On Asymptotics and Resurgent Structures of Enumerative Gromov-Witten Invariants

Making use of large-order techniques in asymptotics and resurgent analysis, this work addresses the growth of enumerative Gromov-Witten invariants---in their dependence upon genus and degree of the embedded curve---for several different threefold Calabi-Yau varieties. In particular, while the leading asymptotics of these invariants at large genus or at large degree is exponential, at combined large genus and degree it turns out to be factorial. This factorial growth has a resurgent nature, originating via mirror symmetry from the resurgent-transseries description of the B-model free energy. This implies the existence of nonperturbative sectors controlling the asymptotics of the Gromov-Witten invariants, which could themselves have an enumerative-geometry interpretation. The examples addressed include: the resolved conifold; the local surfaces local P^2 and local P^1 x P^1; the local curves and Hurwitz theory; and the compact quintic. All examples suggest very rich interplays between resurgent asymptotics and enumerative problems in algebraic geometry.


Introduction
Geometrical-counting problems, albeit many times rather natural and simple to formulate, may lead to remarkably rich and interesting structures. Among these, enumerative invariants play an important classification role within algebraic geometry. For example, counting pseudoholomorphic curves inside symplectic manifolds gives rise to the famous Gromov-Witten (GW) invariants. These are invariants associated to the symplectic manifold X , which are rational numbers (implying a "virtual" counting) depending on both genus, g, and degree, d, of the embedded curve. We shall denote them by N g,d . The computation of GW invariants is generically hard, becoming simpler when the manifold is Calabi-Yau (CY) where they are generated by the A-model topological-string free energy. This is a long story which goes back to the discovery of mirror symmetry; see, e.g., [1][2][3][4][5][6][7][8][9][10] for early references, and, e.g, [11][12][13][14][15] for reviews.
Consider the A-model on a CY X , in the large-radius phase (valid when the Kähler parameter t is large). The A-model free energy is then given by an asymptotic, genus expansion where the genus-g contributions to the free energy may be decomposed as [6] F g (t) = d>0 N g,d Q d . (1. 2) The sum over degree d corresponds to a sum over topological sectors as classified by worldsheet instantons (where Q = e −t in units where α = 2π). While this explicitly shows how the topological-string free energy is a generating function for the genus g, degree d, enumerative GW invariants of X , N g,d , the two expansions above have rather different properties: while the fixed-genus (1.2) is a convergent series, with a non-zero radius of convergence, e.g., [16], (1.1) is instead a divergent asymptotic series, with zero radius of convergence, e.g., [17]. The reason for this is the factorial growth of the genus-g contributions with genus, as F g ∼ (2g)!. From the standpoint of defining the string free energy, the asymptotic nature of the perturbative expansion (1.1) implies that F (X ) cannot be properly defined by perturbation theory alone. One way to move forward is to use the theory of resurgence [18]. In this context, the perturbative expansion gets enlarged into a transseries, an object which fully captures all information concerning the observable that it represents, including both perturbative/analytic components (in powers of the string coupling g s ) and nonperturbative/non-analytic components (in powers of the "instanton" factor e −1/gs ). The asymptotic and resurgent nature of the perturbative sequence implies the existence of these instanton-type terms, of which there can be many distinct types and with different strengths. Remarkably, all these seemingly independent perturbative and nonperturbative sectors in the transseries turn out to be related to each other via a tight web of asymptotic resurgence relations. In particular, the leading factorial growth of perturbation theory is a consequence of these asymptotic relations, as is any other subleading growth correcting that factorial term. As a result, one may in fact extract, or decode, nonperturbative information from perturbation theory alone and vice-versa. Moreover, these interrelations have somewhat universal forms, and should be expected to hold across a wide range of different problems.
nonperturbative transseries-solutions to the holomorphic-anomaly equations of the B-model were constructed in [29,31,34]. These references further focused on the example of local P 2 , a noncompact CY threefold, where a very rich nonperturbative structure was uncovered, with diverse instanton actions vying for dominance on the Borel plane as the moduli changed. In our present paper we wish to turn our attention to the A-model instead, and in particular to the enumerative invariants it generates. From the standpoint of computing enumerative invariants, the convergence properties of their generating functions might not seem terribly important at first sight. It is nonetheless the case that these convergence properties will dictate the asymptotic behavior of these invariants, in genus and in degree, and this is one of the main question we address in the present work. Furthermore, within the A-model the GW invariants are the internal ingredients constructing the string free energies, and it seems reasonable to transfer resurgence questions and properties from the free energies to the invariants themselves. In particular, one natural question is to ask exactly how the GW invariants are responsible for the (known) factorial growth of the free energies they build. For example, the convergence of (1.2) roughly implies that, at fixed genus, the largedegree asymptotics of the GW invariants 2 N g,d corresponds at most to a leading exponential growth. On the other hand, the asymptotic nature of (1.1) might seem to imply that, at fixed degree, the large-genus asymptotics of the GW invariants N g,d corresponds instead to a leading factorial growth, giving rise to the factorial growth inside the free energy. But this will turn out not to be the case. The fixed degree, large-genus asymptotics of the GW invariants is not factorial, and we shall see how the factorial growth of the free energy is more subtly encoded at the level of GW invariants.
Note that there are some important differences between addressing resurgent transseries for the B-model free energy, and investigating resurgent asymptotics of A-model enumerative invariants. In the former case, one deals with an asymptotic series, which subsequently gets completed into a transseries by the addition of new, nonperturbative sectors. In the latter case, Going back to the topological-string asymptotic-series for the free energy (1.1), let us describe it in the B-model as F (0) (g s ; z,z). Here, the string coupling g s is also the resurgent variable, and the (0) superscript specifies perturbative. The pair (z,z) may be regarded as just external parameters, or interpreted as complex-structure moduli of the underlying CY threefold. The free energy is asymptotic, of Gevrey-1 type (see, e.g., [46]), for generic values of (z,z). Understanding the resurgent properties of F (0) and the role played by the moduli (z,z) was the main purpose of [29,31]. There, it was shown how to look for a transseries completion to the topological-string free energy of the form σ n e −nA(z)/gs F (n) (g s ; z,z), (2.2) where the (multi) instanton sectors F (n) (g s ) are also given by asymptotic series. In particular, it was found-both generically and in examples-that (2.2) has several nonperturbative sectors, with associated actions A α , all of them holomorphic and determined by the CY geometry. The transseries (2.2) was constructed by combining a nonperturbative interpretation of the holomorphic anomaly equations of [6] with the resurgence relations that transseries generically satisfy, such as, for example, 0 (z,z), as g → +∞. (2.3) Here F (1) 0 is the first coefficient of the one-instanton series F (1) (g s ) and A(z) is one of the instanton actions (the smallest one in absolute value, for the particular value of z). Subleading corrections to (2.3) lead to further multi-loop coefficients, F This route towards the construction of (2.2), further developed in [34], draws a rather complete picture of what a transseries for F (g s ) should look like. In principle, such a transseries should contain all nonperturbative information concerning the B-model, but also, via mirror symmetry [10], all A-model nonperturbative information. It is within this context that we shall set our attention upon structures of interest in algebraic and enumerative geometry, arising from the A-model set-up, in particular the case of enumerative GW invariants.
Let us spell out our strategy. The B-model construction (2.2) depends upon (z,z), the complex-structure moduli. From the standpoint of resurgence, these moduli may be regarded as external parameters, without any resurgent properties by themselves. But upon mirror symmetry, they relate the B-model CY threefold X , with complex structure z, to the A-model mirror-CY threefold X , with Kähler structure t. A functional relation t = t(z) is then provided by the mirror map. This means that one may in fact compute the mirror transseries to (2.2), where its F (0) g (t) components are nothing but the GW generating functions as in (1.2). Let us next focus on these enumerative invariants in greater detail, with the goal of uncovering which resurgent properties they carry, either intrinsic or merely inherited from the free energy.

Enumerative Gromov-Witten Invariants
GW invariants count embeddings of Riemann surfaces of a given genus into a CY threefold X , attending to the homology class of the image of this map. Thus, GW invariants are labelled by g ∈ N, like the topological-string free energies, and β ∈ H 2 (X , Z), (2.4) Akin to (1.2), they show up in the A-model perturbative free-energies through the expansion Here we have used the mirror map to translate from complex structure moduli, z i , to Kähler moduli, t i =: − log Q i (roughly, the mirror map is . In order to simplify things in the following, we shall restrict to examples where b 2 (X ) = 1, in which case the sum over homology classes simplifies to The index d is called the degree of the embedding. See, e.g., [14] for more details on the relation between the enumerative GW invariants and their A-model generating functions. Now (2.6) is a convergent series in Q, which, in particular, implies that it is not resurgent. Its non-vanishing radius of convergence is generically finite, due to a nearby singularity located at the so-called conifold locus [8]. This convergence may already suggest that the factorial growth of the free energies F (0) g in genus must somehow arise from a combined contribution of several different degrees. We shall next try to understand how this might come about.

Growth of Enumerative Invariants in Degree and in Genus
As we introduce most of our main ideas, let us illustrate them with (partial) results from upcoming diverse examples. The simplest such example is naturally attached to the resolved conifold, for which the free energies can be computed exactly (see, e.g., [47] for a review) where Li p (x) is the polylogarithm function. This immediately yields all GW invariants as More interesting geometries we shall later address include the (non-compact) local P 2 and the (compact) quintic CY threefolds, for which there are no such closed-form expressions. Enumerative invariants may, nonetheless, be generated on the computer to see in more detail how they grow in degree and genus. An example of the sort of numbers we have to work with is show in figure 2, in the instance of local P 2 (to be addressed in section 4.1).
log |N g,d |   20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  20 111.  Figure 2: Sample of GW invariants for the local P 2 CY threefold, alongside a visual representation of their growth with respect to degree d (in blue), genus g (in red), and a linear combination of the two (in green). Only in this latter case shall we find a factorial growth.

Growth in Degree
Let us first consider the growth in degree at fixed genus. For the resolved conifold the answer is immediate from (2.8): it is given by the degree d, raised to a linear function of the genus g, namely 2g −3. For other, more intricate geometries the growth is similar but includes further parameters, such as a critical exponent γ which captures distinct topological-string universality classes, i.e., distinct critical behaviors at the phase-transition point (see, e.g., [42] for a discussion). In general one finds 5 [4,6,40] (further including a possibly g-dependent pre-factor). In this expression, e −tc = Q c marks the radius of convergence of F (0) g on the Q-plane. Expression (2.9) implies that the resolved conifold has γ = 0, being in the same universality class as, e.g., the local P 2 or the quintic CY threefolds. For example, for the quintic we have [4,6] where t c = 7.58995 . . .. For local P 2 this growth is illustrated in the leftmost plot of figure 2, at fixed genus g = 30. For very large degree d, the plotted curve must tend to a straight line of slope |t c |. On the other hand, for the family of local curves the critical exponent is instead γ = −1/2 [42], implying a distinct universality class and we shall discuss this example later in section 4.3.
As mentioned earlier, the radius of convergence Q c signals a singularity of the (free energy) generating function. This is the conifold point [8], the point in moduli space where the A-model geometric interpretation breaks down, with a phase transition taking place from the large-radius (geometric) phase to a non-geometric phase. Near this singularity, Nearby Q c all geometries within the same universality class will resemble each other, which implies that the coefficients c g are universal. For example, for γ = 0 there is a double-scaling limit such that which matches the c = 1 string at self-dual radius [8]. For other values of γ the coefficients c g may be more complicated, being solutions to a nonlinear ODE such as Painlevé I, for example.

Growth in Genus
As we turn towards understanding the dependence of GW invariants on genus, at fixed degree, N g,d , it becomes useful to introduce the Gopakumar-Vafa (GV) invariants. These invariants are integer numbers, roughly counting the number of BPS states inside a CY threefold X , and resulting from a reorganization of the A-model free energy as introduced in [48,49]. The complete result involves a Schwinger-type computation which rewrites the free energy as an index that counts string-theoretic BPS states via an M-theory uplift, and which finally yields (2.14) Here, the n (β) r ∈ Z are the GV invariants, labeled by the Kähler class β and a spin index r. The polynomials c(t i ) and (t i ) will play no role in the following.
It is straightforward to check that, generically, the GW invariants may be written explicitly in terms of the GV invariants as In here we already find the d 2g−3 dependence which is characteristic of the resolved conifold. Now, an important property of the GV invariants, which will be useful in the following, is that for each degree d there is a specific genus, G(d), after which all these invariants vanish, i.e., n (d) r = 0 for r > G(d) [49]. This function G(d) is a polynomial in d, and this will simplify the dependence on g in (2.15) by replacing the upper limit in the r-sum, (2. 16) In this expression the only remaining dependence upon the genus, g, lies in the coefficients c r,g and in the power of d/β. Since the coefficients c r,g are independent of the CY geometry, we should expect a generic formula to hold for the large growth of N g,d in genus. For example, as we shall discuss later on, for the quintic threefold and degree d = 4 we find 17) This formula, involving Bernoulli numbers and factorials, is actually exact for g ≥ 2, not just a large-g approximation. The first numbers (3, −6, −192) can be recognized as the GV invariants n 0 , and n (3) 0 for the quintic threefold, whereas the other (120, 336) are more complicated combinations involving higher-genera invariants. As such, in general, we can expect the following formula to hold (see appendix A for a proof) In this expression the dependence on the genus g is explicitone could even plug-in non-integer values of the genus after analytically continuing the Bernoulli numbers. If we fix the degree, as in N g,d , it is then simple to see that the leading growth in genus is exponential, d 2g−3 , with further subleading exponential and inverse-of-factorial corrections in g. The second plot in figure 2 illustrates this genus dependence, at fixed degree d = 30, for local P 2 . The plotted curve is asymptotic to a straight line of slope 2 log d.
Expression (2.18) shows how the contribution of GW invariants, N g,d , to the free energies at a fixed single degree, d, again cannot be responsible for the factorial growth we need to find. In this way, the only option we have left to find the ∼ (2g)! factorial growth of the free energies, encoded in the GW invariants, is to address the combined growth in genus and degree.

Combined Growth in Genus and Degree
Upon a second look at the (already familiar) characteristic behavior of GW invariants in d 2g−3 , it should be straightforward to deduce that when d and g are linearly related, then the factorial growth is immediately realized. The link is the classical Stirling approximation, Consider one more time the example of the resolved conifold in (2.8), and assume the dependence d = a 0 + a 1 g for some values of a 0 and a 1 . Then, to leading order in g, one finds (2.20) The factorial in g is now explicit and it comes from the term d 2g−3 when d = a 0 + a 1 g. On the other hand, recall that the leading growth of the free energy F (0),coni g in this case is [23] F (0),coni with instanton action A = 2πt. To connect this resurgence relation to the one in (2.20), one has to recall the definition of GW invariants (2.6) 22) and then notice that the largest contribution to this sum, for a fixed value of Q on the right-hand side, comes from So we should expect that taking a 1 = 2/t and a 0 = −3/t in (2.20) will reproduce something resembling (2.21). Indeed, one can easily check that we obtain the same instanton action as 4π a 1 = 2πt (where we ignore the exponential factor, e, since it belongs within Q d ). This strategy of selecting the leading contribution from the Q-expansion inside F (0) g (Q) can be pushed further. One way to do so is to approximate the sum over the degree by an integration, and then perform a saddle-point approximation-and this will be a main theme throughout our analyses. Consider the following saddle-point approximation around To apply this generic formula to our problem one just has to identify The only subtlety in this identification is that the saddle-point x 0 is also proportional to the coupling λ, as we saw for the resolved conifold (2.23). In any case, our goal is to solve for e λV (x 0 ) in (2.24). Then, the only obstacle we have in order to do so is knowing the value of V (x 0 ). For the resolved conifold we had an explicit formula and, as such, we knew that it was −t/(a 0 + a 1 g) 2 where x 0 = a 0 + a 1 g and λ chosen the same; but in general there are no such explicit formulae. Nonetheless, let us postulate a completely similar dependence in g, namely where we have chosen the explicit relation λ = a 0 + a 1 g, and introduced the function a 2 (Q). If one makes further use of the leading large-order growth of the free energies [29], we finally obtain (2.28) Note that this large-order relation depends on a 1 and a 2 , functions of Q which define the position and shape of the saddle. For the resolved conifold, and even for other geometries with actions proportional to a Kähler parameter, we find that a 1 = 2/t and a 2 = t. But for general geometries  g . The resolved conifold, portrayed on the left, has a single saddle-point corresponding to the action A = 2πt; whereas for local P 2 , portrayed on the right, an extra saddle-point attached to the conifold action is also present. These saddles may exchange dominance depending on the value of Q, but the set of leading degrees will always be in correspondence with the set of leading instanton actions.
we do not know what these functions are or should be, and one has to run computational experiments in order to judiciously try to fix them. Note that once one approximates the sum over the degree by an integration, then different saddles will correspond to different leading actions, which may depend on the value of Q. For the resolved conifold there is only one leading action and one saddle. But for general geometries we can expect several of them-albeit one is always proportional to the Kähler parameter t. This is illustrated in figure 3, where we have plotted saddles for the resolved conifold and local P 2 (we shall discuss these plots in greater detail later on). The saddles are identified by numerically selecting, at fixed values of g and t but varying d, the GW invariants which contribute the most to the perturbative free energy. Both models clearly show a saddle associated to a Kähler action, with A = 2πt. For local P 2 there is one further saddle, related to a conifold action, to be discussed in section 4.1.

The Main Question
Let us finally address the main question motivating this paper. Are there nonperturbative extensions of the enumerative GW invariants-denote them by "N (n) g,d ", with n an "instanton label"-just like there are nonperturbative extensions F (n) g of the perturbative free energy? And if so, what is their enumerative interpretation, i.e., which counting problem is associated to these new numbers? An argument in favor of an affirmative answer arises from considering the A-model mirror to the B-model resurgent analysis of the free energy, and its associated transseries constructions [29,31]. But while the perturbative F g (Q) collect the GW invariants as a Q-expansion, the higher instanton sectors F Schematically, we want to make sense of the following diagram The left and upper arrows are well understood. The left arrow is just the A-model definition of GW invariants, while the upper arrow was made precise within the B-model set-up in [29,31]. In this paper we try to take the first (exploratory) steps towards the definition of lower and right arrows, but a complete answer can only come with a geometric/enumerative interpretation of these conjectured quantities "N (n) g,d ", which is beyond the scope of the present work. Further note that, as GW invariants themselves have no transseries completions, we do not expect the lower arrow to be defined directly but rather as combination of left, upper, and right arrows (alongside the mirror map). In this way, one will have to extract the "N (n) g,d invariants" directly out of the nonperturbative sectors F (n) g . Now, the Q-expansion of the perturbative sector arises from a worldsheet-instanton expansion and thus naturally relates to a counting problem. But the nonperturbative sectors lack such power-series expansions in Q, implying that any nonperturbative GW invariants hidden inside the nonperturbative free-energies might be difficult to extract and their enumerative interpretation harder to decode. Furthermore, even after performing an asymptotic resurgent analysis of N g,d , we have to disentangle the dependence in t, coming from the parameters a 0 and a 1 , in the linear dependence between degree and genus. At the end of the day, this leaves the right arrow to be defined. What one has to do is to understand, via mirror symmetry, how to relate nonperturbative multi-loop multi-instanton coefficients in the B-model transseries, to the nonperturbative sectors appearing in the asymptotic resurgence relations for the combined genus/degree growth of GW invariants.
Our goal in this paper is to initiate this line of research, computationally exploring diverse CY examples. We try to identify the structure of these new invariants, as they are encoded in the nonperturbative content of the A-model free energy, but shall leave open their subsequent enumerative interpretation for future research.

An Exactly-Solvable Model: The Resolved Conifold
This section addresses our first example, concerning an exactly solvable model: the resolved conifold. This toric variety is a non-compact CY threefold which is the total space of the bundle O(−1) ⊕ O(−1) → P 1 . The perturbative free-energy for the resolved conifold can be computed exactly to all orders in the genus expansion (see, e.g., [47] and references therein). This of course translates to the fact that one may obtain analytical expressions for all its (infinite) GW invariants [50]. For any genus g, the results are 2) where Li p (z) is the polylogarithm of index p. In the following we will drop the contribution from the constant map [51,50] and mostly focus on the large-order contributions g ≥ 2.
Due to the polylogarithm these free energies grow factorially in the genus and lead to an asymptotic, Gevrey-1 perturbative free-energy [46]. The resurgent properties of this series have been studied in detail in [23,32], with the result where A m (t) = 2π (t + 2πim) are the instanton actions. For our purposes, we shall focus on the leading contribution, whose action is A = 2πt, in which case Let us next translate these resurgent properties to the level of GW invariants. The GW invariants for the resolved conifold can be immediately read from the free energies, by simply expanding the polylogarithm in power series. One finds where f CS g includes the Bernoulli dependence and is defined in (2.8). These invariants have such a simple form, given that they are actually generated by a single non-vanishing GV invariant Likewise, the abc-coefficients we introduced in (2.18) vanish except for the one which equals the GV invariant, a 1 = n (1) 0 = 1. This makes this geometry considerably simpler than the ones we shall explore later, allowing for an analytic treatment whose features will also show up later.
As we anticipated in some detail in the previous section, the factorial growth of the free energy arises from the term d 2g−3 when d grows linearly with g. This is completely precise when the degree is a saddle point, in the sense explained earlier. This point, d = (2g − 3)/t, was computed in equation (2.23) and graphically represented in figure 3 (left plot). One caveat about the saddle-point approximation is that generically the saddle-point lands on non-integer values of the degree. In order to be able to do numerical analyses with actual GW invariants, we look at nearby (integer) values of the degree. A practical computational choice, one that we will also use for other geometries, is to set and set t to an even integer (and q must consequentially be an integer). Since in this example there is an analytic expression for all GW invariants, we can use it to obtain the resurgence relation where the P h (q) are polynomials in q of degree 2h with rational coefficients, the first of which are P 0 (q) = 1, (3.10) and in general they are such that they make the following asymptotic expansion hold for any q as (3.14) Note that expression (3.9) conforms to the usual resurgence relations, which, for some generic free-energy perturbative expansion, look like (see, e.g., [19] for an introduction) Indeed, in (3.9) one immediately identifies the ∼ (2g)! growth, alongside the instanton action A = 2πt which is the same action that appears in the free energies. Of course (3.9) also has higher instanton corrections which improve the asymptotics further as in the above expression. These arise from including the (exponentially) subleading terms in ζ(2g) = +∞ n=1 n −2g in the large-g expansion of the Bernoulli numbers, B 2g , in (2.8); and from computing the complete large-d transseries expansion of d td+2q−3 . The result is Some computational tests on the validity of (3.9) are shown in figures 4 and 5. Figure 4 presents a test of the instanton action. We plot the analytical A K = 2πt against numerical tests of this action via (three) Richardson transforms (similar to tests done in, e.g., [21]). An illustration of these transforms for different values of t and q is shown in the inclosed figures. We do this for varying t (the horizontal axis) but also varying q, i.e., each black dot is actually several overlapping black dots, each one the third Richardson transform of the numerical sequence for the instanton action, for that particular value of t and for a range of different values of q. Then figure 5 tests the validity of (3.10) through (3.13) (in fact up to h = 5), this time around for fixed q. Each inverted triangle in the plot is again the third Richardson transform of the tested sequence. All these plots very cleanly illustrate the validity of (3.9).  In section 2 we showed how to relate GW asymptotics to free-energy instanton sectors, in particular relating the first term in the right-hand side of (3.9) with the one-loop one-instanton free energy F (1) 0 ; see (2.28) and the discussion which follows. Ideally, one would now like to do the same for the multi-loop (eventually multi-instanton) one-instanton free energies and their relation to higher terms in (3.9). Unfortunately, already finding a direct relation at two-loops, between F (1) 1 and any higher term in (3.9), turns out not to be possible using the saddle-point approximation from section 2. In fact, our saddle-point approach is non-standard, in the sense that the saddle-point itself grows with d (or g), which essentially obscures a clear-cut relation between free-energy asymptotics and GW asymptotics beyond the first term. For the present example of the resolved conifold we can bypass this problem, working directly with the explicit form of the GW invariants, but this will not be possible for more complicated examples.
Let us end our discussion of the resolved conifold by going back to our diagram (2.29). As we mentioned earlier, one cannot find nonperturbative GW invariants via a Q-expansion of the resurgent asymptotic expansion for the perturbative free energy F g . This is already clear in equation (3.5), where, although the left-hand side does have a regular expansion around Q = 0 from which one reads the GW invariants, the same does not hold true for the right-hand side, where one finds a logarithmic singularity at that same point. In other words, the "resurgence rewriting" of the perturbative free energies, F (0) g (Q), as an asymptotic series in 1/g does not respect, term by term, a regular Q-expansion. Only when we consider all corrections in 1/g and perform their resummation (yielding the polylogarithm, in this case of the resolved conifold) can we recover regularity at Q = 0. Looking directly at the resurgent GW expansion (3.9), one also sees how the right-hand side has a non-regular Q-dependence through t = − log Q. Although the possibility remains that there might be a better variable than Q or t to establish the match against nonperturbative GW invariants, it may also be the case that there is no such variable and reading nonperturbative GW invariants (naturally formulated using a Q-expansion) from resurgence expressions (naturally written using the t variable) is in fact a nontrivial problem which might require some a priori enumerative interpretation to know what to look for. Perhaps the fact that the polynomials (3.10) through (3.13), appearing in the resurgence relation (3.9), have rational coefficients much like the GW invariants themselves, is a clue in that direction.

Computational Explorations in Calabi-Yau Threefolds
We shall now move on towards non-trivial geometries, for which there are no closed-form expressions for enumerative GW invariants. We shall instead resort to computational methods in order to explore their asymptotics and resurgent structures.

The Example of Local P 2
Our first non-trivial example will be a local-surface toric-variety. We start with the non-compact CY threefold known as local P 2 , which is the total space of the line bundle O(−3) → P 2 . This example of local P 2 has a single complex modulus z, and a mirror map of the schematic form Q = e −t = O(z), which eventually allows for a calculation of GW invariants [52,40] (the resulting Kähler modulus being the size of the P 2 ). In fact, the large-order data for the resurgence analysis first arises in the B-model and will thus require translation into A-model expressions. Specifically, the high genus GW invariants for local P 2 will come out of B-model calculations, both perturbative [53] and nonperturbative [29,31], followed by mirror symmetry [10].

Free Energies and Gromov-Witten Invariants
The perturbative free energies are best computed within the B-model using the holomorphic anomaly equations, which are recurrence relations in the genus [6,54]. The GW invariants are then extracted using the mirror map back to the A-model, and removing the anti-holomorphic dependence (inz) which is introduced by this computation. We shall not get into any details, which may be found in [53], but it is perhaps worth mentioning the Picard-Fuchs equation. Its solutions, the periods, are the source to the genus-zero free energy, the mirror map, and also the instanton actions [26]. For local P 2 , the Picard-Fuchs equation is One of its three independent solutions is a constant. Another one, having a log z singularity, can be identified as the mirror map, The last solution, having a log 2 z singularity, can be associated to ∂ t F (0) 0 . Upon integration of this last solution, and use of the mirror map, one finds the genus-zero free energy as One can ignore the coefficients c i and then read the GW invariants, N 0,d , from this Q-series.
Within the B-model, the higher-genus free energies 6 , F g , may be compactly written as polynomials in z and S zz (z,z), an auxiliary variable called the propagator [55]. To extract higher-genus GW invariants one has to use the holomorphic limit of the propagator S zz (in the large-radius frame), Consider for example F (0) 2 (z, S zz ), which follows from the holomorphic anomaly equations as .
(4.5) Taking the holomorphic limit and using the mirror map, z = z(Q), one then obtains the A-model result F Here, the coefficients of the Q-expansion are the N 2,d GW invariants. In this way, the holomorphic anomaly equations systematically compute F h with h = 1, . . . , g −1, and from them one extracts the N g,d GW invariants as described above. An illustrative (i.e., partial) table of GW invariants for local P 2 may be found in appendix B.1. In figure 6 we schematically represent all the GW invariants we have computed and work with in the present paper. 6 Note that the genus-one free energy is calculated separately (see [54] for details), and further has a direct relation to the propagator; namely ∂zF As studied in great detail in [29,31] the free energies for local P 2 , F (0) g , grow factorially fast and render the free-energy expansion asymptotic. The resurgent structure which was uncovered in those references may be summarized as follows. There are several instanton actions, labelled by A 1 , A 2 , A 3 and A K , which give rise to corresponding nonperturbative sectors within the total free-energy transseries. Out of these, two actions are leading at large-order, these are A 1 and A K , meaning that for some values of the complex-structure modulus z they are the actions controlling the leading growth of the F (0) g . Around the large-radius point in moduli space, z = 0, it is A K = 2πt(z) which is leading, and elsewhere it is is the flat coordinate 7 around the conifold point z = −1/27. For obvious reasons, we name A K as the Kähler action and A 1 as the conifold action (in fact also A 2 and A 3 are related to the conifold point, but they will not play any role in the present paper).
In this case, the large-order growth of the free energies may be either depending on the value of Q. The one-loop one-instanton coefficients are computed from an extension of the holomorphic anomaly equations, alongside the above resurgent relations (which were needed in order to fix the holomorphic anomaly). One finds [31] and F The left expression involves S zz hol, [LR] , whose Q-expansion was written in (4.4), but it also involves the holomorphic limit of the propagator in the conifold frame, S zz hol,[c] (see [31]). It is interesting to note how the expression on the right of (4.8) is actually equivalent, up to a factor of 3 = n (1) 0 , to the one we computed earlier for the resolved conifold. One may also write these expressions in the A-model, where their Q-expansions are explicitly non-regular (4.10)

Analysis of Large-Degree Growth
At fixed genus, the GW invariants grow exponentially in the degree as where t c := t(z = −1/27) = 2.90759 . . . − iπ [56,40]. Figure 7 shows a numerical verification of this value for t c . The exponent 2g − 3 of the degree d may be verified numerically from the following large-d sequence The exponent t c in the growth of N g,d is captured from the ratio of two consecutive GW invariants, when the degree is large. We plot that ratio alongside three Richardson extrapolations, which are clearly converging faster towards the expected result (up to a numerical relative error of about 0.06%). Due to the presence of the d 2 factor, and the limit upon available data, the results are not as good as those for t c . Nonetheless, this exponent may also be cleanly verified numerically, as it is shown in figure 8. Finally, in similar fashion, we can determine the power of the logarithm log d from the sequence This is done in figure 9, where it is shown that this exponent may be well fitted to the expected δ = 2g − 2. Unfortunately, our available data does not allow us to numerically compute the genus-dependent pre-factor c g with enough accuracy as to present it here. Ng,d Figure 9: Local P 2 : The exponent δ of the logarithm log d is the leading term in the sequence (4.13). Having data up to d = 325 implies the horizontal axis only reaches d = 17. We plot the first few Richardson transforms for g = 3 (left) and g = 5 (right), converging faster towards the expected result (up to small numerical relative errors of about 3% and 4%, respectively).

Analysis of Large-Genus Growth
As explained in section 2, the large-genus expansion is best expressed in terms of the coefficients a d , b d,n , and c d , as in (2.18), which we reproduce in here and where, for this example, one explicitly has G P 2 (d) = (d − 1)(d − 2)/2. A table with these first few coefficients is shown in appendix B.1. Recall that these are just convenient integer numbers which essentially capture the very same information as either GW or GV invariants. Some of these coefficients, b d,n with n close to G P 2 (d) − 1, can be identified in closed form as where the p −3 (k) are given by the generating function (4.16) For larger values of k one can try to extend the above formula, at the cost of identifying similar coefficients to p −3 (k). A conjectural partial formula for b P 2 d,n is where m 0 = 0, n 0 = 0, α 0,n = p −3 (n), β 0,n = 2np −3 (n), (4.18) This is as far as we were able to reach with the data we have available for local P 2 .
Combined/Diagonal Large-Growth in Genus and Degree As discussed earlier in section 2, and illustrated in figure 3, local P 2 has two (different) growths of combined genus and degree. They are associated to the Kähler (A K = 2πt) and conifold (A c ) instanton actions, and they are, respectively, Note that, as one varies t, the large-order growth of the free energies will be dominated by either A K or A c , or a competition between both (see the analysis in [31]). The situation is slightly different with the GW large-order. Here, along any diagonal one will find a factorial growth. However, from a resurgence standpoint, perhaps the most interesting diagonals are the ones which connect back to the resurgent structure of the free energies [29,31]. For any chosen diagonal, this connection will exist every time there is a value of t which realizes that chosen diagonal as one of the above (specific) slices. If such a value of t exists, then the large-order growth of the enumerative invariants will be dominated by either A K or A c and the connection to the free energies is rather clean. If not, one will instead be upon a "mixed" diagonal with both A K and A c vying for dominance. Below we shall focus only upon the leading diagonals. The first leading degree above was explored and justified analytically for the resolved conifold, and the main features which were found in that example remain in the present one. The second leading degree above depends on t (or Q) through two unknown functions, a 0 (Q) and a 1 (Q). At this stage, these functions may only be accessed via numerical computations; and given limited data, with some significant limitations. In the following we shall summarize the resulting factorial growth of the GW invariants, along the leading diagonals of their (g, d)-table, and the relation of this growth with the resurgent structure of the topological-string free energy.

Kähler Leading Degree
In this case, the only difference with respect to the resolved conifold turns out to be a simple multiplying factor, the GV invariant n (1) 0 = 3 of local P 2 , in which case the analog of (3.9) is now The polynomials P h (q) are precisely the same as in (3.14), and the integer q is introduced to make both g and d integer; see the discussion around equation (3.8).
Computational tests on the validity of (4.22) are shown in figures 10 and 11; with figure 10 testing the Kähler instanton action and figure 11 testing the (universal) validity of the polynomials P h (q) for h = 0 through 5. The precise nature of these computational tests is exactly the same as we did earlier for the resolved conifold, and we refer to that discussion for further details.
This formula (4.22), when restricted to the first approximation h = 0, reproduces the prediction from the saddle-point approximation which was explained around equation (2.28) (and   leading up to it). Indeed, for the case of Kähler leading degree, a 0 = −3/t, a 1 = 2/t, a 2 = t, and F The shift in a 0 (Q) is not very reliable, but the slope in a 1 is quite close to the expected value. This asymptotics arises from the a d=1 contribution in the abc-expansion of the GW invariants in (2.18). Since a d=1 = n (1) 0 one will always find the resolved-conifold asymptotics multiplied by this factor.

Conifold Leading Degree
The second dominant degree, d = a 0 (Q) + a 1 (Q) g, is harder to analyze as everything must now be approached numerically; from the computation of a 0 (Q) and a 1 (Q) to the asymptotics.
The numerical fit to 1/a 1 (Q) is shown in the lower-right plot of figure 12. It is obtained from first fitting straight lines d = α g + β for different (fixed) values of t. Then fitting these results against a linear dependence in t we have obtained On what concerns a 0 (Q), its numerical calculation is shown in the lower-left plot of figure 12, but there is no obvious fit to do here (numerically, the dependence of 1/a 0 (Q) does not seem to be linear in t, yielding a poor r 2 = 0.849, and this will become even more evident in following examples). At this moment we cannot provide an analytical interpretation for these numbers, or even guarantee that the fit to a straight line is justified since the interval in t we have considered might be too small to be significant. Nonetheless, we do report the results.
Because we now lack the precision we had along the Kähler leading degree, we cannot provide a systematic exploration of the GW asymptotics with conifold leading degree. We can, however, identify particular values of t for which the exploration becomes simpler. One such point is found when a 1 (Q) = 1, or t ≈ 5.6993 . . .. In this case we explore the growth of N g,g+∆ for some integer ∆ ∈ Z (implicitly associated to a 0 (Q)). The numerical exploration of this diagonal slice in the GW-table yields the result where A 0 ≈ 0.655995 . . ., and α 0 and α 1 are pure numbers which cannot be computed with much precision. The interesting point is that the value of A 0 can be matched to the saddle-point prediction involving the conifold action, where the √ Q comes from including the factor Q d that multiplies N P 2 g,g+∆ in the saddle-point expression. On the other hand, the one-loop coefficient in (4.26) should correspond to but a 2 (Q) is directly related to the second derivative at the saddle point and thus it cannot be computed from first principles.

The Example of Local
Let us next address another (toric) local surface, the non-compact CY threefold known as local P 1 × P 1 , which is the total space of the line bundle O(−2, −2) → P 1 × P 1 . Generically, local P 1 × P 1 has two complex structure moduli, z 1 and z 2 , implying that the mirror map is similarly twofold, Q 1 = e −t 1 = O(z 1 ) and Q 2 = e −t 2 = O(z 2 ). In order to have reasonable large-order data for the resurgence analysis, in what follows we shall restrict to a slice of this variety where z 1 = z 2 .

Free Energies and Gromov-Witten Invariants
Instead of working in the full two-dimensional moduli space, we shall restrict to the (simpler) one-dimensional diagonal slice where the sizes of both P 1 's in the local P 1 × P 1 geometry are set to be equal. One is thus left with a single modulus. The resulting such theory is closely related to a rather well-known gauge theory, called ABJM gauge theory [57] (see also [58]), and we shall use this name in the following to denote this diagonal slice of local P 1 × P 1 . Of course the general local P 1 × P 1 geometry has two Picard-Fuchs operators; annihilating periods. One immediate simplification of the diagonal slice is to reduce this number to just one, (z∂ z ) 4 − 4z 4 (z∂ z ) 3 + 4 (z∂ z ) 2 + z∂ z f (z) = 0. From its solutions, we can identify the mirror map and the genus-zero free energy, −t = log z + 4z + 18z 2 + 400 3 z 3 + · · · = log Q, (4.30) After dealing with the genus-one free energy, F 1 , one can proceed and use the holomorphic anomaly equations to compute higher-genus free energies 8 , from which the GW invariants are eventually read. For example, in the language of modular forms, where E 2 (τ ) is the second Eisenstein series, c = ϑ 4 3 (τ ) and d = ϑ 4 4 (τ ) are powers of theta functions, and the modular parameter τ is a function of z; see [59] for full details. Also, in this language F (0) 1 = log η(τ ) with η(τ ) the Dedekind eta-function. In appendix B.2 we list the first few GW invariants, and figure 13 schematically represents the ones we have computed and will work with.
The instanton actions for ABJM were extensively discussed in [26] and are associated to special points in moduli space: large Kähler structure (z = ∞) yielding A K = 2πt, conifold point (z = z c ) yielding A c , and orbifold point (z = 0) yielding A o . These three actions are actually linearly dependent with integer coefficients. This implies that the ABJM transseries (whose construction is still an open problem for future research) might either involve only two of these actions (selected upon some criteria of relevance), or it might involve all three of them (in which case one would obtain a resonant transseries as in, e.g., [37,27,28] Figure 14: ABJM: The exponent t c in the growth of N g,d is captured from the ratio of two consecutive GW invariants, when the degree is large. We plot that ratio alongside six Richardson extrapolations, which are clearly converging faster towards the expected result (up to a numerical relative error of about 0.3%).

Analysis of Large-Degree Growth
The large degree, fixed genus, growth is completely analogous to that of local P 2 , The value of the Kähler parameter at the conifold point is now t c := t(z = 1/16) = 8K/π = 2.33248723 . . . [59], where K = +∞ n=0 (−1) n (2n + 1) −2 is the Catalan constant. The exponent δ numerically matches to the expected 2g − 2. This g-dependence of the exponents may be tested using the same large-d sequences as for local P 2 , i.e., the combinations (4.12) and (4.13). These numerical results are illustrated in figures 14, 15 and 16.

Analysis of Large-Genus Growth
Again, the strategy is essentially the same as for the example of local P 2 . As before, the GW invariants can be expanded in terms of abc-coefficients according to equation (2.18), but where The first few coefficients seem to answer to the closed-form formula But with the (limited) available data we cannot confirm that these formulae are complete.

Combined/Diagonal Large-Growth in Genus and Degree
The combined growth in genus and degree is similar to the one for local P 2 . The two leading combinations, again arising from Kähler and conifold instanton actions, will allow us to connect the factorial growth of the perturbative free energies with the factorial growth of GW invariants. This is illustrated in figure 17. As already happened before, the growth associated to the Kähler action, d = (2g − 3)/t, is well understood since the example of the resolved conifold, whereas the one associated to the conifold action, d = a 0 (Q) + a 1 (Q) g, can only be probed numerically.

Kähler Leading Degree
This growth is completely determined, at least up to the first exponentially-subleading instanton corrections, by the first GV invariant of ABJM which in this case is n (4.38) where A K = 2πt and the definition of P h (q) is given in (3.14), i.e., they are precisely the same polynomials which have already appeared for resolved conifold and local P 2 . Computational tests on the validity of (4.38) are shown in figures 10 and 11, with the exact same discussion as for resolved conifold and local P 2 . In fact, all the very same comments we made for local P 2 in section 4.1 also apply now. In particular, the numerical check of a 0 = −3/t and a 1 = 2/t is indeed confirmed as This check is shown in the upper plots of figure 18.

Conifold Leading Degree
In this case, the analysis can only be carried out numerically, due to lack of theoretical knowledge of where a 0 (Q) and a 1 (Q) come from. The strategy is essentially the one already started with local P 2 , and we find  Figure 18: ABJM: Numerical calculation of a 0 (Q) and a 1 (Q) associated to the Kähler instanton action (the two upper plots) and to the conifold instanton action (the two lower plots). We are showing test for their inverses whenever the dependence seems linear, although we were not able to confirm this analytically in the case of the conifold action.

The Example of the Local Curve X p
Our next example deals with local curves. The non-compact CY threefolds to be considered are the total spaces of the rank-two holomorphic vector bundles X p O(p − 2) ⊕ O(−p) → P 1 , with p an integer (but due to the invariance p − 2 ↔ −p, one may choose p ∈ N). When p = 1 one finds the resolved conifold, O(−1) ⊕ O(−1) → P 1 (addressed earlier), and when p = 2 one finds the Dijkgraaf-Vafa geometries O(0) ⊕ O(−2) → P 1 relating to hermitian matrix models [60]. By making use of the topological vertex machinery [61] one may actually compute high genus GW invariants for the local curve directly in the A-model. We shall nonetheless begin with some comments pertaining to the B-model free energy, following [42].

Free Energies and Gromov-Witten Invariants
The B-model free energy has the general structure [42] where the coefficients a g,n are of the form with k a positive integer and b g,n (p) a polynomial in p. They are not known in general and have to be fixed with GW invariants up to degree d = 5(g − 1) (we present some of these coefficients in appendix B.3). The modulus w is related to the Kähler parameter t through the mirror map where the critical point is at It is interesting to notice that, unlike the previous geometries, all these formulae are now exact.
Further notice that the double-scaled theory at the critical point is now in the universality class of 2d gravity (the free energy being related to the Painlevé I equation) which is a distinct universality class from the previous c = 1 examples [42].
As mentioned earlier, one can compute the partition function, and thus the free energy, as a sum over integer partitions directly in the A-model using the topological vertex [61]. We shall not get into any details, which may be found in [42], and simply quote the end results. We have computed 9 GW invariants N Xp g,d with fixed p = 3, 4, 5 and in appendix B.3 we list a few such invariants. Figure 19 schematically represents all the invariants we did compute and will work with herein. In the rest of this section we will mostly omit the p-dependence of the GW invariants for shortness, but our results for the different types of growth will always be for general p unless explicitly stated otherwise. We should also point out that we are including an extra sign in our GW invariants [48,49] N This is essentially required in order to produce integer GV invariants. For g ≤ 9 we have enough data to completely fix the coefficients in (4.42), which means the GW invariants can then be computed to arbitrarily high degree. It is also worth mentioning that for g = 0, 1 there are explicit formulae for the GW invariants [42] where we have set f ≡ (p − 1) 2 , and where higher-genus closed-form generalizations are not known. One general thing which is known [42] is that if written for arbitrary p, as N Xp g,d (p), the GW invariants are polynomials of degree 2g + 2d − 2 in p, with rational coefficients,   Figure 19: Maximum degree and genus of the GW invariants we computed for the local curves X p with p = 3, 4, 5. For g ≤ 9 we have all the required data to fix (4.42) and thus can compute GW invariants for any degree.

Analysis of Large-Degree Growth
The analysis of the fixed-genus, large-degree growth of GW invariants in this example is best achieved within the B-model formulation (4.42). The A-model (topological vertex) calculation is only efficient up to about degree d = 40 and, as we shall see, the large-degree convergence of the GW invariants is very slow. Using data up to genus g = 40 (for p = 3, 4) we have fixed the coefficients a g,n up to g = 9 and then can compute N g≤9,d up to very high degree. To see how this works, rewrite (4.42) as n=1 n k a g,n (w c − 1) n−k . (4.51) A Lagrange inversion turns (4.51) into a Q-expansion (related to w via the mirror map (4.44)), from where GW invariants are easily extracted. We will skip these details and refer the reader to Appendix A of [42] for the definition of Lagrange inversion as well as instructive examples. In the end, one finds the explicit result and where the P (a,b) n (z) are Jacobi polynomials. Finding the large-degree behavior of the GW invariants now reduces to the corresponding large-degree behavior of the Jacobi polynomials. We still had to approach this behavior numerically, essentially because the degree d appears in three different places. Furthermore, the aforementioned slow convergence of the GW invariants will be made clear in the following, as the large-degree expansion turns out to be a power-series expansion in √ d for which our standard techniques of Richardson extrapolation are not very useful. However, restricting the study to a grid of perfect squares, i.e., d = 2 , we then get back the very fast convergence via Richardson transforms, from where one can then comfortably find rational numbers out of decimal expansions. Consider the following combination for which we find a large-degree expansion of the form (4.55) Using g ≡ 5(g − 1) for shortness, the first coefficients are , c where we have defined In general, they will have the structure with c (j) j 0 (f ) a polynomial in f of degree j. We refer to appendix B.3 for a few such explicit results at the lowest orders. From these results it immediately follows that the leading term in P d,g,0 reproduces the known behavior of the GW invariants (2.9) (with α = β = 0 and γ = −1/2), and the coefficients can be shown to match the solution of Painlevé I, in the appropriate double-scaling limit. This leading behavior then has corrections, suppressed by powers of d −1/2 .
The GW invariants for arbitrary p (or f ) thus have the following large-degree expansion 10 This result is illustrated and tested in figure 20. On its left plot we consider the ratio N (pred) g,d /N g,d , where the asymptotic prediction in the numerator consists of using the expansion (4.60) up to the subleading correction j = 0 (blue), j = 2 (green), j = 4 (yellow) and j = 6 (red), up to degree d = 100 and fixed genus g = 3. At degree d = 100 the leading-order term is of the right order of magnitude, but it is still off by about ∼ 75%. Upon inclusion of subleading terms, the ratio then , at fixed genus g = 3 and degree d ≤ 100, and up to subleading correction j = 0 (blue), j = 2 (green), j = 4 (yellow) and j = 6 (red). Right: Number of decimal places of agreement between the analytical p = 3 GW invariants and their asymptotic prediction, for fixed genus g = 6 (and using the same color code).
starts approaching 1, faster and faster. On the right plot of figure 20 we show the number of decimal places of agreement between the GW invariants and their large-degree expansion, with the same color coding. This time we move only along perfect squares and work with fixed genus g = 6. At d = 10000 the leading term is still only good enough for the first decimal place, but then the agreement improves significantly once we start adding subleading corrections.

Analysis of Large-Genus Growth
As discussed earlier, the large-genus fixed-degree growth of the GW invariants may be read directly from the abc-formula (2.18). In the case of the local curve, it simply reads This expression is valid for any degree, unlike in previous examples.

Combined/Diagonal Large-Growth in Genus and Degree
The combined "diagonal" growth will turn out to be similar to the previous local-surface examples (and in fact will lead to some sort of large-order universality for topological strings in different double-scaled universality classes). In fact, also in the local-curve case we cannot (analytically) pinpoint the nonperturbative structure of the GW invariants in a general situation where d = a 0 (Q) + a 1 (Q) g. As usual, the one exception happens at large-radius t → +∞, where the contribution of the GW invariants to the free energy is strongly peaked around d = (2g − 3)/t. These Kähler and critical-point peaks are illustrated in figure 21. The nonperturbative structure of the local-curve free-energy was addressed in [42,20,21], where instantons associated to spectral-curve B-cycles were found to control the large-order behavior of the free energy. But, as we shall see, as one moves towards larger and larger values of t this picture changes. Let us first address this question within the free energy itself, before translating to the GW invariants. For the remainder of this section we work with an approximated free energy where d max (g) is the highest degree for which we have computed N Xp g,d (our data is represented in figure 19). We should stress that, with the leading contributions arising from near d = 2g−3 t , we can always be sure that no significant contributions were left unaccounted for, and, in the end, the high accuracy of the large-order predictions will confirm that there are no issues with our approximation (4.63). What we find at large-order resembles the resolved conifold (3.4), in that the leading factorial growth is governed by a Gaussian-like action A = 2πt with a one-instanton sector that truncates at two-loops 11 . In addition, there is a tower of other contributions that amounts to replacing the Kähler modulus t with t n = t + 2πi n, n ∈ Z =0 , or, equivalently, with shifted instanton actions A n = A + 4π 2 i n. The result is where we have also defined θ m := arg A m = arctan 2πn t . It may be instructive to rewrite the above large-order relation in a more standard resurgence language, similar to (2.3). The difference is that now we have infinitely-many instanton actions A m , where expansions around each sector truncate at two loops. In this way we rewrite (4.64) as Tests of the large-order prediction (4.64) are shown in figure 22, where it proves convenient to normalize the local-curve free energy against the Gaussian contribution, i.e., we use F g ≡ F g −F G g . The large-order growth of this normalized free-energy then corresponds to the second line in (4.64). In the left plot we show the normalized free energies for t = 100 (multiplied by an appropriate factor to make all numbers of O(1); the gray dots), alongside the sum in (4.64) up to m = 2 (green), m = 4 (yellow) and m = 6 (red). One can clearly see that the agreement with the data gets better and better by including more terms in the tower of corrections in (4.64). In the right plot we further illustrate this by showing how small the error is for t = 24. We use the normalized free-energy and define the error as 1 − F pred,m g /F (0) g , where F pred,m g just corresponds to taking the second line of (4.64) up to a given maximum (the colors are the same as the ones used on the left). We see that for m = 6, at g = 100, the error is of the order 10 −60 %.
Let us comment on the relation to the large-order behavior found in [20,21]. There should be a value of the Kähler parameter for which there is an effective competition between the Bcycle action found in [20,21] and the Gaussian-like tower described above. Unfortunately, our data does not allow us to directly look at this interplay, for as one moves towards smaller t the contribution to the free energies is no longer dominated just by the invariants close to d = 2g−3 t . There will also be other relevant contributions at higher degree, which we do not have enough data to account for. Nonetheless, do notice that an exchange in large-order dominance should be precisely related to this emergence of relevant contributions beyond the large-radius "peak".
Given the above large-order behavior of the free energy, we may next deduce its consequences towards the "diagonal" growth of GW invariants. After accounting for the appropriate Gaussian correction, it turns out that the GW large-order is of the exact same type as in earlier examples (in this case, n (1) where the P h (q) are precisely the polynomials which were introduced in (3.9), and which also appeared in the similar large-order results for local P 2 and local P 1 × P 1 . Computational tests on the validity of this expression are shown in figures 10 and 11, with their details and discussion being the same as before. One thus finds that even for theories in different (critical) universality classes, there is some sort of universal large-order behavior taking place at large radius 12 . This is also very clear in the plots in figures 10 and 11. Furthermore, in the case of the local curve X p , this large-order behavior turns out to be independent of p (up to a sign).

The Example of Hurwitz Theory
Let us now address a slightly more algebraic example, that of Hurwitz theory. Generically, it addresses branched covers of algebraic curves, but herein we restrict to so-called simple Hurwitz numbers, denoted by H P 1 g,d 1 d , which count the number of degree-d disconnected coverings of P 1 by a genus-g Riemann surface. These numbers have a combinatorial definition in terms of Young tableaux, but-in line with what we have been doing-they also have a string-theoretic origin. Indeed, Hurwitz theory may be thought of as a topological string theory, as it can be obtained by a particular limit of the A-model on the local curve X p [42]. This limit consists in taking p → +∞, t → +∞, g s → 0, (4.68) while the combinations g H ≡ p g s , are held fixed. A number of results can then be straightforwardly obtained by applying this limit to our results in the previous section.

Free Energies and Gromov-Witten Invariants
The Hurwitz free-energy was shown to satisfy a Toda-like equation in [63], However, regarding Hurwitz theory as a limiting local-curve in the sense explained above, implies one may compute the genus-g free energy directly in the B-model as Here, the new variable χ is related to the local-curve B-model modulus w as  Figure 23: Maximum degree and genus of the GW invariants we computed for Hurwitz theory.
These are related to simple Hurwitz numbers via (4.75). For g ≤ 17 we have all the required data to fix (4.71) and thus can compute GW invariants for any degree.
and in the Hurwitz limit the mirror map becomes In (4.71) one needs to use the appropriate limit of the a g,n (p) coefficients from (4.42), defined as a H g,n = lim p→+∞ p 8(g−1)−2n (−1) n a g,n . (4.74) From explicit results in appendix B.3 one can see that only some of the coefficients contribute in the limit. The coefficients a H g,n also turn out to be related to the perturbative free energies of 2d gravity as, under the appropriate double-scaling limit, the difference equation (4.70) reduces to the Painlevé I equation. A large-order analysis of the Hurwitz free energy was performed in [21] finding that large-order effects were, as expected, governed by the p → +∞ limit of the B-cycle instanton-action that controlled the large-order effects of the local curve.
In the A-model formulation, GW invariants are defined as usual. Furthermore, without surprise, they may also be obtained from the limit (4.75) Here we have written the GW invariants in terms of connected, simple Hurwitz numbers H P 1 g,d 1 d • . In practice, we computed the N H g,d along the same lines as we computed the N Xp g,d , i.e., starting from the partition function, then computing the free energy at fixed degree, and finally expanding in powers of g H . We have computed (4.75) up to the totals schematically shown in figure 23 (also see appendix B.4). For g ≤ 17, data up to d = 50 is enough to fix all the coefficients in (4.71), and we may thus compute GW invariants for any degree. This data will not be crucial in the following, since several results just follow from the "finite p" case addressed before. Nonetheless, the ability to generate more data also allows us to make predictions to higher orders. That is no longer the case for degree d ≥ 5, where we now find These results can also be easily derived by computing the free energy directly for low degree.
For the purpose of illustration, let us show how the agreement between the exact H P 1 g,d 1 d • (for d = 6 and g = 100) and its prediction from (4.82) improves, as we include more terms. Note that this is an integer number with 241 digits, but we only display the first 82. One has:

Combined/Diagonal Large-Growth in Genus and Degree
Uncovering the combined growth in genus and degree for Hurwitz theory is a harder problem than in previous examples. The main reason being that in this example the "large-radius peak", where GW invariants near d = (2g − 3)/t give the main contribution to the free energy, no longer seems to exist. We did still numerically find another "critical-point peak", as shown in figure 24, but with the data we have available we were not able to find a linear relation, nor to uncover what the large-order behavior should be, analytically.

The Example of the Compact Quintic
For our final example, we shall consider a compact geometry, in comparison to the non-compact local geometries we have been addressing up to now. This is actually the first example in which mirror symmetry was explicitly worked out and GW invariants systematically computed [4], the quintic CY threefold. The mirror of the quintic is described by the equation where z captures the complex structure of the CY manifold. We will follow the notation in [65].

Free Energies and Gromov-Witten Invariants
As there is a single modulus, there is also a single Picard-Fuchs equation for the periods of this geometry, namely From its solutions, we find the mirror map and the genus-zero free energy, −t = log z + 770z + 717825z 2 + 3225308000 3 z 3 + · · · ≡ log Q, (4.85) Akin to what happened for the local geometries, here the higher-genus free energies can also be described in compact form in terms of a few generators (or propagators). Each of them has a holomorphic expansion around the large-radius point (Q = 0), from which one can read the GW invariants. For example, In this work we use the free energies which were computed in [65], and which are available online 16 . In appendix B.5 we list a sample of the first GW invariants, and figure 25 schematically represents the ones we used in our upcoming analysis. Note that we now have significantly less data than for the earlier non-compact examples, implying we will not have as many results. Another important point is that there are essentially no studies of nonperturbative sectors for the quintic, mainly due to a lack of data to drive the analysis. However, it is not too difficult to check that there is an instanton action associated to the conifold point (located at z = 5 −5 ), A c = 2πT c , which is proportional to the flat coordinate T c vanishing at the conifold point. A test of this instanton action is shown in figure 26. We should also expect a Kähler instanton action, but we do not have enough free energies available to report definite results.

Analysis of Large-Degree Growth
The large-degree growth at fixed genus was already considered in [6], being in the same universality class as local P 2 or local P 1 × P 1 . In this case, the familiar asymptotic formula holds, where now t c := t(z = 5 −5 ) = 7.58995 . . .. One can numerically check the value of this critical exponent, t c , as well as the powers in d 2g−3 and (log d) 2g−2 , using the same asymptotic techniques described for local P 2 and ABJM. We show these numerical results in figures 27 and 28.

Analysis of Large-Genus Growth
The fixed-degree, large-genus expansion of the GW invariants determines the abc-coefficients for the quintic in the same way it did for other geometries. We already showed an example for degree d = 4 earlier, in equation (2.17). In appendix B.5 we present a sample of other such coefficients. Unfortunately, the scarce data available limits the analysis of the b-coefficients, and we have no other large-genus results to present for this example.

Combined/Diagonal Large-Growth in Genus and Degree
As in previous examples, also for the quintic we find two saddle-points which are illustrated in figure 29. They are associated to the Kähler instanton action, located at d = (2g − 3)/t, and to the conifold instanton action, located at d = a 0 (Q) + a 1 (Q) g, just like before. The exponent t c in the growth of N g,d is captured from the ratio of two consecutive GW invariants, when the degree is large. We plot that ratio alongside three Richardson extrapolations, which are clearly converging faster towards the expected result (up to a numerical relative error of about 0.2%).
Ng,d Figure 28: Quintic: On the left we address the exponent 2g − 3, which is the leading large-order term in f d − 2f d 2 . We have data up to d = 325 so that the horizontal axis can only reach d = 17. The plot illustrates the first few Richardson transforms for g = 3, converging faster towards the expected result (up to a numerical relative error of about 3%). On the right we address the exponent 2g − 2 of the logarithm log d, which is the leading term in the sequence (4.13). We plot the first few Richardson transforms for g = 3, converging faster towards the expected result (up to a numerical relative error of about 15%).

Kähler Leading Degree
The scarce amount of available data does not let us check that the leading degree is d = (2g −3)/t in the same way as we did for local P 2 or ABJM. Nevertheless, we can assume that this indeed holds, and then explore the asymptotics of N quint g,d Q d , just as we did in (4.22). We find that the same formula applies, but with the appropriate GV invariant n    Figure 30: Quintic: Numerical calculation of a 0 (Q) and a 1 (Q) associated to the conifold instanton action. We show the inverse of a 1 because the dependence seems linear, although we are not able to confirm this analytically. The plot for a 0 does not seem to lead to any linear dependence.

Conifold Leading Degree
For the conifold leading degree we can do better. Figure 30 shows the dependence of a 0 and a 1 upon the modulus t. The inverse of the slope, a −1 1 , resembles a straight line

A Analysis of the abc-Coefficients
When the GW invariants, N g,d , may be written in terms of GV invariants, n (d) g , then there is a third representation in terms of some other integers we denoted by a d , b d,n , c d . They appear naturally when considering the large-genus expansion of N g,d .
PROPOSITION A.1. The relation between GW and GV invariants, and abc-coefficients is Here f CS 0 = 1, f CS g = (−1) g−1 B 2g 2g(2g−2)! for g ≥ 1, and B 2g are the Bernoulli numbers. G(d) satisfies n (d) Since the GV invariants are integers, so are the abc-coefficients. Proof. We start from the definition of GV invariants, where in the second equality we have noticed that n (m) h = 0 for h > G(m). The coefficients c h,g generate 2 sin x 2 2h−2 = +∞ h=g c h,g x 2g−2 , and they are explicitly given by Next, we organize the terms in (A.5) from more to less important as g grows, using (A.6). In order to do this, we split the h-sum in (A.5) into h = 0, h = 1, and h ≥ 2. For h ≥ 2 we can assume that also g ≥ 2 and manipulate to arrive at (A.1). The steps are straightforward once one knows the goal, and they simply require the exchange of double sums, such as, e.g., ; or relabelings, such as, e.g., m|d f (m) = m|d f (d/m). The end result, after four of these manipulations, is from where one immediately can read the abc-coefficients.
We finish this appendix with a short note on how the GV and GW invariants, and the abc-coefficients, may be laid out in Dirichlet series for each genus g, in contrast to the usual Taylor-series expansion in the form of free energies. If we define the generating functions The last expression can be inverted to define B 2g−2 (s) = g h=1 c h,g GV h (s).