# Supersymmetric gauge theory

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

theoretical physics , one often analyzes theories withsupersymmetry which also haveinternalgauge symmetries . So, it is important to come up with a supersymmetric generalizationof gauge theories.In four dimensions, the minimal N=1 supersymmetry may be written using asuperspace . This superspace involves four extra fermionic coordinates $heta^1,\; heta^2,ar\; heta^1,ar\; heta^2$, transforming as a two-componentspinor 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 superfield s, that only depend on the variables $heta$ but not their conjugates (more precisely, $overline\{D\}f=0$). However, avector superfield depends on all coordinates. It describes agauge field and itssuperpartner , namely aWeyl fermion that obeys aDirac equation .:$egin\{matrix\}V\; =\; C\; +\; i\; hetachi\; -\; i\; overline\{\; heta\}overline\{chi\}\; +\; frac\{i\}\{2\}\; heta^2(M+iN)-frac\{i\}\{2\}overline\{\; heta^2\}(M-iN)\; -\; heta\; sigma^mu\; overline\{\; heta\}\; v\_mu\; \backslash +i\; heta^2\; overline\{\; heta\}\; left(\; overline\{lambda\}\; +\; frac\{1\}\{2\}overline\{sigma\}^mu\; partial\_mu\; chi\; ight)\; -ioverline\{\; heta\}^2\; heta\; left(\; lambda\; +\; frac\{i\}\{2\}sigma^mu\; partial\_mu\; overline\{chi\}\; ight)\; +\; frac\{1\}\{2\}\; heta^2\; overline\{\; heta\}^2\; left(\; D+\; frac\{1\}\{2\}Box\; C\; ight)end\{matrix\}$

V is the vector superfield (

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

gauge transformation s act as:$V\; o\; V\; +\; Lambda\; +\; overline\{Lambda\}$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

. Here, C, χ, M and N areall set to zero. The residual gauge symmetries are gauge transformations of the traditional bosonictype.Wess-Zumino gauge A chiral superfield X with a charge of q transforms as:$X\; o\; e^\{qLambda\}X$:$overline\{X\}\; o\; e^\{qoverline\{LambdaX$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 $overline\{D\}\_\{dot\{alphaX=0$ leave us with $left\{overline\{D\}\_\{dot\{alpha,\; overline\{D\}\_\{dot\{eta\; ight\}=F\_\{dot\{alpha\}dot\{eta=0$. A similar constraint for antichiral superfields leaves us with $F\_\{alphaeta\}=0$. 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, $overline\{d\}\_\{dot\{alphaX=0$ 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})^{q}X. So, the gauge invariant quantity is $overline\{X\}e^\{-qV\}X$.In gauge I, we still have the residual gauge $e^Lambda$ where $overline\{d\}\_\{dot\{alphaLambda=0$ 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 $e^\{-V\}\; o\; e^\{-overline\{Lambda\}-V-Lambda\}$. Without any additional constraints, the bridge $e^\{-V\}$ wouldn't give all the information about the gauge field. However, with the additional constraint $F\_\{dot\{alpha\}eta\}$, 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 .

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