$mathcal\{N\}=1$ 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 $theta^1,theta^2,bartheta^1,bartheta^2$, 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 + ithetachi - i overline{theta}overline{chi} + frac{i}{2}theta^2(M+iN)-frac{i}{2}overline{theta^2}(M-iN) - theta sigma^mu overline{theta} v_mu &&+itheta^2 overline{theta} left(overline{lambda} + frac{1}{2}overline{sigma}^mu partial_mu chi right) -ioverline{theta}^2 theta left(lambda + frac{i}{2}sigma^mu partial_mu overline{chi} right) + frac{1}{2}theta^2 overline{theta}^2 left( D+ frac{1}{2}Box Cright) 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 + Lambda + 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

$X\; to\; e^\{qLambda\}X$

$overline\{X\}\; to\; e^\{qoverline\{Lambda\}\}X$

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 G^c. 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 $D\_M=d\_M+iqA\_M$. 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 $F\_\{alphabeta\}=0$. 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

• superpotential
• D-term
• F-term
• current superfield.