PALP: A Package for Analysing Lattice Polytopes with applications to toric geometry

Abstract

We describe our package PALP of C programs for calculations with lattice polytopes and applications to toric geometry, which is freely available on the internet. It contains routines for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points. In addition, there are procedures specialised to reflexive polytopes such as the enumeration of reflexive subpolytopes, and applications to toric geometry and string theory, like the computation of Hodge data and fibration structures for toric Calabi-Yau varieties. The package is well tested and optimised in speed as it was used for time consuming tasks such as the classification of reflexive polyhedra in 4 dimensions and the creation and manipulation of very large lists of 5-dimensional polyhedra. While originally intended for low-dimensional applications, the algorithms work in any dimension and our key routine for vertex and facet enumeration compares well with existing packages.

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

“…We find that F (1) follows an expansion of the form (4.16) with polynomial coefficients P (1) d of modular weight w (1) d = 24d, i.e. F (1) transforms with modular weight k 1 = 0 as expected.…”

“…Here the latin indices run over the moduli fields associated to either the complex structure moduli on W whose tangent space is associated to harmonic forms in H 3,1 (W ) (dual to H 1,3 (W )) or Kähler moduli on M whose tangent space is associated to harmonic forms in (2) in a reference complex structure near large radius with the inverse of the pairing on H 2,2 hor (W ) andη (1) with the inverse pairing on H 3,1 (W ) ⊕ H 1,3 (W ), which by (1.1) is block diagonal. This property is maintained throughout the moduli space due to the charge grading.…”

“…In this expression χ is the Euler characteristic of M , ∆ is the discriminant and z(t) is the mirror map in terms of the algebraic coordinates z and the flat coordinates t. The coefficients b i can be fixed by the limiting behaviour of F (1) in the moduli space.…”

“…Note that an analogous ansatz can be used for the genus one string amplitudes. 5 For the latter case the modular weight of each polynomial coefficient P (1) β is given by w (1) β = 6c 1 (B) · β. For the genus zero case we make a special distinction for the two observed kind of amplitudes.…”

“…The reason is that the construction of Calabi-Yau fourfolds as algebraic varieties in a projective ambient space is very simple and toric, or more generally non-abelian gauged linear σ-model descriptions provide immediately trillions of geometries [1]. In fact, geometric classifications of certain compactifications with restricted physical features seem possible even though this has been achieved mostly for elliptic Calabi-Yau threefolds, where it has been argued that there exists only a finite number of topological types in this class [2].…”