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Concept
Harmonic superspace
Summary
In supersymmetry, harmonic superspace
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 , which is a 2-sphere/Riemann sphere.
Harmonic superspace describes N=2 D=4, N=1 D=5, and N=(1,0) D=6 SUSY in a manifestly covariant manner.
There are many possible coordinate systems over S2, but the one chosen not only involves redundant coordinates, but also happen to be a coordinatization of . We only get S2 after a projection over . This is of course the Hopf fibration. Consider the left action of SU(2)R upon itself. 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. Let us denote the indices of this representation by i,j,k,...=1,2. 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. Let us label the basis functions .
The redundancy in the coordinates is given by
Everything can be interpreted in terms of algebraic geometry. The projection is given by the "gauge transformation" where φ is any real number. Think of S3 as a U(1)R-principal bundle over S2 with a nonzero first Chern class. Then, "fields" over S2 are characterized by an integral U(1)R charge given by the right action of U(1)R. For instance, u+ has a charge of +1, and u− of -1. By convention, fields with a charge of +r are denoted by a superscript with r +'s, and ditto for fields with a charge of -r. R-charges are additive under the multiplication of fields.
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