In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries.
Supersymmetric gauge theory generalizes this notion.
Roughly, there are two types of symmetries, global and local.
Bosons carry integer spin values, and are characterized by the ability to have any number of identical bosons occupy a single point in space.
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.
(or typically many operators), known as a supercharge or supersymmetry generator, which acts schematically as
For instance, the supersymmetry generator can take a photon as an argument and transform it into a photino and vice versa.
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.
The gauge vector fields and its spinorial superpartner can be made to both reside in the same representation of the internal symmetry group.
The main difficulty in construction of a SUSY Gauge Theory is to extend the above transformation in a way that is consistent with SUSY transformations.
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).
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 N = 1 supersymmetry may be written using a superspace.
This superspace involves four extra fermionic coordinates
There exists a special kind of superfields, the so-called chiral superfields, that only depend on the variables θ but not their conjugates (more precisely,
It describes a gauge field and its superpartner, namely a Weyl fermion that obeys a Dirac equation.
The gauge transformations act as where Λ is any chiral superfield.
It's easy to check that the chiral superfield is gauge invariant.
A chiral superfield X with a charge of q transforms as Therefore Xe−qVX is gauge invariant.
Here e−qV is called a bridge since it "bridges" a field which transforms under Λ only with a field which transforms under Λ only.
More generally, if we have a real gauge group G that we wish to supersymmetrize, we first have to complexify it to Gc ⋅ e−qV then acts a compensator for the complex gauge transformations in effect absorbing them leaving only the real parts.
In the analytic basis for the tangent space, the covariant derivative is given by
Integrability conditions for chiral superfields with the chiral constraint leave us with A similar constraint for antichiral superfields leaves us with Fαβ = 0.
Call the two different gauge fixing schemes I and II respectively.
Under the residual gauges, the bridge transforms as Without any additional constraints, the bridge e−V wouldn't give all the information about the gauge field.
Now, the bridge gives exactly the same information content as the gauge field.
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.