jaxvacua.periods.periods

jaxvacua.periods.periods#

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

Bases: object

Period class for Calabi-Yau threefold compactifications. Provides functions to compute periods, prepotential, Kähler potential, gauge kinetic matrix, and derived objects at various moduli space limits (LCS, coniLCS, etc.).

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

Period class.

This class defines functions of the periods of Calabi-Yau threefolds \(X\). They are obtained from integrating the holomorphic 3-form \(\Omega\) over a symplectic basis of \(H_3(X,\mathbb{Z})\).

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.

  • 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 (object | 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.

  • lcs_tree_input (object | None) – Pre-built lcs_tree object. If provided, skips model construction. Defaults to None.

  • ncf (int | None) – Number of conifolds for the coniLCS limit. Defaults to None.

  • conifold_curve (object | None) – Conifold curve charges. Defaults to None.

  • prange (int) – Truncation order for the polylogarithm series. Defaults to 500.

  • use_gvs (bool) – Whether to use GV invariants instead of GW invariants for the instanton sum. Defaults to False.

  • 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.

Raises:

ValueError – If model inputs are inconsistent or required data is missing.

h12#

\(h^{1,2}(X,\mathbb{Z})\) of the CY \(X\).

Type:

int

model_type#

Model type. Defaults to None. Currently, "KS" and "CICY" are available.

Type:

str

limit#

Moduli space limit. Defaults to None. Currently, only "LCS" is available.

Type:

str

model_ID#

Internal model ID. Defaults to None.

Type:

int

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

period_input#

Input period function.

Type:

object

prepotential_input#

Input prepotential function.

Type:

object

Methods

A_per(XPer, cXPer)

Computes the value of the mirror CY volume \(\tilde{\mathcal{V}}\) as a function of the periods \(X^I\).

D_period_vector_per(XPer, cXPer[, conj])

Computes the Kähler convariant derivative \(D_I\Pi\). of the period vector \(\Pi\).

F_LCS_per(XPer[, conj])

Calculates the value of the LCS prepotential \(F_{\text{LCS}}\) in terms of the periods \(X^I\).

F_LCS_poly_per(XPer[, conj])

Computes the polynomial part \(F_{\mathrm{poly}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the periods.

F_inst_per(XPer[, conj])

Computes the instanton part \(F_{\mathrm{inst}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the periods.

ISD_matrix(XPer, cXPer)

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

M(XPer, cXPer)

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

N(XPer, cXPer[, conj])

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

PQ_per(XPer, cXPer[, conj])

Computes the matrices \(P_{IJ}\) and \(Q^{I}\,_{J}\).

P_per(XPer, cXPer[, conj])

Computes the matrix \(P_{IJ}\).

Q_inv_per(XPer, cXPer[, conj])

Computes the inverse of the matrix \(Q^{I}\,_{J}\).

Q_per(XPer, cXPer[, conj])

Computes the matrix \(Q^{I}\,_{J}\).

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

Period class.

compute_a_shift_monodromy(shift)

Symplectic monodromy matrix induced by a shift of the prepotential \(a\)-matrix, \(a \to a + S\).

dM(XPer, cXPer)

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

dM_c(XPer, cXPer)

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(XPer, cXPer[, 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(XPer, cXPer[, 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 \(\overline{X}^I\).

gauge_kinetic_matrix(XPer, cXPer[, conj])

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

gauge_kinetic_matrix_periods(XPer, cXPer[, conj])

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

gauge_kinetic_matrix_prepotential(XPer, cXPer)

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

grad_kahler_potential_per(XPer, cXPer[, conj])

Computes derivatives \(\partial_{X^I}K\) of the Kähler potential \(K\) with respect to the periods :mathh`X^I`.

grad_period_vector_per(XPer[, conj])

Computes derivatives \(\partial_{X^I}\Pi\) of the period vector \(\Pi\) with respect to the periods :mathh`X^I`.

kahler_potential_per(XPer, cXPer)

Computes the value of the Kähler potential \(K\) as a function of the periods \(X^I\).

period_vector_per(XPer[, conj])

Computes the period vector \(\Pi\) in terms of the periods \(X^I\).

prepot_grad_grad_per(XPer[, conj])

Computes the second holomorphic derivatives \(F_{IJ}=\partial_{X^I}\partial_{X^J}F\) of the prepotential \(F\) with respect to the periods \(X^I\).

prepot_grad_per(XPer[, conj])

Computes the holomorphic derivatives \(F_{I}=\partial_{X^I}F\) of the prepotential \(F\) with respect to the periods \(X^I\).

prepot_per(XPer[, conj])

Computes the prepotential \(F\) in terms of the periods \(X^I\).

sigma()

Returns the symplectic matrix

Attributes

F_coniLCS_bulk_per

** 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_per

** 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_poly_split_per

** 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_per

** 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_coni_per

** 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_inst_per_coni

** 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_poly_per

** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.

ddF_coniLCS_poly_per

** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.

dddF_coniLCS_poly_per

** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.

ddddF_coniLCS_poly_per

** Class-level descriptor: surfaces a method only when the instance's limit is in the coniLCS family. Otherwise raises AttributeError so hasattr() returns False.

delete_coni_index

** 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 underlying lcs_tree object holding the CY model data.
A_per(XPer, cXPer)#

Computes the value of the mirror CY volume \(\tilde{\mathcal{V}}\) as a function of the periods \(X^I\).

Details

The mirror CY volume \(\tilde{\mathcal{V}}\) is computed from the period vector \(\Pi\) as

\[\tilde{\mathcal{V}} = -\text{i}\, \Pi^\dagger\cdot \Sigma\cdot\Pi\, .\]
Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

Returns:

Array – Value of the mirror CY volume in terms of the period \(X^I\).

Return type:

complex

D_period_vector_per(XPer, cXPer, conj=False)#

Computes the Kähler convariant derivative \(D_I\Pi\). of the period vector \(\Pi\).

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – Values of \(D_I\Pi\).

Return type:

Array

F_LCS_per(XPer, conj=False)#

Calculates the value of the LCS prepotential \(F_{\text{LCS}}\) in terms of the periods \(X^I\).

Details

At Large Complex Structure (LCS), the prepotential can be expressed as

\[F_{\mathrm{LCS}}(X) = F_{\mathrm{poly}}(X) + F_{\mathrm{inst}}(X)\]

where

\[F_{\mathrm{poly}}(X) = -\frac{1}{6X^0}\widetilde{\kappa}_{ijk}X^iX^jX^k+\frac{1}{2}a_{ij}X^iX^j +b_{i}X^i\, X^0 + \dfrac{\text{i}}{2}\tilde{\xi}\, (X^0)^2\, ,\]

and

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

The former is computed via F_LCS_poly_per(), while the latter in F_inst_per(). See the respective function for more details.

Parameters:

  • XPer (Array) – Values of periods.

  • 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_per()

See also: F_inst_per()

F_LCS_poly_per(XPer, conj=False)#

Computes the polynomial part \(F_{\mathrm{poly}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the periods.

Details

The polynomial part \(F_{\mathrm{poly}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) can be expressed in terms of the periods \(X^I=(X^0,X^i)\) as

\[F_{\mathrm{poly}}(X) = -\frac{1}{6X^0}\widetilde{\kappa}_{ijk}X^iX^jX^k+\frac{1}{2}a_{ij}X^iX^j +b_{i}X^i\, X^0 + \dfrac{\text{i}}{2}\tilde{\xi}\, (X^0)^2\, .\]

Here, \(\widetilde{\kappa}_{ijk}\) are the triple intersection numbers of the mirror dual Calabi-Yau threefold \(\widetilde{X}\). Here, 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 \(a_{ij}\) are rational numbers, while the \(b_i\) are integers. The \(\tilde{D}^i\) are the divisors dual to the basis of \(H^{1,1}(\widetilde{X},\mathbb{Z})\). Finally, \(\chi(\widetilde{X})\) is the Euler characteristic of \(\widetilde{X}\). The last term in \(F_{\mathrm{poly}}\) is the so-called \((\alpha')^3\) 1-loop correction on the mirror dual side.

Parameters:

  • XPer (Array) – Values of periods.

  • 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_per()

F_inst_per(XPer, conj=False)#

Computes the instanton part \(F_{\mathrm{inst}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the periods.

Details

The instanton part \(F_{\mathrm{inst}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) can be expressed in terms of the periods \(X^I=(X^0,X^i)\) as

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

Here 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

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

in terms of genus-0 Gromov-Witten (GW) invariants \(N_q\). We typically work with the latter to simplify the calculation. The relation between the two types of invariants is more explicitly given by

\[N_q = \sum_{d|q}\, \dfrac{1}{d^3}\, n_{q/d}^{0}\, .\]

Here, the sum runs over all divisors \(d\) of the curve class \(q\). The \(N_q\) are typically rational numbers, while the \(n_q^{0}\) are integers. The curve classes \(q\) are specified in a basis of the Mori cone \(\mathcal{M}(\widetilde{X})\) of the mirror dual Calabi-Yau threefold \(\widetilde{X}\). The Mori cone is dual to the Kähler cone of \(\widetilde{X}\). The curve classes \(q\) can be expressed in terms of the generators of the Mori cone. The curve classes are also referred to as curve charges in the following. The curve charges are stored in the attribute GW_charges (GV_charges) for the GW (GV) invariants. The corresponding invariants are stored in the attributes GW_inv (GV_inv). The curve charges and invariants are limited to a certain maximum degree \(d=\text{max_deg}\) in the attributes GW_charges_lim, GV_charges_lim, GW_inv_lim, GV_inv_lim. The maximum degree can be specified when initialising the class. The maximum degree is defined with respect to a grading vector which can be specified when initialising the class. If no grading vector is provided, the default grading vector \((1,1,\ldots,1)\) is used. The maximum available degree for the instanton data is stored in the attribute max_available_deg. If the specified maximum degree exceeds the maximum available degree, a warning is raised and the maximum available degree is used instead.

Note

The sum over curve classes is truncated at a certain maximum degree \(d=\text{max_deg}\) in our implementation. This is justified since the instanton contributions are exponentially suppressed at large complex structure. The maximum degree can be specified when initialising the class.

Parameters:

  • XPer (Array) – Values of periods.

  • 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_per()

ISD_matrix(XPer, cXPer)#

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

Details

Computes a matrix representation \(\mathcal{M}\) for the Hodge-\(\star\) operator for Calabi-Yau threefolds. It can be expressed in terms of the real and imaginary parts of the gauge kinetic matrix \(\mathcal{N} = \mathcal{R}+\mathrm{i}\mathcal{I}\) as

\[\begin{split}\mathcal{M} = \left (\begin{array}{cc} -\mathcal{I}^{-1} & \mathcal{I}^{-1}\mathcal{R} \\ \mathcal{R}\mathcal{I}^{-1} & -\mathcal{I}-\mathcal{R}\mathcal{I}^{-1}\mathcal{R} \end{array}\right )\end{split}\]

see Eq. (2.15) in 2110.05511 or alternatively Eq. (3.7) in 2310.06040.

This so-called ISD-matrix \(\mathcal{M}\) possesses the following properties

\[\mathcal{M} = \mathcal{M}^T\; ,\quad \mathcal{M}^T\Sigma\mathcal{M} = \Sigma \; ,\quad \mathcal{M}^{-1}=\Sigma^T\mathcal{M}\Sigma\, .\]

Moreover, it has positive eigenvalues which can be useful to bound the number of vacua satisfying the ISD condition \(DW=0\), see for example 2310.06040 and 2501.03984.

Warning

Our convention differs slightly from that of Eq. (3.7) in 2310.06040 due to the ordering of periods in the period vector. Specifically, we have

\[\mathcal{M}_{\mathrm{here}} = \mathcal{M}_{\mathrm{there}}^{-1}\, .\]
Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

Returns:

Array – Value of the ISD-matrix \(\mathcal{M}\).

Return type:

Array

Aliases:

M()

M(XPer, cXPer)#

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

Details

Computes a matrix representation \(\mathcal{M}\) for the Hodge-\(\star\) operator for Calabi-Yau threefolds. It can be expressed in terms of the real and imaginary parts of the gauge kinetic matrix \(\mathcal{N} = \mathcal{R}+\mathrm{i}\mathcal{I}\) as

\[\begin{split}\mathcal{M} = \left (\begin{array}{cc} -\mathcal{I}^{-1} & \mathcal{I}^{-1}\mathcal{R} \\ \mathcal{R}\mathcal{I}^{-1} & -\mathcal{I}-\mathcal{R}\mathcal{I}^{-1}\mathcal{R} \end{array}\right )\end{split}\]

see Eq. (2.15) in 2110.05511 or alternatively Eq. (3.7) in 2310.06040.

This so-called ISD-matrix \(\mathcal{M}\) possesses the following properties

\[\mathcal{M} = \mathcal{M}^T\; ,\quad \mathcal{M}^T\Sigma\mathcal{M} = \Sigma \; ,\quad \mathcal{M}^{-1}=\Sigma^T\mathcal{M}\Sigma\, .\]

Moreover, it has positive eigenvalues which can be useful to bound the number of vacua satisfying the ISD condition \(DW=0\), see for example 2310.06040 and 2501.03984.

Warning

Our convention differs slightly from that of Eq. (3.7) in 2310.06040 due to the ordering of periods in the period vector. Specifically, we have

\[\mathcal{M}_{\mathrm{here}} = \mathcal{M}_{\mathrm{there}}^{-1}\, .\]
Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

Returns:

Array – Value of the ISD-matrix \(\mathcal{M}\).

Return type:

Array

Aliases:

M()

N(XPer, cXPer, conj=False)#

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

Details

The gauge kinetic matrix \(\mathcal{N}\) is defined in terms of the periods \(X^I\) and \(\mathcal{F}_I\) as

\[\mathcal{N}_{IJ} = (\mathcal{F}_{I} ,\, D_{\bar{\imath}}\overline{\mathcal{F}}_I )\, \cdot\, (X^{J},\, D_{\bar{\jmath}}\overline{X}^J)^{-1} = P_{IK} \, (Q^{-1})^{K}\,_{J}\]

see e.g. Sect. 2 above Eq. (2.12) in 2110.05511. Here we defined the two matrices \(P_{IJ}\) and \(Q^{I}\,_{J}\) as

\[P_{IJ} = (\mathcal{F}_{I} ,\, D_{\bar{\imath}}\overline{\mathcal{F}}_I )_{J} \, ,\quad Q^{I}\,_{J} = (X^{I},\, D_{\bar{\jmath}}\overline{X}^I)_{J}\, .\]

In the presence of a prepotential \(F\), this object can be computed as follows (see e.g. Eq. (2.12) in 2110.05511)

\[\mathcal{N}_{IJ}= \overline{F}_{IJ} + +2\text{i}\, \dfrac{\text{Im}(F_{I L})X^{L} \, \text{Im}(F_{J K})X^{K}}{X^{M}\text{Im}(F_{MN})X^{N}}\]

where \(F_{IJ}=\partial_{X^{I}}\partial_{X^{J}}F\) is the second derivative of the prepotential \(F\) with respect to the periods \(X^I\).

Both formulas for the gauge kinetic matrix \(\mathcal{N}\) have been implemented. The mode of computation depends on the provided input:

  • at LCS, we use the formula from the periods,

  • with input periods, we use the formula from the periods, and

  • otherwise, with input prepotential, we use the formula for the prepotential.

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – Value of the gauge kinetic matrix \(\mathcal{N}\).

Return type:

Array

PQ_per(XPer, cXPer, conj=False)#

Computes the matrices \(P_{IJ}\) and \(Q^{I}\,_{J}\).

Details

Computes the two matrices \(P_{IJ}\) and \(Q^{I}\,_{J}\) in terms of the periods \(X^I\) and \(\mathcal{F}_I\) as

\[P_{IJ} = (\mathcal{F}_{I} ,\, D_{\bar{\imath}}\overline{\mathcal{F}}_I )_{J} \, ,\quad Q^{I}\,_{J} = (X^{I},\, D_{\bar{\jmath}}\overline{X}^I)_{J}\, ,\]

see Eq. (3.5) in 2310.06040.

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

  • Array – Values of \(P_{IJ}\).

  • Array – Values of \(Q^{I}\,_{J}\).

Return type:

Tuple[Array, Array]

P_per(XPer, cXPer, conj=False)#

Computes the matrix \(P_{IJ}\).

Details

Computes the matrix \(P_{IJ}\). in terms of the periods \(X^I\) and \(\mathcal{F}_I\) as

\[P_{IJ} = (\mathcal{F}_{I} ,\, D_{\bar{\imath}}\overline{\mathcal{F}}_I )_{J}\, ,\]

see Eq. (3.5) in 2310.06040.

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – Values of \(P_{IJ}\).

Return type:

Array

Q_inv_per(XPer, cXPer, conj=False)#

Computes the inverse of the matrix \(Q^{I}\,_{J}\).

Details

Computes the inverse of the matrix \(Q^{I}\,_{J}\) defined in terms of the periods \(X^I\) and \(\mathcal{F}_I\) as

\[Q^{I}\,_{J} = (X^{I},\, D_{\bar{\jmath}}\overline{X}^I)_{J}\, ,\]

see Eq. (3.5) in 2310.06040.

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – Values of \((Q^{-1})^{I}\,_{J}\).

Return type:

Array

Q_per(XPer, cXPer, conj=False)#

Computes the matrix \(Q^{I}\,_{J}\).

Details

Computes the matrix \(Q^{I}\,_{J}\) in terms of the periods \(X^I\) and \(\mathcal{F}_I\) as

\[Q^{I}\,_{J} = (X^{I},\, D_{\bar{\jmath}}\overline{X}^I)_{J}\, ,\]

see Eq. (3.5) in 2310.06040.

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – Values of \(Q^{I}\,_{J}\).

Return type:

Array

compute_a_shift_monodromy(shift)#

Symplectic monodromy matrix induced by a shift of the prepotential \(a\)-matrix, \(a \to a + S\).

Details

The LCS prepotential carries a quadratic term \(\tfrac{1}{2}\, a_{ij}\, X^i X^j\), so under \(a \to a + S\) it gains \(\tfrac{1}{2}\, S_{ij}\, X^i X^j\). The dual periods \(\mathcal{F}_I = \partial_I F\) then transform as

\[\mathcal{F}_i \to \mathcal{F}_i + S_{ij}\, X^j \quad (i,j=1,\ldots,h^{1,2})\,, \qquad \mathcal{F}_0 \to \mathcal{F}_0\,,\]

while every \(X^I\) is unchanged (the \(\mathcal{F}_0\) term is invariant because the degree-two piece cancels by Euler’s theorem). On the period vector \(\Pi = (\mathcal{F}_I,\, X^I)\) (dual periods first, see period_vector_per()) this is the unipotent transformation

\[\begin{split}M(S) = \begin{pmatrix} \mathbb{1} & \widehat{S} \\ 0 & \mathbb{1} \end{pmatrix}\,, \qquad \widehat{S} = \begin{pmatrix} 0 & 0 \\ 0 & S \end{pmatrix}\,,\end{split}\]

where \(\widehat{S}\) embeds the \(h^{1,2}\times h^{1,2}\) shift \(S\) with a zero \(X^0\) row and column. \(M(S)\) is symplectic (\(M^T \Sigma M = \Sigma\), with \(\Sigma\) from sigma()) iff \(S\) is symmetric — which every \(a\)-matrix from jaxvacua.conifold.compute_a_matrix() satisfies (\(a_{ij} = \kappa_{ijj}/2 = a_{ji}\)) — and integer iff \(S\) is integer.

Periods, fluxes and the full flux vector all transform by the same \(M(S)\) (\(\Pi \to M\Pi\), \(\text{flux} \to M\,\text{flux}\)), leaving the Kähler potential \(e^{-K} = i\,\Pi^\dagger \Sigma \Pi\) and the superpotential \(W = (f - \tau h)\cdot \Sigma \cdot \Pi\) invariant. This makes \(M(S)\) the dictionary between two \(a\)-conventions (equivalently two integral flux bases), e.g. the conifold_basis=True and conifold_basis=False descriptions of the same geometry.

Parameters:

shift (Array) – Symmetric \((h^{1,2}, h^{1,2})\) shift matrix \(S = a' - a\).

Returns:

  • Array – The \((2(h^{1,2}+1), 2(h^{1,2}+1))\) symplectic monodromy

  • matrix :math:`M (S)

Return type:

Array

See also: sigma(), period_vector_per(),

jaxvacua.conifold.compute_a_matrix()

dM(XPer, cXPer)#

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

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

Returns:

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

Return type:

Array

dM_c(XPer, cXPer)#

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

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

Returns:

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

Return type:

Array

dN(XPer, cXPer, conj=False)#

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

Parameters:

  • XPer (Array) – JAX array of shape (\(h^{1,2}+1\), ) containing the value of the periods.

  • cXPer (Array) – JAX array of shape (\(h^{1,2}+1\), ) containing the complex conjugate value of the periods.

  • 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(XPer, cXPer, 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 \(\overline{X}^I\).

Parameters:

  • XPer (Array) – JAX array of shape (\(h^{1,2}+1\), ) containing the value of the periods.

  • cXPer (Array) – JAX array of shape (\(h^{1,2}+1\), ) containing the complex conjugate value of the periods.

  • 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

gauge_kinetic_matrix(XPer, cXPer, conj=False)#

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

Details

The gauge kinetic matrix \(\mathcal{N}\) is defined in terms of the periods \(X^I\) and \(\mathcal{F}_I\) as

\[\mathcal{N}_{IJ} = (\mathcal{F}_{I} ,\, D_{\bar{\imath}}\overline{\mathcal{F}}_I )\, \cdot\, (X^{J},\, D_{\bar{\jmath}}\overline{X}^J)^{-1} = P_{IK} \, (Q^{-1})^{K}\,_{J}\]

see e.g. Sect. 2 above Eq. (2.12) in 2110.05511. Here we defined the two matrices \(P_{IJ}\) and \(Q^{I}\,_{J}\) as

\[P_{IJ} = (\mathcal{F}_{I} ,\, D_{\bar{\imath}}\overline{\mathcal{F}}_I )_{J} \, ,\quad Q^{I}\,_{J} = (X^{I},\, D_{\bar{\jmath}}\overline{X}^I)_{J}\, .\]

In the presence of a prepotential \(F\), this object can be computed as follows (see e.g. Eq. (2.12) in 2110.05511)

\[\mathcal{N}_{IJ}= \overline{F}_{IJ} + +2\text{i}\, \dfrac{\text{Im}(F_{I L})X^{L} \, \text{Im}(F_{J K})X^{K}}{X^{M}\text{Im}(F_{MN})X^{N}}\]

where \(F_{IJ}=\partial_{X^{I}}\partial_{X^{J}}F\) is the second derivative of the prepotential \(F\) with respect to the periods \(X^I\).

Both formulas for the gauge kinetic matrix \(\mathcal{N}\) have been implemented. The mode of computation depends on the provided input:

  • at LCS, we use the formula from the periods,

  • with input periods, we use the formula from the periods, and

  • otherwise, with input prepotential, we use the formula for the prepotential.

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – Value of the gauge kinetic matrix \(\mathcal{N}\).

Return type:

Array

gauge_kinetic_matrix_periods(XPer, cXPer, conj=False)#

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

Details

The gauge kinetic matrix \(\mathcal{N}\) is defined in terms of the periods \(X^I\) and \(\mathcal{F}_I\) as

\[\mathcal{N}_{IJ} = (\mathcal{F}_{I} ,\, D_{\bar{\imath}}\overline{\mathcal{F}}_I )\, \cdot\, (X^{J},\, D_{\bar{\jmath}}\overline{X}^J)^{-1} = P_{IK} \, (Q^{-1})^{K}\,_{J}\]

see e.g. Sect. 2 above Eq. (2.12) in 2110.05511. Here we defined the two matrices \(P_{IJ}\) and \(Q^{I}\,_{J}\) as

\[P_{IJ} = (\mathcal{F}_{I} ,\, D_{\bar{\imath}}\overline{\mathcal{F}}_I )_{J} \, ,\quad Q^{I}\,_{J} = (X^{I},\, D_{\bar{\jmath}}\overline{X}^I)_{J}\, ,\]

see Eq. (3.5) in 2310.06040.

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – Value of the gauge kinetic matrix \(\mathcal{N}\) from periods.

Return type:

Array

gauge_kinetic_matrix_prepotential(XPer, cXPer, conj=False)#

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

Details

In the presence of a prepotential \(F\), the gauge kinetic matrix \(\mathcal{N}\) can be computed as follows (see e.g. Eq. (2.12) in 2110.05511)

\[\mathcal{N}_{IJ}= \overline{F}_{IJ} + +2\text{i}\, \dfrac{\text{Im}(F_{I L})X^{L} \, \text{Im}(F_{J K})X^{K}}{X^{M}\text{Im}(F_{MN})X^{N}}\]

where \(F_{IJ}=\partial_{X^{I}}\partial_{X^{J}}F\) is the second derivative of the prepotential \(F\) with respect to the periods \(X^I\).

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – Value of the gauge kinetic matrix \(\mathcal{N}\) from prepotential.

Return type:

Array

grad_kahler_potential_per(XPer, cXPer, conj=False)#

Computes derivatives \(\partial_{X^I}K\) of the Kähler potential \(K\) with respect to the periods :mathh`X^I`.

Warning

If we set conj=True, we compute the anti-holomorphic derivative \(\partial_{\overline{X}^I}K\) instead!

Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

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

Returns:

Array – First derivative \(\partial_{X^I}K\) of the Kähler potential \(K\).

Return type:

Array

grad_period_vector_per(XPer, conj=False)#

Computes derivatives \(\partial_{X^I}\Pi\) of the period vector \(\Pi\) with respect to the periods :mathh`X^I`.

Parameters:

  • XPer (Array) – Values of periods.

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

Returns:

Array – Derivatives \(\partial_{X^I}\Pi\) of the period vector \(\Pi\).

Return type:

Array

kahler_potential_per(XPer, cXPer)#

Computes the value of the Kähler potential \(K\) as a function of the periods \(X^I\).

Details

The Kähler potential on complex structure moduli space is defined as

\[K_{cs} = -\text{ln}(\tilde{\mathcal{V}})\]

in terms of the mirror CY volume \(\tilde{\mathcal{V}}\)

\[\tilde{\mathcal{V}} = -\text{i}\, \Pi^\dagger\cdot \Sigma\cdot\Pi\, .\]
Parameters:

  • XPer (Array) – Values of periods.

  • cXPer (Array) – Complex conjugate values of periods.

Returns:

Array – Value of the Kähler potential \(K\).

Return type:

complex

property lcs_tree#

Description: The underlying lcs_tree object holding the CY model data.

Parameters:

None

Returns:

lcs_tree – The lcs_tree object holding the CY model data.

period_vector_per(XPer, conj=False)#

Computes the period vector \(\Pi\) in terms of the periods \(X^I\).

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 \(\mathcal{F}_I=\mathcal{F}_I(Z)\) are then determined by a prepotential \(F(Z)\) through

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

  • XPer (Array) – Values of periods.

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

Returns:

Array – Period vector \(\Pi\).

Return type:

Array

prepot_grad_grad_per(XPer, conj=False)#

Computes the second holomorphic derivatives \(F_{IJ}=\partial_{X^I}\partial_{X^J}F\) of the prepotential \(F\) with respect to the periods \(X^I\).

Note

This matrix is used among others to compute the gauge kinetic matrix entering the ISD condition for SUSY flux vacua, see gauge_kinetic_matrix_prepotential().

Parameters:

  • XPer (Array) – Values of periods.

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

Returns:

ddF (jax array of arrays) – \((h^{1,2}+1)\times(h^{1,2}+1)\)-matrix with entries corresponding to 2nd derivatives of \(F\).

Return type:

Array

See also: gauge_kinetic_matrix().

prepot_grad_per(XPer, conj=False)#

Computes the holomorphic derivatives \(F_{I}=\partial_{X^I}F\) of the prepotential \(F\) with respect to the periods \(X^I\).

Parameters:

  • XPer (Array) – Values of periods.

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

Returns:

Array – Holomorphic derivatives \(F_{I}=\partial_{X^I}F\) of the prepotential \(F\).

Return type:

Array

prepot_per(XPer, conj=False)#

Computes the prepotential \(F\) in terms of the periods \(X^I\).

Details

At Large Complex Structure (LCS), the prepotential can be computed from mirror symmetry as

\[F_{\mathrm{LCS}}(X) = F_{\mathrm{poly}}(X) + F_{\mathrm{inst}}(X)\]

where

\[F_{\mathrm{poly}}(X) = -\frac{1}{6X^0}\widetilde{\kappa}_{ijk}X^iX^jX^k+\frac{1}{2}a_{ij}X^iX^j +b_{i}X^i\, X^0 + \dfrac{\text{i}}{2}\tilde{\xi}\, (X^0)^2\, ,\]

and

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

The former is computed via F_LCS_poly_per(), while the latter in F_inst_per(). This limit in moduli space is implemented in F_LCS_per().

Warning

The moduli space limit around which the prepotential is computed is set by the global parameter self.periods.limit. Currently, only the limits "LCS", "coniLCS_series", and "coniLCS_bulk" are supported.

Note

This function is used to compute the gauge kinetic matrix \(\mathcal{N}_{IJ}\) of second derivatives of the prepotential which is used to check the ISD-condition for flux vacua.

Parameters:

  • XPer (Array) – Values of periods.

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

Returns:

complex – Value of the prepotential \(F\).

Return type:

complex

See also: F_LCS_per()

See also: prepot_grad_grad_per()

See also: gauge_kinetic_matrix()

sigma()#

Returns the symplectic matrix

\[\begin{split}\Sigma=\left (\begin{array}{cc}0 & 1\\ -1&0\end{array}\right )\, .\end{split}\]
Parameters:

None

Returns:

Array – Symplectic pairing matrix.

Return type:

Array