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.

```python
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.
```python
CY3_quintic.disc()

CY3_quintic.topdata()
```

We list the output of above two lines.

```term
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

## period_computation

`period_computation.sage` provides 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.

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&ndash;Fuchs operator:

```python
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
```

```term
'-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.

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

period.period_string
```

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

The reverse direction is supported too: given a `Period`, `find_annihilating_operators` searches for
`PFOperator`s 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]:

```python
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
```

```term
'-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` &mdash; recovered purely from `M1`, `M2`'s shared power series
solution.