jaxvacua.css

jaxvacua.css#

Complex-structure-sector geometry for Type IIB compactifications.

Purpose#

Define the css class, the geometry layer built on top of jaxvacua.periods.periods. It evaluates the Kähler geometry of complex structure moduli space and provides the tensors used by the flux EFT.

Main public API#

css: constructs period data and exposes affine/projective coordinates, Kähler potential, metric, inverse metric, connection, curvature and covariant-derivative helpers.
Gauge-kinetic and ISD-sector matrices used by FluxEFT and FluxVacuaFinder.
Conifold-limit methods attached when limit is in the coniLCS family.

Design notes#

The class is a JAX pytree and keeps expensive model data on the nested periods object. Methods are written for JIT/vectorisation where possible, so callers should prefer the class API over rebuilding period data manually.

Computational graph#

The complex-structure sector consumes the period vector $\Pi(z)$ and the Kähler potential $K(z, \bar z)$ from jaxvacua.periods and produces the moduli-space metric, the ISD matrix and the gauge-kinetic matrix:

\begin{align*} g_{i \bar\jmath}(z, \bar z) &= \partial_i \partial_{\bar\jmath}\, K(z, \bar z), \\[2pt] M_{AB}(z, \bar z) &= e^{K(z, \bar z)}\, \Pi_A(z)\, \overline{\Pi_B(z)}, \\[2pt] \mathcal{N}_{IJ}(z, \bar z) &= \bar{F}_{IJ} + 2 i\, \frac{\bigl(\operatorname{Im} F\bigr)_{IK} X^K\, \bigl(\operatorname{Im} F\bigr)_{JL} X^L} {\bigl(\operatorname{Im} F\bigr)_{KL} X^K X^L}. \end{align*}

In the diagram, inherited inputs (light grey, from the upstream period layer) flow into the layer’s three computed objects; the public outputs (orange) feed downstream into jaxvacua.flux_eft.FluxEFT (via $\mathcal{N}_{IJ}$) and jaxvacua.sampling (via $M_{AB}$).

jaxvacua.css — Complex-structure sector

Layer 1 — Inherited from periods

from periods ↑

$\Pi(X) = \bigl(\mathcal{F}_I,\, X^I\bigr)$

Period vector in projective coordinates $X^I$.
periods.period_vector_per

coordinate choice

$X^I = (1,\, z^i)$, $z^i = X^i / X^0$

Layer 2 — css in affine coordinates $z^i$

$\Pi(z)$

Period vector after the coordinate choice $X^I = (1,\, z^i)$:
$\Pi(z) = \bigl(\mathcal{F}_0(z),\; \mathcal{F}_i(z),\; 1,\; z^i\bigr)$

$K(z, \bar z)$

Kähler potential, computed from $\Pi(z)$:
$K(z, \bar z) \;=\; -\log\!\bigl(-\,i\, \Pi^\dagger\, \Sigma\, \Pi\bigr)$
periods.kahler_potential_per

Outputs

$\mathcal{N}_{IJ}(z, \bar z)$

Gauge-kinetic matrix in affine coordinates,
obtained from periods.gauge_kinetic_matrix
by the same coordinate choice $X^I \to (1,\, z^i)$
css.gauge_kinetic_matrix

$M_{AB}(z, \bar z)$

ISD / Hodge-star matrix built from $\mathcal{N}=R+iI$:
blocks use $I^{-1}$, $I^{-1}R$, and $R I^{-1}$
css.ISD_matrix

$K_{i\bar\jmath}(z, \bar z)$

Kähler metric on the moduli space:
$K_{i\bar\jmath} \;=\; \partial_i\, \partial_{\bar\jmath}\, K$
css.kahler_metric

Complex structure sector class#

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

Coordinate transformations#

`css.moduli_to_periods`(moduli[, conj])	Transforms complex structure moduli to periods for the global choice of gauge.
`css.periods_to_moduli`(XPer)	Transforms periods to complex structure moduli.

Prepotential#

`css.prepot`(moduli[, conj])	Computes the pre-potential for given values of the moduli.
`css.dF`(moduli[, conj])	Computes the holomorphic derivative $\partial_{z^i} F$ of the prepotential $F$ for given values of the moduli.
`css.period_vector`(moduli[, conj])	Returns the period vector $\Pi$ at a given point in moduli space.
`css.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 $z^i$.
`css.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 $z^i$.
`css.F_LCS`(moduli[, conj])	Calculates the value of the LCS prepotential in terms of the complex structure moduli $z^{i}$.
`F_coniLCS_bulk`(self, moduli[, conj])	Calculates the value of the LCS prepotential in terms of the complex structure moduli $z^{i}$.
`F_coniLCS_series`(self, moduli[, conj])	Calculates the full conifold-LCS prepotential $F_{\mathrm{coniLCS}}$ in terms of the complex structure moduli $z^i$.

Kähler potential#

`css.mirror_volume`(moduli, moduli_c)	Returns the value of the mirror dual Calabi-Yau volume.
`css.kahler_potential`(moduli, moduli_c, tau, ...)	Returns the value of the Kähler potential.
`css.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 $\tau$.
`css.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 $\overline{\tau}$.
`css.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 $z^i$.
`css.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 $\overline{z}^i$.
`css.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 $\tau$.
`css.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 $\overline{\tau}$.

Kähler metric#

`css.kahler_metric`(moduli, moduli_c, tau, tau_c)	Computes the Kähler metric $K_{\overline{I}J}$.
`css.inverse_kahler_metric`(moduli, moduli_c, ...)	Returns the inverse Kähler metric $K^{\overline{I}J}$.
`css.inverse_kahler_metric_grad`(moduli, ...)	Returns the gradient of the inverse Kähler metric.
`css.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.
`css.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.
`css.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 $\tau$.
`css.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 $\overline{\tau}$.
`css.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 $\overline{\tau}$.
`css.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 $\tau$.
`css.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.
`css.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 $z^i$.
`css.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 $z^i$.
`css.ddK_tau_tau`(moduli, moduli_c, tau, tau_c)	Returns the second holomorphic derivatives of the Kähler potential with respect to the axio-dilaton $\tau$.
`css.ddK_ctau_ctau`(moduli, moduli_c, tau, tau_c)	Returns the second holomorphic derivatives of the Kähler potential with respect to the axio-dilaton $\tau$.

Gauge kinetic matrix and ISD matrix#

`css.gauge_kinetic_matrix`(moduli, moduli_c[, ...])	Computes the value of the gauge kinetic matrix $\mathcal{N}$.
`css.ISD_matrix`(moduli, moduli_c)	Computes the value of the ISD-matrix $\mathcal{M}$.

`css.kahler_metric`(moduli, moduli_c, tau, tau_c)	Computes the Kähler metric \(K_{\overline{I}J}\).
`css.inverse_kahler_metric`(moduli, moduli_c, ...)	Returns the inverse Kähler metric \(K^{\overline{I}J}\).
`css.inverse_kahler_metric_grad`(moduli, ...)	Returns the gradient of the inverse Kähler metric.
`css.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.
`css.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.
`css.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 \(\tau\).
`css.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 \(\overline{\tau}\).
`css.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 \(\overline{\tau}\).
`css.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 \(\tau\).
`css.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.
`css.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 \(z^i\).
`css.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 \(z^i\).
`css.ddK_tau_tau`(moduli, moduli_c, tau, tau_c)	Returns the second holomorphic derivatives of the Kähler potential with respect to the axio-dilaton \(\tau\).
`css.ddK_ctau_ctau`(moduli, moduli_c, tau, tau_c)	Returns the second holomorphic derivatives of the Kähler potential with respect to the axio-dilaton \(\tau\).