CalabiYau

CalabiYau ¶

CalabiYau(toric_var, nef_partition=None)

A Calabi-Yau manifold in a toric variety.

This class handles various computations relating to the Calabi-Yau manifold itself. It can be used to compute intersection numbers and the toric Mori and Kähler cones, among other things. The Calabi-Yau is either the anticanonical hypersurface of its ambient toric variety or, if a nef partition is specified, the corresponding complete intersection.

Warning

Generally, objects of this class should not be constructed directly by the end user. Instead, they should be created with the get_cy method of ToricVariety or the get_cy method of Triangulation.

Tip

This package is focused on computations on Calabi-Yau 3-fold hypersurfaces, but there is experimental support for Calabi-Yaus of other dimensions, for non-favorable ones, and for complete intersections. See experimental features for more details.

Parameters:

toric_var (ToricVariety) –

The ambient toric variety of the Calabi-Yau. For hypersurfaces, its defining triangulation must be fine and use either all the points of the polytope not interior to facets, or all of its points.
nef_partition (list[tuple[int, ...]] | None, default: None ) –

A nef partition of the polytope, given as a list of tuples of indices of points of the polytope, which defines a complete intersection Calabi-Yau. If not specified, the anticanonical hypersurface is constructed.

Raises:

ValueError –

If nef_partition does not define a nef partition, if the triangulation is not fine (for hypersurfaces), or if the triangulation does not use a valid set of points (for hypersurfaces).
Exception –

If the experimental features are not enabled and a nef partition is given, the Calabi-Yau is not a 3-fold, or the polytope is not favorable.

Examples:

We construct a Calabi-Yau from a fine, regular, star triangulation of a polytope. Since this class is not intended to be initialized by the end user, we create it with the get_cy method of Triangulation. In this example we obtain the quintic hypersurface in \(\mathbb{P}^4\).

>>> from cytools import Polytope
>>> p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])
>>> t = p.triangulate()
>>> t.get_cy()
A Calabi-Yau 3-fold hypersurface with h11=1 and h21=101 ...

Methods:

ambient_dimension –

Return the complex dimension of the ambient toric variety.
ambient_variety –

Return the ambient toric variety.
chi –

Compute the Euler characteristic of the Calabi-Yau.
clear_cache –

Clear the cached results of any previous computation.
compute_curve_volumes –

Compute the volumes of the generators of the toric Mori cone.
compute_cy_volume –

Compute the volume of the Calabi-Yau at a location in the Kähler cone.
compute_divisor_volumes –

Compute the volumes of divisors at a location in the Kähler cone.
compute_gvs –

Compute the genus-zero Gopakumar-Vafa invariants of the Calabi-Yau.
compute_gws –

Compute the genus-zero Gromov-Witten invariants of the Calabi-Yau.
compute_inverse_kahler_metric –

Compute the inverse Kähler metric at a location in the Kähler cone.
compute_kahler_metric –

Compute the Kähler metric at a location in the Kähler cone.
compute_kappa_matrix –

Compute the matrix \(\kappa_{ijk}t^k\) at a location in the Kähler cone.
compute_kappa_vector –

Compute the vector \(\kappa_{ijk}t^j t^k\) at a location in the Kähler cone.
curve_basis –

Return the current basis of curves of the Calabi-Yau.
dimension –

Return the complex dimension of the Calabi-Yau.
divisor_basis –

Return the current basis of divisors of the Calabi-Yau.
glsm_charge_matrix –

Compute the GLSM charge matrix of the Calabi-Yau.
glsm_linear_relations –

Compute the linear relations among the columns of the GLSM charge matrix.
h11 –

Return the Hodge number \(h^{1,1}\) of the Calabi-Yau.
h12 –

Return the Hodge number \(h^{1,2}\) of the Calabi-Yau.
h13 –

Return the Hodge number \(h^{1,3}\) of the Calabi-Yau.
h22 –

Return the Hodge number \(h^{2,2}\) of the Calabi-Yau.
hpq –

Return the Hodge number \(h^{p,q}\) of the Calabi-Yau.
intersection_numbers –

Return the intersection numbers of the Calabi-Yau.
is_smooth –

Return whether the Calabi-Yau is smooth.
is_trivially_equivalent –

Check if the Calabi-Yaus are trivially equivalent.
mori_cone_cap –

Return the Mori cone cap of the Calabi-Yau.
polytope –

Return the polytope whose triangulation gives rise to the ambient variety.
prime_toric_divisors –

Return the indices of the prime toric divisors that intersect the Calabi-Yau.
second_chern_class –

Compute the second Chern class of the Calabi-Yau.
set_curve_basis –

Set a basis of curves of the Calabi-Yau.
set_divisor_basis –

Set a basis of divisors of the Calabi-Yau.
toric_effective_cone –

Return the effective cone inferred from toric geometry.
toric_kahler_cone –

Return the Kähler cone inferred from toric geometry.
toric_mori_cone –

Return the Mori cone inferred from toric geometry.
triangulation –

Return the triangulation giving rise to the ambient toric variety.

CalabiYau

CalabiYau ¶

ambient_dimension ¶

ambient_variety ¶

chi ¶

clear_cache ¶

compute_curve_volumes ¶

compute_cy_volume ¶

compute_divisor_volumes ¶

compute_gvs ¶

compute_gws ¶

compute_inverse_kahler_metric ¶

compute_kahler_metric ¶

compute_kappa_matrix ¶

compute_kappa_vector ¶

curve_basis ¶

dimension ¶

divisor_basis ¶

glsm_charge_matrix ¶

glsm_linear_relations ¶

h11 ¶

h12 ¶

h13 ¶

h22 ¶

hpq ¶

intersection_numbers ¶

is_smooth ¶

is_trivially_equivalent ¶

mori_cone_cap ¶

polytope ¶

prime_toric_divisors ¶

second_chern_class ¶

set_curve_basis ¶

set_divisor_basis ¶

toric_effective_cone ¶

toric_kahler_cone ¶

toric_mori_cone ¶

triangulation ¶

Invariants ¶

coo property ¶

cutoff property ¶

dok property ¶

grading_vec property ¶

gvs property ¶

gws property ¶

size property ¶

charges ¶

cone ¶

gv ¶

gw ¶

invariant ¶

coo `property` ¶

cutoff `property` ¶

dok `property` ¶

grading_vec `property` ¶

gvs `property` ¶

gws `property` ¶

size `property` ¶