DEFORMATION AND QUANTISATION CONDITION OF THE Q -TOP RECURSION

. We consider a deformation of a family of hyperelliptic reﬁned spectral curves and investigate how deformation eﬀects appear in the hyperelliptic reﬁned topological re-cursion as well as the Q -top recursion. We then show a coincidence between a deformation condition and a quantisation condition in terms of the Q -top recursion on a degenerate elliptic curve. We also discuss a relation to the corresponding Nekrasov-Shatashivili eﬀective twisted superpotential


Introduction
The purpose of the present paper is twofold.One is to describe the so-called variational formula in the framework of the hyperelliptic refined topological recursion as well as the Q-top recursion proposed in [KO22;Osu23].The other is to reveal an intriguing coincidence between a deformation condition and a quantisation condition in terms of the Q-top recursion as an application of the variational formula.

Motivations and Backgrounds
Since motivations and backgrounds of a refinement of topological recursion are discussed in [KO22;Osu23] in detail, we only give a brief review of recent developments in this direction.
As defined in [Osu23] (and in [KO22] for a special class of genus-zero curves), a hyperelliptic refined spectral curve S κ,µ consists of three data: a compactified and normalised Torellimarked hyperelliptic curve C = (Σ, x, y) of genus g1 , complex parameters κ associated with the Torelli markings, and complex parameters µ associated with non-ramified zeroes and poles of a differential ydx2 .We often drop 'hyperelliptic' for brevity.Taking a refined spectral curve as initial data, the refined topological recursion constructs an infinite sequence of multidifferentials ω g,n on Σ n labeled by n ∈ Z ≥0 and g ∈1 2 Z ≥0g is different from the genus of Σ. [KO22; Osu23] proved or conjectured properties of ω g,n .Several results based on matrix models have also been discussed in e.g.[CE06b; BMS11; Che11; CEM11; MS17].
The multidifferentials ω g,n polynomially depend on the refinement parameter Q, up to Q 2g .It is easy to see that the Q-independent part precisely corresponds to the Chekhov-Eynard-Orantin topological recursion [CE06a; CEO06; EO07].As shown in [Osu23], it turns out that the Q-top degree part also give rise to a self-closed recursion, and we call it the Q-top recursion.That is, the Chekhov-Eynard-Orantin topological recursion and the Q-top recursion are a subsector of the full refined topological recursion, and we respectively denote differentials in each subsector by ω CEO g,n and ̟ g,n to notationally distinguish from ω g,n .For a family of hyperelliptic curves C(t) with some complex parameters t, one can consider the corresponding family of refined spectral curves S κ,µ (t) (with mild restrictions, e.g.ramification points should not collide each other under deformation of parameters).As a consequence, ω g,n also depend on the parameters t, and one may ask: how do ω g,n vary under a deformation with respect to t?
In the unrefined setting, this point has already been addressed in [EO07; Eyn17], and we know how ω CEO g,n (t) varies which is known as the variational formula3 .It can be thought of as a generalisation of the Seiberg-Witten relation [SW94a;SW94b].However, it turns out that there is a subtlety and difficulty when one tries to apply the original Eynard-Orantin proof to the refined setting.Thus, we provide an equivalent interpretation of the variational formula (Definition 3.1) which becomes easier to apply to the refined topological recursion.With this perspective, we are able to state a refined analogue of the variational formula.

Summary of main results
The first achievement of the present paper is to prove the variational formula for the refined topological recursion, when Σ = P 1 (Theorem 3.5).However, since we have to fix several notations and technical aspects in order to remove the subtlety mentioned above, it is hard to state the variational formula here and we leave all the details to Section 3. Roughly speaking, it states that a certain deformation δ t * ω g,n with respect to t ∈ t is related to an integral of ω g,n+1 as follows: where (γ, Λ) is defined in Definition 3.3.Let us emphasise that, in contrast to the unrefined setting, the variational formula (1.1) holds only when a refined spectral curve S κ,µ (t) satisfies a certain condition which we call the refined deformation condition (Definition 3.4).See Section 3 for more details.Note that some properties of the refined topological recursion are still conjectural when Σ = P 1 [Osu23], hence the variational formula also remains conjectural in this case.We also note that [CEM11] discuss a similar formula in a different refined setting.
Another achievement of the present paper is to uncover an intriguing coincidence between the refined deformation condition and what we call the Q-top quantisation condition defined as follows.It is shown in [Osu23] that the Q-top recursion naturally constructs a secondorder ordinary differential operator, called the Q-top quantum curve.For a refined spectral curve S κ,µ (t) whose underlying curve is given by y where ǫ 1 is a formal parameter, Q k (x) is a rational function of x determined by {̟ h } h for 2h < k, and the logarithmic derivative of ψ Q−top (x) is a formal sum of ǫ 2g−1 1 • ̟ g,1 over g.In the context of topological recursion, one may sometime require a condition on quantisation that the set of poles of Q k (x) should be a subset of poles of Q 0 (x).Therefore, we say that a refined spectral curve S κ,µ (t) satisfies the Q-top quantisation condition, if the Q-top quantum curve respects the pole structure of Q 0 (x) (Definition 4.5) -existence of a quantum curve in the full refined setting is proven only for a special class of genus-zero curves [KO22] and in this case one can analogously consider the refined quantisation condition.
In order to deliver a clear picture about the coincidence between the refined deformation condition and the Q-top quantisation condition, let us focus on the following example.For t ∈ C * , we consider a one-parameter family of curves C t = (P 1 , x, y) where meromorphic functions (x, y) satisfy: This is the curve associated with the zero-parameter solution of the Painlevé I equation, and t plays the role of the Painlevé time [IS16].Since ydx has a simple zero at the preimages of x = q 0 , the corresponding refined spectral curve S µ (t) carries one parameter µ ∈ C, and ω g,n depend both on t and µ.
In this example, it turns out that S µ (t) satisfies the refined deformation condition if and only if µ is set to a special value µ = µ 0 .(Proposition 4.4).On the other hand, one can show that Q k≥2 (x; t, µ) has a pole at x = q 0 for a generic µ, which is a zero of Q 0 (x; t).However, it turns out that when µ = µ 0 , such poles disappear for all k, and thus, the Q-top quantisation condition is satisfied (Proposition 4.6).Therefore, we observe that the refined deformation condition and the Q-top quantisation condition precisely agree, even though they originated from two different requirements.It is interesting to see whether this coincidence holds in other curves, e.g.curves discussed in [IMS18] in relation to other Painlevé equations.
When µ = µ 0 , the variational formula gives a relation between Q k (x; t, µ 0 ) in (1.2) and a derivative of F Q-top g := ̟ g,0 with respect to t -the former appears in the Q-top quantisation and the latter is a consequence of a deformation of a refined spectral curve: Theorem 1.1 (Theorem 4.7).Consider the above family of refined spectral curves S µ 0 (t) satisfying the refined deformation condition and also the Q-top quantisation condition.Then, the associated Q-top quantum curve is given in the following form: It is crucial to remark that there is no ǫ 2 1 ∂/∂t term in (1.4), in contrast to the quantum curve derived in [IS16; Iwa20] within the framework of the Chekhov-Eynard-Orantin topological recursion.Instead, a similar differential operator to (1.4) has appeared in the context of conformal blocks in the semi-classical limit, or the so-called Nekrasov-Shatashivili limit e.g.[LN21; LN22; Bon+23].Note that they consider a genus-one curve whose singular limit becomes (1.3), and we expect that the form of (1.4) remains the same for the corresponding genus-one curve.Importantly, their arguments and Theorem 1.1 suggest a conjectural statement that F Q−top g agrees with the so-called Nekrasov-Shatashivili effective twisted superpotential W eff g [NS09], when a refined spectral curve is chosen appropriately: where Z Nek is the corresponding Nekrasov partition function [Nek03] and the equality should be considered as a formal series in ǫ 1 .See e.g.[NRS11; HK18; HRS21] for more about Nekrasov-Shatashivili effective twisted superpotentials.Note that for the curve associated with the Painlevé I equation, the Nekrasov partition function is not defined from an irregular conformal block perspective, whereas F Q−top g is perfectly well-defined.We hope that the present paper together with the notion of the Q-top recursion [Osu23] sheds light on verifying the above statement and also triggers a new direction between topological recursion, the Qtop recursion, and invariants in the Nekrasov-Shatashivili limit (e.g. a role of ̟ g,n≥2 ).

Definitions
We briefly review the refined topological recursion proposed in [KO22;Osu23].We refer to the readers [Osu23, Section 2] for more details.
Definition 2.1 ([KO22; Osu23]).A hyperelliptic refined spectral curve S µ,κ consists of the collection of the following data: • (Σ, x, y): a connected compact Riemann surface of genus g with two meromorphic functions (x, y) satisfying y 2 − Q 0 (x) = 0, (2.1) where Q 0 (x) is a rational function of x which is not a complete square.We denote by σ : Σ → Σ the hyperelliptic involution of x : Σ → P 1 and by R the set of ramification points of x, i.e. set of σ-fixed points.
• (A i , B i , κ i ): a choice of a canonical basis A i , B i ∈ H 1 (Σ, Z) and associated parameters κ i ∈ C for i ∈ {1, .., g}, • ( P + , µ p ): a choice of a decomposition P + ⊔ σ( P + ) = P and associated parameters µ p ∈ C for all p ∈ P + where P is the set of unramified zeroes and poles of ydx.
Let us fix some notation before defining the refined topological recursion.First of all, throughout the present paper, g, h are in 1 2 Z ≥0 , n, m in Z ≥0 , i, j in {1, .., g} and a, b in {0, .., n}.We denote by B the fundamental bidifferential of the second kind, and for a choice of representatives A i of A i for each i, we denote by η p A the fundamental differential of the third kind for p ∈ Σ normalised along each A i -cycle.We write p a ∈ Σ for each a, J := (p 1 , .., p n ) ∈ (Σ) n , and J 0 := {p 0 } ∪ J ∈ (Σ) n+1 .Assuming p a ∈ R ∪ σ(P + ) for all a, we denote by C + a connected and simply-connected closed contour such that it contains all points in J 0 ∪ P + and no points in R ∪ σ(J 0 ∪ P + ).With the assumption on p a , one can always find such a contour and we drop the n-dependence on C + for brevity.Similarly, we denote by C − a connected and simply-connected closed contour containing all points in R ∪ σ(J 0 ∪ P + ) but not points in J 0 ∪ P + .We call p ∈ R ineffective if ydx is singular at p, and effective otherwise.We denote by R * the set of effective ramification points.We denote by P 0,∞ + ∪ σ(P 0,∞ + ) the set of unramified zeroes and poles of ydx respectively, and denote by C p − a connected and simply-connected closed contour inside C − but not containing points in σ(P ∞ + ).Finally, we fix Q ∈ C and we call it the refinement parameter.Definition 2.2 ([KO22; Osu23]).Given a hyperelliptic refined spectral curve S µ,κ , the hyperelliptic refined topological recursion is a recursive definition of multidifferentials ω g,n+1 on (Σ) n+1 by the following formulae: where and the * in the sum denotes that we remove terms involving ω 0,1 .
As discussed in [Osu23], it is easy to see for each g, n that ω g,n+1 polynomially depends on Q up to Q 2g , and the recursion for the Q-top degree part is self-closed, i.e. they can be constructed without the information of lower degree parts.We call it the Q-top recursion, and explicitly it is defined as follows: ).Given a hyperelliptic refined spectral curve S µ,κ , the Q-top recursion is a recursive definition of multidifferentials ̟ g,n+1 on (Σ) n+1 by the following formulae: (2.9) where (2.12) Note that there is no Since the Q-top recursion is a subsector of the refined topological recursion, Theorem 2.3 holds for ̟ g,n+1 too, as long as Σ = P 1 .We note that it is meaningful to define the Q-top recursion independently and study it on its own.For example, as discussed in [Osu23], the Q-top recursion would be relevant to the Nekrasov-Shatashivili limit which is an active research area in mathematics and physics.In particular, [Osu23] proved the following property for any Σ, not limited to Σ = P 1 : Theorem 2.6 ([Osu23]).̟ g,1 are well-defined residue-free differentials on Σ whose poles only lie in R * ∪ σ(P 0 + ), and there exists an ordinary second order differential equation of the following form: where Q k (x) is a rational function of x explicitly constructed by ̟ h,1 for 2h < k, and ψ Q-top is a formal series in ǫ 1 defined by (2.14) The associated differential operator (2.13) is called the Q-top quantum curve.Except for a special class of genus-zero curves investigated in [KO22], existence of the refined quantum curve in full generality is still an open question.
When the underlying hyperelliptic curve depends on complex parameters t = {t 1 , .., t n }, one can consider a t-parameter family S κ,µ (t) of refined spectral curves as long as t are in a domain such that no points in R ∪ P collide.All the above definitions and theorems hold for S κ,µ (t).In the next section, we will consider how ω g,n+1 (t) behave while one varies t.
Before turning to the variational formula, let us define the free energy F g , except F 0 , F1 2 , F 1 which will be defined later: Definition 2.7 ([KO22; Osu23]).For g > 1, the genus-g free energy F g , F Q-top g of the refined topological recursion and the Q-top recursion is defined respectively as follows: (2.15) (2.16)

Variation
The variational formula is proven in [EO07] and originally it is explained as follows.Consider a one-parameter family of spectral curves S(t) in the unrefined setting.Then, x and y as functions on Σ depend on the parameter t and so do all ω g,n+1 (t).Then, [EO07] considers a special type of deformation, namely, variation for fixed x.This may sound contradictory with the fact that x depends on t, but what it really means is the following.
Set Q = 0.By choosing one of the branched sheet, one projects ω g,n+1 down to P 1 away from ramification points and treat them locally as multidifferentials on P 1 .The variation for fixed x means that we apply the partial derivative with respect to t for these multidifferentials on P 1 with the understanding that ∂ ∂t dx a = 0, and apply the local inverse x −1 to pull them back to differentials on Σ.That is, the variation symbol δ EO t in [EO07] acting on ω g,n+1 means (c.f.[IKT23; Bon+22]): where on the right-hand side we think of x as independent of t and instead p t depends on both t and x.We will denote by * the action of the variation in order to distinguish from the standard product symbol • which we are using throughout the paper.The standard partial derivative notation ∂ t is commonly used in e.g.[Eyn17; EGF19; MO22] but we avoid this notation to emphasise that the operation is not just a partial derivative.
We will provide another equivalent description of the variation operation without considering the projection and inverse.The motivation of introducing such a new perspective is for the clarity of the proof of the variational formula when Q = 0.The original proof by Eynard and Orantin is based on a graphical interpretation whose analogue does not exist in the refined setting, at least at the moment of writing.As a consequence, we need to directly evaluate the variation of the refined recursion formula (2.5), and in this case, taking the projection and the inverse becomes subtle because C ± contains J 0 and σ(J 0 ).Definition 3.1.Given S µ,κ (t), the topological recursion variational operator δ (n) t is a differential operator acting on meromorphic functions on (Σ) n defined by where (p 1 , .., p n ) ∈ (Σ\R) n and d pa denotes the exterior derivative with respect to p a .We extend the action of δ Note that this definition is valid not only for hyperelliptic curves but also for any algebraic curves.It can be generalised to a multi-parameter family in an obvious way.δ (n) t is defined only when each p a ∈ R which resonates with the fact that one has to choose a branch in the Eynard-Orantin description.Importantly, the above definition implies and for a differential w on (P 1 ) n , its pullback to (Σ) becomes clearer when one thinks of the underlying hyperelliptic curve from the Hitchin perspective [DM14; DM18; Eyn17].A Hitchin spectral curve (of rank 2) is given by a triple (Σ o , ϕ, π) where π : Σ o → P 1 , ϕ is a quadratic differential on P 1 , and Σ o is embedded in T * P 1 as (3.7) Our Σ would be obtained after normalisation and compactification of Σ o .By interpreting π = x and ϕ = (ydx) ⊗2 , variation for fixed x means that one varies the quadratic differential ϕ while keeping the projection π = x invariant.
Given an unrefined spectral curve S(t), let us assume existence of a pair (γ, Λ) such that γ is a path in Σ\R and Λ is a function holomorphic along γ satisfying (3.8) Then, [EO07] showed that the following relation holds for g, n ∈ Z ≥0 by using the graphical interpretation of the unrefined topological recursion formula, which is known as the variational formula: The difficulty to generalise the variational formula into the refined setting arises due to the more complicated pole structure of {ω g,n+1 } g,n .Nevertheless, if we restrict the pair (γ, Λ) to certain classes as below, a refined analogue still holds when Σ = P 1 , and we expect that it works for any Σ in general.
For s ∈ P ∞ \R and r ∈ P ∞ ∩ R, let x(s) = x s , x(r) = x r and suppose ω 0,1 behaves locally show a construction of each Λ s,k , Λ r,k , at least locally.Note that their pole is at most of order m s − 1, m r − 1 respectively.

Definition 3.3 ([Eyn17]
).Given S κ,µ (t), (γ, Λ) is said to be a generalised cycle if it falls into one of the following kinds: .,g} and Λ = 1 II : Let p ∈ Σ be an m p -th order pole of ω 0,1 where m p ≥ 2.Then, for k ∈ {1, .., m p − 1}, Λ p,k is given as in (3.11), and γ p,k is a union of contours encircling p and σ(p) in the opposite orientation if p ∈ R, and γ p,k is a contour encircling p if p ∈ R. III : Let p ∈ Σ be a location of a residue of ω 0,1 which necessarily means p ∈ R.Then, γ p is an open path from σ(p) to p within a fundamental domain, and Λ p = 1.
The corresponding parameters t (γ,Λ) defined by the expansion (3.10) are called 2nd kind times or 3rd kind times, whereas 1st kind times are defined by 1st, 2nd, and 3rd kind times are respectively called filling fractions, temperatures, and moduli of the poles in [EO07].All generalised cycles (γ, Λ) are anti-invariant under σ when it applies to integration.2nd and 3rd kind times are often refered to as KP times and their relation to KP systems are discussed in [Eyn17].
We consider a refined spectral curve S κ,µ (t) such that t 1 , .., t |t| ∈ t are defined as above, which are independent of each other, and we denote by (γ 1 , Λ 1 ), .., (γ |t| , Λ |t| ) associated generalised cycles.In this setting, the variational formula (3.9) holds in the unrefined setting as shown in [EO07].However, when Q = 0, it turns out that an analogous statement holds if S κ,µ (t) satisfies an additional condition, which we call the refined deformation condition: Definition 3.4.Consider S κ,µ (t) parameterised by times of the 1st, 2nd, and 3rd kind t = (t 1 , .., t |t| ).We say that S κ,µ (t) satisfies the refined deformation condition with respect to t l for l ∈ {1, .., |t|} if the following holds: We say that S κ,µ (t) satisfies the refined deformation condition if the above holds for all l.
Note that in the unrefined setting the variational formula (3.9) for (g, n) = (0, 1) automatically holds if ω 0,2 = B.Even if ω 0,2 is defined differently, it is then observed in e.g.[Bon+22] that the variational formula still works for the rest of ω g,n+1 , as long as the variational relation (3.9) holds for (g, n) = (0, 1).In other words, it has to be rather imposed as a supplemental condition in addition to (3.8).The refined deformation condition (Definition 3.4) is analogous to this observation.
Finally, we will state the variational formula in the refined setting, whose proof is entirely given in Appendix A.1 and A.2 because it is lengthy: Theorem 3.5.When Σ = P 1 , assume that S κ,µ (t) satisfies the refined deformation condition with respect to t l for l ∈ {1, .., |t|}.Then, ω g,n+1 and F g (g > 1 for F g ) satisfy: Conjecture 3.6.Theorem 3.5 holds for any Σ.

Examples
We Proof.The proof is done by explicit computations.Since they are genus-zero curves, a rational expression of x, y is given in e.g.[IKT19] in terms of a coordinate z on P 1 , from which one can construct the variational operator δ t for all t ∈ t with respect to z.Then, all one has to do is to compute ω 1 2 ,1 (z 0 ) and ω 1 2 ,2 (z 0 , z 1 ) from the refined topological recursion and explicitly check the refined deformation condition.See Appendix A.3 where we present explicit computations for a few examples.
One can use the variational formula as the defining equation for F1 2 and F 1 as followssince all ω 0,n is independent of the refinement parameter Q, we can define F 0 as [EO07] does: Definition 4.2.For a refined spectral curve S µ (t) associated with a hypergeometric type curve, F1 2 and F 1 are defined as a solution of the following differential equations for all k, l ∈ {1, .., |t|}: where F1 2 is defined up to linear terms in t l and F 1 is defined up to constant terms.
Since Λ = 1 for the 3rd kind, we immediately obtain the following: Corollary 4.3.For a refined spectral curve S µ (t) associated with a hypergeometric type curve, we have the following for 2g − 2 + n ≥ 1: Corollary 4.3 becomes useful to derive a relation between refined BPS structures [Bri19; BBS20] and the refined topological recursion, as a generalisation of [IKT23; IKT19; IK22].For a general refined spectral curve S κ,µ (t), not limited to hypergeometric type curves, we will define F1 2 , F 1 in a similar way to Definition 4.2.See Remark A.7.

A degenerate elliptic curve
Let us consider the case where x and y satisfy the following algebraic equation: A convenient rational expression of x, y in terms of a coordinate z on Σ = P 1 is where for brevity, we set q z := √ 3q 0 .It appears in a singular limit (as an algebraic curve) of the following elliptic curve, y2 = 4x 3 − g 2 x − g 3 , (4.5) where for generic g 2 , g 3 we can write x, y in terms of the Weierstrass ℘-function as x = ℘ and y = ℘ ′ .In [IS16; Iwa20], the curve (4.3) or (4.5) is chosen as a spectral curve of the Chekhov-Eynard-Orantin topological recursion, and a relation between the free energy and a τ -function of the Painlevé I equation is proven.
As shown in [IS16], t in (4.3) plays the role of a 2nd kind time, and the corresponding generalised cycle can be decoded from the following equations where c is one of the roots of 2c 2 − 6c + 3 = 0.The second term in Λ t is irrelevant in the last equation in (4.6), and it is indeed absent in [IS16], though it is necessary for the second equation.Now one may ask: does every S µ (t) satisfy the refined deformation condition similar to hypergeometric type curves (Proposition 4.1)?Here is the answer to that question: Proposition 4.4.Let S µ (t) be a refined spectral curve defined as above.Then, it satisfies the refined deformation condition if and only if µ = 1.
Proof.The proof is again by explicit computations, similar to Proposition 4.1.That is, we explicitly write the variational operator δ (1) t in terms of t and z, and confirm when (3.13) is satisfied.Since everything can be expressed as rational functions, it is easy to find that µ = 1 is the only solution.See Appendix A.3 for computations.
Note that, unlike ω g,n+1 for 2g − 2 + n ≥ 0, poles of ω 1 2 ,1 (z 0 ) are all simple and they are located not only at z 0 = 0, −q z but also at z 0 = q z , ∞ whose residues are given as: Therefore, the refined deformation condition is satisfied exactly when ω 1 2 ,1 becomes regular at P + .Even if we choose P + = {−q z } instead, this aspect remains correct.That is, the refined deformation condition for this curve is equivalent to the condition such that ω 1 4.2.1.Q-top quantum curve Theorem 2.6 shows that the Q-top recursion can be utilised to quantise a refined spectral curve.For a general refined spectral curve S κ,µ (t), not limited to the above example, we introduce the following terminology: Definition 4.5.We say that a refined spectral curve S κ,µ (t) satisfies the Q-top quantisation condition if for each k the set of poles of We return to our example, and consider the Q-top quantisation condition for S µ (t).
Proposition 4.6.The above refined spectral curve S µ (t) satisfies the Q-top quantisation condition if and only if µ = 1.
Proof.The proof is again by computations.The formula in [Osu23] gives (4.9) The if part is easy to see.By setting set µ = 1, then (4.7) implies that ω 1 2 ,1 becomes regular at z = q z hence Q Q-top k≥2 becomes regular at x = q 0 .See Appendix A.3 for the only-if part.
Therefore, the refined deformation condition and the Q-top quantisation condition agree for this example.Note that any refined spectral curve of hypergeometric type satisfies the Q-top, and in fact the refined quantisation condition.We expect that no additional condition will appear in the full refined quantisation, and it is interesting to see whether this coincidence holds for other curves, e.g.curves related to other Painlevé equations [IMS18].
To close, we prove that the Q-top quantum curve for S µ=1 (t) is written in terms of the Q-top free energy F Q-top g whose proof will be given in Appendix A. [LN21; LN22] discuss a similar equation in the context of accessory parameters and conformal blocks in the Nekrasov-Shatashivili limit.Thus, we conjecture that the Q-top free energy F Q-top g coincides with the Nekrasov-Shatashivili effective twisted superpotential [NS09] even when Σ = P 1 as long as an appropriate refined spectral curve is chosen.
Theorem 4.7.For S µ=1 (t) described above, the Q-top quantum curve is given as: where and F Q-top 1 are defined as a solution of the following differential equation: On the other hand, since z(p) can be thought of as a constant in terms of δ (p 0 ,p 1 ) t , we find where δ (p 0 ,p 1 ) t * (z(o)) can be nonzero if o depends on p 0 , p 1 , or t.Nevertheless, the second term in (A.7) will have no contributions after taking the residue, and we have shown that δ (p 0 ,p 1 ) t commutes with Res p=o .One may interpret this result such that a closed contour encircling p = o can be chosen independently from the time t.
Our last task is to transform δ , that is, the variational operator becomes effective with respect to the variable of integration p as well.In fact, by the chain rules, we find ∂x(p) ∂t .
(A.8) Then since f and ω are both meromorphic, the last term vanishes after taking residue.Lemma A.1 can be easily generalised to δ (p 0 ,p 1 ,..,pn) t for any n.We next recall useful results given in [EO07] (see also [Rau59]): where p ∈ Σ is independent of t and p 0 , .., p n and C + is defined in Section 2.
Then, by the induction ansatz, we have Thus, this contribution is non-singular at p 0 = p ± thanks to the first statement of this lemma.On the other hand, the contribution from R g,n+2 can be written as • ω g,n+1 (q, J) + reg at p 0 = p ± . (A.21) The first term vanishes no matter if t is of the 2nd kind or 3rd kind due to the pole structure of Λ(q).Therefore, we conclude that (A.17) is regular at p 0 = p ± .
Let us assume that the variational formula holds up to χ = k for some k ≥ −1, and we consider the case for (g, n) with χ = 2g − 2 + n = k + 1.Then by applying the variational operator to the recursion formula in the form of the first line of (A.18), Lemma A.3 imply where at the second equality we used the induction ansatz on δ (n+1) t * Rec g,n+1 (p, J) and also we applied the recursion formula in the third line to obtain ω g,n+1 (p, J).
Let us simplify (A.22).Consider a decomposition C + = C 0 ∪ C γ such that C γ contains p + inside but no other poles of the integrand.Then, C 0 and γ do not intersect and one can freely exchange the order of integration.In particular, one obtains: Note that the first term in (A.24) is the remnant contribution from C γ whereas the second term is the counter effect of applying the refined recursion formula (A.18) to obtain •ω g,n+2 (p, q, J) on the left-hand side of (A.23).As shown in (A.20) in Lemma A.4, the integrand of the first term in (A.24) as a differential in p becomes regular at p = p ± , hence it vanishes.Furthermore, since ω 0,2 (p, σ(q)) is the only term that has a pole at p = q in the integrand in (A.24), the second term can be written as This always vanishes for any generalised cycle due to the pole order of Λ(q) at q = p ± (see Definition 3.3).This completes the proof for ω g,n+1 .
A.2. Proof of Theorem 3.5: for F g Notice that the above proof was based on the pole structure of the refined topological recursion formula, or equivalently, refined loop equations.Since F g does not appear in the recursion formula, we need a different approach to prove for F g .4 .
For g > 1, we directly take the derivative of the definition of F g which gives where we used Lemma A.1, and we used the variational formula for δ (1) t * ω g,1 at the second equality.Then, since C p − does not contain any point in P ∞ , we can exchange the order of integration with respect to p and q in (A.26).After some manipulation by using the dilaton equation (2.7), we find φ(p) • ω g,2 (p, q), (A.27) where the right-hand side is the counter effect of applying the dilaton equation, similar to (A.24).Therefore, what we have to show is that the right-hand side of (A.27) vanishes.This is straightforward when Q = 0 because ω g,n+1 (p 0 , J)| Q=0 have no poles at p 0 = σ(p i ).However, since the pole structure is different in the refined setting, the proof involves more careful considerations.

A.2.1. Proof for the 2nd kind
We first consider the case where t is a 2nd kind time (Definition 3.3).That is, for m ≥ 2, we assume that ω 0,1 has a pole at p ± of order m, Λ(q) is meromorphic at q = p ± of order l where l ∈ {1, .., m − 1}, and γ is a small contour encircling p ± in the prescribed orientation.Therefore, the integral simply reduces to taking residue at q = p ± , and as a consequence, it is sufficient to check the order of the zero of Res p=σ(q) φ(p) • ω g,2 (p, q).This is a clear contrast from the 3rd kind cases at which one has to consider open-contour integrals.
Our task is to show the following property which immediately implies the variational formula for the 2nd kind: Proposition A.5.Let us define a multidifferential I g,n+1 as follows: I g,n+1 (q, J) := Res p=σ(q) φ(p) • ω g,n+2 (q, J, p).
(A.28) A.2.2.Proof for the 3rd kind We will show an analogous proposition to Proposition A.5 but in a slightly different form.First recall that ω g,n+1 (p 0 , J) for 2g − 2 + n ≥ 0 has no residue with respect to p 0 .Thus, the following residue makes sense: I * g,n+1 (q, J) := Res where the integral is taken with respect to the last variable.
Proof.For I * 0,1 (q), we have ω 0,1 (p) • η p (q). (A.36) Thus, I * 0,1 picks up the singular part of ω 0,1 at q = p + and q = p − (c.f.[Osu23, Section 2]).Thus, it becomes regular after dividing by ω 0,1 (q).For I g,n+1 (q, J) for 2g − 2 + n ≥ −1, since the proposition only concerns a local behaviour at q = p ± , potentially singular terms may appear only from the pole of ω g,n+2 (q, J, p) at p = σ(q) and we only focus on these poles similar to above discussions.Then, for the rest of the proof we apply the same technique as the proof of Lemma A.4 and Proposition A.5.That is, we treat the contributions from ω 0,2 and ω 1 2 ,1 differently, and check the singular behaviour at q = p ± by induction.Since arguments will be almost parallel to the one given in Lemma A.4 and Proposition A.5, we omit it.
[IKT19]tion 4.1.Every refined spectral curve S µ (t) associated with a hypergeometric type curve in the form of[IKT19]satisfies the refined deformation condition.