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Gauge theory with supersymmetry

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.

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 gauge transformation , where is a vector field and is the gauge function. 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. 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 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.