F-theory and orientifolds¶

F-theory uplifts of Calabi-Yau orientifolds.

This subpackage constructs O3/O7 orientifolds of Calabi-Yau hypersurfaces in toric varieties (CY_orientifold) and their F-theory uplifts, described as toric complete intersections (F_Theory_Uplift). It also provides functions to generate orientifolds and uplifts from the Kreuzer-Skarke database (fetch_orientifolds, fetch_F_Theory_uplifts) and to read a precomputed dataset of nef-partition uplifts (fetch_nef_partition_uplifts).

Modules:

FT_CY –

Orientifolds of Calabi-Yau hypersurfaces and their F-theory uplifts.
Uplift_functions –

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

Classes:

CY_orientifold –

O3/O7 orientifold of a Calabi-Yau hypersurface in a toric variety.
F_Theory_Uplift –

F-theory uplift of an O3/O7 Calabi-Yau orientifold.

Functions:

fetch_F_Theory_uplifts –

Yield F-theory uplifts of orientifolds of polytopes from the databases.
fetch_nef_partition_uplifts –

Load precomputed F-theory uplifts that define nef partitions.
fetch_orientifolds –

Yield orientifolds of Calabi-Yau hypersurfaces from the polytope databases.

CY_orientifold ¶

CY_orientifold(fan_polytope_or_points=None, xi=None, resolve_A1_singularities=True, construct_nef_decomposition=True)

O3/O7 orientifold of a Calabi-Yau hypersurface in a toric variety.

The orientifold involution \(\sigma\) is the toric \(\mathbb{Z}_2\) action given by the torus element \(\xi/2\), where \(\xi\) is an integer vector in the lattice \(N\) of the ambient fan. The quotient of the ambient variety by this action is a toric orbifold, whose rays are the images of the original rays in the refined lattice \(N + \mathbb{Z}\,\xi/2\), rescaled to be primitive. The orientifold is described by this orbifold together with the O3/O7 line bundle \(L\) on it, whose generic section cuts out the quotient \(B = X/\sigma\) of the Calabi-Yau hypersurface \(X\), i.e. the base of the F-theory uplift (see F_Theory_Uplift).

Toric rays are referred to by one-indexed labels, following the CYTools convention. Note that the orbifold rays need not be in the same order as the rays of the Calabi-Yau ambient fan.

Parameters:

fan_polytope_or_points (Polytope | VectorConfiguration | Fan | ArrayLike, default: None ) –

The toric data of the ambient variety of the Calabi-Yau hypersurface. If a Polytope is given, its nonzero lattice points (including the points interior to facets) are used as the rays. A VectorConfiguration, an array, or a list of lists specifies the rays directly. If a Fan is given, its rays and cones define the ambient fan; otherwise, the fan is obtained by triangulating the rays (see CY_ambient_toric_fan).
xi (ArrayLike | None, default: None ) –

The integer vector \(\xi\) defining the \(\mathbb{Z}_2\) action. It can only be omitted if the input is a trilayer polytope, in which case it is set to minus the distinguished vertex whose GLSM charges are half of the anticanonical class.
resolve_A1_singularities (bool, default: True ) –

Whether to resolve the A1 singularities of the orbifold along codimension-two fixed loci by adding blowup rays. This only has an effect when the input is a Fan and the orbifold fan was not replaced by the nef-decomposition refinement.
construct_nef_decomposition (bool, default: True ) –

Whether to attempt, for regular orientifolds, to replace the orbifold fan by a refinement of the normal fan of the Newton polytopes of \(L\) and \(6(-K - L)\), where \(-K\) is the anticanonical divisor of the orbifold, so that these divisors become Cartier and nef. See yields_nef_decomposition.

Raises:

ValueError –

If xi is not specified and the input is not a trilayer polytope.
TypeError –

If fan_polytope_or_points has an unsupported type.

Examples:

We construct the orientifold of the degree-8 hypersurface in \(\mathbb{P}_{11114}\). Its polytope is trilayer, so xi can be inferred.

>>> from cytools import Polytope
>>> from cytools.f_theory import CY_orientifold
>>> p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-4]])
>>> o = CY_orientifold(p)
>>> o.xi()
array([ 0,  0,  0, -1])
>>> o.is_regular()
True
>>> o.line_bundle()
array([0, 8, 0, 0, 0, 1])
>>> sorted(o.o_planes())
[(2,), (6,)]

Methods:

CY_ambient_toric_fan –

Return the toric fan of the Calabi-Yau ambient variety, before orbifolding.
NHC –

Return the toric divisors that support non-Higgsable clusters.
Newton_Polytope –

Return the Newton polytope of the O3/O7 line bundle \(L\).
ambient_dim –

Return the dimension of the lattice containing the orbifold rays.
dim –

Return the dimension of the Calabi-Yau ambient variety.
intersection_numbers_orbifold –

Return the intersection numbers of the orbifold ambient variety.
is_regular –

Return whether the orientifold is regular.
line_bundle –

Return the O3/O7 line bundle \(L\) on the orbifold ambient variety.
normal_fan –

Return the normal fan used for the nef-decomposition construction.
o_planes –

Return the ambient cones whose toric strata are fixed by \(\sigma\).
orbifold_toric_fan –

Return the toric fan of the orbifold ambient variety.
polytope –

Return the polytope used to construct the orientifold.
vectors_CY_ambient –

Return the rays of the Calabi-Yau ambient fan, before orbifolding.
vectors_orbifold –

Return the rays of the orbifold ambient fan.
xi –

Return the vector \(\xi\) defining the \(\mathbb{Z}_2\) action.
yields_nef_decomposition –

Return whether the orientifold data yields a Cartier and nef decomposition.

F-theory and orientifolds¶

CY_orientifold ¶

CY_ambient_toric_fan ¶

NHC ¶

Newton_Polytope ¶

ambient_dim ¶

dim ¶

intersection_numbers_orbifold ¶

is_regular ¶

line_bundle ¶

normal_fan ¶

o_planes ¶

orbifold_toric_fan ¶

polytope ¶

vectors_CY_ambient ¶

vectors_orbifold ¶

xi ¶

yields_nef_decomposition ¶

F_Theory_Uplift ¶

CY_ambient_toric_fan ¶

Cayley_M ¶

Cayley_N ¶

M_conv_toric_fan ¶

NHC ¶

NHC_singular_uplift ¶

ambient_dim_base ¶

basis_homology_M ¶

basis_homology_N ¶

blowups ¶

chi ¶

dim_base ¶

divisor_intersection_M ¶

divisor_intersection_N ¶

h11 ¶

h21 ¶

h31 ¶

h31_trivial ¶

intersection_numbers_M_conv ¶

intersection_numbers_orbifold ¶

intersection_numbers_singular_uplift_ambient ¶

intersection_numbers_smooth_uplift_ambient ¶

is_nef_decomposition ¶

is_nef_partition ¶

is_partition ¶

is_regular ¶

line_bundle_base_M ¶

line_bundle_base_N ¶

line_bundle_orbifold ¶

line_bundle_weierstrass_M ¶

line_bundle_weierstrass_N ¶

nef_partition_M ¶

nef_partition_N ¶

normal_fan ¶

orbifold_toric_fan ¶

orientifold ¶

points_not_interior_to_codim_1_and_2_face_M ¶

points_not_interior_to_codim_1_and_2_face_N ¶

pol_B_M ¶

pol_B_N ¶

pol_M_conv ¶

pol_M_sum ¶

pol_N_conv ¶

pol_N_sum ¶

pol_W_M ¶

pol_W_N ¶

polytope ¶

singular_uplift_ambient_toric_fan ¶

smooth_uplift_ambient_toric_fan ¶

vectors_CY_ambient ¶

vectors_orbifold ¶

vectors_singular_uplift_ambient ¶

vectors_smooth_uplift_ambient ¶

xi ¶

fetch_F_Theory_uplifts ¶

fetch_nef_partition_uplifts ¶

fetch_orientifolds ¶