In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries. Supersymmetric gauge theory generalizes this notion.
A gauge theory is a field theory with gauge symmetry. Roughly, there are two types of symmetries, global and local. A global symmetry is a symmetry applied uniformly (in some sense) to each point of a manifold. A local symmetry is a symmetry which is position dependent. Gauge symmetry is an example of a local symmetry, with the symmetry described by a Lie group (which mathematically describe continuous symmetries), which in the context of gauge theory is called the gauge group of the theory.
Quantum chromodynamics and quantum electrodynamics are famous examples of gauge theories.
In particle physics, there exist particles with two kinds of particle statistics, bosons and fermions. Bosons carry integer spin values, and are characterized by the ability to have any number of identical bosons occupy a single point in space. They are thus identified with forces. Fermions carry half-integer spin values, and by the Pauli exclusion principle, identical fermions cannot occupy a single position in spacetime. Boson and fermion fields are interpreted as matter. Thus, supersymmetry is considered a strong candidate for the unification of radiation (boson-mediated forces) and matter.
Q
Q|boson\rangle=|fermion\rangle
Q|fermion\rangle=|boson\rangle
For instance, the supersymmetry generator can take a photon as an argument and transform it into a photino and vice versa. This happens through translation in the (parameter) space. This superspace is a
{Z2}
l{W}=l{W}0 ⊕ l{W}1
l{W}0
l{W}1
The motivation for a supersymmetric version of gauge theory can be the fact that gauge invariance is consistent with supersymmetry.The first examples were discovered by Bruno Zumino and Sergio Ferrara, and independently by Abdus Salam and James Strathdee in 1974.
Both the half-integer spin fermions and the integer spin bosons can become gauge particles. The gauge vector fields and its spinorial superpartner can be made to both reside in the same representation of the internal symmetry group.
Suppose we have a
U(1)
V\mu → V\mu+\partial\muA
V\mu
A
The Wess–Zumino gauge (a prescription for supersymmetric gauge fixing) provides a successful solution to this problem. Once such suitable gauge is obtained, the dynamics of the SUSY gauge theory work as follows: we seek a Lagrangian that is invariant under the Super-gauge transformations (these transformations are an important tool needed to develop supersymmetric version of a gauge theory). Then we can integrate the Lagrangian using the Berezin integration rules and thus obtain the action. Which further leads to the equations of motion and hence can provide a complete analysis of the dynamics of the theory.
In four dimensions, the minimal supersymmetry may be written using a superspace. This superspace involves four extra fermionic coordinates
\theta1,\theta2,\bar\theta1,\bar\theta2
Every superfield, i.e. a field that depends on all coordinates of the superspace, may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called chiral superfields, that only depend on the variables but not their conjugates (more precisely,
\overline{D}f=0
V=C+i\theta\chi-i\overline{\theta}\overline{\chi}+\tfrac{i}{2}\theta2(M+iN)-\tfrac{i}{2}\overline{\theta2}(M-iN)-\theta\sigma\mu\overline{\theta}v\mu+i\theta2\overline{\theta}\left(\overline{λ}-\tfrac{i}{2}\overline{\sigma}\mu\partial\mu\chi\right)-i\overline{\theta}2\theta\left(λ+\tfrac{i}{2}\sigma\mu\partial\mu\overline{\chi}\right)+\tfrac{1}{2}\theta2\overline{\theta}2\left(D+\tfrac{1}{2}\BoxC\right)
is the vector superfield (prepotential) and is real . The fields on the right hand side are component fields.
The gauge transformations act as
V\toV+Λ+\overline{Λ}
where is any chiral superfield.
It's easy to check that the chiral superfield
W\alpha\equiv-\tfrac{1}{4}\overline{D}2D\alphaV
is gauge invariant. So is its complex conjugate
| \overline{W} | |||
|
A non-supersymmetric covariant gauge which is often used is the Wess–Zumino gauge. Here, and are all set to zero. The residual gauge symmetries are gauge transformations of the traditional bosonic type.
A chiral superfield with a charge of transforms as
X\toeqΛX, \overline{X}\toeq\overline{Λ
Therefore is gauge invariant. Here is called a bridge since it "bridges" a field which transforms under only with a field which transforms under only.
More generally, if we have a real gauge group that we wish to supersymmetrize, we first have to complexify it to then acts a compensator for the complex gauge transformations in effect absorbing them leaving only the real parts. This is what's being done in the Wess–Zumino gauge.
Let's rephrase everything to look more like a conventional Yang–Mills gauge theory. We have a gauge symmetry acting upon full superspace with a 1-superform gauge connection A. In the analytic basis for the tangent space, the covariant derivative is given by
DM=dM+iqAM
| \overline{D} | |||
|
X=0
leave us with
| \left\{\overline{D} | |||
|
,
| \overline{D} | |||
|
| \right\}=F | |||||||
|
=0.
A similar constraint for antichiral superfields leaves us with . This means that we can either gauge fix
| A | |||
|
=0
| \overline{d} | |||
|
X=0
In gauge I, we still have the residual gauge where
| \overline{d} | |||
|
Λ=0
e-V\toe-\overline{Λ-V-Λ}.
Without any additional constraints, the bridge wouldn't give all the information about the gauge field. However, with the additional constraint
| F | |||||
|
In theories with higher supersymmetry (and perhaps higher dimension), a vector superfield typically describes not only a gauge field and a Weyl fermion but also at least one complex scalar field.