F-theory helper functions¶

These are the lower-level toric geometry functions used to construct orientifolds and F-theory uplifts. Most users will only need the classes and functions documented in F-theory and orientifolds.

Helper functions for Calabi-Yau orientifolds and their F-theory uplifts.

This module collects the lower-level toric geometry routines used by CY_orientifold and F_Theory_Uplift: GLSM charges, Newton polytopes and sections of toric divisors, Cartier and nef checks, normal fans and their refinements, \(\mathbb{Z}_2\) torus actions, Hodge numbers of codimension-two complete intersections, and assorted array utilities.

Unless stated otherwise, cones are tuples of one-indexed ray labels, and a toric divisor \(D = \sum_\rho a_\rho D_\rho\) is given by its coefficient vector \((a_\rho)\) in the basis of toric prime divisors, ordered like the rays.

Functions:

BL –

Return the base locus of a toric divisor.
Cartier_index –

Return the Cartier index of a toric Weil divisor.
Gorenstein_index –

Return the index of a reflexive Gorenstein cone.
Newton_Polytope –

Return the Newton polytope of a toric divisor.
O3O7_line_bundle –

Return the orientifold line bundle on the toric orbifold.
O7_cones –

Return the cones with a given number of rays whose rays are all O7-planes.
Z2_fixed_locus –

Return the cones whose toric strata are fixed by a finite torus action.
array_to_latex –

Convert a 2D array into a LaTeX pmatrix.
attempt_to_make_Cartier –

Attempt to refine a fan so that a divisor becomes Cartier and nef.
attempt_to_make_nef –

Attempt to find a triangulation on which a divisor is nef.
base_locus –

Return the minimal sets of toric coordinates on which all sections vanish.
basis –

Return the indices of rows that form a basis of the row span.
basis_H2_toric_fan –

Return ray labels whose divisor classes form a basis of the class group.
compute_partition –

Shift two toric divisors so that they partition the anticanonical divisor.
contains_row –

Check whether a vector appears as a row of a 2D array.
contains_rows –

Check whether every row of one 2D array appears as a row of another.
divisor_intersections –

Compute the classes of the toric curves on a complete intersection.
dual_face_Cayley_polytope –

Return the face of the dual Cayley polytope dual to a given face.
find_cone –

Return the face of a simplicial cone that contains a given vector.
find_cone_general –

Return the face of a cone that contains a given vector.
find_trilayer_vertex_polytope –

Return the distinguished vertex of a trilayer polytope.
find_trilayer_vertex_vertices –

Return the distinguished vertex of a trilayer polytope from its vertices.
flatten –

Flatten a nested list by a given number of levels.
generic_section_factorizes –

Check whether the generic section of a toric divisor factorizes.
get_index –

Return the indices of the rows of arr equal to target.
get_indices –

Return the indices of the rows of arr that appear among targets.
get_lower_dimensional_cones –

Return all faces with a given number of rays of a collection of cones.
get_same_rows –

Return the rows of A that also appear as rows of B.
glsm_from_points –

Return a GLSM charge matrix for a point configuration.
h11_2_part –

Compute \(h^{1,1}\) of a Calabi-Yau defined by a two-part nef partition.
h21_2_part –

Compute \(h^{2,1}\) of a fourfold defined by a two-part nef partition.
inequivalent_Z2_actions –

Return the \(\mathbb{Z}_2\) torus actions up to lattice symmetries.
integral_gale_transform –

Return an integral Gale transform of a point configuration.
is_Cartier –

Check whether a toric Weil divisor is Cartier.
is_Gorenstein –

Check whether a cone is Gorenstein.
is_ample –

Check whether a toric divisor is ample.
is_effective –

Check whether a toric divisor has a nonzero global section.
is_nef –

Check whether a toric divisor is nef.
is_partition –

Check whether two toric divisors can be shifted to a partition of \(-K\).
is_reflexive_Gorenstein –

Check whether a cone and its dual are both Gorenstein.
lattice_refinement –

Return the change of basis to a refined lattice.
linebundle_weights_from_Newton_Polytope –

Return the divisor coefficients determined by a Newton polytope.
make_simplicial –

Return a simplicial refinement of a fan.
moving_cone –

Return the moving cone of a toric variety.
nested_sum –

Return all sums of one element from each list, as a nested list.
normal_fan –

Construct the normal fan of a polytope or of a Minkowski sum of polytopes.
points_from_glsm –

Return a point configuration with the given GLSM charges.
points_not_interior_to_facets_and_codim2_faces –

Return the lattice points not interior to faces of codimension one or two.
refine_fan –

Insert new rays into a simplicial fan by star subdivisions.
row_difference –

Return the rows of A that do not appear as rows of B.
same_rows –

Check whether two 2D arrays have the same rows, ignoring order.
sections –

Return the exponent vectors of the monomial sections of a toric divisor.
solve_over_integers –

Find an integer solution of the linear system \(M x = -b\).
sums_to_anticanonical –

Check whether the sum of two toric divisors is linearly equivalent to \(-K\).
toric_orbifold –

Return the rays of the quotient of a toric variety by a finite torus action.
trilayer_5d_Ftheory_uplift –

Construct the polytope of the F-theory uplift of a trilayer orientifold.
trilayer_normal_form –

Bring a trilayer polytope into trilayer normal form.

BL ¶

BL(fan, lb)

Return the base locus of a toric divisor.

This computes the sections of the divisor with sections and passes them to base_locus, restricting to the cones of fan. The divisor must have at least one section.

Parameters:

fan (Fan) –

The toric fan.
lb (ndarray) –

The divisor coefficients.

Returns:

list[tuple[int, ...]] –

The minimal cones \(\sigma\) of the fan such that all sections vanish on the orbit closure \(V(\sigma)\). The base locus is the union of these orbit closures.

Examples:

On the Hirzebruch surface \(\mathbb{F}_1\), the exceptional curve \(D_2\) is its own base locus, while the fiber \(D_1\) is base point free.

>>> import numpy as np
>>> from cytools.vector_config import VectorConfiguration
>>> from cytools.f_theory.Uplift_functions import BL
>>> fan = VectorConfiguration([[1, 0], [0, 1], [-1, 1], [0, -1]]).triangulate()
>>> BL(fan, np.array([0, 1, 0, 0])), BL(fan, np.array([1, 0, 0, 0]))
([(2,)], [])

F-theory helper functions¶

BL ¶

Cartier_index ¶

Gorenstein_index ¶

Newton_Polytope ¶

O3O7_line_bundle ¶

O7_cones ¶

Z2_fixed_locus ¶

array_to_latex ¶

attempt_to_make_Cartier ¶

attempt_to_make_nef ¶

base_locus ¶

basis ¶

basis_H2_toric_fan ¶

compute_partition ¶

contains_row ¶

contains_rows ¶

divisor_intersections ¶

dual_face_Cayley_polytope ¶

find_cone ¶

find_cone_general ¶

find_trilayer_vertex_polytope ¶

find_trilayer_vertex_vertices ¶

flatten ¶

generic_section_factorizes ¶

get_index ¶

get_indices ¶

get_lower_dimensional_cones ¶

get_same_rows ¶

glsm_from_points ¶

h11_2_part ¶

h21_2_part ¶

inequivalent_Z2_actions ¶

integral_gale_transform ¶

is_Cartier ¶

is_Gorenstein ¶

is_ample ¶

is_effective ¶

is_nef ¶

is_partition ¶

is_reflexive_Gorenstein ¶

lattice_refinement ¶

linebundle_weights_from_Newton_Polytope ¶

make_simplicial ¶

moving_cone ¶

nested_sum ¶

normal_fan ¶

points_from_glsm ¶

points_not_interior_to_facets_and_codim2_faces ¶

refine_fan ¶

row_difference ¶

same_rows ¶

sections ¶

solve_over_integers ¶

sums_to_anticanonical ¶

toric_orbifold ¶

trilayer_5d_Ftheory_uplift ¶

trilayer_normal_form ¶