Calabi-Yau-Period-Geometry/README.md

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([3, 2, 1], 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 periods.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 periods.sage; since the two depend on each other, load periods.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.

Frobenius bases

A PFIdeal also computes its complete Frobenius basis — all solutions, logarithmic ones included — at the origin of its current coordinates; for another point, move it to the origin with change_coordinates first. The exponents are the indicials, the roots of the system's indicial ideal, which is completed by linear algebra where the operators' z-free parts alone don't determine it. The optional hodge_numbers — the numbers of solutions of top log-degree 0, 1, ..., n at a MUM point, e.g. [1, 2, 2, 1] for a two-parameter threefold — certify that completion through their sum, the rank, and bound the powers of logarithms. For P_{2,2,2,1,1}[8] at its MUM point and at the point (z0, z1) = (0, 1/4) on its strong coupling divisor:

D1 = PFOperator(
    "theta0^2*(theta0 - 2*theta1) - 4*z0*(4*theta0 + 3)*(4*theta0 + 2)*(4*theta0 + 1)",
    no_variables=2,
)
D2 = PFOperator("theta1^2 - z1*(2*theta1 - theta0 + 1)*(2*theta1 - theta0)", no_variables=2)
ideal = PFIdeal([D1, D2], hodge_numbers=[1, 2, 2, 1])
strong_coupling = ideal.change_coordinates("z0", "z1 - 1/4")

ideal.indicials()
strong_coupling.indicials()
basis = strong_coupling.frobenius_basis(order=2)
basis[5].period_string

[((0, 0), 6)]
[((0, 0), 4), ((0, 1/2), 2)]
INFO:__main__:Completed the indicial ideal at z = 0 at multiplier degree 2 with 1 extra operator(s): colength 6.
INFO:__main__:Exponent class of (0, 0): 4 solution(s) with leading terms ['1', 'log(z0)', 'log(z0)^2', 'log(z0)^3'].
INFO:__main__:Exponent class of (0, 1/2): 2 solution(s) with leading terms ['sqrt(z1)', 'sqrt(z1)*(log(z0) + log(z1))'].
INFO:__main__:Found a Frobenius basis of 6 solution(s) at order 2.
'-1/225*(7200*z0*z1*log(z0) - 720*z1^2*log(z0) + 7200*z0*z1*log(z1) - 720*z1^2*log(z1) + 33600*z0*z1 - 732*z1^2 + 300*z1*log(z0) + 300*z1*log(z1) + 250*z1 - 225*log(z0) - 225*log(z1))*sqrt(z1)'

Within each exponent class the basis is canonical: every solution has its own leading term with coefficient 1, at which the class' other solutions vanish. Coefficients are those of plain powers of log(z_i), without factors of 2πi.