Relations between Stokes constants of unrefined and Nekrasov-Shatashvili topological strings

In this paper we demonstrate that the Stokes constants of unrefined free energies and the Stokes constants of Nekrasov-Shatashvili free energies of topological string on a non-compact Calabi-Yau threefold are identical, possibly up to a sign, for any Borel singularity which is not associated to a compact two-cycle that intersects only with non-compact four-cycles. Since the Stokes constants of Nekrasov-Shatashvili free energies are conjectured to coincide with those of quantum periods and therefore have the interpretation of BPS invariants, our results give strong support that the Stokes constants of unrefined free energies may also be identified with BPS invariants.


Introduction
String theory has rich non-perturbative structures.For instance, it was already pointed out in [1] for bosonic string and later in [2] for fermionic string that the free energy as a perturbative series in string coupling g s is divergent as the coefficients have factorial growth, i.e.
which implies the power series has zero radius of convergence.Such a factorial growth of coefficients also signals that the full and exact free energy would require exponentially small non-perturbative corrections.These non-perturbative corrections were interpreted as effects from D-branes [3], although detailed calculations of D-branes were only performed recently [4,5] to reproduce the exponentially small non-perturbative corrections in the cases of minimal string theory.In general the non-perturbative corrections to string free energies are difficult to study.One division of string theory where one might have a better chance of understanding the non-perturbative corrections is topological string theory.Topological string is constructed by topological twists on the worldsheet of perturbative string, and it captures the important BPS sectors of type II superstring compactified on a Calabi-Yau threefold.On the other hand, topological string is relatively simple.It permits a rather rigorous mathematical definition through the Gromov-Witten theory, and the perturbative free energies in topological sting can be computed via various methods, including holomorphic anomaly equations [6][7][8], topological vertex [9,10], topological recursion [11], blowup equations [12], and etc. Making use of these methods, hundred of terms of perturbative free energies can be computed in the case of non-compact Calabi-Yau threefolds, and more than sixty terms have recently been computed for certain family of compact Calabi-Yau threefolds in [13], building on [14], including the famous quintic model, which provides a treasure of perturbative data unimaginable in critical string theory, making tests of various proposals of non-perturbative corrections possible.It is due to these facts that it was mused in "A panorama of physical mathematics c. 2022" [15] that if we wish to make progress on understanding what is string theory including all of its non-perturbative aspects, it is more tractable and more realistic to try to answer this question first in topological string.
Among different proposals to study non-perturbative corrections to topological string free energies, one of the most promising is to use the resurgence theory [16] 1 .The key idea is that non-perturbative corrections to a divergent perturbative series appear in the form of trans-series, and the full and exact solution can be written as the Borel-Laplace resummation, a distingshed method to resumm a divergent power series, of certain linear combination of these trans-series in addition to the perturbative series.Furthermore, in order for the resummed linear combination to be a well-defined function, the trans-series representing non-perturbative corrections must be encoded in and therefore can be extracted from the perturbative series, through various Stokes automorphisms characterised by a collection of numbers called Stokes constants.The resurgence theory therefore offers a roadmap to systematically study non-perturbative contributions to topological string free energy, making use of the already available rich data of perturbative expansion.
Resurgence techniques were first applied to study topological string in [20][21][22], where for instance it was checked numerically that indeed topological string free energies grow like (1.1), and the non-perturbative effects that control such a growth behavior were analysed in simple models.Later studies focused on the simplest model, the resolved conifold, where both the trans-series and the Stokes constants could be computed [23] (see also [24][25][26] building on [23]).For more complicated geometries, these computations are much more difficult.Nevertheless, based on a novel method proposed in [27,28] to systematically calculate the trans-series using holomorphic anomaly equations [6,7], the trans-series in any non-perturbative sectors were calculated in closed form [29,30] for both compact and non-compact Calabi-Yau threefolds.In a different line of researches, Tom Bridgeland proposed that the BPS invariants or generalised DT invariants of a non-compact Calabi-Yau threefold can be used to construct a Riemann-Hilbert problem whose solution of τ function expands to the topological string free energy [31,32].These BPS invariants are countings of stable D-brane bound states.Various subsets of BPS invariants are already known to be related to topological string.For instance, the Gopakumar-Vafa formula [33,34] expresses the perturbative free energy in terms of the D2-D0 BPS invariants, and the  generating function of D6-D2-D0 BPS invariants, also known as the PT invariants, is also related to the topological string free energy [35].Bridgeland's proposal provides another interesting link between basically Stokes constants of topological string free energies and BPS invariants, but concrete construction could only be made for the resolved conifold (see also [24]), where the only non-trivial BPS invariants are countings of D2-D0 bound states.Various hints exist for more complicated models, for instance checks of integrality of Stokes constants in [29,30,36] in certain models, but no conclusive statements can be made, and many questions still remain open, including: if the relation between Stokes constants and BPS invariants can be generalised, and if true, what are the concrete formulas, and whether the full set or only a subset of BPS invariants can be recovered from the Stokes constants.
In this short note, we will make an important step towards answering these questions through a slightly different but related route, summarised by Fig. 1.1.When the target space is a suitable noncompact Calabi-Yau threefold X, topological string is characterised by a Riemann surface called the mirror curve equipped with a canonical 1-form associated to X by mirror symmetry.At the same time, it is equivalent to a 5d N = 1 gauge theory on S 1 ×R 4 , which reduces to an N = 2 gauge theory when S 1 shrinks to a point, corresponding to certain scaling limit of the mirror curve [37].It was found in [38] that the canonical 1-form describes a metric on the mirror curve, and BPS states of either the topological string or the supersymmetric field theory can be described by geodesic 1-cycles on the mirror curve with respect to this metric.This observation was developed into a full-blown method to calculate BPS invariants in [39] known either as the spectral network (4d field theory) [39,40] or exponential network (5d field theory) [41][42][43].
On the other hand, it was pointed out that the mirror curve can be promoted via quantisation to either a differential operator (4d field theory) or a difference operator (5d field theory), called the quantum mirror curve, and the periods of the mirror curves are promoted to quantum periods which are divergent power series in through the exact WKB method.These quantum periods have remarkable resurgent properties.When the quantum mirror curve is a second order differential operator (4d rank one gauge theory), it was proved that the trans-series that appear in Stokes automorphisms of quantum periods are associated also with geodesic 1-cycles [44] and they should be naturally mapped to the BPS states.Furthermore, the Stokes automorphism takes a form, known as the Delabaere-Dillinger-Pham formula [44][45][46], which resembles the Kontsevich-Soibelman automorphism, a crucial ingredient of the wall-crossing formula [47,48], also known as the spectrum generator of BPS invariants [49].The Stokes constants, which are coefficients of the Stokes automorphisms, should then be identified with the BPS invariants, which are coefficients of the Kontsevich-Soibelman automorphisms.This identification was checked in various 4d rank one gauge theories [50,51] and is expected to hold in higher rank theories.This provides the upper horizontal arrows in Fig. 1.1.
Quantum periods can be studied through another approach different from the exact WKB method.Inspired by Nekrasov's partition function for 4d and 5d gauge theories on the Omega background [52], the topological string free energy was refined to depend on not a single perturbative expansion parameter g s but two 1 , 2 , and it reduces to the unrefined case in the limit 1 + 2 = 0.In another special limit 2 → 0, called the Nekrasov-Shatashvili limit [53], the topological string free energy provides a set of relations between quantum A-and B-periods called the quantum special geometry [54].The Wilson loop amplitudes in various gauge representations in the NS limit then provide another set of relations between these quantum periods.Together the NS free energies and the NS Wilson loop amplitudes completely determine the quantum periods [55], and the Stokes automorphisms of quantum periods can also be derived from those of NS free energies and Wilson loop amplitudes.A pattern of the latter was recognised, from which one concludes that the Stokes automorphisms of quantum periods in 5d gauge theories follow the same DDP formulas as 4d gauge theories [55] (see also [56]), further confirming the horizontal arrows in the second arrow in Fig. 1.1.In addition, Stokes constants of NS free energies are themselves identified with Stokes constants of quantum periods, providing the vertical arrows on the right hand side in Fig. 1.1.
In this note, we will argue that the Stokes constants of unrefined free energies and those of NS free energies of topological string can also be identified, possibly up to a sign, given by the key formula (4.9), thus providing the lower horizontal arrows in Fig. 1.1.The key idea is to use the blowup equations for refined topological string free energy [12,57], which in a special limit provides the sought for relationship between unrefined and NS free energies known as the compatibility formula [12,58,59].Once the lower horizontal arrows are established, one can make direct identification betwene Stokes constants of unrefined topological string free energy and BPS invariants of the Calabi-Yau threefold, given by eq.(4.10), represented by the dashed vertical arrows on the left hand side in Fig. 1.1.The identified BPS invariants include all the D4-D2-D0 stable bound states.
We emphysize that our argument only works on topological string on non-compact Calabi-Yau threefolds where NS free energies and quantum periods as intermediary steps can be defined.A direct argument that make the left vertical arrow that works also for compact Calabi-Yau threefolds would be very desirable, possibly along the line of [60].
The remainder of the paper is structured as follows.In section 2, we sketch the ingredients of resurgence theory enough for understanding the derivation and the statements in this paper.We then summarise the known results on Stokes automorphisms for both unrefined free energies [29,30] and NS free energies [55].In section 3, we introduce and slightly extend the blowup equations for refined topological string, which are important for later sections.We give the key argument in section 4 that makes the connection between the Stokes constants of unrefined and NS free energies.Finally, we summarise and discuss open problems in section 5.In Appendix A, we re-derive and generalise the relationship between Stokes constants of NS free energies and those of quantum periods, which is crucial for closing the circle in Fig. 1.1.

Resurgence theory in a nutshell
We give a lightning overview of the resurgence theory [16].We refer to the lectures [17][18][19] for details.Suppose we have a perturbative series ϕ(z) of the Gevrey-1 type, i.e.
which is divergent with zero radius of convergent, the resurgence theory tells us that if we wish to find the full and exact description of the quantity in the form of a function f (z) that admits ϕ(z) as an asymptotic expansion, we must include non-perturbative corrections, which are actually encoded in and therefore can be extracted from the perturbative series itself.
In order to uncover the hidden non-perturbative corrections, we introduce the Borel transform, This is a convergent series with a positive radius of convergence in the complex ζ-plane C ζ , also known as the Borel plane, and it can be analytically continued to the entire complex plane with possible singularities, known as the Borel singularities.
with θ = arg z.This is known as the Borel-Laplace resummation of ϕ(z).
According to the resurgence theory, each of the discrete singular points in the Borel plane in fact represents a non-perturbative saddle point3 whose action is given by the position A of the singular point, and the perturbative series ϕ (A) (z) in the non-perturbative sector can be uncovered by the remarkable formula (2.4)Here we change (2.3) slightly and define lateral Borel-Laplace resummations, as shown in Fig. 2.1, And in (2.4) we choose z so that The constant s A is known as the Borel residue.If we have a string of singular points kA (k = 1, 2, . ..) along the ray ρ arg A = e i arg A R + , known as the Stokes ray, the right hand side of (2.4) should be modified to include contributions from all these non-peturbative saddles s kA e −kA/z s θ − ϕ (kA) (z). (2.7) All resurgent series form an algebra, and the analytic formula (2.7) can be represented alternatively as an algebraic operator in the algebra of resurgent series.Introducing Stokes automorphism S θ associated to the Stokes ray ρ θ then S θ is, as its name suggests, an automorphism so that for two power series ϕ, ψ (2.11) Upon acting on the series ϕ, one has where the constants S kA are Stokes constants, and they are combinatoric combinations of s kA .More importantly, it can be proved that S θ are proper derivations, in the sense that they satisfy the following properties • Leibniz rule: if ϕ, ψ are two power series • Chain rule: if ψ(x, z) is a parametric power series in z with an auxiliary parameter x, and ψ(z) another power series in z

Resurgent structure of topological string
Consider topological string with the target space a non-compact Calabi-Yau threefold X.
Let the number of linearly independent compact 2-cycles and compact 4-cycles be b 2 and b 4 respectively.We collect the complexified Kahler moduli of 2-cycles t i (i = 1, . . ., b 2 ) and of 4-cycles t D, ( = 1, . . ., b 4 ) in a vector The moduli space of the Calabi-Yau threefold enjoy special properties known as the special geometry relations, among which we can define the prepotential F 0 , a function of t i , so that where C i are entries of the integer valued b 2 × b 4 intersection matrix between compact 2cycles and compact 4-cycles.F 0 is the genus zero component of the free energy, and higher genus free energies F g can be constructed by coupling the worldsheet theory to gravity.Mathematically, the free energies are defined as the generating function of Gromov-Witten invariants, the counting of stable holomorphic maps from genus g Riemann surfaces to 2-cycles in the Calabi-Yau X, i.e.
where t(β) is the complexified volume of the 2-cycle β.Collectively, the perturbative free energy of topological string is Through mirror symmetry, the components of the vector Π are identified with complex structure moduli of the mirror threefold Y , which are periods of the holomorphic (3, 0) form over integral 3-cycles in Y , or equivalently with periods of the canonical 1-form over integral 1-cycles in the mirror curve Σ that the threefold Y can reduce to.Hence Π is also called the period vector.The distinction between t i and t D, corresponds to a choice of symplectic basis of H 1 (Σ) consisting of A-cycles and B-cycles so that the oriented intersection is6 (2.20) And t i , t D, are correspondingly called the A-and B-periods.Such a choice is not unique, and a different choice of A-and B-cycles, known as a different frame, leads to different Aand B-periods.The frame given by (2.17) is called the large radius frame.We denote Aand B-periods in a generic frame Γ by and the special geometry relation (2.17) becomes accordingly where C Γ i is the intersection matrix of A-and B-cycles in frame Γ.Clearly, the prepotential F Γ 0 , as well as the genus g free enegies F Γ g , depend on the choice of frame Γ.Although this is not always the case, throughout our paper, we choose frames so that A-and B-cycles are integral cycles, so that changing a frame amounts to a symplectic transformation in Sp(b 3 (Y ), Z) of the interal periods.To find how free energies change across different frames, it was noted in [61] that free energies F Γ g for g ≥ 2 and exp(F Γ 1 ) are almost holomorphic modular forms, whose modular parameter is and a change of frame is equivalent to a modular transformation of these almost holomorphic modular forms.
In the remainder of this section, we will drop the superscript Γ for frame to reduce the notational clutter.Regardless of which frame one is at, the genus g free energy F g (t) is a well-defined function, while the perturbative series and therefore there should be non-perturbative corrections which can be analysed by resurgence techniques.It has been found that the locations of singularities of the Borel transform always coincide with classical integral periods up to normalisation [23,24,[26][27][28][29]62].More precisely 7 , where ℵ = 4π 2 i, p , q i , s are certain integer numbers.We will not discuss the cases with p = q i = 0 as they are trivial.
The alien derivative of the free energy F (0) (t, g s ) at a singular point A is proportional to the instanton amplitude associated to this singularity.In particular, if we have a sequence of singular points kA, k = 1, 2, . . .along a Stokes ray ρ argA , and let us denote the instanton amplitude at the singularity kA by F (k) , the alien derivatives at these singular points read [29,30] • (2.26) Here, both the perturbative free energy and the instanton amplitudes depend on the holomorphic frame of evaluation.In particular, the expression of the instanton amplitude depends greatly on the type of frame.If a frame, known as an A-frame, is chosen such that A = A (p,q,s) is an A-period, i.e. p = 0, then the instanton amplitude simplifies greatly and we have where the subscript A refers to the A-frame.If we are not in an A-frame, the instanton amplitude has more complicated form, but we will not need them here.The Stokes constants S top kA are very interesting, as it was found empirically in [29,30] that they satisfy certain intriguing properties.They are integers, and they seem to be frame independent as well as the same for all the singular points kA.Among these properties, the frame independence may be due to the following reason.It is known that the holomorphic and frame dependent free energies F g (t) can be lifted to anholomorphic and frame independent free energies F g (t, t) [6,7], and choosing a frame is done by sending t to some fixed value, which can be interpreted as choosing a different gravitational background [63].We speculate that (2.26) also holds when both sides are lifted to anholomorphic amplitudes top (t, t, g s ). (2.28) The frame independence of the Stokes constants is then equivalent to the conjecture that they are background independent.The purpose of this note is to uncover the nature of the Stokes constants S top kA by relating them to the Stokes constants of the refined free energy in the Nekrasov-Shatashvili limit.The perturbative free energy of topological string can be refined to which reduces to the conventional topological string in the limit Another interesting limit we can take is the Nekrasov-Shatashvili limit The components F NS n (t) are also almost holomorphic modular forms and they transform accordingly in a change of frame as well.
The NS free energies are also Gevrey-1 series in , and we can similarly perform resurgence analysis.We find the singularities of the Borel transform are also located in (2.25) [55] 8 .The alien derivatives of NS free energy at such a singularity is found to be Both perturbative and instantonic free energies are frame dependent.In the case where A is an A-period, i.e. in an A-frame, one finds [55] In the cases where A is not an A-period, i.e.A is given by (2.25) with p = 0, we can shift the definition of prepotential F 0 so that difference of the factor of i is because the authors of [55] used the convention Then the instanton ampitudes of NS free energy are [55] where the quantity G(t, ) is defined by Yet again, it was found empirically in [29] that the Stokes constants S NS A are integers, the same for all kA, and seem to be frame independent.The forms of the alien derivatives (2.32) and the properties of the Stokes constants S NS A have profound consequences.As pointed out in [55] (see also [56]) and will be reviewed in Appendix A, they imply, together with the Stokes transformation properties of Wilson loop amplitudes, that the quantum periods satisfy the DDP type of formulas for Stokes automorphism [44][45][46], so that the Stokes constants of quantum periods can be identified with BPS invariants.More importantly, the Stokes constants S NS A are identified with those of quantum periods [55], so that S NS A themselves are given by BPS invariants.More precisely, the coefficients (p, q, s) in the composition of A in (2.25) are brane charges.For instance in the large radius frame, (p, q, s) are respectively the D4-, D2-, and D0-brane charges.S NS A is then the counting of BPS states of stable D-brane bound states with brane charges γ(A) = (p, q, s); in other words, S NS A = Ω(γ(A)). (2.37) We will show in Section 4 that the Stokes constants S top A of unrefined free energies coincide up to a sign with S NS A of NS free energies as in (4.9).
3 Blowup equations of refined topological string

Blowup equations in large radius frame
It was conjecturd [12,57] based on [58,59] and checked in many examples that the blowup equations for supersymmetric gauge theories [64][65][66][67] can be generalised and are satisfied by free energies of topological string on a local Calabi-Yau threefold X.And it was pointed out in [12] that blowup equations can be used to solve D2-D0 type BPS invariants.This line of research was later expanded in [68][69][70][71][72][73][74][75][76].See also related works in [77][78][79][80][81].The blowup equations will play a crucial role for relating the Stokes constants of unrefined and NS free energies of topological string.Let us work in the large radius frame.The number of linearly independent compact 2-cycles and 4-cycles in Calabi-Yau threefold X are respectively b 2 and b 4 .Denote by C the b 2 × b 4 intersection matrix between compact 2-cycles and 4-cycles.The complex Kahler moduli of compact 2-cycles are t.Then it was conjectured that there exist b 2 -dimensional integer valued vectors r satisfying the checkerboard pattern conditions, also known as flux quantisation conditions, for non-vanishing D2-D0 brane BPS invariants N such that the refined free energy of topological string satifies the so-called blowup equations, 2) where |n| = n 1 + . . .+ n b 4 , and Here the vector r is in addition subject to the equivalence relation as (3.2) does not change under this transformation.Besides the crucial factor Λ(m, 1 , 2 ) depends not on all the Kahler moduli but only the mass parameters.We will be interested in the special cases where Λ vanishes identically.These are called vanishing blowup equations.One subtlety concerning the blowup equations as claimed in [12,57,59] is that the refined free energies that appear in (3.2) should be twisted in the sense that where B, known as the B-field, is a Z 2 valued b 2 -dimensional vector defined by The twisted free energy was introduced so that when a gauge theory description is available it coincides with the logarithm of the Nekrasov partition funtion.Here F inst ref is the instanton contributions, while F pert ref is the sum of classical and 1-loop contributions and it is given collectively by (3.7) where a ijk are triple intersection numbers of divisors in X and b NS i , b i are some other intersection numbers.The three terms on the right hand side in (3.7) come from F (0,0) , F (0,1) , F (1,0) respectively.As it stands, F pert ref defined in (3.7) does not have quadratic contributions and it is calculated from the special geometry relation (2.17) using the Frobenius basis of periods.If, however, we integrate the special geometry relation (2.17) using the integral basis of periods as we do in this paper, we would have that O(1), (3.8) with an appropriate representation of B. As we will later see in (3.17), the blowup equations only depend on F (0,0) , F (0,1) , F (1,0) through so that the difference in (3.8) is irrelevant.In light of this relation, we can use the blowup equations (3.2) with the understanding that we can use refined free energies of topological string based on an integral basis of periods for the moduli without twist after making the shift t → t − B/2. (3.10) Let us illustrate (3.8) with the simple example of the local P 2 model.This model has a one dimensional moduli space M parametrised by a global modulus z.In the large radius frame, the integral periods are [61,82] where Π 0 is the Frobenius basis given by where with ψ being digamma function.The special geoemtry relation is [61] The prepotential obtained by integrating the special geometry relation using the integral periods t D and t is while the prepotential obtained by replacing t D , t with the Frobenius periods X (1,1) , X (1) , as practised in [12], is and they satisfy (3.8) after taking into account that we can take B = 1 in local P 2 [12].

Blowup equations in a generic integral frame
The blowup equations (3.2) are formulated for free energies in the large radius frame.Nevertheless, it is possible to change the frame and write down the blowup equations in other integral frames as well.One way of doing this is using the anholomorphic blowup equations proposed in [81] and choosing the appropriate holomorphic limit.Another way is expand the blowup equations in terms of 1 + 2 and 1 2 e F (0,1) −F (1,0) Here we use the notation and Λ (n,g) are components of Λ(m, 1 , 2 ) through the expansion At each order of 1 + 2 and 1 2 , the left hand side is a linear sum of which are theta constants with modulus τ ∝ F (0,0) and its higher dimensional generalisations.The coefficients of the linear sum are products of e F (0,1) −F (1,0) and F (n,g) which are almost holomorphic modular forms of τ .The identity (3.17) at each order of 1 + 2 and 1 2 expansion is an equation of almost holomorphic modular forms, and they have been checked for various examples in [12].A frame transformation is then akin to a modular transformation at each order of (3.17) and they can be reassembled into the blowup equation in the corresponding new frame.
The blowup equation in an arbitrary integral frame takes a form similar to (3.2), with The ingredient R Γ including its coefficients C Γ and r Γ as well as Λ Γ (m, 1 , 2 ) may change over different frames.We will only be interested in vanishing blowup equations so the change of Λ Γ (m, 1 , 2 ) is trivial as it stays zero across all frames.On the other hand, C Γ and r Γ in R Γ should change appropriately so that each component in the expansion (3.17) transform consistently under modular transformations.The properties of C Γ and r Γ would be crucial in later sections.We will only consider frames defined by integral basis of periods, and in these cases we argue that we always have Indeed the sum over n ∈ Z b 4 is a summation over discrete magnetic flux over the exceptional divisor P 2 in the spacetime C 2 ∼ = R 4 blown up at the origin BC 2 in the field theory description [64][65][66][67], and each component of the flux vector is associated to an irreducible compact 4-cycle in the Calabi-Yau X [12,57,59,73].Therefore, in the case of large radius frame where the moduli t Γ i = t i are associated with integral 2-cycles, C Γ = C is defined as the integer valued intersection matrix of 2-cycles and 4-cycles.In a generic integral frame, each modulus t Γ i is associated with either an integral 2-cycle or an integral 4-cycle.In the former case, the corresponding row of C Γ is the integer valued intersection numbers with 4-cycles; in the latter case, the corresponding row of C Γ should be the integer valued decomposition coefficients in terms of a basis of integral and irreducible 4-cycles.
We emphysize that in a generic frame, C Γ is not identified with the intersection matrix C Γ given in (2.22).Similar to the large radius case, the vector r Γ is defined up to the equivalence relation We also comment that even though we do not have a physics argument, the vector r Γ also seems to be integer valued in an arbitrary frame defined by integral periods.We demonstrate the integrality of both C Γ and r Γ through two examples below.

Local P 2
We first consider the simple example of the local P 2 model.The first two orders of the expansion of the vanishing blowup equations (3.21) with Λ Γ = 0 in terms of 1 + 2 and 1 2 , similar to (3.17), are where Θ Γ k (t Γ ) are the theta constants Consider first the large radius frame, where we will drop all the superscript Γ.These two equations have been checked in [12].Indeed, we have and it is easy to see that (3.25a) is satisfied as the summand of Θ 0 is an odd function of n.Furthermore, let us introduce the theta constants relevant for the local P 2 model with They have modular weight 3/2 and enjoy the properties where κ 1,2 are roots of unity.Then the free energies F (0,1) (t), F (1,0) (t) are [82,83] where the modular parameter is Note that here F (0,1) (t) is the holomorphic limit of the anholomorphic with τ 2 = Im τ .Using expressions of F (0,1) , F (1,0) in (3.31), (3.25b) can be integrated to which can be checked to high degrees of q = exp(2πiτ ) expansion.As mentioned before, the local P 2 model has a one dimensional moduli space M(K P 2 ) parametrised by a global parameter z.The moduli space of the local P 2 model has a conifold singularity at z = − 1  27 , at which the period t D vanishes.It is appropriate then to adopt the conifold frame where t D is chosen as the A-period when we are close to the conifold frame, and t as the B-period.In the conifold frame, the special geometry relation is The modular parameter is so that the first few free energies written as almost holomorphic modular forms are (up to a constant term) and whose holomorphic limit is Here we consider another example of the local P 1 × P 1 model.This model has one gauge modulus and one mass parameter.We restrict ourselves to the case of trivial mass parameter, corresponding to constraining the two P 1 's to have the same complexified Kahler modulus t.In this case, the model also has a one dimensional moduli space M(K P 1 ×P 1 ) parametrised by a global parameter z.
The moduli space M(K P 1 ×P 1 ) of the massless local P 1 × P 1 model has a conifold singularity at z = 1  16 , at which the period t D vanishes.It is then suitable to choose the conifold frame where t D is selected as the A-period when we are close to the conifold point, and t as the B-period.In the conifold frame, the special geoemtry relation is and the modular parameter is Using that where κ 1,2 are roots of unity, it is easy to find the first few free energies are (up to a constant term) whose holomorphic limit is We find then again that (3.25a) holds if and only if This is only valid for C c = 1, which is consistent with our prediction for C c as the A-period t D in the conifold frame is associated to the irreducible compact 4-cycle.Together with (3.55) we collect the following facts of integrality in the conifold frame for massless local

Relation between unrefined and NS Stokes constants
In this section, we reveal the intimate connection between the Stokes constants of unrefined and NS free energies.The starting point is the observation in [12,59] that in the limit the so-called compactibility formulas which relate the unrefined and the NS free energies of topological string.The same is true for vanishing blowup equations in an arbitrary integral frame, and the corresponding compactibility formula reads 0 = where R is given in (3.22).In this section, we will drop the superscript Γ indicative of frame to reduce notational clutter.
Let A be a point of the type (2.25) in the Borel plane.We take the compactibility formula (4.1) in an A-frame where A is a classical A-period so that p = 0, and apply the alien derivative • ∆ kA .After using the Leibniz rule (2.13), we find that 0 = Since both the unrefined and NS free energies are evaluated in an A-frame, their alien derivatives are simple, given by (2.26),(2.27),(2.32),(2.33).We then use the commutation relation (2.15) to find and use the chain rule (2.14) to find where we have used that q • C • n ∈ Z due to the integrality condition (3.23).We plug these equations in the second line of (4.2) and drop any component which vanishes due to the compactibility formula, and we arrive at the crucial equation We make the distinction between two cases.If the singularity A is such that If, on the other hand, the condition (4.6) is not satisfied, we argue that one can at least find one vector r for vanishing blowup equations so that the second line of (4.5) does not vanish.Following (3.17), the leading term in the expansion of the second line of (4.5) is Recall from (3.25a) that the leading term in the vanishing blowup equations is and it has been conjetured in [12] that in the large radius frame, once C is known, any integer valued vector r with which (4.8) holds is a suitable r vector for vanishing blowup equations (3.2) with Λ = 0.Among these suitable r vectors, a special one is the one that makes the summand of (4.8) an odd function of n.We denote such an r vector by r odd , and we conjecture that it exists in any integral frame, as the structure of vanishing blowup equations are similar across different frames.In the local P 2 model discussed in Section 3.2.1,r odd = 3 in the large radius frame, and r odd = 1 in the conifold frame.Similarly, in the local P 1 × P 1 model discussed in Section 3.2.2,r odd = 2 in the large radius frame, and r odd = 1 in the conifold frame.Such an r odd can also be found in all the examples discussed in [12].Now if we take r = r odd in (4.7), it will no longer be zero as the linear term q • C • n changes the summand from an odd function to an even function of n.
As the second line of (4.5) does not vanish, we can naturally conclude that the Stokes constants of unrefined free energy and those of NS free energy be the same up to a sign S top kA = (−1) k(q•r odd −1) S NS A .(4.9)This is our key formula.And thanks to (2.37), it implies that i.e. the Stokes constant of unrefined free energy of topological string for the Borel singularity kA, which is not associated with flavor 2-cycles, coincide with the BPS invariant Ω(γ(A)) where γ(A) is the brane charge associated with A, possibly up to a sign as indicated by the dotted equality sign .

Discussion
In this note, we are interested in the relationship between Stokes constants of unrefined perturbative free energies and those of refined perturbative free energies in the Nekrasov-Shatashvili limit of topological string theory on a non-compact Calabi-Yau threefold.It was observed in [29,55] with the example of local P 2 that both perturbative series have Borel singularities located at classical integral periods of mirror Calabi-Yau in the B-model, and the Stokes constants of the two perturbative series at the same Borel singularity might be related.We confirm this observation and demonstrate, using the formulation of the blowup equations in a generic integral frame taken to certain special limit, that this should be true on a generic non-compact Calabi-Yau threefold.More precisely, as long as the Borel singularity does not correspond to some 2-cycles that intersect only with non-compact 4cycles in the A-model, the Stokes constants of the unrefined and the NS perturbative free energies must be the same up to a sign.It was argued in [24,31,32] that the Stokes constants of unrefined topological string free energy for non-compact Calabi-Yau threefold should be related to BPS invariants, although as far as concrete constructions are concerned only the simplest Calabi-Yau threefold, the resolved conifold, is considered.Similar statements for more generic non-compact Calabi-Yau [29,36] and even for compact Calabi-Yau threefolds [30] have been proposed.In this paper we give strong support for these statements.In fact, since the Stokes constants of NS free energies can be shown [55] to coincide with the Stokes constants of quantum periods, and therefore can be interpreted as BPS invariants, the results of this paper imply immediately that the Stokes constants of unrefined free energies on a non-compact Calabi-Yau threefold can similarly be identified as counting of BPS states, i.e. stable D4-D2-D0 brane bound states in type IIA string.Similary conclusions have been reached in [60], where a closed-form formula for Stokes automorphisms of unrefined topological string free energies has been provided, which resembles the DDP formula of quantum periods.
This paper opens many new directions to explore.First of all, our demonstration of the BPS interpretation of Stokes constants of unrefined free energies hinges on two crucial conjectures, the blowup equations for refined topological string free energies in a generic integral frame, and the identification of Stokes constants of NS free energies as BPS invariants.More evidences or maybe proofs are needed for these two conjectures.
Furthermore, the argument in this paper is indirect and is only valid for non-compact Calabi-Yau threefolds.A direct argument or even proof, potentially also valid for compact Calabi-Yau threefold, possibly along the line of [60], will be very desirable.
Finally, the result of this paper suggests using resurgence techniques to systematically study BPS invariants of Calabi-Yau threefolds in different loci of moduli space and to study their stability structures.It would be interesting to compare with the BPS spectrum of local F 0 in [84] and of local P 2 in [85] computed with other techniques.

Figure 1 . 1 :
Figure 1.1:The relations between BPS invariants, Stokes constants of quantum periods, of NS topological string, and of unrefined topological string.
If the singularities are discrete points and form a closed subset Ω ⊂ C ζ and in addition ϕ(ζ) allows analytic continuation along any path in C ζ \Ω, ϕ(ζ) is called a resurgent function, and ϕ(z) called a resurgent series.In this case, we can define the Laplace transform of ϕ(ζ) along any direction which does not pass through any Borel singularities 2

Figure 2 . 1 :
Figure 2.1: The lateral Borel resummations which sandwich a Stokes ray that pass through the singular point A.

. 10 )∆
Another even more powerful way to encode the formula (2.7) is to introduce the alien derivatives • kA associated to each Borel singularity related to the Stokes automorphism S θ by S θ = exp ∞ k=1 • ∆ kA .

. 16 )
where the last entry can be regarded as the trivial Kahler modulus of a point.In general b 2 ≥ b 4 , and if b 2 > b 4 , we can make the distinction of b 4 linear combinations of t i associated to 2-cycles that intersect with compact 4-cycles, and another b 2 − b 4 linear combinations of t i associated to 2-cycles that have zero intersection numbers with compact 4-cycles.These linear combinations are called true moduli and mass parameters, denoted by t * andt k+b 4 * = m k ( = 1, . . ., b 4 , k = 1, . . .b 2 − b 4) and we call the corresponding 2-cycles gauge 2-cycles and flavor 2-cycles, as they are related to Coulomb moduli and flavor masses in the associated field theory.From superstring theory point of view, the Kahler moduli in the vector Π are also interpreted as the central charges of D4-, D2-, and D0-branes supported on these cycles.

1 2
(τ c )η 3 (τ c )). (3.39)Now (3.25a)only holds if R c = C c (n equivalence relation (3.24) for r c .Using (3.37),(3.39), the identity (3.25a) can also be integrated to n∈Z ) 2 τ c = Const.b(τ c ) (3.41) which is only valid for C c = 1.This is consistent with our prediction for C c as the A-period t D in the conifold frame is associated to the irreducible compact 4-cycle.Together with (3.40), we can collect the following facts of integrality in the conifold frame for local P 2 , C c = 1, r c = 1.(3.42) 3.2.2Local P 1 × P 1 equivalence relation (3.24) for r c .Using (3.52),(3.54),we find that (3.25b) can also be integrated to n∈Z (−1) n .6)the second line vanishes and (4.5) gives no constraint between the two types of Stokes constants.Geometrically, condition (4.6) corresponds to flavor 2-cycles in Calabi-Yau threefold X, which are 2-cycles that have zero intersection numbers with compact 4-cycles.Flavor 2-cycles are not expected to support BPS states.