Higher Airy structures and topological recursion for singular spectral curves

We give elements towards the classification of quantum Airy structures based on the $W(\mathfrak{gl}_r)$-algebras at self-dual level based on twisted modules of the Heisenberg VOA of $\mathfrak{gl}_r$ for twists by arbitrary elements of the Weyl group $\mathfrak{S}_{r}$. In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion \`a la Chekhov-Eynard-Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard-Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves, giving a general ELSV-type representation for the topological recursion amplitudes on smooth curves, and formulate precise conjectures for application in open $r$-spin intersection theory.

, Airy structures consist of a system of linear PDEs depending on a parameter ℏ and satisfying a compatibility condition, so that they admit a simultaneous solution which is unique when properly normalised. This solution is called "partition function" and is encoded in Taylor coe cients , indexed by integers ≥ 1 and (half-)integers ≥ 0, which are determined by a topological recursion, i.e. a recursion on 2 − 2 + . The interest in Airy structures comes from the numerous applications of the topological recursion in enumerative geometry, see e.g. [Eyn a].
The purpose of Part I of this work is to construct new Airy structures from representations of the W ( )-algebra. The latter is realised as a sub-VOA of the Heisenberg VOA F C . Twisting the free eld representation of F C by an arbitrary element of the Weyl group ∈ gives rise to an (untwisted) representation of W ( ). Applying a "dilaton shift", [BBCCN ] constructed all the Airy structures that can arise when is an -cycle or an ( − 1)-cycle. Our work explores the possibility of constructing Airy structures from arbitrary ∈ : we will obtain in Theorem . conditions on and the dilaton shifts that are su cient for the success of this construction. This gives rise to many new Airy structures and thus partition functions, for which it would be desirable to nd enumerative interpretations. This approach and the result are presented in Section while Section is devoted to the proof of the main Theorem . .
In Part II, we show that the topological recursion for all these Airy structures can be equivalently formulated via residue (hence, period) computations on possibly singular spectral curves. Roughly speaking, a spectral curve is a branched cover of complex curves : → 0 equipped with a meromorphic function and a bidi erential 0,2 on 2 . The original formulation of the topological recursion, with a spectral curve as input, was developed by Chekhov, Eynard and Orantin [EO ; EO ] in the case of smooth curves with simple rami cations. The output of this CEO recursion is a family of multidi erentials , on that have poles at rami cation points of , are symmetric under permutation of the copies of , and obtained recursively by residue computations on . As observed already in [EO ] and revisited by [KS ; ABCO ], the corresponding Airy structure is based on the Virasoro algebra and related to the W ( 2 )-algebra; for each , , , or , contain the same information packaged in a di erent way. The de nition of the CEO topological recursion was extended in [BHLMR ; BE ] to smooth curves with higher order rami cation points, and its correspondence with W ( )-Airy structures when is an -cycle was established in [BBCCN ]. An important application of this correspondence is a conceptual proof of symmetry of the , based on representation theory of W-algebras. This led the discovery of a non-trivial criterion on the order of at rami cation points for the symmetry to hold. From [BHLMR ], it is easy to propose a de nition of the CEO recursion also valid for singular spectral curves (see Equation ( )), i.e. if has several irreducible components intersecting at rami cation points of arbitrary order. It is however unclear (and in fact not always true) that this de nition leads to symmetric , .
In Proposition . and Theorem . , we show that this de nition naturally arises from the W ( )-Airy structures with arbitrary permutation encoded the rami cation pro le over a branchpoint in a normalisation of , and dilaton shifts specifying the order of at the rami cation points. Smooth spectral curves correspond to having a single cycle. Besides, the basic properties of Airy structures guarantee that the corresponding , are symmetric. The results of Part I therefore give su cient conditions on the rami cation type and the order of at rami cation points for the symmetry to hold, see De nitions . to . . The central result of Part II is then Theorem . giving the correspondence between those Airy structures and our extension of the CEO topological recursion.
In terms of spectral curves, the CEO-like topological recursion provides the (unique when properly normalised) solution to the so-called "abstract loop equations". The latter express that certain polynomial combinations of the ( , ) , are holomorphic at the rami cation points. This in fact provides tools that have been used to establish applications of the topological recursion e.g. in matrix models [BEO ], in Hurwitz theory [BKLPS ; DKPS ], and to the reconstruction of WKB expansions [BBE ; IMS ; BEM ]. The setup of abstract loop equations was developed in [BEO ; BS ] for smooth curves with simple rami cations and extended in [Kra , Section . ] to higher order rami cations. In Section . , we de ne a notion abstract loop equations for arbitrary spectral curves (De nition . ) and prove they admit at most one normalised solution (Proposition . ), which must then be given by our extension of the CEO recursion (Equation ( )). The question of existence of a solution is then reduced to proving this formula yields symmetric , . As we establish the equivalence of the abstract loop equations with the di erential constraints built from W ( )-algebra representations (Proposition . ), it is su cient to check that the latter form an Airy structure to establish symmetry.
Our results give a de nition of topological recursion for many spectral curves that could not be treated before, for instance ( ) In general, nodal points such that tends to distinct non-zero values on both sides are admissible. if : ( , ) = 0 are admissible for ∈ {1, 2} do not intersect in C 2 at zeroes of d or singular points of , then : 1 ( , ) 2 ( , ) = 0 is admissible. The complete list of conditions de ning admissible spectral curves according to our work can be found in Section . , and they only regulate the behavior of and at the points where the cardinality of the bers of jump (zeroes of d and singular points). Examples of non-admissible curves are: , coprime and > 1, > 0 , ( − 2 ) 2 = 0 , In general, the following cases are not admissible for us: non-reduced curves, curves where d and d have a common zero, reducible curves where there is a point at which at least two irreducible components 1 and 2 meet and where d has a zero and is regular on 1 and is not identically zero on 2 . It would be desirable to understand if the admissibility conditions can be weakened even more with a suitable modi cation of the topological recursion residue formula.
We expect all the partition functions of the Airy structures present in this article to admit an enumerative interpretation, i.e. that , or , can be computed via intersection theory on a certain moduli space of curves. This was achieved by Eynard in [Eyn ; Eyn b] in the case of simple rami cation points on smooth curves and holomorphic, in a form that has a structure similar to the ELSV formula [ELSV ], and found applications in Gromov-Witten theory [EO ] and Hurwitz theory [SSZ ; KLPS ]. The case of having a simple pole was later treated by Chekhov and Norbury [Nor ; CN ]. Part III explores the generalisations of this link to other spectral curves that can directly be reached by combining known results with the results of Parts I and II. This stresses the role of Laplace-type integrals on the spectral curve. Although it is not essential in the theory, in the case of global spectral curves the Laplace transform of 0,2 enjoy a factorisation property reminiscent to the use of -matrices in the theory of Frobenius manifolds. This was known by [Eyn b, Appendix B] for smooth spectral curves with simple rami cations, and we show in Corollary . that it extends to singular spectral curves in a slightly di erent form.
In Section . . , building on [BBCCN ] and Theorem . we generalise Eynard's formula to all smooth spectral curves with arbitrary rami cation and of order 1 at the rami cation points -this involves Witten spin classes. This answers a question of Shadrin to the rst-named author. In Section , we apply our general results to the W ( 3 )-constraints of Alexandrov [Ale ] for the open intersection theory developed in [PST ; Bur ; BT ; ST a]. The open intersection numbers can be packaged satis es our extension of the CEO topological recursion applied to the curve ( 2 − 2 ) = 0 (corollary . ). Using modi ed W-constraints, Safnuk had derived in [Saf ] a residue formula associated to this curve, but its structure is di erent and not easily generalisable. On the other hand, we can easily conjecture a residue formula for the open -spin theory. This conjecture is equivalent to the W ( ) constraints rst mentioned in [BBCCN , Section . ]: we expose it in more detail, in particular specifying the normalisations necessary for the comparison, and give some support in its favor by comparison with [BCT ].
Remark . . At the time of writing, there are several foundational conjectures in open intersection theory. In Section . and Section . , we formulate the ones that are directly relevant for us and explain the logical dependence of our statements on these conjectures.
In fact, one of the initial motivation of our work was to generalise the structure of Safnuk's residue formula [Saf ] to higher , and seek along this line for a de nition of topological recursion for curves with many irreducible components. Our conclusion is that, although we do not know how to generalise the structure of Safnuk's recursion, there is a simpler and general de nition of the recursion, which retrieves open intersection theory when applied to the reducible curve ( 2 − 2 ) = 0.
The W ( )-representations that we consider have an explicit though lengthy expression. We extract from them a few concrete calculations: • General formulas for 0,3 , 1, 1 2 and 1,1 , and partial computations for 0,4 . Their symmetry give necessary constraints for and the dilaton shifts. They are weaker than the su cient conditions under which we have an Airy structure according to Theorem . . However, we are inclined to think that our result is generically optimal, i.e. that the su cient conditions in Theorem . are also necessary conditions for generic value of the dilaton shift, that could be implied by the symmetry of 0, for higher ≥ 5.
While this work was in the nal writing stage, we learned that results similar to our Theorem . but restricted to the case of cycles of equal lengths (perhaps with a xed point) are obtained in an independent work of Bouchard and Mastel [BM ]. The question of de ning a Chekhov-Eynard-Orantin topological recursion on singular curves mentioned in their work is solved by Part II of our work. A partition of ∈ N, denoted , is = ( 1 , . . . , ℓ ) such that 1 + · · · + ℓ = . We will sometimes (but not always) require ≥ +1 , in which case we say that is a descending partition. Often it is convenient to express equal blocks = +1 = · · · = + as . Moreover, to any descending partition one can associate a Young diagram Y in a bijective way. For example all notations = (4, 3, 3, 1) ←→ = (4, 3 2 , 1) ←→ Y = characterise the same descending partition 11. The size of is | | = and its length is ℓ ( ) = max{ | > 0}.
If is a nite set, we write L to say that L is a partition of , that is an unordered tuple of pairwise disjoint non-empty subsets of whose union is . We denote ||L|| the number of sets in the partition L.
All of our vector spaces or algebraic spaces are over C. We denote C a 1-dimensional complex vector space equipped a non-zero linear form .
In this Section, we recall the de nition of Airy structures and their partition function and its adaptation to the in nite-dimensional setting for which it will be used. We present the main results of Part I, i.e. we exhibit Airy structures that can be constructed from W ( )-algebra modules, while the proofs are carried on in Section . . A S

. . Finite dimension
We rst present the de nition when is a nite-dimensional C-vector space. For convenience, let us x a basis ( ) ∈ of and ( ) ∈ be the dual basis of * . We consider the graded algebra of di erential operators D ℏ , also called Weyl algebra. It is the quotient of the free algebra generated by ℏ 1 2 , ( ) ∈ and (ℏ ) ∈ , modulo the relations generated by and equipped with the grading We will write = + O ( ) for two elements , ∈ D ℏ if they agree up to at least degree − 1.
De nition . . A family ( ) ∈ of elements of D ℏ is an Airy structure on in normal form with respect to the basis ( ) ∈ if = and it satis es: • The degree 1 condition: for all ∈ , we have • The subalgebra condition: there exist , ∈ D ℏ such that for all , ∈ A family ( ) ∈ of elements of D ℏ is an Airy structure if there exists two matrices N ∈ C × and M ∈ C × such that the family˜= ∈ N , indexed by ∈ is an Airy structure in normal form.
Being an Airy structure does not depend on a choice of basis, but being an Airy structure in normal form does. Airy structures in De nition . would be called in [BBCCN ] "crosscapped higher quantum Airy structure". "Crosscapped" refers to the presence of half-integer powers of ℏ and we comment it in Section . . . "Higher" means that compared to the de nition in [KS ; ABCO ], it can contain terms of degree higher than 2. "Quantum" is used to distinguish it in [KS ; ABCO ] from the classical Airy structures where the Weyl algebra is replaced by the Poisson algebra of polynomial functions on * . We simpli ed the terminology as the restriction to maximum degree 2 and the classical Airy structures will not play any role in this article and handling half-integer powers of ℏ does not lead to any complication in the theory.
The essential property of an Airy structure is that it speci es uniquely a formal function on .
Theorem . . [KS , Theorem . . ], [BBCCN , Proposition . ] If ( ) ∈ is an Airy structure on , then the system of linear di erential equations is called the partition function (resp. free energies). Equation ( ) implies a recursive formula for , on 2 − 2 + > 0. We will typically be interested in the coe cients of decomposition of the free energies on a given basis ( ) ∈ of linear coordinates of , for which we use the notation: When the Airy structure has normal form with respect to this basis, the explicit recursion to obtain the , [ 1 , . . . , ] from appears in [BBCCN , Sections . and . ]. We reproduce it in Equation ( ) at the only place where it is used in the article. We will be led to work with Airy structures that are not given in normal form (cf. Section . ), but for which there is an equivalent formulation of the recursion in terms of spectral curves (Part II), that is often more e cient for calculations (cf. Section ).

. . In nite dimension
In this article we need to handle Airy structures for certain in nite-dimensional vector spaces. This requires some amendments of the previous de nitions which we now explain.
A ltered vector space is a vector space together with a collection of subspaces 0 ⊆ F 1 ⊆ F 2 ⊆ · · · ⊆ , called the ltration. Throughout this paper we will assume that for any ltered vector space , the F are nite-dimensional, and that = >0 F . Two ltrations F , F on a vector space are equivalent if for any > 0 there exists > 0 such that F ⊆ F and for any > 0 there exists > 0 such that F ⊆ F . In particular, all ltrations on a given vector space satisfying our extra assumptions are equivalent.
A ltered set is a set together with a collection of subsets ∅ ⊆ 1 ⊆ 2 ⊆ · · · ⊆ . Again, we assume all are nite and =
• for each ∈ , M , vanish for all but nitely many ∈ ; • N and M are inverse to each other in the sense of ( ) -where the sums are nite due to the previous two points ; • the family˜= ∈ N , indexed by ∈ -which is a ltered family of elements of D ℏ by the rst point -is an Airy structure on in normal form with respect to ( ) ∈ .
The notion of Airy structure does not depend on the choice of ltered basis, and only depends on the equivalence class of the ltration of . We will sometimes omit to specify the ltration when it is evident. The existence and uniqueness of the partition function (Theorem . ) extends to this in nite-dimensional setting and for each , , and , the summation of ( ) with 1 restricted to is nite, that is , ∈ Sym ( * ).

. T W ( )
The Airy structures constructed in this paper are obtained by considering twisted modules of Heisenberg vertex operator algebras (VOAs), taking subalgebras of the associated algebras of modes, and using a dilaton shift to break homogeneity. The idea of this construction dates back to [Mil ]. It was developed more systematically in [BBCCN , Sections and ] and we refer to that paper for details. Here we summarise the main points of the construction.
. . The Heisenberg VOA and W ( )-algebras De nition . . Let be a nite-dimensional vector space with non-degenerate inner product ·, · . The Heisenberg Lie algebra associated to is given bŷ The associated Weyl algebra is de ned as a quotient of its universal enveloping algebra: H (ˆ)/( − 1). The Fock space F is the representation of H generated by a vector |0 and relations |0 = 0 for ∈ , ≥ 0. The Heisenberg VOA is the vertex operator algebra with underlying vector space F equipped with vacuum |0 , state-eld correspondence : F → End F ˜± 1 given by and conformal vector = 1 2 −1 −1 |0 for an orthonormal basis ( ) of . The W-algebra associated to the general linear Lie algebra at the self-dual level is denoted W ( ). It can be constructed as a sub-VOA of the Heisenberg VOA F attached to its Cartan subalgebra C ⊂ . We identify with its dual using the Killing form and take ∈ to correspond to the roots under this identi cation. Note that they are orthonormal.
We introduce the modes , and their ( -form valued) generating series (˜) with the formulas Remark . . Contrarily to [BBCCN ], we do not include a factor of −1 in the de nition of . Our convention that , is the coe cient of˜− ( + ) . This coincides with the convention taken in [BBCCN , Section . . ] but di ers from the convention˜− ( +1) used in the rest of [BBCCN ]. We also nd convenient to consider the generating series of modes of conformal weight to be -di erential forms. The variable˜is often denoted in the VOA literature. However, in Part II, we will see that this variable can be interpreted as the pullback under the normalisation morphism of a function usually denoted for a spectral curve. For consistency, we therefore chose to use the letter˜, and use the letter for local coordinates on (the normalisation of) the spectral curve.
. . The mode algebra and its subalgebras Let A be the associative algebra of modes of W ( ), see [FBZ , Section . ]. Furthermore, let (A) denote the set of possibly in nite sums of ordered monomials in A whose degree and conformal weight is bounded ; we equip it with the bracket ℏ −1 [·, ·], making it a Lie algebra.
De nition . . We say that a subset ⊂ [ ] × Z of modes generates a Lie subalgebra of (A) if the left A-ideal generated by ⊕ ( , ) ∈ , is a Lie subalgebra of (A). An equivalent condition is the existence of ( , ) ( , ),( , ) ∈ (A) such that , .
More precisely, it is rst required that the right-hand side de nes an element of (A), and then that it coincides with the left-hand side.
Given a partition , we set and we de ne the index set Theorem . ] For any descending partition , generates a Lie subalgebra of (A).

. . Twisted modules
Let ∈ be an arbitrary element of the Weyl group of . It is a permutation of the elements and can thus be decomposed into ≥ 1 cycles = 1 · · · with each cycle of length ≥ 1 such that 1 + · · · + = . If = 1 then is a transitive element. After relabelling of the elements of the basis of the Cartan, we can assume that acts as : while keeping all other xed, where we introduced the notation [ ] =1 . It is then easy to check that 2i / is an eigenvector of the action of on , with eigenvalue . We de ne the currents via the state eld-correspondence with fractional mode expansion on these eigenvectors introducing the di erential operators ]. Note that the formal variables˜and are unrelated. The state-eld correspondence can be extended to the whole space F by using formula ( ). This turns (T , ) into a twisted representation of F , whose restriction to W ( ) becomes an (untwisted) representation of W ( ). For details see [Doy ; BBCCN ]. We de ne the twist modes where are the strong generators of W ( ) from ( ). They are di erential operators acting on T .
Lemma . . [BBCCN , Proposition . and Lemma . ]. For an automorphism with cycles of respective lengths the -twisted modes read where the , are de ned as : , ( ) with coe cients Ψ ( ) (· · · ) ∈ Q admitting a representation in terms of sums over th roots of unity: In terms of generating series, this lemma can be restated as If we also take a generating series in by de ning this can be written compactly as Introducing the ltered vector space we see from the condition of summations in ( )-( ) that each , belongs to the completed Weyl algebra D ℏ according to the de nitions in Section . . . Even more, any family ( , ) ( , ) ∈ for which min > −∞ is a ltered family of elements of D ℏ . Our goal is to construct Airy structures from these operators. The degree one condition requires the operators to have the form ℏ + O (2). Unfortunately, It follows from the Baker-Campbell-Hausdor formula that conjugating withˆmeans shifting the s as in ( ). The action on the completed Weyl algebra D ℏ is then well-de ned. Then, certain subsets of the modes , yield Airy structures. Theorem . . [BBCCN , Theorem . ] Let ∈ be transitive and ∈ [ + 1] with = ±1 mod and 0 = 0. Let us de ne to be the descending partition .

( )
Then, the family of operators −1 , forms an Airy structure in normal form on = >0 C , seen as a ltered vector space when equipped with the ltration = ≤ C .
The partition chosen in ( ) to de ne the mode set determines the subalgebra associated to this Airy structure by using Theorem . . The corresponding Young diagrams are depicted in table .

. . A generalisation to arbitrary twists
Let ∈ be a permutation with cycles of respective lengths 1 , . . . , , so that 1 + · · · + = . Then the di erential operators , act on the space C ℏ 1 2 ( ) ∈ [ ], >0 . Again, we will break up the homogeneity of , by performing a dilaton shift. There are independent families of variables ( ) >0 labelled by ∈ [ ] in which we can perform the shift. Two types of shifts will in fact lead to Airy structures: • simultaneous shifts in each of the sets of variables.
• simultaneous shifts in all but one set of variables and the label of the unshifted set of variables correspond to a xed point = 1.

Assume that
• 1 = −1 mod 1 . • = 1 for any ∉ {1, }; • = 1 mod . and de ne to be the descending partition where we set . Then, the family is an Airy structure (not necessarily in normal form) on the ltered vector space given in ( ).
We call the case = ∞ the exceptional case and the other case the standard case. The proof of the Theorem will be presented in Section . The exceptional = 2 case was already obtained in [BBCCN , Theorem . ].
In case = 1 the last block is simply absent. Going through all cases one thus nds that every descending partition is either of the form depicted in table or of the form ( . . ). This implies that all the subalgebras mentioned in Theorem . support two Airy structures: one standard and one exceptional.

. . Arbitrary dilaton shifts and changes of polarisation
In order to connect with the theory of the Chekhov-Eynard-Orantin topological recursion, we ought to be able to conjugate the , with more general operators, inducing dilaton shifts in several of the variables and also making a change in polarisation. This section is completely parallel to [BBCCN , Section . . ].
First, let us consider a general dilaton shift exp ∑︁ with arbitrary scalars ℎ,1 − for ℎ ∈ {0, 1 2 }. The seemingly complicated way to denote these scalars will become natural in Part II, see e.g. Equation ( ). E ectively, this shifts − → − + 0,1 − + ℏ Theorem . . De ning by ( ) and with the same conditions for , ( , , , ) =1 and the same range for ( , ) as in Theorem . , the operators , in Equation ( ) form an Airy structure on . This theorem is proved in Section and the result will be reformulated in terms of spectral curves in Section .

. . Necessary conditions
Theorem . gives su cient conditions for the operators , , ∈ [ ], ≥ 1 − ( ) + ,1 ( ) with as in ( ) to form an Airy structure. By checking the symmetry of 0,3 , 0,4 and 1 2 ,2 , we could prove that most of these conditions are also necessary, and we believe that a more thorough analysis would actually lead to the conclusion that they are all necessary for generic values of ( , ) .
The proof of this Proposition can be found in Section . . We in fact use the reformulation of the W-algebra di erential constraint in terms of topological recursion on spectral curves (Theorem . in Part II), as the calculation of the , s -which are generating series for the , s -appears simpler than with the di erential constraints ( ) themselves. In the course of the proof of Theorem . in Section , we will see that coprimality of and and non-resonance condition for the (Remark . ), as well as non-vanishing of all but maybe one dilaton shift (Remark . ) are obvious necessary conditions to obtain Airy structures with our method. The assumption =1 = 0 is not always necessary and we obtain ner information on this in Proposition . , but we adopted it here to simplify the statement of Proposition . .

. . Half-integer or integer powers of ℏ?
There are several reasons to allow half-integer powers of ℏ in Airy structures instead of just integer power. Our construction admits natural extra degrees of freedom when has at least two cycles, namely the parameters in Theorem . . This is relevant for applications to open intersection theory, where we have to allow indices to be both integer or half-integer -see Section . and Theorem . . As it does not lead to any complication, we write the whole article allowing ∈ 1 2 N. In the terminology of [BBCCN ], we are dealing with crosscapped Airy structures. If all monomials in only feature integer powers of ℏ then , in ( ) vanishes for half-integers . It is therefore straightforward to specialise our results to allow only integer , as it is more common in topological recursions. Let us however note that half-integer already made their appearance in certain other applications of the topological recursion, such as non-hermitian matrix models [CE ], enumeration of non-orientable discretised surfaces [CEM ], Chern-Simons theory with gauge groups SO( ) or Sp(2 ) [BE a], etc. . S , Lemma . . Consider one of the Airy structure of Theorem . or Theorem . . For any , ≥ 0 such that 2 − 2 + ( + 1) > 0, any , 1 , . . . , ∈ [ ] and 1 , . . . , > 0, we have

( )
Proof. We express the constraint =2, =0 · = 0, as it is always part of the Airy structure. From ( )-( ) and taking into account 0 = ℏ 1 2 and the evaluations To get =2, =0 we have to apply the dilaton shifts − → − − , which simply results in adding a term =1 to ( ). Expressing the constraint and using the assumption =1 = 0, we see that =2, =0 · = 0 implies that is annihilated by the operator By the representation ( ) of the s, this yields ( ) for the coe cients ( ) of the partition function.
The partition functions of these Airy structures for = (1 · · · ) or (1 · · · − 1) ( ) enjoy an extra property of homogeneity, which turn ( ) into an analog of the dilaton equation. Corollary . . Assume = 2, 1 = −1 mod 1 , ( 2 , 2 ) = (1, ∞) and 1 = 1 1 . Then, the coe cients of the partition function of the Airy structure described in Theorem . (also appearing in [BBCCN , Theorem . ]) satisfy , 1 ··· 1 1 ··· = 0 whenever =1 ≠ 1 (2 − 2 + ), and the dilaton equation Proof of Corollary . . In this case we only need to consider ∈ N. The Airy structure is in normal form up to an overall normalisation, and we can decompose for ∈ ( ] and ≥ 1 − ( ) Corollary . ] gives the following formula for the coe cients of the partition function: [ℓ] means that is a set of non-empty subsets of [ℓ] which are pairwise disjoint and whose union is ℓ, and for ∈ we denote q ( ) ∈ . Then, ( ] is a family of (possibly empty) pairwise disjoint subsets of ( ] indexed by ∈ , whose union is ( ]. The double prime over the summation means that the terms involving 0,1 [ ] or 0,2 [ , ] are excluded from the sum. We note that the summation condition is equivalent to Since ℓ + 2 ≥ 2, this is indeed a recursion on 2 − 2 + ≥ 0 to compute , starting from the value of 0,2 . 0,2 obviously satis es the homogeneous property. Assume the , for 0 ≤ 2 − 2 + < 2 − 2 + satisfy homogeneity. So, the summands that may contribute to ( ) are such that ( ) Writing 1 = Π( , ) and applying the dilaton shift − → − − 1 to ( ) we know that ( ) [ 1 |q] is a linear combination of terms inside which Together with ( ), this proves homogeneity of , . By induction, homogeneity is established for all , .
We then apply Lemma . . In our case, there is a single , = 1 and = 0. Using homogeneity to simplify the right-hand side of ( ), we obtain Proof of Corollary . . The argument is similar and we only point the minor di erences that must be taken into consideration. Although half-integer and are now allowed, this does not spoil the sum constraints appearing in the recursive formula for , and which were used in the argument. According to ( ) and ( ), we have for ∈ ( 1 + 1] where by convention 1 1 +1, = 0. The analysis of the previous proof applies to the term 1 , . Due to the equation 1, · = ( 1 1 + 2 ) = 0, we can obtain a recursion in normal form involving only , 1 ··· 1 * ··· * by substituting which is the same as ( ) and is all what we need to repeat the previous proof and establish homogeneity.
We then specialise Lemma . to our case, that is 1 = 1 1 and 2 = 0, while 2 = − 1 . Setting = 1 for all ∈ [ ] in ( ) and using homogeneity to simplify the right-hand side, we deduce , +1 For the two above cases, we also have a string equation when 1 = 1 + 1.

P T . .
This Section is mainly devoted to the proof of Theorems . and . . These theorems state that certain collections of operators ( , ) ( , ) ∈ de ned in Equations ( ) and ( ) are Airy structures. In fact we only prove the second theorem, and note that the rst is a special case. We rst prove this in the standard case, proceeding as follows.
(I) The operators of an Airy structure must be of the form + O (2). It is thus necessary to rst identify the degree zero and degree one term of , in order to check this condition. (II) In general, one will nd that ( ) for some matrix M and constants , . This means that for generic , we may expect terms proportional to − = . We will thus construct an index set ⊂ [ ] × Z such that for all ( , ) ∈ we have , = 0 and M ( , ),( , ) = 0 if ≤ 0. (III) Nevertheless, even restricted to the degree one term ( ) is in general a linear combination of many s for > 0. In order to bring the operators into the normal form of an Airy structure + O (2) where each ( , ) appears in a unique operator, we will show that the matrix M restricted to is invertible under certain constraints on the dilaton shifts. One can then obtain the operators˜ ∑︁ which are of desired form. (IV) The modes (˜) ( ∈ [ ], >0) satisfy the subalgebra condition if and only if the ( , ) ( , ) ∈ do. The latter is satis ed when is induced by a descending partition of as speci ed in Theorem . . This criterion thus allows for an easy check whether the mode set constructed in (III) satis es the subalgebra condition ( ) of a higher quantum Airy structure. Together the results of (III) and (IV) will directly imply Theorem . and hence Theorem . in the standard case. The steps (I) to (III) will be carried out in Section . and step (IV) is performed in Section . . In Section . we treat in less details the exceptional case, as it resembles the standard case in many ways.
First, let us recall some notation. Let ∈ be a permutation with cycles of respective length such that = 1 + · · · + . We can then de ne the dilaton shifted modes which in the following are the central objects of study. Here , are the -twisted modes from Lemma . . Compared to Equation ( ), we do not introduce the coe cients 1 2 ,1 and 0,2 here yet, as it turns out these are not important for most of the proof. As in Equation ( ), it will be useful to gather these in a generating function.
We will also recall from Equation ( ) To treat these currents more uniformly, we introduce˜= =1˜, the union of copies of a formal neighbourhood˜of the origin in the complex plane. We use the notation ( ) for the coordinate in the˜. We de ne a function˜:˜→ C by˜( ) = . Cf. Part II for more on this viewpoint. We then de ne a uni ed current The factor of in the denominator is a convention making ( ) simpler. We sometimes omit from the notation and simply denote ∈˜. With these notations, we see that the dilaton shift induces We also use the shorthand notation = − 1 0,1 − for the leading coe cient.
. T In this subsection and the next one we assume that all are nite. Let us begin by identifying the degree one component of . Let 1 be the projection to degree one. ) having the property that, for each ( , ), there exists , such that for any ∈ [ ] and ≥ , we have M ( , ),( , ) = 0.
Note that the right-hand side is a symmetric function in the elements of˜− 1 ( ), and therefore it contains only integral powers of .
Proof. From Equation ( ), we see that ( ) is a linear combination of currents After the dilaton shift, these will only contribute to the degree one component of ( ) and therefore on the right-hand side we need to take the contribution of the shifts in all factors but one. The de nition ( ) of the Ψ (0) is nothing but a sum over subsets of Galois conjugates of the function˜, so we obtain The sum over ⊆ [ ] in ( ) 'globalises' this sum of subsets from one component˜to all of˜. As stated before, the degree one projection extracts the dilaton shifts of all but one (the choice of ) of these factors, which proves ( ). We obtain the matrix M by expanding this equation in and collecting the contributions of . The vanishing property comes from the fact that 0,1 contains only nonnegative positive powers of .
We will restrict the range of indices on both sides and show that the matrix M is invertible in order to bring the di erential operators into the normal form of an Airy structure. But rst, we would like to see that this matrix is invertible without restricting it to any subspace yet. De ne ( ) We are looking for the inverse to this operation. Remember from ( ) that for any ∈ [ ] we write Lemma . . Assume gcd( , ) = 1 for all ∈ [ ] and ≠ for any distinct , such that = .
Then the currents can be recovered from 1 ( , ) as follows: .

Proof. If we plug Equation ( ) back in Equation ( ), we get
Because we took in a small neighbourhood of zero (but of course not zero itself), the conditions of the lemma ensure that all 0,1 ( ) for in the same bre have di erent values. Therefore, the only contribution to the residue comes from = .
Remark . . For Lemma . and Remark . to work, all we really need is that 0,1 takes distinct values on all elements of the bre of˜near the rami cation point = 0, i.e. the map (˜, 0,1 ) :˜→ * P 1 is an embedding on a punctured neighbourhood of the rami cation point. The conditions gcd( , ) = 1 and ≠ for ≠ such that = ensure this. In the undeformed case, the setting of Theorem . , i.e. monomial 0,1 , this is in fact necessary as well as su cient.
In the case ≠ such that = and = , the situation is similar. First note that by rede ning the local coordinate on˜, we may actually assume = . Writing the local coordinates as ∈ã nd ∈˜, we then have 0,1 ( ) = 0,1 ( ) for all ∈ [ ]. By the same argument as above, 1 ( ) then depends on and only in the combination + , so again we can never retrieve the individual and .
For the deformed case, see Remark . .
Since the 0,1 are power series with exponents bounded below (by on component˜), we can see that for any , the set of such that , gets a contribution from with ≤ 0 in Equation ( ) is bounded from above. Therefore, the following de nition makes sense.
De nition . . For ∈ [ ], we de ne min ( ) to be the smallest such that for all ≥ , 1 ( , ) given by Equation ( ) lies in the linear span of with ∈ [ ] and > 0 solely.
It turns out that min ( ) can be approximated as follows.
Lemma . . If 1 1 ≥ · · · ≥ and gcd( , where for a subset ⊆ [ ] we denoted r := ∈ and s := Proof. Recall that is the exponent in As we also get factors of d on the right-hand side of Equation ( ) (one from and − 1 from the 0,1 , we will concentrate on the remaining powers of . From Lemma . , we see that in order to determine an upper bound max ∈ Q for the exponents in ( ) for which we get a non-vanishing contribution from with ≤ 0 it su ces to inspect = 0 and to take only the leading order of all 0,1 s into account. More speci cally, to make in ( ) as large as possible, we need to choose the 0,1 s e ciently: an 0,1 on branch has leading order ( ) d = d . So, to minimise the power of , we need to take minimal, i.e. maximal, i.e.
minimal. From this it follows that the maximal such that * , ≤ 0 can contribute, is found by rst taking all 1 factors 0,1 with arguments on component = 1, then the 2 on component = 2, up until we get to − 1 factors.
. Of course, it might still happen that the coe cient of the power − max vanishes due to cancelling contributions of di erent combinations of 0,1 s. However, in any case max ∈ Q is an upper bound for ∈ Q for which * with ≤ 0 can still contribute. Therefore Finally, let us argue that the right-hand side of the above equation is nothing but r,s ( ) as de ned in ( shall be shown to be an Airy structure for some certain ( , ) =1 . For future reference let us therefore de ne the following index set.
De nition . . We de ne the index set r,s to be With the fact that min ( ) ≤ r,s ( ) and Equation ( ) we have two di erent characterisations of r,s . The rst property tells us that for ( , ) ∈ r,s the degree one projection of , is a linear combination of with > 0 only while the characterisation in ( ) will be important later in order to check whether the modes satisfy the subalgebra condition. In the following we will need yet another characterisation of r,s . Proof.
and , ∈ Z if and only if there exists an ∈ Z such that = + and = − we see that ( , ) has to be of the form for some ∈ Z. Remember that we need to prove that ≥ r,s ( ). First assume ≥ 0 and choose ≥ such that ∈ ( [ −1] , [ ] ]. Since > 0 we know that In the second and third line we used that ≥ for ≤ and in the fourth line we plugged in the expression for and used ( ). In the case − [ −1] = 1 we have to be more careful due to the additional contribution from the Kronecker delta in r,s ( ). We nd In this calculation we used the arguments from the prior one and identi ed − [ −1] = 1 in line three and ve. This closes the case ≥ 0. The case < 0 is left to the reader. Now let ( , ) ∈ r,s . We choose ∈ [ ] and ( , ) ∈ [ ] × Z such that = + [ −1] and = − [ −1] . Then we immediately nd that Π ( , ) = Π ( , ) + Δ which means it su ces to show that Π ( , ) > 0. But this follows from ≥ r,s ( ) which written out is nothing but using Lemma . . Plugging = − [ −1] into the above expression one nds that satis es ( ) which is equivalent to our claim that Π ( , ) > 0.
Remember that we de ned the matrix M ( , ),( , ) to be the collection of coe cients of the projection to degree 1. It is given abstractly in Lemma . . So far we found out that M admits a two-sided inverse and moreover we characterised those modes featuring only derivatives in degree one. The next step is now to combine both results in order to bring the operators ( , ) ( , ) ∈ r,s into the normal form of an Airy structure. Proof. From Lemma . and Remark . we know that the matrix M encoding the coe cients of the degree one projection of the modes , admits a two-sided inverse, if we keep the full range of indices ( , ) ∈ [ ] × Z and ( , ) ∈ [ ] × Z. However, we actually need such a relation between the semi-in nite column vectors For this, let us also introduce the semi-in nite column vectors We can then write Equation ( ) symbolically as H = M · J. The vanishing properties of the matrix M guarantee that this product is well-de ned (i.e. the evaluation of each entry involves only nite sums). By de nition of I r,s , this splits as Writing N for the inverse of M, the relation M · N = id implies that M −− · N −− = id − (with obvious notation). As these are semi-in nite matrices, we cannot conclude yet that N − − ·M − − = id − . This conclusion will nevertheless come from the analysis of N . According to Lemma . , seen N as a linear operator we have .

( )
By a straightforward calculation, we see that, if ∈˜and ∈˜, then, as → 0, Considering the term related to , , we see after some elementary addition of exponents that For this to contribute only to J − , we need Π ( , ) − Δ > 0, which by Lemma . is given for ( , ) ∈ r,s . Thus N +− = 0 and by a similar reasoning as before, N −− · M −− = id − . From ( ), one can check the desired vanishing property for the entries of N −− and therefore, applying the matrix N −− to the semi-in nite column vector with entries is well-de ned and gives the semi-in nite column vector J − . To completely prove the degree one property, we need to check that for any ( , ) ∈ r,s , the degree zero component of , is vanishing. Following the proof of Lemma . one nds that the projection to degree zero of , for arbitrary and is Similarly to Lemma . , we then see that in order to get a non-vanishing contribution to 0 ( , ), we need , where the di erence with that Lemma is the substitution of − 1 by . By the proof of that lemma, Hence 0 ( , ) = 0 for ( , ) ∈ r,s , and this concludes the proof.
Remark . . As in Remark . , the actual requirement for Lemma . to work in some way is that 0,1 takes distinct values on all elements of a bre close to the rami cation point. However, in case several 0,1 s agree up to leading order, the index set r,s would need to be adjusted to account for this: we would get min ( ) < r,s ( ) the order of vanishing of N − − d would change. However, if we adjust r,s , our argument for the subalgebra condition Lemma . does not work anymore. On the geometric side of topological recursion, Proposition . also needs this particular range of indices, and would have to be adjusted substantially.
There may still be cases where this can be made to work, but goes beyond the scope of this work. In any case, more conditions would be necessary, as the speci c example of a spectral curve with two components in the vein of Section , with 0,1 = d yields a non-symmetric 1,2 and therefore cannot correspond to an Airy structure.
. T Having proven the degree one condition for the modes , ( , ) ∈ r,s with index set r,s as de ned in ( ) we need to check whether these modes form a graded Lie subalgebra as demanded for Airy structures. In order to do so we use Theorem . stating that if is induced by a descending partition then ( , ) ( , ) ∈ generate a graded Lie subalgebra. By this we mean that there In our case at hand we want to check whether r,s ∪ {(1, 0)} is induced by a descending partition. We will later then exclude 1,0 from the associated mode set by setting this mode to zero. Explicitly, this means that we want to classify the cases in which there exists a descending partition such that ( ) = 1 − r,s ( ) + ,1 . Writing = + [ −1] for ∈ [ ] again assuming 1 1 ≥ · · · ≥ we can write out r,s ( ) using Lemma . and obtain The case = 1 was studied in [BBCCN ], resulting in the following correspondence.
This lemma has the following generalisation to the case > 1.
Lemma . . Let ≥ 2. Given 1 1 ≥ · · · ≥ with and coprime for all , then there exists a descending partition = ( 1 , . . . , ℓ ) of = 1 + · · · + such that ( ) is satis ed if and only if the following holds In this case is given by Proof. First, let us prove that (i), (ii) and (iii) are necessary such that where we write = + [ −1] as always, admits a descending partition such that ( ) = ( ). We begin by presenting a few general statements regarding the construction of following the lines of [BBCCN , Proposition B. ]. Assume : [ ] → N is weakly increasing with (1) = 1 and Let 1 < · · · < ℓ−1 be a complete list of jumps of in the sense that Additionally, set 0 0 and ℓ . If we further de ne However, the partition ( 1 , . . . , ℓ ) is in general not descending. What one should take away from this construction is that measures the length of the interval between the ( − 1)th and th jump of . Let us get back to our case at hand where is given by ( ). The rst constraint on 1 , . . . , and 1 , . . . , comes from the requirement that ( + 1) − ( ) ∈ {0, 1}. At the value = 1 + [ −1] with > 1 we nd for > 1 that implying that necessarily < . For = 1 and 1 < < one nds that implying = 1. Notice that we do not get any restrictions for in the case = 1 since actually does not depend on . Regarding 1 , by writing 1 = 1 1 + 1 with 1 ∈ [0, 1 ) we nd for < 1 that This implies that either 1 = 0 in which case 1 ∈ [0, 1 ) may be arbitrary or 1 = 1 and which can only hold if 1 ∈ {0, 1} since otherwise for increasing the right-hand side jumps earlier from zero to one than the left-hand side which violates the inequality. The two cases translate into the constraint that 1 ≤ 1 + 1. If 1 = 1 this is also true since in this case (2) − (1) = 1 − 1 implies that 1 ∈ {1, 2}. To summarise, the demand that ( + 1) − ( ) ∈ {0, 1} gives us the constraints In the case 1 = 1 + 1 one can argue that the above conditions already imply that (i) to (iii) hold. Indeed, since we assume that 1 1 ≥ for all ∈ ( ] the case 1 = 1 + 1 forces < 1 for all ≥ 2. Thus (II) forces = 2 and due to (III) necessarily = 1. This case is clearly covered by (i) to (iii). Now assume 1 ≤ 1 . By assumption there exists a descending partition = ( 1 , . . . , ℓ ) with ( ) = ( ). In order to nd a description of in terms of ( ), notice the similarity between ( ) and ( ) for xed and ∈ [ ]. Except for the constant shift [ ] and the ,1 the two maps coincide, which means that except for the transition values = [ −1] → [ −1] + 1 they jump at the same value . At the transition points we nd for all < that as discussed in ( ). This means that if we let = ( 1 , . . . , ) denote the partition satisfying and set (1) if = 1 then the partition is the one satisfying ( ) = ( ). Note that in case = 1 the th line . . . , 1 − 1, 2 , . . . , −1 , + 1, . . . must be replaced with (. . . , , . . .). Since we assume to be a descending partition, has to be descending as well for all ∈ [ ]. Consequently Lemma . tells us that = ±1 mod . Note that in order to apply Lemma . here we use that the range for value of the is constrained by (I)-(II)-(III). Now at the transition between two parts of the partition we see that the constraint + 1 ≥ +1 1 − 1 is always satis ed, because if we consider the explicit form of given in ( ) we nd that Here we used that by assumption ≥ for all ≤ . There are similar arguments for the case where or +1 is equal one. One obtains further restrictions on the choice of considering the constraint that Assume for example that 1 > 2 and 1 = +1 mod 1 , i.e. 1 = 1 1 + 1. Then ( ) tells us that 1 1 = 1 + 1 and 1 2 = . . . = 1 1 = 1 . But this contradicts ( ) for = 1 since 1 1 −1 < 1 1 + 1. Consequently we are left with 1 = −1 mod 1 which is nothing but (i). It is straightforward to see that condition ( ) checked for arbitrary induces (ii) and (iii).
Following the previous analysis of the necessary conditions it is straightforward to see that if We now have everything at hand to prove the standard case of Theorem . .
Proof of Theorem . , standard case. Recall that the situation of the theorem is as follows: Up to now, we have only considered the conjugation withˆ1, so let us nish the argument for that case rst. The selected modes , = 1 exactly correspond to the modes ( , ) ( , ) ∈ r,s where r,s is the index set de ned in ( ) by performing the identi cation of index sets via Lemma . . Thus, Lemma . tells us that after a change of basis the modes ( ) satisfy the degree one condition. Since by assumption the modes ( ) satisfy the subalgebra condition if the modes ( , ) ( , ) ∈ do. Here is de ned as in ( ). Now using that , is obtained from , via conjugation the claim immediately follows from Theorem . .
For the general case, conjugating also withˆ2 andΦ, note rst of all that conjugation preserves commutation relations, so the subalgebra condition still holds. For the degree one condition, note that conjugation byˆ2 gives the shifts − −→ − + ℏ 1 2 1 2 ,1 − , which preserves degrees, and only acts on with < 0, which do not occur in 1 ˆ1 · , ·ˆ− 1 1 ) by the previous parts of the computation. Likewise, conjugation byΦ acts as in Equation ( ), which again preserves degrees and only a ects with < 0, so it also preserves the degree one condition.
. T Contrary to the case considered before let us now allow = ∞ for ∈ [ ]. Let us writẽ and˜− for the collection of all components˜on which = ∞.
Proof. The proof of this Lemma is verbatim to the one of Lemma . taking into account that 0,1 ( ) = 0 for all ∈ [ ] with = ∞.
Remark . . Let us make two important observations. First notice that if we write + and moreover that for all > + + 1 we have Especially, from the last identity we deduce that, in order to end up with an Airy structure, it is necessary to have at most one ∈ [ ] for which = ∞. Moreover, necessarily for this we need = 1. Otherwise, there is no hope to obtain an Airy structure.
Motivated by Remark . in the following we will assume that only ( , ) = (1, ∞) while for all other ∈ [ − 1] we have ≠ ∞, what we call the exceptional case in Theorem . . Moreover, let us assume that as before 1 1 ≥ . . . ≥ −1 −1 . Rather than working with expression ( ) we will mainly use that by ( ) we have where , is obtained from , by formally setting * equal to zero. Of course, , may be computed via ( ) replacing with − 1, i.e. these are modes of the standard case. Therefore, as the modes , are build up from modes considered in Equation ( ) and an additional factor * , we can use the analysis of the standard case from the previous section in order to prove Theorem . in the exceptional case as well.
Proof of Theorem . , exceptional case. As in the standard case we know that conjugation witĥ In order to bring the operators into the normal form of an Airy structure, let us make use of our observation earlier made in Remark . that This can be rephrased in the sense that 1 where A is an upper triangular matrix whose diagonal entries may be read o from ( ) by taking only the leading order contributions of the 0,1 s into account. Thus, one can nd a two-sided inverse fo A, and applying it to the semi-in nite vector ( , ) ( , ) ∈ r,s we get (˜, ) ( , ) ∈ r,s for which 1 ˜, = − r,s ( )+1 .
By taking again suitable linear combinations, we can use the above modes in order to eliminate all from 1 ˜, for < , i.e. get operators ( , ) ( , ) ∈ r,s with ( ) transform into This expression is exactly the degree one projection of the operators considered in Lemma . where we shifted in all cycles. From Lemma . we know that they can be brought in normal form, provided we can argue at last that the degree zero projection of these modes is vanishing. This is indeed the case: since 0 , = 0 , and we know by ( ) that the degree zero of , vanishes as long as ≥ r,s ( ), we see the same holds for , .

P II -S
In this second part, we translate the di erential constraints coming from the W-algebra representations of Section into constraints on the order of poles of certain combinations of multidi erentials , on a spectral curve built from the coe cients , . The latter constraints are called "abstract loop equations". In a second step, we show that the unique solution to the abstract loop equations is provided by an adaptation of Bouchard-Eynard topological recursion to the setting of singular spectral curves. In fact, this provides us with the right de nition of the topological recursion à la Chekhov-Eynard-Orantin in this setting, together with the proof that it is well-de ned.
F A . F We will start by reconsidering Section . . in the case of consisting of a single cycle, of length . In this case, we can omit all -indices, and consider the standard representation of the Heisenberg algebra of . It is useful to write˜= . We split the current as follows: Choose a primitive th root of unity and let ( ) = { , , . . . , −1 }. Set std 0,2 ( 1 , 2 ) = We can rewrite ( ) as The e ect of the dilaton shiftˆin ( ) is to replace + ( ) with + ( ) + ℏ Using the Baker-Campbell-Hausdor formula, it is easy to see that the net e ect of the change of polarisationΦ is to replace std 0,2 with 0,2 ( 1 , 2 ) and to replace − ( ) with For uniformity we also de ne for ≥ 0 We then obtain

( )
We prefer to convert this expression into a sum over subsets ⊆ ( ) of cardinality . Then, we have to sum over partitions 1 . . . set P ( ( )) whose elements are sets of disjoint pairs in ( ), and writing P ∈P if P ∈ P ( ( )), we obtain . F We now return to the general situation of Section . . . Let be a permutation of [ ] with cycles of lengths labelled by ∈ [ ]. For each ∈ [ ], we have generators of the Heisenberg algebra of : , whose currents we split as + ( ) and − ( ) in the same way as in Section . . We obtain modes , indexed by ∈ [ ] and ∈ Z for a representation of the ( ) algebra given by ( ). To match Section . , we introduce for each ∈ [ ] formal variables such that˜= . These thus depend on , but they will appear in generating series with superscript so that one can infer directly from the formula which power one should use to relate it to the global variable˜.
At this stage we are naturally led to introduce a curve which is the union of copies of a formal disk for each ∈ [ ]:˜= When necessary to avoid confusion, points in˜will be denoted to indicate in which copy of the formal disk we consider them. One can consider˜as a branched cover˜−→ Spec C given by ↦ → on the th copy of the -formal disk. The smooth (but reducible) curve˜is in fact the normalisation :˜→ of the singular curve The branched cover˜:˜→ factors through : → . This is the local picture we will globalise later in Section by considering more general branched covers where ,˜are regular curves and is a possibly singular curve whose normalisation is˜. For the moment we stick to the local setting.
Let us again consider a general dilaton shift and change of polarisation 0,2 − − , and the conjugated operator To express (˜), we introduce the basis of meromorphic 1-forms d on˜, indexed by ∈ [ ] and ∈ Z. It is de ned by We also introduce the meromorphic forms 0,1 , 1 2 ,1 and bidi erential 0,2 on˜: For ∈ Z, we introduce the 1-form on˜d * = d .
We recall that the index ∈ [ ] of the component to which a point ∈˜belongs is implicit in the data of . Similarly to Section . , the e ect of the dilaton shift is to replace + ( ) with while the e ect of the change of polarisation is to replace std 0,2 ( 1 , 2 ) with 0,2 1 2 1 2 and − ( ) with For uniformity we also set We can repeat the argument of Section . with several s, de ning the ber over˜in( and getting The partition function is annihilated by the di erential operators above a certain index in the W if and only if the satisfy certain bounds on their pole orders as˜→ 0. Because is a function (i.e. it does not contain a di erential part), it commutes with 0,2 , 0,1 , 1 2 ,1 , and J + . The only non-trivial computation is where each term J − ( ) obtained like this has to act on a later [J − ( ), ], as it annihilates 1. The J − commute among each other, so we get a partition of − into sets of operators acting on a single copy of . We obtain where : + − → [ ] associates to the index ∈ [ ] such that ∈ ( ), and we identi ed | and k with the tuples ( ) ∈ and ( ) ∈ . We decompose in homogeneous terms with respect to the exponent of ℏ and the number of : In order to completely rephrase this in terms of spectral curves, we need to get rid of the and replace them with d s. For every , prepare a tuple [ ] = ( ) =1 of points on˜and de ne To compute it, we introduce the multidi erential forms for ≥ 0 and ≥ 1 such that 2 − 2 + > 0 Besides, under this action, we get which is the series expansion of 0,2 with | | < | |.
We then notice that the sums over˜ , , and ℓ in ( ) recombine into .
We now observe that the factors 0,1 , 1 2 ,1 , 0,2 can be treated uniformly by summing over partitions L and allowing ( , | | + ) = (0, 1), ( 1 2 , 1), (0, 2), which were exactly the terms for which 2 − 2 + | | + ≤ 0. We get The translation of these di erential constraints in terms of the correlators = ( , ) , de ned in ( ) is called "abstract loop equations". It says that for any ≥ 0, we have In other words, E ( ) , (˜, [ ] ) is meromorphic and has a pole of order strictly less than r,s ( ) + at the point˜= 0 in . If we letẼ ( ) , ( , [ ] ) be its pullback to a meromorphic -di erential on˜, this is tantamount to requiring that, for any ∈ [ ] T We are going to formalise what we have found in the context of global, possibly singular spectral curves. This will lead us to de ne the appropriate notion of abstract loop equations in Section . , and to show in Section . that its unique solution is given by an appropriate topological recursion à la Chekhov-Eynard-Orantin, that is by computing residues on the normalisation of the singular curve.
. S De nition . . A spectral curve is a triple C = ( , , ), where is a reduced analytic curve over C and , are meromorphic functions on , such that all bers of are nite.
Note that is not necessarily connected, compact, or irreducible. We will work with its normalisation :˜→ , which is a smooth curve. We have meromorphic functions˜= • and˜= • de ned on˜. Let ⊂ C be the set of points that have a neighbourhood such that the cardinality of the bre of is constant on \ { } and strictly smaller at itself. It is the collection of branchpoints of˜and images of locally reducible points away from ∞. We also denote = −1 ( ) and˜=˜− 1 ( ). We assume that is nite. As a result,˜and are also nite. Note that, since we assumed that all bers of are nite, the same is true of˜and there cannot be an irreducible component of˜whereĩ s constant.
If ∈ , we let ⊂ be a small neighborhood of that is invariant under local Galois transformations and Without loss of generality we can assume that ⊂ C. If ∈˜, we de ne Note that˜is in bijection with the set of branches in˜above , and we denote |˜|. For each ∈˜, we introduce a small neighborhood˜of in˜, such that (˜) = , as well as˜ =˜\ { }. We have of course˜= ∈˜˜.
By taking a smaller neighborhood, we can always assume that the (˜) ∈˜a re pairwise disjoint. As anticipated in Section . , if we want to insist that a point ∈˜belongs to˜, we will denote it . The bers can be decomposed We denote = | ( )| which is independent of ∈˜ and = | ( )| which is independent of ∈˜ . In particular
If is a small loop in around ( ), it induces a Galois transformation in the cover˜|˜, that is for each ∈˜ a permutation of ( ), which on˜( ) restricts to a cyclic transformation of order . This integer represents the order of rami cation at ∈˜of˜|˜.
Remark . . If |˜| = 1, is irreducible locally at , hence smooth at . We can then use the same symbol to denote the point ∈ and the unique point above it in˜. If |˜| > 1, is reducible locally at , hence singular at . If |˜| = 2, is a node. For ∈˜, we have = 1 if and only if is not a rami cation point of˜. We say that the spectral curve is smooth if all rami cation points in are smooth.
As in Section . , when working with local coordinates it should be clear from the context whichĩ s involved. Specifying such coordinates requires the choice of a th root of unity for˜− ( ). We assume such a choice is xed. We also choose a primitive th root of unity, denoted . If ∈˜ , the set of coordinates of the points in ( ) is Let us write locally at ∈ the Laurent series expansion of the function˜ ∼ ∑︁ In the next paragraph we will need to study the order of vanishing of these functions at˜. This is given by the following lemma.
Lemma . . If one of the following conditions is satis ed (i) there exist distinct , ∈˜such that = = +∞; or (ii) there exists at least one ∈˜such that = +∞ and > 1, then ( ) vanishes identically on˜for the involved in these conditions. Otherwise, for any ∈˜, we have ( ) ∈ O ( ) when ∈˜approaches , where If furthermore either (iii) there exist distinct , ∈˜such that , are nite, = and = ; or (iv) there exists ∈˜such that is nite and gcd( , ) > 1, then ( ) ∈ O ( +1 ). If none of the above conditions are satis ed, then there exists a non-zero scalar , such that ( ) − , ∈ O ( +1 ).
Proof. If = +∞,˜is identically zero for ∈˜. Conditions (i) and (ii) both imply there is a ∈ ( ) such that˜( ) ≡ 0 as well, so one of the factors in ( ) vanishes identically. We now assume that (i) and (ii) are not satis ed. Let us add for the moment the assumption that all are nite. We compute and observe that the scalar prefactor in the rst term is non-zero if and only if and are coprimein that case it is equal to −1 . For ∈˜distinct from , we have where · · · are higher order terms, and We have , = 0 if and only if ( , ) obey the condition (iii). Multiplying ( ) with the product of ( ) over all ≠ , we deduce that ( ) = , + O ( +1 ) and , = 0 if and only if the conditions (iii) and (iv) are satis ed, with the exponent as claimed. This concludes the proof in absence of an in nite . Now let us assume there exists a unique − ∈˜such that − = +∞. As we assume that (i) and (ii) are not satis ed, we must have − = 1. If ≠ − and we take ∈˜, we only need to pay attention to the factor ( ) for = − , and in fact Equation ( ) remains valid, hence ( ) = , + O ( +1 ) with the same expression for and the same discussion for the (non-)vanishing of , . Notice that by de nition of the order we must have − = max(˜) so − does not appear in Δ . If ∈˜−, the factor ( ) is absent in ( ) and the other factors ≠ − are as in ( ). But, as − only appears in via the rst term of the rst line of ( ), which can be consistently set to 0 since − = 1, the formula for remains valid.
De nition . . A fundamental bidi erential of the second kind on C is an element ∈ 0 ˜×˜; 2 (2Δ) 2 , with biresidue 1 on the diagonal Δ ⊂˜×˜, where˜is the sheaf of di erentials on˜. A crosscap di erential on C is the data of a (possibly empty) divisor on˜\˜and De nition . . A family of correlators is a family of multidi erentials = ( , ) ∈ 1 2 N, ≥1 on˜such that 0,1 =˜d˜, 0,2 is a fundamental bidi erential of the second kind on C, 1 2 ,1 is a crosscap di erential, and for 2 − 2 + > 0 , ∈ 0 ˜; (˜( * ˜)) . It satis es the projection property if for 2 − 2 + > 0, Note that ( ) is automatically satis ed for ( , ) = (0, 2). Di erentials satisfying the projection property cannot have residues, and if they are holomorphic, they must vanish. We can always assume by taking smaller neighborhoods that the divisor of the crosscap di erential is supported outside ∈˜.
De nition . . Let be a family of correlators, and ≥ 1, ∈ 1 2 N and ≥ 0. The genus , -disconnected, -connected correlator is de ned by We de ne W , , by the same formula, but omitting any summand containing some 0,1 .
If ∈ [ ], we let :˜( ) −→ be the smooth curve obtained by taking the bered product of copies of˜:˜ → C, deleting the big diagonal Δ( ), and quotienting by the (free) action of . Points in˜( ) are exactly subsets of cardinality of ( ) for some ∈˜ . We have natural holomorphic maps where q forgets the order of elements of an -tuple and x ({ 1 , . . . , }) =˜( 1 ) = · · · =˜( ). Let I :˜\ Δ ( ) →˜be the natural inclusion. We introduce ˜ * E ( ) ; , , where all operations do not concern the last variables. More concretely, ( ) The symmetry factor ! disappeared since W , , is symmetric in its rst variables. Note that reading ( ) in the local coordinate 0 of 0 ∈˜ each term may be multivalued -i.e. fractional powers of ( 0 ) could appear -however the sum is single-valued as it is the pullback along˜of a 1-form on .
De nition . . We say that a family of correlators satis es the master loop equations if for any ∈ 1 2 N and ≥ 0 such that 2 − 2 + ( + 1) > 0, for any ∈ and ∈ [ ], any ∈˜, when 0 ∈˜ approaches , we have The relevance of this notion comes from the fact that the master loop equation can be solved by the topological recursion. Remark . . From the proof, we see that if one of the conditions (i) and (ii) appearing in Lemma . is satis ed, the recursion kernel is ill-de ned as the denominator vanishes identically in the neighborhood of some . Besides, if one of the conditions (iii) or (iv) is satis ed, the same thing could occur or at least the order of vanishing of the denominator is nite but higher than the one speci ed by De nition . . In the latter case, one can still ask for the analogue of Proposition . simply by modifying the master loop equation to require that the rst sum in ( ) is O (d ).
We note that the right-hand side of ( ) involves only , with 2 − 2 + < 2 − 2 + ( + 1). For a xed un = ( 0,1 , 1 2 ,1 , 0,2 ), there exists at most one way to complete it into a system of correlators satisfying the master loop equation and the projection property: the , are then determined by ( ) inductively on 2 − 2 + > 0. However, such a system of correlators may actually fail to exist at all. Indeed, ( ) gives a non-symmetric role to 0 compared to 1 , . . . , , therefore the , +1 ( 0 , . . . , ) that ( ) compute may fail to be symmetric, and so would not respect De nition . .

. A
We now address the aforementioned problem of existence of the solution to the master loop equations, thanks to the results obtained in Section . We rst introduce a seemingly di erent notion of "abstract loop equations" valid in the setting of Section . . It will turn out that they give the right generalisation of "abstract loop equations" proposed in [BS ] for smooth spectral curves. We will show that, under admissibility conditions on the spectral curves that pertain to our constructions of Airy structures in Section , the abstract loop equations have a solution satisfying the projection properties, and imply the master loop equation. Therefore, this solution must be given by the topological recursion formula ( ), and this proves a posteriori that this de nition is well-posed, i.e. it produces inductively only multidi erentials that are symmetric under permutations of all their variables. A direct proof of symmetry by residue computations on˜seems rather elusive.
Let C be a spectral curve as in Section . . We introduce integers ( ) for each ∈ and ∈ [ ] matching Lemma . . If ∈ [ ], we rst decompose it into = r [ ) + for the unique ∈˜such that r [ ) < ≤ r [ ] . Then, ∈ [ ] and we have De nition . . We say that a family of correlators satis es the abstract loop equations if for any ∈ 1 2 N and ≥ 0 such that 2 − 2 + ( + 1) > 0, for any ∈ and ∈ [ ] when 0 → ( ) we have This condition is equivalent to the property that, for any ∈˜we have when 0 ∈˜ approaches .
Proposition . . Assume that none of the conditions (i), (ii), (iii), (iv) appearing in Lemma . are satis ed. Then, the abstract loop equations imply the master loop equations.
Proof. We treat the case where is nite for all ∈ . The case where there could exist ,− ∈( which is then unique) such that ,− = +∞ is left as exercise to the reader. For each ∈ and ∈ [ ], the abstract loop equations imply that for any ∈˜we have when 0 ∈˜ approaches Comparing with De nition . , the result will be proved after we justify that is always nonnegative. We recall the de nition of in ( ) We decompose = r [ ) + with the unique ∈˜such that r [ ) < ≤ r [ ] and ∈ [r ], and we denote min˜. Inserting the de nition of ( ) from ( ), we obtain We are going to use often the inequality Checking nonnegativity of ( ) is done by a case discussion.
By de nition of the order, for all ∈ [ , ) we have ≥ , therefore ( ) ≥ 0. For ≥ 2, we can use ≥ and ( ) and obtain ( ) ≥ 0 as well. • If , we rather have For = 1 and = this simpli es to and thanks to the inequality ≥ for all ∈ [ ) we deduce ( ) ≥ > 0. For = 1 and , we have ) .
Due to the inequality ≥ for all ∈ [ , ) we have again ( ) ≥ 0. For ≥ 2, we use the inequality ( ) to write ) and due to the ordering we nd again ( ) ≥ 0.
Remark . . In the proof we see that for any ∈ , there exists ∈˜and ∈ [ ] such that ( ) = 0. Therefore, we do use all the vanishing provided by the abstract loop equations to derive the master loop equations.
Combining with Proposition . , we obtain the following result. The notion of abstract loop equation was rst introduced [BEO ; BS ] for smooth curves with simple rami cations and was shown there to be a mechanism implying directly the topological recursion. This was extended to higher order rami cations on smooth curves having˜holomorphic near in [BE b; BBCCN ; Kra ], and to the more general case where d is holomorphic near in [BBCCN ]. The novelty of Propositions . and . here is the treatment of possibly singular curves.
. T In this paragraph, we express the abstract loop equations in a more algebraic way, that will make the bridge to Airy structures. The converse route was anticipated in Section .
Let C be a spectral curve. We can attach to it a local spectral curve matching the de nitions in Section . . Namely, we let˜l oc = ∈˜˜l oc ,˜l oc Spec C .
For each ∈˜, de ne˜l oc Spec C(( )) and let L 0 ˜l oc ;˜loc C(( )).d be a copy of the space of formal Laurent series, and We denote by Loc : 0 ˜;˜( * ˜) → L the linear map associating to a meromorphic di erential its all-order Laurent series expansion near using the local coordinate in˜, and Loc = ∈˜L oc .
We de ne elements d ∈ L, indexed by ∈˜and ≥ 0 We introduce the standard bidi erential of the second kind on˜, that is std 0,2 Let now be a family of correlators on C. We can encode the correlators , with 2 − 2 + ≥ 0 by the following Laurent series expansion ( ) Using the fundamental bidi erential of the second kind, we introduce another family of di erentials d − , now globally de ned on˜and indexed by ∈˜and > 0 Notice that it is such that, for any , ∈L Assuming that satis es the projection property, by symmetry we can apply this property to each variable to obtain the existence of a nite decomposition for 2 − 2 + > 0 where , k are scalars.
De nition . . The partition function associated to a satisfying the projection property is de ned as .
We now would like to translate the abstract loop equations on into constraints for its partition function. For this purpose, we introduce for each ∈ a copy ; , of the di erential operators in Equation ( ) indexed by ∈ [ ] and ∈ Z forming a representation of the ( )-VOA using as twists permutations which is a product of disjoint cycles of respective orders ( ) ∈˜. They are described in terms of the Heisenberg generators indexed by ∈˜and ∈ Z where we use = Res = 1 2 ,1 ( ) coming from the crosscap di erential. Then, we construct the dilaton shift and the change of polarisa-tionˆ= De nition . . To a spectral curve C equipped with a crosscap di erential 1 2 ,1 and a fundamental bidi erential of the second kind 0,2 , we associate the system of di erential operators indexed by ∈ , ∈ [ ] and ∈ Z ; , , is as in Equation ( ), and is the monodromy permutation at . We also introduce the set I ( , , ) ∈ , ∈ [ ], ≥ ( ) − ,1 .
Proposition . . Assume that none of the conditions (i), (ii), (iii), (iv) appearing in Lemma . are satis ed, and let be a system of correlators satisfying the projection property. Then, the abstract loop equations for are equivalent to the following system of di erential equations for its partition function: Proof. If | | = 1 this is the computation done in Section . . Given the formalism that we introduced, it is straightforward to adapt it to handle several s, where the ; , now form a representation of the direct sum over ∈ of the W ( )-VOAs.
It is now easy to combine the construction of Airy structures in Theorem . with Propositions . and . to obtain our second main result. We recall that we had de ned De nition . . We say ∈ is regularly admissible if • is irreducible locally at , that is |˜| = 1.
In that case, in all the previous de nitions and constructions in the neighborhood˜we replacẽ ( ) with˜( ) −˜( ). In particular, we take = + 1, and the value of˜( ) plays absolutely no role in all the results we have mentioned.
De nition . . We say ∈ is irregularly admissible if • for any ∈˜such that > 1,˜has a pole at but˜d˜is regular at . In particular, this imposes ∈ [1, ). • for any distinct , ∈˜such that ( , ) = ( , ), we have ≠ .
• if |˜| > 1, there exist distinct + , − ∈˜such that ± = ∓1 mod ± and + These conditions always imply that for any , we have gcd( , ) = 1; in other words the plane curve (˜,˜,˜) is locally irreducible at . Here, the second condition avoids the pathology of (iv) in Lemma . and the next results. The third condition is then equivalent to avoiding the pathology (iii) in Lemma . , because = and ( , ) coprime, ( , ) coprime imply that ( , ) = ( , ). The fourth and fth conditions match those in Theorem . if > 1, and the case = 1 corresponds to Theorem . .

De nition . . We say ∈ is exceptionally admissible if
• there exists a unique − ∈˜such that − = +∞, and it has − = 1.
• the three rst properties in De nition . that do not involve − are satis ed.
Allowing in nite , the rst condition guarantees that we avoid the pathologies (i) and (ii) in Lemma . , which make the denominator of the recursion kernel be identically zero in some open set. The last two conditions match those in Theorem . .
De nition . . A spectral curve C = ( , , ) is admissible if all ∈ are either regularly, irregularly or exceptionally admissible. The tuple ( , ) ∈˜i s called the type of the rami cation point ∈ .
Theorem . . Let C be an admissible spectral curve equipped with a fundamental bidi erential of the second kind 0,2 and with a crosscap di erential 1 2 ,1 . Then there exists a unique way to complete ( 0,1 , 1 2 ,1 , 0,2 ) into a system of correlators satisfying the projection property and the abstract loop equations (or the master loop equations). Moreover, , is computed by the topological recursion ( ) by induction on 2 − 2 + > 0, and the result of this formula is symmetric in all its variables. The , determined by its decomposition are the coe cients of expansion of the partition function of the Airy structure introduced in De nition . .
For smooth curves with simple rami cations -i.e.˜= , |˜| = 1 and = 2 for all ∈ -the symmetry is proved in [EO , Theorem . ]. For admissible smooth curves, Theorem . is proved in [BBCCN , Theorem . ]. For singular curves, the admissibility condition we have adopted is not far from being optimal for this formulation of the abstract loop equation/formulas like ( ). It may not be impossible to de ne a topological recursion for more general spectral curves, but either the formula ( ) will have to be di erent or the exponents in the master loop equations/abstract loop equations should be increased, in a consistent way so that there still exist a unique symmetric solution.
. D If we erase some or all of the components of an exceptionally admissible local spectral curve C indexed by the − ∈ such that − = ∞, we still obtain an admissible local spectral curve C . We prove below a decoupling result if 0,2 has no cross-terms with these components and 1 2 ,1 vanishes on these components. This decoupling means that the computing , on C and restricting to C gives the same result as restricting ( 0,1 , 0,2 , 1 2 ,1 ) to C and then computing , by the topological recursion on C .
Proposition . . Let ( , , ) be an exceptionally admissible spectral curve, equipped with a fundamental bidi erential of the second kind 0,2 and with a crosscap di erential 1 2 ,1 . Let be a non-empty subset of exceptionally admissible rami cation points, and denote˜ the set of − ∈˜such that − ∈˜for some ∈ and − = ∞. Assume that for any − ∈˜ and ∈˜\˜ we have Then, if we denote the outcome of topological recursion and the Loc -projection of the system of correlators obtained from ( , , , 0,2 , 1, 1 2 ) by the topological recursion ( ), satis es the topological recursion on the local spectral curve (˜l oc ) = ∈˜\˜ ẽ quipped with the restriction of , , 0,2 , 1 2 ,1 onto (˜l oc ) . Proof. We detail the proof in the case of a single rami cation point with − = ∞. The general case follows because topological recursion is local. It su ces to work from the start with the normalised local spectral curve attached to ( , , ). We write˜l oc − for the connected component of˜l oc associated to − and˜l oc + for all other components. First let us prove all , with exactly one argument in˜l oc − are zero. We will prove this by induction on the Euler characteristic. The base cases hold, as 0,1 and 1 2 ,1 vanish on˜l oc − and 0,2 does not have cross-terms. For the induction step, let us recall the topological recursion formula ( ). , .
In this case, = {0}, and˜0 = {0 ∈˜l oc + , 0 ∈˜l oc − }. Let us analyse this formula with 0 ∈˜l oc − and all ∈˜l oc + for ∈ [ ]. First, because the 0,2 does not contain cross-terms, we must have ∈ − , so also = 0 ∈˜l oc − . Because˜|˜loc − is injective, 0 ( ) contains no other point in˜l oc − . Hence for all terms in the sum, ∩˜l oc − = { }, and by the induction hypothesis, all W ( ; [ ] ) are zero (they contain a factor , with 2 − 2 + < 2 − 2 + and one argument in˜l oc − ). Now, we will prove the proposition using a similar induction. The base cases, 0,1 and 0,2 (and the trivial 1 2 ,1 ) do indeed not mix several components. For the induction step, we again look at the topological recursion formula. Let us look at the terms contributing in the case 0 and all [ ] are in˜l oc + . As before, ∈˜l oc + . Furthermore, 0 ( ) contains exactly one element, say , in˜l oc − , so any contains at most one such element. Any term in the sum not containing also contributes to the topological recursion on˜l oc + , and all terms including must vanish by the rst part of this proof, as they have a factor , with exactly one argument -namely -in˜l oc − .
In this section, we calculate some of the rst correlators in the unique way of Theorem . , but for not necessarily admissible spectral curves. In this generality, there is no guarantee for the correlators to be symmetric, and we will nd that indeed they are not for certain choices of parameters. As correlators coming from Airy structures are symmetric by construction, these calculations give necessary condition for our collections of di erential operators to form Airy structures. These conditions are summarised in Section . . .

. T
Let us consider the spectral curve with a unique rami cation point at which irreducible components labelled by˜=˜intersect, and de ned for ∈˜and on the th component of the normalisation by the formulas We equip it with the bidi erential and the crosscap di erential 0,2 We assume that gcd( , ) = 1, and that for = and ≠ we must have ≠ . The correlators In this section, we compute these correlators, and show that the symmetry of 0,3 and 1 2 ,2 poses constraints on the parameters ( , , , ) ∈˜. We also obtain similar constraints from partial calculation of 0,4 . In light of Propositions . and . , these are necessary constraints to obtain Airy structures from the construction presented in Section , thus proving Proposition . .

. . Genus zero
In this Section, we calculate 0,3 and obtain constraints on the parameters of the Airy structures that are necessary for the symmetry of the correlators 0,3 and 0,4 . In theory, the same approach could get constraints from 0, , which we believe get progressively closer to the su cient conditions from Theorem . . However, the computations also get quite involved, and we have not calculated these in full generality.
Proposition . . Assume and are coprime for all ∈˜. Then 0,3 is symmetric if and only if the following holds (i) = ±1 mod for all ∈˜. (ii) For all 1 ≠ 2 with > 2 such that either 1 = 1 mod 1 and 2 = 1 mod 2 or 1 = −1 mod 1 and 2 = −1 mod 2 one has 1 1 ≠ 2 2 . When these conditions are satis ed, then 0,3 is given by Proposition . . Let , , and be distinct such that = = . Then Before we prove these two results, we rst give some more general considerations for calculating the genus zero correlators.
Recalling Equations ( ) and ( ), we see that the recursion kernel can be split in factors coming from di erent irreducible components of the spectral curve. First, note that in ( 1 ; , ), we always need 1 and to lie in the same irreducible component, and then we have .
Because 0,2 in our current situation does not mix irreducible components, this shows immediately that to get symmetric correlators, we need 0,3 1 2 3 = 0 unless = = : if one (say ) is di erent from the other two, we can use the recursion with respect to its variable, and get ∈˜, so both terms in ( ) would involve an 0,2 between two di erent irreducible components. This vanishing can then be used to calculate 0,4 with arguments in exactly three di erent irreducible components. All terms involving (2) would also involve a vanishing 0,3 , while the same argument as for 0,3 above shows that the contribution of (3) to such 0,4 should vanish -it also involves only 0,2 . In fact, this same argument could be applied to 0,4 1 2 3 4 (we only need one irreducible component to be di erent from all others), but this turns out not to give a new constraint, so we omit it here.
Remark . . This argument can be used inductively to show that all 0, with exactly one argument on a given irreducible component must vanish. This is analogous to the proof of Proposition . , but also uses that all arguments of the recursion kernel must couple to a di erent correlator, as we restrict to genus zero. However, in general the recursive computation of these correlators does require ( ) of order up to the degree of˜, and therefore becomes quite complicated.
For the remainder of this Section, we restrict to recursion kernels coupled to 0,2 's, which is su cient for our calculations. We also assume 1 ∈˜. From the shape of the recursion kernel, we obtain several possible contributions (combining factors from the kernel and the correlators): ( ) There will always be one term ∈˜. ( ) For any other ∈˜, we get a contribution where is a primitive th root of unity and we need to sum over all subsets We need to take the series expansion of each of these near = 0. For ( ), this is d 1 The summations for cases ( ) and ( ) get quite complicated for general sizes of the subset, but for size one, they are computable. For case ( ), we get For case ( ), there are three di erent subcases, depending on the sign of − : .
Proof of Proposition . . Above the proposition, we already argued that 0,3 vanishes unless all arguments are on the same branch. Let us rst calculate the value of 0,3 with all arguments on the same branch . For this, we get the contribution from case ( ) and the one-argument version of case ( ), and then take the residue: For = 1 this expression vanishes and hence is symmetric. In case > 1 we know from [BBCCN , Proposition B. ] that for ∈ [ + 1], this is symmetric if and only if ±1 mod . If however, > 1 and > + 1 it is easy to see that the correlators can never be symmetric. Let us assume 0,3 is symmetric. Then for all 1 , 2 , 3 > 0 satisfying 1 + 2 + 3 = we must have From this we deduce that ℓ ( − − 1) = − 1. This means that there must exist an such that = − − 1 + ( − 1). This in turn implies that = 1 which contradicts our starting assumption. We therefore conclude that the symmetry condition for 0,3 1 2 3 is exactly captured by (i).

This vanishes if and only if
for all > 0 and ℓ ≥ 0. Then plugging in we see that the statement is equivalent to for all > 0 and ℓ ≥ 0. Using now that by assumption 0 < − the latter holds if and only if for all > 0. The above constraint is both necessary and su cient for 0,3 1 2 3 to vanish.
Let us prove that ( ) is automatically satis ed for all ≥ if = ±1 mod . First, assume that = 1 mod . In this case we have for all ≥ where we used that ≤ .
The case = −1 mod can be covered using similar arguments. Thus, it su ces to inspect ( ) for < in which case the constraint reduces to In order to derive the second line we substituted − → . In the following going through all cases, we will prove that, under the assumption of (i), property ( ) is satis ed if and only if (ii) holds.
Checking this expression for the two possible choices of we nd the following. This shows that indeed ( ) is satis ed, i.e. 0,3 1 2 3 = 0 in this particular case. -= + 1: We can assume that > 2 since ∈ {1, 2} is covered in the case considered before. Then ( ) is equivalent to the constraint which is always satis ed unless = , which is nothing but property (ii). Note that due to > we always have ≥ . • = + − 1: Again we may assume that > 2 since ∈ {1, 2} is a special case of the one considered before. Notice that for > 2 the constraint ( ) is equivalent to ≥ + 1 .
( ) -= + − 1: In this case, Clearly, this is at least + 1 if and only if > , which is exactly (ii) in this case.
-= + 1: Because > , we get > . Thus implying that ( ) and (ii) are both automatically satis ed. To sum up, we found that, under the assumption that (i) holds, 0,3 1 2 3 = 0 if and only if (ii) is satis ed. This closes the case > .

Hence, this vanishes if and only if
for all > 0 and ℓ > 0. Plugging in the explicit expression for ℓ ( ) this constraint translates into for all > 0 using that − > 0. Let us consider the case = + 1. First, we assume > , and write = − . Then so if the inequality holds for , it also holds for . It follows that we only need to consider ≤ . The case = should be treated separately. In this case, This is greater than − 2 if and only if + 1 ≥ ( − 1). If = − 1, this always holds, as then + 1 < , while if = + 1, this is implied by (ii). Now assume ≤ < . Then and so ( ) is automatically satis ed if (i) holds. The case = + − 1 can be treated with similar arguments and is therefore omitted. Thus, under the assumption that (i) holds we deduce that 0,3 1 2 3 vanishes if and only if ( ) holds for all ∈ [ ). It is straightforward to see that this is in turn equivalent to the statement that ( ) Let us stress the similarity between the constraint ( ) and the above one. It is therefore not surprising that one can prove that ( ) is equivalent to property (ii) under the assumption that (i) holds. Since one can use similar arguments as in the case of ( ) we omit the further analysis of this constraint.
Proof of Proposition . . In order to calculate this via the topological recursion formula, we need to take the residue of the product of three contributions: once ( ), with = 2, and twice ( a), once for ( , 3 ) and once for ( , 4 ). The rst factor has a rst order zero in , while the other two each have an order − 1 pole in , with non-zero coe cients at each lower order. As all of these coe cients also incorporate powers of [4] , it is easily seen that the terms contributing to the residue may not cancel, as long as 2( − 1) − 1 ≥ 1. This proves the formula. As 0,4 ( , , , ) is always zero, due to the structure of the recursion kernel, the last statement follows as well.
Note that the cases above in which Δ = 1 are exactly those considered in the beginning of the proof.
Proof of Proposition . . Let us start by computing 1 2 ,2 1 2 . Inspecting Equation ( ) it should be clear that one may proceed as in the computation of 0, . While the second term in in the bracket in ( ) gives a contribution 1 2 ,2 1 2 = . . . + Res where in the second line we also included the contribution from ( ). Let us further on use the notation 1 2 ,2 Inspecting ( ) we notice that for ≤ 2 the components 1 2 ,2 1 2 are always symmetric under the exchange of arguments. To be more precise, for = 1 the components all vanish and for = 2 they get a contribution form the rst and second line of ( ) solely. Hence, using that ℓ (1) = for = 2 we see that in this case 1 2 ,2 1 2 is indeed given by ( ). Now let us assume that > 2. Then due to property (ii) of Proposition . the contribution = in ( ) has to vanish and one ends up with 1 2 ,2 1 2 = − 1 + 2 , 2 Since − ≠ 0 for ≠ we can analyse the symmetry constraints coming from the rst and second line of ( ) individually. First, assume = + 1. Then if 1 + 2 = we may use that This expression is symmetric under the exchange of 1 and 2 if and only if Let us proceed by analysing the symmetry constraints coming from coe cients with 1 + 2 < , i.e. we consider the second line of ( ). Clearly the case = 3 is symmetric. Therefore now assume that > 3. In this case we can only expect a non-vanishing 1 2 ,2 1 2 if | − | ∈ [ − 3] for some ≠ . Due to Lemma . we know that for all either | − | = 1 or | − | ≥ − 2 which limits the cases in which 1 2 ,2 1 2 ≠ 0 for 1 + 2 < extremely. To be more precise, Lemma . tells us that | − | = 1 if and only if = 1 and = . Therefore for 2 < 1 + 2 < we have Together with ( ) this explains symmetry condition (iii). Regarding the formula for 1 2 ,2 1 2 stated in ( ) notice that for = 2 we have In order to obtain the rst two terms in the above expression one applies the symmetry constraint (iii) on ( ) and ( ). The third term is due to contributions ≠ in ( ) for which | − | = − 2. Lemma . tells us that this is the case for exactly those ≠ with = 2, < , and = . This now explains the origin of all terms occurring in ( ) which closes the analysis of the case = + 1. Now assume that = + − 1 and > 2. First let us inspect 1 2 ,2 1 2 given by ( ) for 1 + 2 = . In this case we may use that in order to nd that 1 2 ,2 1 2 = 1 2 ( + 1) − 2 + ∑︁ ≠ > for 1 + 2 = . This is symmetric if and only if Now let us turn to the case 1 + 2 < . Again, we may only expect a contribution from ≠ in the second line of ( ) possibly leading to a non-symmetric term if | − | = 1. Now Lemma . says that | − | = 1 if and only if = 1 and = . Thus, for 2 < 1 + 2 < we nd that , which is symmetric if and only if . This condition together with ( ) are of course nothing but (iv). Since the derivation of formula ( ) is in line with the one of ( ) we omit a further discussion. Now let us consider 1 2 ,2 1 2 for ≠ . First, assume = . Note that in this case property (ii) of Proposition . forces = ≤ 2. Using the same approach as in the derivation of ( ) one nds that Let us rst simplify this expression before turning to 1 2 ,2 2 1 . Comparing ( ) with ( ) we can deduce that property (ii) of Proposition . ensures that 1 2 ,2 1 2 = 0 unless 1 = 1. And moreover also 1 2 ,2 1 2 = 0 unless 1 + ( 2 + ℓ ( 2 )) − ℓ ( 2 ) = .
( ) In equation ( ) we have seen that the above relation cannot be satis ed for 2 ≥ . On the other hand for 1 ≤ 2 < plugging in the explicit expression ( ) for ℓ ( 2 ) we nd that ( ) is equivalent to ( ) Let us rst analyse the case = + 1. Since − > 0 and because of ( ) condition ( ) can only be satis ed for 2 = 1. Otherwise this would contradict property (ii) of Proposition . . However, plugging 2 = 1 into ( ) gives = + − 1 , i.e. = −1 mod and = . The case where = + − 1 can be treated similarly. We can assume that > 2 since ≤ 2 is a special case of the one considered before. Now since − ≥ 0 (compare with ( )) we always have implying that ( ) cannot be satis ed for any 2 > 0. Thus, 1 2 ,2 1 2 = 0 for all > 0 if > 2 and = + − 1. To sum up, assuming that (i) and (ii) from Proposition . hold we have One can obtain a similar expression for 1 2 ,2 2 1 . First, one nds that Then comparing the above expression with ( ) one can simplify the expression for 1 2 ,2 2 1 as we did with 1 2 ,2 1 2 before. One nds that We study separately the two types of terms in ( ) where the presence of 0,2 ( , ) forces the three variables in the recursion kernel to belong to the same component˜of˜. To handle the sum, we use the following identity for not divisible by ( ) The contribution of each term of the right-hand side of ( ) to ( ) can be computed similarly to item ( a). After computation of the residue we nd Then, we have So, we obtain the same formula for I 1,1 in both cases.
Remark . . This computation can also be done directly from the Airy structure, as and 1,1 [ ] is the constant term of order ℏ in the unique operator of the Airy structure of the form ℏ + O (2). In fact, as ( ) coincides with the topological recursion on the sole component˜of the spectral curve, the value of 1,1 [ ] | =0 must coincide at = 1 with the one computed in [BBCCN , Lemma B. ]. This is indeed the case, but we note the calculation by this other method involves the sum which is much simpler than ( ) although it leads to the same result.
To obtain II 1,1 we may split the contribution of the various as we did in the derivation of Equation ( ). The details are omitted and we only give the outcome: As before we only require that gcd( , ) = 1, and that for = and ≠ we must have ≠ . In the following we will compute correlators 0,3 and 1 2 ,2 and characterise the cases in which they are symmetric. These correlators are still given by formula ( ) and we will learn that in the exceptional case they mostly behave as in the standard case which was discussed in the preceding sections.
Lemma . . The correlators 0,3 and 1 2 ,2 are symmetric if and only if the conditions (i), (ii) of Proposition . and (iii), (iv), and (v) of Proposition . are satis ed. In this case the correlators are still given by the formulas stated in Proposition . and Proposition . , where any Kronecker delta involving − = ∞ evaluates to 0. Moreover, the statement of Proposition . also stays valid in the exceptional case.
Remark . . One should remark that all expressions occurring in (i) to (v) make sense even with − = ∞ for a single − ∈˜if we understand − = 1 mod − .
Proof of Lemma . . We begin with the computation of 0,3 . First of all we notice that in case 1 , 2 , 3 satisfy ≠ − the correlator 0,3 1 2 3 1 2 3 is computed as in the standard case which was considered in Proposition . since the bidi erential does not mix the components. So rst we can deduce that restriction of 0,3 to the non-exceptional components is symmetric if and only if ( , ) ≠ − satisfy (i) and (ii) and second we know that in this case 0,3 is given by ( ).
The following two cases are discussed quickly: if 1 , 2 , 3 ∈˜are all pairwise distinct the nature of the recursion kernel forces 0,3 1 2 3 1 2 3 to vanish. If 1 = 2 = 3 = − it vanishes as well since the bidi erential is not mixing the components and ( ) ∩˜− = ∅ if ∈˜−. So the only remaining case which needs to be covered is the one where the arguments of 0,3 lie on exactly two distinct components of which one is˜−. So let ≠ − . On the one hand we know that 0,3 − 3 2 1 = 0 while on the other hand ( ) tells us that where we used that In order to have symmetric correlators we therefore need ( ) to vanish. Clearly, this is the case if and only if 2 + > . For > 1 this is always the case as (i) forces that ≤ + 1. For = 1 however the correlator vanishes if and only if ≤ 2. It turns out this symmetry condition is already included in (ii). Assume = 1 and > 2. Then = 0 = − − and both = 1 mod and − = 1 mod − which means this case is forbidden by (ii).
As with the correlator we considered before, also has to vanish in order to have a symmetric 0,3 . This is the case if and only if ( − )/ + 2 > 0 for all > 0 satisfying | − . First let us assume > 1. Then (i) forces ≤ + 1 which implies that ( − )/ + 2 ≥ ( − 1)/ + 1 > 0 and thus as required 0,3 − − 1 2 3 = 0. Now let us assume = 1. In this case which vanishes if and only if ≤ 2. We conclude that as for the correlator considered before the condition for 0,3 − − 1 2 3 to be symmetric is expressed by (ii). This concludes the analysis of 0,3 . Now let us turn to 1 2 ,2 . In the following set 1 2 ,2 1 2 ,2 | −=0 , i.e. with 1 2 ,2 we denote the restriction of 1 2 ,2 to the non-exceptional components. Then for distinct , ∈˜\ { − } we have 1 2 ,2 1 2 = 1 2 ,2 1 2 as the bidi erential is not mixing the components. Thus, from the analysis done in Proposition . we deduce that the condition ensuring the symmetry of this correlator is (v) and in the symmetric case it may be evaluated using formula ( ). However, if the arguments lie on the same component ≠ − we obtain an additional contribution 1 2 ,2 1 2 = 1 2 ,2 1 2 + − Res coming from the rst term in the bracket in ( ). This is symmetric for ≤ 2 while for > 3 we may use that 1 2 ,2 1 2 is computed via ( ) in order to get 1 2 ,2 1 2 = ∑︁ 1 , 2 ≥1 .
Comparing ( ) with the above equation and following the characterisation of the symmetry of ( ) it should be clear that the condition for the symmetry of 1 2 ,2 1 2 above is encoded in (iii) and (iv). Now let us turn to the computation of correlators with arguments lying on the exceptional component. It is a straightforward calculation to nd that , ( ) which is always symmetric. Notice at this point that this is in accordance with (iii) and (iv) as the two conditions do not imply any constraints for − . Now let us brie y argue that formula ( ) exactly produces the result for 1 2 ,2 − − 1 2 we obtained above. Since − = − − = 0 the rst term in ( ) vanishes. The two terms after are empty sums, hence vanishing. Thus the last term in ( ) is the only one potentially leading to a non-zero contribution and indeed one nds that the sum condition in this sum coincides with the one in ( ). We therefore deduce that formula ( ) even applies for the exceptional case.
It is also straightforward to compute for ≠ − . We therefore deduce the symmetry condition that for all ≠ − with = + 1 necessarily = − − . This condition is however covered in (v). Notice moreover that formula ( ) applied to the case at hand indeed produces ( ).
At last, let us show that the statement of Proposition . is true in the exceptional case as well. For this notice that if we choose , , ∈˜pairwise distinct with = = then already < ∞.
Thus, 0,4 1 2 3 4 gets the same contributions as in the standard case which means that the further discussion of the symmetry of this correlator must be in line with the proof of Proposition . .

. N
We now prove Proposition . . In Section . we used that Theorem . gives us conditions for the values of ( , , , ) ∈˜s u cient for the existence of a family of correlators satisfying the master loop equation and the projection property on that curve. This was done by associating a set of di erential constraints ( ) to the curve and showing that these imply the master loop equation. Note that in order to prove the latter implication we only assumed that none of the conditions (i)-(iv) appearing in Lemma . hold. So all further conditions on the input data solely come from Theorem . which are su cient for the set of operators in Proposition . to be an Airy structure.
( ) Here we chose the notation and the bidi erential of the second kind exactly so that the di erential operators associated to this spectral curve introduced in De nition . are those considered in Section . . . As for certain values of ( , , , ) ∈˜w e ended up with non-symmetric multidi erentials , , we can deduce that in this case the associated set of di erential operators cannot be an Airy structure. Hence, condition (i)-(v) found in Proposition . and Proposition . are necessary conditions for ( ) to be an Airy structure. In order to be even closer to the notation of Section . . , let us choose a lexicographic ordering : [ ] →˜satisfying ≺ +1 . Proposition . addresses the question which constraints the symmetry conditions (i)-(v) found in Proposition . and Proposition . put on the values of ( , ) ∈ĩ n case we want the set of di erential operators ( ) to be an Airy structure for all 1 , . . . , −1 ∈ C * and, if ≠ ∞, also ∈ C * satisfying ( ) and all 1 , . . . ,  (iii) and (iv) also force ∈ {1, 2} using that we can choose the s arbitrary except that they must satisfy ( ).
As one can use similar arguments in order to show that also = 1 mod we omit the further discussion of this case. Now let us analyse the implications coming from condition (v) in case of generic s. If = 2 condition (v) is always satis ed since the requirement that = − for ≠ is nothing but property ( ). Therefore, let us assume that > 2. In this case (v) exactly forbids that = and = = 2 for ≠ . Hence, taking into account the other constraints we have already derived for ( , ) ∈˜w e deduce that for > 2 necessarily = = 1 whenever ( , ) = ( , ) for some ≠ .
Note that due to Lemma . the above discussion covers both the standard and the exceptional case.

P III -A
We expect that the coe cients , of the partition function for all basic Airy structures (those of Section . . and Section . . ) and the , of the corresponding topological recursion can be represented as integrals of distinguished cohomology classes on M , or its cousins. As soon as such a representation is known for one spectral curve admitting a single rami cation point and whose type belong to a certain set, it is relatively easy to extend it to any spectral curve having rami cation points whose type belong to this set. The reason is that the corresponding Airy structure is built from the basic Airy structures by direct sums, change of polarisations and further dilaton shifts, cf. Section . . . In terms of partition functions, this is sometimes called "Givental decomposition". We develop this idea for two types for which the link to M , is already known: • the type ( , ) = ( , + 1) is related to Witten -spin theory. The = 2 subcase is Eynard's formula [Eyn ; Eyn b], and by generalising it to any we answer a question of Shadrin to the rst-named author. • the type ( 1 , 1 , 2 , 2 ) = ( , + 1, 1, ∞) is related to open -spin intersection theory as discussed in Section . The type ( , ) for other is discussed in Section . assuming the existence of a special class on M , which has only been constructed for ( , ) = (2, 1) so far [Nor ; CN ]. It turns out that Laplace-type integrals play an important role in such representations, and we rst study them in the preliminary Section . . The method is general: if in the future an enumerative interpretation is found for a larger set of types, it is rather automatic to follow the strategy at work in these two examples and extend our representations to any spectral curve having rami cations whose types belong to this larger set. R . T L Let ≥ 1. If ≥ 1 − is an integer, we introduce the -fold factorial, either by induction or equivalently in terms of the Gamma function where we have de ned De nition . . We introduce two isomorphisms.
Abusing notation slightly we also write ± for the maps extended to the domain C(( ))d by de ning them to be zero on any monomial not in the original domain of de nition.
The rst map can be realised by integrating over paths from 0 to ∞ in the -plane.
Lemma . . We have where the constants are ∑︁ The second map can be realised by contour integration. To this end, let be the Hankel contour giving the Gamma function representation ( Figure ) 1 that is, goes from −∞ − i0 to −i0, then round the origin to +i0 and ends in −∞ + i0. Under the branched covering ↦ → ( ) = , this contour has di erent lifts ( ) =1 , which we label so that comes from the asymptotic direction − i (2 +1) (+∞ + i0), approaches the origin and then ends in the asymptotic direction − i (2 −1) (+∞ − i0). These contours belong to the lattice of rank ( − 1) of Lefschetz thimbles with the constants already appearing in ( ).
Proof. Let > 0, ∈ [ ] and consider the integral for complex ∈ C. Here, ↦ → is de ned in the usual way as the analytic function on C \ R − such that for ∈ R * − , we have lim →0 + ( ± i ) = ±i | | . Let us rst consider Re > 0. In this case, as the integrand is regular we can squeeze the Hankel contour to the half-axes of angles − i (2 +1) for ∈ [ ]. After a change of variable˜= − , we nd sin ( + 1) Γ + 1 .

( )
We note that by de nition in ( ), ( ) is an entire function of ∈ C. This is also true for the righthand side of ( ): the Gamma function has simple poles when +1 ∈ −2N which are compensated by a zero in the prefactor. Then, by analytic continuation the identity ( ) remains true for all ∈ C. We apply it to = −( + 1) where is a positive integer that we decompose as = + with ∈ [ ] and ≥ 0. Then Coming back to the de nition ( ) of , we get . L

. . Total Laplace transform
Let be a curve with normalisation :˜→ , equipped with a meromorphic 1-form d . We shall rely on the notations introduced in Section . . Recall that we write d˜= * d , that is the set of zeroes of d (including singular points) and˜= −1 ( ). Denote the order of a zero of d at ∈˜by − 1 (this could be zero if ( ) is singular). Around each ∈˜, we have a local coordinate such that ∫ · d˜= .
We de ne the vector spaces = ∈˜, =1 C. , , and equip with the pairing ( , ⊗ , ) = , + , . The fact that , are null vectors for is a convention: it simpli es the formulas in Section . . but has no e ect elsewhere. When needed, we shall decompose integers ∈ Z as =

+ˆ,ˆ∈ [ ]
and the index ∈˜that one should use will be clear from the context. In order to get rid of fractional powers of the Laplace variable, we introduce the isomorphism * : where the role of * , is now played by , . Remark . . From Lemmata . and . , we see that the natural variables Laplace dual to = are not and −1 , but and −1 . Moreover, by a di erent choice of constants (replacing multiple factorials by values of the Γ function at rational numbers) the transforms could have been given by a single integral. We give the Laplace transform in this way to conform to conventions in the literature using -fold factorials for the case of a smooth spectral curve, and its relation to the Witten -spin class.

. . Two generating series
Assume that we are given a holomorphic 1-form 0,1 in a neighboorhood of˜in˜. .
Assume that we are given a fundamental bidi erential of the second kind 0,2 on the smooth curvẽ . where the second line follows from the decomposition ( ).
These de nitions in particular apply to admissible spectral curves equipped with a fundamental bidi erential of the second kind, but make sense in this greater generality. Their relevance will become clear in Section . .

. . Factorisation property for B
We prove in this section a factorisation property for B when˜is compact. Such a property appeared in the case where is smooth and d has simple zeroes in [Eyn b, Appendix B].
In light of Remark . , and for this section only, we use the 'right' Laplace variables and (where depends on the branch point), and to this end we give the following de nition. The idea of the proof is to derive a recursion for these forms using the action of d · d˜ -this is Equation ( ) below. We will rst prove it for the polar part near the rami cation points, and use that ˜i s smooth and compact and d˜meromorphic on˜to get an equality of globally de ned meromorphic forms. This implies a recursion for the regular part of the expansion ( ), i.e. the coe cients 0,2 , and this will imply the desired relation for B( 1 , 2 ).
Since d˜= −1 d near the rami cation point , we have As d˜is meromorphic, the 1-form is holomorphic on˜. Since˜is compact and smooth, 1 (˜, C) is a nite-dimensional symplectic space equipped with the intersection pairing, and K ∈ 1 (˜, C) ∫ 0,2 (·, ) = 0 is a Lagrangian subspace. From the de nition ( ), we see that integrating any d − for > 0 and ∈˜along a cycle in K gives zero. The same is true for the rst term in ( ) since it is an exact form. As the period map induces a non-degenerate pairing 0 (˜,˜) ⊗ K → C and ( ) is sent to 0, we deduce the identity between meromorphic forms We now would like to apply the local Laplace transform + [·] ( ) to this relation. Recall that + by de nition kills the polar part. So, by direct computation on the basis elements with De nition . , we have Replacing and by / and / , respectively, and summing over , ∈˜, this implies the desired formula ( ). Since the left-hand side is a formal power series in , , it can be specialised at = − . This is possible on the right-hand side if and only if ( ) is satis ed. Corollary . . Assume˜is compact and d is meromorphic. Then, we have The structure of the rst line is familiar from Givental formalism and from [Eyn b, Appendix B], but the other three are new.
. R W For ≥ 2, we denote spin , For comparison, let us examine the basic properties of · · · ★ . Firstly, due to the constraint ( 1 + 2 ) ★ = 0 for > 0 and the de nition ( ), each insertion of • ( ) amounts to the insertion of (−1/ ) while incrementing the number by 1. Secondly, the constraint =2, =0 ★ = 0 gives, by computations similar to those of Section . , the string equation , and thus after taking ( ) into account, ( ) describes the only non-vanishing intersection number for ( , ; , ) = (0, 1; 1, 1). Thirdly, we have from corollary . the dilaton equation We predict that the partition function ★ describes the full open -spin intersection theory in any genera and with arbitrary descendants.
Conjecture . . There is a geometric de nition of the open -spin intersection numbers, and it satis es • 1 ( 1 ) · · · • ( ) 1 · · · spin , ; , = • 1 ( 1 ) · · · • ( ) 1 · · · ★ , ; , . A weaker prediction involving only quantities whose de nition is available at the time of writing, is that the Bertola-Yang BY partition function satis es ( )-constraints with zero mode values 1 = − 2 = 1. Including the expected normalisations, this would translate into the following. Conjecture . . We have for , > 0 In support of the conjectures, we see that the basic properties listed for · · · ★ match the ones listed for · · · in the range of parameters in which the comparison is possible. The dilaton equation of [BCT , Proposition . ] matches the restriction of ( ) to ( , ) = (0, 1) and = 0. The string equation of [BCT , Proposition . ] matches the same restriction of ( ), and observe that in absence of boundary descendants the second sum in the right-hand side of ( ) is absent. Their extension to > 0 expected in [BCT , Section . ] also matches our proposal. TRR relations involving in a linear way the open -spin intersection numbers mentioned in ( ) and the closed -spin intersection numbers are given in [BCT , Theorem . ]. The information of ( ) should be encoded for us in ★ 1 2 , , as it is easy to see that the -constraints indeed imply some quadratic relation with a similar structure involving ★ 0, in a non-linear way and ★ 1 2 , in a linear way. As ★ 0, contains only information from the closed sector and satis es -constraints on its own, the structure somehow resembles the TRR relation, but establishing an exact match is left to future work.