Definitions

# Supersymmetric gauge theory

## $mathcal\left\{N\right\}=1$ 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, $overline\left\{D\right\}f=0$). 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 ($overline\left\{V\right\}=V$). 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

$W_alpha equiv -frac\left\{1\right\}\left\{4\right\}overline\left\{D\right\}^2 D_alpha V$
is gauge invariant. So is its complex conjugate $overline\left\{W\right\}_\left\{dot\left\{alpha\right\}\right\}$.

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

$X to e^\left\{qLambda\right\}X$
$overline\left\{X\right\} to e^\left\{qoverline\left\{Lambda\right\}\right\}X$
The following term is therefore gauge invariant
$overline\left\{X\right\}e^\left\{-qV\right\}X$

$e^\left\{-qV\right\}$ is called a bridge since it "bridges" a field which transforms under Λ only with a field which transforms under $overline\left\{Lambda\right\}$ 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 $D_M=d_M+iqA_M$. Integrability conditions for chiral superfields with the chiral constraint $overline\left\{D\right\}_\left\{dot\left\{alpha\right\}\right\}X=0$ leave us with $left\left\{overline\left\{D\right\}_\left\{dot\left\{alpha\right\}\right\}, overline\left\{D\right\}_\left\{dot\left\{beta\right\}\right\} right\right\}=F_\left\{dot\left\{alpha\right\}dot\left\{beta\right\}\right\}=0$. A similar constraint for antichiral superfields leaves us with $F_\left\{alphabeta\right\}=0$. This means that we can either gauge fix $A_\left\{dot\left\{alpha\right\}\right\}=0$ or $A_\left\{alpha\right\}=0$ but not both simultaneously. Call the two different gauge fixing schemes I and II respectively. In gauge I, $overline\left\{d\right\}_\left\{dot\left\{alpha\right\}\right\}X=0$ and in gauge II, $d_alpha overline\left\{X\right\}=0$. 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, $overline\left\{X\right\}X$ 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 $overline\left\{X\right\}e^\left\{-qV\right\}X$.

In gauge I, we still have the residual gauge $e^Lambda$ where $overline\left\{d\right\}_\left\{dot\left\{alpha\right\}\right\}Lambda=0$ and in gauge II, we have the residual gauge $e^\left\{overline\left\{Lambda\right\}\right\}$ satisfying $d_alpha overline\left\{Lambda\right\}=0$. Under the residual gauges, the bridge transforms as $e^\left\{-V\right\}to e^\left\{-overline\left\{Lambda\right\}-V-Lambda\right\}$. Without any additional constraints, the bridge $e^\left\{-V\right\}$ wouldn't give all the information about the gauge field. However, with the additional constraint $F_\left\{dot\left\{alpha\right\}beta\right\}$, 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.