Curvature

cymyc.curvature

Calculation of various curvature-related quantities for a general Kahler manifold \(X\). This module may also be used to compute curvature information on a real manifold without complex structure, by omitting the imaginary part of the local coordinates for \(p \in X\).

Note the following:

• Curvature quantities involve \(n\)-th order derivatives of some generic function fun. These are computed with autodiff - a good usage pattern is to make a partial closure to bind all arguments to fun except the coordinate dependence.
• These functions expect local coordinates z in a c_dim-dimensional space, with the real and imaginary parts concatenated to form a real-valued 2*c_dim vector, p = [Re(z); Im(z)].

Info

When transforming functions which compute derivatives of some fun passed as an input, you will need to either:

• In the case of jit, specify that fun is a static argument.
• Wrap fun in a jax.tree_util.Partial closure to make it compatible with Jax transformations.

See the below example.

Example

import jax
import jax.numpy as jnp
fun = lambda x: jnp.sum(jnp.cos(x))

>>> jax.jit(del_z, static_argnums=(1,))(p, fun)  # ok
>>> jax.jit(del_z)(p, jax.tree_util.Partial(fun))  # ok
>>> jax.jit(del_z)(p, fun)  # TypeError

del_z(p: Float[Array, i], fun: Callable[[Float[Array, ...]], Array], *args) -> Complex[Array, '... i']

Holomorphic derivative of a function.

Parameters:

Name	Type	Description	Default
p	array_like	2 * complex_dim real coords with float type at which fun is evaluated. Consists of the concatenation of real and imaginary parts along the last axis.	required
fun	callable	Locally defined function fun: \(\mathbb{R}^m -> \mathbb{C}^{a,b,c...}\) sending real-valued inputs to complex-valued outputs	required

Returns:

Name	Type	Description
dfun_dz	array_like	Holomorphic derivative of fun.

Notes

Computes holomorphic Wirtinger derivative, w.r.t. complex \(p = x + iy\).

\[ \frac{\partial f}{\partial z} = \frac{1}{2}\left( \frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y} \right) \]

Examples:

>>> p = jnp.ones((8,))
>>> fun = lambda x: jnp.sum(jnp.cos(x))
>>> del_z(p, fun)
Array([-0.42073548+0.42073548j, -0.42073548+0.42073548j,
       -0.42073548+0.42073548j, -0.42073548+0.42073548j], dtype=complex64)

del_bar_z(p: Float[Array, i], fun: Callable[[Float[Array, ...]], Array], *args) -> Complex[Array, '... i']

Anti-holomorphic derivative of a function.

Parameters:

Name	Type	Description	Default
p	array_like	2 * complex_dim real coords at which fun is evaluated. Shape [i].	required
fun	callable	Locally defined function fun: \(\mathbb{R}^m -> \mathbb{C}^{a,b,c...}\) sending real-valued inputs to complex-valued outputs	required

Returns:

Name	Type	Description
dfun_dz_bar	array_like	Anti-holomorphic derivative of fun.

del_z_bar_del_z(p: Float[Array, i], fun: Callable[[Float[Array, ...]], Array], *args, wide: bool = False) -> Complex[Array, '... i i']

Computes the composition \(\partial \circ \overline{\partial}\) of a given function. Note this coincides with the Hessian \(\nabla df\) on a complex manifold.

Parameters:

Name	Type	Description	Default
p	array_like	2 * complex_dim real coords at which fun is evaluated. Shape [i].	required
fun	callable	Locally defined function fun: \(\mathbb{R}^m -> \mathbb{C}^{a,b,c...}\) sending real-valued inputs to complex-valued outputs	required
wide	bool	Flag to use reverse-mode autodiff if function is wide, i.e. if the output of fun is a scalar.	False

Returns:

Name	Type	Description
dfun_dz_bar_dz	array_like	\(\bar{\partial} \partial f\). Shape [..., \mu, \bar{\nu}]. Note holomorphic index comes first.

christoffel_symbols_kahler(p: Float[Array, i], metric_fn: Callable[[Array], Array], pullbacks: Complex[Array, 'cy_dim i'] = None) -> Complex[Array, 'j j j']

Returns holomorphic Levi-Civita connection coefficients, with optional support for variety \(\iota: X \hookrightarrow \mathbb{P}^n\). Schematically, if \(g\) is the metric tensor, $$ \Gamma \sim g^{-1} \cdot \partial g~. $$

Parameters:

Name	Type	Description	Default
p	array_like	2 * complex_dim real coords at which fun is evaluated. Shape [i].	required
metric_fn	callable	Function representing metric tensor in local coordinates \(g : \mathbb{R}^m -> \mathbb{C}^{a,b...}\).	required

Returns:

Name	Type	Description
gamma_holo	array_like	Holomorphic Christoffel symbols of the Kahler metric. \(\Gamma^{\lambda}_{\mu \nu}\). Shape [...,k,i,j], symmetric in (i,j). The Kahler conditions imply \(\Gamma^{\lambda}_{\mu \nu}\) and its conjugate are the only nonzero connection coeffs.

Other Parameters:

Name	Type	Description
pullbacks	array_like	Pulllback matrices from ambient to projective variety. If supplied, computes Christoffels on the variety.

riemann_tensor_kahler(p: Float[Array, i], metric_fn: Callable[[Array], Array], pullbacks: Complex[Array, 'cy_dim i'] = None, return_aux: bool = False) -> Complex[Array, 'j j j j'] | Sequence[Array]

Returns Riemann curvature tensor on a Kähler manifold, with optional support for variety \(\iota: X \hookrightarrow \mathbb{P}^n\). This can also be used to construct the Riemann tensor on a real manifold by omission of the imaginary part of p. Schematically, $$ \textsf{Riem} \sim \partial \Gamma + \Gamma \Gamma ~. $$

Parameters:

Name	Type	Description	Default
p	array_like	2 * complex_dim real coords at which fun is evaluated. Shape [i].	required
metric_fn	callable	Function representing metric tensor in local coordinates \(g : \mathbb{R}^m -> \mathbb{C}^{a,b...}\). Warning This function explicitly instantiates the complex Hessian of metric_fn - this may result in memory issues if vmap-ing over a large batch. Try reducing the batch size or reducing the complexity of metric_fn if memory-constrained.	required

Returns:

Name	Type	Description
riemann	array_like	(1,3) Riemann tensor corresponding to the Kahler connection \(R^{\kappa}_{\lambda \mu \overline{\nu}}\). See page 335, (8.97) of Nakahara.

Other Parameters:

Name	Type	Description
pullbacks	array_like	Pulllback matrices from ambient to projective variety. If supplied, computes Riemann tensor on the variety.

ricci_tensor_kahler(p: Float[Array, i], metric_fn: Callable[[Array], Array], pullbacks: Complex[Array, 'cy_dim i'] = None) -> Complex[Array, 'j j']

Returns Ricci curvature tensor on a Kähler manifold, with optional support for variety \(\iota: X \hookrightarrow \mathbb{P}^n\).

Parameters:

Name	Type	Description	Default
p	array_like	2 * complex_dim real coords at which fun is evaluated. Shape [i].	required
metric_fn	callable	Function representing metric tensor in local coordinates \(g : \mathbb{R}^m -> \mathbb{C}^{a,b...}\).	required

Returns:

Name	Type	Description
ricci	array_like	(0,2) Ricci tensor corresponding to the Kahler connection $ R_{\mu \bar{\nu}}$.

Other Parameters:

Name	Type	Description
pullbacks	array_like	Pulllback matrices from ambient to projective variety. If supplied, computes Ricci tensor on the variety.

See Also

ricci_form_kahler : Computes Ricci form as \(\partial_{\mu} \overline{\partial}_{\overline{\nu}} \log g\).

ricci_form_kahler(p: Float[Array, i], metric_fn: Callable[[Array], Array], pullbacks: Complex[Array, 'cy_dim i'] = None) -> Complex[Array, 'j j']

Returns Ricci form \(\rho\) on a Kähler manifold, with support for variety \(\iota: X \hookrightarrow \mathbb{P}^n\). Componentwise, \(\rho_{\mu\bar{\nu}} = i R_{\mu \bar{\nu}}\), and $$ \rho = \partial \overline{\partial} \log \det g~. $$

Parameters:

Name	Type	Description	Default
p	array_like	2 * complex_dim real coords at which fun is evaluated. Shape [i].	required
metric_fn	callable	Function representing metric tensor in local coordinates \(g : \mathbb{R}^m -> \mathbb{C}^{a,b...}\).	required

Returns:

Name	Type	Description
ricci	array_like	(1,1) Ricci form corresponding to the Kahler connection $ R_{\mu \bar{\nu}}$.

Other Parameters:

Name	Type	Description
pullbacks	array_like	Pulllback matrices from ambient to projective variety. If supplied, computes Ricci tensor on the variety.

ricci_scalar(p: Float[Array, i], metric_fn: Callable[[Array], Array], pullbacks: Complex[Array, 'cy_dim i'] = None) -> Complex[Array, '']

Returns Ricci scalar on a Kähler manifold, with support for variety \(\iota: X \hookrightarrow \mathbb{P}^n\).

Parameters:

Name	Type	Description	Default
p	array_like	2 * complex_dim real coords at which fun is evaluated. Shape [i].	required
metric_fn	callable	Function representing metric tensor in local coordinates \(g : \mathbb{R}^m -> \mathbb{C}^{a,b...}\).	required

Returns:

Name	Type	Description
R	array_like	Ricci scalar, \(R = g^{\mu \bar{\nu}}R_{\mu \bar{\nu}}\). Shape [...]

Other Parameters:

Name	Type	Description
pullbacks	array_like	Pulllback matrices from ambient to projective variety. If supplied, computes Ricci tensor on the variety.