Crepant resolutions and open strings II

We recently formulated a number of Crepant Resolution Conjectures (CRC) for open Gromov-Witten invariants of Aganagic-Vafa Lagrangian branes and verified them for the family of threefold type A-singularities. In this paper we enlarge the body of evidence in favor of our open CRCs, along two different strands. In one direction, we consider non-hard Lefschetz targets and verify the disk CRC for local weighted projective planes. In the other, we complete the proof of the quantized (all-genus) open CRC for hard Lefschetz toric Calabi-Yau three dimensional representations by a detailed study of the G-Hilb resolution of $[C^3/G]$ for $G=\mathbb{Z}_2 \times \mathbb{Z}_2$. Our results have implications for closed-string CRCs of Coates-Iritani-Tseng, Iritani, and Ruan for this class of examples.


Introduction
In a recent paper [2], we proposed two versions of a Crepant Resolution Conjecture for open Gromov-Witten invariants of Aganagic-Vafa orbi-branes inside semi-projective toric Calabi-Yau 3-orbifolds: • a general Bryan-Graber-type comparison between disk potentials after analytic continuation (the disk CRC);

• a stronger identification of the full open string partition function at all genera and arbitrary boundary components for hard Lefschetz targets (the quantized open CRC).
We recall these statements more precisely in Section 2. Both conjectures were proved in [2] for the case of the crepant resolutions of type A threefold singularities, but they are expected to hold in wider generality. In particular, the disk CRC should hold true for general (non-hard Lefschetz) toric CY3 that are projective over their affinization; moreover, the proof of the quantized open CRC in [2] left out one exceptional example of (toric) hard Lefschetz crepant resolution. The purpose of this paper is to offer further evidence of the general validity of the disk CRC, as well as to conclude the proof of the quantized open CRC for hard Lefschetz toric three dimensional representations.
The first problem we tackle is the disk CRC for non-hard Lefschetz targets. We concentrate our attention to local weighted projective planes: our poster-child is the partial crepant resolution π : K P(1,1,n) → C 3 /Z n+2 , where π contracts the image of the zero section to give the quotient singularity 1 n+2 (1, 1, −2). In particular, we establish the following Theorem 1 [(Theorem 3.6 and Corollary 3.7)]: the disk CRC holds for Y = K P(n,1,1) and X = [C 3 /Z n+2 ].
On a somewhat orthogonal direction, we complete the study of hard Lefschetz crepant resolutions of three dimensional representations by considering the G-Hilb resolution of [C 3 /G] for G = Z 2 × Z 2 -the so-called closed topological vertex geometry studied in [4]. In [5], it was shown in detail in the specific example of the A 1 threefold singularity that the local CRC for [C 3 /Z 2 ] glues to a crepant resolution statement for K P 1 ×P 1 → [O(−1) P 1 ⊕ O(−1) P 1 /Z 2 ]. Theorem 2, the results of [2], and a suitable generalization of the gluing theorem of [5]  : H X → H Y between the Givental spaces of X and Y [10]; here ρ denotes a choice of analytic continuation path. Further, Iritani's theory of integral structures [21] makes a prediction for U X ,Y ρ based exclusively on the classical geometry of the targets. In this section we briefly summarize some of the recent extensions of the Coates-Iritani-Tseng CRC that this work relates to, and that are relevant for our formulation of the CRC for open Gromov-Witten invariants. Background, motivation, and extensive discussions of the setup presented here can be found in our previous paper [2, Sec. 2 and App. A]; the reader who is not familiar with the closed string CRC and its higher genus analogues is referred to the survey papers [11,22].

2.A. The disk CRC
In [2], the authors formulate an Open Crepant Resolution Conjecture (OCRC) as a comparison diagram relating geometric objects in the Givental spaces of the targets, following the philosophy of [10]. Let W be a three-dimensional CY toric orbifold, p a fixed point such that a neighborhood is isomorphic to [C 3 /G], with G Z n 1 × . . . × Z n l . The local group action is defined by the character vectors ( m 1 , m 2 , m 3 ) and a Calabi-Yau 2-torus action T (C * ) 2 is specified by weights (w 1 , w 2 , w 3 ) ∈ H • T (pt). Fix a Lagrangian boundary condition L which we assume to be on the first coordinate axis in this local chart. Define n eff = lcm{n j / gcd(m 1 j , n j ) |j = 1, . . . , l} to be the size of the effective part of the action along the first coordinate axis. There exist a map from an orbi-disk mapping to the first coordinate axis with winding d and twisting 2 k if the compatibility condition is satisfied. Via the Atiyah-Bott isomorphism, the Chen-Ruan cohomology ring of [C 3 /G] is naturally identified with a part of H • T (W ), with generators 1 p,k . Denoting by 1 k p the Poincaré dual of 1 p,k , we define the disk tensor at p as: where Γ k W is the 1 p,k coefficient of Iritani's homogenized Gamma function ( [2], Eqn. (27)). The global disk tensor for W is then defined as the sum of the disk tensors at the points adjacent to the Lagrangian L in the toric diagram of W . Note that z is thought of as the descendant parameter and hence D + W (z; w) is naturally a tensor on H W , the Givental space of W .
The winding neutral disk potential is defined to be the contraction of the J function of W with the disk tensor. Lowering indices in the J function with the Poincaré pairing, we can write this as the composition: The winding neutral disk potential is a section of Givental space that contains information about disk invariants at all winding, in the sense that disk invariants of winding d appear in the specialization of upon analytic continuation of quantum cohomology parameters.
The analytic functions h W arise from the discrepancy between the small J-function and the canonical basis-vector of solutions of the Picard-Fuchs system: a precise definition and discussion appears in [2, App. A.1.1]. Here we only remark that the functions h W are completely determined by classical geometric data. Because of the close relationship between the disk tensor and the Gamma factors of the central charge in Iritani's theory of integral structures [21,2], we have a prediction for the transformation O in terms of the toric geometry of the targets.

Proposal 2. (The transformation O)
Choose a grade restriction window W in the GIT problem to identify the K-theory lattices of X and Y , and for W = X , Y , define: Then the transformation O in Proposal 1 has the form: where we denote by CH W = z − 1 2 deg CH W the matrix of Chern characters (homogenized with respect to the cohomological degree "deg") in the bases given by W.
In [2], we show that Proposal 1 follows from the Coates-Iritani-Tseng's CRC. Proposal From Proposal 1 one can extract comparison statements about generating functions for disk invariants. The strongest statement can be made when the Lagrangian boundary condition intersects a leg whose isotropy is preserved in the crepant transformation.

Proposal 3. (Scalar disk potentials)
Let L be a Lagrangian boundary condition on X that intersects a torus invariant line whose generic point has isotropy group G L , and such that if we denote L be the corresponding boundary condition in Y , then L also intersects a torus invariant line with generic isotropy group G L . For W = X , Y , define the scalar disk potential 4 : 3 ↑ The fact that Γ -integral structures match with the natural B-model integral structures under mirror symmetry was proved in [21] for compact toric orbifolds. A general proof of the fully equivariant version of Iritani's K-theoretic CRC has been announced by Coates-Iritani-Jiang. 4 ↑ We choose to define the scalar disk potential as a generating function for the absolute value of disk invariants. In the course of the verifications of Proposal 3, one may observe that the scalar potentials could be matched on the nose with the use of appropriate matrices of roots of unity -that in the end contribute just signs, albeit with some non-trivial pattern. We have deliberately forgone to keep track of these phenomena, especially in light of the choice-of-signs the theory of open invariants is everywhere laden with.
Then, upon identifying the insertion variables via the change of variable prescribed by the closed CRC, we have:

2.B. Hard Lefschetz targets: the quantized OCRC
When X satisfies the hard Lefschetz condition 5 , a natural generalization of the CRC to higher genus GW invariants is achieved by canonical quantization [10,11]: the all-genus Gromov-Witten partition functions are viewed as elements of the respective Fock spaces [19,18] In the context of torus-equivariant Gromov-Witten theory of orbifolds with zero-dimensional fixed loci, the hard Lefschetz quantized CRC can be proven in two steps [2, Prop. 6.3], as follows.
(1) Combining the Coates-Givental/Tseng quantum Riemann-Roch theorem [9,27] with Givental's quantization formula in a neighborhood of the large radius points of W identifies a "canonical" Rcalibration defined locally by the genus 0 GW theory of W ; (2) Conjecture 2.2 then follows from establishing the equality, upon analytic continuation, of the canonical R-calibrations of X and Y on the locus where the quantum product is semi-simple.
The main consequence drawn in [2] for open Gromov-Witten invariants is a CRC statement for all genera and number of holes.

Proposal 4. (The quantized OCRC [2]
) Let X → X ← Y be a Hard Lefschetz diagram for which the higher genus closed CRC holds. Define the genus g, -holes winding neutral potential F g, where J W g, encodes genus g, -point descendent invariants: 5 ↑ This is age(φ) = age(I * (φ)) for all φ ∈ H orb (X ), where I : IX → IX is the canonical involution on the inertia stack.

3.A. Classical geometry
The family of geometries we study arises as the GIT quotient with torus action on the coordinates (x 1 , x 2 , x 3 , x 4 ) specified by the charge matrix The quotients obtained as the character χ varies are the toric varieties whose fans are represented in Figure  1. The right hand side of Figure 1 corresponds to χ > 0 . The irrelevant ideal is and the resulting geometry Y is the total space of O(−n − 2) P(n,1,1) ; [x 1 : x 2 : x 3 ] serve as (quasi)homogeneous coordinates for the base, and x 4 is an affine fiber coordinate. Torus fixed points and invariant lines are: We have L 1 P 1 , L 2 , L 3 P(1, n), P 2 , P 3 [pt], P 1 BZ n . The fibers over the fixed points P 2 and P 3 are non-gerby. The fiber over P 1 is non-gerby when when n is odd; when n is even, it has a Z 2 -subgroup as a stabilizer.
When χ is negative we have the fan on left hand side of Figure 1, which gives the irrelevant ideal Quotienting by x 4 0 gives a residual Z n+2 action on C 3 with weights (n, 1, 1); the resulting orbifold [C 3 /Z n+2 ] will be denoted by X . Moving across χ = 0 (20) where we denoted by [x 1 , . . . , x n ] the equivalence class in the appropriate quotient, is a birational contraction of the image of the zero section s : P(n, 1, 1) → K P(n,1,1) . Figure 1: A height 1 slice of the fans of [C 3 /Z n+2 ] (left) and local P(n, 1, 1) (right) for n = 2.

3.A.a. Bases for cohomology
We consider a Calabi-Yau 2-torus action on Y and X , descending from an action on C 4 with geometric weights (α 1 , α 2 , −(α 1 + α 2 ), 0). Note that we consider the geometric weights as elements of H 2 (BT ): an integer α corresponds to the first Chern class of the representation t → t α . The tangent weights at the torus fixed points are depicted in the toric diagrams in Figure 2.
As a module over H(BT ), the equivariant Chen-Ruan cohomology ring of Y = K P(n,1,1) is spanned by

3.B. Quantum geometry
Genus-zero Gromov-Witten invariants of X and Y can be computed using the quantum Riemann-Roch theorems of Coates-Givental [9] and Tseng [27] applied to the Gromov-Witten theories of BZ n+2 and P(n, 1, 1), respectively. We have the following Then, for W either X or Y and w either x or y, Proof. This is [6, Theorem 3.5 and 3.7].
Since the I-functions of X and Y belong to the cone and behave like z + O(1) at large z, they coincide with suitable restrictions of the respective big J-functions to a subfamily of quantum cohomology parameters.

Corollary 3.2. Denote by q the Novikov variable associated to p and write
for an orbifold cohomology class φ ∈ H orb T (X ). Then the following equalities hold:

3.B.a. Analytic continuation and U X ,Y ρ
A standard method [10,8] to relate the Lagrangian cones of X and Y upon analytic continuation hinges on the following three-step procedure: (1) find a holonomic linear differential system of rank equal to dim H • (Y ) = dim H • orb (X ) jointly satisfied, upon appropriate identification of the quantum parameters, by the components of the I-functions of X and Y as convergent power series around the respective boundary point; (2) determine the relation between the I-functions upon analytic continuation along a path ρ connecting the two boundary points; (3) invoke a reconstruction theorem to recover from the latter the content of big quantum cohomology and the full-descendent theory in genus zero [7,13].
Step (3) has been achieved in full generality for toric Deligne-Mumford stacks in [7]. The first step is also standard [17]; we spell out the details below for the sake of completeness. The main intricacy here lies in Step (2), as the rank of the system is parametrically large in n and the usual Mellin-Barnes method [6,20] is technically more subtle to apply; we present a workaround in the discussion leading to Lemma 3.4.

Lemma 3.3.
Let D Y the (n + 2) th order linear differential operator where θ y = zy∂ y and define D X to be the differential operator obtained by replacing y = x −n−2 in Eq. (28). Then, Proof. The statement follows from a straightforward calculation from Eqs. (23) and (24).
The linear operator D W is the Picard-Fuchs operator of W = X , Y : Lemma 3.3 establishes that the torus-localized components of the I-functions of X and Y furnish two bases solutions of the linear system D W f = 0, respectively in the neighbourhood of the Fuchsian points y = 0 and ∞. Relating the cones of X and Y thus boils down to finding the change-of-basis matrix connecting the two set of solutions upon analytic continuation from one boundary point to the other. Let I X k (x, z) denote the coefficient of 1 k/(n+2) in Eq. (24), and define in the same vein It is immediately noticed that k=0 as a basis of solutions of D X f = 0. On the other hand, localizing Eq. (23) to the T -fixed points and resumming in d for |y| < n n (n+2) n+2 we obtain where and p F q ({A}; {B}; y) denotes the generalized hypergeometric series which is convergent for |w| < 1.
In order to continue to x = y −n−2 1 we will need the following analytic continuation theorem for p F q ({A}; {B}; y), which generalizes the classical Kummer continuation formula at infinity for the Gauss function.
Proof. The argument follows almost verbatim the steps leading to the well-known result for q = 1. Φ(w) q+1 F q ({A}; {B}; w) satisfies the generalized hypergeometric equation with θ = w∂ w . The same analysis at w = ∞ as for the Gauss equation reveals that A i are local exponents of Eq. (37), Now, Φ(w) can be represented as the multiple Euler-Pochhammer integral [16] where γ = [C 0 , C 1 ] is the commutator of simple loops around t = 0 and t = 1. Taking the limit w → ∞ along ρ and using the Euler Beta integral, gives

Remark 3.5. (On general toric wall-crossings)
The arguments we used for the examples of this Section have a wider applicability to wall-crossings in toric Gromov-Witten theory, including the multi-parameter case. On general grounds, I-functions -and their extended versions [7] -are multiple hypergeometric functions of Horn type [20,21]. When crossing a single wall in the B-model moduli space, however, the analytic continuation is effectively taking place in one parameter only. Restricting to the sublocus where all the spectator variables are set to zero reduces the multiple Horn series to a single-variable series which, upon manipulations of Gamma factors in the summand as in the next section, can always be cast in the form of a generalized hypergeometric function p F q ({A}, {B}, w) with q ≥ p − 1. Whenever the series has a finite radius of convergence as in the Calabi-Yau case, we have p = q + 1, for which Lemma 3.4 applies. The general case is obtained similarly.

3.B.b. Grade restriction window and the K-theoretic CRC
Let us now turn to Conjecture 2.1 for this family of geometries. Throughout this section, we work with the natural basis {1 k n+2 } k=0,1,...,n−1 for H • T (X ) and with the localized basis yields a natural bijection between the K-lattices of X and Y . We make the notational convention of taking all indexing sets to range from 0 to n + 1, with the sole purpose of leaving the coefficients corresponding to identities/trivial objects in the first row/column of any matrix we write. With these choices the matrices representing the (homogenized, involution pulled-back) Chern characters for X and Y are Theorem 3.6. Conjecture 2.1 holds with the restriction window W above and the analytic continuation path ρ as in Lemma 3.4. Proof. Consider the linear map V : and Euler's identity, Γ (x)Γ (1 − x) = π/ sin(πx); the final result is a trigonometric matrix with coefficients [V ] i j being Laurent polynomials in e 2πiα k , k = 1, 2, 3. Right-multiplication by the Chern character matrix of X and telescoping the resulting sums over roots of unity returns CH Y , as given in Eq. (47).

3.C. The OCRC
As discussed in Section 2.A, the first implication we draw from Theorem 3.6 is a comparison theorem for winding neutral disk potentials. Corollary 3.7. Proposals 1 and 2 hold for Y = K P(n,1,1) and This can be employed to obtain more concrete identifications of scalar disk potentials, as we now show.

3.C.a. Scalar disk potentials: non-special legs
In the case where the Lagrangian on Y is on a leg that attached to a non-stacky point, the equality of scalar disk potentials follows in a simple fashion for all n. When the Lagrangian is on the leg that attached to the stacky point, we need to consider separately the case n-odd, where the quotient on the leg is effective, and n-even, where there is a residual Z 2 isotropy.
We consider non-special legs first. We have the following Proof. In this case the tensors Θ from (5) are: We compute the transformation O as in Eq. (6); note it has nonzero coefficients only for l = n. We then specialize z = (n+2)α 2 d to obtain a map we denote O d , The expression in Eq. (53) is summed over the index j ranging from 0 to n + 1. When k is not congruent to d modulo n + 2, the exponential part is a sum of roots of unity that adds to 0. When k ≡ d modulo n + 2, O k d,n = ±1. Hence our OCRC, Corollary 3.7, together with Eq. (53) gives Disk invariants of winding d for X are the coefficients of the classes 1 k n+2 with k ≡ d modulo n + 2 after specializing z = (n+2)α 2 d in F disk L,X . Summing over all d, we obtain the equality of scalar potentials as stated in Theorem 3.8.

3.C.b. Scalar disk potentials for the special leg: n odd
Theorem 3.9. Let n be an odd integer. Consider a Lagrangian boundary condition L on X which intersects the first coordinate axis, and denote by L the proper transform in Y . Then, upon identifying the insertion variables via the change of variable prescribed by the closed CRC, we have the equality of scalar disk potentials: Proof. In this case the tensors Θ from (5) are: We compute the transformation O as in Eq. (6). We then specialize z = The expression in Eq. (58) is summed over the index j ranging from 0 to n + 1. The degree-twisting compatibilities are: The Chinese remainder theorem then states that both compatibilities are satisfied when d ≡ kn + l(n + 2) mod n(n + 2).
When (59) is satisfied, the difference in the arguments in the sine functions is an integer multiple of π, hence O k d,l = ±1. When only the compatibility for Y is satisfied, then the exponential part of Eq. (58) consists of a sum of (n + 2) roots of unity that add to 0. All other entries of the matrix representing O d do not matter for our purposes. For a fixed d, there is a unique pair (k,l) satisfying both twisting conditions, and Eq. (58) gives: Disk invariants of winding d for X are the coefficients of the class 1¯k n+2 after specializing z = (n+2)α 1 d in F disk L,X , whereas for Y they are obtained as the coefficients of the class 1¯l n after the same specialization of z in F disk L,Y . Hence, summing over all d, Eq. (60) yields the equality of scalar potentials as stated in Theorem 3.9.
3.C.c. Scalar disk potentials for the special leg: n even Theorem 3.10. Let n be an even integer. Consider a Lagrangian boundary condition L on X which intersects the first coordinate axis, and denote by L the proper transform in Y . Then, upon identifying the insertion variables via the change of variable prescribed by the closed CRC, we have the equality of scalar disk potentials: Proof. The transformation O in this case is the same as in Section 3.C.c. However we specialize to z = (n+2)α 1 2d to obtain O d : The expression in Eq. (62) is summed over the index j ranging from 0 to n + 1. The degree-twisting compatibilities are: Modular arithmetic again tells us that for any d there are four pairs of solutions to the above system of congruences, corresponding to the solutions to: Note that if (k 0 , l 0 ) is a solution of (63), then the other solutions are (k 0 , l 1 ), (k 1 , l 0 ), (k 1 , l 1 ), where k 1 = k 0 + n+2 2 and l 1 = l 0 + n 2 . Without loss of generality we denote (k 0 , l 0 ) and (k 1 , l 1 ) the solutions such that 2d ≡ kn + l(n + 2) mod n(n + 2) and we observe that O Just as before, for l = l 0 , l 1 and all other k's, the corresponding coefficients in the matrix O d vanish. This gives the equalities: We recognize the disk invariants of winding d for X (resp. Y ) in the sum of the left hand sides (resp. right hand sides) of Eq. (64) and Eq. (65). Hence, summing over all d, Eq. (60) yields the equality of scalar potentials as stated in Theorem 3.10.

Example 2: the closed topological vertex 4.A. Classical geometry
The closed topological vertex arises from the GIT quotient construction [12] 0 where The resulting geometry is a quotient C 6 // χ (C ) 3 , where the characters of the torus action on the affine coordinates x 1 , . . . , x 6 of C 6 are encoded in the rows of M.
In two distinct chambers, the GIT quotient yields the toric varieties whose fans are given by cones over Figure 3. The picture on the left hand side corresponds to the orbifold chamber: we delete the unstable locus and then quotient by Eq. (67): using the torus action to make x 4 , x 5 and x 6 equal to 1 gives a residual effective µ 3 2 /µ 2 Z 2 × Z 2 action 6 on C 3 with coordinates x 1 , x 2 , x 3 . We denote by X [C 3 /(Z 2 × Z 2 )] the resulting orbifold, and by X its coarse moduli space.
The picture on the right hand side corresponds instead to the distinguished large radius chamber that gives rise to Nakamura's Hilbert scheme of (Z 2 × Z 2 )-clusters: we delete the set (2,4,6), (3,5,6), (4,5,6) x and then quotient by the (C ) 3 action in Eq. (67); we will denote by Y the corresponding smooth toric variety. This is the trivalent geometry on the right-hand-side of Figure 4: the local geometry of three (−1, −1) curves inside a Calabi-Yau threefold intersecting at a point.

4.A.a. Bases for cohomology
We equip Y and X with a Calabi-Yau 2-torus action descending from the action on C 6 with geometric weights (α 1 , α 2 , −α 1 − α 2 , 0, 0, 0). This descends to give an effective T (C * ) 2 action on Y and X which preserves their canonical bundle; the resolution diagram is T -equivariant.
Bases for the equivariant cohomology of Y and X can be presented as follows. Let L i ⊂ Y , i = 1, 2, 3 denote the torus-invariant projective lines The cohomology of Y is generated as a module by the duals ω i = [L i ] ∨ ∈ H 2 (Y ) of the fundamental classes in Eqs. (71)-(73), plus the identity class 1 Y ∈ H 0 (Y ). The action on C 6 above yields canonical lifts of i * L j ω i = c 1 (O L j (δ ij )) to equivariant cohomology. Denoting by q the intersection of the three fixed lines, p i the other torus fixed point of L i , and by capital letters the corresponding cohomology classes, the Atiyah-Bott isomorphism sends: The T -equivariant Poincaré pairing η Y (φ 1 , φ 2 ) = P i φ 1 | P i φ 2 | P i e −1 (N P i /Y ), in the basis (Q, P 1 , P 2 , P 3 ) for H • T (Y ), takes the block-diagonal form On X , the torus equivariant cohomology is spanned by the T -equivariant cohomology classes 1 g , labeled by the corresponding group elements g = (0, 0), (0, 1), (1, 0) and (1, 1).

4.A.b. The grade restriction window
Consider the natural restriction window W consisting of the trivial representation of (C * ) 3 and the three one dimensional representations whose characters are given by the first three columns of the matrix M in Eq. (67). These descend to the four irreducible representations of X , whose nontrivial characters are still encoded by the first three columns of M via iπ-exponentiation; and to the bundles O and O L j (δ ij ) on Y . Using W to identify the K-lattices, the natural basis of irreducible representations for H • T (X ) and the fixed point basis for H • T (Y ), the matrix representing the (homogenized, involution pulled-back) Chern character for X and Y are

4.B. Quantum geometry
The primary T -equivariant Gromov-Witten invariants of Y were computed for all genera and degrees in [23].
As far as X is concerned, its quantum cohomology was determined in [3] by an explicit calculation of Z 2 × Z 2 Hurwitz-Hodge integrals. Introduce linear coordinates x i,j on the T -equivariant Chen-Ruan cohomology of X by H orb where the Bryan-Graber change of variables t(x) reads

4.C. One-dimensional mirror symmetry
In the analysis of the disk and quantized CRC for the type A resolutions in [2], a prominent role was played by a realization of the D-modules underlying quantum cohomology in terms of a single-field logarithmic Landau-Ginzburg model, or, in the language of [25], the Frobenius dual-type structure on a genus-zero double Hurwitz space. This was motivated by a connection of the Gromov-Witten theory for these targets with a class of reductions of the 2-Toda hierarchy [1]. A similar connection with integrable systems holds for the closed topological vertex as well; the general story will appear elsewhere, but its consequences for the purposes of the paper are discussed below. Define Fix now a branch C of the logarithm and denote by M α 1 ,α 2 M 0,6 × C * the smooth complex fourdimensional manifold of multi-valued functions λ(q) of the form A given point λ ∈ M α 1 ,α 2 is a perfect Morse function in q with four critical points q cr i , i = 1, . . . , 4; its critical values, give a system of local coordinates on M α 1 ,α 2 , which is canonical up to permutation. Define now holomorphic tensors η ∈ Γ (Sym 2 T * M α 1 ,α 2 ), c ∈ Γ (Sym 3 T * M α 1 ,α 2 ) on M α 1 ,α 2 via Whenever η is non-degenerate, this defines a commutative, unital product ∂•∂ on Γ (T M α 1 ,α 2 ) by "raising the indices": η(∂, ∂ • ∂ ) = c(∂, ∂ , ∂ ).
Proof. Associativity and semi-simplicity of the product follow immediately from the fact that the canonical coordinate fields, ∂ u i , are a basis of idempotents of Eq. (89). A straightforward computation of the residues in Eq. (88) in the coordinate chart t i shows that Eq. (88) is a flat metric and the variables t i are a flat coordinate system for η; similarly, a direct evaluation of Eq. (89) shows that the algebra admits a potential function, which coincides with Eq. (81).

Then a system of flat coordinates for ∇ (z)
X is given by the periods and we denoted by C x a simple loop encircling counterclockwise the point q = x.
This is [2,Prop. 5.2], where the superpotential and primitive differential λ and φ there are identified respectively with e λ and ψ(q)dq here: the contours γ i give a basis of the first homology of the complex line twisted by a set of local coefficients given by the algebraic monodromy of e λ/z around the singular points Z i , 0 and ∞. The reason behind this particular choice of basis, as well as the normalization factor in front of the integral, will be apparent in the course of the asymptotic analysis of Section 4.D.d.

Remark 4.3.
In the language of [25], the Frobenius manifold F α 1 ,α 2 is the Frobenius dual-type structure on the genus zero double Hurwitz space H 0,κ with ramification profile κ = (α 1 , α 1 , α 2 , α 2 , −α 1 − α 2 , α 1 − α 2 ), with e λ as its superpotential and the third kind differential ψ(q)dq as its primitive one-form; the integrals Eq. (92) were called the twisted periods of F α 1 ,α 2 in [2]. The corresponding Principal Hierarchy [13] is a four-component reduction of the genus-zero Whitham hierarchy with three punctures [24]. The special case α 1 = α 2 = α is particularly interesting, as in that case F α,α is the dual (in the sense of Dubrovin [14]) of a conformal charge one Frobenius manifold with non-covariantly constant identity; flat coordinates for the two Frobenius structures are in bijection with Darboux coordinates for a pair of compatible Poisson brackets for the Principal Hierarchy, which thus gives rise to a (new) bihamiltonian integrable system of independent interest. We will report on it in a forthcoming work.

4.C.a. Computing U X ,Y ρ
Encoding the coefficients of Γ X (z) and Γ Y (z) as entries of diagonal matrices, the prediction for the symplectomorphism U X ,Y ρ from Iritani's theory of integral structure is obtained by composing as we now turn to verify. Let Y be the ball of radius around e t = 0, measured w.r.t. the Euclidean metric (ds 2 ) = i (de t i ) 2 in exponentiated flat coordinates, and define the path in Y 1 Beside Π i , systems of flat coordinates for the deformed flat connection ∇ (z) are given by the components of the J-functions of X and Y ; the discrepancy between them encodes the morphism of Givental spaces that identifies the Lagrangian cones of X and Y under analytic continuation along the path ρ: As in [2], U X ,Y ρ can be computed in two steps, by expressing J • in terms of the periods Π, where J X α and J Y j are the components of the J-functions of X and Y respectively in the inertia basis of X and in the localized basis of Y ; we have labeled elements of Z 2 × Z 2 by a single index α = 0, 1, 2, 3 for g = (0, 0), (1, 0), (0, 1) and (1, 1) respectively. Throughout the rest of this Section, in order to simplify formulas, we define µ i α i /z.

Proposition 4.4. We have
−e iπ(µ 1 +µ 2 ) sin(πµ 1 ) sin(πµ 2 ) 0 e iπ(µ 1 +µ 2 ) sin(πµ 1 ) sin(πµ 2 ) −1 −(−1) µ 1 sin(π(µ 1 +µ 2 )) sin(πµ 2 ) (−1) 2µ 1 (−1) µ 1 sin(π(µ 1 +µ 2 )) sin(πµ 1 ) where and B(x, y) denotes Euler's β-function Proof. J X α (x, z) is characterized as the unique system of flat coordinates of ∇ (z) which is linear with no inhomogeneous term in e x 0 /z and satisfies at the orbifold point x = 0. Then, The integrals appearing on the r.h.s. of Eq. (106) can be explicitly evaluated in terms of the Euler β-integral; this is illustrated in detail in Appendix A.A, and returns Eqs. (101)-(103). Similarly, J Y j (t, z) is characterized as the unique system of flat coordinates of ∇ (z) (linear with vanishing inhomogeneous term in e t 0 /z ) that diagonalizes the monodromy of ∇ (z) at large radius as where the r.h.s. is determined by the localization of ω i at p j as in Eqs. (74)-(76). Then A is determined by the decomposition of the periods in terms of eigenvectors of the monodromy at large radius, that is, by their asymptotic behavior as Re(t) → −∞. The details of the large radius asymptotics of Π i are quite involved and are deferred to Appendix A.B; the final result is Eq. (98).

4.D. Quantization and the all-genus CRC
For j = 1, . . . , 4, define 1-forms formally analytic in z, R j = R ij (u, z)e u j /z du i , satisfying the following set of conditions: R2: ∇ (z) R j = 0 as a formal Taylor series in z, By condition R2 and their prescribed singular behavior at z = 0, R j are formal (asymptotic) flat sections of the Dubrovin connection uniquely defined up to right multiplication by constants, and let ∆ i (u) be the normalized inverse-square-length of the coordinate vector field ∂ u i in the Frobenius metric, Eq. (88). We will also denote by ψ W the Jacobian matrix of the change-of-variables from the canonical frame, Eq. (87), to the flat coordinate systems t and x for W = Y and X respectively, with columns normalized by √ ∆. where The Gromov- where χ αj is the character table of Z 2 × Z 2 , V (0) is the trivial part of the representation V (thought of as a vector bundle on the classifying stack), and where t u denotes the shifted descendent times t p u = t p + τ W (u)δ p0 . Moreover, the Coates-Iritani-Tseng/Ruan quantized CRC, holds if and only if the Gromov-Witten R-calibrations agree on the semi-simple locus,

4.D.a. Saddle-point asymptotics
Formal power series solutions in z of ∇ (z) R = 0 are obtained from the saddle-point asymptotics of Eq. (92) at z = 0. The latter is an essential singularity of the horizontal sections of the Dubrovin connection, and their asymptotic analysis at z = 0 relies on a choice of phase for the parameters α 1 , α 2 , z -namely, a choice of Stokes sector. A technically convenient choice is to restrict our study to the wedge S + = {(µ 1 , µ 2 )|Re(µ 1 ) > 0, Re(µ 2 ) < −Re(µ 1 )}; as individual correlators depend rationally on µ 1 , µ 2 , our statements will hold in full generality by analytic continuation in the space of equivariant parameters. Proof. Asymptotic horizontal sections R i (u, z) are given by the classical Laplace asymptotics of the integrals where the Lefschetz thimble L i is given by the union of the downward gradient lines of Re(λ) emerging from its i th critical point. Let us first consider the situation at the orbifold point, which is schematized in Figure 5. We compute from Eq. (86) with σ (1) = σ (4) = 0, σ (3) = σ (2) = 1. It is straightforward to check that the constant phase paths of e λ/z emerging from q cr i are the straight lines arg(q) = ±π(σ (i) + 1/4) that terminate at the nearest algebraic zero of e λ/z or at infinity, as in Figure 5. Moreover, for our choice of phases of the weights in S + , the contour integrals of e λ/z ψ around the Pochhammer contours γ i retract [2, Rmk 5.5] to line integrals on the straight line segments connecting the zeroes of e λ/z inside γ i . At the orbifold point, these are precisely the Lefschetz thimbles L i : then, the saddle-point expansion of the differentials R i = ψ X αj R ji (u, z)e u i /z dx α dI i = dΠ i satisfies conditions R1-R2 above. We claim that up to right multiplication by N i ∈ C[[z]], R i this satisfies R3 and coincides with the Gromov-Witten R-calibration of X . Indeed, as shown in Appendix A.A, in the trivialization given by x α the differential of the periods of e λ/z at x = 0 reduce to Euler Beta integrals, whose steepest-descent asymptotics is determined by Stirling's expansion for the Γ function: Then: and by Eqs. (106), (119), and (112) we obtain ψ X aj R ji x=0 = 2π e eq (V (0) ) α D X a χ ai (121) so that R = √ 2πR X . In particular, since by Eq. (112) R satisfies the unitarity condition at x = 0, and because parallel transport under the Dubrovin connection is an isometry of the pairing in R3, it satisfies condition R3 for all u. At large radius, by condition R1 and the asymptotic behavior of J Y (t, z) around Re(t) → −∞ (Eq. (107)), we must have that ]. Its calculation via the steepest descent analysis of Eq. (116) at large radius requires extra care since e t = 0 is a singular point for ∇ (z) : in this limit, the critical points of the superpotential either coalesce at zero or drift off to infinity, The essential divergences in the saddle-point computation of N Y from Eq. (116) can be treated as follows: first rescale the integration variables in Eq. (116) by e −t 2 /2 , e −t 1 −t 2 /2 , e t 2 /2 and e −t 2 /2 for i = 1, 2, 3, 4 respectively; then integrate over the steepest descent path, isolating the essential divergence at the large radius point, and finally take the resulting (finite) limit Re(t) → −∞: notice that the last two steps do not commute in general, as poles are generally created along the integration contour in the large radius limit. The final result reduces, for all i, to the computation of the saddle-point asymptotics of Beta integrals. Explicitly, we get where ∆ cl = lim Re(t)→−∞ ∆(u) and B as (x, y) denotes the Stirling expansion of the Euler Beta function. Then, lim and thus R X = R Y , concluding the proof.
Finally, for Π 1 we use that where we have resummed w.r.t. x 2 , applied Lemma 3.4 for q = 1, and isolated the leading contribution in so that which concludes the proof.