Topological String on elliptic CY 3-folds and the ring of Jacobi forms

We give evidence that the all genus amplitudes of topological string theory on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degree. The denominators of these forms have a simple universal form with the property that the poles of the meromorphic form lie only at torsion points. The modular parameter corresponds to the fibre class while the role of the string coupling is played by the elliptic parameter. This leads to very strong all genus results on these geometries, which are checked against results from curve counting. The structure can be viewed as an indication that an N=2 analog of the reciprocal of the Igusa cusp form exists that might govern the topological string theory on these Calabi-Yau manifolds completely.

C Reducing the ambiguity with the involution symmetry 53

Introduction and Summary
Topological strings on non-compact Calabi-Yau geometries are solvable and have a very interesting structure with a wealth of connections to gauge theories, integrable models, large N-dualities, Chern-Simons theories, supersymmetric localisation and matrix models.
In fact topological string techniques had considerable influence on these areas and solved important problems, see [17] for a recent example. As a benchmark problem in topological string theory with a similar wealth of connections to quantum gravity problems, remains the solution of the topological string on compact Calabi-Yau manifolds. In this paper we report on some progress in this matter. According to [6] the genus g topological string amplitudes F g (S ij , S j , S; z) are inhomogeneous polynomials of weighted degree 3g − 3 in the anholomorphic propagators S ij , S j , S, which have respective weights (1,2,3), with rational functions of the moduli z as coefficients. The holomorphic anomaly determines all of F g , but the weight zero piece f g , whose determination is the key conceptual obstacle to solving the topological string on compact Calabi-Yau spaces. A careful analysis of the conifold gap condition [42], regularity at the orbifold and Castelnuovo's criterium for the vanishing of higher genus curves reveals that the f g can be determined up genus 51 1 for the quintic [42]. While for non-compact models the same conditions lead to a complete solution of the closed topological string on these geometries [27] 2 , on compact Calabi-Yau manifolds one needs additional boundary condition.
One purpose of this article is to investigate the additional symmetries and boundary conditions that are specific for elliptic fibered Calabi-Yau threefolds 3 . Beside the motivation to solve the topological string in general the elliptically fibered cases are very interesting in their own right as the topological string amplitudes predict terms in the low energy effective actions of F-theory and calculate indices of 6d conformal theories in particular the E-string and its generalizations with gauge theories [26].
One possible constraint could come from the SL(2, Z) modularity of the amplitudes in the Kähler parameter of the elliptic fibre that has been argued in [52] [3] based on a monodromy analysis in one global case [11] and calculations for more general elliptic threefolds [50]. It also extends observations for local elliptic surfaces inside Calabi-Yau spaces [53] and [32,33,29].
We find that the constraint from the fibre modularity is equivalent to the constraint that comes from a Z 2 involution symmetry I, that acts on the moduli space of elliptic fibrations as an R symmetry, i.e. it is not realized as a symplectic transformation like the monodromies of the model. This symmetry I acts also on the F g by (−1) (g−1) and on the propagators in a way described in section 3.4.3. Knowing these actions we can restrict the possible constants in the rational ambiguity f g to roughly one fourth. Using this information, the conifold gap and the regularity at the other points in the moduli space, especially the orbifold point we can solve the topological string for all classes to genus 9 on an elliptic fibration over P 2 , with one section. This is already the strongest available result for compact multiparameter Calabi-Yau manifolds and needs only very mild results on the vanishing of the BPS numbers.
Our main result is however as follows. Consider the topological string partition function Z = exp(F ) = exp( ∞ g=0 λ 2g−2 F g ) expanded in the base degree(s). E.g. for the P 2 case one has 4 where τ is the complexified Kähler class of the fiber λ is string coupling and Q = e 2πit B with t d B the suitable defined complexified Kähler class of the base, see (4.1). Then Z d B is a quotient of even weak Jacobi forms of the following form 5 , (1.2) where η(τ ) is the Dedekind function and ϕ d B (τ, z) is an even weak Jacobi form of index , however we can argue that the Castelnuovo bounds discussed in section 5.2 bring down the growth to d 4 B . Combined with the restriction from the constraints from the conifold gap and the regularity of the F g in the interior of the moduli space and with the independence of these conditions assumed 6 , this allows to solve the model up to genus g = 189 for arbitrary base and fibre classes, or up to d B ≤ 20 for arbitrary genus and fibre classes.
We will show in section 4.4 that the Jacobi forms fulfill a heat kernel type form of the holomorphic anomaly equation for Z d B that is equivalent to a limit of the wave function equation proposed by Witten [69] for the topological string partition function in interpreting the holomorphic anomaly equation of [6] as consequence of a background independent quantization of H 3 (M, C). This heat kernel type form of the holomorphic anomaly equation further summarizes all the holomorphic anomaly equations for the fibre modulus, which were proven in [52], i.e. the ansatz (1.2) identically satisfies the holomorphic anomaly equation for the fibre modulus.
We argue that the special form of the denominator in our ansatz naturally is crucial to satisfy the Castelnuovo bounds. The scaling of the z argument in the denominator can interpreted as coming from multiple string windings of the base similar as in the elliptic genus of E-strings [28,10,49] or more general strings of 6d SCFTs [26] with gauge symmetries. On the other hand, the paper [49] constructs a 2d quiver gauge theory for E-strings and [26] for theD 4 string that can in principle compute the elliptic genus of any finite number of E-strings with the techniques of [5] 7 .
The prediction from the B-model calculations and from (1.1,1.2) are sucessfully checked in section 5.2 against results from enumerative geometry. Since the moduli space of the BPS states tends to becomes smoother for fixed degree and high g we can make an infinite number of nontrivial checks on the form of Z d B and the determination of its coefficients following the methods proposed in [48].
A remarkable mathematical characterization of the form (1.2) is is that, again due to the form of the denominator, the poles of Z d B lie only at the points z ∈ Qτ + Q, i.e. at torsion points of the elliptic curve. As such they have a modified theta expansion that involves mock modular forms [71]. These structures played rôle in the understanding of Wall-crossing for N = 4 BPS states [14] based on the reciprocal of the Igusa cusp and related meromophic forms.
The general structure (1.1,1.2) can be viewed as an extension of the only compact case 3-fold case K3 × T 2 for which all BPS states associated 8 to moduli space of curves are known [48]. Due to the SU (2) holonomy the Type II compactifications has N = 4 supersymmetry. The prediction is based on the formula for the elliptic genus of symmetric products of K3 [16] and M-theory/Type IIB duality. It has been recently proven in [63]. In this case Z is related to the reciprocal of the Igusa cusp form of a genus two curve with weight 10 which has a Fourier expansion in p = exp(2πiσ) as 9 Z(Ω) = 1 Φ 10 (Ω) = m=−1 ψ −10,m (τ, z)p m , with Ω = τ z z σ . (1.3) Here Ω parametrizes the Siegel upper half space H 2 and Φ 2,m = ∆(τ )ψ −10,m are meromophic 7 It would be very interesting to see if such 2d quiver theory duals exist also for the compact threefold case, i.e. 6d theories that do include gravity. 8 In order to make this interpretation one has to lift zero modes that come from the T 2 and make the N = 2 invariants trivially zero. 9 We follow the notation of [14].
Of course a key question for a full N = 2 extension of the discussed structure is whether the formulas (1.1,1.2) come from an underlying modular object as (1.3,1.5). One obvious difference is the quadratic versus linear growth of the index in the weak Jacobi forms. Nevertheless we like to discuss in section 3.4.2 the formulation of projective special Kähler manifolds in N = 2 supergravity that might give a hint for the construction of such an object. We hope to report on progress on that question in [37].
In this article we focus for simplicity on elliptic fibrations with only I 1 singular fibers in the Kodaira classification, i.e. no gauge symmetries, but eventually flavor symmetries, in the F theory interpretation. We note however that for the gauge theory case, i.e. elliptic fibrations with higher Kodaira fibre singularities, the work [26] gives already the crucial hint for the generalization of the numerator in (1.2). In this case, we will find additional Jacobi theta functions in the numerator, whose different elliptic arguments correspond to the additional Kähler classes of the resolution. This Jacobi theta function will introduce additional zeros in the numerator, which signal the gauge symmetry enhancements, and should be fixed by this structure. The formulas we find also suggest a possible refinement, which we shortly indicate in (5.1.2).

The elliptic Calabi-Yau manifolds and their mirrors
In this section we describe the toric construction of elliptically fibered Calabi-Yau spaces with only I 1 Kodaira fibers. Mirror symmetry is manifest in this formalism. We collect the key data of our main example and describe shortly the general structure of the genus zero sector.
∆ is reflexive if ∆ * is a lattice polyhedron. Note (∆ * ) * = ∆ so if ∆ is reflexive (∆, ∆ * ) is a reflexive pair and each polyhedron contains the origin as its unique inner point. The Calabi-Yau n-fold M n is given by the canonical hypersurface in the toric ambient space P ∆ with coordinate ring y j , while the mirror Calabi-Yau n-fold W n is given by the constraint P * = 0 in P ∆ * with coordinate ring x i , where P * = 0 is defined analogously to (2.1) with ∆ and ∆ * exchanged. The notation∆ (∆ * ) means the polyhedra ∆ (∆ * ) with the integer points interior to codimension one faces omitted.
To give the ambient space P ∆ a fibration structure 10 such that the embedded Calabi-Yau n-fold defined as hypersurface has a fibration by a Calabi-Yau m-fold, we combine a base polyhedron ∆ B * and a reflexive fibre polyhedron ∆ F * into an n + 1 dimensional polyhedron ∆ * as follows If ∆ * F and ∆ * B are reflexive then (∆, ∆ * ), given by the complex hull of the indicated points, is a reflexive pair. Reflexivity of ∆ * F is required by the CY condition on the fibre. For ∆ * B n−m it is not a necessary condition, see [41] for more details on this construction. We defined s ij = ν F i , ν F * j + 1 ∈ N and scaled ∆ B → s ij ∆ B . Here we indicated the dimensions of some polyhedra by subscripts; elliptic fibrations correspond to m = 1.
For n = 3 and m = 1 we get many examples by choosing any of the 16 reflexive polyhedra in 2d as ∆ * F and ∆ * B respectively and specifying in addition ν F * i ∈ ∆ * F as well as twisting parameters [9], which not indicated in (2.2).

2.2
The X 18 (1, 1, 1, 6, 9) 3-fold, an elliptic fibration over P 2 Our main example is the smooth elliptic fibration over P 2 . This case 11 is a two parameter model discussed in [30] and in greater detail in [11] and illustrates the general construction described above.
Pick for the base P 2 , whose toric polyhedron is the convex hull of the points ∆ * B = conv({(1, 0), (0, 1), (−1, −1)}), for the fibre polynomial ∆ * F = conv({(1, 0), (0, 1), (−2, −3)}) and for ν * F 3 = (−2, −3). Then ν F 3 = (−1, −1) and s 33 = 6. We list the points which give rise to the coordinate ring of P = P ∆ /(Z 18 × Z 6 ), all points ν * i ∈∆ * and the two vectors of linear relations among them, which correspond to the Mori cone of P ∆ * , as well the toric divisors D 3) The where H and L are the divisors dual to the curves defined by the Mori vectors l (E) and l (B) and the Kähler classes J E and J B . The intersection with the second Chern class The mirror manifold is given by the zero locus in the space P. z := x 4 , x := x 5 = and y := x 6 are the conventional names of variables in the Weierstrass form of the elliptic fibre. In P there are toric C * actions on the coordinates x i , i = 1, . . . , 6, which can be used to eliminate all a i , but the two complex structure variables (a, b) of W . This is because two C * actions are divided out from the coordinate ring of P. One can introduce manifestly C * invariant combinations the a i as complex structure variables of W , namely In the case at hand z 1 := z E = to the base class. Using the C * actions on the period integrals Π(z) = γ 3 Ω with (a = a 0 ) Ω = γ aµ P * , (2.9) 12 In the notation of [30] these intersections are encoded in the ring R = 9J 3 E + 3J 2 E JB + JEJ 2 B . H, L are the notations for the divisors used in [11].

Integral symplectic basis and genus zero topological string amplitudes
The PF operators determine the 3-point couplings. Its solutions in a special basis determine the metric on the moduli space, which is a Kähler manifold G i = ∂ t i ∂ t K , whose Kähler potential K is given by where (k = h 1,1 (W ) + 1) is the symplectic pairing on H 3 (W, Z) and Π(z) = (X I = A I Ω, F I = B I Ω) T is the period vector w.r.t. to the corresponding symplectic basis (A I , B I ), I = 0, . . . , k − 1 of H 3 (W, Z). The two structures are related by special geometry, which implies the existence of a prepotential, the genus zero amplitude F (0) , with (2.15) Here (2.16) and X I are homogeneous coordinates and t i (z) are inhomogeneous coordinates which serve as a mirror map. In particular the complexified volumes of the curves in the Mori cone t i are the flat coordinates at large radius near z i = 0, ∀i = 1, . . . , h 11 (M ) , where the third equal sign in (2.15) is valid, X 0 is the unique holomorphic period normalized to ijk are the classical intersection numbers. There is a freedom in choosing the n ij , but because of the odd intersections and the requirement of integer monodromy around z i = 0 , they cannot be set to zero as they must be half integral. Following [11] or the topological description the quadratic terms in F (0) in the appendix of [42] we may take them to n 11 = 9 2 and n 12 = 3 2 for the X 18 (1,1,1,6,9) model. Like for all toric hypersurfaces in this example the Picard-Fuchs equations are of the generalized hyperelliptic type and all logarithmic solutions can be derived from with X 0 (z) = ω 0 (z, 0), by taking derivatives with respect to ρ α , in particular see the Appendix in [31] for the explicit structure of the higher derivatives.

Involution symmetry and BCOV formalism
In this section we mainly discuss the involution symmetry, its realization on the topological string amplitudes and its consequences. In the course of the exposition we also review the relevant aspects of the BCOV formalism. Many of the actual calculations are quite technical and relegated to the appendices. We comment on the relation of the involution symmetry to the monodromies in subsection 3.2 and on the local limit in subsection 3.2.1. The section 3.4.1 illustrates the idea of the BCOV ring and contains many modular properties needed in section 4. Subsection 3.4.2 is more speculative and tries to give a perspective on a possible global modular object related to the all genus amplitude.

The involution symmetry
Fibrations with fibre polytope ∆ * F have an involution symmetry I acting on the moduli space, which is independent of the chosen base B. The action of I on the X 18 (1,1,1,6,9) moduli space that we describe below appeared in [11]. I acts non-trivial on the higher genus amplitudes F g (S ij , S i , S) and the anholomorphic propagators S ij , S j , S. The F g are inhomogeneous polynomials of degree 3g −3 with weights 1, 2, 3 and rational functions in the moduli. This follows from the Feynman rules for the genus g amplitudes in [6]. The rational function f g that does not multiply any propagator is not determined by the holomorphic anomaly equation, that otherwise fix the F g recursively in g. The function f g is restricted by the pole behavior of the F g at the boundary divisors of the moduli space to contain only finite parameters for each g. This is called the holomorphic ambiguity and its determination is a main problem to solve the topological string on compact Calabi-Yau manifolds. If we know the action of I on F g and the propagators we can further restrict f g . In fact this requirement cuts down the possible parameters in f g to roughly one fourth. Moreover we find that the symmetry I is equivalent to the constraints that the fibre modularity imposes on f g .
To see the involution symmetry explicitly define the monomial m = ( i x i ) 6 , where x i are the coordinates of the base. Then where g(z, x, a B ) is a polynomial compatible with the scaling, which does not contain m. Now requiring that √ 3a and c 3 = √ 3a 3 and acts on the parameters as I : (a, b) → (ia, b + a 6 ) . This involution operation acts on the (3, 0) form Ω given in (2.9) by because P * as well as the measure µ are invariant. Since Ω defines vacuum line bundle L and the higher genus amplitudes F g transforms as section F g ∈ L 2g−2 we conclude that the involution symmetry maps F (g) toF (g) with In the the z i coordinates the involution acts as The PF operators (2.10) are invariant up to some trivial factors under the involution. On the other hand, the involution exchanges the two discriminants up to some factors as (3.7)

Monodromy group versus involution symmetry
The involution symmetry multiplies the periods w.r.t. Ω with i, so it is in particular not a symplectic transformation (in fact M T I ηM I = −η) and cannot be related to an actual monodromy action.
The monodromy group is most quickly described as follows. There are Neveu-Schwarz B-field shifts that leave the instanton action invariant t i → t i + 1, i = 1, .., h 1,1 (M ). Their monodromy is completely fixed by (2.15) after specifying the topological data as in subsections 2.2,2.3 for the main example. We call these monodromies and the corresponding 6 × 6 matrices T 1 (= T E ) and T 2 (= T B ).
Further there is a cycle ν 1 , which corresponds to the B 3 ∼ S 3 base of the Strominger-Yau-Zaslow T 3 → B 3 fibration, that vanishes at the conifold ∆ 1 = 0. In the X 6 (1, 2, 3) elliptic fibrations such as the X 18 (1,1,1,6,9) one has a second conifold discriminant ∆ 2 = 0, where ∆ 1 and are exchanged, up to irrelevant factors, by the involution symmetry as discussed in the last section. Let us call the corresponding vanishing cycle ν 2 and use the integral symplectic basis of cycles that corresponds to the period vector (2.15). Using some basic analytic continuation one calculates the vanishing cycles in this basis as 13 The Lefschetz monodromy theorem for 3-folds states that the monodromy around a conifold divisor, where by definition an S 3 sphere ν vanishes, on each cycle γ ∈ H 3 (M, Z) is given by the symplectic reflection S ν (γ) = γ − γ, ν ν , (3.9) where , denotes the symplectic pairing. By this formula and (3.8) we can calculate the monodromy in the basis (2.15). We call the corresponding monodromies and 6 × 6 matrices C 1 and C 2 respectively. The T i and C i generate the monodromy group of the X 18 (1,1,1,6,9) model in fact redundantly, as can be seen by the Van Kampen relations analyzed for this case in great detail in [11]. In particular one has that an order 18 element A is given by 14 and the two conifold monodromies C 1 and C 2 are conjugated to each other by the order 6 element A 3 11) 13 In fact ν1 corresponds always to F0, while ν2 corresponds to a i FB,i, where the ai defined above (4.1) .
14 We use the same notation A for this element as in [11]. The other notions ate related by C1 = T , C2 = B, T1 = T∞ and T2 = D∞. which exchanges ν 1 ↔ ν 2 and is explicitly One can conclude that the monodromy group is generated by A and C 1 [11]. Note that T 1 and A 3 acts on t 1 as [11] These operations generate an Sl(2, Z) action on the elliptic fibre parameter, which we call therefore often τ in the following. As we mentioned already invariance under the involution symmetry I is equivalent to the fibre modularity. This could be guessed from the fact that it is A 3 that really imposes the nontrivial part of fibre modularity -the shift symmetry is present for all Kähler moduli -and conjugates the conifold monodromies just as I exchanges the two conifold discriminants. The equivalence of the restrictions imposed the by invariance under I and the fibre modularity will be strictly proven in appendix D.

The local limit
As we have seen there is an SL(2, Z) action on the elliptic fibre modulus. This symmetry governs the amplitudes order by order in the exponential Q B = exp(2πit B ) of the flat coordinate representing the base curve in P 2 . There is an other action of the congruent subgroup Γ 0 (3) = a b c d ∈ SL(2, Z) c = 0 mod 3 of SL(2, Z) on the base modulus.
To see it we must make the volume of the fibre large, lim(t E ) → i∞, which corresponds to z E ∼ q = exp(2πit E ) ∼ z E ∼ 0. This is known as the local limit. In the A-model language it focuses on the O(−3) → P 2 geometry by decompactifying the elliptic fibre of X 18 (1,1,1,6,9). The periods of the local geometry are given by integrals of a meromorphic differential λ over cycles on an elliptic curve C B that is the the mirror geometry. These pe- fulfill local rigid special geometry [36]. The occurrence of the 1 indicates that the CY (3,0) form Ω becomes in the local limit the meromorphic one form λ which has a non-vanishing residuum at a pole λ on C B . From (2.10) one sees that the periods in the local limit Π loc are governed by a specialization of the second differential operator L = θ 3 2 + z 2 2 i=0 (3θ 2 + i), which is another way to see the constant solution.
However we want to understand precisely, which periods Π = Γ Ω on the compact CY 3-fold become the local periods in the limit or equivalently how the Γ 0 (3) action is embedded in the action of Sp (6, Z). To see this notice that we have to make a linear change in the ∂ t i F (0) , i = 1, 2 part of the basis Π (2.15) to keep with (3.14) in the limit z E = 0 three fine periods X 0 = 1, t B ,F 2 . If we now conjugate the monodromies that we found in the last section by the corresponding element C = we obtain the following monodromies Here we have printed the elements d c a b of the Γ 0 (3) subgroup, see [1], in bold face and underlined the shifts due to the non-vanishing residua of λ. A consequence of this symmetry is the fact that in the strict local limit the amplitudes , can be expressed in the indicated generators of the ring of almost holomorphic ring of forms of Γ 0 (3) and has been completely solved [27]. Here u is the complex modulus of the local geometry [27], which is identified with the b parameter in (2.6). Of course this base modularity that holds in the strict large fibre limit, and acts not directly on the periods of the 3-fold geometry, extends much less trivially to the whole two parameter family than the fibre modularity, that we discuss further in section 4.

Involution symmetry at genus one
First we consider the genus one amplitude F (1) which fulfills the holomorphic anomaly equation [6] where j|φī|k =Cīk = C j k ī η j j η kk and j|φ i |k = C ijk are the 3 point functions discussed above, with the relation C jk ı = e 2K G j G kkCīk . The solution has the well known form The topological data for the X 18 (1, 1, 1, 6, 9) model are given in section 2.2. The leading asymptotic of F (1) anholomorphic near large volume limit is The values of the second Chern-Classes (2.5) determine the constants s 1 = 114, s 2 = 48.
In the holomorphic limit, the determinant of the Kahler metric transforms as det(G) = det( ∂z i ∂x j ) det(G). A simple calculation shows that the invariance of the genus one amplitude F (1) under the involution transformation (3.6) impose the constrain on the constants s 1 = −30 + 3s 2 , which is consistent with the evaluation of the second Chern class on J i .

Involution symmetry at higher genus
The higher genus amplitudes are defined recursively from the holomorphic anomaly equations [6] The F (g) can be integrated using an anholomorphic potential S forC jk ı , whose existence is consequence of special geometry or more generally the tt * geometry [6]. One has The F (g) were obtained in [6] either using the special geometry commutator nl  and partial integration or a Feynman graph expansion of a master functional, which fulfills a heat equation type of equation. The most vantage point of view is to find directly a solution for this master integral, which we achieve at least iteratively in the base modulus.

The propagators and rigid special geometry
It was shown in [70] as a consequence of special geometry that the covariant derivatives D i in (3.19) close on anholomorphic generators, that are closely related to the propagators, up to rational functions of z. We call the corresponding ring generated by these anholomorphic generators over the rational functions in z the BCOV ring.
This principal structure of the BCOV ring can be understood by the analogy to the ring of almost holomorphic forms k M k (Γ 0 ) = C[Ê 2 , E 4 , E 6 ] of Γ 0 = SL(2, Z) (or congruent subgroups thereof) that is graded by the weight k. Let us denote the modular transformation where the last equal sign holds straightforwardly only for k > 2. Here σ k (n) is the sum of the k-th power of the positive divisors of n and ζ(k) = r≥0 The Bernoulli numbers B k k = 1 are defined by the generating function If the sum in (3.21) converges k > 2 it is obvious that E k (τ γ ) = (cτ +d) k E k (τ ) transforms as a weight k form under modular transformations. For k = 2 the second equal sign in (3.21) can be viewed as a regularization of the sum on the right [8]. In this case the modular transformation is broken to Since 1 Im(τγ ) = (cτ +d) 2 Im(τ ) − 2ic(cτ + d) = |cτ +d| 2 Im(τ ) one can define an almost holomorphic object which transforms as a modular form of weight 2.
In the solution of the holomorphic anomaly equationÊ 2 (τ ) plays the rôle of the propagator and the Maass derivative D τ :M k →M k+2 on weight k forms, with d τ = 1 2πi d dτ , plays the role of the covariant derivative D i . The an-holomorphic version of the Ramanujan identities (3.27) show that it closes on k M k (Γ 0 ). Note that by the isomorphy ( [46] Prop. 1) of the ring k M k (Γ 0 ) and the one quasi modular forms kM k (Γ 0 ) = C[E 2 , E 4 , E 6 ] one can take the holomorphic limit in (3.27) by replacingÊ 2 → E 2 and D τ → d τ without losing information about the ring structure.
The relation between the BCOV ring of the X 18 (1, 1, 1, 6, 9) Calabi-Yau manifold and the ring of almost holomorphic forms is more then an analogy. In the limit of large fibre 3.2.1 the special Kähler structure of supergravity reduces to local rigid special geometry and the BCOV ring of X 18 (1,1,1,6,9) In the limit of large base the relevant ring becomes k M k (Γ 0 ) and the structure of the holomorphic anomaly equation can be be even more neatly combined into the ring of weak Jacobi forms, which include all powers of the topological string coupling as we will discuss in section 4.

Projective special Kähler manifolds
The form (1.5) suggest that the all genus amplitude in K3 × T 2 is related to a modular form associated to a degenerate Riemann surface of genus 3. In this section we recall some structure of projective special Kähler manifolds, see [20] for a modern review, that relates for N=2 compactifications of type II string the complex structure moduli of Calabi-Yau 3 folds to a family of Riemann surfaces of genus h 21 + 1.
Of course we could first address the simpler question, how the objects of rigid special Kähler geometry described in the last section extend to the moduli space of Riemann surfaces of higher genus. However this is done in [54], with the result that a holomorphic analog of E 2 does not exist, but a meromorphic one does.
In the formalism of projective special Kähler manifolds the formal description of the formulae for the propagators and the closing of the covariant derivatives on the propagators, i.e the analogs of the relations (3.27,B.2), as well as the recursive integration w.r.t. to the anholomorphic generators all simplify relative to the formalism [70] and resemble closely the formalism in the rigid case [24].
However the price one has to pay is very complicated anholomorphic embedding of the complex moduli space of the Calabi-Yau manifold including the string coupling λ, that is related to the projective scaling, into the Siegel upper half plane H h 21 +1 of Riemann surfaces of genus h 21 + 1, which is given by [15] see [59] for a concise review 15 . Here upper case letter refer to the projective coordinates of the complex moduli space given by a set of A-cycle periods X I (z), I = 0, . . . , h 21 (W ) in the integral symplectic basis discussed in section 2.3. F IJ (z) = ∂ 2 F ∂X I ∂X J with F (z) = (X 0 ) 2 F (0) the prepotential in homogeneous coordinates. The positivity, or rather negativity in the supergravity conventions Im(N IJ ) < 0, is discussed in [68]. It is physically enforced by the condition that kinetic terms in vector multiplets and the gauge kinetic terms L = − 1 4 g −2 IJ F I nm F J nm − θ IJ 32π 2 F IF J + . . . of the graviphoton F 0 nm and the vector bosons F i nm , i = 1, . . . , h 21 in vector multiplets with g −2 IJ = i 4 (N IJ −N IJ ) and θ IJ = 2π 2 (N IJ +N IJ ) must be positive definite.
Note that N IJ transforms under real or integer symplectic transformations, w.r.t. to (2.14)  This suggest that the special monodromy family of Riemann surface of genus 3 and a better understanding of the map (3.28) might be essential to understand whether the formulae (1.1,1.2) come from an underlying modular object as in the N = 4 case.

Involution symmetry on the propagators
For the concrete calculation we follow [70] and work in the in-homogenous coordinates and use the BCOV propagators with a shift by derivatives of the Kähler potential as the generators [2]. We denote the propagators after this shift (B.1) as S ij , S i , S. The definition of the propagators involves a choice of rational functions in the moduli z [6,50], which is not unique. It has to be accompanied by a consistent choice of rational functions of z in the closing relations (B.2) [2] that generalize (3.27). The derivation these rational functions for non-hypergeometric, but more interesting Apery like Picard-Fuchs equations that arise in non-abelian gauged linear σ models requires a minimal choice that uses the properties of differential equations [44]. For the X 18 (1, 1, 1, 6, 9) model, we summarize two canonical choices and the derivation of the action under the involution symmetry in the appendix B.
Let us now state the action of the involution symmetry on the propagators. Suppose f (z 1 , z 2 ) is a rational function of z 1 , z 2 , we denote with the tilde symbolf = f (x 1 , x 2 ), the transformed function by replacing the z i 's in the arguments with x i 's in the involution (3.6). Thef can be considered as a function of either x i 's or z i 's using the transformation (3.6). For example under the involution, the three point functions (2.12) transform as a tensor except for a minus sign due to (3.5) We find in B that the action of the involution on the propagators has the following form The precise form of the rational functions f i , f 0 depends on the choice of the rational functions in the choice of the propagator as well as in (B.2). Fixing this choice as in (B.3) [2] we find the solution

Involution symmetry on the higher genus amplitudes
The topological string amplitudes transform under the involution symmetry (3.6) by replacing the propagators with the transformed propagators (3.32 and the complex structure coordinates z i 's with x i 's. We would like to understand how the transformed amplitude is related to the original amplitude and how many constraints the symmetry implies. At higher genus g ≥ 2, the BCOV holomorphic anomaly equation [6] can be written as partial derivatives with respect to the an-holomorphic propagators. Assuming the algebraic independence of K i 's, we can write the partial derivatives in terms of lower genus amplitudes. Here the K i appear as in the covariant derivative of higher genus amplitude D i F (g) = ∂ i F (g) − (2g − 2)K i F (g) . We need to be a little careful for the genus one case, as the K i already appear at the ordinary derivative log ∆ a ), (3.34) where we have computed the derivative of Kahler metric in terms of Christoffel connections and use the first equation in (B.2). The equations for partial derivative are where the c g is defined as (3.36) We have also used the notation ∂ to denote i.e. on the right hand in (3.35), we use the formula (3.34) for ∂ i F (1) omitting the −( χ 24 −1)K i term.
Since the propagators are symmetric S ij = S ji , we can choose to use only the S ij with i ≤ j. In the case of i = j, the right hand side of first equation in (3.35) need to be multiplied by an extra factor of 2 to take account of the double contribution.
Let us make some remarks about (3.5). When we combine the amplitude with its involution transformation as in (3.5), we find the dependence on the propagators S ij , S i , S cancels with our proposal for the transformed propagators (3.32) and the shifts (3.33), which is a consistency check of the formulae (3.32), (3.33).
To make an ansatz for the holomorphic ambiguity f (g) at genus g, one considers a rational function of z 1 , z 2 with a pole of (∆ 1 ∆ 2 ) 2g−2 . When we combine the correct holomorphic ambiguity with its transformationf (g) + (−1) g f (g) as in (3.5), we find the degree of the pole at discriminant ∆ 1 ∆ 2 is reduced for g > 2, and the pole completely cancels for the genus g = 2 case. Meanwhile, a new pole at ∆ 3 = 1 − 432z 1 appears due to the involution transformation (3.6) 16 .
The use of the involution symmetry greatly reduce the number of unknown constants in the holomorphic ambiguity. In the naive ansatz in [3], the holomorphic ambiguity has the form p is a polynomial of degree 7(g − 1) in z 1 and degree 5(g − 1) in z 2 . So naively there are (7g − 6)(5g − 4) unknown constants. For example, at genus 2 and 3, the naive ansatz has 48 and 165 unknown constants. We find the equation (3.5) imposes 36 and 125 conditions on the naive ansatz, so we are left with only 12 and 40 unknown constants to be fixed by boundary conditions. We present an detailed analysis on the reduction of unknowns in appendix C. Together with the conifold gap discussed in Appendix E, this allows to fix the ambiguity to genus 9.

Fiber modularity
In this section we reformulate the holomorphic anomaly equation expressing the fibre modularity [52] in a convenient way that allows us to summarize the structure in terms of even weak Jacobi forms. We exemplify the geometry setting with the elliptic fibration over F 1 , which exhibits geometric limits that yields the BPS numbers the K3 fibre, the BPS numbers on an embedded 1 2 K3 and maybe most remarkably by blowing down the 1 2 K3, the compact elliptic fibration over P 2 . The even weak Jacobi forms are hence a unified mathematical language to address heterotic type duality [60], the six dimensional superconformal theories [22][53] and compact elliptic Calabi-Yau spaces.

The modular anomaly equation
In the simplest case the base B n−1 of the elliptic fibration is Fano (or almost Fano) 17 , the degenerate elliptic fibers are only of Kodaira type I 1 and the elliptic fibre has only one holomorphic section and no rational sections. In the language of F -theory the complex structure is so generic that the 12-2n dimensional theory is completely higgsed.
The cohomology and the C.T.C Wall data of M 3 is then fixed by the cohomology of the base and the fibre type [52]. One key feature of topological string amplitudes that comes from the described geometrical setting is the Γ F ∈ SL(2, Z) modularity of the amplitudes. Here Γ F is the modular group acting projectively on the fibre modulus. In particular for one holomorphic section Γ F is the full 18 SL(2, Z). In this case the b 2 (M 3 ) Kähler classes of M 3 split in b 2 (B n−1 ) and one class, which corresponds to the elliptic fibre. Lett i be the complexified volumes of the curve classes [C] i , i = 1, . . . , b 2 (B 2 ) in the base and τ the complexified volume of the elliptic fibre. One defines with q = exp(2πiτ ) a parameter, which 16 The genus two and three holomorphic ambiguity formulae in [3] miss a constant 511 144 and − 2105053 1959552 . 17 To construct already many concrete examples we may assume that Bn−1 is a toric variety P ∆ * B n−1 , i.e. specified by an n − 1 dimensional reflexive polyhedron ∆ * B n−1 . 18 It gets restricted to congruence subgroup ΓF ∈ SL(2, Z) if the fibration has more rational sections or multi sections. becomes exponentially small, when the volume Im(τ ) of the elliptic fibre becomes large and similarlyQ β ≡ . Let a i are the intersections of the curves [C] i with the canonical class K B of the base. E.g. for B 2 a P 2 one has a 1 = 3, while for B 2 the Hirzebruch surfaces F n one has (a 1 , a 2 ) = (2, 2 − n) etc, see [52] for the other toric bases. In order to make the SL(2, Z) duality in the fibre direction more manifest it is convenient to redefine exp(2πit α β α ). We can write an expansion of the disconnected topological string amplitudes also known as the free energy in the large volume limit The classical terms are at genus zero a cubic polynomial c 3 (t) in the Kähler parameter t that can be read from (2.16) without the constant term. Similarly c 1 (t) can be read from (3.18). Constant terms are proportional to the Euler number and are included via the 0 in the sum over H 2 (B n , Z) in (4.4). This a formal expansion in the string coupling λ, but each genus g amplitude F (g) (τ, Q) has a finite radius of convergence in the (q, Q) parameters, determined by the discriminant of the Picard-Fuchs equations. We decompose the genus g amplitudes in fibre q and base moduli Q as There are two closely related properties partially characterizing the F (g) β (q).
• The F (g) β (τ ) are, up to a potential sign transformation, meromorphic modular forms in q of weight 2g − 2. More precisely the F The first factor Z osc β (τ ) the partition function of bosons, i.e. given by Z osc β = 1 η n(β) . Here i = 1, . . . , h 2 (B n−1 ). Since n(β)/12 can be an odd integer (4.5) can take an additional sign under modular transformations.
The second factor P (g) β (τ ) is a polynomial in the generators of almost holomorphic modular forms with respect to the subgroup Γ F of SL(2, Z). In particular for our main examples with the X 6 (1, 2, 3) elliptic fibre type Γ F = SL(2, Z) and the generators are given by almost holomorphic weight 2 Eisenstein seriesÊ 2 and the holomorphic Eisenstein series E 4 and E 6 . The w weight and hence the degree of the polynomial P (g) β (q) grow linearly with g and β as (4.7) • They satisfy a holomorphic anomaly equation, which for elliptic fibrations one section and only I 1 fibers is completely determined by topological data of the base [52] , which depends only on the holomorphic generators of Γ F , i.e. in particular not onÊ 2 , and is called the holomorphic or modular ambiguity.
• We define a similar split expansion in fibre and base degree for the partition function We identify now λ = z with the elliptic parameter of the weak Jacobi forms 20 and assign modular weight −1 to it so that Z and Z β (τ, λ) have modular weight zero. From (4.8) follows Basic properties of the weak Jacobi forms are summarized in the next section, see equation (4.26), yield then that the index of As a corollary to the holomorphic anomaly equation (4.8) we have the following. If the base has itself a fibration B n−2 · F = 1 and F 2 = 0, we can decompactify the fibre and consider the topological string on the local geometry of a rational fibration over the B n−2 .

The Hirzebruch surface F 1 as base
In this section we present the base F α = F 1 as an explicit example and discuss the relation between the holomorphic anomaly equation in various dimensions. Our base has two divisor classes a minus −α curve which we call S and the fibre class F . The canonical class of the base is K B = 2S + (2 + α)F . One has the intersections S 2 = −α, F S = 1 and F 2 = 0. We denote β = mF + nS and a short form F m,n so that our anomaly formula (4.8) reads This formula has interesting specialization, e.g. for the α = 1 case we get • In the n = 0 case corresponds to the corollary mentioned above. The two dimensional surface is a specially polarized K3. In this case we get a very simple recursion in the genus ∂F 12) but no recursion in the base degree. An all genus result can be obtained for all m using the heterotic one loop result.
• The m = 0 case corresponds likewise to the corollary. The rational elliptic surface is the 1 2 K3. The solution has been discussed first in [53] and in the context of the (refined) holomorphic anomaly equation in [41,28,10]. A quiver description has been found in [49].
• m = n corresponds to the blow down of the Hirzebruch surface F 1 to P 2 . For this geometry we get hence The solution will be discussed in section 4.6.

The ring of weak Jacobi forms
Jacobi forms ϕ : H×C → C depend on a modular parameter τ ∈ H and an elliptic parameter z ∈ C. They transform under the modular group [18] and under quasi periodicity in the elliptic parameter as Here k ∈ Z is called the weight and m ∈ Z >0 is called the index of the Jacobi form.
The Jacobi forms have a Fourier expansion φ(τ, z) = n,r c(n, r)q n y r , where q = e 2πiτ , y = e 2πiz (4.17) Because of the translation symmetry one has c(n, r) =: C(4nm − r 2 , r), which depends on r only modulo 2m. For a holomorphic Jacobi form c(n, r) = 0 unless 4mn ≥ r 2 , for cusp forms c(n, r) = 0 unless 4mn > r 2 , while for weak Jacobi forms one has only the condition c(n, r) = 0 unless n ≥ 0. According to [18] 21 , a weak Jacobi form of given index m and even modular weight k is freely generated over the ring of modular forms of level one, i.e. polynomials in R = E 4 (τ ), We summarize the weights and index of some important forms in the table 1. The first Table 1: are generators of the ring of weak Jacobi forms with even weights, ϕ d B (τ, z) captures the all genus amplitudes for the X 18 (1, 1, 1, 6, 9) CY 3-fold and Z d B (τ, z) is the ratio of weak Jacobi forms.
generators are defined in (3.21). Our conventions for the elliptic theta function and weak Jacobi forms A and B are the followings 22 The weak Jacobi form of index φ −2,1 has simple product form using the Jacobi triple product for θ 1 and x = 2 sin λ ) .
(4.21) 21 A review of the theory can be found in [14]. We try to follow the notation used there. 22 Our conventions for the θ functions associated to the spin structure on the torus are (4.19) and the Jacobi theta functions θ1 = iΘ . and the weight zero index one form is one half of the elliptic genus of the K3 χ(K3; q, y) = 2φ 0,1 (τ, z) = 2y + 20 + 2 y + 20

Weak Jacobi Forms and holomorphic anomaly equation
There is a simple connection between the ring of quasi modular forms, which have ring isomorphism to the ring of almost holomorphic forms [46], which are crucial in the solution of the holomorphic anomaly equations in rank one Seiberg Witten theories [38] and local Calabi-Yau spaces [1].
In this section we will show by a very simple argument that the master holomorphic anomaly equation (4.10) for fibre modularity, which reads for the main example X 18 (1, 1, 1, 6, 9) Because of (4.15) and (3.24) given a weak Jacobi form ϕ k,m (τ, z) one can always define modular form of weight k as follows It follows that the weak Jacobi forms ϕ k,m (τ, z) have a Taylor expansion in z with coefficients that are quasi-modular forms as [18,14] (4.25) Here the = d τ = d 2πidτ and ξ ν ∈M k+2ν (Γ 0 ), i.e. the coefficients of can be expressed as polynomials of Eisenstein series E 2 (τ ), E 4 (τ ) and E 6 (τ ). Moreover from (4.24) one has Prop. 1 of [46] 23 implies the claim (4.23). Moreover by (3.25) we can write this as holomorphic anomaly equation We note as examples of (4.25) the first coefficients in the expansion of φ −2,1 (τ, z) and φ 0,1 (τ, z) are Therefore φ −2,1 (τ, z) and φ 0,1 (τ, z) can be thought of as quasi-modular forms and we can see that they satisfy the modular anomaly equation Let us finish the section with a comparison of (4.23,4.27) with Witten's wave equation for the the topological string partition function that reads [69] ∂ If we apply this equation to Z defined in (4.9) with (t )ā =τ and Q β = e 2πid B t B , we get in the large base because of the special from of the intersection matrix of elliptically fibered Calabi-Yau 3 folds only derivatives in the base direction t B for t b and t c . Identifying λ with z we see already that the index will grow quadratically in d B . A more detailed analysis as in [52] shows also the shift by K B so that the large base limit of (4.30) becomes equivalent to all equations (4.10).

Exact formulae for base degree zero
For base degree d B = 0, since the BPS numbers n (g) d E ,0 = 0 for genus g ≥ 2, we get from (5.1) the formula where χ is the Euler number. This formula holds not just for the cases of elliptic fibrations with only I 1 Kodaira fibers [52], whose toric description is reviewed in section 2, but also for the case discussed in [26] with higher Kodaira fibre and non-abelian gauge symmetries. We recognize in the multi-cover formula (4.31) the well-known formula for Eisenstein series (4.32) So we see for χ = −540 that P For the d B = 0 genus one amplitude for our case, we can compute the base degree zero amplitude P (1) 0 from the formula (3.17) and modularity formulas proven in [37]. Taking z 2 → 0 of theses, we compute the determinant of Kähler metric as det . We can hence write the genus one amplitude in the z 2 → 0 limit as A careful calculation show that For the d B = 0 genus the zero amplitude P (0) 0 has formally weight −2. We therefore consider the three point coupling which is the well defined observable after fixing the the conformal Killing symmetries on the world-sheet. The classical contribution is a cubic polynomial of the flat coordinates (2.16), which encode the classical triple intersection numbers. Similar to genus one formula (4.34), we should shift the base flat coordinate according to (4.1) and find which confirms the effective modular weight of −2 since each derivative increases modular weight by 2.

Exact formulae for higher base degrees
Our main claim is that the all genus partition function for the topological string elliptic fibration over P 2 realized e.g. as degree 18 hypersurface in the weighted projective space is given by . Scaling z in ϕ k,m (τ, z) ∈ J k,m by s corresponds to a Hecke-like operator U s : J * k,m → J * k,s 2 m . The weights and indices of the relevant weak Jacobi forms are summarized in Table 4 This determines for base degree 1 all genus BPS invariants by (4.36,4.9,4.5) and the multicovering formula (5.1). Up to g = 6 we and d E = 6 we list them in the table (2). We notice that they match all the numbers that have been obtained to genus 8 using the direct integration of the holomorphic anomaly condition, the involution symmetry and the conifold gap condition. A detailed discussion on the perfect matching with the accessible enumerative invariants can be found in section 5.  Note that(4.36) implies Hilbert-Scheme like infinite product formulae for the generating functions. E.g. for d B = 1 one gets where B 2k are the Bernoulli numbers. This infinite product formula alone eliminates the in the holomorphic ambiguity (E.8) and allows already to topological string amplitudes up to g ≤ 18 for this model.
For the d B = 2 three of 17 coefficients can be already fixed by demanding that there is no pole z −4 = λ −4 in P 2 (τ, z). Note that this pole has to be canceled by the (Z 1 ) 2 contribution in P 2 (τ, z). This explains the first term in (4.42). The vanishing bound which we will demonstrate in Section 5.2.1 is a Castelnuovo-like criterium, namely that fixes eleven other constants and implies all vanishing on the right from the edge. The actual nonzero values of the BPS numbers on the generic part of the edge for d E ≥ 5 are then calculated geometrically and yield no further constraints. In order to fix the remaining three constants one needs the information of any three nonzero numbers in the same row or the same column way from the numbers (4.41). Notice that three numbers n 2d E −5 d E ,2 , d E = 5 + i, i = 0, 1, 2 do yield only one constraint. Random patterns of nonzero numbers away (4.41) will lead in general to independent equations. The result e.g. fixed from three genus zero numbers gives 0  6  2700 -574560 74810520 -49933059660 7772494870800  31128163315047072  1  0  15  -8574  2126358  521856996  1122213103092  879831736511916  2  0  0  -36  20826  -5904756  -47646003780  -80065270602672  3  0  0  0  66  -45729  627574428  3776946955338  4 0 Together with the data up to genus eight tabulated in appendix A, this allows to fix ϕ 4 with 20 and ϕ 5 with 2 nontrivial checks respectively. The tables of base degree d B = 4 and d B = 5 to genus 27 and 41 respectively can be found likewise in the appendix. We appended the expression for ϕ d , d = 1, .., 5 in a for algebraic programs readable form to this TeX file. One can make further checks on the results bases on considerations in section (5.2.1). For examples we see that for each d B there is a critical value in the (g, d E ) plane so that the BPS states on the line in the (g, d E ) takes the value which can be confirmed as well as many other predictions by direct curve counting in section 5.2.1. These formulas (4.38, 4.42) are valid for all genera, so we can consider their implications for Gopakumar-Vafa (GV) invariants at large genus. We use the Mathematica program to compute the GV invariants to genus 8, and the results are listed in the tables in the Appendix. A well-known feature of the GV invariants from algebraic geometric arguments [48] is that, for a given 2nd-homology class, in our case a pair of fiber and base degrees  (17)  . To save space we only give the first and the last two significant digits and the number of omitted digits in brackets.
vanishes for sufficiently large genus g. The largest genus with non-vanishing GV invariant is called the top genus for the given degrees (d E , d B ). For example, from the GV tables in the Appendix A we can see that the top genus for the degrees (d E , 1) is always d E , while the top genus for (d E , 2) is d E for the cases of d E ≤ 4, and is 6, 8 for d E = 5, 6 respectively. The situation is summarized in Figure 1 and the bounds are verified from algebraic geometry in section 5.2.
We consider whether such vanishing conditions help to fix the unknown coefficients in the ansatz of the weak Jacobi form. It turns out for the first case of d B = 1, a general ansatz (4.36) ensures the vanishings of n g (d E ,1) = 0 for g > d E , therefore this does not help to fix the two coefficients in the numerator in the formula (4.38). To understand this structure, let us analyze the formula for φ −2,1 (τ, z) in (4.20), which can be also written in the infinite product form. For comparing with GV invariants, we define the variable x = (2 sin( z 2 )) 2 , then we find So we see that excluding the total factor of x, in the power series expansion of the infinite product, the power of q is always no less than that of x. Using the series expansion (1 + x) −2 = ∞ n=0 (−1) n (n + 1)x n , we find the inverse can be written as where f n−1 (x) is a polynomial of x of degree n − 1. Topological string free energy in the d B = 1 case has no multi-cover contributions from lower degrees, and according to the general ansatz (4.36) is simply φ −2,1 (τ, z) −1 multiplying a modular form of weight 16 and indicates the value at which the boundary slope becomes generic and the numbers of BPS states on the line are given by (4.46). the factor of ( q η(q) 24 ) ) 3 2 , which only increase the q power in the series expansion. Here the factor q 3 2 comes from the shift of base Kahler parameter such that the partition function has a nice modular form. From the well-known GV expansion of the topological free energy, the coefficient of q d E x g−1 is exactly the GV invariant n g (d E ,1) . Therefore we see from the expansion (4.48) that indeed the GV invariant n g (d E ,1) vanishes for g > d E . Furthermore, using the formula (4.38) we can easily calculate the top genus numbers n g (g,1) = 3(−1) g (g+1). This agrees with the numbers in the tables of GV invariants and can be also derived from algebraic geometric arguments in section 5.2.1. Now we consider the general cases of higher base degrees. Similar to the case of φ −2,1 (τ, z) in (4.47), we can also expand φ 0,1 (τ, z) as power series of x, q where f n (x) symbolizes a degree n polynomial of x. Overall the difference of the powers of x and q in both the series φ −2,1 (τ, z) and φ 0,1 (τ, z) is always no larger than 1. So we can consider the sub-family of ansatz with only φ −2,1 (τ, z) in the denominator, namely where f 18d B +2k (E 4 , E 6 ) symbolizes a modular form of weight 18d B + 2k, which is a polynomials of E 4 and E 6 with many unknown coefficients, so that the total modular weight and index are the same as the more general ansatz (4.36) with the product d B k=1 φ −2,1 (τ, kz) in the denominator. From the power series of φ −2,1 (τ, z) and φ 0,1 (τ, z) in x, q and including the factor of q 3d B 2 from the shift of base Kahler parameter, we deduce that the sub-family of ansatz (4.50) contribute only to the GV invariants n g We can choose a particular solution from the general ansatz (4.36), so that the GV invariants vanish at sufficiently large genus, then the addition of extra ansatz of the form (4.50) does not affect the conditions. So we conclude that the sub-family of ansatz (4.50) can not be fixed by vanishing GV conditions.
The reverse is also true, namely, any remaining ambiguities which can not be fixed by vanishing GV conditions are necessarily of the form (4.50). To see this, we note that (2 sin( kz 2 )) 2 can be written as a polynomial of x ≡ (2 sin( z 2 )) 2 of degree k, more explicitly where f k−1 (x) is a degree k − 1 polynomial, related to the Chebyshev polynomials. For example, for k = 2, we have (2 sin z) 2 = x(4 − x). The constant term f k−1 (0) in the polynomial is always k 2 since we have (2 sin( kz 2 )) 2 x ∼ k 2 in the limit x ∼ z ∼ 0. When we expand 1 φ −2,1 (τ,kz) for k > 1 as power series of x, q, we see that the power of x has no upper bound for a given power of q, unlike the case of k = 1. So if the factor f k−1 (x) is not cancelled in the denominator, the ansatz would contribute to the GV invariants n g (d E ,d B ) of arbitrarily large genus g for the given degrees (d E , d B ), and we can always fix them with vanishing GV invariants, even without the precise knowledge of the top genus. Furthermore, we require that the power of x in the generating function of the GV invariants is always no less than −1, since the lowest genus is zero. Those ansatz that contribute to x −n with n > 1 can be always fixed by this requirement. Overall we have at most one factor x in the denominator for those ansatz that are not affected by these considerations. Thus all ansatz except for those of the form (4.50) can be fixed in this way. This is checked by actual calculations. One can easily count that the sub-family of ansatz (4.50) has 2, 3, 10, 26 unknown coefficients for base degrees d B = 1, 2, 3, 4 respectively. After we impose the conditions that the generating function of GV invariants start from genus zero and vanish at sufficiently large genus for a given fiber degree, these are the indeed the remaining numbers of unfixed coefficients in the more general ansatz (4.36) with the denominator d B k=1 φ −2,1 (τ, kz). We can also see the scaling of the top genus for a given base degree d B and for large fiber degree d E . We can expand more explicitly where f k−1 (x) is the polynomial in (4.51) and g n−1 is a degree n − 1 polynomial. There are also multi-cover contributions from lower degrees. For example, we can consider k-cover contributions from the base degree one formula. The denominator can be expanded as where the polynomials f k−1 , g n−1 are the same as in (4.52). We see that the first terms, which would have unbounded powers of x in series expansion, are the same. At the end, the polynomial f k−1 (x) must be cancelled in the denominator, so that the resulting GV invariants have a top genus. In this way we see that the minimal requirements for the denominator of the ansatz at base degree d B is indeed d B k=1 φ −2,1 (τ, kz), otherwise there will be non-vanishing GV invariants at arbitrarily large genus for fixed degrees from the multi-cover contributions. Since the factor f k−1 (x) in the denominator of the first term in (4.52) is eventually canceled, this term contributes a finite polynomial of x in the generating function of GV invariants and does not affect the leading scaling behavior of the top genus for large fiber degree. At the base degree d B , the leading contribution to the top genus comes from the second term in (4.52) for k = d B with large n, and scales like g top ∼ d B d E . However it is more tricky to determine the exact formula in this way for the cases of d B > 1. We find that the exact top genus numbers are not universal for the general ansatz (4.36), and the top genus formula d B d E + 1 2 (−3d 2 B + d B + 2) for large d E derived from algebraic arguments only appears for the the particular solution in topological string theory.
This approach can be combined with the B-model holomorphic anomaly approach to compute higher genus topological string amplitudes. Using the involution symmetry and the boundary conditions at the conifold point, we find that the exact formula at base degree d B can provide sufficient boundary data to fix the B-model formula at genus 9(d B +1), valid for all base and fiber degrees, see the Appendixes for more details. On the other hand, in order to fix the exact formula at base degree d B , we need to fix the ansatz (4.50) with some non-vanishing GV invariants. In particular, consider the term in 2 + 1, we can repeat this procedure to fix the exact formula with increasing base degrees. If no other obstacle arises, we foresee the eventual termination of this recursive process only at d B = 20. In this way we can in principle determine the exact formula up to base degree d B = 20 (for all genera and fiber degrees), and the topological string free energy up to genus 189 (for all base and fiber degrees). In practice we provide here the solution of the topological strings up to d B = 5 for all genera and genus g = 8 for all degrees.

BPS invariants
After a technical recapitulating of the the physical definition of the BPS invariants in Section 5.1, the main part of the section is devoted to the direct geometrical calculation of the unrefined BPS invariants or Gopakumar-Vafa invariants for the X 18 (1, 1, 1, 6, 9) Calabi-Yau 3-fold in section 5.2.

Physical definition of the BPS invariants
In this section we recall the definition of the integer BPS invariants from F(λ, t) or Z(λ, t).
In view of the similarity of the all genus expansions discussed in section 4.6 and the product expansion of (1.4) based on the Borcherds lifting of automorphic forms of O(2, n, Z), see [7] and [56] the formulae (5.5) and (5.11) should give further hints how to characterize Z. We comment on possible refinements in 5.1.2 in analogy to the E-string.

Unrefined BPS invariants
The unrefined BPS states multiplicities n g β ∈ Z are defined by their contribution to a Schwinger-Loop integral in 5d N = 2 supergravity theory [23] 24 t α β α ) and y m 2 − y − m 2 = 2i sin mλ 2 . One can interpret β as BPS charge and α t α β α as the central charge mass term. In the 5-field theory the fugacity λ traces the left spin of the BPS states.
The n g β ∈ Z may be calculated by identifying the above expression to the perturbative topological string expansion on M where λ is now the topological string coupling constant. The Gromov-Witten invariants r g β ∈ Q can be mathematically defined and in suitable cases calculated using localization. 24 Caligraphic letters should refer to quantities in the holomorphic limit.
In this article we use mirror symmetry, modularity, the holomorphic anomaly equation and direct curve counting to obtain the F (g) (t). With x k k m and the Bernoulli numbers are defined in (3.23). Using further ζ(−n) = − B n+1 n+1 for n ∈ N + we get which matches the constant map contribution [19] for n 0 0 = − χ 2 . Exponentiating (5.1) one gets [50,47]

Refined BPS invariants
Let us define as in [40] in analogy with (4.2,4.4) We decompose the genus g amplitude as then holomorphic anomaly equation for E-strings [41] becomes (5.8) Making an analogous fibre and base separation as in (4.9) of the E-string geometry 1 2 K3 surface with the rational elliptic fibration E → P 1 , one finds [28] for the massless E-string the analogous equation to (4.10), compare [?] We note that in the Nekrasov-Shatashvili limit the solution will be a weak Jacobi form.
Indeed the index growth linearly as in the N = 4 case discussed in the introduction. In the general case we expect as solution a meromorphic weak Jacobi theta function. The refined BPS invariants are labelled by β ∈ H 2 (M, Z) and j L , j R , the 5d left and right spin then (5.1) generalizes with u = e i 1 and v = e i 2 to .

Geometry of curves
In this section, we describe the geometry of a number of families of curves in the X 18 (1, 1, 1, 6, 9) model, from which a good number of GV invariants can be extracted. For simplicity, we will simply refer to this model as X in this section.
We begin by reviewing some results and notation from [11], which will be used to describe the geometry.
The exceptional divisor E is isomorphic to P 2 . Projection to the first three coordinates of the weighted projective space presents X as a Weierstrass elliptic fibration π : X → P 2 . Let L = π −1 ( ) be the pullback of a line ⊂ P 2 to X, and let H = 3L + E. Then π * (x 1 ), π * (x 2 ), and π * (x 3 ) are sections of L, π * (x 4 ) is a section of 2H, and π * (x 5 ) is a section of 3H. 25 See [13] for a mathematical definition of the N β j L j R in the local case using the virtual Bialynicki-Birula decomposition of the moduli space Pandharipande-Thomas invariants.
For the benefit of those readers more familiar with the toric description, we refer to (2.3) where the two descriptions of the divisors are related: the toric divisor D x 0 is identified with E, the divisor classes of the toric divisors D x i are all equal to L for i = 1, 2, 3, the toric divisor D x is in the class 2H, and the toric divisor D y is in the class 3H.
The Kähler cone of X is generated by H and L. The triple intersections are Dually, the Mori cone is generated by the class of an elliptic fiber f and the class of a line˜ ⊂ E P 2 .
Now, let C ⊂ X be a connected curve (not necessarily reduced or irreducible, but having no embedded points). The invariants of C are where the last is the arithmetic genus of C. The curves of fixed (g; d E , d B ) are parametrized by a Hilbert scheme, and we will be able to describe many of these as well as the corresponding GV invariants. As a comment on the mathematical rigor of our calculations, we are actually studying PT moduli spaces and computing PT invariants, which are mathematically equivalent to the GV invariants invariants by [64]. However, we will frequently simplify the exposition by using the language of Hilbert schemes as in [48] and then describing corrections required by the use of stable pairs. The reader is referred to [64] for more details, and to [13] for an amplification of the comparison between the methods of [13] and [48] We say that (d E , d B ) is the degree of C, and sometimes use the notation C d E ,d B for a curve of degree (d E , d B ). A fiber f has degree (1, 0) and a line˜ ⊂ E has degree (0, 1).
We are able to completely describe the moduli space of connected curves of degree (d E , d B ) and genus g for various d E , d B , and g. In this section we describe the geometric principles that these descriptions are based on. In the next section, we will apply these principles to compute various moduli spaces with d B ≤ 5 and use this description to compute the associated Gopakumar-Vafa invariants, and make some comments about general d B .
Then π(C) is a plane curve of degree d B , including multiplicity.
Proof. We compute the degree of π(C) as where ⊂ P 2 is a line.

Corollary 1.
A curve of degree (d E , 0) is a union of d E fibers (including multiplicity).
Proof. By Lemma 1, π(C) is a finite point set, hence C is a union of fibers. Since each fiber has degree (1, 0), the result follows.

Lemma 2.
If d E < 3d B , then for any curve C of degree (d E , d B ) at least one component of C (with its reduced structure) is contained in E. In particular, if in addition C is irreducible, then d E = 0 and C red ⊂ E, where C red is the reduced (multiplicity 1) structure on C.
Proof. To see this, we first note that if a curve C is not contained in E, then C · E ≥ 0, as the intersection number is just a count of the intersection points of C and E with multiplicity. Then we compute If this is negative, it follows immediately that some component C of C must be contained in E. If in addition C is irreducible, it follows that C red ⊂ E. Since H · E = 0, it follows that d E = C · H = 0 as claimed.
Remark. If C rd ⊂ E, it is possible for C itself to not be contained in E due to thickenings in a direction transverse to E.
Proposition 1. If C has degree (0, d B ), then its genus satisfies with equality holding if and only if C is a plane curve of degree d B after identifying E with The second half of the proposition says that if we try to thicken components of C outside of E, the genus would be less than if we thicken inside E.
Proof. We have C · H = d E = 0. Since H is ample on the weighted hypersurface in P(1, 1, 1, 6, 9) before the blowup, the only way for the intersection C · H to be 0 is for C red to be contained in E. Since any P 2 in a Calabi-Yau threefold has a neighborhood which is isomorphic to a neighborhood of P 2 inside local P 2 , the result follows from the corresponding result for local P 2 .
Remark. We will see in the next subsection that for each d B ≥ 2, non-reduced curves occur generically in moduli spaces of curves with particular d E and g.
We now set out to find the largest possible genus of a connected curve of degree (d E , d B ). Part of our strategy in studying curves is to first study irreducible curves and then study how the components can glue together.
By Lemma 1, a curve C of degree (d E , d B ) is contained in a surface π −1 (D) where D ⊂ E P 2 is a plane curve of degree less than or equal to d B . The degree will be strictly less than d B if and only if some component of π(C) has multiplicity greater than 1.
By analogy with the case of curves in P 3 , where the genus of a curve of degree d is maximized when the curve is contained in a plane, one might expect that in our situation the genus is maximized if the degree of D is one, i.e. when C ⊂ π −1 ( ) for some line ⊂ E. This turns out to be the case for irreducible curves with d E > 0.
with equality holding if and only if C ⊂ π −1 ( ) for some . In that case, C ⊂ S is the zero locus of a section of O S (d E L + d B E). The moduli space of all curves C (not necessarily irreducible) given by all and all sections of Part of the proof involves an analysis of certain curves in surfaces of the form S = π −1 ( ) which will be given below. Another part of the proof is analogous to the proof of the Castelnuovo bound for the genus of a curve of fixed degree in P 3 . The proof itself is omitted.
We now describe curves in a smooth S = π −1 ( ). Suppose that S ∈ |L|, i.e. S = π −1 ( ) for some line ⊂ P 2 . Then for C ⊂ S, π(C) is either a point p (in which case C is an elliptic fiber of π, hence completely understood), or π(C) = . Conversely, if C is any curve in X for which π(C) = , then C ⊂ S = π −1 ( ).
We now describe all divisors on S associated to line bundles on S which are restrictions of line bundles on X. We denote restrictions to S by a subscript, so that the basic divisor classes on S are E S and L S . For divisors D and D on X we have for the intersection on S of their restrictions D S · D S = D · D · L, (5.17) where the intersection on the right hand side is taken in X. We now turn to computing the genus and dimension. For simplicity of exposition, we assume that S is smooth. The adjunction formula says By the projection formula we get H 0 (S, K S ) = H 0 (S, O S (L S )) = H 0 ( , O (1)) and so p g = dim H 0 (S, K S ) = 2. Here we are using the language of classical algebraic geometry, where p g = dim H 0 (S, K S ) is the geometric genus of a complex surface. It is not hard to see that q := h 1 (S, O S ) = 0. So The genus is given by The curves C ⊂ S are a projective space of dimension h 0 (S, O(C)) − 1. If d E > 0 and d B > 1 then C − K S = C − L S is ample, so by Kodaira vanishing we have h i (S, O(C)) = 0 for i > 0. So we only need to compute χ(O(C)): Since the moduli of these S is just P 2 corresponding to the family of lines in P 2 , the moduli space of such curves C is a P χ(O(C))−1 -bundle over P 2 , with χ(O(C)) given by (5.23).

Examples and Computations
In this section, we apply the results of the previous section to describe many moduli spaces of curves and compute the associated Gopakumar-Vafa invariants. We denote the moduli space of curves of degree (d E , d B ) and genus g by M g d E ,d B . If M g d E ,d B is smooth and there are no curves of degree (d E , d B ) and genus strictly greater than g, we have a formula that may need correction as discussed in Section 5.2.1. Now let C g d E ,d B be the universal curve, also supposed smooth, and suppose that there are no other curves of degree (d E , d B ) and genus greater than or equal to g − 1. Then we have, by [48] This formula may also need correction. Examples: We begin with d E = 0 and any d B > 0. Then we assume d E > 0 and treat 0 ≤ d B ≤ 5 in turn.
2)/2, then Proposition 1 says that M g 0,d B is identical to the moduli space of plane curves of degree d E in local P 2 . In particular, the GV invariants n g 0,d B for g ≤ p a (d B ) are identical with the well-known GV invariants of local P 2 . This observation provides the geometric verification of the GV invariants in the entire first columns of Tables 5-13.
We will see later that a similar argument can be used to compute or verify many of the GV invariants in some of the subsequent columns. Now let C have degree (d E , d B ) with d E > 0 and we set out to bound the genus. It is well known that for an elliptically fibered Calabi-Yau threefold X over a base B, the only nonzero Gopakumar-Vafa invariants in multiples of the fiber class f are We therefore get n 0 d E ,0 = 540 (d E > 0), n 1 1,0 = 3, (5.27) in agreement with the d B = 0 rows of Tables 5-13.
We compute some Gopakumar-Vafa invariants for d B = 1 by geometry and always find complete agreement with Table 2 and additional calculations we have done beyond the table.
We put D = π(C) and note that D = is a line, by Lemma 1. Then C has a unique irreducible component C mapping by π onto D. Since any other components of C map to points, they must be unions of fibers, which have d B = 0. Thus C has d B = 1. By Proposition 1 we have g(C) ≤ 0. Hence g(C) = 0 and C is a line in E, again by Proposition 1. Since C has degree (0, 1), we see that the remaining components have degree (d E , 0), hence consists of d E fibers by the d B = 0 case. Thus C is the union of a line in E and d E fibers, and necessarily has genus d E . ¿From this description, we see that M d E d E ,1 is the relative Hilbert scheme of d E points on lines in the plane, a P d E -bundle over P 2 . Hence by (5.24) we have The second equation follows since the same description shows that all curves of degree (d, 1) have genus d < d E .
These results agree with those presented in the d B = 1 row of Tables 5-13. We have also checked these results for much larger values of d E , and always find agreement.
We compute some Gopakumar-Vafa invariants for d B = 2 by geometry and always find complete agreement with Table 3 and additional calculations we have done beyond the  table. We D = π(C) and note that D has degree 1 or 2.
If D is a line , then C ⊂ π −1 ( ). By Proposition 2 we get g = 2d E − 4. If D = 1 ∪ 2 is union of lines, then just as in the d B = 1 case we have that C contains irreducible components C 1 and C 2 mapping isomorphically by π to 1 and 2 respectively. Since each C i has degree (0, 1), the remaining components of C have degree (d E , 0), hence consist of d E fibers. Thus C consists of the reducible conic C 1 ∪ C 2 in E with d E fibers attached.
If D is an irreducible conic, then C has a unique component C mapping to C with degree 1. Hence by smoothness of C we see that C is isomorphic to C. By Proposition 1, C is a conic in E. Since C has degree (0, 2), we see just as in the last case that C is the union of C and d E fibers. The last two cases combine into one moduli space of curves, parametrizing conics in E (both smooth and singular) with d E fibers attached. This description shows that moduli space is isomorphic to the relative Hilbert scheme of d E points on plane conics. This space is smooth for d E ≤ 3 [48].
Thus the moduli space of all curves of degree (d E , 2) has two components: the one just described as the relative Hilbert scheme of d E points on plane conics, and the other component parametrizing curves in a surface S = π −1 ( ) as in Proposition 2.  [48] or [13], we have by (5.24) n 0 0,2 = −6, n 1 1,2 = 15, n 2 2,2 = −36, n 3 3,2 = 66 (5.29) in agreement with the numbers in Tables 5-8. The same argument also shows that all numbers to the left of these numbers are zero.
For d E ≥ 5, the above discussion shows that curves of genus g = 2d E −4 are all contained in surfaces S = π −1 ( ) and are described by Proposition 2. So M 2d E −4 d E ,2 is a P 2d E −5 bundle over P 2 , hence by (5.24) we get in agreement with the d B = 2 lines of Tables 11-13 for d E ≥ 5. We have also checked these results for much larger values of d E and always find agreement. Our last set of checks come from the application of (5.25) to curves of type ((1, 2)). From the genus formula max(d E , 2d E − 4) we see that the hypotheses of (5.25) hold if d E < 2d E − 5, so that d E ≥ 6. We need to describe the universal curve C 2d E −4 d E ,2 . Given a point of the universal curve, which we denote by (C, p), we have a natural map C 2d E −4 d E ,2 → X obtained by forgetting C. We claim that the fiber is a P 2d E −6 -bundle over P 1 , hence the universal curve is smooth.
To see this, note that the line for which C ⊂ S = π −1 ( ) must contain the point q = π(p). Therefore the lines used to describe curves in the fiber over p form a P 1 rather than a P 2 . The fiber is itself fibered over this P 1 , and it only remains to find the fiber of this last fibration. Now fixing , hence fixing S, the possible curves C are those corresponding to sections of H 0 (S, O(d E L S + d B E S )) which vanish at p, which is a hyperplane in H 0 (S, O(d E L S + d B E S )). So the fiber is a P 2d E −6 rather than a P 2d E −5 , as claimed.
We can now apply (5.25) to compute which simplifies to which agrees with the results in Table 3 as well as larger values of d E . Furthermore, all numbers to the left of these numbers are zero, which also checks.
The general procedure should now be clear. We let D = π(C) and we study the cases where D has irreducible components of each possible degree and multiplicity. In other words the degree d B curve D can split up into components D j of degree d j and multiplicity m j . So the possible components of moduli can be indexed by partitions of d B into unordered lists of pairs ((d j , m j )) with repetition allowed satisfying Correspondingly, C splits up into a union of components C j , mapping by π to D j with multiplicity m j , possibly together with fibers. We investigate the maximum genus of each C j . The genus of C can then be maximized by configuring the curves C j and any fibers so that they intersect in as many points (including multiplicity) as possible.
The analysis gets more intricate for each degree. Rather than carry out this program for d E ≤ 5, we content ourselves with presenting some Gopakumar-Vafa invariants that can be computed by these methods, giving the corresponding ((d j , m j )), and matching to the tables.
For any d B ≥ 2 some strata combine naturally, as we already saw for d B = 2. Let d B = k j=1 be a partition of d B and consider the list ((d j , 1) k j=1 ). These strata combine to describe the Hilbert scheme of d E points on plane curves of degree d B . The stratification is just an artifact of our method, corresponding to splitting up the moduli space according to the degrees of the components of the plane curve.
( (1, 2), (1, 1)) is the union of a curve in a surface of the form S = π −1 ( ), a line in E, and possibly some fibers. It is not hard to see by extending the analysis below that the maximum genus is attained when there are no additional fibers. So C is the union of a curve C d E ,2 in S and a line C 0,1 ⊂ E (which has degree (0, 1)). Here and in the sequel, a subscript denotes the degree of a curve. We have already seen that C d E ,2 has genus 2d E − 4.
The genus of C is therefore (2d E − 4) + m − 1, where m is the number of intersection points Z of C d E ,2 and . We claim that either m = 1 or we are actually in the ((1, 3)) case.
then C 0,1 must be equal to that line. In that case C ⊂ S, and we are really in the ((1, 3)) case. So m = 1 and C has genus 2d E − 4.
Combining the three cases, we have The maximum genus is attained in the first case for d E ≤ 5, in the second case for d E ≥ 7, and in the third case for 5 ≤ d E ≤ 7.
If d E ≤ 4, then the maximum genus d E + 1 can only be attained in the first case, and M d E +1 d E ,3 is the relative Hilbert scheme of d E points on plane cubics, which is smooth by [48]. So we compute from the relative Hilbert scheme is a P 3d E −13 -bundle over P 2 by Proposition 2. Hence, for d E ≥ 8 we have n 3d E −11 For these d E we also have n 3d E −11 We have checked these results for large values of d E and always find agreement.
For d E = 6, the maximum genus 2d E − 4 = 8 can only be realized in the third case ((1, 2), (1, 1)). By the discussion above, we have C = C 6,2 ∪ C 0,1 , where C 6,2 lies in a surface S = π −1 ( ) and the line C 0,1 intersects C 6,2 in exactly one point. But C 6,2 · E = 0 by (5.16). On the other hand, C 6,2 cannot be disjoint from E, since it intersects C 0,1 ⊂ E nontrivially. Therefore C 6,2 has a component which lies entirely in E, necessary a line or a conic. If it contains a conic plane curve, then since it also contains the line C 0,1 then it contains a (reducible) cubic plane curve. So we are really in the case of a cubic with d E fibers already considered. So C 6,2 contains a line. The remaining components are a curve C 6,1 , which by the d B = 1 case are a line and 6 fibers. If the two lines are distinct, then they form a conic and we are in the previous case. Otherwise the line has multiplicity 2, and the curve C 6,2 is the union of the line E S with multiplicity 2 (in S) and 6 fibers. The line C 0,1 is arbitrary. So M 8 6,3 is the product of P 2 and the relative Hilbert scheme of 6 points in lines in P 2 . Since this relative Hilbert scheme is a P 6 -bundle over P 2 , this gives n 8 6,3 = 63, (5.36) in agreement with Table 13. Furthermore, all of the numbers to the left are zero, again in agreement.
We can also apply (5.25) for curves of type ((1, 3)), which have genus 3d E − 11. The hypotheses hold if d E ≥ 9 and then we get which agrees with the results in Table 4 as well as larger values of d E . Furthermore, all numbers to the left of these numbers are zero, which also checks.
We compute some Gopakumar-Vafa invariants for d B = 4 by geometry and always find complete agreement with Table 14 and additional calculations we have done beyond the table. By now we have provided enough examples that we can simply present our results and the reader will be able to check details.
Combining the cases ((4, 1)), ((3, 1), (1, 1)), ((2, 1), (2, 1)), ((2, 1), (1, 1), (1, 1)), and ((1, 1), (1, 1), (1, 1), (1, 1)), we consider plane quartic curves in E with d E fibers attached. These curves have genus d E + 3. If d E ≤ 6, any curve of degree (d E , 4) and genus d E + 3 can be seen to be of this type, and furthermore there are no curves of that degree and higher genus. We conclude that M d E +3 d E ,4 is the relative Hilbert scheme of d E points on plane quartic curves, a P 14−d E bundle over Hilb d E (P 2 ), and is smooth for d E ≤ 5 [48]. We therefore get  The same argument confirms in the usual way that all numbers to the left of these are zero, in agreement with the tables. For curves of type ((1, 4)), i.e. curves contained in a surface S = π −1 ( ), we have g = 4d E − 21, and M 4d E −21 is a P 4d E −24 -bundle over P 2 by Proposition 2. If d E ≥ 11, then any curve of this degree and genus is of this type, and there are no curves of this degree and higher genus. This gives in complete agreement with our calculations. Also n 4d E −21 d,4 = 0 for d < d E . For d E = 7 we have curves of type ((1, 2), (2, 1)), which consist of a curve C 7,2 of degree (7, 2) union a conic C 0,2 ⊂ E. The curve C 7,2 lies in a surface S = π −1 ( ) and has genus 2d E − 4 = 10. Since C 0,2 is to meet C 7,2 in two points but C 7,2 · E = 1, we see that C 7,2 must contain a line in E. Repeating an argument from the d B = 3 case, we further see that C 7,2 must contain a line doubled in S together with d E fibers. Our geometric description shows that the moduli space of C 7,2 is the relative Hilbert scheme of 7 points on lines, a P 7 -bundle over P 2 . There are no curves of degree (7, 2) and higher genus, so we can now multiply be the P 5 moduli of C 0,2 and compute the Euler characteristic, giving n 16 7,4 = 144, (5.41) in agreement with our other calculations. In addition we have n 16 d,4 = 0 for d < 7, also in agreement.
We can also apply (5.25) for curves of type ((1, 4)), which have genus 4d E − 21. The hypotheses hold if d E ≥ 12 and then we get  Table 14 as well as larger values of d E . Furthermore, all numbers to the left of these numbers are zero, which also checks.
We compute some Gopakumar-Vafa invariants for d B = 5 by geometry and always find complete agreement with Table 15 and additional calculations we have done beyond the  table. We first consider the curves which are unions of plane quintics and d E fibers, which have genus d E + 6. We also compute that for d E ≤ 7, all curves of degree (d E , 5) and genus d E + 6 are of this type. Therefore M d E +6 d E ,5 is the relative Hilbert scheme of d E points on plane quintics, which is smooth for d E ≤ 6. We also compute that there are no curves of that degree and higher genus. For curves of type ((1, 5)), i.e. those contained in a surface S = π −1 ( ) we have g = 5d E − 34 and moduli space a P 5d E −38 -bundle over P 2 . For d E ≥ 14 we compute that all curves of degree (d E , 5) are of this type, and that there are no curves of higher genus. This gives n 5d E −34 in complete agreement with our calculations. Also the Gopakumar-Vafa invariants to the left of these numbers are all zero, also in agreement. We can also apply (5.25) for curves of type ((1, 5)), which have genus 5d E − 34. The hypotheses hold if d E ≥ 15 and then we get which agrees with the results in Table 14 as well as larger values of d E . Furthermore, all numbers to the left of these numbers are zero, which also checks.

Acknowledgments
We thanks Babak Haghighat, Guglielmo Lockhart and Cumrun Vafa for discussions about related structures in M-and E-string theories and Martin Westerholt-Raum and Don Zagier for crucial comments on Siegel modular-and weak Jacobi forms. We like to thanks Gaetan Borot, Hans Jockers and Marcos Mariño for comments on the draft. This work started when MH was affiliated with Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU). He thanks Kavli IPMU for supports. MH also thanks the BCTP, the HCM and the organizers of "The 2nd Workshop on Developments in M-theory" in Korea for hospitality. MH is supported by the "Young Thousand People" plan by the Central Organization Department in China, and by the Natural Science Foundation of China. S.K. is supported by NSF DMS-12-01089. A.K. thanks for support by KL 2271/1-1 and NSF DMS-11-01089.

B Derivation of the involution symmetry on the propagators
For the compact models we have the an-holomorphic propagators S ij , S i , S, where i, j runs over the complex structure moduli. It is convenient to make a change of variables with the derivative of Kähler potential K i = ∂ i K by the following In the following we refer to the propagators S ij , S i , S as the ones after the change of variables in [2]. The propagators are defined by relating their anti-holomorphic derivatives to three point couplings. One can integrate these relations and also special geometry relation, and we get Here the holomorphic ambiguities s k ij , h jk i , h ij , h j i , h i are some rational functions from the integration constants of the anti-holomorphic derivatives. We can compute the Kahler potential K and Christoffel connections Γ k ij in the holomorphic limit from Picard-Fuchs equations. There are some freedom to choose some of the ambiguities such that the set of equations has a consistent solution for the other ambiguities and the propagators S ij , S i , S.
We first discuss the gauge choice made in [3], which are the followings We will need to know how the propagators transform under the involution. Our guiding principle is the followings. We replace the coordinates z i 's with x i 's and all quantities with their transformations in the equations (B.2). We then require the resulting equations are equivalent to the original ones. Here for the holomorphic ambiguities we can simply replace the z i coordinates with x i 's in the expressions (B.3). Since the Picard-Fuchs operators are invariant under the involution, we argue the Kahler potential is invariantK = K, and the Christoffel symbols transform as We find that it is possible to solve for the transformed propagatorsS ij ,S i ,S such that the transformed equations of (B.2) are equivalent to the original ones.
It is easy to check that with the choices of the ambiguities s i jk in (B.3), the transformations ares The shift exactly cancels that of the Christoffel symbols in the first equation in (B.2), therefore if S ij transform as tensor with a minus sign, the transformed equation is equivalent to the original one by a coordinate transformation.
To solve for the shifts in the transformed propagatorsS i , we combine the involution transformation with the coordinate transformation of the second equation in (B.2). We find the shift in the gauge s k ij exactly cancels the one from the derivative of S jk propagator and all propagators cancel out, so the shifts f i (i = 1, 2) are determined by the equations We see that if h jk i transformed as a tensor with a minus sign, the shifts would have vanished. The shifts are results of the non-tensorial transformation of the gauge choice h jk i . Similarly we combine the involution transformation with the coordinate transformation of the third equation in (B.2). Utilizing the solution (3.33), we again find all dependence on propagators cancel, and the shift f 0 is determined by and we find the the last equation in where as an example we choose One can easily check that after the shifts, the ambiguity h jk i transforms as a tensor with a minus sign under the involution transformation. So according to (B.6) this new gauge choice will eliminate the shifts f 1 , f 2 in the involution transformation of the propagators S i . The shift in h jk i will change the propagators S i , S and the other ambiguities h j i , h i , h ij as well, which can be straightforwardly determined.
Similarly we can further shift the ambiguity h j i to cancel the shift f 0 in the involution transformation of the propagator S.
The gauge choice which gives no shifts for the involution transformation of the propagators is conceptually simple, and give a much better understanding of the involution symmetry as auto-equivalence of the underlying derived categories. However, as it turns out such a gauge choice would introduce pole of 1−432z 1 in the ambiguities h j i , h i . Without detailed calculations, this point can be seen by noting that the shifts (3.33) have poles in 1 − 432z 1 which does not appear in the gauge choice in (B.3), so in order to cancel the shifts, we would likely need to introduce poles of 1 − 432z 1 in the ambiguities. As a consequence, the holomorphic ambiguities at higher genus would have also poles at 1 − 432z 1 . The total topological string amplitudes at higher genus should be regular at 1 − 432z 1 , so the singular part of the holomorphic ambiguities at the pole 1 − 432z 1 can be fixed and the gauge choice does not really enlarge the space of holomorphic ambiguities at higher genus but complicates the calculations.
In practical calculations it is simpler to use the gauge (B.3), for which it turns out the holomorphic ambiguities at higher genus have no pole at 1 − 432z 1 , but only at the conifold divisors ∆ 1 and ∆ 2 . In the followings we will still use the gauge choice (B.3).

C Reducing the ambiguity with the involution symmetry
We can compute the number of remaining unknown constants for general genus g after imposing the condition (3.5) more precisely. First we can find a particular holomorphic ambiguity such that the total amplitude with the propagator dependent part satisfies the constrain (3.5). Then there are freedom to add an additional piece f (g) to the holomorphic ambiguity which satisfies the same symmetry conditioñ The space of such f (g) form a linear space. The number of remaining unknown constants is the dimension of this linear space. We can define the following linear vector spaces of holomorphic ambiguities V (g,m,n) 0 It is easy to check they are indeed linear vector spaces, i.e., any linear combinations of elements of the space is also an element in the space. We would like to construct the explicit linear basis for these spaces. Clearly dim(V (g,m,n) 0 ) = m + 1. A convenient linearly independent basis is We can work out the involution transformation We discuss two cases for k ≥ 3n − (6g−6). On the other hand, any linear combinations of the basis with k < 3n−(6g−6) in (C.3) is not in the space V (g,m,n) 1 since there will be poles of ( 1 432 − z 1 ) in the numerator. Therefore the space V (g,m,n) 1 is generated by the linear combinations of , where k = 3n−(6g −6), · · · , m. In this case the dimension of V is generated by the linear combinations of , where k = 0, 1, · · · , m + 3n − 6g + 6. In In both cases we can write the formula For any element f ∈ V (g,m,n) 1 we can write Clearly the first term 1 2 (f + (−1) g−1f ) ∈ V (g,m,n) 2 and the second term 1 2 (f + (−1) gf ) ∈ V (g,m,n) 3 . We also note that V From the transformation (C.4), we can check that for f ∈ V for the case of n + g an even integer.
We can estimate the number of unknown constants for the holomorphic ambiguity at large genus g after imposing the involution symmetry, assuming the degree of z 1 is 7(g − 1) in the numerator This is about one quarter of the (7g − 6)(5g − 4) unknown constants in the naive ansatz at large genus.

D Fibre modularity versus involution symmetry
In appendix we will prove that the constraints imposed on the amplitudes and especially on the holomorphic ambiguity from the involution symmetry is equivalent to the constraints from the modularity in the fibre direction discussed in section 4. In addition the appendix provides many identities that are useful for further discussion of the model [37]. From the solutions 27 of the (2.10) around the large volume point (z 1 , z 2 ) ∼ (0, 0), we find The above relations can be readily inverted and we can write z 1,2 as power series of q 1,2 .
If we keep the gauge X 2g−2 0 explicit and note that q B is defined without the shift (4.1) then (4.4,4.5) read while for the case of g = 0 we do not need the factor of X −2 0 in the conventional definition of the prepotential.
From section 4 follows for the X 18 (1, 1, 1, 6, 9) model that the P (g) k (q E ) are quasi-modular forms of SL(2, Z), of modular weight 18k + 2g − 2 and satisfy (4.13) as a specialization of (4.8). We would like to understand whether and to what extent this constraint from the modularity of the fiber fixes the holomorphic ambiguity at higher genus. Suppose at genus g ≥ 2 we have found a particular holomorphic ambiguity such that the topological string amplitude F (g) satisfies the fiber modularity constrain. Then we may add an additional holomorphic ambiguity f (g) , whose expansion is k (q E ) to be also a quasi-modular form of weight 18k + 2g − 2. Furthermore, since the sum P (g) k (q E ) also need to satisfy the equation (4.13), we find In particular for the case k = 0, 1, this equation is understood as ∂p (g) k ∂E 2 = 0, i.e. p (g) 0 and p (g) 1 are always modular forms. 27 We lower the index on the A-periods X i in order to avoid cluttering with exponentials.
It turns out that not all rational functions f (g) satisfy the fiber modularity constraint (D.4, D.5), so the fiber modularity does impose some conditions on holomorphic ambiguities. However we find that there exist some non-vanishing rational functions f (g) satisfying these conditions, so the fiber modularity does not completely fix the holomorphic ambiguity. We see that the BCOV holomorphic anomaly equation alone does not implies the modularity of the fiber at higher genus.
However it turns out that the modularity of the fiber is equivalent to the involution symmetry. It gives more precise bounds for the degrees in the polynomials in the numerators of the holomorphic ambiguity and further reduces the number of unknown constants in the holomorphic ambiguity. To understand this equivalence, we consider a holomorphic ambiguity f ∈ V (g,+∞,n) 0 . Since f ∼ z n 2 ∼ q n B in the small z limit, it is easy to see that the first non-vanish coefficient in the expansion in base degree (D.4) is that of q n According to (D.5) the first coefficient p n (q E ) has no E 2 dependence and is purely a modular form, i.e. polynomials of E 4 and E 6 . We define the linear space We will prove the following proposition Proof. The proof will proceed in two steps. We shall first show V ). We will see how the number [ 19 3 (g − 1)] appears in the process.
In order to prove the Proposition 3, we only need the leading order term in the small z 2 expansion. The first Picard-Fuchs equation in (2.10) in the z 2 → 0 limit has been studied before in [32,52]. We find the power series solution and A-model parameter q E are determined by the exact relations where we have taken the z 2 → 0 limit for X 0 and q E . We will drop the argument q E = q 1 in the modular forms and η function for simplicity. For the other logarithmic solution we need to keep the log term in z 2 → 0 limit where ξ(z 1 ) = 180z 1 + O(z 2 1 ) is a power series. Using the first Picard-Fuchs equation in (2.10) we find [θ 2 1 − 12z 1 (6θ 1 + 1)(6θ 1 + 5)]ξ(z 1 ) = 3θ 1 X 0 (D.10) There is an exact solution for ξ(z 1 ) using (D.8) and the well-known Ramanujan derivative relations for quasi-modular forms Of course one can add any linear combination of two solutions of PF equation in the z 2 → 0 limit, which are E 4 (q 1 ) 1 4 and log(q 1 )E 4 (q 1 ) 1 4 . The above solution is the only one with the correct asymptotic behavior ξ(z 1 ) ∼ z 1 for small z 1 . So the A-model parameter q B is We first discuss the case that n + g is an odd integer.
then the leading term in the expansion (D.6) is where we have used the relations (D.8) and ignore constant factors. Since n + g is an odd integer, the exponent of E 4 is an integer. Furthermore η 24k 1 is a modular form of weight 18n + 2g − 2. The case of n + g an even integer is similar. We need to use the formula 1 − 864z 1 = ). We discuss two cases. (∆ 1 ∆ 2 ) 2g−2 where h(z 1 ) is a polynomial of z 1 . We find the leading term in the expansion (D.6) We consider the zero at 1 − 432z 1 ∼ 0, which correspond to E 6 + E 1 . For the scaling factor X 0 in the topological string amplitude we take the one with the lowest exponent X 0 ∼ z − 1 6 1 . So the scaling exponent including the conifold divisors is (D. 21) and the regularity of X 2g−2 0 F (g) at z 1 ∼ ∞ requires the degree of z 1 at the numerator in the holomorphic ambiguity be no bigger than 19 3 (g − 1). The non-zero scaling exponent in X 0 around z 1 ∼ ∞ is characteristic of compact Calabi-Yau models, and we will see that it also is responsible for preventing the solution of the model at large genus as in one-parameter models like the quintic.
Here we fix the modular forms in low orders by perturbative calculations. In general we conjecture the following In the next subsection D.2, we shall prove this modularity conjecture. We note that we only need to prove the modularity of the coefficients of the expansion of z 1 (1 − 432z 1 ) and 1 − 864z 1 , which immediately imply that of ∆ 1 ∆ 2 (1−432z 1 ) 3 , since we can write (D. 28) We also notice that the higher order modular form coefficients in the expansion formula (D.26) contain a factor of η 24 = 1728 . This condition is sufficient to ensure the modularity of expansion (D. 24), and therefore that of (D.25) as well. To see this, suppose where f 18n−6 are some modular forms of weight 18n − 6. We can compute (D.24) easily Now it is easy to see that if our conjecture (2) is correct, then for f ∈ W (g,n) 0 , it has the following expansion where f n are modular forms of q E of modular weight 18n + 2g − 2. The next step is to expand Z parameter in terms of the base coordinate q B . For convenience we denote the normalized Kahler coordinate We can check that the coefficients c n of Q n are quasi-modular forms, and satisfy the following equation as in (D.5) where the P (0) n are the quasi-modular forms of the prepotential expanded in terms of base degree, for example We define the following space Here we do not specify the modular weights of c n , so unlike the previous cases, the space W 1 is not a linear space. Rather, we will see it is a graded module over modular forms. Since we know the modular weight of P (0) n is 18n − 2, the modular weights of all c n 's are actually determined by the first non-vanishing coefficient according to equation (D.34). It is clear if f ∈ W 1 and g is a modular form, then f g ∈ W 1 .
The equation (D.34) can be equivalently written as where we define the generating function Furthermore, If f ∈ W 1 , then any algebraic function of f would also satisfies the equation (D. 37). However, f may not have the structure of Q-expansion defined by W 1 space in (D. 36), and the coefficients c n may be only algebraic functions of quasi-modular forms. The equation (D.37) can be further written in a simpler way which will be useful later. In the left hand side of (D.37), we regard f as function of f (Q, E 2 , E 4 , E 6 ) when taking partial derivative with respect to E 2 . Alternatively, we can also use inverse function and regard Q as function of Q(f, E 2 , E 4 , E 6 ). In this perspective we have ∂f ∂E 2 = 0 and ∂Q ∂E 2 = 0. We can take partial derivative of E 2 on f = ∞ n=1 c n Q n , and find So we can write equation (D.37) as Based on the perturbative calculations (D.33), we conjecture If our conjecture is correct, then obviously it is also true that Z n ∈ W 1 for any positive integer n. Using the expansion (D.31) and keeping track of the modular weights, we can see that the conjectures (2) (3) and calculations in this subsection would imply the validity of the main conjecture (1).

D.2 Proof of Vonjecture 2 and Conjecture 3.
In this subsection we shall prove the conjectures (2) and (3) for our compact model. The proof uses some ingredients in the proof of genus zero modular anomaly equation for the non-compact half K3 model in [32,52], but it is much more complicated for compact model.
To solve the Picard-Fuchs equations (2.10), it is convenient to define the variableq, which is related to z 1 by with the asymptotic behaviorq ∼ z 1 ∼ 0 in small z 1 limit. This is the Kahler parameter in local large base limit z 2 → 0, but in the compact model, it is different from the physical Kähler parameter q 1 which depends on both z 1 and z 2 . To be clear in this subsection we will keep the argumentsq and q 1 in the modular functions, and we also define the logarithmic flat coordinates t 1 = log(q 1 ) andt = log(q). We can write the ansatz for the solution to the equations (2.10) in small z 2 power series expansion It is well known in the local z 2 → 0 limit, there are two linearly independent solutions, which can be written as E 4 (q) 1 4 and log(q)E 4 (q) 1 4 , in terms of the modular parameterq defined in (D.41). So the initial function c 0 (z 1 ) in the expansion is E 4 (q) 1 4 or log(q)E 4 (q) 1 4 . The first and second PF equations in (2.10) become where θ z 1 = z 1 ∂ z 1 . We see that the second equation provides a recursion relation to compute the higher order coefficients c n (z 1 ). Furthermore, one can check that the two equations are consistent, i.e. the c n+1 from recursion in the second equation also satisfies the first equation for c n with by replacing n with n + 1. The recursion implies A particular modularity structure for such recursion was derived in [32,52]. Here we repeat the analysis. From the relation (D.41) it is easy to compute We can write some derivative formulae , ∂t E 6 (q) 12E 4 (q) 3 4 .

(D.46)
First we consider the case c 0 (z 1 ) = E 4 (q) 1 4 . From the derivative formulae (D.46) one can deduce inductively that the recursion function where P 6n (E 2 , E 4 , E 6 ) is a quasi-modular form of weight 6n and is linear in E 2 (q). We can write (E 4 (q) 3 2 + E 6 (q)) n (a n E 2 (q) + b n ), (D.48) where a n and b n are modular forms of weight 6n − 2 and 6n, without E 2 dependence. Some low order formulae are the followings The recursion relation for the modular forms a n and b n are a n+1 = −(2n − 1 6 )E 6 (q)a n + 1 6 We then consider the case c 0 (z 1 ) =tE 4 (q) (E 4 (q) 3 2 + E 6 (q)) n t (a n E 2 (q) + b n ) + 12a n (D.50) So we find the power series solution and one logarithmic log(z 1 ) solution to the Picard-Fuchs equations (2.10) We compute the flat coordinate of the compact model The expression quantifies the deformation of t 1 away fromt as asymptotic expansion in small z 2 , where the coefficients are rational functions of quasi-modular forms oft where P 18n−2 (q) are some quasi-modular forms of weight 18n−2. We will need the following nice formula ∂ E 2 (q) (t 1 −t) −1 = 1 12 . (D.54) Here we regard z 2 as free variable which is independent ofq, so that ∂ E 2 (q) z 2 = 0. Before we proceed further, we shall prove a useful general formula on E 2 derivative. Suppose P k is a rational function of quasi-modular forms, with modular weight k, then we have the formula where we use the notation t = log(q) = 2πiτ . A similar formula was used in [34] to derive of holomorphic anomaly equation in the Nekrasov-Shatashvili limit, and in [35] to studying the relation between holomorphic and modular anomaly equations in the Seiberg-Witten theory. The derivation of (D.55) uses the covariant derivative. It is well known that E 2 is not a modular form, but with an an-holomorphic shiftÊ 2 = E 2 − 6i π(τ −τ ) it becomes a modular form. To preserve the almost holomorphic structure, we shall use the covariant or Maass derivative , (D. 56) where k is the modular weight. The derivative with respect to E 2 can be related to the anti-holomorphic derivative∂τ = (∂τÊ 2 )∂Ê 2 = 6 πi(τ −τ ) 2 ∂Ê 2 (D. 57) We can compute the derivativē ∂τ D τ P k = (∂ τ + k (τ −τ ) )∂τ P k + (∂τ k (τ −τ ) )P k (D.58) Taking the holomorphic limit and cancel out the infinitesimal factor (τ −τ ) −2 we arrive at the formula (D.55). The above formula (D.55) is valid for any rational function P k of quasi-modular forms, with possible fractional powers, which satisfies the homogeneous relation with modular weight P k (λ 2 E 2 , λ 4 E 4 , λ 6 E 6 ) = λ k P k (E 2 , E 4 , E 6 ). (D.59) However it is a little tricky for the logarithmic function, which does not satisfy the above relation (D.59). Naively the logarithmic function has zero modular weight and the E 2 derivative should commute with τ derivative. But this is not the case, as we can easily compute with the formula (D. 55), (D.60) We will not need this logarithmic formula in this subsection, but it will be useful later. We can take derivative n times on a rational function of quasi-modular forms, and use formula (D.55) to compute ∂ E 2 ∂ n t P k = ∂ n t ∂ E 2 P k + n(k + n − 1) 12 ∂ n−1 t P k (D.61) Suppose P k (q 1 ) is a rational function with possible fractional powers of quasi-modular forms of q 1 , with modular weight k. We can expand it as a power series of z 2 (1−432) 3 with coefficients as rational functions of (quasi)-modular forms ofq, using the expansion (D.52) P k (q 1 ) = ∞ n=0 (t 1 −t) n n! ∂ ñ t P k (q), (D. 62) We can compute the derivative of E 2 (q) and use the two formulae (D.54, D.61) = ∂ E 2 (q 1 ) P k (q 1 ) + k 12 (t 1 −t)P k (q 1 ). (D. 63) We see the derivative vanishes when P k (q 1 ) has zero modular weight k = 0 and is modular, i.e. no E 2 dependence. Now we are have the necessary ingredients for proving the conjecture (2). First we consider the case of 1 − 864z 1 = E 6 (q) E 4 (q) 3 2 . We shall expand in terms of the modular parameter q 1 of the compact model E 6 (q) We can determine the coefficients f n recursively by expanding the modular forms of q 1 in terms ofq and compare with the left hand side. It is clear that f n (q 1 ) are rational functions of quasi-modular forms of q 1 of modular weight 18n + 6, with possible negative powers of E 4 , but no fractional powers. To show it has no E 2 dependence, we take derivative of E 2 (q) on both sides and use the formula (D.63) 0 = ∞ n=1 ∂ E 2 (q 1 ) f n (q 1 ) where we have use the fact that the coefficients have zero modular weight. Since z 2 is a free variable, we find ∂ E 2 (q 1 ) f n (q 1 ) = 0, i.e. it is modular.
We also need to check that there is no negative powers of E 4 (q 1 ) in f n (q 1 ). From the expansion (D.53) we see that the coefficients of z n 2 (1−432z 1 ) 3n always have the factor of E 4 (q 1 ) − 9n 2 . There could be also negative powers of E 4 from the derivative ∂t in (D.62) since each derivative increases the negative power of E 4 by one. However, we note that since a n has modular weight is 6n − 2, it can not be purely a power of E 6 (q) and always has a factor E 4 (q). So the (t 1 −t) n factor in (D.62) contributes a factor E 4 (q) n which cancel the one from taking derivative. Therefore f n (q 1 ) is a modular form.

E Expansion around a point on the conifold divisor
We now study the expansion of the higher genus topological string amplitudes around a point on the conifold divisor in terms of the local flat coordinates, and use the gap condition to fix the holomorphic ambiguity. Since the involution symmetry exchanges the two conifold divisors, it is sufficient to consider the simpler one ∆ 2 = 1 + 27z 2 . The simplest point to consider is the intersection point of the divisor z 1 = 0 and ∆ 2 = 1 + 27z 2 . This is a normal intersection, and we can choose the local coordinates as (z c1 , z c2 ) = (z 1 , z 2 + 1 27 ) (E.1) We solve the PF equations (2.10) perturbatively around this point. There should be 6 linearly independent solutions. We only need 3 of them with at most one logarithmic leading term in order to compute the flat coordinates. The three solutions are two power series solutions X 0 , X 1 and one logarithmic solution X 2 as the followings The local flat coordinates are computed as the ratio of the solutions t c1 = X 1 X 0 and t c2 = X 2 X 0 . For the power series solution, the flat coordinate goes like t c1 ∼ z c2 = z 2 + 1 27 , which is the expected asymptotic behavior of near a conifold point. For the logarithmic solution, the flat coordinate goes like t c2 ∼ log(z c1 ), which is similar to the behavior near large volume point, and we should use the exponential of the flat coordinate as the expansion parameter. So we define the expansion parameters in terms of flat coordinates t c := t c1 = X 1 X 0 , q c := exp(t c2 ). (E.5) We shall compute the asymptotic expansion of the propagators S ij , S i , S, where we have written the propagators in z i coordinate and for simplicity we denote S ij = S z i z j and S i = S z i . Here the propagators should simply transform as tensors. In the case of involution symmetry, there is a minus sign and shifts for S i and S because the involution map the point to a different branch of moduli space. But this should not happen locally.
have sufficient boundary conditions to fix the B-model formula up to any genus no bigger than 9(d B + 1). To see this, we further define a vector space where we denote the ± subscript universally as (−1) g+n+1 depending on the sign. Here we have also set m = [ 19(g−1) 3 ] in (E.8). The space Y (g,d B ) is exactly the space of remaining holomorphic ambiguities at genus g, after we impose the boundary conditions at the conifold and orbifold loci, and use the exact formulas for base degrees up to d B . Now we consider g = 9(d B + 1), then according to (C.10, C.11), the dimension of the first linear space in the direct sum (E.10) is The dimensions of the other spaces in (E.10) are smaller and also vanish, so we have Dim(Y (9(d B +1),d B ) ) = 0, i.e. all holomorphic ambiguities can be fixed at genus 9(d B + 1).
On the other hand, consider one genus higher at g = 9(d B + 1) + 1, then the dimension of the first linear space is computed similarly Dim(V