Harmonic superspace

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 ${\displaystyle SU(2)_{R}/U(1)_{R}\approx S^{2}\simeq \mathbb {CP} ^{1}}$, 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,^[2] but the one chosen not only involves redundant coordinates, but also happen to be a coordinatization of ${\displaystyle SU(2)_{R}\approx S^{3}}$. We only get S² after a projection over ${\displaystyle U(1)_{R}\approx S^{1}}$. 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 ${\displaystyle u^{\pm i}}$.

${\displaystyle \left(u^{+i}\right)^{*}=u_{i}^{-}}$.

The redundancy in the coordinates is given by

${\displaystyle u^{+i}u_{i}^{-}=1}$.

Everything can be interpreted in terms of algebraic geometry. The projection is given by the "gauge transformation" ${\displaystyle u^{\pm i}\to e^{\pm i\phi }u^{\pm i}}$ 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 ${\displaystyle Q^{i\alpha }}$, and the corresponding fermionic coordinates are ${\displaystyle \theta ^{i\alpha }}$. 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

${\displaystyle \theta ^{\pm \alpha }=u_{i}^{\pm }\theta ^{i\alpha }}$.

We can define the covariant derivatives ${\displaystyle D_{\alpha }^{\pm }}$ with the property that they supercommute with the SUSY transformations, and ${\displaystyle D_{\alpha }^{\pm }f(u)=0}$ where f is any function of the harmonic variables. Similarly, define

${\displaystyle D^{++}\equiv u^{+i}{\frac {\partial }{\partial u^{-i}}}}$

and

${\displaystyle D^{--}\equiv u^{-i}{\frac {\partial }{\partial u^{+i}}}}$.

A chiral superfield q with an R-charge of r satisfies ${\displaystyle D_{\alpha }^{+}q=0}$. A scalar hypermultiplet is given by a chiral superfield $q^{+}$. We have the additional constraint

${\displaystyle D^{++}q^{+}=J^{+++}(q^{+},\,u)}$.

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

Relation to quaternions

The group ${\displaystyle SU(2)_{R}}$ can be identified with the Lie group of quaternions with unit norm under multiplication. ${\displaystyle SU(2)_{R}}$, and hence the quaternions act upon the tangent space of extended superspace. The bosonic spacetime dimensions transform trivially under ${\displaystyle SU(2)_{R}}$ while the fermionic dimensions transform according to the fundamental representation.^[3] 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.

See also

• Superspace
• Projective superspace

References

1. Galperin, Alexander Samoilovich; E. A. Ivanov; V. I. Ogievetsky; E. S. Sokatchev (2001). Harmonic Superspace. Cambridge University Press. p. 306. ISBN 978-0-521-80164-5.
2. Needless to say, other coordinate systems are also possible, and nothing physical is dependent upon the choice of coordinates, but the u coordinates have the advantage of being simple and convenient to use.
3. In 10D ${\displaystyle {\mathcal {N}}=(1,0)}$ SUSY with four spatial dimensions compactified over a hyperkähler manifold, half of the SUSY generators are broken, and the remaining generators can be expressed using harmonic superspace. The four compactified spatial dimensions transforms as a fundamental representation under ${\displaystyle SU(2)_{R}}$.