jaxvacua.css.css

jaxvacua.css.css#

class css(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, gauge_choice=1 + 0j, prange=5, use_gvs=True, save_file=False, **kwargs)#

Bases: object

__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, gauge_choice=1 + 0j, prange=5, use_gvs=True, save_file=False, **kwargs)#

This class defines the complex structure sector in Type IIB orientifold compactifications. It contains a class object of type jaxvacua.periods.periods to compute periods, the prepotential, the gauge kinetic matrix etc. The moduli space limit (LCS, coniLCS, etc.) is selected via the limit parameter.

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 | str | 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 (Optional[cytools.CalabiYau]) – 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.

  • save_file (bool) – Save files for new models. Defaults to False.

  • **kwargs – Extra inputs.

dimension_H3#

Dimension of \(H^{3}(X,\mathbb{Z})\).

Type:

int

_dimension_H3_tot#

Dimension of \(H^{3}(X,\mathbb{Z})\) plus dimension of \(H_{3}(X,\mathbb{Z})\).

Type:

int

gauge_choice#

Choice of gauge for projective coordinates.

Type:

complex

gauge_choice_conj#

Choice of gauge for conjugate projective coordinates.

Type:

complex

h12#

Hodge number h12 of the Calabi-Yau setting the number of complex structure moduli.

Type:

int

Methods

A(moduli, moduli_c)

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

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 \(z^i\).

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_tilde(moduli, moduli_c)

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

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

This class defines the complex structure sector in Type IIB orientifold compactifications. It contains a class object of type jaxvacua.periods.periods to compute periods, the prepotential, the gauge kinetic matrix etc. The moduli space limit (LCS, coniLCS, etc.) is selected via the limit parameter.

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.

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.

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 \(\overline{X}^I\).

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 \(z^i\).

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 \(\bar{B} = (\bar{z}^i, \bar{\tau})\).

gauge_kinetic_matrix(moduli, moduli_c[, conj])

Computes the value of the gauge kinetic matrix \(\mathcal{N}\).

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.

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.

prepot(moduli[, conj])

Computes the pre-potential for given values of the moduli.

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.

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.

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.
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: period_vector()

See also: kahler_potential()

F(moduli, conj=False)#

Computes the pre-potential for given values of the moduli.

Note

We return the value of the pre-potential in terms projective coordinates

\[z^{i}=\frac{X^i}{X^0}\, .\]

Per default, we work in the gauge choice \(X^0=1\), but other gauge choices can be provided as inputs.

Note

We provide the option to compute the pre-potential and some additional functions in terms of the periods directly, see in particular jaxvacua.periods.periods.F_LCS_per(), jaxvacua.periods.periods.prepot_per() and jaxvacua.periods.periods.period_vector_per().

Warning

The moduli space limit around which the pre-potential is computed is set by the global parameter self.periods.limit. Currently, only self.periods.limit="LCS" is supported.

Parameters:

  • moduli (Array) – Complex structure moduli values.

  • conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

complex – Value of the prepotential \(F(z^i)\).

Return type:

complex

Errors:

ValueError: If the moduli space limit is not identified.

Aliases:

F()

See also: prepot_per()

F_LCS(moduli, conj=False)#

Calculates the value of the LCS prepotential in terms of the complex structure moduli \(z^{i}\).

Details

At LCS, we can write the prepotential as

\[F_{\text{LCS}}(z^1,\ldots , z^{h^{1,2}})=F_{\text{poly}}(z^1,\ldots , z^{h^{1,2}}) + F_{\text{inst}}(z^1,\ldots , z^{h^{1,2}})\]

see e.g. Eq. (2.13) in 1312.0014 for an equivalent formula. Here, the polynomial piece is given by

\[F_{\text{poly}}(z^1,\ldots , z^{h^{1,2}})=\dfrac{1}{6}\kappa_{ijk}\, z^i z^j z^k+\dfrac{1}{2} a_{ij}z^i z^j+b_i\, z^i+\dfrac{\text{i}}{2}\tilde{\xi}\, ,\]

Here, \(\widetilde{\kappa}_{ijk}\) are the triple intersection numbers of the mirror dual Calabi-Yau threefold \(\widetilde{X}\). Further, we defined

\[\begin{split}a_{ij} = \dfrac{1}{2}\begin{cases} \widetilde{\kappa}_{iij} & i\geq j\\[0.3em] \widetilde{\kappa}_{ijj} & i<j \end{cases} \, , \quad b_i = \dfrac{1}{24} \int_{\tilde{D}^i}\, c_2(\widetilde{X})\, , \quad \tilde{\xi}=\frac{\zeta(3)\, \chi(\widetilde{X})}{(2\pi)^3}\, .\end{split}\]

The instanton contributions read

\[F_{\mathrm{inst}}(z) = -\frac{1}{(2\pi\mathrm{i})^3}\, \sum_{q\in\mathcal{M}(\widetilde{X})}\, n_q^{0}\, \text{Li}_3\left (\text{e}^{2\pi \text{i} q_i z^i}\right )\; , \quad \text{Li}_3\left (x\right )=\sum_{m=1}^{\infty}\, \dfrac{x^{m}}{m^{3}}\, .\]

in terms of genus-0 Gopakumar-Vafa (GV) invariants \(n_q^0\) and the 3rd polylogarithm \(\text{Li}_3(x)\). Alternatively, the instanton contributions can be expressed as

\[F_{\text{inst}}(z^1,\ldots , z^{h^{1,2}})=\sum_{q\in\mathcal{M}(\widetilde{X})}\, N_q\, \text{e}^{2\pi \text{i} q_i z^i}\]

in terms of genus-0 Gromov-Witten (GW) invariants \(N_q\). Both pieces are computed in separate functions, see F_LCS_poly() and F_inst() for details.

Parameters:

  • moduli (Array) – Complex structure moduli values.

  • conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

complex – Value of the LCS prepotential \(F_{\text{LCS}}\).

Return type:

complex

See also: F_LCS_poly(), F_inst()

F_LCS_poly(moduli, conj=False)#

Computes the polynomial contribution \(F_{\mathrm{poly}}\) to the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the complex structure moduli \(z^i\).

Details

At Large Complex Structure (LCS), the polynomial part \(F_{\mathrm{poly}}\) of the prepotential \(F_{\mathrm{LCS}}\) is given by

\[F_{\text{poly}}(Z)=\dfrac{1}{6}\widetilde{\kappa}_{ijk}\, z^i z^j z^k +\dfrac{1}{2} a_{ij}z^i z^j+b_i\, z^i +\dfrac{\text{i}}{2}\tilde{\xi}\, .\]

Here, \(\widetilde{\kappa}_{ijk}\) are the triple intersection numbers of the mirror dual Calabi-Yau threefold \(\widetilde{X}\). Further, we defined

\[\begin{split}a_{ij} = \dfrac{1}{2}\begin{cases} \widetilde{\kappa}_{iij} & i\geq j\\[0.3em] \widetilde{\kappa}_{ijj} & i<j \end{cases} \, , \quad b_i = \dfrac{1}{24} \int_{\tilde{D}^i}\, c_2(\widetilde{X})\, , \quad \tilde{\xi}=\frac{\zeta(3)\, \chi(\widetilde{X})}{(2\pi)^3}\, .\end{split}\]
Parameters:

  • moduli (Array) – Complex structure moduli values.

  • conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

complex – Value of the polynomial contribution \(F_{\mathrm{poly}}\) to the LCS prepotential \(F_{\mathrm{LCS}}\).

Return type:

complex

See also: F_LCS()

F_inst(moduli, conj=False)#

Returns the instanton part \(F_{\mathrm{inst}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the complex structure moduli \(z^i\).

Details

The sum over world-sheet instantons in the Type IIA language is given by

\[F_{\text{inst}}(z^1,\ldots , z^{h^{1,2}})=-\frac{1}{(2\pi\mathrm{i})^3}\, \sum_{q\in\mathcal{M}(\widetilde{X})}\, n_q^{0}\, \text{Li}_3\left (\text{e}^{2\pi \text{i} q_i z^i}\right )\; , \quad \text{Li}_3\left (\right )=\sum_{m=1}^{\infty}\, \dfrac{x^{m}}{m^{3}}\]

where the sum is performed over all effective curve classes \(q\in\mathcal{M}(\widetilde{X})\) in the Mori cone \(\mathcal{M}(\widetilde{X})\) of the mirror dual manifold \(\widetilde{X}\). Here, the \(n_q^{0}\) are the genus-0 Gopakumar-Vafa (GV) invariants which can be computed systematically using methods described in hep-th/9308122.

The infinite sum appearing in the poly-logarithm \(\text{Li}_3\) can be rewritten to arrive at

\[\sum_{q\in\mathcal{M}(\widetilde{X})}\, n_q^{0}\, \text{Li}_3\left (\text{e}^{2\pi \text{i} q_i z^i}\right ) = \sum_{q\in\mathcal{M}(\widetilde{X})}\, N_q\, \text{e}^{2\pi \text{i} q_i z^i}\]

in terms of genus-0 Gromov-Witten (GW) invariants \(N_q\). We typically work with the latter to simplify the calculation.

Parameters:

  • moduli (Array) – Complex structure moduli values.

  • conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

complex – Value of the instanton part \(F_{\mathrm{inst}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\).

Return type:

complex

See also: F_LCS()

See also: F_LCS_poly()

ISD_matrix(moduli, moduli_c)#

Computes the value of the ISD-matrix \(\mathcal{M}\).

Note

This function descends from jaxvacua.periods.periods.ISD_matrix() 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 – Value of the ISD-matrix \(\mathcal{M}\).

Return type:

Array

Aliases:

M()

See also: jaxvacua.periods.periods.ISD_matrix()

K(moduli, moduli_c, tau, tau_c)#

Returns the value of the Kähler potential.

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:

\[K(z^i,\overline{z}^i,\tau,\overline{\tau})=-\log\bigl [-\text{i}(\tau-\overline{\tau})\bigr ]-\log\bigl [\mathcal{A}(z^i,\overline{z}^i)\bigr ]\]

where \(\mathcal{A}(z^i,\overline{z}^i)\) is the mirror dual CY volume defined as

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

The period vector can be computed via period_vector() or jaxvacua.periods.periods.period_vector_per().

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 – Value of the Kähler potential.

Return type:

complex

Aliases:

K(), kahler_potential()

See also: A()

M(moduli, moduli_c)#

Computes the value of the ISD-matrix \(\mathcal{M}\).

Note

This function descends from jaxvacua.periods.periods.ISD_matrix() 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 – Value of the ISD-matrix \(\mathcal{M}\).

Return type:

Array

Aliases:

M()

See also: jaxvacua.periods.periods.ISD_matrix()

N(moduli, moduli_c, conj=False)#

Computes the value of the gauge kinetic matrix \(\mathcal{N}\).

Note

This function descends from jaxvacua.periods.periods.gauge_kinetic_matrix() 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 – Value of the gauge kinetic matrix \(\mathcal{N}\).

Return type:

Array

Aliases:

N()

See also: jaxvacua.periods.periods.gauge_kinetic_matrix()

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: period_vector()

See also: kahler_potential()

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

See also: dddK(), inverse_kahler_metric(), riemann_tensor()

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.

dF(moduli, conj=False)#

Computes the holomorphic derivative \(\partial_{z^i} F\) of the prepotential \(F\) for given values of the moduli.

Parameters:

  • moduli (Array) – Complex structure moduli values.

  • conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of the holomorphic derivatives \(\partial_{z^i}F\) of the prepotential \(F(z^i)\).

Return type:

Array

See also: prepot()

dGamma(moduli, moduli_c, tau, tau_c)#

Returns the holomorphic derivative of the Christoffel symbols \(\partial_B \Gamma^E_{AC}\).

Details

Differentiates \(\Gamma^E_{AC} = K^{E\bar{F}}\,\partial_A K_{C\bar{F}}\) with respect to the holomorphic direction \(z^B\) (or \(\tau\)).

This quantity appears in the holomorphic second derivative of the inverse Kähler metric (and thus in the holomorphic Hessian \(\partial_A\partial_B V\)) via

\[\partial_A\partial_B K^{I\bar{J}} = -(\partial_B\Gamma^I_{AC})\,K^{C\bar{J}} + \Gamma^I_{AC}\,\Gamma^C_{BD}\,K^{D\bar{J}}\,.\]
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 – \(\partial_B \Gamma^E_{AC}\) with shape (n, n, n, n) and index ordering [E, A, C, B].

Return type:

Array

See also: christoffel_symbols()

dIKM_c(moduli, moduli_c, tau, tau_c)#

Returns the anti-holomorphic derivative of the inverse Kähler metric \(\partial_{\bar{B}} K^{I\bar{J}}\).

Details

From the identity \(K_{I\bar{F}}\,K^{I\bar{J}} = \delta_{\bar{F}}^{\bar{J}}\) one obtains

\[\partial_{\bar{B}} K^{I\bar{J}} = - K^{I\bar{F}}\,(\partial_{\bar{B}} K_{C\bar{F}})\,K^{C\bar{J}}\,.\]

This quantity appears in the anti-holomorphic variation of the \(F\)-term scalar \(S = D_{\bar{I}}\bar{W}\,K^{I\bar{J}}\,D_J W\).

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 – \(\partial_{\bar{B}} K^{I\bar{J}}\) with shape (n, n, n) and index ordering [I, J̄, B̄].

Return type:

Array

See also: dddK_c(), inverse_kahler_metric()

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 \(\tau\).

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 \(\partial_I K\).

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

ddIKM(moduli, moduli_c, tau, tau_c)#

Returns the mixed second derivative of the inverse Kähler metric \(\partial_A\partial_{\bar{B}} K^{I\bar{J}}\).

Details

Differentiating \(\partial_{\bar{B}} K^{I\bar{J}} = -K^{I\bar{F}}(\partial_{\bar{B}}K_{C\bar{F}})K^{C\bar{J}}\) with respect to \(z^A\) yields three terms via the product rule:

\[\partial_A\partial_{\bar{B}} K^{I\bar{J}} = \underbrace{-\Gamma^I_{AQ}\,(\partial_{\bar{B}} K^{Q\bar{J}})}_{\text{A1}} \;\underbrace{-\;K^{I\bar{F}}\,R_{C\bar{F}A\bar{B}}\,K^{C\bar{J}}}_{\text{A2}_R} \;\underbrace{-\;K^{I\bar{F}}\,(K^{M\bar{N}} K_{3h,C\bar{N}A}\,K_{3a,M\bar{F}\bar{B}})\,K^{C\bar{J}}}_{\text{A2}_\Gamma} \;\underbrace{+\;K^{I\bar{F}}\,K_{3a,C\bar{F}\bar{B}}\,\Gamma^C_{AD}\,K^{D\bar{J}}}_{\text{A3}}\]

where \(K_{3h}\) and \(K_{3a}\) are the holomorphic and anti-holomorphic third Kähler derivatives (dddK(), dddK_c()), \(\Gamma^E_{AC}\) are the Christoffel symbols (christoffel_symbols()), and \(R_{i\bar{\jmath}k\bar{l}}\) is the Riemann tensor (riemann_tensor()).

The Riemann tensor enters through the holomorphic derivative of the anti-holomorphic Kähler-metric derivative \(\partial_A(\partial_{\bar{B}} K_{C\bar{F}}) = K_{C\bar{F}A\bar{B}}\), which decomposes as \(R + \Gamma\)-contraction via the metric formula for the Riemann tensor.

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 – \(\partial_A\partial_{\bar{B}} K^{I\bar{J}}\) with shape (n, n, n, n) and index ordering [I, J̄, B̄, A].

Return type:

Array

See also: riemann_tensor(), christoffel_symbols(), dIKM_c()

ddK_ctau_ctau(moduli, moduli_c, tau, tau_c)#

Returns the second holomorphic derivatives of the Kähler potential with respect to the axio-dilaton \(\tau\).

Details

The returned array has the components

\[K_{\tau\tau}(z^i,\overline{z}^i,\tau,\overline{\tau})=\partial_{\tau}\partial_{\tau}K(z^i,\overline{z}^i,\tau,\overline{\tau})\, .\]

Note

These holomorphic derivatives are used e.g. in DDW() to compute the second Kähler covariant derivative of the superpotential.

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 – Second holomorphic derivative of the Kähler potential with respect to \(\tau\).

Return type:

complex

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 \(\overline{\tau}\).

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 – Second derivatives \(K_{\bar{\imath}\overline{\tau}}=\partial_{\overline{z}^i}\partial_{\overline{\tau}}K\) of the Kähler potential \(K\).

Return type:

Array

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 \(z^i\).

Details

The returned array has the components

\[K_{ij}(z^i,\overline{z}^i,\tau,\overline{\tau})=\partial_{z^i}\partial_{z^{j}}K(z^i,\overline{z}^i,\tau,\overline{\tau})\, .\]

Note

These holomorphic derivatives are used e.g. in DDW() to compute the second Kähler covariant derivative of the superpotential.

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 – \(h^{1,2}\times h^{1,2}\) matrix of the second holomorphic derivatives of the Kähler potential with respect to \(z^i\).

Return type:

Array

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 \(\tau\).

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 – Second derivatives \(K_{\tau\overline{j}}=\partial_{\tau}\partial_{\overline{z}^{j}}K\) of the Kähler potential \(K\).

Return type:

Array

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.

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 – Second derivatives \(K_{\bar{\jmath}i}=\partial_{\overline{z}^j}\partial_{z^i}K\) of the Kähler potential \(K\).

Return type:

Array

See also: dK_z()

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.

Note

The returned scalar is given by

\[K_{\tau\overline{\tau}}(z^i,\overline{z}^i,\tau,\overline{\tau})=\partial_{\tau}\partial_{\overline{\tau}}K\, .\]

For the standard Kähler potential \(K \supset - \log (-\text{i}(\tau-\overline{\tau}))\), we find

\[K_{\tau\overline{\tau}} = -\frac{1}{(\tau-\overline{\tau})^2}= \frac{1}{4\mathrm{Im}(\tau)^2}\, .\]
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 – Second derivatives \(\partial_{\tau}\partial_{\overline{\tau}}K\) of the Kähler potential \(K\).

Return type:

complex

ddK_tau_tau(moduli, moduli_c, tau, tau_c)#

Returns the second holomorphic derivatives of the Kähler potential with respect to the axio-dilaton \(\tau\).

Details

The returned array has the components

\[K_{\tau\tau}(z^i,\overline{z}^i,\tau,\overline{\tau})=\partial_{\tau}\partial_{\tau}K(z^i,\overline{z}^i,\tau,\overline{\tau})\, .\]

Note

These holomorphic derivatives are used e.g. in DDW() to compute the second Kähler covariant derivative of the superpotential.

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 – Second holomorphic derivative of the Kähler potential with respect to \(\tau\).

Return type:

complex

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 \(\overline{\tau}\).

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 – Second derivatives \(K_{i\overline{\tau}}=\partial_{z^i}\partial_{\overline{\tau}}K\) of the Kähler potential \(K\).

Return type:

Array

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.

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 – Second derivatives \(K_{i\bar{\jmath}}=\partial_{z^i}\partial_{\overline{z}^j}K\) of the Kähler potential \(K\).

Return type:

Array

See also: dK_z()

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 \(\tau\).

Details

The returned array has the components

\[K_{i\tau}(z^i,\overline{z}^i,\tau,\overline{\tau})=\partial_{z^i}\partial_{\tau}K(z^i,\overline{z}^i,\tau,\overline{\tau})\, .\]

Note

These holomorphic derivatives are used e.g. in DDW() to compute the second Kähler covariant derivative of the superpotential.

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 – \(h^{1,2}\times 1\) matrix of the second holomorphic derivatives of the Kähler potential with respect to \(z^i\) and \(\tau\).

Return type:

Array

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 \(z^i\).

Details

The returned array has the components

\[K_{ij}(z^i,\overline{z}^i,\tau,\overline{\tau})=\partial_{z^i}\partial_{z^{j}}K(z^i,\overline{z}^i,\tau,\overline{\tau})\, .\]

Note

These holomorphic derivatives are used e.g. in DDW() to compute the second Kähler covariant derivative of the superpotential.

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 – \(h^{1,2}\times h^{1,2}\) matrix of the second holomorphic derivatives of the Kähler potential with respect to \(z^i\).

Return type:

Array

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 \(A = (z^i, \tau)\).

Details

Computes the holomorphic derivative of the Kähler metric \(K_{C\bar{F}} = \partial_C\partial_{\bar{F}} K\) with respect to each holomorphic direction \(A\):

\[(\texttt{dddK})_{C\bar{F}A} = \partial_A K_{C\bar{F}} = \partial_A \partial_C \partial_{\bar{F}} K\,.\]

These third derivatives appear in the Christoffel symbols of the Kähler connection via

\[\Gamma^E_{AC} = K^{E\bar{F}}\,\partial_A K_{C\bar{F}} \,.\]
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 – Third Kähler derivative \(\partial_A K_{C\bar{F}}\) with shape (n, n, n) and index ordering [C, F̄, A] where n = h^{1,2} + 1.

Return type:

Array

See also: kahler_metric(), christoffel_symbols()

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 \(\bar{B} = (\bar{z}^i, \bar{\tau})\).

Details

Computes the anti-holomorphic derivative of the Kähler metric \(K_{C\bar{F}}\):

\[(\texttt{dddK\_c})_{C\bar{F}\bar{B}} = \partial_{\bar{B}} K_{C\bar{F}} = \partial_{\bar{B}} \partial_C \partial_{\bar{F}} K\,.\]

These appear in the anti-holomorphic derivative of the inverse Kähler metric and in the Riemann curvature tensor.

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 – Third Kähler derivative \(\partial_{\bar{B}} K_{C\bar{F}}\) with shape (n, n, n) and index ordering [C, F̄, B̄] where n = h^{1,2} + 1.

Return type:

Array

See also: kahler_metric(), riemann_tensor()

gauge_kinetic_matrix(moduli, moduli_c, conj=False)#

Computes the value of the gauge kinetic matrix \(\mathcal{N}\).

Note

This function descends from jaxvacua.periods.periods.gauge_kinetic_matrix() 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 – Value of the gauge kinetic matrix \(\mathcal{N}\).

Return type:

Array

Aliases:

N()

See also: jaxvacua.periods.periods.gauge_kinetic_matrix()

inverse_kahler_metric(moduli, moduli_c, tau, tau_c, mode='block diagonal')#

Returns the inverse Kähler metric \(K^{\overline{I}J}\).

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.

  • mode (str) – Description

Returns:

Array – Inverse Kähler metric \(K^{\overline{I}J}\).

Return type:

Array

See also: kahler_metric()

inverse_kahler_metric_grad(moduli, moduli_c, tau, tau_c, mode='block diagonal')#

Returns the gradient of the inverse Kähler metric.

Note

This function is currently not being used, but might turn out to be useful when taking derivatives of the F-terms \(F^i = K^{i\overline{j}}D_{\overline{j}}\overline{W}\). The output corresponds to the tuple

\[(\partial_{z^i}K^{\bar{J}L},\partial_{\overline{z}^i}K^{\bar{J}L},\partial_{\tau}K^{\bar{J}L},\partial_{\overline{\tau}}K^{\bar{J}L})\, .\]
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.

  • mode (str) – Description

Returns:

Array – Gradient of the inverse Kähler metric \(\partial_{I}K^{\overline{J}L},\partial_{\overline{I}}K^{\overline{J}L}\).

Return type:

Tuple[Array, Array, complex, complex]

See also: kahler_metric()

See also: inverse_kahler_metric()

kahler_metric(moduli, moduli_c, tau, tau_c, mode='block diagonal')#

Computes the Kähler metric \(K_{\overline{I}J}\).

Details

We assemble the various components of second derivatives obtained from ddK_z_cz(), ddK_z_ctau(), ddK_cz_tau() and ddK_tau_ctau() in such a way that

\[\begin{split}K_{A\overline{B}}=\left (\begin{array}{cc} K_{i\overline{j}} & K_{i\overline{\tau}} \\[0.3em] K_{\tau\overline{j}} & K_{\tau\overline{\tau}} \end{array} \right )\end{split}\]

Depending on the use case, we assume a block diagonal structure of the Kähler metric in which case the mixed second derivatives vanish \(K_{i\overline{\tau}}=K_{\tau\overline{j}}=0\).

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.

  • mode (str) – Mode to compute the Kahler metric. Defaults to "block diagonal" meaning that there are no mixing terms between complex structure moduli and the axio-dilaton in the Kähler potential. If set to None instead, the full set of 2nd derivatives of the Kähler potential is computed.

Returns:

Array – Kähler metric \(K_{\overline{I}J}\).

Raises:

ValueError – Wrong mode for computation. Only mode=None and mode="block diagonal" are supported.

Return type:

Array

kahler_potential(moduli, moduli_c, tau, tau_c)#

Returns the value of the Kähler potential.

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:

\[K(z^i,\overline{z}^i,\tau,\overline{\tau})=-\log\bigl [-\text{i}(\tau-\overline{\tau})\bigr ]-\log\bigl [\mathcal{A}(z^i,\overline{z}^i)\bigr ]\]

where \(\mathcal{A}(z^i,\overline{z}^i)\) is the mirror dual CY volume defined as

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

The period vector can be computed via period_vector() or jaxvacua.periods.periods.period_vector_per().

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 – Value of the Kähler potential.

Return type:

complex

Aliases:

K(), kahler_potential()

See also: A()

property 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.

Returns:

lcs_tree – The topological data tree.

mirror_volume(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: period_vector()

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: monodromy_matrix_single()

monodromy_matrix_single(b)#

Computes the monodromy matrix \(T_b\) for the shift \(z^b \to z^b + 1\).

The period vector transforms as \(\Pi \to T_b \cdot \Pi\).

At LCS, uses the analytical formula from the intersection numbers, a-matrix, and b-vector. For other limits, falls back to a numerical computation via the period vector.

Note

The monodromy matrices receive no instanton corrections: the instanton sum \(F_{\mathrm{inst}}\) involves \(e^{2\pi i q_a z^a}\) which is invariant under \(z^a \to z^a + 1\).

Parameters:

b (int) – Index of the modulus to shift (0-based: b = 0, ..., h12-1).

Returns:

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

Return type:

ndarray

See also: monodromy_matrix(), verify_monodromy()

period_vector(moduli, conj=False)#

Returns the period vector \(\Pi\) at a given point in moduli space.

Details

We introduce a symplectic basis of \(\{\Sigma_{I},\Sigma^I\} \subset H_3(X,\mathbb{Z})\) together with the corresponding Poincaré dual forms \(\{\alpha^I,\beta_I\}\). We then define the periods by integrating the holomorphic \(3\)-form \(\Omega\) over these cycles,

\[X^I=\int_{\Sigma_{I}}\Omega=\int_{X} \Omega\wedge \alpha^I\, ,\quad \mathcal{F}_I=\int_{\Sigma^I}\Omega=\int_{X} \Omega \wedge \beta_I \, .\]

We collect the periods \(X^I,\mathcal{F}_I\) in the period vector \(\Pi\), that is,

\[\begin{split}\Pi=\left (\begin{array}{c} \mathcal{F}_I\\ X^I \end{array}\right )\, I=0,1,\ldots,h^{1,2}\, .\end{split}\]

The periods \(X^I\) serve as homogeneous complex coordinates on a local patch of the complex structure moduli space of \(X\). Away from the locus \(X^0=0\), we introduce projective coordinates

\[z^i =\dfrac{X^i}{X^0}\, , i=1,\ldots,h^{2,1}(X)\, ,\]

and normalise \(\Omega\) such that \(X^0=1\). The dual periods \(\mathcal{F}_I=\mathcal{F}_I(z^i)\) are then determined by a prepotential \(F(z^i)\) through

\[\mathcal{F}_i(z^i)=\partial_{z^i} F \, ,\quad \mathcal{F}_0 =2F-z^i\partial_{z^i}F\, .\]

The period vector can then be computed from the derivatives of the prepotential \(F(z^i)\) as follows

\[\begin{split}\Pi(z^1,\ldots , z^{h^{1,2}})=\left (\begin{array}{c} 2F-F_i z^i\\ F_i\\ 1\\ z^i \end{array}\right )\, i=1,\ldots,h^{1,2}\, .\end{split}\]

Here, we make use of standard identities for computing periods in terms of the projective coordinates directly.

Note

To compute the period vector in terms of the periods directly, please use jaxvacua.periods.periods.period_vector_per().

Parameters:

  • moduli (Array) – Complex structure moduli values.

  • conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

Array – Value of the period vector \(\Pi\).

Return type:

Array

See also: prepot()

See also: dF()

See also: jaxvacua.periods.periods.period_vector_per()

periods_to_moduli(XPer)#

Transforms periods to complex structure moduli.

Parameters:

XPer (Array) – Values of the periods.

Returns:

Array – Value of the complex structure moduli.

Return type:

Array

prepot(moduli, conj=False)#

Computes the pre-potential for given values of the moduli.

Note

We return the value of the pre-potential in terms projective coordinates

\[z^{i}=\frac{X^i}{X^0}\, .\]

Per default, we work in the gauge choice \(X^0=1\), but other gauge choices can be provided as inputs.

Note

We provide the option to compute the pre-potential and some additional functions in terms of the periods directly, see in particular jaxvacua.periods.periods.F_LCS_per(), jaxvacua.periods.periods.prepot_per() and jaxvacua.periods.periods.period_vector_per().

Warning

The moduli space limit around which the pre-potential is computed is set by the global parameter self.periods.limit. Currently, only self.periods.limit="LCS" is supported.

Parameters:

  • moduli (Array) – Complex structure moduli values.

  • conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

complex – Value of the prepotential \(F(z^i)\).

Return type:

complex

Errors:

ValueError: If the moduli space limit is not identified.

Aliases:

F()

See also: prepot_per()

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.

Details

The Riemann tensor of a Kähler manifold can be expressed in terms of the Kähler metric and its derivatives as

\[R_{i\bar{\jmath}k\bar{l}} = \partial_k\partial_{\bar{l}} K_{i\bar{\jmath}} - K^{m\bar{n}}\, (\partial_k K_{i\bar{n}})\, (\partial_{\bar{l}} K_{m\bar{\jmath}})\,.\]

The first term is the fourth mixed Kähler derivative \(K_{i\bar{\jmath}k\bar{l}} = \partial_k\partial_{\bar{l}} K_{i\bar{\jmath}}\), while the second subtracts the Christoffel-symbol contraction.

The tensor satisfies the Kähler symmetry

\[R_{i\bar{\jmath}k\bar{l}} = R_{k\bar{\jmath}i\bar{l}}\,.\]

It enters the Hessian of the \(\mathcal{N}=1\) supergravity scalar potential at generic points in moduli space (see e.g. hep-th/0411183, Eq. 2.3).

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 – Riemann tensor \(R_{i\bar{\jmath}k\bar{l}}\) with shape (n, n, n, n) and index ordering [i, j̄, k, l̄].

Return type:

Array

See also: christoffel_symbols(), dddK(), dddK_c()

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