Supersymmetric gauge theory

Contents

\mathcal{N}=1 SUSY in 4D (with 4 real generators)

In theoretical physics, one often analyzes theories with supersymmetry which also have internal gauge symmetries. So, it is important to come up with a supersymmetric generalization of gauge theories. In four dimensions, the minimal N=1 supersymmetry may be written using a superspace. This superspace involves four extra fermionic coordinates \theta^1,\theta^2,\bar\theta^1,\bar\theta^2, transforming as a two-component spinor and its conjugate.

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 \theta but not their conjugates (more precisely, \overline{D}f=0). However, a vector superfield depends on all coordinates. It describes a gauge field and its superpartner, namely a Weyl fermion that obeys a Dirac equation.

\begin{matrix} V &=& C %2B i\theta\chi - i \overline{\theta}\overline{\chi} %2B \frac{i}{2}\theta^2(M%2BiN)-\frac{i}{2}\overline{\theta^2}(M-iN) - \theta \sigma^\mu \overline{\theta} v_\mu \\ &&%2Bi\theta^2 \overline{\theta} \left( \overline{\lambda} %2B \frac{1}{2}\overline{\sigma}^\mu \partial_\mu \chi \right) -i\overline{\theta}^2 \theta \left( \lambda %2B \frac{i}{2}\sigma^\mu \partial_\mu \overline{\chi} \right) %2B \frac{1}{2}\theta^2 \overline{\theta}^2 \left( D%2B \frac{1}{2}\Box C\right) \end{matrix}

V is the vector superfield (prepotential) and is real (\overline{V}=V). The fields on the right hand side are component fields.

The gauge transformations act as

V \to V %2B \Lambda %2B \overline{\Lambda}

where Λ is any chiral superfield.

It's easy to check that the chiral superfield

W_\alpha \equiv -\frac{1}{4}\overline{D}^2 D_\alpha V

is gauge invariant. So is its complex conjugate \overline{W}_{\dot{\alpha}}.

A nonSUSY covariant gauge which is often used is the Wess-Zumino gauge. Here, C, χ, M and N are all set to zero. The residual gauge symmetries are gauge transformations of the traditional bosonic type.

A chiral superfield X with a charge of q transforms as

X \to e^{q\Lambda}X
\overline{X} \to e^{q\overline{\Lambda}}X

The following term is therefore gauge invariant

\overline{X}e^{-qV}X

e^{-qV} is called a bridge since it "bridges" a field which transforms under Λ only with a field which transforms under \overline{\Lambda} 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. This is what's being done in the Wess-Zumino gauge.

Differential superforms

Let's rephrase everything to look more like a conventional Yang-Mill gauge theory. We have a U(1) 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 D_M=d_M%2BiqA_M. Integrability conditions for chiral superfields with the chiral constraint \overline{D}_{\dot{\alpha}}X=0 leave us with \left\{\overline{D}_{\dot{\alpha}}, \overline{D}_{\dot{\beta}} \right\}=F_{\dot{\alpha}\dot{\beta}}=0. A similar constraint for antichiral superfields leaves us with F_{\alpha\beta}=0. This means that we can either gauge fix A_{\dot{\alpha}}=0 or A_{\alpha}=0 but not both simultaneously. Call the two different gauge fixing schemes I and II respectively. In gauge I, \overline{d}_{\dot{\alpha}}X=0 and in gauge II, d_\alpha \overline{X}=0. Now, the trick is to use two different gauges simultaneously; gauge I for chiral superfields and gauge II for antichiral superfields. In order to bridge between the two different gauges, we need a gauge transformation. Call it e-V (by convention). If we were using one gauge for all fields, \overline{X}X would be gauge invariant. However, we need to convert gauge I to gauge II, transforming X to (e-V)qX. So, the gauge invariant quantity is \overline{X}e^{-qV}X.

In gauge I, we still have the residual gauge e^\Lambda where \overline{d}_{\dot{\alpha}}\Lambda=0 and in gauge II, we have the residual gauge e^{\overline{\Lambda}} satisfying d_\alpha \overline{\Lambda}=0. Under the residual gauges, the bridge transforms as e^{-V}\to e^{-\overline{\Lambda}-V-\Lambda}. Without any additional constraints, the bridge e^{-V} wouldn't give all the information about the gauge field. However, with the additional constraint F_{\dot{\alpha}\beta}, there's only one unique gauge field which is compatible with the bridge modulo gauge transformations. Now, the bridge gives exactly the same information content as the gauge field.

Theories with 8 or more SUSY generators

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.

See also