A Generalized Construction of Calabi-Yau Models and Mirror Symmetry

We extend the construction of Calabi-Yau manifolds to hypersurfaces in non-Fano toric varieties, requiring the use of certain Laurent defining polynomials, and explore the phases of the corresponding gauged linear sigma models. The associated non-reflexive and non-convex polytopes provide a generalization of Batyrev's original work, allowing us to construct novel pairs of mirror models. We showcase our proposal for this generalization by examining Calabi-Yau hypersurfaces in Hirzebruch n-folds, focusing on n=3,4 sequences, and outline the more general class of so-defined geometries.

generalized the well-known Calabi-Yau complete intersections of hypersurfaces in products of projective spaces [2,3,4] so as to allow certain of the hypersurface defining equations to have negative degrees over some of the factor projective spaces, and so necessarily use Laurent (rational) monomials amongst the defining equations [1,5]. Given the novelty of such models and the physics phenomena they exhibit already within the (m 0) infinite sequence of hypersurfaces in Hirzebruch n-folds F (n) m , 1 in Section 2 we analyze the gauged linear σ-model (GLSM) world-sheet field-theory [8,9] corresponding to these sequences, focusing on n = 3, 4. We also map out the associated enlarged (complete) Kähler moduli space, i.e., "phases," by considering all possible triangulations of the spanning polytope (and its convex hull) associated to the embedding non-Fano toric variety.
In order to analyze the GLSM ground states, which form a toric variety, the secondary fan [8,10,11,12,13], in Section 3 we generalize the toric methods [14,15,16,17,18,19] to cases where Laurent superpotentials naturally appear in the defining equations of the (Calabi-Yau) subvarieties. In particular, our extension includes a large class of non-convex and possibly self-crossing (VEX) polytopes and corresponding fans, which contain flipped, i.e., reversely oriented cones (faces). We then show that such generalizations: (a) produce Laurent monomials in the defining equations of transversal 2 Calabi-Yau hypersurfaces, and (b) automatically realize natural pairs of mirror Calabi-Yau n-folds generalizing earlier results [20,21,22], associated to "trans-polar" pairs of VEX polytopes.
In Section 4 we explicitly compute the Euler and Hodge numbers of the Calabi-Yau hypersurfaces in F (n) m to demonstrate that these key numerical invariants of the trans-polar pairs of oriented VEX polytopes (a) evaluate exactly as they do for convex polytopes, and (b) exhibit all requisite aspects of the mirror relations. Section 5 summarizes our results and concluding comments, while computational details are collected in the appendices. While this proof-of-concept paper illustrates the various toric geometry techniques by focusing on Hirzebruch n-folds [5] and their Calabi-Yau hypersurfaces, more general examples and further details may be found in the companion paper [23].

The gauged linear sigma model
Recent work [1,5] has shown that there are significant merits to constructing Calabi-Yau algebraic varieties at least some of the defining equations of which contain Laurent monomials, 1 Using methods of classical algebraic geometry [6,7], we have found that the classical topological data of certain sequences of such constructions exhibit a periodicity [5], which is broken by quantum effects. Since classical physics models on the Calabi-Yau hypersurfaces in the Hirzebruch n-fold F (n) m are equivalent to those in F (n) m+n , respectively, the transformation F (n) m → F (n) m+n is a classical (discrete) symmetry. Its breaking by quantum effects such as instanton numbers [5] then represents a novel (stringy) quantum anomaly.
2 A function f (x) is transversal if f (x) = 0 and df (x) = 0 have no common solution other than the "baselocus" (such as x = 0); we then also refer to the zero-locus f −1 (0) := {x : f (x) = 0} as being transversal. Throughout this paper, we focus on this distinctly algebraic quality, and defer its relation to the subtler complex-analytic property of smoothness (related to Cauchy-Riemann conditions and similar) for later. and that standard methods of algebraic geometry and cohomological algebra can be adapted to compute the requisite classical data. For applications in string theory and its M-and F-theory extensions, it is desirable to find a world-sheet field theory model with such target spaces.
For well over two decades now, the standard vehicle to this end is Witten's gauged linear sigma model (GLSM) [8,24,25], where fermionic integration leaves a potential for the scalar fields of the general form: Here σ a is the scalar field from the a th gauge twisted-chiral superfield, x i and F i are respectively the scalar and auxiliary component fields from the i th "matter" chiral superfield X i , Q a i is the charge of the i th chiral superfield with respect to the a th U (1) gauge interaction, and the r a are the contributions from the Fayet-Iliopoulos terms. In supersymmetric theories and especially when acting on chiral superfields, gauge groups are typically complexified and the GLSM naturally has U (1, C) C * actions -which are the "torus actions" in the toric geometry of the space of ground-states in the GLSM.

Laurent superpotentials
For illustration, consider the GLSM models with the superpotential 3 where m, n > 1 are integers and X 0 is the chiral superfield that in some ways serves as a Lagrange multiplier; we focus on n = 2, 3, 4, but generalizations are straightforward. Such superpotentials are strictly invariant with respect to the U 1 (1)×U 2 (1) gauge symmetry with the charges Manifestly, for m > 2, the a ij -terms become Laurent monomials; as we will see below in more detail, this turns out to be closely related to the models considered in Ref. [1,5]. For now, we discuss the GLSM with the potential (1a) in its own right, being especially interested in the novelty of the m > 2 cases. The standard requirement for the superpotential to be chiral is straightforwardly satisfied: owing to the fact that X 0 and all X i 's are chiral superfields, and regardless of the fact that the chiral superfields X n+1 and X n+2 appear with negative powers for m > 2: As we will show below, the background values (vev's) of the lowest component fields in X 0 , X i are always restricted so that the vev's of f (X) and X 0 ∂f (X) ∂X i remain finite, even in the cases when X n+j → 0. Eq. (4) then insures that the superpotential (2) is itself a chiral superfield and all manifest supersymmetry methods apply; we therefore proceed as usual. In addition, this also makes this superpotential regular in all component fields, and so insures that (2) specifies a well-defined field theory. 4

The ground state
The potential (1a) is a sum of positive-definite terms, each of which has to vanish separately in the ground state. The first four groups of constraints stem from the vanishing of the F -terms, for which the equations of motion give F i = ∂W ∂x i : The vanishing of the last term in (1a) imposes: This identifies the "normal mode" linear combinations: U 3 (1) generated by mQ 1 +Q 2 with respect to which x 1 is neutral, and U 4 (1) generated by (m−2)Q 1 +nQ 2 with respect to which x 0 is neutral. Finally, the vanishing of the D-terms (1a) impose:

Phases
We now turn to analyze the D-term constraints (6), following [13]. The U (1) charges Q a i of the chiral superfields, (X 0 , X i ), determine the two-dimensional secondary fan (phase diagram) given in Figure 1 5 . In particular, we can find the phase-boundaries by determining the Figure 1: The phase diagram of the GLSM with the Calabi-Yau n-fold ⊂ F (n) m "geometric" phase; the " * " entries are generally nonzero and are outside the Stanley-Reisner ideal.
Thus, there are four different phases, as depicted in Figure 1. We now analyze them in turn, using that a ground state solution must also satisfy the F -term constraints (5).
Phase I: r 1 , r 2 > 0. The F -term constraints are solved by having x 0 = 0 and f (x) = 0. From the D-term analysis above, the excluded region in the field-space is exactly the Stanley-Reisner (or irrelevant [18]) ideal for the Hirzebruch n-fold F (n) m (mtwisted P n−1 -bundle over P 1 ). Since the x n+j cannot both vanish (5e) implies that σ 2 = 0. Eq. (5e) then simplifies and implies that σ 1 = 0 since the x i , i = 1, . . . , n cannot all be zero. Thus, f (x) = 0 defines a Calabi-Yau (n−1)-fold hypersurface in F (n) m . Direct computation shows that the polynomial f (x) is transversal for generic choices of a ij , a j , so that its n+2 gradient components ∂f ∂x i , ∂f ∂x n+j vanish simultaneously with f (x) itself only within the excluded region (7), see Appendix A for more details.
Phase III: m−2 n r 1 < r 2 < −mr 1 . The D-term constraints imply that now both x 0 = 0 and x 1 = 0 must be excluded : Thus, all the F -term constraints, (5b)-(5a) are solved by setting x i = 0 for i = 2, · · · , n+2, and (5e) simplifies to The vevs x 0 = 0 = x 1 break U 1 (1)×U 2 (1) → Z m(n−1)+2 ; e.g., x 1 = 0 sets σ 1 = mσ 2 , producing m(1−n)−2 σ 1 2 |x 0 | 2 = 0. Thus, the vacuum solution in phase III is that of a Z m(n−1)+2 Landau-Ginzburg orbifold of f (x) = 0, acting with charges (mQ 1 + Q 2 . Phase IV: 0 < r 2 < m−2 n r 1 . The D-term analysis implies that x 0 = 0 must now be excluded , whereupon the first two F -term constraints, (5b) and (5c), imply that x 1 = . . . = x n = 0. The remaining F -term constraints, (5a) and (5d) turn out to be satisfied, leaving x n+1 , x n+2 unconstrained. Since x 1 = 0, the 2nd D-term constraint (6b) produces 2 j=1 |x n+j | 2 = r 2 + m−2 n r 1 , which is positive in this phase, so that x n+1 = x n+2 = 0 must also be excluded , and we obtain: Since x n+1 , x n+2 cannot both vanish, their vevs break U 2 (1) completely, while x 0 = 0 breaks U 1 (1) → Z n since Q 1 (x 0 ) = −n; correspondingly, (5e) reduces to |(−n)σ 1 ||x 0 | = 0. This then is a hybrid phase in which a Landau-Ginzburg Z n orbifold of {(x 1 , · · · , x n ) : f (x) = 0} is fibered over the base-P 1 = P(x n+1 , x n+2 ). At this point let us make the following comment on the vanishing of the x i = 0, i > 0 in the Landau-Ginzburg phase above and the existence of a superpotential with Laurent monomials (2). By going through phase IV, where x i = 0 for i = 1, . . . , n correspond to the fiber having collapsed to a Landau-Ginzburg orbifold, the Laurent monomials are absentin fact, f (x) vanishes identically in phase IV. By transitioning to phase III, we then have x 1 = 0 while x 2 = · · · = x n+2 = 0, which is the true Landau-Ginzburg orbifold. On the other hand, when transitioning from phase II, through the boundary (iii), into phase III, we have to specify the multi-limit x n i (13) shows that x 2 , · · · , x n should vanish sufficiently faster than x n+j , except perhaps one of the x i for which then we must insure that a i1 . The phases and their boundaries are summarized in Table 1. The vevs of x 0 and x 1 Table 1: The cyclic listing of the phases (I-IV) and boundaries (i-iv) of the GLSM (2) Generically, the U (1) 2 is completely broken, but is "restored" to a discrete subgroup at special points.
change continuously as (r 1 , r 2 ) are varied through the cycle of phases I-II-III-IV-I, so that the secondary fan depicted to the left in Figure 1 is complete as given.

The toric geometry of the ground state
We now turn to examine the toric geometry of GLSM ground states defined by Laurent superpotentials such as (2), and exhibit a justification for the inclusion of Laurent monomials in superpotential such as (1a) and its generalizations. In particular, Laurent superpotentials such as (2) motivate a refinement of the standard methods of toric geometry [14,15,16,17,19,18], and we first sketch a few basic facts to establish notation and conventions, adapting from [16,17,18].
Every compact toric variety may be specified by a complete rational polyhedral fan Σ within a lattice N , which in turn has a lattice spanning polytope 7 ∆ , the vertices ν ρ of which are the minimal N -integral generators of Σ, and ∆ is (coarsely) star triangulated by Σ about (0 ∈ Σ) ∈ rel int ∆ . Every toric variety also has a Newton polytope ∆ (in a lattice M dual to N ), the M -integral points of which correspond to anticanonical sections, specified as monomials in the so-called Cox variables [26] x ρ where , denotes the Euclidean scalar product, and the "+1" in the exponent indicates sections of the anticanonical (K * ) +1 . For convex Newton polytopes, the spanning polytope ∆ turns out to be the so-called polar polytope: For reflexive convex polyhedra, the polar operation is involutive: The "global" nature of the definition (15) also implies that where Conv(∆ ) is the convex hull (envelope) of ∆ , rather than ∆ itself. That is, (15) obscures every non-convex detail in ∆ . More importantly, for every reflexive convex polyhedron ∆ , the x ρ -monomials given by (∆ ) • always turn out to be regular, and so cannot provide for the Laurent monomials appearing in (2b) for m > 2. This "non-convexity hiding" nature (16) of the polar operation (15) turns out to be closely correlated with the systematic omission of Laurent x ρ -monomials in (14), which is inadequate for constructing Calabi-Yau hypersurfaces in non-Fano n-folds such as considered recently [1,5]. We thus seek a twin generalization of both the polar operation (15) and of convex polytopes.

The generalization
We propose a twin definition of a class of "VEX polytopes" (to include all convex polytopes but also certain non-convex ones 8 ), and a "trans-polar" operation amongst them such that: A. For every convex polytope ∆, the trans-polar equals the polar: ∆ = ∆ • . B. The trans-polar of every VEX polytope is also a VEX polytope. C. For every VEX polytope ∆, (∆ ) = ∆.
Here d(θ) effectively counts the number of unit volume simplices, see also Appendix B for more details. Requirement A is satisfied by design for all convex polytopes: Step 1 halts with ∆ itself, Step 2 produces ∆ = ∆ • , and there is nothing left for Step 3 . Requirements B and C then define the class of VEX polytopes as the maximal closure under the trans-polar operation. By defining the deficit of (15): requirement C above is equivalent to requiring the trans-polar operation of Construction 3.1 to have no deficit on VEX polytopes. For any toric variety X and its spanning polytope ∆ X , we define the extension part of the (complete) Newton polytope 10 : We now proceed to give a more explicit form of the conditions on the VEX polytopes and the construction of mirror manifolds in terms of the associated toric varieties.
Although we have no conclusive explicit definition of VEX polytopes, they are necessarily defined within a lattice L Z n and must have: 1. a single L-integral point, "0", in the interior of the polytope; 2. only L-integral vertices, and each at minimal Euclidean distance from "0"; 3. no k-face (k > 0) in a k-plane containing "0"; 4. a star-triangulation in which L-integral points are only at the apex "0" and the base of each star-simplex.
We note that for 2-dimensional polytopes condition 2 implies condition 3, but not so in higher dimensions; also, without convexity, condition 1 does not imply condition 4. This generalizes mirror symmetry as applied to Calabi-Yau hypersurfaces constructed from reflexive pairs (∆ X , ∆ X ), in which the mirror manifold of a generic Calabi-Yau hypersur-faceẐ f ⊂ X, specified by f (x) = 0 in a toric variety X with a reflexive (and convex) Newton polytope ∆ X is (an MPCP desingularization of) the Calabi-Yau hypersurfaceẐ g ⊂ X, specified by g(y) = 0 in the toric variety Y the Newton polytope of which is ∆ Y = ∆ X [21,28]. Define a class of VEX polytopes wherein every pair of trans-polar polytopes defines a pair of trans-polar toric varieties, so that a Calabi-Yau hypersurfaceẐ g ⊂ Y is a natural mirror of a Calabi-Yau hypersurfacê Z f ⊂ X. The point of this proof-of-concept note is to show that this includes a large collection of non-convex polytopes such as those of Hirzebruch n-folds.
Finally, we note that through step 3 of Construction 3.1, a choice of an orientation of ∆ induces an orientation of its trans-polar, ∆ .
To construct the trans-polar of ∆ F 3 , we start by noting that the facets Φ 1 , · · · , Φ 6 ⊂ ∆ are all convex. Following Construction 3.1, we find The vertices µ 1 , µ 2 indeed belong to (∆ F 3 ) • , but µ 3 , · · · , µ 6 lie beyond the fractional vertices of (∆ F 3 ) • : they delimit the extension (18), with the quadrangular facet Θ 1 that lies in the is more precisely delimited by the trans-polar images of the edges adjacent to ν 1 : With the standard " −1" conditions in (15), the computations (22) would have produced an empty set, since the standard polar operation (15) fails to take into account the nonconvexity of ν 1 . We remedy this by "flipping" the defining inequalities in correlation with (non-)convexity: The extending facet Θ 1 may be defined by first rewriting the n, u −1 condition in (15) as ( v, u +1) 0, and then flipping the sign of the left-hand side according to the (non-)convexity of v: Here F = 1 indicates that the usual condition for the polar (" −1") is reversed-owing to the non-convexity of ν 1 itself; the first two conditions (F = 2) are however flipped a second time (and so back to the original inequality) owing to the fact that the edges [ν 1 , ν 2 ] and [ν 1 , ν 3 ] are themselves non-convex. In passing, [ν 1 , ν 2 , ν 3 ] is the sub-polytope of the P 2 -fiber in F m . Proceeding in this way (step 3 of Construction 3.1) produces the complete Newton polytope ∆ F 3 shown to the right in Figure 3. In particular, the facets Θ 2 , Θ 3 are self-crossing, owing to the fact that ν 2 , ν 3 ∈ ∆ F 3 are each adjacent to three convex and one non-convex edge.
The spanning polytope ∆ F 3 admits a uniform outward orientation, which then induces an orientation of the Newton polytope ∆ F 3 . At each vertex ν ρ ∈ ∆ F 3 , this orders the adjacent facets; for example,  The spanning polytope ∆ F 3 and the Newton polytope ∆ F 3 with several polar pairs of elements indicated. The outward orientation of ∆ F 3 at a vertex ν ρ orders the adjacent facets Φ i ∩ · · · ∩ Φ k , and so induces the (reverse) ordering of Φ i , and the compatible orientation for Θ ρ .
outward and opposite inward orientations, respectively. This orientation will be essential in the combinatorial formulae for the Euler and Hodge numbers, see section 4 and Appendix B. Finally, the standard formula (14) associates the Laurent monomials Straightforward computation shows that generic Laurent polynomials formed with both (20) and (24) are transversal: following the GLSM analysis in Section 2, the polynomial f (x) and its gradient ∂f (x)/∂x i vanish simultaneously only in the Landau-Ginzburg orbifold phase, where x 2 , x 3 , x 4 , x 5 → 0.

Combinatorial calculations
Let us first consider Batyrev's Euler characteristic formula [21,29] χ whereẐ f is the MPCP desingularization of the anticanonical ∆ X -regular (transversal) hypersurface f (x) = 0 in a toric variety X and ∆ X is its Newton polytope. For every face θ ⊂ ∆ X in the Newton polytope, θ * ⊂ ∆ X is the dual face 13 in the spanning polytope such that dim(θ) + dim(θ * ) = dim(Ẑ f ). In fact, there is a natural generalization of (25) in which we sum over all the codimension k faces, where for dim(θ) = −1, θ is the unique interior point in ∆ X and θ * = ∆ X , and similarly for dim(θ) = dim X, where θ = ∆ X and so θ * is the unique interior point in ∆ X . Furthermore, because we restrict to star-triangulations, it follows that since dim(θ * ) = dim X − 1 and dim(θ * ) = 0 and hence the two sums range over codimensionone faces in ∆ and ∆ , respectively. Thus, the contribution from k = −1 and k = 0 cancel, as do the k = dim X−1 and k = dim X terms. We now will demonstrate that the trans-polar pair of polytopes ∆ X , ∆ X := (∆ X ) provides for computing the basic topological characteristics -and that the orientations discussed between (20) and (24) turn out to be crucial. In light of this, we propose the following conjecture generalizing Batyrev's construction [21,29] 14 : Conjecture 4.1. The Euler number, χ(Ẑ f ), and Hodge numbers, h 1,1 (Ẑ f ) and h n−2,1 (Ẑ f ) for a Calabi-Yau (n−1)-foldẐ f , which is the MPCP desingularization of the anticanonical ∆ X -regular (transversal) hypersurface f (x) = 0 in a toric variety X with ∆ X its VEX Newton polytope, are given by (25), and respectively, where l(θ) (l(θ * )) is the number of internal points in the face θ ⊂ ∆ X (θ * ⊂ ∆ X ), which may be negative if θ (θ * ) has opposite orientation. The similarly constructed Calabi-Yau (n−1)-foldẐ g , which is the MPCP desingularization of the anticanonical ∆ Y -regular (transversal) hypersurface g(y) = 0 in a toric variety Y with ∆ Y = ∆ X its VEX Newton polytope, is the mirror manifold toẐ f , with the roles of ∆ X and ∆ X interchanged, such that h 1,1 (Ẑ g ) = h n−2,1 (Ẑ f ) and h n−2,1 (Ẑ g ) = h 1,1 (Ẑ g ), and thus χ(Ẑ g ) = (−1) n−1 χ(Ẑ f ).
We first focus on the example K3 ⊂ F 3 from Section 3.2, followed by the corresponding Calabi-Yau threefolds, where we also calculate h 1,1 and h 2,1 ; additional examples may be found in the companion paper [23].

K3
Adapting (25) to the case of the K3,Ẑ f hypersurface f (x) = 0 ⊂ F m , the indicated summation should extend only over 1-dimensional faces (edges) θ ⊂ ∆ Fm in the Newton polytope (the dual of which, θ * ⊂ ∆ Fm , are edges in the spanning polytope): All edges θ * ⊂ ∆ Fm have unit degree for m = 2, in which case the sum reduces to the degrees of the edges in the Newton polytope ∆ Fm = (∆ Fm ) . The m 2 cases are well understood and convex so that the trans-polar operation reduces to the familiar polar (15). We then focus on m 3. As is evident from Figure 3 and 6 below, ∆ Fm has a total of nine edges: • the one tallest vertical edge [ν 2 , ν 3 ] = Θ 2 ∩ Θ 3 = [µ 1 , µ 2 ] has degree 2 + 2m;  Figure 6) shows that for m 3 these two edges manifestly extend in the direction opposite from the m 2 cases. 14 Tallying these contributions produces in (30): In fact, the computation is also true for m = 2; not only is the end result independent of m, but the method itself extends. Finally, note that since (30) is completely symmetric in exchanging θ and θ * , it is clear that the mirror K3,Ẑ g , to the hypersurface f (x) = 0 ⊂ F m , is defined as a hypersurface g(y) = 0 in the toric variety Y constructed by exchanging the roles of ∆ and ∆.
The Euler characteristic: We first calculate the Euler number along the lines of the K3 in the previous subsection, evaluating the two terms in Batyrev's expression (25) for an (n = 4)-dimensional ambient toric variety (see also [21,Theorem 4.5.3]): Here θ ⊂ ∆ The Hodge numbers: We next turn to calculating the Hodge numbers h 2,1 and h 1,1 following Batyrev's formulae [21,Theorem 4.3.7]: and [21,Theorem 4.4.2]: Here, θ * ⊂ ∆ Fm is the facet dual to θ ⊂ ∆ Fm , and vice versa. To avoid the ambiguity of counting internal points in negative-degree faces, we rewrite l * (θ) and l * (θ * ) using the general formulae (51) and (49) to re-express the summands in terms of various k-face degrees. Thus, Batyrev's formulae (37) and (36) take the following form: where N k (N * k ) refers to the number of k-faces in ∆ Fm (∆ Fm ) and c(θ (2) ) is the effective circumference of θ (2) , see (51). The reader can consult Appendix B.2 for the details of the calculation the result of which is that h 2,1 = 86 and h 1,1 = 2, independent of m and as with the Euler number in agreement with the gCICY result [1,5]
We note that the ratio of the sizes of the geometric and the quantum symmetry groups equals the ratio of the degrees of the polytopes 18 : The analogous relationship persists also for any n ≥ 2, and for m 0. Given our choice of vertices and monomials (40), this chain of equalities is in complete agreement with the detailed analysis in Section 3 of Ref. [22]. The fact that these relationships continue to hold also for the n = 2, 3, 4 and m 0 sequences of Laurent polynomials such as (40) we find to provide additional corroboration of our extension of toric methods in Section 3 and their application in Section 4, and of Construction 3.1 and Conjecture 4.1 in particular.
We close by noting that even without a detailed analysis of the Landau-Ginzburg orbifolds or the more complete mirror pair of GLSM's, the discrete symmetries are essential both in constructing the Hilbert spaces of the Landau-Ginzburg models and in significantly restricting the Yukawa couplings via the Wigner-Eckart theorem. In fact, the symmetries of the defining (Laurent) polynomials f (x) and g(y) will play such an important role in all phases of the corresponding GLSM's. We find it therefore significant that the phase symmetries fully conform to the transposition prescription [20,22].

Conclusions and Outlook
In this paper we have found that there is a natural generalization of the GLSM to Laurent superpotentials in which the phase-diagram of the enlarged Kähler moduli space, i.e., the secondary fan is constructed from the triangulations of the spanning polytope ∆ X (and its convex hull) of non-Fano toric varieties. Our construction 3.1 specifies the "trans-polar" operation, which extends the standard "polar" operation so as to apply to all VEX polytopes, including the above ∆ X and the Newton polyhedron ∆ X := (∆ X ) , as well as the original class of reflexive polytopes considered by Batyrev [21].
In particular, for polytopes corresponding to the Hirzebruch n-folds [5] F (n) m for m 0 (collectively denoted in the subsequent listing as X), we have also shown that the spanning polytope ∆ X and the Newton polytope ∆ X := (∆ X ) admit a mutually compatible orientation. The orientation provides a sign for every face in each polytope and each cone in the fan that it spans. Furthermore, ∆ X and ∆ X both admit oriented star-triangulations (compatible with the orientation of the polytopes), which provides a sign to the degree of every star-simplex. Allowing the degree of a star-simplex to be negative is crucial for correctly calculating the Euler and Hodge numbers from the combinatorial data in the pair of oriented polytopes (∆ X , ∆ X ) and their oriented star-triangulations by adapting and generalizing Batyrev's formula (25). Note that the extension, xtn(∆ X ), (18) within the (complete) Newton polytope ∆ X is essential not only for the computation of the above topological data, but also in that the Laurent monomials corresponding to the integral points of xtn(∆ X ) render the generic anticanonical polynomial transversal ("∆ X -regular" [21]). Finally, as ∆ X spans the fan of X, ∆ X := ∆ X defines a trans-polar toric n-fold X , the fan of which is spanned by ∆ X . Hence, swapping ∆ X ↔ ∆ X evidently induces the expected mirror effect on the Euler and Hodge numbers. This indicates the pair of (MPCP desingularized) anticanonical hypersurfaces in the toric varieties specified by the trans-polar pair of polytopes (∆ X , ∆ X := ∆ X ) as prime candidates for mirror manifolds. Further evidence of mirror symmetry also follows from the exchange of "geometric" and "quantum" phase symmetries [20] for the Landau-Ginzburg orbifold phases obtained from a minimal choice of superpotentials, f min (X) and g min (Y ), related by transposition of the matrix of exponents of anti-canonical monomials.
The present work has two natural extensions. First, it would be desirable to put the results presented herein on a rigorous mathematical footing, and in particular provide a proof of our main Conjecture 4.1, including to what extent it is valid. Second, and the main motivation of the program at hand, is to understand the enlarged Kähler moduli space in order to calculate the non-perturbative corrections, i.e., Gromow-Witten invariants. This would allow us to determine whether the observed (mod n) symmetry within the class of Calabi-Yau hypersurfaces in the Hirzebruch n-folds is indeed broken.
Including other monomials from the Newton polytope makes the system more generic, and is on general grounds expected not to worsen the above behavior.
Note that the GLSM analysis in Section 2 is more detailed as it catalogues the branches of the base-locus of the superpotential x 0 · f (x), not just f (x).

B Combinatorial Calculations B.1 K3
Adapting again Batyrev's n ≥ 4 formula [21,29], we write: where the first part is the contribution from the Picard group, the second counts the "toric" deformations of the complex structure, and the final term is a correction term which can be thought of as counting either non-polynomial deformations of the complex structure or non-toric Kähler deformations. We now address these terms in turn.
Picard term: It should be manifest from Figure 3 This result perfectly agrees with the homology algebra computation: H 2 (F m , Z) is indeed 2-dimensional, generated by the Kähler forms of P 3 × P 1 when realizing F m as a degree-(1, m) hypersurface in P 3 × P 1 , and both generators are for all m 0 inherited by the Calabi-Yau hypersurface K3 ⊂ F m [5].
Toric deformations: In order to calculate the second (and third) term in (46) it is necessary to understand the contribution of the extension part of the Newton polytope ∆ Fm and how it varies with m. In particular we need to understand how to count the effective number of integral points interior to the various codimension one and two faces. Since Θ 1 and portions of Θ 2 and Θ 3 are negatively oriented (see Section 3.2), this will require some care. We find it expedient to relate the number of points in the relative interior of a face to the degree of that face: 0-dim.: d(θ (0) ) = 0!· Vol 0 (θ (0) ) = 1 for θ (0) a vertex.
Note that a negatively oriented edge of (signed) length − (formally) therefore has −( + 1) points in its relative interior; a negatively oriented unit-size edge (formally) has −2 points in its relative interior.
Thus, it follows that the effective number of points in the (complete) Newton polytope is Alternatively, we can simply count the number of points in ∆ Fm : 1. The "standard part of ∆ Fm ↔ H 0 (F m , K * ) has 22 30 + ϑ m 3 · 4(m−3) integral points. This is in perfect agreement with the result of standard homological algebra [5,Eq. (A.28)]. 2. The m 3 extensions (Θ 1 , the red (dark-shaded) quadrangles in Figure 6) consist of two parts: (a) The top and bottom edge of Θ 1 are Θ 1 ∩ Θ 4 and Θ 1 ∩ Θ 5 , respectively. Those integral points count positively as belonging to Θ 4 , Θ 5 , but negatively as belonging to the negative-degree Θ 1 , and so cancel out. (b) The outlined 4(m−3) integral points within the (negative-degree) Θ 1 , including the ones on the side that are shared with the flip-oriented edge of the self-crossing Θ 2 and Θ 3 . These latter 4(m−3) integral points contribute negatively and precisely cancel the excess integral points in the rising tip of the standard part of ∆ Fm .
The net result is that the oriented Newton polytope ∆ Fm := (∆ Fm ) encodes an effective number of 30 elements of H 0 (F m , K * ). Summarizing the calculation, the contributions from the growing "tip" of the positively oriented portion of the Newton polytope and the growing negatively oriented extension cancel in just the same way also for Calabi-Yau hypersurfaces in Hirzebruch 2-and 4-folds [23]. Thus, the number of toric deformations is Correction term: Finally, the "correction term," θ l * (θ) l * (θ * ) ranging over codimension-2 faces θ ⊂ ∆ Fm , identically vanishes. To see this, note that θ * are edges in the spanning polytope ∆ Fm , all of which have positive unit degree, and no internal points by (49); with all l * (θ * ) = 0, the sum vanishes. Putting (48), (58) and zero for the third, "correction" term in (46), we obtain: h 1,1 (K3 ⊂ F m ) = 2 + 18 + 0 = 20, as expected for a K3 surface.

B.2 Calabi-Yau three-folds
We now turn to calculating the degrees of the various faces in the pair of polytopes (∆ Fm , ∆ Fm ), for m ≥ 3.
In turn, the corner edges of the extension, Owing to this double negative, these points count as deformations of the complex structure of F m itself, and are being canceled precisely by the surplus reparametrizations [5]. This renders the Hirzebruch 4-folds effectively rigid. We thus remain with the effective number of 105 anticanonical sections, 18 reparametrizations and one overall scaling of the equation defining the hypersurface, producing the expected result: 105 − 18 − 1 = 86 = h 2,1 (Y m ) for the Calabi-Yau 3-fold Y m ⊂ F m [5].