Subspaces in associated to a pair of pure spinors
The Lie superalgebra has -grading:Suppose that we are also given a pair of pure spinors and . Then we can refine the decomposition of Eq. (9) by further splitting each , as a linear space, into a direct sum of linear subspaces.
1 Notations
For any , we will denote:
where was defined in Eq. (4).
2 Tangent and normal space to pure spinor cones
The projector
where is adjusted to satisfy (11). In fact is the projection
to the tangent space along the space which is orthogonal to with
respect to the metric defined by :
The projector is defined similarly. We have the following exact sequence of linear spaces:
The of Eq. (10) is given by the following expression:
Notice that is actually both super-traceless and traceless; it is
the same as (with the overline extending over “”).
3 Refinement of
Consider the decomposition:
Here is a 4-dimensional subspace -orthogonal to and commuting with :
and is -orthogonal to and commuting with :
4 Refinement of and
Similarly we can split and as follows:
Therefore, as a linear space:
5 More definitions
For and ,
let denote the map:
This is a direct sum of two completely independent linear maps: a map and a map
For a pair we decompose:
where a special representative of the cokernel is used:
Similarly, any (assumed to be both TL and STL) can be decomposed:
Explicitly: