Harmonic Superspace

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 .

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The redundancy in the coordinates is given by

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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.

The SUSY charges are, and the corresponding fermionic coordinates are . 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. The product is somewhat twisted in that the fermionic coordinates are also charged under U(1)_R. This charge is given by

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We can define the covariant derivatives with the property that they supercommute with the SUSY transformations, and where f is any function of the harmonic variables. Similarly, define

and

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A chiral superfield q with an R-charge of r satisfies . A scalar hypermultiplet is given by a chiral superfield . We have the additional constraint

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According to the Atiyah-Singer index theorem, the solution space to the previous constraint is a two dimensional complex manifold.