Preliminary treatment

Using superspace and superfields

Preliminary treatment

The action is invariant under the supersymmetry transformations, given in infinitesimal form by

${\displaystyle \delta _{\epsilon }S={\bar {\epsilon }}\psi }$

${\displaystyle \delta _{\epsilon }P={\bar {\epsilon }}\gamma _{5}\psi }$

${\displaystyle \delta _{\epsilon }\psi =[\partial \!\!\!/-m-\lambda (S+P\gamma _{5})](S+P\gamma _{5})\epsilon }$

where ${\displaystyle \epsilon }$ is a Majorana spinor-valued transformation parameter and ${\displaystyle \gamma _{5}}$ is the chirality operator.

The alternative form is invariant under the transformation

${\displaystyle \delta _{\epsilon }\phi ={\sqrt {2}}\epsilon \chi }$

${\displaystyle \delta _{\epsilon }\chi ={\sqrt {2}}i\sigma ^{\mu }{\bar {\epsilon }}\partial _{\mu }\phi -{\sqrt {2}}\epsilon {\frac {\partial W^{\dagger }}{\partial \phi ^{\dagger }}}}$.

Without developing a theory of superspace transformations, these symmetries appear ad-hoc.

Superfield treatment

If the action can be written as ${\displaystyle S=\int d^{4}xd^{4}\theta K(x,\theta ,{\bar {\theta }})}$ where ${\displaystyle K}$ is a real superfield, that is, ${\displaystyle K^{\dagger }=K}$, then the action is invariant under supersymmetry.

Then the reality of ${\displaystyle K={\bar {\Phi }}\Phi }$ means it is invariant under supersymmetry.

Superconformal symmetry

The massless Wess–Zumino model admits a larger set of symmetries, described at the algebra level by the superconformal algebra. As well as the Poincaré symmetry generators and the supersymmetry translation generators, this contains the conformal algebra as well as a conformal supersymmetry generator ${\displaystyle S_{\alpha }}$.

The conformal symmetry is broken at the quantum level by trace and conformal anomalies, which break invariance under the conformal generators ${\displaystyle D}$ for dilatations and ${\displaystyle K_{\mu }}$ for special conformal transformations respectively.

R-symmetry

The ${\displaystyle {\text{U}}(1)}$ R-symmetry of ${\displaystyle {\mathcal {N}}=1}$ supersymmetry holds when the superpotential ${\displaystyle W(\Phi )}$ is a monomial. This means either ${\displaystyle W(\phi )={\frac {1}{2}}m\phi ^{2}}$, so that the superfield ${\displaystyle \Phi }$ is massive but free (non-interacting), or ${\displaystyle W(\Phi )={\frac {1}{3}}\lambda \phi ^{3}}$ so the theory is massless but (possibly) interacting.

This is broken at the quantum level by anomalies.

Super QCD

For general ${\displaystyle N}$, a superpotential of the form ${\displaystyle W(\Phi _{a},{\tilde {\Phi }}_{a})=m{\tilde {\Phi }}_{a}\Phi _{a}}$ has a ${\displaystyle {\text{SU}}(N)}$ symmetry when ${\displaystyle \Phi _{a},{\tilde {\Phi }}_{a}}$ rotate with opposite charges, that is under ${\displaystyle U\in {\text{SU}}(N)}$

${\displaystyle \Phi _{a}\mapsto U_{a}{}^{b}\Phi _{b}}$

${\displaystyle {\tilde {\Phi }}_{a}\mapsto (U^{-1})_{a}{}^{b}{\tilde {\Phi }}_{b}}$.

This symmetry can be gauged and coupled to supersymmetric Yang–Mills to form a supersymmetric analogue to quantum chromodynamics, known as super QCD.

Supersymmetric sigma models

If renormalizability is not insisted upon, then there are two possible generalizations. The first of these is to consider more general superpotentials. The second is to consider ${\displaystyle K}$ in the kinetic term

${\displaystyle S=\int d^{4}x\,d^{2}\theta ^{2}\,d^{2}{\bar {\theta }}^{2}K(\Phi ,{\bar {\Phi }})}$

to be a real function ${\displaystyle K=K(\Phi ,{\bar {\Phi }})}$ of ${\displaystyle \Phi ^{i}}$ and ${\displaystyle {\bar {\Phi }}^{\bar {j}}}$.

The action is invariant under transformations ${\displaystyle K(\Phi ,\Phi ^{\dagger })+\Lambda (\Phi )+{\bar {\Lambda }}({\bar {\Phi }})}$: these are known as Kähler transformations.

Considering this theory gives an intersection of Kähler geometry with supersymmetric field theory.

By expanding the Kähler potential ${\displaystyle K(\Phi ,{\bar {\Phi }})}$ in terms of derivatives of ${\displaystyle K}$ and the constituent superfields of ${\displaystyle \Phi ,{\bar {\Phi }}}$, and then eliminating the auxiliary fields ${\displaystyle F,{\bar {F}}}$ using the equations of motion, the following expression is obtained:

${\displaystyle S_{K}=\int d^{4}x\left[g_{i{\bar {j}}}(\partial _{\mu }\phi ^{i}\partial ^{\mu }{\bar {\phi }}^{\bar {j}})+g_{i{\bar {j}}}{\frac {i}{2}}(\nabla _{\mu }\psi ^{i}\sigma ^{\mu }{\bar {\psi }}^{\bar {j}}-\psi ^{i}\sigma ^{\mu }\nabla _{\mu }{\bar {\psi }}^{\bar {j}})+{\frac {1}{4}}R_{i{\bar {j}}k{\bar {l}}}(\psi ^{i}\psi ^{k})({\bar {\psi }}^{\bar {j}}{\bar {\psi }}^{\bar {l}})\right]}$

where

• ${\displaystyle g_{i{\bar {j}}}}$ is the Kähler metric. It is invariant under Kähler transformations. If the kinetic term is positive definite, then ${\displaystyle g_{i{\bar {j}}}}$ is invertible, allowing the inverse metric ${\displaystyle g^{i{\bar {j}}}}$ to be defined.

• The Christoffel symbols (adapted for a Kähler metric) are ${\displaystyle \Gamma ^{i}{}_{jk}=g^{i{\bar {l}}}\partial _{j}g_{k{\bar {l}}}}$ and ${\displaystyle {\bar {\Gamma }}^{\bar {i}}{}_{{\bar {j}}{\bar {k}}}=g^{l{\bar {i}}}\partial _{\bar {j}}g_{l{\bar {k}}}.}$

• The covariant derivatives ${\displaystyle \nabla _{\mu }\psi ^{i}}$ and ${\displaystyle \nabla _{\mu }{\bar {\psi }}^{\bar {j}}}$ are defined

${\displaystyle \nabla _{\mu }\psi ^{i}=\partial _{\mu }\psi ^{i}+\Gamma ^{i}{}_{jk}\psi ^{j}\partial _{\mu }\phi ^{k}}$

and

${\displaystyle \nabla _{\mu }{\bar {\psi }}^{\bar {i}}=\partial _{\mu }\psi ^{\bar {i}}+{\bar {\Gamma }}^{\bar {i}}{}_{{\bar {j}}{\bar {k}}}{\bar {\psi }}^{\bar {j}}\partial _{\mu }{\bar {\phi }}^{\bar {k}}}$

• The Riemann curvature tensor (adapted for a Kähler metric) is defined ${\displaystyle R_{i{\bar {j}}k{\bar {l}}}=g_{m{\bar {j}}}\partial _{\bar {l}}\Gamma ^{m}{}_{ik}=\partial _{k}\partial _{\bar {l}}g_{i{\bar {j}}}-g^{m{\bar {n}}}(\partial _{k}g_{i{\bar {n}}})(\partial _{\bar {l}}g_{m{\bar {j}}})}$.