Open WDVV Equations and Virasoro Constraints

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.


Introduction
The WDVV equations, also called the associativity equations, is a system of nonlinear partial differential equations for one function, depending on a finite number of variables. Let N ≥ 1 and η = (η αβ ) be an N × N symmetric non-degenerate matrix with complex coefficients. The WDVV equations is the following system of PDEs for a function F(t 1 , . . . , t N ) defined on some open subset M ⊂ C N : where (η αβ ) := η −1 and we use the convention of sum over repeated Greek indices. Suppose that the function F satisfies the additional assumption ∂ 3 F ∂t 1 ∂t α ∂t β = η αβ . 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 (1996Dubrovin ( , 1999.
Consider formal variables t α p , 1 ≤ α ≤ N , p ≥ 0, where we identify t α 0 = t α . There is a natural way to associate to the function F a descendent potential F, which is a function of the variables t α p , such that the difference F| t * ≥1 =0 − F is at most quadratic in the variables t 1 , . . . , t N and the following equations are satisfied: 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 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 where d (·) denotes the full differential. Let ε be a formal variable and t α p := t α p − δ α,1 δ p,1 . If our Frobenius manifold is conformal, meaning that the function F satisfies a certain homogeneity condition, then in Dubrovin and Zhang (1999)  satisfying the commutation relations and such that the following equations, called the Virasoro constraints, are satisfied: (1.4) We recall the details in Sect. 2. In Gromov-Witten theory, for each g ≥ 0 one defines the generating series F g (t * * ) of intersection numbers on the moduli space of maps from a Riemann surface of genus g to a target variety, F 0 = F. The Virasoro conjecture says that the following equations are satisfied: L m e g≥0 ε 2g−2 F g = 0, m ≥ −1. (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 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. 2018Buryak et al. , 2019. The solutions of Eqs. (1.6), (1.7), considered in these works, also satisfy the additional condition There is an open analog of Eq. (1.3). Let s p , p ≥ 0, be formal variables, where we identify s 0 = s. The open topological recursion relations are the following PDEs for a function F o , depending on the variables t α p and s p , and such that the difference − F o is at most linear in the variables t 1 , . . . , t N and s: (1.10) As we already mentioned in Remark 1.1, these equations appeared in the works (Pandharipande et al. 2014;Buryak et al. 2018Buryak et al. , 2019. The simplest Frobenius manifold has dimension 1 and the potential F = F pt = (t 1 ) 3 6 . A natural descendent potential F 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 "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 pt,o = t 1 s + 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 F pt,o of the open TRR Eqs. (1.9), (1.10). Moreover, they introduced the operators where L pt m are the Virasoro operators for our Frobenius manifold, and proved the equations (Pandharipande et al. 2014, Theorem 1.1) (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 F and the Virasoro operators L m , m ≥ −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 F o of the open TRR Eqs. (1.9), (1.10) and differential operators L m , m ≥ −1, of the form such that the equations 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 , . . . , 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 η 1μ ∂ F ∂t μ , . . . , η N μ ∂ F ∂t μ , F o 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 F . . .,and already in [PST14] the authors conjectured that the following equations should hold: where F 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

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 ≥ 1 and let M be a simply connected open neighbourhood of a point (t 1 orig , . . . , t N orig ) ∈ C N . Denote by O the sheaf of analytic functions on C N . Consider a solution F ∈ O(M) of the WDVV Eq. (1.1), satisfying ∂ 3 F ∂t 1 ∂t α ∂t β = η αβ . 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 orig , . . . , t N orig ) ∈ C N meaning that O(M) denotes in this case the ring of formal power series in the variables (t α − t α orig ).

Descendent Potential
In order to construct a descendent potential, one has to choose an additional structure, called a calibration of the Frobenius manifold. Denote A calibration is a collection of functions α,0 β,d ∈ O(M), 1 ≤ α, β ≤ N , d ≥ −1, satisfying the following properties: The space of all calibrations is non-empty and is parameterized by (Dubrovin 1999, Lemma 2.2 and Exercise 2.8).
Let us choose a calibration. One can immediately see that ∂ α,0 1,0 ∂t β = δ α β , which implies that α,0 1,0 − t α is a constant. Let us make the change of coordinates t α → α,0 1,0 , so that we have now α,0 1,0 = t α . Let v 1 , . . . , v N be formal variables and consider the system of partial differential equations called the principal hierarchy. We see that the equations for the flow ∂ ∂t 1 0 look as ] the solution of the principal hierarchy specified by the initial condition The descendent potential corresponding to the Frobenius manifold together with the chosen calibration is defined by where we recall that t α p = t α p − δ α,1 δ p,1 (Dubrovin and Zhang 1999, Section 3). The difference F| t * ≥1 =0 − F is at most quadratic in the variables t 1 , . . . , t N and the function F satisfies Eq. (1.2) together with the equation 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).

Virasoro Constraints
The Frobenius manifold is said to be conformal if there exists a vector field E of the form The number δ is often called the conformal dimension. Denote In the conformal case there exists a calibration, satisfying the property for some matrices R n , n ≥ 1, satisfying [μ, R n ] = n R n , ηR n η −1 = (−1) n−1 R T n .
One can actually see that the matrices R n are determined uniquely by the functions α,0 β,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 For an arbitrary N × N matrix A = (A α β ) define matrices P m (A, R), m ≥ −1, by the following recursion relation: Alternatively, the matrix P m (A, R) can be defined by 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 The Virasoro operators L m , m ≥ −1, of our conformal Frobenius manifold are given by and Eq. (1.4) hold (Dubrovin and Zhang 1999, page 428).

Remark 2.2
One can easily see that the last term δ m,0 1 4 tr 1 4 − μ 2 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.

Open Virasoro Constraints
(1.6), (1.7) and condition (1.8). Let us assume that the function F o satisfies the homogeneity condition This condition holds in the examples, considered in the papers (Horev and Solomon 2012;Pandharipande et al. 2014;Buryak et al. 2018Buryak et al. , 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 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 and define a diagonal (N + 1) × (N + 1) matrix μ by μ := diag(μ 1 , . . . , μ N +1 ).

Definition 3.1 A calibration of the function F o is a sequence of functions
One can easily see that, for a calibration of the function F o , matrices R n are uniquely determined by the functions β,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 Let us make the change of coordinates t N +1 → 1,0 , so that we have now 1,0 = t N +1 .
Let v 1 , v 2 , . . . , v N +1 be formal variables and consider the system of partial differential equations Let ( v top ) α be the solution, specified by the initial condition Then we define the open descendent potential F o by We will prove the lemma in Sect. 4.4. Let L m , m ≥ −1, be the Virasoro operators for our conformal Frobenius manifold and define operators L m , m ≥ −1, by (3.5)

Theorem 3.4 We have
Therefore, the operators L m , given by (3.5), have the form (1.14).

Remark 3.6 One can easily see that the expression
for some functions f i depending on the variables t α p , 1 ≤ α ≤ N + 1, p ≥ 0. Note that the vanishing of the function f −2 is equivalent to the Virasoro equations for the descendent potential F, Remark 3.7 One can see that the terms δ m,0 for the operator 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.

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.

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. with unit e on each tangent space, analytically depending on the point p ∈ M, such that the one-parameter family of connections ∇ + z• is flat and torsionless for any z ∈ C, and ∇e = 0.
From the flatness and the torsionlessness of ∇ + z• one can deduce the commutativity and the associativity of the algebras (T p M, •). Moreover, if one choses flat coordinates t α , 1 ≤ α ≤ N , N = dim M, for the connection ∇, with e = ∂ ∂t 1 , then it is easy to see that locally there exist analytic functions give the structure constants of the algebras (T p M, •), From the associativity of the algebras (T p M, •) and the fact that the vector ∂ ∂t 1 is the unit it follows that The A flat F-manifold, given by a vector potential (F 1 , . . . , F N ), is called conformal, if there exists a vector field of the form (2.1) such that Remark 4.2 A Frobenius manifold with a potential F and a metric η defines the flat F-manifold with the vector potential F α = η αμ ∂ F ∂t μ . If the Frobenius manifold is conformal, then the associated flat F-manifold is also conformal. This follows from the property has a basis of idempotents π 1 , . . . , π N , satisfying π k • π l = δ k,l π k . Moreover, locally around such a point one can choose coordinates u i such that ∂ ∂u k • ∂ ∂u l = δ k,l ∂ ∂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).

Extensions of Flat F-manifolds and the Open WDVV Equations
Consider a flat F-manifold structure, given by a vector potential (F 1 , . . . , F N +1 ) on an open subset M × U ∈ C N +1 , where M and U are open subsets of C N and C, respectively. Suppose that the functions F 1 , . . . , F N don't depend on the variable t N +1 , varying in U . Then the functions F 1 , . . . , 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 × 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 , . . . , t N ) ∈ O(M) and a metric η, F α = η αμ ∂ F ∂t μ , 1 ≤ α ≤ N . It is easy to check that a function

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.

General Case
Let M be a simply connected open neighbourhood of a point (t 1 orig , . . . , t N orig ) ∈ C N and consider a flat F-manifold structure on M given by a vector potential (F 1 , . . . , F N ), Let us describe the space of all calibrations of our flat F-manifold. Denote by ∇ the family of connections, depending on a formal parameter z, given by where X and Y are vector fields on M. Then Eq. (4.5) is equivalent to the flatness, with respect to ∇, of the 1-forms From the flatness of the connection ∇ it follows that a calibration of our flat F-manifold exists. In order to describe the whole space of calibrations, introduce N × N matrices Let us introduce matrices 0 By definition, we put 0 −1 := Id. From Eq. (4.6) we get (4.8)

Conformal Case
Suppose now that our flat F-manifold is conformal. Introduce a diagonal matrix Q by Q := diag(q 1 , . . . , q N ).

Proposition 4.4 There exists a calibration such that
for some matrices R n , n ≥ 1, satisfying [Q, R n ] = n R n .
Proof Similarly to the work (Dubrovin 1999, pages 310, 312), the proposition is proved by considering a certain flat connection on M × C * .
Introduce a family of connections ∇ λ , depending on a complex parameter λ, on M × C * by where X and Y are vector fields on M × C * having zero component along C * .

Remark 4.5
Note that for the flat F-manifold, associated to a conformal Frobenius manifold M, the connection ∇ δ Proof Direct computation analogous to the one from (Dubrovin 1999, proof of Proposition 2.1).
A differential form ζ α (t * , z)dt α on M × C * is flat with respect to the connection ∇ λ if and only if the following equations are satisfied: where U α β := E μ c α μβ . Denote by ζ the row vector (ζ 1 , . . . , ζ N ), then the last two equations can be written as where U := (U α β ). 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 orig , . . . , t N orig ) × C * ⊂ M × C * . Let U 1 := U| t α =t α orig and consider the equation 12) for an N × N matrix ξ . Then there exists a transformation ξ = G(z)ξ T with G(z) ∈ Mat N ,N (C) [[z]], G(0) = Id, that transforms Eq. (4.12) to where matrices R n satisfy [Q, R n ] = n R 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 λ is obvious.
Therefore, the matrix ξ = z R z λ−Q (G T (z)) −1 satisfies Eq. (4.12). Using Eq. (4.10), we can extend the constructed function ξ on the punctured line (t 1 orig , . . . , t N orig ) × C * ⊂ M × C * to a function ζ on the whole space M × C * . The function ζ has the form (4.14) Since the connection ∇ λ is flat, the function ζ satisfies Eq. (4.11). Equation (4.10) for a function ζ of the form (4.14) implies that the sequence of matrices d 0 , d ≥ −1, is a calibration of our flat F-manifold. Equation (4.11) gives ⎛ Taking the coefficient of z d , d ≥ 0, in this equation, we get d−1 By (4.10), the left-hand side is equal to E θ ∂ d 0 ∂t θ . This completes the proof of the proposition.
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 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 η 0 d η −1 = ( d 0 ) T and η R n η −1 = (−1) n−1 R T n .

Calibrations of Extensions of Flat F-manifolds
For an (N + 1) × (N + 1) matrix A denote by π N (A) the N × N matrix formed by the first N raws and the first N columns of A.

Then there exists a calibration of the flat F-manifold M × U with matrices d 0 and R n satisfying the properties
Proof Consider the construction of a calibration of the conformal flat F-manifold M ×U from the proof of Proposition 4.4 in more details. So we consider the differential equation Id + n≥1 G n z n , transforming this differential equation to the form where [ Q, R n ] = n R n , is determined by the recursion relation (Dubrovin 1999, equation (2.53)) (−1) n−1 R T n = δ n,1 U T 1 + n G n + [ Q, G n ] If one has computed the matrices G i and R i for i < n, then Eq. (4.17) determines the matrix R n and the elements ( G n ) α β with q α − q β = −n. The elements ( G n ) α β with q α − q β = −n can be chosen arbtitrarily. Note that ( with q α − q β = −n to be zero, we can guarantee that ( G n ) N +1 ≤N = 0. Let U 1 := π N ( U 1 ) and Q := π N ( Q). We know that the N × N matrices G n , given by Therefore, there exist matrices R n and G n , n ≥ 1, satisfying Eq. (4.17) and the properties π N ( R n ) = R n , π N ( G n ) = G n and ( R n ) ≤N N +1 = ( G n ) N +1 ≤N = 0. Note that the property [ Q, R n ] = n R n implies that the diagonal elements of R n are also zero.
We construct the matrices d 0 as a solution of Eq. (4.6), satisfying the initial condition

Descendent Vector Potentials of a Flat F-manifold
Consider a flat F-manifold, given by a vector potential (F 1 , . . . , and let us choose a calibration. One can immediately see that Let us make the change of coordinates t α → α,0 1,0 , so that we have now α,0 1,0 = t α . Let v 1 , . . . , 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): (4.19) The flows of the principal hierarchy pairwise commute. Since α,0 1,0 = t α , we can identify x = t 1 0 .
Clearly the functions v α = t α 0 satisfy the subsystem of system (4.19), given by the flows ∂ [[t * ≥1 ]] the solution of the principal hierarchy specified by the initial condition is a symmetry of the principal hierarchy (4.19). Since, obviously, we get Eq. (4.20). The rescaling combined with the shift along t 1 1 , given by is also a symmetry of the principal hierarchy. We compute We also adopt the convention −1 q = q −1 := δ q,0 Id, q ≥ 0. One can easily check that

Lemma 4.11 We have
Proof By (4.22), we have The fact that the flows of the principal hierarchy pairwise commute implies that . This completes the proof of the lemma.
We finally define the descendent vector potentials (F 1, p , . . . , F N , p ), p ≥ 0, associated to our flat F-manifold and its calibration, by Let us also adopt the convention Proposition 4.12 1. We have 2. By the first part of the proposition, ∂ β,0 ∂t γ = c α βγ , the second part of the proposition is also proved.

Remark 4.13
Consider a Frobenius manifold, its calibration and the associated flat F-manifold. Then the functions F α, p are related to the descendent potential F of the Frobenius manifold by F α, p = η αμ ∂F ∂t μ 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 η 1μ ∂ F ∂t μ , . . . , η N μ ∂ F ∂t μ , F o . Immediately from the definitions and Lemma 4.9 we see that if (F 1, p , . . . , F N +1, p ), p ≥ 0, are the decendent vector potentials of this flat F-manifold, then F α, p = η αμ ∂F ∂t μ p , 1 ≤ α ≤ N , and F N +1,0 = F o . Therefore, Lemma 3.3 follows from Proposition 4.12 and Eq. (4.22).
Recall that R = i≥1 R i . Let λ be a complex parameter and define Define the following expressions, depending on the parameter λ: Proposition 4.14 We have A α m = 0, 1 ≤ α ≤ N , m ≥ −1.

Remark 4.15 For a Frobenius manifold the expressions A α m have the following interpretation:
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 top ) α and ( top ) α, p β,q by v α and α, p β,q , respectively. We will also denote the function c α βγ | t θ →(v top ) θ by c α βγ and the function (1−q α )(v top ) α +r α by E α .
Define an operator B = (B α β ), depending on the parameter λ, by
We begin with the following lemma.
We then compute B 0 where e denotes the unit vector (1, 0, . . . , 0). (4.27) On the other hand, we have (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: 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): On the other hand, the expression in line (4.29) is equal to As a result, we get For m ≥ n ≥ −1 denote C m,n := Lemma 4. 19 We have C m,n = 0, m ≥ n ≥ −1.
Proof We proceed by induction on n. By Eq. (4.7), we have Suppose that m ≥ n ≥ 0. We compute Using property (4.15), we transform the last expression in the following way: One can see that the last two expressions cancel each other and, as a result, that, by the induction assumption, is equal to zero. The lemma is proved.
Lemmas 4.18 and 4.19 imply that B A m = ∂ x A m+1 for m ≥ −1. Introduce a differential operator L :

Lemma 4.20 We have L A
Proof Using formulas (4.20), (4.25) and the formula , the lemma is proved.

Lemma 4.21
We have A α m t * ≥1 =0 = 0, m ≥ −1. Proof During the proof of this lemma we return to the initial notation, where α, p β,q is a function of t 1 , . . . , t N . Note that, by Eqs. (4.24) and (4.25), we have F α, , which, by Lemma 4.19, is equal to zero.
Let us now prove that A α m = 0 by induction on m. We have Suppose that m ≥ 0. Let us express A α m as a power series in the variables t β d defined by From the induction assumption, Lemma 4.20 and the fact that We also know that A α m | 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 t
Proof Denote by P the set of all finite sequences λ = (λ 1 , . . . , λ l ), l ≥ 0, λ i ∈ Z, such that λ 1 ≥ · · · ≥ λ l ≥ 0. We also denote l(λ) := l and |λ| := λ i . Then we can express the power series f in the following way: where for an element λ = (λ 1 , . . . , λ l ) ∈ P we denote t α λ := t α λ 1 · · · t α λ l . Note that if we assign to a monomial t 1 ] spanned by monomials of degree d. Using the lexicographical order on monomials t 1 λ 1 · · · t N λ N , it is easy to check that the map k≥0 t α k+1 ∂ ∂ t α k : V d → V d+1 is injective for d ≥ 1; and for d = 0 its kernel consists of constants. This completes the proof of the lemma.

Proof of Theorem 3.4
Let us apply Proposition 4.14 to the flat F-manifold associated to the function F o . We choose λ = 3−δ 2 , then we have The first term here is equal to The next four terms correspond to the four summations in the expression (3.5) for the operators L m . Since μ N +1 = 1 2 , the last term is equal to zero. Thus, that proves Theorem 3.4.

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 F 0 (t * * ) = F, F 1 (t * * ), F 2 (t * * ), . . ., such that for the differential operators L m , given by (2.3), Eq. (1.5) hold (Givental 2001(Givental , 2004Teleman 2012). If one considers the Gromov-Witten theory of a given target variety, then the functions 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 . ., such that for the differential operators L m , given by (3.5), the equations 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 L m .
Proof Denote the parts of the expression for the operator L m from lines (2.3), (2.4) and (2.5) by L 1 m , L 2 m and L 3 m , correspondingly. One can see that the operators L m can be written in the following way: Let us first prove that [L −1 , L n ] = (−1 − n)L n−1 , for n ≥ 0. For this we compute A term in this sum is equal to zero unless μ α < 1 2 ⇔ p + d ≤ m − 2, μ β < 1 2 ⇔ q − d ≤ n − 1 and μ α + μ β = 0 ⇔ p + q = m + n − 2. The first two condition give p + q ≤ m + n − 3, that contradicts the third condition. Thus, ε 2 p,q≥0 I p;q ∂ 2 ∂s p ∂s q = 0, as required. This completes the proof of the proposition.

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.

Extended r-Spin Theory
Let us fix an integer r ≥ 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 -spin (t 1 , . . . , t r −1 ) with the metric η, given by η αβ = δ α+β,r , and the Euler vector field The conformal dimension is δ = r −2 r . The potential F r -spin is a polynomial in the variables t 1 , . . . , 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 F r -spin (t * * ), corresponding to our Frobenius potential F r -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 where we use the notation for a complex number x and an integer n ≥ 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 ext (t 1 , . . . , t r ) of the open WDVV equations, satisfying condition (1.8) and the homogeneity condition Recall that we identify t r = s. Note that the function

Solutions Given by the Canonical Coordinates
Consider a conformal Frobenius manifold given by a potential F = F(t 1 , . . . , t N ), a metric η and an Euler vector field E. Suppose that the Frobenius manifold is semisimple and let u 1 , . . . , u N be the canonical coordinates. It is well-known that in the canonical coordinates the Euler vector field E looks as for some constants a i ∈ C. is F o = t 1 s + e t 2 2 φ(t 2 , s), for some function φ(t 2 , s), satisfying 2e t 2 2 ∂φ ∂t 2 = D 2 t 2 + Ds + E. Making the transformation φ → φ − (D 2 (t 2 + 2) + Ds + E)e − t 2 2 , we come to a function F o of the form  For α = 0 we get the functions