jaxvacua.conifold.conifold_utils.get_basis_change

get_basis_change(coninop)

Constructs the unimodular integer basis transformation that maps a conifold curve charge vector to the canonical form \((1, 0, \ldots, 0)\).

Details

Given the integer charge vector \(q \in \mathbb{Z}^{h^{1,1}}\) of the conifold curve (with \(\gcd(q) = 1\)), the function builds the \(h^{1,1}\times h^{1,1}\) unimodular matrix \(\Lambda \in \mathrm{GL}(h^{1,1}, \mathbb{Z})\) satisfying

\[\Lambda \, q = (1,\, 0,\,\ldots,\, 0)^T \,.\]

\(\Lambda\) is assembled from two complementary pieces:

1. First row — the Bézout coefficients \(b\) computed by extended_euclidean(), satisfying \(b \cdot q = 1\).

2. Remaining rows — generators of the lattice \(\{v : v \cdot q = 0\}\), computed by orthogonal_lattice().

The resulting \(\Lambda\) is passed to lcs_tree to change the complex-structure-moduli basis so that \(z^1\) directly parameterises the approach to the conifold singularity.

Parameters:

coninop (Array) – Integer charge vector \(q \in \mathbb{Z}^n\) of the conifold curve, satisfying \(\gcd(q) = 1\).

Returns:

np.ndarray – Unimodular integer matrix \(\Lambda\) of shape \((n, n)\) satisfying \(\Lambda\, q = (1, 0,\ldots,0)^T\).

See also: extended_euclidean(), orthogonal_lattice()