Fubini study

cymyc.fubini_study

Computation of the Fubini-Study metric - the unique \(U(n+1)\) Riemannian metric on \(\mathbb{P}^n\) + associated functions.

fubini_study_metric(p: Float[Array, i], normalization: Complex = jax.lax.complex(1.0, 0.0), cdtype: DTypeLike = np.complex64)

Returns Fubini-Study metric in \(\mathbb{P}^n\) evaluated at p in inhomogeneous coordinates.

Parameters:

Name	Type	Description	Default
p	Float[Array, i]	2*n real inhomogeneous coords at which metric is evaluated. Shape [i].	required

Returns:

Name	Type	Description
g_FS	array_like	Hermitian metric in local coordinates, \(g_{\mu \bar{\nu}}\). Shape [i,j].

fubini_study_metric_homo(p: Float[Array, i], normalization: Complex = jax.lax.complex(1.0, 0.0), cdtype: DTypeLike = np.complex64)

Returns Fubini-Study metric in \(\mathbb{P}^n\) evaluated at p in homogeneous coordinates.

Parameters:

Name	Type	Description	Default
p	Float[Array, i]	2*(n+1) real homogeneous coords at which metric is evaluated. Shape [i].	required

Returns:

Name	Type	Description
g_FS	array_like	Hermitian metric in local coordinates, \(g_{\mu \bar{\nu}}\). Shape [i,j]. Warning Note the returned metric is expressed in homogeneous coordinates and will not be of full rank.

fubini_study_metric_homo_pb(p: Float[Array, i], dQdz_info: tuple, cy_dim: int, normalization: Complex = jax.lax.complex(1.0, 0.0), ambient_out: bool = False, cdtype: DTypeLike = np.complex64)

Returns FS metric on hypersurfaces \(X\) immersed in \(\mathbb{P}^n\) evaluated at p in homogeneous coordinates, i.e. \([x_1 : x_2: \cdots : x_{n+1}]\). This is the ambient FS metric in CP^n pulled back by the inclusion map: \(\iota: X \hookrightarrow \mathbb{P}^n\).

Parameters:

Name	Type	Description	Default
p	array_like	2*(n+1) real homogeneous coords at which metric is evaluated. Shape [i].	required

Returns:

Name	Type	Description
g_FS_pb	array_like	Hermitian metric pulled back to \(X\) in local coordinates, \(g_{\mu \bar{\nu}}\). Shape [i,j].

fubini_study_metric_homo_pb_cicy(p: Float[Array, i], pullbacks: Complex[Array, 'cy_dim i'], n_coords: int, ambient: tuple, k_moduli: Array = None, cdtype: DTypeLike = np.complex64)

Returns ambient Fubini-Study metric for a CICY in product of projective spaces pulled back by the inclusion map \(\iota: X \hookrightarrow \mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_K}\).

Parameters:

Name	Type	Description	Default
p	Float[Array, i]	2*(n+1) real homogeneous coords at which metric is evaluated. Shape [i].	required
pullbacks	array_like	Pullback tensor from ambient to projective variety.	required
n_coords	int	Dimension of ambient combined projective space.	required
ambient	tuple	Dimension of each projective space factor.	required
k_moduli	(array_like, optinal)	Kahler moduli for each projective space factor.	None

Returns:

Name	Type	Description
g_FS_pb	array_like	Hermitian metric pulled back to \(X\) in local coordinates, \(g_{\mu \bar{\nu}}\). Shape [i,j]. Warning Note the returned metric is expressed in homogeneous coordinates and will not be of full rank.

fubini_study_inverse(p: Float[Array, i], cdtype: DTypeLike = np.complex64)

Returns analytic inverse in inhomogeneous coords using the Woodbury matrix identity.

Parameters:

Name	Type	Description	Default
p	array_like	2*n real inhomogeneous coords at which metric is evaluated. Shape [i].	required

Returns:

Name	Type	Description
g_FS_inv	array_like	Inverse of Hermitian metric in inhomogeneous coordinates. Shape [i,j]