{'fraction_non_alphanumeric': 0.05194197473093121, 'fraction_numerical': 0.023982218062704726, 'mean_word_length': 3.8364922206506367, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Topological Strings, Instantons and Asymptotic Forms of Gopakumar-Vafa Invariants

We calculate the topological string amplitudes of Calabi-Yau toric threefolds corresponding to 4D, N=2, SU(2) gauge theory with N_f=0,1,2,3,4 fundamental hypermultiplets by using the method of the geometric transition and show that they reproduce Nekrasov's formulas for instanton counting. We also determine the asymptotic forms of the Gopakumar-Vafa invariants of the Calabi-Yau threefolds including those at higher genera from instanton amplitudes of the gauge theory.


Introduction 2 Topological String Amplitude
In this section, we calculate the topological string amplitudes of Calabi-Yau toric threefolds that reproduce four-dimensional N = 2 supersymmetric gauge theories with gauge group SU (2) and with N f = 0, 1, 2, 3, 4 fundamental hypermultiplets.
First we briefly review the calculation of the topological string amplitudes of Calabi-Yau threefolds X following [2] when X is the canonical bundle of a smooth toric surface classified in [12]. Recall that a two-dimensional integral polytope (the section of the fan at the height 1) of X has only one interior integral point (0, 0) and this point and each integral point v i (1 ≤ i ≤ k) on the boundary span an interior edge (here we define k to be the number of the interior edges and take v 1 , v 2 , . . . in the clockwise direction). Therefore the corresponding web diagram consists of a polygon with k-edges and external lines attached to it. We take the orientation of edges on the polygon in the clockwise direction, and that of the external lines in the outgoing direction. Then the integer m i of the framing for an interior edge dual to v i is given by Here γ i is the self-intersection number of the P 1 associated to v i and computed from the The derivation of m i is included in appendix. Then we assign a partition Q i to each interior edge and the partition of zero to each external edge. Finally we obtain the topological string amplitude by multiplying all quantities associated to vertices and edges and by summing over all partitions. The brief summary of the rule is as follows: to a trivalent vertex we associate the three-point amplitude C R 1 ,R 2 ,R 3 if the orientation of all the edges are outgoing, (−1) l(R 1 ) C R 1 t ,R 2 ,R 3 if the orientation of one edge with a partition R 1 is incoming, etc, where R 1 , R 2 , R 3 are partitions assigned to three edges attached to the vertex; to an interior edge we associate (−1) ml(R) q − mκ(R) 2 e −l(R)t where m is the integer coming from the framing and R is the partition assigned to the edge and t is the Kähler parameter of the corresponding P 1 .
Here l(R) := i µ i for a partition R = (µ 1 , µ 2 , . . .) and κ(R) := i µ i (µ i − 2i + 1). Thus the topological string amplitude for X is written as Here we have defined γ k+1 := γ 1 , Q k+1 := Q 1 . This result was derived in [13,14,4]. The relation between the topological string amplitude Z and the Gromov-Witten invariants is that log Z| q=e √ −1gs is the generating function of the Gromov-Witten invariants where g s is the genus expansion parameter. This statement has been proved for the canonical bundle of fano toric surfaces [14] 1 .
Next we derive more compact formulas for the toric Calabi-Yau threefolds that corresponding to 4D, N = 2 SU(2) gauge theories with N f = 0, 1, 2, 3, 4 fundamental hypermultiplets. The Calabi-Yau threefolds are the canonical bundles of the Hirzebruch surfaces F 0 , F 1 , or F 2 blown up at N f points. There exist 3,2,3,3,2 such Calabi-Yau threefolds for N f = 0, 1, 2, 3, 4. 2 The fans and the web diagrams for these threefolds are shown in figures 1 and 3. We take as a basis of the second homology where C B (resp. C F ) is the base P 1 (resp. fiber P 1 ) and C E i is an exceptional curve. The intersections for 1 ≤ i, j ≤ N f . The values of b are 1 or 2 and will be listed later. Here t B , t F , t E i (1 ≤ i ≤ 3) denote the Kähler parameters of the base P 1 , the fiber P 1 , the i-th (−1)-curve and Now we compute the topological string amplitudes by using the same strategy as [7,8,4].
Let us take N f = 2 cases shown in figure 1 as examples. We first cut the polygon in the web diagram into two upper and lower parts (as shown by dotted line) and compute the amplitudes separately. Then we glue the two amplitudes together along the vertical edges 1 P 2 and (2)(3)(5)(7) in figures 1, 3. 2 Although there are 16 smooth toric surfaces classified in [12], three among them (1,10, 16 in figure 1 in [12]) do not correspond to the four-dimensional gauge theories. For N f = 0, 1, 2, 4 see figure 3.
x 3 x 4 Figure 2: The web diagrams corresponding to H (k) to obtain the whole topological string amplitude. They are written as follows: Here R 1 (resp. R 2 ) is a partition assigned to the right (resp. left) vertical edge. H (k) are amplitudes corresponding to the web diagrams in figure 2 and defined by Using the expression for C R 1 ,R 2 ,R 3 written in terms of the skew-Schur functions and the identities (see appendix), H (k) The second (resp. third ) line should be set to 1 for k = 1 (resp. for k ≤ 2). Here is the conjugate partition of R 1 (resp. R 2 ) and (i, j) ∈ R means that there is a box in the place of i-th row and j-th column in the partition R regarded as a Young diagram. We have used an identity in appendix to obtain the expression (14). The final form of the topological string amplitudes for the N f = 2 cases become The numbers b are 1, 2, 1 and (m 1 , m 2 ) are (0, 0), (−1, 1), (0, 0) for (7)(8) (9). In Z ≥1 we have used the identity The generating function of the Gromov-Witten invariants is obtained from the topological string amplitude by taking the logarithm and substituting e √ −1gs into q: N g,d B ,d F ,d 1 ,d 2 denotes the genus zero, 0-pointed Gromov-Witten invariant for an integral One can read off the Gromov-Witten invariants with d B = 0 from Z 0 , because only log Z 0 gives the terms with degree zero in q B : (8)(9), 0 for (7).
Hence the nonzero Gopakumar-Vafa invariants for a second homology class are as follows (we slightly change the notation and write the Gopakumar-Vafa invariant as n g α for a second homology class α): and The invariants at We will interpret these results from the viewpoint of relation to the Seiberg-Witten prepotential of SU(2) gauge theory in the next section.
The Gopakumar-Vafa invariants with d B ≥ 1 are given by log Z ≥1 . We remark two properties. The one is that the Gopakumar-Vafa invariant is nonzero only when One could read this fact from the expression (15) as follows. The summand is the polynomial . Therefore the degree in q i −1 is always equal or smaller than the degree in q B , and this fact persists when we take the logarithm. Thus the Gromov-Witten invariants are zero unless the condition (20) is satisfied, and so are the Gopakumar-Vafa invariants. The other property is that the Gopakumar-Vafa invariants are symmetric with respect to q 1 , q 2 .
This follows from the invariance of Z ≥1 under the exchange of q 1 and q 2 .
Finally, we summarize the topological string amplitude for all the cases corresponding to SU(2) gauge theory with N f = 0, 1, 2, 3, 4 hypermultiplets shown in figures 1, 3. (21) b is the self-intersection of the base P 1 and m 1 (m 2 ) is the integer of the framing of the right (left) vertical edge: Note that the properties of the Gopakumar-Vafa invariants mentioned in N f = 2 cases hold in all N f cases: the nonzero Gopakumar-Vafa invariants for a second homology class in table 1 and one can easily see that in all cases n 0 and they are symmetric with respect to d 1 , . . . , d N f . Note also that Z ≥1 's of (7) and (9) (resp. (14) and (15)) are the same, which means that the Gopakumar-Vafa invariants in cases (7) and (9) (resp. (14) and (15)) with d B ≥ 1 are completely the same.

Nekrasov's Formula
In this section we show that the topological string amplitude Z gives the one-loop corrections in the prepotential and Nekrasov's formula [9] for the 4D, N = 2 SU(2) gauge theory with N f fundamental hypermultiplets at a certain limit. The argument in this section closely follows that in [7,8,4].
First let us identify the parameters in the two sides: Then the limit we should take is the limit β → 0. Here a = a 1 = −a 2 is the vacuum expectation value of the complex scalar field in a gauge multiplets, m i (1 ≤ i ≤ N f )'s are mass parameters of the fundamental hypermultiplets. g s has been introduced as the genus expansion parameter.
Next, let us show that the t B independent part Z 0 in the topological string amplitude gives the perturbative one-loop correction terms. We again take N f = 2 cases (7)(8)(9) as examples. Note that (8)(9), 0 for (7).
Each trilogarithm corresponds to one logarithmic term in the Seiberg-Witten prepotential.
We can see this correspondence in the following way. If we take the third derivative in a, the right-hand side of (23) becomes In passing to the second line, we have used the formula ∞ k=0 x k = (1 − x) −1 . This is exactly the third derivative of the one-loop correction in the SU(2) Seiberg-Witten prepotential with N f fundamental hypermultiplets. Note that the last term in (23) does not cause any problem because such term depends only on mass parameters, not on a.
Now we show that the logarithm of t B -dependent part Z ≥1 in (21) gives the instanton correction terms in the gauge theory. More precisely we show that Z ≥1 becomes Nekrasov's formula in the limit β → 0. We introduce the following function for the sake of convenience: whereã, , β are defined by q = e −β and x = e −βã . Then the following equations hold: Therefore we can rewrite Z ≥1 as follows.
We have rewritten R 2 in the summation as R 2 t and a 1,2 = a 1 − a 2 = 2a = −a 2,1 . Next we look into the limit β → 0. The first line of (28) becomes

The second line becomes
The third line and the fourth become Thus if we take the limit β → 0 with the topological string amplitude (21) becomes Nekrasov's formula for instanton counting [9]: Here the meaning ofq is thatq = Λ 4−N f for N f = 1, 2, 3 andq = e

Asymptotic Form of Gopakumar-Vafa Invariants
In this section we derive the asymptotic forms of the Gopakumar-Vafa invariants of the Calabi-Yau toric threefolds that correspond to the SU(2) gauge theory with N f = 0, 1, 2, 3, 4 hypermultiplets. We derive first the asymptotic forms of the Gromov-Witten invariants and then those of the Gopakumar-Vafa invariants.
Let us state the result: for a second homology class where n C k = n! k!(n−k)! is the binomial coefficient. F g,k is defined by  This asymptotic form is valid in the region Note that the asymptotic form of the Gopakumar-Vafa invariants consists of two factors.
The one is the asymptotic form of N f = 0 case which is a monomial in d F with the prefactor given by the instanton amplitude of the gauge theory. This factor is common to all the N f cases and the genus dependence appears only through this part. The other factor is the binomial part which represents the dependence on d i 's (1 ≤ i ≤ N f ).
For concreteness, we compute log Z If one expand this further by , one could obtain the coefficients 2 4k+2g−2 F g,k in table 2.
In the rest of this section we explain the derivation of the asymptotic form. In the previous section, we showed that the topological string amplitudes reproduce Nekrasov's partition functions at the field theory limit (30). By taking the logarithm of the equation (31), the left-hand side is written as On the other hand, the logarithm of the right-hand side takes the following form: Here B g,k is a polynomial in N f variables and it has the degree k as a polynomial in each variable. Moreover, the highest degree term is where F g,k is the same for all N f = 0, 1, 2, 3, 4. This is clear from the expression of Nekrasov's partition function in eq. (31). Therefore it is given by (33). We can also understand this from the well-known fact that a gauge theory with fundamental hypermultiplets reproduces the pure Yang-Mills theory by decoupling hypermultiplets.
Let us adopt the following ansatz: By substituting the ansatz into (36) and identifying the parameters as (30) , we obtain The last factors have appeared from Then comparing (37)(38) and (40) as a series in and Λ, we obtain the condition which We have actually used only (38) in the comparison and we will discuss this point shortly.
In passing to the second line, we have identified a with t F /2β. It is clear that the powers of β cancel out and this relation is independent of the limit β → 0. Then we replace the summation over d F with the integration and regard the left-hand side of (41) as the Laplace as a function in d F to a function in t F . Therefore by performing the inverse Laplace transform on the right-hand side, we obtain the following: This asymptotic form is valid only in the region In the derivation, we have used only the term with the highest degree in m 1 , . . . , m N f in (37).
As it turned out, this is suffice, because the contributions from the terms with lower degrees (in m i ) are smaller: it would be monomials in d F with degree smaller than 4d B + 2g − 3.
Thus such terms can be neglected since we consider the region where d F is large.
Finally let us consider the asymptotic form of the Gopakumar-Vafa invariants. It is the same as the Gromov-Witten invariants, because the contribution from the lower degree and lower genus Gromov-Witten invariants can be neglected. However, the region where the asymptotic form is valid becomes more restricted. It is Here we have added the factor (g + 1) in the right-hand side of (43) because the number of lower degree/genus terms is d B (g + 1) and this number must be small enough compared to d F .

Example
In this section we explicitly compute the Gromov-Witten invariants and compare them with the asymptotic forms derived in the previous section. We take as an example the case (7)(9) which correspond to the SU(2) gauge theory with N f = 2 fundamental hypermultiplets. The ratios between the Gopakumar-Vafa invariants and the asymptotic forms are shown in figure 4 for d B = 1, 2, g = 0, 1, 2.
By expanding these as series in q F , q 1 −1 , q 2 −1 we obtain the Gopakumar-Vafa invariants listed in table 3.

Conclusion
In this article, we have computed the topological string amplitudes of the canonical bundles of toric surfaces which are the Hirzebruch surfaces blown up at N f = 0, 1, 2, 3, 4 points and showed that in a certain limit, they reproduce the Nekrasov's formulas for 4D N = 2 SU(2) gauge theories with N f fundamental hypermultiplets.
We have also derived the asymptotic form of the Gopakumar-Vafa invariants at all genera from the instanton amplitudes of the gauge theory. From the result in [15] and ours, we expect that the asymptotic form of the Gopakumar-Vafa invariants in the SU(n + 1) cases are given as the product of the two factors: the one is the asymptotic form of N f = 0 case derived in [15] and the other is just the same binomial part as the SU(2) case.

Acknowledgement
The author would like to thank H. Kanno for valuable discussions. She is also grateful to M. Naka and K. Sakai for collaboration in previous works.

A Formulas
We list some formulas in this section.
We use letters R, R i , R ′ , Q etc. for partitions. As mentioned before, l(R) := i µ i for a partition R = (µ 1 , µ 2 , . . .), κ(R) : is the conjugate partition of R 1 (resp. R 2 ) and (i, j) ∈ R means that there is a box in the place of i-th row and j-th column in the partition R regarded as a Young diagram.
The three point amplitude is [2,4,5,6] In the first line the summation is over pairs of partitions Q 1 , Q 3 . The coefficient N R 1 ,R 3 Q 1 ,Q 2 is defined as follows: N R ′′′ R,R ′′ is the tensor product coefficient. In the second line, s R , s R/Q are the Schur function and the skew Schur function [16]: where x = (x 1 , . . . , x l(R) ). The tensor product coefficient N R ′′′ R,R ′′ can be computed from the formula or by using the Littlewood-Richardson rule.
The formulas essential to the calculation of H (k) Note that the summations in the right-hand sides are infinite but those in the left-hand sides are finite.
The proof of the last expression (57) is the same as the proof of theorem 1.11 in [17]. The proofs of other formulas (53)-(56) can be found in [6].

B Calculation of the integer m i
In this section, we present the calculation of (1).
where v in and v out are two-component vectors described The second term is minus the volume of the triangle spanned by v i , v i−1 , which is −1 because the surface is smooth. To compute the first term, note that the following equation holds by (2): (2)(3)(4) are Hirzebruch surfaces and correspond to N f = 0, (5) (6) to N f = 1, (7)(8) (9) to N f = 2 and (14) (15) to N f = 4. The orientation of the interior edges are taken in the clockwise direction. N f = 2 cases are separately shown in figure 1.   Figure 4: The ratio between the Gopakumar-Vafa invariants n g d B ,d F ,d 1 ,d 2 and the asymptotic form in the case of (7)(9) for (d B , g) = (1, 0), (2, 0), (2, 1), (2,2).