Flat F-manifolds, F-CohFTs, and integrable hierarchies

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple flat F-manifold and additional data in genus $1$, obtained in our previous work. Our construction of these dispersive deformations is quite explicit and we compute several examples. In particular, we provide a complete classification of rank $1$ hierarchies of DR type at the order $9$ approximation in the dispersion parameter and of homogeneous DR hierarchies associated with all $2$-dimensional homogeneous flat F-manifolds at genus $1$ approximation.


Introduction
Since Witten's conjecture [Wit91] and its proof by Kontsevich [Kon92], there have been growing and fruitful interactions between the area of integrable hierarchies of PDEs and algebraic geometry of the moduli spaces of algebraic curves. In this context, and in connection with topological field theory, Dubrovin introduced in the 90s the notion of Frobenius manifold [Dub96], a differential-geometric structure that encodes genus-zero information af a cohomological field theory (CohFT) on the moduli space of stable curves, besides having far reaching connections with other areas of mathematics.
From the point of view of integrable systems, given a Dubrovin-Frobenius manifold, there exists an associated integrable hierarchy of Hamiltonian quasilinear PDEs called Dubrovin's principal hierarchy, or simply principal hierarchy. An important problem in the theory of integrable systems consists in constructing a full dispersive hierarchy starting from its dispersionless limit.
In the framework of moduli spaces, the principal hierarchy associated to a Dubrovin-Frobenius manifold and its dispersive deformation should satisfy additional constraints coming from the intersection theory of the CohFT. In the semisimple case, there exist two different (but conjecturally Miura-equivalent [Bur15,BDGR18,BGR19]) constructions defining such dispersive deformations: (1) The Dubrovin-Zhang construction [DZ01] is based on the idea that the partition function of the corresponding CohFT in all genera is the logarithm of the tau-function of a special solution (called the topological solution) to a full dispersive hierarchy (the DZ hierarchy). One can construct the hierarchy itself starting from this tau-function, and it turns out that the the principal hierarchy is the dispersionless limit of DZ hierarchy. Moreover the full DZ hierarchy and the principal hierarchy are related by a special change of dependent variables, called a quasi-Miura transformation, which can be uniquely determined in the semisimple case from genus zero information.
(2) The double ramification construction, introduced by one of the authors in [Bur15], is based on the definition of an infinite set of commuting Hamiltonian densities [BR16a] in terms of intersection numbers of the CohFT, the double ramification cycles and other natural tatutological classes on the moduli space of curves.
For both constructions and in the (homogeneous) semisimple case, the reconstruction of the full dispersive hierarchy from its dispersionless limit (the principal hierarchy of the Dubrovin-Frobenius manifold encoding the genus 0 part of the CohFT) is possible thanks to the Givental-Telemann reconstruction theorem for the CohFT itself from its genus 0 part [Tel12,Giv01].
Notice that, by construction, the dispersionless limits of both the DZ and DR hierarchies coincide with the principal hierarchy of the Dubrovin-Frobenius manifold underlying the CohFT.
In the last 20 years, it has been observed that many constructions related to Dubrovin-Frobenius manifolds can be extended to a more general setting ([Sab98, Get04, Man05, LPR09, SZ11, AL13a, Lor14, KMS15, AL17, DH17, BR18, KMS18, AL19, BB19, ABLR20]). For instance, it was observed in [LPR09] that the notion of principal hierarchy does not require the existence of an invariant flat metric. This leads naturally to the consideration of the generalization of Dubrovin-Frobenius manifolds, called F-manifolds with compatible flat structure [Man05] or simply flat F-manifolds [LPR09], obtained by replacing a flat metric with a flat torsionless connection and keeping all the axioms of Dubrovin-Frobenius manifolds apart from those involving explicitly the metric and not just the associated Levi-Civita connection. In flat coordinates for the flat connection, the flows of the principal hierarchy are systems of conservation laws. In the case of Dubrovin-Frobenius manifolds, the presence of an invariant flat metric has to deal with the presence of a local Hamiltonian structure.
In this paper we construct (homogeneous) double ramification hierarchies starting from a (homogeneous) CohFT. In particular, in the semisimple case, leveraging on the results of [ABLR20], this provides dispersive deformations of the principal hierarchy associated to a semisimple (homogeneous) flat F-manifold. The existence of these dispersive integrable deformations relies on: (1) a generalization of the notion of cohomological field theory, called F-cohomological field theory (or F-CohFT for short) introduced in [BR18, ABLR20]; (2) a reconstruction theorem for a semisimple (homogeneous) F-CohFT starting from a flat F-manifold and additional data in genus 1 [ABLR20]; (3) the definition of an infinite set of commuting flows (the DR hierarchy) in terms of intersection numbers of the F-CohFT, the double ramification cycles, the top Hodge class, and psi classes on the moduli space of stable curves.
The paper is organized as follows.
Section 1 is devoted to the construction of the DR hierarchy of an F-CohFT (see also [BR18]). The main properties of this hierarchy are given in terms of densities of local vector fields on the formal loop space and a special basis for their integrals of motion. We also consider the additional properties of the hierarchy in the case of a homogeneous F-CohFT.
In Section 2, after recalling the definition of a flat F-manifold and the construction of its associated principal hierarchy, we present our main result: given an arbitrary semisimple flat F-manifold and an associated principal hierarchy, we construct a family of dispersive integrable deformations of the principal hierarchy. These deformations, called the descendant DR hierarchies, come from the family of DR hierarchies associated to a family of F-CohFTs parameterized by a semisimple point of our flat F-manifold. The descendant DR hierarchy depends on a choice of a certain vector field on the flat F-manifold, which we call a framing. We prove that the descendant DR hierarchies corresponding to different framings are not related to each other by a Miura transformation that is close to identity.
In Section 3, we discuss the role of (descendant) DR hierarchies in the problem of classification of integrable deformations of integrable dispersionless systems of conservation laws. One can impose various constraints for such integrable deformations, and we discuss the corresponding results (mostly at the approximation up to some finite power of ε) for flat F-manifolds of dimension 1 and 2 in Section 3.2 and 3.1.1. In Section 3.3, we briefly mention the problem of computing general integrable deformations of principal hierarchies of flat F-manifolds. It was conjectured in [AL18] that the equivalence classes of such deformations are labeled by certain functional parameters called Miura invariants. In the case of Dubrovin-Frobenius manifolds and bihamiltonian deformations, these invariants are equivalent to central invariants, which are known to classify deformations of semisimple local bihamiltonian structures of hydrodynamic type ( [DLZ06,CPS18]).

Double ramification hierarchy of an F-CohFT
In this section, we associate to any F-CohFT with a vector space V an infinite sequence of commuting vector fields on the formal loop space of V , i.e., an infinite sequence of compatible systems of evolutionary PDEs of rank N := dim V (in particular, in the form of conservation laws). This construction is a generalization of the double ramification hierarchy of [Bur15,BR16a] to the context of F-CohFTs and enjoys most of its properties (for instance, recursion formulas for the higher symmetries), but loses in general the Hamiltonian nature.
1.1. F-cohomological field theories. We recall from [BR18,ABLR20] the definition of an F-cohomological field theory on the moduli space M g,n of stable curves of genus g with n marked points. We will denote by H * (X) the cohomology ring with coefficients in C of a topological space X. When considering the moduli space of stable curves, X = M g,n , the even part H even (M g,n ) in the cohomology ring H * (M g,n ) can optionally be replaced by the Chow ring A * (M g,n ).
Definition 1.1. An F-cohomological field theory (or F-CohFT) is a system of linear maps where V is an arbitrary finite dimensional vector space, together with a special element e ∈ V , called the unit, such that, chosen any basis e 1 , . . . , e dim V of V and the dual basis e 1 , . . . , e dim V of V * , the following axioms are satisfied: (i) The maps c g,n+1 are equivariant with respect to the S n -action permuting the n copies of V in V * ⊗ V ⊗n and the last n marked points in M g,n+1 , respectively.
There is an obvious generalization of the notion of an F-CohFT where the maps c g,n+1 take value in H even (M g,n+1 ) ⊗ K, where K is a C-algebra. We will call such objects F-cohomological field theories with coefficients in K.
Definition 1.2. An F-CohFT c g,n+1 : V * ⊗ V ⊗n → H even (M g,n+1 ) is called homogeneous if there exists an operator Q ∈ End(V ), a vector r ∈ V , and a complex constant γ such that Qe = 0 and the following condition is satisfied: where Deg ∈ End(H * (M g,n )) is the operator acting on H i (M g,n ) by the multiplication by i 2 , π : M g,n+2 → M g,n+1 is the map that forgets the last marked point, ⊗r : is the operator of tensor multiplication from the right by r, and Q t ∈ End(V * ) is the transposed operator. The constant γ is called the conformal dimension of our F-CohFT. Remark 1.3. Our definition of a homogeneous F-CohFT is slightly more general, than the one from the paper [ABLR20] where the operator Q was required to be diagonalizable. However, it is easy to see that all the results from [ABLR20] about homogeneous F-CohFTs are true with the new definition (see also Section 2.1 with a new definition of a homogeneous flat F-manifold). An example of a homogeneous F-CohFT with a nondiagonalizable operator Q will appear in Section 3.
1.2. Vector fields on the the formal loop space. Let A and Λ be the spaces of differential polynomials and local functionals in formal (even) variables u α k , 1 ≤ α ≤ N, k ≥ 0, and ε, with the differential grading deg ∂x u α k = k, deg ∂x ε = −1, where the definitions and the notations are taken from [Ros17, Section 2.1].
The space of densities of local multivector fields (on the formal loop space of V ) is the supercommutative associative algebra , where the new formal variables θ α,k , 1 ≤ α ≤ N, k ≥ 0, are odd (anti-commuting among themselves and commuting with ε and u α k ) with deg ∂x θ α,k := k, u α := u α 0 , and θ α := θ α,0 , and the symbol * , as an index, denotes any of the allowed values for that index. The algebra A • is endowed with the super grading, denoted by deg θ , which is defined by deg θ θ α,k := 1 and deg θ u α k = deg θ ε := 0. The sub-vector space of A • homogeneous of super degree i ≥ 0 is denoted by A i and called the space of densities of local i-vector fields. We have A = A 0 , while A 1 is called the space of densities of local vector fields. The homogeneous component of the space A i of differential degree k will be denoted by ( The operator ∂ x is extended from A to A • as the super-derivation where, here and in what follows, we perform summation over repeated Greek indices. The space of local multivector fields is defined as and, for i ≥ 0, the space of local i-vector fields Λ i is the image of A i in the quotient. If f ∈ A • , its image in Λ • is denoted by f = f dx. As before, Λ = Λ 0 , and Λ 1 is called the space of local vector fields. Naturally, the spaces Λ i inherit the differential grading deg ∂x . For any 1 ≤ α ≤ N, we define the (super) variational derivatives which are well defined on Λ • since they vanish on Im The Schouten-Nijenhuis bracket [·, ·] : Taking the integral with respect to x of formula (1.4), using that δ(x − y)g(y)dx = g(y), reproduces indeed formula (1.2), and further integration with respect to y gives (1.3).
As usual, for i = j = 1, the above Schouten-Nijenhuis brackets are called the Lie brackets. For i = 1 and j = 0, the Schouten-Nijenhuis brackets reduce simply to the differentiation of (a density of) a local functional along (a density of) a vector field, from which we see that the symbol θ α,k can be interpreted as the operator ∂ k x • δ δu α : Λ → A.
Given a local vector field X ∈ Λ 1 , there is a unique representative X ∈ A 1 of X such that X = X α θ α with X α ∈ A. This representative is given by X = δX δθα θ α . The system of evolutionary PDEs associated to X is Two systems of evolutionary PDEs the generators u * * and θ * , * of A • transform according to the formulae where u α ( u * * ; ε) is obtained by inverting u α = u α (u * * ; ε) order by order in ε. For a local vector field, these formulae give from which we obtain that a system of evolutionary PDEs (1.5) transforms into , α = 1, . . . , N.
Performing the change of formal variables one can rewrite a density of a local multivector field f (u * * , θ * , * ; ε) ∈ ( A m ) [d] as a formal Fourier series f = n,s≥0 a 1 ,...,an∈Z b 1 ,...,bm∈Z f a 1 ,...,an,b 1 ,...,bm α 1 ,...,αn,β 1 ,...,βm;s ε s p α 1 a 1 . . . p αn an q β 1 ,b 1 . . . q βm,bm e i( n j=1 a j + m where the coefficient f a 1 ,...,an,b 1 ,...,bm α 1 ,...,αn,β 1 ,...,βm;s , as a function of the indices a 1 , . . . , a n , b 1 , . . . , b m , is a homogeneous polynomial of degree s + d. Formal Fourier series of this type form a supercommutative associative algebra where the formal variables q * , * are odd. Moreover, the local multivector field f corresponds to the constant term of the Fourier series. Similarly to the variables θ * , * , one should interpret the variable q α,a to represent the vector ∂ ∂p α −a . This is coherent with the following formulae for the variational derivatives in the variables p * * and q * , * : acting on local multivector fields to give densities of local multivector fields. Accordingly, using the formal Fourier expansion δ(x) = a∈Z e iax for the formal Dirac delta function, it is easy to obtain the formula for the Schouten-Nijenhuis bracket (1.4) on A i × A j in the new variables: from which analogues of (1.2) and (1.3) are easily obtained by integration in x and then y.
1.3. Densities of local vector fields for the DR hierarchy. Denote by ψ i ∈ H 2 (M g,n ) the i-th psi class, which is the first Chern class of the line bundle over M g,n formed by the cotangent lines at the i-th marked point. Denote by E the rank g Hodge vector bundle over M g,n whose fibers are the spaces of holomorphic one-forms on stable curves. Let λ j := c j (E) ∈ H 2j (M g,n ), these classes are called the Hodge classes.
For any a 1 , . . . , a n ∈ Z, n i=1 a i = 0, denote by DR g (a 1 , . . . , a n ) ∈ H 2g (M g,n ) the double ramification (DR) cycle. We refer the reader, for example, to [BSSZ15] for the definition of the DR cycle on M g,n , which is based on the notion of a stable map to CP 1 relative to 0 and ∞. If not all the multiplicities a i are equal to zero, then one can think of the class DR g (a 1 , . . . , a n ) as the Poincaré dual to a compactification in M g,n of the locus of pointed smooth curves (C; p 1 , . . . , p n ) Consider the Poincaré dual to the double ramification cycle DR g (a 1 , . . . , a n ) in the space M g,n . It is an element of H 2(2g−3+n) (M g,n ), and abusing notation it is also denoted by DR g (a 1 , . . . , a n ).
The restriction DR g (a 1 , . . . , a n ) M ct g,n , where M ct g,n is the moduli space of stable curves of compact type, is a homogeneous polynomial in a 1 , . . . , a n of degree 2g with the coefficients in H 2g (M ct g,n ). This follows from Hain's formula [Hai13] for the version of the DR cycle defined using the universal Jacobian over M ct g,n and the result of the paper [MW13], where it is proved that the two versions of the DR cycle coincide on M ct g,n (the polynomiality of the DR cycle on M g,n is proved in [JPPZ17]). The polynomiality of the DR cycle on M ct g,n together with the fact that λ g vanishes on M g,n \ M ct g,n (see, e.g., [FP00, Section 0.4]) imply that the cohomology class λ g DR g (− n j=1 a j , a 1 , . . . , a n ) ∈ H 4g (M g,n+1 ) is a degree 2g homogeneous polynomial in the coefficients a 1 , . . . , a n .
The double ramification hierarchy associated to the given F-CohFT is the infinite system of local vector fields Y β,d , 1 ≤ β ≤ N, d ≥ −1, associated with the above densities or, in terms of evolutionary PDEs, the system DRg(− n j=1 a j ,0,a 1 ,...,an) Let us adopt the convention P α β,−1 := δ α β . Notice that the system of evolutionary PDEs (1.12) carries strictly less information than the corresponding densities (1.11). We have the following result.
Let us prove part (i). If d 1 = −1 or d 2 = −1, then the statement easily follows from the definitions. For d 1 , d 2 ≥ 0, the statement is analogous to [BR16b, Lemma 3.3], and we use [BSSZ15, Corollary 2.2], describing the intersection of the psi classes with the DR cycle, together with the fact that that λ g vanishes on M g,n \M ct g,n . Let n ≥ 0 and consider integers a 1 , . . . , a n+3 with the vanishing sum. For a subset I = {i 1 , . . . , i |I| } ⊂ [n + 3], i 1 < i 2 < . . . < i |I| , denote by A I the string a i 1 , a i 2 , . . . , i |I| . For I, J ⊂ [n + 3] \ {2, 3} with I ⊔ J = [n + 3] \ {2, 3}, and for g 1 , g 2 > 0 with 2g 1 + |I| > 0, 2g 2 + |J| > 0, let us denote by DR g 1 (a 2 , A I , −k) ⊠ DR g 2 (a 3 , A J , k) the cycle in M g 1 +g 2 ,n+3 obtained by gluing the two DR cycles at the marked points labeled by the integers −k and k, respectively. Here, the coefficient a j , 1 ≤ j ≤ n + 3, is attached to the marked point j. Then we have One then needs to intersect this relation with the class −a 1 e −ia 2 x e −ia 3 y ψ d 1 2 ψ d 2 3 c g,n+3 (e α 1 ⊗ ⊗ n+3 i=2 e α i ), where, as usual, the covector e α 1 is attached to the marked point 1 and each vector e α i is attached to the marked point i. Thanks to the gluing axiom of the F-CohFT, by the definitions (1.10) and (1.14), and after setting α 2 = β 1 and α 3 = β 2 , the left-hand side of equation (1.15) produces the right-hand side of the equation in part (i) and depending on whether, in the above sum, the marked point 1 belongs to the subset I or J, we obtain either of the two terms in the Lie bracket on the left-hand side of the equation in part (i).
Part (ii) is immediately obtained from (i) upon integration in both x and y.

Part (vi) immediately follows from parts (iv), (v), the properties Ker
For part (vii), we compute ∂ x ∂P β 1 1 1,1 1.4. Densities of integrals of motion for the DR hierarchy. The DR hierarchy of a CohFT is a Hamiltonian integrable system [Bur15,BR16a], so the Hamiltonians both generate the commuting vector fields and provide integrals of motion for the hierarchy. In the non-Hamiltonian F-CohFT case, integrals of motion have a separate geometric definition in terms of intersection numbers on the moduli space of curves. For 1 ≤ β ≤ N and d ≥ 0, we define the following system of formal Fourier series: which, thanks to the polynomiality property of the DR cycle, can be rewritten as differential polynomials g β,d ∈ A [0] as DRg (− n j=1 a j ,a 1 ,...,an) To this definition, we add the extra densities of conserved quantities g β,−1 := u β , 1 ≤ β ≤ N, and the "primary" local vector field Y := − g β,0 θ β,1 dx or, in other words, ∂ x g β,0 = δY δθ β , 1 ≤ β ≤ N.
Finally, for 1 ≤ β 1 , β 2 ≤ N and d 1 , d 2 ≥ 0, let us define the generating series (1.21) To this definition, for future convenience, we add g β 1 ,−1 The proof of (i) is completely analogous to the proof of (i) in Theorem 1.5. For d 1 = −1 or d 2 = −1, the statement easily follows from the definitions. Suppose d 1 , d 2 ≥ 0. Let n ≥ 0 and consider integers a 1 , . . . , a n+2 with the vanishing sum. Let us write the same relation as (1.15), but with the psi classes taken at other marked points: Intersecting this relation with the class (−i)e −a 1 ix e −a 2 iy ψ d 2 1 ψ d 1 2 c g,n+2 (e α 1 ⊗⊗ n+2 j=2 e α j ) and forming the corresponding generating series, we obtain part (i) (after setting α 2 = β 1 and α 1 = β 2 ).
The proof of (ii) to (iv) follows strictly the arguments in the proof of the corresponding parts in Theorem 1.5.
The proof of part (v) is the same as the proof of part (vi) in Theorem 1.5.
For the proof of (vi), consider the equation of part (iii) with d 2 = −1. Multiplying it by θ β 2 , summing over β 2 , and integrating over x we obtain, on the left-hand side, and, on the right-hand side, Part (vii) is proved in an analogous fashion starting from (iv).
Proof. The proof is a simple consequence of equation (1.1) together with dimension counting for the intersection numbers involved in the definitions of g α,d , Y α,d , and Y and the fact that where π : M g,n+1 → M g,n forgets the last marked point and δ 0 i,n+1 is the closure in M g,n+1 of the locus of stable curves whose dual graph is a tree with two vertices, one of which has genus 0 and exactly two legs marked by i and n + 1.
In [BRS20], the authors presented an explicit conjectural formula for a bihamiltonian structure of the DR hierarchy corresponding to a homogeneous CohFT. This in particular gives a recursion of certain type, called a bihamiltonian recursion, expressing the flows ∂ ∂t α d+1 , 1 ≤ α ≤ N, of the hierarchy in terms of the flows ∂ ∂t α d , 1 ≤ α ≤ N. For a general homogeneous F-CohFT, we don't expect the corresponding DR hierarchy to have a Hamiltonian structure. However, we will now present a conjectural generalization of the bihamiltonian recursion in this setting.
Following [BRS20], we associate with a differential polynomial f ∈ A a sequence of differential operators indexed by α = 1, . . . , N and k ≥ 0: Consider an arbitrary homogeneous F-CohFT and the corresponding DR hierarchy. Define an operator R = (R α β ) by where the notation E γ L 0 β (g α,0 ) (respectively, L 0 β (g α,0 ) x ) means that we apply the operator E γ (respectively, ∂ x ) to the coefficients of the operator L 0 β (g α,0 ).
The proof of part (2) follows closely the proof of [BRS20, Proposition 2.1].

Principal hierarchy of a flat F-manifold and dispersive deformations
In this section, using the results from the previous section, we construct a family of dispersive integrable deformations of a principal hierarchy associated to an arbitrary semisimple flat F-manifold. Moreover, we prove that different hierarchies from this family are not equivalent to each other by a Miura transformation that is close to identity.

Flat F-manifolds.
Here we recall the notion of a flat F-manifold ( [Get04,Man05], see also [AL18] and [LPR09]) and its main properties. The algebras (T p M, •) are commutative and associative. Let t α , 1 ≤ α ≤ N, N = dim M, be flat coordinates for the connection ∇. Locally, there exist analytic functions F α (t 1 , . . . , t N ), 1 ≤ α ≤ N, such that the second derivatives Also, in the coordinates t α the unit e has the form e = A α ∂ ∂t α for some constants A α ∈ C. Moreover, the following equations are satisfied: A point p ∈ M of an N-dimensional flat F-manifold (M, ∇, •, e) is called semisimple if T p M has a basis π 1 , . . . , π N satisfying π α • π β = δ α,β π α . Moreover, locally around such a point one can choose coordinates u i such that ∂ ∂u α • ∂ ∂u β = δ α,β ∂ ∂u α . These coordinates are called canonical coordinates. In particular, this means that the set of semisimple points is open in M. In the canonical coordinates, we have e = α ∂ ∂u α . A flat F-manifold (M, ∇, •, e) is called semisimple if the set of semisimple points is dense in M.
A flat F-manifold given by a vector potential F is called homogeneous if there exists a vector field E of the form satisfying [e, E] = e and such that Note that this equation can be written more invariantly as Lie E (•) = •, where Lie E denotes the Lie derivative. The vector field E is called the Euler vector field. Around a semisimple point, the Euler vector field has the following form in canonical coordinates: Remark 2.2. As we already mentioned in Remark 1.3, our definition of a homogeneous flat F-manifold is slightly more general than the one from [ABLR20], but all the results from that paper remains valid. In the semisimple case, the flatness of the dual structure is equivalent to the condition ∇∇E = 0 [AL17] (see [KMS18] for the regular case). Thus, in the structure of a semisimple homogeneous flat F-manifold is equivalent to the structure of a semisimple bi-flat F-manifold.
Given an F-CohFT c g,n+1 : V * ⊗ V ⊗n → H even (M g,n+1 ), dim V = N, and a basis e 1 , . . . , e N ∈ V , with e = A α e α , an N-tuple of functions (F 1 , . . . , F N ) satisfying equations (2.2) and (2.3) can be constructed as the following generating functions: thus yielding an associated flat F-manifold structure on a formal neighbourhood of 0 in V (see, e.g., [ABLR20, Proposition 3.2]). The flat F-manifold associated to a homogeneous F-CohFT is homogeneous with the Euler vector field (2.4) where q α β e α := Qe β and r α e α := r.
2.2. Principal hierarchy of a flat F-manifold. Given a flat F-manifold (M, ∇, •, e), one can construct an integrable dispersionless hierarchy called a principal hierarchy associated to (M, ∇, •, e) (see [LPR09]). This construction generalizes the notion of a principal hierarchy associated to a Dubrovin-Frobenius manifold. Before presenting the construction, let us introduce a small generalization of the space of densities of local multivector fields.
Let U be an open subset of C N , with coordinates u 1 , . . . , u N . Denote by O(U) the space of analytic functions on U. Consider the following space: ]. Clearly, the space A • can be considered as the space A • U where U is a formal neighborhood of 0. The space A • U will also be called the space of densities of local multivector fields. It is easy to see that all the constructions from Section 1.2 (except, probably, the constructions related to the change of variables (1.8)) work with the more general space A • U . The space of local multivector fields corresponding to A • U will be denoted by Λ It is immediate to see from (2.5) that X α,−1 , α = 1, . . . N, are flat vector fields for ∇, while the vector fields X α,d are obtained via the recurrence relation ∇X α,d+1 = X α,d •. If U is connected, then the collection of flat sections X α (z) is determined uniquely up to a transformation of the form ] is invertible. If M is simply connected, then flat sections X α (z) can be constructed on the whole M. Consider now a flat F-manifold structure on M ⊂ C N given by a vector potential F , together with a calibration X α (z). The principal hierarchy associated to our calibrated flat F-manifold is the following system of PDEs: where X α β,d ∂ ∂t α := X β,d . We see that the system (2.6) has the form of a system of conservation laws. Moreover, this is a system of quasilinear evolutionary PDEs, which is dispersionless and integrable, in the sense that all the flows pairwise commute (see [LPR09]).
Suppose that M is a formal neighbourhood of 0 ∈ C N . There exist unique flat sections X α (z) on M satisfying the condition X α,−1 = ∂ ∂t α and the condition that X α,d vanish at 0 for d ≥ 0. The corresponding principal hierarchy is called the ancestor principal hierarchy.
Proposition 2.5. Consider an F-CohFT and the associated flat F-manifold and the DR hierarchy. Then the dispersionless part of the DR hierarchy coincides with the ancestor principal hierarchy of the flat F-manifold.
Proof. This immediately follows from the construction of the DR hierarchy and [ABLR20, Proposition 3.2] (see also an analogous statement in [Bur15, Section 4.2.2]).
We see that this proposition can be immediately used for a construction of dispersive deformations of ancestor principal hierarchies. In order to construct dispersive deformations of arbitrary principal hierarchies, we need a generalization of the construction of the DR hierarchy, which we will introduce in the next section.
2.3. Dispersive deformations of a principal hierarchy: descendant DR hierarchies. In order to construct dispersive deformations of a principal hierarchy associated to an arbitrary semisimple flat F-manifold, we first need to study analytic families of F-CohFTs depending on a semisimple point of a flat F-manifold.
Consider a semisimple flat F-manifold structure on M ⊂ C N defined by a vector potential F . Recall that for an arbitrary semisimple point on M, on its connected open neighborhood U, one has the following objects (we use the notations from [ABLR20, Section 1.2]): Note that the matrix H is defined uniquely up to the transformation H → AH, where A is a constant nondegenerate diagonal matrix. After such a transformation, the matrices Ψ, Γ, and R k transform as follows: Recall also that if we fix H, then the matrices R k are defined uniquely up to the transformation where D i , i ≥ 1, are arbitrary constant diagonal matrices.
Using the notations from [ABLR20, Section 4.4], for any G 0 ∈ C N , let us define an analytic family of F-CohFTs parameterized by a point t ∈ U by where R(z) := i≥0 R i z i . Note that if G 0 = 0, then the maps c G 0 ,t g,n+1 are zero for g ≥ 1.
Proposition 2.6. 1. For any t 0 = (t 1 0 , . . . , t N 0 ) ∈ U, a vector potential of the flat F-manifold corresponding to the F-CohFT c G 0 ,t 0 is equal to F (t * − t * 0 ). 2. For any fixed t 0 ∈ U, the Taylor expansion of c G 0 ,t at t 0 coincides with the formal shift of c G 0 ,t 0 , i.e., c G 0 ,t 0 +τ Proof. 1. We know that under the transformation H → AH, where A is a nondegenerate constant diagonal matrix, R(z) transforms as R(z) → AR(z)A −1 , and therefore Thus, for a fixed t the family {c G 0 ,t } G 0 ∈C N doesn't depend on a choice of H. Let us choose H such that H i (t 0 ) = 1, then c G 0 ,t 0 = Ψ −1 (t 0 )R −1 (−z, t 0 ).c triv,G 0 . The fact that a vector potential of the associated flat F-manifold is equal to F (t * − t * 0 ) was proved in [ABLR20, Section 4.4] (see equation (4.3) there).
2. An elementary computation shows that c G 0 ,t = Ψ −1 R −1 (−z).c H,H −1 G 0 , where by H we denote the vector (H 1 , . . . , H N ). The statement of part 2 of the proposition is equivalent to the property which was proved in [ABLR20, proof of Proposition 4.11].
To our family of F-CohFTs c G 0 ,t , t ∈ U, one can associate a natural vector field X = X α ∂ ∂t α on U where X α is the degree zero part of c G 0 ,t 1,1 (e α ) ∈ H * (M 1,1 ). Note that This motivates the following definition.
Using this language, we can say that our family of F-CohFTs c G 0 ,t induces a framing on U.
Suppose that all the points of our flat F-manifold M are semisimple and X is a framing on M. We see that for any point t 0 ∈ M the above construction gives a family of F-CohFTs around t 0 such that the induced framing coincides with X . This family is not unique because the matrix R(z) is defined uniquely only up to the transformation (2.7). Suppose that M is simply connected. Then it is easy to see that there is a consistent choice of matrix R(z) in all the charts such that the local families glue in a global family of F-CohFTs parameterized by t ∈ M. Let us denote this global family by c X ,t . This global family is not unique: in order to fix the ambiguity, one can, for example, fix a choice of matrix R(z) at some fixed point of M. Note that if X = 0, then the maps c X ,t g,n+1 are zero for g ≥ 1.
Let us now apply the construction of the DR hierarchy to the F-CohFTs c X ,t . We obtain a family of densities Y t β,d ∈ ( A 1 ) [1] , where the superscript t signals that the densities Y t β,d analytically depend on t ∈ M. It is convenient to consider the generating series of densities Y t β,d : Lemma 2.8. We have Proof. This follows from the definition of the densities Y t β,d , the property ∂ ∂t β (c X ,t ) g,n+1 = π 1 * • (c X ,t ) g,n+2 • (⊗e β ) (which is equivalent to part 2 of Proposition 2.6), and the fact that π * where the class δ 0 i,n+1 was defined in the proof of Proposition 1.7.
Consider now a calibration X α (z) of our flat F-manifold M. Define densities Y t β,d ∈ ( Lemma 2.9. We have Proof. This immediately follows from Lemma 2.8 and the property The previous lemma implies that for a fixed t ∈ M the density Y t β,d is the Taylor expansion of the density Y desc β,d at u γ = t γ , i.e., Y t β,d = Y desc β,d u γ →t γ +u γ , as elements of ( A 1 ) [1] . Therefore, since for any t ∈ M the densities Y t β,d produce a hierarchy of pairwise commuting flows, the densities Y desc β,d also produce a hierarchy of pairwise commuting flows. This hierarchy is called the descendant DR hierarchy.
In more details, the equations of the descendant DR hierarchy are given by ∂u α and P t;α β,d are the differential polynomials (1.13) corresponding to the F-CohFT c X ,t . Also, we adopt the convention P desc;α β,−1 := X α β,−1 . Note that we have We immediately see that P desc;α β,d ε=0 = X α β,d t γ =u γ , and therefore the dispersionless part of the descendant DR hierarchy coincides with the principal hierarchy. For X = 0, the descendant DR hierarchy coincides with the principal hierarchy.
Statements analogous to the ones from Theorem 1.5 are true for the descendant DR hierarchy. We present here the proof of a couple of them.
To summarize the above constructions, given the following data: • a flat F-manifold structure on M ⊂ C N given by a vector potential F such that M is simply connected and all the points of M are semisimple; • its calibration; • a framing on M; we have constructed a dispersive integrable deformation of the principal hierarchy . In the next section, we will prove that the dispersive deformations corresponding to different framings are not related to each other by a Miura transformation that is close to identity.
Definition 2.11. Two dispersive deformations of the principal hierarchy of a calibrated flat F-manifold are called equivalent if they are related by a Miura transformation that is close to identity.
It is easy to see that under the Miura transformation the Miura matrix of our system of PDEs transforms as follows: Now consider the descendant DR hierarchies corresponding to different framings X and X . Let us denote the Miura matrices of a flow ∂ ∂t α d from these two hierarchies by S (α,d) (z) and S (α,d) (z), respectively. Clearly, S (α,d) (0) = S (α,d) (0). Suppose that the hierarchies are related by a Miura transformation that is close to identity. Denote its symbol by T (z). For the calibration of our flat F-manifold, without loss of generality, we can assume that X α β,−1 = δ α β . Consider the expansions For the descendant DR hierarchy corresponding to the framing X , we have P t;α 1 1,1 = P t;α 1 1,1 + P t;α µ,0 X µ 1 1,0 + X α 1 1,1 , P t;α µ,0 = P t;α µ,0 + X α µ,0 , which implies that the matrix S = (S α β ) := S (1 1,1) 2 − N µ=1 X µ 1 1,0 S (µ,0) 2 is given by which is equal to 2Coef a 2 DR 1 (a,−a) λ 1 = 1 12 times the degree zero part of c X ,t 1,2 (e α ⊗ e β ). By the construction of the cohomological field theory c X ,t , the degree zero part of c X ,t 1,2 (e α ⊗ e β ) is Ψ −1 is a diagonal matrix for any 1 ≤ α ≤ N and d ≥ 0, the diagonal part of Ψ S Note that during the proof of the theorem we have obtained the following explicit relation between a framing and the differential polynomials defining the flows ∂ ∂t 1 1 1 and ∂ ∂t µ 0 of a corresponding descendant DR hierarchy.
Lemma 2.13. Consider a flat F-manifold, a calibration satisfying X α β,−1 = δ α β , a framing X = X α ∂ ∂t α , and a corresponding descendant DR hierarchy. Then we have 2.5. Homogeneous dispersive deformations. As at the beginning of Section 2.3, consider a semisimple flat F-manifold structure on M ⊂ C N defined by a vector potential F , a semisimple point, canonical coordinates u i on an open neighborhood U of this point, the diagonal matrix of one-forms D, a diagonal nondegenerate matrix H, and matrices R k . Suppose that our flat F-manifold is homogeneous with an Euler vector field E of the form (2.4). By [ABLR20, Proposition 1.14], the diagonal matrix i E D is constant, i E D = −diag(δ 1 , . . . , δ N ) = −∆, δ i ∈ C. Moreover, we have E α ∂ ∂t α H = ∆H, and by [ABLR20, Proposition 1.16] we can fix a choice of matrices R k by the additional conditions E α ∂ ∂t α R k = −kR k + [∆, R k ] for k ≥ 1. By [ABLR20, proof of Theorem 4.10], for an arbitrary 1 ≤ l ≤ N and an eigenvector G 0 of the matrix ∆ corresponding to the eigenvalue δ l the family of F-CohFTs c G 0 ,t satisfies the property This implies that for any t ∈ U the F-CohFT c G 0 ,t is homogeneous of conformal dimension −2δ l . Note that the corresponding framing X on U satisfies the property [E, X ] = (−2δ l − 1)X .
Suppose that M is connected, then it is clear that up to permutations of the components the vector (δ 1 , . . . , δ N ) doesn't depend on a semisimple point. We come to the following natural definition.
Suppose that all the points of M are semisimple. As in the previous section, we can now glue the local families of F-CohFTs in a global family. Note that given a framing X on M satisfying [E, X ] = (−2δ l − 1)X we can now construct a unique global family c X ,t , t ∈ M, of F-CohFTs fixing the choice of matrices R k using the Euler vector field.
Summarizing the considerations of this section, we obtain the following result.
Theorem 2.15. Consider a homogeneous flat F-manifold structure on a connected open subset M ⊂ C N defined by a vector potential F . Suppose that all the points of M are semisimple. Let γ = (γ 1 , . . . , γ N ) be the vector of conformal dimensions. Let 1 ≤ l ≤ N and let X be a framing on M such that [E, X ] = (γ l − 1)X . Then the family of F-CohFTs c X ,t satisfies the property In particular, for any t ∈ M the F-CohFT c X ,t is homogeneous of conformal dimension γ l .
Let us now discuss properties of the descendant DR hierarchies in the homogeneous case. Under the assumptions of the theorem, suppose also that M is simply connected. By [BB19, Proposition 4.4], there exists a calibration X α (z) and complex matrices R i , i ≥ 1, such that where X(z) := d≥−1 X α β,d z d+1 and R(z) := i≥1 R i z i . Such a calibration is called homogeneous. Consider now the associated descendant DR hierarchy.
Let us introduce a generating series P desc (z) by Proposition 2.16. We have Proof. Let us introduce generating series P t (z) and P t (z) by We have to check that For this, we compute where C γ := (C α γβ ), and we recall that C α βγ = ∂ 2 F α ∂t β ∂t γ . Since zE γ C γ X(z) = E γ ∂ ∂t γ X(z) = z ∂ ∂z X(z) + [X(z), Q] + X(z) R(z), the expression in line (2.9) is equal to as required.

Towards a classification of dispersive deformations
In this section we consider the problem of classification of dispersive integrable deformations of principal hierarchies for flat F-manifolds and observe the central role played in it by the DR hierarchies. We propose two a priori different classes of deformations and we classify them, up to some finite order in ε, for 1 and 2 dimensional flat F-manifolds, respectively. Up to that approximation, we observe that both classes contain essentially the DR hierarchies considered in Section 2.
3.1. Dispersive deformations of DR type and the rank 1 case.
3.1.1. Integrable systems of DR type. Given a local vector field X ∈ ( Λ 1 ) [1] , consider the operator D X : with the initial conditions Y α (z = 0) = −θ α,1 . Then a new vector of solutions with the same initial conditions can be found by the following transformation: Theorem 3.1. Assume that X ∈ ( Λ 1 ) [1] satisfies the following properties: , where δ δu 1 1 = A α δ δu α and A α are some complex constants. Then, up to a transformation of type (3.1), we have The proof follows closely the proof of [BDGR19, Theorem 5.1-5.2] with Lie brackets of densities of local vector fields replacing Poisson brackets of differential polynomials.
Remark 3.2. When we restrict to ε = 0, a particular local vector field satisfying condition (a) of Theorem 3.1 is given by X = −(D − 2) F α (u 1 , . . . , u N )θ α,1 dx where the functions F α (t 1 , . . . , t N ) are solutions to the oriented WDVV equations (2.2), (2.3). It is easy to check that for such X solutions Y α (z) are given by Y α (z) = − d≥−1 X β α,d θ β,1 z d+1 where the functions X β α,d form a calibration of the flat F-manifold satisfying X β α,−1 = δ β α (see Section 2.2). Therefore, the functions Y α (z) are the generating series of densities of local vector fields of the principal hierarchy of the flat F-manifold. Note that condition (b) for our X is equivalent to ∂F α ∂t 1 1 = t α , which can always be fulfilled by adding to F α appropriate linear terms.
Theorem 3.4. The double ramification hierarchy (1.11) associated to an F-CohFT is a hierarchy of double ramification type.
Proof. Hypotheses (a) and (b) of Theorem 3.1 follow immediately from claims (iii) and (v), respectively, of Theorem 1.5.
3.1.2. Classification of rank 1 hierarchies of DR type. Thanks to Theorem 3.1 and Remark 3.2, it makes sense to use equation (3.2) to find all possible deformations of DR type of a principal hierarchy associated to a given flat F-manifold, at low order in the dispersion parameter ε. These deformations will, in particular, include the ones coming from all F-CohFTs with the given genus 0 part.
Remark 3.5. It is easy to check that the r.h.s of equation (3.3) is a total x-derivative. Comparing with the results of [ALM15] we see that a similar result can be obtained starting from generic scalar conservation laws of the form ; the reduction to this form by means of a Miura transformation is always possible and it is unique, and imposing the following conditions: • Commutativity of the flows.
• String property: ∂ ∂u P d+1 = P d for d ≥ −1, where P −1 := 1. According to the conjecture formulated in [ALM15], it should be possible to write all the coefficients appearing in the deformation as functions of the coefficients of the quasilinear part. Moreover, the coefficients of the quasilinear part should be constant (due to the string property) and arbitrary. This is consistent with the formula (3.3) since the additional free parameters appearing at the order ε 8 are related to the coefficients of the quasilinear part by constraints obtained considering higher order conditions. Even more intriguing is the isolated deformation (3.4), which, up to reabsorbing the constant C into the factor ε, is the celebrated Burgers equation, which is dissipative and hence non-Hamiltonian. The appearence of terms with odd powers of ε in a hierarchy of DR type rules out the possibility that it is the double ramification hierarchy of an F-CohFT. However, considering that flat F-manifolds are known to appear in genus 0 open Gromov-Witten and Saito theory [PST14, BCT18, BCT19, BB19] it is tempting to conjecture that Burgers equation (3.4) and its higher symmetries might control some version of F-CohFT on the space of Riemann surfaces with boundaries, where curves can indeed possess half-integer genus accounting for odd powers of the genus parameter ε.
The fact that Burgers equation (3.4) and its higher symmetries form a hierarchy of DR type can be proved rigorously at all order in ε as follows.
Theorem 3.6. The vector field X = (uu x + εu xx )θdx of the Burgers equation defines a hierarchy of DR type, i.e., it satisfies conditions (a) and (b) of Theorem 3.1.
Proof. Let us first present a reformulation of the Schouten-Nijenhuis bracket [·, ·] : Λ 1 × A 1 → A 1 in terms of formal differential operators. Consider an arbitrary local vector field X = Xθdx ∈ Λ 1 and a density Y = k≥0 Y k θ k ∈ A 1 . The local vector field X defines a flow on the space of differential polynomials by and we consider also formal differential operators L X and L Y defined by Directly from the definition (1.2), we obtain the following identity: where we apply the differentiation ∂ ∂t to the operator L Y coefficient-wise.
Let us prove condition (a) of Theorem 3.1 by showing that a required solution Y (z) = k≥−1 Y k z k+1 of equation (3.2) is given by Since L X = −u∂ x + ε∂ 2 x , equation (3.2) is equivalent to where D := n≥0 u n ∂ ∂un + ε ∂ ∂ε , and we apply D to L Y (z) coefficient-wise. Note that DL Y (z) = z ∂ ∂z L Y (z) . Therefore, equation (3.6) is equivalent to Note that the last equation follows from the elementary identity ∂ Condition (b) of Theorem 3.1 immediately follows from the equation δ δu X = −uθ 1 + εθ 2 .
such that the following properties are satisfied: (1) Commutativity of the flows: the flows ∂ ∂t β d pairwise commute, (2) The dispersionless limit of the system (3.7) coincides with the principal hierarchy of the given calibrated flat F-manifold, (3) String property: ∂ ∂u 1 1 P α β,d+1 = P α β,d for d ≥ −1, where P α β,−1 := δ α β , (4) Dilaton property: In this section, working out the N = 2 case, we observe how descendant DR hierarchies appear in the problem of classification of dispersive integrable deformations of principal hierarchies of flat F-manifolds of the above form, which we refer to as a homogeneous deformation with string and dilaton properties. The role played by conditions (3), (4), and (5) is central in producing finite dimensional spaces of deformations even without having to quotient with respect to equivalence up to Miura transformations of the dependent variables.
Remark 3.7. Axioms (1), (3), and (4) above correspond closely to properties (iii), (iv), and (v) of Theorem 3.1 for hierarchies of local vector fields of DR type. Homogeneity (5) corresponds to property (iii) of Proposition 1.7 for homogeneous DR hierarchies. Finally, condition (2) above is satisfied by hierarchies of DR type, see Remark 3.2. This means that homogeneous dispersive deformations with string and dilaton properties contain homogeneous descendant hierarchies of DR type whose local vector fields have only even powers of ε. It's not a priori clear that the converse is true and it would be interesting to investigate this point.

3.2.2.
Classification of semisimple homogeneous flat F-manifolds in dimension 2. In the semisimple case, using canonical coordinates u 1 , ..., u N , the structure of a homogeneous flat F-manifold can be recovered from a solution of the following system ([AL19]): For N = 2, the above system reduces to (3.8) and (3.9), and the general solution is where ǫ 1 and ǫ 2 are arbitrary constants. Note that the corresponding vector of conformal dimensions is equal to (2ǫ 2 , 2ǫ 1 ). In order to compute a vector potential, we need to introduce flat coordinates u, v (these correspond to t 1 , t 2 in Section 2.1). We have to distinguish 3 cases: I. ǫ 1 + ǫ 2 = 0, 1. In this case, flat coordinates are where c = 2 ǫ 1 −ǫ 2 ǫ 1 +ǫ 2 , m = 1 1−ǫ 1 −ǫ 2 = 0, 1, and a vector potential is The unit is ∂ ∂v , the Euler vector field is E = 1 m u ∂ ∂u +v ∂ ∂v , and γ = (2−c)(m−1) If m is a half-integer, these are the vector potentials of the bi-flat F-manifold structures defined on the orbit space of the dihedral group I 2 (2m) [AL17]. If also c = 0, the above vector potential comes from the Dubrovin-Frobenius manifold structure defined on the orbit space of the dihedral group.
II. ǫ 1 = c, ǫ 2 = 1 − c, c = 0 (see the remark about the case c = 0 below). Using the flat coordinates we obtain The unit is 1 c ∂ ∂u , the Euler vector field is E = u ∂ ∂u − ∂ ∂v , and γ = (2 − 2c, 2c). For c = 1 2 , the above vector potential comes from the genus 0 Gromov-Witten potential of the complex projective line.
In the case c = 0, choosing the flat coordinates u = u 2 and v = − ln (u 1 − u 2 ), we obtain F 1 = u 2 2 and F 2 = uv − e −v . This flat F-manifold is isomorphic to the flat F-manifold (3.10) with c = 1 and shifted by v → v + πi.
If c = 0, then choosing as flat coordinates the canonical coordinates u = u 1 and v = u 2 we obtain The unit is ∂ ∂u + ∂ ∂v , the Euler vector field is E = u ∂ ∂u + v ∂ ∂v , and γ = (0, 0).

3.2.3.
Integrable deformations of rank 2 homogeneous principal hierarchies. We now want to classify all homogeneous deformations with string and dilaton properties of principal hierarchies associated to the homogeneous two-dimensional flat F-manifolds considered above. In our computations below, we have observed the following remarkable facts: • If such a deformation exists and is nontrivial at the ε 2 approximation, then γ must be equal to γ 1 or γ 2 .
• For γ = γ i , at the ε 2 approximation, any such deformation coincides with the descendant DR hierarchy constructed using an appropriate framing. In particular, any such deformation at the approximation up to ε 2 can be extended to a deformation at all orders of ε.
Let us consider all three cases from Section 3.2.2 in detail.
Case I. For simplicity, we consider the case 1 m = Z, which guarantees that there is a unique homogeneous calibration of standard type such that R i = 0 for i ≥ 1. Recall that the vector of conformal dimensions is (γ 1 , γ 2 ) = (2−c)(m−1) 2m , (2+c)(m−1) 2m . We have three subcases.
Case I1. If γ 1 = γ 2 and γ = γ 1 , we obtain Here A is an arbitrary complex constant. This deformation is given by the descendant DR hierarchy corresponding to the framing (X 1 , Case I2. If γ 1 = γ 2 and γ = γ 2 , we obtain Case I3. If γ 1 = γ 2 (which is equivalent to c = 0) and γ coincides with them, we get a two-parameter family of deformations formed by linear combinations of the deformations from Cases I1 and I2.
Case II3. If γ 1 = γ 2 (which is equivalent to c = 1 2 ) and γ coincides with them, we get a two-parameter family of deformations formed by linear combinations of the deformations from Cases II1 and II2.
Case III. There is a unique homogeneous calibration of standard type such that R i = 0 for i ≥ 1. Recall that the vector of conformal dimensions is (γ 1 , γ 2 ) = (−2c, 2c).

General integrable deformations and open problems.
In Section 3.2.1, we considered the problem of classification of dispersive deformations, containing only even powers of ε and satisfying properties (1)-(5), of principal hierarchies of two-dimensional homogeneous semisimple flat F-manifolds. We observed that at the approximation up to ε 2 all such deformations are given by the descendant DR hierarchies.
In this section, we consider more general dispersive deformations of the same principal hierarchies: first, we allow odd powers of ε in the dispersive deformation (3.7), and, second, we require that only properties (1)-(2) are satisfied. In other words, we require only integrability, i.e., pairwise commutativity of the flows. In the table below, we summarize the results of computations of such deformations at the approximation up to ε 2 (the results for Case I were already obtained in [AL18]). When we refer to a functional parameter relative to an integrable deformation, we mean that at a specified order the equivalence classes of deformations depend on an arbitrary function. Recall (see Definition 2.11) that two deformations are said to be equivalent if they are related by a Miura transformation that is close to identity.
For special values of the functional parameters, we recover the genus one approximations of the descendant DR hierarchies from Section 3.2.3. Unfortunately, for generic choices of the functional parameters the existence of a full dispersive hierarchy is an open problem. Toward this direction, let us point out that in [AL18] it was conjectured that up to equivalence integrable deformations are labelled by a simple set of invariants called Miura invariants. Consider a