jaxvacua.freezer.Freezer

class Freezer(model)

Bases: ABC

Abstract base class for a reduced effective field theory obtained by integrating out a set of heavy moduli.

Given a full model with moduli \((z_{\text{heavy}}, z_{\text{light}}, \tau)\) and fluxes, the reduced EFT expresses the heavy moduli as functions of the light fields via their leading-order EOM:

\[z_{\text{heavy}} = z_{\text{heavy}}(z_{\text{light}}, \tau, \text{fluxes})\]

and provides the superpotential, its derivatives, etc. evaluated on this solution.

Subclasses must implement:

• heavy_indices: which moduli are frozen out

• solve_heavy: solve for heavy moduli given light fields

• _real_light_to_full: convert real light-field coordinates to full array

__init__(model)

Initialise the Freezer base class.

Parameters:

model (Any) – A flux EFT model object providing superpotential, DW, DW_x, dDW_x, _convert_real_to_complex, and period data via lcs_tree.

Methods

DW_light(z_light, z_light_c, tau, tau_c, ...)	Covariant derivatives \(D_i W\) with respect to the light moduli, with heavy moduli on-shell.
DW_x_light(x_light, fluxes, **kwargs)	Gradient of the superpotential \(\partial_{x^a} W\) in real coordinates for the light moduli, with heavy moduli on-shell.
V_x_light(x_light, fluxes[, noscale])	Scalar potential \(V\) evaluated at the light-field coordinates, with heavy moduli on-shell.
__init__(model)	Initialise the Freezer base class.
dDW_x_light(x_light, fluxes, **kwargs)	Hessian \(\partial_{x^a}\partial_{x^b} W\) in real coordinates for the light moduli.
dV_x_light(x_light, fluxes[, noscale])	Gradient of the scalar potential \(\nabla_\phi V\) with respect to the real light-field coordinates, with heavy moduli on-shell.
ddV_x_light(x_light, fluxes[, noscale])	Hessian of the scalar potential \(\partial_{\phi^\alpha}\partial_{\phi^\beta} V\) with respect to the real light-field coordinates, with heavy moduli on-shell.
reconstruct_full_moduli(z_light, tau, ...)	Reconstruct the full moduli array by solving for the heavy moduli and inserting them at the correct positions.
solve_heavy(z_light, tau, fluxes, **kwargs)	Solve the leading-order EOM for the heavy moduli as functions of the light moduli, axio-dilaton, and fluxes.
superpotential(z_light, tau, fluxes, **kwargs)	Superpotential of the reduced theory.

Attributes

heavy_indices	Description: Indices of the heavy moduli within the full moduli array.
light_indices	Description: Indices of the light moduli (complement of heavy_indices).
n_heavy	Description: Number of heavy moduli.
n_light	Description: Number of light moduli.

DW_light(z_light, z_light_c, tau, tau_c, fluxes, **kwargs)

Covariant derivatives \(D_i W\) with respect to the light moduli, with heavy moduli on-shell.

Parameters:

• z_light (Array) – Light moduli values.

• z_light_c (Array) – Conjugate light moduli values.

• tau (complex) – Axio-dilaton.

• tau_c (complex) – Conjugate axio-dilaton.

• fluxes (Array) – Full flux vector.

Returns:

Array – \(D_i W\) for the light moduli and \(D_\tau W\).

Return type:

Array

DW_x_light(x_light, fluxes, **kwargs)

Gradient of the superpotential \(\partial_{x^a} W\) in real coordinates for the light moduli, with heavy moduli on-shell.

This is the analogue of model.DW_x but restricted to the light degrees of freedom.

Parameters:

• x_light (Array) – Real variables for light moduli and axio-dilaton.

• fluxes (Array) – Full flux vector.

Returns:

Array – Real gradient restricted to light directions.

Return type:

Array

V_x_light(x_light, fluxes, noscale=True, **kwargs)

Scalar potential \(V\) evaluated at the light-field coordinates, with heavy moduli on-shell.

Parameters:

• x_light (Array) – Real coordinates for light moduli and axio-dilaton.

• fluxes (Array) – Full flux vector.

• noscale (bool) – If True, uses the no-scale scalar potential \(V = e^K K^{I\bar J} D_I W D_{\bar J}\bar W\). Defaults to True.

Returns:

float – Value of \(V\) with heavy moduli at their on-shell values.

Return type:

float

dDW_x_light(x_light, fluxes, **kwargs)

Hessian \(\partial_{x^a}\partial_{x^b} W\) in real coordinates for the light moduli.

Parameters:

• x_light (Array) – Real variables for light moduli and axio-dilaton.

• fluxes (Array) – Full flux vector.

Returns:

Array – Hessian restricted to light directions.

Return type:

Array

dV_x_light(x_light, fluxes, noscale=True, **kwargs)

Gradient of the scalar potential \(\nabla_\phi V\) with respect to the real light-field coordinates, with heavy moduli on-shell.

Details

Let \(\phi^\alpha = (a^1, v^1, \ldots, a^{n_{\rm light}}, v^{n_{\rm light}}, c_0, s)\) denote the real light-field coordinates, where \(z^i = a^i + \mathrm{i}\,v^i\) and \(\tau = c_0 + \mathrm{i}\,s\). This function returns the restriction

\[\nabla_\phi V \big|_{\phi^\alpha} = \partial_{\phi^\alpha} V(x_{\rm full}(\phi))\]

where \(x_{\rm full}(\phi)\) substitutes the on-shell heavy moduli via _real_light_to_full().

Parameters:

• x_light (Array) – Real coordinates for light moduli and axio-dilaton.

• fluxes (Array) – Full flux vector.

• noscale (bool) – If True, uses the no-scale scalar potential. Defaults to True.

Returns:

• Array – Gradient \(\partial_{\phi^\alpha} V\), restricted to

• light directions, of shape `` (2 * n_light + 2,)

Return type:

Array

ddV_x_light(x_light, fluxes, noscale=True, **kwargs)

Hessian of the scalar potential \(\partial_{\phi^\alpha}\partial_{\phi^\beta} V\) with respect to the real light-field coordinates, with heavy moduli on-shell.

Details

Using the same real light-field coordinates \(\phi^\alpha = (a^1, v^1, \ldots, a^{n_{\rm light}}, v^{n_{\rm light}}, c_0, s)\) as in dV_x_light(), this function returns the block of the full Hessian restricted to the light directions:

\[(\nabla\nabla V)_{\alpha\beta} = \partial_{\phi^\alpha}\partial_{\phi^\beta} V\]

with heavy moduli substituted by their on-shell values.

Parameters:

• x_light (Array) – Real coordinates for light moduli and axio-dilaton.

• fluxes (Array) – Full flux vector.

• noscale (bool) – If True, uses the no-scale scalar potential. Defaults to True.

Returns:

• Array – Hessian block restricted to light directions, of shape

• `` (2 * n_light + 2, 2 * n_light + 2)

Return type:

Array

abstract property heavy_indices: Tuple[int, ...]

Description: Indices of the heavy moduli within the full moduli array.

property light_indices: Tuple[int, ...]

Description: Indices of the light moduli (complement of heavy_indices).

property n_heavy: int

Description: Number of heavy moduli.

property n_light: int

Description: Number of light moduli.

reconstruct_full_moduli(z_light, tau, fluxes, **kwargs)

Reconstruct the full moduli array by solving for the heavy moduli and inserting them at the correct positions.

Parameters:

• z_light (Array) – Light moduli values.

• tau (complex) – Axio-dilaton value.

• fluxes (Array) – Full flux vector.

Returns:

Array – Full moduli array of length h12.

Return type:

Array

abstractmethod solve_heavy(z_light, tau, fluxes, **kwargs)

Solve the leading-order EOM for the heavy moduli as functions of the light moduli, axio-dilaton, and fluxes.

Parameters:

• z_light (Array) – Values of the light complex structure moduli.

• tau (complex) – Axio-dilaton value.

• fluxes (Array) – Full flux vector.

Returns:

Array – Values of the heavy moduli.

Return type:

Array

superpotential(z_light, tau, fluxes, **kwargs)

Superpotential of the reduced theory.

Parameters:

• z_light (Array) – Light moduli values.

• tau (complex) – Axio-dilaton.

• fluxes (Array) – Full flux vector.

Returns:

complex – \(W(z_{\text{light}}, \tau)\) with heavy moduli on-shell.

Return type:

complex