Open WDVV Equations and Virasoro Constraints

Abstract

In their fundamental work, Dubrovin and Zhang, generalizing the Virasoro equations for the genus 0 Gromov–Witten invariants, proved the Virasoro equations for a descendent potential in genus 0 of an arbitrary conformal Frobenius manifold. More recently, a remarkable system of partial differential equations, called the open WDVV equations, appeared in the work of Horev and Solomon. This system controls the genus 0 open Gromov–Witten invariants. In our paper, for an arbitrary solution of the open WDVV equations, satisfying a certain homogeneity condition, we construct a descendent potential in genus 0 and prove an open analog of the Virasoro equations. We also present conjectural open Virasoro equations in all genera and discuss some examples.

1 Introduction

The WDVV equations, also called the associativity equations, is a system of non-linear partial differential equations for one function, depending on a finite number of variables. Let $N\ge 1$ and $\eta =(\eta _{\alpha \beta })$ be an $N\times N$ symmetric non-degenerate matrix with complex coefficients. The WDVV equations is the following system of PDEs for a function $F(t^1,\ldots ,t^N)$ defined on some open subset $M\subset \mathbb {C}^N$:

$$\begin{aligned} \frac{{\partial }^3 F}{{\partial }t^\alpha {\partial }t^\beta {\partial }t^\mu }\eta ^{\mu \nu }\frac{{\partial }^3 F}{{\partial }t^\nu {\partial }t^\gamma {\partial }t^\delta }=\frac{{\partial }^3 F}{{\partial }t^\delta {\partial }t^\beta {\partial }t^\mu }\eta ^{\mu \nu }\frac{{\partial }^3 F}{{\partial }t^\nu {\partial }t^\gamma {\partial }t^\alpha },\quad 1\le \alpha ,\beta ,\gamma ,\delta \le N,\nonumber \\ \end{aligned}$$

(1.1)

where $(\eta ^{\alpha \beta }):=\eta ^{-1}$ and we use the convention of sum over repeated Greek indices. Suppose that the function F satisfies the additional assumption $\frac{{\partial }^3 F}{{\partial }t^1{\partial }t^\alpha {\partial }t^\beta }=\eta _{\alpha \beta }$. Then the function F defines a structure of Frobenius manifold on M and is also called the Frobenius manifold potential. Such a structure appears in different areas of mathematics, including the singularity theory and curve counting theories in algebraic geometry (Gromov–Witten theory, Fan–Jarvis–Ruan–Witten theory). A systematic study of Frobenius manifolds was first done by Dubrovin (1996, 1999).

Consider formal variables $t^\alpha _p$, $1\le \alpha \le N$, $p\ge 0$, where we identify $t^\alpha _0=t^\alpha$. There is a natural way to associate to the function F a descendent potential${\mathcal {F}}$, which is a function of the variables $t^\alpha _p$, such that the difference $\left. {\mathcal {F}}\right| _{t^*_{\ge 1}=0}-F$ is at most quadratic in the variables $t^1,\ldots ,t^N$ and the following equations are satisfied:

$$\begin{aligned} \frac{{\partial }^3{\mathcal {F}}}{{\partial }t^\alpha _{a+1}t^\beta _b{\partial }t^\gamma _c}=\frac{{\partial }^2{\mathcal {F}}}{{\partial }t^\alpha _a{\partial }t^\mu _0}\eta ^{\mu \nu }\frac{{\partial }^3{\mathcal {F}}}{{\partial }t^\nu _0{\partial }t^\beta _b{\partial }t^\gamma _c},\quad 1\le \alpha ,\beta ,\gamma \le N,\quad a,b,c\ge 0. \end{aligned}$$

(1.2)

These equations are called the topological recursion relations (TRR). In Gromov–Witten theory, where the function F is the generating series of intersection numbers on the moduli space of maps from a Riemann surface of genus 0 to a target variety, a natural descendent potential ${\mathcal {F}}$ is given by the generating series of intersection numbers with the Chern classes of certain line bundles over the moduli space. Note that the system of Eq. (1.2) can be equivalently written as

$$\begin{aligned} d\left( \frac{{\partial }^2{\mathcal {F}}}{{\partial }t^\alpha _{a+1}{\partial }t^\beta _b}\right) =\frac{{\partial }^2{\mathcal {F}}}{{\partial }t^\alpha _a{\partial }t^\mu _0}\eta ^{\mu \nu }d\left( \frac{{\partial }^2{\mathcal {F}}}{{\partial }t^\nu _0{\partial }t^\beta _b}\right) ,\quad 1\le \alpha ,\beta \le N,\quad a,b\ge 0, \end{aligned}$$

(1.3)

where $d \left( \cdot \right)$ denotes the full differential.

Let $\varepsilon$ be a formal variable and $\widetilde{t}^\alpha _p:=t^\alpha _p-\delta ^{\alpha ,1}\delta _{p,1}$. If our Frobenius manifold is conformal, meaning that the function F satisfies a certain homogeneity condition, then in Dubrovin and Zhang (1999) the authors constructed differential operators $L_m$, $m\ge -1$, of the form

$$\begin{aligned} L_m=\,&\sum _{p,q\ge 0}\left( \varepsilon ^2a_m^{\alpha ,p;\beta ,q}\frac{{\partial }^2}{{\partial }t^\alpha _p{\partial }t^\beta _q}+b_{m;\alpha ,p}^{\beta ,q}\widetilde{t}^\alpha _p\frac{{\partial }}{{\partial }t^\beta _q}+\varepsilon ^{-2}c_{m;\alpha ,p;\beta ,q}\widetilde{t}^\alpha _p\widetilde{t}^\beta _q\right) \\&+{\mathrm {const}}\cdot \delta _{m,0},\quad a_m^{\alpha ,p;\beta ,q},b_{m;\alpha ,p}^{\beta ,q},c_{m;\alpha ,p;\beta ,q}\in \mathbb {C}, \end{aligned}$$

satisfying the commutation relations

$$\begin{aligned} {[}L_i,L_j]=(i-j)L_{i+j},\quad i,j\ge -1, \end{aligned}$$

and such that the following equations, called the Virasoro constraints, are satisfied:

$$\begin{aligned} {\mathrm {Coef}}_{\varepsilon ^{-2}}\left( \frac{L_m e^{\varepsilon ^{-2}{\mathcal {F}}}}{e^{\varepsilon ^{-2}{\mathcal {F}}}}\right) =0,\quad m\ge -1. \end{aligned}$$

(1.4)

We recall the details in Sect. 2.

In Gromov–Witten theory, for each $g\ge 0$ one defines the generating series ${\mathcal {F}}_g(t^*_*)$ of intersection numbers on the moduli space of maps from a Riemann surface of genus g to a target variety, ${\mathcal {F}}_0={\mathcal {F}}$. The Virasoro conjecture says that the following equations are satisfied:

$$\begin{aligned} L_m e^{\sum _{g\ge 0}\varepsilon ^{2g-2}{\mathcal {F}}_g}=0,\quad m\ge -1. \end{aligned}$$

(1.5)

One can see that Eq. (1.4) is the genus 0 part of Eq. (1.5). The Virasoro conjecture is proved in a wide class of cases, but is still open in the whole generality.

More recently, a remarkable system of PDEs, extending the WDVV Eq. (1.1), appeared in the literature. Let s be a formal variable. The open WDVV equations are the following PDEs for a function $F^o(t^1,\ldots ,t^N,s)$:

$$\begin{aligned}&\frac{{\partial }^3F}{{\partial }t^\alpha {\partial }t^\beta {\partial }t^\mu }\eta ^{\mu \nu }\frac{{\partial }^2F^o}{{\partial }t^\nu {\partial }t^\gamma }+\frac{{\partial }^2F^o}{{\partial }t^\alpha {\partial }t^\beta }\frac{{\partial }^2F^o}{{\partial }s{\partial }t^\gamma }\nonumber \\&\quad =\frac{{\partial }^3F}{{\partial }t^\gamma {\partial }t^\beta {\partial }t^\mu }\eta ^{\mu \nu }\frac{{\partial }^2F^o}{{\partial }t^\nu {\partial }t^\alpha }+\frac{{\partial }^2F^o}{{\partial }t^\gamma {\partial }t^\beta }\frac{{\partial }^2F^o}{{\partial }s{\partial }t^\alpha },\quad 1\le \alpha ,\beta ,\gamma \le N, \end{aligned}$$

(1.6)

$$\begin{aligned}&\frac{{\partial }^3F}{{\partial }t^\alpha {\partial }t^\beta {\partial }t^\mu }\eta ^{\mu \nu }\frac{{\partial }^2F^o}{{\partial }t^\nu {\partial }s}+\frac{{\partial }^2F^o}{{\partial }t^\alpha {\partial }t^\beta }\frac{{\partial }^2F^o}{{\partial }s^2}\nonumber \\&\quad =\frac{{\partial }^2F^o}{{\partial }s{\partial }t^\beta }\frac{{\partial }^2F^o}{{\partial }s{\partial }t^\alpha },\quad 1\le \alpha ,\beta \le N. \end{aligned}$$

(1.7)

These equations first appeared in Horev and Solomon (2012, Theorem 2.7) in the context of open Gromov–Witten theory. The open WDVV equations also appeared in the works (Pandharipande et al. 2014; Buryak et al. 2018, 2019). The solutions of Eqs. (1.6), (1.7), considered in these works, also satisfy the additional condition

$$\begin{aligned} \frac{{\partial }^2 F^o}{{\partial }t^1{\partial }t^\alpha }=0,\qquad \frac{{\partial }^2 F^o}{{\partial }t^1{\partial }s}=1. \end{aligned}$$

(1.8)

Remark 1.1

The works (Pandharipande et al. 2014; Buryak et al. 2018, 2019 don’t mention the open WDVV explicitly, but, as it is explained in Buryak (2018, Section 4), the open WDVV equations follow immediately from the open TRR equations (Pandharipande et al. 2014, Theorem 1.5), (Buryak et al. 2018, Theorem 4.1), (Buryak et al. 2019, Lemma 3.6).

There is an open analog of Eq. (1.3). Let $s_p$, $p\ge 0$, be formal variables, where we identify $s_0=s$. The open topological recursion relations are the following PDEs for a function ${\mathcal {F}}^o$, depending on the variables $t^\alpha _p$ and $s_p$, and such that the difference $\left. {\mathcal {F}}^o\right| _{\begin{array}{c} t^*_{\ge 1}=0\\ s_{\ge 1}=0 \end{array}}-F^o$ is at most linear in the variables $t^1,\ldots ,t^N$ and s:

$$\begin{aligned}&d\left( \frac{{\partial }{\mathcal {F}}^o}{{\partial }t^\alpha _{p+1}}\right) =\,\frac{{\partial }^2{\mathcal {F}}}{{\partial }t^\alpha _p{\partial }t^\mu _0}\eta ^{\mu \nu }d\left( \frac{{\partial }{\mathcal {F}}^o}{{\partial }t^\nu _0}\right) +\frac{{\partial }{\mathcal {F}}^o}{{\partial }t^\alpha _p}d\left( \frac{{\partial }{\mathcal {F}}^o}{{\partial }s}\right) ,&1\le \alpha \le N,\quad p\ge 0, \end{aligned}$$

(1.9)

$$\begin{aligned}&\qquad \qquad \qquad d\left( \frac{{\partial }{\mathcal {F}}^o}{{\partial }s_{p+1}}\right) =\,\frac{{\partial }{\mathcal {F}}^o}{{\partial }s_p}d\left( \frac{{\partial }{\mathcal {F}}^o}{{\partial }s}\right) ,\quad p\ge 0. \end{aligned}$$

(1.10)

As we already mentioned in Remark 1.1, these equations appeared in the works (Pandharipande et al. 2014; Buryak et al. 2018, 2019).

The simplest Frobenius manifold has dimension 1 and the potential $F=F^{\mathrm {pt}}=\frac{(t^1)^3}{6}$. A natural descendent potential ${\mathcal {F}}^{\mathrm {pt}}$, associated to it, is given by the generating series of the integrals of monomials in the psi-classes over the moduli space of stable curves of genus 0. Here “${\mathrm {pt}}$” means “point”, because such integrals can be considered as the Gromov–Witten invariants of a point. One can easily see that the function $F^o=F^{{\mathrm {pt}},o}=t^1 s+\frac{s^3}{6}$ satisfies the open WDVV equations and condition (1.8). In Pandharipande et al. (2014) the authors, using the intersection theory on the moduli space of stable pointed disks, constructed a solution ${\mathcal {F}}^{{\mathrm {pt}},o}$ of the open TRR Eqs. (1.9), (1.10). Moreover, they introduced the operators

$$\begin{aligned} \mathcal {L}^{\mathrm {pt}}_m&:=\,L^{\mathrm {pt}}_m+\varepsilon ^{-1}\delta _{m,-1}s+\left( \sum _{i\ge 0}\frac{(i+m+1)!}{i!}s_i\frac{{\partial }}{{\partial }s_{m+i}}+\delta _{m,0}\frac{3}{4}\right) \nonumber \\&+\,\varepsilon \frac{3(m+1)!}{4}\frac{{\partial }}{{\partial }s_{m-1}},\quad m\ge -1, \end{aligned}$$

(1.11)

where $L^{\mathrm {pt}}_m$ are the Virasoro operators for our Frobenius manifold, and proved the equations (Pandharipande et al. 2014, Theorem 1.1)

$$\begin{aligned} {\mathrm {Coef}}_{\varepsilon ^{-1}}\left( \frac{\mathcal {L}^{\mathrm {pt}}_m e^{\varepsilon ^{-2}{\mathcal {F}}^{\mathrm {pt}}+\varepsilon ^{-1}{\mathcal {F}}^{{\mathrm {pt}},o}}}{e^{\varepsilon ^{-2}{\mathcal {F}}^{\mathrm {pt}}+\varepsilon ^{-1}{\mathcal {F}}^{{\mathrm {pt}},o}}}\right) =0,\quad m\ge -1, \end{aligned}$$

(1.12)

which they called the open Virasoro constraints.

Remark 1.2

Strictly speaking, in Pandharipande et al. (2014) the authors constructed a function ${\mathcal {F}}^{{\mathrm {PST}}}$, related to our function ${\mathcal {F}}^{{\mathrm {pt}},o}$ by $\left. {\mathcal {F}}^{{\mathrm {PST}}}={\mathcal {F}}^{{\mathrm {pt}},o}\right| _{s_{\ge 1}=0}$. The function ${\mathcal {F}}^{{\mathrm {pt}},o}$ can be reconstructed from the function ${\mathcal {F}}^{{\mathrm {PST}}}$, using the differential equations $\frac{{\partial }{\mathcal {F}}^{{\mathrm {pt}},o}}{{\partial }s_n}=\frac{1}{(n+1)!}\left( \frac{{\partial }{\mathcal {F}}^{{\mathrm {pt}},o}}{{\partial }s}\right) ^{n+1}$. The Virasoro equations, proved in Pandharipande et al. (2014), look as

$$\begin{aligned}&{\mathrm {Coef}}_{\varepsilon ^{-1}}\left( \frac{\mathcal {L}^{{\mathrm {PST}}}_m e^{\varepsilon ^{-2}{\mathcal {F}}^{\mathrm {pt}}+\varepsilon ^{-1}{\mathcal {F}}^{{\mathrm {PST}}}}}{e^{\varepsilon ^{-2} {\mathcal {F}}^{\mathrm {pt}}+\varepsilon ^{-1}{\mathcal {F}}^{{\mathrm {PST}}}}}\right) =0,\nonumber \\&\mathcal {L}^{{\mathrm {PST}}}_m:=L_m^{\mathrm {pt}}+\varepsilon ^m s\frac{{\partial }^{m+1}}{{\partial }s^{m+1}}+\frac{3m+3}{4}\varepsilon ^m\frac{{\partial }^m}{{\partial }s^m},\quad m\ge -1. \end{aligned}$$

(1.13)

The fact, that Eqs. (1.12) and (1.13) are equivalent, was noticed in Buryak (2016, Section 5.2).

In our paper, we generalize formula (1.12) for an arbitrary conformal Frobenius manifold and a solution of the open WDVV equations. We consider an arbitrary conformal Frobenius manifold, an associated descendent potential ${\mathcal {F}}$ and the Virasoro operators $L_m$, $m\ge -1$. Let $F^o$ be a solution of the open WDVV equations, satisfying condition (1.8) and a certain homogeneity condition, that we will describe later. We will construct a solution ${\mathcal {F}}^o$ of the open TRR Eqs. (1.9), (1.10) and differential operators $\mathcal {L}_m$, $m\ge -1$, of the form

$$\begin{aligned} \mathcal {L}_m&=L_m+\varepsilon ^{-1}\left( \delta _{m,-1}s+\sum _{p\ge 0}d_{m;\alpha ,p}\widetilde{t}^\alpha _p\right) \nonumber \\&\quad +\left( \sum _{i\ge 0}\frac{(i+m+1)!}{i!}s_i\frac{{\partial }}{{\partial }s_{m+i}}+\sum _{p,q\ge 0}e_{m;\alpha ,p}^q\widetilde{t}^\alpha _p\frac{{\partial }}{{\partial }s_q}+\delta _{m,0}\frac{3}{4}\right) \nonumber \\&\quad +\varepsilon \left( \sum _{p\ge 0}f_m^{\alpha ,p}\frac{{\partial }}{{\partial }t^\alpha _p}+\frac{3(m+1)!}{4}\frac{{\partial }}{{\partial }s_{m-1}}\right) \nonumber \\&\quad +\varepsilon ^2\sum _{p,q\ge 0}g_m^{\alpha ,p;q}\frac{{\partial }^2}{{\partial }t^\alpha _p{\partial }s_q},\quad d_{m;\alpha ,p}, e_{m;\alpha ,p}^q, f_m^{\alpha ,p}, g_m^{\alpha ,p;q}\in \mathbb {C}, \end{aligned}$$

(1.14)

such that the equations

$$\begin{aligned} {\mathrm {Coef}}_{\varepsilon ^{-1}}\left( \frac{\mathcal {L}_m e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}{e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}\right) =0,\quad m\ge -1, \end{aligned}$$

hold. The details are given in Sect. 3, with the main result, formulated in Theorem 3.4.

It occurs that, given a function $F = F(t^1,\dots ,t^N)$, defining a Frobenius manifold and a solution $F^o$ of the open WDVV equations, satisfying (1.8), the $(N+1)$-tuple of functions $\left( \eta ^{1\mu }\frac{{\partial }F}{{\partial }t^\mu },\ldots ,\eta ^{N\mu }\frac{{\partial }F}{{\partial }t^\mu },F^o\right)$ forms a vector potential of a flat F-manifold. This was observed by Paolo Rossi. We say that this flat F-manifold extends the Frobenius manifold given. In Sect. 4 we prove Virasoro type constraints for flat F-manifolds and derive Theorem 3.4 as a special case of this result.

Let us return to the particular case, considered in the paper (Pandharipande et al. 2014). The construction of the intersection theory on the moduli space of stable pointed disks, given there, can be generalized to higher genera. This has been announced by Solomon and Tessler, some of the details of their construction are presented in Tessler (2015). As a result, one gets a sequence of functions ${\mathcal {F}}^{{\mathrm {pt}},o}_0={\mathcal {F}}^{{\mathrm {pt}},o},{\mathcal {F}}^{{\mathrm {pt}},o}_1,{\mathcal {F}}^{{\mathrm {pt}},o}_2,\ldots$, and already in [PST14] the authors conjectured that the following equations should hold:

$$\begin{aligned} \mathcal {L}^{\mathrm {pt}}_m e^{\sum _{g\ge 0}\varepsilon ^{2g-2}{\mathcal {F}}^{\mathrm {pt}}_g+\sum _{g\ge 0}\varepsilon ^{g-1}{\mathcal {F}}^{{\mathrm {pt}},o}_g}=0,\quad m\ge -1, \end{aligned}$$

(1.15)

where ${\mathcal {F}}^{\mathrm {pt}}_g$ is the generating series of the intersection numbers of psi-classes on the moduli space of curves of genus g. This conjecture was proved in Buryak and Tessler (2017).

In Sect. 5 we discuss a conjectural generalization of Eq. (1.15) for an arbitrary conformal Frobenius manifold and a solution of the open WDVV equations.

Finally, in Sect. 6 we present examples of solutions of the open WDVV equations, for which our main result can be applied. In a forthcoming paper (Basalaev and Buryak 2019) we will discuss solutions of the open WDVV equations for the Coxeter groups.

2 Virasoro Constraints for Frobenius Manifolds

In this section we review the construction of a descendent potential associated to a solution of the WDVV equations and recall the Virasoro constraints.

Let us fix $N\ge 1$ and let M be a simply connected open neighbourhood of a point $(t^1_{\mathrm {orig}},\ldots ,t^N_{\mathrm {orig}})\in \mathbb {C}^N$. Denote by $\mathcal {O}$ the sheaf of analytic functions on $\mathbb {C}^N$. Consider a solution $F\in \mathcal {O}(M)$ of the WDVV Eq. (1.1), satisfying $\frac{{\partial }^3 F}{{\partial }t^1{\partial }t^\alpha {\partial }t^\beta }=\eta _{\alpha \beta }$. In order to include the case, when F is a formal power series, in our considerations, we allow M to be a formal neighbourhood of $(t^1_{\mathrm {orig}},\ldots ,t^N_{\mathrm {orig}})\in \mathbb {C}^N$ meaning that $\mathcal {O}(M)$ denotes in this case the ring of formal power series in the variables $(t^\alpha -t^\alpha _{{\mathrm {orig}}})$.

2.1 Descendent Potential

In order to construct a descendent potential, one has to choose an additional structure, called a calibration of the Frobenius manifold. Denote

$$\begin{aligned} c^\alpha _{\beta \gamma }:=\eta ^{\alpha \mu }\frac{{\partial }^3 F}{{\partial }t^\mu {\partial }t^\beta {\partial }t^\gamma }. \end{aligned}$$

A calibration is a collection of functions $\Omega ^{\alpha ,0}_{\beta ,d}\in \mathcal {O}(M)$, $1\le \alpha ,\beta \le N$, $d\ge -1$, satisfying the following properties:

$$\begin{aligned}&\Omega ^{\alpha ,0}_{\beta ,-1}=\delta ^\alpha _\beta , \\&\frac{{\partial }\Omega ^{\alpha ,0}_{\beta ,d}}{{\partial }t^\gamma }=c^\alpha _{\gamma \mu }\Omega ^{\mu ,0}_{\beta ,d-1}, \quad d\ge 0,\\&\sum _{\begin{array}{c} p+q=d\\ p,q\ge -1 \end{array}}(-1)^{p+1}\Omega ^{\alpha ,0}_{\mu ,p}\eta ^{\mu \nu }\Omega ^{\beta ,0}_{\nu ,q}=0, \quad d\ge -1. \end{aligned}$$

The space of all calibrations is non-empty and is parameterized by elements $G(z)\in \mathrm {Mat}_{N,N}(\mathbb {C})[[z]]$, satisfying $G(0)={\mathrm {Id}}$ and $G(-z)\eta ^{-1} G^T(z)=\eta ^{-1}$ (Dubrovin 1999, Lemma 2.2 and Exercise 2.8).

Let us choose a calibration. One can immediately see that $\frac{{\partial }\Omega ^{\alpha ,0}_{1,0}}{{\partial }t^\beta }=\delta ^\alpha _\beta$, which implies that $\Omega ^{\alpha ,0}_{1,0}-t^\alpha$ is a constant. Let us make the change of coordinates $t^\alpha \mapsto \Omega ^{\alpha ,0}_{1,0}$, so that we have now $\Omega ^{\alpha ,0}_{1,0}=t^\alpha$.

Let $v^1,\ldots ,v^N$ be formal variables and consider the system of partial differential equations

$$\begin{aligned} \frac{{\partial }v^\alpha }{{\partial }t^\beta _d}={\partial }_x \left( \left. \Omega ^{\alpha ,0}_{\beta ,d}\right| _{t^\gamma =v^\gamma }\right) ,\quad 1\le \alpha ,\beta \le N,\quad d\ge 0, \end{aligned}$$

called the principal hierarchy. We see that the equations for the flow $\frac{{\partial }}{{\partial }t^1_0}$ look as $\frac{{\partial }v^\alpha }{{\partial }t^1_0}=v^\alpha _x$. This allows to identify $t^1_0=x$. Denote by $(v^{\mathrm {top}})^\alpha \in \mathcal {O}(M)[[t^*_{\ge 1}]]$ the solution of the principal hierarchy specified by the initial condition

$$\begin{aligned} \left. (v^{\mathrm {top}})^\alpha \right| _{t^\beta _d=\delta ^{\beta ,1}\delta _{d,0}x} =\delta ^{\alpha ,1}x. \end{aligned}$$

We have $\left. (v^{\mathrm {top}})^\alpha \right| _{t^*_{\ge 1}=0}=t^\alpha _0$.

Define functions $\Omega _{\alpha ,p;\beta ,q}\in \mathcal {O}(M)$ and $\Omega _{\alpha ,p;\beta ,q}^{\mathrm {top}}\in \mathcal {O}(M)[[t^*_{\ge 1}]]$, $p,q\ge 0$, by

$$\begin{aligned} \Omega _{\alpha ,p;\beta ,q}:=\sum _{i=0}^q(-1)^{q-i} \Omega _{\alpha ,p+q-i}^{\mu ,0}\eta _{\mu \nu } \Omega ^{\nu ,0}_{\beta ,i-1},\qquad \Omega _{\alpha ,p;\beta ,q}^{\mathrm {top}}:=\left. \Omega _{\alpha ,p;\beta ,q}\right| _{t^\gamma =(v^{\mathrm {top}})^\gamma }. \end{aligned}$$

The descendent potential corresponding to the Frobenius manifold together with the chosen calibration is defined by

$$\begin{aligned} {\mathcal {F}}:=\frac{1}{2}\sum _{p,q\ge 0}\widetilde{t}^\alpha _p\widetilde{t}^\beta _q\Omega ^{\mathrm {top}}_{\alpha ,p;\beta ,q}\in \mathcal {O}(M)[[t^*_{\ge 1}]], \end{aligned}$$

where we recall that $\widetilde{t}^\alpha _p=t^\alpha _p-\delta ^{\alpha ,1}\delta _{p,1}$ (Dubrovin and Zhang 1999, Section 3). The difference $\left. {\mathcal {F}}\right| _{t^*_{\ge 1}=0}-F$ is at most quadratic in the variables $t^1,\ldots ,t^N$ and the function ${\mathcal {F}}$ satisfies Eq. (1.2) together with the equation

$$\begin{aligned} \sum _{p\ge 0}\widetilde{t}^\alpha _{p+1}\frac{{\partial }{\mathcal {F}}}{{\partial }t^\alpha _p}+\frac{1}{2}\eta _{\alpha \beta }t^\alpha t^\beta =0, \end{aligned}$$

which is called the string equation (Dubrovin and Zhang 1999, Sections 3, 4).

Remark 2.1

Strictly speaking, in Dubrovin (1999) and Dubrovin and Zhang (1999) the authors consider the case of a conformal Frobenius manifold, but one can easily see that the results, discussed in this section, together with their proofs presented in Dubrovin (1999), Dubrovin and Zhang (1999), hold for all not necessarily conformal Frobenius manifolds. We have borrowed the term “calibration” from the paper (Dubrovin et al. 2016).

2.2 Virasoro Constraints

The Frobenius manifold is said to be conformal if there exists a vector field E of the form

$$\begin{aligned} E=\underbrace{((1-q^\gamma )t^\gamma +r^\gamma )}_{=:E^\gamma }\frac{{\partial }}{{\partial }t^\gamma },\quad q^\gamma ,r^\gamma \in \mathbb {C},\quad q^1=0, \end{aligned}$$

(2.1)

satisfying

$$\begin{aligned} L_E F=(3-\delta )F+\frac{1}{2}A_{\alpha \beta }t^\alpha t^\beta +B_\alpha t^\alpha +C,\quad \text {for some } \delta ,A_{\alpha \beta },B_\alpha ,C\in \mathbb {C}. \end{aligned}$$

The number $\delta$ is often called the conformal dimension. Denote

$$\begin{aligned} \mu ^\alpha :=q^\alpha -\frac{\delta }{2},\qquad \mu :={\mathrm {diag}}(\mu ^1,\ldots ,\mu ^N). \end{aligned}$$

In the conformal case there exists a calibration, satisfying the property

$$\begin{aligned} E^\theta \frac{{\partial }}{{\partial }t^\theta }\Omega ^{\alpha ,0}_{\beta ,d}=(d+1+\mu ^\beta -\mu ^\alpha )\Omega ^{\alpha ,0}_{\beta ,d}+\sum _{i=1}^{d+1}\Omega ^{\alpha ,0}_{\mu ,d-i}(R_i)^\mu _\beta ,\quad d\ge -1, \end{aligned}$$

(2.2)

for some matrices $R_n$, $n\ge 1$, satisfying

$$\begin{aligned} {[}\mu ,R_n]=nR_n,\quad \eta R_n \eta ^{-1}=(-1)^{n-1}R_n^T. \end{aligned}$$

One can actually see that the matrices $R_n$ are determined uniquely by the functions $\Omega ^{\alpha ,0}_{\beta ,d}$. Note that only finitely many of the matrices $R_n$ are non-zero. The space of all calibrations, satisfying property (2.2), can be explicitly described, see (Dubrovin 1999, Section 2). A calibration of a conformal Frobenius manifold will always be assumed to satisfy property (2.2).

Let us choose a calibration of our conformal Frobenius manifold and denote

$$\begin{aligned} R:=\sum _{i\ge 1}R_i. \end{aligned}$$

For an arbitrary $N\times N$ matrix $A=(A^\alpha _\beta )$ define matrices $P_m(A,R)$, $m\ge -1$, by the following recursion relation:

$$\begin{aligned} P_{-1}(A,R)=\,&{\mathrm {Id}},\\ P_{m+1}(A,R)=\,&R P_m(A,R)+P_m(A,R)\left( A+m+\frac{1}{2}\right) ,\quad m\ge -1. \end{aligned}$$

Alternatively, the matrix $P_m(A,R)$ can be defined by

$$\begin{aligned} P_m(A,R)=:\prod _{i=0}^m\left( R+A+i-\frac{1}{2}\right) :, \end{aligned}$$

where the symbol  :  :  means that, when we take the product, we should place all R’s to the left of all A’s. Given an integer p, define a matrix $[A]_p$ by

$$\begin{aligned} ([A]_p)^\alpha _\beta := {\left\{ \begin{array}{ll} A^\alpha _\beta ,&{}\quad \text {if } q^\alpha -q^\beta =p,\\ 0,&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

The Virasoro operators $L_m$, $m\ge -1$, of our conformal Frobenius manifold are given by

$$\begin{aligned} L_m:&=\frac{\varepsilon ^{-2}}{2}\sum _{d_1,d_2\ge 0}(-1)^{d_1}\left( [P_m(\mu +d_2+1,R)]_{m+d_1+d_2+1}\right) ^\mu _\beta \eta _{\alpha \mu } \widetilde{t}^\alpha _{d_1}\widetilde{t}^\beta _{d_2}\end{aligned}$$

(2.3)

$$\begin{aligned}&\quad +\sum _{\begin{array}{c} d\ge 0\\ 0\le k\le m+d \end{array}}\left( [P_m(\mu +d+1,R)]_{m+d-k}\right) ^\beta _\alpha \widetilde{t}^\alpha _d\frac{{\partial }}{{\partial }t^\beta _k}\end{aligned}$$

(2.4)

$$\begin{aligned}&\quad +\frac{\varepsilon ^2}{2}\sum _{\begin{array}{c} d_1+d_2\le m-1\\ d_1,d_2\ge 0 \end{array}}(-1)^{d_2+1}\left( [P_m(\mu -d_2,R)]_{m-1-d_1-d_2}\right) ^\alpha _\mu \eta ^{\mu \beta }\frac{{\partial }^2}{{\partial }t^\alpha _{d_1}{\partial }t^\beta _{d_2}}\nonumber \\&\quad +\delta _{m,0}\frac{1}{4}{\mathrm {tr}}\left( \frac{1}{4}-\mu ^2\right) ,\quad m\ge -1, \end{aligned}$$

(2.5)

and Eq. (1.4) hold (Dubrovin and Zhang 1999, page 428).

Remark 2.2

One can easily see that the last term $\delta _{m,0}\frac{1}{4}{\mathrm {tr}}\left( \frac{1}{4}-\mu ^2\right)$ in the expression for the Virasoro operators $L_m$ doesn’t play a role in Eq. (1.4). However, this term plays a role in the Virasoro constraints in all genera (1.5). We discuss it in more details in Sect. 5.

3 Open Virasoro Constraints

Here we present our construction of an open descendent potential associated to a solution of the open WDVV equations and present our main result—the open Virasoro constraints, given in Theorem 3.4.

Consider a conformal Frobenius manifold M together with its calibration from Sect. 2.2. Let U be a simply connected open neighbourhood of a point $s_{{\mathrm {orig}}}\in \mathbb {C}$. Suppose that a function $F^o(t^1,\ldots ,t^N,s)\in \mathcal {O}(M\times U)$ satisfies the open WDVV Eqs. (1.6), (1.7) and condition (1.8). Let us assume that the function $F^o$ satisfies the homogeneity condition

$$\begin{aligned}&E^\gamma \frac{{\partial }F^o}{{\partial }t^\gamma }+\underbrace{\left( \frac{1-\delta }{2}s+r^{N+1}\right) }_{=:E^{N+1}}\frac{{\partial }F^o}{{\partial }s}\nonumber \\&\quad =\frac{3-\delta }{2}F^o+D_\alpha t^\alpha +\widetilde{D}s+E,\quad \text {for some } r^{N+1},D_\alpha ,\widetilde{D},E\in \mathbb {C}. \end{aligned}$$

(3.1)

This condition holds in the examples, considered in the papers (Horev and Solomon 2012; Pandharipande et al. 2014; Buryak et al. 2018, 2019). Actually, in these examples the constant $r^{N+1}$ is zero, but this is not needed in our considerations.

In order to construct an open descendent potential ${\mathcal {F}}^o$ we need an additional structure, similar to a calibration of a Frobenius manifold. It will be convenient for us to denote the variable $s_d$ by $t^{N+1}_d$. We adopt the conventions

$$\begin{aligned} \Omega ^{\alpha ,0}_{N+1,d}:=0,\quad 1\le \alpha \le N,\quad d\ge -1,\quad \mu ^{N+1}:=\frac{1}{2}, \end{aligned}$$

and define a diagonal $(N+1)\times (N+1)$ matrix $\widetilde{\mu }$ by

$$\begin{aligned} \widetilde{\mu }:={\mathrm {diag}}(\mu ^1,\ldots ,\mu ^{N+1}). \end{aligned}$$

Definition 3.1

A calibration of the function $F^o$ is a sequence of functions $\Phi _{\beta ,d}\in \mathcal {O}(M\times U)$, $1\le \beta \le N+1$, $d\ge -1$, satisfying the properties

$$\begin{aligned} \Phi _{\beta ,-1}=\,&\delta _{\beta ,N+1},&\end{aligned}$$

(3.2)

$$\begin{aligned} \frac{{\partial }\Phi _{\beta ,d}}{{\partial }t^\gamma }=\,&\sum _{1\le \mu \le N}\frac{{\partial }^2 F^o}{{\partial }t^\gamma {\partial }t^\mu }\Omega ^{\mu ,0}_{\beta ,d-1}+\frac{{\partial }^2 F^o}{{\partial }t^\gamma {\partial }t^{N+1}}\Phi _{\beta ,d-1},&d\ge 0, \end{aligned}$$

(3.3)

$$\begin{aligned} E^\mu \frac{{\partial }\Phi _{\beta ,d}}{{\partial }t^\mu }=\,&\left( d+\frac{1}{2}+\mu ^\beta \right) \Phi _{\beta ,d}+\sum _{i=1}^{d+1}\Phi _{\mu ,d-i}(\widetilde{R}_i)^\mu _\beta ,&d\ge -1, \end{aligned}$$

(3.4)

for some $(N+1)\times (N+1)$ matrices $\widetilde{R}_n$, $n\ge 1$, satisfying

$$\begin{aligned} (\widetilde{R}_n)^\alpha _\beta = {\left\{ \begin{array}{ll} (R_n)^\alpha _\beta ,&{}\quad \text {if } 1\le \alpha ,\beta \le N,\\ 0,&{}\quad \text {if } \beta =N+1, \end{array}\right. } \qquad [\widetilde{\mu },\widetilde{R}_n]=n\widetilde{R}_n. \end{aligned}$$

One can easily see that, for a calibration of the function $F^o$, matrices $\widetilde{R}_n$ are uniquely determined by the functions $\Phi _{\beta ,d}$.

Lemma 3.2

The space of calibrations of the function $F^o$ is non-empty.

The proof of the lemma will be given in Sect. 4.3.3.

Consider a calibration of the function $F^o$. One can easily see that $\frac{{\partial }\Phi _{1,0}}{{\partial }t^\gamma }=\delta _{\gamma ,N+1}$, which implies that $\Phi _{1,0}-t^{N+1}$ is a constant. Let us make the change of coordinates $t^{N+1}\mapsto \Phi _{1,0}$, so that we have now $\Phi _{1,0}=t^{N+1}$.

Let $\widetilde{v}^1,\widetilde{v}^2,\ldots ,\widetilde{v}^{N+1}$ be formal variables and consider the system of partial differential equations

Let $(\widetilde{v}^{\mathrm {top}})^\alpha$ be the solution, specified by the initial condition

$$\begin{aligned} \left. (\widetilde{v}^{\mathrm {top}})^\alpha \right| _{t^\beta _d=\delta ^{\beta ,1} \delta _{d,0}x}=\delta ^{\alpha ,1}x. \end{aligned}$$

Clearly, $(\widetilde{v}^{\mathrm {top}})^\alpha =(v^{\mathrm {top}})^\alpha$, for $1\le \alpha \le N$. Define

$$\begin{aligned} \Phi _{\beta ,d}^{\mathrm {top}}:=\left. \Phi _{\beta ,d}\right| _{t^\gamma =(\widetilde{v}^{\mathrm {top}})^\gamma }. \end{aligned}$$

Then we define the open descendent potential ${\mathcal {F}}^o$ by

$$\begin{aligned} {\mathcal {F}}^o:=\sum _{d\ge 0}\widetilde{t}^\alpha _d\Phi ^{\mathrm {top}}_{\alpha ,d}. \end{aligned}$$

Lemma 3.3

The function ${\mathcal {F}}^o$ satisfies Eqs. (1.9), (1.10) and the difference $\left. {\mathcal {F}}^o\right| _{\begin{array}{c} t^*_{\ge 1}=0\\ s_{\ge 1}=0 \end{array}}-F^o$ is at most linear in the variables $t^1,\ldots ,t^N$ and s.

We will prove the lemma in Sect. 4.4.

Let $L_m$, $m\ge -1$, be the Virasoro operators for our conformal Frobenius manifold and define operators $\mathcal {L}_m$, $m\ge -1$, by

$$\begin{aligned} \mathcal {L}_m:&=L_m+\varepsilon ^{-1}\sum _{d\ge 0}\sum _{1\le \alpha \le N+1}\left( \left[ P_m(\widetilde{\mu }+d+1,\widetilde{R})\right] _{m+d+1}\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d \nonumber \\&\quad +\left( \sum _{\begin{array}{c} d\ge 0\\ 0\le k\le m+d \end{array}}\sum _{1\le \alpha \le N+1}\left( \left[ P_m(\widetilde{\mu }+d+1,\widetilde{R})\right] _{m+d-k}\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d\frac{{\partial }}{{\partial }t^{N+1}_k}+\delta _{m,0}\frac{3}{4}\right) \nonumber \\&\quad +\varepsilon \left( \sum _{0\le k\le m}\sum _{1\le \alpha ,\mu \le N}(-1)^{k+1}\left( \left[ P_m(\widetilde{\mu }-k,\widetilde{R})\right] _{m-k}\right) ^{N+1}_\alpha \eta ^{\alpha \mu }\frac{{\partial }}{{\partial }t^\mu _k}+\frac{3(m+1)!}{4}\frac{{\partial }}{{\partial }t^{N+1}_{m-1}}\right) \nonumber \\&\quad +\varepsilon ^2\sum _{\begin{array}{c} d_1,d_2\ge 0\\ d_1+d_2\le m-1 \end{array}}\sum _{1\le \alpha ,\mu \le N}(-1)^{d_2+1}\left( \left[ P_m(\widetilde{\mu }-d_2,\widetilde{R})\right] _{m-1-d_1-d_2}\right) ^{N+1}_\alpha \eta ^{\alpha \mu }\frac{{\partial }^2}{{\partial }t^{N+1}_{d_1}{\partial }t^\mu _{d_2}}. \end{aligned}$$

(3.5)

Theorem 3.4

We have

$$\begin{aligned} {\mathrm {Coef}}_{\varepsilon ^{-1}}\left( \frac{\mathcal {L}_m e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}{e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}\right) =0,\quad m\ge -1. \end{aligned}$$

(3.6)

Remark 3.5

Since $\widetilde{R}^\alpha _{N+1}=0$, we have

$$\begin{aligned} \left( \left[ P_m(\widetilde{\mu }+d+1,\widetilde{R})\right] _{m+d+1}\right) ^{N+1}_{N+1}&=\delta _{m,-1}\delta _{d,0},&m\ge -1,\,\, d\ge 0,\\ \left( \left[ P_m(\widetilde{\mu }+d+1,\widetilde{R})\right] _{m+d-k}\right) ^{N+1}_{N+1}&=\delta _{k,m+d}\frac{(d+m+1)!}{d!},&m\ge -1,\,\, d\ge 0, \,\, 0\le k\le m+d. \end{aligned}$$

Therefore, the operators $\mathcal {L}_m$, given by (3.5), have the form (1.14).

Remark 3.6

One can easily see that the expression $\frac{\mathcal {L}_m e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}{e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}$ has the form

$$\begin{aligned} \frac{\mathcal {L}_m e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}{e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}=\sum _{i\ge -2}\varepsilon ^i f_i(t^*_*), \end{aligned}$$

for some functions $f_i$ depending on the variables $t^\alpha _p$, $1\le \alpha \le N+1$, $p\ge 0$. Note that the vanishing of the function $f_{-2}$ is equivalent to the Virasoro equations for the descendent potential ${\mathcal {F}}$,

$$\begin{aligned} f_{-2}={\mathrm {Coef}}_{\varepsilon ^{-2}}\left( \frac{L_m e^{\varepsilon ^{-2}{\mathcal {F}}}}{e^{\varepsilon ^{-2}{\mathcal {F}}}}\right) =0. \end{aligned}$$

Remark 3.7

One can see that the terms $\delta _{m,0}\frac{3}{4}$ and $\varepsilon \frac{3(m+1)!}{4}\frac{{\partial }}{{\partial }t^{N+1}_{m-1}}$ in the expression for the operator $\mathcal {L}_m$ don’t play a role in Eq. (3.6). However, they play a role in our conjectural open Virasoro constraints in all genera, which we discuss in Sect. 5.

In the next section we derive Virasoro type equations for flat F-manifolds and then get Theorem 3.4 as a special case of this result.

4 Virasoro Type Constraints for Flat F-manifolds

In this section we recall the definition of a flat F-manifold and show how such an object can be associated to a solution of the open WDVV equations. We then present a construction of descendent vector potentials, corresponding to a flat F-manifold, and prove Virasoro type constraints for them. The open Virasoro constraints from Theorem 3.4 are derived as a corollary of this result.

4.1 Flat F-manifolds

Here we recall the definition and the main properties of flat F-manifolds. We refer a reader to the papers (Manin 2005; Arsie and Lorenzoni 2018) for more details.

Definition 4.1

A flat F-manifold $(M,\nabla ,\circ )$ is the datum of an analytic manifold M, an analytic connection $\nabla$ in the tangent bundle TM, an algebra structure $(T_p M,\circ )$ with unit e on each tangent space, analytically depending on the point $p\in M$, such that the one-parameter family of connections $\nabla +z\circ$ is flat and torsionless for any $z\in \mathbb {C}$, and $\nabla e=0$.

From the flatness and the torsionlessness of $\nabla +z\circ$ one can deduce the commutativity and the associativity of the algebras $(T_pM,\circ )$. Moreover, if one choses flat coordinates $t^\alpha$, $1\le \alpha \le N$, $N=\dim M$, for the connection $\nabla$, with $e = \frac{{\partial }}{{\partial }t^1}$, then it is easy to see that locally there exist analytic functions $F^\alpha (t^1,\ldots ,t^N)$, $1\le \alpha \le N$, such that the second derivatives

$$\begin{aligned} c^\alpha _{\beta \gamma }=\frac{{\partial }^2 F^\alpha }{{\partial }t^\beta {\partial }t^\gamma } \end{aligned}$$

(4.1)

give the structure constants of the algebras $(T_p M,\circ )$,

$$\begin{aligned} \frac{{\partial }}{{\partial }t^\beta }\circ \frac{{\partial }}{{\partial }t^\gamma }=c^\alpha _{\beta \gamma }\frac{{\partial }}{{\partial }t^\alpha }. \end{aligned}$$

From the associativity of the algebras $(T_p M,\circ )$ and the fact that the vector $\frac{{\partial }}{{\partial }t^1}$ is the unit it follows that

$$\begin{aligned}&\displaystyle \frac{{\partial }^2 F^\alpha }{{\partial }t^1{\partial }t^\beta } = \delta ^\alpha _\beta , \quad 1\le \alpha ,\beta \le N, \end{aligned}$$

(4.2)

$$\begin{aligned}&\displaystyle \frac{{\partial }^2 F^\alpha }{{\partial }t^\beta {\partial }t^\mu } \frac{{\partial }^2 F^\mu }{{\partial }t^\gamma {\partial }t^\delta } = \frac{{\partial }^2 F^\alpha }{{\partial }t^\gamma {\partial }t^\mu } \frac{{\partial }^2 F^\mu }{{\partial }t^\beta {\partial }t^\delta },\quad 1\le \alpha ,\beta ,\gamma ,\delta \le N. \end{aligned}$$

(4.3)

The N-tuple of functions $(F^1,\ldots ,F^N)$ is called the vector potential of the flat F-manifold.

Conversely, if M is an open subset of $\mathbb {C}^N$ and $F^1,\ldots ,F^N\in \mathcal {O}(M)$ are functions, satisfying Eqs. (4.2) and (4.3), then these functions define a flat F-manifold $(M,\nabla ,\circ )$ with the connection $\nabla$, given by $\nabla _{\frac{{\partial }}{{\partial }t^\alpha }}\frac{{\partial }}{{\partial }t^\beta }=0$, and the multiplication $\circ$, given by the structure constants (4.1).

A flat F-manifold, given by a vector potential $(F^1,\ldots ,F^N)$, is called conformal, if there exists a vector field of the form (2.1) such that

$$\begin{aligned} E^\mu \frac{{\partial }F^\alpha }{{\partial }t^\mu }=(2-q^\alpha )F^\alpha +A^\alpha _\beta t^\beta +B^\alpha ,\quad \text {for some } A^\alpha _\beta ,B^\alpha \in \mathbb {C}. \end{aligned}$$

(4.4)

Remark 4.2

A Frobenius manifold with a potential F and a metric $\eta$ defines the flat F-manifold with the vector potential $F^\alpha =\eta ^{\alpha \mu }\frac{{\partial }F}{{\partial }t^\mu }$. If the Frobenius manifold is conformal, then the associated flat F-manifold is also conformal. This follows from the property

$$\begin{aligned} (q^\alpha +q^\beta -\delta )\eta _{\alpha \beta }=0. \end{aligned}$$

A point $p\in M$ of an N-dimensional flat F-manifold $(M,\nabla ,\circ )$ is called semisimple if $T_pM$ has a basis of idempotents $\pi _1,\dots ,\pi _N$, satisfying $\pi _k \circ \pi _l = \delta _{k,l} \pi _k$. Moreover, locally around such a point one can choose coordinates $u^i$ such that $\frac{{\partial }}{{\partial }u^k}\circ \frac{{\partial }}{{\partial }u^l}=\delta _{k,l}\frac{{\partial }}{{\partial }u^k}$. These coordinates are called the canonical coordinates. In particular, this means that the semisimplicity is an open property of a point. The flat F-manifold M is called semisimple, if a generic point of M is semisimple.

Remark 4.3

In the semisimple case, a conformal flat F-manifold is a special case of a bi-flat F-manifold, see (Arsie and Lorenzoni 2017, Theorem 4.4).

4.2 Extensions of Flat F-manifolds and the Open WDVV Equations

Consider a flat F-manifold structure, given by a vector potential $(F^1,\ldots ,F^{N+1})$ on an open subset ${M\times U\in \mathbb {C}^{N+1}}$, where M and U are open subsets of $\mathbb {C}^N$ and $\mathbb {C}$, respectively. Suppose that the functions $F^1,\ldots ,F^N$ don’t depend on the variable $t^{N+1}$, varying in U. Then the functions $F^1,\ldots ,F^N$ satisfy Eq. (4.3) and, thus, define a flat F-manifold structure on M. In this case we call the flat F-manifold structure on $M\times U$ an extension of a flat F-manifold structure on M.

Consider the flat F-manifold, associated to a Frobenius manifold, given by a potential $F(t^1,\ldots ,t^N)\in \mathcal {O}(M)$ and a metric $\eta$, $F^\alpha =\eta ^{\alpha \mu }\frac{{\partial }F}{{\partial }t^\mu }$, $1\le \alpha \le N$. It is easy to check that a function $F^o(t^1,\ldots ,t^N,s)\in \mathcal {O}(M\times U)$ satisfies Eqs. (1.6), (1.7) and (1.8) if and only if the $(N+1)$-tuple $(F^1,\ldots ,F^N,F^o)$ is a vector potential of a flat F-manifold. Recall that here we identify $s=t^{N+1}$. This defines a correspondence between solutions of the open WDVV equations, satisfying property (1.8), and flat F-manifolds, extending the Frobenius manifold given. This observation belongs to Paolo Rossi.

4.3 Calibration of a Flat F-manifold

In this section we introduce the notion of a calibration of a flat F-manifold. This is done analogously to the case of Frobenius manifolds.

4.3.1 General Case

Let M be a simply connected open neighbourhood of a point $(t^1_{\mathrm {orig}},\ldots ,t^N_{\mathrm {orig}})\in \mathbb {C}^N$ and consider a flat F-manifold structure on M given by a vector potential $(F^1,\ldots ,F^N)$, $F^\alpha \in \mathcal {O}(M)$. A calibration of our flat F-manifold is a collection of functions $\Omega ^{\alpha ,d}_{\beta ,0}\in \mathcal {O}(M)$, $1\le \alpha ,\beta \le N$, $d\ge -1$, satisfying $\Omega ^{\alpha ,-1}_{\beta ,0}=\delta ^\alpha _\beta$ and the property

$$\begin{aligned} \frac{{\partial }\Omega ^{\alpha ,d}_{\beta ,0}}{{\partial }t^\gamma }=c^\mu _{\gamma \beta }\Omega ^{\alpha ,d-1}_{\mu ,0}, \quad d\ge 0. \end{aligned}$$

(4.5)

Let us describe the space of all calibrations of our flat F-manifold. Denote by $\overline{\nabla }$ the family of connections, depending on a formal parameter z, given by

$$\begin{aligned} \overline{\nabla }_X Y:=\nabla _X Y+z X\circ Y, \end{aligned}$$

where X and Y are vector fields on M. Then Eq. (4.5) is equivalent to the flatness, with respect to $\overline{\nabla }$, of the 1-forms

$$\begin{aligned} \left( \sum _{d\ge 0}\Omega ^{\alpha ,d-1}_{\beta ,0}z^d\right) d t^\beta ,\quad 1\le \alpha \le N. \end{aligned}$$

From the flatness of the connection $\overline{\nabla }$ it follows that a calibration of our flat F-manifold exists. In order to describe the whole space of calibrations, introduce $N\times N$ matrices $\Omega ^d_0$, $d\ge -1$, by $(\Omega ^d_0)^\alpha _\beta :=\Omega ^{\alpha ,d}_{\beta ,0}$. Then Eq. (4.5) can be written as

$$\begin{aligned} \frac{{\partial }}{{\partial }t^\gamma }\left( \sum _{d\ge 0}\Omega ^{d-1}_0z^d\right) =z\left( \sum _{d\ge 0}\Omega ^{d-1}_0z^d\right) C_\gamma , \end{aligned}$$

(4.6)

where $C_\gamma :=(c_{\gamma \beta }^\alpha )$. From this it becomes clear that the generating series $\sum _{d\ge 0}\Omega ^{d-1}_0z^d$ is determined uniquely up to a transformation of the form

$$\begin{aligned} \sum _{d\ge 0}\Omega ^{d-1}_0z^d\mapsto G(z)\left( \sum _{d\ge 0}\Omega ^{d-1}_0z^d\right) ,\quad G(z)\in \mathrm {Mat}_{N,N}(\mathbb {C})[[z]],\quad G(0)={\mathrm {Id}}. \end{aligned}$$

Let us introduce matrices $\Omega ^0_d=(\Omega ^{\alpha ,0}_{\beta ,d})$, $d\ge 0$, by the equation

$$\begin{aligned} \left( {\mathrm {Id}}+\sum _{d\ge 1}(-1)^d\Omega ^0_{d-1}z^d\right) \left( {\mathrm {Id}}+\sum _{d\ge 1}\Omega ^{d-1}_0 z^d\right) ={\mathrm {Id}}. \end{aligned}$$

(4.7)

By definition, we put $\Omega ^0_{-1}:={\mathrm {Id}}$. From Eq. (4.6) we get

$$\begin{aligned} \frac{{\partial }\Omega ^0_d}{{\partial }t^\gamma }=C_\gamma \Omega ^0_{d-1}, \quad d\ge 0. \end{aligned}$$

(4.8)

4.3.2 Conformal Case

Suppose now that our flat F-manifold is conformal. Introduce a diagonal matrix Q by

$$\begin{aligned} Q:={\mathrm {diag}}(q^1,\ldots ,q^N). \end{aligned}$$

Proposition 4.4

There exists a calibration such that

$$\begin{aligned} E^\theta \frac{{\partial }\Omega ^d_0}{{\partial }t^\theta }=(d+1)\Omega ^d_0+[\Omega ^d_0,Q]+\sum _{i=1}^{d+1}(-1)^{i-1}R_i\Omega ^{d-i}_0,\quad d\ge -1, \end{aligned}$$

(4.9)

for some matrices $R_n$, $n\ge 1$, satisfying $[Q,R_n]=nR_n$.

Proof

Similarly to the work (Dubrovin 1999, pages 310, 312), the proposition is proved by considering a certain flat connection on $M\times \mathbb {C}^*$.

Introduce a family of connections $\widetilde{\nabla }^\lambda$, depending on a complex parameter $\lambda$, on $M\times \mathbb {C}^*$ by

$$\begin{aligned}&\widetilde{\nabla }^\lambda _X Y:=\nabla _X Y+z X\circ Y,\qquad \widetilde{\nabla }^\lambda _{\frac{{\partial }}{{\partial }z}}Y:=\frac{{\partial }Y}{{\partial }z}+E\circ Y+\frac{\lambda -Q}{z}Y,\\&\quad \widetilde{\nabla }^\lambda _X\frac{{\partial }}{{\partial }z}=\widetilde{\nabla }^\lambda _{\frac{{\partial }}{{\partial }z}}\frac{{\partial }}{{\partial }z}:=0, \end{aligned}$$

where X and Y are vector fields on $M\times \mathbb {C}^*$ having zero component along $\mathbb {C}^*$.

Remark 4.5

Note that for the flat F-manifold, associated to a conformal Frobenius manifold M, the connection $\widetilde{\nabla }^{\frac{\delta }{2}}$ coincides with the flat connection $\widetilde{\nabla }$ on $M\times \mathbb {C}^*$ from the paper (Dubrovin 1999, page 310).

Lemma 4.6

The connection $\widetilde{\nabla }^\lambda$ is flat.

Proof

Direct computation analogous to the one from (Dubrovin 1999, proof of Proposition 2.1). $\square$

A differential form $\zeta _\alpha (t^*,z)dt^\alpha$ on $M\times \mathbb {C}^*$ is flat with respect to the connection $\widetilde{\nabla }^\lambda$ if and only if the following equations are satisfied:

$$\begin{aligned} \frac{{\partial }\zeta _\alpha }{{\partial }t^\beta }=\,&z c^\gamma _{\beta \alpha }\zeta _\gamma ,\nonumber \\ \frac{{\partial }\zeta _\alpha }{{\partial }z}=\,&\mathcal {U}^\gamma _\alpha \zeta _\gamma +\frac{\lambda -q^\alpha }{z}\zeta _\alpha , \end{aligned}$$

where $\mathcal {U}^\alpha _\beta :=E^\mu c^\alpha _{\mu \beta }$. Denote by $\zeta$ the row vector $(\zeta _1,\ldots ,\zeta _N)$, then the last two equations can be written as

$$\begin{aligned}&\displaystyle \frac{{\partial }\zeta }{{\partial }t^\beta }=z \zeta C_\beta , \end{aligned}$$

(4.10)

$$\begin{aligned}&\displaystyle \frac{{\partial }\zeta }{{\partial }z}=\zeta \left( \mathcal {U}+\frac{\lambda -Q}{z}\right) , \end{aligned}$$

(4.11)

where $\mathcal {U}:=(\mathcal {U}^\alpha _\beta )$.

Let us construct a certain matrix solution of the system (4.10), (4.11). We first consider Eq. (4.11) along the punctured line $(t^1_{\mathrm {orig}},\ldots ,t^N_{\mathrm {orig}})\times \mathbb {C}^*\subset M\times \mathbb {C}^*$. Let $U_1:=\mathcal {U}|_{t^\alpha =t^\alpha _{{\mathrm {orig}}}}$ and consider the equation

$$\begin{aligned} \frac{{\partial }\xi }{{\partial }z}=\xi \left( U_1+\frac{\lambda -Q}{z}\right) \quad \Leftrightarrow \quad \frac{{\partial }\xi ^T}{{\partial }z}=\left( U_1^T+\frac{\lambda -Q}{z}\right) \xi ^T, \end{aligned}$$

(4.12)

for an $N\times N$ matrix $\xi$. Then there exists a transformation $\xi '=G(z)\xi ^T$ with $G(z)\in \mathrm {Mat}_{N,N}(\mathbb {C})[[z]]$, $G(0)={\mathrm {Id}}$, that transforms Eq. (4.12) to

$$\begin{aligned} \frac{{\partial }\xi '}{{\partial }z}=\left( \frac{\lambda -Q}{z}+\sum _{n\ge 1}(-1)^{n-1}R_n^T z^{n-1}\right) \xi ', \end{aligned}$$

(4.13)

where matrices $R_n$ satisfy $[Q,R_n]=nR_n$ (see e.g. Dubrovin 1999, Lemma 2.5). The fact that G(z) and the matrices $R_n$ can be chosen not to depend on $\lambda$ is obvious. The matrix $\xi '=z^{\lambda -Q}z^{\overline{R}^T}$, where $\overline{R}:=\sum _{n\ge 1}(-1)^{n-1}R_n$, satisfies equation (4.13). Therefore, the matrix $\xi =z^{\overline{R}}z^{\lambda -Q}(G^T(z))^{-1}$ satisfies Eq. (4.12).

Using Eq. (4.10), we can extend the constructed function $\xi$ on the punctured line $(t^1_{\mathrm {orig}},\ldots ,t^N_{\mathrm {orig}})\times \mathbb {C}^*\subset M\times \mathbb {C}^*$ to a function $\zeta$ on the whole space $M\times \mathbb {C}^*$. The function $\zeta$ has the form

$$\begin{aligned} \zeta =z^{\overline{R}} z^{\lambda -Q}\left( \sum _{d\ge 0}\Omega ^{d-1}_0z^d\right) , \end{aligned}$$

(4.14)

where $\Omega ^d_0\in \mathrm {Mat}_{N,N}(\mathcal {O}(M))$, $\Omega ^{-1}_0={\mathrm {Id}}$ and $\left. \left( \sum _{d\ge 0}\Omega ^{d-1}_0z^d\right) \right| _{t^\alpha =t^\alpha _{{\mathrm {orig}}}}=(G^T(z))^{-1}$. Since the connection $\widetilde{\nabla }^\lambda$ is flat, the function $\zeta$ satisfies Eq. (4.11).

Equation (4.10) for a function $\zeta$ of the form (4.14) implies that the sequence of matrices $\Omega ^d_0$, $d\ge -1$, is a calibration of our flat F-manifold. Equation (4.11) gives

$$\begin{aligned}&\left( \frac{\lambda -Q}{z}+\sum _{n\ge 1}(-1)^{n-1}R_nz^{n-1}\right) \left( \sum _{d\ge 0}\Omega ^{d-1}_0z^d\right) +\sum _{d\ge 0}(d+1)\Omega ^d_0z^d\\&\quad =\left( \sum _{d\ge 0}\Omega ^{d-1}_0z^d\right) \left( \mathcal {U}+\frac{\lambda -Q}{z}\right) . \end{aligned}$$

Taking the coefficient of $z^d$, $d\ge 0$, in this equation, we get

$$\begin{aligned} \Omega ^{d-1}_0\mathcal {U}=(d+1)\Omega ^d_0+[\Omega ^d_0,Q]+\sum _{i=1}^{d+1}(-1)^{i-1}R_i\Omega ^{d-i}_0,\quad d\ge 0. \end{aligned}$$

By (4.10), the left-hand side is equal to $E^\theta \frac{{\partial }\Omega ^d_0}{{\partial }t^\theta }$. This completes the proof of the proposition. $\square$

A calibration of a conformal flat F-manifold will always be assumed to satisfy property (4.9).

From Eq. (4.9) it is easy to deduce that

$$\begin{aligned} E^\theta \frac{{\partial }\Omega ^0_d}{{\partial }t^\theta }=(d+1)\Omega ^0_d+[\Omega ^0_d,Q]+\sum _{i=1}^{d+1}\Omega ^0_{d-i} R_i, \quad d\ge -1. \end{aligned}$$

(4.15)

Remark 4.7

We see that a calibration of a conformal Frobenius manifold is the same as a calibration of the associated flat F-manifold, satisfying the additional properties $\eta \Omega ^0_d\eta ^{-1}=(\Omega ^d_0)^T$ and $\eta R_n\eta ^{-1}=(-1)^{n-1}R_n^T$.

4.3.3 Calibrations of Extensions of Flat F-manifolds

For an $(N+1)\times (N+1)$ matrix A denote by $\pi _N(A)$ the $N\times N$ matrix formed by the first N raws and the first N columns of A.

Lemma 4.8

Consider a flat F-manifold, given by a vector potential $(F^1,\ldots ,F^N)$, $F^\alpha \in \mathcal {O}(M)$, and its extension with a vector potential $(F^1,\ldots ,F^{N+1})$, $F^{N+1}\in \mathcal {O}(M\times U)$, where the open subsets $M\in \mathbb {C}^N$ and $U\in \mathbb {C}$ are simply connected. Suppose that the flat F-manifold $M\times U$ is conformal with an Euler vector field $E=\sum _{\alpha =1}^{N+1}((1-q^\alpha )t^\alpha +r^\alpha )\frac{{\partial }}{{\partial }t^\alpha }$. Let us also fix a calibration of the flat F-manifold M, given by matrices $\Omega ^d_0$ and $R_n$.

Then there exists a calibration of the flat F-manifold $M\times U$ with matrices $\widetilde{\Omega }^d_0$ and $\widetilde{R}_n$ satisfying the properties

$$\begin{aligned} \pi _N(\widetilde{\Omega }^d_0)=\Omega ^d_0,\qquad \pi _N(\widetilde{R}_n)=R_n,\quad \widetilde{\Omega }^{\le N,d}_{N+1,0}=(\widetilde{R}_n)^{\le N+1}_{N+1}=0. \end{aligned}$$

(4.16)

Proof

Consider the construction of a calibration of the conformal flat F-manifold $M\times U$ from the proof of Proposition 4.4 in more details. So we consider the differential equation

$$\begin{aligned} \frac{{\partial }\widetilde{\xi }^T}{{\partial }z}=\left( {\widetilde{U}}_1^T+\frac{\lambda -\widetilde{Q}}{z}\right) \widetilde{\xi }^T \end{aligned}$$

for an $(N+1)\times (N+1)$ matrix $\widetilde{\xi }$, where $({\widetilde{U}}_1)^\alpha _\beta :=(E^\mu c^\alpha _{\mu \beta })|_{t^\theta =t^\theta _{{\mathrm {orig}}}}$, $1\le \alpha ,\beta \le N+1$, and $\widetilde{Q}:={\mathrm {diag}}(q^1,\ldots ,q^{N+1})$. A transformation $\xi '=\widetilde{G}(z)\widetilde{\xi }^T$, $\widetilde{G}(z)={\mathrm {Id}}+\sum _{n\ge 1}\widetilde{G}_n z^n$, transforming this differential equation to the form

$$\begin{aligned} \frac{{\partial }\xi '}{{\partial }z}=\left( \frac{\lambda -\widetilde{Q}}{z}+\sum _{n\ge 1}(-1)^{n-1}\widetilde{R}_n^T z^{n-1}\right) \xi ', \end{aligned}$$

where $[\widetilde{Q},\widetilde{R}_n]=n\widetilde{R}_n$, is determined by the recursion relation (Dubrovin 1999, equation (2.53))

$$\begin{aligned}&(-1)^{n-1}\widetilde{R}_n^T=\delta _{n,1}{\widetilde{U}}_1^T+n\widetilde{G}_n+[\widetilde{Q},\widetilde{G}_n]\nonumber \\&\quad +\sum _{k=1}^{n-1}\left( \widetilde{G}_{n-k}\delta _{k,1}{\widetilde{U}}_1^T-(-1)^{k-1}\widetilde{R}_k^T\widetilde{G}_{n-k}\right) ,\quad n\ge 1. \end{aligned}$$

(4.17)

If one has computed the matrices $\widetilde{G}_i$ and $\widetilde{R}_i$ for $i<n$, then Eq. (4.17) determines the matrix $\widetilde{R}_n$ and the elements $(\widetilde{G}_n)^\alpha _\beta$ with $q^\alpha -q^\beta \ne -n$. The elements $(\widetilde{G}_n)^\alpha _\beta$ with $q^\alpha -q^\beta =-n$ can be chosen arbtitrarily. Note that $({\widetilde{U}}_1)^{\le N}_{N+1}=0$. Therefore, if $(\widetilde{R}_i)^{\le N}_{N+1}=(\widetilde{G}_i)^{N+1}_{\le N}=0$ for $i<n$, then $(\widetilde{R}_n)^{\le N}_{N+1}=0$ and, choosing the elements $(\widetilde{G}_n)^{N+1}_\alpha$ with $q^\alpha -q^\beta =-n$ to be zero, we can guarantee that $(\widetilde{G}_n)^{N+1}_{\le N}=0$.

Let $U_1:=\pi _N({\widetilde{U}}_1)$ and $Q:=\pi _N(\widetilde{Q})$. We know that the $N\times N$ matrices $G_n$, given by $\left. \left( \sum _{d\ge 0}\Omega ^{d-1}_0z^d\right) \right| _{t^\alpha =t^\alpha _{{\mathrm {orig}}}}=({\mathrm {Id}}+\sum _{n\ge 1}G_n^T z^n)^{-1}$, together with the matrices $R_n$ satisfy the equations

$$\begin{aligned} (-1)^{n-1}R_n^T= & {} \delta _{n,1}U_1^T+n G_n+[Q,G_n]\\&+\sum _{k=1}^{n-1}\left( G_{n-k}\delta _{k,1}U_1^T-(-1)^{k-1}R_k^T G_{n-k}\right) ,\quad n\ge 1. \end{aligned}$$

Therefore, there exist matrices $\widetilde{R}_n$ and $\widetilde{G}_n$, $n\ge 1$, satisfying Eq. (4.17) and the properties $\pi _N(\widetilde{R}_n)=R_n$, $\pi _N(\widetilde{G}_n)=G_n$ and $(\widetilde{R}_n)^{\le N}_{N+1}=(\widetilde{G}_n)^{N+1}_{\le N}=0$. Note that the property $[\widetilde{Q},\widetilde{R}_n]=n\widetilde{R}_n$ implies that the diagonal elements of $\widetilde{R}_n$ are also zero.

We construct the matrices $\widetilde{\Omega }^d_0$ as a solution of Eq. (4.6), satisfying the initial condition $\left. \left( \sum _{d\ge 0}\widetilde{\Omega }^{d-1}_0z^d\right) \right| _{t^\alpha =t^\alpha _{{\mathrm {orig}}}}=({\mathrm {Id}}+\sum _{n\ge 1}\widetilde{G}_n^T z^n)^{-1}$, and one can easily check that $\pi _N(\widetilde{\Omega }^d_0)=\Omega ^d_0$ and $\widetilde{\Omega }^{\le N,d}_{N+1,0}=0$. $\square$

Let us now apply this lemma to the conformal flat F-manifold, associated to a solution of the open WDVV equations, satisfying property (1.8) and the homogeneity condition (3.1). We assume that the matrices $\Omega ^d_0$ and $R_n$ give a calibration of the Frobenius manifold. By Lemma 4.8, there exists a calibration of the flat F-manifold $M\times U$, given by matrices $\widetilde{\Omega }^d_0$ and $\widetilde{R}_n$, satisfying properties (4.16). Consider the matrices $\widetilde{\Omega }^0_d$ and define functions $\Phi _{\beta ,d}$, $1\le \beta \le N$, $d\ge -1$, by

$$\begin{aligned} \Phi _{\beta ,d}:=\widetilde{\Omega }^{N+1,0}_{\beta ,d}. \end{aligned}$$

(4.18)

Lemma 4.9

• 1. The functions $\Phi _{\beta ,d}$ together with the matrices $\widetilde{R}_n$ give a calibration of the function $F^o$. As a corollary, Lemma 3.2 is true.

• 2. Equation (4.18) defines a correspondence between calibrations of the flat F-manifold $M\times U$, satisfying properties (4.16), and calibrations of the function $F^o$.

Proof

• 1. Property (3.2) is obvious. Equations (4.8) and (4.15) for the matrices $\widetilde{\Omega }^0_d$ give exactly properties (3.3) and (3.4), respectively.

• 2. The fact that Eqs. (4.8) and (4.15) for the matrices $\Omega ^0_d$ together with Eqs. (3.3) and (3.4) give the required equations for the matrices $\widetilde{\Omega }^0_d$ is obvious. $\square$

4.4 Descendent Vector Potentials of a Flat F-manifold

Consider a flat F-manifold, given by a vector potential $(F^1,\ldots ,F^N)$, $F^\alpha \in \mathcal {O}(M)$, and let us choose a calibration. One can immediately see that $\frac{{\partial }\Omega ^{\alpha ,0}_{1,0}}{{\partial }t^\beta }=\delta ^\alpha _\beta$, which implies that $\Omega ^{\alpha ,0}_{1,0}-t^\alpha$ is a constant. Let us make the change of coordinates $t^\alpha \mapsto \Omega ^{\alpha ,0}_{1,0}$, so that we have now $\Omega ^{\alpha ,0}_{1,0}=t^\alpha$.

Let $v^1,\ldots ,v^N$ be formal variables and consider the principal hierarchy associated to our flat F-manifold and its calibration (see e.g. Arsie and Lorenzoni 2018, Section 3.2):

$$\begin{aligned} \frac{{\partial }v^\alpha }{{\partial }t^\beta _d}={\partial }_x\left( \left. \Omega ^{\alpha ,0}_{\beta ,d}\right| _{t^\gamma =v^\gamma }\right) ,\quad 1\le \alpha ,\beta \le N,\quad d\ge 0. \end{aligned}$$

(4.19)

The flows of the principal hierarchy pairwise commute. Since $\Omega ^{\alpha ,0}_{1,0}=t^\alpha$, we can identify $x=t^1_0$.

Clearly the functions $v^\alpha =t^\alpha _0$ satisfy the subsystem of system (4.19), given by the flows $\frac{{\partial }}{{\partial }t^\beta _0}$. Denote by $(v^{\mathrm {top}})^\alpha \in \mathcal {O}(M)[[t^*_{\ge 1}]]$ the solution of the principal hierarchy specified by the initial condition

$$\begin{aligned} \left. (v^{\mathrm {top}})^\alpha \right| _{t^*_{\ge 1}=0}=t^\alpha _0. \end{aligned}$$

Recall that $\widetilde{t}^\alpha _d=t^\alpha _d-\delta ^{\alpha ,1}\delta _{d,1}$.

Lemma 4.10

We have

$$\begin{aligned}&\displaystyle \sum _{d\ge 0}\widetilde{t}^\gamma _{d+1}\frac{{\partial }(v^{\mathrm {top}})^\alpha }{{\partial }t^\gamma _d}+\delta ^{\alpha ,1}=0, \end{aligned}$$

(4.20)

$$\begin{aligned}&\displaystyle \sum _{d\ge 0}\widetilde{t}^\gamma _d\frac{{\partial }(v^{\mathrm {top}})^\alpha }{{\partial }t^\gamma _d}=0. \end{aligned}$$

(4.21)

Proof

From the property $\frac{{\partial }\Omega ^0_d}{{\partial }t^1}=\Omega ^0_{d-1}$, $d\ge 0$, it is easy to deduce that the system

$$\begin{aligned} \frac{{\partial }v^\alpha }{{\partial }\tau _{-1}}=\sum _{d\ge 0}\widetilde{t}^\gamma _{d+1}\frac{{\partial }v^\alpha }{{\partial }t^\gamma _d}+\delta ^{\alpha ,1},\quad 1\le \alpha \le N, \end{aligned}$$

is a symmetry of the principal hierarchy (4.19). Since, obviously, $\left. \frac{{\partial }(v^{\mathrm {top}})^\alpha }{{\partial }\tau _{-1}}\right| _{t^*_{\ge 1}=0}=0$, we get Eq. (4.20).

The rescaling combined with the shift along $t^1_1$, given by

$$\begin{aligned} \frac{{\partial }v^\alpha }{{\partial }\tau _0}=\sum _{d\ge 0}\widetilde{t}^\gamma _d\frac{{\partial }v^\alpha }{{\partial }t^\gamma _d},\quad 1\le \alpha \le N, \end{aligned}$$

is also a symmetry of the principal hierarchy. We compute

$$\begin{aligned} \left. \frac{{\partial }(v^{\mathrm {top}})^\alpha }{{\partial }\tau _0}\right| _{t^*_{\ge 1}}=t^\alpha _0-\left. \frac{{\partial }(v^{\mathrm {top}})^\alpha }{{\partial }t^1_1}\right| _{t^*_{\ge 1}=0}=t^\alpha _0-\frac{{\partial }\Omega ^{\alpha ,0}_{1,1}}{{\partial }t^1_0}=0, \end{aligned}$$

concluding that Eq. (4.21) is true. $\square$

Define matrices $\Omega ^p_q=(\Omega ^{\alpha ,p}_{\beta ,q})$, $p,q\ge 0$, by

$$\begin{aligned} \Omega ^p_q:=\sum _{i=0}^q(-1)^{q-i}\Omega ^{p+q-i}_0\Omega ^0_{i-1}{\mathop {=}\limits ^{{\text {eq. (4.7)}}}}\sum _{i=0}^p(-1)^{p-i}\Omega ^{i-1}_0\Omega ^0_{p+q-i}. \end{aligned}$$

We also adopt the convention $\Omega ^{-1}_q=\Omega ^q_{-1}:=\delta _{q,0}{\mathrm {Id}}$, $q\ge 0$. One can easily check that

$$\begin{aligned} d\Omega ^p_q=\,&\Omega ^{p-1}_0 d\Omega ^0_q=d\Omega ^p_0\cdot \Omega ^0_{q-1},\quad p,q\ge 0, \end{aligned}$$

(4.22)

$$\begin{aligned} \frac{{\partial }\Omega ^p_q}{{\partial }t^1}=\,&\Omega ^{p-1}_q+\Omega ^p_{q-1},\quad p,q\ge 0,\quad p+q\ge 1. \end{aligned}$$

(4.23)

Let

$$\begin{aligned} (\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}:=\left. \Omega ^{\alpha ,p}_{\beta ,q}\right| _{t^\gamma \mapsto (v^{\mathrm {top}})^\gamma }. \end{aligned}$$

Lemma 4.11

We have $\frac{{\partial }(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}}{{\partial }t^\gamma _r}=\frac{{\partial }(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\gamma ,r}}{{\partial }t^\beta _q}$, $p,q,r\ge 0$.

Proof

By (4.22), we have $\frac{{\partial }(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}}{{\partial }t^\gamma _r}=(\Omega ^{\mathrm {top}})^{\alpha ,p-1}_{\mu ,0}\frac{{\partial }(\Omega ^{\mathrm {top}})^{\mu ,0}_{\beta ,q}}{{\partial }t^\gamma _r}$. The fact that the flows of the principal hierarchy pairwise commute implies that $\frac{{\partial }(\Omega ^{\mathrm {top}})^{\mu ,0}_{\beta ,q}}{{\partial }t^\gamma _r}=\frac{{\partial }(\Omega ^{\mathrm {top}})^{\mu ,0}_{\gamma ,r}}{{\partial }t^\beta _q}$. This completes the proof of the lemma. $\square$

We finally define the descendent vector potentials $({\mathcal {F}}^{1,p},\ldots ,{\mathcal {F}}^{N,p})$, $p\ge 0$, associated to our flat F-manifold and its calibration, by

$$\begin{aligned} {\mathcal {F}}^{\alpha ,p}:=\sum _{q\ge 0}(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}\widetilde{t}^\beta _q\in \mathcal {O}(M)[[t^*_{\ge 1}]],\quad p\ge 0. \end{aligned}$$

Let us also adopt the convention

$$\begin{aligned} {\mathcal {F}}^{\alpha ,p}:=(-1)^{p+1}\widetilde{t}^\alpha _{-p-1},\quad \text {if p<0}. \end{aligned}$$

Proposition 4.12

• 1. We have

  $$\begin{aligned}&\displaystyle \frac{{\partial }{\mathcal {F}}^{\alpha ,p}}{{\partial }t^\beta _q}=(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}, \quad p\ge 0, \end{aligned}$$

  (4.24)
  $$\begin{aligned}&\displaystyle \sum _{q\ge 0}\widetilde{t}^\beta _{q+1}\frac{{\partial }{\mathcal {F}}^{\alpha ,p}}{{\partial }t^\beta _q}+{\mathcal {F}}^{\alpha ,p-1}=0, \quad p\in \mathbb {Z}. \end{aligned}$$

  (4.25)

• 2. The difference ${\mathcal {F}}^{\alpha ,0}|_{t^*_{\ge 1}=0}-F^\alpha$ is at most linear in the variables $t^1,\ldots ,t^N$.

Proof

• 1. We compute

  $$\begin{aligned} \frac{{\partial }{\mathcal {F}}^{\alpha ,p}}{{\partial }t^\beta _q}= & {} \sum _{r\ge 0}\frac{{\partial }(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\gamma ,r}}{{\partial }t^\beta _q}\widetilde{t}^\gamma _r+(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q} {\mathop {=}\limits ^{\scriptstyle \text {Lemma 4.11}}}\sum _{r\ge 0}\frac{{\partial }(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}}{{\partial }t^\gamma _r}\widetilde{t}^\gamma _r\\&+(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q} {\mathop {=}\limits ^{\scriptstyle \text {eq. (4.21)}}}(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}. \end{aligned}$$

  For $p<0$ Eq. (4.25) is obvious. Suppose $p\ge 0$, then we have

  $$\begin{aligned} \sum _{q\ge 0}\widetilde{t}^\beta _{q+1}\frac{{\partial }{\mathcal {F}}^{\alpha ,p}}{{\partial }t^\beta _q} =\,&\sum _{q,r\ge 0}\frac{{\partial }(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\gamma ,r}}{{\partial }t^\beta _q}\widetilde{t}^\beta _{q+1}\widetilde{t}^\gamma _r+\sum _{q\ge 0} \widetilde{t}^\beta _{q+1}(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q} {\mathop {=}\limits ^{\scriptstyle \text {eq. (4.20)}}}\\ =\,&-\sum _{r\ge 0}\frac{{\partial }(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\gamma ,r}}{{\partial }(v^{\mathrm {top}})^1}\widetilde{t}^\gamma _r+\sum _{q\ge 0}\widetilde{t}^\beta _{q+1} (\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q} {\mathop {=}\limits ^{\scriptstyle \text {eq. (4.23)}}}\\ =\,&-\sum _{r\ge 0}(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\gamma ,r} \widetilde{t}^\gamma _{r+1}-\sum _{r\ge 0}(\Omega ^{\mathrm {top}})^{\alpha ,p-1}_{\gamma ,r} \widetilde{t}^\gamma _r+\sum _{q\ge 0}\widetilde{t}^\beta _{q+1}(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}\\ =\,&-{\mathcal {F}}^{\alpha ,p-1}. \end{aligned}$$

• 2. By the first part of the proposition, $\frac{{\partial }}{{\partial }t^\beta _0}\left( {\mathcal {F}}^{\alpha ,0}|_{t^*_{\ge 1}=0}\right) =\Omega ^{\alpha ,0}_{\beta ,0}$. Since $\frac{{\partial }\Omega ^{\alpha ,0}_{\beta ,0}}{{\partial }t^\gamma }=c^\alpha _{\beta \gamma }$, the second part of the proposition is also proved. $\square$

Remark 4.13

Consider a Frobenius manifold, its calibration and the associated flat F-manifold. Then the functions ${\mathcal {F}}^{\alpha ,p}$ are related to the descendent potential ${\mathcal {F}}$ of the Frobenius manifold by ${\mathcal {F}}^{\alpha ,p}=\eta ^{\alpha \mu }\frac{{\partial }{\mathcal {F}}}{{\partial }t^\mu _p}$.

Consider a conformal Frobenius manifold together with a calibration and a solution $F^o$ of the open WDVV equations, satisfying properties (1.8), (3.1), also with a calibration. We have the associated flat F-manifold with the vector potential $\left( \eta ^{1\mu }\frac{{\partial }F}{{\partial }t^\mu },\ldots ,\eta ^{N\mu }\frac{{\partial }F}{{\partial }t^\mu },F^o\right)$. Immediately from the definitions and Lemma 4.9 we see that if $({\mathcal {F}}^{1,p},\ldots ,{\mathcal {F}}^{N+1,p})$, $p\ge 0$, are the decendent vector potentials of this flat F-manifold, then ${\mathcal {F}}^{\alpha ,p}=\eta ^{\alpha \mu }\frac{{\partial }{\mathcal {F}}}{{\partial }t^\mu _p}$, $1\le \alpha \le N$, and ${\mathcal {F}}^{N+1,0}={\mathcal {F}}^o$. Therefore, Lemma 3.3 follows from Proposition 4.12 and Eq. (4.22).

4.5 Virasoro Constraints

We consider a conformal flat F-manifold, given by a vector potential $(F^1,\ldots ,F^N)$, $F^\alpha \in \mathcal {O}(M)$, and an Euler vector field (2.1), its calibration, described by matrices $\Omega ^d_0$, $d\ge -1$, and $R_n$, $n\ge 1$, and the associated descendent vector potentials $({\mathcal {F}}^{1,p},\ldots ,{\mathcal {F}}^{N,p})$, $p\ge 0$.

Recall that $R=\sum _{i\ge 1}R_i$. Let $\lambda$ be a complex parameter and define

$$\begin{aligned} \mu ^\alpha :=q^\alpha +\lambda -\frac{3}{2},\quad \mu :={\mathrm {diag}}(\mu ^1,\ldots ,\mu ^N). \end{aligned}$$

Define the following expressions, depending on the parameter $\lambda$:

$$\begin{aligned} A^\alpha _m:=\,&\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1} \left( [P_m(\mu -d_2,R)]_{m-1-d_1-d_2}\right) ^\gamma _\mu (\Omega ^{\mathrm {top}})^{\alpha ,0}_{\gamma ,d_1}{\mathcal {F}}^{\mu ,d_2},\quad \\&1\le \alpha \le N,\quad m\ge -1. \end{aligned}$$

Proposition 4.14

We have $A^\alpha _m=0$, $1\le \alpha \le N$, $m\ge -1$.

Remark 4.15

For a Frobenius manifold the expressions $A^\alpha _m$ have the following interpretation:

$$\begin{aligned} A^\alpha _m|_{\lambda =\frac{3-\delta }{2}}=\eta ^{\alpha \mu }\frac{{\partial }}{{\partial }t^\mu _0}\left[ {\mathrm {Coef}}_{\varepsilon ^{-2}}\left( \frac{L_m e^{\varepsilon ^{-2}{\mathcal {F}}}}{e^{\varepsilon ^{-2}{\mathcal {F}}}}\right) \right] , \end{aligned}$$

where $L_m$ are the Virasoro operators described in Sect. 2.2. Therefore, we consider Proposition 4.14 as a generalization of the Virasoro constraints (1.4) for an arbitrary conformal flat F-manifold.

Proof of Proposition 4.14

During the proof of the proposition, for the sake of shortness, we will denote the functions $(v^{\mathrm {top}})^\alpha$ and $(\Omega ^{\mathrm {top}})^{\alpha ,p}_{\beta ,q}$ by $v^\alpha$ and $\Omega ^{\alpha ,p}_{\beta ,q}$, respectively. We will also denote the function $c^\alpha _{\beta \gamma }|_{t^\theta \mapsto (v^{\mathrm {top}})^\theta }$ by $c^\alpha _{\beta \gamma }$ and the function $(1-q^\alpha )(v^{\mathrm {top}})^\alpha +r^\alpha$ by $E^\alpha$.

Define an operator ${\mathcal {B}}=({\mathcal {B}}^\alpha _\beta )$, depending on the parameter $\lambda$, by

$$\begin{aligned} {\mathcal {B}}^\alpha _\beta :=E^\mu c^\alpha _{\mu \beta }{\partial }_x+v^\gamma _x(\lambda -1+q^\beta )c^\alpha _{\gamma \beta }=E^\mu c^\alpha _{\mu \beta }{\partial }_x+v^\gamma _x\left( \mu ^\beta +\frac{1}{2}\right) c^\alpha _{\gamma \beta }. \end{aligned}$$

$\square$

Remark 4.16

For a Frobenius manifold the operator ${\mathcal {B}}|_{\lambda =\frac{3-\delta }{2}}{\partial }_x^{-1}$ coincides with the recursion operator $\mathcal {R}$ from the paper (Dubrovin and Zhang 1999, equation (3.36)).

We begin with the following lemma.

Lemma 4.17

We have ${\mathcal {B}}\Omega ^0_d={\partial }_x\left( \Omega ^0_{d+1}\left( \mu +d+\frac{3}{2}\right) +\sum _{i=1}^{d+2}\Omega ^0_{d+1-i}R_i\right)$, $d\ge -1$.

Proof

Note that ${\mathcal {B}}^\alpha _\beta ={\partial }_x\cdot (E^\mu c^\alpha _{\mu \beta })+v^\gamma _x(\lambda +q^\alpha -2)c^\alpha _{\gamma \beta }$. This follows from the property $E^\mu \frac{{\partial }c^\alpha _{\beta \gamma }}{{\partial }v^\mu }=(q^\beta +q^\gamma -q^\alpha )c^\alpha _{\beta \gamma }$ which is equivalent to the homogeneity condtition (4.4). We then compute ${\mathcal {B}}\Omega ^0_d={\partial }_x\left( E^\mu \frac{{\partial }\Omega ^0_{d+1}}{{\partial }v^\mu }+(\lambda +Q-2)\Omega ^0_{d+1}\right)$, that, by Eq. (4.15), implies the lemma. $\square$

Denote by $A_m$ the column vector $(A^1_m,\ldots ,A^N_m)$.

Lemma 4.18

We have

$$\begin{aligned}&{\mathcal {B}}A_m-{\partial }_x A_{m+1}\\&\quad =\left( \frac{1}{2}-\mu \right) \sum _{d_1,d_2\ge -1}(-1)^{d_2+1}\Omega ^0_{d_1}[P_m(\mu -d_2,R)]_{m-d_1-d_2}\Omega ^{d_2}_0\cdot e,\quad m\ge -1, \end{aligned}$$

where e denotes the unit vector $(1,0,\ldots ,0)$.

Proof

We compute

$$\begin{aligned} {\mathcal {B}}^\alpha _\mu A^\mu _m&=\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1} \left( [P_m(\mu -d_2,R)]_{m-1-d_1-d_2}\right) ^\gamma _\nu {\mathcal {B}}^\alpha _\mu \Omega ^{\mu ,0}_{\gamma ,d_1}\cdot {\mathcal {F}}^{\nu ,d_2}\nonumber \\&\qquad +\sum _{d\ge -1}\left( [P_m(\mu +1,R)]_{m-d}\right) ^\gamma _1E^\theta \frac{{\partial }\Omega ^{\alpha ,0}_{\gamma ,d+1}}{{\partial }v^\theta }\nonumber \\&\qquad +\sum _{d_1\ge -1,\,d_2\ge 0}(-1)^{d_2+1}\left( [P_m(\mu -d_2,R)]_{m-1-d_1-d_2}\right) ^\gamma _\nu E^\theta \frac{{\partial }\Omega ^{\alpha ,0}_{\gamma ,d_1+1}}{{\partial }v^\theta }\Omega ^{\nu ,d_2}_{1,0}\nonumber \\&=\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1} \left( [P_m(\mu -d_2,R)]_{m-1-d_1-d_2}\right) ^\gamma _\nu {\mathcal {B}}^\alpha _\mu \Omega ^{\mu ,0}_{\gamma ,d_1}\cdot {\mathcal {F}}^{\nu ,d_2} \end{aligned}$$

(4.26)

$$\begin{aligned}&\qquad +\sum _{d_1,d_2\ge -1}(-1)^{d_2+1} \left( [P_m(\mu -d_2,R)]_{m-1-d_1-d_2}\right) ^\gamma _\nu E^\theta \frac{{\partial }\Omega ^{\alpha ,0}_{\gamma ,d_1+1}}{{\partial }v^\theta }\Omega ^{\nu ,d_2}_{1,0}. \end{aligned}$$

(4.27)

On the other hand, we have

$$\begin{aligned} {\partial }_x A^\alpha _{m+1}&=\sum _{d_1\ge 0,\,d_2\in \mathbb {Z}}(-1)^{d_2+1} \left( [P_{m+1}(\mu -d_2,R)]_{m-d_1-d_2}\right) ^\gamma _\mu {\partial }_x\Omega ^{\alpha ,0}_{\gamma ,d_1}\cdot {\mathcal {F}}^{\mu ,d_2} \end{aligned}$$

(4.28)

$$\begin{aligned}&\qquad +\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}\left( [P_{m+1}(\mu -d_2,R)]_{m-d_1-d_2}\right) ^\gamma _\mu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\mu ,d_2}_{1,0}. \end{aligned}$$

(4.29)

Let us show that the expression in line (4.26) is equal to the expression in line (4.28). Using Lemma 4.17, we rewrite the first one as follows:

$$\begin{aligned}&\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1}{\mathcal {F}}^{\nu ,d_2} \left( [P_m(\mu -d_2,R)]_{m-1-d_1-d_2}\right) ^\gamma _\nu {\partial }_x \left( \left( \mu ^\gamma +d_1+\frac{3}{2}\right) \Omega ^{\alpha ,0}_{\gamma ,d_1+1}\right. \\&\qquad \left. +\sum _{i=1}^{d_1+1} \Omega ^{\alpha ,0}_{\theta ,d_1+1-i}(R_i)^\theta _\gamma \right) \\&\quad =\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1}{\mathcal {F}}^{\nu ,d_2} \left( \left[ P_m(\mu -d_2,R) \left( \mu -d_2+m+\frac{1}{2}\right) \right] _{m-1-d_1-d_2}\right) ^\gamma _\nu {\partial }_x\Omega ^{\alpha ,0}_{\gamma ,d_1+1}\\&\qquad +\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}\sum _{i=1}^{d_1+1}(-1)^{d_2+1} {\mathcal {F}}^{\nu ,d_2}([R_iP_m(\mu -d_2,R)]_{m-1-d_1-d_2+i})^\theta _\nu {\partial }_x\Omega ^{\alpha ,0}_{\theta ,d_1+1-i}\\&\quad =\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1}{\mathcal {F}}^{\nu ,d_2}\left( \left[ P_m(\mu -d_2,R) \left( \mu -d_2+m+\frac{1}{2}\right) +\right. \right. \\&\qquad +R P_m(\mu -d_2,R)\bigg ]_{m-1-d_2-d_2}\bigg )^\gamma _\nu {\partial }_x\Omega ^{\alpha ,0}_{\gamma ,d_1+1}\\&\quad =\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1} {\mathcal {F}}^{\nu ,d_2}([P_{m+1}(\mu -d_2,R)]_{m-1-d_1-d_2})^\gamma _\nu {\partial }_x\Omega ^{\alpha ,0}_{\gamma ,d_1+1}, \end{aligned}$$

and we see that the last expression coincides with the expression in line (4.28).

Using Eq. (4.15), we transform the expression in line (4.27):

$$\begin{aligned}&\sum _{d_1\ge 0,\,d_2\ge -1}(-1)^{d_2+1}\left( [P_m(\mu -d_2,R)]_{m-d_1-d_2}\right) ^\gamma _\nu \bigg ((\mu ^\gamma -\mu ^\alpha +d_1+1)\Omega ^{\alpha ,0}_{\gamma ,d_1}\\&\qquad \left. +\sum _{i=1}^{d_1+1}\Omega ^{\alpha ,0}_{\theta ,d_1-i} (R_i)^\theta _\gamma \right) \Omega ^{\nu ,d_2}_{1,0}\\&\quad =\sum _{d_1\ge 0,\,d_2\ge -1}(-1)^{d_2+1}\left( [P_m(\mu -d_2,R) (\mu -\mu ^\alpha +m-d_2+1)]_{m-d_1-d_2}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\nu ,d_2}_{1,0}\\&\qquad +\sum _{d_1\ge 0,\,d_2\ge -1}\sum _{i=1}^{d_1+1}(-1)^{d_2+1} \left( [R_iP_m(\mu -d_2,R)]_{m-d_1-d_2+i}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1-i}\Omega ^{\nu ,d_2}_{1,0}\\&\quad =\left( \frac{1}{2}-\mu ^\alpha \right) \sum _{d_1\ge 0,\,d_2\ge -1} (-1)^{d_2+1}\left( [P_m(\mu -d_2,R)]_{m-d_1-d_2}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\nu ,d_2}_{1,0}\\&\qquad +\sum _{d_1\ge 0,\,d_2\ge -1}(-1)^{d_2+1}\left( \left[ P_m(\mu -d_2,R) \left( \mu +m-d_2+\frac{1}{2}\right) \right] _{m-d_1-d_2}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\nu ,d_2}_{1,0}\\&\qquad +\sum _{d_1,d_2\ge -1}(-1)^{d_2+1} \left( [RP_m(\mu -d_2,R)]_{m-d_1-d_2}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1} \Omega ^{\nu ,d_2}_{1,0}. \end{aligned}$$

On the other hand, the expression in line (4.29) is equal to

$$\begin{aligned}&\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}\left( \left[ P_m(\mu -d_2,R) \left( \mu +m-d_2+\frac{1}{2}\right) \right] _{m-d_1-d_2}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\nu ,d_2}_{1,0}\\&\qquad +\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}\left( [RP_m(\mu -d_2,R)]_{m-d_1-d_2}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\nu ,d_2}_{1,0}. \end{aligned}$$

As a result, we get

$$\begin{aligned}&{\mathcal {B}}^\alpha _\mu A^\mu _m-{\partial }_x A^\alpha _{m+1}\\&\quad =\left( \frac{1}{2} -\mu ^\alpha \right) \sum _{d_1\ge 0,\,d_2\ge -1}(-1)^{d_2+1} \left( [P_m(\mu -d_2,R)]_{m-d_1-d_2}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\nu ,d_2}_{1,0}\\&\qquad \quad -\sum _{d_2\ge -1}(-1)^{d_2+1}\left( \left[ P_m(\mu -d_2,R) \left( \mu +m-d_2+\frac{1}{2}\right) \right] _{m+1-d_2}\right) ^\alpha _\nu \Omega ^{\nu ,d_2}_{1,0}\\&\quad =\left( \frac{1}{2}-\mu ^\alpha \right) \sum _{d_1,d_2\ge -1}(-1)^{d_2+1} \left( [P_m(\mu -d_2,R)]_{m-d_1-d_2}\right) ^\gamma _\nu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\nu ,d_2}_{1,0}, \end{aligned}$$

as required. $\square$

For $m\ge n\ge -1$ denote

$$\begin{aligned} C_{m,n}:=\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}\Omega ^0_{d_1} [P_n(\mu -d_2,R)]_{m-d_1-d_2}\Omega ^{d_2}_0. \end{aligned}$$

Lemma 4.19

We have $C_{m,n}=0$, $m\ge n\ge -1$.

Proof

We proceed by induction on n. By Eq. (4.7), we have

$$\begin{aligned} C_{m,-1}=\sum _{\begin{array}{c} d_1,d_2\ge -1\\ d_1+d_2=m \end{array}}(-1)^{d_2+1} \Omega ^0_{d_1}\Omega ^{d_2}_0=0,\quad m\ge -1. \end{aligned}$$

Suppose that $m\ge n\ge 0$. We compute

$$\begin{aligned} C_{m,n}&=\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}\Omega ^0_{d_1} \left[ P_{n-1}(\mu -d_2,R)\left( \mu -d_2+n-\frac{1}{2}\right) \right. \\&\quad \left. +R P_{n-1}(\mu -d_2,R)\right] _{m-d_1-d_2}\Omega ^{d_2}_0\\&=\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}\Omega ^0_{d_1} \left[ \left( \mu -m+n+d_1-\frac{1}{2}\right) P_{n-1}(\mu -d_2,R)\right. \\&\quad \left. +R P_{n-1} (\mu -d_2,R)\right] _{m-d_1-d_2}\Omega ^{d_2}_0. \end{aligned}$$

Using property (4.15), we transform the last expression in the following way:

$$\begin{aligned}&\left( \mu -m+n-\frac{3}{2}\right) C_{m,n-1}+\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}E^\theta \frac{{\partial }\Omega ^0_{d_1}}{{\partial }v^\theta }[P_{n-1}(\mu -d_2,R)]_{m-d_1-d_2}\Omega ^{d_2}_0\\&\quad -\sum _{d_1,d_2\ge -1}\sum _{i=1}^{d_1+1}(-1)^{d_2+1}\Omega ^0_{d_1-i} \left[ R_iP_{n-1}(\mu -d_2,R)\right] _{m-d_1-d_2+i}\Omega ^{d_2}_0\\&\quad +\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}\Omega ^0_{d_1}\left[ R P_{n-1}(\mu -d_2,R)\right] _{m-d_1-d_2}\Omega ^{d_2}_0. \end{aligned}$$

One can see that the last two expressions cancel each other and, as a result,

$$\begin{aligned} C_{m,n}=\left( \mu -m+n-\frac{3}{2}\right) C_{m,n-1}+E^\theta \frac{{\partial }\Omega ^0_0}{{\partial }v^\theta }C_{m-1,n-1}, \end{aligned}$$

that, by the induction assumption, is equal to zero. The lemma is proved. $\square$

Lemmas 4.18 and 4.19 imply that ${\mathcal {B}}A_m={\partial }_x A_{m+1}$ for $m\ge -1$. Introduce a differential operator $L:=\sum _{d\ge 0}\widetilde{t}^\gamma _{d+1}\frac{{\partial }}{{\partial }t^\gamma _d}$.

Lemma 4.20

We have $L A^\alpha _m=(-m-1)A^\alpha _{m-1}$, $m\ge 0$.

Proof

Using formulas (4.20), (4.25) and the formula $\frac{{\partial }\Omega ^0_d}{{\partial }v^1}=\Omega ^0_{d-1}$, $d\ge 0$, we compute

$$\begin{aligned} LA^\alpha _m=\,&-\sum _{d_1\ge 0,\,d_2\in \mathbb {Z}}(-1)^{d_2+1}([P_m(\mu -d_2,R)]_{m-1-d_1-d_2})^\gamma _\mu \Omega ^{\alpha ,0}_{\gamma ,d_1-1}{\mathcal {F}}^{\mu ,d_2}\\&-\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1}([P_m(\mu -d_2,R)]_{m-1-d_1-d_2})^\gamma _\mu \Omega ^{\alpha ,0}_{\gamma ,d_1}{\mathcal {F}}^{\mu ,d_2-1}\\ =\,&\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1}([P_m(\mu -d_2-1,R)-P_m(\mu -d_2,R)]_{m-2-d_1-d_2})^\gamma _\mu \Omega ^{\alpha ,0}_{\gamma ,d_1}{\mathcal {F}}^{\mu ,d_2}. \end{aligned}$$

Since $P_m(\mu -d_2-1,R)-P_m(\mu -d_2,R)=(-m-1)P_{m-1}(\mu -d_2,R)$, the lemma is proved. $\square$

Lemma 4.21

We have $\left. A^\alpha _m\right| _{t^*_{\ge 1}=0}=0$, $m\ge -1$.

Proof

During the proof of this lemma we return to the initial notation, where $\Omega ^{\alpha ,p}_{\beta ,q}$ is a function of $t^1,\ldots ,t^N$. Note that, by Eqs. (4.24) and (4.25), we have ${\mathcal {F}}^{\alpha ,p}|_{t^*_{\ge 1}=0}=\Omega ^{\alpha ,p+1}_{1,0}$, $p\ge 0$. Therefore, we have

$$\begin{aligned} \left. A^\alpha _m\right| _{t^*_{\ge 1}=0}=\,&\sum _{d\ge -1}([P_m(\mu +1,R)]_{m-d})^\gamma _\mu t^\mu _0\Omega ^{\alpha ,0}_{\gamma ,d}\\&-\sum _{d\ge -1}([P_m(\mu +2,R)]_{m+1-d})^\gamma _1\Omega ^{\alpha ,0}_{\gamma ,d}\\&+\sum _{d_1\ge -1,\,d_2\ge 0}(-1)^{d_2+1}([P_m(\mu -d_2,R)]_{m-1-d_1-d_2})^\gamma _\mu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\mu ,d_2+1}_{1,0}\\ =\,&-\sum _{d_1,d_2\ge -1}(-1)^{d_2+1}([P_m(\mu +1-d_2,R)]_{m-d_1-d_2})^\gamma _\mu \Omega ^{\alpha ,0}_{\gamma ,d_1}\Omega ^{\mu ,d_2}_{1,0}\\ =\,&-(C_{m,m})^\alpha _1\Big |_{\begin{array}{c} t^*_{\ge 1}=0\\ \lambda \mapsto \lambda +1 \end{array}}, \end{aligned}$$

which, by Lemma 4.19, is equal to zero. $\square$

Let us now prove that $A^\alpha _m=0$ by induction on m. We have

$$\begin{aligned} A^\alpha _{-1}=\sum _{\begin{array}{c} d\ge -1,\,d_2\in \mathbb {Z}\\ d_1+d_2=-2 \end{array}}(-1)^{d_2+1}\Omega ^{\alpha ,0}_{\gamma ,d_1} {\mathcal {F}}^{\gamma ,d_2}=t^\alpha _0+\sum _{d\ge 0}\Omega ^{\alpha ,0}_{\gamma ,d}\widetilde{t}^\alpha _{d+1} =t^\alpha _0+L{\mathcal {F}}^{\alpha ,0} {\mathop {=}\limits ^{\scriptstyle \text {eq.(4.25)}}}0. \end{aligned}$$

Suppose that $m\ge 0$. Let us express $A^\alpha _m$ as a power series in the variables $\widehat{t}^\beta _d$ defined by

$$\begin{aligned} \widehat{t}^\beta _d:={\left\{ \begin{array}{ll} t^\beta _0-t^\beta _{{\mathrm {orig}}},&{}\quad \text {if } d=0,\\ t^\beta _d,&{}\quad \text {if } d\ge 1. \end{array}\right. } \end{aligned}$$

From the induction assumption, Lemma 4.20 and the fact that ${\mathcal {B}}A_{m-1}={\partial }_x A_m$ it follows that

$$\begin{aligned} \sum _{d\ge 0}\widehat{t}^\gamma _{d+1}\frac{{\partial }A^\alpha _m}{{\partial }\widehat{t}^\gamma _d}=0. \end{aligned}$$

We also know that $A^\alpha _m|_{\widehat{t}^*_*=0}=0$. We see that the proof of the proposition is completed by the following lemma.

Lemma 4.22

Let f be a formal power series in the variables $\widehat{t}^\alpha _d$ such that $\sum _{d\ge 0}\widehat{t}^\alpha _{d+1}\frac{{\partial }f}{{\partial }\widehat{t}^\alpha _d}=0$ and $f|_{\widehat{t}^*_*=0}=0$. Then $f=0$.

Proof

Denote by $\mathcal {P}$ the set of all finite sequences $\lambda =(\lambda _1,\ldots ,\lambda _l)$, $l\ge 0$, $\lambda _i\in \mathbb {Z}$, such that $\lambda _1\ge \cdots \ge \lambda _l\ge 0$. We also denote $l(\lambda ):=l$ and $|\lambda |:=\sum \lambda _i$. Then we can express the power series f in the following way:

$$\begin{aligned} f=\sum _{\lambda ^1,\ldots ,\lambda ^N\in \mathcal {P}}a_{\lambda ^1,\ldots ,\lambda ^N}\widehat{t}^1_{\lambda ^1}\cdots \widehat{t}^N_{\lambda ^N},\quad a_{\lambda ^1,\ldots ,\lambda ^N}\in \mathbb {C}, \end{aligned}$$

where for an element $\lambda =(\lambda _1,\ldots ,\lambda _l)\in \mathcal {P}$ we denote $\widehat{t}^\alpha _\lambda :=\widehat{t}^\alpha _{\lambda _1}\cdots \widehat{t}^\alpha _{\lambda _l}$. Note that if we assign to a monomial $\widehat{t}^1_{\lambda ^1}\cdots \widehat{t}^N_{\lambda ^N}$ degree $\sum |\lambda ^i|$, then the operator $\sum _{d\ge 0}\widehat{t}^\alpha _{d+1}\frac{{\partial }}{{\partial }\widehat{t}^\alpha _d}$ becomes homogeneous of degree 1. Denote by $V_d$ the subspace of $\mathbb {C}[[\widehat{t}^*_*]]$ spanned by monomials of degree d. Using the lexicographical order on monomials $\widehat{t}^1_{\lambda ^1}\cdots \widehat{t}^N_{\lambda ^N}$, it is easy to check that the map $\sum _{k\ge 0}\widehat{t}^\alpha _{k+1}\frac{{\partial }}{{\partial }\widehat{t}^\alpha _k}:V_d\rightarrow V_{d+1}$ is injective for $d\ge 1$; and for $d=0$ its kernel consists of constants. This completes the proof of the lemma. $\square$

4.6 Proof of Theorem 3.4

Let us apply Proposition 4.14 to the flat F-manifold associated to the function $F^o$. We choose $\lambda =\frac{3-\delta }{2}$, then we have

$$\begin{aligned} A^{N+1}_m=\sum _{d_1\ge -1,\,d_2\in \mathbb {Z}}(-1)^{d_2+1}\left( [P_m(\widetilde{\mu }-d_2,\widetilde{R})]_{m-1-d_1-d_2}\right) ^\gamma _\mu \Phi ^{\mathrm {top}}_{\gamma ,d_1}{\mathcal {F}}^{\mu ,d_2}. \end{aligned}$$

Since $\widetilde{R}^\alpha _{N+1}=0$, we have $\left( [P_m(\widetilde{\mu }-d_2,\widetilde{R})]_{m-1-d_1-d_2}\right) ^\gamma _{N+1}=0$ for $1\le \gamma \le N$. Therefore,

$$\begin{aligned} A^{N+1}_m=\,&\sum _{d_1\ge 0,\,d_2\in \mathbb {Z}}\sum _{1\le \gamma ,\mu \le N}(-1)^{d_2+1}\left( [P_m(\widetilde{\mu }-d_2,\widetilde{R})]_{m-1-d_1-d_2}\right) ^\gamma _\mu \frac{{\partial }{\mathcal {F}}^o}{{\partial }t^\gamma _{d_1}}{\mathcal {F}}^{\mu ,d_2}\\&+\sum _{d\ge 0}\left( [P_m(\widetilde{\mu }+d+1,\widetilde{R})]_{m+d+1}\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d\\&+\sum _{d,k\ge 0}\left( [P_m(\widetilde{\mu }+d+1,\widetilde{R})]_{m+d-k}\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d\frac{{\partial }{\mathcal {F}}^o}{{\partial }s_k}\\&+\sum _{k\ge 0}\sum _{1\le \alpha ,\mu \le N}(-1)^{k+1}\left( [P_m(\widetilde{\mu }-k,\widetilde{R})]_{m-k}\right) ^{N+1}_\alpha \eta ^{\alpha \mu }\frac{{\partial }{\mathcal {F}}}{{\partial }t^\mu _k}\\&+\sum _{d_1,d_2\ge 0}\sum _{1\le \alpha ,\mu \le N}(-1)^{d_2+1}\left( [P_m(\widetilde{\mu }-d_2,\widetilde{R})]_{m-1-d_1-d_2}\right) ^{N+1}_\alpha \frac{{\partial }{\mathcal {F}}^o}{{\partial }s_{d_1}}\eta ^{\alpha \mu }\frac{{\partial }{\mathcal {F}}}{{\partial }t^\mu _{d_2}}\\&+\sum _{d_1\ge -1,\,d_2\ge 0}(-1)^{d_2+1}\left( [P_m(\widetilde{\mu }-d_2,\widetilde{R})]_{m-1-d_1-d_2}\right) ^{N+1}_{N+1}\Phi ^{\mathrm {top}}_{N+1,d_1}{\mathcal {F}}^{N+1,d_2}. \end{aligned}$$

The first term here is equal to ${\mathrm {Coef}}_{\varepsilon ^{-1}}\left( \frac{L_m e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}{e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}\right)$. The next four terms correspond to the four summations in the expression (3.5) for the operators $\mathcal {L}_m$. Since $\mu ^{N+1}=\frac{1}{2}$, the last term is equal to zero. Thus,

$$\begin{aligned} A^{N+1}_m={\mathrm {Coef}}_{\varepsilon ^{-1}}\left( \frac{\mathcal {L}_m e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}{e^{\varepsilon ^{-2}{\mathcal {F}}+\varepsilon ^{-1}{\mathcal {F}}^o}}\right) =0, \end{aligned}$$

that proves Theorem 3.4.

5 Open Virasoro Constraints in All Genera

There is a canonical construction that associates to a given semisimple conformal Frobenius manifold and its calibration a sequence of functions ${\mathcal {F}}_0(t^*_*)={\mathcal {F}},{\mathcal {F}}_1(t^*_*),{\mathcal {F}}_2(t^*_*),\ldots$, such that for the differential operators $L_m$, given by (2.3), Eq. (1.5) hold (Givental 2001, 2004; Teleman 2012). If one considers the Gromov–Witten theory of a given target variety, then the functions ${\mathcal {F}}_g(t^*_*)$ are the generating series of intersection numbers on the moduli space of maps from a Riemann surface of genus g to the target variety.

We conjecture that, under possibly some additional assumptions, there is a canonical way to associate to a solution $F^o$ of the WDVV equations, satisfying properties (1.8), (3.1), and its calibration a sequence of functions ${\mathcal {F}}^o_0(t^{\le N}_*,s_*)={\mathcal {F}}^o,{\mathcal {F}}_1(t^{\le N}_*,s_*),{\mathcal {F}}_2(t^{\le N}_*,s_*),\ldots$, such that for the differential operators $\mathcal {L}_m$, given by (3.5), the equations

$$\begin{aligned} \mathcal {L}_m e^{\sum _{g\ge 0}\varepsilon ^{2g-2}{\mathcal {F}}_g+\sum _{g\ge 0}\varepsilon ^{g-1}{\mathcal {F}}^o_g}=0,\quad m\ge -1, \end{aligned}$$

are satisfied. At the moment the conjecture is verified only in the case, corresponding to the intersection theory on the moduli space of Riemann surfaces with boundary (Pandharipande et al. 2014; Buryak and Tessler 2017).

As a step towards the proof of this conjecture, we verify the following commutation relations between the operators $\mathcal {L}_m$.

Proposition 5.1

We have $[\mathcal {L}_m,\mathcal {L}_n]=(m-n)\mathcal {L}_{m+n}$, $m,n\ge -1$.

Proof

Denote the parts of the expression for the operator $L_m$ from lines (2.3), (2.4) and (2.5) by $L_m^1$, $L_m^2$ and $L_m^3$, correspondingly. One can see that the operators $\mathcal {L}_m$ can be written in the following way:

$$\begin{aligned} \mathcal {L}_m= & {} L_m+\mathcal {L}_m^1+\mathcal {L}_m^2+\mathcal {L}_m^3+\mathcal {L}_m^4+\varepsilon ^{-1}\delta _{m,-1}s +\delta _{m,0}\frac{3}{4}\\&+\sum _{d\ge 0}\frac{(d+m+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+m}}+\varepsilon \frac{3(m+1)!}{4}\frac{{\partial }}{{\partial }s_{m-1}}, \end{aligned}$$

where

$$\begin{aligned} \mathcal {L}_m^1&=\varepsilon ^{-1}\sum _{d\ge 0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =-\frac{1}{2}-m-d \end{array}}\left( \prod _{i=0}^m(\widetilde{R}-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d,\\ \mathcal {L}_m^2&=\sum _{d,k\ge 0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =\frac{1}{2}-m-d+k \end{array}}\left( \prod _{i=0}^m(\widetilde{R}+k+1-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d\frac{{\partial }}{{\partial }s_k},\\ \mathcal {L}_m^3&=\varepsilon \sum _{k\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\mu \le N\\ \mu ^\alpha =\frac{1}{2}-m+k \end{array}}(-1)^{k+1}\left( \prod _{i=0}^m(\widetilde{R}-i)\right) ^{N+1}_\alpha \eta ^{\alpha \mu }\frac{{\partial }}{{\partial }t^\mu _k},\\ \mathcal {L}_m^4&=\varepsilon ^2\sum _{d_1,d_2\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\mu \le N\\ \mu ^\alpha =\frac{3}{2}-m+d_1+d_2 \end{array}}(-1)^{d_2+1}\left( \prod _{i=0}^m(\widetilde{R}+d_1+1-i)\right) ^{N+1}_\alpha \eta ^{\alpha \mu }\frac{{\partial }^2}{{\partial }s_{d_1}{\partial }t^\mu _{d_2}}. \end{aligned}$$

Let us first prove that $[\mathcal {L}_{-1},\mathcal {L}_n]=(-1-n)\mathcal {L}_{n-1}$, for $n\ge 0$. For this we compute

$$\begin{aligned}&{[}\mathcal {L}_{-1},L_n]=(-1-n)L_{n-1}, \quad {[}\mathcal {L}_{-1},\mathcal {L}^1_n]=\varepsilon ^{-1}\sum _{d\ge 1}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =\frac{1}{2}-n-d \end{array}} \left( \prod _{i=0}^n(\widetilde{R}-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d, \end{aligned}$$

(5.1)

$$\begin{aligned}&{[}\mathcal {L}_{-1},\mathcal {L}^2_n]=\sum _{d\ge 1,\,k\ge 0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =\frac{3}{2}-n-d+k \end{array}} \left( \prod _{i=0}^n(\widetilde{R}+k+1-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d\frac{{\partial }}{{\partial }s_k} \end{aligned}$$

(5.2)

$$\begin{aligned}&\qquad \qquad \qquad -\varepsilon ^{-1}\sum _{d\ge 0} \sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =\frac{1}{2}-n-d \end{array}} \left( \prod _{i=0}^n(\widetilde{R}+1-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d \end{aligned}$$

(5.3)

$$\begin{aligned}&\qquad \qquad \qquad -\sum _{d,k\ge 0} \sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =\frac{3}{2}-n-d+k \end{array}} \left( \prod _{i=0}^n(\widetilde{R}+k+2-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d\frac{{\partial }}{{\partial }s_k}, \end{aligned}$$

(5.4)

$$\begin{aligned}&[\mathcal {L}_{-1},\mathcal {L}^3_n]=\varepsilon ^{-1} \sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =\frac{1}{2}-n \end{array}} \left( \prod _{i=0}^n(\widetilde{R}-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _0 \end{aligned}$$

(5.5)

$$\begin{aligned}&\qquad \qquad \qquad +\varepsilon \sum _{k\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\mu \le N\\ \mu ^\alpha =\frac{3}{2}-n+k \end{array}}(-1)^{k+1} \left( \prod _{i=0}^n(\widetilde{R}-i)\right) ^{N+1}_\alpha \eta ^{\alpha \mu } \frac{{\partial }}{{\partial }t^\mu _k}, \end{aligned}$$

(5.6)

$$\begin{aligned}&{[}\mathcal {L}_{-1},\mathcal {L}^4_n]=\varepsilon ^2\sum _{d_1,d_2\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\mu \le N\\ \mu ^\alpha =\frac{5}{2}-n+d_1+d_2 \end{array}}(-1)^{d_2+1} \left( \prod _{i=0}^n(\widetilde{R}+d_1+1-i)\right) ^{N+1}_\alpha \eta ^{\alpha \mu }\frac{{\partial }^2}{{\partial }s_{d_1}{\partial }t^\mu _{d_2}} \end{aligned}$$

(5.7)

$$\begin{aligned}&\qquad \qquad \qquad +\sum _{d\ge 0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha = \frac{3}{2}-n+d \end{array}}\left( \prod _{i=0}^n(\widetilde{R}+d+1-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _0\frac{{\partial }}{{\partial }s_d} \end{aligned}$$

(5.8)

$$\begin{aligned}&\qquad \qquad \qquad -\varepsilon \sum _{d\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\mu \le N\\ \mu ^\alpha =\frac{3}{2}-n+d \end{array}}(-1)^{d+1}\left( \prod _{i=0}^n(\widetilde{R}+1-i)\right) ^{N+1}_\alpha \eta ^{\alpha \mu }\frac{{\partial }}{{\partial }t^\mu _d}\end{aligned}$$

(5.9)

$$\begin{aligned}&\qquad \qquad \qquad -\varepsilon ^2\sum _{d_1,d_2\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\mu \le N\\ \mu ^\alpha =\frac{5}{2}-n+d_1+d_2 \end{array}}(-1)^{d_2+1}\left( \prod _{i=0}^n (\widetilde{R}+d_1+2-i)\right) ^{N+1}_\alpha \eta ^{\alpha \mu }\frac{{\partial }^2}{{\partial }s_{d_1} {\partial }t^\mu _{d_2}}, \end{aligned}$$

(5.10)

and

$$\begin{aligned} \left[ \mathcal {L}_{-1},\sum _{d\ge 0}\frac{(d+n+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+n}}\right] =\,&-\delta _{n,0}\varepsilon ^{-1}s+(-1-n)\sum _{d\ge 0}\frac{(d+n)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+n-1}},\\ \left[ \mathcal {L}_{-1},\varepsilon \frac{3(n+1)!}{4}\frac{{\partial }}{{\partial }s_{n-1}}\right] =\,&-\delta _{n,1}\frac{3}{2}-\varepsilon \frac{3(n+1)!}{4}\frac{{\partial }}{{\partial }s_{n-2}}. \end{aligned}$$

It remains to note that the sum of the expressions in lines (5.1), (5.3), (5.5) is equal to ${(-1-n)\mathcal {L}^1_{n-1}}$, the sum of the expressions in lines (5.2), (5.4), (5.8) is equal to $(-1-n)\mathcal {L}^2_{n-1}$, the sum of the expressions in lines (5.6), (5.9) is equal to $(-1-n)\mathcal {L}^3_{n-1}$ and the sum of the expressions in lines (5.7), (5.10) is equal to $(-1-n)\mathcal {L}^4_{n-1}$.

Let us now prove the proposition for $m,n\ge 0$. The commutator $[\mathcal {L}_m,\mathcal {L}_n]$ has the form

$$\begin{aligned} {[}\mathcal {L}_m,\mathcal {L}_n]=\,&(m-n)L_{m+n}+\varepsilon ^{-1}\sum _{d\ge 0}\sum _{1\le \alpha \le N}A_{\alpha ,d}\widetilde{t}^\alpha _d\\&+\left( \sum _{p,q\ge 0}\sum _{1\le \alpha \le N}B_{\alpha ,p}^q\widetilde{t}^\alpha _p\frac{{\partial }}{{\partial }s_q}+\sum _{p,q\ge 0}C_p^q s_p\frac{{\partial }}{{\partial }s_q}+D\right) \\&+\varepsilon \left( \sum _{d\ge 0}\sum _{1\le \alpha \le N}E^{\alpha ,d}\frac{{\partial }}{{\partial }t^\alpha _d}+\sum _{d\ge 0}G^d \frac{{\partial }}{{\partial }s_d}\right) \\&+\varepsilon ^2\left( \sum _{p,q\ge 0}\sum _{1\le \alpha \le N}H^{\alpha ,p;q}\frac{{\partial }}{{\partial }t^\alpha _p{\partial }s_q}+\sum _{p,q\ge 0}I^{p;q}\frac{{\partial }}{{\partial }s_p{\partial }s_q}\right) . \end{aligned}$$

Let us consider separately the terms on the right-hand side of this expression.

1. Term$\varepsilon ^{-1}\sum _{d\ge 0}\sum _{1\le \alpha \le N}A_{\alpha ,d}\widetilde{t}^\alpha _d$:

$$\begin{aligned} \varepsilon ^{-1}\sum _{d\ge 0}\sum _{1\le \alpha \le N}A_{\alpha ,d}\widetilde{t}^\alpha _d=[L_m^2,\mathcal {L}_n^1]+[L_m^1,\mathcal {L}_n^3]-[L_n^2,\mathcal {L}_m^1]-[L_n^1,\mathcal {L}_m^3]. \end{aligned}$$

We compute

$$\begin{aligned} {[}L_m^2,\mathcal {L}_n^1]=\varepsilon ^{-1}\sum _{d,k\ge 0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =-\frac{1}{2}-m-n-d \end{array}}\sum _{\begin{array}{c} 1\le \beta \le N\\ \mu ^\beta =-\frac{1}{2}-n-k \end{array}}\left( \prod _{i=0}^m(R-n-i)\right) ^\beta _\alpha \left( \prod _{i=0}^n(\widetilde{R}-i)\right) ^{N+1}_\beta \widetilde{t}^\alpha _d. \end{aligned}$$

Similarly, we get

$$\begin{aligned} {[}L_m^1,\mathcal {L}_n^3]=\varepsilon ^{-1}\sum _{d\ge 0,\,k<0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =-\frac{1}{2}-m-n-d \end{array}}\sum _{\begin{array}{c} 1\le \beta \le N\\ \mu ^\beta =-\frac{1}{2}-n-k \end{array}}\left( \prod _{i=0}^m(R-n-i)\right) ^\beta _\alpha \left( \prod _{i=0}^n(\widetilde{R}-i)\right) ^{N+1}_\beta \widetilde{t}^\alpha _d. \end{aligned}$$

As a result,

$$\begin{aligned} {[}L_m^2,\mathcal {L}_n^1]+[L_m^1,\mathcal {L}_n^3]=\varepsilon ^{-1}\sum _{d\ge 0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =-\frac{1}{2}-m-n-d \end{array}}\left( (\widetilde{R}-n)\prod _{i=0}^{m+n}(\widetilde{R}-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d, \end{aligned}$$

which finally gives

$$\begin{aligned}&{[}L_m^2,\mathcal {L}_n^1]+[L_m^1,\mathcal {L}_n^3]-[L_n^2,\mathcal {L}_m^1]-[L_n^1,\mathcal {L}_m^3]\\&\quad =(m-n)\varepsilon ^{-1}\sum _{d\ge 0} \sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =-\frac{1}{2}-m-n-d \end{array}} \left( \prod _{i=0}^{m+n}(\widetilde{R}-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d=(m-n)\mathcal {L}_{m+n}^1, \end{aligned}$$

as required.

2. Term$\sum _{p,q\ge 0}\sum _{1\le \alpha \le N}B_{\alpha ,p}^q\widetilde{t}^\alpha _p\frac{{\partial }}{{\partial }s_q}$:

$$\begin{aligned} \sum _{p,q\ge 0}\sum _{1\le \alpha \le N}B_{\alpha ,p}^q\widetilde{t}^\alpha _p\frac{{\partial }}{{\partial }s_q}=\,&[L_m^2,\mathcal {L}_n^2]+\left[ \mathcal {L}_m^2,\sum _{d\ge 0}\frac{(d+n+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+n}}\right] \\&+[L_m^1,\mathcal {L}_n^4]-\\&-[L_n^2,\mathcal {L}_m^2]-\left[ \mathcal {L}_n^2,\sum _{d\ge 0}\frac{(d+m+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+m}}\right] \\&-[L_n^1,\mathcal {L}_m^4]. \end{aligned}$$

We compute

$$\begin{aligned}&{[}L_m^2,\mathcal {L}_n^2]+\left[ \mathcal {L}_m^2,\sum _{d\ge 0}\frac{(d+n+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+n}}\right] \\&\quad =\sum _{d,k,l\ge 0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =\frac{1}{2}-m-n-d+k \end{array}}\sum _{\begin{array}{c} 1\le \beta \le N+1\\ \mu ^\beta =\frac{1}{2}-n+k-l \end{array}}\\&\qquad \times \left( \prod _{i=0}^m(\widetilde{R}+k+1-n-i)\right) ^\beta _\alpha \left( \prod _{i=0}^n(\widetilde{R}+k+1-i)\right) ^{N+1}_\beta \widetilde{t}^\alpha _d\frac{{\partial }}{{\partial }s_k}, \end{aligned}$$

and then check that this sum, after replacing the summation $\sum _{l\ge 0}$ by the summation $\sum _{l<0}$, is equal to the commutator $[L_m^1,\mathcal {L}_n^4]$. This gives

$$\begin{aligned}&{[}L_m^2,\mathcal {L}_n^2]+\left[ \mathcal {L}_m^2,\sum _{d\ge 0}\frac{(d+n+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+n}}\right] +[L_m^1,\mathcal {L}_n^4]\\&\quad =\sum _{d,k\ge 0}\sum _{\begin{array}{c} 1\le \alpha \le N\\ \mu ^\alpha =\frac{1}{2}-m-n-d+k \end{array}}\left( (\widetilde{R}+k+1-n)\prod _{i=0}^{m+n}(\widetilde{R}+k+1-i)\right) ^{N+1}_\alpha \widetilde{t}^\alpha _d\frac{{\partial }}{{\partial }s_k}. \end{aligned}$$

As a result, $\sum _{p,q\ge 0}\sum _{1\le \alpha \le N}B_{\alpha ,p}^q\widetilde{t}^\alpha _p\frac{{\partial }}{{\partial }s_q}=(m-n)\mathcal {L}_{m+n}^2$, as required.

3. Term$\sum _{p,q\ge 0}C_p^q s_p\frac{{\partial }}{{\partial }s_q}$:

$$\begin{aligned} \sum _{p,q\ge 0}C_p^q s_p\frac{{\partial }}{{\partial }s_q}&=\left[ \sum _{p\ge 0}\frac{(p+m+1)!}{p!}s_p\frac{{\partial }}{{\partial }s_{p+m}},\sum _{q\ge 0}\frac{(q+n+1)!}{q!}s_q\frac{{\partial }}{{\partial }s_{q+n}}\right] \\&=(m-n)\sum _{d\ge 0}\frac{(d+m+n+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+m+n}}, \end{aligned}$$

as required.

4. ConstantD:

$$\begin{aligned} D=\,&[\mathcal {L}_m^1,\mathcal {L}_n^3]-[\mathcal {L}_n^1,\mathcal {L}_m^3],\quad \text {where}\\ [\mathcal {L}_m^1,\mathcal {L}_n^3]=\,&\sum _{d\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\beta \le N\\ \mu ^\alpha =-\frac{1}{2}-m-d\\ \mu ^\beta =\frac{1}{2}-n+d \end{array}}\left( \prod _{i=0}^m(\widetilde{R}-i)\right) _\alpha ^{N+1} \eta ^{\alpha \beta }\left( \prod _{i=0}^n(\widetilde{R}-i)\right) _\beta ^{N+1}. \end{aligned}$$

Because of the property $\mu \eta +\eta \mu =0$, the last expression is equal to zero unless $m+n=0$, which implies that $m=n=0$. Thus, $D=0$, as required.

5. Term$\varepsilon \sum _{d\ge 0}\sum _{1\le \alpha \le N}E^{\alpha ,d}\frac{{\partial }}{{\partial }t^\alpha _d}$:

$$\begin{aligned} \varepsilon \sum _{d\ge 0}\sum _{1\le \alpha \le N}E^{\alpha ,d}\frac{{\partial }}{{\partial }t^\alpha _d}=[L_m^2,\mathcal {L}_n^3]+[L_m^3,\mathcal {L}_n^1]-[L_n^2,\mathcal {L}_m^3]-[L_n^3,\mathcal {L}_m^1]. \end{aligned}$$

We first compute

$$\begin{aligned} {[}L_m^2,\mathcal {L}_n^3]=\,&\varepsilon \sum _{d,k\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\beta ,\gamma \le N\\ \mu ^\beta =m+n-k-\frac{1}{2}\\ \mu ^\alpha =-\frac{1}{2}+n-d \end{array}}(-1)^d\left( \prod _{i=0}^n(\widetilde{R}-i)\right) ^{N+1}_\gamma \eta ^{\gamma \alpha }\left( \prod _{i=0}^m(R+n+i)\right) ^\beta _\alpha \frac{{\partial }}{{\partial }t^\beta _k}\nonumber \\ =\,&\varepsilon \sum _{d,k\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\beta ,\gamma \le N\\ \mu ^\beta =m+n-k-\frac{1}{2}\\ \mu ^\gamma =\frac{1}{2}-n+d \end{array}}(-1)^{k+1}\left( \prod _{i=0}^n(\widetilde{R}-i)\right) ^{N+1}_\gamma \left( \prod _{i=0}^m(R-n-i)\right) ^\gamma _\alpha \eta ^{\alpha \beta }\frac{{\partial }}{{\partial }t^\beta _k}, \end{aligned}$$

(5.11)

where the second equality is obtained using the property $\eta R_i\eta ^{-1}=(-1)^{i-1}R_i^T$. Then one can compute that the commutator $[L_m^3,\mathcal {L}_n^1]$ is equal to the expression in line (5.11) with the summation $\sum _{d\ge 0}$ replaced by $\sum _{d<0}$. This implies that

$$\begin{aligned}&[L_m^2,\mathcal {L}_n^3]+[L_m^3,\mathcal {L}_n^1]=\varepsilon \sum _{k\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\beta \le N\\ \mu ^\beta =m+n-k-\frac{1}{2} \end{array}}(-1)^{k+1}\\&\quad \left( \prod _{i=0}^n(\widetilde{R}-i)\prod _{i=0}^m(\widetilde{R}-n-i)\right) ^{N+1}_\alpha \eta ^{\alpha \beta }\frac{{\partial }}{{\partial }t^\beta _k}, \end{aligned}$$

and, as a result,

$$\begin{aligned}&{[}L_m^2,\mathcal {L}_n^3]+[L_m^3,\mathcal {L}_n^1]-[L_n^2,\mathcal {L}_m^3]-[L_n^3,\mathcal {L}_m^1]\\&\quad =(m-n)\varepsilon \sum _{k\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\beta \le N\\ \mu ^\beta =m+n-k-\frac{1}{2} \end{array}}(-1)^{k+1} \left( \prod _{i=0}^{m+n}(\widetilde{R}-i)\right) ^{N+1}_\alpha \eta ^{\alpha \beta } \frac{{\partial }}{{\partial }t^\beta _k}=(m-n)\mathcal {L}_{m+n}^3, \end{aligned}$$

as required.

6. Term$\varepsilon \sum _{d\ge 0}G^d\frac{{\partial }}{{\partial }s_d}$:

$$\begin{aligned} \varepsilon \sum _{d\ge 0}G^d\frac{{\partial }}{{\partial }s_d}= & {} (m-n)\varepsilon \frac{3(m+n+1)!}{4}\frac{{\partial }}{{\partial }s_{m+n-1}}\\&+\,[\mathcal {L}_m^2,\mathcal {L}_n^3]+[\mathcal {L}_m^1,\mathcal {L}_n^4] -[\mathcal {L}_n^2,\mathcal {L}_m^3]-[\mathcal {L}_n^1,\mathcal {L}_m^4]. \end{aligned}$$

We compute

$$\begin{aligned} {[}\mathcal {L}_m^2,\mathcal {L}_n^3]= & {} \varepsilon \sum _{d\ge 0}(-1)^d\sum _{\begin{array}{c} 1\le \alpha ,\beta \le N\\ \mu ^\alpha =-\frac{1}{2}+n-d\\ \mu ^\beta =\frac{1}{2}-n+d \end{array}}\\&\times \left( \prod _{i=0}^m(\widetilde{R}+m+n-i)\right) ^{N+1}_\alpha \eta ^{\alpha \beta } \left( \prod _{i=0}^n(\widetilde{R}-i)\right) ^{N+1}_\beta \frac{{\partial }}{{\partial }s_{m+n-1}}. \end{aligned}$$

A term in this sum is equal to zero unless $\mu ^\alpha <\frac{1}{2}\Leftrightarrow d>n-1$ and $\mu ^\beta<\frac{1}{2}\Leftrightarrow d<n$, that never happens. Therefore, $[\mathcal {L}_m^2,\mathcal {L}_n^3]=0$, and, similarly, one can check that $[\mathcal {L}_m^1,\mathcal {L}_n^4]=[\mathcal {L}_n^2,\mathcal {L}_m^3]=[\mathcal {L}_n^1,\mathcal {L}_m^4]=0$. Hence, $\varepsilon \sum _{d\ge 0}G^d\frac{{\partial }}{{\partial }s_d}=(m-n)\varepsilon \frac{3(m+n+1)!}{4}\frac{{\partial }}{{\partial }s_{m+n-1}}$, as required.

7. Term$\varepsilon ^2\sum _{p,q\ge 0}\sum _{1\le \alpha \le N}H^{\alpha ,p;q}\frac{{\partial }}{{\partial }t^\alpha _p{\partial }s_q}$:

$$\begin{aligned}&\varepsilon ^2\sum _{p,q\ge 0}\sum _{1\le \alpha \le N}H^{\alpha ,p;q}\frac{{\partial }}{{\partial }t^\alpha _p{\partial }s_q}\\&\quad =[L_m^2,\mathcal {L}_n^4]+[L_m^3,\mathcal {L}_n^2]+\left[ \mathcal {L}_m^4,\sum _{d\ge 0}\frac{(d+n+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+n}}\right] \\&\qquad -[L_n^2,\mathcal {L}_m^4]-[L_n^3,\mathcal {L}_m^2]-\left[ \mathcal {L}_n^4,\sum _{d\ge 0}\frac{(d+m+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+m}}\right] . \end{aligned}$$

We proceed with the computation

$$\begin{aligned} {[}L_m^2,\mathcal {L}_n^4]&=\varepsilon ^2\sum _{d_1,d_2,d\ge 0}(-1)^d\sum _{\begin{array}{c} 1\le \alpha ,\beta ,\mu \le N\\ \mu ^\alpha =\frac{3}{2}-n+d_1+d\\ \mu ^\beta =-\frac{3}{2}+m+n-d_1-d_2 \end{array}} \left( \prod _{i=0}^n(\widetilde{R}+d_1+1-i)\right) ^{N+1}_\alpha \\&\quad \times \eta ^{\alpha \mu }\left( \prod _{i=0}^m(R-1-d_1+n+i)\right) ^\beta _\mu \frac{{\partial }^2}{{\partial }s_{d_1}{\partial }t^\beta _{d_2}}. \end{aligned}$$

Using the relation $\eta R_i\eta ^{-1}=(-1)^{i-1}R_i^T$, we convert this sum to

$$\begin{aligned}&\varepsilon ^2\sum _{d_1,d_2,d\ge 0}(-1)^{d_2+1}\sum _{\begin{array}{c} 1\le \alpha \le N+1\\ \mu ^\alpha =\frac{3}{2}-n+d_1+d\\ \mu ^\beta =-\frac{3}{2}+m+n-d_1-d_2 \end{array}}\nonumber \\&\quad \times \left( \prod _{i=0}^n(\widetilde{R}+d_1+1-i)\right) ^{N+1}_\alpha \left( \prod _{i=0}^m(\widetilde{R}+1+d_1-n-i)\right) ^\alpha _\mu \eta ^{\mu \beta }\frac{{\partial }^2}{{\partial }s_{d_1}{\partial }t^\beta _{d_2}}.\nonumber \\ \end{aligned}$$

(5.12)

Then one can check that the expression $[L_m^3,\mathcal {L}_n^2]+\left[ \mathcal {L}_m^4,\sum _{d\ge 0}\frac{(d+n+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+n}}\right]$ is equal to the expression (5.12), with the summation $\sum _{d\ge 0}$ replaced by $\sum _{d<0}$. As a result,

$$\begin{aligned}&{[}L_m^2,\mathcal {L}_n^4]+[L_m^3,\mathcal {L}_n^2]+\left[ \mathcal {L}_m^4,\sum _{d\ge 0}\frac{(d+n+1)!}{d!}s_d\frac{{\partial }}{{\partial }s_{d+n}}\right] \\&\quad =\varepsilon ^2\sum _{d_1,d_2\ge 0}(-1)^{d_2+1}\sum _{\mu ^\beta =-\frac{3}{2}+m+n-d_1-d_2} \left( (\widetilde{R}+d_1+1-n)\prod _{i=0}^{m+n}(\widetilde{R}+d_1+1-i)\right) ^{N+1}_\mu \\&\qquad \times \eta ^{\mu \beta }\frac{{\partial }^2}{{\partial }s_{d_1}{\partial }t^\beta _{d_2}}, \end{aligned}$$

which gives $\varepsilon ^2\sum _{p,q\ge 0}\sum _{1\le \alpha \le N}H^{\alpha ,p;q}\frac{{\partial }}{{\partial }t^\alpha _p{\partial }s_q}=(m-n)\mathcal {L}_{m+n}^4$, as required.

8. Term$\varepsilon ^2\sum _{p,q\ge 0}I^{p;q}\frac{{\partial }^2}{{\partial }s_p{\partial }s_q}$:

$$\begin{aligned} \varepsilon ^2\sum _{p,q\ge 0}I^{p;q}\frac{{\partial }^2}{{\partial }s_p{\partial }s_q}=[\mathcal {L}_m^4,\mathcal {L}_n^2]-[\mathcal {L}_n^4,\mathcal {L}_m^2], \end{aligned}$$

where

$$\begin{aligned} {[}\mathcal {L}_m^4,\mathcal {L}_n^2]= & {} \varepsilon ^2\sum _{p,q,d\ge 0}\sum _{\begin{array}{c} 1\le \alpha ,\beta \le N\\ \mu ^\alpha =\frac{3}{2}-m+p+d\\ \mu ^\beta =\frac{1}{2}-n+q-d \end{array}}(-1)^{d+1} \left( \prod _{i=0}^m(\widetilde{R}+p+1-i)\right) ^{N+1}_\alpha \\&\times \eta ^{\alpha \beta }\left( \prod _{i=0}^n(\widetilde{R}+q+1-i)\right) ^{N+1}_\beta \frac{{\partial }^2}{{\partial }s_p{\partial }s_q}. \end{aligned}$$

A term in this sum is equal to zero unless $\mu ^\alpha <\frac{1}{2}\Leftrightarrow p+d\le m-2$, $\mu ^\beta <\frac{1}{2}\Leftrightarrow q-d\le n-1$ and $\mu ^\alpha +\mu ^\beta =0\Leftrightarrow p+q=m+n-2$. The first two condition give $p+q\le m+n-3$, that contradicts the third condition. Thus, $\varepsilon ^2\sum _{p,q\ge 0}I^{p;q}\frac{{\partial }^2}{{\partial }s_p{\partial }s_q}=0$, as required. This completes the proof of the proposition. $\square$

6 Examples of Solutions of the Open WDVV Equations

In this section we present several examples of solutions of the open WDVV equations, satisfying conditions (1.8) and (3.1) and, thus, Theorem 3.4 can be applied to them.

6.1 Extended r-Spin Theory

Let us fix an integer $r\ge 2$. There is a conformal Frobenius manifold that controls the integrals of the so-called Witten class over the moduli space of stable curves of genus 0 with an r-spin structure. This Frobenius manifold has dimension $r-1$ and is described by a potential $F^{r\text {-spin}}(t^1,\ldots ,t^{r-1})$ with the metric $\eta$, given by $\eta _{\alpha \beta }=\delta _{\alpha +\beta ,r}$, and the Euler vector field

$$\begin{aligned} E=\sum _{\alpha =1}^{r-1}\frac{r+1-\alpha }{r}t^\alpha \frac{{\partial }}{{\partial }t^\alpha },\qquad \sum _{\alpha =1}^{r-1}E^\alpha \frac{{\partial }F^r\text {-spin}}{{\partial }t^\alpha }=\frac{2r+2}{r}F^r\text {-spin}. \end{aligned}$$

The conformal dimension is $\delta =\frac{r-2}{r}$. The potential $F^{r\text {-spin}}$ is a polynomial in the variables $t^1,\ldots ,t^{r-1}$. For more details, we refer a reader, for example, to Pandharipande et al. (2019), Buryak et al. (2019).

The generating series of the descendent integrals with Witten’s class over the moduli space of curves of genus 0 with an r-spin structure gives the descendent potential ${\mathcal {F}}^{r\text {-spin}}(t^*_*)$, corresponding to our Frobenius potential $F^{r\text {-spin}}$. This descendent potential corresponds to a calibration with all the matrices $R_i$ being zero. Thus, the Virasoro operators $L_m$ are given by

$$\begin{aligned} L_m=\,&\sum _{\begin{array}{c} 1\le \alpha \le r-1\\ a\ge 0 \end{array}}\frac{(\alpha +a r)^{\overline{m+1}}}{r^{m+1}}(t^\alpha _a-\delta ^{\alpha ,1}\delta _{a,1}) \frac{{\partial }}{{\partial }t^\alpha _{a+m}}+\frac{\varepsilon ^2}{2}\sum _{\begin{array}{c} \alpha +\beta =r\\ a+b=m-1 \end{array}} \frac{\alpha ^{\overline{a+1}}\beta ^{\overline{b+1}}}{r^{m+1}}\frac{{\partial }^2}{{\partial }t^\alpha _a{\partial }t^\beta _b}\\&+\delta _{m,-1}\frac{1}{2\varepsilon ^2}\sum _{\alpha +\beta =r}t^\alpha _0t^\beta _0 +\delta _{m,0}\frac{r^2-1}{24r}, \end{aligned}$$

where we use the notation

$$\begin{aligned} x^{\overline{n}}:={\left\{ \begin{array}{ll} \prod _{i=1}^n(x+i-1),&{}\quad \text {if } n\ge 1,\\ 1,&{}\quad \text {if } n=0, \end{array}\right. } \end{aligned}$$

for a complex number x and an integer $n\ge 0$.

In Jarvis et al. (2001) the authors considered a generalization of the r-spin theory, that we call the extended r-spin theory, and in Buryak et al. (2019) the authors noticed (see Remark 1.1) that such a generalization produces a solution $F^{{\mathrm {ext}}}(t^1,\ldots ,t^r)$ of the open WDVV equations, satisfying condition (1.8) and the homogeneity condition

$$\begin{aligned} \sum _{\alpha =1}^{r-1}E^\alpha \frac{{\partial }F^{{\mathrm {ext}}}}{{\partial }t^\alpha }+\frac{1-\delta }{2}t^r\frac{{\partial }F^{{\mathrm {ext}}}}{{\partial }t^r}=\frac{3-\delta }{2}F^{{\mathrm {ext}}}. \end{aligned}$$

Recall that we identify $t^r=s$. Note that the function $F^{{\mathrm {ext}}}$ also controls the open r-spin theory, constructed in [BCT18]. The generating series ${\mathcal {F}}^{{\mathrm {ext}}}(t^*_*)$ of the descendent integrals in the extended r-spin theory is the open descendent potential, corresponding to the function $F^{{\mathrm {ext}}}$ and a calibration with all the matrices $\widetilde{R}_i$ being zero. Thus, the associated open Virasoro operators $\mathcal {L}_m$ are

$$\begin{aligned} \mathcal {L}_m=L_m+\sum _{d\ge 0}\frac{(d+m+1)!}{d!}t^r_d\frac{{\partial }}{{\partial }t^r_{d+m}}+\varepsilon \frac{3(m+1)!}{4}\frac{{\partial }}{{\partial }t^r_{m-1}}+\delta _{m,-1}\varepsilon ^{-1}t^r+\delta _{m,0}\frac{3}{4}. \end{aligned}$$

6.2 Solutions Given by the Canonical Coordinates

Consider a conformal Frobenius manifold given by a potential $F = F(t^1,\dots ,t^N)$, a metric $\eta$ and an Euler vector field E. Suppose that the Frobenius manifold is semisimple and let $u^1,\dots ,u^N$ be the canonical coordinates. It is well-known that in the canonical coordinates the Euler vector field E looks as

$$\begin{aligned} E=\sum _{i=1}^N(u^i+a^i)\frac{{\partial }}{{\partial }u^i}, \end{aligned}$$

for some constants $a^i\in \mathbb {C}$.

Proposition 6.1

For any $1\le k\le N$ the function $F^o=u^k\cdot s$ satisfies the open WDVV equations together with condition (1.8) and the homogeneity condition

$$\begin{aligned} \sum _{\alpha =1}^N E^\alpha \frac{{\partial }F^o}{{\partial }t^\alpha }+\frac{1-\delta }{2}s\frac{{\partial }F^o}{{\partial }s}=\frac{3-\delta }{2}F^o+a^ks. \end{aligned}$$

(6.1)

Proof

Recall that the canonical coordinates satisfy the equations

$$\begin{aligned} c_{\alpha \beta }^\nu \frac{{\partial }u^k}{{\partial }t^\nu } = \frac{{\partial }u^k}{{\partial }t^\alpha }\frac{{\partial }u^k}{{\partial }t^\beta },\quad 1\le \alpha ,\beta \le N, \end{aligned}$$

(6.2)

which immediately imply Eq. (1.7). Equation (1.6) for the function $F^o=u\cdot s$ are equivalent to

$$\begin{aligned} c_{\alpha \beta }^\nu \frac{{\partial }^2 u^k}{{\partial }t^\nu {\partial }t^\gamma }+\frac{{\partial }^2 u^k}{{\partial }t^\alpha {\partial }t^\beta }\frac{{\partial }u^k}{{\partial }t^\gamma }=c_{\gamma \beta }^\nu \frac{{\partial }^2 u^k}{{\partial }t^\nu {\partial }t^\alpha } + \frac{{\partial }^2 u^k}{{\partial }t^\gamma {\partial }t^\beta }\frac{{\partial }u^k}{{\partial }t^\alpha }. \end{aligned}$$

(6.3)

Differentiating Eq. (6.2) with respect to $t^\gamma$, we get

$$\begin{aligned} \frac{{\partial }c_{\alpha \beta }^\nu }{{\partial }t^\gamma } \frac{{\partial }u^k}{{\partial }t^\nu } = \frac{{\partial }^2 u^k}{{\partial }t^\gamma {\partial }t^\alpha } \frac{{\partial }u^k}{{\partial }t^\beta } + \frac{{\partial }u^k}{{\partial }t^\alpha } \frac{{\partial }^2 u^k}{{\partial }t^\gamma {\partial }t^\beta } - c_{\alpha \beta }^\nu \frac{{\partial }^2 u^k}{{\partial }t^\nu {\partial }t^\gamma }. \end{aligned}$$

Combining this equation with the similar equation for $\dfrac{{\partial }c_{\gamma \beta }^\nu }{{\partial }t^\alpha } \dfrac{{\partial }u^k}{{\partial }t^\nu }$ and noting that $\dfrac{{\partial }c_{\gamma \beta }^\nu }{{\partial }t^\alpha } = \dfrac{{\partial }c_{\alpha \beta }^\nu }{{\partial }t^\gamma }$, we get Eq. (6.3).

Property (1.8) follows from the fact that $\frac{{\partial }}{{\partial }t^1}=\sum _{i=1}^n\frac{{\partial }}{{\partial }u^i}$. The homogeneity property (6.1) is obvious. $\square$

6.3 Open Gromov–Witten Theory of ${\mathbb {P}}^1$

Consider the 2-dimensional Frobenius manifold given by the Gromov–Witten theory of ${\mathbb {P}}^1$:

$$\begin{aligned} F(t_1,t_2) = \frac{1}{2}t_1^2t_2+e^{t_2}. \end{aligned}$$

The Euler vector field is $E = t_1\frac{{\partial }}{{\partial }t_1} + 2 \frac{{\partial }}{{\partial }t_2}$ and $\delta = 1$. Let us find all solutions $F^o(t_1,t_2,s)$ of the open WDVV Eqs. (1.6), (1.7), satisfying condition (1.8) and the homogeneity condition

$$\begin{aligned} t_1\frac{{\partial }F^o}{{\partial }t_1}+2\frac{{\partial }F^o}{{\partial }t_2}=F^o+D_1 t_1+D_2 t_2+\widetilde{D}s+E. \end{aligned}$$

We consider such solutions up to adding a constant and linear terms in the variables $t_1,t_2$ and s. Then we can assume that $D_1=0$. The general form of such a function $F^o$ is $F^o = t_1 s + e^{\frac{t_2}{2}} \phi (t_2,s)$, for some function $\phi (t_2,s)$, satisfying $2e^{\frac{t_2}{2}}\frac{{\partial }\phi }{{\partial }t_2}=D_2 t_2+\widetilde{D}s+E$. Making the transformation $\phi \mapsto \phi -(D_2(t_2+2)+\widetilde{D}s+E)e^{-\frac{t_2}{2}}$, we come to a function $F^o$ of the form

$$\begin{aligned} F^o = t_1 s + e^{\frac{t_2}{2}} \phi (s), \end{aligned}$$

(6.4)

where the function $\phi$ depends only on s. Such a function $F^o$ satisfies the homogeneity property

$$\begin{aligned} t_1\frac{{\partial }F^o}{{\partial }t_1}+2\frac{{\partial }F^o}{{\partial }t_2}=F^o \end{aligned}$$

and condition (1.8).

The system of open WDVV Eqs. (1.6), (1.7) for a function $F^o$ of the form (6.4) is equivalent to the equation $(\phi ')^2 - \phi \phi '' = 4$ and, solving this ordinary differential equation, we get the following two-parameter family of solutions:

$$\begin{aligned} F^o_{\alpha ,\beta } = t_1s\pm 2\alpha ^{-1}e^{\frac{t_2}{2}}\sinh \left( \alpha (s+\beta )\right) ,\quad \alpha ,\beta \in \mathbb {C}. \end{aligned}$$

For $\alpha =0$ we get the functions

$$\begin{aligned} F^o_{0,\beta }=\left. \left( t_1s\pm 2\alpha ^{-1}e^{\frac{t_2}{2}}\sinh \left( \alpha (s+\beta )\right) \right) \right| _{\alpha =0}=t_1s\pm 2 e^{\frac{t_2}{2}}(s+\beta ), \end{aligned}$$

which correspond to the solutions given by the canonical coordinates $u_1=t_1+2e^{\frac{t_2}{2}}$ and $u_2=t_1-2e^{\frac{t_2}{2}}$ of our Frobenius manifold.

Remark 6.2

We believe that the function $F^o_{\alpha ,\beta }$ with a correctly chosen calibration and the corresponding open descendent potential should control the genus 0 open Gromov–Witten invariants of ${\mathbb {P}}^1$, which don’t have a rigorous geometric construction at the moment.