EKC Construction: h11=2 Example

This notebook demonstrates constructing the extended Kahler cone (EKC) of a Calabi-Yau threefold with \(h^{1,1} = 2\), using the GV-based method of arXiv:2212.10573 and arXiv:2303.00757.

The cybir package implements this pipeline and integrates with CYTools.

import cybir
from cybir import CYBirationalClass, patch_cytools
import cytools
import numpy as np

# Activate CYTools integration
patch_cytools()

Warning: cache /Users/elijahsheridan/Library/Caches/CYTools/twoface_ineqs.pkl.gz is broken (Ran out of input), removing it...

Load a Calabi-Yau threefold

We fetch an \(h^{1,1} = 2\) polytope from the CYTools database, triangulate it, and extract a Calabi-Yau hypersurface.

# Load an h11=2 polytope from CYTools database
p = cytools.fetch_polytopes(h11=2, limit=1)[0]
cy = p.triangulate().get_cy()
print(f"h11 = {cy.h11()}, h21 = {cy.h21()}")

h11 = 2, h21 = 29

Construct the Extended Kahler Cone

The CYBirationalClass.from_gv classmethod runs the full pipeline:

1. Compute GV invariants up to max_deg

2. Set up the root phase from the CYTools geometry

3. BFS over Mori cone walls, classifying and flopping

ekc = CYBirationalClass.from_gv(cy, max_deg=10, verbose=True)
print(ekc)

CYBirationalClass(constructed, phases=1, contractions=2)

Inspect Phases

Each phase is a CalabiYauLite object with intersection numbers, second Chern class, Mori cone generators, and Kahler cone generators.

for phase in ekc.phases:
    print(f"{phase.label}: int_nums shape = {phase.int_nums.shape}")
print(f"\nTotal phases: {len(ekc.phases)}")

CY_0: int_nums shape = (2, 2, 2)

Total phases: 1

Inspect Contractions

Each wall between phases is an ExtremalContraction with a classified type (flop, symmetric flop, asymptotic, CFT, or su(2)) and the contraction curve.

for c in ekc.contractions:
    print(f"  {c.contraction_type.display_name()}: curve = {c.contraction_curve}")

  asymptotic: curve = [1 0]
  asymptotic: curve = [0 1]

Phase Graph Structure

The CYGraph encodes which phases are connected by contractions. Self-loops represent terminal walls (asymptotic, CFT, su(2)).

print(ekc.graph)
for phase in ekc.phases:
    neighbors = ekc.graph.neighbors(phase.label)
    print(f"{phase.label} -> {[n.label for n in neighbors]}")

CYGraph(phases=1, contractions=2)
CY_0 -> ['CY_0']

Coxeter Data

If any symmetric-flop walls are found, their reflection matrices form a Coxeter group. The Coxeter matrix encodes the orders of pairwise products.

if ekc.coxeter_matrix is not None:
    print("Coxeter matrix:")
    print(ekc.coxeter_matrix)
else:
    print("No Coxeter reflections found (all walls are terminal or generic flops)")

No Coxeter reflections found (all walls are terminal or generic flops)

Coxeter Orbit Expansion (optional)

If the Coxeter group from symmetric-flop reflections is non-trivial and finite, apply_coxeter_orbit enumerates the full group and applies every element to each fundamental-domain phase, producing the complete birational geometry.

ekc.apply_coxeter_orbit()
print(f"After orbit expansion: {len(ekc.phases)} phases")
if ekc.coxeter_type:
    print(f"Coxeter type: {ekc.coxeter_type}")
    print(f"Group order: {ekc.coxeter_order}")

After orbit expansion: 1 phases