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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.

- $$

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

- $$

It's easy to check that the chiral superfield

- $W\_alpha\; equiv\; -frac\{1\}\{4\}overline\{D\}^2\; D\_alpha\; V$

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

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

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

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

$e^\{-qV\}$ 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.

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.

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.

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Last updated on Tuesday June 03, 2008 at 23:02:39 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday June 03, 2008 at 23:02:39 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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