \hat{\mathcal A}^+. \end{aligned}$$

For general $p\ge 1$, we can prove (2.34) by using (2.37) and induction on p. The proposition is proved.$\square$

3 Deformations of Virasoro Symmetries: Formulation

In this section, we first recall the theory of variational bihamiltonian cohomology developed in [23] and then explain how to use it to study Virasoro symmetries of the deformed Principal Hierarchies. We also use the example of the deformation of the Riemann hierarchy to illustrate our approach to the study of Virasoro symmetries.

3.1 Variational bihamiltonian cohomologies

In [23], we established a cohomology theory on the space $\mathrm {Der}^\partial (\hat{\mathcal A})$ consisting of derivations on $\hat{\mathcal A}$ that commute with $\partial _x$. This theory provides us suitable tools to study Virasoro symmetries of deformations of the Principal Hierarchies. We recall the basic definitions and results in this subsection.

Let us define the space $\mathrm {Der}^\partial (\hat{\mathcal A})$ by

$$\begin{aligned} \mathrm {Der}^\partial (\hat{\mathcal A})= \{X\in \mathrm {Der}(\hat{\mathcal A})\mid [X,\partial _x] = 0\}, \end{aligned}$$

it admits a gradation induced from $\mathrm {Der}(\hat{\mathcal A})$ and we denote $\mathrm {Der}^\partial (\hat{\mathcal A})^p_d = \mathrm {Der}^\partial (\hat{\mathcal A})\cap \mathrm {Der}(\hat{\mathcal A})^p_d$.

Lemma 8

${\mathrm {Der}^\partial (\hat{\mathcal A})}^p_d = 0$ for $p\le -2$ or $d<0$.

Proof

Let us choose a local coordinate system $(w^\alpha ,\phi _\alpha )$ on ${\hat{M}}$. Assume $X\in \mathrm {Der}^\partial (\hat{\mathcal A})$ with super degree $p\le -2$ or $d< 0$. Then by definition this means

$$\begin{aligned} X(w^\alpha ) = X(\phi _\alpha ) = 0. \end{aligned}$$

Since $[X,\partial _x] = 0$, we immediately see that $X({w^{\alpha ,s}}) = 0$ and $X(\phi _\alpha ^s) = 0$ for $s\ge 0$. Hence $X = 0$ and the lemma is proved.$\square$

Let $P^{[0]}$ be a Hamiltonian structure of hydrodynamic type and $(P_0^{[0]},P_1^{[0]})$ be a semisimple bihamiltonian structure of hydrodynamic type. Then by using (2.3) we have a complex $(\mathrm {Der}^\partial (\hat{\mathcal A}),D_{P^{[0]}})$ and a double complex $(\mathrm {Der}^\partial (\hat{\mathcal A}),D_{P_0^{[0]}},D_{P_1^{[0]}})$. We define the following cohomology groups:

$$\begin{aligned} H^p_d\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P^{[0]}\bigr )= & {} \frac{{\mathrm {Der}^\partial (\hat{\mathcal A})}^p_d\cap \ker D_{P^{[0]}}}{{\mathrm {Der}^\partial (\hat{\mathcal A})}^p_d\cap {{\,\mathrm{Im}\,}}D_{P^{[0]}}},\quad p,\,d\ge 0, \end{aligned}$$

(3.1)

$$\begin{aligned} BH^p_d\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]},P_1^{[0]}\bigr )= & {} \frac{{\mathrm {Der}^\partial (\hat{\mathcal A})}^p_d\cap \ker D_{P_0^{[0]}}\cap \ker D_{P_1^{[0]}}}{{\mathrm {Der}^\partial (\hat{\mathcal A})}^p_d\cap {{\,\mathrm{Im}\,}}D_{P_0^{[0]}}D_{P_1^{[0]}}}, \quad p,\,d\ge 0.\nonumber \\ \end{aligned}$$

(3.2)

Note that the spaces ${\mathrm {Der}^\partial (\hat{\mathcal A})}^{-1}_d \ne 0$ for $d\ge 0$ and they must be taken into account while computing the cohomology groups. For example, the space $H^0_d(\mathrm {Der}^\partial (\hat{\mathcal A}),P^{[0]})$ is given by

$$\begin{aligned} H^0_d\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P^{[0]}\bigr ) = \frac{\ker (D_{P^{[0]}}:{\mathrm {Der}^\partial (\hat{\mathcal A})}^{0}_d\rightarrow {\mathrm {Der}^\partial (\hat{\mathcal A})}^{1}_{d+1})}{{{\,\mathrm{Im}\,}}(D_{P^{[0]}}:{\mathrm {Der}^\partial (\hat{\mathcal A})}^{-1}_{d-1}\rightarrow {\mathrm {Der}^\partial (\hat{\mathcal A})}^{0}_{d})}. \end{aligned}$$

By using the canonical symplectic structure on ${\hat{M}}$, we can identify the space $\mathrm {Der}^\partial (\hat{\mathcal A})$ with the space ${\bar{\Omega }}$ of local functionals of variational 1-forms, and the vector fields $D_{P^{[0]}}$, $D_{P_0^{[0]}}$ and $D_{P_1^{[0]}}$ induce differentials on ${\bar{\Omega }}$ via Lie derivatives. This is the reason why we call the above cohomology groups the variational cohomology groups. In [23], the cohomology groups (3.1) and (3.2) are computed by converting them to the cohomology groups on the space ${\bar{\Omega }}$. The details of the computation of these cohomology groups are not used in the present paper, so we omit them and refer the readers to [23]. The following result plays an essential role in the present paper.

Theorem 10

([23]). We have the following results on the cohomology groups (3.1) and (3.2):

• 1. $H^p_d\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P^{[0]}\bigr ) = 0$ for $p\ge 0$, $d> 0$.

• 2. $BH^0_{\ge 2}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]},P_1^{[0]}\bigr ) = 0$.

• 3. $BH^1_{\ge 4}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]},P_1^{[0]}\bigr ) = 0$.

• 4. $BH^1_3\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]},P_1^{[0]}\bigr ) \cong \oplus _{i=1}^nC^\infty ({\mathbb {R}})$.

Moreover, if we denote the action of a cocycle $X\in {\mathrm {Der}^\partial (\hat{\mathcal A})}^1_3$ on the i-th canonical coordinate $u^i$ by

$$\begin{aligned} X(u^i) = \sum _{j=1}^n\sum _{k=0}^3 X_{i,j}^k\theta _j^{3-k},\quad X_{i,j}^k\in \hat{\mathcal A}^0_k, \end{aligned}$$

then the cohomology class [X] is determined by the following functions:

$$\begin{aligned} c_1 = \frac{X_{1,1}^0}{(f^1)^2},\quad c_2 = \frac{X_{2,2}^0}{(f^2)^2},\quad \dots ,\quad c_n = \frac{X_{n,n}^0}{(f^n)^2}. \end{aligned}$$

Here each function $c_i$ depends only on the i-th canonical coordinate $u^i$, and $f^i$ is the function defined in (2.7).

3.2 Virasoro symmetries of the Principal Hierarchies

Virasoro symmetries as well as Virasoro constraints are central conceptions in the study of modern mathematical physics, see, e.g. [11, 16, 17, 30]. In this subsection, we recall the construction of Virasoro symmetries of the super tau-cover of the Principal Hierarchy following [22]. In [11], a family of infinitely many symmetries $\frac{\partial }{\partial s_m^{even}}$ for $m\ge -1$ of the tau-cover of the Principal Hierarchy associated with a Frobenius manifold M was constructed. This family of symmetries are called the Virasoro symmetries due to the property

$$\begin{aligned} \left[ \frac{\partial }{\partial s_k^{even}}, \frac{\partial }{\partial s_l^{even}}\right] = (l-k)\frac{\partial }{\partial s_{k+l}^{even}},\quad k,l\ge -1. \end{aligned}$$

These symmetries can be represented by a family of quadratic differential operators $L_m^{even}$ of the form

$$\begin{aligned} L^{even}_m&= {} \sum _{p,q\ge 0}a_m^{\alpha ,p;\beta ,q}\frac{\partial ^2}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}+{b}_{m;\alpha ,p}^{\beta ,q} t^{\alpha ,p}\frac{\partial }{\partial t^{\beta ,q}}+c_{m;\alpha ,p;\beta ,q} t^{\alpha ,p} t^{\beta ,q}\\&\quad +\frac{1}{4}\delta _{m,0}\mathrm {tr}\left( \frac{1}{4}-\mu ^2\right) ,\end{aligned}$$

where $a_m^{\alpha ,p;\beta ,q}$, ${b}_{m;\alpha ,p}^{\beta ,q}$, $c_{m;\alpha ,p;\beta ,q}$ are some constants determined by the monodromy data of M and one may refer to [11] for details. These operators satisfy the Virasoro commutation relation

$$\begin{aligned}{}[L_k^{even},L_l^{even}] = (k-l)L_{l+k}^{even}. \end{aligned}$$

In this paper, we only need the explicit expressions for $L_{-1}^{even}$ and $L_2^{even}$ which are given by

$$\begin{aligned} L_{-1}^{even}&= \frac{1}{2} \eta _{\alpha \beta }t^{\alpha ,0}t^{\beta ,0}+\sum _{p\ge 1}t^{\alpha ,p}\frac{\partial }{\partial t^{\alpha ,p-1}}, \end{aligned}$$

(3.3)

$$\begin{aligned} L_2^{even}&= a^{\alpha \beta }\frac{\partial ^2}{\partial t^{\alpha ,1}\partial t^{\beta ,0}}+b^{\alpha \beta }\frac{\partial ^2}{\partial t^{\alpha ,0}\partial t^{\beta ,0}}+{\mathcal {L}}_2^{even} + c_{2;\alpha ,p;\beta ,q}t^{\alpha ,p}t^{\beta ,q}, \end{aligned}$$

(3.4)

where the constants have the expressions

$$\begin{aligned} a^{\alpha \beta }= & {} \eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\beta \right) \left( \frac{1}{2}+\mu _\alpha \right) \left( \frac{3}{2}+\mu _\alpha \right) , \end{aligned}$$

(3.5)

$$\begin{aligned} b^{\alpha \beta }= & {} \frac{1}{2} \eta ^{\beta \gamma }(R_1)^\alpha _\gamma \left( \frac{1}{4}+3\mu _\beta -3\mu _{\beta }^2\right) , \end{aligned}$$

(3.6)

and the operator ${\mathcal {L}}_2^{even}$ is given by

$$\begin{aligned} {\mathcal {L}}_2^{even}&= \sum _{p\ge 0}\left( p+\frac{1}{2}+\mu _\alpha \right) \left( p+\frac{3}{2}+\mu _\alpha \right) \left( p+\frac{5}{2}+\mu _\alpha \right) t^{\alpha ,p}\frac{\partial }{\partial t^{\alpha ,p+2}} \nonumber \\ {}&\quad +\, \sum _{p\ge 0}\sum _{1\le r\le p+2}\left( 3\left( p+\frac{1}{2}+\mu _\alpha \right) ^2+6\left( p+\frac{1}{2}+\mu _\alpha \right) +2\right) \nonumber \\ {}&\qquad \times \left( R_r\right) ^\beta _\alpha t^{\alpha ,p}\frac{\partial }{\partial t^{\beta ,p-r+2}}\nonumber \\ {}&\quad +\, \sum _{p\ge 0}\sum _{2\le r\le p+2}\left( 3p+\frac{9}{2}+3\mu _\alpha \right) \left( R_{r,2}\right) ^\beta _\alpha t^{\alpha ,p}\frac{\partial }{\partial t^{\beta ,p-r+2}}\nonumber \\ {}&\quad +\, \sum _{p\ge 1}\sum _{3\le r\le p+2}\left( R_{r,3}\right) ^\beta _\alpha t^{\alpha ,p}\frac{\partial }{\partial t^{\beta ,p-r+2}}. \end{aligned}$$

(3.7)

The explicit expressions for the matrices $R_{k,l}$ and constants $c_{2;\alpha ,p;\beta ,q}$ are not used in this paper, so we omit them.

We have the following theorem for the Virasoro symmetries of the tau-cover of the Principal Hierarchy.

Theorem 11

([11]). Let us define the following time-dependent flows for $m\ge -1$:

$$\begin{aligned} \frac{\partial f_{\lambda ,k}}{\partial s_m^{even}}&= \frac{\partial }{\partial t^{\lambda ,k}}\left( \sum a_m^{\alpha ,p;\beta ,q}f_{\alpha ,p}f_{\beta ,q}+\sum b_{m;\alpha ,p}^{\beta ,q} t^{\alpha ,p}f_{\beta ,q}+\sum c_{m;\alpha ,p;\beta ,q} t^{\alpha ,p} t^{\beta ,q}\right) ,\\ \frac{\partial v^{\lambda }}{\partial s_m^{even}}&= \eta ^{\lambda \gamma }\frac{\partial }{\partial t^{1,0}}\frac{\partial f_{\gamma ,0}}{\partial s_m^{even}}. \end{aligned}$$

Then the following commutation relation holds true:

$$\begin{aligned} \left[ \frac{\partial }{\partial s_k^{even}}, \frac{\partial }{\partial t^{\alpha ,p}}\right] = 0,\quad \left[ \frac{\partial }{\partial s_k^{even}}, \frac{\partial }{\partial s_m^{even}}\right] = (m-k)\frac{\partial }{\partial s_{k+m}^{even}},\quad k,m\ge -1. \end{aligned}$$

We also have the following theorem for the Virasoro symmetries of the super tau-cover of the Principal Hierarchy.

Theorem 12

([22]). Let us define

$$\begin{aligned} \frac{\partial }{\partial s_m^{odd}} = \sum _{p\ge 0}(p+c_0)\tau _p\frac{\partial }{\partial \tau _{p+m}},\quad m \ge -1, \end{aligned}$$

where $c_0$ is an arbitrary constant, and let us define $\frac{\partial }{\partial \tau _{-1}}$ to be zero, then the following flows are symmetries of the super tau-cover of the Principal Hierarchy associated with a Frobenius manifold:

$$\begin{aligned} \frac{\partial f_{\alpha , p}}{\partial s_m}&= \frac{\partial f_{\alpha , p}}{\partial s^{even}_m}+\frac{\partial f_{\alpha , p}}{\partial s^{odd}_m}, \quad \frac{\partial \Phi ^n_{\alpha ,p}}{\partial s_m}=\frac{\partial }{\partial \tau _n}\left( \frac{\partial f_{\alpha ,p}}{\partial s_m}\right) ,\\ \frac{\partial v^\alpha }{\partial s_m}&= \frac{\partial v^\alpha }{\partial s^{even}_m}+\frac{\partial v^\alpha }{\partial s^{odd}_m},\quad \frac{\partial \sigma _{\alpha ,p}}{\partial s_m}=\frac{\partial }{\partial \tau _p}\left( \frac{\partial f_{\alpha ,0}}{\partial s_m}\right) . \end{aligned}$$

Moreover, these flows satisfy the commutation relation

$$\begin{aligned}\left[ \frac{\partial }{\partial s_k},\frac{\partial }{\partial s_m}\right] =(m-k) \frac{\partial }{\partial s_{k+m}},\quad k, m\ge -1.\end{aligned}$$

Remark 2

Let us explain why there is an arbitrary constant involved in the Virasoro symmtries of the super tau-cover of the Principal Hierarchy. If we assign the odd time variables $\tau _k$ a degree $c_k$, then we can modify the zeroth Virasoro symmetry of the tau-cover of the Principal Hierarchy, which is a homogeneous condition, to the following symmetry of the super tau-cover:

$$\begin{aligned} \frac{\partial }{\partial s_0} = \frac{\partial }{\partial s_0^{even}}+\sum _{k\ge 0}c_k\tau _k\frac{\partial }{\partial \tau _k}. \end{aligned}$$

By requiring that the above flow is a symmetry of the super tau-cover of the Principal Hierarchy we arrive at $c_k = c_0+k$, here $c_0$ is an arbitrary constant.

Let us explain the motivation to introduce the non-local odd variables $\sigma _{\alpha ,k}$ for $k\ge 1$ and the super extension of the tau-cover of the deformed Principal Hierarchy. For a given tau-symmetric bihamiltonian deformation of the Principal Hierarchy of a semisimple Frobenius manifold, we want to deform the Virasoro symmetries given in Theorem 11, i.e., to construct the flows $\frac{\partial }{\partial {\tilde{s}}_m^{even}}$ as deformations of $\frac{\partial }{\partial s_m^{even}}$, such that

$$\begin{aligned} \left[ \frac{\partial }{\partial {\tilde{s}}_k^{even}}, \frac{\partial }{\partial {\tilde{t}}^{\alpha ,p}}\right] = 0,\quad \left[ \frac{\partial }{\partial {\tilde{s}}_k^{even}}, \frac{\partial }{\partial {\tilde{s}}_m^{even}}\right] = (m-k)\frac{\partial }{\partial {\tilde{s}}_{k+m}^{even}},\quad k,m\ge -1, \end{aligned}$$

here we denote by $\frac{\partial }{\partial {\tilde{t}}^{\alpha ,p}}$ the flows of the deformed Principal Hierarchy. Due to the Virasoro commutation relation, we only need to find the flows $\frac{\partial }{\partial \tilde{s}_{-1}^{even}}$, $\frac{\partial }{\partial {\tilde{s}}_2^{even}}$, and use them to generate all other flows $\frac{\partial }{\partial {\tilde{s}}_m^{even}}$. It is proved in [9] that the symmetry $\frac{\partial }{\partial {\tilde{s}}_{-1}^{even}}$ always exists and therefore it remains to construct the flow $\frac{\partial }{\partial {\tilde{s}}_{2}^{even}}$ which satisfies the following equations:

$$\begin{aligned} \left[ \frac{\partial }{\partial {\tilde{s}}_{2}^{even}}, \frac{\partial }{\partial {\tilde{t}}^{\alpha ,p}}\right] = 0,\quad \alpha = 1,\dots ,n,\quad p\ge 0. \end{aligned}$$

Since there are infinitely many equations, it is not easy to solve them. From the study of the theory of variational bihamiltonian cohomologies, it follows that the problem of solving the above equations can be converted to solve the following two equations:

$$\begin{aligned} \left[ \frac{\partial }{\partial {\tilde{s}}_{2}^{even}}, \frac{\partial }{\partial \tau _0}\right] =\left[ \frac{\partial }{\partial {\tilde{s}}_{2}^{even}}, \frac{\partial }{\partial \tau _1}\right] = 0. \end{aligned}$$

Since the odd flows $\frac{\partial }{\partial \tau _0}$ and $\frac{\partial }{\partial \tau _1}$ are not contained in the original tau-cover, we need a super extension of it. However, the above equations do not hold true at the dispersionless level. In [22], we prove that one can remedy this problem by adding infinitely many odd time variables $\tau _m$ and odd flows $\frac{\partial }{\partial \tau _m}$ for $m\ge 0$ to the Virasoro operator $L_k^{even}$, as described in Theorem 12.

The introduction of the odd flows $\frac{\partial }{\partial \tau _m}$ is motivated by the following simple observation. Let M be an n-dimensional Frobenius manifold, then we have

$$\begin{aligned} \frac{\partial v^\alpha }{\partial \tau _1} = {\mathcal {R}}\frac{\partial v^\alpha }{\partial \tau _0},\quad {\mathcal {R}} = {\mathcal {P}}_1\circ {\mathcal {P}}_0^{-1}, \end{aligned}$$

here ${\mathcal {P}}_0$ and ${\mathcal {P}}_1$ are the Hamiltonian operators of the bihamiltonian structure (2.13). The non-local operator ${\mathcal {R}}$ is called the recursion operator and we can define the odd flows recursively as follows:

$$\begin{aligned} \frac{\partial v^\alpha }{\partial \tau _{m+1}} = {\mathcal {R}}\frac{\partial v^\alpha }{\partial \tau _m},\quad m\ge 1. \end{aligned}$$

Due to the non-local nature of the recursion operator, we see that generally the action of the flow $\frac{\partial }{\partial \tau _m}$ on $v^\alpha$ can not be represented by elements of $\hat{\mathcal A}$ for $m\ge 2$. To overcome this non-locality problem we introduce, as it is typically done in the theory of integrable system, the odd variables $\sigma _{\alpha ,k}$ for $k\ge 1$ to describe the actions of the odd flows $\frac{\partial }{\partial \tau _m}$. The constructions of $\sigma _{\alpha ,k}$ are given by (2.18) and the flows $\frac{\partial }{\partial \tau _m}$ are defined in (2.21).

Remark 3

The idea of introducing non-local odd variables $\sigma _{\alpha ,k}$ to study the non-local Hamiltonian structures is presented and illustrated via some examples in [18], see also [27].

3.3 Formulation of the deformation problem

In this subsection, we first state the main problem of this paper, then explain the motivation and strategy of our proof of the Main Theorem 1.

From now on, we fix a semisimple Frobenius manifold M of dimension n and let $(P_0,P_1)$ be a deformation of the bihamiltonian structure (2.13) with constant central invariants. Then $(P_0,P_1)$ determines a unique deformation of the Principal Hierarchy associated with M. After a suitable Miura type transformation we may assume, as we do in Sect. 2.4, that

$$\begin{aligned} P_0 = \frac{1}{2}\int \eta ^{\alpha \beta }\sigma _\alpha \sigma _\beta ^1 = [Z,P_1],\quad Z = \int \sigma _1, \end{aligned}$$

and $P_1$ has no $\hat{\mathcal F}^2_2$ components. We also introduce odd variables $\sigma _{\alpha ,m}$ for $m\ge 0$ as we explained in Sect. 2.3.

Let us denote by $\hat{\mathcal A}^{++}$ the following $\hat{\mathcal A}^+$-module

$$\begin{aligned} \hat{\mathcal A}^{++}:= \hat{\mathcal A}^+\left[ \Phi _{\alpha ,p}^m,\,m\ge 1\right] . \end{aligned}$$

From Proposition 9 it follows that $\hat{\mathcal A}^{++}=\hat{\mathcal A}^+$ if the Frobeius manifold M satisfies the condition

$$\begin{aligned} k-\frac{1}{2}+\mu _\alpha \ne 0,\quad \forall k\ge 1,\quad \forall \alpha = 1,\dots ,n. \end{aligned}$$

Let $\hat{\mathcal A}^{Vir}$ be the following $\hat{\mathcal A}^{++}$-module:

$$\begin{aligned} \hat{\mathcal A}^{Vir} = \hat{\mathcal A}^{++}[f_{\alpha ,p}][[\, t^{\alpha ,p},\tau _m]], \end{aligned}$$

here the time variables $\tau _m$ are odd. We will consider the space $\mathrm {Der}^\partial (\hat{\mathcal A}^{Vir})$, which consists of derivations of the space $\hat{\mathcal A}^{Vir}$ that commute with $\partial _x$. Here we extend the action of $\partial _x$ to the space $\hat{\mathcal A}^{Vir}$ in the following natural way:

$$\begin{aligned} \partial _x \Phi _{\alpha ,p}^m = \frac{\partial h_{\alpha ,p}}{\partial \tau _m},\quad \partial _x f_{\alpha ,p} = h_{\alpha ,p},\quad \partial _x t^{\alpha ,p} = \delta ^{\alpha ,1}\delta ^{p,0},\quad \partial _x\tau _m = 0. \end{aligned}$$

Our problem can be stated as follows: to find a unique derivation $X\in \mathrm {Der}^\partial (\hat{\mathcal A})^0_{\ge 0}$ such that the flow $\frac{\partial }{\partial s_2}\in \mathrm {Der}^\partial (\hat{\mathcal A}^{Vir})$ defined by

$$\begin{aligned} \frac{\partial v_\lambda }{\partial s_2} =\&a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) \nonumber \\&+Xv_{\lambda }+{\mathcal {L}}_2v_\lambda , \end{aligned}$$

(3.8)

$$\begin{aligned} \frac{\partial \sigma _{\lambda ,0}}{\partial s_2} =\&a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) \nonumber \\&+X\sigma _{\lambda ,0} +M^\zeta _{\lambda }\sigma _{\zeta ,2}+N^\zeta _{\lambda }\sigma _{\zeta ,1}+\mathcal {L}_2\sigma _{\lambda ,0}, \end{aligned}$$

(3.9)

satisfies the conditions

$$\begin{aligned} \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial \tau _0}\right] = \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial \tau _1}\right] = 0, \end{aligned}$$

(3.10)

and we require that the leading term of X is determined by the Virasoro symmetry $\frac{\partial }{\partial s_2}$ of the super tau-cover of the Principal Hierarchy given in Theorem 12. The actions of the flow $\frac{\partial }{\partial s_2}$ on $f_{\alpha ,p}$ and $\Phi _{\alpha ,p}^m$ are omitted here for simplicity. These actions can be derived from (3.20) and the details will be given later at the end of this subsection. We also define that

$$\begin{aligned} \frac{\partial t^{\alpha ,p}}{\partial s_2} = \frac{\partial \tau _m}{\partial s_2} = 0. \end{aligned}$$

Moreover, the operator ${\mathcal {L}}_2$ is given by

$$\begin{aligned} {\mathcal {L}}_2 = {\mathcal {L}}_2^{even}+\sum _{p\ge 0}(p+c_0)\tau _p\frac{\partial }{\partial \tau _{p+2}}, \end{aligned}$$

and $M^\zeta _\lambda ,\ N^\zeta _\lambda \in \hat{\mathcal A}^0$ are some differential polynomials whose definitions will be given later. The flows $\frac{\partial }{\partial \tau _0}$ and $\frac{\partial }{\partial \tau _1}$ are also extended to the space $\hat{\mathcal A}^{Vir}$ naturally by using the super tau-cover of the deformed Principal Hierarchy and by defining

$$\begin{aligned} \frac{\partial t^{\alpha ,p}}{\partial \tau _i} = 0,\quad \frac{\partial \tau _m}{\partial \tau _i} = \delta _{i,m},\quad i = 0,1. \end{aligned}$$

If we find such a derivation X, then we can prove that the flow $\frac{\partial }{\partial s_2}$ is a symmetry of the super tau-cover of the deformed Principal Hierarchy, i.e.,

$$\begin{aligned} \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial t^{\alpha ,p}}\right] = 0,\quad \alpha = 1,\dots , n,\quad p\ge 0. \end{aligned}$$

The above commutator should be understood as the natural commutator defined in the space $\mathrm {Der}^\partial (\hat{\mathcal A}^{Vir})$ and the actions of the flows $\frac{\partial }{\partial t^{\alpha ,p}}$ are naturally extended to $\hat{\mathcal A}^{Vir}$. Therefore a priori we have

$$\begin{aligned} \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial t^{\alpha ,p}}\right] \in \mathrm {Der}^\partial (\hat{\mathcal A}^{Vir}). \end{aligned}$$

However, if we can show that

$$\begin{aligned} \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial t^{\alpha ,p}}\right] \in \mathrm {Der}(\hat{\mathcal A})^0_{\ge 2}, \end{aligned}$$

(3.11)

i.e., the actions of the above commutator can be restricted to the space $\hat{\mathcal A}$, then we conclude the vanishing of (3.11) from the property that

$$\begin{aligned}BH^0_{\ge 2}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),{P_0^{[0]}},{P_1^{[0]}}\bigr ) = 0,\end{aligned}$$

and the fact that the commutator (3.11) is a cocycle. Here we use the definition (3.2) and Lemma 8 to arrive at the fact that

$$\begin{aligned}BH^0_{\ge 2}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),{P_0^{[0]}},{P_1^{[0]}}\bigr ) = \mathrm {Der}^\partial (\hat{\mathcal A})^0_{\ge 2}\cap \ker D_{P_0^{[0]}}\cap \ker D_{P_1^{[0]}}. \end{aligned}$$

By using the definition (3.8) and (3.9) of $\frac{\partial }{\partial s_2}$ and after a simple computation, we arrive at

$$\begin{aligned} \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial t^{\alpha ,p}}\right] v_\lambda \in \hat{\mathcal A},\quad \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial t^{\alpha ,p}}\right] \sigma _{\lambda ,0}\in \hat{\mathcal A}^+, \end{aligned}$$

so the condition (3.11) is actually a locality condition, i.e., it is equivalent to

$$\begin{aligned} \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial t^{\alpha ,p}}\right] v_\lambda \in \hat{\mathcal A},\quad \left[ \frac{\partial }{\partial s_2}, \frac{\partial }{\partial t^{\alpha ,p}}\right] \sigma _{\lambda ,0}\in \hat{\mathcal A}, \end{aligned}$$

(3.12)

which is the condition we actually need to check.

Let us explain how to find a unique $X\in \mathrm {Der}^\partial (\hat{\mathcal A})$ such that the conditions in (3.10) hold true. To this end, we first rewrite the conditions in (3.10) into the equations for X as follows:

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, X\right] = I_0,\quad \left[ \frac{\partial }{\partial \tau _1}, X\right] = I_1, \end{aligned}$$

(3.13)

where $I_0$ and $I_1$ are some derivations that will be given later. The uniqueness of X is a consequence of the equations (3.13) and the fact that $BH^0_{\ge 2}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]},P_1^{[0]}\bigr ) = 0$, since the leading term of X is fixed. We will prove the existence of X by taking the following steps.

Step 1. To check the locality condition (3.12) and to prove that

$$\begin{aligned} I_0,\ I_1\in \mathrm {Der}^\partial (\hat{\mathcal A}), \end{aligned}$$

(3.14)

which is also a locality condition.

Step 2. To check the closedness condition

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, I_0\right] = 0. \end{aligned}$$

(3.15)

After we finish Step 2, we can find a derivation $X^\circ \in \mathrm {Der}^\partial (\hat{\mathcal A})$ by using the property $H^1_{>0}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]}\bigr ) = 0$ such that

$$\begin{aligned} I_0 = \left[ \frac{\partial }{\partial \tau _0}, X^\circ \right] , \end{aligned}$$

and the leading term of $X^\circ$ is determined by the Virasoro symmetry of the super tau-cover of the Principal Hierarchy. We define ${\mathcal {C}} = X-X^\circ \in \mathrm {Der}^\partial (\hat{\mathcal A})^0_{\ge 2}$, then the equations (3.13) for X are transformed to the following equations for ${\mathcal {C}}$:

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, {\mathcal {C}}\right] = 0,\quad \left[ \frac{\partial }{\partial \tau _1}, {\mathcal {C}}\right] = I_1-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right] . \end{aligned}$$

(3.16)

Step 3. To check the closedness conditions

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, I_1-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right] \right] = 0,\quad \left[ \frac{\partial }{\partial \tau _1}, I_1\right] = 0. \end{aligned}$$

(3.17)

Step 4. To check that the differential degree 3 component of the derivation $I_1-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right]$ vanishes in the cohomology group $BH^1_{3}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]},P_1^{[0]}\bigr )$.

We call the above fact the vanishing of the genus one obstruction for the following reason. By using the first equation in (3.16) and the vanishing of $H^0_{\ge 2}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]}\bigr )$, we see that there exists a unique ${\mathcal {T}}\in \mathrm {Der}^\partial (\hat{\mathcal A})^{-1}_{\ge 1}$ such that

$$\begin{aligned} {\mathcal {C}} = \left[ \frac{\partial }{\partial \tau _0}, {\mathcal {T}}\right] . \end{aligned}$$

The derivation ${\mathcal {T}}$ must also satisfy the second equation in (3.16)

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _1}, \left[ \frac{\partial }{\partial \tau _0}, {\mathcal {T}}\right] \right] = I_1-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right] . \end{aligned}$$

(3.18)

If the differential degree 3 component of the derivation $I_1-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right]$ does not vanish in the cohomology group $BH^1_{3}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]},P_1^{[0]}\bigr )$, then such ${\mathcal {T}}$ does not exist.

However if the genus one obstruction vanishes, there exists a derivation ${\mathcal {T}}$, whose differential degree 1 part is unique, such that the Eq. (3.18) is valid at the approximation of differential degree 3. Then by using $BH^1_{\ge 4}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]},P_1^{[0]}\bigr ) = 0$ and the closedness conditions (3.17), we can solve ${\mathcal {T}}$ from (3.18) degree by degree. In this way, we can find a derivation X such that the equations in (3.10) hold true.

Step 5. To lift the symmetry $\frac{\partial v_\lambda }{\partial s_2}$ to a symmetry of the tau-cover of the deformed Principal Hierarchy, and to define all the other flows $\frac{\partial }{\partial s_m}$ of the Virasoro symmetries for $m\ge 0$. Note that the symmetry $\frac{\partial }{\partial s_{-1}}$ is constructed in [9]. We remark that we can also lift the symmetry (3.8) and (3.9) to a symmetry of the super tau-cover of the deformed Principal Hierarchy, but it is not necessary for the consideration of our problem.

In the remaining part of this subsection, we explain how the equations (3.8) and (3.9) are derived from the Eq. (1.1) of the main theorem. Let ${\mathcal {Z}}$ be a tau-function of the tau-cover (2.32) of the deformed Principal Hierarchy, i.e.,

$$\begin{aligned} f_{\alpha ,p} = \frac{\partial \log {\mathcal {Z}}}{\partial t^{\alpha ,p}},\quad w^\alpha = \eta ^{\alpha \beta }\frac{\partial ^2\log {\mathcal {Z}}}{\partial t^{1,0}\partial t^{\beta ,0}} \end{aligned}$$

(3.19)

give a solution of the tau-cover (2.32). Our goal is to find a symmetry of the following form

$$\begin{aligned} \frac{\partial {\mathcal {Z}}}{\partial s_2} = L_2^{even}{\mathcal {Z}}+ O_2{\mathcal {Z}}, \end{aligned}$$

(3.20)

where $L_2^{even}$ is the operator (3.4) and $O_2$ is a differential polynomial. If we assume that this is indeed a symmetry, then by using (3.19) we obtain the flow

here $W_\lambda$ are some differential polynomials. Now recall that $v_\lambda$ and $w_\zeta$ are related by a Miura type transformation, hence by using the equation

$$\begin{aligned} \frac{\partial v_\lambda }{\partial s_2} = \sum _{s\ge 0}\frac{\partial v_\lambda }{\partial w_\zeta ^{(s)}}\partial _x^s\frac{\partial w_{\zeta }}{\partial s_2} \end{aligned}$$

we know that there exist differential polynomials $X^0_\lambda \in \hat{\mathcal A}^0$ such that

$$\begin{aligned} \frac{\partial v_\lambda }{\partial s_2}&=\,a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) \nonumber \\&\quad +\, X_{\lambda }^0+{\mathcal {L}}_2^{even}v_\lambda . \end{aligned}$$

(3.21)

As we have discussed at the end of Sect. 3.2, we also need to write down the actions of the flow $\frac{\partial }{\partial s_2}$ on the odd variables $\sigma _{\lambda ,0}$. To this end, we must replace $\mathcal L_2^{even}$ by ${\mathcal {L}}_2$ to include the odd time variables $\tau _m$ and odd flows $\frac{\partial }{\partial \tau _m}$. By using (2.36) it is easy to see that

$$\begin{aligned} \frac{\partial \varvec{f_0}}{\partial \tau _0} = {\varvec{A}}\varvec{\sigma _0}, \end{aligned}$$

where $\varvec{f_0} = (f_{1,0},\dots ,f_{n,0})^T$, $\varvec{\sigma _0} = (\sigma _{1,0},\dots ,\sigma _{n,0})^T$ and ${\varvec{A}}$ is a matrix of differential operator of the form

$$\begin{aligned} {\varvec{A}} = \sum _{g\ge 0}\sum _{k=0}^{2g}\varvec{A_{g,k}}\partial _x^k,\quad \varvec{A_{g,k}} \in M_n(\hat{\mathcal A}^0_{2g-k}). \end{aligned}$$

Note that $\varvec{A_{0,0}}$ is the identity matrix, therefore ${\varvec{A}}$ is invertible as a differential operator, i.e., there exists ${\varvec{B}} = \sum _{g\ge 0}\sum _{k=0}^{2g}\varvec{B_{g,k}}\partial _x^k$ such that $\varvec{AB} = \varvec{BA} = {\varvec{I}}$, and in particular $\varvec{B_{0,0}}$ is the identity matrix. Thus we can represent the odd variables $\sigma _{\lambda ,0}$ in the form

$$\begin{aligned} \sigma _{\lambda ,0} = \frac{\partial f_{\lambda ,0}}{\partial \tau _0}+\sum _{g\ge 1}\sum _{k=0}^{2g} \left( {\varvec{B_{g,k}}}\right) _\lambda ^\zeta \partial _x^k\frac{\partial f_{\zeta ,0}}{\partial \tau _0}. \end{aligned}$$

This identity leads us to the following definition of the evolutions of the odd variables $\sigma _{\lambda ,0}$ along the flow $\frac{\partial }{\partial s_2}$:

$$\begin{aligned} \frac{\partial \sigma _{\lambda ,0}}{\partial s_2} = \frac{\partial }{\partial \tau _0}\frac{\partial f_{\lambda ,0}}{\partial s_2}+\sum _{g\ge 1}\sum _{k=0}^{2g} \frac{\partial }{\partial s_2}\left( {\varvec{B_{g,k}}}\right) _\lambda ^\zeta \partial _x^k\frac{\partial f_{\zeta ,0}}{\partial \tau _0}+\sum _{g\ge 1}\sum _{k=0}^{2g} \left( {\varvec{B_{g,k}}}\right) _\lambda ^\zeta \partial _x^k \frac{\partial }{\partial \tau _0}\frac{\partial f_{\zeta ,0}}{\partial s_2}. \end{aligned}$$

By using Proposition 9 and Lemma 7, it follows from the explicit expressions (3.5), (3.6) and (3.7) that the odd variables $\Phi _{\alpha ,p}^m$ appearing in $\frac{\partial }{\partial \tau _0}\frac{\partial f_{\lambda ,0}}{\partial s_2}$ can be represented by elements of $\hat{\mathcal A}^+$. So there exist differential polynomials $M^\zeta _\lambda$, $N^\zeta _\lambda \in \hat{\mathcal A}^0$ and $X_\lambda ^1\in \hat{\mathcal A}^1$ such that

$$\begin{aligned} \frac{\partial \sigma _{\lambda ,0}}{\partial s_2} =&\, a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) \\ {}&+X^1_{\lambda } +M^\zeta _{\lambda }\sigma _{\zeta ,2}+N^\zeta _{\lambda }\sigma _{\zeta ,1}+\mathcal L_2\sigma _{\lambda ,0}. \end{aligned}$$

Finally we define a derivation $X\in \mathrm {Der}^\partial (\hat{\mathcal A})^0$ such that $Xv_{\lambda } = X_\lambda ^0$, where $X_\lambda ^0$ is the differential polynomial introduced in (3.21), and $X\sigma _{\lambda ,0} = X^1_\lambda$. Therefore our problem of finding such a derivation X is a necessary condition of the main theorem.

3.4 Example: one-dimensional Frobenius manifold

In this subsection we present an example to illustrate how the general framework described in the previous subsection works. We consider the one-dimensional Frobenius manifold M, it has the following potential and Euler vector field:

$$\begin{aligned} F = \frac{1}{6} v^3,\quad E = v\partial _v. \end{aligned}$$

Due to the dimension reason, we will omit the Greek indices, for example, we will use $v^{(s)}$ and $\sigma _m^s$ instead of $v^{1,s}$ and $\sigma _{1,m}^s$. The Principal Hierarchy associated with M is the Riemann hierarchy

$$\begin{aligned} \frac{\partial v}{\partial t_p} = \frac{v^p}{p!}v_x,\quad p\ge 0, \end{aligned}$$

whose bihamiltonian structure is given by

$$\begin{aligned} P_0^{[0]} = \frac{1}{2}\int \sigma _0\sigma _0^1,\quad P_1^{[0]} = \frac{1}{2}\int v\sigma _0\sigma _0^1. \end{aligned}$$

It is proved in [7, 24] that every deformation $(P_0,P_1)$ of $(P_0^{[0]},P_1^{[0]})$ with a constant central invariant is equivalent to the bihamiltonian structure given by

$$\begin{aligned} P_0 = \frac{1}{2}\int \sigma _0\sigma _0^1,\quad P_1 = \frac{1}{2}\int v\sigma _0\sigma _0^1 + \varepsilon ^2c\sigma _0\sigma _0^3 \end{aligned}$$

via a certain Miura type transformation. Here the dispersion parameter $\varepsilon$ is added for clearness, and the central invariant of $(P_0,P_1)$ is $\frac{c}{3}$. In particular, when $c = \frac{1}{8}$ the corresponding deformed Riemann hierarchy is the KdV hierarchy that controls the 2D topological gravity [19, 30].

We have the following flows for the super tau-cover of the deformed Riemann hierarchy:

$$\begin{aligned} \frac{\partial v}{\partial t_1}&= vv_x+\frac{2}{3}\varepsilon ^2cv^{(3)},\quad \frac{\partial \sigma _0}{\partial t_1} = v\sigma _0^1+\frac{2}{3}\varepsilon ^2c\sigma _0^3;\\ \frac{\partial v}{\partial t_2}&= \frac{1}{2} v^2v_x+\varepsilon ^2c\left( \frac{4}{3} v_xv_{xx}+\frac{2}{3} vv^{(3)}\right) +\frac{4}{15}\varepsilon ^4c^2v^{(5)},\\ \frac{\partial \sigma _0}{\partial t_2}&= \frac{1}{2} v^2\sigma _0^1+\varepsilon ^2c\left( \frac{2}{3} v_{xx}\sigma _0^1+\frac{2}{3} v_x\sigma _0^2+\frac{2}{3} v\sigma _0^3\right) +\frac{4}{15}\varepsilon ^4c^2\sigma _0^5;\\ \varepsilon \frac{\partial f_0}{\partial \tau _0}&= \sigma _0,\quad \varepsilon \frac{\partial f_1}{\partial \tau _0} = 2\sigma _1-v\sigma _0-\frac{4}{3}\varepsilon ^2c\sigma _0^2;\\ \varepsilon \frac{\partial f_2}{\partial \tau _0}&= \frac{4}{3}\sigma _2-\frac{2}{3}v\sigma _1-\frac{1}{6} v^2\sigma _0-\varepsilon ^2c\left( \frac{2}{3}v_{xx}\sigma _0+\frac{4}{3}v_x\sigma _0^2+\frac{4}{3}v\sigma _0^2\right) -\frac{16}{15}\varepsilon ^4c^2\sigma _0^4. \end{aligned}$$

Here the odd variables $\sigma _m$ satisfy the recursion relation

$$\begin{aligned} \sigma _{m+1}^1 = v\sigma _{m}^1+\frac{1}{2}v_x\sigma _m+\varepsilon ^2 c \sigma _m^3,\quad m\ge 0. \end{aligned}$$

We also have the following Hamiltonian densities for the deformed Riemann hierarchy:

$$\begin{aligned} h_1&= \frac{v^2}{2}+\frac{2}{3} \varepsilon ^2cv_{xx},\quad h_2 = \frac{v^3}{6}+\varepsilon ^2c\left( \frac{1}{3} v_x^2+\frac{2}{3}vv_{xx}\right) +\frac{4}{15}\varepsilon ^4c^2v^{(4)}. \end{aligned}$$

Note that

$$\begin{aligned} L_2 = \frac{3}{8}\varepsilon ^2\frac{\partial ^2}{\partial t_1\partial t_0}+{\mathcal {L}}_2,\quad {\mathcal {L}}_2 = \sum _{p\ge 0}\frac{\Gamma \left( \frac{7}{2}+p\right) }{\Gamma \left( \frac{1}{2}+p\right) }t_p\frac{\partial }{\partial t_{p+2}}+(p+c_0)\tau _p\frac{\partial }{\partial \tau _{p+2}}. \end{aligned}$$

Then the equations (3.8) and (3.9) for this example have the form

$$\begin{aligned} \frac{\partial v}{\partial s_2}&=\ \frac{3}{8}\varepsilon \left( v_xf_1+\frac{\partial v}{\partial t_1}f_0\right) +Xv+{\mathcal {L}}_2v,\\ \frac{\partial \sigma _0}{\partial s_2}&=\ \frac{3}{8}\varepsilon \left( \sigma _0^1f_1+\frac{\partial \sigma _0}{\partial t_1}f_0\right) +\left( \frac{5}{2}+c_0\right) \sigma _2-\frac{v}{2}\sigma _1+X\sigma _0+{\mathcal {L}}_2\sigma _0. \end{aligned}$$

We are to find the derivation $X\in \mathrm {Der}^\partial (\hat{\mathcal A})^0$ such that the flow $\frac{\partial }{\partial s_2}$ commutes with $\frac{\partial }{\partial \tau _0}$ and $\frac{\partial }{\partial \tau _1}$. These conditions yield the following equations for X:

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, X\right] = I_0,\quad \left[ \frac{\partial }{\partial \tau _1}, X\right] = I_1, \end{aligned}$$

where the derivations $I_0$ and $I_1$ are given by

$$\begin{aligned} I_0v&=\ vv_x\sigma _0+\frac{7}{2}v^2\sigma _0^1+\varepsilon ^2c\left( v^{(3)}\sigma _0+\frac{13}{2}v_{xx}\sigma _0^1+8v_x\sigma _0^2+6v\sigma _0^3\right) +3\varepsilon ^4c^2\sigma _0^5, \end{aligned}$$

(3.22)

$$\begin{aligned} I_0\sigma _0&=\ v\sigma _0\sigma _0^1-\varepsilon ^2c\left( \frac{1}{2} \sigma _0^1\sigma _0^2-\sigma _0\sigma _0^3\right) , \end{aligned}$$

(3.23)

$$\begin{aligned} I_1v&=\ \frac{5}{4}v^2v_x\sigma _0+\frac{5}{2}v^3\sigma _0^1+\varepsilon ^2c\left( \frac{7}{2} v_xv_{xx}\sigma _0+2vv^{(3)}\sigma _0+\frac{45}{4}v_x^2\sigma _0^1+\frac{31}{2}vv_{xx}\sigma _0^1\right) \nonumber \\&\quad +\, \varepsilon ^2c\left( 26vv_x\sigma _0^2+\frac{19}{2}v^2\sigma _0^3\right) +\varepsilon ^4c^2\left( v^{(5)}\sigma _0+\frac{17}{2}v^{(4)}\sigma _0^1+\frac{45}{2}v^{(3)}\sigma _0^2\right) \nonumber \\&\quad +\, \varepsilon ^4c^2\left( \frac{59}{2}v_{xx}\sigma _0^3+\frac{43}{2}v_x\sigma _0^4+9v\sigma _0^5\right) +3\varepsilon ^6c^3\sigma _0^7,\end{aligned}$$

(3.24)

$$\begin{aligned} I_1\sigma _0&=\ \frac{5}{4}v^2\sigma _0\sigma _0^1+\varepsilon ^2c\left( \frac{5}{2} v_{xx}\sigma _0\sigma _0^1+\frac{5}{2} v_x\sigma _0\sigma _0^2-\frac{1}{2}v\sigma _0^1\sigma _0^2+2v\sigma _0\sigma _0^3\right) \nonumber \\&\quad -\, \varepsilon ^4c^2\left( \frac{1}{2}\sigma _0^1\sigma _0^4-\sigma _0\sigma _0^5\right) . \end{aligned}$$

(3.25)

It follows from (3.22) and (3.23) that we can choose $X^\circ \in \mathrm {Der}^\partial (\hat{\mathcal A})$ such that $[\frac{\partial }{\partial \tau _0},X^\circ ] = I_0$, whose actions on v and $\sigma _0$ are given by

$$\begin{aligned} X^\circ v&= v^3+\varepsilon ^2c\left( \frac{5}{4} v_x^2+3vv_{xx}\right) +\varepsilon ^4c^2v^{(4)},\\ X^\circ \sigma _0&= -\frac{1}{2} v^2\sigma _0-\varepsilon ^2c\left( v_{xx}\sigma _0+\frac{5}{2} v_x\sigma _0^1+3v\sigma _0^2\right) -2\varepsilon ^4c^2\sigma _0^4. \end{aligned}$$

Then according to the general discussions given in the previous subsection, the derivation ${\mathcal {C}} = X-X^\circ$ satisfies the equations

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, {\mathcal {C}}\right] = 0,\quad \left[ \frac{\partial }{\partial \tau _1}, {\mathcal {C}}\right] = I_1-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right] . \end{aligned}$$

Finally we can solve the above equations and obtain the unique derivation ${\mathcal {C}}$ that is defined by

$$\begin{aligned} \mathcal Cv&= \varepsilon ^2c\left( 3v_x^2+3vv_{xx}\right) +2\varepsilon ^4c^2v^{(4)},\\ {\mathcal {C}}\sigma _0&= \varepsilon ^2c\left( 3v_x\sigma _0^1+v\sigma _0^2\right) +2\varepsilon ^4c^2\sigma _0^4. \end{aligned}$$

By forgetting all the odd variables we obtain the following symmetry for the deformed Riemann hierarchy:

$$\begin{aligned} \frac{\partial v}{\partial s_2} = \frac{3}{8}\varepsilon \left( v_xf_1+\frac{\partial v}{\partial t_1}f_0\right) +v^3+\varepsilon ^2c\left( \frac{17}{4}v_x^2+6vv_{xx}\right) +3\varepsilon ^4c^2v^{(4)}+\mathcal L_2^{even} v. \end{aligned}$$

It is easy to check that the action of this symmetry on the tau function ${\mathcal {Z}}$ of the deformed Riemann hierarchy can be represented by

$$\begin{aligned} \frac{\partial {\mathcal {Z}}}{\partial s_2} = L_2^{even}\mathcal Z+\left( 3c-\frac{3}{8}\right) \left( \frac{v^2}{2}+\frac{2}{3}\varepsilon ^2cv_{xx}\right) {\mathcal {Z}}. \end{aligned}$$

(3.26)

In particular, when $c = \frac{1}{8}$, this symmetry is given by a linear action on ${\mathcal {Z}}$.

4 Deformation of Virasoro Symmetries: Existence and Uniqueness

In this section, we present details of the proof of the main theorem following the framework described in Sect. 3.3.

4.1 Locality conditions

We start by verifying the locality condition (3.14).

Let us first find the differential polynomials $M_\lambda ^\zeta$ and $N_\lambda ^\zeta$ in (3.9) to ensure that $I_0$ is local. By using the relation (2.36) and the equations (3.8) and (3.9), we arrive at

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial s_2}\right] v_\lambda = c_0\sigma _{\lambda ,2}^1+a^{\alpha \beta }\frac{\partial f_{\alpha ,1}}{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}-\partial _x\left( M_\lambda ^\zeta \sigma _{\zeta ,2}+N_\lambda ^\zeta \sigma _{\zeta ,1}\right) +loc. \end{aligned}$$

(4.1)

Here and henceforth we will use loc to denote the local terms, i.e., terms belonging to $\hat{\mathcal A}$. We need to find $M_\lambda ^\zeta$ and $N_\lambda ^\zeta$ such that the right hand side of (4.1) is local.

Lemma 9

The following identities hold true:

$$\begin{aligned} \frac{\partial }{\partial \sigma _{\beta ,0}}\frac{\delta P_1}{\delta \sigma _{\lambda ,0}} = \eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial v^\lambda }{\partial t^{\alpha ,0}},\quad \frac{\partial }{\partial \sigma _{\beta ,0}}\frac{\delta P_1}{\delta v^\lambda } = \eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}. \end{aligned}$$

Proof

By using the identity (2.29) we obtain the identity

$$\begin{aligned} \left[ P_1, H_{\alpha ,-1}\right] = \left( \frac{1}{2}+\mu _\alpha \right) \left[ P_0, H_{\alpha ,0}\right] . \end{aligned}$$

By taking the variational derivatives of both sides of the above identity and by using (2.4), (2.5) and (2.31), we arrive at the result of the lemma.$\square$

Lemma 10

We have the following relation:

$$\begin{aligned} \sigma _{\lambda ,2}^1 = \left( \frac{1}{2}+\mu _\varepsilon \right) \eta ^{\zeta \varepsilon }\frac{\partial v_\lambda }{\partial t^{\varepsilon ,0}}\sigma _{\zeta ,1}+loc. \end{aligned}$$

(4.2)

Proof

Let us denote by ${\mathcal {P}}_1$ the Hamiltonian operator of $P_1$ and represent it in the form

$$\begin{aligned} {\mathcal {P}}_1 = \sum _{k\ge 0}{\mathcal {P}}_{1,k}\partial _x^k,\quad \mathcal P_{1,k}\in M_n(\hat{\mathcal A}). \end{aligned}$$

Then from the recursion relation

$$\begin{aligned} \eta ^{\gamma \lambda }\sigma _{\lambda ,2}^1 = {\mathcal {P}}_1^{\gamma \lambda }\sigma _{\lambda ,1} = \sum _{k\ge 0}{\mathcal {P}}_{1,k}^{\gamma \lambda }\sigma _{\lambda ,1}^k, \end{aligned}$$

it is easy to see that

$$\begin{aligned} \sigma _{\lambda ,2}^1 = \eta _{\lambda \gamma }\mathcal P_{1,0}^{\gamma \zeta }\sigma _{\zeta ,1}+loc. \end{aligned}$$

Now it follows from the definition of the Hamiltonian operator that

$$\begin{aligned} {\mathcal {P}}_{1,0}^{\gamma \zeta } = \frac{\partial }{\partial \sigma _{\zeta ,0}}\frac{\delta P_1}{\delta \sigma _{\gamma ,0}}. \end{aligned}$$

Therefore by using Lemma 9 we arrive at the identity (4.2) and the lemma is proved.$\square$

Proposition 10

There exist unique differential polynomials $M^\zeta _\lambda$ and $N^\zeta _\lambda$ such that (4.1) is local. More explicitly, we have

$$\begin{aligned} M^\zeta _\lambda = \left( \frac{5}{2}+\mu _\lambda +c_0\right) \delta ^\zeta _\lambda ,\quad \partial _xN^\zeta _{\lambda } = \left( \frac{1}{2}+\mu _\alpha \right) \eta ^{\alpha \zeta }\left( \mu _\zeta -\mu _\lambda -1\right) \frac{\partial v_\lambda }{\partial t^{\alpha ,0}}, \end{aligned}$$

here $c_0$ is the arbitrary constant appearing in the operator ${\mathcal {L}}_2$.

Proof

To ensure the vanishing of the coefficients of $\sigma _{\zeta ,2}$ in the right hand side of (4.1), $M^\zeta _\lambda$ should be a constant, so it is determined by the leading terms of the right hand sides of (3.8) and (3.9), which are fixed by the Virasoro symmetry $\frac{\partial }{\partial s_2}$ of the super tau-cover of the Principal Hierarchy. Hence we obtain

$$\begin{aligned}M^\zeta _\lambda = \left( \frac{5}{2}+\mu _\lambda +c_0\right) \delta ^\zeta _\lambda . \end{aligned}$$

By using Proposition 9 we have

$$\begin{aligned} \left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial f_{\alpha ,1}}{\partial \tau _0} = \sigma _{\alpha ,1}+loc. \end{aligned}$$

(4.3)

Then by using this equation, the recursion relations (2.18) and the identity (4.2), we see that the vanishing of the coefficients of $\sigma _{\zeta ,1}$ in the right hand side of (4.1) gives the equation

$$\begin{aligned} \partial _xN^\zeta _{\lambda } = \left( \frac{1}{2}+\mu _\alpha \right) \eta ^{\alpha \zeta }\left( \mu _\zeta -\mu _\lambda -1\right) \frac{\partial v_\lambda }{\partial t^{\alpha ,0}}. \end{aligned}$$

(4.4)

Both sides of (4.4) are total x-derivatives, so we can integrate (4.4) to obtain $N^\zeta _{\lambda }$ upto a constant, which is uniquely determined from the Virasoro symmetry of the super tau-cover of the Principal Hierarchy. The proposition is proved.$\square$

Now by a direct computation of the condition

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial s_2}\right] = 0, \end{aligned}$$

we obtain the following explicit expressions for $I_0$, a derivation which is defined by (3.13):

$$\begin{aligned} I_0v_\lambda&= a^{\alpha \beta }\left( f_{\beta ,0}'\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+f_{\alpha ,1}'\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) -a^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+\frac{\partial f_{\alpha ,1}}{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) \nonumber \\&\quad +\, b^{\alpha \beta }\left( f_{\beta ,0}'\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}+f_{\alpha ,0}'\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) -b^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}}+\frac{\partial f_{\alpha ,0}}{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) \nonumber \\&\quad +\, \left( \frac{5}{2}+\mu _\lambda \right) \sigma _{\lambda ,2}^1+W\sigma _{\lambda ,0}+\partial _x\left( N^\zeta _\lambda \sigma _{\zeta ,1}\right) , \end{aligned}$$

(4.5)

$$\begin{aligned} I_0\sigma _{\lambda ,0}&= -a^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+\frac{\partial f_{\alpha ,1}}{\partial \tau _0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) -b^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}+\frac{\partial f_{\alpha ,0}}{\partial \tau _0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) \nonumber \\&\quad +\, \left( \frac{5}{2}+\mu _\lambda \right) \frac{\partial \sigma _{\lambda ,0}}{\partial \tau _2}-\frac{\partial }{\partial \tau _0}\left( N^\zeta _\lambda \sigma _{\zeta ,1}\right) , \end{aligned}$$

(4.6)

here and henceforth we will use $f_{\alpha ,p}'$ to denote the differential polynomial $\frac{\partial f_{\alpha ,p}}{\partial t^{1,0}}$ and use W to denote the coefficient of $t^{1,0}$ of the operator ${\mathcal {L}}_2$. More explicitly, we have

$$\begin{aligned} W&=\, \left( \frac{1}{2}+\mu _1\right) \left( \frac{3}{2}+\mu _1\right) \left( \frac{5}{2}+\mu _1\right) \frac{\partial }{\partial t^{1,2}}\nonumber \\&\quad +\, \sum _{k=1}^2\left( 3\left( \frac{1}{2}+\mu _1\right) ^2+6\left( \frac{1}{2}+\mu _1\right) +2\right) \left( R_k\right) ^\beta _1 \frac{\partial }{\partial t^{\beta ,2-k}}\nonumber \\&\quad +\, \left( \frac{9}{2}+3\mu _1\right) \left( R_{2,2}\right) ^\beta _1\frac{\partial }{\partial t^{\beta ,0}}. \end{aligned}$$

(4.7)

Proposition 11

The derivation $I_0$ given by (4.5), (4.6) is local.

Proof

The locality of (4.5) follows from the definition of $N_\lambda ^\zeta$, so we only need to check the locality of (4.6). By using the definition of the odd flows we know that

$$\begin{aligned} \frac{\partial \sigma _{\lambda ,0}}{\partial \tau _2} =T_{0,2}\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1} = \sum _{s\ge 0}\sigma _{\beta ,1}^s\frac{\partial }{\partial \sigma _{\beta ,0}^s}\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1}, \end{aligned}$$

therefore it follows from Lemma 9 that

$$\begin{aligned} \frac{\partial \sigma _{\lambda ,0}}{\partial \tau _2} = \sigma _{\beta ,1}\eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}+loc. \end{aligned}$$

Then we arrive at the locality of (4.6) by using (4.3) and the following obvious fact:

$$\begin{aligned} \frac{\partial N^\zeta _{\lambda }}{\partial \tau _0} = \left( \frac{1}{2}+\mu _\alpha \right) \eta ^{\alpha \zeta }\left( \mu _\zeta -\mu _\lambda -1\right) \frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}. \end{aligned}$$

The proposition is proved.$\square$

Let us proceed to prove the locality of $I_1$. Similar to the expression (4.5) and (4.6), we can write down the explicit expression for $I_1$, which can be found in the next subsection. However the locality of $I_1$ can not be derived from this expression directly, so we turn to prove the following equivalent conditions:

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _1}, \frac{\partial }{\partial s_2}\right] v_\lambda \in \hat{\mathcal A},\quad \left[ \frac{\partial }{\partial \tau _1}, \frac{\partial }{\partial s_2}\right] \sigma _{\lambda ,0}\in \hat{\mathcal A}. \end{aligned}$$

(4.8)

By a direct computation we have

$$\begin{aligned}&\quad \left[ \frac{\partial }{\partial \tau _1}, \frac{\partial }{\partial s_2}\right] v_\lambda \nonumber \\ {}&=\ a^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _1}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+\frac{\partial f_{\alpha ,1}}{\partial \tau _1}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) \nonumber \\ {}&\quad +\, b^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _1}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}}+\frac{\partial f_{\alpha ,0}}{\partial \tau _1}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) +(1+c_0)\frac{\partial v_\lambda }{\partial \tau _3}\nonumber \\ {}&\quad -\, \sum _{s\ge 0}\left( \left( \frac{5}{2}+c_0+\mu _\zeta \right) \sigma _{\zeta ,2}^s+\partial _x^s\left( N^\varepsilon _\zeta \sigma _{\zeta ,1}\right) \right) \frac{\partial }{\partial \sigma _{\zeta ,0}^s}\frac{\partial v_\lambda }{\partial \tau _1}+loc.\end{aligned}$$

(4.9)

Lemma 11

There exists a unique differential polynomial $Z^\beta _{\alpha }$, for each pair of indices $1\le \alpha ,\beta \le n$, such that

$$\begin{aligned} \left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial f_{\alpha ,1}}{\partial \tau _1} = \sigma _{\alpha ,2}+Z^\beta _{\alpha }\sigma _{\beta ,1}+loc, \end{aligned}$$

where $Z^\beta _{\alpha }$ satisfies the following equation

$$\begin{aligned} \partial _xZ^\beta _{\alpha } = -\left( \frac{1}{2}+\mu _\varepsilon \right) \eta ^{\beta \varepsilon }\frac{\partial v_\alpha }{\partial t^{\varepsilon ,0}}. \end{aligned}$$

(4.10)

Proof

The existence and uniqueness of $Z^\beta _\alpha$ can be obtained from Proposition 9, hence we only need to derive (4.10), which can be obtained by using the identity (4.2) and the fact that $\partial _x\left( \sigma _{\alpha ,2}+Z^\beta _{\alpha }\sigma _{\beta ,1}\right)$ is local. The lemma is proved.$\square$

From the definition of the odd flows and the recursion relation (2.18) we have the equation,

$$\begin{aligned} \frac{\partial v_\lambda }{\partial \tau _3} = \sum _{s\ge 0}\sigma _{\zeta ,2}^s\frac{\partial }{\partial \sigma _{\zeta ,0}^s}\frac{\partial v_\lambda }{\partial \tau _1}, \end{aligned}$$

which, together with Lemma 11 and the identity (4.2), enables us to rewrite (4.9) in the form

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _1}, \frac{\partial }{\partial s_2}\right] v_\lambda = U_\lambda ^\zeta \sigma _{\zeta ,2}+V_\lambda ^\zeta \sigma _{\zeta ,1}+loc, \end{aligned}$$

where the differential polynomials $U_\lambda ^\zeta$ and $V_\lambda ^\zeta$ are given by

$$\begin{aligned} U_\lambda ^\zeta&= \eta ^{\zeta \beta }\left( \frac{1}{2}+\mu _\beta \right) \left( \frac{3}{2}+\mu _\zeta \right) \frac{\partial v_\lambda }{\partial t^{\beta ,0}}-\left( \frac{3}{2}+\mu _\zeta \right) \frac{\partial }{\partial \sigma _{\zeta ,0}}\frac{\partial v_\lambda }{\partial \tau _1},\\ V_\lambda ^\zeta&= a^{\alpha \zeta }\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+(b^{\alpha \zeta }+b^{\zeta \alpha })\frac{\partial v_{\lambda }}{\partial t^{\alpha ,0}}-\sum _{s\ge 0}\partial _x^s\left( N^\zeta _{\gamma }-\left( \frac{3}{2}+\mu _\gamma \right) Z^\zeta _{\gamma }\right) \frac{\partial }{\partial \sigma _{\gamma ,0}^s}\frac{\partial v_\lambda }{\partial \tau _1}. \end{aligned}$$

Lemma 12

We have $U_\lambda ^\zeta = V_\lambda ^\zeta = 0$.

Proof

The vanishing of $U_\lambda ^\zeta$ follows directly from Lemma 9. To prove the vanishing of $V_\lambda ^\zeta$ , let us consider the functional

$$\begin{aligned} F^\zeta = \eta ^{\alpha \zeta }\left( \frac{1}{2}+\mu _\alpha \right) \left( \frac{1}{2}+\mu _\zeta \right) \int h_{\alpha ,1}. \end{aligned}$$

By using (4.4) and (4.10) we can check that there exists a constant $C^\zeta _\lambda$ such that

$$\begin{aligned} N^\zeta _{\lambda }-\left( \frac{3}{2}+\mu _\gamma \right) Z^\zeta _{\lambda } = \frac{\delta F^\zeta }{\delta v_\lambda }+C^\zeta _\lambda . \end{aligned}$$

Thus from (2.5) it follows that

$$\begin{aligned} \sum _{s\ge 0}\partial _x^s\left( N^\zeta _{\gamma }-\left( \frac{3}{2}+\mu _\gamma \right) Z^\zeta _{\gamma }\right) \frac{\partial }{\partial \sigma _{\gamma ,0}^s}\frac{\partial v_\lambda }{\partial \tau _1} = -\frac{\delta }{\delta \sigma _{\lambda ,0}}\left[ F^\zeta , P_1\right] +C^\zeta _\gamma \frac{\partial }{\partial \sigma _{\gamma ,0}}\frac{\partial v_\lambda }{\partial \tau _1}. \end{aligned}$$

By using the bihamiltonian recursion relation (2.29) and Lemma 9 we see that the right hand side of the above equation is a linear combination of the flows $\frac{\partial }{\partial t^{\alpha ,p}}$ acting on $v_\lambda$, hence so is $V^\zeta _\lambda$. On the other hand, the Virasoro symmetry $\frac{\partial }{\partial s_2}$ of the super tau-cover of the Principal Hierarchy implies that $V^\zeta _\lambda \in \hat{\mathcal A}^0_{\ge 2}$, so by using the theory of bihamiltonian cohomology [7], we know that $V^\zeta _\lambda$ must vanish. The lemma is proved.$\square$

We have verified the first relation in (4.8), now let us proceed to prove the second one. We have

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _1}, \frac{\partial }{\partial s_2}\right] \sigma _{\lambda ,0}&=\ a^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+\frac{\partial f_{\alpha ,1}}{\partial \tau _1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) \nonumber \\&\quad +\, b^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}+\frac{\partial f_{\alpha ,0}}{\partial \tau _1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) \nonumber \\&\quad +\, \left( \frac{5}{2}+c_0+\mu _\lambda \right) \frac{\partial \sigma _{\lambda ,2}}{\partial \tau _1}+\frac{\partial N_\lambda ^\zeta }{\partial \tau _1}\sigma _{\zeta ,1}+(1+c_0)\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _3}\nonumber \\&\quad -\, \sum _{s\ge 0}\left( \left( \frac{5}{2}+c_0+\mu _\zeta \right) \sigma _{\zeta ,2}^s+\partial _x^s\left( N^\varepsilon _\zeta \sigma _{\zeta ,1}\right) \right) \frac{\partial }{\partial \sigma _{\zeta ,0}^s}\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1}+loc. \end{aligned}$$

By using the equations

$$\begin{aligned} \frac{\partial \sigma _{\lambda ,0}}{\partial \tau _3}&= \sum _{s\ge 0}\sigma _{\gamma ,2}^s\frac{\partial }{\partial \sigma _{\gamma ,0}^s}\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1}+\frac{\partial \sigma _{\lambda ,1}}{\partial \tau _2},\quad \frac{\partial \sigma _{\lambda ,1}}{\partial t^{\alpha ,p}} = \frac{\partial }{\partial \tau _1}\frac{\delta h_{\alpha ,p+1}}{\delta v^\lambda },\\ \frac{\partial \sigma _{\lambda ,1}}{\partial \tau _2}&= \sigma _{\zeta ,1}\left( \frac{1}{2}+\mu _\varepsilon \right) \eta ^{\varepsilon \zeta }\frac{\partial }{\partial \tau _1}\frac{\delta h_{\varepsilon ,1}}{\delta v^\lambda }+loc, \end{aligned}$$

we obtain that

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _1}, \frac{\partial }{\partial s_2}\right] \sigma _{\lambda ,0} = \sigma _{\zeta ,2}\tilde{U}_\lambda ^\zeta +\sigma _{\zeta ,1}{\tilde{V}}_\lambda ^\zeta +loc, \end{aligned}$$

where the differential polynomials ${\tilde{U}}_\lambda ^\zeta$, $\tilde{V}_\lambda ^\zeta$ are given by

$$\begin{aligned} {\tilde{U}}_\lambda ^\zeta&=\, \eta ^{\zeta \beta }\left( \frac{1}{2}+\mu _\beta \right) \left( \frac{3}{2}+\mu _\zeta \right) \frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}-\left( \frac{3}{2}+\mu _\zeta \right) \frac{\partial }{\partial \sigma _{\zeta ,0}}\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1},\\ {\tilde{V}}_\lambda ^\zeta&=\, a_{\alpha \zeta }\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+(b_{\alpha \zeta }+b_{\zeta \alpha })\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}-\frac{\partial }{\partial \tau _1}\left( N^\zeta _{\lambda }-\left( \frac{3}{2}+\mu _\gamma \right) Z^\zeta _{\lambda }\right) \\ {}&\quad -\, \sum _{s\ge 0}\partial _x^s\left( N^\zeta _{\gamma }-\left( \frac{3}{2}+\mu _\gamma \right) Z^\zeta _{\gamma }\right) \frac{\partial }{\partial \sigma _{\gamma ,0}^s}\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1}. \end{aligned}$$

These differential polynomials actually vanish, the reason is similar to the one for the vanishing of $U_\lambda ^\zeta$ and $V_\lambda ^\zeta$ given in the proof of Lemma 12. Hence the locality condition (3.14) is verified.

Finally let us consider the locality condition (3.12). The first condition

$$\begin{aligned}\left[ \frac{\partial }{\partial t^{\delta ,j}}, \frac{\partial }{\partial s_2}\right] v_\lambda \in \hat{\mathcal A}\end{aligned}$$

follows from the definition (3.8). To verify the second locality condition, we consider the equation

$$\begin{aligned}&\quad \left[ \frac{\partial }{\partial t^{\delta ,j}}, \frac{\partial }{\partial s_2}\right] \sigma _{\lambda ,0}\nonumber \\ {}&=\ \left( \frac{5}{2}+c_0+\mu _\lambda \right) \frac{\partial \sigma _{\lambda ,2}}{\partial t^{\delta ,j}}+\frac{\partial N_\lambda ^\zeta }{\partial t^{\delta ,j}}\sigma _{\zeta ,1}\nonumber \\ {}&\quad -\, \sum _{s\ge 0}\partial _x^s\left( N^\zeta _{\gamma }\sigma _{\zeta ,1}+\left( \frac{5}{2}+c_0+\mu _\gamma \right) \sigma _{\gamma ,2}\right) \frac{\partial }{\partial \sigma _{\gamma ,0}^s}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\delta ,j}}+loc.\end{aligned}$$

(4.11)

Proposition 12

The right hand side of the Eq. (4.11) is local.

Proof

By using the identity

$$\begin{aligned} \frac{\partial }{\partial \sigma _{\beta ,0}}\frac{\delta }{\delta v^\alpha } = \frac{\delta }{\delta v^\alpha }\frac{\delta }{\delta \sigma _{\beta ,0}} \end{aligned}$$

that is proved in [25], we obtain

$$\begin{aligned} \frac{\partial }{\partial \sigma _{\beta ,0}}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,p}} = \frac{\partial }{\partial \sigma _{\beta ,0}}\frac{\delta X_{\alpha ,p}}{\delta v^\lambda } = 0, \end{aligned}$$

from which it follows that the flows $\frac{\partial \sigma _{\lambda ,2}}{\partial t^{\delta ,j}}$ can be written as

$$\begin{aligned} \frac{\partial \sigma _{\lambda ,2}}{\partial t^{\delta ,j}} =T_2\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\delta ,j}} = \sum _{s\ge 0}\sigma _{\beta ,2}^s\frac{\partial }{\partial \sigma _{\beta ,0}^s}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\delta ,j}}= B^\zeta _\lambda \sigma _{\zeta ,1}+loc, \end{aligned}$$

where $B_\lambda ^\zeta$ are certain differential polynomials. From the identity (4.2) we know that

$$\begin{aligned} B^\zeta _\lambda = -\frac{\partial Z_\lambda ^\zeta }{\partial t^{\delta ,j}}, \end{aligned}$$

and we can represent the right hand side of (4.11) in the form

$$\begin{aligned} V^\zeta _{\delta ,j;\lambda }\sigma _{\zeta ,1}+loc, \end{aligned}$$

where $V^\zeta _{\delta ,j;\lambda }$ is a differential polynomial given as follows:

$$\begin{aligned} V^\zeta _{\delta ,j;\lambda }&=\, \frac{\partial }{\partial t^{\delta ,j}}\left( N^\zeta _{\lambda }-\left( \frac{5}{2}+c_0+\mu _\gamma \right) Z^\zeta _{\lambda }\right) \\&\quad -\, \sum _{s\ge 0}\partial _x^s\left( N^\zeta _{\gamma }-\left( \frac{5}{2}+c_0+\mu _\gamma \right) Z^\zeta _{\gamma }\right) \frac{\partial }{\partial \sigma _{\gamma ,0}^s}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\delta ,j}}. \end{aligned}$$

Define the following functional

$$\begin{aligned} F^\zeta = \eta ^{\alpha \zeta }\left( \frac{1}{2}+\mu _\alpha \right) \left( \frac{3}{2}+c_0+\mu _\zeta \right) \int h_{\alpha ,1}, \end{aligned}$$

then by applying (2.4) we obtain the following identity:

$$\begin{aligned} V^\zeta _{\delta ,j;\lambda } = \frac{\delta }{\delta v^\lambda }\left[ F^\zeta , X_{\delta ,j}\right] , \end{aligned}$$

here $X_{\delta ,j}$ is the vector field defined in (2.26). Since the functional $F^\zeta$ is a conserved quantity of the deformed Principal hierarchy, we have $\left[ F^\zeta , X_{\delta ,j}\right] = 0$. The proposition is proved.$\square$

4.2 Closedness conditions

In this subsection we prove the closedness condition (3.15) and (3.17). The verification of (3.15) is straightforward by using the explicit expressions (4.5) and (4.6), so we omit the details here.

Let us prove the closedness condition (3.17). We fix a choice of $X^\circ \in \mathrm {Der}^\partial (\hat{\mathcal A})$ such that $X^\circ$ satisfies the condition

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, X^\circ \right] = I_0, \end{aligned}$$

and that the differential degree zero part of $X^\circ$ is given by the Virasoro symmetry $\frac{\partial }{\partial s_2}$ of the super tau-cover of the Principal Hierarchy.

We first write down the explicit expression for $I_1$ defined in (3.13). Define a derivation ${\hat{I}}_1\in \mathrm {Der}(\hat{\mathcal A})^0$ by the formulae ${\hat{I}}_1v_\gamma = {\hat{I}}_1\sigma _{\gamma ,0} = 0$ and

$$\begin{aligned} {\hat{I}}_1v_\gamma ^{(n)}&~=~ n\partial _x^{n-1}W(v_\gamma )+a^{\alpha \beta }\partial _x^n\left( f_{\beta ,0}\frac{\partial v_\gamma }{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial v_\gamma }{\partial t^{\beta ,0}}\right) \\&\quad +b^{\alpha \beta }\partial _x^n\left( f_{\beta ,0}\frac{\partial v_\gamma }{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial v_\gamma }{\partial t^{\beta ,0}}\right) \\&\quad -\, a^{\alpha \beta }\left( f_{\beta ,0}\partial _x^n\frac{\partial v_\gamma }{\partial t^{\alpha ,1}}+f_{\alpha ,1}\partial _x^n\frac{\partial v_\gamma }{\partial t^{\beta ,0}}\right) \\&\quad -b^{\alpha \beta }\left( f_{\beta ,0}\partial _x^n\frac{\partial v_\gamma }{\partial t^{\alpha ,0}}+f_{\alpha ,0}\partial _x^n\frac{\partial v_\gamma }{\partial t^{\beta ,0}}\right) ,\\ {\hat{I}}_1\sigma _{\gamma ,0}^n&~ = ~ n\partial _x^{n-1}W(\sigma _{\gamma ,0})+a^{\alpha \beta }\partial _x^n\left( f_{\beta ,0}\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\beta ,0}}\right) \\&\quad +b^{\alpha \beta }\partial _x^n\left( f_{\beta ,0}\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\beta ,0}}\right) \\&\quad -\, a^{\alpha \beta }\left( f_{\beta ,0}\partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,1}}+f_{\alpha ,1}\partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\beta ,0}}\right) \\&\quad -b^{\alpha \beta }\left( f_{\beta ,0}\partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,0}}+f_{\alpha ,0}\partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\beta ,0}}\right) \\&\quad +\, \left( \frac{3}{2}+\mu _\gamma \right) \left( \sigma _{\gamma ,2}^n+\partial _x^nZ^\varepsilon _\gamma \sigma _{\varepsilon ,1}\right) +\partial _x^n\left( N^\varepsilon _\gamma \sigma _{\varepsilon ,1}\right) -\partial _x^nN^\varepsilon _\gamma \sigma _{\varepsilon ,1}, \end{aligned}$$

here $n\ge 1$ and W is the derivation defined in (4.7). Note that this derivation does NOT commute with $\partial _x$. It is easy to see that ${\hat{I}}_1$ is indeed local. By a careful computation, we obtain the following expression for $I_1$:

$$\begin{aligned} I_1v_\lambda&=\, {\hat{I}}_1\left( \frac{\partial v_\lambda }{\partial \tau _1}\right) +A^\alpha \frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+B^\alpha \frac{\partial v_\lambda }{\partial t^{\alpha ,0}},\\ I_1\sigma _{\lambda ,0}&=\, {\hat{I}}_1\left( \frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1}\right) +A^\alpha \frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+B^\alpha \frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}\\&\quad +\, \left( \frac{3}{2}+\mu _\lambda \right) \left( {\frac{\partial \sigma _{\lambda ,1}}{\partial \tau _2}-\sigma _{\beta ,1}\eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial \sigma _{\lambda ,1}}{\partial t^{\alpha ,0}}}\right) , \end{aligned}$$

where $A^\alpha$ and $B^\alpha$ are local differential polynomials given by

$$\begin{aligned} A^\alpha&= -a^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _1}-\sigma _{\beta ,1}\right) ,\\ B^\alpha&= -\eta ^{\alpha \beta }\left( \frac{1}{2} +\mu _\alpha \right) \left( \frac{3}{2}+\mu _\beta \right) \left( \left( \frac{1}{2}+\mu _\beta \right) \frac{\partial f_{\beta ,1}}{\partial \tau _1}-\sigma _{\beta ,2}-Z^\varepsilon _\beta \sigma _{\varepsilon ,1}\right) \\&\quad \quad -(b^{\alpha \beta }+b^{\beta \alpha })\left( \frac{\partial f_{\beta ,0}}{\partial \tau _1}-\sigma _{\beta ,1}\right) . \end{aligned}$$

We start by proving the first identity given in (3.17), which can also be written as

$$\begin{aligned}\left[ \frac{\partial }{\partial \tau _0}, I_1\right] +\left[ \frac{\partial }{\partial \tau _1}, \left[ \frac{\partial }{\partial \tau _0}, X^\circ \right] \right] = 0.\end{aligned}$$

The following lemmas can be proved by a lengthy but straightforward computation.

Lemma 13

Let $Q\in \hat{\mathcal A}$ be any local differential polynomial, then we have

$$\begin{aligned} \left[ {\hat{I}}_1, \partial _x\right] Q&=\ W(Q)+a_{\alpha \beta }\left( f_{\beta ,0}'\frac{\partial Q}{\partial t^{\alpha ,1}}+f_{\alpha ,1}'\frac{\partial Q}{\partial t^{\beta ,0}}\right) + b_{\alpha \beta }\left( f_{\beta ,0}'\frac{\partial Q}{\partial t^{\alpha ,0}}+f_{\alpha ,0}'\frac{\partial Q}{\partial t^{\beta ,0}}\right) \\&\quad +\, \sum _{n\ge 0}\sigma _{\zeta ,1}^1\partial _x^n\left( Y^\zeta _{\gamma }-\left( \frac{3}{2}+\mu _\gamma \right) Z^\zeta _{\gamma }\right) \frac{\partial }{\partial \sigma _{\gamma ,0}^n}Q\\ {}&\quad +\, \left( \frac{3}{2}+\mu _\gamma \right) \partial _x\left( \sigma _{\gamma ,2}+Z^\varepsilon _\gamma \sigma _{\varepsilon ,1}\right) \frac{\partial }{\partial \sigma _{\gamma ,0}}Q. \end{aligned}$$

Lemma 14

The following identities hold true:

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, I_1\right] v_\lambda&=\ \left[ \frac{\partial }{\partial \tau _1}, \left[ \frac{\partial }{\partial \tau _0}, {\hat{I}}_1\right] \right] v_\lambda +\frac{\partial A^\alpha }{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+\frac{\partial B^\alpha }{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}},\\ \left[ \frac{\partial }{\partial \tau _0}, I_1\right] \sigma _{\lambda ,0}&=\ \left[ \frac{\partial }{\partial \tau _1}, \left[ \frac{\partial }{\partial \tau _0}, {\hat{I}}_1\right] \right] \sigma _{\lambda ,0}+\frac{\partial A^\alpha }{\partial \tau _0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+\frac{\partial B^\alpha }{\partial \tau _0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}\\&\quad +\, \left( \frac{3}{2}+\mu _\lambda \right) \left( {\frac{\partial \sigma _{\lambda ,1}}{\partial \tau _2}-\sigma _{\beta ,1}\eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial \sigma _{\lambda ,1}}{\partial t^{\alpha ,0}}}\right) . \end{aligned}$$

Lemma 15

We have the following decomposition:

$$\begin{aligned}\left[ \frac{\partial }{\partial \tau _0}, {\hat{I}}_1+X^\circ \right] = D+\frac{\partial }{\partial \tau _2},\end{aligned}$$

where D is a derivation $\hat{\mathcal A}\rightarrow \hat{\mathcal A}^+$ whose actions are given by the formulae

$$\begin{aligned} Dv_\gamma ^{(n)}&= \, -a^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _0}\partial _x^n\frac{\partial v_\gamma }{\partial t^{\alpha ,1}}+\frac{\partial f_{\alpha ,1}}{\partial \tau _0}\partial _x^n\frac{\partial v_\gamma }{\partial t^{\beta ,0}}\right) \\ {}&\quad -b^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _0}\partial _x^n\frac{\partial v_\lambda }{\partial t^{\alpha ,0}}+\frac{\partial f_{\alpha ,0}}{\partial \tau _0}\partial _x^n\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) \\&\quad +\, \sigma _{\beta ,1}\eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\alpha \right) \left( \frac{1}{2}+\mu _\beta \right) \partial _x^n\frac{\partial v_\gamma }{\partial t^{\alpha ,0}}, \end{aligned}$$

$$\begin{aligned} D\sigma _{\gamma ,0}^n&= \, -a^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _0}\partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,1}}+\frac{\partial f_{\alpha ,1}}{\partial \tau _0}\partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\beta ,0}}\right) \\ {}&\quad -b^{\alpha \beta }\left( \frac{\partial f_{\beta ,0}}{\partial \tau _0}\partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,0}}+\frac{\partial f_{\alpha ,0}}{\partial \tau _0}\partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\beta ,0}}\right) \\&\quad +\, \sigma _{\beta ,1}\eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\alpha \right) \left( \frac{1}{2}+\mu _\beta \right) \partial _x^n\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,0}}+\frac{\partial \sigma _{\beta ,0}}{\partial \tau _1}\partial _x^n\left( N^\beta _\gamma -\left( \frac{3}{2}+\mu _\gamma \right) Z^\beta _\gamma \right) \\&\quad -\, \delta _{n,0}\left( \frac{3}{2}+\mu _\gamma \right) \frac{\partial }{\partial \tau _0}\left( \sigma _{\gamma ,2}+Z^\varepsilon _\gamma \sigma _{\varepsilon ,1}\right) . \end{aligned}$$

Proposition 13

The first identity of the closedness condition (3.17) holds true, i.e.,

$$\begin{aligned}\left[ \frac{\partial }{\partial \tau _0}, I_1\right] +\left[ \frac{\partial }{\partial \tau _1}, \left[ \frac{\partial }{\partial \tau _0}, X^\circ \right] \right] = 0.\end{aligned}$$

Proof

It follows from Lemma 14 that

$$\begin{aligned}&\,\left[ \frac{\partial }{\partial \tau _0}, I_1\right] v_\lambda +\left[ \frac{\partial }{\partial \tau _1}, \left[ \frac{\partial }{\partial \tau _0}, X^\circ \right] \right] v_\lambda \\&\quad =\, \left[ \frac{\partial }{\partial \tau _1}, \left[ \frac{\partial }{\partial \tau _0}, {\hat{I}}_1+X^\circ \right] \right] v_\lambda +\frac{\partial A^\alpha }{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+\frac{\partial B^\alpha }{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}}. \end{aligned}$$

To prove the vanishing of the right hand side of the above equation, we need to verify the following identity due to Lemma 15:

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _1}, D\right] v_\lambda +\frac{\partial A^\alpha }{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+\frac{\partial B^\alpha }{\partial \tau _0}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}} = 0, \end{aligned}$$

which can be checked directly by using the definition of D. Similarly we can prove that

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, I_1\right] \sigma _{\lambda ,0}+\left[ \frac{\partial }{\partial \tau _1}, \left[ \frac{\partial }{\partial \tau _0}, X^\circ \right] \right] \sigma _{\lambda ,0} = 0. \end{aligned}$$

The proposition is proved.$\square$

It remains to prove the second identity of (3.17).

Lemma 16

The following identities hold true:

$$\begin{aligned} \left[ I_1, \frac{\partial }{\partial \tau _1}\right] v_\lambda&= \, \sum _{s\ge 0}G^\gamma _s\frac{\partial }{\partial v_\gamma ^{(s)}}\frac{\partial v_\lambda }{\partial \tau _1}+F^\gamma _s\frac{\partial }{\partial \sigma _{\gamma ,0}^{s}}\frac{\partial v_\lambda }{\partial \tau _1}\\&\quad -\, A^\alpha \frac{\partial }{\partial \tau _1}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}-B^\alpha \frac{\partial }{\partial \tau _1}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}}+\frac{\partial B^\alpha }{\partial \tau _1}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}},\\ \left[ I_1, \frac{\partial }{\partial \tau _1}\right] \sigma _{\lambda ,0}&=\, \sum _{s\ge 0}G^\gamma _s\frac{\partial }{\partial v_\gamma ^{(s)}}\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1}+F^\gamma _s\frac{\partial }{\partial \sigma _{\gamma ,0}^{s}}\frac{\partial \sigma _{\lambda ,0}}{\partial \tau _1}\\&\quad -\, A^\alpha \frac{\partial }{\partial \tau _1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}-B^\alpha \frac{\partial }{\partial \tau _1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}+\frac{\partial B^\alpha }{\partial \tau _1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}, \end{aligned}$$

where $G^\gamma _s$ and $F^\gamma _s$ are differential polynomials defined by

$$\begin{aligned} G_s^\gamma = \left[ \frac{\partial }{\partial \tau _1}, {\hat{I}}_1\right] v_\gamma ^{(s)}+\partial _x^s\left( I_1v_\gamma \right) ,\quad F_s^\gamma = \left[ \frac{\partial }{\partial \tau _1}, {\hat{I}}_1\right] \sigma _{\gamma ,0}^s+\partial _x^s\left( I_1\sigma _{\gamma ,0}\right) . \end{aligned}$$

Lemma 17

The differential polynomials $G^\gamma _s$ and $F^\gamma _s$ defined in Lemma 16 have the following expressions:

$$\begin{aligned} G_s^\gamma&=\, A^\alpha \partial _x^s\frac{\partial v_\gamma }{\partial t^{\alpha ,1}}+B^\alpha \partial _x^s\frac{\partial v_{\gamma }}{\partial t^{\alpha ,0}},\\ F_s^\gamma&=\, A^\alpha \partial _x^s\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,1}}+B^\alpha \partial _x^s\frac{\partial \sigma _{\gamma ,0}}{\partial t^{\alpha ,0}}\\&\quad +\, \delta _{s,0}\left( \frac{3}{2}+\mu _\gamma \right) \left( \frac{\partial \sigma _{\gamma ,1}}{\partial \tau _2}-\sigma _{\beta ,1}\eta ^{\alpha \beta }\left( \frac{1}{2}+\mu _\alpha \right) \frac{\partial \sigma _{\gamma ,1}}{\partial t^{\alpha ,0}}\right) . \end{aligned}$$

Proof

By using the definition of $G^\gamma _s$ we can obtain the identity

$$\begin{aligned} G^\gamma _{s+1}-\partial _xG^\gamma _s = \frac{\partial }{\partial \tau _1}\left( \left[ {\hat{I}}_1, \partial _x\right] v_\gamma ^{(s)}\right) +\left[ \partial _x, {\hat{I}}_1\right] \frac{\partial v_\gamma ^{(s)}}{\partial \tau _1}. \end{aligned}$$

Therefore it follows from Lemma 13 that

$$\begin{aligned} G^\gamma _{s+1}-\partial _xG^\gamma _s = -\partial _xA^\alpha \partial _x^s\frac{\partial v_\gamma }{\partial t^{\alpha ,1}}-\partial _xB^\alpha \partial _x^s\frac{\partial v_\gamma }{\partial t^{\alpha ,0}}. \end{aligned}$$

Hence $G^\gamma _s$ can be solved recursively starting from the initial condition

$$\begin{aligned} G^\gamma _0 = I_1v_\gamma -{\hat{I}}_1\frac{\partial v_\gamma }{\partial \tau _1} = A^\alpha \frac{\partial v_\gamma }{\partial t^{\alpha ,1}}+B^\alpha \frac{\partial v_\gamma }{\partial t^{\alpha ,0}}. \end{aligned}$$

The expressions of the differential polynomials $F_s^\gamma$ can be obtained similarly. The lemma is proved.$\square$

Proposition 14

The second identity of the closedness condition (3.17) holds true, i.e.,

$$\begin{aligned}\left[ \frac{\partial }{\partial \tau _1}, I_1\right] = 0.\end{aligned}$$

Proof

The proof of the proposition is straightforward by combining the results of Lemmas 16 and 17.$\square$

4.3 Vanishing of the genus one obstruction

In this subsection, we will work with the canonical coordinates of the Frobenius manifold. Let us start by recalling some useful formulae related to the canonical coordinates. For the details one may refer to the work [5, 12].

We denote by $u^1,\dots ,u^n$ the local canonical coordinates of a semisimple Frobenius manifold M and denote by $(u^i;\theta _i)$ the corresponding coordinates of ${\hat{M}}$. Under this system of local coordinates, the bihamiltonian structure (2.13) takes the form (2.7), i.e.,

$$\begin{aligned} P_0^{[0]}= & {} \frac{1}{2}\int \sum _{i,j=1}^n\left( \delta _{i,j} f^i\theta _i\theta _i^1+ A^{ij}\theta _i\theta _j\right) ,\quad \\ P_1^{[0]}= & {} \frac{1}{2}\int \sum _{i,j=1}^n\left( \delta _{i,j} u^if^i\theta _i\theta _i^1+ B^{ij}\theta _i\theta _j\right) . \end{aligned}$$

Introduce functions

$$\begin{aligned} \psi _{i1} = \frac{1}{\sqrt{f^i}},\quad i=1,\dots ,n, \end{aligned}$$

(4.12)

where the sign of the square root can be arbitrarily chosen, and define

$$\begin{aligned} \psi _{i\alpha } = \frac{1}{\psi _{i1}}\frac{\partial v_\alpha }{\partial u^i},\quad \gamma _{ij} = \frac{1}{\psi _{j1}}\frac{\partial \psi _{i1}}{\partial u^j},\quad i\ne j. \end{aligned}$$

Then it is proved in [5] that

$$\begin{aligned}&\frac{\partial v_\alpha }{\partial u^i} = \psi _{i1}\psi _{i\alpha },\quad \frac{\partial u^i}{\partial v^\alpha } = \frac{\psi _{i\alpha }}{\psi _{i1}}, \end{aligned}$$

(4.13)

$$\begin{aligned}&\frac{\partial \psi _{i\alpha }}{\partial u^k} = \gamma _{ik}\psi _{k\alpha },\quad i\ne k,\quad \frac{\partial \psi _{i\alpha }}{\partial u^i} = -\sum _{k\ne i}\gamma _{ik}\psi _{k\alpha }. \end{aligned}$$

(4.14)

Let $V_{ij} = \mu _\alpha \eta ^{\alpha \beta }\psi _{i\alpha }\psi _{j\beta },$ then we have the identity

$$\begin{aligned} \gamma _{ij}(u^j-u^i) = V_{ij}. \end{aligned}$$

(4.15)

Now we are going to describe the deformation problem given in Sect. 3.3 in terms of the canonical coordinates. Introduce the odd variables $\theta _{i,m}$ by the recursion relations (2.16), then it follows from the the relation (2.17) that the equations (3.8) and (3.9) can be represented in the form

$$\begin{aligned} \frac{\partial u^i}{\partial s_2}&= a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial u^i}{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial u^i}{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial u^i}{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial u^i}{\partial t^{\beta ,0}}\right) \nonumber \\ {}&\quad +Xu^i+{\mathcal {L}}_2u^i,\end{aligned}$$

(4.16)

$$\begin{aligned} \frac{\partial \theta _{i,0}}{\partial s_2}&= a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial \theta _{i,0}}{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial \theta _{i,0}}{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial \theta _{i,0}}{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial \theta _{i,0}}{\partial t^{\beta ,0}}\right) \nonumber \\ {}&\quad +X\theta _{i,0} +\sum _jA^j_i\theta _{j,2}+B^j_i\theta _{j,1}+{\mathcal {L}}_2\theta _{i,0}. \end{aligned}$$

(4.17)

In the canonical coordinates, the Hamiltonian operator $\mathcal P_1$ of $P_1$ has the form

$$\begin{aligned} {\mathcal {P}}_1^{ij} =u^if^i\delta _{ij}\partial _x+\frac{1}{2} \partial _x\left( u^if^i\right) \delta _{ij}+B^{ij}+Q^{ij}\partial _x^3+\dots , \end{aligned}$$

where $B^{ij}$ is defined in (2.6), $Q^{ij}$ is given by the formula (cf., e.g., [13])

$$\begin{aligned} Q^{ij} = 3c_i\left( f^i\right) ^2\delta _{ij}+\frac{1}{2}\left( u^i-u^j\right) \left( f^j\partial _jf^ic_i-f^i\partial _if^jc_j\right) , \end{aligned}$$

(4.18)

and $c_i$ is the i-th central invariant. Here and henceforth we omit the terms that do not contribute to the relevant computations given later. Let us also represent the evolutions of $\theta _i$ along the flows $\frac{\partial }{\partial t^{\alpha ,p}}$ as follows:

$$\begin{aligned} \frac{\partial \theta _i}{\partial t^{\alpha ,p}} = T_{\alpha ,p}^i\theta _i^1+\dots +\sum _j K_{\alpha ,p;j}^i\theta _j^3+\dots ,\quad T_{\alpha ,p}\in \hat{\mathcal A}^0_0,\ K_{\alpha ,p;j}^i\in \hat{\mathcal A}^0_3. \end{aligned}$$

Note that the coefficients of $\theta _j^1$ of the leading term of $\frac{\partial \theta _i}{\partial t^{\alpha ,p}}$ are zero for $j\ne i$, this is due to the fact that the leading term of the flow $\frac{\partial }{\partial t^{\alpha ,p}}$ is diagonal, one may refer to [9] for details.

Let us turn to the proof of the vanishing of the genus one obstruction. Due to the closedness condition (3.15), we can choose a derivation $X^\circ$ such that when we take $X = X^\circ$ in (4.16) and (4.17) we have

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial s_2}\right] = 0. \end{aligned}$$

We require that the leading term of $X^\circ$ is given by the Virasoro symmetry $\frac{\partial }{\partial s_2}$ of the super tau-cover of the Principal Hierarchy, hence it follows from the result of [11] that the actions of $X^\circ$ on $u^i$ and $\theta _i$ take the form

$$\begin{aligned} X^\circ u^i&= (u^i)^3+\sum _jL^i_ju^{j,2}+\dots ,\quad L^i_j\in \hat{\mathcal A}^0_0, \\ X^\circ \theta _i&= \sum _jM^j_i\theta _j+\sum _jJ^j_i\theta _j^2+\dots ,\quad M^j_i,J^j_i\in \hat{\mathcal A}^0_0. \end{aligned}$$

Lemma 18

We have the following identity:

$$\begin{aligned} J^i_i-L^i_i&=\, \frac{2}{f^i}c_0(u^i)^2Q^{ii}-\sum _j\frac{A^j_i}{f^j}(u^i+u^j)Q^{ij}-\sum _j\frac{B^j_i}{f^j}Q^{ij}\\&\quad -\, b^{\beta ,q}_{2;1,0}K^i_{\beta ,q;i}-a^{\alpha ,p;\beta ,q}_2\left( \left( f_{\beta ,q}'\right) _0K^i_{\alpha ,p;i}+\left( f_{\alpha ,p}'\right) _0K^i_{\beta ,q;i}\right) , \end{aligned}$$

where $b^{\beta ,q}_{2;\alpha ,p}$ and $a^{\alpha ,p;\beta ,q}_2$ are the constants that appear in the operator ${\mathcal {L}}_2$, and $\left( f_{\alpha ,p}'\right) _0$ denotes the differential degree zero component of $f_{\alpha ,p}'$.

Proof

We can prove this lemma by looking at the differential degree 3 component of the left hand side of the equation

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial s_2}\right] u^i = 0, \end{aligned}$$

and by computing the coefficient of $\theta _i^3$. The lemma is proved.$\square$

Remark 4

Note that $c_0$ is the arbitrary constant that appears in the operator ${\mathcal {L}}_2$, which is different from central invariants $c_1,\dots ,c_n$.

According to the discussion given in Sect. 3.3, we need to show the triviality of the cohomology class of the differential degree 3 component of the derivation $\left[ \frac{\partial }{\partial \tau _1}, \mathcal C\right] = I_1-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right]$. Due to Theorem 10, it suffices to prove that in the differential degree 3 component of

$$\begin{aligned} I_1u^i-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right] u^i, \end{aligned}$$

the coefficient of $\theta _i^3$ vanishes. By using Lemma 18 and a straightforward computation, we obtain the following lemma.

Lemma 19

In the differential degree 3 component of $I_1u^i-\left[ \frac{\partial }{\partial \tau _1}, X^\circ \right] u^i,$ the coefficient of $\theta _i^3$ reads

$$\begin{aligned}&\sum _{j}Q^{ij}\left( M^i_j+A^i_j(u^i)^2+B^i_ju^i\right) +E^3Q^{ii}-(6+c_0)(u^i)^2Q^{ii}\nonumber \\&\quad +3Q^{ii}b^{\beta ,q}_{2;1,0}T_{\beta ,q}^i+3Q^{ii}a^{\alpha ,p;\beta ,q}_2\left( \left( f_{\beta ,q}'\right) _0T_{\alpha ,p}^i+\left( f_{\alpha ,p}'\right) _0T_{\beta ,q}^i\right) , \end{aligned}$$

(4.19)

here $E^3$ is the cubic power of the Euler vector field which is given by

$$\begin{aligned} E^3 = \sum _i(u^i)^3\frac{\partial }{\partial u^i}. \end{aligned}$$

Lemma 20

We have the identities

$$\begin{aligned} M^i_i+A^i_i(u^i)^2+B^i_iu^i&=\, (3+c_0)(u^i)^2-\frac{1}{f^i}E^3f^i-b^{\beta ,q}_{2;1,0}T_{\beta ,q}^i\\ {}&\quad -a^{\alpha ,p;\beta ,q}_2\left( \left( f_{\beta ,q}'\right) _0T_{\alpha ,p}^i+\left( f_{\alpha ,p}'\right) _0T_{\beta ,q}^i\right) , \end{aligned}$$

and $M^j_i+A^j_i(u^j)^2+B^j_iu^j = 0$ for $i\ne j$.

Proof

We can prove this lemma by considering the differential degree 1 component of the left hand side of the equation

$$\begin{aligned} \left[ \frac{\partial }{\partial \tau _0}, \frac{\partial }{\partial s_2}\right] u^i = 0, \end{aligned}$$

and by computing the coefficient of $\theta _j^1$. The lemma is proved.$\square$

In order to prove the vanishing of the expression (4.19), we need to check, due to Lemma 20 and the expression (4.18) for $Q^{ii}$, the following identity:

$$\begin{aligned} E^3f^i+2f^i\left( b^{\beta ,q}_{2;1,0}T_{\beta ,q}^i+a^{\alpha ,p;\beta ,q}_2\left( \left( f_{\beta ,q}'\right) _0T_{\alpha ,p}^i+\left( f_{\alpha ,p}'\right) _0T_{\beta ,q}^i\right) \right) = 3(u^i)^2f^i. \end{aligned}$$

Proposition 15

For $m\ge -1$, we have

$$\begin{aligned}&E^{m+1}f^i+2f^i\left( b^{\beta ,q}_{m;1,0}T_{\beta ,q}^i\right. \\&\quad {+\left. a^{\alpha ,p;\beta ,q}_m\left( \left( f_{\beta ,q}'\right) _0T_{\alpha ,p}^i+\left( f_{\alpha ,p}'\right) _0T_{\beta ,q}^i\right) \right) } = (1+m)(u^i)^mf^i,\end{aligned}$$

here $E^{m+1}$ is the $(m+1)$-th power of the Euler vector field E which is given by

$$\begin{aligned} E^{m+1} = \sum _i(u^i)^{m+1}\frac{\partial }{\partial u^i}. \end{aligned}$$

Proof

Let us consider the following generating functions:

$$\begin{aligned}&\sum _{m\ge -1}\frac{1}{\lambda ^{m+2}}E^{m+1}f^i = \sum _j\frac{1}{\lambda -u^j}\frac{\partial f^i}{\partial u^j},\quad \sum _{m\ge -1}\frac{1}{\lambda ^{m+2}}(1+m)(u^i)^mf^i = \frac{f^i}{(\lambda -u^i)^2},\\&\sum _{m\ge -1}\frac{1}{\lambda ^{m+2}}\left( b^{\beta ,q}_{m;1,0}T_{\beta ,q}^i+a^{\alpha ,p;\beta ,q}_m\left( \left( f_{\beta ,q}'\right) _0T_{\alpha ,p}^i+\left( f_{\alpha ,p}'\right) _0T_{\beta ,q}^i\right) \right) \\&\quad =\, \frac{1}{2}\frac{1}{(\lambda -u^i)^2}+\sum _{j\ne i}\frac{\psi _{j1}}{\psi _{i1}}\frac{V_{ij}}{(\lambda -u^i)(\lambda -u^j)}, \end{aligned}$$

where the last generating function is computed in [12], one may refer to Lemma 3.10.18 and the proof of Theorem 3.10.29 of [12] for details. Then the proposition is proved by using (4.12), (4.14) and (4.15).$\square$

Finally we have the following theorem.

Theorem 13

For a given semisimple Frobenius manifold and a tau-symmetric bihamiltonian deformation of its Principal Hierarchy, there exists a unique deformation $\frac{\partial }{\partial s_2}\in \mathrm {Der}^\partial (\hat{\mathcal A}^{Vir})$ of the Virasoro symmetry of the super tau-cover of the Principal Hierarchy such that it is a symmetry of the deformed super tau-cover. Moreover, the actions of $\frac{\partial }{\partial s_2}$ on the local variables are given by

$$\begin{aligned} \frac{\partial v_\lambda }{\partial s_2}&=\, a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial v_\lambda }{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial v_\lambda }{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial v_\lambda }{\partial t^{\beta ,0}}\right) \nonumber \\&\quad +\, Xv_{\lambda }+{\mathcal {L}}_2v_\lambda , \end{aligned}$$

(4.20)

$$\begin{aligned} \frac{\partial \sigma _{\lambda ,0}}{\partial s_2}&=\, a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial \sigma _{\lambda ,0}}{\partial t^{\beta ,0}}\right) \nonumber \\&\quad +\, X\sigma _{\lambda ,0} +\left( \frac{5}{2}+c_0+\mu _\lambda \right) \sigma _{\lambda ,2}+N^\zeta _{\lambda }\sigma _{\zeta ,1}+\mathcal L_2\sigma _{\lambda ,0}, \end{aligned}$$

(4.21)

where $X\in \mathrm {Der}^\partial (\hat{\mathcal A})^0$ and $N^\zeta _{\lambda }\in \hat{\mathcal A}^0_{\ge 0}$ is the differential polynomial described in Lemma 10.

4.4 Lifting to the tau-covers

In order to lift the symmetry (4.20) to the tau-cover of the deformed Principal Hierarchy, we first need to rewrite (4.20) in terms of the normal coordinates $w^1,\dots , w^n$ of M. We start by proving the following lemmas.

Lemma 21

Let $g_\lambda \in \hat{\mathcal A}^0_{\ge 1}$ be the differential polynomials given in (2.35) which satisfy the identities

$$\begin{aligned} h_{\lambda ,0} = v_\lambda +\partial _xg_\lambda ,\quad \lambda = 1,\dots ,n. \end{aligned}$$

Then we have:

• 1. $g_1 = 0$.

• 2. For any $\alpha = 1,\dots ,n$ and $p\ge 0$,

  $$\begin{aligned} \Omega _{\alpha ,p;\lambda ,0} = \frac{\delta h_{\alpha ,p+1}}{\delta v^\lambda }+\frac{\partial g_\lambda }{\partial t^{\alpha ,p}}. \end{aligned}$$

Proof

Due to Proposition 7, we have $D_{X_{1,0}} = \partial _x$ and $X_{1,0} = -[H_{1,0},P_0]$, from which it follows that

$$\begin{aligned} v_\alpha = \frac{\delta H_{1,0}}{\delta v^\alpha }. \end{aligned}$$

By taking $\alpha = 1$, we obtain the first property by using the definition (2.28). The second one is obvious due to Theorem 8. The lemma is proved.$\square$

Lemma 22

There exists a derivation $X^\circ \in \mathrm {Der}^\partial (\hat{\mathcal A})$ such that its leading term is given by the Virasoro symmetry $\frac{\partial }{\partial s_2}$ of the Principal Hierarchy, and it satisfies the equations $\left[ \frac{\partial }{\partial \tau _0}, X^\circ \right] = I_0$ and

$$\begin{aligned} X^\circ h_{\lambda ,0}+{\hat{I}}_1h_{\lambda ,0}&=\, a^{\alpha \beta }\left( f_{\alpha ,1}'\Omega _{\beta ,0;\lambda ,0}+f_{\beta ,0}'\Omega _{\alpha ,1;\lambda ,0}+\frac{\partial ^2\Omega _{\alpha ,1;\beta ,0}}{\partial t^{\lambda ,0}\partial t^{1,0}}\right) \nonumber \\&\quad +\, b^{\alpha \beta }\left( f_{\alpha ,0}'\Omega _{\beta ,0;\lambda ,0}+f_{\beta ,0}'\Omega _{\alpha ,0;\lambda ,0}+\frac{\partial ^2\Omega _{\alpha ,0;\beta ,0}}{\partial t^{\lambda ,0}\partial t^{1,0}}\right) \nonumber \\&\quad +\, b^{\beta ,q}_{2;1,0}\Omega _{\beta ,q;\lambda ,0}+b^{\beta ,q}_{2;\lambda ,0}\Omega _{\beta ,q;1,0}+2c_{2;\lambda ,0;1,0}. \end{aligned}$$

(4.22)

Proof

We first find a particular solution ${\tilde{X}}^\circ$ of the equation $\left[ \frac{\partial }{\partial \tau _0}, X\right] = I_0$, then we modify it by a solution $\tilde{{\mathcal {C}}}$ of the homogeneous equation $\left[ \frac{\partial }{\partial \tau _0}, {{\mathcal {C}}}\right] = 0$ such that $X^\circ :=\tilde{X}^\circ +\tilde{{\mathcal {C}}}$ satisfies (4.22).

Let us define ${\tilde{X}}^\circ \in \mathrm {Der}^\partial (\hat{\mathcal A})$ as follows:

$$\begin{aligned} {\tilde{X}}^\circ v_\lambda&=\ a^{\alpha \beta }\left( f_{\alpha ,1}'\frac{\delta h_{\beta ,1}}{\delta v^\lambda }+f_{\beta ,0}'\frac{\delta h_{\alpha ,2}}{\delta v^\lambda }\right) +b^{\alpha \beta }\left( f_{\alpha ,0}'\frac{\delta h_{\beta ,1}}{\delta v^\lambda }+f_{\beta ,0}'\frac{\delta h_{\alpha ,1}}{\delta v^\lambda }\right) \\&\quad +\, b^{\beta ,q}_{2;1,0}\frac{\delta h_{\beta ,q+1}}{\delta v^\lambda }+b^{\beta ,q}_{2;\lambda ,0}\frac{\delta h_{\beta ,q+1}}{\delta v^1}+2c_{2;\lambda ,0;1,0};\\ {\tilde{X}}^\circ \sigma _{\lambda ,0}&=\ a^{\alpha \beta }\left( \frac{\partial f_{\alpha ,1}}{\partial \tau _0}\frac{\delta h_{\beta ,1}}{\delta v^\lambda }+\frac{\partial f_{\beta ,0}}{\partial \tau _0}\frac{\delta h_{\alpha ,2}}{\delta v^\lambda }\right) +b^{\alpha \beta }\left( \frac{\partial f_{\alpha ,0}}{\partial \tau _0}\frac{\delta h_{\beta ,1}}{\delta v^\lambda }+\frac{\partial f_{\beta ,0}}{\partial \tau _0}\frac{\delta h_{\alpha ,1}}{\delta v^\lambda }\right) \\&\quad +\, b^{\beta ,q}_{2;\lambda ,0}\frac{\partial f_{\beta ,q}}{\partial \tau _0}-\left( \frac{5}{2}+\mu _\lambda \right) \sigma _{\lambda ,2}-N^\zeta _\lambda \sigma _{\zeta ,1}. \end{aligned}$$

Firstly, from the definition of $N^\zeta _\lambda$ it follows that $\tilde{X}^\circ$ is indeed local. We also note that the leading terms of ${\tilde{X}}^\circ v_\lambda$ and ${\tilde{X}}^\circ \sigma _{\lambda ,0}$ coincide with the local terms of the Virasoro symmetry $\frac{\partial v_\lambda }{\partial s_2}$ and $\frac{\partial \sigma _{\lambda ,0}}{\partial s_2}$ of the Principal Hierarchy. By a direct computation, it is easy to check that $\left[ \frac{\partial }{\partial \tau _0}, {\tilde{X}}^\circ \right] = I_0$.

Next we want to determine $\tilde{{\mathcal {C}}}\in \mathrm {Der}^\partial (\hat{\mathcal A})$ such that $X^\circ = {\tilde{X}}^\circ +\tilde{{\mathcal {C}}}$ satisfies (4.22). By using the definition of ${\tilde{X}}^\circ v_\lambda$, Lemmas 13 and 21, it is straightforward to show that

$$\begin{aligned} \tilde{{\mathcal {C}}}(h_{\lambda ,0}) = a^{\alpha \beta }\frac{\partial ^2\Omega _{\alpha ,1;\beta ,0}}{\partial t^{\lambda ,0}\partial t^{1,0}} + b^{\alpha \beta }\frac{\partial ^2\Omega _{\alpha ,0;\beta ,0}}{\partial t^{\lambda ,0}\partial t^{1,0}}-\partial _x{\hat{I}}_1(g_\lambda )-\partial _x{\tilde{X}}^\circ (g_\lambda ), \end{aligned}$$

(4.23)

which uniquely determines the actions of $\tilde{{\mathcal {C}}}$ on $v_\lambda$.

Finally we need to check such $\tilde{{\mathcal {C}}}$ satisfies $\left[ \frac{\partial }{\partial \tau _0}, \tilde{{\mathcal {C}}}\right] = 0$. By using the next lemma we know that it suffices to show that $\int \tilde{{\mathcal {C}}} v_\lambda$ = 0, which is obvious from (4.23) and (2.35). The lemma is proved.$\square$

Lemma 23

Let $U_1,\dots , U_n$ be differential polynomials with $U_\lambda \in \hat{\mathcal A}^0_{\ge 2}$. Then there exists $\mathcal C\in \mathrm {Der}^\partial (\hat{\mathcal A})^0_{\ge 1}$ such that $\left[ \frac{\partial }{\partial \tau _0}, {\mathcal {C}}\right] = 0$ and ${\mathcal {C}} v_\lambda = U_\lambda$ if and only if $\int U_\lambda = 0$.

Proof

Let ${\mathcal {C}}\in \mathrm {Der}^\partial (\hat{\mathcal A})^0$ be a derivation such that $\left[ \frac{\partial }{\partial \tau _0}, {\mathcal {C}}\right] = 0$. Then by using the triviality of the variational Hamiltonian cohomology $H^0_{\ge 1}\bigl (\mathrm {Der}^\partial (\hat{\mathcal A}),P_0^{[0]}\bigr )$, we know the existence of a certain ${\mathcal {K}}\in \mathrm {Der}^\partial (\hat{\mathcal A})^{-1}$ such that $\left[ \frac{\partial }{\partial \tau _0}, {\mathcal {K}}\right] = {\mathcal {C}}$. Let us denote ${\mathcal {K}}\sigma _{\lambda ,0} = V_\lambda \in \hat{\mathcal A}^0$. Then it is easy to see that

$$\begin{aligned} \mathcal Cv_\lambda = \left[ \frac{\partial }{\partial \tau _0}, {\mathcal {K}}\right] v_\lambda = \mathcal K\sigma _{\lambda ,0}^1 = \partial _xV_\lambda . \end{aligned}$$

Therefore we have $\int U_\lambda = \int \partial _xV_\lambda = 0$.

Conversely, if $U_\lambda = \partial _xV_\lambda$ for some $V_\lambda \in \hat{\mathcal A}^0$, we can define a unique derivation ${\mathcal {C}}\in \mathrm {Der}^\partial (\hat{\mathcal A})^0$ by

$$\begin{aligned} {\mathcal {C}} v_\lambda = \partial _xV_\lambda ,\quad {\mathcal {C}}\sigma _{\lambda ,0} = \frac{\partial V_\lambda }{\partial \tau _0}, \end{aligned}$$

then it is easy to check that $\left[ \frac{\partial }{\partial \tau _0}, {\mathcal {C}}\right] = 0$ and the lemma is proved.$\square$

Now let us denote ${\mathcal {C}} = X - X^\circ$, where the derivation X is described in Theorem 13 and $X^\circ$ satisfies (4.22). Then we can rewrite the Virasoro symmetry $\frac{\partial }{\partial s_2}$ in terms of the normal coordinates as follows:

$$\begin{aligned} \frac{\partial w_\lambda }{\partial s_2}&=\ a^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial w_\lambda }{\partial t^{\alpha ,1}}+f_{\alpha ,1}\frac{\partial w_\lambda }{\partial t^{\beta ,0}}\right) + b^{\alpha \beta }\left( f_{\beta ,0}\frac{\partial w_\lambda }{\partial t^{\alpha ,0}}+f_{\alpha ,0}\frac{\partial w_\lambda }{\partial t^{\beta ,0}}\right) \\&\quad +\, a^{\alpha \beta }\left( f_{\alpha ,1}'\Omega _{\beta ,0;\lambda ,0}+f_{\beta ,0}'\Omega _{\alpha ,1;\lambda ,0}+\frac{\partial ^2\Omega _{\alpha ,1;\beta ,0}}{\partial t^{\lambda ,0}\partial t^{1,0}}\right) \\&\quad +\, b^{\alpha \beta }\left( f_{\alpha ,0}'\Omega _{\beta ,0;\lambda ,0}+f_{\beta ,0}'\Omega _{\alpha ,0;\lambda ,0}+\frac{\partial ^2\Omega _{\alpha ,0;\beta ,0}}{\partial t^{\lambda ,0}\partial t^{1,0}}\right) \\&\quad +\, b^{\beta ,q}_{2;1,0}\Omega _{\beta ,q;\lambda ,0}+b^{\beta ,q}_{2;\lambda ,0}\Omega _{\beta ,q;1,0}+2c_{2;\lambda ,0;1,0}+\mathcal C(h_{\lambda ,0})+{\mathcal {L}}_2 w_\lambda . \end{aligned}$$

According to Lemma 23, there exist differential polynomials $Q_\lambda \in \hat{\mathcal A}^0_{\ge 1}$ such that ${\mathcal {C}}(h_{\lambda ,0}) = \partial _x Q_\lambda$. From the fact that

$$\begin{aligned} \frac{\partial }{\partial t^{\gamma ,0}}\frac{\partial w_\lambda }{\partial s_2} = \frac{\partial }{\partial t^{\lambda ,0}}\frac{\partial w_\gamma }{\partial s_2}, \end{aligned}$$

we have the equation

$$\begin{aligned} \frac{\partial Q_\lambda }{\partial t^{\gamma ,0}} = \frac{\partial Q_\gamma }{\partial t^{\lambda ,0}}, \end{aligned}$$

(4.24)

which means that $\int Q_1$ is a conserved quantity of the flow $\frac{\partial }{\partial t^{\lambda ,0}}$ for all $\lambda = 1,\dots ,n$. We want to find a differential polynomial $Q\in \hat{\mathcal A}_{\ge 0}$ such that $Q_\lambda = \frac{\partial Q}{\partial t^{\lambda ,0}}$. To this end we need the following lemma.

Lemma 24

(Lemma 4.12 and Theorem A.2 of [9]). If the Frobenius manifold with dimension $n \ge 2$ is irreducible, then there exist constants $c^\alpha$ for $\alpha = 1,\dots ,n$ such that the leading term of the derivation

$$\begin{aligned} D = c^\alpha \frac{\partial }{\partial t^{\alpha ,0}}\in \mathrm {Der}^\partial (\hat{\mathcal A})\end{aligned}$$

is non-degenerate, i.e., in terms of the canonical coordinates $u^i$, its leading term $D^{[0]}\in \mathrm {Der}^\partial (\hat{\mathcal A})^0_1$ can be represented by

$$\begin{aligned} D^{[0]}u^i = A^i(u)u^{i,1},\quad i=1,\dots ,n \end{aligned}$$

such that the condition $\frac{\partial A^i}{\partial u^i}\ne 0$ holds true for all $i=1,\dots ,n$. Moreover, any conserved quantity $H\in \hat{\mathcal F}^0_{\ge 1}$ of D is trivial.

The irreducibility of a Frobenius manifold is a mild condition and without loss of generality, we may always assume that a Frobenius manifold is irreducible (see [5] for details). Therefore it follows from the above lemma that if the dimension of the Frobenius manifold $\dim M\ge 2$, there exists $Q\in \hat{\mathcal A}^0_{\ge 0}$ such that $Q_1 = \partial _x Q$. By setting $\gamma = 1$ in the identity (4.24), we obtain that

$$\begin{aligned} \partial _x Q_\lambda = \frac{\partial Q_1}{\partial t^{\lambda ,0}} = \partial _x \frac{\partial Q}{\partial t^{\lambda ,0}}\in \hat{\mathcal A}^0_{\ge 2}, \end{aligned}$$

from which we conclude that $Q_\lambda = \frac{\partial Q}{\partial t^{\lambda ,0}}.$

Remark 5

The existence of Q can be also obtained by proving a Poincaré lemma for general semi-Hamiltonian systems [28] using the idea given in Appendix of [9], whose proof is a little bit complicated, so we use the method presented above.

Theorem 14

Let ${\mathcal {Z}}$ be a tau-function of the tau-cover (2.32) of the deformed Principal Hierarchy. Then there exists a differential polynomial $O_2\in \hat{\mathcal A}^0_{\ge 0}$ which yields a symmetry of the tau-cover given by

$$\begin{aligned} \frac{\partial {\mathcal {Z}}}{\partial s_2} = L_2^{even}{\mathcal {Z}}+O_2{\mathcal {Z}}, \end{aligned}$$

where the evolutions of $f_{\alpha ,p}$ and $w_\lambda$ along the flow $\frac{\partial }{\partial s_2}$ are given by

$$\begin{aligned} \frac{\partial f_{\alpha ,p}}{\partial s_2} = \frac{\partial }{\partial t^{\alpha ,p}}\frac{\partial \log \mathcal Z}{\partial s_2},\quad \frac{\partial w_\lambda }{\partial s_2} = \frac{\partial }{\partial t^{1,0}}\frac{\partial f_{\lambda ,0}}{\partial s_2}. \end{aligned}$$

Proof

When $\dim M\ge 2$ we take $O_2 = Q$ which is just given above. When $\dim M = 1$, $O_2$ is given in (3.26). The theorem is proved.$\square$

Now let us proceed to determine all other Virasoro symmetries $\frac{\partial }{\partial s_m}$ of the tau-cover of the deformed Principal Hierarchy. In what follows we will denote ${\mathcal {F}} = \log {\mathcal {Z}}$ and denote

$$\begin{aligned}&{\mathcal {G}}_m = a_m^{\alpha ,p;\beta ,q}\left( \frac{\partial \mathcal F}{\partial t^{\alpha ,p}}\frac{\partial {\mathcal {F}}}{\partial t^{\beta ,q}}+\frac{\partial ^2{\mathcal {F}}}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}\right) +c_{m;\alpha ,p;\beta ,q}t^{\alpha ,p}t^{\beta ,q};\\ {}&\mathcal L_m^{even} = b^{\beta ,q}_{m;\alpha ,p}t^{\alpha ,p}\frac{\partial }{\partial t^{\beta ,q}},\quad m\ge -1. \end{aligned}$$

It is proved in [9] that

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s_{-1}} = {\mathcal {G}}_{-1}+\mathcal L_{-1}^{even}{\mathcal {F}} \end{aligned}$$

induces a symmetry of the tau-cover of the deformed Principal Hierarchy. On the other hand, by using Theorem 14 we know that

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s_{2}} = {\mathcal {G}}_{2}+O_2+\mathcal L_{2}^{even}{\mathcal {F}} \end{aligned}$$

also induces a symmetry of the tau-cover of the deformed Principal Hierarchy. From the commutation relation

$$\begin{aligned}{}[L_{-1}^{even},L_2^{even}] = -3 L_1^{even}, \end{aligned}$$

it follows that

$$\begin{aligned} \left[ \frac{\partial }{\partial s_{-1}}, \frac{\partial }{\partial s_2}\right] {\mathcal {F}} = 3\left( \mathcal G_1+{\mathcal {L}}_1^{even}{\mathcal {F}}\right) +\frac{\partial O_2}{\partial s_{-1}}-\mathcal L_{-1}^{even}O_2. \end{aligned}$$

Since $\frac{\partial O_2}{\partial s_{-1}}-{\mathcal {L}}_{-1}^{even}O_2$ is a differential polynomial, we can define

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s_1} =\frac{1}{3}\left[ \frac{\partial }{\partial s_{-1}}, \frac{\partial }{\partial s_2}\right] {\mathcal {F}} = \mathcal G_1+O_1+{\mathcal {L}}_1^{even}{\mathcal {F}},\quad O_1 =\frac{1}{3}\left( \frac{\partial O_2}{\partial s_{-1}}-{\mathcal {L}}_{-1}^{even}O_2\right) . \end{aligned}$$

Then it is obvious that $\frac{\partial {\mathcal {F}}}{\partial s_1}$ induces a symmetry of the tau-cover of the deformed Principal Hierarchy. Similarly we define the symmetry

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s_0} = \frac{1}{2} \left[ \frac{\partial }{\partial s_{-1}}, \frac{\partial }{\partial s_1}\right] {\mathcal {F}} = \mathcal G_0+O_0+{\mathcal {L}}_0^{even}{\mathcal {F}},\quad O_0 =\frac{1}{2}\left( \frac{\partial O_1}{\partial s_{-1}}-{\mathcal {L}}_{-1}^{even}O_1\right) . \end{aligned}$$

Now let us define $\frac{\partial {\mathcal {F}}}{\partial s_m}$ for $m\ge 3$ recursively in the following way. Assume we have defined

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s_m} = {\mathcal {G}}_m+O_m+\mathcal L_m^{even}{\mathcal {F}} \end{aligned}$$

such that it induces a symmetry of the tau-cover of the deformed Principal Hierarchy for $m\ge 2$, then we have

$$\begin{aligned} \frac{\partial }{\partial s_1}\frac{\partial {\mathcal {F}}}{\partial s_m} = \frac{\partial \mathcal G_m}{\partial s_1}+\frac{\partial O_m}{\partial s_1}+{\mathcal {L}}_m^{even}\left( \mathcal G_1+O_1+{\mathcal {L}}_1^{even}{\mathcal {F}}\right) . \end{aligned}$$

It follows from the definition of ${\mathcal {G}}_m$ that

$$\begin{aligned} \frac{\partial {\mathcal {G}}_m}{\partial s_1}&=\, \frac{\partial }{\partial s_1}\left( a_m^{\alpha ,p;\beta ,q}\left( \frac{\partial {\mathcal {F}}}{\partial t^{\alpha ,p}}\frac{\partial {\mathcal {F}}}{\partial t^{\beta ,q}}+\frac{\partial ^2{\mathcal {F}}}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}\right) +c_{m;\alpha ,p;\beta ,q}t^{\alpha ,p}t^{\beta ,q}\right) \\&=\, a_m^{\alpha ,p;\beta ,q}\left( f_{\alpha ,p}\frac{\partial }{\partial t^{\beta ,q}}({\mathcal {G}}_1+O_1+{\mathcal {L}}_1^{even}{\mathcal {F}})+f_{\beta ,q}\frac{\partial }{\partial t^{\alpha ,p}}({\mathcal {G}}_1+O_1+{\mathcal {L}}_1^{even}{\mathcal {F}})\right) \\&\quad +\, a_m^{\alpha ,p;\beta ,q}\frac{\partial ^2}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}({\mathcal {G}}_1+O_1+{\mathcal {L}}_1^{even}{\mathcal {F}})\\&=\, a_m^{\alpha ,p;\beta ,q}\left( f_{\alpha ,p}\frac{\partial O_1}{\partial t^{\beta ,q}}+f_{\beta ,q}\frac{\partial O_1}{\partial t^{\alpha ,p}}+\frac{\partial ^2O_1}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}\right) +\dots , \end{aligned}$$

here $\dots$ stands for remaining terms that are independent of $O_1$. In a similar way we can compute $\frac{\partial }{\partial s_m}\frac{\partial \mathcal F}{\partial s_1}$. By using the commutation relation

$$\begin{aligned}{}[L_1^{even},L_m^{even}] = (1-m)L_{m+1}^{even} \end{aligned}$$

we obtain the following result:

$$\begin{aligned}&\left[ \frac{\partial }{\partial s_1}, \frac{\partial }{\partial s_m}\right] {\mathcal {F}} =\ (m-1)\left( {\mathcal {G}}_{m+1}+{\mathcal {L}}_{m+1}^{even}{\mathcal {F}}\right) \\&\quad +\, \frac{\partial O_m}{\partial s_1}-a_1^{\alpha ,p;\beta ,q}\left( f_{\alpha ,p}\frac{\partial O_m}{\partial t^{\beta ,q}}+f_{\beta ,q}\frac{\partial O_m}{\partial t^{\alpha ,p}}+\frac{\partial ^2O_m}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}\right) -{\mathcal {L}}_1^{even} O_m\\&\quad -\, \frac{\partial O_1}{\partial s_m}+a_m^{\alpha ,p;\beta ,q}\left( f_{\alpha ,p}\frac{\partial O_1}{\partial t^{\beta ,q}}+f_{\beta ,q}\frac{\partial O_1}{\partial t^{\alpha ,p}}+\frac{\partial ^2O_1}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}\right) +{\mathcal {L}}_m^{even} O_1. \end{aligned}$$

Therefore we obtain the symmetry $\frac{\partial {\mathcal {F}}}{\partial s_{m+1}}$ by defining

$$\begin{aligned} \frac{\partial \mathcal F}{\partial s_{m+1}}=\frac{1}{m-1}\left[ \frac{\partial }{\partial s_1}, \frac{\partial }{\partial s_m}\right] {\mathcal {F}} = {\mathcal {G}}_{m+1}+O_{m+1}+{\mathcal {L}}_{m+1}^{even}{\mathcal {F}}, \end{aligned}$$

where $O_{m+1}$ is the differential polynomial given by

$$\begin{aligned}&\quad O_{m+1} \\ {}&=\ \frac{1}{m-1}\left( \frac{\partial O_m}{\partial s_1}-a_1^{\alpha ,p;\beta ,q}\left( f_{\alpha ,p}\frac{\partial O_m}{\partial t^{\beta ,q}}+f_{\beta ,q}\frac{\partial O_m}{\partial t^{\alpha ,p}}+\frac{\partial ^2O_m}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}\right) -{\mathcal {L}}_1^{even} O_m\right) \\ {}&\quad - \frac{1}{m-1}\left( \frac{\partial O_1}{\partial s_m}-a_m^{\alpha ,p;\beta ,q}\left( f_{\alpha ,p}\frac{\partial O_1}{\partial t^{\beta ,q}}+f_{\beta ,q}\frac{\partial O_1}{\partial t^{\alpha ,p}}+\frac{\partial ^2O_1}{\partial t^{\alpha ,p}\partial t^{\beta ,q}}\right) -{\mathcal {L}}_m^{even} O_1\right) .\end{aligned}$$

Thus we obtain recursively an infinite set of symmetries of the tau-cover of the deformed Principal Hierarchy. Their actions on ${\mathcal {F}}$ can be represented by

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s_m} = {\mathcal {G}}_m+O_m+\mathcal L_m^{even}{\mathcal {F}}. \end{aligned}$$

Next we show that we can further adjust $O_m$ by adding certain constants such that these symmetries satisfy the Virasoro commutation relation

$$\begin{aligned} \left[ \frac{\partial }{\partial s_k}, \frac{\partial }{\partial s_l}\right] = (l-k)\frac{\partial }{\partial s_{k+l}},\quad k,l\ge -1. \end{aligned}$$

Lemma 25

There is a unique choice of constants $\kappa _m$ for $m\ge -1$ such that the flows

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s_m} = {\mathcal {G}}_m+O_m+\kappa _m+\mathcal L_m^{even}{\mathcal {F}} \end{aligned}$$

satisfy the Virasoro commutation relation

$$\begin{aligned} \left[ \frac{\partial }{\partial s_k}, \frac{\partial }{\partial s_l}\right] = (l-k)\frac{\partial }{\partial s_{k+l}},\quad k,l\ge -1. \end{aligned}$$

(4.25)

Proof

Let us first fix an arbitrary choice of $O_m$ and denote

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial {\hat{s}}_m} = {\mathcal {G}}_m+O_m+\mathcal L_m^{even}{\mathcal {F}}. \end{aligned}$$

Then we obtain the differential polynomials ${\tilde{O}}_{k+l}$ such that

$$\begin{aligned} \left[ \frac{\partial }{\partial {\hat{s}}_k}, \frac{\partial }{\partial {\hat{s}}_l}\right] {\mathcal {F}} = (l-k)\left( {\mathcal {G}}_{l+k}+{\tilde{O}}_{l+k}+\mathcal L_{l+k}^{even}{\mathcal {F}}\right) . \end{aligned}$$

But both $\frac{\partial }{\partial {\hat{s}}_{l+k}}$ and $\left[ \frac{\partial }{\partial {\hat{s}}_k}, \frac{\partial }{\partial {\hat{s}}_l}\right]$ are symmetries of the tau-cover of the deformed Principal Hierarchy, so we conclude that

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s}:= \frac{\partial {\mathcal {F}}}{\partial {\hat{s}}_{k+l}}-\frac{1}{l-k}\left[ \frac{\partial }{\partial {\hat{s}}_k}, \frac{\partial }{\partial {\hat{s}}_l}\right] {\mathcal {F}} = O_{k+l}-{\tilde{O}}_{k+l} \end{aligned}$$

is also a symmetry of the tau-cover of the deformed Principal Hierarchy. The action of this symmetry on the normal coordinates has the expression

$$\begin{aligned} \frac{\partial w_\lambda }{\partial s} = \frac{\partial ^2}{\partial t^{\lambda ,0}\partial t^{1,0}}\left( O_{k+l}-{\tilde{O}}_{k+l}\right) . \end{aligned}$$

Thus $\frac{\partial w_\lambda }{\partial s}\in \hat{\mathcal A}_{\ge 2}$, and therefore such a symmetry must vanish due to the result of the bihamiltonian cohomology [7]. Hence we conclude that

$$\begin{aligned} O_{k+l}-{\tilde{O}}_{k+l} = c_{k,l} \end{aligned}$$

for some constant $c_{k,l}$, and this means that

$$\begin{aligned} \left[ \frac{\partial }{\partial {\hat{s}}_k}, \frac{\partial }{\partial {\hat{s}}_l}\right] = (l-k)\frac{\partial }{\partial {\hat{s}}_{k+l}} -(l-k) c_{k,l},\quad k,l\ge -1. \end{aligned}$$

(4.26)

Let us denote by ${\mathfrak {W}}_1$ the Lie algebra of formal vector fields on a line, which is an infinite dimensional Lie algebra with a basis

$$\begin{aligned} e_m = z^{m+1}\frac{d}{dz},\quad m\ge -1. \end{aligned}$$

Then the relation (4.26) implies that the Lie algebra $\{\frac{\partial }{\partial {\hat{s}}_m}\}$ defines a central extension of ${\mathfrak {W}}_1$. It is computed in [15] that $H^2({\mathfrak {W}}_1,{\mathbb {R}}) = 0$ and hence every central extension is trivial. Therefore we can modify each $O_m$ by adding an appropriate constant $\kappa _m$ such that the modified flows

$$\begin{aligned} \frac{\partial {\mathcal {F}}}{\partial s_m} = {\mathcal {G}}_m+O_m+\kappa _m+\mathcal L_m^{even}{\mathcal {F}} \end{aligned}$$

satisfy the commutation relations (4.25). Moreover, the choice of $\kappa _m$ is unique since $H^1({\mathfrak {W}}_1,{\mathbb {R}}) = 0$ (see [15]). The lemma is proved.$\square$

Thus we have proved the following theorem.

Theorem 15

For every tau-symmetric bihamiltonian deformation of the Principal Hierarchy associated with a semisimple Frobenius manifold, the deformed integrable hierarchy possesses an infinite set of Virasoro symmetries. The actions of these symmetries on the tau function ${\mathcal {Z}}$ are represented by

$$\begin{aligned} \frac{\partial {\mathcal {Z}}}{\partial s_m} = L_m^{even}{\mathcal {Z}}+O_m{\mathcal {Z}},\quad m\ge -1, \end{aligned}$$

where $O_m\in \hat{\mathcal A}$ are certain differential polynomials, and the flows $\frac{\partial }{\partial s_m}$ satisfy the commutation relations

$$\begin{aligned} \left[ \frac{\partial }{\partial s_k}, \frac{\partial }{\partial s_l}\right] = (l-k)\frac{\partial }{\partial s_{k+l}},\quad k,l\ge -1. \end{aligned}$$

Example 6

Let M be the 2-dimensional Frobenius manifold defined on the orbit space of the Weyl group of type $B_2$. Its potential and Euler vector field are given by

$$\begin{aligned} F = \frac{1}{2} v^2u+\frac{4}{15}u^5,\quad E = v\partial _v+\frac{1}{2}u\partial _u. \end{aligned}$$

Here $v = v^1$ and $u = v^2$ are the flat coordinates of M. We denote by $\sigma _1$ and $\sigma _2$ the dual coordinates of the fiber of ${\hat{M}}$, then the bihamiltonian structure $(P_0^{[0]},P_1^{[0]})$ associated with M is given by

$$\begin{aligned} P_0^{[0]} = \frac{1}{2}\int \sigma _1\sigma _2^1+\sigma _2\sigma _1^1,\quad P_1^{[0]} = \frac{1}{2}\int 8u^3\sigma _1\sigma _1^1+\frac{1}{2} u\sigma _2\sigma _2^1+2v\sigma _1\sigma _2^1-\frac{1}{2}v_x\sigma _1\sigma _2. \end{aligned}$$

Let us first write down the Virasoro symmetry $\frac{\partial }{\partial s_1}$ of the tau-cover of the Principal Hierarchy associated with M. Here we choose the symmetry $\frac{\partial }{\partial s_1}$ instead of $\frac{\partial }{\partial s_2}$ just for simplicity. The Virasoro operator $L_1^{even}$ has the expression

$$\begin{aligned} L_1^{even} = \frac{3}{16}\frac{\partial ^2}{\partial t^{1,0}\partial t^{2,0}}+ {\mathcal {L}}_1^{even}, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {L}}_1^{even}=\sum _{p\ge 0}\left( p+\frac{1}{4}\right) \left( p+\frac{5}{4}\right) t^{1,p}\frac{\partial }{\partial t^{1,p+1}}+\left( p+\frac{3}{4}\right) \left( p+\frac{7}{4}\right) t^{2,p}\frac{\partial }{\partial t^{2,p+1}}. \end{aligned}$$

Then the action of $\frac{\partial }{\partial s_1}$ on the genus zero free energy ${\mathcal {F}}^{[0]}$ of the tau-cover of the Principal Hierarchy is given by

$$\begin{aligned} \frac{\partial {\mathcal {F}}^{[0]}}{\partial s_1} = \frac{3}{16}f_{1,0}f_{2,0}+\mathcal L_1^{even}{\mathcal {F}}^{[0]}. \end{aligned}$$

Consider the bihamiltonian structure $(P_0,P_1)$ of the Drinfel’d–Sokolov hierarchy [3] associated with the untwisted affine Kac–Moody algebra $B_2^{(1)}$. After performing a suitable Miura type transformation we have $P_0 = P_0^{[0]}$, and the Hamiltonian operator ${\mathcal {P}}_1$ of $P_1$ has the expression

$$\begin{aligned} {\mathcal {P}}_1 = \begin{pmatrix}8u^3\partial _x+12u^2u_x &{} v\partial _x+\frac{1}{4}v_x\\ v\partial _x+\frac{3}{4} v_x&{} \frac{1}{2}u\partial _x+\frac{1}{4}u_x\end{pmatrix}+\varepsilon ^2\begin{pmatrix}D_1 &{} D_2\\ D_3&{} D_4\end{pmatrix}+O(\varepsilon ^4), \end{aligned}$$

where the differential operators $D_i$ are given by $D_2 = u\partial _x^3+\frac{3}{4} u_x\partial _x^2$, $D_4 = \frac{5}{8} \partial _x^3$ and

$$\begin{aligned} D_1&=\, 14u^2\partial _x^3+42uu_x\partial _x^2+\left( 20u_x^2+16uu_{xx}+\frac{1}{2}v_{xx}\right) \partial _x\\ {}&\quad +12u_xu_{xx}+6uu^{(3)}+\frac{1}{4} v^{(3)},\\ D_3&=\,u\partial _x^3+\frac{9}{4}u_x\partial _x^2+\frac{3}{2}u_{xx}\partial _x+\frac{1}{4}u^{(3)}. \end{aligned}$$

The bihamiltonian structure $(P_0,P_1)$ is a deformation of $(P_0^{[0]},P_1^{[0]})$ with central invariants $c_1 = \frac{1}{6}$, $c_2 = \frac{1}{12}$ (see [8, 24]), and it determines a unique deformation of the Principal Hierarchy associated with M. We can find the Virasoro symmetry $\frac{\partial }{\partial s_1}$ of the tau-cover of the deformed Principal Hierarchy by using the results developed in the present paper. It turns out that the action of $\frac{\partial }{\partial s_1}$ on the tau-function ${\mathcal {Z}}$ can be represented by

$$\begin{aligned} \frac{\partial {\mathcal {Z}}}{\partial s_1} = L_1^{even}{\mathcal {Z}}+\left( \frac{1}{2} u^2+\frac{1}{4}v+\frac{1}{4}\varepsilon ^2 u_{xx}\right) {\mathcal {Z}}. \end{aligned}$$

(4.27)

A similar result is also given in the Example 5.5 of [31] by using the Kac–Moody–Virasoro algebra.

5 Conclusion

In the present paper, we prove the existence of an infinite set of Virasoro symmetries for a given tau-symmetric bihamiltonian deformation of the Principal Hierarchy associated with a semisimple Frobenius manifold. These symmetries can be represented in terms of the tau-function ${\mathcal {Z}}$ of the integrable hierarchy in the form

$$\begin{aligned} \frac{\partial {\mathcal {Z}}}{\partial s_m} = L_m{\mathcal {Z}}+O_m\mathcal Z,\quad m\ge -1. \end{aligned}$$

(5.1)

Note that the differential polynomials $O_m$ depend on the choice of the representative, in the equivalence class of Miura type transformations, of the deformations of the bihamiltonian structure of hydrodynamic type $(P_0^{[0]},P_1^{[0]})$. It is proved in [9] that, for two different choices of representatives $(P_0,P_1)$ and $({\tilde{P}}_0,{\tilde{P}}_1)$, the corresponding normal coordinates

$$\begin{aligned} w^\alpha =\eta ^{\alpha \beta }\frac{\partial ^2 \log {\mathcal {Z}}}{\partial t^{1,0}\partial t^{\beta ,0}},\quad {\tilde{w}}^\alpha =\eta ^{\alpha \beta }\frac{\partial ^2 \log \tilde{{\mathcal {Z}}}}{\partial t^{1,0}\partial t^{\beta ,0}},\quad \alpha =1,\dots ,n \end{aligned}$$

of the deformed Principal Hierarchy are related by a Miura type transformation

$$\begin{aligned} {\tilde{w}}^\alpha =w^\alpha +\eta ^{\alpha \beta }\frac{\partial ^2 G}{\partial t^{1,0}\partial t^{\beta ,0}}, \end{aligned}$$

where $G\in \hat{\mathcal A}$ is a differential polynomial, and the tau-functions are related by the equation

$$\begin{aligned} {\tilde{\mathcal {Z}}} =\exp (G){\mathcal {Z}}. \end{aligned}$$

(5.2)

Conversely, any differential polynomial $G\in \hat{\mathcal A}$ defines a Miura type transformation for the deformed bihamiltonian structure and the integrable hierarchy in the manner described above.

After a Miura type transformation induced from (5.2), the Virasoro symmetries (5.1) are transformed to the form

$$\begin{aligned} \frac{\partial \tilde{{\mathcal {Z}}}}{\partial s_m} = L_m\tilde{{\mathcal {Z}}}+\tilde{O}_m\tilde{{\mathcal {Z}}},\quad m\ge -1, \end{aligned}$$

where the differential polynomials ${\tilde{O}}_m$ can be computed from $O_m$ and G.

We are going to study the problem of linearization of Virasoro symmetries in subsequent work, i.e., to study whether it is possible to find a suitable differential polynomial G such that all the functions ${\tilde{O}}_m$ vanish.

Let us exam the possibility of linearizing the Virasoro symmetries given in the example of Sect. 3.4 for the one-dimensional Frobenius manifold. We want to find a certain Miura type transformation given by (5.2) which linearizes the Virasoro symmetry (3.26) and leaves the expression of the Virasoro symmetry

$$\begin{aligned} \frac{\partial {\mathcal {Z}}}{\partial s_{-1}} = L_{-1}^{even}{\mathcal {Z}} \end{aligned}$$

unchanged. It follows from these requirements that the differential degree zero component $G_0$ of G must satisfy the equations

$$\begin{aligned} \frac{\partial G_0}{\partial v} = 0,\quad v^3\frac{\partial G_0}{\partial v} = -\left( 3c-\frac{3}{8}\right) \frac{v^2}{2}, \end{aligned}$$

which do not possess any solution unless $c = \frac{1}{8}$. When $c = \frac{1}{8}$, the linearized Virasoro symmetries for this example are well known [30], and the central invariant of the corresponding deformed bihamiltonian structure is $\frac{1}{3} c=\frac{1}{24}$.

We can do a similar computation for Example 6. We want to find a Miura type transformation given by (5.2) to linearize the Virasoro symmetry (4.27) and to preserve the expression of the Virasoro symmetry

$$\begin{aligned} \frac{\partial {\mathcal {Z}}}{\partial s_{-1}} = L_{-1}^{even}{\mathcal {Z}}. \end{aligned}$$

Then the differential degree zero component $G_0$ of G must satisfy the equations

$$\begin{aligned} \frac{\partial G_0}{\partial v} = 0,\quad uv\frac{\partial G_0}{\partial u} = -\left( \frac{1}{2} u^2+\frac{1}{4}v\right) , \end{aligned}$$

which have no solution. Therefore the Virasoro symmetries given by the bihamiltonian structure $(P_0,P_1)$ in this example cannot be linearized.

In general we have the following theorem, whose proof will be given in the paper [21].

Theorem 16

The Virasoro symmetries for a given tau-symmetric bihamiltonian deformation of the Principal Hierarchy associated with a semisimple Frobenius manifold is linearizable if and only if the central invariants of the corresponding deformed bihamiltonian structure are all equal to $\frac{1}{24}$.