jaxvacua.conifold.coni.find_conifolds

find_conifolds(polytope, FRSTs=None, n_conifolds=2, verbosity=0)

Identifies conifold limits in complex structure moduli space by searching for flop transitions in the triangulations of a toric polytope.

Details

A conifold transition corresponds to a flop: an edge inside a 2-face of the dual polytope that (i) can be flipped in the triangulation (Whitehouse flip) and (ii) whose associated curve is an extremal generator of the Mori cone.

The search proceeds in three stages:

1. Face-pair selection — pairs \((f_1, f_2)\) of primal 1-faces and dual 2-faces are retained when:

   • \(f_1\) has exactly \(n_{\rm cf}-1\) interior points (the conifold divisors).

   • \(f_2\) has no interior points (for \(n_{\rm cf}>1\)).

   • \(f_2\) is not a simplex (\(\geq 4\) vertices), so a flip is geometrically possible.

2. Triangulation loop — for each FRST of the dual polytope, the simplices restricted to each candidate 2-face are extracted. Interior edges (shared by exactly two simplices) are enumerated, and flippable ones are identified by checking whether the convex hull of the four involved points has four vertices (a quadrilateral).

3. Mori cone check — a candidate flop edge produces a charge vector \(q\) (see Conifold.conifold_charge()); only edges for which \(q\) is an extremal ray of the Mori cone cap are accepted as genuine conifolds.

Parameters:

• polytope (Polytope) – Toric polytope of the CY model.

• FRSTs (Optional[List[Triangulation]]) – Pre-computed fine, regular, star triangulations of the dual polytope. If None, all NTFE classes are computed automatically. Defaults to None.

• n_conifolds (int) – Number of conifold divisors — i.e. the number of interior points of the 1-face \(f_1\). Defaults to 2.

• verbosity (int) – 0 — silent; 1 — prints candidate count; 2 — prints each conifold found. Defaults to 0.

Returns:

• list of Conifold – All Conifold instances found across all

• provided triangulations.