Supersymmetry is a postulated symmetry between bosonic and fermionic fields in quantum field theories and string theories.

The theory of Supsersymmetry has been incorporated in the Standard Model (MSSM), Yang-Mills Theory (Super-Yang-Mills Theory), and most famously String Theory (Superstring theory).

While Supersymmetry remains experimentally unconfirmed, one of its greatest achievements is that the MSSM (which also appears in realistic M-Theory vacua) predicts a Higgs of mass 125 GeV (which was measured by the LHC recently.), which is contrary to the Standard Model, which predicts such a mass to be rather unlikely.

There are two types of supersymmetry; worldsheet supersymmetry, and spacetime supersymmetry.

The Ramond-Neveu-Schwarz Formalism has explicit worldsheet supersymmetry. Since the RNS Action is given by adding the Polyakov Action to the Dirac Action, it is given by:

\({{\mathsf{\mathcal{L}}}_ {RNS}}=\frac{T}{2} h^{\alpha \beta} \left( \partial_\alpha X^\mu \partial_\beta X^\nu +i\hbar c_0 \bar{\psi_\mu} \not{\partial} \psi^\mu \right) g_{\mu\nu}\)

The supersymmetryic transformations on the worldsheet can therefore be (almost trivially, by taking variations of this above action) shown to be:

\( \begin{align}
\delta {X^\mu } \to \bar \epsilon {\psi ^\mu } ; \\
\delta {\psi ^\mu } \to - i \not \partial {X^\mu }\epsilon \\
\end{align}\)

The Green-Schwarz Formalism, or the Superspace Formalism, are with explicit spacetime supersymmetry. The supersymmetryic transformations on spacetime are (which is rather intuitive if you compare this to the RNS Worldsheet supersymmetry transformations) given by:

\(\begin{align}
\delta {\Theta ^{Aa}} \leftrightarrow {\varepsilon ^{Aa}} ; \\
\delta {X^\mu } \leftrightarrow {{\bar \varepsilon }^A}{\gamma ^\mu }{\Theta ^A} \\
\end{align} \)