Calabi-Yau-Period-Geometry

This project collects code used for analysing Calabi–Yau families. It allows for computation of discriminant loci and topological data of manifolds defined as hypersurfaces or complete intersections in toric ambient spaces.

toric_topdata

Supported initialisations are represented by the following examples.

elliptic_curve_D = ToricPolytopeProjectiveSpace([1, 2, 3], model_name="elliptic_curve_D")

CY3_quintic = ToricPolytopeProjectiveSpace([1, 1, 1, 1, 1], model_name="quintic")
CY3_bicubic = ToricPolytopeCICY([[3, 3]], model_name="bi-cubic")
CICY3_two_parameter_manual_nef = ToricPolytope(
    [
        [1, 0, 0, 0, 0, 0],
        [0, 1, 0, 0, 0, 0],
        [0, 0, 1, 0, 0, 0],
        [0, 0, 0, 1, 0, 0],
        [0, 0, 0, 0, 1, 0],
        [0, 0, 0, 0, 0, 1],
        [-1, -1, 0, 0, 0, 0],
        [0, 0, -1, -1, -1, -1],
    ],
    nef_partition=[[0, 1, 2, 3], [4, 5, 6, 7]],
)

CICY5_two_parameter = ToricPolytopeCICY([[6, 1], [0, 2]])

The discriminant factors and topological data, e.g. for the quintic, can then be computed with the methods below.

CY3_quintic.disc()

CY3_quintic.topdata()

We list the output of above two lines.

INFO:__main__:Discriminant factors: [[z1 + 1/3125, 0]]
INFO:__main__:

--- Polytope and GLSM table ------------------
 1|1|1|1|1|| 1

-1|1|0|0|0|| 0
-1|0|1|0|0|| 0
-1|0|0|1|0|| 0
-1|0|0|0|1|| 0
--------------
 1,1,1,1,1; -5

--- L-vectors (GLSM charges) -----------------
[[1, 1, 1, 1, 1, -5]]

--- Intersection numbers CY ------------------
{(0, 0, 0): 5}

--- Intersection ring ------------------------
5*t0^3

--- Intersection ring no multiplicities ------
5*t0^3

--- Chern polynomials ------------------------
[[0], [10*t0^3], [-40*t0^3]]

--- Integrated Chern classes -----------------
[[0], [50], [-200]]
DEBUG:__main__:

--- Kähler cone generators (ambient space) ---
['[z4]']

--- Intersection numbers ambient space -------
{(0, 0, 0, 0): 1}

The result is saved in the folder data/topdata as a JSON file — giving a model name helps keeping track of these outputs. Note that for Calabi–Yau dimensions larger than four, the additional

Operators and periods

operators.sage and period.sage provide classes for working with Picard–Fuchs operators and their period solutions: PFOperator, PFIdeal and Period, together with Ansatz variants of the first and last (PFOperatorAnsatz, PeriodAnsatz) used to search for unknown operators or periods of a given z- and theta-degree. The operator classes live in operators.sage, the period classes in period.sage; since the two depend on each other, load period.sage, which also loads operators.sage.

A PFOperator is parsed from a string in the variables z0, ..., z<n-1> and theta0, ..., theta<n-1>, the logarithmic derivatives theta_i = z_i d/dz_i. For example, the quintic's Picard–Fuchs operator:

L = PFOperator(
    "theta0^4 - 3125*z0*theta0^4 - 6250*z0*theta0^3 - 4375*z0*theta0^2 - 1250*z0*theta0 - 120*z0",
    no_variables=1,
)

L.simplify().operator_string

'-5*(5*theta0 + 4)*(5*theta0 + 3)*(5*theta0 + 2)*(5*theta0 + 1)*z0 + theta0^4'

An operator (or a PFIdeal of several) can be solved for its power series solution at given indicial exponents and order.

ideal = PFIdeal([L])
period = ideal.find_power_series_solution(indicials=[0], order=3)[0]

period.period_string

'168168000*z0^3 + 113400*z0^2 + 120*z0 + 1'

The reverse direction is supported too: given a Period, find_annihilating_operators searches for PFOperators of a given z- and theta-degree that annihilate it, by solving an Ansatz of unknown coefficients via linear algebra. Both directions extend to several moduli, e.g. for the two-parameter model P_{2,2,2,1,1}[8]:

M1 = PFOperator(
    "theta1*(-2*theta0 + 2*theta1 - 1) + 2*(theta0 - 2*theta1 - 1)*(theta0 - 2*theta1)*z1",
    no_variables=2,
)
M2 = PFOperator(
    "theta0^2*(2*(theta0 - 2*theta1)*z1 - theta1) - 16*(2*theta0 + 1)*(4*theta0 + 1)*(4*theta0 + 3)*z0*z1",
    no_variables=2,
)
ideal = PFIdeal([M1, M2])

period = ideal.find_power_series_solution(indicials=[0, 1 / 2], order=6)[0]
recovered = period.find_annihilating_operators(z_degree=1, theta_degree=2)

period.period_string
recovered[0].simplify().operator_string

'-1/45045*(60886425600*z0^3*z1^3 - 2767564800*z0^2*z1^4 + 100638720*z0*z1^5 - 14192640*z1^6 + 830269440*z0^2*z1^3 - 30750720*z0*z1^4 + 4193280*z1^5 - 242161920*z0^2*z1^2 + 9884160*z0*z1^3 - 1281280*z1^4 - 3459456*z0*z1^2 + 411840*z1^3 + 1441440*z0*z1 - 144144*z1^2 + 60060*z1 - 45045)*sqrt(z1)'
'-2*(theta0 - 2*theta1)*(theta0 - 2*theta1 - 1)*z1 + (2*theta0 - 2*theta1 + 1)*theta1'

recovered[0] is, up to scale, M1 — recovered purely from M1, M2's shared power series solution.