jaxvacua.conifold.zcf_solver

Conifold-modulus solver and bulk-EFT helpers.

Purpose

Implement the z_cf physics used after integrating out the conifold modulus in coniLCS models, including bulk superpotential terms, log-coefficient evaluation and reconstruction of the conifold modulus.

Main public API

• Building blocks: W_bulk, dK_cf_bulk and log_prefactor.

• Topological splitting and log-coefficient routes: _split_conifold_bulk, _W_log_coeff_pfv, _W_log_coeff_manual, _W_log_coeff_autodiff and W_log_coeff.

• Kähler-covariant correction and reconstruction helpers: log_coeff_K_corr, _zcf_from_log_coeff, compute_zcf, compute_zcf_x and zcf_handling.

Design notes

The leading F-term equation has the form log_prefactor * log(-2*pi*i*z_cf) + W_log_coeff = 0, optionally with the Kähler correction added to the log coefficient. Public functions are plain self methods attached to FluxEFT by the conifold package.

The bulk effective theory after integrating out the conifold modulus \(z_{\rm cf}\). All public functions take flux; private helpers are tightly coupled to the flux pathway.

Mathematical structure (notes/main.tex eq:Wtilde1Explicit, eq:zcf_corrected):

\[\partial_{z_{\rm cf}} W_{\rm coni} = \texttt{log\_prefactor} \cdot \ln(-2\pi i\, z_{\rm cf}) + \texttt{W\_log\_coeff} + \mathcal{O}(z_{\rm cf})\]

with

\[\texttt{log\_prefactor} = (M^1 - \tau H^1) \cdot n_{\rm cf} / (2\pi i)\,,\]

so that

\[z_{\rm cf} = -\frac{1}{2\pi i}\, \exp\!\Bigl(-\bigl(\texttt{W\_log\_coeff} \,[+\,\texttt{log\_coeff\_K\_corr}]\bigr) /\,\texttt{log\_prefactor}\Bigr).\]

Building blocks

W_bulk(self, z, tau, flux[, conj, normalise])
dK_cf_bulk(self, z, cz)
log_prefactor(self, tau, flux[, conj])	Coefficient of \(\ln(-2\pi i z_{\rm cf})\) in \(\partial_{z_{\rm cf}} W_{\rm coni}\):

W̃₁ assembly (three independent routes)

The closed-form manual and the autodiff route autodiff agree numerically; pfv is a deliberate racetrack approximation that becomes exact at the racetrack-stationary point.

W_log_coeff(self, z, tau, flux[, mode, conj])

Public dispatcher for the regular \(O(z_{\rm cf}^0)\) coefficient \(\widetilde W_1\) of \(\partial_{z_{\rm cf}} W_{\rm coni}\) near the conifold.

Kähler correction + final exponentiation

log_coeff_K_corr(self, z, cz, tau, ctau, flux)

Kähler-covariant correction to W_log_coeff:

Top-level dispatchers (attached to FluxEFT)

compute_zcf(self, z_bulk, cz_bulk, tau, ...)	Unified solver for the conifold modulus \(z_{\rm cf}\) after integrating out the heavy direction.
compute_zcf_x(self, x_bulk, flux[, mode, ...])	Unified solver for the conifold modulus \(z_{\rm cf}\) after integrating out the heavy direction.
zcf_handling(self, x_bulk, flux[, mode, ...])	Handle the conifold modulus depending on the mode chosen.