jaxvacua.flux_eft

Flux effective field theory for Type IIB compactifications.

Purpose

Define FluxEFT, the flux-physics layer above css. It evaluates the Gukov-Vafa-Witten superpotential, F-terms, scalar potential, tadpole and stability data for integer three-form flux backgrounds.

Main public API

• FluxEFT: extends css with flux splitting, superpotential, Kähler-covariant derivatives, real-coordinate derivatives, Hessians and mass matrices.

• Tadpole, ISD and physicality helpers used by the vacuum-search algorithms.

• Fundamental-domain and monodromy utilities used to compare equivalent solutions.

Design notes

This module contains the reusable EFT model. Search, sampling and classification workflows live in jaxvacua.flux_vacua_finder and operate directly on this class through inheritance.

Computational graph

This layer assembles the integer flux background into the GVW superpotential, its covariant F-term derivatives, the no-scale scalar potential, and the D3-tadpole contribution. The defining relations are

\begin{align*} W_{\text{GVW}}(z, \tau;\, \text{flux}) &= \int_X G_3 \wedge \Omega = \bigl(F - \tau H\bigr) \cdot \Pi(z), \\[2pt] D_I W &= \partial_I W + (\partial_I K)\, W, \quad I = (z^i,\, \tau), \\[2pt] V_{\rm ns} &= e^{K}\!\left( g^{i\bar\jmath}\, D_i W\, \overline{D_j W} + (\operatorname{Im}\tau)^{-2} |D_\tau W|^2 \right), \\[2pt] N_{\text{flux}} &= f^T \Sigma\, h, \end{align*}

with \(G_3 = F_3 - \tau H_3\) the complex three-form flux and \(\Sigma\) the symplectic intersection form on \(H^3(X, \mathbb{Z})\). The full supergravity \(-3|W|^2\) contribution can be included by setting noscale=False.

The diagram below splits the inputs into two visually distinct groups: those inherited from the upstream period and complex-structure layers (light grey, solid border) and the external user-supplied flux integers and axio-dilaton (white, dashed border). Public outputs of the layer are highlighted in orange.

jaxvacua.flux_eft.FluxEFT — Computational graph

Inputs (inherited from periods/css  +  external runtime)

Inherited from periods / css

$\Pi(z)$

period vector

$K,\; K_{i\bar\jmath}$

Kähler potential and metric

External (user-supplied at runtime)

flux $(F_3, H_3)$

$\in \mathbb{Z}^{2(h^{1,2}+1)}$

$\tau$

axio-dilaton

Superpotential & F-terms

tadpole  $N_{\rm flux}$

$N_{\rm flux} = f^T \Sigma\, h$ for flux vector $(f,h)$
$\Longrightarrow\; N_{\rm flux} \leq Q_{\rm O3}$ (D3-charge)
FluxEFT.tadpole

$W = (F_3 - \tau H_3) \cdot \Pi(z)$

GVW superpotential: $W = \int_X G_3 \wedge \Omega$ with $G_3 = F_3 - \tau H_3$
FluxEFT.superpotential

$D_I W = \partial_I W + (\partial_I K)\, W$

$I = (z^i,\, \tau)$ — Kähler-covariant derivative on the full sector
$D_I W = 0$ defines a SUSY vacuum
FluxEFT.DW

Scalar potential

$V_{\rm ns} \;=\; e^{K}\, K^{I\bar J}\, D_I W\, \overline{D_J W}$

Default no-scale flux potential on the $(z^i,\tau)$ sector; set noscale=False to include the full supergravity $-3|W|^2$ term
FluxEFT.scalar_potential · FluxEFT.V

Differentials of $V$

$\partial_I V$

Gradient of the scalar potential
(target for gradient-flow vacuum search)
FluxEFT.dV · FluxEFT.dV_x

$\partial_I \partial_J V$

Full Hessian on the moduli + axio-dilaton sector
FluxEFT.hessian · FluxEFT.ddV

$M^2_{IJ}$

Bosonic mass matrix evaluated at $D_I W = 0$,  $m^2 = \operatorname{eig}(M^2)$
FluxEFT.mass_matrix

EFT flux class

FluxEFT([h12, model_ID, model_type, limit, ...])

A class representing the flux sector in a 4D EFT obtained from Type IIB compactification on orientifolds of CY threefolds with 3-form flux backgrounds.

Superpotential and F-terms

FluxEFT.superpotential(moduli, tau, fluxes)	Calculates the value of the superpotential for given flux, moduli and axio-dilaton.
FluxEFT.superpotential_gauge_invariant(...)	Calculates the value of the gauge invariant version of the flux superpotential.
FluxEFT.DW(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic Kähler covariant derivatives of the superpotential with respect to the complex structure moduli \(z^{i}\) and the axio-dilaton \(\tau\).
FluxEFT.DW_z(moduli, moduli_c, tau, tau_c, ...)	Calculates the Kähler covariant derivative of the superpotential \(W\) with respect to the complex structure moduli \(z^{i}\).
FluxEFT.DW_tau(moduli, moduli_c, tau, tau_c, ...)	Calculates the Kähler covariant derivative of the superpotential with respect to the axio-dilaton \(\tau\).
FluxEFT.DW_x(x, fluxes)	Calculates the real and imaginary parts of the \(F\)-term conditions.
FluxEFT.DcDW(moduli, moduli_c, tau, tau_c, ...)	Returns a matrix containing the mixed second Kähler derivatives \(D_{\overline{I}}D_{J}W\).
FluxEFT.DDW(moduli, moduli_c, tau, tau_c, fluxes)	Returns a matrix the second holomorphic Kähler derivatives \(D_I D_J W\) of the superpotential.
FluxEFT.canonical_fterms(moduli, moduli_c, ...)	Returns the canonically normalised \(F\)-term conditions \(F_I,\, F^I\).

Scalar potential

FluxEFT.scalar_potential(moduli, moduli_c, ...)	Returns the value of the \(F\)-term scalar potential.
FluxEFT.V(moduli, moduli_c, tau, tau_c, fluxes)	Returns the value of the \(F\)-term scalar potential.
FluxEFT.V_x(x, fluxes[, noscale])	Returns the value of the real part of the F-term scalar potential \(V\) for input of the real scalar fields.
FluxEFT.ddV(moduli, moduli_c, tau, tau_c, fluxes)	Returns the second derivatives \((\partial_I\partial_J V,\,\partial_I\partial_{\bar J}V)\) of the \(F\)-term scalar potential.
FluxEFT.ddV_x(x, fluxes[, noscale])	Returns the second derivatives of the F-term scalar potential \(V\) with respect to the real scalar fields for real inputs.
FluxEFT.mass_matrix(moduli, moduli_c, tau, ...)	Returns the mass matrix after canonical normalisation of the kinetic terms.
FluxEFT.hessian(moduli, moduli_c, tau, ...)	Returns the Hessian of the scalar potential.

ISD condition

FluxEFT.ISD_condition(moduli, moduli_c, tau, ...)	Checks whether the fluxes satisfy the ISD condition \(\star G_3=\text{i}G_3\).
FluxEFT.ISD_matrix(moduli, moduli_c)	Computes the value of the ISD-matrix \(\mathcal{M}\).
FluxEFT.projection_fluxes(moduli, tau, fluxes)	Computes the Hodge decomposition of the 3-form flux \(G_3 = F_3 - \tau H_3\) into its \((p,q)\)-components on the Calabi-Yau threefold.

Flux utilities

FluxEFT.map_to_fd_tau(tau, fluxes[, ...])	Map of the axio-dilaton value and the flux vector to the fundamental domain (FD) of \(\text{SL}(2,\mathbb{Z})\).
FluxEFT.tadpole(fluxes)	Calculates the D3-charge for given fluxes.
FluxEFT.flux_to_pfv(flux)	Returns M- and K-vector specifying PFV from full flux vector.
FluxEFT.pfv_to_flux(M, K)	Returns full flux vector from M- and K-vector specifying PFV.
FluxEFT.pfv_to_moduli(M, K, tau)	Returns values of the complex structure moduli at the level of the PFV.