S_{ij}^{AB} = sigma_i^{AC'}*sigma_j^B_C' - sigma_j^{BC'}*sigma_i^A_C'

    and

barS_{ij}^{A'B'} = sigma_i^{CA'}*sigma_j_C^B' - sigma_j^{CB'}*sigma_i^_C^A'.

    The tensor indices of the bivector solder forms are raised with the metric g defined by sigma.

Example 1.

First create a vector bundle over M with base coordinates [t, x, y, z] and fiber coordinates [z1, z2, w1, w2].

Define a metric g on M. Note that our spinor conventions have the metric with signature [+1, -1, -1, -1].

Define an orthonormal frame on M with respect to the metric g.

Calculate the solder form sigma from the frame F.

Calculate the bivector solder form S from sigma.

Example 2.

The contraction of two bivector solder forms on their tensor indices can be expressed in terms of the Kronecker delta spinor.

S_{ij}^{AB}*S^{ij}_{CD} = 4*(delta^A_C*delta^B_D + delta^B_C*delta^A_D)

We check this identity using the solder form from Example 1.  First we calculate the left-hand side.

To calculate the right-hand side we construct the symmetrized tensor product of 2 Kronecker delta spinors and multiply by 8 (because SymmetrizeIndices will include a factor of 1/2).

Check that the LHS and RHS are the same.

Example 3.

The contraction of two bivector soldering forms on their tensor indices can be expressed in terms of the metric and the permutation tensor

S_{ij}^{AB}*S_{hk}_{AB} = 2*(g_{ih}*g_{jk} - g_{jh}*g_{ik} - I*epsilon_{ijhk}).

We check this identity using the solder form from Example 1.  First we calculate the left-hand side.

To calculate the right-hand side we first construct the tensor product of the metric tensor with itself.

We re-arrange the indices of G to obtain the first two terms on the right-hand side.

We construct the epsilon tensor using the commands MetricDensity and PermutationSymbol.

Evaluate the right-hand side of the identity and check that it agrees with the left-hand side.

Example 4.

The bivector solder form is anti-self-dual, that is,

S_{ij}^{AB} = -I/2*epsilon_{ijhk}*S^{hk}^{AB}

We check this identity using the solder form from Example 1.  The left-hand side is just the solder form S1 from Example 1.

To evaluate the right-hand side we begin with the contravariant form of the bivector solder form.

Construct the epsilon tensor and contract with S4 and to obtain the left-hand side.

DifferentialGeometry, Tensor, ContractIndices, KroneckerDeltaSpinor, MetricDensity, PermutationSymbol, SpacetimeConventions, SolderForm, SymmetrizeIndices, RearrangeIndices