Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds

We study the mirrors of five-parameter Calabi-Yau threefolds first studied by Hulek and Verrill in the context of observed modular behaviour of the zeta functions for Calabi-Yau manifolds. Toric geometry allows for a simple explicit construction of these mirrors, which turn out to be familiar manifolds. These are elliptically fibred in multiple ways. By studying the singular fibres, we are able to identify the rational curves of low degree on the mirror manifolds. This verifies the mirror symmetry prediction obtained by studying the mirror map near large complex structure points. We undertake also an extensive study of the periods of the Hulek-Verrill manifolds and their monodromies. On the mirror, we compute the genus-zero and -one instanton numbers, which are labelled by 5 indices, as $h^{1,1}=5$. There is an obvious permutation symmetry on these indices, but in addition there is a surprising repetition of values. We trace this back to an $S_{6}$ symmetry made manifest by certain constructions of the complex structure moduli space of the Hulek-Verrill manifold. Among other consequences, we see in this way that the moduli space has six large complex structure limits. It is the freedom to expand the prepotential about any one of these points that leads to this symmetry in the instanton numbers. An intriguing fact is that the group that acts on the instanton numbers is larger than $S_6$ and is in fact an infinite hyperbolic Coxeter group, that we study. The group orbits have a 'web' structure, and with certain qualifications the instanton numbers are only nonzero if they belong to what we term 'positive webs'. This structure has consequences for instanton numbers at all genera.


Preamble
In this paper, we study mirror symmetry for a family of Calabi-Yau manifolds associated to the root lattice A 4 . This family was first investigated in relation to the modularity of its zeta-function by Hulek and Verrill [1]. These manifolds comprise a five-parameter family birational to varieties parametrised by a = (a 0 , a 1 , a 2 , a 3 , a 4 , a 5 ) ∈ P 5 which we call singular Hulek-Verrill varieties and denote 1 by HV a . They are embedded in the projective torus T 4 = P 4 \ {X 1 · · · X 5 = 0} as the vanishing loci of (X 1 + X 2 + X 3 + X 4 + X 5 ) These varieties on the projective torus T 4 admit a toric compactification, which we will review briefly in §2. Of particular interest are small projective resolutions HV of HV, which have smooth projective Calabi-Yau models [1]. We concentrate mostly on analysing these, and call them simply Hulek-Verrill manifolds.
These manifolds receive attention in the physics literature, since the periods of these manifolds (and their analogues in each dimension) are related to the banana Feynman graphs [2]. The particular manifolds (and quotients of) HV (1,1,1,1,1,1/ϕ) exhibit rank-two attractor points with interesting number theoretic properties, which were identified in [3].
The mirror-symmetric counterpart to the work of [3] involves a IIA setup. In [4], nonperturbative solutions were given to the attractor equations which involved instanton numbers, or Gromov-Witten invariants, giving a hint of microstate counting. This motivates us to study the geometry of the mirror Hulek-Verrill manifold focusing especially on aspects related to counting microstates of D4-D2-D0 brane systems on the manifold.
In studying the periods of HV we are naturally led to consider integrals of products of Bessel functions, similar to those considered in [5,6]. We find additional motivation for the present work in the connection between the manifolds HV and this topic.
While this paper was in preparation we received [2], which has overlap with the present work.

Outline of the paper
The analysis of the Hulek-Verrill manifolds presented in this paper occasionally becomes very technical. To avoid getting bogged down in details, we will give below a brief overview of the contents and main results of each section. In addition, we strive to keep different sections relatively independent so that the reader can focus on the details of results they find interesting.

A comment on indices
To give concise accounts of the different subjects that we touch upon, we adopt a specific index convention in §3, §4, and §6, and also within individual subsections of §2. While these conventions are strictly followed in their respective sections, they are not the same in different sections of the paper. The conventions are explicitly given in Table 1.

Section
Index Convention §2 Varies by subsection. §3 Greek indices run from 0 to 5. Latin indices run from 1 to 5. §4 Greek indices run from 0 to 5. Latin indices run from 1 to 5. §6 Latin indices run from 0 to 4. Distinct indices are understood to take distinct values.

Toric geometry of Mirror Hulek-Verrill manifolds
In §2, we briefly review the toric construction of the singular Hulek-Verrill manifolds HV as first discussed in [1]. Then we proceed to find a toric description of its small resolution. These are given as toric compactifications of intersections of two polynomials on a torus T 5 . We denote these manifolds by HV (a 0 ,...,a 5 ) , or more compactly by HV. We use the method of Batyrev and Borisov [7,8] to find the toric description of the mirror Hulek-Verrill manifolds HΛ. Somewhat surprisingly, these mirror manifolds turn out to be familiar spaces [9,10], given by the complete intersection Parenthetically, we note that this manifold is itself a remarkable split of the tetraquadric, (1. 4) Subfamilies exist that admit a Z 5 × Z 2 × Z 2 symmetry, or a subgroup thereof. The symmetry has a simple description: denoting the coordinates in each of these projective spaces by Y i,0 and Y i,1 , the symmetries act for all i as We write the most general expressions for the polynomials defining manifolds invariant under these symmetries. In particular, the manifold invariant under Z 5 × Z 2 × Z 2 is given as the simultaneous where m abcde are Z 5 invariant multidegree (1, 1, 1, 1, 1) polynomials: (1.7) It will turn out to be occasionally useful to consider the singular mirror Hulek-Verrill manifolds HΛ, which can be obtained by using the contraction procedure of [11], or equivalently by blowing down 24 degree 1 lines parallel to one of the P 1 's. In this way we obtain a family of singular varieties, which are birational to mirrors of the singular Hulek-Verrill manifolds HV found by using Batyrev's method [12].

Periods of the five-parameter family
The next section §3 deals with the periods of HV, which describe the variation of the Hodge structure as a function of moduli space coordinates. We study the five-parameter family (2.3). The overall scaling of coordinates a µ does not affect the vanishing locus, and thus we can identify the moduli space 2 with P 5 . The manifolds are singular on the loci where one of the coordinates vanishes, E µ = (a 0 , a 1 , a 2 , a 3 , a 4 , a 5 ) ∈ P 5 a µ = 0 , (1.8) and on the conifold locus Often it is necessary to work on an affine patch, for which we most often choose a 0 = 1, with Latin indices then running from 1 to 5. Results obtained in this patch apply in any patch a i = 1, after making a suitable permutation of indices.
On seeking differential equations obeyed by this period, we are led to the system These constitute a partial Picard-Fuchs system, giving 32 solutions among which we find the 12 periods 3 . These are the components of the vector 0 = ( 0 0 , 0 1,i , 0 2,j , 0 3 ) T , i, j = 1, . . . , 5 . (1.12) By a simple separation-of-variables argument, it can be shown that integrals of Bessel functions of the following form furnish a basis of solutions: where B i (ζ) is either K 0 (ζ) or iπI 0 (ζ). Naïvely there are 2 6 = 64 integrals of this type, but it turns out that at a generic point in the moduli space there are exactly 32 such integrals that are convergent. The analytic continuation of each integral outside of its domain of convergence is a linear combination of integrals of the form (1.13) that converge in the new region.
There is an additional equation which, in addition to those above, completely fixes the periods. After setting a 0 = 1, this takes the form of a polynomial in Θ with coefficients that are polynomials in a µ . In principle this operator is determined by the recurrence methods of [14], but for fully general a i these recurrence relations cannot be solved in a practical amount of time. It is possible, however, to choose constants s i and specialise the parameters to a i = s i ϕ, giving lines in the moduli space, and write a differential operator in terms of ϕ that governs the variation of the periods along these lines. In many cases, it is possible to find this remaining operator on these lines, and in our examples this operator obtained via the methods of [14] in fact factorises 4 . We give an example of such an operator in §3.3.
Despite lacking the explicit form of the general Picard-Fuchs system, we can fix the 12 periods among the 32 solutions of the partial system by imposing boundary conditions. These are found by matching the asymptotics of the solutions to the asymptotics near the large complex structure point predicted by mirror symmetry. We also give explicit series expansions for these periods near the large complex structure point.

Mirror map and large complex structure
The large complex structure points are located at the loci where all but one of the coordinates a µ vanish. Near the large complex structure point with a 0 = 0, the period vector in the integral basis can be written in terms of the prepotential F as The Y abc are topological quantities which we compute in Appendix C and the n p are the genus 0 instanton numbers of multi-degree p. We find the following relation between the period vector Π 0 in the integral basis and the period vector 0 in the Frobenius basis, found in §3: Π 0 = Tµ 0 , (1.14) with matrices Here I 5 denotes the unit matrix while 0 m×n and 1 m×n are matrices of the indicated dimension, all of whose entries are 0 or 1 respectively.
With the period vectors in the integral basis in hand, we can compute the instanton numbers by studying the Yukawa couplings y ijk . These are given by the formula but also have the following expansions in terms of the instanton numbers: (1. 16) Due to the permutation symmetry of the parameters a i , we can express many quantities in terms of the elementary symmetric polynomials. This results in a significantly less complicated series expressions which are far more amenable to computations. While we are still unable to reach the number of terms available in one-parameter computations, we find all the instanton numbers up to a total degree of 15, which we collect in Table 7.
In addition, we are able to compute the genus 1 instanton numbers by constructing the genus 1 prepotential using the expressions in [15]. Rather pleasantly, the prepotential turns out to be conceptually simpler than on the quotients studied in [3]. This is largely due to the fact that the distinct singular points on the moduli space of the quotient are replaced by the irreducible singular locus ∆ = 0 on the moduli space of HV. The limiting factor is the number of genus 0 instanton numbers we are able to compute, since those are needed to extract the genus 1 numbers form the prepotential. We are thus able to compute the genus 1 instanton numbers up to total degree 15, and we give these in Table 8.

Monodromies
In §5, we turn to computing the monodromies around the singular loci a 0 a 1 a 2 a 3 a 4 a 5 = 0 and ∆ = 0. As hinted by the fact that (1.13) is a function of √ a µ , this is most conveniently done by first classifying the singularities in coordinates √ a µ . Then the singular locus ∆ = 0 becomes a reducible union of codimension-1 hyperplanes of the form The monodromies around these loci can be found by numerically integrating the Picard-Fuchs equations on a path circling around these loci. Alternatively, one can find the linear relations between analytically continued Bessel function integrals in different regions, and use this to compute the monodromies. While the former approach is too difficult with arbitrary paths due to the complicated nature of the complete Picard-Fuchs system, we can integrate along various lines on which the Picard-Fuchs operator can be found as discussed above. By studying various different lines and using symmetry, we can use the resulting "reduced" monodromy matrices to piece together the full monodromies.
What makes this computation simpler than it first appears is the fact that the monodromy matrix around a conifold locus should be expressible in terms of a single vector: Here w is a 12-component vector that gives the integral basis components of the three-cycle vanishing at the conifold locus. Consequently, the vector w should also observe the symmetries relevant to each locus.
At first, we study the periods in the patch a 0 = 1, although later we find it useful to consider other patches as well. To find the partial monodromy matrices, we study lines of the form (a 0 , . . . , a 5 ) = (1, s 1 ϕ, . . . , s 5 ϕ), (1.19) where s 1 , . . . , s 5 are constants. To make the numerical computations tractable, we take at least two s i equal. To be concrete, consider the simple case where s 1 = s 2 = s 3 = s 4 = s 5 . Then, by symmetry and there are 6 independent periods, which form a vector Π 0 . (1.20) In the general case the monodromy matrices M can be written as where u i are 12-component column vectors Since some of the periods are equal on the line (a 0 , . . . a 5 ) = (1, s 1 ϕ, s 2 ϕ, . . . , s 2 ϕ), we cannot find the full monodromy matrces M directly by computing monodromies around the singular points on the line. Instead, we find reduced monodromy matrices M which give the monodromy of the vector Π 0 . These matrices take the form M = (û 0 ,û 1 ,û 2 +û 3 +û 4 +û 5 ,û 6 ,û 7 ,û 8 +û 9 +û 10 +û 11 ) , (1.23) where theû i are 6 component column vectorŝ By considering several lines and using symmetry arguments to simplify the computations, we are able to gain enough information to completely fix the full monodromy matrices.
Around a conifold locus, given the vector w w = (w 0 , w 1 , w 2 , . . . , w 2 , w 7 , w 8 , w 9 , . . . , w 9 ) , (1.25) the reduced 6 × 6 matrix M takes the form The reduced intersection matrix Σ is given by In this way we find 16 of the 32 vectors corresponding to the vanishing loci: w {0} = ( 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0) , w {0,1} = (−2, 0, 0, 0, 0, 0, 1, −1, 0, 0, 0, 0) , Near the large complex structure point at a 0 = a 2 = · · · = a 5 = 0, we have, in the natural integral basis, the period vector Π 1 , which is obtained by replacing the a 1 -dependence in Π 0 by a 0 and vice versa. By symmetry, in this basis, the monodromy around this locus is To find the corresponding monodromy matrix in the original basis of Π 0 , we just need to find the relation between these two bases. We find the matrix T Π 1 Π 0 (5.19) which takes us from one base to another. With this, we are able to find the monodromy matrix M {1} in the original basis: The other monodromy matrices of the form M {i} , M {i,j} and M {i,j,k} are found in a similar manner.

Counting curves on the mirror Hulek-Verrill manifold
In §6 we use use elementary geometric methods in tandem with the Kodaira classification of singular elliptic fibres [16,17] to directly count curves of certain multi-degrees on generic manifolds in the family HΛ.
Counting of these curves is based on the observation that HΛ can be viewed as an elliptic fibration with base P 1 × P 1 . While the generic fibre is an elliptic curve, it is possible to find the discriminant locus corresponding to base points above which the fibres are singular. According to Kodaira classification, the fibres over nodes of the discriminant locus are unions of two rational curves. By classifying the these fibres, we find all rational curves of degrees ≤ 3, and some of the higher-degree curves.
As the discriminant of the elliptic fibration is relatively simple for tetraquadrics, it is often beneficial to consider the singular manifolds HΛ i obtained by blowing down 24 lines along i'th copy of P 1 in the ambient space. On a generic manifold HΛ i , the discriminant locus has 200 nodes, of which  Table 2: Some symbols that are used throughout the paper with references to where they are defined.

Toric Geometry and Mirror Symmetry
We review the construction of Hulek and Verrill's manifold [1] following in part [3]. The starting point of their analysis is the five-parameter family HV (a 0 ,...,a 5 ) of singular varieties embedded in the projective torus T 4 = P 4 \ {X 1 · · · X 5 = 0} as the vanishing locus of These varieties can be compactified by using the standard methods of toric geometry (see for example [18]), giving in general a variety with 30 singularities. Outside of the discriminant locus 5 these have small resolutions, which constitute a smooth family that we call Hulek-Verrill manifolds HV (a 0 ,...,a 5 ) .
Particularly interesting examples of such manifolds are provided by a highly symmetric oneparameter subfamily, where a 0 = 1 and a 1 = · · · = a 5 = ϕ. These are characterised by a Z 5 × Z 2 symmetry, with the group action on the coordinates generated by where the addition is understood modulo 5. The action on the manifold is free outside of the points ϕ ∈ { 1 25 , 1 9 , 1} in moduli space where fixed points are present. This allows one to take a quotient with respect to these symmetries to get a one-parameter family of Calabi-Yau manifolds, which are smooth for moduli outside these isolated points.
As noted in [1], the varieties on T 4 defined by (2.1) are birational to complete intersection varieties in P 5 defined as the vanishing locus of two polynomials: This innocuous-looking transformation turns out to be useful for finding the (non-singular) mirror manifolds HΛ of the (non-singular) Hulek-Verrill Manifolds HV. Combined with the methods of Batyrev and Borisov [7,8,12], which we briefly review in §2.2, they allow us to find the mirror Calabi-Yau manifold as a subvariety of a suitable toric variety.
By standard methods of toric geometry, we can find the mirror manifolds HΛ and HΛ of HV and HV. As expected, we find that HΛ is singular and birational to HΛ. Figure 1  Coordinates on T Polytope face labels ρ n τ n Table 3: Quantities associated to the lattices N , N , N * = M and N * = M .

The polytopes corresponding to singular varieties
We group the symbols denoting various polytopes, Cox coordinates, and other related information by their associated lattices in Table 3. The lattices N and M associated to the singular varieties HV and HΛ are four-dimensional, and consequently for them the index i runs from 1 to 4. The lattices N and M are five-dimensional and for them the indices take values i = 1, . . . , 5.

Five-dimensional description
The polynomial P (X; a) contains 21 monomials in coordinates X 1 , . . . , X 5 . Writing these monomials using multi-index notation defines 21 vectors v n = (v 1 n , v 2 n , v 3 n , v 4 n , v 5 n ), n = 0, . . . , 20, in Z 5 The vectors v n make up the set (0, 0, 0, 0, 0) ∪ e i − e j | i, j = 1, . . . , 5 , i = j .  The dual lattice can be realised as a sublattice of M = N * Z 5 , with the basis given by The situation is a little more involved on the discriminant locus, for details see [1].
where e i and e i are the canonical bases of N Z 5 and M Z 5 . With these definitions we have that the canonical inner product gives a non-degenerate pairing: To find a convenient four-dimensional description for these lattices, we project N → N Z 4 and (2.10) Four-dimensional description of ∆ An equivalent way of arriving at the form of the four-dimensional polytope starts with going to an affine patch, say X 5 = 1, where the polynomial P (X; a) contains 21 monomials that are now of the form 1 , For the labelling of the faces, see appendix A. The 20 triangular prisms break up into two Z 5 × Z 2 transitive orbits, under the actions A and B given in (2.15), and the tetrahedra form one such orbit. The facets meet as displayed in Figure 2.
The polytope ∆ defines a fan whose cones are exactly those supported by the faces of ∆. This fan, however, is not simplicial, and consequently we wish to find a triangulation of ∆, which corresponds to a smooth fan. We find that there are two triangulations that respect the Z 5 × Z 2 symmetry. In the four-dimensional description, the action is a composition of the Z 5 × Z 2 in five dimensions and the projection to four dimensions. This gives and their images under Z 2 × Z 5 , together with the 10 simplicial cones supported by the tetrahedra.
The dual polytope ∆ * The polytope ∆ has a dual reflexive polytope ∆ * which is bounded by 20 planes These planes intersect ∆ * in 20 cubical faces. For the labelling of the faces (which manifests the explicit duality between these faces and the vertices of ∆), see appendix A. It follows that ∆ * is a convex hull of 31 lattice points that we label v 0 , . . . , v 30 . For explicit numbering, see again appendix A.
Note that the action of ς ∈ S 5 on N is subtle: we have to consider the action of S 5 on the fivedimensional lattice and then project this to back to the four-dimensional lattice. Doing this, one is left with the following action on the basis (2.20) The triangulation data serves as input for Batyrev's formula [12] for the Hodge numbers of smooth members of the families of Calabi-Yau manifolds corresponding to the polytopes ∆ and ∆ * : where pts( Θ) and int( Θ) denote the number of lattice points and interior lattice points of Θ. Θ and Θ * are faces of ∆ and ∆ * , respectively. These formulae are manifestly compatible with mirror symmetry. From the toric descriptions for the manifolds HV and HΛ, we find the Hodge numbers

The Method of Batyrev and Borisov
To find the small resolutions HV and HΛ of the singular manifolds related to the polytopes discussed above, we use the toric geometry methods pioneered by Batyrev and Borisov [7,8,12]. We briefly review this approach 6 .
Given a variety defined as a vanishing locus of the set of n Laurent polynomials {P i } n i=1 , one can study the intersection of affine hypersurfaces V (P i ) corresponding to the polynomials P i form a nef-partition of a reflexive polytope ∆, we can define an ambient space P ∆ * ⊃ T corresponding to the fan associated to ∆ * . The toric variety P ∆ * has a desingularisation P ∆ * , corresponding to a maximal projective triangulation of ∆ * . The surfaces V (P i ) have closures V (P i ) ⊂ P ∆ * and V (P i ) ⊂ P ∆ * , and we can define the closures of the intersections M = V (P 1 ) ∩ ... ∩ V (P n ) and M = V (P 1 ) ∩ ... ∩ V (P n ). It can be shown [7] that if M is non-empty and irreducible, and also dim M ≥ 3, then M defined in this way is a smooth manifold 7 .
To find the mirror variety of the smooth manifold constructed in this way, we note that by the definition of a nef-partition where Mink denotes the Minkowski sum. In addition, we can define the convex hull of the union of the polytopes ∆ i : One can show [7] that the polytope ∇ * so defined is also a reflexive polytope. In particular, it has a well-defined dual polytope ∇. This, and the dual polytope ∆ * of ∆, can be shown to be expressible in terms of n smaller polytopes {∇ i } n i=1 : where the sum is again a Minkowski sum, and {∇ i } n i=1 gives a nef-partition of ∇. Now we can define the mirror manifold of M as follows: first we use the polytopes ∇ i to define a set of polynomials {Q i } n i=1 and a desingularisation P ∇ * corresponding to a maximal projective triangulation of ∇ * .
Then the mirror manifold W of M can be expressed as the closure V (Q 1 ) ∩ · · · ∩ V (Q n ) of the variety {Q 1 = · · · = Q n = 0} ⊂ T. Due to the way W is constructed, it follows that it is smooth and irreducible if and only if M is [7].
There is an algorithm for computing the Hodge numbers of varieties defined in this way [19,20]. In the case of complete intersection varieties, it is more complicated than Batyrev's original formulae for the Hodge numbers [12]. We will not review the details here, and simply note that some computer algebra packages, such as PALP [21], provide an implementation of the algorithm.

The polytopes corresponding to small resolutions
Small polytopes ∆ 1 , ∆ 2 To find the toric descriptions of the non-singular manifolds HV and HΛ, we study the polytopes ∆ 1 , ∆ 2 ⊂ Z 5 . Their vertices correspond to monomials in the polynomials P 1 and P 2 , defined in (2.3), that define on P 5 a variety birational to HV. We work directly in an affine patch where X 0 = 1. Then the two polytopes can be expressed as These, and the other polytopes defined this subsection, are schematically represented in two dimensions in Figure 3. Using these two, we can construct two larger polytopes as their Minkowski sum and the convex hull of their union From the definition of convex hull, it follows immediately that the vertices of ∇ * are exactly ±e i with i = 1, . . . , 5. Its 32 faces are the four-dimensional simplices of the form τ n = Conv ( 1 e 1 , 2 e 2 , 3 e 3 , 4 e 4 , 5 e 5 ) , i ∈ {−1, 1} , (2.27) given by intersections of ∇ * with bounding planes The polytope ∆ contains in total 31 lattice points, Thus it can be written as a convex hull of 30 lattice points Its only internal point is the origin, and it has 62 faces that are hypercubes, given by intersections with planes It can be shown that {∆ 1 , ∆ 2 } is a nef-partition of ∆.

The Hulek-Verrill manifolds and their mirrors
Having studied the relevant lattice geometry, we are ready to turn to the toric geometry associated to the triangulations of the fans corresponding to the triangulated polytopes we have found in the previous sections. We will give both the singular manifolds HV and HΛ and their resolutions HV and HΛ as a vanishing locus of a set of polynomials inside the relevant ambient toric variety. We also find some basic properties of these manifolds, which will be relevant in the following sections.
The quantities associated to each manifold are summarised in Table 4.

The singular Hulek-Verrill Manifold HV
The ambient toric variety P ∆ * in which HV can be embedded corresponds to the polytope ∆ * . To the vertices we associate Cox coordinates ξ 1 , . . . , ξ 30 . The ambient variety can then be given by the usual construction as The scalings (C * ) 26 correspond to linear relations between the vectors corresponding to the vertices of ∆ * . F is the union of sets given by the simultaneous vanishing of Cox coordinates associated to rays not lying in the same cone. Excising this from C 30 prior to quotienting in (2.38) ensures a well-defined toric variety 8 .
To study the Calabi-Yau manifold HV ⊂ P ∆ * , we identify the coordinates X 1 , . . . , X 4 with the coordinates Ξ 1 , . . . , Ξ 4 on the torus, which we define in terms of Cox coordinates in appendix A. Then the Calabi-Yau manifold can be written as as a subset i =j (2.39) We are chiefly concerned with the five-parameter subfamily where the polynomial in (2.39) takes the form P given in (1.1). The generic manifold in this family contains 30 nodal singularities on HV \ T 4 , which can be seen by considering the local patches corresponding to the triangulation of the polytope ∆ * [1]. They have, however, small projective resolutions HV (a 0 ,...,a 5 ) , which are smooth Calabi-Yau manifolds. We will discuss the toric description of these manifolds later in this section.

The singular mirror Hulek-Verrill Manifold HΛ
We can use Batyrev's construction [12] to find the mirror manifolds of the singular Hulek-Verrill manifolds. The manifolds that are of interest to us turn out to be singular. However, they are birational to the mirror manifolds of the small resolutions mentioned above. The construction of the resolved manifold in this way is somewhat complicated, but in §2.4 we give another method of finding this resolution.
We have already found the vertices of the dual polytope ∆ * in (2.18). These, together with the interior points, correspond to the monomials Each of the indices i, j, k, l are unequal and take values in {1, 2, 3, 4}. The intersection of a generic mirror singular Hulek-Verrill manifold with the torus T 4 is given by the closure of the vanishing One obtains this by taking the most general polynomial with monomials (2.41) and multiplying through by Y 1 Y 2 Y 3 Y 4 , which gives the same variety on T 4 .
Given the triangulation (2.16) of ∆ discussed in §2.1, we can consider the local affine patches A σ i corresponding to the simplicial cones σ i . Equivalently, we can choose suitable 4-tuples of the Cox coordinates η i to act as the local coordinates on patches isomorphic to A 4 . It is only necessary to study the six local patches related to the fans given in (2.16) and a single patch generated by any tetrahedron. The other local patches are obtained from these by Z 5 × Z 2 symmetry.
As an example, let us consider the cone σ 1 . The coordinates associated to the generators of this cone are Since the generators corresponding to these coordinates belong to the same simplicial cone, we can set the other coordinates to unity, and thus identify the local coordinates with those on the torus as We can immediately find the local coordinates on The equalities denote identifications with the coordinates on the affine patch A Aσ 1 . Thus on this patch, we can make the identifications with the torus coordinates as Note that this corresponds to Z 5 acting on the global coordinates as which of course corresponds to the Z 5 action e i → e i+1 of the five-dimensional lattice M , projected down to four dimensions by (2.10).
Writing the polynomial Q in global coordinates gives, for generic values of the moduli, an irreducible multi-degree (2, 4, 4, 4) polynomial. A member of this family is generically smooth, but smooth members are not birational to mirrors of Hulek-Verrill manifolds HΛ (a 0 ,...,a 5 ) .
Instead, it turns out that we must only consider those whose defining polynomials can be written in the form where α, β, γ, and δ are multidegree (1, 1, 1, 1) polynomials in the coordinates Y 1 , . . . , Y 4 . A manifold with this property has exactly 24 singularities, which can be resolved in order to obtain a smooth variety.

The Hulek-Verrill manifold HV
As we have already remarked, Hulek and Verrill noted that the singular variety HV (a 0 ,...,a 5 ) defined by the equation on the toric variety P ∆ * is birational to the subvariety of P 5 defined by the two polynomials It is possible to develop this further by studying the two equations P 1 = P 2 = 0 on the torus T 5 and finding the toric closure of this variety. This can be achieved using the techniques reviewed briefly in §2.2. In §2.1, we have studied the polytopes ∆ 1 and ∆ 2 whose vertices correspond to the monomials in P 1 and P 2 , and found the polytope ∆ * which gives the ambient space P ∆ * . The Cox coordinates and coordinate scalings defining the ambient variety are given in Appendix A.
We can analyse this variety further by specialising to various local patches. We only need to analyse the patches that are not related by symmetry.
The Cox coordinates associated to the generators of the cone σ 1 in (2.37) are Using the leftover scalings to set the other 57 Cox coordinates ξ to unity, we can identify the invariants Ξ 1 , . . . , Ξ 5 as By further identifying these Ξ i with the coordinates X i on the torus, we can write the polynomials P 1 and P 2 as The analogous relations for the remaining cones, σ 2 and σ 3 , can be found in a similar manner.
By studying the equations P 1 = P 2 = dP 1 ∧ dP 2 = 0, it is not difficult to see that generically the variety HV does not have singularities. In agreement with the original analysis of Hulek and Verrill [1], we find that there are singularities if and only if The algorithm in [19,20], implemented in PALP [21], gives the Hodge numbers of this variety as (2.56) Thus we identify these manifolds as the small projective resolutions of the singular manifolds HV.
When a 0 = 1 and a i = ϕ for i = 0, the manifold admits a Z 5 × Z 2 ⊂ S 5 × Z 2 symmetry group, which acts freely outside of the singular locus ∆ = 0. The actions of Z 5 and Z 2 on the coordinates can be written as with the indices understood mod 5. The Hodge numbers of the varieties obtained by taking the quotients are given in Table 5.

The mirror Hulek-Verrill manifold HΛ
The mirror Hulek-Verrill manifold can be defined as the vanishing locus of two polynomials corresponding to the polytopes ∇ 1 and ∇ 2 inside the ambient variety P ∇ * associated to the triangulated polytope ∇ * .
The monomials associated to the vertices of with the indices understood to take distinct values. The monomials associated to ∇ 2 are simply the inverses of these.
Looking at the vertices of ∇ * listed in Appendix A, we see that the ambient variety P ∇ * is nothing but the product (P 1 ) 5 . The Cox coordinates are the homogeneous coordinates on each P 1 , which we often denote by P 1 i with i = 0, . . . , 4 if there is a need to distinguish between different factors in the product (P 1 ) 5 . The coordinates Y i on the torus are identified with the affine coordinates giving the homogeneous coordinates on the i'th copy of P 1 . It is convenient to intoduce the following monomials of homogeneous coordinates where a, b, c, d, e ∈ {0, 1}. Using these, the most general polynomials associated to ∇ 1 and ∇ 2 can be written as For a special choice of coefficients A and B, the vanishing locus of Q 1 and Q 2 admits Z 5 , Z 5 × Z 2 or Z 5 × Z 2 × Z 2 symmetry [9]. These act freely, and thus can be used to obtain smooth quotient manifolds. Denoting the generator of the Z 5 as S, the generator of the first Z 2 as U and the second Z 2 as V , we can take the symmetry transformations to act on the coordinates as where addition is again understood modulo 5. The symmetries S and V can be seen to descend from the Z 5 and Z 2 symmetries acting on the polytope ∇ * . to write down the polynomials invariant under there symmetries, it is convenient to introduce the Z 5 invariant combinations of the monomials M abcde , The polynomials defining the Z 5 symmetric manifolds can be found by specialising the coefficients A and B so that the vanishing locus Q 1 = Q 2 = 0 is invariant under Z 5 , or equivalently by finding the Z 5 orbits of Q 1 and Q 2 . In this manner, we find (2.65) To find the defining polynomials in the Z 5 × Z 2 symmetric case, we can further demand that the vanishing locus of the polynomials is invariant under the Z 2 generated by V , which gives us two polynomials of the form Alternatively, we can demand the the vanishing locus is invariant under the second Z 2 generated by U . In this case, the polynomials can be written as (2.66) Note that the actions of U and V are exchanged under a suitable redefinition of coordinates, and therefore we can choose either of these two forms for the polynomials defining the Z 5 × Z 2 invariant variety. Note also that in the latter case the polynomials Q 1 and Q 2 are not each Z 2 invariant, but instead are mapped to each other under the action on Z 2 , thus keeping their mutual vanishing locus invariant.
Finally, we can consider the variety invariant under the full Z 5 × Z 2 × Z 2 . In this case we can write the defining polynomials as (2.67) It turns out that the varieties defined in this way and their quotients under their respective symmetry groups are smooth Calabi-Yau manifolds, which we can identify as mirror manifolds of the five-parameter family HV (a 0 ,...,a 5 ) . We call these mirror Hulek-Verrill manifolds HΛ. The Hodge number of the corresponding quotient varieties were already found in [9]. We reproduce these in Table 6.  Counting the parameters in the polynomials seems naïvely to produce too many parameters compared to the Hodge numbers. However, by taking into account rescalings; remaining automorphisms of the ambient variety (P 1 ) 5 ; and SL(2, C) transformation of the polynomials; we find that the number of free parameters in the defining polynomials agrees with the Hodge numbers. We leave the details to Appendix D.
Finally, we note that this variety is birational to the singular HΛ. This is most easily seen by observing that the intersection HΛ ∩ T 4 can be obtained from HΛ by blowing up a suitable set of degree 1 rational lines, as we will discuss in detail in §6.

The Periods of HV
The periods of the HV manifold are essential for understanding both the geometry and physics of the Hulek-Verrill manifolds as well as their mirrors. The series expansions of periods about large complex structure points allow for a mirror-symmetry computation of the instanton numbers for the manifold HΛ. In this section, We derive series expressions that we utilise to perform this computation in §4. Additionally, the periods as functions of the complex structure moduli of HV are instrumental in describing string theory compactifications on HV. We hope to return to this point in future work, to study flux vacua in type IIB string theory compactified on HV.
Our approach begins with investigating some differential equations satisfied by the fundamental period 0 , which is long known to admit concise descriptions [1,14]. We find a set of PDEs which, together with asymptotic data coming from mirror-symmetry considerations, allow us to find all periods within the large complex structure regions of moduli space. We go further by using the methods of [14] to study an ODE satisfied by the fundamental, and indeed all, periods. This latter equation is used to analytically continue the periods, and with the data we obtain from this, we can give expressions for the periods in all regions of moduli space.
We derive formulae that express all periods using integrals of products of Bessel functions. To our knowledge, this is the first appearance of such equalities and we anticipate that these also have applications in the study of banana amplitudes. For instance, the expansion (4.16) of [2] expresses the full non-equal mass 4-loop banana integral in the large momentum region of parameter space, where the simplest available expression (their equation (2.10)) does not converge. The authors gave the first few terms of the series expansions of the functions that are used as a basis. The integral expressions that we use to describe the periods also fit this purpose after a change of basis. Appropriate generalisations of our expressions relevant to higher-dimensional Hulek-Verrill manifolds will perform the same task for higher-loop banana diagrams.

Moduli space
The parameters a 0 , · · · , a 5 in the equation (2.3) defining the manifold HV constitute a set of projective coordinates for P 5 . The parameters a 0 , · · · , a 5 appear symmetrically, which we can use to great effect to describe different regions in the moduli space. A convenient atlas for P 5 is given by the six sets where one of the projective coordinates is nonvanishing. In the following sections, we mostly work in the patch where a 0 = 0, but the arguments go through in the other five patches mutatis mutandis. Accordingly, the Latin subscripts i, j, k, . . . are always understood to run from 1 to 5, whereas the Greek subscripts µ, ν, λ, . . . are taken to run from 0 to 5.
It is not difficult to see that the manifold HΛ is singular on the locus We denote the irreducible components in this locus by E µ = (a 0 , a 1 , a 2 , a 3 , a 4 , a 5 ) ∈ P 5 a µ = 0 The intersections of 5 of these hypersurfaces turn out to be large complex structure points, or points of maximal unipotent monodromy, as we will verify in §4 by computing the monodromies around these hypersurfaces explicitly.
As we have reviewed earlier in §2.4, the Hulek-Verrill manifold has conifold singularities on the locus It is often useful to consider the square roots √ a i as coordinates on the moduli space. This of course gives a multiple cover. We can, without loss of generality, choose branches for the square roots with Re[ √ a i ] > 0. The functions that we study are related to those in other branches via monodromy transformations a i → e 2πi a i around the large complex structure point.
In the coordinates √ a i it is convenient to study the vanishing loci of the individual factors in ∆.
Let I be a subset of indices in {0, . . . , 5} and I c be its complement in {0, . . . , 5}. Then we define the following closed components D I corresponding to each set I, sketched in Figure 4:

The fundamental period
The holomorphic period for HV (a 0 ,...,a 5 ) can be found by integrating the holomorphic three form over the torus. We briefly review this procedure. As we consider the torus, we can use the equation (2.50) defining HΛ in order to obtain this period by the Dwork-Katz-Griffiths method [24].
Near the large complex structure point at a 1 = a 2 = · · · = a 5 = 0, one finds the series expansion (3.6) We will next identify a set of differential operators that annihilate this fundamental period, the expectation being that the other periods should satisfy the same equations. Although this set of equations is demonstrably not the full Picard-Fuchs system, we can proceed using the high degree of symmetry and the asymptotics for the periods found from mirror symmetry considerations. In this way, we are able to find expressions for the periods using the 32 solutions to this partial Picard-Fuchs system. As a very non-trivial check, we are able to compute several genus 0 instanton numbers in §4, the first few of which match the numbers that we find from geometric arguments in §6.

The ordinary differential equation obeyed by the fundamental period
Consider the sequence of c n which gives the coefficients in the series (3.6). In principle, one could use a recurrence relation that c n satisfies in order to write an ODE -containing derivatives only with respect to a 0 , but coefficients functions of all a µ -which is satisfied by the fundamental period. Such recurrence relations (which themselves depend on the a i ) were studied by Verrill in [14], wherein a method for determining such a recurrence was given. It was shown that c n is a holonomic sequence, solving a linear recurrence with polynomial coefficients.

Partial differential equations obeyed by the fundamental period
We adopt the following notation for certain differential operators: Note that on a single term a p a −n 0 , where |p| = n, the action of the operator Θ is the same as that of −a 0 ∂ 0 . Using this fact, we find that the fundamental period 0 obeys the following five differential equations: These equations are, after a change of variables, equivalent to the differential equations (4.8) of [2]. In addition, we have equations obtained by taking differences of the above equations, or by directly inspecting (1.10):  There is a similar relation obtained from the other four equations L i f (a) = 0. Together these five relations (3.12) enforce as was to be proved. We remark that while the system of equations L i F = 0 has a unique holomorphic solution, it is shown below that the system has a solution-space of dimension greater than Dim H 3 (HV) = 12. Therefore it cannot be the entire Picard-Fuchs system, for it is not sufficiently constrained. The additional restriction on the solution space comes from the differential equation discussed in §3.3, which is too difficult to write down in full generality.
Next, we argue using the Frobenius method that there are 32 functions taking the form of power series multiplied by logarithms of the a i that solve (3.9). To see this, one sets up an indicial equation. Take a solution ansatz where the = ( 1 , . . . , 5 ) is a five-components multi-index consisting of as-yet undetermined real constants and f n (p + ) is defined by replacing x! by Γ(1 + x) in (3.13). One can compute There are five i with respect to which we can either differentiate zero or one times. In total all such choices give us 2 5 = 32 independent solutions. These solutions can be distinguished by their logarithmic dependencies on the a i .

Separation of variables
Upon expanding the operators θ i , the differential equations L i,j F = 0 become Making a separation-of-variables ansatz F (a) = 5 j=1 G j (a j ) and simplifying 1 . (3.17) Employ the traditional separation of variables logic: both sides of this equation respectively depend only on a i and a j , and so both must equal a constant value. With a certain prescience, we will denote this constant by z 2 /4. Attention should then be turned to the ordinary differential equation that the G i satisfy: This has the following general solution: C 1 (z) and C 2 (z) are arbitrary functions of z with no x-dependence. Therefore, for any choice of distributions C 1 (z), C 2 (z), the equations L i,j F = 0 for i, j = 1, . . . , 5 have solutions where the five functions B j are each taken to be modified Bessel functions I 0 or K 0 . This brings us closer to the periods, but at this stage of our reasoning, only looking at the system L i,j F = 0, there is still a considerable degree of ignorance as to what the function C should be and which combinations of these solutions we should take to give the periods.
We proceed by noting the following expression for the fundamental period, valid in the regime Re We give a proof of this claim in §B. The identity (3.21) suggests that C(z) should be taken to be K 0 ( √ a 0 z). Indeed, by replacing the I 0 functions in the above integral with K 0 functions, we can form 32 functions f that obey the equations L i,j f = 0. These 32 functions can be seen to satisfy the system L i f = 0, and therefore must furnish a basis of series solutions of the system L i f = L i,j f = 0. To be sure, the 32 functions obtained in this way have powers series that form a basis for the linear span of the 32 Frobenius solutions given by the construction in §3.5.
On symmetry grounds, there will be a role for functions obtained by replacing the K 0 with an I 0 in patches a i = 1 in the moduli space. The reason for this is that, from the global perspective, a 0 is not distinguished from the a i .

Determining closed form expressions for all periods
We have seen that the partial Picard-Fuchs system given by (3.10) and (3.9) should have exactly 32 solutions. Furthermore, we have seen that the integrals of Bessel functions of the form furnish a set of solutions to our partial differential equations. The B µ ( √ a µ z) above are replaced by a conveniently normalised modified Bessel function: either K 0 ( √ a µ z) or iπ I 0 ( √ a µ z). Naïvely it seems that this would give us 64 solutions. However, not all of these converge simultaneously.
Indeed, an integral of this form converges in the region of the moduli space where The negative sign for Re[ √ a i ] is chosen when B 0 (z √ a i ) = K 0 (z √ a i ), and the positive sign when . This follows from demanding that the product of Bessel functions decays exponentially in the limit z → ∞ and recalling the asymptotics of the Bessel functions for large z: The boundary between the different regions of convergence is exactly the restriction of the conifold locus ∆ = 0 to the real plane.
On a generic 10 point in the moduli space corresponding to a non-singular manifold, there are exactly 32 convergent integrals of Bessel functions of the form (3.22). This is seen as follows: every curve of the form divides the space into two regions, those "above" and "below". The curve itself belongs to the discriminant locus. There is exactly one Bessel function integral of the form (3.22) that converges almost everywhere above the curve (3.25) and exactly one converging almost everywhere below the curve. As there are 32 such curves we find exactly 32 convergent integrals at any given point. We (3.26) These have the following convenient properties where Re∆ denotes the space of all points that satisfy any of the equations (3.25).
In the subset of each patch where they converge, these Bessel integrals satisfy the partial differential equations (3.10) and (3.9). There are exactly 32 solutions to these equations, so it follows that the periods, which should solve the differential equations, can be expressed in terms of the convergent Bessel function integrals in any patch. In the next subsection we will present an argument, based on known asymptotics, to fix the periods as sums of these Bessel integrals in the regions U {i} and U {0} . To find the correct linear combinations of these integrals to give the periods in other regions we study the ODE of §3.3. Choosing values a i = s i ϕ in this ODE gives a differential equation that the restrictions of the periods to these lines must satisfy. Given enough lines, we can always find enough equations to completely fix the periods in terms of the Bessel integrals.
To find the relation between the bases of periods in different patches, we analytically continue the Bessel integrals from one region to another. In practice, the easiest way to do this is to numerically integrate the Picard-Fuchs equation along a line crossing multiple regions, and then 10 In addition to the restriction of the discriminant locus to the real plane, the Bessel function integrals also diverge on points whose real parts satisfy the equation (3.25). 11 The open sets cover the moduli space apart from points which satisfy (3.25).
find the relations between each pair of bases. By the normalisation of the Bessel function integrals, these matrices relating different bases are integral. In what follows, we will not need most of these relations, hence we do not record them here. However, an important special case that we will be using relates the basis of periods near the large complex structure point in the patch U {0} to the basis in the patch U {i} , where there is another large complex structure point.
For instance, we can study the line (1, ϕ, ϕ 20 , . . . , ϕ 20 ) where the periods satisfy the Picard-Fuchs equation L (6) f = 0, with the operator L (6) given by (3.7). The Bessel function integrals near a i = 0 that satisfy this equation are given by where we have used the following shorthand for the Bessel functions appearing here Given the operator L (6) it is indeed easy to check that these integrals satisfy the equation.
By integrating the Picard-Fuchs operator L (6) numerically, we can find the continuation of the period vector π 0 to the region |ϕ| > 89.7214, giving the following relation between the vectors π 0 and π 1 : (3.32) We have written the Bessel function integrals in π 0 and π 1 in this particular way because these are natural restrictions of the 12 periods to the line (a 0 , . . . , a 5 ) = (1, ϕ, ϕ 20 , . . . , ϕ 20 ). The generic 12-component period vectors are given by π 0 = a 0 iπ ∞ 0 dz z π 0 0 , π 0 1,1 , . . . , π 0 1,5 , π 0 2,1 , . . . , π 0 2,5 , π 0 3 T , (3.33) in which (3.34) The vector π (1) is given by permuting the indices 0 and 1. In terms of these quantities, restricted to the line, we have a natural way of writing the relations (3.32) in a symmetric form. For example, the relation corresponding to the third row of the matrix can be written as The coordinates a 2 , a 3 , a 4 , and a 5 must appear symmetrically in all of these relations. Thus we are able to guess that the relations in the case where all of the coordinates are unequal are We can verify this expectation by studying the line (a 0 , . . . , a 5 ) = (1, ϕ, ϕ 50 , ϕ 100 , . . . , ϕ 100 ), which singles out the period π 0 1,2 , and thus allows verifying the above relation in the case j = 2. The other relations then follow by symmetry. Working in this way, we find that in general the period vectors π 0 and π 1 are related by (3.37)

The periods near large complex structure points
The set U {0} is a neighbourhood of the large complex structure point at E 1 ∩ · · · ∩ E 5 , and the U {i} are neighbourhoods of other large complex structure points. In the region U {0} , according to the discussion above, the convergent integrals are of the form.

38)
A basis for the periods can be given as 12 linear combinations of these functions 12 . We apply a boundary condition such that this set of 12 functions furnish a Frobenius basis for the periods: as one approaches the point a = 0 one function should be holomorphic in the a i ; five should contain a single logarithm; five should be quadratic in logarithms; and one should be cubic in logarithms.
Given that the five-parameter family of Calabi-Yau manifolds in question is symmetric under permutation of the a i , we shall choose combinations of terms (3.38) that share this symmetry.
We normalise the basis so that the leading logarithms have coefficient 1. When we consider the case with a given number of logarithms, we shall add multiples of the solutions with smaller powers of logarithms so that another Frobenius condition is met: the power series that multiply powers of logarithms lower than the highest such power in a solution vanish at a i = 0.
These conditions fix the periods in the Frobenius basis completely. The relation between the periods in the Bessel integral basis π µ and the periods in the Frobenius basis µ is Explicitly, this means that the single-logarithm periods near the large complex structure point at a 1 = · · · = a 5 = 0 are given by The period cubic in logarithms is (3.42)

Series expansions
We collect some series expressions below that are used to express the periods as series. Denote by H n the n th harmonic number, and by ψ (m) the Polygamma function. (3.43) We shall express the periods using the following intermediate series: The Bessel function expressions (3.40)-(3.42) can be expressed near the point a 1 = a 2 = · · · = a 5 = 0 in terms of the following series. Details of the derivations are delegated to appendix §B.

Mirror Map and Large Complex Structure
To determine the mirror map, we recall that near the large complex structure limit the period vector takes the form [25] Here z i are the projective coordinates on the Kähler moduli space of HΛ (a 1 ,...a 5 ) . We often use the corresponding affine coordinates t i def = z i z 0 , so that for example the complexified Kähler class of HΛ (a 1 ,...a 5 ) is given by where e i generate the second integral cohomology H 2 (HΛ, Z). The quantities F 0 and F i are derivatives of the prepotential F, which near the large complex structure point is given in terms of the genus 0 instanton numbers n p by The quantities Y abc are given by topological quantities related to HΛ: We compute these quantities in §C. The Y ijk are given by For the other numbers one finds Note that as a consequence of the highly symmetric nature of the manifold HΛ, none of the couplings depend on the indices i, j, k. It is then convenient to write the non-vanishing quantities Y abc as The large complex structure points are located on loci where all but one of the parameters a i vanish. For concreteness, we are going to concentrate on the large complex structure point at a 1 = · · · = a 5 = 0 in the affine patch a 0 = 1. We denote the integral period vector in this patch by Π 0 . The other cases are related to this one by the permutation symmetry.
As usual, we can identify the affine coordinates t i of the Kähler moduli space with the periods by The last expression gives the asymptotic form in the limit a 1 , . . . , a 5 → 0, and O(a) denotes terms that are of order 1 or higher in any a i . Inverting this map order-by-order one finds the coordinates a i in terms of t i . It is useful to write the resulting map in terms of the elementary symmetric polynomials 13 σ i (q): Near this large complex structure point the periods in the Frobenius basis have the asymptotic form log a m log a n l<m<n log a l log a m log a n On the other hand, the asymptotics of Π 0 can be read directly from the prepotential and are given by By requiring that the asymptotic forms match 14 , we find that the period vectors must be related by

9)
13 Due to the identity q 5 1 − q 4 1 σ1 + q 3 1 σ2 − q 2 1 σ3 + q1σ4 − σ5 = 0, this expression is not unique. Unique expressions are obtained, for example, by using this identity to eliminate occurrences of σ5, or explicit appearances of powers of q1 higher than four.
with matrices

Yukawa Couplings and Instanton Numbers
To find the instanton numbers, we compute the Yukawa couplings where the indices I, J, K run from 1 to 5. The couplings can be computed using the relation between forms on the manifold HΛ and the ring of defining polynomials modulo the Jacobian ideal [26]. Alternatively, one can find y ijk as a series in q by a direct computation. As we are mostly interested in finding the instanton numbers, the latter method is sufficient. We express the Yukawa couplings in terms of the period vectors as where Σ is the matrix giving the standard symplectic inner product We then express the Yukawa coupling in terms of the quantities q i . The y ijk above is computed in the gauge z 0 = 0 0 . To be able to compare this to the expression (4.14) we need to transform to gauge z 0 = 1 in addition to the tensor transformation: Due to the symmetries, there are only three independent Yukawa couplings up to permutation of coordinates. For the purposes of finding the instanton numbers, we need only one of these, say y 111 . Expressing it as series in q, we find Similar expressions hold for y 112 and y 123 . The series expansions for the Yukawa couplings can be written in terms of the instanton numbers as (4.14) By comparing this to the series expansion (4.13), we can identify the first few instanton numbers as listed in Table 7.

Genus 1 instanton numbers
It is possible [15] to define a genus 1 prepotential, which effectively counts the genus 1 curves. In the topological limit it can be expressed as det ∂a ∂t f + const. , (4.15) where f is a holomorphic function which can be fixed by imposing appropriate boundary conditions. In particular, the prepotential F 1 must be regular inside the Kähler moduli space. In the large complex structure limit, F 1 has an expansion Here d p are the genus 1 instanton numbers, and the Euler function is given by To get the correct growth in the large complex structure limit, f must contain a factor of 5 i=1 a −3 i . Outside the loci a µ = 0, we require that F 1 is singular only on the discriminant locus given by the vanishing of In the one-parameter cases, where the singularities appear as points ϕ * in the moduli space, conifold singularities produce a factor of (ϕ − ϕ * ) −1/6 . We assume that a straightforward generalisation of this holds in the multiparameter case, and thus we take c = − 1 6 . With this choice we find the genus 1 instanton numbers up to degree 15, given in Table 8.
Intriguingly, a number of patterns can be identified. We conjecture some here, based on our tables.
• When n I = n J , one has n k·I = n k·J for k ∈ N.
• There are 8

Recovering the results on quotient manifolds
Using these results, instanton numbers on the quotient manifolds HΛ Z 5 and HΛ Z 10 can be recovered. The first few instanton numbers for the one-parameter manifolds are reproduced from [3] in Table 9.
We fix attention here to the Z 5 quotient. The action of Z 5 on the cohomology H 2 (HΛ) is given by (4.20) where addition is understood modulo 5, and we have taken this action to be consistent with the choice (2.57) for the action of Z 5 on HV. This also induces an action on H 4 (HΛ) via Hodge duality. The Z 5 action on the periods of HV is The locus of Z 5 symmetric Hulek-Verrill manifolds is a 1 = · · · = a 5 def = ϕ, and the corresponding mirror manifolds are found on the locus t 1 = · · · = t 5 def = t. Thus one identifies the generator of the second cohomology of the one-parameter manifold with e = e 1 + e 2 + e 3 + e 4 + e 5 . (4.22) The prepotential on the one-parameter family is identified with that of the five-parameter family by  We can identify the other topological numbers Y abc and the instanton numbers in a similar fashion. Since the group Z 5 has no proper subgroups, curves on the manifold must either belong to an orbit of 5 curves or be mapped to themselves. If a curve with Euler character χ is mapped to itself by Z 5 then the quotient map will take said curve to a curve with Euler character χ/5. In particular, the Euler character of a genus 0 curve is 2, and so there cannot be any genus 0 curves fixed by the Z 5 action. Then the relation between the n I of Table 7 and the n k of Table 9 is |I|=k c I n I = n k . For the genus 1 numbers d I the relation is more complex since a genus 1 curve has χ = 0, so there can exist genus 1 curves, invariant under the symmetry group, whose quotient is again a genus 1 curve. The formula analogous to (4.26) is now 27) and serves to compute the numbers d inv k,k,k,k,k of Z 5 invariant genus 1 curves of degree k. A small check is that the numbers d k,k,k,k,k − d inv k,k,k,k,k should be divisible by 5, which they are, to the extent of the tables.
The fact that all the instanton numbers we have computed agree with those computed on the oneparameter families through increasingly intricate relations provides a non-trivial consistency check of the results of sections §3 and §4.  Table 9: n k and d k are respectively the genus 0 and genus 1 degree k instanton numbers for the quotient manifolds. The quantity κ is taken to equal 1 or 2 depending on whether one is working on the Z 10 or Z 5 quotient. This table is reproduced from [3], where these numbers were given up to degree 20, with thousands being computable.

Monodromies
We wish to find the monodromies around the loci E µ and D I defined in (3.2) and (3.4). In the next subsection, we will compute the monodromy around the varieties E i using the series expansions for the periods around the large complex structure point. For the loci D I , we use numerical integration of the Picard-Fuchs equation to find the monodromies. As we do not have the general five-parameter Picard-Fuchs equation and such an equation would in any case be impractical for this purpose, we use the Picard-Fuchs equations for one-parameter subfamilies as discussed in §3.3. Finally, using the relation between the natural basis of periods in the patch a 0 = 0 and a i = 0, we are able to compute the monodromies around E 0 in §5.3.

Monodromies around the large complex structure points E i
The monodromy matrices around the loci E i can be read directly from the asymptotics of the period vector Π 0 in the integral basis. These correspond to coordinate transformations a i → e 2πi a i , or alternatively t i → t i + 1. These transformations give the following monodromies.
The monodromies around other loci E i are obtained by swapping the second and (i + 2)'th column and row and the seventh and (i + 7)'th column and row with each other.

Monodromies around the loci D I
We now set a i = s i ϕ, a 0 = 1 with s i complex constants. ∆ becomes a polynomial of degree 16 in ϕ. This has 16 roots, which are the intersections of the singular locus ∆ = 0 with the plane a i = s i ϕ. We will find particularly simple Picard-Fuchs operators when some of the s i are equal. In these cases some of the periods become equal, hence there exists an operator of degree < 12, whose independent solutions are exactly the distinct periods. These differential equations can be integrated numerically, yielding the monodromy matrices for the independent periods.
Of course the matrices found this way do not give the complete monodromy, as not all of the 12 periods are independent on the lines that we study. However, there is a natural relation between these "reduced" matrices and the full monodromy matrices, which can be used, together with the S 5 symmetry, to find the full monodromy. To exemplify this process, let us consider the case where s 1 = s 2 = s 3 = s 4 = s 5 . This leaves a set of 6 independent periods, as The general monodromy matrix, giving the monodromy transformation of the periods around a singularity ϕ * , can be written as where u i are 12-component column vectors Since some of the periods are equal, we cannot find their individual contributions to this matrix from the reduced monodromy matrix. Instead, the reduced matrix takes the form M ϕ * = (û 0 ,û 1 ,û 2 +û 3 +û 4 +û 5 ,û 6 ,û 7 ,û 8 +û 9 +û 10 +û 11 ) , (5.4) where nowû i are 6 component column vectorŝ Relations like this constrain the full 12 × 12 monodromy matrices. We can construct the full matrices from this data by numerically integrating the Picard-Fuchs equation along several paths in the complex line.
Finally, to make the computation slightly simpler, we use the fact that the singularities at ∆ = 0 correspond to conifolds. It is expected that the monodromies around the conifold loci take the form where w is a 12-component vector that gives the cycle vanishing at the conifold point. Thus we can reduce the problem to finding 16 vectors corresponding to the different components D I of the singular locus.
To get an idea of how the computation proceeds, we briefly explain the computation of some monodromies in a relatively simple example. To be precise, we study the case We have 6 independent periods and so can find, using the procedure outlined in §3.3, a Picard-Fuchs operator of degree 6. This operator has solutions 0 0 (ϕ), 0 1,1 (ϕ), 0 1,2 (ϕ), 0 2,1 (ϕ), 0 2,2 (ϕ), and 0 3 (ϕ). In the ensuing discussion, we shall find use for the shorthands The discriminant expressed in terms of ϕ is in this case, up to a multiplicative constant, Each of these factors corresponds to an intersection of a component D I with the line. In this way, we can associate each factor with such a component: (5.14) As a consistency check, it can be seen that the matrix around ν is given by a product of reduced monodromy matrices: The matrices corresponding to the remaining loci can be found using similar techniques. This is made slightly more complicated by the fact that paths on the lines s 1 = s 2 = s 3 = s 4 = s 5 only circle intersections of multiple components. Perhaps the easiest way to circumvent this is to consider a new case where s 1 = s 2 = s 3 = s 4 = s 5 = s 1 , and permutations thereof. In the case s 1 = s 2 = s 3 , D {0,1,2} intersects the plane a i = s i ϕ in a point that is distinct from the other components. This computation, together with symmetry considerations, leads us to a form for the monodromy matrix where the vanishing cycle is given by  However, the remaining matrices can be constructed from the known 16 by a change of indices 0 ↔ i. By symmetry, the matrices that are related to each other by such a permutation must be equal. We must, however, take into account that the monodromy transformations obtained in this way are expressed in different bases. Changing all to a common basis, which we take to be the symplectic basis where Π 0 is given by (4.1), gives matrices with different entries. Thus, for example where T Π 1 Π 0 , given explicitly in (5.19), is a change of basis matrix from the canonical integral basis in the patch a 0 = 1 to the canonical integral basis in the patch a 1 = 1. We will see another explicit example of this in the next subsection where we use this observation to compute the monodromy around the locus E 0 "at infinity".

Monodromy around infinity, E 0
The remaining singular locus is the locus a 0 = 0, which, in the patch a 0 = 1 corresponds to the monodromy around infinity. Due to the S 5 symmetry, we know that the locus a 0 = 0 is on par with the other loci a i = 0. The only essential difference to the earlier computation is that the basis where the monodromy around a 0 takes the same form as the monodromies around other loci a i is different from the basis we have been using in this section thus far.
To find the appropriate change of basis, we use the matrix T 01 from (3.37), which gives the relation between the period vectors π 1 and π 0 , whose components give the periods as combinations of Bessel function integrals. Using the matrices T i π i and T Π i i , we can change from this basis to the integral basis of Π. Note that due to the symmetry, the relation of the vectors π 1 to the integral period vector Π 1 is same as that of π 0 to Π 0 , so that T Π 1 π 1 = T Π 0 π 0 . The transformation from Π 1 to Π 0 is thus given by The monodromy of Π 0 around a 0 = 0 is, by symmetry, equal to the mondromy of Π 1 around a 1 = 0, which directly allows us to find the monodromy of Π 1 around the locus a 0 = 0: We have used the inverse of the matrix M E 0 because the contour's direction is reversed when changing patches.

Recovering monodromies for the quotient manifolds
Finally, let us briefly comment on the relation of the results presented here to those found for the quotient manifolds in [3]. Specialising to the locus a i = ϕ, a 0 = 1, the discriminant vanishes for ϕ ∈ { 1 25 , 1 9 , 1}. The locus D {0} is associated to the first of these points, the loci D {0,i} to the second, and D {0,i,j} to the last.
On the locus a i = ϕ, a 0 = 1, only four of the elements of Π 0 are independent. Collect these into the reduced period vector Π 0 . This is related to the integral period vector Π Z 10/κ of the quotient manifold HV/Z 10/κ by a matrix T κ .

Counting Curves on the Mirror Hulek-Verrill Manifold
There is an interesting problem in directly counting the numbers of various curves of different degrees on the Hulek-Verrill manifold and its quotients. This serves multiple purposes, such as confirming the predictions of mirror symmetry and counting microstates for some configurations of branes wrapped on various cycles on the manifold. In this section, we will find the rational curves up to degree 5, and verify that their number agrees with the instanton numbers of §4.
It is good to recognise that the manifolds in HΛ can be realised as blowups of singular tetraquadrics HΛ i with 24 nodes, using the procedure of [11]. HΛ i are limits of the family corresponding to the configuration Members of the family HΛ are elliptically fibred manifolds, and we are able to compute the discriminant of the fibration using standard methods [27]. It turns out that the the first few low-degree rational curves appear as irreducible components of singular fibres of the elliptic fibration (see Figure 5).
πm,n πm,n Figure 5: Structure of the fibrations relevant to counting some rational and elliptic curves. L i denote the lines on HΛ that are blown down to obtain the singular mirror Hulek-Verrill manifold HΛ j with the birational map denoted by π j . HΛ j is an elliptically fibred manifold with base P 1 × P 1 , and a generic fibre F j . On the discriminant locus ∆ = 0 of the elliptic fibration, the fibre becomes singular. On a special set of points B, corresponding to nodes of the discriminant locus, the degenerate fibre is a union of two rational curves.
The explicit embeddings of curves depend non-trivially on the coefficients in the defining polynomials, but the curve counts for generic members of the family of mirror manifolds agree. For this reason we will, in place of explicit expressions, discuss properties of a generic member of the family HΛ.
Parts of our discussion are best framed in terms of various embedding maps with different degrees. Amongst these appear numerous context-specific rational functions. For this reason we will often use the symbols r k (z), r k (z), to denote a ratio of two situation-dependent polynomials of degree k. Two instances of these symbols in this section should not automatically be understood as referring to the same function. In this section Latin indices run from 0 to 4. When two different indices appear in an expression, they are understood to refer to distinct numerical values.

Blow-Down and Elliptic Fibration
The configuration matrix of HΛ is of the form considered in [11], which means that we can use the contraction procedure to obtain a quadric manifold HΛ i defined by one equation: We frequently distinguish the five P 1 factors in the product (P 1 ) 5 by subscripts. For example P 1 i denotes the i'th such P 1 , and has projective coordinates Y i,0 , Y i,1 . Throughout this section, we use This makes the equations simpler, and the instances where projective coordinates are needed for statements to be strictly correct are few. Nonetheless, all polynomials in this section can be homogenised using projective coordinates and in this way any minor ambiguities relating to points at infinity are cleared up.
To see in detail how the process depicted in (6.2) works, let us consider the contraction with respect to the coordinate Y i . The equations defining the manifold HΛ can be written as with α i , β i , γ i , δ i each being a linear function of the four coordinates that are not Y i . Note that in (6.3) there is no sum over the repeated i. The pair of conditions (6.3) is equivalent to the single matrix equation Existence of a solution is equivalent to the determinant of the matrix vanishing, that is We denote the variety defined by { Q i = 0} ⊂ (P 1 ) 4 as HΛ i . One can see from (6.5) that HΛ i is a conifold. Since the functions α i , β i , γ i , δ i are multilinear, the corresponding configuration matrix is indeed of the form (6.2).
Note that the varieties HΛ i are birational to HΛ. The projection π i : HΛ → HΛ i defined by gives the birational map between the varieties. Given a point (Y j , Y k , Y m , Y n ) ∈ HΛ i , with α i = 0 or γ i = 0, the equations Q 1 = Q 2 = 0 are solved by the unique point The manifold HΛ is generically a smooth elliptic threefold, while HΛ i is an elliptically fibred singular variety (see Figure 6). To see this explicitly, let us choose the base of the fibration to be P 1 m × P 1 n . We can view the polynomial Q i as a biquadratic whose coefficients depend on Y m and Y n .
where A a,b are functions of the base coordinates Y m , Y n . The exact form of these functions depends on the choice of the Calabi-Yau manifold HΛ. This defines a biquadric subvariety E i;m,n of P 1 j × P 1 k , which is a Calabi-Yau variety of dimension one, and thus an elliptic curve. This has a configuration matrix This is a one-dimensional Calabi-Yau manifold, and so an elliptic curve.
HΛ E i;m,n HΛ i P 1 m × P 1 n .
π i πm,n Any biquadratic in P 1 m × P 1 n can be transformed into the Weierstrass form [27]. To this end, one first computes the quadratic discriminant of (6.7) with respect to Y j .
One computes the two "Eisenstein invariants of plane quartics" defined in [27] for this polynomial: Each b is a function of Y m and Y n These can be used to write the Weierstrass form of the elliptic curve as 10) The discriminant of this elliptic curve is It is useful to observe that the discriminants satisfy the relations ∆ i; m,n = ∆ j; m,n = ∆ k; m,n , (6.12) In other words, for the purposes of computing the discriminant on the base P m × P n , it does not matter which contraction we choose. We plot the zero loci for three ∆ i; m,n in Figure 8.
In case the manifold is symmetric under Z 2 or Z 2 × Z 2 , the discriminant satisfies one or both of the following symmetry relations: (6.14) A sketch of ∆ for such a Z 2 × Z 2 symmetric case is given in Figure 7. The vanishing locus of ∆ corresponds to the singular locus of elliptic fibres. The types of singular fibres on elliptic surfaces have been classified by Kodaira [16,17].  As generically ∆ i; m,n is irreducible, a generic point on the curve ∆ i; m,n = 0 corresponds to a singularity of the type I 1 . In other words the fibre over a generic point over {∆ i; m,n = 0} ⊂ P 1 m ×P 1 n is a nodal curve. This is related to the fibration structure of the manifold. Namely, the generic fibre over the projection HΛ → P 1 n is a K3 surface. Furthermore, a K3 surface can be realised as an elliptic fibration over P 1 m with exactly 24 nodal curves. As ∆ i; m,n is a bidegree 24 polynomial, a generic fibre over P 1 n is an elliptically fibred P 1 m with 24 nodal fibres. In addition to these generic points, the discriminant curve ∆ i; m,n = 0 has singularities. We find that on HΛ i these fall into two categories, corresponding to cases I 2 and II in the Kodaira classification. In the generic case there are 200 points of type I 2 and 192 of type II. These account for all 392 singularities on a generic curve. In accordance with the Kodaira classification, on singularities of type I 2 the polynomials Q i (Y m , Y n ) factorise, with each factor corresponding to an irreducible curve. The two components meet at two points, which are the singularities of the fibre. The only exceptions to this are fibres which contain degree 5 rational curves on HΛ -the second component of such a fibre is a degree 1 rational curve. When this curve is parallel to P i , it is exactly the line which has been blown down to obtain HΛ i , and thus does not appear in the fibres on HΛ i .  In what follows, we mostly study the fibres on the singular varieties HΛ i . However, using the birational map between HΛ i and HΛ we can lift the curves on HΛ i found this way to curves on HΛ. Outside of the exceptional divisors the lift preserves the structure of the fibres. The two-component fibres of Kodaira type I 2 are unions of degree 1,2,3,4, and 5 rational curves. In particular, the singular fibres include all lines, quadrics and cubics. We discuss each of these cases in detail in the following subsections §6.3, §6.4, and §6.5. First, however, it is convenient to briefly review some general aspects of curves on (P 1 ) 5 .
It is often convenient to study the lines and other curves on the singular spaces HΛ i , where their connection to the elliptic fibration can be immediately appreciated. Given a curve C, and a projection π to a base B, then C may project to a curve of B, or project to a point. If C projects to a curve, it is said to be horizontal in the projection π, and if C projects to a point it is said to be vertical with respect to π.
In the following we will study each projection π j , and we will sometimes say that a vertical curve is parallel to the projection and a horizontal curve is orthogonal to the projection. We will study each case in turn, and finally show that the lines can be uniquely associated to a unique degree 5 line and to a node in the discriminant ∆ i; m,n .

Complete Intersection Curves on (P 1 ) 5
It turns out that the curves we consider in the following can be expressed as complete intersections of four polynomials in (P 1 ) 5 . The degrees and Euler characteristics of such curves are susceptible to elementary techniques. Complete intersections on (P 1 ) 5 can be systematically searched for, and doing this we obtain some evidence, consistent with the prediction of mirror symmetry, that there are no more curves than those we find here. We consider one-dimensional varieties defined by four equations The two-form dual to the subvariety p α = 0 is given by where J i is the Kähler, or equivalently volume, form of P 1 i . Then the dual form of the curve The sum runs over all permutations of {0, . . . , 4}. The i'th degree of a curve dual to C is The total Chern class of the curve (6.15) is given by . (6.19) It is straightforward to compute the Euler characteristic from the first Chern class: These formulae give the degrees and genera of various curves in the following sections. The degrees defined in this way will also agree with the degrees of isomorphisms ϕ : P 1 → C.
As we are interested in curves in the Calabi-Yau manifold HΛ, we need to make sure that the curve C lies completely within this manifold. In the language of algebraic geometry, this is equivalent to requiring that the radical of the ideal generated by the polynomials p i contains the polynomials Q 1 and Q 2 which define the HΛ manifold.

Lines
Every degree 1 rational curve in (P 1 ) 5 is given by a set of four linear equations, each in a single variable 15 . These read, for some j ∈ {0, 1, 2, 3, 4} and each s ∈ {0, 1, 2, 3, 4} \ j, In this way y = (y i , y k , y m , y n ) defines a line L j , which is necessarily parallel to P 1 j . Using the data of equations (6.21), the formulae (6.18) and (6.20) tell us that which is exactly as expected for a line. For a line L j to lie on HΛ, the solutions to (6.21) must additionally satisfy Q 1 = Q 2 = 0. A substitution reveals that this condition amounts to α j (y) + β j (y)Y j = 0 , γ j (y) + δ j (y)Y j = 0 . (6.23) Therefore the y must solve α j = β j = γ j = δ j = 0, and so gives a singularity on HΛ j . As has already been mentioned, these equations have 24 solutions for each j. There are therefore 5 × 24 = 120 lines. In the Z 5 symmetric case, the permissible values of y group into Z 5 orbits and the quotient procedure leaves us with 24 lines. Similarly, in the Z 2 symmetric cases, the involution Y i → −Y i (or equivalently Y i,0 ↔ Y i,1 ) identifies two lines. On HΛ/Z 5 × Z 2 there are therefore 12 lines, each descending from a family of 10 lines on the covering space. Finally, the generic Z 5 × Z 2 × Z 2 quotient contains exactly 5 lines.

Orthogonal Lines
For definiteness, let us consider the projection π 4 , the lines L 2 , and take the elliptic fibration E 4;0,1 with base P 0 × P 1 . The lines L 2 on HΛ can be understood to arise as blow-ups of singular points y on HΛ 2 , and can be given by the embedding z → (y 0 , y 1 , z, y 3 , y 4 ) . (6.24) The projection π 4 then takes this line to a line in HΛ 4 , given by the embedding z → (y 0 , y 1 , z, y 3 ) . (6.25) Thus L 2 forms part of the fibre of E 4;0,1 lying over the basepoint (y 0 , y 1 ). This fibre can tautologically be instantiated as the curve defined by the equation Reflecting the fact that this fibre contains a line and hence is reducible, the above polynomial factorises into degree 1 and degree 3 pieces (in homogeneous coordinates). The first factor is of course the equation of the image of the line L 2 on HΛ 4 .
Other maps ψ i;m,n;j are defined similarly, with the privileged role of Y 4 , Y 0 , Y 1 , Y 2 in this construction replaced by Y i , Y m , Y n , Y j . We display the interplay between these maps and projections in Figure 9.
j ψ i;m,n;j πm,n Figure 9: A chain of birational maps allows us to see lines L (a) i , corresponding to a singularity of HΛ j at y (a) explicitly as singular fibres on HΛ i viewed as a fibration over P m × P n . The polynomial Q i (Y m , Y n ) factorises into two factors, one of degree (0, 1), corresponding to the line, and the other of degree (2, 1). This latter factor corresponds to a projection of a degree 5 curve down to HΛ i .

Parallel Lines
Let us now shift our attention to the line L 4 , which is mapped to point 16 y by π 4 . By symmetry, over the point (y 0 , y 1 ) on the base P 0 × P 1 in HΛ, the fibre is given by the union of the line L 4 together with a degree 5 curve C (0,0,2,2,1) , which meets the line in two points. Projecting this fibre down to HΛ 4 maps the line to a point y, and the degree 5 curve to a degree 4 curve C (0,0,2,2) , which intersects itself at the point y. So there exists a birational map P 1 → C (0,0,2,2) z → y which is not, however, an isomorphism due to the self-intersection. Such a curve will not fit Kodaira's classification, which can be traced back to the fact that HΛ 4 is singular. Indeed, the lift of the fibre is an union of two rational curves meeting at two points, and thus corresponds to a node in the discriminant locus of the fibration HΛ. Upon projecting down to HΛ, this becomes a node of the locus ∆ 4; 0,1 = 0. An alternative way of arriving at the same conclusion is by noting that, as we have remarked previously, ∆ 4; 0,1 = ∆ 2; 0,1 , and by a previous subsection, L 4 corresponds to a node of ∆ 2; 0,1 = 0.
A straightforward generalisation of the the results of the last two subsections reveals that the 72 lines L i , L j , and L k , together with the degree 5 curves, account for 72 of the nodes of the discriminant locus ∆ i; m,n = 0. The locus has in total 200 nodes, the rest of which turn out to correspond to curves of degrees 2, 3, and 4, as we will see in what follows.

Quadrics
The analysis of irreducible degree 2 curves proceeds largely along the same lines. Algebraic quadrics on HΛ can be expressed, for a triple k, m, n and with constants q k , q m , q n , as the complete inter- Here, p is an irreducible multi-degree (1, 1, 1, 1, 1) polynomial. With i, j denoting the pair in {0, 1, 2, 3, 4} \ {k, m, n}, the equations (6.30) define a curve C with deg s (C) = δ s,i + δ s,j , χ(C) = 2. (6.31) While this is not the most general form of degree 2 curve on (P 1 ) 5 , we will show that only curves of this form lie in HΛ. To ensure that a curve defined by (6.30) lies in HΛ, we must have that, The square root indicates the radical of the ideal p , which in this case is the ideal itself. As p is irreducible and all three polynomials Q 1 , Q 2 , p are multidegree (1,1,1,1,1), this requires p = CQ 1 or p = CQ 2 , with C a constant. Further, we must have either Q 1 = Q 2 or one of the Q's vanishing at Y k = q k , Y m = q m , Y n = q n . We cannot have both Q's vanishing after this specialisation. In general there are 24 values of {q k , q m , q n } for which these conditions are satisfied. There are 10 ways of choosing the triple k, m, n, and so we find 240 curves of degree 2 on HΛ. In the Z 5 , Z 5 × Z 2 , and Z 5 × Z 2 × Z 2 symmetric cases, these curves come in families of 5, 10, and 20, respectively, so taking the quotient by Z 5 gives exactly 48 curves on HΛ/Z 5 , 24 on HΛ/Z 5 × Z 2 , and 12 on HΛ/Z 5 × Z 2 × Z 2 . This agrees with the results of [3].

Cubics
Cubic curves whose multidegree is a permutation of (1, 1, 1, 0, 0) can be expressed as complete intersections. The most general cubic curves that can be defined by four multilinear equations are of the form This defines a curve C 3 with Curves of this form include all cubics lying in HΛ. To sit in HΛ, the ideal generated by these polynomials must contain the polynomials Q 1 and Q 2 . This condition is equivalent to requiring that there are coefficients d a , e b such that when Y m = c m , Y n = c n For a quintuple (i, j, k, m, n) there are in general exactly 112 solutions to these equations. Summing over the 10 distinct choices of (i, j, k, m, n) gives us 1120 curves of degree 3, which once again come in Z 5 , Z 5 × Z 2 , and Z 5 × Z 2 × Z 2 invariant families in the symmetric cases. Taking the quotients with respect to Z 5 , Z 5 × Z 2 , and Z 5 × Z 2 × Z 2 leave 224, 112, and 56 curves of degree 3 respectively, in agreement with [3].
As was the case with the lines and quadrics, the cubics also appear as singular fibres of elliptic fibrations, and in fact account for the remaining 56 nodes of the discriminant locus ∆ i; m,n = 0. Take again (i, j, k, m, n) = (4, 2, 3, 0, 1) to expedite the discussion, and consider the cubic curves C (0,0,1,1,1) . The projection of this curve to HΛ 4 is a quadric C (0,0,1,1) . As before, this indicates that the polynomial Q 4 (Y 2 , Y 3 ) factorises into two components, both of degree (1, 1). The isomorphisms with P 1 are of the form z → (c 0 , c 1 , z, r 1 (z)) . (6.38) The quantity β 4 α 4 determining the lift to a curve on HΛ is a priori a ratio of two degree 2 polynomials. However, this is a component of a reducible elliptic fibre inside of which we already have a curve of total degree 3, therefore the two polynomials α 4 , β 4 must share a factor so that the lifts are curves C (0,0,1,1,1) . The isomorphisms with P 1 are given by z → (c 0 , c 1 , z, r 1 (z), r 1 (z)) . (6.39)

Summary
This completes the classifications of fibres over the nodes of the discriminant curves on singular varieties HΛ i (over the base P m × P n ), and their lifts to HΛ. We summarise our findings in Table 11 and Table 12, taking i = 4, m = 0, n = 1 for concreteness.

A. Toric Geometry Data
Here we gather some data related to the polytopes and toric varieties discussed in section §2.

C. Computing the Topological Quantities Y abc
To find the triple intersection numbers Y ijk , we first note that e i ∧ e i = 0 for every i. Therefore the only non-vanishing triple intersection numbers are those with all indices different. To find these numbers, we recall that e i is dual to a hypersurface {Y i − y i = 0} ⊂ HΛ, where y i is a constant. The intersection of two of these hyperplanes gives an elliptic curve, which in turn intersects a third hyperplane generically in two points. Therefore the Y ijk are given by Naïvely, the numbers Y ij0 would equal HΛ c 1 (HΛ) ∧ e i ∧ e j and thus vanish. This argument is not correct, and in fact Y ij0 can in some cases take the value 1/2. Based on the gamma class [28], it is expected that in the one-parameter case one can have Y 110 = 0 exactly when Y 111 is even. On the quotient HΛ/Z 5 the triple intersection number Y 111 is 24, so Y 110 = 0. The fiveparameter prepotential is related to the prepotential for one-parameter manifolds essentially by setting t 1 = · · · = t 5 = t and dividing by 5. Thus we concretely deduce that the quantities Y ij0 do in fact vanish.

D. Parameter Counting
The polynomials (2.65)-(2.67) defining the manifolds HΛ and their various quotients contain a number of parameters, which can be viewed as the complex structure parameters of the family HΛ. Naïvely it would seem that there are more free parameters in the defining polynomials than there are complex structure parameters. However, a more careful consideration will show that upon correctly accounting for redundancies, the parameter counts indeed agree.
Consider, for concreteness, the varieties in the family HΛ which are symmetric under Z 5 × Z 2 , which we take to be those generated by S and V as in (2.63).
We wish to determine the independent parameters in the polynomials Q 1 and Q 2 defining this symmetric variety. There are at least two sources of redundancy. The first is that different polynomials can generate the same ideal. The second arises from automorphisms of the ambient variety P 1 5 .
We begin by considering the most general Z 5 -invariant polynomials: To have a variety that is invariant under the Z 2 transformation V : We demand that the ideal Q 1 , Q 2 is invariant under the action of V . In this case this reduces to demanding that V Q 1 and V Q 2 are linear combinations of Q 1 and Q 2 : for some M ∈ GL(2, C) . The only residual redefinitions of Q 1 and Q 2 are those that keep the diagonalised M fixed, that is rescalings of Q 1 and Q 2 . Leaving these scalings unfixed for the time being, the condition (D.5) can be solved to give Demanding the condition (D.6) fixes most of the automorphisms of (P 1 ) 5 /Z 5 , but there is one remaining family of Möbius automorphisms of the form T : for all i . (D.7) Transformations of this form preserve the condition (D.6). The images of Q 1 and Q 2 can be written down, but the generic form is slightly complicated. We note that By choosing k suitably, we can force the coefficient of m 00000 to vanish. Upon redefining the remaining parameters the polynomials Q 1 and Q 2 become Finally, we can eliminate two parameters by rescaling. This leaves two polynomials with five independent parameters.