Harmonic superspace

Last updated April 06, 2019

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 $SU(2)_{R}/U(1)_{R}\approx S^{2}\simeq \mathbb {CP} ^{1}$ , which is a 2-sphere/Riemann sphere.

In particle physics, supersymmetry (SUSY) is a principle that proposes a relationship between two basic classes of elementary particles: bosons, which have an integer-valued spin, and fermions, which have a half-integer spin. A type of spacetime symmetry, supersymmetry is a possible candidate for undiscovered particle physics, and seen as an elegant solution to many current problems in particle physics if confirmed correct, which could resolve various areas where current theories are believed to be incomplete. A supersymmetrical extension to the Standard Model would resolve major hierarchy problems within gauge theory, by guaranteeing that quadratic divergences of all orders will cancel out in perturbation theory.

In mathematics, the complexification of a vector space V over the field of real numbers yields a vector space V^C over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for V may also serve as a basis for V^C over the complex numbers.

In mathematics, the tensor product $V \otimes W$ of two vector spaces $V$ and $W$ is itself a vector space, endowed with the operation of bilinear composition, denoted by $\otimes$ , from ordered pairs in the Cartesian product $V \times W$ onto $V \otimes W$ in a way that generalizes the outer product. The tensor product of $V$ and $W$ is the vector space generated by the symbols $v \otimes w$ , with $v \in V$ and $w \in W$ , in which the relations of bilinearity are imposed for the product operation $\otimes$ , and no other relations are assumed to hold. The tensor product space is thus the "freest" such vector space, in the sense of having the fewest constraints.

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 $SU(2)_{R}\approx S^{3}$ . We only get S²after a projection over $U(1)_{R}\approx S^{1}$ . 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 $u^{\pm i}$ .

In the mathematical field of differential topology, the Hopf fibration describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function from the $3$ -sphere onto the $2$ -sphere such that each distinct point of the $2$ -sphere comes from a distinct circle of the $3$ -sphere. Thus the $3$ -sphere is composed of fibers, where each fiber is a circle—one for each point of the $2$ -sphere.

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

.

The redundancy in the coordinates is given by

u^{+i}u_{i}^{-}=1

.

Everything can be interpreted in terms of algebraic geometry. The projection is given by the "gauge transformation" $u^{\pm i}\to e^{\pm i\phi }u^{\pm i}$ 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.

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The SUSY charges are $Q^{i\alpha }$ , and the corresponding fermionic coordinates are $\theta ^{i\alpha }$ . 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

\theta ^{\pm \alpha }=u_{i}^{\pm }\theta ^{i\alpha }

.

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

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

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

and

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

.

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

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 $SU(2)_{R}$ can be identified with the Lie group of quaternions with unit norm under multiplication. $SU(2)_{R}$ , and hence the quaternions act upon the tangent space of extended superspace. The bosonic spacetime dimensions transform trivially under $SU(2)_{R}$ 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.

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References

↑ 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.
↑ 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.
↑ In 10D ${\mathcal {N}}=(1,0)$ 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 $SU(2)_{R}$ .

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[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 ${\mathcal {N}}=(1,0)$ 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 $SU(2)_{R}$ .

Harmonic superspace

Contents

Relation to quaternions

See also

Related Research Articles

References