Supersymmetric gauge theory

Supersymmetric gauge theory

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 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{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

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