jaxvacua.flux_eft.FluxEFT

jaxvacua.flux_eft.FluxEFT#

class FluxEFT(h12=None, model_ID=None, model_type='KS', limit='LCS', maximum_degree=0, mirror_cy=None, model_data=None, lcs_tree_input=None, model_file='', use_cytools=False, basis_change=None, ncf=None, conifold_curve=None, conifold_basis=True, grading_vector=None, period_input=None, prepotential_input=None, Q=None, gauge_choice=1 + 0j, prange=5, use_gvs=True, save_file=False, **kwargs)#

Bases: css

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.

__init__(h12=None, model_ID=None, model_type='KS', limit='LCS', maximum_degree=0, mirror_cy=None, model_data=None, lcs_tree_input=None, model_file='', use_cytools=False, basis_change=None, ncf=None, conifold_curve=None, conifold_basis=True, grading_vector=None, period_input=None, prepotential_input=None, Q=None, gauge_choice=1 + 0j, prange=5, use_gvs=True, save_file=False, **kwargs)#

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

Parameters:

h12 (int | None) – The number of moduli for the compactified geometry.
model_type (str) – The type of manifold considered for the compactification. Currently, "KS" and "CICY" are available.
model_ID (int | None) – ID specifying a certain model.
limit (str) – String identifying the type of periods to be considered. Currently, only "LCS" is available.
model_data (dict | None) – Contains model data like triple intersection numbers etc.
instanton_data (list) – List of GV and GW invariants.
maximum_degree (int) – Maximum degree used for the instanton sum.
use_cytools (bool) – Whether or not to use CYTools to compute topological data of Calabi-Yau.
mirror_cy (Any | None) – Mirror Calabi-Yau threefold.
basis_change (Array | None) – Basis transformation to be applied to topological data of Calabi-Yau.
grading_vector (Array | None) – Grading vector to be used for the GV computation.
period_input (Optional[Callable]) – Input for periods.
prepotential_input (Optional[Callable]) – Input for prepotential.
gauge_choice (complex) – Gauge choice parameter.
save_file (bool) – Save files for new models. Defaults to False.
**kwargs (Any) – Additional keyword arguments.

D3_tadpole#

Maximum allowed D3-tadpole.

Type:: int

n_fluxes#

Number of fluxes.

Type:: int

Methods

`A`(moduli, moduli_c)	Returns the value of the mirror dual Calabi-Yau volume.
`DDW`(moduli, moduli_c, tau, tau_c, fluxes[, ...])	Returns a matrix the second holomorphic Kähler derivatives \(D_I D_J W\) of the superpotential.
`DDW_SUSY`(moduli, moduli_c, tau, tau_c, fluxes)	Returns a matrix containing the second holomorphic Kähler derivative of the superpotential assuming that the F-flatness conditions \(D_{i}W=D_{\tau}W=0\) are satisfied.
`DDW_general`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the second Kähler covariant derivatives \(D_I D_J W\) of the superpotential \(W\) with respect to the axio-dilaton \(\tau\) and the complex structure moduli \(z^{i}\).
`DDW_matrix`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the matrix of masses for chiral fermions.
`DDW_matrix_SUSY`(moduli, moduli_c, tau, ...)	Returns the matrix of second Kähler derivatives of the superpotential assuming that the F-term conditions are satisfied for all fields.
`DDW_matrix_general`(moduli, moduli_c, tau, ...)	Returns the matrix of second Kähler derivatives of the superpotential.
`DDW_tau_ctau`(moduli, moduli_c, tau, tau_c, ...)	Returns the Kähler covariant derivative \(D_{\overline{\tau}}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the complex conjugate axio-dilaton \(\overline{\tau}\).
`DDW_tau_cz`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the Kähler covariant derivative \(D_{\overline{z}^i}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).
`DDW_tau_tau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the Kähler covariant derivative \(D_{\tau}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the axio-dilaton \(\tau\).
`DDW_tau_z`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the Kähler covariant derivative \(D_{z^i}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the complex structure moduli \(z^i\).
`DDW_z_ctau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the Kähler covariant derivative \(D_{\overline{\tau}}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex conjugate axio-dilaton \(\overline{\tau}\).
`DDW_z_cz`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the Kähler covariant derivative \(D_{\overline{z}^i}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).
`DDW_z_tau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the Kähler covariant derivative \(D_{\tau}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the axio-dilaton \(\tau\).
`DDW_z_z`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the Kähler covariant derivative \(D_{z^i}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex structure moduli \(z^i\).
`DW`(moduli, moduli_c, tau, tau_c, fluxes[, conj])	Returns the holomorphic Kähler covariant derivatives of the superpotential with respect to the complex structure moduli \(z^{i}\) and the axio-dilaton \(\tau\).
`DW_tau`(moduli, moduli_c, tau, tau_c, fluxes)	Calculates the Kähler covariant derivative of the superpotential with respect to the axio-dilaton \(\tau\).
`DW_x`(x, fluxes)	Calculates the real and imaginary parts of the \(F\)-term conditions.
`DW_z`(moduli, moduli_c, tau, tau_c, fluxes[, ...])	Calculates the Kähler covariant derivative of the superpotential \(W\) with respect to the complex structure moduli \(z^{i}\).
`DcDW`(moduli, moduli_c, tau, tau_c, fluxes[, ...])	Returns a matrix containing the mixed second Kähler derivatives \(D_{\overline{I}}D_{J}W\).
`F`(moduli[, conj])	Computes the pre-potential for given values of the moduli.
`F_LCS`(moduli[, conj])	Calculates the value of the LCS prepotential in terms of the complex structure moduli \(z^{i}\).
`F_LCS_poly`(moduli[, conj])	Computes the polynomial contribution \(F_{\mathrm{poly}}\) to the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the complex structure moduli \(z^i\).
`F_inst`(moduli[, conj])	Returns the instanton part \(F_{\mathrm{inst}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the complex structure moduli \(z^i\).
`H`(moduli, moduli_c, tau, tau_c, fluxes[, ...])	Returns the Hessian of the scalar potential.
`ISD_condition`(moduli, moduli_c, tau, tau_c, ...)	Checks whether the fluxes satisfy the ISD condition \(\star G_3=\text{i}G_3\).
`ISD_matrix`(moduli, moduli_c)	Computes the value of the ISD-matrix \(\mathcal{M}\).
`K`(moduli, moduli_c, tau, tau_c)	Returns the value of the Kähler potential.
`M`(moduli, moduli_c)	Computes the value of the ISD-matrix \(\mathcal{M}\).
`N`(moduli, moduli_c[, conj])	Computes the value of the gauge kinetic matrix \(\mathcal{N}\).
`V`(moduli, moduli_c, tau, tau_c, fluxes[, ...])	Returns the value of the \(F\)-term scalar potential.
`V_tilde`(moduli, moduli_c)	Returns the value of the mirror dual Calabi-Yau volume.
`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.
`W`(moduli, tau, fluxes[, conj, normalise])	Calculates the value of the superpotential for given flux, moduli and axio-dilaton.
`W0`(moduli, tau, fluxes[, conj, normalise])	Calculates the value of the gauge invariant version of the flux superpotential.
`__init__`([h12, model_ID, model_type, limit, ...])	A class representing the flux sector in a 4D EFT obtained from Type IIB compactification on CY threefolds with 3-form flux backgrounds.
`apply_monodromy`(moduli, fluxes, n)	Apply a monodromy shift \(z^a \to z^a + n^a\) to the moduli and transform the fluxes accordingly via the monodromy matrix \(M(n)\).
`canonical_fterms`(moduli, moduli_c, tau, ...)	Returns the canonically normalised \(F\)-term conditions \(F_I,\, F^I\).
`christoffel_symbols`(moduli, moduli_c, tau, tau_c)	Returns the Christoffel symbols \(\Gamma^E_{AC}\) of the Levi-Civita connection on the Kähler moduli space.
`compute_tau_vev`(moduli, flux)	Computes the value of the axio-dilaton \(\tau\) from the fluxes and complex structure moduli as a solution to the corresponding \(F\)-term condition \(D_{\tau}W=0\).
`conifold_monodromy_matrix`([conifold_curve, ...])	Picard-Lefschetz monodromy matrix around a conifold singularity where a 3-cycle \(\gamma = \sum_a c_a A^a\) shrinks to zero.
`dDW`(moduli, moduli_c, tau, tau_c, fluxes[, conj])	Returns the holomorphic derivative \(\partial_{I}D_{J}W\) of the \(F\)-terms \(D_{J}W\) with respect to the complex structure moduli \(z^i\) and the axio-dilaton \(\tau\).
`dDW_c`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\overline{I}}D_{J}W\) of the \(F\)-terms \(D_{J}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\) and the axio-dilaton \(\overline{\tau}\).
`dDW_ctau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\overline{\tau}}D_{J}W\) of the \(F\)-term \(D_{J}W\) with respect to the complex conjugate axio-dilaton \(\overline{\tau}\).
`dDW_cz`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\overline{z}^i}D_{J}W\) of the \(F\)-term \(D_{J}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).
`dDW_real`(moduli, tau, fluxes)	Returns the Jacobian \(\partial_{\phi^\alpha}(D_J W)\) of the \(F\)-terms with respect to the real scalar fields \(\phi^\alpha = (a^i, v^i, c_0, s)\).
`dDW_tau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\tau}D_{J}W\) of the \(F\)-terms \(D_{J}W\) with respect to the the axio-dilaton \(\tau\).
`dDW_tau_ctau`(moduli, moduli_c, tau, tau_c, ...)	Returns the holomorphic derivative \(\partial_{\overline{\tau}}D_{\tau}W\) of the \(F\)-term \(D_{\overline{\tau}}W\) with respect to the complex conjugate axio-dilaton \(\overline{\tau}\).
`dDW_tau_cz`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\overline{z}^i}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).
`dDW_tau_tau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\tau}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the axio-dilaton \(\tau\).
`dDW_tau_z`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{z^i}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the complex structure moduli \(z^i\).
`dDW_x`(x, fluxes)	Returns the first derivatives of the F-term conditions by differentiating with respect to the real fields.
`dDW_z`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{z^i}D_{J}W\) of the \(F\)-terms \(D_{J}W\) with respect to the complex structure moduli \(z^i\).
`dDW_z_ctau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\overline{\tau}}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex conjugate axio-dilaton \(\overline{\tau}\).
`dDW_z_cz`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\overline{z}^i}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).
`dDW_z_tau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{\tau}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the axio-dilaton \(\tau\).
`dDW_z_z`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic derivative \(\partial_{z^i}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex structure moduli \(z^i\).
`dF`(moduli[, conj])	Computes the holomorphic derivative \(\partial_{z^i} F\) of the prepotential \(F\) for given values of the moduli.
`dGamma`(moduli, moduli_c, tau, tau_c)	Returns the holomorphic derivative of the Christoffel symbols \(\partial_B \Gamma^E_{AC}\).
`dIKM_c`(moduli, moduli_c, tau, tau_c)	Returns the anti-holomorphic derivative of the inverse Kähler metric \(\partial_{\bar{B}} K^{I\bar{J}}\).
`dK`(moduli, moduli_c, tau, tau_c)	Returns the holomorphic derivative \(\partial_I K\) of the Kähler potential \(K\) with respect to the complex structure moduli \(z^{i}\) and the axio-dilaton \(\tau\).
`dK_c`(moduli, moduli_c, tau, tau_c)	Returns the holomorphic derivative \(\partial_{\overline{I}} K\) of the Kähler potential \(K\) with respect to the complex conjugate complex structure moduli \(\overline{z}^{i}\) and the axio-dilaton \(\overline{\tau}\).
`dK_ctau`(moduli, moduli_c, tau, tau_c)	Returns the anti-holomorphic derivative \(\partial_{\overline{\tau}}K\) of the Kähler potential \(K\) with respect to the conjugate axio-dilaton \(\overline{\tau}\).
`dK_cz`(moduli, moduli_c, tau, tau_c)	Returns the anti-holomorphic derivative \(\partial_{\overline{z}^i}K\) of the Kähler potential \(K\) with respect to the conjugate complex structure moduli \(\overline{z}^i\).
`dK_tau`(moduli, moduli_c, tau, tau_c)	Returns the holomorphic derivative \(\partial_{\tau}K\) of the Kähler potential \(K\) with respect to the axio-dilaton \(\tau\).
`dK_z`(moduli, moduli_c, tau, tau_c)	Returns the holomorphic derivative \(\partial_{z^i}K\) of the Kähler potential \(K\) with respect to the complex structure moduli \(z^i\).
`dM`(moduli, moduli_c)	Returns the holomorphic derivative \(\partial_{z^i}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\) with respect to the complex structure moduli \(z^i\).
`dM_X`(moduli, moduli_c)	Returns the holomorphic derivative \(\partial_{X^I}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\) with respect to the periods \(X^I\).
`dM_c`(moduli, moduli_c)	Returns the anti-holomorphic derivative \(\partial_{\overline{z}^i}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).
`dM_cX`(moduli, moduli_c)	Returns the anti-holomorphic derivative \(\partial_{\overline{X}^I}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\) with respect to the complex conjugate periods \(\overline{X}^I\).
`dN`(moduli, moduli_c[, conj])	Returns the holomorphic derivative \(\partial_{z^i}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\) with respect to the complex structure moduli \(z^i\).
`dN_X`(moduli, moduli_c[, conj])	Returns the holomorphic derivative \(\partial_{X^I}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\) with respect to the periods \(X^I\).
`dN_c`(moduli, moduli_c[, conj])	Returns the anti-holomorphic derivative \(\partial_{\overline{z}^i}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).
`dN_cX`(moduli, moduli_c[, conj])	Returns the anti-holomorphic derivative \(\partial_{\overline{X}^I}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\) with respect to the complex conjugate periods \(\overline{X}^I\).
`dV`(moduli, moduli_c, tau, tau_c, fluxes[, ...])	Returns the first holomorphic derivative \(\partial_{I}V\) of the \(F\)-term scalar potential \(V\).
`dV_tau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the first holomorphic derivative \(\partial_{\tau}V\) of the \(F\)-term scalar potential \(V\) with respect to the axio-dilaton \(\tau\).
`dV_x`(x, fluxes[, noscale])	Returns the gradients of the F-term scalar potential \(V\) with respect to the real scalar fields for real inputs.
`dV_z`(moduli, moduli_c, tau, tau_c, fluxes[, ...])	Returns the first holomorphic derivative \(\partial_{z^i}V\) of the \(F\)-term scalar potential \(V\) with respect to the complex structure moduli \(z^i\).
`dW`(moduli, tau, fluxes[, conj])	Calculates the holomorphic derivative \(W_I\) of the superpotential \(W\) with respect to all moduli \((z^i, \tau)\).
`dW_tau`(moduli, tau, fluxes[, conj])	Calculates the holomorphic derivative \(W_\tau=\partial_{\tau}W\) of the superpotential \(W\) with respect to the axio-dilaton \(\tau\).
`dW_z`(moduli, tau, fluxes[, conj])	Calculates the holomorphic derivative \(W_i=\partial_{z^i}W\) of the superpotential \(W\) with respect to the complex structure moduli \(z^{i}\).
`ddDW`(moduli, moduli_c, tau, tau_c, fluxes[, ...])	Returns the second derivative of the \(F\)-terms \(\partial_A(D_I W)\).
`ddDW_x`(x, fluxes)	Returns the second derivatives of the F-term conditions by differentiating with respect to the real fields.
`ddIKM`(moduli, moduli_c, tau, tau_c)	Returns the mixed second derivative of the inverse Kähler metric \(\partial_A\partial_{\bar{B}} K^{I\bar{J}}\).
`ddK_ctau_ctau`(moduli, moduli_c, tau, tau_c)	Returns the second holomorphic derivatives of the Kähler potential with respect to the axio-dilaton \(\tau\).
`ddK_cz_ctau`(moduli, moduli_c, tau, tau_c)	Returns the second derivatives \(K_{\bar{\imath}\overline{\tau}}=\partial_{\overline{z}^i}\partial_{\overline{\tau}}K\) of the Kähler potential \(K\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\) and the axio-dilaton \(\overline{\tau}\).
`ddK_cz_cz`(moduli, moduli_c, tau, tau_c)	Returns the second anti-holomorphic derivatives of the Kähler potential with respect to the complex structure moduli \(z^i\).
`ddK_cz_tau`(moduli, moduli_c, tau, tau_c)	Returns the second derivatives \(K_{\tau\overline{j}}=\partial_{\tau}\partial_{\overline{z}^{j}}K\) of the Kähler potential \(K\) with respect to the conjugate complex structure moduli \(\overline{z}^i\) and the axio-dilaton \(\tau\).
`ddK_cz_z`(moduli, moduli_c, tau, tau_c)	Returns the second derivatives \(K_{\bar{\jmath}i}=\partial_{\overline{z}^j}\partial_{z^i}K\) of the Kähler potential \(K\) with respect to the complex structure moduli \(z^i\) and their conjugate.
`ddK_tau_ctau`(moduli, moduli_c, tau, tau_c)	Returns the second derivatives \(\partial_{\tau}\partial_{\overline{\tau}}K\) of the Kähler potential \(K\) with respect to the axio-dilaton \(\tau\) and its conjugate.
`ddK_tau_tau`(moduli, moduli_c, tau, tau_c)	Returns the second holomorphic derivatives of the Kähler potential with respect to the axio-dilaton \(\tau\).
`ddK_z_ctau`(moduli, moduli_c, tau, tau_c)	Returns the second derivatives \(K_{i\overline{\tau}}=\partial_{z^i}\partial_{\overline{\tau}}K\) of the Kähler potential \(K\) with respect to the complex structure moduli \(z^i\) and the conjugate axio-dilaton \(\overline{\tau}\).
`ddK_z_cz`(moduli, moduli_c, tau, tau_c)	Returns the second derivatives \(K_{i\bar{\jmath}}=\partial_{z^i}\partial_{\overline{z}^j}K\) of the Kähler potential \(K\) with respect to the complex structure moduli \(z^i\) and their conjugate.
`ddK_z_tau`(moduli, moduli_c, tau, tau_c)	Returns the second holomorphic derivatives of the Kähler potential with respect to the complex structure moduli \(z^i\) and the axio-dilaton \(\tau\).
`ddK_z_z`(moduli, moduli_c, tau, tau_c)	Returns the second holomorphic derivatives of the Kähler potential with respect to the complex structure moduli \(z^i\).
`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.
`ddV_ctau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the mixed second derivatives \(\partial_{I}\partial_{\bar\tau}V\) of the \(F\)-term scalar potential.
`ddV_cz`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the mixed second derivatives \(\partial_{I}\partial_{\bar{z}^j}V\) of the \(F\)-term scalar potential.
`ddV_tau`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the second derivatives \(\partial_{I}\partial_{\tau}V\) of the \(F\)-term scalar potential.
`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.
`ddV_z`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the holomorphic second derivatives \(\partial_{I}\partial_{z^j}V\) of the \(F\)-term scalar potential.
`ddW`(moduli, tau, fluxes[, conj])	Calculates the full second holomorphic derivatives \(W_{IJ}=\partial_I\partial_J W\) of the superpotential.
`ddW_tau_tau`(moduli, tau, fluxes[, conj])	Calculates \(W_{\tau\tau}=\partial_\tau^2 W\).
`ddW_z_tau`(moduli, tau, fluxes[, conj])	Calculates the mixed second derivative \(W_{i\tau}=\partial_{z^i}\partial_{\tau}W\) of the superpotential.
`ddW_z_z`(moduli, tau, fluxes[, conj])	Calculates the second holomorphic derivatives \(W_{ij}=\partial_{z^i}\partial_{z^j}W\) of the superpotential.
`dddK`(moduli, moduli_c, tau, tau_c)	Returns the third holomorphic-mixed Kähler derivative \(\partial_{A} K_{C\bar{F}}\) with respect to the combined field-space index \(A = (z^i, \tau)\).
`dddK_c`(moduli, moduli_c, tau, tau_c)	Returns the third anti-holomorphic-mixed Kähler derivative \(\partial_{\bar{B}} K_{C\bar{F}}\) with respect to the combined anti-holomorphic field-space index \(\bar{B} = (\bar{z}^i, \bar{\tau})\).
`flux_to_pfv`(flux)	Returns M- and K-vector specifying PFV from full flux vector.
`gauge_kinetic_matrix`(moduli, moduli_c[, conj])	Computes the value of the gauge kinetic matrix \(\mathcal{N}\).
`hessian`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the Hessian of the scalar potential.
`inverse_kahler_metric`(moduli, moduli_c, tau, ...)	Returns the inverse Kähler metric \(K^{\overline{I}J}\).
`inverse_kahler_metric_grad`(moduli, moduli_c, ...)	Returns the gradient of the inverse Kähler metric.
`kahler_metric`(moduli, moduli_c, tau, tau_c)	Computes the Kähler metric \(K_{\overline{I}J}\).
`kahler_potential`(moduli, moduli_c, tau, tau_c)	Returns the value of the Kähler potential.
`map_to_fd`(moduli, tau, fluxes[, axion_fd, ...])	Maps a vacuum solution `(moduli, tau, fluxes)` to the fundamental domain by:
`map_to_fd_tau`(tau, fluxes[, ...])	Map of the axio-dilaton value and the flux vector to the fundamental domain (FD) of \(\text{SL}(2,\mathbb{Z})\).
`mass_matrix`(moduli, moduli_c, tau, tau_c, fluxes)	Returns the mass matrix after canonical normalisation of the kinetic terms.
`mirror_volume`(moduli, moduli_c)	Returns the value of the mirror dual Calabi-Yau volume.
`moduli_to_periods`(moduli[, conj])	Transforms complex structure moduli to periods for the global choice of gauge.
`monodromy_matrix`(n)	Computes the monodromy matrix \(T(\vec{n})\) for a general integer shift \(z^a \to z^a + n^a\).
`monodromy_matrix_single`(b)	Computes the monodromy matrix \(T_b\) for the shift \(z^b \to z^b + 1\).
`period_vector`(moduli[, conj])	Returns the period vector \(\Pi\) at a given point in moduli space.
`periods_to_moduli`(XPer)	Transforms periods to complex structure moduli.
`pfv_to_flux`(M, K)	Returns full flux vector from M- and K-vector specifying PFV.
`pfv_to_moduli`(M, K, tau)	Returns values of the complex structure moduli at the level of the PFV.
`prepot`(moduli[, conj])	Computes the pre-potential for given values of the moduli.
`projection_fluxes`(moduli, tau, fluxes[, mode])	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.
`riemann_tensor`(moduli, moduli_c, tau, tau_c)	Returns the Riemann curvature tensor \(R_{i\bar{\jmath}k\bar{l}}\) of the Kähler moduli space.
`scalar_potential`(moduli, moduli_c, tau, ...)	Returns the value of the \(F\)-term scalar potential.
`superpotential`(moduli, tau, fluxes[, conj, ...])	Calculates the value of the superpotential for given flux, moduli and axio-dilaton.
`superpotential_gauge_invariant`(moduli, tau, ...)	Calculates the value of the gauge invariant version of the flux superpotential.
`tadpole`(fluxes)	Calculates the D3-charge for given fluxes.
`verify_monodromy`(b[, z, tol])	Numerically verify the monodromy matrix by checking \(T_b \cdot \Pi(z) = \Pi(z + e_b)\).

Attributes

`F_coniLCS_bulk`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`F_coniLCS_exp`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`F_coniLCS_series`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`W_bulk`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`W_log_coeff`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`compute_zcf`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`compute_zcf_x`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`conifold_fluxes`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`dF_coniLCS_exp`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`dK_cf_bulk`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`lcs_tree`	Description: The `lcs_tree` object containing the topological data (intersection numbers, second Chern class, GV invariants, etc.) for the underlying Calabi-Yau geometry.
`log_coeff_K_corr`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`log_prefactor`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.
`zcf_handling`	** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.

A(moduli, moduli_c)#

Returns the value of the mirror dual Calabi-Yau volume.

Details

Let \(X\) be a Calabi-Yau threefold and \(\Omega_3\) its holomorphic \((3,0)\)-form. Then this function returns the value of the following integral:

\[\mathcal{A}(z^i,\overline{z}^i)=-\text{i}\, \int_{X}\, \Omega_3\wedge \overline{\Omega}_3 = -\text{i}\, \Pi^\dagger(\overline{z}^i)\cdot \Sigma\cdot \Pi(z^i)\]

This object appears as the argument inside the Kähler potential, see kahler_potential(). Given that it is dual to the volume of the mirror Calabi-Yau threefold \(\widetilde{X}\), it is manifestly positive, provided the Kähler cone conditions of \(\widetilde{X}\) are satisfied.

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.

Returns:

complex – Mirror dual Calabi-Yau volume.

Return type:

complex

Aliases:: A(), mirror_volume(), V_tilde()

See also: kahler_potential()

DDW(moduli, moduli_c, tau, tau_c, fluxes, conj=False, mode=None)#

Returns a matrix the second holomorphic Kähler derivatives \(D_I D_J W\) of the superpotential.

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
mode (str) – The mode to \(DDW\). Currently implemented modes are: * None: general expression from explicit second derivatives. * “SUSY”: computes at a SUSY locus where DW=0 using standard SUGRA formulas.

Returns:

Array – Value of \(D_I D_J W\).

Return type:

Array

DDW_SUSY(moduli, moduli_c, tau, tau_c, fluxes, conj=False, mode='block diagonal')#

Returns a matrix containing the second holomorphic Kähler derivative of the superpotential assuming that the F-flatness conditions \(D_{i}W=D_{\tau}W=0\) are satisfied.

Details

Returns a matrix containing the second holomorphic Kähler derivative of the superpotential assuming that the F-flatness conditions \(D_{i}W=D_{\tau}W=0\) are satisfied. More explicitly, we compute

\[D_I D_I W = \partial_I D_J W + K_I D_J W = \partial_I \partial_J W + (K_{IJ}-K_I K_J) W + K_I D_J W + K_J D_I W \, .\]

At the SUSY minimum, this suggests

\[D_I D_I W\bigl |_{D_I W=0} = \partial_I \partial_J W + (K_{IJ}-K_I K_J) W\, .\]

For the axio-dilaton, we have \(W_{\tau\tau}=0\) for the GVW superpotential and thus

\[D_\tau D_\tau W\bigl |_{D_\tau W=0} = (K_{\tau\tau} - K_\tau K_\tau) W\, .\]

For the standard Kähler potential \(K\supset -\log(-\mathrm{i}(\tau-\bar{\tau}))\), this automatically implies

\[D_\tau D_\tau W\bigl |_{D_\tau W=0} = 0\; , \quad D_\tau D_i W\bigl |_{D_I W=0} = 0\, .\]

Warning

We assume that there is a block-diagonal structure in the Kahler potential such that the mixed second derivatives vanish \(K_{i\tau}=K_{\tau i}=0\). This might not necessarily be true once further corrections to the Kahler potential are included.

Warning

Further, we use that \(W_{\tau\tau}=0\) because the flux superpotential superpotential() is linear in \(\tau\). Again, this might not necessarily be true anymore once non-perturbative instanton effects are present in the superpotential!

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
mode (str) – Whether to assume that the Kähler metric is block-diagonal. Defaults to "block diagonal".

Returns:

Array – Value of \(D_I D_J W\) assuming \(D_IW=0\).

Return type:

Array

DDW_general(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the second Kähler covariant derivatives \(D_I D_J W\) of the superpotential \(W\) with respect to the axio-dilaton \(\tau\) and the complex structure moduli \(z^{i}\).

Details

We compute the Kähler covariant derivatives of the \(F\)-terms \(D_J W\)

\[D_I D_J W = \partial_I D_J W + (\partial_I K) D_J W\]

using automatic differentiation. To do so, we combine the outputs of DDW_z_z(), DDW_tau_tau(), DDW_z_tau() and DDW_tau_z().

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Second Kähler covariant derivatives \(D_I D_J W\) of the superpotential \(W\) with respect to the axio-dilaton \(\tau\) and the moduli \(z^{i}\).

Return type:

Array

DDW_matrix(moduli, moduli_c, tau, tau_c, fluxes, mode='SUSY')#

Returns the matrix of masses for chiral fermions.

Details

The diagonal entries of this matrix (evaluated at a minimum) are proportional to the mass matrix of chiral fermions, see equation (2.5) in 1312.5659. By abuse of terminology, we call this matrix the ‘’fermionic mass matrix’’.

Note

To speed up the computation, it can be advantageous to compute this function using the mode="SUSY" option provided that \(DW=0\) for all fields.

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
mode (str) – Whether or not the point at which the mass matrix is evaluated corresponds to a minimum with \(DW=0\) for all fields. Default is mode="SUSY".

Returns:

(Array)

Return type:

Array

DDW_matrix_SUSY(moduli, moduli_c, tau, tau_c, fluxes)#

Returns the matrix of second Kähler derivatives of the superpotential assuming that the F-term conditions are satisfied for all fields.

Details

Explicitly, we compute

\[M=\left (\begin{array}{cc}\partial_{A}\partial_{B} W + \left ( K_{A B} - K_{A} K_{B} \right ) W & K_{A\overline{B}} W\ K_{A\overline{B}} \overline{W} & \partial_{\overline{A}}\partial_{\overline{B}} \overline{W} + \left ( K_{\overline{A} \overline{B}} - K_{\overline{A}} K_{\overline{B}} \right ) \overline{W} \end{array}\right )\]

which is equivalent to the matrix of second derivatives of \(W\) (see massive_directions_general()) provided that the F-flatness conditions

\[D_{Z^{i}} W = D_{\tau} W = 0\]

are satisfied. In this case, it turns out that

\[D_{A} D_{\overline{B}} W = K_{A\overline{B}} W\]

as well as

\[D_{A} D_{B} W = \partial_{A}\partial_{B} W + \left ( K_{A B} - K_{A} K_{B} \right ) W\]

as computed via DDW_SUSY().

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.

Returns:

See also – massive_directions_general()
See also – DDW_SUSY()

Return type:

Array

DDW_matrix_general(moduli, moduli_c, tau, tau_c, fluxes)#

Returns the matrix of second Kähler derivatives of the superpotential.

Details

Explicitly, we compute

\[M=\left (\begin{array}{cc} D_{A}D_{B} W & D_{A}D_{\overline{B}} W \ D_{\overline{A}}D_{B} \overline{W} & D_{\overline{A}}D_{\overline{B}} \overline{W} \end{array}\right )\]

which is encodes the mass terms for chiral fermions.

In this case, it turns out that

\[D_{A} D_{\overline{B}} W =K_{\overline{B}}W_{A}+\left ( K_{A\overline{B}}+K_{A}K_{\overline{B}}\right ) W\]

This matrix encodes the mass terms for chiral fermions in the 4D effective theory (cf. Eq. (2.5) of 1312.5659).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Axio-dilaton value.
tau_c (complex) – Complex conjugate value for axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.

Returns:

Array – Shape (2n, 2n) matrix with blocks [[DDW, DcDW], [conj(DcDW), conj(DDW)]].

Return type:

Array

DDW_tau_ctau(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the Kähler covariant derivative \(D_{\overline{\tau}}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the complex conjugate axio-dilaton \(\overline{\tau}\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(D_{\overline{\tau}}D_{\tau}W\).

Return type:

Array

DDW_tau_cz(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the Kähler covariant derivative \(D_{\overline{z}^i}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(D_{\overline{z}^i}D_{\tau}W\).

Return type:

Array

DDW_tau_tau(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the Kähler covariant derivative \(D_{\tau}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the axio-dilaton \(\tau\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(D_{\tau}D_{\tau}W\).

Return type:

Array

DDW_tau_z(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the Kähler covariant derivative \(D_{z^i}D_{\tau}W\) of the \(F\)-term \(D_{\tau}W\) with respect to the complex structure moduli \(z^i\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(D_{z^i}D_{\tau}W\).

Return type:

Array

DDW_z_ctau(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the Kähler covariant derivative \(D_{\overline{\tau}}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex conjugate axio-dilaton \(\overline{\tau}\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(D_{\overline{\tau}}D_{z^j}W\).

Return type:

Array

DDW_z_cz(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the Kähler covariant derivative \(D_{\overline{z}^i}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(D_{\overline{z}^i}D_{z^j}W\).

Return type:

Array

DDW_z_tau(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the Kähler covariant derivative \(D_{\tau}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the axio-dilaton \(\tau\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(D_{\tau}D_{z^j}W\).

Return type:

Array

DDW_z_z(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the Kähler covariant derivative \(D_{z^i}D_{z^j}W\) of the \(F\)-term \(D_{z^j}W\) with respect to the complex structure moduli \(z^i\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(D_{z^i}D_{z^j}W\).

Return type:

Array

DW(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the holomorphic Kähler covariant derivatives of the superpotential with respect to the complex structure moduli \(z^{i}\) and the axio-dilaton \(\tau\).

Details

This function combines the outputs of the functions DW_z() and DW_tau()

\[D_I W = \bigl(D_{1} W,\ldots, D_{h^{1,2}} W,D_{\tau} W\bigl)\, .\]

This output is later used in the \(F\)-term scalar potential scalar_potential().

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Axio-dilaton value.
tau_c (complex) – Complex conjugate value for axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Values of the \(F\)-term conditions for the complex structure moduli \(z^{i}\) and the axio-dilaton \(\tau\).

Return type:

Array

See also: superpotential()

See also: kahler_potential()

V_tilde(moduli, moduli_c)#

Returns the value of the mirror dual Calabi-Yau volume.

Details

Let \(X\) be a Calabi-Yau threefold and \(\Omega_3\) its holomorphic \((3,0)\)-form. Then this function returns the value of the following integral:

\[\mathcal{A}(z^i,\overline{z}^i)=-\text{i}\, \int_{X}\, \Omega_3\wedge \overline{\Omega}_3 = -\text{i}\, \Pi^\dagger(\overline{z}^i)\cdot \Sigma\cdot \Pi(z^i)\]

This object appears as the argument inside the Kähler potential, see kahler_potential(). Given that it is dual to the volume of the mirror Calabi-Yau threefold \(\widetilde{X}\), it is manifestly positive, provided the Kähler cone conditions of \(\widetilde{X}\) are satisfied.

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.

Returns:

complex – Mirror dual Calabi-Yau volume.

Return type:

complex

Aliases:: A(), mirror_volume(), V_tilde()

See also: kahler_potential()

V_x(x, fluxes, noscale=True)#

Returns the value of the real part of the F-term scalar potential \(V\) for input of the real scalar fields.

Note

This function is a wrapper which is used to find non-SUSY minima by looking at gradients of the scalar potential directly, see for example _newton_method_flux_vacua_real().

Parameters:

x (Array) – JAX array of shape (\(2(h^{1,2}+1)\),) containing the moduli and axio-dilaton as real and imaginary parts.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
noscale (bool) – If True, uses the no-scale flux scalar potential. Defaults to True.

Returns:

float – Value of \(V\).

Return type:

float

See also: superpotential()

See also: kahler_potential()

apply_monodromy(moduli, fluxes, n)#

Apply a monodromy shift \(z^a \to z^a + n^a\) to the moduli and transform the fluxes accordingly via the monodromy matrix \(M(n)\).

The RR-fluxes \(f\) and NSNS-fluxes \(h\) both transform as \(f \to M(n) \cdot f\), \(h \to M(n) \cdot h\).

Parameters:

moduli (Array) – Complex structure moduli.
fluxes (Array) – Flux vector [f | h].
n (array-like) – Integer shift vector of shape (h12,).

Returns:

Tuple[Array, Array] – (moduli_shifted, fluxes_shifted)

Return type:

Tuple[Array, Array]

canonical_fterms(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the canonically normalised \(F\)-term conditions \(F_I,\, F^I\).

Details

The canonically normalised \(F\)-term conditions are defined as

\[F_I = \mathrm{e}^{K/2}\, D_IW\, ,\quad F^I = \mathrm{e}^{K/2}\, K^{I\bar{J}}\, D_{\bar{J}}\overline{W}\, .\]

They parametrise the contribution to the flux scalar potential scalar_potential()

\[F^I F_I = \mathrm{e}^{K}\, K^{I\bar{J}}\, D_IW\, D_{\bar{J}} \overline{W} \, .\]

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Canonically normalised \(F\)-term conditions \(F_I\).
Array – Canonically normalised \(F\)-term conditions \(F^I\).

Return type:

Tuple[Array, Array]

christoffel_symbols(moduli, moduli_c, tau, tau_c)#

Returns the Christoffel symbols \(\Gamma^E_{AC}\) of the Levi-Civita connection on the Kähler moduli space.

Details

The Christoffel connection on a Kähler manifold is given by

\[\Gamma^E_{AC} = K^{E\bar{F}}\,\partial_A K_{C\bar{F}}\]

where \(K_{C\bar{F}}\) is the Kähler metric and \(K^{E\bar{F}}\) its inverse. The indices \(A, C, E\) run over the combined field space \((z^1,\ldots,z^{h^{1,2}},\tau)\).

For a Kähler manifold the connection is torsion-free, \(\Gamma^E_{AC} = \Gamma^E_{CA}\), and the only non-vanishing components have purely holomorphic lower indices.

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Value of the axio-dilaton.
tau_c (complex) – Complex conjugate value of the axio-dilaton.

Returns:

Array – Christoffel symbols \(\Gamma^E_{AC}\) with shape (n, n, n) and index ordering [E, A, C].

Return type:

Array

compute_tau_vev(moduli, flux)#

Computes the value of the axio-dilaton \(\tau\) from the fluxes and complex structure moduli as a solution to the corresponding \(F\)-term condition \(D_{\tau}W=0\).

Details

The axio-dilaton \(\tau\) can be computed from the fluxes and complex structure moduli as a solution to the \(F\)-term condition

\[D_{\tau}W = \partial_{\tau}W + W \partial_{\tau}K = 0\, .\]

This condition can be written as

\[D_{\tau}W = -\frac{1}{\tau-\overline{\tau}} (f - \overline{\tau} h)\cdot \Sigma \cdot \overline{\Pi}(z^i) = 0\, ,\]

and the corresponding solution is given by

\[\tau = \frac{f\cdot \Sigma \cdot \overline{\Pi}(z^i)}{h\cdot \Sigma \cdot \overline{\Pi}(z^i)}\, ,\]

where \(f=(f_1,f_2)\) and \(h=(h_1,h_2)\) are the RR- and NSNS-fluxes respectively with \(f_i,h_i \in \mathbb{Z}^{h^{1,2}+1}\). Further, \(\Pi(z^i)\) is the period vector depending on the complex structure moduli and \(\Sigma\) is the symplectic matrix defined in period_matrix().

Note

Notice that this equation only has a solution if the denominator is non-zero, i.e. if

\[h\cdot \Sigma \cdot \overline{\Pi}(z^i) \neq 0\, .\]

If this condition is not satisfied, then this function raises a ZeroDivisionError.

Parameters:

moduli (Array) – Complex structure moduli values.
flux (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.

Returns:

complex – Value of the axio-dilaton \(\tau\).

conifold_monodromy_matrix(conifold_curve=None, conifold_index=None)#

Picard-Lefschetz monodromy matrix around a conifold singularity where a 3-cycle \(\gamma = \sum_a c_a A^a\) shrinks to zero.

The monodromy acts on the period vector \(\Pi = (F_0, F_a, X^0, z^a)\) as

\[F_a \;\to\; F_a + c_a \sum_b c_b\, z^b\,,\]

with all other periods invariant. This follows from the Picard-Lefschetz formula \(\delta \to \delta + (\delta \cdot \gamma)\,\gamma\) applied to the symplectic basis of \(H_3(X,\mathbb{Z})\).

Parameters:

conifold_curve (array-like, optional) – Charge vector \(c = (c_1, \dots, c_{h^{2,1}})\) of the vanishing cycle. The vanishing period is \(w = c^T z\). If None, uses self.lcs_tree.conifold.conifold_curve when available.
conifold_index (int, optional) – Index (0-based among \(z^1 \dots z^h\)) of the conifold modulus, equivalent to conifold_curve = e_{index}.

Returns:

np.ndarray – Integer matrix of shape (2*h12+2, 2*h12+2).

Raises:

ValueError – If both conifold_curve and conifold_index are provided, or if neither is given and lcs_tree.conifold.conifold_curve is unavailable.

dDW(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the holomorphic derivative \(\partial_{I}D_{J}W\) of the \(F\)-terms \(D_{J}W\) with respect to the complex structure moduli \(z^i\) and the axio-dilaton \(\tau\).

Note

The output shape is such that output[J-1][I-1] corresponds to \(\partial_{I}D_{J}W\) with \(I,J=1,\ldots, h^{1,2}+1\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(\partial_{I}D_{J}W\).

Return type:

Array

dDW_c(moduli, moduli_c, tau, tau_c, fluxes, conj=False)#

Returns the holomorphic derivative \(\partial_{\overline{I}}D_{J}W\) of the \(F\)-terms \(D_{J}W\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\) and the axio-dilaton \(\overline{\tau}\).

Note

The output shape is such that output[J-1][I-1] corresponds to \(\partial_{\overline{I}}D_{J}W\) with \(I,J=1,\ldots, h^{1,2}+1\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – : Value of axio-dilaton.
tau_c (complex) – Value of complex conjugate axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of \(\partial_{\overline{I}}D_{J}W\).

Return type:

Array

See also: kahler_potential()

dK_c(moduli, moduli_c, tau, tau_c)#

Returns the holomorphic derivative \(\partial_{\overline{I}} K\) of the Kähler potential \(K\) with respect to the complex conjugate complex structure moduli \(\overline{z}^{i}\) and the axio-dilaton \(\overline{\tau}\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Value of the axio-dilaton.
tau_c (complex) – Complex conjugate value of the axio-dilaton.

Returns:

Array – Values of \(\partial_{\overline{I}} K\).

Return type:

Array

See also: kahler_potential()

dK_ctau(moduli, moduli_c, tau, tau_c)#

Returns the anti-holomorphic derivative \(\partial_{\overline{\tau}}K\) of the Kähler potential \(K\) with respect to the conjugate axio-dilaton \(\overline{\tau}\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Value of the axio-dilaton.
tau_c (complex) – Complex conjugate value of the axio-dilaton.

Returns:

complex – Anti-holomorphic derivative \(\partial_{\overline{\tau}}K\) of the Kähler potential \(K\).

Return type:

complex

See also: kahler_potential()

dK_cz(moduli, moduli_c, tau, tau_c)#

Returns the anti-holomorphic derivative \(\partial_{\overline{z}^i}K\) of the Kähler potential \(K\) with respect to the conjugate complex structure moduli \(\overline{z}^i\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Value of the axio-dilaton.
tau_c (complex) – Complex conjugate value of the axio-dilaton.

Returns:

Array – Anti-holomorphic derivative \(\partial_{\overline{z}^i}K\) of the Kähler potential \(K\).

Return type:

Array

See also: kahler_potential()

dK_tau(moduli, moduli_c, tau, tau_c)#

Returns the holomorphic derivative \(\partial_{\tau}K\) of the Kähler potential \(K\) with respect to the axio-dilaton \(\tau\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Value of the axio-dilaton.
tau_c (complex) – Complex conjugate value of the axio-dilaton.

Returns:

complex – Holomorphic derivative \(\partial_{\tau}K\) of the Kähler potential \(K\).

Return type:

complex

See also: kahler_potential()

dK_z(moduli, moduli_c, tau, tau_c)#

Returns the holomorphic derivative \(\partial_{z^i}K\) of the Kähler potential \(K\) with respect to the complex structure moduli \(z^i\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Value of the axio-dilaton.
tau_c (complex) – Complex conjugate value of the axio-dilaton.

Returns:

Array – Holomorphic derivative \(\partial_{z^i}K\) of the Kähler potential \(K\).

Return type:

Array

See also: kahler_potential()

dM(moduli, moduli_c)#

Returns the holomorphic derivative \(\partial_{z^i}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\) with respect to the complex structure moduli \(z^i\).

Parameters:

moduli (Array) – Values of the complex structure moduli.
moduli_c (Array) – Complex conjugate values of the complex structure moduli.

Returns:

Array – Holomorphic derivative \(\partial_{z^i}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\).

Return type:

Array

dM_X(moduli, moduli_c)#

Returns the holomorphic derivative \(\partial_{X^I}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\) with respect to the periods \(X^I\).

Note

This function descends from jaxvacua.periods.periods.dM() upon gauge fixing, i.e., making a choice of homogeneous complex coordinates on the complex structure moduli space of \(X\).

Parameters:

moduli (Array) – Values of the complex structure moduli.
moduli_c (Array) – Complex conjugate values of the complex structure moduli.

Returns:

Array – Holomorphic derivative \(\partial_{X^I}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\).

Return type:

Array

dM_c(moduli, moduli_c)#

Returns the anti-holomorphic derivative \(\partial_{\overline{z}^i}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).

Parameters:

moduli (Array) – Values of the complex structure moduli.
moduli_c (Array) – Complex conjugate values of the complex structure moduli.

Returns:

Array – Anti-holomorphic derivative \(\partial_{\overline{z}^i}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\).

Return type:

Array

dM_cX(moduli, moduli_c)#

Returns the anti-holomorphic derivative \(\partial_{\overline{X}^I}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\) with respect to the complex conjugate periods \(\overline{X}^I\).

Note

This function descends from jaxvacua.periods.periods.dM_c() upon gauge fixing, i.e., making a choice of homogeneous complex coordinates on the complex structure moduli space of \(X\).

Parameters:

moduli (Array) – Values of the complex structure moduli.
moduli_c (Array) – Complex conjugate values of the complex structure moduli.

Returns:

Array – Anti-holomorphic derivative \(\partial_{\overline{X}^I}\mathcal{M}\) of the ISD-matrix \(\mathcal{M}\).

Return type:

Array

dN(moduli, moduli_c, conj=False)#

Returns the holomorphic derivative \(\partial_{z^i}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\) with respect to the complex structure moduli \(z^i\).

Parameters:

moduli (Array) – Values of the complex structure moduli.
moduli_c (Array) – Complex conjugate values of the complex structure moduli.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Holomorphic derivative \(\partial_{z^i}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\).

Return type:

Array

dN_X(moduli, moduli_c, conj=False)#

Returns the holomorphic derivative \(\partial_{X^I}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\) with respect to the periods \(X^I\).

Note

This function descends from jaxvacua.periods.periods.dN() upon gauge fixing, i.e., making a choice of homogeneous complex coordinates on the complex structure moduli space of \(X\).

Parameters:

moduli (Array) – Values of the complex structure moduli.
moduli_c (Array) – Complex conjugate values of the complex structure moduli.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Holomorphic derivative \(\partial_{X^I}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\).

Return type:

Array

dN_c(moduli, moduli_c, conj=False)#

Returns the anti-holomorphic derivative \(\partial_{\overline{z}^i}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\) with respect to the complex conjugate complex structure moduli \(\overline{z}^i\).

Parameters:

moduli (Array) – Values of the complex structure moduli.
moduli_c (Array) – Complex conjugate values of the complex structure moduli.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Anti-holomorphic derivative \(\partial_{\overline{z}^i}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\).

Return type:

Array

dN_cX(moduli, moduli_c, conj=False)#

Returns the anti-holomorphic derivative \(\partial_{\overline{X}^I}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\) with respect to the complex conjugate periods \(\overline{X}^I\).

Note

This function descends from jaxvacua.periods.periods.dN_c() upon gauge fixing, i.e., making a choice of homogeneous complex coordinates on the complex structure moduli space of \(X\).

Parameters:

moduli (Array) – Values of the complex structure moduli.
moduli_c (Array) – Complex conjugate values of the complex structure moduli.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Anti-holomorphic derivative \(\partial_{\overline{X}^I}\mathcal{N}\) of the gauge kinetic matrix \(\mathcal{N}\).

Return type:

Array

dV(moduli, moduli_c, tau, tau_c, fluxes, noscale=True, conj=False, mode='complex')#

Returns the first holomorphic derivative \(\partial_{I}V\) of the \(F\)-term scalar potential \(V\).

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Axio-dilaton value.
tau_c (complex) – Complex conjugate value for axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
noscale (bool) – If True, uses the no-scale flux scalar potential. Defaults to True.
conj (bool) – If True, computes the complex conjugate. Defaults to False. Becomes void if mode="real".
mode (str, optional) – String specifying whether to return complex or real derivatives. Defaults to "complex".

Returns:

Array – Value of \(\partial_{I}V\).

Return type:

Array

dV_tau(moduli, moduli_c, tau, tau_c, fluxes, noscale=True, conj=False)#

Returns the first holomorphic derivative \(\partial_{\tau}V\) of the \(F\)-term scalar potential \(V\) with respect to the axio-dilaton \(\tau\).

Details

Computed via jax.jacrev of scalar_potential() with respect to \(\tau\) (argnums=2). For conj=True, differentiates with respect to \(\bar\tau\) (argnums=3):

\[\partial_\tau V = \mathrm{e}^K \bigl[K_\tau\,(S - \lambda G) + \partial_\tau S - \lambda\,\partial_\tau G\bigr]\]

Parameters:

moduli (Array) – Complex structure moduli values.
moduli_c (Array) – Complex conjugate values for complex structure moduli.
tau (complex) – Axio-dilaton value.
tau_c (complex) – Complex conjugate value for axio-dilaton.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
noscale (bool) – If True, uses the no-scale flux scalar potential. Defaults to True.
conj (bool) – If True, computes \(\partial_{\bar\tau}V\). Defaults to False.

Returns:

complex – Value of \(\partial_{\tau}V\) (scalar).

Return type:

Array

dV_x(x, fluxes, noscale=True)#

Returns the gradients of the F-term scalar potential \(V\) with respect to the real scalar fields for real inputs.

Details

Let us denote the complex scalar fields as

\[z^i = a^i + \text{i} v^i\, ,\quad \tau = c_0 + \text{i} s\, .\]

Then this function computes

\[\nabla V = (\partial_{a^1}V, \partial_{v^1}V,\ldots ,\partial_{a^{h^{1,2}}}V, \partial_{v^{h^{1,2}}}V,\partial_{c_0}V, \partial_{s}V)\, .\]

Parameters:

x (Array) – JAX array of shape (\(2(h^{1,2}+1)\),) containing the moduli and axio-dilaton as real and imaginary parts.
fluxes (Array) – Flux vector of shape (\(4(h^{1,2}+1)\),)
noscale (bool) – If True, uses the no-scale flux scalar potential. Defaults to True.

Returns:

Array – Value of \((\partial_{a^1}V, \partial_{v^1}V,\ldots ,\partial_{a^{h^{1,2}}}V, \partial_{v^{h^{1,2}}}V,\partial_{c_0}V, \partial_{s}V)\).

Return type:

Array

See also: kahler_potential()

moduli_to_periods(moduli, conj=False)#

Transforms complex structure moduli to periods for the global choice of gauge.

Parameters:

moduli (Array) – Values of the complex structure moduli.
conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of the periods.

Return type:

Array

monodromy_matrix(n)#

Computes the monodromy matrix \(T(\vec{n})\) for a general integer shift \(z^a \to z^a + n^a\).

This is the product \(T(\vec{n}) = \prod_b T_b^{n_b}\). The LCS monodromy matrices commute (maximally unipotent monodromy), so the order does not matter.

Parameters:: n (array-like) – Integer shift vector of shape (h12,).
Returns:: np.ndarray – Integer monodromy matrix of shape (2*h12+2, 2*h12+2).
Return type:: ndarray

See also: kahler_potential()

superpotential(moduli, tau, fluxes, conj=False, normalise=False)#

Calculates the value of the superpotential for given flux, moduli and axio-dilaton.

Details

For a Calabi-Yau manifold \(X\), the Gukov-Vafa-Witten superpotential hep-th/9906070 is defined in terms of the complex 3-form flux \(G_3=F_3-\tau H_3\) and the holomorphic 3-form \(\Omega_3\) as

\[W(Z,\tau) = \int_{X} \, G_3\wedge \Omega_3 = (\vec{f}-\tau\vec{h})^T\cdot\Sigma\cdot\Pi(Z)\]

following the conventions of eq. (4) in 1912.10047. The period vector \(\Pi\) and the symplectic pairing \(\Sigma\) are defined in the periods class jaxvacua.periods.periods().

Parameters:

moduli (Array) – Complex structure moduli values.
tau (complex) – Axio-dilaton value.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.
normalise (bool) – If True, rescales superpotential by \(\sqrt{2/\pi}\). Defaults to False.

Returns:

complex – Value of the superpotential.

Return type:

complex

superpotential_gauge_invariant(moduli, tau, fluxes, conj=False, normalise=True)#

Calculates the value of the gauge invariant version of the flux superpotential.

Details

Following the conventions of 1912.10047, we define the gauge invariant version of the superpotential as

\[\tilde{W} = \sqrt{\dfrac{2}{\pi}}\, \mathrm{e}^{K/2}\, W\]

Its absolute value is proportional to the gravitino mass. The normalisation is taken from 1908.04788.

Parameters:

moduli (Array) – Complex structure moduli values.
tau (complex) – Axio-dilaton value.
fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
conj (bool) – If True, computes the complex conjugate. Defaults to False.
normalise (bool) – If True, rescales superpotential by \(\sqrt{2/\pi}\). Defaults to True.

Returns:

complex – Value of the gauge invariant superpotential.

Return type:

complex

See also: superpotential()

See also: kahler_potential()

tadpole(fluxes)#

Calculates the D3-charge for given fluxes.

Details

The D3-charge induced by fluxes is defined as

\[N_{\mathrm{flux}} = \vec{f}\Sigma \vec{h}\, ,\]

where :math`vec{f}, vec{h}` are the NSNS- and RR-fluxes respectively. The symplectic pairing \(\Sigma\) is implemented in the periods class jaxvacua.periods.periods().

Parameters:: fluxes (Array) – Array of fluxes. The ordering starts with RR-fluxes, then the NSNS-fluxes.
Returns:: int – D3-charge induced by 3-form fluxes.
Return type:: float

verify_monodromy(b, z=None, tol=1e-08)#

Numerically verify the monodromy matrix by checking \(T_b \cdot \Pi(z) = \Pi(z + e_b)\).

Parameters:

b (int) – Direction index for the shift.
z (Array, optional) – Test point. If None, a random point is generated.
tol (float) – Tolerance for the check.

Returns:

dict – Dictionary with keys 'T_b', 'max_error', 'passed'.

Return type:

dict

jaxvacua.flux_eft.FluxEFT

Contents

jaxvacua.flux_eft.FluxEFT#