In supersymmetry, harmonic superspace [1] is one way of dealing with supersymmetric theories with 8 real SUSY generators in a manifestly covariant manner.
It turns out that the 8 real SUSY generators are pseudoreal, and after complexification, correspond to the tensor product of a four-dimensional Dirac spinor with the fundamental representation of SU(2)R. The quotient space
Harmonic superspace describes N=2 D=4, N=1 D=5, and N=(1,0) D=6 SUSY in a manifestly covariant manner.
We can then extend this to the space of complex valued smooth functions over SU(2)R. In particular, we have the subspace of functions which transform as the fundamental representation under SU(2)R. The fundamental representation (up to isomorphism, of course) is a two-dimensional complex vector space.
The subspace of interest consists of two copies of the fundamental representation.
Under the right action by U(1)R -- which commutes with any left action—one copy has a "charge" of +1, and the other of -1.
The redundancy in the coordinates is given by Everything can be interpreted in terms of algebraic geometry.
Think of S3 as a U(1)R-principal bundle over S2 with a nonzero first Chern class.
Harmonic superspace is given by the product of ordinary extended superspace (with 8 real fermionic coordinatates) with S2 with the nontrivial U(1)R bundle over it.
with the property that they supercommute with the SUSY transformations, and
Similarly, define and A chiral superfield q with an R-charge of r satisfies
We have the additional constraint According to the Atiyah-Singer index theorem, the solution space to the previous constraint is a two-dimensional complex manifold.
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.
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.