Relation To Quaternions
The group can be identified with the Lie group of quaternions with unit norm under multiplication., and hence the quaternions act upon the tangent space of extended superspace. The bosonic spacetime dimensions transform trivially under while the fermionic dimensions transform according to the fundamental representation. The left multiplication by quaternions is linear. Now consider the subspace of unit quaternions with no real component, which is isomorphic to S2. Each element of this subspace can act as the imaginary number i in a complex subalgebra of the quaternions. So, for each element of S2, we can use the corresponding imaginary unit to define a complex-real structure over the extended superspace with 8 real SUSY generators. The totality of all CR structures for each point in S2 is harmonic superspace.
Read more about this topic: Harmonic Superspace
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