jaxvacua.conifold.conifold_utils.get_bulk_embedding

get_bulk_embedding(q)

Bulk embedding matrix for the conifold charge \(q\): the bulk rows \(\Lambda_{1:}\) of the basis change \(\Lambda\) (from get_basis_change()), transposed to shape \((h^{1,2}, h^{1,2}-1)\).

Details

Used to build a full \(h^{1,2}\)-vector from its \((h^{1,2}-1)\) bulk components. It serves three roles, all with the same matrix because charges are covectors and moduli are vectors of the same lattice:

1. embed bulk moduli/periods: X_bulk -> bulk_embedding @ X_bulk, which carries zero conifold component (\(q\cdot\Lambda_{1:}^T = 0\));

2. contract the intersection/a/b tensors along bulk directions;

3. project a charge/flux covector onto the bulk: charge @ bulk_embedding \(= \Lambda_{1:}\, \tilde q\).

Together with the conifold embedding get_embedding() (\(e_q = \Lambda_0\), the conifold direction of a vector) and the conifold charge \(q\) it satisfies the resolution of identity \(q\, e_q^T + \text{bulk\_proj}\cdot\text{bulk\_embedding}^T = \mathbb{1}\), where the matching bulk projection (the left inverse, \(\text{bulk\_embedding}^T\,\text{bulk\_proj} = \mathbb{1}_{h^{1,2}-1}\)) is get_bulk_projection(). In the conifold-aligned basis (\(q=(1,0,\ldots,0)\)) it reduces to \([0;\mathbb{1}]\).

Parameters:

q (Array) – Integer conifold charge vector of length \(h^{1,2}\).

Returns:

Array – Bulk embedding matrix of shape \((h^{1,2}, h^{1,2}-1)\).

See also: get_bulk_projection(), get_embedding(), get_basis_change()