Characteristic Classes

A characteristic class \(\kappa\) is a natural transformation that associates to each vector bundle \(E \to M\) a cohomology class \(\kappa(E) \in H^*(M;R)\) such that for any continuous map \(f\colon N \to M\) from another topological manifold \(N\), the naturality condition is satisfied:

\[f^*\kappa(E) = \kappa(f^* E) \in H^*(N;R)\]

Roughly speaking, characteristic classes measure the non-triviality of vector bundles.

One way to obtain and compute characteristic classes in the de Rham cohomology with coefficients in the ring \(\CC\) is via the so-called Chern-Weil theory using the curvature of a differentiable vector bundle.

For that let \(\nabla\) be a connection on \(E\), \(e\) a local frame on \(E\) and \(\Omega\) be the corresponding curvature matrix (see: curvature_form()).

Namely, if \(P: \mathrm{Mat}_{n \times n}(\CC) \to \CC\) is an invariant polynomial, the object

\[\left[ P \left( \Omega \right) \right] \in H^{2*}_{\mathrm{dR}}(M, \CC)\]

is well-defined, independent of the choice of \(\nabla\) (the proof can be found in [Roe1988] pp. 31) and fulfills the naturality condition. This is the foundation of the Chern-Weil theory and therefore the following definitions.

Note

This documentation is rich of examples, but sparse in explanations. Please consult the references for more details.

AUTHORS:

  • Michael Jung (2019) : initial version

REFERENCES:

Contents

We consider the following three types of classes:

Additive Classes

In the complex case, let \(f\) be a holomorphic function around zero. Then we call

\[\left[\mathrm{tr}\left( f\left( \frac{\Omega}{2 \pi i} \right) \right)\right] \in H^{2*}_{\mathrm{dR}}(M, \CC)\]

the additive characteristic class associated to \(f\) of the complex vector bundle \(E\).

Important and predefined additive classes are:

  • Chern Character with \(f(x) = \exp(x)\)

In the real case, let \(g\) be a holomorphic function around zero with \(g(0)=0\). Then we call

\[\left[\mathrm{tr}\left( \frac{1}{2} g\left( -\frac{\Omega^2}{4 \pi^2} \right) \right)\right] \in H^{4*}_{\mathrm{dR}}(M, \CC)\]

the additive characteristic class associated to \(g\) of the real vector bundle \(E\).

EXAMPLES:

Consider the Chern character on some 2-dimensional spacetime:

sage: M = Manifold(2, 'M', structure='Lorentzian')
sage: X.<t,x> = M.chart()
sage: E = M.vector_bundle(1, 'E', field='complex'); E
Differentiable complex vector bundle E -> M of rank 1 over the base space
 2-dimensional Lorentzian manifold M
sage: e = E.local_frame('e')

Let us define the connection \(\nabla^E\) in terms of an electro-magnetic potential \(A(t)\):

sage: nab = E.bundle_connection('nabla^E', latex_name=r'\nabla^E')
sage: omega = M.one_form(name='omega')
sage: A = function('A')
sage: omega[1] = I*A(t)
sage: omega.display()
omega = I*A(t) dx
sage: nab.set_connection_form(0, 0, omega)

The Chern character is then given by:

sage: ch = E.characteristic_class('ChernChar'); ch
Characteristic class ch of additive type associated to e^x on the
 Differentiable complex vector bundle E -> M of rank 1 over the base space
 2-dimensional Lorentzian manifold M
sage: ch_form = ch.get_form(nab); ch_form.display_expansion()
ch(E, nabla^E) = [1] + [0] + [1/2*d(A)/dt/pi dt/\dx]

Multiplicative Classes

In the complex case, let \(f\) be a holomorphic function around zero. Then we call

\[\left[\det\left( f\left( \frac{\Omega}{2 \pi i} \right) \right)\right] \in H^{2*}_{\mathrm{dR}}(M, \CC)\]

the multiplicative characteristic class associated to \(f\) of the complex vector bundle \(E\).

Important and predefined multiplicative classes on complex vector bundles are:

  • Chern class with \(f(x) = 1+x\)
  • Todd class with \(f(x) = \frac{x}{1-\exp(-x)}\)

In the real case, let \(g\) be a holomorphic function around zero with \(g(0)=1\). Then we call

\[\left[\det\left( \sqrt{ g \left( -\frac{\Omega^2}{4 \pi^2} \right) } \right) \right] \in H^{4*}_{\mathrm{dR}}(M, \CC)\]

the multiplicative characteristic class associated to \(g\) on the real vector bundle \(E\).

Important and predefined multiplicative classes on real vector bundles are:

  • Pontryagin class with \(g(x) = 1+x\)
  • \(\hat{A}\) class with \(g(x) = \frac{\sqrt{x}/2}{\sinh(\sqrt{x}/2)}\)
  • Hirzebruch class with \(g(x) = \frac{\sqrt{x}}{\tanh(\sqrt{x})}\)

EXAMPLES:

We consider the Chern class of the tautological line bundle \(\gamma^1\) over \(\CC\mathbf{P}^1\):

sage: M = Manifold(2, 'CP^1', start_index=1)
sage: U = M.open_subset('U')
sage: c_cart.<x,y> = U.chart() # homogeneous coordinates in real terms
sage: c_comp.<z, zbar> = U.chart(r'z:z zbar:\bar{z}') # complexification
sage: cart_to_comp = c_cart.transition_map(c_comp, (x+I*y, x-I*y))
sage: comp_to_cart = cart_to_comp.inverse()
sage: E = M.vector_bundle(1, 'gamma^1', field='complex')
sage: e = E.local_frame('e', domain=U)

To apply the Chern-Weil approach, we need a bundle connection in terms of a connection one form. To achieve this, we take the connection induced from the hermitian metric on the trivial bundle \(\CC^2 \times \CC\mathbf{P}^1 \supset \gamma^1\). In this the frame \(e\) corresponds to the section \([z:1] \mapsto (z,1)\) and its magnitude-squared is given by \(1+|z|^2\):

sage: nab = E.bundle_connection('nabla')
sage: omega = U.one_form(name='omega')
sage: omega[c_comp.frame(),1,c_comp] = zbar/(1+z*zbar)
sage: nab.set_connection_form(1, 1, omega, frame=e)

Now, the Chern class can be constructed:

sage: c = E.characteristic_class('Chern'); c
Characteristic class c of multiplicative type associated to x + 1 on the
 Differentiable complex vector bundle gamma^1 -> CP^1 of rank 1 over the
 base space 2-dimensional differentiable manifold CP^1
sage: c_form = c.get_form(nab)
sage: c_form.display_expansion(c_comp.frame(), chart=c_comp)
c(gamma^1, nabla) = [1] + [0] + [1/2*I/(pi + pi*z^2*zbar^2 + 2*pi*z*zbar) dz/\dzbar]

Since \(U\) and \(\CC\mathbf{P}^1\) differ only by a point and therefore a null set, it is enough to integrate the top form over the domain \(U\):

sage: integrate(integrate(c_form[2][[1,2]].expr(c_cart), x, -infinity, infinity).full_simplify(),
....:           y, -infinity, infinity)
1

The result shows that \(c_1(\gamma^1)\) generates the second integer cohomology of \(\CC\mathbf{P}^1\).

Pfaffian Classes

Usually, there is no such thing as “Pfaffian classes” in literature. However, using the matrix’ Pfaffian and inspired by the aforementioned definitions, such classes can be defined as follows.

Let \(E\) be a real vector bundle of rank \(2n\) and \(f\) an odd real function being analytic at zero. Furthermore, let \(\Omega\) be skew-symmetric, which certainly will be true if \(\nabla\) is metric and \(e\) is orthonormal. Then we call

\[\left[\mathrm{Pf}\left( f\left( \frac{\Omega}{2 \pi} \right) \right)\right] \in H^{2n*}(M,\RR)\]

the Pfaffian class associated to f.

The most important Pfaffian class is the Euler class which is simply given by \(f(x)=x\).

EXAMPLES:

We consider the Euler class of \(S^2\):

sage: M = Manifold(2, name='S2', latex_name=r'S^2', start_index=1)
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # M is the union of U and V
sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
sage: xy_to_uv = c_xy.transition_map(c_uv,
....:                                (x/(x^2+y^2), y/(x^2+y^2)),
....:                               intersection_name='W',
....:                               restrictions1= x^2+y^2!=0,
....:                               restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: eU = c_xy.frame() ; eV = c_uv.frame()
sage: TM = M.tangent_bundle()
sage: e_class = TM.characteristic_class('Euler'); e_class
Characteristic class e of Pfaffian type associated to x on the Tangent
 bundle TS2 over the 2-dimensional differentiable manifold S2

To compute a particular representative of the Euler class, we need to determine a connection:

sage: g = M.metric('g') # standard metric on S2
sage: g[eU,1,1], g[eU,2,2] = 4/(1+x^2+y^2)^2, 4/(1+x^2+y^2)^2
sage: g[eV,1,1], g[eV,2,2] = 4/(1+u^2+v^2)^2, 4/(1+u^2+v^2)^2
sage: nab = g.connection()

In case of the the Euler class, skew-symmetric curvature matrices are needed for the Pfaffian. For this, we need to define the curvature matrices by hand:

sage: cmatrix_U = [[nab.curvature_form(i,j,eU) for j in TM.irange()]
....:               for i in TM.irange()]
sage: cmatrix_V = [[nab.curvature_form(i,j,eV) for j in TM.irange()]
....:               for i in TM.irange()]

Fortunately, both curvature matrices are already skew-symmetric:

sage: for i in range(TM.rank()):
....:    for j in range(TM.rank()):
....:        print(cmatrix_U[i][j].display())
curvature (1,1) of connection nabla_g w.r.t. Coordinate frame
 (U, (d/dx,d/dy)) = 0
curvature (1,2) of connection nabla_g w.r.t. Coordinate frame
 (U, (d/dx,d/dy)) = 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx/\dy
curvature (2,1) of connection nabla_g w.r.t. Coordinate frame
 (U, (d/dx,d/dy)) = -4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx/\dy
curvature (2,2) of connection nabla_g w.r.t. Coordinate frame
 (U, (d/dx,d/dy)) = 0
sage: for i in range(TM.rank()):
....:    for j in range(TM.rank()):
....:        print(cmatrix_V[i][j].display())
curvature (1,1) of connection nabla_g w.r.t. Coordinate frame
 (V, (d/du,d/dv)) = 0
curvature (1,2) of connection nabla_g w.r.t. Coordinate frame
 (V, (d/du,d/dv)) = 4/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du/\dv
curvature (2,1) of connection nabla_g w.r.t. Coordinate frame
 (V, (d/du,d/dv)) = -4/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du/\dv
curvature (2,2) of connection nabla_g w.r.t. Coordinate frame
 (V, (d/du,d/dv)) = 0

Now the representative of the Euler class with respect to the connection \(\nabla_g\) induced by the standard metric can be computed:

sage: cmatrices = {eU: cmatrix_U, eV: cmatrix_V}
sage: e_class_form = e_class.get_form(nab, cmatrices)
sage: e_class_form.display_expansion()
e(TS2, nabla_g) = [0] + [0] + [2/(pi + pi*x^4 + pi*y^4 + 2*pi*x^2 +
 2*(pi + pi*x^2)*y^2) dx/\dy]

Let us check whether this form represents the Euler class correctly:

sage: integrate(integrate(e_class_form[2][[1,2]].expr(), x, -infinity, infinity).simplify_full(),
....:           y, -infinity, infinity)
2

As we can see, the integral coincides with the Euler characteristic of \(S^2\) so that our form actually represents the Euler class appropriately.

class sage.manifolds.differentiable.characteristic_class.CharacteristicClass(vbundle, func, class_type='multiplicative', name=None, latex_name=None)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.sage_object.SageObject

An instance of this class represents a characteristic class on some differentiable vector bundle over the field \(\RR\) or \(\CC\).

INPUT:

  • vbundle – vector bundle on which the characteristic class should be defined
  • func – symbolic expression representing the function to which self should be associated to
  • class_type – (default: 'multiplicative') class type of the characteristic class; at this stage, the following options are possible:
    • 'multiplicative' – returns a class of multiplicative type, using the determinant
    • 'additive' – returns a class of additive type, using the trace
    • 'Pfaffian' – returns a class of Pfaffian type, using the Pfaffian
  • name – string representation given to the characteristic class
  • latex_name – (default: None) LaTeX name given to the characteristic class

EXAMPLES:

Get characteristic classes using predefined ones:

sage: M = Manifold(4, 'M')
sage: TM = M.tangent_bundle()
sage: TM.characteristic_class('Pontryagin')
Characteristic class p of multiplicative type associated to x + 1 on the
 Tangent bundle TM over the 4-dimensional differentiable manifold M
sage: TM.characteristic_class('Hirzebruch')
Characteristic class L of multiplicative type associated to
 sqrt(x)/tanh(sqrt(x)) on the Tangent bundle TM over the 4-dimensional
 differentiable manifold M
sage: TM.characteristic_class('AHat')
Characteristic class A^ of multiplicative type associated to
 1/2*sqrt(x)/sinh(1/2*sqrt(x)) on the Tangent bundle TM over the
 4-dimensional differentiable manifold M

The vector bundle’s base field and definition domain of the characteristic class must fit together, otherwise an error message occurs:

sage: TM.characteristic_class('Chern')
Traceback (most recent call last):
...
ValueError: base field must be complex for class 'Chern'

If your favourite class is not predefined yet, the associated function can be put manually:

sage: cl = TM.characteristic_class(1+x^2, name='cl'); cl
Characteristic class cl of multiplicative type associated to x^2 + 1 on
 the Tangent bundle TM over the 4-dimensional differentiable manifold M
class_type()

Return the class type of self.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: TM = M.tangent_bundle()
sage: ch = TM.characteristic_class(exp(x), class_type='additive',
....:                              name='ch', latex_name=r'\mathrm{ch}')
sage: ch.class_type()
'additive'
function()

Return the function corresponding to this characteristic class.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: TM = M.tangent_bundle()
sage: e_class = TM.characteristic_class('Euler')
sage: e_class.function()
x
sage: AHat = TM.characteristic_class('AHat')
sage: AHat.function()
1/2*sqrt(x)/sinh(1/2*sqrt(x))
sage: c = TM.characteristic_class(1+x, name='c')
sage: c.function()
x + 1
get_form(connection, cmatrices=None)

Return the form representing self with respect to the given connection connection.

INPUT:

  • connection – connection to which the form should be associated to; this can be either a bundle connection as an instance of BundleConnection or, in case of the tensor bundle, an affine connection as an instance of AffineConnection
  • cmatrices – (default: None) a dictionary of curvature matrices with local frames as keys and curvature matrices as items; if None, Sage tries to get the curvature matrices from the connection

OUTPUT:

  • mixed form as an instance of MixedForm representing the total characteristic class

Note

Be aware that depending on the characteristic class and complexity of the manifold, computation times may vary a lot. In addition, if not done before, the curvature form is computed from the connection, here. If this behaviour is not wanted and the curvature form is already known, please use the argument cmatrices.

EXAMPLES:

Again, consider the Chern character on some 2-dimensional spacetime:

sage: M = Manifold(2, 'M', structure='Lorentzian')
sage: X.<t,x> = M.chart()
sage: E = M.vector_bundle(1, 'E', field='complex'); E
Differentiable complex vector bundle E -> M of rank 1 over the base
 space 2-dimensional Lorentzian manifold M
sage: e = E.local_frame('e')

And again, we define the connection \(\nabla^E\) in terms of an electro-magnetic potential \(A(t)\):

sage: nab = E.bundle_connection('nabla^E', latex_name=r'\nabla^E')
sage: omega = M.one_form(name='omega')
sage: A = function('A')
sage: omega[1] = I*A(t)
sage: omega.display()
omega = I*A(t) dx
sage: nab.set_connection_form(0, 0, omega)

The Chern character is then given by:

sage: ch = E.characteristic_class('ChernChar'); ch
Characteristic class ch of additive type associated to e^x on the
 Differentiable complex vector bundle E -> M of rank 1 over the base
 space 2-dimensional Lorentzian manifold M

Inserting the connection, the result is a mixed differential form with a priori non-zero components in even degrees:

sage: ch_form = ch.get_form(nab); ch_form
Mixed differential form ch(E, nabla^E) on the 2-dimensional
 Lorentzian manifold M
sage: ch_form.display()
ch(E, nabla^E) = ch_0(E, nabla^E) + zero + ch_1(E, nabla^E)
sage: ch_form.display_expansion()
ch(E, nabla^E) = [1] + [0] + [1/2*d(A)/dt/pi dt/\dx]

Due to long computation times, the form is saved:

sage: ch_form is ch.get_form(nab)
True