- Supersymmetric gauge theory
= 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 , 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 but not their conjugates (more precisely, ). However, avector superfield depends on all coordinates. It describes agauge field and itssuperpartner , namely aWeyl fermion that obeys aDirac equation .:
V is the vector superfield (prepotential) and is real (). The fields on the right hand side are component fields.
The
gauge transformation s act as:where Λ is any chiral superfield.It's easy to check that the chiral superfield:is gauge invariant. So is its complex conjugate .
A nonSUSY covariant gauge which is often used is the
Wess-Zumino gauge . Here, C, χ, M and N areall set to zero. The residual gauge symmetries are gauge transformations of the traditional bosonictype.A chiral superfield X with a charge of q transforms as::The following term is therefore gauge invariant:
is called a bridge since it "bridges" a field which transforms under Λ only with a field which transforms under 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 . Integrability conditions for chiral superfields with the chiral constraint leave us with . A similar constraint for antichiral superfields leaves us with . This means that we can either gauge fix or but not both simultaneously. Call the two different gauge fixing schemes I and II respectively. In gauge I, and in gauge II, . 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, 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 .
In gauge I, we still have the residual gauge where and in gauge II, we have the residual gauge satisfying . Under the residual gauges, the bridge transforms as . Without any additional constraints, the bridge wouldn't give all the information about the gauge field. However, with the additional constraint , 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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