Chern gauss bonnet

cymyc.chern_gauss_bonnet

Calculation of topological invariants (Chern classes, Euler characteristic) for a general Kahler manifold. Note some functions may specialise to the case of a projective variety. Note the following:

• As with curvature, these functions involve \(n\)-th order derivatives of some 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)].
• The Euler characteristic is defined up to an integer factor from the normalisation of the volume form. The canonical choice is to normalise according to the intersection number computation.

riem_real(p: Float[Array, 2 * i], metric_fn: Callable[[Array], Array], *args: Sequence, pullbacks: Optional[Float[Array, 'dim i']] = None, return_down: bool = False) -> Float[Array, '2*dim 2*dim 2*dim 2*dim']

Viewing the \(n\)-dim Kähler manifold as a real \((2n)\)-dimensional manifold, computes the real \((2n)\)-dimensional Riemann tensor corresponding to the given metric tensor.

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
metric_fn	callable	Function representing metric tensor in local coordinates.	required
*args	tuple	Additional arguments to pass to metric_fn.	()
pullbacks	array_like	Pullback matrix from ambient to projective variety. If supplied, computes Riemann tensor on the variety, by default None.	None
return_down	bool	If True, return the Riemann tensor with all indices lowered, by default False.	False

Returns:

Type	Description
array_like	The real (2n)-dimensional Riemann tensor. If return_down is True, returns the tensor with all indices lowered.

chern1(riem: Complex[Array, 'dim dim dim dim']) -> Complex[Array, 'dim dim']

Computes the first Chern class. Let \(\mathcal{R} \in \Omega^2_X\left(\text{End}(T_X)\right)\) be the curvature two-form on \(X\), then $$ c_1 \propto \textsf{Tr}\mathcal{R}~.$$

Parameters:

Name	Type	Description	Default
riem	array_like	(1,3) Riemann tensor corresponding to the Kahler connection \(R^{\kappa}_{\lambda \mu \bar{\nu}}\).	required

Returns:

Type	Description
array_like	First Chern form.

chern2(riem: Complex[Array, 'dim dim dim dim']) -> Complex[Array, 'dim dim dim dim']

Computes the second Chern class, $$ c_2 \propto \left(\textsf{Tr}\mathcal{R}^2 - \textsf{Tr}\mathcal{R} \wedge \textsf{Tr}\mathcal{R} \right)~.$$

Parameters:

Name	Type	Description	Default
riem	array_like	(1,3) Riemann tensor corresponding to the Kahler connection \(R^{\kappa}_{\lambda \mu \bar{\nu}}\).	required

Returns:

Name	Type	Description
c2	array_like	Second Chern form.

chern3(riem: Complex[Array, 'dim dim dim dim']) -> Complex[Array, '']

Computes the third Chern class,

\[ c_3 \propto c_1 \wedge c_2 + c_1 \wedge \textsf{Tr} \mathcal{R}^2 - \textsf{Tr}\mathcal{R}^3~. \]

Parameters:

Name	Type	Description	Default
riem	array_like	(1,3) Riemann tensor corresponding to the Kahler connection \(R^{\kappa}_{\lambda \mu \bar{\nu}}\).	required

Returns:

Name	Type	Description
c3	array_like	The third Chern form, expressed as the coefficient of the complex wedgey part in standard form: \(dz^1 \wedge d\bar{z}^1 \wedge \cdots \wedge dz^n \wedge d\bar{z}^n\).

euler_characteristic(data: Tuple[Float[Array, 2 * i], Float[Array, ''], Float[Array, '']], pullbacks: Optional[Float[Array, 'dim i']], metric_fn: Callable[[Array], Array], cy_dim: int = 3) -> Float[Array, '']

Computes the Euler characteristic from the Pfaffian of the curvature two-form \(\mathcal{R} \in \Omega^2_X(T_X)\), with support for Calabi-Yau \(n\)-folds for \(n=1,2,3\).

\[\chi = \frac{1}{(2\pi)^n} \int_X \textsf{Pf}(\mathcal{R})~.\]

Parameters:

Name	Type	Description	Default
data	tuple	A tuple containing coordinates p, weights, and the volume form dVol_Omega.	required
pullbacks	array_like	Pullback matrix from ambient to projective variety. If supplied, computes the Euler characteristic on the variety, by default None.	required
metric_fn	callable	Function representing metric tensor in local coordinates.	required
cy_dim	int	Complex dimension of Calabi-Yau, default 3.	3

Returns:

Name	Type	Description
chi	array - like	The Euler characteristic.

euler_characteristic_form(data: Tuple[Float[Array, 2 * i], Float[Array, ''], Float[Array, '']], pullbacks: Optional[Float[Array, 'dim i']], metric_fn: Callable[[Array], Array], cy_dim: int = 3) -> Float[Array, '']

Computes the Euler characteristic via integration of the third Chern class over \(X\) for a Calabi-Yau threefold.

\[\chi = \int_X c_3~.\]

Parameters:

Name	Type	Description	Default
data	tuple	A tuple containing coordinates p, weights, and the volume form dVol_Omega.	required
pullbacks	array_like	Pullback matrix from ambient to projective variety. If supplied, computes the Euler characteristic on the variety, by default None.	required
metric_fn	callable	Function representing metric tensor in local coordinates.	required
cy_dim	int	Complex dimension of Calabi-Yau.	3

Returns:

Name	Type	Description
chi	array - like	The Euler characteristic.