Harmonic

cymyc.approx.harmonic.HarmonicFull

Bases: Harmonic

__init__(cy_dim: int, monomials: List[np.array], ambient: ArrayLike, deformations: List[Callable], dQdz_monomials: List[np.array], dQdz_coeffs: List[np.array], metric_fn: Callable, pb_fn: Callable, coeff_fn: Callable, psi: float)

Approximation of harmonic one-forms for Calabi-Yaus with an arbitrary number of complex structure moduli. Note the methods vectorise over all possible complex structure deformations.

Parameters:

Name	Type	Description	Default
cy_dim	int	Dimension of Calabi-Yau manifold.	required
monomials	List[array]	List of defining monomials.	required
ambient	array_like	Dimensions of the ambient space factors.	required
deformations	List[Callable]	List of functions representing complex structure deformations.	required
dQdz_monomials	List[array]	List of monomials corresponding to polynomial Jacobian \(dQ/dz\).	required
dQdz_coeffs	List[array]	List of coefficients corresponding to polynomial Jacobian \(dQ/dz\).	required
metric_fn	Callable	Function representing metric tensor in local coordinates \(g : \mathbb{R}^m -> \mathbb{C}^{a,b...}\).	required
pb_fn	Callable	Function computing pullback matrices from ambient space to projective variety.	required
coeff_fn	Callable	Function returning polynomial coefficients at given point in moduli space.	required
psi	float	Complex structure parameter psi.	required

See also

cymyc.moduli.wp.WP

__call__(p: Float[Array, i], params: Mapping[str, Array]) -> Complex[Array, 'h_21 cy_dim cy_dim']

Constructs all harmonic representatives by \(\overline{\partial}\)-exact correction to a representative from the \(H^{0,1}\) Dolbeault cohomology, \(\xi\);

\[ \eta = \xi + \overline{\partial} \theta~. \]

Here \(\theta\) is taken to be a linear combination of a basis of sections of \(V\).

Parameters:

Name	Type	Description	Default
p	Float[Array, i]	2 * complex_dim real coords on \(X\).	required
params	Mapping[str, Array]	Model parameters stored as a dictionary - keys are the module names registered upon initialisation and values are the parameter values.	required

codifferential_eta(p: Float[Array, i], pullbacks: Complex[Array, 'cy_dim i'], g_pred: Complex[Array, 'dim dim'], params: Mapping[str, Array]) -> Complex[Array, 'h_21 cy_dim']

Computes codifferential of \(\alpha \in H^{(0,1)}(X; T_X)\) with respect to the given metric. This is a smooth section of the holomorphic tangent bundle, \(\bar{\partial}^{\dagger} \eta \in \Gamma(T_X)\).

Parameters:

Name	Type	Description	Default
p	Float[Array, i]	2 * complex_dim real coords on \(X\).	required
pullbacks	Complex[Array, 'cy_dim i']	Pullback matrices from ambient to projective variety.	required
g_pred	Complex[Array, "dim dim"]	Predicted metric \(g_{\mu \overline{\nu}}\) in local coordinates.	required
params	Mapping[str, Array]	Model parameters stored as a dictionary - keys are the module names registered upon initialisation and values are the parameter values.	required

Returns:

Name	Type	Description
codiff	Complex[Array, 'h_21 cy_dim']	Section of tangent bundle.

wp_metric_harmonic(data: Tuple[ArrayLike, ArrayLike, ArrayLike], eta: Complex[Array, 'h_21 cy_dim cy_dim']) -> Complex[Array, 'h_21 h_21']

Takes in harmonic one-form eta and forms interior product with the holomorphic form \(\Omega\), to yield Weil-Petersson metric via cup product,

\[ \mathcal{G}_{a\overline{b}} \propto \int_X \iota_{\eta_a} \Omega \wedge \overline{\iota_{\eta_b} \Omega}~. \]

Parameters:

Name	Type	Description	Default
data	Tuple[ArrayLike, ArrayLike, ArrayLike]	Tuple containing input points, integration weights and canonical volume form \(\Omega \wedge \bar{\Omega}\) in local coords.	required
eta	Complex[Array, 'h_21 cy_dim cy_dim']	Harmonic representative \(\eta\).	required

Returns:

Type	Description
Complex[Array, 'h_21 h_21']	Weil-Petersson metric.

inner_product_Hodge(data: Tuple[ArrayLike, ArrayLike, ArrayLike], eta: Complex[Array, 'h_21 cy_dim cy_dim'], g_pred: Complex[Array, 'cy_dim cy_dim']) -> Complex[Array, 'h_21 h_21'] staticmethod

Hodge star inner product between harmonic forms eta parameterising moduli tangent directions to yield Weil-Petersson metric,

\[ \mathcal{G}_{a\overline{b}} \propto \int_X \eta_a \wedge \overline{\star}_g \eta_b~. \]

This should agree with the cup product calculation provided the Ricci-flat metric is used.

Parameters:

Name	Type	Description	Default
data	Tuple[ArrayLike, ArrayLike, ArrayLike]	Tuple containing input points, integration weights and canonical volume form \(\Omega \wedge \bar{\Omega}\) in local coords.	required
eta	Complex[Array, 'h_21 cy_dim cy_dim']	Harmonic representative \(\eta\).	required
g_pred	Complex[Array, 'cy_dim cy_dim']	Approximate Ricci-flat metric in local coords.	required

Returns:

Type	Description
Complex[Array, 'h_21 h_21']	Weil-Petersson metric.